Advanced Computer Techniques in Applied Electromagnetics

ADVANCED COMPUTER TECHNIQUES IN APPLIED

ELECTROMAGNETICS

Studies in Applied Electromagnetics

and Mechanics

Series Editors: K. Miya, A.J. Moses, Y. Uchikawa, A. Bossavit, R. Collins, T. Honma,

G.A. Maugin, F.C. Moon, G. Rubinacci, H. Troger and S.-A. Zhou

Volume 30

Previously published in this series:

Vol. 29. A. Krawczyk, R. Kubacki, S. Wiak and C. Lemos Antunes (Eds.), Electromagnetic

Field, Health and Environment – Proceedings of EHE’07

Vol. 28. S. Takahashi and H. Kikuchi (Eds.), Electromagnetic Nondestructive Evaluation (X)

Vol. 27. A. Krawczyk, S. Wiak and X.M. Lopez-Fernandez (Eds.), Electromagnetic Fields in

Mechatronics, Electrical and Electronic Engineering

Vol. 26. G. Dobmann (Ed.), Electromagnetic Nondestructive Evaluation (VII)

Vol. 25. L. Udpa and N. Bowler (Eds.), Electromagnetic Nondestructive Evaluation (IX)

Vol. 24. T. Sollier, D. Prémel and D. Lesselier (Eds.), Electromagnetic Nondestructive

Evaluation (VIII)

Vol. 23. F. Kojima, T. Takagi, S.S. Udpa and J. Pávó (Eds.), Electromagnetic Nondestructive

Evaluation (VI)

Vol. 22. A. Krawczyk and S. Wiak (Eds.), Electromagnetic Fields in Electrical Engineering

Vol. 21. J. Pávó, G. Vértesy, T. Takagi and S.S. Udpa (Eds.), Electromagnetic Nondestructive

Evaluation (V)

Vol. 20. Z. Haznadar and Ž. Štih, Electromagnetic Fields, Waves and Numerical Methods

Vol. 19. J.S. Yang and G.A. Maugin (Eds.), Mechanics of Electromagnetic Materials and

Structures

Vol. 18. P. Di Barba and A. Savini (Eds.), Non-Linear Electromagnetic Systems

Vol. 17. S.S. Udpa, T. Takagi, J. Pávó and R. Albanese (Eds.), Electromagnetic

Nondestructive Evaluation (IV)

Vol. 16. H. Tsuboi and I. Vajda (Eds.), Applied Electromagnetics and Computational

Technology II

Vol. 15. D. Lesselier and A. Razek (Eds.), Electromagnetic Nondestructive Evaluation (III)

Vol. 14. R. Albanese, G. Rubinacci, T. Takagi and S.S. Udpa (Eds.), Electromagnetic

Nondestructive Evaluation (II)

Vol. 13. V. Kose and J. Sievert (Eds.), Non-Linear Electromagnetic Systems

Vol. 12. T. Takagi, J.R. Bowler and Y. Yoshida (Eds.), Electromagnetic Nondestructive

Evaluation

Volumes 1–6 were published by Elsevier Science under the series title “Elsevier Studies in

Applied Electromagnetics in Materials”.

ISSN 1383-7281

Advanced Computer Techniques in

Applied Electromagnetics

Edited by

Sławomir Wiak

Institute of Mechatronics and Information Systems,

Technical University of Lodz, Poland

Andrzej Krawczyk

Institute for Labour Protection, Warsaw, Poland

and

Ivo Dolezel

Technical University of Prague, Czech Republic

Amsterdam • Berlin • Oxford • Tokyo • Washington, DC

© 2008 The authors and IOS Press.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system,

or transmitted, in any form or by any means, without prior written permission from the publisher.

ISBN 978-1-58603-895-3

Library of Congress Control Number: 2008931370

Publisher

IOS Press

Nieuwe Hemweg 6B

1013 BG Amsterdam

Netherlands

fax: +31 20 687 0019

e-mail: [email protected]

Distributor in the UK and Ireland Distributor in the USA and Canada

Gazelle Books Services Ltd. IOS Press, Inc.

White Cross Mills 4502 Rachael Manor Drive

Hightown Fairfax, VA 22032

Lancaster LA1 4XS USA

United Kingdom fax: +1 703 323 3668

fax: +44 1524 63232 e-mail: [email protected]

e-mail: [email protected]

LEGAL NOTICE

The publisher is not responsible for the use which might be made of the following information.

PRINTED IN THE NETHERLANDS

Preface

This book contains papers presented at the International Symposium on Electromag-

netic Fields in Electrical Engineering ISEF’07 which was held in Prague, the Czech

Republic, on September 13–15, 2007. ISEF conferences have been organized since

1985 as a common initiative of Polish and European researchers who deal with elec-

tromagnetic field applied to electrical engineering. Until the present the conferences

have been held every two years either in Poland or in one of European academic cen-

tres renowned for electromagnetic research. Technical University of Prague and the

Chech Academy of Sciences make Prague be such a centre. Additionally, Prague is

well-known in the world for its beauty and charm and it is called “Golden Prague”. The

city of Prague is one of the six most frequently visited cities in Europe. Indeed, it is

indisputable that Prague can attract every has the opportunity to visit it.

The long, more then 20-year-old, tradition of ISEF meetings is that they try to tan-

gle quite a vast area of computational and applied electromagnetics. Moreover, ISEF

symposia aim at joining theory and practice, thus the majority of papers are deeply

rooted in engineering problems and simultaneously present high theoretical level. Bear-

ing this tradition, we attempt to touch the core of electromagnetic phenomena.

After the selection process 237 papers were accepted for the presentation at the

Symposium and almost all of them were presented at the conference, both orally and in

the poster sessions. The papers have been divided into the following groups:

• Micro and Special Devices

• Electromagnetic Engineering

• Computational Electromagnetics

• Coupled Problems and Special Applications

• Measurement Monitoring and Testing Techniques

• Bioelectromagnetics

• Magnetic Material Modelling

The papers which were presented at the symposium had been reviewed and as-

sessed by the sessions’ chairmen and the Editorial Board assembled for the post-

conference issue of ISEF’07. All the papers accepted for further publication were di-

vided into three groups: 1) of more computational aspect, 2) of information technology

aspect and 3) of more applicable nature. The latter ones are published in this volume

while the first ones went to COMPEL journal (COMPEL: The International Journal

for Computation and Mathematics in Electrical and Electronic Engineering, vol. 27,

No. 3/2008) and the second group to Springer Verlag (series on Studies in Computa-

tional Intelligence, vol. 119, 2008).

The papers selected for this volume have been grouped in three chapters and seven

sub-chapters. The division introduces some order in the pile of papers and the titles of

chapters mirror the content of the papers to some extent. Names of chapters and sub-

chapters are as follows:

Chapter A Fundamental Problems and Methods

Fundamental Problems

Methods

Advanced Computer Techniques in Applied ElectromagneticsS. Wiak et al. (Eds.)IOS Press, 2008© 2008 The authors and IOS Press. All rights reserved.

v

Chapter B Computer Methods in Applied Electromagnetism

Computational Methods

Numerical Modelling of Devices

Chapter C Applications

Electrical Machines and Transformers

Actuators and Special Devices

Special Applications

The papers gathered in Chapter A are mainly devoted to physics of electromag-

netic materials and mathematical approaches to electromagnetic problems. In the first

sub-chapter papers concern physical phenomena, like magnetostriction, vibrations,

anisotropy, occuring in the various electromagnetic materials from ferromagnetics to

dielectromagnetics. And the second sub-chapter consists of papers concerning methods

of analysis of electromagnetic phenomena in their methodological aspects.

Chapter B contains papers dealing with numerical (or computer) analysis of elec-

tromagnetic devices and phenomena. The first sub-chapter shows how mathematical

methods are realised numerically, i.e. how to make real calculation, based on numbers.

And the papers gathered in the second sub-chapter deal with numerical modelling of

some groups of devices.

Chapter C, in turn, reveals the world of engineering problems, showing how theo-

retical and methodological considerations can be transferred to real engineering prob-

lems. Indeed, the chapter gives the image of real applied electromagnetics. The first

sub-chapter is devoted to the very classical electrical devices, namely transformers and

electrical machines. In spite of avery long tradition of numerical analysis of electro-

magnetic phenomena in such devices, the papers bring some new ideas and approaches.

The second sub-chapter shows newer applications like sensors and actuators, and thus

the area of engineering called mechatronics. Special approaches are needed inthe

analysis of these devices as their size and operation features are quite different fromthe

previous ones. And the last sub-chapter gathers a few papers dealing with very special

applications based, for example, on superconductivity or ferroresonance. Needless to

add that the electromagnetic analysis in such cases requires again new techniques and

methods.

The division of the papers is far from clear distinction of the papers’ topics and

content. It is a very rough distinction which gives prospective readers some suggestion

on how to find a paper of their personal interest.

Summarising this introductory remarks we, the Editors of the book, would like to

express our hope that the book you have in your hands will help the world-wide elec-

tromagnetic community, both academic and engineering, in better understanding elec-

tromagnetism itself and its application to technical problems.

At the end of these remarks let us be allowed to express our thanks to our col-

leagues who have contributed to the book by submitting their papers or/and by peer-

reviewing the papers at the conference as well as in the publishing process. We also

convey our thanks to IOS Press Publisher for their effective collaboration in giving this

very attractive shape of the book and its promoting. Let us also express our strong be-

lief that ISEF conference will maintain strong links with IOS Press in the future.

Ivo Dolezel Andrzej Krawczyk Sławomir Wiak

Chairman of the Scientific Secretary Chairman of the ISEF

Organising Committee Symposium

vi

Contents

Preface v

Ivo Dolezel, Andrzej Krawczyk and Sławomir Wiak

Chapter A. Fundamental Problems and Methods

A1. Fundamental Problems

Power Effect in Magnetic Lamination Taking into Account Elliptical Hysteresis

Approach 3

Kazimierz Zakrzewski

Study of Electromagnetic Field Properties in the Neighbourhood of the Metallic

Corners 8

Stanisław Apanasewicz and Stanisław Pawłowski

Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming 16

Jan Albert, Wolfgang Hafla, André Buchau and Wolfgang M. Rucker

Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in

Metals 21

Dagmar Faktorová

Electromagnetic Field Energy in Ferromagnetic Barriers 26

Ryszard Niedbała, Daniel Kucharski and Marcin Wesołowski

The Influence of Temperature on Mechanical Properties of Dielectromagnetics 34

Barbara Slusarek, Piotr Gawrys and Marek Przybylski

Influence of the Magnetic Anisotropy on Electrical Machines 39

M. Herranz Gracia and K. Hameyer

Analysis of Structural Deformation and Vibration of Electrical Steel Sheet by

Using Magnetic Property of Magnetostriction 47

Wataru Kitagawa, Koji Fujiwara, Yoshiyuki Ishihara and Toshiyuki Todaka

A2. Methods

Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction 53

J. Turowski, Xose M. Lopez-Fernandez, A. Soto Rodriguez

and D. Souto Revenga

Application of Logarithmic Potential to Electromagnetic Field Calculation in

Convex Bars 58

Stanisław Apanasewicz

Multi-Frequency Sensitivity Analysis of 3D Models Utilizing Impedance

Boundary Condition with Scalar Magnetic Potential 64

Konstanty Marek Gawrylczyk and Piotr Putek

vii

Very Fast and Easy to Compute Analytical Model of the Magnetic Field in

Induction Machines with Distributed Windings 72

Manuel Pineda, Jose Roger Folch, Juan Perez and Ruben Puche

Coupling Thermal Radiation to an Inductive Heating Computation 80

Christian Scheiblich, Karsten Frenner, Wolfgang Hafla and

Wolfgang M. Rucker

Consideration of Coupling Between Electromagnetic and Thermal Fields in

Electrodynamic Computation of Heavy-Current Electric Equipment 85

Karol Bednarek

Force Computation with the Integral Equation Method 93

Wolfgang Hafla, André Buchau and Wolfgang M. Rucker

Chapter B. Computer Methods in Applied Electromagnetism

B1. Computation Methods

Numerical Simulation of Non-Linear Electromagnet Coupled with Circuit to

Rise up the Coil Current 101

Slawomir Stepien, Grzegorz Szymanski and Kay Hameyer

Numerical Calculation of Power Losses and Short-Circuit Forces in

Isolated-Phase Generator Busbar 108

Dalibor Gorenc and Ivica Marusic

Numerical Methods for Calculation of Eddy Current Losses in Permanent

Magnets of Synchronous Machines 116

Lj. Petrovic, A. Binder, Cs. Deak, D. Irimie, K. Reichert and C. Purcarea

3-D Finite Element Analysis of Interior Permanent Magnet Motors with Stepwise

Skewed Rotor 124

Yoshihiro Kawase, Tadashi Yamaguchi, Hidetomo Shiota, Kazuo Ida and

Akio Yamagiwa

Advance Computer Techniques in Modelling of High-Speed Induction Motor 130

Maria Dems and Krzysztof Komęza

Computation of the Equivalent Characteristics of Anisotropic Laminated

Magnetic Cores 137

E. Napieralska-Juszczak, D. Roger, S. Duchesne and J.-Ph. Lecointe

Improving Solution Time in Obtaining 3D Electric Fields Emanated from High

Voltage Power Lines 144

Carlos Lemos Antunes, José Cecílio and Hugo Valente

Thermal Distribution Evaluation Directly from the Electromagnetic Field Finite

Elements Analysis 151

A. di Napoli, A. Lidozzi, V. Serrao and L. Solero

Coordination of Surge Protective Devices Using “Spice” Student Version 158

Carlos Antonio França Sartori, Otávio Luís de Oliveira and

José Roberto Cardoso

viii

B2. Numerical Models of Devices

Nonlinear Electromagnetic Transient Analysis of Special Transformers 167

Marija Cundeva-Blajer, Snezana Cundeva and Ljupco Arsov

Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine 175

Y. Tamto, A. Foggia, J.-C. Mipo and L. Kobylanski

Reduction of Cogging Torque in Permanent Magnet Motors Combining Rotor

Design Techniques 179

Andrej Černigoj, Lovrenc Gašparin and Rastko Fišer

Optimum Design of Linear Motor for Weight Reduction Using Response Surface

Methodology 184

Do-Kwan Hong, Byung-Chul Woo, and Do-Hyun Kang

Analytical Evaluation of Flux-Linkages and Electromotive Forces in Synchronous

Machines Considering Slotting, Saliency and Saturation Effects 192

Antonino di Gerlando, Gianmaria Foglia and Roberto Perini

Radiation in Modeling of Induction Heating Systems 202

Jerzy Barglik, Michał Czerwiński, Mieczysław Hering and

Marcin Wesołowski

Time-Domain Analysis of Self-Complementary and Interleaved Log-Periodic

Antennas 212

A.X. Lalas, N.V. Kantartzis and T.D. Tsiboukis

New Spherical Resonant Actuator 220

Y. Hasegawa, T. Yamamoto, K. Hirata, Y. Mitsutake and T. Ota

Chapter C. Applications

C1. Electrical Machines and Transformers

Influence of the Correlated Location of Cores of TPZ Class Protective Current

Transformers on Their Transient State Parameters 231

Elzbieta Lesniewska and Wieslaw Jalmuzny

Machine with a Rotor Structure Supported Only by Buried Magnets 240

Jere Kolehmainen

FEM Study of the Rotor Slot Design Influences on the Induction Machine

Characteristics 247

Joya Kappatou, Kostas Gyftakis and Athanasios Safacas

Concentrated Wound Permanent Magnet Motors with Different Pole Pair

Numbers 253

Pia Salminen, Hanne Jussila, Markku Niemelä and Juha Pyrhönen

Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines 259

Ioan-Adrian Viorel, Larisa Strete, Vasile Iancu and Cosmina Nicula

ix

Dynamic Simulation of the Transverse Flux Reluctance Linear Motor for Drive

Systems 268

Ioan-Adrian Viorel, Larisa Strete and Do-Hyun Kang

Influence of Air Gap Diameter to the Performance of Concentrated Wound

Permanent Magnet Motors 276

Pia Salminen, Asko Parviainen, Markku Niemelä and Juha Pyrhönen

Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different

Geometries 284

V. Fireţeanu

Analysis and Performance of a Hybrid Excitation Single-Phase Synchronous

Generator 294

Nobuyuki Naoe, Akiyuki Minamide and Kazuya Takemata

Numerical Calculation of Eddy Current Losses in Permanent Magnets of BLDC

Machine 299

Damijan Miljavec and Bogomir Zidarič

Analysis of High Frequency Power Transformer Windings for Leakage

Inductance Calculation 307

Mauricio Valencia Ferreira da Luz and Patrick Dular

Influence of the Stator Slot Opening Configuration on the Performance of

an Axial-Flux Induction Motor 313

Asko Parviainen and Mikko Valtonen

Characteristics of Special Linear Induction Motor for LRV 318

Nobuo Fujii, Kentaro Sakata and Takeshi Mizuma

Electromagnetic Computations in the End Zone of Power Turbogenerator 324

M. Roytgarts, Yu. Varlamov and А. Smirnov

C2. Actuators and Special Devices

The Impact of Magnetic Circuit Saturation on Properties of Specially Designed

Induction Motor for Polymerization Reactor 335

Andrzej Popenda and Andrzej Rusek

Electromagnetic Design of Variable-Reluctance Transducer for Linear Position

Sensing 343

J. Corda and S.M. Jamil

The Influence of the Matrix Movement in a High Gradient Magnetic Filter on

the Critical Temperature Distribution in the Superconducting Coil 350

Antoni Cieśla and Bartłomiej Garda

Macro- and Microscopic Approach to the Problem of Distribution of Magnetic

Field in the Working Space of the Separator 356

Antoni Cieśla

Electric Field Exposure Near the Poles of a MV Line 363

D. Desideri, A. Maschio and E. Poli

x

Study on High Efficiency Swithced Reluctance Drive for Centrifugal Pumping

System 370

Jian Li, Junho Cha and Yunhyun Cho

C3. Special Applications

Power Quality Effects on Ferroresonance 381

Luca Barbieri, Sonia Leva, Vincenzo Maugeri and Adriano P. Morando

FEM Computation of Flashover Condition for a Sphere Spark Gap and for

a Special Three-Electrode Spark Gap Design 388

Matjaž Gaber and Mladen Trlep

Recent Developments in Magnetic Sensing 396

Barbaros Yaman, Sadık Sehit and Ozge Sahin

Modelling of Open Magnetic Shields’ Operation to Limit Magnetic Field of

High-Current Lines 403

R. Goleman, A. Wac-Włodarczyk, T. Giżewski and D. Czerwiński

Selected Problems of the Flux Pinning in HTc Superconductors 410

J. Sosnowski

The Effect of the Direction of Incident Light on the Frequency Response of p-i-n

Photodiodes 417

Jorge Manuel Torres Pereira

3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis

Algorithm 425

S. Coco, A. Laudani, F. Riganti Fulginei and A. Salvini

Electro-Quasistatic High Voltage Field Simulation of Large Scale 3D Insulator

Structures Including 2D Models for Conductive Pollution Layers 431

Daniel Weida, Thorsten Steinmetz, Markus Clemens, Jens Seifert and

Volker Hinrichsen

Electromagnetic Aspects of Data Transmission 438

Liliana Byczkowska-Lipińska and Sławomir Wiak

Application of the Magnetic Field Distribution in Diagnostic Method of Special

Construction Wheel Traction Motors 449

Zygmunt Szymański

Author Index 457

xi

This page intentionally left blank

Chapter A. Fundamental Problems

and Methods

A1. Fundamental Problems


Power Effect in Magnetic Lamination

Taking into Account Elliptical Hysteresis

Approach

Kazimierz ZAKRZEWSKI

Institute of Mechatronics and Information Systems, Technical University of Lodz

Stefanowskiego 18/22 str., 90-924 Lodz, Poland

[email protected]

Abstract. On the grounds the author’s early works the analytical formulae for unit

active power (losses) and reactive power in magnetic lamination were presented.

The elliptical hysteresis approach of magnetic loops was assumed.

In particular, these powers have been refereed to magnetization frequency.

The universal functions F1, F

2 for the case of magnetic flux forcing and F

3, F

4 for

magnetic strength application on the booth side of lamination have been deduced

in the work.

Introduction

Despite many efforts, the full satisfactory analytical formulae for active and reactive

power in magnetic laminations used in electromagnetic devices and electromechanical

converters are not elaborated. The microscopic phenomena which we transferred for

macroscopic effects are not exact described analytically. The numerous approximations

which model static or dynamic hysteresis loop [1–3] not enable to calculate the power

losses or reactive power with exactness acceptable in electrical engineering. Author, for

dozen years prefers the use of equivalent elliptical hysteresis loops by approximation of

real magnetization characteristics which enable to take account the hysteresis and first

kind of eddy current losses. Thanks to introduction so called anomaly coefficient An the

analytical results may be adapt to total losses obtained experimentally [5,6].

Forced Magnetic Flux in Lamination

The introduction of elliptic hysteresis loops is connected with assumption that all func-

tion of electromagnetic field are sinusoidal in time.

The authors formulae deduced in [4] will need for the next discussion. In the pa-

per [4] the active and reactive power were refereed to lamination segment with side

surface area equal to 1 m2

, in this work will be recalculated for 1 kG of the lamination

weight. The measured active power may be expressed

( )

3

2

2

,

2

meas n m

m

k

P A Bav f

dφ φ

φ ξ

σ μ γ

= (1)

Advanced Computer Techniques in Applied ElectromagneticsS. Wiak et al. (Eds.)IOS Press, 2008© 2008 The authors and IOS Press. All rights reserved.doi:10.3233/978-1-58603-895-3-3

3

where: meas

Pφ

– measured active power, n

A – anomaly coefficient as function of ampli-

tude Bav (average induction in a cross section of lamination) and frequency f

m

k fπ μ σ= (2)

m

μ – magnetic permeability for Bav, σ – conductivity of lamination material, γ – spe-

cific weight of lamination material, d – thicknees of lamination, m

φ – forced magnetic

flux in lamination cross-section 2

1 d m⋅

m

Bav dφ = ⋅ [Wb/m] (3)

( ) ( )

( ) ( )

sinh sin

cosh cos

a akd b bkd

akd bkd

φξ

−

=

−

(4)

cos sin

2 2

a

δ δ

= + (5)

cos sin

2 2

b

δ δ

= − (6)

δ – angle of elliptical and symmetric hysteresis loop with amplitude Bav.

The value n

A was called anomaly coefficient because takes into account the addi-

tional losses in relation to calculated ones, as a result of elliptical and continuous alter-

nating magnetization inside the lamination material.

In the work [6], the coefficient An for the transformer lamination ET3 0,35 mm

was investigated and described. With a same approximation it is possible to assume the

average values of An coefficients in dependence on frequency in a range (5 ÷ 300) Hz

as a constant values. The function ( )n

A Bav is presented in Table 1.

The forced magnetic flux appears in laminated core of transformer by voltage exci-

tation.

Essential dependence in the transformer praxis is the total losses reference to the

frequency as a measurement method for hysteresis losses extrapolation in the case

0f → .

It will be indicated that frequently used extrapolation by application a direct line is

not correct.

Using (1)

Table 1. Average values of n

A for ET3 lamination

Bav T 0,2 0,5 1,0 1,1 1,3 1,5 1,7

n

A – 2,0 2,0 1,92 1,76 1,45 1,29 1,29

K. Zakrzewski / Power Effect in Magnetic Lamination4

( )meas

n

P P

A Bav

f f

φ φ

= (7)

where

( )2

2

2

m

m

P

kd

f d

φ

φ

π

φ ξ

μ γ

= . (8)

From (8) may be introduce the universal function (not dimensional)

( )1

F kdφξ= . (9)

This formula may be presented in dependence not dimensional value ( )2

kd . The

relation between f and ( )2

kd is

( )2

2

m

kd

f

dπ μ σ

= (10)

In the case of reactive power it is possible to introduce the function

( )2

F kdφ

ψ= (11)

and

( )2

2

2

m

m

Q

kd

f d

φ

φ

π

φ ψ

μ γ

= (12)

where

( ) ( )

( ) ( )

sinh sin

cosh cos

b akd a bkd

akd bkd

φψ

+

=

−

(13)

The reactive power was less investigated than active power. From author’s experi-

ence results that an adaptive coefficient for measured power is equal to HI/H

m relation,

where HI – first harmonic amplitude, H

m – amplitude of magnetic strength during sinu-

soidal change of induction with amplitude Bav in conditions of the real hysteresis loop

magnetization.

measI

m

Q QH

f H f

φ φ

= (14)

The functions F1, F

2 are illustrated in Fig. 1.

K. Zakrzewski / Power Effect in Magnetic Lamination 5

Forced Magnetic Field Strenght on Lateral Surface of Lamination

Another means of practical generation of electromagnetic field in lamination is an exci-

tation by electrical current with forced magnetic field strength on lateral surfaces of

lamination.

The adequate formulae on the base [4] are for active power (losses)

( )

2mH H

m

P

H

f kd

π μ ζ

γ

= (15)

where:

( ) ( )

( ) ( )

sinh sin

cosh cos

H

a akd b bkd

akd bkd

ζ

−

=

+

(16)

for reactive power

( )

2mH H

m

Q

H

f kd

π μ ψ

γ

= (17)

where:

( ) ( )

( ) ( )

sinh sin

cosh cos

H

b akd a bkd

akd bkd

ψ

+

=

+

(18)

Unfortunately, the formulae (15), (17) with adaptation to results of measurements

was not yet investigated (open problem).

Figure 1. Diagrams of functions F1, F

2.

K. Zakrzewski / Power Effect in Magnetic Lamination6

Analogically to F1, F

2 it is possible to introduce the universal functions F

3 =

ζH /(kd), F

4= ψ

H /(kd) which are illustrated in Fig. 2.

Conclusion

The universal functions F1 and F

2 may be used for investigation of losses and reactive

power in a wide range of magnetic flux frequency in lamination (voltage excitation).

Adequately, the functions F3 and F

4 are interesting for this investigation in a case of

forced magnetic field strength on lateral surfaces of lamination (current excitation).

The analytical formulae may be helpful in the praxis for design of different elec-

tromagnetic devices.

References

[1] D.C. Jiles, D.L. Atherton: “Theory of Ferromagnetic Hysteresis”, Journal of Magnetism and Magnetic

Materials 61 (1986), North-Holland, Amsterdam, pp. 48-60.

[2] D. Mayergoyz: “Mathematical Models of Hysteresis”, Springer-Verlag, New York 1991.

[3] J.K. Sykulski (editor): “Computational Magnetics” Chapman and Hall, 1995 London, Glasgow, Wein-

heim, New York, Tokyo, Melbourne.

[4] K. Zakrzewski: “Berechnung der Wirk und Blindleistung in einem ferromagnetischen Blach unter

Berücksichtigung der Komplexen magnetischen Permeabilität”, Wiss. Z.TH Ilmenau (1970), H.5,

s. 101-105.

[5] K. Zakrzewski: “Method of calculations of unit power losses and unit reactive power in magnetic lamina-

tions in a wide range change of induction and frequency”, Proceedings of ISEF’99 – 12th International

Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering, ISEF’99,

Pavia, Italy, September 23-25 1999, s. 208-211.

[6] K. Zakrzewski, W. Kubiak, J. Szulakowski: „Wyznaczanie współczynnika anomalii strat w blachach

magnetycznych anizotropowych, Prace Naukowe IMNPE Politechniki Wrocławskiej, Studia i Materiały

Nr 20, Wrocław 2000, s. 299-305.

Figure 2. Diagrams of functions F3, F

4.

K. Zakrzewski / Power Effect in Magnetic Lamination 7

Study of Electromagnetic Field Properties

in the Neighbourhood of the Metallic

Corners

Stanisław APANASEWICZ and Stanisław PAWŁOWSKI

Technical University in Rzeszów, Chair of Electrodynamics and

Electrical Machinery Systems, Poland

Study of electromagnetic field in the neighbourhood of the metallic corners is an aim of

the present paper. Concave and convex corners are considered.

Introduction

It is known that in the case of depth of electromagnetic field penetration in the metal

that is small in comparison with radius of curvature of metal surface, calculation of the

field distribution can be simplified. Impedance boundary conditions are applied. Aim

of that is simplification of Helmholtz equation in the metallic area: term with second

derivative in the direction tangential to the boundary is rejected from this equation and

the second derivative in the transverse direction is left. Such a simplified equation is

solved in the open way and it causes impedance boundary conditions in the air area.

Normal derivative of z component of vector potential A is proportional to A ie

r

A A

n

α

μ

∂

=

∂

. In the neighbourhood of the corners such simplifications are not possible

and in such cases calculation of electromagnetic field distribution is more complicated.

Primary aim established by authors of the present paper is to find adequate simpli-

fications if there is metallic area of large curvature, radius of which is small in com-

parison to the depth of field penetration. It turned out that this problem looks differ-

ently for the concave and convex corners. Therefore, our first task is to study the essen-

tial differences occurring in these both cases. At first, we analyze the corners creating

angles π/2 lub π/3; however we omit the analysis of rounded corners. We consider

three types of field excitation: a) electrostatic, b) incidence of straight flat wave, c) ex-

citation of eddy currents in the metal by fields generated by sinusoidal currents flowing

in wires in air area. In the mentioned case, solution of adequate Helmholtz and Laplace

equations can be presented in the form of Fourier integrals and determination of inte-

grand comes down to solve an equation of Fredholm integral equations of the second

kind.

In this paper we are restricted to two first variants.

Advanced Computer Techniques in Applied ElectromagneticsS. Wiak et al. (Eds.)

IOS Press, 2008© 2008 The authors and IOS Press. All rights reserved.

doi:10.3233/978-1-58603-895-3-8

8

Study of Electrostatic Field

We assume that electrostatic field is generated by charges Q and –Q placed for the

simplicity symmetrically in relation to the metallic walls in points (a,a) and (b,b)

(Fig. 1a and 1b).

Electrostatic field is described by scalar potential ( , )x yϕ ϕ= that fulfils Laplace’s

equation. This potential can be presented in the form of two components:

0 i

ϕ ϕ ϕ= + (1)

First of them presents the field generated by charges Q, –Q and the second presents

influence of metallic walls. Term ϕ0 is known and it is presented in the following way:

2 2

1 2

0 2 2

0 0 0

cos sin( ) ( )

ln , 0

4 2( ) ( )

x

g x g yQ x a y a Q

e d x

x b y b

τ

τ τ

ϕ τ

πε πε τ

∞

+− + −

= = <

− + −

∫ (2)

where:

1

cos cos

sb sa

g e sb e sa

− −

= −

2

sin sin

sb sa

g e sb e sa

− −

= −

However component ϕi can be presented in the following way:

0

1

( ) sin sin , 0, 0x y

i

D e y e x d x yτ τ

ϕ τ τ τ τ

τ

∞

− −

⎡ ⎤= + > >⎣ ⎦∫ (3)

=γ

∞=γ

a

b

a b x

y

0=γ

∞=γ

a

b

a b x

y

Q

Q− Q−

Q

a) Concave corner b) Convex corner

Figure 1. Sketch of studied corners.

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties 9

In the case of concave corner, D function is the only function that should be calcu-

lated. Potential ϕ must be equal zero on the surface of metal: ( ,0) (0, ) 0x yϕ ϕ= =

what is tantamount to the zeroing of tangential components of electrostatic field

1 2

( , ,0) ( , ,0)x y

E E E ϕ ϕ= = − −

, so the following condition should be fulfilled

2 2 2 2

00

( )cos ( )

2 ( ) ( )

Q x b x a

D xd g x

x b b x a a

τ τ τ

πε

∞

⎡ ⎤− −

= − =⎢ ⎥

− + − +⎣ ⎦

∫ (4)

hence

0

2

( )cosD g x xdxτ

π

∞

= ∫ (5)

D function and ϕi

potential can be calculated in the overt form. Ultimately, omit-

ting laborious computational transformations, we can present solution in the form of

mirror images:

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )i

x y b b a a b b b b a a a aϕ ϕ ϕ ϕ ϕ ϕ ϕ∗ ∗ ∗ ∗ ∗ ∗

= − − + − − + − + − − − − − (6)

where 2 2

0

( , ) ln ( ) ( )

4

Q

a b x a y bϕ

πε

∗

⎡ ⎤= − + −⎣ ⎦

.

Fictitious Q charges are located in points (a,–a), (b,–b), (–a,–a), (–b,–b), (–a,a),

(–b,b).

In the event of convex corner in the area x > 0, y > 0 we accept ϕi in the form of

Formula (3) and in the area x < 0, y >0:

0

1

( ) sin ( ) sinx y

i

R e y R e x d

τ τ

ϕ τ τ τ τ τ

τ

∞

−

∗

⎡ ⎤= +⎣ ⎦∫ (7)

We assume that ϕi and ϕ

ix are continuous on the line x = 0, y > 0 in order to calcu-

late functions D, R, R*. Additionally, on the line y = 0, x < 0 the condition

0

0i

ϕ ϕ+ =

must be fulfilled.

On the basis of these conditions, we obtain:

[ ]2 2

0

2

, ( ) ( )

ds

R D D R D s R s

s

τ

π τ

∞

∗

= + = −

+

∫ (8)

2 2 2 2

00

cos ( ) , 0

2 ( ) ( )

Q x b x a

R xd g x x

x b b x a a

τ τ

πε

∞

∗

⎡ ⎤− −

= − = <⎢ ⎥

− + − +⎣ ⎦

∫

hence:

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties10

[ ]2 2

( ) ( ) ( )

ds

D D s R s

s

τ

τ

π τ

∗

= −

+

(9)

2 2 2 2 2

0 0

( ) cos

( ) ( )

Q s b s a

R sds

s b b s a a

τ τ

π ε

∞

∗

⎡ ⎤+ +

= −⎢ ⎥

+ + + +⎣ ⎦

∫ (10)

By elimination of R* from Eqs (9), (10) we obtain integral equation for determina-

tion of D function:

02 2

0

0 2 2

0

( ) ( ) ( )

( ) ( )

ds

D D s D

s

ds

D R s

s

τ

τ τ

π τ

τ

τ

π τ

∞

∞

∗

= −

+

=

+

∫

∫

(11)

Equation (11) is solved numerically.

Enclosed Figs 2–4 illustrate results of calculations.

0 1 2 3 4 5

-2000

-1000

0

1000

2000

3000

4000

5000

for cocave corner

for convex corner

E1

(V

/m)

x=y (m)

Figure 2. The E1 component of electrical field distribution on y = x line.


1

2

3

4

5

-3000

-2000

-1000

0

1000

2000

3000

4000

1

2

3

4

5

E2

(V/m

)

y(m

)

x (m)

Figure 3. The E2 component distribution in the case of concave corner.

-4

-2

0

2

4

-1000

0

1000

2000

3000

4000

1

2

3

4

5

E2

(V/m

)

y

(m)

x (m)

Figure 4. The E2 component distribution in the case of convex corner.


Study of Diffraction of Straight Waves on the Concave and Convex Corners

We consider the following wave structure of the electromagnetic field:

1 2

(0,0, ( , )) , ( , ,0)E E x y H H H= =

(12)

2

1 2

0 0

, , ,

yx

jE jE

E k E H H k

c

ω

ωμ ωμ

Δ = − = = − =

In the event of concave corner we have also elementary solution of the problem. If

sinusoidal straight wave with amplitude E* falls at an angle of ϕ (in relation to x axis)

on the wall of metallic area (what can be verified easily) complete E field can be de-

termined by one elementary formula:

[ ]2 2cos ( cos sin ) cos ( sin cos )E E k x y k y xϕ ϕ ϕ ϕ∗

= + + − (13)

Course of wave is illustrated on Figs 5 and 6.

However in the event of convex corner, situation is completely different. Solution

of Helmholtz equation (12) requires solution of integral equation.

We present solution of Eq. (12) in the following form:

0 i

E E E= + (14)

1 2

0

2 4

0

5 6

0

( ) sin ( ) sin , 0, 0

( ) sin ( ) sin , 0, 0

( ) sin ( ) sin , 0, 0

py px

py px

i

py px

W e x W e y d x y

E W e x W e y d x y

W e x W e y d x y

τ τ τ τ τ

τ τ τ τ τ

τ τ τ τ τ

∞

− −

∞

−

∞

−

⎧

⎡ ⎤+ ≥ ≥⎪ ⎣ ⎦

⎪

⎪⎪

⎡ ⎤= + ≤ ≥⎨ ⎣ ⎦

⎪

⎪

⎪ ⎡ ⎤+ ≤ ≤⎣ ⎦

⎪⎩

∫

∫

∫

(15)

Figure 5. Concave corner – no shadow; elemen-

tary solution.

Figure 6. Convex corner – there is a shadow zone,

field is described by integral solution.


( cos sin )

0

jk y x

E E e

ϕ ϕ− −

∗

= (16)

where E* – amplitude of falling wave, ϕ – angle between direction of falling wave and

y axis. E0 term represents complex amplitude of falling wave and E

i represents the reac-

tion of the metallic wall.

We assume the conditions presented below in order to calculate Wi integrands:

a) Continuity of Ei and components of magnetic field H

1, H

2 on lines x = 0,

y ≥ 0 and y = 0, x ≥ 0.

b) Zeroing of E on the metallic surface: E(0, y) = 0, y ≤ 0 and E(x, 0) = 0, x ≤ 0.

We omit detailed calculations and present final result:

W1=W

5, W

2=W

4

2 6

1 2 2 2

0

( )

( )

s W W ds

p W

s k

τ

τ

π τ

∞

−

=

+ −

∫ (17)

1 3

2 2 2 2

0

( )

( )

s W W ds

p W

s k

τ

τ

π τ

∞

−

=

+ −∫ (18)

[ ]6 2 2 2

2

( cos ) ( cos )

cos

W E j k k

k

τ

δ τ ϕ δ τ ϕ

π τ ϕ∗

⎧ ⎫

= + − − +⎨ ⎬−⎩ ⎭

(19)

[ ]3 2 2 2

2

( sin ) ( sin )

sin

W E j k k

k

τ

δ τ ϕ δ τ ϕ

π τ ϕ∗

⎧ ⎫

= + − − +⎨ ⎬−⎩ ⎭

(20)

where δ – δ-Dirac’s function.

After elimination of W3

and W6 functions from expressions (17) and (18), we ob-

tain system of two integral equations:

2 1

1 1 2 22 2 2 2 2 2

0 0

( ) ( )

( ) , ( )

sW s ds sW s ds

pW pW

s k s k

τ τ

ξ τ ξ τ

π πτ τ

∞ ∞

= + = +

+ − + −

∫ ∫ (21)

where:

2 2

1 2 2 2

cos

( )

sin

k jk

E

k

τ τ ϕ

ξ τ

π τ ϕ∗

⎡ ⎤− +

= − ⎢ ⎥

−⎢ ⎥⎣ ⎦

,

2 2

1 2 2 2

sin

( )

cos

k jk

E

k

τ τ ϕ

ξ τ

π τ ϕ∗

⎡ ⎤− −

= − ⎢ ⎥

−⎢ ⎥⎣ ⎦

.

Instead of system of two equations with two unknowns W1, W

2 (21) one can solve

two equations with one unknown assuming u1= W

1+W

2, u

2= W

1–W

2.

These equations have the following form:


1

1 1 22 2 2

0

( )

( ) ( ) ( )

su s ds

pu t

s k

τ

ξ τ ξ τ

π τ

∞

= + +

+ −

∫ ,

2

1 1 22 2 2

0

( )

( ) ( ) ( )

su s ds

pu t

s k

τ

ξ τ ξ τ

π τ

∞

= + −

+ −

∫ .

Observed Characteristics of the Field in the Concave and Convex Corners

a) Characteristics of electromagnetic field in the corners are significantly differ-

ent depending on the sign of curvature. In the presented examples in the con-

cave corner we obtain solutions of adequate equations in the elementary form;

however in the event of convex corner solution of these equations causes inte-

gral equations, which can be solved only numerically.

b) Infinite values of field in the corners are not observed, what takes place in

some other cases (for example if potentials are set on boundaries of the area).

But extreme values of field are observed in corners.

c) If corners presented on Fig. 1 are rounded off with use of open arc, it turns out

that solution of applied Helmholtz equations (in the case of eddy currents) is

easier for the concave corner (centre of the rounding arc is outside of the me-

tallic area) than for the convex case (mentioned centre is inside the metallic

area).


Eddy Currents and Lorentz Forces

in Pulsed Magnetic Forming

Jan ALBERT, Wolfgang HAFLA, André BUCHAU and Wolfgang M. RUCKER

Institute for Theory of Electrical Engineering

Pfaffenwaldring 47, 70569 Stuttgart, Germany

[email protected]

Abstract. This paper deals with the numerical computation of eddy currents and

the forces they cause. These effects are of special interest when considering pulsed

magnetic forming, which is a technique to form thin metal sheets or pipes. A high

transient current in a coil near to the work piece excites eddy currents and the as-

sociated Lorentz forces press the work piece into a die.

Introduction

Pulsed magnetic forming (PMF) is a quite old technique developed in the first half of

the twentieth century. In recent years some new ideas concerning different applications

and the use of field formers lead to increased efforts in the research of pulsed magnetic

forming. The use of field formers results in an extended lifetime of the excitation coils

but also offers the possibility to use the same installation for a large amount of applica-

tions when using differently shaped field formers [1]. Numerical simulations take a big

part in the development of field formers because they offer a cheap and effective way

to show the advantages and drawbacks of the suggested field former constructions and

can therefore significantly reduce development costs. In this paper several ways of the

numerical computation of the occurring eddy currents and Lorentz forces are consid-

ered.

Computation of Eddy Currents

The governing equation for eddy current problems neglecting displacement currents

and considering only non-ferromagnetic, and conductive materials is

0

1

curlcurlC

t

σ

μ

∂

+ =

∂

A A J , (1)

where A is the magnetic vector potential, Jc the impressed current density, σ the con-

ductivity and µ0 the magnetic permeability. Equation (1) can directly be implemented

as a formulation for FEM programs. A drawback of simulating PMF with a finite ele-

ment method (FEM) lies in the large amount of nodes that is required to describe the

geometrical problem properly. Particularly meshing the air domain takes much effort.



doi:10.3233/978-1-58603-895-3-16

16

Other problems result from the fact that the work piece deforms and one has to create a

new mesh for every time-step. So, using a boundary element method (BEM), which

requires meshing of the surface of the bodies, only, or at least an integral equation

method (IEM) with the advantage of an unmeshed air region is suggested. Due to the

fact that one has to deal only with nodes on the surfaces, BEM-formulations are more

complex than FEM-formulations and have to be solved with a larger effort per node,

but the total number of nodes is significantly lower, and the computational costs reduce

for large problems especially for bodies with a small relation between surface and vol-

ume.

Three Examples of Integral Equation Method Formulations

In recent years different BEM formulations have been developed. Depending on the

considered problem one has to choose between several formulations and has to pick out

the one that fits best to the actual requirements.

A Direct BEM Formulation Based on Two Field Quantities

Equations (2) and (3) are presented as an example for a pure BEM-formulation. In free

space the governing equation for the magnetic field is

( )

0

1

( )

2

1

( ) grad ( , ) curl ( ) grad ( , ) d

j

( ) grad ( , ) d

e

s t s e s s t s e s

c e e s

G G

G

Γ

Ω

Γ

ωμ

Ω

+

⎧ ⎫⎛ ⎞⎪ ⎪× × + ⋅⎨ ⎬⎜ ⎟

−⎪ ⎪⎝ ⎠⎩ ⎭

= ×

∫

∫

H r

n H r r r n E r r r

J r r r

, (2)

where Г is the surface of the eddy current body and Ω the volume of the coil, Jc is the

impressed current density inside the coil. So, on the right side of Eq. (2) we have the

Biot-Savart integral, which describes the magnetic field of the exciting coil. For the

eddy current region one obtains an equation for the electric field [2]

( ) ( ) 0

1

( ) ( ) grad ( , ) j ( ) ( , ) d 0

2e s t s e s s t s e s

G G

Γ

ωμ Γ− × × + − × =∫E r n E r r r n H r r r ,

(3)

where H and E are the complex magnetic and electric field strengths with Ht and E

t as

their tangential components, G is Green’s function and n is the normal vector pointing

outwards the considered region, re and r

s are the coordinate vectors of the evaluation

and the source location, respectively. A drawback of this formulation is based on the

fact that it deals with problems in the frequency domain, while in PMF one has to con-

sider transient currents. Of course, this problem can be solved with a Fourier transfor-

mation at high computation costs.

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming 17

A Minimum Order Boundary Integral Equation

In an indirect formulation developed in the 1970’s a solution vector consisting of the

values of the imaginary volume current density J at the surface of the eddy current

body and the imaginary magnetic surface charge density M

σ at the same surface is

computed [3].

(1 j)1 1

( ) ( ) grad e d

2

1 1

( ) grad d 2 ( )

2

se

k

e s s

se

M s s s coil e

se

Γ

Γ

Γ

π

σ Γ

π

− +

⎛ ⎞⎛ ⎞

+ × ×⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞

− × = − ×⎜ ⎟⎜ ⎟

⎝ ⎠

∫

∫

r

J r n J r

r

r n n H r

r

(4)

3

(1 j)

1

( ) ( ) d

2

1 1

( ) grad e d 2 ( )

2

se

se

M e M s s

se

k

s s s coil e

se

Γ

Γ

σ σ Γ

π

Γ

π

− +

⎛ ⎞

⎜ ⎟+ ⋅

⎜ ⎟

⎝ ⎠

⎛ ⎞⎛ ⎞

+ ⋅ × = − ×⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∫

∫

r

r

r r n

r

n J r n H r

r

(5)

Here, Hcoil

is the primary complex magnetic field raised by the exciting coil and com-

puted with Biot-Savart law,

1

2

k σμω= , rse

is the difference beetween rs and r

e, and

the other expressions have the same meaning as in the equations above. The total mag-

netic field consists of the primary magnetic field and the secondary field Hsec

raised by

effects in the eddy current body

sectotal coil

= +H H H . (6)

In free space one obtains

sec, ,

gradFS M FS

ϕ= −H , (7)

with

,

( )1

( ) d

4

M s

M FS e

seΓ

σ

ϕ Γ

π

= ∫

r

r

r

. (8)

The magnetic field in free space can be described by scalar variables σM

, because it

is irrotational; σM

are the sources of the magnetic field strength, but they should not be

mistaken as the magnetic charges in a ferromagnetic body (they are still present even if

µr= 1), though both are treated equivalent.

In the eddy current body the magnetic field is rotational, and, therefore, must be

described by a vector variable, which is the imaginary surface current density J at the

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming18

surface of the eddy current region. The magnetic field inside the eddy current region

can be obtained by

(1 j)

sec,

1 1

( ) curl ( ) e d

4

se

k

EB e s

seΓ

Γ

π

− +

⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∫r

H r J r

r

(9)

The physical values of the eddy current density can directly be achieved by apply-

ing Maxwell’s law

, sec,

curl( ) curl( )phys EB total EB coil

= = +J H H H (10)

The advantage of this formulation is found in the small size of the matrix, due to

the four degrees of freedom. One decisive drawback is the fact that a Laplace transfor-

mation is necessary for considering transient problems.

A Volume Integral Equation Method for Transient Computations

From the induction law follows

0

t

∂

+ − =

∂

E A v× B (11)

Under the assumption that motion raised induction can be neglected, and that the

total magnetic vector potential A consists of the impressed coil field Acoil

and the re-

duced vector potential Aeddy

raised by eddy currents one obtains (12) and (13)

( ) 0coil eddy

t

∂

+ + =

∂

E A A , (12)

( ) ( )

d ( )

4

e s

coil e

se

t tΩ

μ

Ω

σ π

⎛ ⎞∂ ∂

+ = −⎜ ⎟⎜ ⎟

∂ ∂⎝ ⎠

∫

J r J r

A r

r

. (13)

This formulation has the major advantage that either a time stepping method – a

so-called Euler method – or a frequency domain method can be implemented with ease.

Unfortunately, this method requires a large amount of boundary and volume elements,

but is suited for matrix compression techniques like the fast multipole method [4].

Other enhancements, for example inclusion of motion raised induction effects, are pos-

sible but within the first studies motion raised induction shall be neglected, because it is

assumed that, in PMF, the induced forces of the first few moments are decisive for the

result. At this time the work piece is not really moving yet.

Computation of the Occurring Forces

Due to the fact that in PMF bodies have to be considered while they are deforming one

cannot use Maxwell´s stress tensor, which is the most popular way to compute forces

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming 19

raised by electromagnetic fields. Instead, a method has to be found, which returns a

force density. The direct computation of the Lorentz forces is an accurate way to obtain

a force density f, which can be used as an input parameter for a simulation of the struc-

tural deformation

eddy

f = J × B (14)

The necessary field values of the eddy current density Jeddy

and the magnetic flux

density B can directly be obtained from FEM results or from post-processing the BEM

results.

Conclusion

To predict the deformation of a work piece caused by a PMF process one has to couple

electromagnetic and structural mechanics simulations. Several FEM, BEM, or hybrid

products are applicable to this problem. For small problems FEM simulations are pre-

ferred. The larger a problem gets, e.g. when regarding several field formers, BEM

simulations become more attractive, due to the reduced number of nodes and the de-

creased memory requirements.

References

[1] R. Hahn, Werkzeuge zum impulsmagnetischen Warmfügen von Profilen aus Aluminium- und Magnesi-

umlegierungen, Dissertation, Berlin, 2004.

[2] C. J. Huber, Numerische Berechnung dreidimensionaler elektromagnetischer Felder mit Integral-

gleichungsverfahren, Dissertation, TU Graz, 1998.

[3] I. D. Mayergoyz, “Boundary integral equations of minimum order for the calculation of three-

dimensional eddy current problems”, IEEE Trans. Mag., vol. 18, no. 2, pp. 536-539, march 1982.

[4] G. Rubinacci, A. Tamburrino, S. Ventre, F. Villone, “A fast 3-D multipole method for eddy-current

computation”, IEEE Trans. Mag., vol. 40, no. 2, pp. 1290-1293, march 2004.

J. Albert et al. / Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming20

Applicability of Microwave Frequencies

to the Evaluation of Deeper Defects

in Metals

Dagmar FAKTOROVÁ

University of Žilina, Faculty of Electrical Engineering, Department of Measurement

and Applied Electrical Engineering, Univerzitná 1, Žilina, 010 26, Slovak Republic

[email protected]

Abstract. The paper deals with non-destructive (NDT) microwave measurement

of defects in metal samples exploiting the waveguide features at the defect depth

evaluation. In this article some results concerning their evaluations regarding to

microwave access are shown. More measurements were performed to evaluate the

geometry of defects in metal samples. Apart from established methods two new

unusual microwave connections are presented and the results with their use at de-

fects examination are given and compared with the previous results. Their advan-

tages are discussed and some proposals for their utilizations are given.

Introduction

Application of microwave technique in defectoscopy at metal materials is relatively a

new area, in which the research character of works is prevailing, while new access are

emerging and their utilization spectrum is increasing. The testing ability of dielectric

materials with microwaves is given by the natural property as their ability to penetrate

through such materials and we have devoted the adequate attention in this field,

too, [1].

In another works we aimed ourselves to finding geometry and some additional de-

fect properties in metals. After the experimentally confirmation the fact that the defect

can be examinate as a special waveguide section we engaged in examination of its im-

pedance properties enabling to obtain more precision information about the defect, [2].

The next experiments were directed at the defect depth settling by utilizing of mi-

crowave knowledge. For this purpose were predominately used classical microwave

measuring technique exploiting reflected signal properties either registered directly by

some elements of the microwave line (e.g. ferrite circulator) or by means of standing

wave ratio (SWR) measurement and subsequent adoption of measured values (imped-

ance character, minimum standing wave shift and like that), [2].

As the microwave technique disposes of the extensive range of measuring meth-

ods, we directed our attention in next experiments at giving precision to measured re-

sults and on the one hand at the shape of the reflected signal from the defect and on the

other hand at its depth.


21

Experimental Background

For the sake of information of about the link-up this article to our works in the micro-

wave NDT area we will briefly mention the experimental results which led to the

achieved results presentation.

The basic pieces of knowledge were the measurements aimed at the influence of

the defect depth on the reflected signal amplitude, [3]. For this purpose two measure-

ments for comparison were carried out:

1. measurement of the reflected signal on a lossless waveguide,

2. measurement of the reflected signal on a sample with the defect.

The reflected signal was in both cases recorded through the ferrite circulator and

the measurements were performed for such short-lime piston positions on the lossless

waveguide which corresponded to the respective defect depth. The comparison of the

both measurements is in the Fig. 1.

One can concesive of the decreasing amplitude of the reflected signal at the defect

depth settling at ( )g

λ

2 1

4

n + , (g

λ is length of wave in waveguide and n is integral

number), distant maxima and consequently also about up to what depths the reliable

information about the defect with particular material properties can be obtained at the

given measurement precision.

Experimental Results

After verifying our theoretical assumptions, [2] we proposed two new methods for the

defect evaluation and have also tested them. The first method is based on a special mi-

crowave line connection using the balancing principle with the magic T as a passive

element from microwave technique with wide possibilities, [4]. More detailed informa-

f=10,1 GHz

0

20

40

60

80

100

0 20 40 60 80 100 120 140

hĺbka defektu [mm]

amplitúda[a.u.]

defect depth [mm]

am

plitu

de

[a

. u

.]

lossless

waveguide

defect

Figure 1. Dependence of reflected signal amplitude from the defect depth in metal and from the position of

the shorted-line piston in lossless waveguide.

D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals22

tion about magic T can be found e.g. also in [4]. We will recapitulate only its properties

from the point of view of our measurement. Magic T, Fig. 2 is symmetrical with regard

to the plane crossing through the axes of the ports C and D. Microwave generator will

be connected to the port C and the indicator to the port D. If the ports A and B are

loaded with equal loads, then the energy from generator is divided in port A and B and

in the port D energy does not show.

If the ports A and B are not the same, then a part of energy in consequence of non

symmetry in the fields gets into D and the reading on the indicator will depend from

the degree of the load difference in ports A and B. The same is valid when the signals

are reflected from loads in ports A and B. We have made use of these properties in two

ways:

a) for obtaining similar information about the defect as at classical measure-

ments, [3],

b) for obtaining direct information about the defect depth.

The measurements were performed in the connection illustrated in Fig. 3.

In the case according to Fig. 3a with the matched load in the port A and the indica-

tor in the port D, the signal indicated in the port D will be proportional to the reflection

coefficient from the port B. This way in our measurement port B was terminated with

the open-waveguide probe and the sample with a defect (width 0,5 mm, depth 8,5 mm)

was shifted in front of it. Corresponding graph is plotted in the Fig. 4, curve a.

Measurement in connection according to Fig. 3b was performed for the purpose of

getting a facility for simple measuring of the crack depth. To the port A was connected

an artificial defect simulating the real one with the continuously changing depth. To the

port B were gradually attached samples with known depths and for each depth the

bridge was balanced (port D without signal) and corresponding reading recorded. This

way the calibration was obtained and it is plotted in the Fig. 5, (dotted line – measured,

full line – idealized).

Among other things we were interested in the possibility cavity resonator using for

the examination of defects in metal samples. For this purpose we have used our experi-

ences with cavity resonators, [2] where also basic theoretical information are presented.

We supposed that at suitable resonator binding on the defect as an impedance element

it will influence the resonator’s quality factor Q and its resonant frequency.

A

D

B

C

matched

load

input of

mw signal

crystal

detector

input of

mw signal

calibrated

load

open

waveguide

sample

with defect

a)b)

DC DC

A

B

A

B

Figure 2. Magic T. Figure 3. Magic T connection for the defect depth measurement.

D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals 23

For this purpose we have carried out an experiment with the resonator connected

on standard microwave line. The resonator was connected on standard microwave line

in the transmission way and one resonator aperture was terminated with an open

waveguide probe. Resonator was tuned in resonance. In front of the probe was shifted a

metal sample with the defect (width 0,5 mm, depth 8,5 mm). The behavior of the reso-

nator was followed by means of a signal brought out through a loop antenna. The

measured values are plotted in the Fig. 4, curve b. We have plotted in Fig. 4 for com-

parison also the curve c obtained by previous familiar way (through the ferrite circula-

tor).

Conclusions

Further to the theoretical basis and our previous experiments [2,3] there are in this pa-

per presented new experiments which can provide new possibilities for defect in metal

f =10,1 GHz

0

20

40

60

80

100

120

-15 -10 -5 0 5 10 15

probe position [mm]

am

plitu

de

[a

.u

.]

a b c

Figure 4. Dependence of reflected signal amplitude from the defect.

1

1,5

2

2,5

3

3,5

4

0 5 10 15 20

defect depth [mm]

calibration data [m

m]

Figure 5. Calibration curve for magic T.

D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals24

samples evaluation. Especially the “magic T” method seems to be very effective

whether the question is the shape of crack determining or very quick method not de-

manding any computation but only using suitable calibrated aid. From the graph com-

paring three methods the resonator method is workable in spite of the fact that in Fig. 4

it does not show markable difference. The presented sensitivity was given by the qual-

ity factor of the used resonator, and can be improved by using a resonator with high

sensitivity (approximately of 15000–20000) and this is not its only feature because also

the detuning of the resonator can be very sensitive representation of the crack depth.

Acknowledgement

The author would like to thank MSc. Pavol Žirko director of High School for Agricul-

ture and Fishing in Mošovce for technical help at realization of experiments.

References

[1] D. Faktorová, Microwave Nondestructive Testing of Dielectric Materials, Advances in Electrical and

Electronic Engineering, ISSN 1336-1376, Vol. 5, No. 1-2, pp. 230-233, 2006.

[2] D. Faktorová, Using of Microwaves at Investigation of Solid Materials Inhomogenities, Proceedings of

APCNDT 2006, 12th Asia – Pacific Conference on Non-Destructive Testing, Auckland, New Zealand,

November 5-10, 2006, http://www.ndt.net/article/apcndt2006/index.htm, 4 pp.

[3] D. Faktorová, Interaction of Solid Materials Inhomogenities with Microwaves, Proceedings of EDS

2006, Electronic Devices and Systems, Brno, Czech Republic, ISBN 80-214-3246-2, September 14-15,

2006, pp. 388-393.

[4] B.M. Maškovcev, K.N. Cibizov, B.F. Emelin, Teorija volnovodov, Moskva-Leningrad: Nauka, 1966.

D. Faktorová / Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals 25

Electromagnetic Field Energy in

Ferromagnetic Barriers

Ryszard NIEDBAŁA, Daniel KUCHARSKI and Marcin WESOŁOWSKI

The Institute of Electrical Power Engineering, Warsaw University of Technology

Abstract. This paper refers to electromagnetic field propagation around high-

current circuits. The effects of its influence on conducting bodies that are seals and

casings of electric devices are discussed. These barriers which are the construc-

tional parts are made of ferromagnetic materials – good absorbers of energy trans-

formed into heat in their structures. Intensity of absorbance depends on the dis-

tance between the barrier and source of radiant energy. Limitation of diffusing en-

ergy by absorbing it requires a high accuracy quantitative analysis to assure a high

value of absorption coefficient and optimal power density of heat sources. The

analysis results of basic geometries are shown.

Introduction

Production, transmission, processing or reception of electric energy are closely con-

nected with formation and propagation of electromagnetic field (EM) around sources,

feeders and receivers of energy. In most cases those disturbances are identified with

electromagnetic interferences. From electromagnetic compatibility standpoint, there is

a strong tendency to limit the influence of EM interferences. Considering biological

and thermal danger for living organisms and mechanisms (especially electrical) caused

by electromagnetic field, the most beneficial solution is to clear the environment from

EM influences.

In this paper high-current systems characterized by slow time-changing fields oc-

currence in their environment with intensities much exceeding permissible levels will

be examined. One of the efficient method to reduce EM energy emitted to the environ-

ment is to applicate the conducting barriers fulfilling the function of the casings, bush-

ings or constructional elements of the devices. Those barriers are placed in vicinity of

the compact electrical power devices generating EM fields.

Effects of attenuation of the electromagnetic energy are particulary intensive in

barriers with closed electric circuits [3]. This closure is the basic stipulation for the

screens to obtain the best attenuation properties. In those systems the field from in-

duced currents in barriers will counteract the disrupting field. Otherwise, currents in

barriers will close themselves in a plane of forced electric field and the energy will not

be attenuated [3]. This solution practically does not reduce spreading of EM energy.

Most barriers in vicinity of sources of EM waves are made of materials with high

value of magnetic permeability. On one hand it raises relativity of barriers sizes, while

on the other hand it causes absorption of bigger energy which makes this effect more

superficial. Attenuation of EM field in steel and magnetic materials requires special

approach to determine power value as very often it is the reason of overheating the

constructional elements and protective equipment.



doi:10.3233/978-1-58603-895-3-26

26

Absorbtion of Electromagnetic Energy by the Ferromagnetic Barriers

Application of conducting barriers with accurate matching (material and geometrical)

parameters leads to the situation where energy of magnetic field in those barriers is

practically attenuated. It is said [1–3] that conducting bodies are good screens of mag-

netic field, but not the electric ones.

However, barriers which are placed in vicinity of generating systems interact with

them and that interaction causes absorption of additional energy changed into heat.

These losses cause increased load of main circuit what can be the reason of unstable

work of electric power systems.

Electromagnetic field around high – current circuits can be attenuated practically

only by using barriers of thickness bigger than skin depth, what enforce necessity of

application massive objects for low frequency. Application of thinner barriers is con-

fined to rare cases of living organisms or measuring equipment protection. Thin barri-

ers can absorb comparatively big portions of energy.

It is hard to explicitly determine the conducting barriers properties because of its

dependence on geometric and material parameters. These barriers are boundary sur-

faces separating two regions marked by two different energetic states. Total attenuation

rate is defined by equation:

r a d

k = k k k⋅ ⋅ (1)

wherer

k ,a

k ,d

k are reflection, absorption and transmittance coefficients, attached to

incident electromagnetic wave.

The main component of energy attenuate is absorption in conducting material. In

many cases, absorbed energy becomes real thermal hazard to structural parts of cas-

ings. Typical casing systems with open circuit conductors [3] are especially exposed to

that interaction. Absorbed energy is almost twice in constructions with open electric

circuits in comparing to close-circuited, but attenuation effect is much smaller [3].

Close-circuited barriers, described in this paper are characterized by different effects.

In these systems reflection coefficient’s ( )r

k value equal 1, because of reflection en-

ergy effect inside barrier’s structures, hence after multiple reflection, all energy is ab-

sorbed by the conducting material. These barriers have a very good attenuation rate for

magnetic fields, but not for electric ones. In Fig. 1 the relative magnetic and electric

field intensities leaving steel and copper barriers are shown.

Relative electric field intensity is similar for both materials, and it depends only on

barrier’s thickness. In the case of magnetic field intensity, relative value depends addi-

tionally on material properties. Value of attenuation rate is higher for copper but re-

quires larger thickness of conducting body (from several to several dozen larger skin

depth).

It is necessary to mention that the absorbance of energy strongly depends on

physical barrier’s thickness. For thick body, absolute value of absorbed energy is less

then in thin one. In Fig. 2 real, reactive, apparent power and power factor distribution

versus relative barrier’s thickness are presented.

The most popular constructional material is steel. So that, it is necessary to take

into account some special parameters of this material, especially nonlinearity of mag-

netic permeability. There are some methods, based on the idea of isotropic and inho-

mogeneous material, which has constant parameters during a period. These methods

R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers 27

allow to take into account the nonlinear property of ferromagnetic material by average

energy density during a period time used to construct effective permeability [4–6].

These methods force us to make fast calculations of electromagnetic field, especially

important, when one have to couple it with thermal analysis.

These methods use the mathematical model described by Helmholtz equation (2).

This is not very precisely as it is difficult and/or sometimes impossible to calculate the

effective magnetic permeability.

2

0H jωμγ∇ − = (2)

In time-domain analysis one can use the mathematical model described by Eq. (3).

It gives the possibility to recalculate current value and magnetic permeability every

time step. The power absorbed in barriers and momentary values of inductance currents

and voltages will be compared.

Figure 1. Magnetic and electric field attenuation vs. relative casings thickness (a/δ), for copper and steel.

Thickness is relative to skin-depth of given material. Value of relative field intensity is a proportion between

field intensity on both surfaces.

Figure 2. Real, reactive and apparent power, and power factor vs. normalized barrier’s thickness.

R. Niedbała et al. / Electromagnetic Field Energy in Ferromagnetic Barriers28

2

0

H

H μγ =

τ

∂

∇ −

∂

(3)

Effect of Nonlinear Permeability for Energy Absorbing

In the paper, simple geometry was considered to indicate effect of magnetic permeabil-

ity for power losses in closed conducting barrier. Therefore calculations of cylindrical

ferromagnetic shield surrounding axially placed electrical lines were made. In this case,

variability of current and shield distance, have similar effect for electrical power prop-

erties. Calculations were made for 0.5 m diameter. Results are presented for a unit

length.

( )( )ln ,

,

B H A

A A

τ

μ τ

−

= (4)

Where H is a function of time (τ) and space (A).

Magnetic permeability was approximated by Eq. (4) which is exact for magnetic

permeability after inflection point.

Type of Sources and Their Energetic Result

In the example only a fraction of transmitted electrical power would be absorbed by

ferromagnetic shield and current-fed is assumed. This is dangerous case because of

quadratically rising energy loss in ferromagnetic medium. Inverse behavior is observed

in electrical devices that are voltage-fed.

In Fig. 3 currents and unit voltage timing diagram in the shield are presented.

Voltage un

was calculated for magnetic permeability as a function of space and time

µ = f(τ,A), but ulwas calculated for surface magnetic permeability µ = µ

s. It results in

lower voltage amplitude and greater phase shift and it gives considerably less amount

of shield’s absorbed energy. Figure 4 gives a picture of power electric changes, and

character of electromagnetic screen load, added to kind of transmission line load.

Figure 3. Effect of nonlinear permeability for voltage with source current 10 kA.


Shapes, which area represents value of shield energy absorbed during one period, are

shown on graph of voltage vs. current (Fig. 4). Errors of calculation shield’s real power

are presented in Table 1.

Implementation of function µ = f(τ,A) results in strong deformation of quadrati-

cally dependent power with current. This is shown in Fig. 5 (continuous line). Power

vs. current for medium with constant permeability, chosen for value of surface mag-

netic field intensity (µ = µs) and H equal to 0, are shown by dotted lines.

Figure 4. Voltage vs. current for different excitation current with constants and variable permeability.

Table 1. Power calculation errors for μ = var i μ = const.

I[A] Pn[W] P

l[W] ΔP[%]

2500 498.2 419 –19

5000 1504.3 1243 –21

7500 2841.1 2408 –18

10000 4505.6 3918 –15

Figure 5. Comparison of shield absorbed power, calculated with μ = var and μ = const.


Errors for different currents (Table 1) show that one could choose substitutive

magnetic permeability which minimized deviations of real power. In Fig. 6 time-

dependant voltage calculated with magnetic permeability chosen in such way were

presented. Although voltage ul

shifting is compensated with amplitude, screen load

character is different.

1 100l

n

P

ΔP=

P

⎛ ⎞

− ⋅⎜ ⎟

⎝ ⎠

(5)

Time-dependant voltage were presented in Fig. 7 for different thicknesses, with the

same excitation current. Errors generated with nonlinear magnetic permeability were

established for thickness value equal a few skin-depth.

For thin ferromagnetic barriers, distribution of power and error of calculated power

are presented in Fig. 8. Differences between power errors in function of thickness and

current (Table 1) should be emphasized.

Figure 6. Effect of nonlinear magnetic permeability for time-depending voltage with source current 7500 A.

Figure 7. Shapes of voltage vs. time, for different substitutive magnetic permeability.


Figure 8. Surface screen power and their error in relation with screen thickness.

Discussion and Conclusion

Conducting barriers surrounding electrical objects that radiate electromagnetic fields

are one of most efficient way to reduce their influence on environment. Wrong made

casings can absorb great amount of energy, thus dangerously increasing their tempera-

ture without much affecting incident wave. It is important to make electrical linking

between casings, causing induced current to flow between or outside them. In this case

they absorb energy, also they are shielding objects in range of EM source. It is essential

to make accurate electromagnetic calculations, due to the fact that absorbed energy, in

this kind of shields, often decide on their material and geometric parameters.

In the paper, calculations of electromagnetic barriers done with substitutive mag-

netic permeability were proved not correct. Computations, which give a practically

usable results, should be done in time-domain. In this case, value of magnetic perme-

ability could be approximated by magnetization curves, giving more exact results.

It was shown that error of calculation depends on the way of approximation of

magnetic permeability. This may be function of thickness or time, thus it is causing

different rate of error.

An interesting alternative for one-layered barrier is compound made of two differ-

ent physical materials: the first made of nonmagnetic, very good conducting such as

copper, and the second – ferromagnetic. This structure is absorbing nearly all energy by

nonmagnetic material. This kind of barrier are efficient electromagnetic screens, ab-

sorbing less energy, and not exposed to excessive heating.

References

[1] H. Waki, H. Igarashi and T. Honma “Analysis of Magnetic Shielding Effect of Layered Shields Based on

Homogenization”, IEEE Transactions on Magnetics., vol. 42, No. 4, 2006.

[2] J. Čuntala, “Simulation of Electromagnetic Shielding in Comsol Multiphisics Environment”,

http://www2.humusoft.cz/www/akce/comsol06/cuntala.pdf.

[3] R. Niedbała “Absorpcja Energii Elektromagnetycznej przez Ekrany Magnetyczne”, Przegląd

Elektrotechniczny, Number 2, 2003, pp. 127-130.


[4] O. Biro, K. Preis and K. R. Richter, “Various FEM Formulation for the Calculation of Transient 3D Eddy

Currents in Nonlinear Media”, IEEE Transactions on Magnetics, vol. 31, No. 3, 1995.

[5] K. Preis, L. Bardi, O. Biro and K. R. Richter “Nonlinear Periodic Eddy Currents in Single and Multicon-

ductor System”, IEEE Transactions on Magnetics, vol. 32, No. 3, 1996.

[6] G. Paoli and O. Biro “Time Harmonics Eddy Currents in Non-Linear Media”, COMPEL: The Interna-

tional Journal for Computation and Mathematics in Electrical and Electronic Engineering, Volume 17,

Number 5, 1998, pp. 567-575(9).


The Influence of Temperature on

Mechanical Properties of

Dielectromagnetics

Barbara SLUSAREK, Piotr GAWRYS and Marek PRZYBYLSKI

Tele & Radio Research Institute, Ratuszowa 11 St., 03-450 Warsaw, Poland

[email protected], [email protected], [email protected]

Abstract. Development of technology and notably development of new generation

magnetic materials caused substitution of cast magnets with magnets manufactured

in the process of powder metallurgy. Development in magnetic materials is ob-

served not only hard magnetic materials, but also soft magnetic materials. Particu-

lar intensive development can be observed in powder materials, which more and

more often substitute traditional materials, e.g. electrical steel. The main factor de-

ciding about application of soft magnetic elements is its magnetic properties. In

many applications mechanic properties are equally important as magnetic parame-

ters. Physical properties of materials change with the change in temperature. The

main goal of research is to know changes of mechanical properties of dielectro-

magnetics with changes of temperature.

Introduction

Magnetic circuit is a fragment of space containing interconnected elements executed in

ferromagnetic materials, which make a closed way for the flow of magnetic induction

flux generated with a permanent magnet or a coil. Magnetic core is the part of magnetic

circuit, used to assign the way of magnetic flux generated by a permanent magnet or a

coil.

Development in magnetic materials causes changes and development in magnetic

circuits of electric machinery, and in a broader perspective of electromagnetic trans-

ducers. Until recently electric motors were commonly fitted with magnetic circuits em-

ploying cast permanent magnets and magnetic core of electrical steel.

Development of technology and notably development of new generation magnetic

materials caused substitution of cast magnets with magnets manufactured in the process

of powder metallurgy. Development in magnetic materials is observed not only in hard

magnetic materials, but also in soft magnetic materials. Particular intensive develop-

ment can be observed in powder materials, which more and more often substitute tradi-

tional materials, e.g. electrical steel.

First powder magnetic cores were created in the 1880’s. They were executed in

crumbled iron, insulated with wax. At the same time however sheet magnetic cores

were developed, which was substituted powder cores of worse magnetic parameters for

a long time.

In recent years we witness a comeback of magnetic cores of powder composites,

due to i.e. development of special low power electric machinery. New designs of elec-



doi:10.3233/978-1-58603-895-3-34

34

trotechnic transducers often require magnetic circuits where application of electrotech-

nic sheets is difficult for structural or technical reasons.

Manufacture of powder magnetic cores with better parameters became feasible

thanks to new generations of magnetic materials and new powder metallurgy technolo-

gies.

Soft magnetic parts of magnetic circuits can be manufactured using two basic

processes of powder metallurgy: sintering or binding with plastic. This latter method

finds wider and wider application for circuits with alternating magnetic flux, notably of

frequency higher than 50 Hz, as this method has many advantages. Binding of powder

with plastic method is used in manufacture of permanent magnets too, which allows to

obtain integrated magnetic circuits with hard and soft magnetic layers.

Application of powder metallurgy has many advantages, such as i.e. low losses of

material, low consumption of labor and energy, and resulting low unit cost of product.

That is the reason behind intensification of works aiming at improvement of magnetic

circuits manufactured with binding magnetic powder with plastic method. Soft mag-

netic elements manufactured using that method are called dielectromagnetics, whereas

hard magnetic elements are called dielectromagnets. Magnetic core manufactured from

soft magnetic powder composite allows three dimensional distribution of magnetic flux

in magnetic core. Full adaptation of magnetic flux density extends the range of design

solutions and allows for better use of powder magnetic material. The strength of such

magnetic cores is ability to obtain complex shapes and shaping their physical proper-

ties. Presence of dielectric, which acts as a binding and insulating agent, reduces losses

due to eddy currents. Thermal isotropy and good thermal conductivity improves dissi-

pation of heat from external surface. Recycling of powder magnetic core of electric

machinery is easier than electrical steel. Separation of iron powder from copper is eas-

ier in the case of powder magnetic core than in the case of sheet magnetic core.

Soft magnetic powder for magnetic circuits should have the following properties:

− highest possible saturation magnetization

− highest possible magnetic permeability

− highest possible resistivity

− best possible compressibility

− highest possible mechanic resistance of non-hardened form

− lowest possible coercive field strength

− lowest possible magnetic loss

− lowest possible price.

Since as of present there is no such powder that meets each and every requirement

as specified above, designers of electric machinery with magnetic circuits have to de-

cide which feature of given material is most critical to given solution, and match pow-

der parameters to parameters of designed magnetic core at design stage.

During first works on manufacture of soft magnetic elements with binding mag-

netic powder with plastic method powders were applied designated originally for sin-

tering method.

As of present leading manufacturers of powder materials manufacture soft mag-

netic materials designated for magnetic circuits manufactured with powder binding

method.

The process of manufacturing soft magnetic elements with powder binding method

consists in pressing and then hardening the moulding. Physical properties of soft mag-

B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics 35

netic element, such as magnetic, mechanic or electric properties, depend mostly on type

of used powder, pressing pressure, and therefore density, mixture content, and harden-

ing process parameters, such as temperature or hardening time. It is possible to shape

their physical properties, as in the case of bound permanent magnets, through applica-

tion of mixture of various powder and plastics. That allows proper matching of mag-

netic core and machine. Soft magnetic powder designated for manufacture of magnetic

circuits are acronymized as SMC – Soft Magnetic Composites. They are iron powder

isolated with one thin organic or inorganic layer or two. Such barrier reduces losses due

to eddy currents. Obviously a decisive parameter for application of soft magnetic ele-

ment is its magnetic properties.

In many applications mechanic properties are equally important as magnetic pa-

rameters. In many electric machines magnetic circuit is also a structural element. Ap-

plication area of appliances with powder magnetic circuits is continuously expanding.

They are used in multitude of environmental conditions, often in high temperatures,

often also in low temperatures. It is well-known that physical properties of materials

change with the change in temperature. It is also the case with magnetic materials.

Physical properties of materials as specified in catalogs refer to room temperature.

Knowledge of mechanic properties of dielectromagnetics of iron powder designated for

magnetic circuits of electric machinery shall allow their designers to consider changes

in mechanic properties in temperature at the stage of design. That was the main goal of

research [1–7].

Experiments

The main goal of experiments is to find correlation between changes of mechanical

properties with changes of temperature for dielectromagnetics. The changes of me-

chanical properties with temperature are shown on compressive, tensile and bending

strength of dielectromagnetics.

High purity iron powder Somaloy 500 was used in research. Iron powder grains

were coated with 0.6% of lubricant and LB1 binder. All test specimens were executed

at the same process parameters.

Examination of dielectromagnetics samples of bending, compressive and tensile

strength were performed using universal testing machine. Samples for bending strength

tests were beam-shaped, dimensions 76 × 12 × 6 mm. Samples for compression strength

tests were cylindrical in shape, 10mm in diameter, 14 mm in height. Samples for tensile

strength tests were 90 mm long and have square cross-section of 5,6 × 5,6 mm and

measuring length of 25 mm. They were prepared according with ISO 2740 standard.

Compression strength, bending strength and tensile strength tests were performed at

temperatures from –40o

C to +100o

C in a thermal chamber.

Magnetic properties were measured on ring samples of inner diameter 45 mm,

outer diameter 55mm and 5mm height according to PN EN-604040-6.

Results and Discussion

Magnetic properties were measured and are shown on Fig. 1. Figure 1a) represents

magnetization curve and Fig. 1b) dynamic amplitude magnetic permeability.

Maximum dynamic amplitude magnetic permeability is 230 as Fig. 1 shows.

B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics36

Tests of mechanic properties were performed on specimens of dielectromagnetics.

Figure 2 contains results of mechanic properties surveys of tested specimens of dielec-

tromagnetics.

Figure 2 presents how temperature influence on mechanic properties such as com-

pressive, tensile and bending strength. As the curve on Fig. 2 shows that compression

strength of measured samples decrease with increase of temperature but in negative

temperatures compression strength improves. The same phenomenon is observed for

bending strength of samples, but the changes of properties are lower. Whereas the in-

fluence of temperature on tensile strength of samples of dielectromagnetics is inconsid-

erable.

Figures 3a) and 3b) show for example compressive stress and bending stress in de-

pendence of displacement of cross-bar of universal testing machine for dielectromag-

netics at temperature –40°C.

Summary

As it can be concluded from data presented in Fig. 2 comparison of dielectromagnetics’

compression strength and bending strength proves that those parameters deteriorate

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

1,3

1,4

1,5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

Hm

[A/m]

Bm

[T

]

0

20

40

60

80

100

120

140

160

180

200

220

240

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

Hm

[A/m]

μr[-]

Figure 1. a) Magnetization curve for Somaloy 500 + 0,6% LB1, f = 50 Hz b) Dynamic amplitude magnetic

permeability for Somaloy 500 + 0,6% LB1, f = 50 Hz.

0

40

80

120

160

200

240

280

320

-40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100

Temperature [°C]

Rc

, T

RS

, R

m [M

Pa

]

Compression strength - Rc Bending strength - TRS Tensile strength Rm

Figure 2. Mechanic properties for Somaloy 500 + 0,6% LB1.

B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics 37

with increase in temperature. At the same time decrease in temperature below room

temperature causes significant improvement of compression strength and bending

strength. Changes of temperature causes not very significant changes in tensile strength

of dielectromagnetics. Research proved that decrease in temperature also resulted in

decrease in tensile strength. Research is in progress.

References

[1] Kordecki A., “Dielektromagnetyki magnetowodów maszyn elektrycznych” Prace Naukowe Instytutu Uk-

ładów Elektromaszynowych Politechniki Wrocławskiej Nr 44, Seria Monografie Nr 11, 1994 (in Polish).

[2] Węgliński B., “Perspektywy rozwojowe w dziedzinie kompozytów proszkowych na magnetowody przet-

worników elektrycznych” Prace Naukowe Instytutu Układów Elektromaszynowych Politechniki Wro-

cławskiej Nr 24, Seria Monografie Nr 2, 1977 (in Polish).

[3] Węgliński B., “Magnetycznie miękkie kompozyty proszkowe na osnowie żelaza” Prace Naukowe Insty-

tutu Układów Elektromaszynowych Politechniki Wrocławskiej Nr 32, Seria Monografie Nr 5, 1981 (in

Polish).

[4] Ślusarek B., Długiewicz L., “Application of soft and hard magnetic powders in small electric machines”,

Advanced in Powder Metallurgy & Particulate Materials – 2006, ISBN: 0-9762057-6-9, San Diego, June

2006.

[5] Hultman L., Persson M., Engdahl P., “Soft magnetic composities for advanced machine design” PM Asia

April 2005 Shanghai.

[6] Dougan M. J., Torres Y., Mateo L., Llanes L., “The fatigue behaviour of soft magnetic composite pow-

ders” Euro PM 2004 Vienna.

[7] Gelinas C., Viarouge P., Cros J., “Use of soft magnetic composite materials in industrial applications”

Euro PM 2004 Vienna.

0

50

100

150

200

250

300

350

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8

Displacement [mm]

Co

mp

re

ss

iv

e s

tre

ss

[M

Pa

]

0

20

40

60

80

100

120

140

160

0 0,1 0,2 0,3 0,4 0,5

Displacement [mm]

Be

nd

in

g s

tre

ss

[M

Pa

]

a) b)

Figure 3. a) The curve of dielectromagnet sample compression test at temperature –40°C, b) The curve of

dielectromagnet sample bending test at temperature –40°C.

B. Slusarek et al. / The Influence of Temperature on Mechanical Properties of Dielectromagnetics38

Influence of the Magnetic Anisotropy

on Electrical Machines

M. HERRANZ GRACIA and K. HAMEYER

Institute of Electrical Machines, RWTH Aachen University,

Schinkelstraße 4 D-52056 Germany

[email protected]

Abstract. Non grain-oriented electrical steel has an inherent anisotropy, which is

normally neglected in the calculation of electrical machines. Moreover, the mag-

netic anisotropy is usually measured in small material samples. Due to the cutting

effect, the magnetic anisotropy in the machine is not the same as in the sample. In

this paper, the magnetic anisotropy is considered as a global problem. A method to

measure it is presented and its influence on the electromagnetic and acoustic be-

havior is considered through the example of an induction motor.

Introduction

Non grain-oriented electrical steel and hence electrical machines have an inherent ani-

sotropy due to the variation of the magnetic properties in rolling and perpendicular to

the rolling direction. Moreover, the cutting process has a different influence in these

two directions. Therefore, the standard procedure of measurement of the anisotropy in

small samples of the material [1] and not directly in the machine is not appropriate for

a detailed study. The influence of the anisotropy in the magnetic losses of the machine

has already been widely studied [2]. But the anisotropy acts also like an eccentricity

with double periodicity and, therefore, generates field harmonics in the air gap, which

have an influence on the torque and the radial force on the stator i.e. the acoustic be-

havior of the machine. These effects have been treated only briefly in the litera-

ture [2,3]. This paper presents a more global view on the problematic of the magnetic

anisotropy. A test setup to measure the anisotropy directly on the stator of the machine

is presented. Measurements are conducted in twenty stator samples of an induction

motor to study the variability of the magnetic anisotropy. The influence in the electro-

magnetic and acoustic behavior of the machine is studied then analytically and by FE

simulation. The aim of this paper is to predict better the parasitic effects in the machine

(torque ripple and radial force on the stator teeth) and so increase the reliability of the

machine.

Test Setup

A variation of the differential method presented in [4] is used here to measure the mag-

netic anisotropy of the stator of an induction motor. A 2-pole rotor is built in such a

way that the magnetic flux passes the stator through the teeth with the same angle to

the rolling direction of the iron lamination. The purpose of this test setup is to distin-


39

guish even small variation on the magnetic properties of the stator teeth. The magnetic

anisotropy of the stator yoke is not going to be measured because the magnetic flux

flows along 180° on it and, therefore, the magnetic properties for the different angles

are averaged. The rotor outer radius is chosen to be equal to the one in the original ma-

chine and the rotor winding is dimensioned to generate an air gap flux density as in the

original machine. Three different variants (see Fig. 1) for the rotor pole pitch were con-

sidered:

• Variant 1: A rotor pole pitch is equal to a stator tooth pitch. This variant has

the advantage that it allows to measure each tooth separately but, as it can be

observed on Figure 1, the saturation level on the rotor is so high that the sen-

sibility of the measurement would be strong limited. Moreover, leakage flux

would adulterate the results.

• Variant 2: A rotor pole pitch is equal to a stator pole pitch. This variant pro-

duces in the stator teeth a similar magnetization as in the original machine but

it has two important drawbacks. Firstly, the yoke of the machine is strong

saturated because the original machine has four and not two poles. Secondly,

nine teeth would be measured at the same time, what difficults to measure

small magnetic variations between the teeth.

• Variant 3: A rotor pole pitch is three times a stator tooth pitch. In this variant

both the yoke of the stator and of the rotor remain unsaturated, so that the

magnetization of the teeth is the most important parameter of the system.

Three stator teeth are measured simultaneously but this resolution is consid-

ered to be sufficient for our study. For this reason this variant is chosen.

(a) (b)

(c)

Figure 1. Rotor variants for the test setup: (a) Variant 1; (b) Variant 2; (c) Variant 3.

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines40

The rotor is rotated by a stepper motor at 0.025s–1

and the magnetic anisotropy is

measured through the change of the current in the rotor winding, which is fed with a

constant 50-Hz voltage. This frequency component is filtered on-line in order to ac-

quire only the changes in the rms. value of the current. The measurement is performed

for six voltage levels (5, 10, 15, 20, 25 and 30 V) to study the variation of the magnetic

anisotropy at different saturation levels.

Measurements

Figure 2 shows the measurement in space and frequency domain for a stator with 36

stator slot and 26 cutting notches. Both orders appear clearly in the measurements but

they do not mask the 2nd order, which corresponds to the magnetic anisotropy. For

α = 0° the rotor is aligned with the rolling direction, where the magnetic resistance is

minimal. Then the inductivity of the rotor winding is at a minimum. Therefore, the

current is at a maximum. It can be stated that this test setup is feasible to measure the

magnetic anisotropy of the stator core. It is used now to compare the magnetic anisot-

ropy of two different production series (A and B). Ten stators for each series have been

measured (see Fig. 3). As comparison parameter, the normalized 2nd order of the cur-

rent is chosen. Series B shows a near to constant magnetic anisotropy in all the stators

of 1% rms. or a peak-to-peak value of 2.8%. This is the expected behavior for a pro-

duction series. On the other hand, Series A shows a very high variability on the mag-

netic anisotropy. Some stators have values up to 2.8% rms. (7.9% peak-to-peak) and

other ones doe not reach even the value of 0.3% rms. (0.8% peak-to-peak). This fact

seems to indicate that either magnetic steel from different origins were used in the

same production series or the tools used in the manufacturing process has produces

much different cutting effects on the stators because of the erosion.

Effect on Parasitic Effects

The measurement results have shown that the magnetic anisotropy can differ largely in

a same machine. This fact should be considered for the prediction of the performance

of the machine and its parasitic effects to ensure the reliability of the machine. The

influence of the magnetic anisotropy on the losses of the machine has been treated ex-

Figure 2. Measurement of the magnetic anisotropy in the space and frequency domain.

M. Herranz Gracia and K. Hameyer / Influence of the Magnetic Anisotropy on Electrical Machines 41

tensively in the literature ([3]). Therefore, this work will concentrate in two other para-

sitic effects: torque ripple and radial forces on the stator teeth.

These effects are going to be studied here analytically and through FE Simulation.

Thanks to the fact that the studied machine has two pole pairs, it is possible to model

the anisotropy only in the stator teeth. Moreover, it is assumed that there is a magnetic

difference between teeth but not inside one of them, so that the magnetic anisotropy is

modeled through a different magnetization curve for each tooth. This assumption sim-

plifies hugely the simulation. The measurement has shown that the magnetic anisotropy

of the stator can be between 0% and 10% (peak-to-peak). In this work, the torque ripple

and the radial force in the stator teeth will be compared for simulations of the machine

with values of the magnetic anisotropy from 0% to 25%.

The magnetic anisotropy acts like a static eccentricity repeated twice on the air

gap. The permeability function of the magnetic anisotropy can then be written as

0 01

( , ) ,

( , ) 1 cos(2 )anis

x t

x t Anis x

μ μ

δ δ ϕ

Λ = =

′′ − ⋅ −

(1)

where δ″ is the equivalent air gap and Anis the peak-to-peak value of the magnetic ani-

sotropy. This permeability function can be decomposed in its Fourier series as follows.

0

1,2...

( , ) cos( (2 ))x t xλ λ

λ

λ ϕ

=

Λ = Λ + Λ −∑ (2)

Regarding the first field harmonic, the air gap flux density due to the magnetic ani-

sotropy is

Figure 3. Comparison of the anisotropy measurements of stator laminations of the series A and B.


2 2 1

cos[( 2) ( )],p p p

b B p xμ λ

ϕ ϕ± ± =

= ± − − (3)

where p is the number of pole pairs in the machine. This means, that the magnetic ani-

sotropy generates an air gap flux density with the space harmonic p ± 2.

Effect of the Magnetic Anisotropy on the Torque Ripple

Figure 4 shows the frequency spectrum of the torque for different values of the mag-

netic anisotropy. Two different phenomenons can be observed:

• The 15th, 21st, 26th, 36th, 62nd and 73rd harmonic order appear with the

magnetic anisotropy and the magnitude of them is proportional to the mag-

netic anisotropy. This new orders appear from the combination of the air gap

flux density due to the magnetic anisotropy (Eq. (3)) and other flux density

harmonics. For example, the 21st and the 26th order can be easily identified

as the combination of the first harmonic of the rotor slots and the magnetic

anisotropy.

• The mean value and the 47th, 49th and 57th order are maximal without mag-

netic anisotropy and they decrease proportional in magnitude with the mag-

netic anisotropy. The reason is that the worsening of the magnetic properties

in the teeth not parallel to rolling direction acts as an enlargement of the air

gap. Therefore, the main component of the magnetic field decreases and also

all the orders of the torque ripple caused by it.

In the worst case, the torque mean value is 3% less as without magnetic anisotropy

and the torque ripple 8% higher.

Effect of the Magnetic Anisotropy on the Radial Forces on the Stator Teeth

The electromagnetic radial forces on the stator teeth are the main responsible for the

acoustic noise of electrical machines. As well as the magnitude, the frequency and the

mode r of these forces are decisive. It has been proved [5] that low modes and specially

r = 2 are the most critical ones. Therefore, the analysis of the radial forces on the stator

teeth is not done in the time-space domain but after a 2-dimensional FFT in the mode-

order domain. Figure 5 shows the radial force with the modes 0, 2, 4 and 6 for different

values of the magnetic anisotropy.

Figure 4. Frequency spectrum of the torque ripple for different values of the magnetic anisotropy.


As the torque ripple, the results show two different trends:

• The orders and modes, which are generated from the fundamental component

of the field, are weakened with the anisotropy. An example of this is the 21st

order with mode 2.

Figure 5. Radial forces on the stator teeth for the modes 0, 2, 4 and 6.


• New harmonic orders and modes appear due to the magnetic anisotropy and

its magnitude is increased along with the value of the magnetic anisotropy.

The 12th and the 36th harmonic order with mode 2 are examples of this.

These orders are expected to have a big influence on the acoustic behavior of

the machine because mode 2 has mechanically the highest amplification coef-

ficient.

An interesting tendency can be observed for different orders. There is a transfer of

force from the original mode rorig

to rorig

– 2. For example for the 31st order, the elec-

tromagnetic force without anisotropy was much higher with mode 6 as with mode 4. As

the anisotropy increases, the electromagnetic force with mode 4 increases and with

mode 6 decreases. The origin of this transfer is the combination of the original radial

force wave with the new component of the flux density caused by the magnetic anisot-

ropy. As it can be seen in Eq. (3), this new wave is invariant in the time and is repeated

twice in the space. Therefore, the combination of this wave with the original ones pro-

duces waves with the same order as the original waves but with mode ±2.

_mag anis orig

υ υ= (4)

_

2mag anis orig

r r= ± (5)

Conclusions

This paper has shown that the magnetic anisotropy of non grain-oriented electrical steel

has an influence in the electromagnetic and acoustic behavior of an induction machine,

although this effect is neglected regularly. The torque ripple in the machine increases

up to 8% due to the emergence of new harmonic orders. The influence of the magnetic

anisotropy can be especially critical in the acoustic behavior because the magnetic ani-

sotropy produces radial force waves with the same order as the machine without anisot-

ropy but with smaller modes.

Furthermore, a test procedure is presented to measure the magnetic anisotropy di-

rectly in the stator of the machine. Measurement has shown that the value of the mag-

netic anisotropy can vary from near 0 to 10% (peak-to-peak). This variability should be

taken into account for the prediction of the parasitic effects i.e. for the reliability stud-

ies on the machine.

References

[1] T. Nakata et al., “Measurement of Magnetic Characteristic along Arbitrary Directions of Grain-Oriented

Silicon Steel up to high Flux Densities”, IEEE Transactions on Magnetics, Vol. 29, No. 6, pp. 3544-3548,

November 1993.

[2] S. Urata, et al., “Magnetic Characteristic Analysis of the Motor Considering 2-D Vector Magnetic Prop-

erty”, IEEE Transactions on Magnetics, Vol. 42, No. 4, pp. 615-618, April 2006.

[3] B. Hribernik, “Influence of Cutting Strains and Magnetic Anisotropy of Electrical Steel on the Air Gap

Flux Distribution of an Induction Motor”, Journal of Magnetism and Magnetic Materials, Vol. 41,

pp. 427-430, 1984.


[4] W. Wilczynski, “Influence of Manufacturing Condition of Magnetic Cores on their Magnetic Properties”,

Proc. of the Conf. on Soft Magnetic Materials, 20-22 April 1998.

[5] C. Schlensok et al., “Structure-Dynamic Analysis of an Induction Machine Depending on Stator-Housing

Coupling”, Proc. of the International Electric Machines and Drives Conference, IEMDC 2007, 3-5 May

2007.


Analysis of Structural Deformation and

Vibration of Electrical Steel Sheet by Using

Magnetic Property of Magnetostriction

Wataru KITAGAWA, Koji FUJIWARA, Yoshiyuki ISHIHARA and

Toshiyuki TODAKA

Department of Electric Engineering, Doshisha University

1-3, Tataramiyakodani, Kyotanabe, Kyoto, 610-0321, Japan

telephone: +81-774-65-6327, e-mail: [email protected]

Abstract. Recently, it is examined with many papers about magnetostriction of

electrical steel sheet and magnetostriction of transformer model. In this paper, de-

formation and vibration of electrical steel sheet by magnetostriction was analyzed

and measured. There results were compared and examined. As results, it was re-

ported that natural mode was provoked to force of magnetostriction.

Introduction

It is examined that observation of the magnetostriction and the vibration of the electri-

cal steel sheet in some papers [1–4]. However, the displacement analysis of the defor-

mation of the iron core disfigured by magnetostriction has rarely been calculated.

The method of preventing the noise and the trouble by the resonance of the vibra-

tion is the most general to do the design that the natural frequency of the equipment

doesn’t enter ranges of driving vibrations. However, the character frequency exists a lot

according to the object equipment, and might not resonate even if the character fre-

quency is a category of driving frequency [5,6].

So transformation by magnetostriction of the electromagnetic steel sheets is ana-

lyzed, and correlation with oscillation gets possible to be made clear by confirming

form.

In this paper, displacement of an electromagnetic steel sheet by magnetostriction

was analyzed using structural analysis technique by finite element method and it was

compared with measurement value, and the adequacy was inspected, and comparing by

eigenvalue analysis and frequency response analysis examined it.

Analysis and Measurement Technique

The analysis technique of deformation of electric steel sheet by the magnetostriction

which used structural analysis technique by two dimensional finite element method is

proposed in this paper.

Basic equation of structural analysis is the following equation.


47

fKu =

(1)

K: Rigidity matrix, u: Displacement vector, f: External force vectoring.

The nodal force can be calculated from distortion using the following equation.

T

0

S

dS= ∫p B Dε (2)

T

0 0 0

0 , 0x y x y

B B ε ε= =B ε (3)

1 0

1 0

(1 )(1 2 )

0 0 (1 2 ) / 2

E

ν ν

ν ν

ν ν

ν

−⎡ ⎤

⎢ ⎥= −

⎢ ⎥+ −

⎢ ⎥−⎣ ⎦

D (4)

E : Young’s modulusν : Poisson’s ratio.

T

1 1 2 2 3 3 4 4

[ | | | | | | | | | | | | | | | |]y x x y y x x y

p p p p p p p p= − − + + − − + +p (5)

p: Nodal force of arbitrary element.

The direction of p is established so that volume of constituent keeps uniformity.

When magnetostriction is supposed in proportion to square of magnetic flux den-

sity, it is following equation.

2

0

2

0

x x x

y y y

B

B

ε α

ε α

⎫= ⎪

⎬= ⎪⎭

(6)

Figure 1(c’) is shown, the direction of the nodal force p is supposed that the infla-

tion force works in case of the horizontal direction and the compressive force works in

case of the perpendicular direction of the magnetic flux because the direction shown in

Fig. 1(c) is expand the elements. It is shown by the following Eq. (7).

T

1 1 2 2 3 3 4 4

[ | | | | | | | | | | | | | | | |]y x x y y x x y

p p p p p p p p= − − + + − − + +p (7)

As the measurement of frequency response, the sample which it was hanged from

ceiling exited using excitation frame like a infinity solenoid. The sample is bound with

B coil, and acceleration pickup is glued together, and magnetic flux density and accel-

eration are detected. Let excitation frequency change with this state, and acceleration in

0.1 T uniformity is measured.

As the sample, the thickness of no grain oriented magnetic steel sheet is 50 mm,

iron loss is less than 13.00 W/Kg (JIS:50A1300) is used.

Results

Analysis result be shown Fig. 2, and eigenvalue and natural frequency exist. And it is

vibrated by resonant frequency 1656 Hz of the magnetostriction that 2 times of excita-

W. Kitagawa et al. / Analysis of Structural Deformation and Vibration of Electrical Steel Sheet48

tion frequency are fundamental wave in the mode 8 from frequency response shown

with Fig. 3 is supposed.

Natural frequency to measure from calculated consequence is supposed. Accelera-

tion and excitation frequency response of sample are shown in Fig. 4.

Measurement result shows that analysis result is agreed, and eigenvalue mode 8 is

provoked by 1700 Hz, and it is thought that resonant is provoked. The shape by reso-

nance are shown in Fig. 5.

Figure 1. Decision of direction of joint force.

(a) Mode 7 (2410.3 Hz) (b) Mode 8 (3299.6 Hz) (c) Mode 9 (3423.1 Hz) (d) Mode10 (4422.9 Hz)

Figure 2. Natural Mode of Vibration (Analyzed).

Figure 3. Frequency Response (Analyzed). Figure 4. Frequency Response (Measured).

W. Kitagawa et al. / Analysis of Structural Deformation and Vibration of Electrical Steel Sheet 49

It is able to easily predict which frequency influences resonance and oscillation by

structural analysis and natural vibration analysis and be useful for noise prediction of

electric steel sheet.

Conclusion

Measurement result accords with analysis result well, and eigenvalue mode 8 is guided

by 1700 Hz, and it is thought that resonance event is caused. And, as the possible cause

that peak of response exists in 3650 Hz, magnetostriction waveform does frequency

component of 2 times of excitation frequency with fundamental wave, but shade in-

cludes component too of excitation frequency, and natural frequency of existing sample

accords in this component and 3400 Hz vicinity, and it is supposed that resonance

event produced it.

Magnetostriction is converted into force, and structural analysis of electric steel

sheets by magnetostriction is possible by structural analysis. It is able to easily predict

which frequency influences resonance and oscillation by structural analysis and natural

vibration analysis and be useful for noise prediction of electric steel sheets.

References

[1] A.J. Moses: “Measurement of Magnetostriction and Vibration with Regard to Transformer Noise”, IEEE

Trans. Magn., Vol. MAG-10, No. 2, pp. 154-156 (1974).

[2] K. Kuhara, S. Kawamura, Y. Hori, and M. Sasaki: “Vibration Analysis of Transformer Core”, 72nd An-

nual Conference of the Japan Society of Mechanical Engineers, Vol. IV, pp. 109-110 (1998) (in Japa-

nese).

[3] Y. Hori, S. Abe, M. Sasaki, and K. Kuhara: “Vibration characteristics of transformer”, The Papers of

Technical Meeting on Magnetics, IEE Japan, MAG-99-79, pp. 17-22 (1999) (in Japanese).

[4] M. Imamura, T. Sasaki, and H. Nisimura: “AC magnetostriction in Si-Fe single crystals close to (110)”,

IEEE Trans. Magnetics., Vol. MAG-19, No. 1, pp. 20-27 (1983).

[5] T. Sasaki, S. Takada, S. Saeki, F. Ishibashi, and S. Noda: “Magnetostriction of Erectromagnetic Steel

Sheets under AC Magnetization Superimposed with Higher Harmonics”, T. IEEE Japan, Vol. 112-A,

No. 6, pp. 539-544 (1992-6) (in Japanese).

[6] T. Sasaki, T. Takada, F. Ishibashi, I. Suzuki, S. Noda, and M. Imamura: “Magnetostrictive vibration of

electrical steel sheet under non-sinusoidal magnetizing condition”, IEEE Trans. Magn., Vol. MAG-23,

No. 5, pp. 3077-3079 (1987).

(a) Change of the Sheet by magnetostriction (× 105

) (b) Natural Mode 8 (3299.6 Hz)

Figure 5. Shape by Resonance.

W. Kitagawa et al. / Analysis of Structural Deformation and Vibration of Electrical Steel Sheet50

Chapter A. Fundamental Problems

and Methods

A2. Methods


Rapid Design of Sandwich Windings

Transformers for Stray Loss Reduction

J. TUROWSKIa

, Xose M. LOPEZ-FERNANDEZb

, A. SOTO RODRIGUEZc

and

D. SOUTO REVENGAc

a

Dept. of Intelligent Information Systems, WSHE – Lodz Poland,

consultant for University of Vigo Spain

[email protected]

b

Dep. of Electrical Engineering, University of Vigo – Spain

[email protected]

c

Earlier Socrates Diploma Students from Vigo University in the Technical University

of Lodz, Poland; now Engineers in EFACEC Transformer Works, Porto, Portugal

Abstract. Rapid design is one of the main imperatives of modern manufacturing,

followed from principles of mechatronics [1], and is handy tool of regular optimi-

zation of structure. In this work it is presented such rapid design method for spe-

cific class of power transformers with sandwich windings. To accelerate the design

process an expert system and rapid interactive procedure was applied. At such ap-

proach the more scientific and design experience is located into the Knowledge

Base of the Expert System, the more, rapid, easier and cheaper is a regular design

and optimization of a machine. Thanks to analytical preparation, approximation

and linearization coefficients the programming and calculation is discharged from

cumbersome iteration and other formal disturbances. Hybrid analytically-

reluctance Network Method three-dimensional RNM−3D [2] has proved here as

one of the best for rapid design of such complex structures like modern transform-

ers with extreme electromagnetic filed concentration, its crushing forces, eddy cur-

rent loss and overheating hazard.

Calculation Model for Rapid Design

Rapid design, time to market, innovativeness belong to the main imperatives of modern

industry, based on principles of mechatronics [1]. The application of expert system of

design is one of the ways to reach these aims (Fig. 1). At such system of programming,

the more scientific and design experience is located into the Knowledge Base, the

more, rapid, easier and cheaper is a regular design and optimization of a machine.

For the solution of problem specified in the title, into the Knowledge Base it was

introduced the same scientifically proved analytical formulae, approximation and lin-

earization like in the book [2]. Namely:

1) Basic Turowski’s formula for power losses (W) in solid and/or screened steel

walls:

ΔP =

2 22

01

2 2

e m St

x

e ms e m ms m p rs ms St

A A A

p H dA p H dA a H dA

ωμ

γ

μ

⎡ ⎤

+ +⎢ ⎥

⎢ ⎥⎣ ⎦

∫∫ ∫∫ ∫∫ =

= I2

(a + bI) (1)


53

where pe

<< 1, pm

<< 1 are screening coefficients ([5] p. 200 and 198); Ae, A

m, A

St –

surfaces covered with corresponding (e-electromagnetic, m-magnetic) screens or not

screened (St-steel), Hms

– magnetic field strength on a metal surface.

2) Magnetic nonlinearity μ(H) inside solid iron, for field Hms

stronger than 5 A/cm,

was considered with the help of average linearization coefficients ap≈ 1,4 for active

power and aq≈ 0,85 for reactive power.

3) Magnetic nonlinearity μ(H) along the steel wall surface was considered with the

help of analytical approximation 2

r

Hμ ≈ cHb

and corresponding exponent coeffi-

cient x in (1). The value of x varies for different transformers, but is typically between

1,1 and 1,14 ([5], p. 345).

4) Eddy current reaction of solid metal wall has been taken into account with the

help of complex reluctances:

1 1

(0,37 0,61)

s

bR jR jμ μ

ωσ

μ

+ = + – for solid steel or

( )

2 2

2 0

sinh 2

cosh 1

Cu

d

d d

R

α α

μ α μ−

≅ ≅ –

when Cu or Al electromagnetic screens are applied. In a first approach one can adopt

RCu

→ ∞.

5) For reluctance of laminated magnetic screens (shunts), in comparison with di-

electric and solid metal elements, in a first approach one can adopt value RFe≅ 0.

6) Excessive heating hazard follows from (1) where value Hms

on the steel surface

is responsible for the loss density and therefore for a local heating. From a thermal

equilibrium equation we can find a permissible tangential field component on the steel

surface:

Hms,perm

= 1962(8

1 3,29 10 1c

−

+ ⋅ − ) (2)

for permissible temperature tperm

, where c= f(σ,ω,tperm

). Over the Hms,perm

value exces-

sive local heating hazard (Hot-Spot) of solid steel elements due to induced eddy-

currents can appear. Eq. (2) was the basis of electromagnetic overheating criteria pro-

Figure 1. Block diagram of expert system type (a) design of a machine: 1 – Large portion of introduced

knowledge and experience – simple, inexpensive and rapid solution, e.g. 1 sec; 2 – small portion of knowl-

edge and experience – difficult, expensive and labour-consuming solution, e.g. 3 months (see J.

Turowski [1]).

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction54

posed at 1972 CIGRE Plenary Session [4] and used until now for possible Hot-Spot

localisation, also in packages RNM-3D [3] for intelligent localisation of hot-spot.

Design of adequate calculation model and selection of proper calculation method

also belong to the crucial components of the Knowledge Base, which decide on the

success.

Physics of it is based on principals of technical application of Maxwell’s electro-

magnetic theory [5].

Transformers with Sandwich Windings

After testing different available calculation programs, the hybrid, three-dimensional,

analytically-numerical software of the RNM−3D class [2] has been selected. It has

proved [6] as one of the best and popular tools for rapid design, broadly used

(J. Turowski [3], tab. V) in most of transformer works, world over.

In most popular cases of three-phase transformers RNM-3D calculate stray field

and losses in CPU time shorter than 1 second, whereas FEM-3D – from 30 to 300

hours [1,3] and is very complicated in regular industrial design. However, the

RNM−3D is devoted only to common transformers with cylindrical windings (Fig. 2a).

At the same time exist an important group of transformers with sandwich windings

(Fig. 2b,c), which needs similar rapid design method for the 3D leakage field region.

Main objective of this is such design of winding coils and tank screening in order

that to reduce of short circuit destruction forces, eddy current concentrations as well as

stray loss, and excessive local heating hazard.

Such rapid design programs are especially important in present power system reli-

ability problems and acute market competition, when one needs to design new version

of construction and examine its reliability.

It reveals particularly, when special transformers are needed with a specific pa-

rameters and unique construction, like furnace, low impedance and other “tailor made”

solutions.

a) b) c)

Figure 2. Cross section of transformers windings: a) Core type, cylindrical winding φσm

– maximum stray

flux, Bm(x) – stray flux density distribution, l

av/π – average winding diameter, I

1N

1, I

2N

2 – ampere-turns of

HV and LV windings. b) Stray flux density distribution in sandwich winding ([7] p.127). c) Sandwich wind-

ing of furnace transformer 16 MVA, 35 kV/ 384, 342, 225, 243 V at different tap connection. At connection

7-6 – Delta, at 7-8 – Y ([8] p. 101).

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction 55

For this task new package “RNM-3Dsandw

” was developed, which is basis for such

regular rapid design.

Theoretical basis is the same like in the equivalent reluctance method RNM-3D

[2,3] i.e., where in linear region elementary reluctances were calculated from the ge-

ometry Ri=

is

oμ

il

and in metal parts – from the Maxwell equations:

∇ x Hm = σE

m and ∇ x E

m = – jωμ H

m and complex Poynting’s Vector

S = P1 + jQ

1 in [VA/m

2

]

considering iron linearization coefficients, eddy current reaction, electromagnetic wave

interference inside electromagnetic screens, etc.

Only model investigated was different (Fig. 3), as element of total cross-section

from Fig. 2b.

For power loss calculation the formula (1) was used.

Hence from (1) thanks to model symmetry total loss in tank Ptotal

= 8 Ptotal(1/8)

,

2

(1/ 8) 0 2 2

arctg arctg

2 2

arctg arctg

total a

B A B A B

P P

B AA B

B c

A A

υ

⎧ ⎛ ⎞ ⎡⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + −⎨ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢

+⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎣⎝ ⎠⎩

⎫⎤⎛ ⎞ ⎛ ⎞− − ⎬⎜ ⎟ ⎜ ⎟⎥

⎝ ⎠ ⎝ ⎠⎦⎭

(3)

where A, B, c are geometrical parameters from the model in Fig. 3a.

The package of “RNM-3Dsandw

” (Fig. 3b) has the same structure like in RNM-3D.

Calculation (Fig. 4) is partly automatised like in RNM-2Dexe.

Results

As a result it was obtained new rapid semiautomatic tool of regular analysis and design

in few seconds per one design variant of: Graphs of magnetic field Hms

(x,y) – Fig.5,

a) b)

Figure 3. Model of single-phase sandwich transformer: a) Design drawing, b) RNM model.

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction56

loss density P1 (x, y) – Fig. 6 and hot-spots if H

ms > H

ms,perm (2) on the tank surface;

Graphs of magnetic flux density Bm (x,y) for calculation eddy-loss and forces in wind-

ings.

Total loss P = ∫∫A

P1(x,y,z = 0) dx dy on the whole surface A of the tank for screen-

ing or shunting optimisation, considering cost 3000 to 10 000 US$/kW of capitalized

power losses.

References

[1] J. Turowski, Mechatronics Impact upon Electrical Machines and Drives. Proceedings of Internat. Aegean

Conference on Electric Machines and Power Electronics – ACEMP’04. May 26-28, 2004. Istanbul, Tur-

key, pp. 65-70. Invited plenary lecture.

[2] J. Turowski, Reluctance Networks. Chapter 4, pp.145-178 in the book “Computational Magnetics”.

Chapman & Hall. London 1990, editor J. Sykulski (Extended translation from Polish: J. Turowski editor,

“Ossolineum”, Wroclaw, 1990).

[3] J. Turowski, “Stray Losses, Screening, and Local Excessive Heating Hazard in Large Power Transform-

ers”. Chapter in CD book “Transformers in Practice”, ISBN: 978-84-609-7515-9, © 2006 Xose M. Lo-

pez-Fernandez.

[4] M. Kozlowski, J. Turowski, Stray losses and local overheating hazard in Transformers. CIGRE. Paris

1972. Rep.12-10.

[5] J. Turowski, Elektrodynamika Techniczna. Warszawa, WNT 1993.

[6] J. Turowski, I. Kraj, K. Kulasek, Industrial Verification of Rapid Design Methods in Power Transform-

ers. International Conference Transformer’01, 5-6.09.2001, Bydgoszcz, Poland.

[7] E. Jezierski, Transformers. Theoretical bases (in Polish), WNT Warszawa 1965.

[8] E. Jezierski et al., Transformers. Construction and design (in Polish), WNT Warszawa 1963.

Figure 4. Developed interactive design structure of RNM-3Dsandw

with Java Solver.

Figure 5. Distribution of Hms

on the tank surface. Figure 6. Power Losses P1 on the tank Surface.

J. Turowski et al. / Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction 57

Application of Logarithmic Potential

to Electromagnetic Field Calculation

in Convex Bars

Stanisław APANASEWICZ

Technical University in Rzeszów, Chair of Electrodynamics and

Electrical Machinery Systems

Description of method of calculation of electromagnetic field induced by sinusoidal

alternating current (or direct current) flowing in the infinitely long cylinder with con-

vex intersection is aim of this paper. Potential of simple layer and Newtonian potential

are applied. Adequate integral equations were introduced for calculation of integrands

in these potentials. In the event of alternating current impedance boundary condition is

considered.

List of More Significant Designations Used

(s , τ) – curvilinear co-ordinate system on the cross-section of bar;

s – length of the arc,

τ – perpendicular to the L line, s0– length of the L curve.

0 0

: ( ) , ( )L x x s y y s= = – parametric description of the L line; 2 2

0 0

1x y′ ′+ =

2 2 2

1 0 0

( , , ) ( ( )) ( ( ))R x y s x x s y y s= − + −

2 2 2

2

( , , , ) ( ) ( )R x y x y x x y y′ ′ ′ ′= − + −

( )0 00 0 0 0 0

( ) ( ), ( ) ( ) ,R x t x s y t y s R R= − − =

0 0 0 0

( , ) , ( , )S x y N y x′ ′ ′ ′= = −

– vectors: tangential and normal (in the point (x0, y

0)) to

the L line.

( ) ( )

( ) ( )

00 0 0 0 0 0

0 2 2 2

00 0 0 0

( ) ( ) ( ) ( ) ( ) ( )

( , )

( ) ( ) ( ) ( )

x t x s y t y t y s x tR N

W s t

R x t x s y t y s

′ ′− − −⋅

= =

− + −

Calculation of Electromagnetic Field in the Case of Sinusoidal Current

We take the following assumptions to solve depicted problem:

a) Depth of field penetration in the metal is small in comparison with radius of

curvature of the bar surface.

b) Form of the bar cross-section is symmetrical in relation to the x axis (please

see the explanatory drawing enclosed)



doi:10.3233/978-1-58603-895-3-58

58

c) Vector potential (complex amplitude) (0,0, )A A=

has only one component

( , )A A x y= in Cartesian coordinates.

This function fulfils Laplace’s equation or Helmholtz equation:

2

, In the metallic area

, outside of it 0

A

A

α⎧

Δ = ⎨

=⎩

(1)

First assumption causes that application of simplified boundary condition called

impedance condition is possible. So, in the metallic area, one can introduce curvilinear

coordinate (s, τ);

locally, function A fulfils equation like in the case of Cartesian coordinates:

2

ss

A A A Aττ

αΔ = + = (2)

In this equation, similarly like in the skin area (in the boundary layer), derivatives

2

2

,

s s

∂ ∂

∂ ∂

are small in comparison with derivatives

2

2

,

τ τ

∂ ∂

∂ ∂

.

So, Eq. (2) reduces itself and we have:

2

A Aττ

α= (3)

Solution of this equation has the following form:

*

( )A A s e

ατ−

= (4)

Tangential component Bs of magnetic induction and magnetic field strength H

s

have the following form:

* *

( ) , ( )s s

B A s e H A s e

ατ ατ

α

α

μ

− −

= − = − (5)

x

0

s

L

Ωτ

),( τsB

),0( τA

Figure 1. Diagram of the studied system.

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation 59

on the bar surface τ = 0 magnitudes Hs and

*

A are continuous, so at the side of air the

following condition must be fulfilled:

0

0

0

,

s

s s

H A H ds i

α

μ

= − =∫ (6)

Continuity of function A ensures continuity of tangential component of electric

field.

Solution of Laplace equation (1) is sought in the form of logarithmic potential of

simple layer.

0

10

1

( , ) ( ) ln

s

A x y g s ds

R

= ∫ (7)

We determine unknown integrand g = g(s) from the boundary condition (6).

Known function A enables calculation of magnetic induction vector B =

1 2

( , ,0)B B rot A=

:

0 0

0 0

1 22 2

1 10 0

( ) ( )

( ) , ( )

s s

y x

y y s x x s

B A g s ds B A g s ds

R R

− −

= = − = − =∫ ∫ (8)

On the bar surface in the point (x0(t), y

0(t)) normal component B

n and tangential

component Bs of magnetic induction can be calculated in accordance with the following

formulas:

1 0 2 0 0 0

1

grad ,s y x s

A A

B B x B y A x A y A N H

n nμ

∂ ∂

′ ′ ′ ′= + = − = − ⋅ = − = −

∂ ∂

(9)

1 0 2 0 0 0

gradn y x

B B y B x A y A x A S′ ′ ′ ′= − = + = ⋅

(10)

As a result of that, the boundary condition (6) achieves the following form:

0 0

0 00 0

1 1

( ) ln ( ) ln

s s

r r

A

A g s ds g s ds

n n R R

α α

μ μ

∂ ∂

= ≡ =

∂ ∂∫ ∫

Normal derivative

0

1

ln

n R

∂

∂

in accordance with known characteristic of a potential

of simple layer is determined by the following formula:

0 0

0

2

0 00 0

1

( ) ln ( ) ( )

s s

A R N

g s ds g s ds g t

n n R R

π

∂ ∂ ⋅

= = −

∂ ∂∫ ∫

(11)

As a result of that, the boundary condition (6) achieves the following form:

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation60

0

0

( ) ( , ) ( )

s

g s W s t ds g tπ=∫ (12)

0

0

1

( , ) ( , ) ln

r

W s t W s t

R

α

μ

= −

g integrand is a periodical function with s0 period and is on the strength of assumption

even in relation to s. We look for it in the form of the Fourier series.

0

1 0

2

( ) cosn

n

n

g s g g s

s

π∞

=

= +∑ (13)

At the same time, free term g0 is known; namely: in the large distance from the bar

A function from the formula (7) has the following form

0 0

0 0

0 2 2

0 0

1 1 1

( , ) ( , ) ln ln ( ) ln

4

s s

i

A x y A x y g ds g s ds

R Rx y

μ

π

= = = =

+

∫ ∫

therefore:

0

0 0 0 0

0 0 0

00

,

2 2

s

i i

gds g s g

s

μ μ

π π

= = =∫ (14)

We obtain from Eqs (12) and (13):

0

0 0 0 0

1 1 10 00

1 2 2

cos ( , ) cos ( ) ( )

s

n k n n

n n n

n k

g g s W s t dt g g t g W t g W t

s s

π π

π

∞ ∞ ∞

= = =

⎡ ⎤

+ = + = +⎢ ⎥

⎣ ⎦

∑ ∑ ∑∫

(15)

0

00

1 2

( , )cos

s

n

n

W W s t sds

s

π

π

= ∫

From Eqs (15) we obtain the final system of equations for determination of gn co-

efficients:

0 0 0,0 ,0

10

1

n n

n

g g W g W

s

∞

=

⎡ ⎤= +

⎢ ⎥

⎣ ⎦

∑

0 0, ,

10

2

k k n n k

n

g g W g W

s

∞

=

⎡ ⎤= +

⎢ ⎥

⎣ ⎦

∑ (16)

0 0 0

,

0 0 00 0 0

2 2 2

( )cos ( , ) cos cos

s s s

n k n

k k n

W W t tdt W s t t s ds dt

s s s

π π π

= =∫ ∫ ∫


In order to carry out numerical calculations, we reduce existing infinite series to L

first terms and obtain system of algebraic equations:

0,0

0 ,0

10 0

1

0 0, ,

10

1

1

2

, 1,2,..., 1

L

n n

n

L

k k n n k

n

W

g g W

s s

g g W g W k L

s

=

−

=

⎧ ⎡ ⎤

− =⎪ ⎢ ⎥

⎪ ⎣ ⎦

⎨

⎡ ⎤⎪= + = −

⎢ ⎥⎪⎣ ⎦⎩

∑

∑

(17)

Calculation of Electromagnetic Field in the Case of Direct Current

If direct current i = i0

= const flows in the bar, component of A potential will fulfil

Poisson and Laplace’s equation (instead of Eq. (1)):

0

, in the metallic area

, outside of it

0

i

J

A S

μ

μ

⎧− = −⎪

Δ = ⎨

⎪⎩

(18)

Solution of the above equation can be accepted in the form of two logarithmic po-

tentials (potential of simple layer and Newtonian potential):

2 1

1 1

ln ( ) ln

2L

J

A dx dy g s ds

R R

μ

πΩ

′ ′= +∫∫ ∫ (19)

In accordance with properties of logarithmic potentials, A function in accordance

with (18) is continuous function on the whole plane (x, y). At the same time, all deriva-

tives of the first term are continuous too; however normal derivative of the second term

is discontinuous on the L line (formula (11)). So, normal component of induction is

continuous on the L line. One should select function g(s) in (19) in such a way to

achieve the continuous tangential component of the magnetic field strength Hs. on

the L.

We may look for g function in the form of Fourier series (13).

g0 constant can be determined analogically as in the previous case (formula (14)).

Namely, we have for great values (x, y):

0

0

2 2 2 2

0 0

2 2

1 1

( , ) ( ) ln ln

2 2

1

ln

2

L

iJ

A x y dx dy g s ds g

x y x y

i

x y

μμ

π π

μ

π

Ω

⎡ ⎤ ⎛ ⎞′ ′= + = +⎢ ⎥ ⎜ ⎟

⎝ ⎠+ +⎣ ⎦

=

+

∫∫ ∫

hence:

0 0

0

(1 )

2r

i

g

μ

μ

π

= − (20)

S. Apanasewicz / Application of Logarithmic Potential to Electromagnetic Field Calculation62

On the basis of formula (19) and use of described characteristics of potentials on

the line L, we will obtain the condition of continuity of tangential component of mag-

netic field on the line L. This condition is expressed in the following way:

0

1 1

1 ( ) ( , ) 1 ( )

2r rL

J

W dx dy g s W s t ds g t

μ

π

μ π μ

∗

Ω

⎡ ⎤⎛ ⎞ ⎛ ⎞

′ ′− + = − +⎜ ⎟ ⎜ ⎟⎢ ⎥

⎝ ⎠ ⎝ ⎠⎣ ⎦

∫∫ ∫ (21)

Equality (21) after taking into consideration (13) can be brought to the following

form:

0 0 0, 0

1 1 0

1 2

( ) ( ) ( ) cos

( 1)

r

k k n

k nr

n

W t g u t g u t g g

s

μ π

μ π

∞ ∞

∗∗

= =

− ⎡ ⎤+ + = +

⎢ ⎥+ ⎣ ⎦

∑ ∑ (22)

where ( )

2

J

W t W dx dy

μ

π

∗∗ ∗

Ω

′ ′= ∫∫ ,

0

0, 0

00

2

( ) ( , )cos

s

k

k

u t W s t s ds

s

π

= ∫ .

From equality (22) we obtain final system for determination of unknowns gk:

0 0,0 0 ,0

1

0 0, ,

1

(1 )

2

k k

k

n n n k k n

k

g pu p h g v

g p h g u g v

∞

=

∞

=

⎡ ⎤− = +

⎢ ⎥

⎣ ⎦

⎡ ⎤= + +

⎢ ⎥

⎣ ⎦

∑

∑

(23)

where:

0

0 00

1 2

, ( ) cos

(1 )

s

r

n

r

n

p h W t t dt

s s

μ π

π μ

∗∗

−

= =

+

∫ ,

0

, 0,

00

2

( )cos

s

k n k

n

v u t t dt

s

π

= ∫ .

System (23) is equivalent of system (16). On the basis of systems (21) and (22)

one can state that function g(s) ≡ 0 for μr= 1. It means that in this case Newtonian

logarithmic potential (first term in the expression (19)) is a solution of the whole prob-

lem.


Multi-Frequency Sensitivity Analysis of

3D Models Utilizing Impedance Boundary

Condition with Scalar Magnetic Potential

Konstanty Marek GAWRYLCZYK and Piotr PUTEK

Szczecin University of Technology, 70-310 Szczecin, Sikorskiego 37,

E-mail: [email protected], [email protected]

Abstract. In this work the inverse problem solution with iterative Gauss-Newton

algorithm and Truncated Singular Value Decomposition (TSVD) is shown. For the

goal function a norm l2 was chosen. To solve the inverse problem, which consists

of the identification of conductivity distribution in a 3D model, the multi-

frequency sensitivity analysis has been applied. The correctness of sensitivity cal-

culation has been proved utilizing three different methods, namely Tellegen’s

method of adjoint model, differentiation of stiffness and mass matrix, as well as

sensitivity approximation by means of difference quotient. Regarding the effec-

tiveness of those methods, the first one is preferred because of shortest computa-

tional time.

Introduction

The objective of this work is to solve the inverse problem of crack shape recognition

arising in eddy-current method of non-destructive testing. For this purpose, the Gauss-

Newton algorithm (GN) with TSVD (Truncated Singular Value Decomposition) based

on sensitivity information obtained with the finite element method was applied. The

inverse job itself consists in iterative optimization of pre-defined goal function. While

this function depends on measured field values, the goal is to find such material pa-

rameters distribution, that result in numerical simulation converge with these of meas-

urement.

To carry-out the iterative optimization one needs to calculate the Hessian matrix

and the gradient of goal function. This may be obtained with sensitivity analysis of

electromagnetic field versus material parameters.

Eddy Current Nondestructive Testing

Testing with eddy-currents belongs to non-destructive and contact-less quantity meth-

ods. Most important is the recognition of crack shape in conducting materials on the

surface or inside the material. For example, they may find application in the recogni-

tion of crack shape either in conducting or ferromagnetic materials.

At first the conception of method relies on placing the tested, electrically conduct-

ing object on an area of variable electromagnetic field, and then processing the ob-

tained in this way information, which is included in measured signal and on model



doi:10.3233/978-1-58603-895-3-64

64

structure. The sinusoidal current at considerably different frequency is field source in

case of multi-frequency method.

The authors propose solving the above-mentioned problem in an iterative way us-

ing the gradient method. However, the information on gradient provides sensitivity

analysis.

The Considered Test Problem

In the case of 3D problems the squared model of absolute probe was applied. The size

of the used detector was shown in the Fig. 1.

The model of the analyzed object, after providing the spatial discretization using

first order tetrahedral finite elements, was presented in Fig. 2. The lift-off parameter

was equal to lf = 1 mm. The interaction between the detector and the conducting area

was simulated by means of SIBC (Surface Impedance Boundary Condition) [5].

Figure 1. Absolute probe above conducting plate with search area.

Figure 2. Computation model.

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models 65

For calculating the field distribution in the considered model, the quantity of coil

flow equal to 1 mAΘ n I= ⋅ = was assumed. Described model of probe took a possibil-

ity of applying just one component of electrical vector potential T, what one may de-

fine such as [2]

( )

( )

( )

( )

( )

( )

6 6

x

6 6

0 y

6

0

6 20z

x x

6 20z

S y y

2

0

5 10 110 [A/m] in area 0,

( , ) 5 10 110 [A/m] in area 0,

110 [A/m] in area 0, ,

5 10 [A/m ] in area 0,

5 10 [A/m ] in area 0,

0 [A/m ] in area 0, ,

z

x S b

T x y y S b

S b

T

S b

y

T

S b

x

S b

⎧ − ⋅ + ⋅ ×

⎪

= − ⋅ + ⋅ ×⎨

⎪⋅ ×

⎩

∂⎧= ⋅ ×−

⎪∂

⎪

∂⎪= − = ⋅ ×⎨

∂⎪

×⎪

⎪⎩

1

1

J 1

(1)

where symbols such as Sx

× (0,b), Sy

× (0,b), S0

× (0,b) accruing in Eq. (1) were ex-

plained in Fig. 3 and Fig. 4.

According to the assumed symmetry in the analyzed model of used nondestructive

testing system in practical computation, only half of the space of the model was con-

cerned, for which the symmetry plane is x0z. For simulation of the measurement proc-

ess, as well as the reconstruction of thee analyzed conducting plane, the same mesh

was used. Hence, the applied mesh consisted of NE = 4800 finite elements, ND = 9471

nodes, and a band of mass and stiffness matrix was equal to SB = 360. The calculation

was provided on a computer, which was fitted out by processor Intel Centrino

1.86 GHz, and memory stick RAM 512 MB. The time of a single calculation for a

fixed pick-up position averaged about 50 s.

The score of simulated measurement process, during which the absolute probe was

moved along the x axis with the size of movements Δk = 0.5 mm for 25 number of

Figure 3. Model of coil winding. Figure 4. Distribution of T0z

(x,y).

K.M. Gawrylczyk and P. Putek / Multi-Frequency Sensitivity Analysis of 3D Models66

movements and 5 – number of harmonic signals from 75 kHz to 200 kHz with varied

level of noise equal –30, –20, –15 [dB]1

was presented in Fig. 5, Fig. 6, Fig. 7 and

Fig. 8. Measurement voltage using absolute probe Up includes information on defect

U0, but also the information without defect U

n. Mathematically, one could describe it as

0 P n

U U U= − (2)

According to the supposition in flaw’s presence on analyzed model, the measure-

ment signal’s amplitude was bigger for the higher frequency of excitation current. The

vector of the measurement voltage consisted of 150 complex type samples correspond-

ing to the position of the absolute probe for the assumed spectrum of frequencies.

1

To add the noise level the function available in toolbox Matlab™ Communication was applied.

Figure 5. Compensated voltage U0 for different

positions of absolute probe.

Figure 6. Voltage U0 in case of noise level –30 [dB].

Figure 7. Voltage U0 in case of noise level –20 [dB]. Figure 8. Voltage U

0 in case of noise level

–15 [dB].


Adjoint Model in Tellegen’s Method

For the used field problem formulation by means of coupled potentials T – φ, deriva-

tion of the adjoint model from the Lorenz principle was done in [1,2]. The final score

in the form of sensitivity equation is given by [6]

1 1 22

,

S

V V

dV dVγ∂ = ∂∫ ∫J E E E

(3)

with appropriate assumption of boundary conditions (Neumann and Dirichlet) to the

aim of vanishing of integrals

2 1 2 1( ) 0,

n

S

dS

H E E H 1×∂ + ×∂ =∫

(4)

where Js2

is the current density in the original model, E1, E

2, mean electric intensity

vectors in both models, original and adjoint, respectively. Moreover, using distribution

of delta function, the left hand side of Eq. 4 can be rearranged to the form

p

1 0 0 0 12

δ( , , ) ,S p

V L S L

dV x x y y z z dS d d U∂ = − − − ∂ = ∂ = ∂∫ ∫ ∫ ∫J E E l E l

(5)

whereas the right hand side using classical conception of SIBC can be described as

1 2 1 2j .

V V

dV dVγ γ ωμγ∂ = −∂∫ ∫E E H H

(6)

Hence, Eq. 6 after simple rearranging allows one to calculate sensitivity from a

more useful form

( )

( )

( )

( )

0

0

0

2

2 ( )1 2 1 2

2

2

2 ( )1 2 1 2

2

1 2 1 2

2

e

,

,

2

1 j / , 2 / , 1,2,... ,

e

e

z z

e

e

V

z ze

e

zS

e

e

S

e e e e

U

dV

x x y y

dS e dz

x x y y

dS

x x y y

j e E

α

α

ϕ ϕ ϕ ϕα

γγ

ϕ ϕ ϕ ϕα

γ

ϕ ϕ ϕ ϕα

γ

α ωμγ δ δ ωμγ

− −

∞

− −

⎛ ⎞∂ ∂ ∂ ∂∂

= +⎜ ⎟∂ ∂ ∂ ∂∂ ⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂

= +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂

= +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

= = + = =

∫

∫ ∫

∫

(7)

where H is the magnetic intensity vector, inside conducting plate defined as

1,2 1,2

1,2( ) e ,

z

x yz

x y

α

ϕ ϕ−

∂ ∂⎛ ⎞

= −⎜ ⎟∂ ∂⎝ ⎠

H 1 1

(8)

and φ1,2

means scalar magnetic potential, γe

is conductivity in considered finite element

of search area.


Gauss-Newton Algorithm with TSVD

The way of calculation of induced voltage sensitivity versus conductivity in every fi-

nite element of search area decides on numerical efficiency of the proposed identifica-

tion algorithm. From this point of view, the described Tellegen’s method of adjoint

model is favourable because of the possibility to calculate in one cycle the voltage sen-

sitivity versus all conductivities. This information is essential to the implementation of

the Gauss-Newton’s optimization method. Sensitivity information is a crucial compo-

nent of a goal function gradient. The goal function in that case is in the classical

norm l2.

The Sensitivity Analysis for the Specified Goal Function

Definition of the goal function is essential for the optimization process due to the use of

the deterministic gradient method. In the presented algorithm the following goal func-

tion has been assumed using a vector of complex referenced voltages:

*

1

1( ( )) ( ( ) )( ( ) )

2i i i i

j

FG ξ ξ ξ= − −∑U U Uo U Uo(9)

where: ( )j

ξU is the jth

component of voltage modeled by algorithm, j

Uo means the

referenced voltage for the jth

position of measurement coil, iξ relates to the optimized

quantity (the conductivity of the ith

finite element).

First, integrating voltage sensitivity versus each specified parameter i

ξ (i = 1…I)

on area of the measurement coil for every position (j = 1…J), and then determining

quantity ijs according to formula (7) one can obtain:

1 11 12 1 1

2 21 22 2 2

1 2

...

...

... ... ... ... ... ...

...

I

I

J J J JI I

U s s s

U s s s

U s s s

ξ

ξ

ξ

Δ Δ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ Δ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

=

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ Δ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(10)

where: quantity | | | |j j j

UΔ = −U Uo denotes the difference of ( )j j=U ξ U .

In the events of Jacobian of optimized function having form of rectangular matrix

( J I≥ ) with disadvantageous features, that is

− the singular values decay gradually to zero,

− the ratio between the largest and the smallest nonzero singular values is

large [4],

the identification of model parameters belongs to the wide class of the discrete ill-

posed problems, and requires the application of the special regularization method e.g.

TSVD (Truncated Singular Value Decomposition). To combine TSVD with the pro-

posed algorithm the filtering function [3,4] removing the singular vectors correspond-

ing with small singular values σi, was defined

.

0

( )

1TSVD

f

σ δ

σ

σ δ

≤⎧

= ⎨>

⎩

(11)


Numerical Examples

a) b) c) d) γ [S/m]

Figure 9. Course of identification for the noise level equal –30 [dB]: a) the assumed conductivity distribu-

tion, b) the conductivity distribution in 2th

iteration, c) the conductivity distribution in 6th

iteration, d) the

conductivity distribution after 16 iterations.

a) b) c) d) γ [S/m]

Figure 10. Course of identification for the noise level equal –20 [dB]: a) the assumed conductivity distribu-

tion, b) the conductivity distribution in 2th


iteration, d) the

conductivity distribution after 10 iterations.

a) b) c) d) γ [S/m]

Figure 11. Course of identification for the noise level equal –30 [dB]: a) the considered/known conductivity

distribution, b) the conductivity distribution in 3th


iteration, d)

the conductivity distribution after 9 iterations.


Conclusions

In this work the simple algorithm for solution of 3D inverse problems of conductivity

estimation basing on scalar magnetic potential and impedance boundary condition was

proposed. For the solution of current distribution inside conductors the full 3D-

formulation with four degree of freedom at each node is necessary. The proposed

methods of sensitivity evaluation is applicable also in this case. This work was sup-

ported by the Polish Government (No. of awarded grant 3 T10A 045 28).

References

[1] Gawrylczyk K.M.: Adaptiven Algorithmen auf der Basis der Methode der Finiten Elemente, Szczecin

University of Technology, 1992.

[2] Putek P.: Phd dissertation, The defects’ identification in conducting material basing on the multi-

frequency sensitivity method in finite element method, Szczecin University of Technology, Szczecin

2007.

[3] Gawrylczyk K.M., Putek P.: Multi-frequency sensitivity analysis in FEM application for conductive ma-

terials flaw identification, ISEF’05, Sept. 15-17, Baiona, Spain.

[4] Gawrylczyk K.M., Putek P.: Adaptive meshing algorithm for recognition of material cracks, COMPEL

Vol. 23, No. 3, 2004, pp. 677-684.

[5] Deeley E.M.: Surface Impedance Near Edges and Corners in Tree-Dimensional Media, IEEE Transaction

on Magnetics, Vol. 26, No. 2, March 1990, pp. 712-714.

[6] Dyck D.N., Lowther D.A.: A Method of Computing the Sensitivity of Electromagnetic Quantities to

Changes in Material and Sources, IEEE Trans. On Mag., Vol. 30, No. 5, 1994, pp. 3415-3418.


Very Fast and Easy to Compute Analytical

Model of the Magnetic Field in Induction

Machines with Distributed Windings

Manuel PINEDA, Jose ROGER FOLCH, Juan PEREZ and Ruben PUCHE

Department of Electrical Engineering, Universidad Politécnica de Valencia

Cno de Vera s/n. 46022 Valencia, Spain

[email protected], [email protected], [email protected], [email protected]

Abstract. Torque and e.m.f. of an induction motor can be derived from the air-gap

flux density. The paper shows a new method for computing the flux density distri-

bution of constant air-gap width machines, neglecting magnetic saturation, by

making use of very efficient techniques widely used in the field of discrete signals

processing: the Fast Fourier Transform (FFT) and the Discrete Circular Convolu-

tion. The mutual inductances between the phases of the machine are obtained with

a single, very simple formula, in terms of the machine’s windings distribution and

the geometric dimensions, which is solved with the FFT. As the method can han-

dle arbitrary winding conductor distributions, it is highly suitable to the analysis of

the magnetic field and electromagnetic torque in machines with stator or rotor

faults, such as inter-turn short circuits or broken bars.

Introduction

Transient analysis of rotating electrical machines by use of the well established d-q

model neglects the harmonic contents generated by phase windings. In [1] a model of

the electrical machine which takes into account the effect of the spatial harmonics is

presented. In [2] the Multiple Coupled Circuit Model of a symmetrical, general m−n

induction machine is established based on the phase self and mutual inductances, de-

rived on harmonic bases. The accuracy of the analysis depends on the number of har-

monics included in the calculation. In [3] the Winding Function Approach (WFA) for

the calculation of inductances in machines with small and constant air gap, taking into

account the space harmonics, is presented, and has been used for modeling of induction

machines [4] and fault analysis in [5–7]. In [8] the effect of slot skewing and the linear

rise of the air gap MMF across the slot are introduced in the WFA.

WFA is a method of general validity for calculating the inductances of rotating

electrical machines, but has some drawbacks: to account for coil pitch, slot skewing or

the rise of the air gap MMF across the slot, different winding functions must be used in

each case. Besides, the winding function of a phase must be assembled from the wind-

ing functions of the coils that constitute the phase. Complex integrals must be solved in

this process, which may be very cumbersome in the case of arbitrary winding distribu-

tions. As it is stated in [3], this task typically consumes a high amount of time, so that

only discrete curves of inductance versus rotor position are calculated and linear inter-

polation is applied at intermediate rotor positions. In this paper, a completely different

method of solving problem is undertaken, characterized by:



doi:10.3233/978-1-58603-895-3-72

72

1. The conductor, instead of the coil, is used as the basic element to compute

both the air gap MMF and the flux linkage of a phase, so that arbitrarily com-

plex windings layouts can be easily modeled.

2. There is no need of coils or phase winding functions. Only the physical distri-

bution of the conductors of the phases and a single, characteristic function of

the system (the yoke flux of a conductor placed at the origin and fed with a

unit current) are needed to establish a single resulting equation, which gives

the mutual inductance of two phases for every relative position between them.

3. The resulting equation is expressed as a discrete circular convolution, which is

computed in the spatial frequency domain in a very fast way using the FFT.

It is a very general and rather unconscious assumption to associate the FFT exclu-

sively to signal analysis in the time domain. Nevertheless, it proves to be also ex-

tremely powerful when applied to the treatment of quantities in the spatial domain,

such as the air gap MMF and the yoke flux. The expressions derived for these quanti-

ties have a mathematical structure analogous to the ones found in the analysis of time

signals, so that the tools used in this field, as the FFT, can be successfully applied to

compute phase inductances. This approach results in an extremely fast method: the

computation of the mutual inductance between two phases, in 3600 different relative

positions, considering 1800 harmonics, skewing and linear rise of the air gap MMF

across the slot is computed in less than a tenth of a second, on a 2800 MHz Pentium IV

processor. And this time is independent of the complexity of the windings layout.

System Equations

The following equations system can be written for an induction machine with m stator

and n rotor phases with arbitrary layout (that is, even with winding fault conditions like

inter-turn short circuits or broken bars):

[ ] [ ] [ ] [ ]S S S S

U R I d dt= + Ψ (1)

[0] [ ] [ ] [ ]r r r

R I d dt= + Ψ (2)

[ ] [ ] [ ] [ ] [ ]s ss s sr r

L I L IΨ = + (3)

[ ] [ ] [ ] [ ] [ ]T

r sr s rr r

L I L IΨ = + (4)

1 2

[ ] [ ... ]T

S s s sm

U u u u= (5)

1 2

[ ] [ ... ]T

S s s sm

I i i i= (6)

1 2

[ ] [ ... ]T

r r r rn

I i i i= (7)

[ ]

[ ] [ ]T sr

e S r

L

T I I

∂

=

∂θ

(8)

M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field 73

2

2e L

d d

T T J J

dt dt

Ω θ

− = = (9)

where [U] is the voltage matrix, [I] is the current matrix, [R] is the resistance matrix,

[Ψ] is the flux linkage matrix and [L] is the matrix of inductances. Subscripts s and r

stand for stator and rotor. Te is the electromechanical torque of the machine, T

L is the

load torque, J is the rotor inertia, Ω is the mechanical speed and θ is the mechanical

angle. To compute (3), (4) and (8), self and mutual phase inductance matrices must be

calculated. Due to the presence of the derivatives in (1), (2) and (8), it is necessary to

achieve a very good accuracy in this process (especially if different fault conditions are

to be detected and diagnosed in a sure way), so that air gap MMF harmonics must be

considered. End turn and slot leakage inductances need to be pre-calculated, and are

treated as constants in (3) and (4), as usual in the technical literature.

Proposed Method for Computing the Mutual Inductance Between Two Phases via

Discrete Circular Convolution and FFT

The inductance between two phases, A and B, is calculated in this paper through the

following process:

1. Phase A is fed with a constant unit current, and its yoke flux is obtained, as

given in [9].

2. Flux linkage of phase B due to the yoke flux of phase A is determined, which

corresponds to the mutual inductance between the phases. (If B = A, we get

the phase magnetizing self inductance).

Both steps are treated with the same mathematical tool, a circular convolution

computed with the FFT, giving the mutual inductances between phases A and B for

every relative angular position in a single equation.

Yoke Flux Produced by an Arbitrary Winding

The distribution curve of the air gap MMF, F(φ), is defined by the relation:

( ) ( )·r

F H gϕ = ϕ (10)

where φ is the angular coordinate, Hr(φ) is the mean value of the radial component of

the magnetic field intensity in the air gap at φ and g is the air gap width. The air gap

MMF F0(φ) (Fig. 1) that produces a single conductor placed at φ = 0 and fed with a

unit current, with of infinite iron permeability, as given in [10]:

0

1

( ) 1

2

F

ϕ⎛ ⎞ϕ = −⎜ ⎟

π⎝ ⎠

(11)

In the case of a constant and small air gap width, the yoke flux Φy0

(φ) generated by

(11) is (Fig. 2a):

M. Pineda et al. / Very Fast and Easy to Compute Analytical Model of the Magnetic Field74

0

0

0

0

· ·

( )· 0

( )

( ) 2

y

y

r

F d

g

π

ϕ

⎧μϕ ϕ ≤ ϕ < π⎪

Φ ϕ = ⎨

⎪Φ ϕ− π π ≤ ϕ < π

⎩

∫

(12)

where is stack’s length, r is the average radius of the air gap, and g its width. Substi-

tution of (11) gives:

20

0

· ·

( ) ·( )

4y

r

g

μ

Φ ϕ = ϕ− π

π

(13)

If the conductor is placed at another angular position α, its yoke flux, Φyα

(φ), is ob-

tained by shifting the curve Φy0

(φ) to the new position, 0

( ) ( )y yα

Φ ϕ = Φ ϕ−α , as shown

in Fig. 2b. The yoke flux ΦyA

(φ) generated by a phase A with an arbitrary conductors

distribution nA(φ), fed with a unit current, can now be obtained from (13), by applying

linear superposition.

( )

2

0 0

0

( ) ( )· ( ) · ( )y y A y AA

n d n

π

Φ ϕ = Φ ϕ−α α α = Φ ⊗ ϕ∫ (14)

Figure 1. Air gap MMF F0(φ) generated by a conductor placed at the origin and fed with a unit current.

(a) (b)

Figure 2. Yoke flux generated by a conductor fed with a unit current placed at (a) the origin (b) coordinate α.


The particular mathematical structure in (14) is very well known in the control and

signal theory. It is called a circular convolution, represented with the symbol ⊗. It must

be evaluated at every angular coordinate φ, which can be very cumbersome for a com-

plex winding layout, nA(φ). Nevertheless, there is an alternative and very fast way

to compute it, based on the following property of the Fourier Transform (FT): the FT

of the convolution of two functions is equal to the product of their FTs, that is,

FT(f ⊗ g) = FT(f) · FT(g). Applying this property to (14) gives the following algo-

rithm:

0 0

0 0

ˆ( ) ( )

ˆ ˆˆ( ) · ( ) ( ) ( )

ˆ( ) ( )

FT

IFT

AÁFT

A A

n

n n

⎫Φ ϕ →Φ ξ ⎪

Φ ξ ξ = Φ ξ → Φ ϕ⎬

⎪ϕ → ξ ⎭

(15)

This algorithm reduces the computation of integral (14) to a simple product of two

functions in the spatial frequency domain, but it has a serious drawback: two FTs and

one IFT must be computed. However, there is an extremely efficient algorithm to ob-

tain the FT of a function, the Fast Fourier Transform (FFT), and its inverse (IFFT), but

they can only be applied to discrete sequences, not to continuous functions. To convert

the functions in (15) into discrete sequences, the circular length of the machine air gap

is divided into N equally spaced intervals, each of them spanning an angle Δφ = 2π/N.

A high value of N is necessary to achieve a good accuracy (for example, N = 3600

yields a precision of 0.1º). These discrete sequences are represented as column vectors

of N elements, generated as follows:

• The discrete sequence of function Φ0(φ), Φ

0, is generated by sampling at the

beginning of each interval.

2

0

· · · 1

[ ] ·

2

r k

k

g N

μ π ⎛ ⎞Φ = −⎜ ⎟

⎝ ⎠y0

(16)

• The discrete sequence corresponding to the distribution of conductors nA(φ),

nA, is generated by assigning to each interval the sum of all the conductors

that it contains:

( 1)·

·

[ ] ( )· 0

k

A A

k

n k n d k N

+ Δϕ

Δϕ

= ϕ ϕ ≤ <∫ (17)

Algorithm (15) can now be formulated in terms of these discrete sequences as:

( )( )( )IFFT FFT FFT=

0

Φ Φ .*

yA y A

n (18)

where the symbol .* denotes an element by element product of two vectors. It must be

remarked that, in (18), the sequence Φy0

and its FFT are the same for every induction

machine, except for a constant factor that depends on its geometrical dimensions. Be-

sides, the sequence nA can easily represent any arbitrary winding layout. Slot width of

skewing, for example, can be taken into account just by distributing the conductors

uniformly in the intervals that spans the slot opening of the skew slot, respectively.


Flux Linkage of a Phase with an Arbitrary Distribution of Conductors

The flux linkage of a phase with an arbitrary distribution of conductors is obtained by

simply adding up the values of the yoke flux at the yoke sections corresponding to each

one of its conductors. Figure 3 shows the basis of this method: the flux linkage Ψab

of

an arbitrary coil a-b can be calculated by replacing the coil by two equivalent annular

ones, (a-a’, b-b’) and summing up the yoke flux that crosses them.

The flux linkage ΨBA

of a phase B with an arbitrary conductors’ distribution, nB(φ),

due to the yoke flux generated by another phase A, ΦyA

(φ), is obtained by applying the

aforementioned method to each of its coils:

2

0

( )· ( )·BA B y

A

n d

π

ψ = ϕ Φ ϕ ϕ∫ (19)

If phase B is now ideally or actually (for instance, due to a change in the rotor po-

sition) shifted by an angle ε with respect to A, (21) can be applied by simply using a

shifted distribution of the conductors of phase B

2

0

( ) ( )· ( )·BA B y

A

n d

π

ψ ε = ϕ− ε Φ ϕ ϕ∫ (20)

If phase A is fed with unit current, (20) is the mutual inductance between phases A

and B as a function of ε:

2

0

( ) ( ) ( )· ( )·BA BA B y

A

L n d

π

ε = Ψ ε = ϕ− ε Φ ϕ ϕ∫ (21)

The integral in (21) can be difficult to calculate for complex conductors distribu-

tions, but it can be computed with an algorithm similar to (18):

( ) ( )( )( )*

.*IFFT FFT FFT= ΦBA yA B

L n (22)

where the superscript *

stands for complex conjugate. LBA

is a column vector, whose

kth element is the mutual inductance of phases A and B, for a relative angle between

Figure 3. Flux linkage of an arbitrary coil: a) actual coil b) replaced by two equivalent annular coils.


them equal to k·2π/N. Furthermore, (18) and (22) can now be combined, yielding a

single equation

( ) ( ) ( )( )( )*

0

.* .*

A

IFFT FFT FFT FFT= ΦBA y B

L n n (23)

which can be very easily computed with modern scientific software. For example, in

MATLAB, it can be written as Lba = ifft( fft(na) .* fft(Yf0) .* conj( fft(nb))).

Analytical Evaluation of the Currents and Electromagnetic Torque

The proposed method has been applied to compute the inductances, starting current and

electromagnetic torque of an 11 Kw induction motor, fully characterized in [11]. Fig-

ure 4 shows the starting current (simulated and experimental) during a start-up

transient, with a constant 1.5 Nm load torque, and the computed electromagnetic torque

for that transient.

Conclusion

A new and completely different approach for the calculation of winding inductances in

induction machines has been presented in this paper. The election of the conductor as

the winding basic element, the yoke flux as the main flux quantity, and the formulation

of inductances in terms of a discrete circular convolution, computed with the FFT, are

the key elements of the new method. After discretization of the air gap into N equally

spaced intervals, the mutual inductances of two phases corresponding to N relative an-

gular positions, taking into account the first N/2 air gap MMF harmonics, are obtained

simultaneously with a single equation, solved via FFT. The method involves only three

discrete sequences, namely the distributions of the conductors of the two phases and a

characteristic function of the machine: the yoke flux generated by a conductor placed at

the origin with a unit current flowing through it. The computation of the mutual induc-

tance of two phases, for N = 3600, takes less than a tenth of a second on a 2600 MHz

Pentium IV processor. Arbitrary, complex winding layouts, the linear rise of the air gap

MMF across the slot, and slot skewing can be easily modeled and taken into account

without increasing at all this computing time. As the method can handle arbitrary phase

IR (A) I

R (A) T (Nm)

time (s) time (s) time (s)

(a) (b) (c)

Figure 4. Transient current of a stator phase during start-up (a) simulated (b) experimental, and computed

electromagnetic torque (c).


conductor distributions, it is highly suitable to the analysis of machines with stator or

rotor faults, such as inter-turn short circuits or broken bars.

References

[1] F. Taegen and E. Hommes, “Das allgemeine Gleichungssystem des Käfigläufermotors unter Berücksi-

chtigung der Oberfelder. Teil I: Allgemeine Theorie”, Archiv für Elektrotechnik, vol. 55, no. 1,

pp. 21-31, Jan. 1972.

[2] H. R. Fudeh and C. M. Ong, “Modeling and Analysis of Induction Machines containing Space Har-

monics. Part I”, IEEE Transactions on Power Apparatus and Systems, vol. 102, no. 8, August 1983.

[3] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a Concentrated Winding Induction Machine for

Adjustable Speed Drive Applications Part 1 (Motor Analysis)”, IEEE Trans. Energy Convers., vol. 6,

no. 5, pp. 679-683, Dec. 1991.

[4] X. Luo, Y. Liao, H. A. Toliyat, A. El-Antably, and T. Lipo, “Multiple Coupled Circuit Modeling of In-

duction Machines”, IEEE Trans. Ind. Appl., vol. 31, no. 2, pp. 311-318, March/April 1995.

[5] H. A. Toliyat and T. A. Lipo, “Transient Analysis of Cage Induction Machines under Stator, Rotor Bar

and End Ring Faults”, IEEE Trans. Energy Convers., vol. 10, no. 2, pp. 241-247, June 1995.

[6] J. Milimonfared, H. M. Kelk, A. Der Minassians, S. Nandi, and H. A. Toliyat, “A Novel Approach For

Broken Rotor Bar Detection in Cage Induction Motors”, IEEE Trans. Ind. Appl., vol. 35, no. 5.,

pp. 1000-1006, Sept./Oct. 1999.

[7] S. Nandi and H. A. Toliyat, “Novel Frequency-Domain-Based Technique to Detect Stator Interturn

Faults in Induction Machines using Stator-Induced Voltages after Switch-Off”, IEEE Trans. Ind. Appl.,

vol. 38, pp. 101-109, Jan./Feb. 2002.

[8] G. Joksimovic, M. Djurovic, and A. Obradovic, “Skew And Linear Rise of MMF Across Slot Model-

ing. Winding Function Approach”, IEEE Trans. Energy Convers., vol. 14, pp. 315-320, Sept. 1999.

[9] L. Serrano-Iribarnegaray, Fundamentos de Máquinas Eléctricas Rotativas. Barcelona: Marcombo,

1989, p. 195.

[10] B. Hague, The Principles of Electromagnetism Applied to Electrical Machines”. New York: Dover

Publications, Inc., 1929.

[11] M. Pineda-Sánchez, “Máquinas Eléctricas con Armónicos de Devanado: Desarrollo y Comparación de

Distintos Métodos de Análisis, de Complejidad Gradualmente Creciente, hasta Incluir Permeabilidad

del Hierro Finita, Ranurado, Excentricidad y Desplazamiento De Corrientes”, Ph.D. dissertation, Dept.

Ing. Elec., Universidad Politécnica de Valencia, Valencia, 2004.


Coupling Thermal Radiation

to an Inductive Heating Computation

Christian SCHEIBLICH, Karsten FRENNER, Wolfgang HAFLA and

Wolfgang M. RUCKER

Institute for Theory of Electrical Engineering, Pfaffenwaldring 47,

70569 Stuttgart, Germany,

[email protected]

Abstract. Inductive ovens heat with eddy currents due to a low frequency elec-

tromagnetic field. Within a first approximative approach, the effects of thermal

conduction and thermal radiation are taken into account. Therefore, a finite ele-

ment method for calculating the thermal conduction is coupled to a boundary ele-

ment method for calculating the resulting thermal radiation. For the presented ap-

plication, a steel tube surrounded by an axisymmetric coil is chosen and the nor-

malized thermal radiation distribution resulting from the thermal conductive trans-

fer and the eddy current density is examined.

Introduction

To heat materials in a very fast and efficient way, the principle of an inductive oven is

of first choice. The main physical effects are eddy currents in case of an induced elec-

tromagnetic field and a resulting object-based heat flow due to thermal conduction.

Treating temperatures higher than 500° Celsius, heat transfers caused by thermal radia-

tion cannot be neglected. Therefore, thermal radiative transfers should be examined and

displayed.

For the presented application, the setup of a steel tube surrounded by an axisym-

metric coil is chosen (Fig. 1). The tube model consists of a volume mesh of 2014 first

order tetrahedrons and a boundary mesh of 758 first order triangles. The coil is meshed

with 297 first order tetrahedrons and generates an electromagnetic field of low fre-

quency. The eddy currents in the tube appear mainly in the influence area of the coil

and do not intrude into the inner regions of the tube at the chosen frequency. One can

compute the heating sources from the eddy current density and the electric conductiv-

ity. The heating sources represent the boundary conditions for the thermal diffusion

equation. Accordingly, heat sources as boundary conditions for the thermal radiation

equation, are also available within this step. Finally, it is possible to obtain the thermal

radiation distribution of tube’s surface.

Inductive Heating

The problem considered are eddy currents at a low frequency ω , displacement cur-

rents are ignored. The magnetic permeability μ and the electric conductivity κ are



doi:10.3233/978-1-58603-895-3-80

80

assumed to be constant. The mathematical model for time harmonic quasi-static eddy

current problems can be derived from Maxwell’s equations resulting in

1

curl curl jωκ

μ

+ =c

A A J , (1)

where A is the complex magnetic vector potential and c

J is the current density of the

source coil. The eddy current density e

jωκ = −A J is computed with a finite element

method (FEM) coupled to a boundary element method (BEM) [1]. Therefore, no vol-

ume mesh to model the air is necessary.

In Fig. 2 the arising eddy current density in the tube by a maximum value of

2.72 kA per square meter – in the mid of the tube – and by the minimum value of 50 A

per square meter – dark, at the end of the tube – is shown. The values appear at a fre-

quency of 500 Hz, one turn for the coil and a current density of 100 kA per square me-

ter feeding the coil. For one single time step, the magnitude of the eddy current density

e

J determines the heat source distribution,

2

e

edd

P

κ

=

J

. (2)

The presented heat source distribution for the thermal conduction process is evalu-

ated from one selected time step of the eddy current simulation from which a tempera-

ture distribution can be retrieved.

Figure 1. The model of a steel tube surrounded by a coil.

C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation 81

Figure 2. Arising eddy currents at tube’s surface.

Conductive Heat Transfer

To receive a temperature distribution dif

T one has to solve Fourier’s thermal diffusion

equation,

divgrad

dif

dif edd rad

T

T P P

t

α

∂

= + +

∂

. (3)

edd

P and rad

P are the heat source distributions from the eddy current density and

the thermal radiation. The thermal diffusivity α can be retrieved by the specific ther-

mal conductivity relative to the ability of thermal energy storage [2]. Solving equation

(3) one can obtain radiation sources at the boundaries from the temperature distribution

dif

T to compute the thermal radiation distribution. Denote that in this first approach a

single time step from the conductive heat transfer is taken and, further, the temperature

distribution and the associated thermal radiation distribution are normalized to one.

Radiative Heat Transfer

The thermal radiation in an inductive oven is limited to grey diffuse reflectors. Emis-

sion, absorption, and reflection are the considered physical effects. To compute the

radiosity j

B a boundary element method is used [3], where the surfaces of the field

C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation82

problem have to be considered. This is tackled by the radiosity equation, which is well

known from graphics applications,

( )4

i i j ij i ij

j

T B Fσε δ ρ= −∑ , (4)

where 4

i iTσε is the emitted energy for a given temperature

i

T , σ and i

ε are the

Stefan-Boltzmann constant and the emission factor, ij

δ is the Kronecker symbol, re-

spectively. The reflectance i

ρ multiplied by the possible rate of energy ij j

F B scattered

from surface element i

A towards surface element j

A represents the incident flux of a

surface. ijF is known as the view factor,

2

cos cos1

d d

i j

i j

ij ij i j

i A Aij

F V A A

A

θ θ

π

= ∫ ∫r

. (5)

The view factor includes a visibility function ij

V that results either in zero for no

sight or one for sight between two focused surface elements i

A and j

A . The resulting

radiosity j

B is the thermal radiation ,j rad

P at each surface element j

A and can be di-

rectly added to the thermal diffusion equation (3) as a heating source at the boundary of

the body. The overall thermal radiation shown in Fig. 3 is represented by the colour

scheme at the surface of the steel tube. It was normalized to one, therefore, the result-

Figure 3. Thermal radiation displayed for tube’s half boundary.

C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation 83

ing values range from 0.9 due to the loss – dark, outside the tube – to 1.22 – white,

inside the tube. Using a time step based process these results can affect the diffusion

equation (3) and display the influence of thermal radiation processes to the temperature

distribution.

Conclusion

Within the presented approach, eddy currents are computed for a three-dimensional

model. When obtaining thermal heat sources at higher temperatures due to eddy cur-

rents, one has to consider thermal conductive and thermal radiative processes. Appear-

ing radiative heat transfers are computed for one time step of the conductive transfer.

This allows denoting the thermal radiation distribution for any area of the focused

model and can be used to influence the next time step. Further, it is possible to calcu-

late the heat flow to other objects and so, the heat distribution in non-conductive ob-

jects.

Outlook

To obtain the thermal radiation in the presented application it is necessary to solve a

large fully populated system matrix originated from the BEM. Therefore, an enormous

amount of physical memory is required while BEM applications increase the memory

requirements with the second power to the number of unknowns. To reduce this growth

a compression of the system matrix may be examined. Due to the smoothness of the

kernel function (5), a wavelet-based compression of separated areas of the computed

model may reduce the growth of the BEM system matrix (4) nearly to linear complex-

ity and allows for solving problems with higher precision.

References

[1] V. Rischmüller, J. Fetzer, M. Haas, S. Kurz, W. M. Rucker, Computational efficient BEM-FEM coupled

analysis of 3D nonlinear eddy current problems using domain decomposition, Proceedings of 8th Interna-

tional IGTE Symposium, Graz, 1998.

[2] F. P. Incropera, D. P. DeWitt, Fundamentals of Heat and Mass Transfer, Fifth Edition, John Wiley &

Sons, 2002.

[3] C. Scheiblich, K. Frenner, W. M. Rucker, Computation of Radiative Heat Transfer with The Boundary

Element Method for Inductive Heating, Proceedings of 12th International IGTE Symposium, Graz, 2006.

C. Scheiblich et al. / Coupling Thermal Radiation to an Inductive Heating Computation84

Consideration of Coupling Between

Electromagnetic and Thermal Fields

in Electrodynamic Computation of

Heavy-Current Electric Equipment

Karol BEDNAREK

Institute of Electrical Engineering and Electronics, Poznan University of Technology,

Piotrowo 3a, 60-965 Poznan, Poland

E-mail: [email protected]

Abstract. The paper presents a model of electrodynamic computation (current

density, power loss, temperatures) of three-phase heavy current busways with the

use of integral equation method. The computation makes allowance of coupling

between the electromagnetic and thermal fields. Results of calculation are shown

and compared with the measurement trials performed with physical objects.

Introduction

An important factor of the procedures of heavy-current electric equipment designing is

accurate definition of the electrodynamic parameters decisive for their proper opera-

tion. A very important element conducive to improving accuracy of electromagnetic

calculation consists in precise consideration of physical factors affecting the process,

e.g. by appropriate account of coupling between the electromagnetic and thermal inter-

actions.

The paper presents a model of electrodynamic calculation of three-phase heavy-

current lines (i.e. power busways) with consideration of coupling between electromag-

netic and thermal fields. More accurate calculation of electrodynamic parameters ob-

tained in consequence of the above is confirmed by measurement of the physical ob-

jects. The software package developed this way was used in designing of the process of

heavy-current busways. This allows for significant financial savings at the stages of

their manufacturing and exploitation processes.

Taking into account the symmetry of the system and radial heat transmission the

use of a 3D system would be unjustified (no significant variations along the path).

Therefore, a 2D system was used for computation purposes, that is important for the

optimization process (with many times repeated calculations).

The Model of Electrodynamical Calculation

A three-phase heavy-current screened busway composed of three oval conductors

(Fig. 1) of cross section Sc was considered. The conductors were disposed symmetri-


85

cally each 120º apart, in a screen of inner radius RS and outer radius R

O. Another char-

acteristic dimensions are: the large a and small b diameter of the conductor, thickness g

of its walls, height h of the conductor suspension.

The basic parameter required for purpose of further electrodynamic analysis is the

distribution of the currents of phase conductors and the ones induced in the screen. As

a basis for determining the distribution the relationships of the magnetic vectorial po-

tential are used, written down for particular sub-areas of the system. In case of the con-

sidered 2D system the magnetic vectorial potential A(r,ϕ,z) has only one component in

z-axis, depending solely on the (r,ϕ) coordinates [2,4].

In consequence, A(r,ϕ,z) = 1z A(r,ϕ) and meets the following relationships in par-

ticular areas (Fig. 1):

− in the I area (inside the screen), i.e. for 0 ≤ r ≤ RS:

I 1 2

(r (r, ) + (r, ),ϕ) = ϕ ϕA A A (1)

According to the relationship A1(r,ϕ) originates from the currents flowing in the

phase conductors and may be expressed by the formula:

[ ]

C

i

O i i

i = 1S

3 x

(r, ) = J(r', ') a sin i( ') + b cos i( ') r' dr' d '

4 i

∞

ϕ μ ϕ ϕ−ϕ ϕ − ϕ ϕ

π

∑∫A1

(2)

The potential A2 (r,ϕ) is due to the currents induced in the screen and fulfills the

Laplace equation:

2

( r, ) 02

∇ ϕ =A (3)

− in the II area (material of the screen), i.e. for RS≤ r ≤ R

O:

2

II O S S II

(r, ) = j (r, )∇ ϕ ω μ μ γ ϕA A (4)

Figure 1. Geometry of the considered system (with marked sub-areas).

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields86

− in the III area (outside the screen), i.e. for r ≥ RO:

2

III

(r, ) = 0∇ ϕA (5)

In the above relationships: Sc – is the cross-section area of a single phase conduc-

tor; J – current density of R-phase; ω – pulsation (ω = 2πf); μ0 – magnetic permeability

of vacuum; μS

– relative magnetic permeability of the screen material; γS

– conductivity

of the screen material. The coefficients xi, a

i, and b

i take the values [2]:

⎪

⎩

⎪

⎨

⎧

⎪

⎩

⎪

⎨

⎧

⎩

⎨

⎧

≠

−−

−=

≥

≤

3l i for 1

3l = i for 0

= b

3l = i for j

3l = i for j

3l = i for 0

a

r' r for

r

r'

r' r for

r'

r

=x ii

,

2

1,

(6)

for l = 1, 2, 3, ….

Moreover, the following boundary conditions should be satisfied inside particular

areas:

− for r = RS:

AI(r,ϕ) = A

II(r,ϕ) and H

Iϕ(r,ϕ) = H

IIϕ(r,ϕ) (7)

− for r = RO:

AII

(r,ϕ) = AIII

(r,ϕ) and HIIϕ

(r,ϕ) = HIIIϕ

(r,ϕ) (8)

Distribution of current density J(r,ϕ) in the phase conductor is derived from an ap-

proximate solution of integral equations [2,4] obtained in result of the use of known

relationships for the electromagnetic field E = –jωA and J = γE:

[ ]O O O C O O

S

3

J(r, ) J(r , ) j J(r', ' ) K(r', ', r, ) K(r', ', r , ) dr' d ' 0

4

ϕ − ϕ + ωμ γ ϕ ϕ ϕ − ϕ ϕ ϕ =

π

∫

(9)

S

J(r', ' )r' dr' d ' Iϕ ϕ =∫ (10)

where (ro,ϕ

o) is a reference point, γ

C is conductivity of conductor material, I – intensity

of the current flowing in the R-phase, while K(r',ϕ',r,ϕ) – is a kernel of the integral

equation. The coefficients occurring in equations are described in [2].

It results from symmetry of the system that distribution of current density of two

remaining phase conductors (S and T) is the same as in R but shifted by +120° and

–120°, respectively.

The presented system of integral equations may be solved in approximate manner

using a moment method, being a variation of Ritz method [2,4]. In order to apply this

method the cross-section S of the conductor is divided into N elements of the areas ΔSm

(with m = 1,2,…,N).

K. Bednarek / Consideration of Coupling Between Electromagnetic and Thermal Fields 87

Current density is expanded in functional space:

N

m m

m = 1

J (r, ) = J f ϕ ∑ (11)

while Jm = J (r

m,ϕ

m) = const (for m = 1, 2, …, N) and f

m are base functions defined as

follows:

m

m

1 for S

f =

0 for the remaining elements

Δ⎧

⎨

⎩

(12)

In result, Jm is an approximate value of current density in a ΔS

m – element. Making

use of the moment method the system of integral equations is then replaced by a system

of N linear algebraic equations in the form:

1,3 1,N 1 1,1 1,2

2,3 2,N 22,1 2,2

3,3 3,N 33,1 3,2

N-1,3 N-1,NN-1,1 N-1,2

3 N1 2

l l Jl l ......

l l Jl l ......

l l Jl l ......

......... .................. ......... ............

l ll l ......

S SS S ......

⎡ ⎤

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

Δ ΔΔ Δ⎢ ⎥⎣ ⎦

N-1

N

0

0

0

J 0

J I

⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(13)

where:

[ ]

m

m,n m,n N,n O C m m N N

S

3

l j K(r', ' ,r , ) K(r', ' ,r , ) dr' d '

4Δ

= δ − δ + ω μ γ ϕ ϕ − ϕ ϕ ϕ

π

∫

(14)

for: m = 1,2,3,...,N–1 and n = 1,2,3,…,N, δm,n

is Kronecker delta, K(r',ϕ',r,ϕ) – a ker-

nel of the integral equation defined in the form presented in [2].

Solution of the system (13) provides approximate values Jm (for m = 1, 2, 3,…, N)

of current density in particular elements of cross-section ΔS1, ΔS

2, …, ΔS

N. Total cur-

rent I flowing in the phase conductor may be considered as a set of m conductors

transmitting the currents, Im = J

mΔS

m, (m = 1,2,…,N).

Distribution of the current JS induced in the shield is obtained with the use of the

analogical relationships.

Knowledge of approximate distribution of the current density vector enables de-

termining the value of active power losses in conductors and shield. The losses of ac-

tive power in phase conductors PC (falling to unit length) and in the shield P

S may be

determined e.g. from the Joule law [2,4]:

2

C

C S

3

P J(r', ') r'dr'd ' ,= ϕ ϕ

γ

∫

2

S S

S Ss

3

P J (r', ') r'dr'd '= ϕ ϕ

γ

∫ (15)


Taking into account the approximate distribution of current density, the relation-

ship takes a form:

N

2

C m m m m

m = 1C

3

P J (r , ) S

γ

= ϕ Δ∑ , m

N

2

S S m m m

m = 1S

1

P J (r , ) ΔS

γ

= ϕ∑ (16)

Knowledge of the active power losses and the distribution of power density emit-

ted in the conductors and in the shield is necessary for determining thermal conditions

of the system.

Distribution of power density output in the shield is expressed by the relationship:

2

S

S

J (r', ')

ρ(r, )

γ

ϕ

ϕ = (17)

Results of many calculations made for aluminium shield of 3-mm thickness have

shown that ρ(r,ϕ) is a function strongly dependent on ϕ and symmetrically distributed

every 120° [2,3], while the variable r only slightly affects ρ.

Total thermal power emitted from phase conductors dissipates radially, approxi-

mately uniformly [1–6]. At the outer surface of the shield the following boundary con-

dition is met:

C

S S

S

P T

for r = R

2 R r

∂

= − λ

π ∂

(18)

Inside the shield the temperature meets Poisson equation:

S

r

r

2ρ( ,ϕ)

∇ Τ( ,ϕ) =

λ

(19)

From outer surface of the shield the thermal power is emitted to the environment in

result of convection and radiation:

[ ]CR O O

T(r, )

T(r, ) T for r R

r

∂ ϕ

λ = − α ϕ − =

∂

(20)

with surface film conductance: αCR

= αC + α

R, and α

C – surface film conductance re-

sulting from convection, αR – surface film conductance resulting from radiation, T

O –

temperature of the environment. Temperature excess above that of the environment

meets the Poisson equation [1,3]:

2

C 2

T(l)

p(l)

l

∂

λ = −

∂

, where

n

j mj

j=1

1 2

p(l) P (l l ) T(l)

g g

α

= δ − −∑ (21)


is a difference between the power emitted from the branch through n inner sources,

and the power carried away to the environment with constant α coefficient, δ(l – lmj

) is

delta function, g is the thickness of conductor wall.

Methods of solving the above questions and determining the coefficients occurring

in the equations are discussed in detail in the works [1–6].

The calculation model used here makes allowance for coupling between the elec-

tromagnetic and thermal fields. The coupling occurs on the conductance of the conduc-

tors and shield. The temperature changes cause the changes in conductivity of the

shield and conductors and in result power losses, and vice versa. These dependencies

may be expressed as:

T = f (ρ,γ,λ) and ρ = f (T,γ) (22)

The calculations must be carried out with iterative methods. Before the calcula-

tions an assumption must be made about temperature of the conductors and the shield.

The assumption must be checked at the end of the calculations. Once the error exceeds

0.5 K a new assumption must be made and the calculation repeated [1,2,5,6].

Results of Calculation and Measurement

The thermal analyses are performed for an air-insulated screened three-phase heavy-

current busway. Temperature distribution is determined in aluminum conductors and

the shield. The following data were assumed for the calculation purposes: rated current

1.6 kA, a = 78 mm, b = 50 mm, g = 11.5 mm, h = 92 mm, RS = 247 mm, R

O = 250 mm,

conductivity: of shield material 28⋅106

S/m and of conductor material 31⋅106

S/m,

emissivity coefficient 0.4, temperature of the environment 297.5 K, thermal conductiv-

ity 220 W/(m⋅K). Results of the calculation are shown in Fig. 2a. The calculated tem-

perature distribution is presented with the accuracy of two digits after decimal point,

only with a view to depicting the scale of temperature differences between particular

points of the system. Such accuracy is useless in case of practical temperature compu-

tation as the temperatures so determined depend on many factors (significantly affect-

ing the results and difficult to consider), like the condition of the conductor and shield

surfaces (influencing the values of surface film conductance), arrangement of the

busway (even slight deviation from horizontal arrangement may affect the convection),

or external conditions (random air motion in proximity of the busways or consideration

of insolation in case of open areas). The calculation accuracy achieved in practice con-

siderably depends on the accuracy of the coefficients used for computation (provided

by manufacturers of the materials and depending, for example, on the time-varying

surface conditions of the elements) and, in the same time, on some random factors (e.g.

random motion of the air around).

Temperatures were calculated for various current values and the variants with and

without the coupling of the electromagnetic and thermal fields. The results of the calcu-

lation are compared to the measurements performed on physical objects, that is shown

in Fig. 2b.

The software package of electrodynamic computation has been used in the optimi-

zation process of the heavy-current busways. Results of the optimization calculation

are presented in the papers [2,3].


a) b)

Figure 2. Results of calculation and measurement: a) temperature distribution T[K] of the system (results of

calculation), b) temperatures T[K] of the conductors changes in the system as functions of the currents I[A].

Conclusion

Consideration of the coupling between the electromagnetic and thermal fields in the

mathematical model of electrodynamic calculation for the heavy-current busways im-

proves accuracy of the results. In case of the calculation carried out without the cou-

pling the calculation error is smaller than 2.5 per cent, while consideration of the cou-

pling reduces the error below 1 percent. Accuracy of the thermal calculation depends

on many factors, among which the following might be mentioned: correct choice of the

assumed mathematical model, the accuracy of the coefficients determining the type of

the material and the condition of the elements’ surfaces (e.g. the colour or roughness

change due to external factors), and proper consideration of the external conditions

(e.g. random air motion, consideration of insolation or other heat sources in the envi-

ronment).

Better accuracy of the results may be achieved by the use of 3D models, allowing

for consideration of any design details. Such calculation models are conducive to re-

markable growth of dimensions of equation systems obtained this way and the increase

in duration of the computation process. Nevertheless, such an approach becomes bur-

densome in case of optimization processes in which the calculation is many times re-

peated (in order to repeated determination of the objective function). It should be no-

ticed that for purposes of initial approximation of a system to be optimized very high

accuracy of the parameters is not required. Moreover, in symmetrical systems the elec-

trodynamic parameters in the third dimension very often remain nearly unchanged.

Hence, the 3D approach is unjustified.

References

[1] G. F. Hewitt, G. L. Shires, T. R. Bott, Process heat transfer, New York CRC Press, Boca Raton, 1994.

[2] K. Bednarek, Electrodynamical and optimization problems of oval three-phase heavy current lines,

Boundary Field Problems and Computer Simulation, 46th thematic issue, series 5: Computer Science,

Scientific Proceedings of Riga Technical University, Riga 2004, pp. 6-18.

[3] K. Bednarek, Heat and optimization problems in heavy current electric equipment, ISEF’2005, Baiona

(Spain), September 2005, pp. CFSA-2.17_1-4.


[4] K. Bednarek, Determination of temperature distribution in oval three-phase shielded heavy current lines,

UEES’2004, International Conference on Unconventional Electromechanical and Electrical Systems,

Alushta (Ukraine), September 2004, pp. 649-655.

[5] M. F. Modest, Radiative heat transfer, ed. II, Academic Press, N. York, Oxford, Tokyo, 2003.

[6] Y. Jaluria, K.E. Torrance, Computational Heat Transfer. Hemisphere Pub. Corp., Washington, N. York,

London, 1986.


Force Computation with the Integral

Equation Method

Wolfgang HAFLA, André BUCHAU and Wolfgang M. RUCKER

Institute for Theory of Electrical Engineering, Pfaffenwaldring 47,

70569 Stuttgart, Germany

[email protected]

Abstract. There is a great multitude of different approaches to compute magnetic

forces, each of them giving different results. Therefore, in this paper three popular

force computation methods are investigated for nonlinear magnetostatic problems

with the integral equation method. Computed results are compared with measure-

ments from two setups. First, the recently presented TEAM Workshop Problem

No. 33.a is used since it allows for verification of force densities. Second, the

computed total magnetic force acting on a body is verified with the TEAM Work-

shop Problem No. 20.

Introduction

In recent years the integral equation method (IEM) has become applicable to solve

magnetostatic problems since it has been used in combination with matrix compression

techniques, such as the fast multipole method. This approach leads to a reduction of

computational costs from 2

( )NO to approximately ( )NO , where N is the number of

unknowns. Since computation of magnetic forces is often required, e.g. during the de-

sign process of electrical machines. There is a large number of different approaches to

compute these forces, and unfortunately, many of them give different results. We,

therefore, investigated three different force computation formulations with respect to

their accuracy and applicability when using them with the IEM. With the investigated

methods, a magnetic pressure p is computed. The total force is then obtained by inte-

grating p . By investigation of the TEAM Workshop Problem 20 [1] it is shown that it

all formulations are of approximately the same accuracy. This is due to the high perme-

ability of this setup. Fortunately, the new TEAM Workshop Problem No. 33.a [2] has

been presented lately. The unique property of this setup is that it contains a sample with

an extremely low permeability. It is shown that with this setup it is possible to decide

which force computation formulation is the most applicable for magnetostatic prob-

lems.

IEM and Force Computation Formulations

Magnetostatic field problems as shown in Fig. 1 are considered. Free currents J within

domainJ

Ω cause a source field that magnetizes magnetic material in the domain M

Ω


93

andF

Ω . The air domain is 0

Ω . The magnetostatic problem is solved with the IEM for-

mulation [3]

( ) ( ) ( )( ) ( ) ( )0

d

M F

J

' ' ' G , ' V 'ψ μ χ ψ ψ

Ω +Ω

+ ⋅∇ ⋅∇ =∫r r r r r r , (1)

where ψ is the total magnetic scalar potential, J

ψ is the magnetic scalar potential of

the source field J

H , χ is the magnetic susceptibility, r and 'r are observation and

source points, and G is Green’s function of free space. The IEM is accelerated with

the fast multipole method [4]. Nonlinear problems are tackled by direct iteration [5].

After ψ has been solved with (1) the magnetic field H is computed within M

Ω

and F

Ω with

ψ= −∇H (2)

and in the air domain with

( ) ( )

( ) ( )( )

( )3 5

1

1 3

4

M F

r J

' '

dV '

ψ ψ

μ

πΩ +Ω

⎛ ⎞∇ ∇

= − − +⎜ ⎟

⎝ ⎠

∫

r r r - r'

H r H r

r - r' r - r'

. (3)

The force that acts on F

Ω can be computed from the magnetic pressure p . It is

obtained by evaluation of Maxwell’s stress tensor. For evaluation points 0

\F

∈ Ω Ωr ,

( ) ( ) ( )( ) ( )

( )( )2

1 0

2

p μ

⎡ ⎤

= ⋅ − ⋅⎢ ⎥

⎣ ⎦

H r

r n r H r H r n n . (4)

If ∈M

r ∂Ω

( ) ( ) ( )2 3, n t

p p p= ±r r r (5)

with

0

Ω

F

Ω

r

μ

J

Ω

J

M

Ω

r

μ

Figure 1. Considered field problems.

W. Hafla et al. / Force Computation with the Integral Equation Method94

( )2

0

1 1

1

2

n n

r

p Bμ

μ

⎛ ⎞= −

⎜ ⎟

⎝ ⎠

r , (6)

( ) ( )2

0

1

1

2

n r t

p Hμ μ= −r , (7)

where the plus and minus sign is obtained when (5) is derived from Maxwell’s stress

tensor method and the energy principle, respectively. t

H denotes the tangential com-

ponent of H in the air domain.

For inhomogeneous materials, additional volume forces

( )2

1

2

H χ= − ∇f r (8)

have to be considered. This gives three different force computation methods. First, the

force F can be obtained by integrating p over a closed surface A placed in 0

Ω that

encloses the body

( ) ( )2 3 1

d,

A

p= ∫F r r A

. (9)

Also, the surface M

A = ∂Ω can be used

( ) ( ) ( )2 3 2 3

d d

M

, ,

A

p V

Ω

= +∫ ∫F r r A f r

. (10)

Since the force on a body is well-defined, either 2

p or 3

p has to give wrong re-

sults. In practice, however, this is hard to decide. This is due to the fact that magnetic

setups are usually made of highly permeable ferromagnetic materials. In these cases the

magnetic field is almost perpendicular on the material’s surface and therefore t n

p p ,

i.e. (6) and (7) give almost the same results.

Numerical Results

The TEAM workshop problems No. 20 and No. 33a which are shown in Figs 2a and

2b, respectively, have been investigated. Both field problems were discretized with

second-order tetrahedrons.

TEAM Workshop Problem No. 20

The force acting on the center pole of the setup has been computed with the three in-

vestigated formulations and compared with measurements. Hereby, different ampere-

turns were used in order to investigate the accuracy of the formulations at different

saturation states of the setup. For computation of 1

F the integration surface that sur-

W. Hafla et al. / Force Computation with the Integral Equation Method 95

rounds the center pole shown in Fig. 2a has been used. It was discretized with 2868

second-order triangles. The results shown in Fig. 3 indicate that the two Maxwell for-

mulations 1

F and 2

F give the most accurate results. Especially at higher ampereturns

the energy formulation 3

F gives less accuracy than 1

F and 2

F . The two formulation

2

F and 3

F can obviously be used for this setup though in principle, they are valid for

linear materials only.

TEAM Workshop Problem No. 33a

Due to the high permeability of the above setup, the results obtained with 2

F and 3

F

are of the relatively same accuracy. Therefore the TEAM Workshop Problem No. 33a

has been investigated. Hereby, the relative permeability of the sample is only 2.5. This

leads to large tangential field strength components and a vanishing volume force den-

sity. Since the magnetic pressures 2

p and 3

p differ therefore significantly, this setup

allows for deciding which formulation is more applicable for force computation. The

results are depicted in Fig. 4. Interestingly, there is no need to compare the result with

measurements since the magnetic pressure computed with the energy formulation is

obviously wrong. The pressure doesn’t cause a pulling but a pushing force. This is,

because due to the large values of t

H the pressure t

p is larger than n

p so when (5) is

computed3

0p < .

Conclusions

The integral equation method is applicable for force computation with nonlinear mag-

netostatic problems. Depending on whether a force formulation is based on Maxwell’s

stress tensor or on the energy principle, different analytical results are obtained. Since

A) B)

Figure 2. Investigated setups: (A) TEAM Workshop Problem No. 20 and (B) TEAM Workshop Problem

No. 33a.

W. Hafla et al. / Force Computation with the Integral Equation Method96

the magnetic force that is acting on a magnetizable body is well-defined it is clear at

least one of the formulations has to give wrong results. By investigating the setup with

extremely low relative permeability it has been shown that only the Maxwell formula-

tion seem to give accurate results.

References

[1] O. Bíró, “Solution of TEAM Benchmark Problem #20 (3-D Static Force Problem),” in Proceedings of the

Fourth International TEAM Workshop, pp. 23-25, Miami, USA, 1993.

[2] O. Barré, P. Brochet, M. Hecquet, “Experimental validation of magnetic and electric local force

formulations associated to energy principle,” IEEE Trans. Mag, vol. 42, issue 4, pp. 1475-1478, April

2006.

[3] L. Han, L. Tong, “Integral equation method using total scalar potential for the solution of linear or

nonlinear 3D magnetostatic field with open boundary,” IEEE Transactions on Magnetics, 30(5):

2897-2900, 1994.

[4] A. Buchau, W. M. Rucker, O. Rain, V. Rischmüller, S. Kurz, and S. Rjasanow, “Comparison Between

Different Approaches for Fast and Efficient 3D BEM Computations,” IEEE Transactions on Magnetics,

39(3):1107-1110, 2003.

[5] W. Hafla, A. Buchau, F. Groh, and W. M. Rucker, “Efficient Integral Equation Method for the Solution

of 3D Magnetostatic Problems,” IEEE Transactions on Magnetics, vol. 41, no. 5, pp. 1408-1411, 2005.

Figure 3. Forces with TEAM workshop problem No. 20 computed with different formulations.

A) B)

Figure 4. Magnetic pressures computed with A) Maxwell formulation2

( )p and B) energy formulation3

( )p .

W. Hafla et al. / Force Computation with the Integral Equation Method 97


Chapter B. Computer Methods in Applied

Electromagnetism

B1. Computation Methods


Numerical Simulation of Non-Linear

Electromagnet Coupled with Circuit

to Rise up the Coil Current

Slawomir STEPIENa

, Grzegorz SZYMANSKIa

and Kay HAMEYERb

a

Chair of Computer Engineering, Poznan University of Technology, Poznan, Poland

b

Institut fur Elektrische Maschinen, IEM RWTH – Aachen, Aachen, Germany

Abstract. In this paper a numerical model and simulation of non-linear field – cir-

cuit system dynamics is presented. The system includes electromagnet coupled

with two RC circuits. The RC circuit performs a current rise acceleration and im-

proves the electromagnet dynamics. In proposed system a value of capacitance C

impact on electromagnet dynamics is numerically determined.

Introduction

The applications of electromagnet drives are widespread. They are used in electronic

and electromechanical consumer products, telecommunication, automatics, computer

technology, etc. The electromagnet actuators can be excited from impulse voltages

generated by special electronic drivers. Then device response as a plunger displace-

ment can be controlled by width or frequency modulation of input voltages [4]. In sev-

eral cases an improvement of electromagnet dynamics is necessary [7]. This improve-

ment is related with reduction of the motion time. Then solution with external circuit

composed of serial resistance and parallel capacitor connected to actuator winding is

proposed. The analysis is based on field – circuit – movement model [1,3,5,6].

Figure 1 shows cross section of the cylindrical electromagnet geometry and con-

figuration of the external circuit part. The RC circuit is connected to each coil and sig-

nificantly changes an input impedance of the system. The inductivity and electromotive

force change in time during plunger motion. The RC circuit connected to coil input

makes rise of the coil current faster when additional resistance R and capacitance C are

well matched and voltage excitation is increased.

In this article the problem consists of finding relationship between value of capaci-

tance C and time needed to set plunger in extreme position. Thus modeling and nu-

R

C

coil 1 coil 2

plunger

spring

u1 u2

R

C

Figure 1. An electromagnet geometry with coil input circuits.


101

merical simulation of mentioned system is presented. There are given advantages and

disadvantages of the proposed method to improve the device dynamics. Also as a new

contribution an interesting direct coupled field – circuit model of the device is widely

presented and discussed.

Modeling Technique

The actuator is modelled in 3D domain using Maxwell equations to formulate the field

model. FEM approximation is used to discretise the domain. Magnetic vector potentials

and electric scalar potentials are used as variables of electromagnetic field [2,7]. The

boundary value problem is defined by Eqs (1)–(2):

( )( ) jAv

A

A =×∇×−⎟

⎠

⎞

⎜

⎝

⎛∇+

∂

∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

×∇×∇ σσ

μ

V

t

1

(1)

( )( ) 0=⎟

⎠

⎞

⎜

⎝

⎛∇−

∂

∂

−×∇×⋅∇ V

t

σσσ

A

Av (2)

where A is a magnetic vector potential defined in mesh nodes, V is a electric scalar

potential defined in nodes of conductive regions, j is a current density, σ represents

conductivity, μ represents permeability and v is a velocity of moving body.

To define external electric circuit connected to the actuator windings, an electric

circuit equations are defined by Eqs (3)–(4):

( )111

1l

1l

2

2

uiRR

dt

d

dt

d

RC

LL

=⋅+++

∫∫AdlAdl (3)

( )222

2l

2l

2

2

uiRR

dt

d

dt

d

RC

LL

=⋅+++

∫∫AdlAdl (4)

where R is a resistance of external circuit, C is capacitance of the external circuit,

1L

R and 2L

R represent resistance of windings, 1L

i and 2L

i are winding currents, 1

u

and 2

u are a circuits excitations.

Obtained second order differential equations shows that in each circuit exist two

elements which are able to magazine energy and can transfer it between themselves.

These second order equations can be reduced to first order system of equations by us-

ing additionally defined scalar variables w. In this way a state space formulation of

circuit equations [1,5] is given by:

11

1

1

1

1

1l

11

u

RC

i

RC

RR

w

RCdt

dw

w

dt

d

L

L

+

+

−=

=

−

∫Adl

(5)

S. Stepien et al. / Numerical Simulation of Non-Linear Electromagnet102

22

2

2

2

2

2l

11

u

RC

i

RC

RR

w

RCdt

dw

w

dt

d

L

L

+

+

−=

=

−

∫Adl

(6)

In above equations a voltage sources have an impact on winding currents 1L

i and

2L

i , winding voltages 1

w and 2

w ,

dt

dw

1

and

dt

dw

2

which can be defined as velocity of

windings voltage change or linkage flux acceleration related to the windings.

After discretisation in time and space a global matrix system is obtained. The

boundary value equations (1)–(2) produces matrices related to vector potential A and

scalar potential V. In non-linear case the matrix related to A also depends on μ . Dis-

cretisation of (5)–(6) produces additional rows in global matrix system. The equa-

tions (1)–(2) and (5)–(6) have one common variable A and for this reason the field

equations can be coupled with circuit equations directly. Then current density j is

eliminated from (1), because winding currents are a functions of vector potential A.

Taking above into account a coupled field – circuit system of matrix equations be-

comes:

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+

+

⋅

⋅

⋅

=

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+

−

+

−

+

+

+

⋅

+

+

+

+

+

+

+

+

+

ttt

t

ttt

t

t

t

tt

L

tt

tt

L

tt

tt

tt

L

L

wu

RC

t

wu

RC

t

i

w

i

w

RC

RR

t

RC

t

t

RC

RR

t

RC

t

t

22

2

11

1

2

2

1

1

2

2

1

1

10000

0000

00100

0000

000

000

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

AL

AL

AF

AD

V

A

L

L

0GHF

0ED)(C μ

(7)

where matrices C(μ), D, E, F, G, and H are obtained from discretisation of following

terms:

( )( )∫∫

×∇×⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

×∇×∇ −

ΩΩ

dΩ(..)σdΩ(..)

1

v

μ

, dΩ(..)

1

Ω

∫Δ

σ

t

,∫

∇

Ω

dΩ(..)σ, dS

∫Δ

S

(..)

1

σ

t

,

∫∇

S

(..)dSσ and ( )( )

∫×∇×−

S

(..) dSvσ. Rows

1

L and 2

L are derived from discretisa-

tion of

∫

1l

Adl and

∫

2l

Adl .

The matrices and vectors defined in equation system (7) are composed of real coef-

ficients and have following dimensions: n

RA∈ ,m

RV∈ ,nn×

∈RDC, ,mn×

∈RE ,

nm×

∈RHF, ,n×

∈

1

2

,

1

RLL and 1

212121

,,,,, R∈wwuuii

LL

.

S. Stepien et al. / Numerical Simulation of Non-Linear Electromagnet 103

This complex system of equations is non-symmetric and solved using bi-conjugate

gradient method (BiCG) for large and sparse matrix equations. The non-linearity prob-

lem is solved using Newton-Raphson method based on B-H curve continuos approxi-

mation [7]. At each time step, the magnetic force is determined using Maxwell stress

tensor. The force is a non-linear function of potentials A and calculated locally in each

element of discretisation. The total force which acts on movable body is a sum of local

forces [7]. The force density is given by following formula:

Tf ⋅∇= (8)

where T denotes Maxwell’s stress tensor. Then total force is defined as:

dΩfF∫Ω

= (9)

The total force is determined as a vector of three components [ ]T

zr

FFF

ϕ

=F .

As shown in Fig. 1, the motion of movable armature takes place only along axis z, thus

to solve the 1 DOF problem displacement z

s and velocity z

v are chosen as state of

mechanical motion and componentz

F is used as excitation.

z

z

z

z

z

F

m

v

s

m

b

m

k

v

s

dt

d

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

−−=⎥

⎦

⎤

⎢

⎣

⎡

1

010

(10)

where k is a spring stiffness and b is a damping coefficient. Discrete form of equations

(10) is obtained using Eulers recurrence method.

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

−

+

+

+

+

tt

z

t

z

t

z

tt

z

tt

z

F

m

tv

s

v

s

m

b

t

m

k

t

t

ΔΔ

Δ

ΔΔΔ

Δ

11

1

(11)

Summing up this mathematical description of analysed system, in each iterative

step take place calculation of field – circuit equation system (7), force (9) and motion

equation (11).

Numerical Analysis

The developed field – circuit model of presented system has been used to investigate

the relationship between capacitance of the external RC circuit and time of the plunger

motion. The external circuit resistance is assumed R = 2Ω and capacity is taken from

set C = 100 nF, 1 μF, 10 μF, 47 μF, 100 μF, 470 μF, 1000 μF. Presented device is

made from Armco.

The specific gravity of armature is equal to 7800 kg/m3

, substitute stiffness

k = 800 N/m of springs, and damping coefficient b = 2 Ns/m. Voltage square-wave


excitation is applied on device, where actuator windings have 424 turns and resistance

RL

= 2,05 Ω. Bellow is presented the B-H curve of the Armco.

To demonstrate the dynamics improvement of analysed actuator, two kind of simu-

lations are compared: firstly is examined system free of external circuit where the volt-

ages with amplitude u = 12 V and frequency f = 250 Hz are applied directly on actuator

windings and next actuator is equipped with external circuits (as shown in Fig. 1). In

the second case voltages which control the plunger motion are twice as big and guaran-

tee the same windings current flow in steady state. The voltages in both cases are

shown bellow.

Firstly the voltage 2

u cause a current flow in right winding and produces force

which moves plunger in right direction (s = 3 mm). When 2

u is zero, then voltage 1

u

controls the plunger in left position (s = –3 mm). Finally when both voltages are zero

then movable armature should reach middle position, in other starting point of plunger

motion. The results of analysis two extreme cases (without external circuit and with

external circuit for C = 100 nF) are given bellow.

Figure 2. Mesh of analysed actuator.

Figure 3. Measured Armco B-H curve.


As compared in Fig. 5 time needed to move plunger from position s = 0 mm to

s = 3 mm is decreased two times by usage external RC circuit. This rule also is being in

force when plunger is moved from right s = 3 mm to left s = –3 mm. In this case the

time is somewhat larger then previously. Unfortunately, when actuator is autonomous,

in other free of the excitations, an energy concentrated in capacitance does not allow to

plunger return in position s = 0 mm immediately. The return is delayed.

The situations were examined for various capacities C = 100 nF, 1 μF, 10 μF,

47 μF, 100 μF, 470 μF, 1000 μF. To estimate capacity impact on motion time of the

plunger during actuator starting to position s = 3 mm is examined as function ts = f(C).

As can be seen in above plots, the usage of external circuits improves the electro-

magnet dynamics during starting phase. The reason is due to the fact that the inertia of

input circuits is reduced. However, the dynamics also depends on input capacity. As

shown in Fig. 6, the motion time reduction is the best for small values of capacity.

Greater capacitance delays plunger motion. So, the delays of motion can be controlled

by value of capacity, too.

a)

b)

Figure 4. An input square-wave voltages: a) without RC circuit b) with RC circuit.

Figure 5. Actuator responses on applied voltages.


a)

b)

Figure 6. Electromagnet starting: a) displacement b) capacity – stating time relationship.

Conclusion

This work presents numerical analysis of field – circuit system considering motion and

magnetic non-linearity. The simulation experiment shows that additional circuit with

capacitance improves dynamics of an electromagnet. However there exist an exact rela-

tionship between value of capacity and plunger displacement.

In this paper as a new contribution a strongly coupled field – circuit model which

occurs with state space description suggests that presented methodology works in cal-

culations. The proposed method can be used to design a electromagnetic drives and

may contribute to the improvement of the drives dynamics.

References

[1] A. de Oliveira, R. Antunes, P. Kuo-Peng, N. Sadowski and P. Dular, Electrical machine analysis consid-

ering field – circuit – movement and skewing effects, COMPEL: The International Journal for Computa-

tion and Mathematics in Electrical and Electronic Engineering, Vol. 23, No. 4, pp. 1080-1091, 2004.

[2] D. Lowther, Automating the Design of Low Frequency Electromagnetic Devices – A Sensitive Issue,

COMPEL – International Journal for Computation and Mathematics in Electrical and Electronic Engi-

neering, Vol. 22, No. 3, pp. 630-642, 2003.

[3] K. Hameyer, Field – circuit coupled models in electromagnetic simulation, Journal of Computational and

Applied Mathematics, Vol. 168, pp. 125-133, 2004.

[4] L. Nowak, J. Mikolajewicz, Field-circuit model of the dynamics of electromechanical device supplied by

electronic power converters, COMPEL – International Journal for Computation and Mathematics in

Electrical and Electronic Engineering, Vol. 23, No. 4, pp. 977-985, 2004.

[5] H. De Gersem, R. Mertens, D. Lahaye, S. Vandewalle and K. Hameyer, Solution Strategies for Transient,

Field-Circuit Coupled Systems, IEEE Trans. Mag., Vol. 36, No. 4, pp. 1531-1534, 2000.

[6] S. Lepaul, J. Sykulski, C. Biddlecombe, A. Jay, and J. Simkin, Coupling of motion and circuits with

electromagnetic analysis. IEEE Trans. Mag. Vol. 33, No. 3, pp. 1602-1605, 1999.

[7] S. Stępień, A. Patecki, Modeling and Position Control of Voltage Forced Electromechanical Actuator,

COMPEL – International Journal for Computation and Mathematics in Electrical and Electronic Engi-

neering, Vol. 25, No. 2, pp. 412-426, 2006.


Numerical Calculation of Power Losses

and Short-Circuit Forces in Isolated-Phase

Generator Busbar

Dalibor GORENCa

and Ivica MARUSICb

a

Koncar – Electrical Engineering Institute, Fallerovo setaliste 22 Zagreb, Croatia

[email protected]

b

HEP Production Ltd.PA HPP SOUTH Split, Zakucac HPP, Croatia

[email protected]

Abstract. Metal enclosed generator busbars are applied in electric power plants

for the transmission of electric power from generator to transformer. The losses in

conductors and enclosures due to rated current are the basis for the calculation of

steady state temperature-rises. On the other hand, the maximum force on the con-

ductor due to short-circuit current is an important parameter in determination of

mechanical stresses in the conductors and supports. In this paper, using a program

based on finite element method (FEM), power losses and short-circuit forces are

calculated in arrangement of isolated-phase generator busbar.

Introduction

In isolated-phase busbar (IPB), each phase conductor is enclosed by an individual

metal enclosure, separated from adjacent conductor enclosures by air (Fig. 1, Fig. 2).

The enclosures are electrically continuous and short-circuited at both ends, and

grounded at one end. The conductor currents induce longitudinal currents in the enclo-

sures of almost the same magnitude but of the opposite direction to the conductor cur-

rents [1]. Thus, only a small percentage of the total magnetic field extends outside the

bonded enclosures. This results in a considerable reduction of the electromagnetic

forces between phases under short-circuit conditions [2,3] and avoids eddy currents in

neighboring steel structures under normal operating conditions.

Field Equations

The calculation of power losses and short-circuit forces is based under the following

assumptions:

− longitudinal dimension of busbar is significantly greater then the cross section

and a two-dimensional field analysis may be applied;

− characteristics of materials are constant;

− displacements currents are neglected.

The electromagnetic field, under described assumptions, is described by the fol-

lowing equations:



doi:10.3233/978-1-58603-895-3-108

108

H J∇× =

(1)

B

E

t

∂

∇× = −

∂

(2)

0B∇⋅ =

(3)

where H

is the magnetic field, J

is the current density, E

is the electric field, and B

is the magnetic flux density. For linear materials, the constitutive equations are:

B Hμ= ⋅

(4)

J Eσ= ⋅

(5)

where μ is permeability, and σ is conductivity. It is convenient to define a vector mag-

netic potential A

by:

B A= ∇×

(6)

These equations can be combined into vector Helmholtz equation with the mag-

netic vector potential as the unknown variable:

1

s

A

A J

t

σ

μ

∂

∇× ∇× = −

∂

(7)

In Eq. (7), the current density has been split into prescribed sources s

J

and the in-

duced currents /A tσ ∂ ∂

. In two-dimensional problems only z components of A and J

exist. Vector Helmholtz equation can be simplified to scalar one:

1z

z s

A

A J

t

σ

μ

∂

−∇ ∇ = −

∂

(8)

Using a FEM based program MagNet, Eq. (8) is solved numericaly in a frequency

or time domain, depending on whether the problem is steady-state or transient.

Phase conductor

Short-circuit plate

Enclosure

enclosure

phase conductor

x

y

phase A phase B phase C

Figure 1. IPB with continuous enclosures. Figure 2. Cross section of IPB.

D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces 109

Model of Busbar

The model of busbar is given on Fig. 3. Each phase conductor consists of an octogonal

duct made up of two sections held together at regular intervals by welded spacers, thus

maintaining a constant clearance. In order to calculate the current distribution in each

conductor section, it is assumed that they are electrically separated along the z-axis.

The phase conductor is concentrically enclosed in a welded circular enclosure. The

material of conductors and enclosures is aluminum with conductivity of 34.2 m/Ωmm2

,

and permeability μ0. The conductors are connected to the external current source while

the enclosures are short circuited at both ends and grounded at one end (Fig. 4). Total

Table 1. Main geometry data of the analyzed model

Width of

conductor

(mm)

Thickness of

conductor

(mm)

Outer diameter

of enclosure

(mm)

Thickness of

enclosure

(mm)

Centre-line distance

between conductors

(mm)

260 12.5 720 4 1000

Figure 3. Model of busbars.

Figure 4. Circuit connection of conductors and enclosures.

D. Gorenc and I. Marusic / Numerical Calculation of Power Losses and Short-Circuit Forces110

current in each phase conductor is known and is given by expression (9)–(11) or

(12)–(14). The domain of computation is the rectangular area with busbar and enclos-

ing air, dimensions 4.4 × 2.4 m. The boundary condition was A = 0 on the borders of

the area (Dirichlet) and the number of triangular elements was 165064.

Calculation of Power Losses

Under normal operating conditions, driving currents in phase conductors are varying

sinusoidally in time with a frequency of 50 Hz. In a frequency domain they can be ex-

pressed as phasors:

0

1

2

j

i I e= ⋅ (9)

120

2

2

j

i I e

− ⋅

= ⋅

(10)

240

3

2

j

i I e

− ⋅

= ⋅

(11)

Here, I is the RMS-value of the rated current in conductors.

Results

Figure 6 shows the distribution of losses (W/m3

) in upper part of conductor and enclo-

sure of phase B for the rated current of I = 7000 A. Total losses in conductors and en-

closures per meter of busbar are presented in Table 2. Ratio of ac to dc resistance

shows that the skin effect factor for the analyzed conductor design is equal to 1.12.

Figure 5. Finite element mesh.


Calculation of Short-Circuit Forces

Three-phase symmetrical short-circuit was analyzed because it causes the greatest dy-

namic stress. Under short-circuit condition, the fault currents in phase conductors are

function of time as follows [3]:

( ) ( )1

( ) 2 cos cos

t

T

k

i t I e tϕ ω ϕ

−⎛ ⎞

⎜ ⎟

⎝ ⎠

⎡ ⎤

= ⋅ ⋅ ⋅ − +⎢ ⎥

⎢ ⎥⎣ ⎦

(12)

( ) ( )2

( ) 2 cos cos

t

T

k

i t I e tϕ γ ω ϕ γ

−⎛ ⎞

⎜ ⎟

⎝ ⎠

⎡ ⎤

= ⋅ ⋅ − ⋅ − + −⎢ ⎥

⎢ ⎥⎣ ⎦

(13)

( ) ( )3

( ) 2 cos cos

t

T

k

i t I e tϕ γ ω ϕ γ

−⎛ ⎞

⎜ ⎟

⎝ ⎠

⎡ ⎤

= ⋅ ⋅ + ⋅ − + +⎢ ⎥

⎢ ⎥⎣ ⎦

(14)

where: kI is the RMS-value of the steady state short-circuit current, ϕ is the switching

angle, T is the time constant of direct current component and γ is the phase angle be-

tween currents (γ = 120°).

Results

In the case of unshielded conductors (enclosures conductivity σ = 0), the maximum

force acts on the central conductor, and appears about 10 ms after the beginning of

Figure 6. Distribution of losses in conductor and enclosure of phase B (W/m3

).

Table 2. Total losses in conductors and enclosures (W/m)

A-phase

conductor

B-phase

conductor

C-phase

conductor

A-phase

enclosure

B-phase

enclosure

C-phase

enclosure

167 167 167 155.3 159.2 163.5


short-circuit (Fig. 7). The switching angle for the maximum force is ϕ = 0°. In the case

of IPB, maximum force acts also on the central conductor but for switching angle equal

to ϕ = 30° [3]. Due to enclosure shielding effect, the maximum force appears 95 ms

after the beginning of short-circuit (Fig. 8) and is reduced by the factor of 7 as com-

pared with arrangement without enclosures.

However, two section of the B-phase conductor will be mutually attracted by the

force which is of the same order of magnitude as the conductor force without enclo-

sures. This fact should be considered in determination of distance between welded

spacers along the phase conductor.

Forces on enclosures have almost the same magnitude as corresponding conductor

forces but act in the opposite direction.

Figure 7. Force on conductor B, without enclosures (Ik= 50 kA, T = 45 ms and ϕ = 0°).

Figure 8. Force on conductor B, with enclosures (Ik= 50 kA, T = 45 ms and ϕ = 30°).


Figure 9. Force on conductor A, with enclosures (Ik= 50 kA, T = 45 ms and ϕ = 30°).

Figure 10. Force on conductor C, with enclosures (Ik= 50 kA, T = 45 ms and ϕ = 30°).

Conclusion

Presented numerical procedure enables fast and accurate calculation of power losses

and short-circuit forces for any geometry of metal enclosed busbars, thus taking into

account the following factors:

− detailed shapes and dimensions of the conductors and the enclosures,

− distributions of current among multiple conductors in one phase,

− magnetic field interaction between conductors and enclosures,

− proximity effect with conductors in other phases.

Similar methodology can be applied for three-phase segregated and nonsegregated

type of busbars in which all three phase conductors are in a common metal enclosure

with or without metal barriers between phases. In this type of busbars, the enclosure


has a shape of rectangle and the phase conductor may consist of one or more conduc-

tors of different shapes and dimensions.

References

[1] W.F. Skeats, N. Swerdlow, Minimizing the Magnetic Field Surrounding Isolated-Phase Bus by

Electrically Continuous Enclosures, AIEE Transactions PAS 81, pp. 655-667, February 1963.

[2] W.R.Wilson, L.L.Mankoff, Short-Circuit Forces in Isolated-Phase Busses, AIEE Transactions PAS,

April 1954.

[3] P. Dokopoulos, D. Tambakis, Analysis of Transient Forces in Metal Clad Generator Buses, IEEE Trans.

on Energy Conversion, Vol. 6, No. 3, September 1991.

[4] M.R. Shah, G. Bedrosian, J. Joseph, Steady-State Loss and Short-Circuit Force Analysis of a Three-

Phase Bus Using a Coupled Finite Element+Circuit Approach, IEEE Transactions on Energy Conversion,

Vol. 14, No. 4, December 1999.


Numerical Methods for Calculation

of Eddy Current Losses in Permanent

Magnets of Synchronous Machines

Lj. PETROVICa

, A. BINDERa

, Cs. DEAKa

, D. IRIMIEc

, K. REICHERTd

and

C. PURCAREAb

a

Inst. of Electrical Energy Conversion, Darmstadt University of Technology,

Landgraf-Georg-Strasse 4, D-64283 Darmstadt, Germany

E-mail: [email protected], [email protected],

[email protected]

b

Inst. of Power Electronics & Control of Drives, Darmstadt University of Technology,

Landgraf-Georg-Strasse 4, D-64283 Darmstadt, Germany

c

Faculty of Electrical Engineering, Technical Univ. Cluj-Napoca,

Str. C. Daicoviciu 15, 400020, Romania

d

emeritus ETH Zuerich, now: Schartenfelsstr. 1B, CH-5430 Wettingen, Switzerland

Abstract. Eddy current losses in rotor permanent magnets (PM) of synchronous

machines are calculated for sinusoidal stator currents and for PWM inverter sup-

ply. Three calculation methods are compared in the FE environment: a) time-

stepping method, b) quasi-static method, c) semi-analytical post-processing. These

2D methods are with end effect coefficients, and they consider the time variation

of currents and of the rotor position. Whereas method a) includes the variation of

flux-density over the magnet cross section and the reaction field of the eddy cur-

rents, method b) is neglecting the reaction field. Method c) in several variants fea-

tures either neglecting of the eddy current reaction field or an averaging of the flux

density along the magnet width or height. Neglecting the reaction field is possible

for materials with low conductivity and low permeability like rare-earth magnets

for low to medium frequencies up to several kHz. The quasi-static methods need

less computation time, but depend on the machine geometry like stator MMF wave

length, slot pitch, segmented vs. massive magnets and small or big magnet height.

The comparison of methods a), b), c) is given for two different stator geometries of

permanent magnet synchronous machines with open vs. semi-closed slots and sur-

face-mounted vs. buried magnets.

Introduction

In PM synchronous machines (PMSM) the stator slot openings cause already at no-load

a flux variation in the rotor magnets, which leads to an induced voltage and eddy cur-

rent losses. At load the stator magneto-motive force (MMF) space harmonics, excited

by the sinusoidal stator current, increase these losses. The additional stator current rip-

ple due to PWM inverter supply amplifies this effect. For calculating these losses, a

two-dimensional (2D) finite element method (FEM) may be used, if the 3D end effects,

like the axial segmentation, are considered by an appropriate end effect coefficient.

a) The time-stepping method solves the Maxwell equations for an arbitrary cur-

rent waveform per stator phase and a moving rotor, neglecting only the dis-



doi:10.3233/978-1-58603-895-3-116

116

placement current term in the first Maxwell equation (Ampere’s law), which is

of no relevance in the frequency range of interest of up to several kHz [3].

b) The magneto-static finite element method has a shorter computation time,

but neglects the eddy current reaction field. In rare earth magnets the low

permeability and conductivity in the considered frequency range gives rather

small eddy current densities, hence justifying the method. This method can be

faster and is less expensive.

c) In case that a quasi-static 2D finite element code is not offering a current den-

sity calculation acc. to b), an additional semi-analytical post-processing is

possible to evaluate these losses. These semi-analytical methods are consid-

ered in several variants in this paper and are compared with the results of a)

and b) [1].

Comparison of Calculation Methods

In the following for simplification rectangular magnets are considered with a magnet

width bM

in x-direction, a height hM

in y-direction, and a magnet length lM

in z-direction

(Fig. 1a). The results can be generalized for more arbitrary magnet geometries like

shell or bread-loaf shapes in an axial-symmetrical cylindrical coordinate system. For

2D calculations the axial end effect must be taken into account. The “equivalent” con-

ductivity is smaller than real conductivity of the material, if one considers the eddy-

currents to be caused by the “voltage forced” situation [2]. The normal component

(y-component) of the air gap magnetic flux density, excited e.g. by a sinusoidal varying

current time harmonic ˆcos( )

s

I tω⋅ , shows as a Fourier series of different field waves,

which penetrate the magnet, beside the synchronously moving wave usually a domi-

nant harmonic wave y( ) cos( 2 / )

s

B y x tπ λ ω⋅ ⋅ − , which causes most of the eddy cur-

rents. Considering its wave length λ in relationship to the magnet width bM

and the

axial length of the magnet lM

, the equivalent decrease of the magnet conductivity κM

due to the longer eddy current path in 3D is given with the end effect coefficient k as

M,eq M, 0 1k kκ κ= ⋅ ≤ ≤ [4].

2

1 th

2

M

M

l

k

l

πζ

π ζ

⋅⎛ ⎞

= − ⋅⎜ ⎟

⋅ ⎝ ⎠

/ 2 for / 2M

bζ λ λ= ≤ or for / 2M M

b bζ λ= >

(1)

a) The time-step calculation solves the first three Maxwell equations and the con-

stitutive laws within the magnet in a 2D x-y-coordinate system with ( , , )H x y t

,

( , , )J x y t

, ( , , )B x y t

, ( , , )E x y t

and ( , ,0)x y

H H H=

, ( , ,0)x y

B B B=

, (0,0, )z

J J=

,

(0,0, )z

E E=

as

M,eq M

rot rot / div 0 H J E B t B J E B Hκ μ= = −∂ ∂ = = =

, (2)

Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses 117

resulting in div 0J =

and the losses per magnet

M M

M

M

0 0 0

b hT

l

P dt dx dy

T

= ⋅∫ ∫ ∫

2

M,eq( , , ) /

z

J x y t κ , where 2 /T π ω= is the longest time period of the calculated eddy

current density. For the time-step calculation the FE program FLUX2D was used [3].

b) The quasi-static method is neglecting rotH J=

. With the vector potential A

and rotB A=

the eddy-current density in each finite element of the mesh is derived

from (2) as

M,eq

/J dA dtκ= − ⋅

. (3)

c) The semi-analytical post-processing of a magneto-static finite element field

solution can be done in different ways.

c) (i) A): For “small” magnets (e.g. segmented magnets) with bM

< λ/2 the varia-

tion of the flux density along the magnet width is replaced by its average value

0

1

( , ) ( , , )

Mb

yy

M

B y t B x y t dx

b

= ∫ . Then the flux variation ( , ) ( , )yM M

y t B y t b lΦ = ⋅ at each

coordinate plane y = const for 0 ≤ y ≤ hM

can be given as a Fourier sum ( , )y tΦ =

1

( ) sin( )k

k

y k tΦ ω

∞

=

⋅∑ . The eddy current loss formula for a plane material of thickness bM

including the eddy current reaction field (4), (5) is applied [5]. If one takes the result

only at the top of the magnet y = hM

, where the y-component of the field is maximum,

one overestimates the losses by a +60% [1], but by averaging with Simpson’s formula

(6) the results agree well with methods a) and b). The losses due to Bx are very small,

as the flux density in rotor magnet is oriented mainly radial.

M,eq3 2 2

M M M M , m,k

1

( ) ( ) ( )

24y k

k

P y h b l k B y K

κ

ω

∞

=

= ⋅ ⋅ ⋅ ⋅∑ , M M

( ) ( ) /( )y k k

B y y b lΦ= ⋅ (4)

m,k

sh sin3

ch cos

k k

k k k

K

ξ ξ

ξ ξ ξ

−

= ⋅

−

M M M,eq

/ 2k

b kξ μ κ ω= ⋅ ⋅ ⋅ ⋅ (5)

( ( 0) 4 ( / 2) ( )) / 6MM M M M M

P P y P y h P y h= = + = + = (6)

c) (i) B): As alternative to (4) for “narrow” magnets the variation of

M M

( , ) ( , ) /( )y

B y t y t b lΦ= ⋅ with time is taken directly into account, neglecting the in-

fluence of the reaction field of eddy currents:

3

2M

M M,eq( ) ( ( , ) / )

12

yM M

b

P y l h dB y t dt κ= ⋅ ⋅ ⋅ ⋅ (7)

c) (ii): For “broad” magnets with bM

> λ/2 (e.g. massive magnet pieces per pole)

the averaging of the flux density will lead to wrong results. Hence method b) is

Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses118

adopted, but as several magneto-static FEM solvers do not give direct access to the

vector potential for post-processing, Az had to be reconstructed by

M,eq

0

( , , ) ( , , ) ( , , ) ( , , ) /

x

z y z z

A x y t B x y t dx J x y t dA x y t dtκ= ⋅ → = − ⋅∫ , (8)

before the losses can be evaluated according to the above noted method. In our case the

loss evaluation was again done on three circumferential levels 0, hM

/2 and hM

. Method

(ii) appears to be the most exact one of the methods c), because (i) does not give cor-

rect values for bM

> λ/2. For methods b) and c) the FE program FEMAG was used [6],

and for c) alternatively also FLUX2D.

Investigated Machine Topology

Two PMSM, called Motor A and Motor B (Fig. 1 b, c), with tooth-coil three-phase

winding for 45 kW, 1000/min, 430 Nm, which can operate at constant voltage 400 V

up to 3000/min via field weakening with negative d-current, were investigated [1].

With tooth-coil windings the content of MMF harmonics in PMSM is increased, caus-

ing increased magnet losses at load. Both machines are water-jacket cooled, and have

been built in our lab, featuring 16 poles, a laminated stator and rotor iron stack of the

length lFe

= 80 mm and NdFeB rotor magnets. Per pole 7 segmented magnets in cir-

cumference and 6 in axial direction are used (dimensions bM

× lM

= 3.6 x 30 mm) with

a magnet height hM

= 4.8 mm (Motor A) and 4.7 mm (Motor B). The magnet conduc-

tivity is κM

= 7·105

S/m, the permeability μM

= 1.05μ0. Motor A has open stator slots,

q = ¼ slots per pole and phase with profiled copper coils on the wider teeth and buried

magnets in the rotor. Motor B has semi-closed stator slots, q = ½ slots per pole and

phase with round wire copper coils on each tooth of identical width and surface

mounted magnets. A second rotor with surface mounted magnet shells of magnet

height 4.7 mm and four rows of axial shell length 45 mm with the same pole coverage

0.78 was built, fitting to the stator of Motor B, called Motor B′. For Motor A a rotor

with one massive magnet per pole (dimensions bM

× lM

= 25.2 × 45 mm, four axial

magnet rows) was designed (Motor A′), but was only simulated, not manufactured. In

FLUX2D the massive magnets per pole are modelled by redefining the face regions of

the 7 magnet segments per pole to be 1 entity (Fig. 2 a, b). For Motors A and B κM,eq

=

6.25·105

S/m was used, for Motors A′ and B′ κM,eq

= 4.37·105

S/m.

a) b) c)

Figure 1. a) Cross-section of magnet, b) Cross section of Motor A and c) Motor B.


Eddy Current Loss Calculation Results in the Permanent Magnets

Investigation of Radial Variation of Flux Density Over a Magnet Segment

Methods a) and c) (i) A) are compared for Motor A in order to clarify the influence of

the change of flux density in radial direction within the “narrow” magnets on the

eddy current losses, using the FEM program FLUX2D. The Fourier sum of Method

c) (i) A) considered 47 flux harmonics, but already 5 harmonics would have been suffi-

cient. The sinusoidal currents for operation at load are impressed into the stator wind-

ing via three current sources. The initial phase angle of the current was adjusted with

respect to the initial rotor position to get the rated torque 430 Nm. In Table 1 the calcu-

lated losses PM,left

are given for the most left magnet segment per pole, for which either

the flux variation at the top (label: T) y = hM

, or the average of the loss calculation for

three levels y = hM

(top), y = hM

/2 (middle), y = hM

(bottom) (label: TMB) were consid-

ered. Considering the field variation of By only at the top of the magnet gives by +60%

too large losses, as the decrease of the field to the bottom of the magnet is neglected.

Therefore the presented loss values in [1] are by about 60% too big. Considering

“TMB”, the losses are by 20% too small, as the influence of the field component Bx is

neglected. This influence is seen in the central magnet in Fig. 2a. The eddy current

losses in all magnets PM

of Motor A are shown in Table 2 for no-load (Is = 0) at

1000/min and 3000/min, which vary with n2

, hence with a factor 9 in difference. At

rated power 45 kW at 1000/min (Isq

-current operation) and at 45 kW, 3000/min (field

weakening, Isd

-Isq

-current operation) the losses differ only by a factor 2.4 due to the

weakened field at 3000/min.

In the same way the losses were evaluated with Method c) (i) B), showing very

good coincidence with Method c) (i) A), which proves also numerically, that the

influence of the self field of the eddy currents on the losses in segmented magnets with

a) b)

Figure 2. Calculated eddy current density with Method a) in the magnets at full load 45 kW, 1000/min:

a) for Motor B (segmented magnets), b) for Motor B′. Here the 7 magnets are regarded as one magnet entity

by the FEM program. Eddy current density range: a) J = –1.3 … + 0.8 A/mm2

; b) J = – 1.8 … + 1.1 A/mm2

.

Table 1. Calculated losses in the most left magnet segment of a pole for Motor A at 1000 rpm, rated sinusoi-

dal current, 45 kW. Results of Method c) (i) A) with field variation at the top of the magnet (T) and averaged

for top, middle and bottom (TMB) of the magnet in comparison to Method a)

Method c) (i) A) T c) (i) A) TMB a)

Losses PM,left

[W] 0.190 0.097 0.121


bM

< λ/2 is small. Each magnet segment of Motor B was subdivided into 10 finite ele-

ments in x-direction to evaluate the flux density profile By(x, t) for a given rotor posi-

tion (Fig. 3). For example, at no-load Motor B has the dominating wave length of the

field harmonic as the slot pitch λ = τQ, which is obviously bigger than b

M by a factor of

at least 2. The average value M M

( , ) ( , ) /( )y

B y t y t b lΦ= ⋅ , which is used for the calcula-

tion, is also shown in Fig. 3.

Eddy Current Losses in “Broad” Magnets

For “broad” magnets bM

> λ/2 like in Motor A′ and Motor B′ the calculation Methods

c) (i) (either A) or B)) will not give correct results, but yield 40 … 50 (!) times bigger

results. This is due to the different eddy current distribution in massive magnets, which

is clearly visible in Fig. 2b, where the seven magnet segments are regarded as one piece

(bM

= 25.2 mm, hM

= 4.7 mm) with the condition

0 0

( , ) 0

M Mb h

zJ x y dx dy⋅ ⋅ =∫ ∫ . Hence only

Method a) and b) can be used for the calculation. Both Methods a) and b) consider the

influence of the Bx- and B

y-component on the eddy current losses, but method b) ne-

glects the self field of the eddy currents. Hence it gives about 30% too big results for

“broad” magnets (Motor B′) at no-load (Table 3). Alternatively to Method b) the sim-

plified Method c) (ii) may be used, which neglects the influence of Bx and is only aver-

aging the variation of A in the planes y = hM

(top), y = hM

/2 (middle), y = hM

(bottom).

At load both Methods a) and b) give nearly the same results. As Method b) was imple-

mented in the FEM program FEMAG [6] and Method a) in the program FLUX2D,

there is also a slight influence of the program package itself on this difference. For

“narrow” magnets the results of the above described Method c) (i) B) are also close to

Table 2. As Table 1, but calculated losses for all magnet segments of Motor A

Method c) (i) A) T c) (i) A) TMB a)

n [1/min] 1000 3000 1000 3000 1000 3000

no-load 0.42 3.8 0.24 2.2 0.19 1.7

PM

[W]

load 36 89 21 51 24 57

Figure 3. Motor B at no load, 1000 rpm: Calculated radial component of flux density in each of seven mag-

net segments: a) 10 values per magnet segment (dark line), b) 1 average value per magnet segment (light

line).


the results of Method a). Method c) (ii) gives similar results as Method b), but with a

deviation of up to 30%. Due to the reconstruction of the vector potential A from the

flux density B it is more inaccurate. Nevertheless this method is helpful, if the used

FEM program does not feature any eddy current loss calculation for magnets.

Inverter-Caused Eddy Current Losses in the Magnets

In the case of inverter supply, the feeding by current sources can be applied only, if the

current ripple due to the switching of the inverter is known in advance. If only the in-

verter data are given, a coupled circuit FE simulation must be used, which is done with

Method a). Hence the IGBT transistors of the inverter are modelled in FLUX2D as low

resistances in the “on-state” and with high resistance in the “off-state”. Two extremely

different sets for these resistance values both for the IGBT transistors (T) and the free-

wheeling diodes (D) were applied to investigate the influence of transistor data on the

eddy current losses in Motor A and Motor A′. The switching frequency was 2 kHz for

both cases. Hence the dominant current harmonics occur at frequency side bands

around 2 kHz and 4 kHz, but also over-modulation influence is visible. The harmonic

spectrum of calculated currents for both cases is given in Table 5 for Motor A. For Mo-

tor A′ it was nearly identical. The smaller 1st harmonic in Case 2 influences the result

more than the bigger higher harmonics. An increase of eddy current losses due to in-

verter-caused harmonics of 25 … 30% occurs in both cases. The massive magnets suf-

fer from 30 times (!) higher eddy current losses.

Case 1: RON,T

= RON,D

= 0.1 mΩ, ROFF,T

= ROFF,D

= 1 MΩ

Case 2: RON,T

= 50 mΩ, RON,D

= 10 mΩ, ROFF,T

= 0.1 MΩ, ROFF,D

= 50 kΩ.

Conclusions

The time-stepping Method a) is time-consuming, but allows with a coupled circuit cal-

culation a reliable determination of the eddy current losses in permanent magnets also

Table 3. Calculated eddy current magnet losses for Motor B and B′ at no-load (zero current) and at sinusoidal

current operation 45 kW; with Method a), Method b) and Methods c) (i) B) and c) (ii)

PM

[W] 1000 /min 3000 /min

Method b) c) (ii) c) (i) B) a) b) c) (ii) c) (i) B) a)

Motor B no-load 7.6 5.8 7.2 8.4 68 52 65 76

Motor B′ no-load 35 34 – 27 319 307 – 240

Motor B load 45 kW 28 19 22 28 83 59 70 89

Motor B′ load 45kW 207 223 – 211 454 489 – 486

Table 4. Inverter-caused losses in the Motors A and A′ for nominal power 45 kW at 1000 /min; Method a)

Stator winding supply Sinusoidal Inverter (Case 1) Inverter (Case 2)

PM

/ W Motor A/Motor A′ 25.5/784 33.0/984 32.5/–

Table 5. Fourier analysis of the phase current for inverter-operation of Motor A at 45 kW, 1000 /min

fk/Hz Case 133 665 931 1463 1729 2261 2527 3857 4123

k

ˆ

I /A 1/2 166/157 5.4/12.9 2.2/8.1 1.4/5.1 6.2/7.3 5.1/3.6 1.4/3.8 0.8/2.0 0.9/2.0


at inverter supply. The quasi-static position-stepping (Method b)) neglects the influence

of the self field of the eddy currents in the magnets. Due to the low conductivity and

permeability the penetration depth at the interesting frequencies is bigger than the

magnet dimensions. So it delivers nearly the same loss results. In addition three semi-

analytic post-processing methods to determine eddy current losses are presented

(Methods c) (i) A), (i) B), and (ii)). The first two are only valid for “narrow” magnets,

where the variation of flux density over the magnet width is small, whereas the third is

also useful for “broad” magnets, as long as the self field of the eddy currents is negligi-

ble. The influence of the chosen calculation method on the resulting losses is shown for

four permanent magnet synchronous machines with a rating 45 kW, 1000/min and

tooth coils.

Acknowledgements

The authors acknowledge the support of German Research Foundation (DFG) for fi-

nancing the project FOR575.

References

[1] Deak, C; Binder; A.; Magyari, K.: “Magnet Loss Analysis of Permanent-Magnet Synchronous Motors

with Concentrated Windings”, ICEM 2006, Chania, Greece, 2006, 6 pages, CD-ROM.

[2] Atkinson, G.; Mecrow, B.; Jack, A.; et al.: “The Analysis of Losses in High-Power Fault-Tolerant Ma-

chines for Aerospace Applications”, IEEE Trans. Ind. Appl. 42, 2006, p. 1162-1170.

[3] FLUX 9.30 User’s Guide, April 2006, www.cedrat.com.

[4] Russell, R. L.; Norsworthy, K. H.: “Eddy currents and wall losses in screened-rotor induction motors”,

Proc. IEE, p. 163-175, April 1958.

[5] Schuisky, W.: Die Berechnung elektrischer Maschinen, Spinger, Wien, 1960.

[6] Reichert, K.: FEMAG, interactive program to calculate and analyze 2-dimensional and axis-symmetric

Magnetic and Eddy-Current fields, User’s Manual, February 2007, http://people.ee.ethz.ch/~femag.


3-D Finite Element Analysis of Interior

Permanent Magnet Motors with Stepwise

Skewed Rotor

Yoshihiro KAWASEa

, Tadashi YAMAGUCHIa

, Hidetomo SHIOTAa

,

Kazuo IDAb

and Akio YAMAGIWAc

a

Department of Information Science, Gifu University, 1-1, Yanagido,

Gifu, 501-1193, Japan


b

Daikin Industries, Ltd., 1000-2, Ohtani, Okamoto-cho, Kusatsu,

Shiga, 525-8526, Japan


c

Daikin Air-Conditioning And Environmental Laboratory, LTD., 1000-2, Ohtani,

Okamoto-cho, Kusatsu, Shiga, 525-8526, Japan


Abstract. In this paper, the effects of the stepwise skew on the torque waveform

of an interior permanent magnet motor are analyzed by using the 3-D finite ele-

ment method. The usefulness of the stepwise skew is confirmed through the cal-

culated torque waveforms and measured ones.

Introduction

The interior permanent magnet motors (IPM motors) are widely used as high-efficiency

motors in various usage. It is important for the IPM motors to reduce the noise and

vibration as well as to improve efficiency [1,2]. In the IPM motors, it is thought that the

torque ripple is one of the reasons for the noise and vibration. There are some tech-

niques like the skew of rotor in order to reduce the torque ripple. The interlaminar gap

should be considered in the 3-D finite element analysis for skewed IPM motors. Be-

cause the axial component of flux density vectors in the cores is computed very large

when the interlaminar gap in the cores is not taken into account. In this paper, we ana-

lyzed the effects of the stepwise skew of IPM motors on the torque waveform by using

the 3-D finite element method (3-D FEM) with gap elements to take the interlaminar

gap in the rotor and stator cores into account. The usefulness of the computation is con-

firmed through the comparison between the calculated torque waveforms and measured

ones [3].



doi:10.3233/978-1-58603-895-3-124

124

Analysis Method

Magnetic Field Analysis

The fundamental equation of the magnetic field can be written using the magnetic vec-

tor potential A as follows:

MJA rot)rotrot(00

νν += (1)

where ν is the reluctivity, J0 is the exciting current density, ν

0 is the reluctivity of the

vacuum, M is the magnetization of permanent magnet.

Gap Elements

It is necessary to take into account the interlaminar gap between the electrical steel

sheets, which is very small, in order to calculate axial component of the flux density

caused by the stepwise skew accurately. However, if the very small air gap is divided

by the conventional meshes, it costs a lot of physical memory and CPU time. Therefore,

we take into account the interlaminar gap using the gap elements [4].

The weighted residual Gi of Galerkin’s method for the gap elements are given by:

∫∫=

sS

iidSDG )rot(rot

0AN ν (2)

where D is the length of the gap, Ss is the region of the gap element, and N

i is the interpo-

lation function.

Nodal Force Calculation

Nodal force method is to calculate a local magnetic force in the 3-D FEM. The force Fn

on each node n can be calculated as follows [5]:

∫−=

V

nn

VdNT )grad(F (3)

where V is the total volume of elements related the node n, T is the Maxwell stress ten-

sor, and Nn is the interpolation function of elements related the node n. The torque T

m is

given as follows:

( ) rFT λ⋅= ∑

Ω

n

m ,×

=

×

R r

R r

λ (4)

where Ω is all the nodes contained in the rotor region (the rotor and the half of air gap),

λ is the unit vector of a rotary direction, r is the directional vector towards the node n,

and R is the axis directional vector [6].

Y. Kawase et al. / 3-D Finite Element Analysis of Interior Permanent Magnet Motors 125

Analyzed Model and Conditions

Figure 1 shows the photographs of an IPM motor. Figure 2 shows the analyzed model

of an IPM motor. Figure 3 shows the appearances of stepwise skew. Figure 3(a) shows

the no-skewed rotor model. Figures 3(b) and 3(c) show the stepwise skewed rotor

model. Figure 4 shows the 3-D finite element mesh. The gap elements are inserted in

each x-y plane in the stator and rotor cores. It is assumed that the space factor of the

electrical steel sheets of the stator and rotor core is 96%. The analyzed region is 1/4 of

the whole region because of the symmetry and the periodicity. Table 1 shows the

analysis conditions.

(a) rotor core and

magnets

(b) stator core and coil

Figure 1. Photographs of an IPM motor. Figure 2. Analyzed model (no-skewed rotor model).

(a) no-skewed rotor

model

(b) skewed rotor

model A

(2 magnets)

(c) skewed rotor

model B

(4 magnets)

Figure 3. Appearances of stepwise skew. Figure 4. 3-D finite element mesh

(except coil and air).

Table 1. Analysis conditions

Number of poles 4

Exciting current (A) 14

Coil

Number of turns (turn/slot) 22

Magnetization of magnet (T) 1.2

Frequency of coil current (Hz) 60

Revolution speed (min–1

) 1,800

Y. Kawase et al. / 3-D Finite Element Analysis of Interior Permanent Magnet Motors126

Results and Discussions

Figure 5 shows the distributions of flux density vectors in o-a section. The distributions

of flux density vectors of Figs 5(i) and 5(ii) in each model look almost the same by

comparing between no-skewed rotor model and with skewed one. Therefore, Fig. 5(iii)

shows the difference between Figs. 5(i) and 5(ii). From Fig. 5(iii), it is found that the

axial components of flux density vectors between stepwise skewed magnets of rotor

core are very large.

Figure 6 shows the waveforms of cogging torque with and without gap elements.

In the no-skewed rotor model, the waveforms of cogging torque with gap elements and

without one is about the same. On the other hand, in the skewed rotor model, it is found

that the peak of cogging torque is different between with gap elements and without one.

Consequently, it is found that the usufulness of considering gap elements is clarified.

Figure 5. Distributions of flux density vectors (o-a section).


Figure 7 shows the waveforms of torque, which is normalized by the average of

the torque of no-skewed rotor model. From Fig. 7, it is found that the calculated wave-

forms of torque agree very well with measured ones. It is also found that the torque

ripples of skewed rotor models A and B are reduced to about 1.5% of the no-skewed

rotor model by the stepwise skew in both calculation and measurement.

Table 2 shows the discretization data and CPU time.

Figure 6. Waveforms of cogging torque with and without gap elements.

Figure 7. Waveforms of torque.

Table 2. Discretization data and CPU time

Number of elements 814,716

Number of nodes 141,470

Number of edges 967,355

Number of unknown variables 939,928

Number of time steps 180

Computer used: Pentium 4 (3.0GHz) PC

Y. Kawase et al. / 3-D Finite Element Analysis of Interior Permanent Magnet Motors128

Conclusion

The effects of the stepwise skew of IPM motors on the torque waveforms were ana-

lyzed using the 3-D FEM with gap elements to take the interlaminar gap in the rotor

and stator cores into account. It was found that the ripples of torque can be reduced by

the stepwise skew. It was also found that there is ample effect for the stepwise skew by

2 magnets to reduce the torque ripple in this model. The validity of the stepwise skew

was confirmed through the calculations and measurements.

References

[1] Y. Kawase, T. Yamaguchi, S. Sano, M. Igata, K. Ida and A. Yamagiwa, “3-D eddy current analysis in a

silicon steel sheet of an interior permanent magnet motor”, IEEE Trans. on Magnetics, vol. 39, no. 3,

pp. 1448-1451, May, 2003.

[2] Y. Kawase, N. Mimura and K. Ida, “3-D electromagnetic force analysis of effects of off-center of rotor in

interior permanent magnet synchronous motor”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1858-1862,

July, 2000.

[3] A. Yamagiwa, K. Nishijima, Y. Sanga, Y. Kawase, T. Yamaguchi and T. Yano, “Reduction of Motor

Vibration by Stepwise Skewed Rotor”, Japan Industry Applications Society Conference, No. 3-92, 2005.

[4] T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, “3-D magnetic field analysis using special ele-

ments”, IEEE Trans. Magn. vol. 26, no. 5, pp. 2379-2381, 1990.

[5] A. Kameari, “Local force calculation in 3D FEM with edge elements”, International Journal of Applied

Electromagnetics in Materials, vol. 3, pp. 231-240, 1993.

[6] Y. Kawase, H. Kikuchi and S. Ito, “3-D Nonlinear Transient Analysis of Dynamic Behavior of the Clap-

per Type DC Electromagnet”, IEEE Trans. Magn. vol. 27, no. 5, pp. 4238-4241, 1991.


Advance Computer Techniques in

Modelling of High-Speed Induction Motor

Maria DEMS and Krzysztof KOMĘZA

Institute of Mechatronics and Information Systems, Technical University of Lodz,

ul. Stefanowskiego 18/22, 90-924 Lodz, Poland


Abstract. In the paper, circuit and field-circuit analyses of high-speed small size

induction motors are presented. The circuit analysis is possible only for the first

harmonic of the supply voltage. For the real shape of the voltage which has many

higher harmonics accurate field – circuit analysis is necessary, but this method is

very time consuming. The circuit and field-circuit analyses were done for 2-D

structure of the motor and for all values of applied frequencies. The results of cal-

culation of magnetizing current are compared with the measurement.

Introduction

Nowadays the high-speed induction motors are widely used in many industrial installa-

tions and also in aircraft industry. Many of them are designed as converter-fed induc-

tion machines. Some electrical drives with not so sophisticated speed control have volt-

age shape which has many higher harmonics. The field-circuit method makes accurate

computation of high-speed induction motors characteristics possible. Unfortunately,

this method is very time consuming. When the motor is supplied by a PWM inverter

the length of time step must be smaller than time given by carrier frequency of the in-

verter. Furthermore the Newton-Raphson iteration is carried out at each time step to

consider the magnetic saturation. Therefore in design and optimisation process of these

motors improved classical circuit methods are very interesting. In the paper, the differ-

ent field-circuit and circuit methods are presented and the results are compared with the

measurement.

Object of Investigation

A high-speed construction of the small induction motor was designed basing on the

classical structure of the four-pole induction motor model size 80. The supply voltage

was 230 V for the frequency 200 Hz and stator windings were delta connected. The

number of series turns of stator windings was 216. This motor had stator core shape

with cut of parts making stator yoke width not constant; the external maximal diameter

of the stator core was Dse max = 120 mm, and external minimal diameter Dse min =

114 mm [ ]. In the field – circuit analysis the real shape of the stator core was taken

into account, but in the circuit analysis the calculations were made for the average

value of the external stator diameter Dse av = 117 mm, The motor construction is with

closed rotor slots.



doi:10.3233/978-1-58603-895-3-130

130

Field-Circuit Analysis of the Motor

The field-circuit analysis of the high-speed induction motors can be made using differ-

ent levels of accuracy. For each circuit, including the parts described with the field

equations and external elements described with the resistance Rz and inductance L

z, the

circuit equation has the following form:

j

j

jzzVd

dt

di

LiRu Δ++= ∑ (1)

Applying this to the combination of the presented equations leads to the following

system:

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

∂

∂

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

U

0

J

i

ΔV

A

L00

00H

00Q

i

ΔV

A

RD0

DW0

0HG

T

T

t

(2)

where

i 0

1

J

ij i j ij i kk ij i j

k k

i

G N N dS H N dS W dS Q N N dS

J N dS

γ γ γ

μ

= ∇ ∇ = = =

=

∫ ∫ ∫ ∫

∫

(3)

R and L are diagonal matrixes of the resistances and external inductances. The

equations system can be presented in a more general form:

B

X

SRX =

∂

∂

+

t

(4)

For quasi-static model the assumption that all field variables are varying sinusoi-

dally is made. Since the potential and the currents are varying sinusoidally, they can be

expressed as the real part of complex functions. The equations system now becomes:

i+ =RX SX B (5)

This system can be solved using complex arithmetic.

The quasi-static solvers calculate element permeability using amplitude of the

magnetic flux density. This can introduce some errors in highly saturated small ma-

chines despite the transient calculation of the magnetisation current needed.

In this case, the system can be solved using differential schema with time step

equal to Θ

( ) ( ) 01111

=Θ+Θ−+⎥⎦

⎤

⎢⎣

⎡

Δ

+Θ+⎥⎦

⎤

⎢⎣

⎡

Δ

−Θ−++ nnnn

BBX

t

S

RX

t

S

R (6)

M. Dems and K. Komeza / Modelling of High-Speed Induction Motor 131

Nowadays, the economy of the production causes the use of the construction with

not round stator core in motor manufacturing. This type of construction has very high

saturation of the motor stator yoke parts with decreased width.

Additional problems appeared when the motor has been supplied by the PWM in-

verter. In this case the supply voltages have complicated shape which depends on the

type and algorithm used by the control system. In this case two methods are available:

the first when only the first harmonic of the supply voltage is taken into account and

the second when the real voltage shape is used. The last method is of course the most

accurate but is very time consuming according to the fact that the transient state has to

be considered before the steady state is achieved.

Figure 1 shows the vector plot and magnitude of magnetic flux density calculated

by mentioned methods for 200 Hz supply. It can be noticed that the magnitude of mag-

netic flux density for this case is rather small despite the fact that above 120 Hz the

voltage is constant.

Figure 2 presents the calculated terminal current values versus time for real voltage

shape at the motor compared with measured values, for 200 Hz supply. The main

Figure 1. Vector plot and distribution of magnitude of magnetic flux density for 200 Hz supply.

-3

-2

-1

0

1

2

3

0,1632 0,1642 0,1652 0,1662 0,1672 0,1682 0,1692

current [A]

time [s]

measured

calculated

Figure 2. Comparison of calculated and measured current values versus time for 200 Hz supply.

M. Dems and K. Komeza / Modelling of High-Speed Induction Motor132

source of differences is the influence of capacitances they are not incorporated in field-

circuit model.

In Figs 3 and 4, two different times instant for 100 Hz supply are presented. From

these it can be seen that saturation of the motor changes significantly in time depend on

the resultant flux position.

In Fig. 5 the voltage and current versus time, calculated and measured, for 100 Hz

supply are presented. In this Figure, the first harmonic of the supply voltage is also

presented. Values of no-load currents, calculated with the first harmonic supply, by

field-circuit and circuit methods correspond roughly.

The comparisons with the measurement show that the current values calculated

with this assumption are very narrow.

Figure 3. Vector plot and distribution of magnitude of magnetic flux density for 100 Hz supply, for the first

time instant.

Figure 4. Vector plot and distribution of magnitude of magnetic flux density for 100 Hz supply, for the sec-

ond time instant.


-800

-600

-400

-200

0

200

400

600

800

0,18 0,19 0,2 0,21 0,22 0,23

voltage [V]

time [s]

-5

-4

-3

-2

-1

0

1

2

3

4

5

0,066 0,068 0,07 0,072 0,074 0,076 0,078 0,08

current [A]

calculated

measured

time [s]

Figure 5. Voltage and first harmonic of the voltage and comparison of calculated and measured current val-

ues versus time for 100 Hz supply.

Table 1. Flux density in the motor core for different values of the frequency

Frequency 50 Hz 100Hz 150 H 200Hz

Flux density in the stator yoke [T] 1,674 1,709 1,376 1,031

Flux density in the stator tooth [T] 1,481 1,476 1,185 0,892

Flux density in the rotor tooth [T] 1,478 1,473 1,182 0,890

Flux density in the rotor yoke [T] 0,828 0,825 0,662 0,497

0

50

100

150

200

250

300

0 20 40 60 80 100 120 140 160 180 200

frequency [Hz]

effective voltage [V]

measured

first harmonic of the voltage

0,0

0,5

1,0

1,5

2,0

2,5

0 20 40 60 80 100 120 140 160 180 200

frequency [Hz]

magnetizing current [A]

measured

calculated for first harmonic of the voltage

Figure 6. Effective voltage and magnetizing current vs. frequency measured and calculated for first harmonic

of the supply voltage.

Calculation of Magnetizing Current of the Motor Using Circuit Method

For this motor the calculation using improved circuit method enables the calculation

for higher frequencies of the magnetising current for different conditions of the supply

of the motor. From 10 Hz to 120 Hz the linear increase of the supply voltage and fre-

quency was made, and in result we obtain the value of flux density, and torque equal

constant. For the frequency higher than 120 Hz the value of the voltage was constant.

The values of magnetic flux density in the motor core calculated using the circuit

method (STAT) are shown in Table 1. Figure 6 shows the effective value of supply

voltage, measured, and the first harmonic of this voltage, which was used for calcula-

tions. In this figure also magnetizing current of the motor, measured and calculated for

first harmonic of the supply voltage was shown.


Computing of Operating Curves Using Circuit Model

For this induction motor the parameters and curves of current, power factor, efficiency

and rotor slip versus output power were computed, for different values of the fre-

quency. The results of these calculations for the different values of the frequency for

linear increase of the supply voltage are shown in Fig. 7.

Figure 8 shows the same curves for the frequency higher than 120 Hz and constant

value of the supply voltage.

From Fig. 7 and Fig. 8 we can state that the highest value of the efficiency and

power factor we obtain for the frequency equal to 200 Hz. It is caused by the lowest

value of the magnetising current and in result also total stator current of the motor.

Conclusion

In the calculations of the high-speed small power induction motors supplied by inverter

the higher harmonics and also nonlinear phenomena can be taken into account only in

the field-circuit models but they are still very time consuming. Therefore, in design and

optimisation process of these motors improved classical circuit methods are very inter-

esting, but their use is possible for the first harmonic of the supply voltage only.

0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,00 0,25 0,50 0,75 1,00 1,25 1,50

current [A]

100Hz, 120Hz

50Hz

relative power output

30Hz

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,00 0,25 0,50 0,75 1,00 1,25 1,50

power factor

100Hz, 120Hz

50H


30H

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0,00 0,25 0,50 0,75 1,00 1,25 1,50

efficiency

100Hz, 120 Hz

50Hz


30H

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,10

0,00 0,25 0,50 0,75 1,00 1,25 1,50

slip

100Hz


50Hz

120Hz

30Hz

Figure 7. Current, power factor, efficiency and rotor slip versus output power for the high-speed induction

motor, for different values of the frequency, for linear increase supply voltage.


0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,00 0,25 0,50 0,75 1,00 1,25 1,50

current [A]

120Hz

150H


200Hz

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,00 0,25 0,50 0,75 1,00 1,25 1,50

power factor

120Hz150Hz


200Hz

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0,00 0,25 0,50 0,75 1,00 1,25 1,50

efficiency

100Hz

150Hz


200Hz

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,00 0,25 0,50 0,75 1,00 1,25 1,50

slip

150Hz


120Hz

200Hz

Figure 8. Current, power factor, efficiency and rotor slip versus output power for the high-speed induction

motor, for different values of the frequency, for constant value of the supply voltage.

References

[1] Dems M., Komęza K, Wiak S., Stec T., Kikosicki M., Application of circuit and field-circuit methods in

designing process of small induction motors with stator cores made from amorphous iron, COMPEL, The

International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 25,

No. 2, 2006, s. 283-296.

[2] Dems M., Modeling of electromechanical transient processes in induction motors with closed slots of the

rotor, Archives of Electrical Engineering, nr. 3, 1997, pp. 333-353.

[3] Dems M., Komęza K., A comparison of circuit and field-circuit models of electromechanical transient

processes of the induction motor with power controller supply, Proceedings COMPUMAG’2001, Lyon–

Evian, France, 2-5 July, 2001, pp. 206-207.

[4] Dems M., Rutkowski Z., “STATz_F Software for calculation of electromagnetic parameters and charac-

teristics of induction motors”, Technical University of Lodz, Poland.

[5] Gąsiorowski T., “Experiences of FSE “BESEL” S.A. in production of induction motors supplied from

frequency inverters”, Proceedings of VII Symposium PPEE 99, Ustroń 22-25 mars 1999, pp. 2-6.

[6] PC OPERA-2D – version 11, Software for electromagnetic design from Vector Fields, 2006.


Computation of the Equivalent

Characteristics of Anisotropic Laminated

Magnetic Cores

E. NAPIERALSKA-JUSZCZAK, D. ROGER, S. DUCHESNE and J.-Ph. LECOINTE

Laboratoire Systèmes Electrotechniques et Environnement, Technoparc Futura,

62400 Béthune, France

[email protected]

Abstract. The method presented in this paper allows computing the equivalent

characteristic of anisotropic steel sheets used to stack the magnetic circuits. The re-

luctivity of the anisotropic sheets is a function of the flux density B and the angle

between the flux density vector and the rolling direction. This paper focuses par-

ticularly the problem of modeling different kinds of overlaps and apparent air-gaps

is solved by the homogenization technique which is based on the assumption that,

in the layer structure, the magnetic field energy tends to achieve its minimum. The

presented method has been verified by comparing the computational results with

the measurements of real sets. The paper presents the analysis of magnetic proper-

ties of overlaps taking into account the magnetic characteristics of the steel.

1. Introduction

The effects caused by magnetic fluxes in anisotropic cores are difficult to handle, both

theoretically and numerically. Modeling of the overlapping regions of thin laminations,

of thickness 0.2–0.3 mm and surface area of tens or hundreds of square centimeters,

taking account of the insulation of micrometer thickness, is practically impossible for

realistic systems and thus it poses significant challenges. The authors have undertaken

research related to such anisotropic magnetic cores used in transformers and electrical

machines. In particular, they are addressing the issues of the influence of the position

and pattern of equivalent air-gaps, under different overlapping arrangements, on iron

losses and noise due to magnetostriction. This research programme necessitates crea-

tion of several mathematical models capable of simulating the transfer of flux between

laminations under different anisotropy angles and various air-gap positions.

This paper presents the results from the first stage of the project. A homogeniza-

tion technique has been developped to approximate the overlapping of the laminations.

This has enabled to replace the real three-dimensional structures by far simpler ho-

mogenous two-dimensional models. The approach relies on the natural tendency of the

energy of the magnetic field to achieve a minimum in the non-homogenous laminated

structure [1–4]. When calculating the flux density vector in each lamination using the

minimisation principle, the functional has been assumed in terms of the energy in the

whole structure, while the constraint is the relationship between the flux density vector

in the whole structure and the relevant vectors in every lamination.

At last, the distribution (sub-division) of the total flux density vector between

component laminations of a particular overlapping structure will lead to:


137

• a method of calculating the equivalent (homogenised) reluctivity at any point

of the laminated sheet and/or the air-gap;

• a method of establishing equivalent characteristics for various structures;

• a method of calculating the magnetic field distribution in a transformer core

under different overlapping schemes;

• a method of calculating the magnetic field distribution in an electrical ma-

chine, when the core is anisotropic, at various anisotropy angles.

2. Method Presentation

Using the homogenization technique makes it possible to replace real 3D structures by

simpler homogeneous 2D structures. In the homogenization technique, concepts of

macrostructure and microstructure are of importance. In the considered problem, the

macrostructure is a complete assembly of layers, while the microstructure is a repeat-

able structure of two or more layers made of sheets with different rolling directions. In

the case of apparent air gaps, the microstructure is a set of sheet layers and air layers.

The usual procedure consists in defining a basic volume V able to replace the whole

structure [5,6]. Thus, the main purpose in this paper is to replace the set of microstruc-

tures by the equivalent volume V without changing the flux distribution in each layer.

The first step of the method is to define the components of the resulting flux den-

sity vector B

. The relation between B

and the flux density vectors b

in the repetitive

structures representing a full set of the overlapped layers is given by formula (1) where

V is the macrostructure basic volume.

1

V

B bdv

V

= ∫

(1)

If the integral is replaced by the sum for the basic volume dx*dy*H, Eq. (2) is ob-

tained. Denotes n is the number of different not repetitive layers in the structure, hi is

the thickness of the layer i and H is the thickness of the macrostructure.

1

1n

i i

i

B B dx dy h

dx dy H=

= ⋅ ⋅ ⋅

⋅ ⋅

∑

(2)

The components Bix and B

iy (Eq. 3a and 3b) of the flux density B

in point P of a

multilayer set representing overlapping of sheets (i = 1, 2, 3,…n) are computed. The

calculation is based on the total flux penetrating through the limiting surfaces of the

volume V.

2

ix ix

ix

i

B

dy h

φ φ′ ′′+

=

⋅ ⋅

(3a)

2

iy iy

iy

i

B

dx h

φ φ′ ′′+

=

⋅ ⋅

(3b)

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores138

ix

φ ′ and ix

φ ′ are the fluxes going inside the volume; ix

φ ′′ and ix

φ ′′ are the fluxes going

outside the volume (Fig. 1).

The fluxes going through the limiting surfaces of the volume V have to be equal to

the sum of fluxes in all particular layers. Therefore the components Bx and B

y of the

resulting flux density vector are given by Eqs (4a) and (4b). The flux density compo-

nent in the direction z is smaller then 10% of Bx and B

y, so this component can be omit-

ted.

1

2

n

ix ix

i

x

B

dy H

φ φ

=

′ ′′+

=

⋅ ⋅

∑

(4a)

1

2

n

iy iy

i

yB

dx H

φ φ

=

′ ′′+

=

⋅ ⋅

∑

(4b)

In order to replace the three-dimensional system by a 2D system, it is assumed that

the analyzed volume is made of an equivalent homogenous material in which the dis-

tribution of magnetic field is identical to the resulting distribution in the real structure.

The second step of the method is to define the goal function to minimize the mag-

netic energy. The total energy of the macrostructure results from the sum (Eq. 5) of the

magnetic energy Wµi

stored in each volume Vlay

of the layer i (Eq. 6). µi and

i

B

are

respectively the permeability and the flux density vector of the layer i.

( )min

,

µ µi i i

W W µ B=∑

(5)

1

2

i lay i i

W V B Hμ

=

(6)

φ’x φ’’

x

φ’’y

φ’y

axis x

Figure 1. Fluxes going inside and outside the volume V.

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores 139

To calculate flux density vectors in the particular layer, a minimization method is

applied. The applied goal function is the minimum of the magnetic energy in the vol-

ume V, while the relationship between the resultant flux density vector and each vector

of the flux density in particular layers is the restriction. The goal function depends on

the computed structure and it has to be prepared taking into account the interlacing, the

insulation between the laminations and the air-gaps. On the one hand, the air-gaps de-

pend on the inaccuracy of the sheet overlapping. On the other hand, it is possible to

take advantage of the air gaps to get the required orientation of the magnetic flux. In-

deed, apparent air-gaps exist on the edges and inside the structure made of overlapped

sheets. Air gaps and sheets form a non-homogenous layer structure. That is why two

kinds of materials – steel sheets with permeability μFe

and air with permeability μ0 –

appear in parallel, forming together the apparent air gap. The division of the resulting

flux density B

between the steel and air depends as well on the rule of minimum of

energy stored in magnetic field. The resulting flux density vector in apparent air-gap is

given by the expression (7) were 0

B

is the flux density in the air, Fe

B

is the flux den-

sity in the sheet.

0res Fe

B B B= +

(7)

The magnetic energy stored in the structure results from the sum of the energy

stored in the air gaps and in the sheets. If Vlay

denotes the volume of the simple layer,

nFe

the number of sheet layers, n0 the number of air layers, ν

0 and ν

Fe the reluctivity

respectively in air and in sheets, then the goal function for the structure is given by the

formula (8).

( )

2 2

0 0 0

min 0

min

,

2

lay Fe Fe Fe lay

µ Fe

V n B V n B

W B B

ν ν⎡ ⎤+

= ⎢ ⎥

⎢ ⎥⎣ ⎦

(8)

Third step of the method consists in establishing the family of the anisotropic char-

acteristics of the whole structure. Successive values of the flux density from 0 to the

steel saturation are imposed. Every point of the characteristics is determined with the

following algorithm. First the flux density vectors for the structure are calculated with

the minimization task. Then, the expression (9) makes it possible to calculate the reluc-

tivity of macrostructure. At last, the field intensity H Bν= is calculated.

( )

2 2

0 0 0

min 2

,

Fe Fe Fe

µ

n B n B

B W

nB

ν ν

ν

+

=

(9)

3. Comparison of Calculation and Measurement Results

Calculations for two types of different structures are made. The proposed method is

composed the following steps:

• the homogenized reluctivity at any point of the laminated sheet or the air gap

is calculated,

• the equivalent characteristics are established,


• the properties of the overlapped structure are compared with the magnetic

properties of the steel forming these overlaps. It gives, in few seconds, a first

idea of the quality of the overlapping without doing a Finite Element (FE)

simulation which requires a long computation time,

• the magnetic field distribution in every tested structure is calculated using a

FE method, taking into account a nonlinear anisotropic reluctivity,

• the division of the total flux density vector between the component lamina-

tions is analysed,

• the results of simulations are compared with measurements.

The first tested structure is made of layers alternatively shifted of 90°: the first

layer is placed in the rolling direction (anisotropy angle 0°), the second layer has an

anisotropy angle of 90°. This structure corresponds to the conventional two-cycle inter-

lacing (90° transformer core overlap). Figure 2 shows the family of the equivalent

magnetizing characteristics B(H). The anisotropy angle α of the whole structure is arbi-

trary defined as the angle of between the resulting flux density vector and the rolling

direction of the first layer. The characteristics of the equivalent structure for α = 0°,

30°, 45°, 60° and 90° are presented at Fig. 2. The characteristic of the anisotropic steel

used to form the structure are also presented for the anisotropy angles 0°, 60°, 90°. It

makes possible to estimate the quality of the whole structure.

Figure 3 presents the total flux density B

at the point p(x,y) of the structure (solid

line) and the division of B

between the layer 1 and the layer 2 (interrupted lines) ver-

sus ωt (with ω the grid pulsation). Figure 4 gives the comparison of the calculated re-

sults and measurement results for both layers at p(x,y) [8–10]. The calculations were

made using measured vectors of the resultant flux density.

A second structure makes it possible to test the influence of the air gaps. Its ge-

ometry is presented at Fig. 5. Every layer is made of a sheet in the rolling direction

(anisotropy angle of 0°), an air gap and a sheet placed perpendicularly to the rolling

direction (anisotropy angle of 90°). The position of the air gap and the repartition of the

two kinds of magnetic materials are different for every layer.

The characteristics calculated for every layers of this structure and compared with

the characteristics of the anisotropic steel are shown at Fig. 6. Figure 7 shows the fam-

ily of the equivalent magnetizing characteristics B(H) for the layer 4. One can observe

that the equivalent layer material has anisotropic properties, like the elementary steel

Figure 2. Family of characteristics B(H) for the different anisotropy directions.


sheets. The characterization is different for the low and high values of the flux density

and the numerical board between them is about 1.1 T. The properties of the equivalent

layer are worse than the sheet properties for α between 0° and 45°. However, equiva-

lent layer properties are better if α is superior to 45°.

To check the proposed method, not only the simple structure has been tested. Dif-

ferent types of transformers and complex systems have been also studied, for example

a transformer supplying a converter system during normal working or working under

different kinds of faults.

Figure 3. Resultant flux density in the structure

divided between layers.

Figure 4. Comparison of measurements and calcu-

lations.

Air gap 90° 0°

Layer 1

Layer 2

Layer 3

Layer 4

Figure 5. Presentation of the structure.

Figure 6. Family of characteristic B-H for the directions 0° for the 4 layers.


4. Conclusion

The measurement results confirm the good accuracy of the proposed method. The aver-

age fractional error betweens calculation and measurements is between 5–10%, de-

pending of the structure. This method makes it possible to substitute the complex 3-D

structures by far simples two-dimension structures. For example, it gives the influence

of the joints and air-gaps on the required magnetic flux distribution. The method can be

applied to model all kinds of transformer core overlapping. It allows the calculation of

the flux density vectors in any layer of the transformer cores with different overlapped

limb and yoke sheets. It should provide a considerable help for the designers since it

allows them to arrange cores of the same dimensions but of optimum structure. The

presented method can be also applied to design rotating electrical machines equipped

with a rotor or a stator made of anisotropic materials.

References

[1] J. Gyselinck, R.V. Sabariego, P. Dular, “A nonlinear time-domain homogenization technique for lami-

nated iron cores in three-dimensional finite-element models”, IEEE Trans. on mag., Vol. 42, Issue 4,

April 2006, pp. 763–766.

[2] Hiroyuki Kaimori, Akihisa Kameari, Koji Fujiwara, “FEM Computation of Magnetic Field and Iron

Loss in Laminated Iron Core Using Homogenization Method”, IEEE Trans. on mag, Vol. 43, Issue 4,

April 2007, pp. 1405–1408.

[3] A.J. Bergqvist, S.G. Engdahl, “A homogenization procedure of field quantities in laminated electric

steel”, IEEE Trans. on mag, Vol. 37, Issue 5, Part 1, Sept. 2001, pp. 3329–3331.

[4] L. Krahenbuhl, P. Dular, T. Zeidan, F. Buret, “Homogenization of lamination stacks in linear magneto-

dynamics”, IEEE Trans. on Mag, Vol. 40, Issue 2, Part 2, March 2004, pp. 912–915.

[5] A. De Rochebrune, J. Dedulle, J. Sabonnadiere, “A Technique of homogenization applied to the model-

ing of transformers”, IEEE Trans. on Mag., vol. 26, No

2, 1991, pp. 520–523.

[6] J.M. Dedulle, G. Meunier, A. Foggia, J.C. Sabonnadiere, D. Shen, “Magnetic fields in nonlinear anisot-

ropic grain-oriented iron-sheet”, IEEE Trans. Mag., Vol. 26, N°. 2, 1990, pp. 524–527.

[7] E. Napieralska-Juszczak, M. Pietruszka, “Semi-analytical method of modelling the magnetising curves

for anisotropic sheets”, 4th Int. Workshop on Electric and Magnetic Fields, Marseille France, 1998,

pp. 451–456.

[8] M. Pietruszka, E. Napieralska-Juszczak, “Lamination of T-joints in the transformer core”, IEEE Trans.

on Mag, Vol. 32, Issue 3, Part 1, May 1996, pp. 1180–1183.

[9] A.J. Moses, “Rotational magnetization-problems in experimental and theoretical studies of electrical

steels and amorphous magnetic materials”, IEEE Trans. on Mag, Vol. 30, N°2, 1994, pp. 902–906.

[10] M. Pietruszka, “A method to compute the magnetic field in anisotropic 3-phase transformer cores with

arbitrary overlapping structures”, D.Sc Thesis, Poland, 1995 (ISSN 0137-4834).

Figure 7. Family of characteristics B(H) for the layer 4 for the different anisotropy directions.


Improving Solution Time in Obtaining

3D Electric Fields Emanated from High

Voltage Power Lines

Carlos LEMOS ANTUNESa,b

, José CECÍLIOb

and Hugo VALENTEc

a

Lab. CAD/CAE, Electrical Engineering Dept., University of Coimbra, Pólo II,

3030 – 290 Coimbra, Portugal

b

APDEE – Assoc. Port. Prom. Desenv. Eng. Electrotécnica, Rua Eládio Alvarez,

Ap. 4102, 3030 – 281 Coimbra, Portugal

c

REN – Rede Eléctrica Nacional, Av. Estados Unidos da América 55,

1749 – 061 Lisboa, Portugal

E-mail: [email protected]; [email protected]

Abstract. In this paper it is presented an algorithm to reduce the computational

time in obtaining the electric field distribution in a plane of analysis due to High

Voltage Power Lines. It is used a two dimensional interpolation based on a spline

function using as known nodal values, the field solution at nodes of a coarser plan

grid.

Introduction

The LMAT_SIMEL [1] is a software program that calculates the 3D electric field dis-

tribution on specified nodes, emanating from general 3D Line(s) configurations. The

electric field is calculated using a 3D integral numerical approach and makes use of the

image method. The conductors are considered filamentary wires of arbitrary geometric

configuration with known imposed voltages: phase-earth or zero if it corresponds to the

guard conductor and the catenary is approximated by straight lineal segments. The

electric field can be calculated along any path or on any plane. The earth is considered

as a perfect conductor at zero voltage reference value and its influence is taken into

account using the method of images. The influence of vegetation and terrain elevations

is not taken into consideration.

The grid discretization of the solution plane is very important to obtain a good or

smoother solution for electric field distribution which may lead to a considerable com-

putational time. To reduce this computational time, we have used a two dimensional

interpolation function to estimate the field solution at intermediate nodes, from the field

solution obtained in a coarser plane grid.

Formulation

The phasor electric field ˆ

E at any point P(x, y, z) due to a Line, is calculated by



doi:10.3233/978-1-58603-895-3-144

144

( )1

20

10

ˆ1

ˆˆ

4

SN

i

i

i'

s

E L ds a

r r

λ

πε=

= ⋅ ⋅ ⋅ ⋅

−

∑ ∫ (1)

where the point P(x, y, z) is defined by r and the phasor charge density ˆ

i

λ in the seg-

ment i is located at '

r , with ˆa as the unit vector in direction ( )'

r r− .

It is seen that the phasor linear charge density has to be previously calculated for

all the Line(s) and their images.

For each line segment the charge distribution is approached by a cubic spline poly-

nomial as (2).

( )2 3

0 1 2 3

ˆ

s c c s c s c sλ = + + + (2)

where

2 3 2 3 2 3 2 3

0 1 2 32 3 2 2 3 2

1 1 1 1 1 1 1 1

3 2 2 3 2

1 , ,

s s s s s s s s

c c s c and c

L L L L L L L L

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= − + = − + = − = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

s is an adimensional parameter ( 0s = at the beginning of the segment and 1s = at the

end of that segment).

For each Line node i it is required the calculation of the phasor charge density ˆ

i

λ

and its derivative ˆ

i

'

λ , which is based on the following equation:

( )

( )

( )

( )

0

0 10 1

1 11 1

0 0

0 0

ˆ ˆˆ ˆ

ˆ

4 4

o

P

'

' '

'

' '

k sk s

k S k St t

V ds ds

r r r r

λ λλ λ

πε πε

⎡ ⎤⎡ ⎤

⎡ ⎤ ⋅ ⎢ ⎥⎡ ⎤ ⋅ ⎢ ⎥ ⎣ ⎦⎣ ⎦⎢ ⎥

⎣ ⎦ ⎣ ⎦= +

− −∫ ∫

(3)

It was used a two dimensional interpolation function [2,3] to estimate the interme-

diate values from two known values. The interpolation is essential to obtain a smoother

or good representation of electric field distribution, between two known field values.

Given a rectangular grid ,k l

x y and the associated set of numbers klz which cor-

respond to the known field values, with 1 , 1k m l n≤ ≤ ≤ ≤ , we have to find a bivari-

ate function ( ),z f x y= that interpolates the data (field solution), i.e., ( ),

k l kl

f x y z=

for all values of k and l . The grid points must be sorted monotonically, i.e.

1 2 m

x x x< < ⋅⋅⋅ < with a similar ordering of the y-ordinates.

To generate a bivariate interpolation on the rectangular grids and calculate the

value in the points specified in the arrays i

x and i

y it is used a spline interpolation,

like (4), for example for x:

( )3 2

P x a x b x c x d= ⋅ + ⋅ + ⋅ + (4)

The corresponding mathematical spline must have a continuous second derivative

and satisfy the same interpolation constraints. The breakpoints of a spline are also re-

ferred to as knots.

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields 145

The first derivative ( )'

P x of our piecewise cubic function is defined by different

formulas on either side of a knot k

x . Both formulas yield the same value k

d at the

knots, so ( )'

P x is continuous.

Case Studies

It is presented as illustration examples two case studies regarding the electric field

emanated from High Voltage Power Lines. Case 1 corresponds to a single Line and

Case 2 corresponds to two Lines orthogonally placed to each other. For both cases the

electrical conditions are the same, with 220 kV and 1140 A per phase conductor and

the catenary of the Line(s) approached by 30 straight lineal segments. Both Lines have

100 m length. The solution plane is defined by a span of Line and it was considered as

reference, a grid defined by one meter space between nodes in the solution plane. This

discretization corresponds to a grid with 10000 nodes and produces a smoother solution

of the electric field. It is seen in Fig. 1 the electric field distribution (smoother solution)

in the solution plane for Case 1, and in Fig. 2 for Case 2.

To obtain the electric field for Case 1 and Case 2, with 10000 nodes a considerable

computational time was required namely 3

32.49 10 sec× and 4

6.51 10 sec 18.2hours× ≈

respectively. The computational time to obtain the charge density and the correspond-

ing derivatives has to be added to this time for obtain the total computational time.

The idea was then to produce an electric field solution as accurate as possible but

with considerable much less computational time.

Results

Three different grids with two meter space, five meter space and ten meter space be-

tween nodes were used. The field solutions at these nodes were exactly the same as for

the finer grid and the derived field solution for the other nodes of the finer grid were

processed by the interpolation function.

It is shown in Fig. 3 the different computational time t versus the nº of nodes n for

Case 1.

020

4060

80100

0

50

1000

1000

2000

3000

4000

x

Field distribution in the plane

y

Eef

020

4060

80100

0

50

1000

1000

2000

3000

4000

5000

x

Field distribution In the plane

y

Eef

Figure 1. Electric Field smoother solution (Case 1). Figure 2. Electric Field smoother solution (Case 2).

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields146

The computational time function ( )t t n= can be approximated by Eq. 5.

( ) 3.2425 60.53t n n= ⋅ + (5)

To access the accuracy of the field solution, a nodal local error parameter [ ]%n

ε

was calculated, as:

[ ]%

n ref

n

ref

E E

E

ε

−

= (6)

where n

E is the electric filed value obtained at the nodes for the coarser grid and ref

E

is the corresponding electric field value obtained with the reference grid (one meter

space between nodes).

In Fig. 4 it is shown in the form of coloured plot the visualization of a 2D projec-

tion of the error distribution at the nodes in the plane of analysis for one grid defined by

10 meter space between nodes for Case 1.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

3

3.5x 10

4

n=nº of nodes

t=C

ompu

tatio

nal t

ime

[s]

Figure 3. Computational time (Case 1).

0

4.21e-002

8.41e-002

1.26e-001

1.68e-001

2.10e-001

2.52e-001

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100Error (%) - 10x10 grid

x

y

Figure 4. 2D projection of the error distribution in plane (Case 1).


In the form of graphic it is shown in Fig. 5 the variation of [ ]%n

ε for a line y =

70 m for the three different grid discretization, and in Fig. 6 it is shown the variation of

[ ]%n

ε for a line x = 70 m.

As it is seen the error [ ]%n

ε varies in the range of [0, 0.252]%, and the computa-

tional time to get the electric field solution is 384.7 sec, thus 84 times lower than the

time to obtain the field solution for 10000 nodes.

For Case 2, it is shown in Fig. 7 the computational time t versus the nº of nodes n

corresponding to these different grids.

The computational time function ( )t t n= can be approximated by Eq. 7.

( ) 6.4917 198.2292t n n= ⋅ + (7)

In Fig. 8 it is shown the visualization of a 2D projection of the error distribution in

plane of analysis for one grid defined by 10 meter space between nodes for Case 2.

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

x

Err

or [

%]

Error in x direction for y=70

2x2

5x5

10x10

Figure 5. 2D error distribution in plane (Case 1).

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

-3

x

Err

or [

%]

Error in y direction for x=70

2x2

5x5

10x10

Figure 6. 2D error distribution in plane (Case 1).


In this case the error [ ]%n

ε varies in the range of [0, 10]%. As this error is high,

the effective value varies in the range of [0, 500] V/m, it was analysed the error distri-

bution in plane of analysis for one grid defined by 5 meter space between nodes (400

nodes). The 2D projection of this error distribution it is shown in Fig. 9.

In this case the error [ ]%n

ε varies in the range of [0, 1] %. The nodal error is big-

ger than for Case 1, but still very low and quite acceptable. The computational time to

get the electric field solution is 3

2.7949 10 sec× , thus 19 times lower than the time to

obtain the field solution for 10000 nodes.

The computational time to get the electric field solution for 100 nodes is lower

than the time to obtain the field solution for 400 nodes, but the error is bigger.

The authors suggest that the user should take as minimal grid configuration, the

5 meter space between nodes (400 nodes) to obtain a quite acceptable field accuracy

solution.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

6

7x 10

4

n=nº of nodes

t=C

ompu

tatio

nal t

ime

[s]

Figure 7. Computational time (Case 2).

0

5.58e-002

1.12e-001

1.67e-001

2.23e-001

2.79e-001

1.00e+001

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90


x

y

Figure 8. 2D projection of the error distribution in plane (Case 2).


0

5.53e-003

1.11e-002

1.66e-002

2.21e-002

2.77e-002

9.96e-001

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90


x

y

Figure 9. Error distribution in plane for one grid defined by 5 meter space between nodes (Case 2).

Conclusions

It was presented an algorithm to reduce the computational time in obtaining the electric

field distribution in a plane of analysis due to High Voltage Power Lines. The nodal

error in the field solution is quite negligible when comparing solution obtained with

finer grids in plane of analysis.

This algorithm is implemented in the LMAT_SIMEL software, which is part of a

more complete package LMAT_SIMX that allows the analysis and simulation of Elec-

trical and Magnetic Fields emanated from very High Voltage Power Lines, developed

by the authors.

Acknowledgments

The authors gratefully acknowledge REN-Redes Energéticas Nacionais SGPS, SA for

the financial support received under Project COIMBRA_EMF.ELF.

References

[1] Carlos Lemos Antunes, José Cecílio, Hugo Valente, “LMAT_SIMEL – The Electric Field numerical cal-

culator of the package LMAT_SIMX for Very High Voltage Power Lines”, accepted for presentation at

ISEF 2007 – XIII International Symposium on Electromagnetic Field in Mechatronics, Electrical and

Electronic Engineering, Prague, Czech Republic, September 13-15, 2007.

[2] www.mathworks.com/access/helpdesk/help/techdoc/ref/index.html.

[3] www.mathworks.com.


Thermal Distribution Evaluation Directly

from the Electromagnetic Field Finite

Elements Analysis

A. DI NAPOLI, A. LIDOZZI, V. SERRAO and L. SOLERO

University ROMA TRE, Via della Vasca Navale 79 – Roma, Italy

[email protected]

Abstract. This paper deals with the thermal field evaluation achieved directly

from the finite elements representation of the relative electromagnetic field. The

proposed strategy can be applied to any generic electrical device; in this work it

has been proved on a permanent magnets electrical machine. This methodology

has been implemented in two different ways. At first laminar and turbulent mo-

tions have been considered from thermal convection point of view and then con-

vection has been reduced to pure conduction heat transfer.

Introduction

A finite element evaluation algorithm devoted to analyze both electric machines and

power switches has been implemented by means of ANSYS@

software.

At first the electromagnetic field inside the structure has been evaluated and plot-

ted. The structure has been considered as a discretized surface and both boundary con-

dition and currents have been set.

Induction and eddy currents values have been achieved and then dc-current Joule

effect losses and the additional losses have been computed.

Thermo-electric analysis has been accomplished considering convection and con-

duction heat transmission. Two different methodologies have been implemented. At

first, thermal convection has been studied under the assumption of both laminar and

turbulent motion. After that, convection heat transfer has been simulated under laminar

motion, where a particular coefficient has allowed reducing the heat exchange to pure

conduction, simplifying the simulation model and making the simulation faster.

When convection is studied as normal conduction, simulation software ANSYS al-

lows achieving directly the temperature distribution starting from the electric currents,

so avoiding previous steps concerning the evaluation of the electromagnetic field. In

this manner the simulation time is strongly reduced together with memory occupancy.

Magnetic Analysis

Proposed finite elements analysis has been applied to a permanent magnet synchronous

machine, where main data are shown in Table 1 and Table 2. Maxwell Eq. (1) written


151

using the vector A

has allowed studying the system as a plane if the observation is far

enough from the machine end regions.

2A

A J

t

μ ∂

∇ = +μ

ρ ∂

(1)

Concerning the boundary conditions, magnetic induction field (B) paths are sup-

posed to be tangent to both inner and outer circumferences shown in Fig. 1 as the seg-

ment BD and AC respectively.

Finite elements analysis has been implemented to only one by six of the machine

section; having the machine three pole-pairs it shows an odd symmetry. Avoiding any

saturation effect being the amplitude of the magnetic field quite small, a linear analysis

has been performed.

Figure 2 shows the achieved mesh useful to evaluate machine inner magnetic

fields. Inside the regions where the parasitic currents phenomena are absent, the current

density from the previous magnetic fields evaluation has allowed to evaluate the dc-

current losses. In the other regions the heat sources has been accomplished from the

eddy currents.

Table 1. PM machine data

Outer diameter D m 0.0125

Air-gap diameter Di m 0.0068

Air-gap thickness δ m 0.001

PM residual induction (NdFeB) Bres

T 1.15

Table 2. PM machine evaluated data

Irms

(A) Torque (Nm) Id (A) Iq (A)

75 20 69 30

125 40 110 58

Figure 1. Permanent magnet machine representation

and boundary conditions.

Figure 2. Permanent magnet machine mesh.

A. di Napoli et al. / Thermal Distribution Evaluation152

Thermal Analysis

Thermal analysis is based on the heat sources which can be defined for every mesh

point. Machine geometry and its mesh must be kept unchanged when both magnetic

and thermal analysis are based on the PLANE77 solver.

The data achieved by the magnetic analysis are then used to perform the thermal

investigation. Collected data, especially heat sources, have been stored in a data-base

and used in the machine mesh.

Heat transmission is controlled by the following expression

2q 1 T

T

t

⋅

∂

∇ + =

λ α ∂

(2)

where ρ is the mass density [kg/m3

], Cp is the specific heat [J/(kgK)], λ is the thermal

conductivity [W/(m2

K)], q is the sources heat power density [W/m3

] and finally α is the

thermal diffusion.

Along lines AC and BD shown in Fig. 2, boundary conditions have been defined

as the expression reported in Eq. (3):

( )dT

h T T

dn

∞

−λ = − (3)

where h takes into account the convection heat transfer from stator outer surface to

ambient which is supposed to be T∞

= 30 °C. Surfaces AB and CD, still shown in

Fig. 2, have been considered adiabatic.

Rotor position affects both power losses and mesh thermal characteristic; tempera-

ture is accomplished by the averaging different rotor position since thermal time con-

stant is greater than rotor revolution time.

The most difficult task is determining the convection coefficients; these parameters

describe heat and mass exchange within the fluid. To this purpose, non-dimensional

numbers achieved by analytical and experimental results explain fluid behaviors. Under

the assumption of natural convection, convection coefficient can be evaluated starting

from the Nusselt number, which can be achieved from Prandlt and Grashof numbers.

Convection heat transfer between stator outer surface (BD line in Fig. 3) and air-

gap has been deeply analyzed where the first surface is usually common with still sur-

faces.

Outer Surface

When the heat transfer is based on natural convection, which is true if the stator outer

surface and the free air heat exchange is taken into account, main heat transfer numbers

are shown as follow.

p

C

Pr

⋅μ

=

λ

= 0.710 (4)

A. di Napoli et al. / Thermal Distribution Evaluation 153

( )3

2

g T T D

Gr∞

β −

=

ν

where

1

T∞

β = =30 °C (5)

n

Nu C (Pr Gr)= ⋅ ⋅ (6)

where Pr is Prandlt’s number, Gr is the Grashof’s number, Nu is the Nusselt’s num-

ber and D is the machine diameter. The operating point has been selected as p

C =

1.007 [KJ/KgK], μ = 19.2010–6

[Kg/sm], λ = 27.16*10–3

[W/mK].

Under the assumption of laminar motion where Pr·Gr is lower than 109

and turbu-

lent for higher values, Nusselt equation parameters can be determined as follow:

102

< (Pr Gr)⋅ < 104

C = 0.85, n = 0.188

104

< (Pr Gr)⋅ < 109

C = 0.53, n = 1/4

10 < (Pr Gr)⋅ < 1012

C = 0.13, n = 1/3

After that, the convection coefficient h can be simply achieved considering a

smooth motor outer surface, and it allows the defining of the outer surface boudary

condition.

Nu

h

D

⋅λ

= = 10 W/m2

K

Air-Gap

Both convection and conduction within the air-gap are affected by the roughness of the

rotor surface (K. Ball, B. Farouk e V. Dixit. [1]), the stator system and rotor rotational

speed. In case of laminar motion, Reynolds number is small and the air-gap thermal

conductivity coefficient is near to the still air.

In case of turbulent motion, conduction coefficient is replaced by a new one that

takes into account also convection transfer.

2,9084 0,4614 ln(3,3361 )

eff

0,0019 Re

− ⋅ ⋅η

λ = ⋅η⋅ ⋅ (7)

Figure 3. Map of magnetic flux, Id = 68.7 A and

Iq = 30 A.

Figure 4. Map of magnetic flux, Id = 110 A and

Iq = 57,9 A.


where η is shown in Eq. (8)

o

i

r

R

η = (8)

with 0.4 < η < 1. r0 is the rotor outer radius and R

i is the stator inner radius. Reynodls

number is given by

0 m

Re r /= ⋅ω ⋅δ ν (9)

where ωm

is the rotor angular speed [rad/s], δ is the air-gap lenght [m] and ν kinetic

viscosity [m2

/s].

When smooth surface is assumed, the transition between laminar motion and tur-

bulent motion is given by the Taylor’s number,

2 3

m m

2

r

Ta

ω δ

=

ν

(10)

where

m

o

i

r

r

ln

r

δ

=

⎛ ⎞

⎜ ⎟

⎝ ⎠

is the average logarithmic radius (11)

o

r and i

r are respectively the stator inner and outer radius.

Following Becker and Koye theory, when Ta is lower than 1700 heat transfer is

mainly devoted to laminar motion and Nusselt number is equal to 2, otherwise the fol-

lowing expression should be used:

u

h 2g

N 2

⋅

= =

λ

for Ta < 1700 and i

g / r 0→

0.367

u

N 0.128 Ta= ⋅ for 1800 < Ta < 12000

0.241

u

N 0.41 Ta= ⋅ for 12000 < Ta < 6

4 10⋅

(12)

Air-gap heat transfer coefficient includes both conduction and convection and it

can be written as

Nu

h

2g

⋅λ

= .

Machine under test has 3 pole-pairs and it is fed with 50 Hz electrical, then

2 f / p 104.6 rad / sΩ = π⋅ = , m

r = 67.5 m (5.14); g = 0.001 m; ν = 172.6 10–7

m2

/s.

Taylor number is equal to Ta = 2479, therefore turbulent motion in the air-gap

determines the heat transfer which a Nusselt number equal to 2.25 and then h =

29.25 W/m2

K.

Concerning the air-gap, the data shown in [4] have been used, when the laminar

motion and thermal conductivity coefficient is closer to quiescent free air.


Figure 5. Thermal performance of stator and rotor

structure.

Figure 6. Thermal performance of stator structure.

Figure A1. Example of the structure under study.

Appendix

In this section the comparison of the two proposed methodologies is shown. As exam-

ple, it has been applied to the study of the thermal behavior of a simple structure shown

in Fig. A1, where areas A1 and A3 are composed by iron (thermal conductivity Fe

λ =

45 [W/mK] and size 0.03 × 0.03 [m]) and A2 area is air (adduction coefficient aria

λ =

0.026 (T = 30°C) [W/mK] and size 0.03 × 0.03 [m]). Boundary condition are T1= 40°C

and T2= 30 °C.

At first, the air has been considered as a solid material with its thermal conductiv-

ity. After that, air has been considered as a fluid with its phisical characteristics. The

achieved results have been compared.

Once the materials, the boundary conditions and the mesh have been selected,

thermal analysis has been carried out considering pure conduction heat transfer. The

second step was the introduction of the thermal convection, fluid characteristics as well

as the density temperature dependence.

A comparison between the proposed methodologies is shown in Figs A.2 and A.3,

where air temperature close to the borders is the same in both cases. Heat transfer is

mainly due to thermal conduction being temperatures very close. Thermal convection


heat transfer can be taken into account by means of the adduction coefficient since the

convection motion speed is quite low.

References

[1] Christos Mademlis, Nikos Margaris, and Jannis Xypteras, Magnetic And Thermal Performance Of A

Synchronous Motor Under Loss Minimization Control, Proc. Of the IEEE International Symposium On

Industrial Electronics July 1995.

[2] Y.K. Chin, D.A. Staton, Transient Thermal Analysis using both Lumped Circuit Approach and Finite

Element Method of a Permanent Magnet Traction Motor, AfriCon 2004, pp. 1027–1035.

[3] Ansys, Ansys Thermal Analysis Guide, November 2004.

[4] J.M. Owen, Fluif Flow and Heat transfer in Rotating Disk Systems, Proc. Heat and Mass Transfer in Ro-

tating Machinery, pp. 81–116, Springer Verlag, 1984.

Figure A.2. Solution of the thermal analysis by

means of the adduction coefficient.

Figure A.3. Air-iron temperature.


Coordination of Surge Protective Devices

Using “Spice” Student Version

Carlos Antonio França SARTORI, Otávio Luís DE OLIVEIRA and

José Roberto CARDOSO

Escola Politécnica da Universidade de São Paulo, Departamento de Engenharia

Elétrica PEA/EPUSP, Av. Prof. Luciano Gualberto Trav. 3, 158. O5508-900.

São Paulo, SP. Brazil


Abstract. This paper presents a methodology concerning Surge Protective Device

(SPD) Coordination Studies; taking into account the available tools of the student

version of the “Spice”. As an approach to satisfy the related limitations of this ver-

sion, preliminary analytical studies are carried out; allowing us to selected a list of

the available Spice SPD models to be applied in the simulations. Some applica-

tions regarding residential installations were chosen, and their results are presented

and compared with the ones presented in literature.

1. Introduction

Nowadays, the electrical and electronic equipment and systems have been used in

many branches of our modern society. On the other hand, the electromagnetic phenom-

ena that they are exposed presents resulting effects which characteristics can be higher

than their immunity levels, representing potential sources of Electromagnetic Interfer-

ence (EMI). Concerning the electrical phenomena related to surges; it should be men-

tioned that suitable protection systems have to be designed. In particular, the adoption

of Surge Protection Devices (SPD) is recommended, and the SPD Coordination studies

are a project requirement: These studies have as objective to guarantee the suitable en-

ergy to be dissipated by SPD, besides the clamped voltages to satisfy the equipment

immunity levels. Moreover, the improper usage of protective devices can result in sev-

eral fails or damages to the electrical systems and equipment. In fact, the selection of

those devices is not a simple task, and it requires many parameters to be considered like

the kind of the surges waveform, its intensity and the associated energy, frequency of

occurrence, the SPD configuration, the equipment immunity levels, etc. [1–5]. Based

on the aforementioned scenario, it should be mentioned that the use of a single analyti-

cal methods can result in a complex and time-consuming work, and the application of

computational tools appear as an interesting and a suitable approach. There are a lot of

computational tools that can assist in solving electric circuits, but many of them are

relatively expensive. For this reason, we have proposed, as part of the methodology, the

use of a student version of a circuit simulator. Due to the wide use and literature avail-

ability, the so-called “Spice” was the software that the authors have chosen to be ap-

plied [6,7].



doi:10.3233/978-1-58603-895-3-158

158

2. Methodology

Basically, the proposed methodology presents as main feature, preliminary analytical

evaluations that allows obtaining pre-defined SPD models [3]. These SPD models will

be, then, taken into account in the further Spice simulations. The details of the SPD,

equipment models, and of the full method are briefly detailed in this section. Regarding

the general aspects of the method, the following steps can describe it:

1. Definition of the surge characteristics according to Lightning Protection

Zones (LPZ) [8];

2. Representation of the electrical system to be protected;

3. Definition of the immunity level of equipment [9];

4. Definition of the SPD characteristics;

5. Definition the pre-selected SPDs and configuration to be used in the computa-

tional simulation [3];

6. Implementation of the devices selected in the previous step in the “Spice”

simulator;

7. Computational simulation of the electrical system in study;

8. Verification of the Surge Coordination.

The flowchart concerning the proposed methodology is presented in Fig. 1.

2.1. Definition of the Electromagnetic Environment

The IEC 61312-3 standard presents the parameters that helps us to classify the electro-

magnetic environment of a structure to be protected [8]. These aforementioned areas or

electromagnetic environments are called Lightning Protection Zone (LPZ). The Fig. 2

shows this principle concerning a pre-selected structure. It should be mentioned that the

Figure 1. Methodology Flowchart.

C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version 159

maximum value of voltage and current surges, the test wave-form, are defined accord-

ing to the LPZ.

2.2. SPD Characteristics

Regarding the SPD, it could be mentioned the following characteristics [2,3]:

1. Under the rating voltage, the device will not conduct, although we can ob-

serve a small current, called leakage current, in this condition;

2. For higher voltage, an electric current flows through the device, but the volt-

age across it will not increase significantly (Clamp Voltage);

3. The energy capability of the SPD must be compatible with the surges energy

level of the electrics systems;

4. After the suppression of the surge, the SPD returns to the condition described

in 1;

5. The resulting clamp voltage must be smaller than the required immunity level

of the equipment.

A typical device used in this study is the well-known varistor.

2.3. “Spice” Varistor Model

The SPD models can be built based on the aforementioned characteristics and obtained

directly from the available manufacturer technical literature. Regarding the varistor

models, they can be considered as an association of 4 components, representing the

physical phenomena that occur in the real varistor. Figure 3 presents an electric model

of these devices. In the proposed varistor “Spice” model, initially, a series resistance

(Rserie) is assumed, whose value is constant and equal to 100 nΩ. The components,

Lserie and Cparalelo, represent series inductance and parallel capacitance of the device.

These values change in accordance with the real model of the device. The variable re-

Figure 2. Lightning Protection Zone division.

C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version160

sistance (Rvariavel) is related to representation of the nonlinear characteristic of them.

The dependence between voltage (v) and current (i) is simulated by a voltage generator,

controlled by the current, and modeled by:

( )1 2 3 4

log V p p p p= + + + (1)

where,

p1 = b

1;

p2 = b

2log(i);

p3 = b

3 exp (–log(i));

p4 = b

4 exp (–log(i)).

The parameters b1, b

2, b

3, b

4 are related to specific characteristics of each varistor.

2.4. Details of the SPD Model Implementation in Spice

Although the aspects concerning the software Spice have already been presented in

various scientific papers, it should be emphasized some aspects of it. The Spice can be

understood as formed by sub-routines, each one related to a specific part of the circuit

simulation task. Firstly, it should be emphasized the structure and which files are used;

in order to allow the model implementation in its library, like the varistor ones, in order

to be available though the Schematics. In this case, the first file of interest is the

“VAR.SLB” that contains the information regarding the graphical part of the devices:

name, nodes, electrical parameters, mathematical characteristics, etc. This file should

be associated to “PSpice” through the program Schematics, and it does not present any

restrictions for the student versions. In fact, this file defines only the generic device of

the component, called, e.g., “VAR”. The second file of interest is the “VAR.LIB”. This

file has the mathematical definition of the characteristics of the component. It is where

the parameters of each type of varistor (Cparalelo, Lserie, b1, b

2, b

3, and b4) are defined.

This file should also be associated to “PSpice” through the program Schematics, and

Figure 3. Spice varistor Model.


now the restrictions of the student version take an important role on this process. The

Spice student version admits no more than 15 definitions of components in

“VAR.LIB”. Preliminary analytical studies are proposed, as an approach to satisfy the

related constrains of this version, allowing us to select a list of the available Spice SPD

models to be applied in the simulations. Thus, the file “VAR.LIB” can be edited, and

the non-selected device definitions deleted from the available list of components.

3. Application and Results

Several results regarding different residential circuits were obtained. In order to vali-

date the proposed SPD coordination approach, a configuration presented in literature

was selected [4]: A residential one with eight branch circuits, which circuits were mod-

eled as Transmission Lines (Z0

= 100 Ω, C = 50 pF/m and L = 0,5 µH/m). Figure 4

shows a schematic representation of circuits, and the corresponding Spice models. A

surge waveform 20 kA (10 × 350 μs) was assumed in the simulations. Some results,

related to different SPD arrangements, and energy dissipation, are presented in Table 1.

Six different cases were simulated varying the configuration of the SPD, using four

types of devices (S20K130, S20K150, S20K250 and S20K625). The energy dissipated

on the SPD was compared with the device withstand energy, as well as the resulting

clamp voltages were checked with the equipment immunity levels. Concerning the

convergence simulation requirements, an analysis of sensitivity for the parameters

named “ABSTOL”, “RELTOL” and “Step Ceiling” were carried out in order to opti-

SPD 1

SPD 2

Figure 4. Schematic representation of the circuits.


mize it. Notice that these parameters can affect the simulations, since they are directly

related to approach used, like the Modified Node Analysis of the circuits [6,7]. The

parameters used in each simulation are presented in Table 2.

4. Discussion

Case 0 can be considered as a “control case”. It represents a residence without a surge

protection system, and it is suitable for making comparisons with the cases in which the

SPDs were adopted. Note that the results here presented are focused on the SPD energy

dissipation. Cases 1 and 3 represent single SPD systems, whose SPD are positioned in

the entry of the energy distribution system. The Case 2 counts with a second SPD, and

it can be observed that the energy wasted in device S20K625 are smaller when com-

pared to the Case 1 and Case 3. Case 4 presents the best alternative among the pro-

posed protection configuration, presenting suitable energy coordination. The same con-

clusion can be observed in [4]. Case 5 does not present a good coordination between

the SPDs. The Case 5 SPDs have a relatively low and close operation voltage rating,

and the amount of energy dissipated in SPD 2 has increased when compared to the pre-

vious SPD configurations. Some differences on the results were observed when they

are compared with the ones given in [4]: For example, the values of energy obtained in

the Case 2 are smaller than the ones presented in this reference. On the other hand, in

the Case 5, the energy observed on SPD 1 is lower, while the value of energy found for

SPD 2 shows a value that is slightly above of the one presented in [4]. These differ-

ences can be attributed to the fact that was not possible to assume the same SPD char-

acteristics in these works. As an example, the one of the devices used in the reference

work that presents a nominal voltage equal to 200V was not available in the library of

components that we have used, and it was substituted by another SPD (S20K130),

Table 1. Resulting SPD energy

Configuration #

SPD 1

Model

SPD 2

Model

Energy

SPD 1 (J)

Energy

SPD 2 (J)

0 - - - -

1 20K625 - 23216 -

2 20K625 20K130 1255 496,261

3 20K150 - 1565,7 -

4 20K150 20K250 1467,2 89,566

5 20K150 20K130 1242,4 260,802

Table 2. Parameters used in simulations

Case Print Step Final Time Step Ceiling ABSTOL RELTOL

0 0.1us 500us 0.1us 1pA 0.001

1 0.1us 4.5ms 0.1us 1pA 0.5

2 0.1us 1.6ms 0.1u 1pA 0.5

3 0.1us 3ms 0.1us 1pA 0.5

4 0.1us 3ms 0.1us 1pA 0.5

5 0.1us 3ms 0.01us 1pA 0.5


which Vn is 205 V. This can affect significantly the results due to the nonlinear behav-

ior of those devices. That is, depending on the point of operation of the SPD, concern-

ing its characteristic curve, small variations of terminal voltage can result in a wide

difference in the resulting current. Besides that, as mentioned before, the parameters

assumed to satisfy the convergence requirements, can also explain some of the differ-

ences on the results.

5. Conclusion

An approach to satisfy the limitations of the Spice student version were presented

based, preliminary, on analytical studies that allows us to selected a list of the available

Spice SPD models to be applied in the simulations. Some applications regarding resi-

dential installations were chosen, and their results were here presented and discussed.

Although some adequacies were adopted to satisfy the convergence constraints of the

numeral method, the present methodology appears as a potential one to be used in the

studies related to low voltage SPD coordination. It is emphasized the importance of this

study, due to the relative low immunity levels of the equipments that is used in all

branches of our society. An analysis of the influence of different load and circuit mod-

els are proposed as part of the future development of this work.

References

[1] Lai, J.S.; Martzloff, F.D. “Coordinating Cascaded Surge Protection Devices: High-Low versus Low-

High”, IEEE Transaction on Industry Applications, Vol. 29, No. 4, pp. 680-687, July/August 1993.

[2] Paul, D.; Srinivasa I.V. “Power Distribution System Equipment Overvoltage Protection”, IEEE Trans-

action on Industry Applications, Vol. 30, No. 5, pp. 1290-1297, September/October 1994.

[3] Paul, D. “Light Rail Transit DC Traction Power System Surge Overvoltage Protection”, IEEE Transac-

tion on Industry Applications, Vol. 38, No. 1, pp. 21-28, January/February 2002.

[4] Standler, Ronald B. “Calculations of Lightning Surge Currents Inside Buildings”, 1992 IEEE Interna-

tional Symposium on EMC, Proceedings, pp. 195-199, Aug. 1992.

[5] Standler, Ronald B. “Transient on the Mains in a Residential Environment” IEEE Transactions on Elec-

tromagnetic Compatibility, Vol. 31, No. 2, pp. 170-176, May 1989.

[6] Tuinenga, P.W. “Spice: A guide to Circuit Simulation & Analysis using PSpice”, Prentice-Hall, 1988.

[7] Herniter, Marc E. “Schematic Capture with PSpice”. Macmillan College Publishing Company, 1994.

[8] IEC/TS 61312-3: 2000. International Electrotechnical Commission – “Protection against lightning elec-

tromagnetic impulse – Part 3: Requirements of surge protective devices (SPDs)”.

[9] IEC 61000-4-5: 2001. Electromagnetic compatibility (EMC) of electrical and electronic equipment –

Part 4: Testing and measurement techniques – Section 5: Surge Immunity test.


Chapter B. Computer Methods in Applied

Electromagnetism

B2. Numerical Models of Devices


Nonlinear Electromagnetic Transient

Analysis of Special Transformers

Marija CUNDEVA-BLAJER, Snezana CUNDEVA and Ljupco ARSOV

Ss. Cyril & Methodius, Faculty of Electrical Engineering and Information

Technologies, Karpos II b.b, POBox 574, R. Macedonia


Abstract. In the paper original methodology for nonlinear electromagnetic tran-

sient analysis of special transformers will be given. A universal nonlinear trans-

former model will be developed by using the finite element method study results.

The electromagnetic field analysis will be done by applying the original program

package FEM-3D developed at the Faculty of Electrical Engineering and Informa-

tion Technologies-Skopje (FEIT). The FEM results will experimentally be verified

on a resistance welding transformer through actual test results recorded in a labo-

ratory. The same methodology and model will be used for the metrological tran-

sient analysis of 20 kV combined current-voltage instrument transformer.

Introduction

The special transformers, like resistance welding transformer (RWT) or combined cur-

rent-voltage instrument transformer (CCVIT) are complex non-linear electromagnetic

systems which operate in transient working regimes. The transients of the RWT are

introduced by the nature of the welding process. The CCVIT must comply with the

rigorous metrological specifications of the IEC 60044-2 standard [1] during the tran-

sient regimes. The RWT and CCVIT electromagnetic phenomena are described by the

voltage equilibrium equations:

1 1 1

11

d

u i r

dt

b

ψ

ω

= ⋅ + ⋅ (1)

2

2 22

'1

' ' '

b

d

u i r

dt

ψ

ω

= ⋅ + ⋅

(2)

where, 11

λωψ ⋅=

b

,22

λωψ ⋅=

b

, λ1 and λ

2 are the resultant fluxes created by the pri-

mary and secondary winding currents, respectively and b

ω is the basic (industrial) fre-

quency at which the reactances will be calculated by using the FEM. The RWT and

CCVIT are non-linear bounded electromagnetic systems with prescribed boundary

conditions and the electromagnetic field distribution is most suitably expressed by the

system of non-linear partial differential equations of the Poisson’s type:


167

( ) ( ) ( ) ( , , )

A A A

B B B j x y z

x x y y z z

ν ν ν

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3)

where the magnetic vector potential A

as an auxiliary quantity is introduced, B

is the

magnetic flux density, ν

the magnetic reluctivity and ( , , )j x y z

the volume current

density. The analyzed devices are heterogeneous and (3) can be solved by numerical

methods, only. The magnetic field analysis is done by an original and universal pro-

gram package FEM-3D developed at the FEIT-Skopje, [2]. The results of the FEM

analysis will be input data in a dynamic transformer model for transient analysis. The

core saturation will be incorporated in the model as in [3] by using the relationship

between measured saturated and unsaturated values of mutual flux.

Transformer Model for Transient Study

The transient performance of transformers is influenced by a number of factors with

most notable the exponentially decaying dc component of the primary current. Its pres-

ence influences the build-up of core flux, a phenomenon which is likely to cause satu-

ration and subsequently substantial errors in the magnitude and phase angle of the gen-

erated signals. Core saturation mainly affects the value of the mutual inductance and, to

a much lesser extent, the leakage inductances. Though small, the effects of saturation

on the leakage reactances are rather complex and would require construction details of

the transformer that are not generally available. In the FEM numerical approach trans-

former leakage inductance can be calculated by computing the normal flux at no load

along a contour defined over the winding i.e. the normal leakage flux '

σ

Φ . Multiplying

the normal flux with the actual transformer length in the z direction lz yields:

z

'

l⋅=

σσ

ΦΦ (4)

Then the total leakage flux is defined as:

N⋅=

σσ

Φψ (5)

where N are the number of turns excited. The leakage inductance is calculated by:

0

I/Lσσ

ψ= (6)

where I0 is the no load current flowing through the primary winding.

In many dynamic simulations, the effect of core saturation may be assumed to be

confined to the mutual flux path. Different iron core models are described in the litera-

ture and their summary is described in [4].

In this paper a nonlinear SIMULINK transformer model has been developed and it

is presented in Fig. 1. The effects of core saturation in the dynamic simulations have

been incorporated using the relationship between saturated and unsaturated values of

the mutual flux linkage as described in [3].

M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers168

Study Case-RWT: Experimental Verification of the Model

Prior to setting up a transformer model suitable for transient studies, a set of test results

for commercial RWT were performed in the laboratory at FEIT. According to the

manufacturer data and the measurements performed, the resistance welding transformer

has the following rated data: primary voltage 380 V; secondary no-load voltage

(1,41–4,63) V; conventional power 24 kVA; rated frequency 50 Hz; thyristor con-

trolled switching; number of primary tap positions 9. The transformer is a single phase

with shell type core. Core saturation has been determined from the open circuit mag-

netization curve of the investigated transformer. The leakage reactances of the RWT

were calculated by using the FEMM program [5,6] and they served as input in the dy-

namic model shown in Fig. 1. The results from the numerical finite elements calcula-

tion are presented in Table 1.

The derived SIMULINK results have been experimentally verified on the resis-

tance welding transformer through actual test results recorded in a laboratory. A small

selection of the derived results is presented in Table 2.

The test results are satisfactory with the exception of a slightly different magnitude

of the simulated iron losses Pfe and simulated reactive power Q

fe. The simulation results

for the currents are nearly identical to the laboratory tests. With these results the valid-

ity of the derived model has been proved.

The transient performance of the RWT has been modeled by defining zero transi-

tion of the winding voltage β = 0. Sample results for the transient performance of the

i2'v2p•

Dpsi

psim

psi2'

psi1 i1

4

Out_i2'

3

Out_i1

2

Out_psim

1

Out_psi1

v1

Initial izeand plot

uslovi

s

1

psi2'_

s

1

psi1_

y

To Workspace

Scope4

Scope3

Scope1Scope

Mux Mux5

Mux

Mux4

Mux

Mux3

Mux

Mux2

Mux

Mux1

Mux

Mux

Memory1

Load Module

(u[1]-u[2])/xpl2

Fcn5

f(u)

Fcn4

xM*(u[1]/xl1+u[2]/xpl2-u[3]/xm)

Fcn3

wb*(u[2] -(rp2/xpl2)*(u[1]-u[3]))

Fcn2

wb*(u[2]-(r1/xl1)*(u[3]-u[1]))

Fcn

Plots FFT

FFT

Dpsi=f(psisat)

Clock

V

IPQ

Active & ReactivePower

Figure 1. SIMULINK transformer model for transient analysis.

Table 1. FEMM numerical results

normal leakage

flux

total leakage

flux

Leakage

inductance

magnetic field

energy

Magnetizing

inductance

Φσ [Vs/turns] ψ

σ [Vs] L

σ [H] W [J] Lm [H]

5,656⋅10–6

531,6⋅10–6

118,13⋅10–6

1,877 0,185

M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers 169

analyzed RWT at nominal position 8 are shown in Fig. 2. From the results the exponen-

tially decaying dc component of the primary current can be clearly observed. The cur-

rent waveform displays large peak at the beginning (up to 100 times the rated value).

Study Case-CCVIT: Application of the Verified Model

The developed transformer model verified on the RWT study case is applied for tran-

sient analysis of the combined instrument transformer CCVIT with (voltage measure-

ment core VMC ratio 20000 V 100 V

:

3 3

and current measurement core CMC ratio

100 A: 5 A). The CCVIT is with a complex electromagnetic construction and its ge-

ometry has been given in details in [7]. The electromagnetic parameters of the CCVIT

are most important for its transient analysis, as they are the input data in the above

developed SIMULINK transformer model. The CCVIT quasi-steady-state electromag-

netic field analysis has been done by the original and universal program package

FEM-3D, [2], developed at the Faculty of Electrical Engineering and Information

Technologies in Skopje. The detailed CCVIT FEM-3D analysis has been given in [8].

The main electromagnetic characteristics of the CCVIT derived by FEM-3D, necessary

for the transient analysis are given in Figs 3–8. In Fig. 3 the main flux characteristic per

turn ϕmu

in the upper middle cross-section of the VMC magnetic core via the VMC

Table 2. Comparison of the RWT experimental and simulated results

Tap position 5 Tap position 8

Experiment Simulation Experiment Simulation

I0 [A] noload current 0,65 0,80 5,26 5,40

I1 [A] primary rms current 55,4 58,0 125 134

I1max [A] primary magnitude current 78,3 83,0 177 187

Pfe [W] active core losses 115 100 298 180

Qfe [VA] reactive core losses 217 285 1529 2000

Figure 2. RWT switching transients versus time [s] at most rigorous phase angle β = 0.


0,0

10,0

20,0

30,0

40,0

0,0 0,5 1,0 1,5

Relative VMC input voltage

Uu

/Uur

[r. u.]

Ma

in

flu

x p

er tu

rn

only VMC

Ii/Iir=0

Ii/Iir=0,2

Ii/Iir=0,4

Ii/Iir=0,6

Ii/Iir=0,8

Ii/Iir=1,0

Ii/Iir=1,2

0,0

10,0

20,0

30,0

40,0

0,0 0,5 1,0 1,5

Relative CMC input current

Ii/I

ir [r. u.]

Ma

in

flu

x p

er tu

rn

only CMC

Uu/Uur=0

Uu/Uur=0,2

Uu/Uur=0,4

Uu/Uur=0,6

Uu/Uur=0,8

Uu/Uur=1,0

Uu/Uur=1,2

Figure 3. CCVIT main flux characteristics in [μVs]

in the upper middle cross-section of the VMC mag-

netic core via the VMC input voltage and the CMC

input current as a parameter.

Figure 4. CCVIT main flux characteristics in [μVs]

in the upper middle cross-section of the CMC mag-

netic core via the CMC input current and the VMC

input voltage as a parameter.

0

5

10

15

20

25

30

35

40

0,0 0,4 0,8 1,2


Uu

/Uur

[r. u.]

Lea

ka

ge flu

x p

er tu

rn

only VMC

Ii/Iir=0

Ii/Iir=0,2

Ii/Iir=0,4

Ii/Iir=0,6

Ii/Iir=0,8

Ii/Iir=1,0

Ii/Iir=1,2

0

5

10

15

20

25

30

0,0 0,4 0,8 1,2


Uu

/Uur

[r. u.]

Lea

ka

ge flu

x p

er tu

rn

only VMC

Ii/Iir=0

Ii/Iir=0,2

Ii/Iir=0,4

Ii/Iir=0,6

Ii/Iir=0,8

Ii/Iir=1,0

Ii/Iir=1,2

Figure 5. VMC primary winding (24000 turns)

leakage flux characteristics per turn in [μVs] via the

VMC relative input voltage and the CMC relative


Figure 6. VMC secondary winding (120 turns) leak-

age flux characteristics per turn in [μVs] via the

VMC relative input voltage and the CMC relative


0

10

20

30

40

50

60

70

0,0 0,4 0,8 1,2


Ii/I

ir [r. u.]

Lea

ka

ge flu

x p

er tu

rn

only CMC

Uu/Uur=0

Uu/Uur=0,2

Uu/Uur=0,4

Uu/Uur=0,6

Uu/Uur=0,8

Uu/Uur=1,0

Uu/Uur=1,2

0

5

10

15

20

25

0,0 0,4 0,8 1,2


Ii/I

ir [r. u.]

Lea

ka

ge flu

x p

er tu

rn

only CMC

Uu/Uur=0

Uu/Uur=0,2

Uu/Uur=0,4

Uu/Uur=0,6

Uu/Uur=0,8

Uu/Uur=1,0

Uu/Uur=1,2

Figure 7. CMC primary winding (6 turns) leakage

flux characteristics per turn in [μVs] via the CMC

relative input current and the VMC relative input

voltage as a parameter.

Figure 8. CMC secondary winding (120 turns) leak-

age flux characteristics per turn in [μVs] via the

CMC relative input current and the VMC relative

input voltage as a parameter.


relative input voltage Uu/U

ur and the CMC relative input current I

i/Iir as a parameter are

displayed. In Fig. 4 the main flux characteristic per turn ϕmi

in the upper middle cross-

section of the CMC magnetic core via the CMC relative input current Ii/Iir

and the

VMC relative input voltage Uu/U

ur as a parameter are displayed.

The CCVIT transient analysis is done by coupling with the FEM-3D results. The

magnetic field distribution results, e.g. leakage reactances characteristics are input data

into the non-linear mathematical model of the CCVIT. The complex non-linear analy-

sis has been done for rated loads of the both measurement cores and rated frequency of

50 Hz. The input voltage phase angle is β = 0.

The CCVIT is a measurement device therefore its metrological parameters are of

greatest interest. By using the SIMULINK transformer model the most important met-

rological CCVIT parameters have been calculated for the most rigorous moment the

first forth of the signal period at the worst, from metrological point of view, regime at

β = 0: the VMC voltage error pu= –17,5% and CMC current p

i= –19%.

Figure 9. Switching transient primary CCVIT current of the VMC via time [s] at most rigorous phase angle

β = 0.

Figure 10. Switching transient primary CCVIT current of the CMC via time [s] at most rigorous phase angle

β = 0.

Figure 11. Time dependence of the VMC primary

current (RMS value) at rated load of the both cores

and β = 0.

Figure 12. Time dependence of the CMC primary

current (RMS value) at rated load of the both cores

and β = 0.


0,00

0,02

0,04

0,06

0,08

0,10

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Relative VMC primary voltage

Uu/U

ur [r. u.]

I1umax[A]

only VMC

Ii/Iir=0.0

Ii/Iir=0.2

Ii/Iir=0.4

Ii/Iir=0.8

Ii/Iir=1.0

Ii/Iir=1.2

Figure 13. Maximal VMC primary plug-in current via the VMC relative voltage, the relative CMC current is

a parameter at β = 0 and rated loads of both cores.

0

20

40

60

80

100

120

0,0 0,5 1,0 1,5

Relative to rated steady- state regime CMC primary

current Ii/I

ir [r. u.]

I1imax[A] only CMC

Uu/Uur=0.0

Uu/Uur=0.2

Uu/Uur=0.4

Uu/Uur=0.8

Uu/Uur=1.0

Uu/Uur=1.2

Figure 14. Maximal CMC primary plug-in current via the CMC relative current (through the CMC steady-

state regime current), the relative VMC voltage is a parameter at β = 0 and rated loads of both cores.

Conclusions

The developed transient performance transformer model has been verified as accurate

on the resistance welding transformer and it has been further applied for transient

analysis of a combined instrument transformer. The confirmed transformer model has

been coupled with finite element method results. The methodology in the paper is uni-

versal and can be applied for other complex electromagnetic devices.

References

[1] IEC (International Electrotechnical Commission) 60044-2, 1980: Instrument transformers, Part 3: Com-

bined transformers, Geneve, 1980.

[2] M. Cundev, L. Petkovska, The Weighted Residuals Method for Electromagnetic Field Problems in Elec-

trical Machines, Proc. of the 32nd UPEC’97 Conference, Vol. 2, UMIST, pp. 934-937, 1997.


[3] C.M. Ong “Dynamic Simulation of Electric Machinery Using MATLAB/SIMULINK”, Prentice Hall

PTR, Upper Saddle River, New Jersey, 1998.

[4] Working Group C-5 of the Systems Protection Subcommittee of the IEEE Power System Relaying

Committee, “Mathematical models for current, voltage, and coupling capacitor voltage transformers”,

IEEE Transactions on power delivery, Vol. 15, No. 1, Jan. 2000, pp. 62-72.

[5] D. Meeker, Finite Element Method Magnetics – User’s Manual 3.0, 1998-2000.

[6] S. Cundeva, L. Petkovska., V. Filiposki., A Methodology for Coupled Steady-State Electromagnetic-

Thermal Modeling of Resistance Welding Transformer, Proceedings of XI International Symposium on

Electromagnetic Fields in Electrical Engineering ISEF’03, Maribor, Slovenia, 2003, pp. 751-756.

[7] M. Cundeva, L. Arsov, G. Cvetkovski, Genetic Algorithm coupled with FEM-3D for metrological opti-

mal design of combined current-voltage instrument transformer, COMPEL: The International Journal

for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23, No. 3, 2004,

pp. 670-676.

[8] M. Cundeva-Blajer, L. Arsov, FEM-3D for Metrological Optimal Design and Transient Analysis of

Combined Instrument Transformer, Przeglad Elektrotechniczny, R. 83 NR 7-8/2007, pp. 96-99.


Some Methods to Evaluate Leakage

Inductances of a Claw Pole Machine

Y. TAMTO

a

, A. FOGGIA

a

, J.-C. MIPO

b

and L. KOBYLANSKI

b

a

LEG, UMR5529 INPG/UJF-CNRS, BP46, 38402 Saint Martin d’Hères Cedex, France

b

VALEO Equipements Electriques & Moteurs, 2 Rue André Boulle-BP150,

94017 Créteil Cedex, France

Abstract. To increase claw pole machine efficiency, we evaluate leakages in the

magnetic circuit. These leakages are mainly due to end windings, air gap and slot

leakages. Thus we obtain an equivalent circuit model. This model will be helpful

to simulate the machine under load conditions and to make possible its optimiza-

tion. The aim of this paper is then to present some methods of determination of

leakage inductance, by computation and under test.

Introduction

Methods for leakage inductance determination under test and computation, with the

short circuit, the open-circuit and the zero-power-factor characteristic, to build the

Potier diagram are well known. These methods were developed mainly for high power

alternator. In the case of claw pole machine, zero-power-factor test and computation

are difficult to run because of the value of inductances needed to simulate zero-power-

factor. This is why we explore other ways of leakage evaluation: we used the singular

configuration of the claw pole alternator under load conditions, and the stator alone,

without the rotor.

1. The Potier Method

This model takes into account the magnetic saturation and is used in the special case of

alternators with smooth poles. Figure 1 represents the Potier diagram in the general

case.

Under load, there is not phase difference between V and I because the load is a di-

ode rectifier bridge charging a battery. We then compute λ with load characteristic, by

solving Potier vector equations (1) and using finite element computation. Tests results

will validate the parameters previously obtained.

2 2 2

2 2 2

( ) ( )

( ) 2 cos( )

( )

2

r

fr f

V RI I

I I I I

I

arctg

V RI

ωλ

α α γ

π λω

γ

Ε = + +

= + −

= +

+

(1)


175

Er: Voltage behind leakage inductance

λ : Leakage inductance

α : Potier coefficient of equivalence

Ifr: Load excitation corresponding to Er

I f: Load excitation

V: voltage /phase

I: Current/phase

R: Resistance/phase

Figure1. Vector diagram of Potier.

2. Example

The main characteristics of one of the claw pole machine used for tests are on the fol-

lowing table

Winding star

Number of pole pairs 6

Number of turns of rotor winding 380

Rotor Resistance 2,66 ohms

Number of phases 3

Number of turns per phase 6

Resistance per phase 27 milliohms

Number of stator slots 36

Air-gap thickness 0.365 mm

No load characteristic allow to express Ifr

from Er,

the parameter α is the slope of

short circuit graph. Then with load tests we find λ by using electrical values at one

phase boundaries at different speeds. Results for Potier parameters are:

0,033

48 H

α

λ μ

=

=

The proposed method is more accurate for this type of alternator because test under

load is more easy to carry out than the use of zero factor characteristic. But the compu-

tation time is long. So, we try to evaluate leakages by the removed rotor method.

3. Removed Rotor Method

The rotor is removed and the stator is supplied by a three-phase current source, the am-

plitude of current varies from a very low value to the nominal one. The idea is that by

removing the rotor, we take the main flux off; it only remains the leakage flux. The

leakage inductance is then deduced from the electric quantities at the terminals of the

stator windings. Another calculation of the leakages is done by using the electromag-

netic energy stored.

Y. Tamto et al. / Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine176

Removed Rotor Test

A tree phase current source is needed to feed stator phases. We made frequency and

module variable. For each set of variables, the temperature of the windings is measured

to evaluate the right resistance.

2 2

X a Z R

V

Z

I

= −

=

(2)

Xa = ω λ: Leakage reactance; Z: Phase impedance; V: Voltage; I: Current; R: Resis-

tance of the phase at T1 temperature R T1

: = R T0

. (1 + α.(T1 – T0)); α = 4 10

–3

°C

–1

for

copper.

Removed rotor tests

0

48

96

50 250 450 650

Frequency( Hz)

Leakag

e in

ud

ctan

ce (u

H)

Figure 2. Leakage inductance.

Finite Element Computation

The machine to computaee has three phases and 6 pole pairs. We choose the area under

the stator teeth with tangent property for the magnetic field. A triplet of current is im-

posed in the phases [A, B, C] and they are distributed as follows: [IA, –I

A/2, –I

A/2].

Then leakage inductance is calculated by (3).

A

E

2

*)3/4(

Ι

=λ

(3)

E: electromagnetic energy. IA: current in phase A.

Y. Tamto et al. / Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine 177

Figure 3. Design without Rotor.

Results

IA (A) Energy (J) Leakage Inductance (μH)

5 9,01E-04 48,1

10 3,61E-03 48,2

20 1,45E-02 48,3

30 3,26E-02 48,3

40 5,80E-02 48,4

50 9,07E-02 48,4

60 1,31E-01 48,4

70 1,78E-01 48,4

80 2,32E-01 48,4

90 2,94E-01 48,4

We can notice that value of leakage doesn’t change with phase current and are the

same for removed rotor computations and tests.

Conclusion

The values of leakage inductances obtained by method exposed are nearly equals, this

study validates removed rotor method for determination of leakage inductance and we

can determinate alternator outputs with an accuracy of 99%. This method is speed and

easy to test and to computate. In the aim of an optimization for example it’s useful to

simplify the claw pole machine by its electrical model of Potier, this model allow to

create a virtual test bench of the claw pole environment with numerical language.

References

[1] IEEE Std 115-1995, pp. 50-66.

[2] J. Dos Ghali, Essais Spéciaux sur les Machines Electriques.

Y. Tamto et al. / Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine178

Reduction of Cogging Torque in Permanent

Magnet Motors Combining Rotor Design

Techniques

Andrej ČERNIGOJ

a

, Lovrenc GAŠPARIN

a

and Rastko FIŠER

b

a

Iskra Avtoeletrika d.d., Polje 15, 5290 Šempeter pri Gorici, Slovenia

[email protected], [email protected]

b

University of Ljubljana, Faculty of Electrical Engineering, Tržaška cesta 25,

1000 Ljubljana, Slovenia

[email protected]

Abstract. High performance motor drive applications require permanent magnet

synchronous motors (PMSM) that produce smooth torque with low torque ripple

components. This paper quantifies various sources of torque ripple and is focused

on rotor PMSM design techniques that can be used for reducing the cogging

torque. For each chosen design technique a validation with finite element method

(FEM) analysis is given. Finally the comparison and evaluation of design princi-

ples and their combination are presented and commented.

Introduction

The ability of PMSM to produce smooth torque and high power density is important in

high performance motor drive applications. Nevertheless, due to the mechanical con-

struction and material properties there are various causes responsible for producing

undesired torque ripple like: reluctance torque, cogging torque, and harmonics in the

back emf.

Several well-known design techniques [1–4] can be used to minimize this parasitic

harmonic torque components, but most of them consequently reduce the output torque

as well. Presented paper discusses and quantifies compromises among reducing cog-

ging torque in connection with preserving the main output torque at high value in order

to optimize the PMSM design for a specific drive application.

Instantaneous Torque of PMSM

Instantaneous torque of a PMSM has two components: constant component T0 and pe-

riodic component Tr(α), which is a function of an electric angle α and presents torque

pulsations called torque ripple (Fig. 1).

( ) ( )0 r

T T Tα α= + (1)


179

There are three origins of torque ripple in PMSMs:

• the difference between permeances in the air-gap in the d and q magnetic axis

produces reluctance torque,

• cogging effect is the interaction between rotor magnetic flux produced by

permanent magnets and variable permeance of the air-gap due to stator slots

and produces cogging torque,

• distortion of sinusoidal distribution of the magnetic flux density in the air-gap

produces field harmonic electromagnetic torque.

Rotor Design Techniques for Cogging Torque Reduction

Elimination of cogging torque using available design techniques, while keeping the

output torque at the same level is a challenging task. At the beginning the proper

slot/pole combination must be selected for effective reduction of cogging torque. Be-

side this, for a given motor, the following rotor design techniques can be considered

respectively or in any mutual combination: magnet span variation, magnet pole shift-

ing, magnet shape and magnetization pattern optimisation, and magnet step skew or

skewing.

Because of the complexity of the given task, the full parametric FEM model of

36-slot and 6-pole PMSM was chosen. Due to slot/pole combination basic design will

express a significant cogging torque, thus different rotor design techniques for cogging

torque reduction can be studied and evaluated.

Selection of Magnet Span with Magnet Pole Shifting

Among several possibilities an effective way to reduce cogging torque, while maintain-

ing the output torque, is to optimize the magnet span αm and shift magnetic pole angle

γ, as shown in Fig. 2. To simplify the representation and enhance clearness only curves

of maximal cogging torque Tcog max

versus magnetic pole shift angle γ for several mag-

net spans αm are shown in Fig. 3. The minimal cogging torque values appear around the

shift angle γ = 56° and are two to three times smaller than in the case of symmetrical

magnetic pole distribution at γ = 60°. The influence of magnet span αm on cogging

torque values can also be observed and it is obviously that the minimum appears at

αm

= 50°, thus optimizing of the PM motor design is an iterative process with many

variable parameters.

Figure 1. The instantaneous torque of a PMSM motor T(α).

A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors180

Figure 2. Principle of magnetic poles shifting γ ≠ δ (left), and 2D FEM model of PM rotor (right).

Figure 3. Maximal cogging torque versus magnet pole shift angle γ.

Shape and Magnetization Pattern of Permanent Magnets

Air-gap flux density distribution is strongly dictated by the shape and the magnetiza-

tion pattern of applied permanent magnets. Furthermore this has a substantial influence

on cogging torque, harmonic contents and magnetic saturation. Figure 4 shows various

shapes and magnetization patterns of arc magnets usually used for surface mounted

PMSMs.

The influence of shape and magnetization pattern on the air-gap flux densitiy dis-

tribution is presented in Fig. 5. Notice that a bread loaf (lens-shaped) magnet shape

flux densitiy distribution is the closest to the sinusoidal distribution.

The analysis of shape and magnetization pattern of permanent magnets on cogging

torque was carried out for all three shapes of PMs. Using a bread loaf magnet shape a

minimal value of cogging torque compared to the constant component of the output

torque T0 is achieved, but the output torque T

0 is also considerable reduced in respect to

the initial basic PMSP model, as presented in Fig. 6. Shifted magnet poles with selec-

tion of magnet span at the same time results in efficient reduction of cogging torque

while keeping the level of the output torque T0 high (Table 1). Such approach repre-

sents more efficient solution in cogging torque minimization.

A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors 181

Figure 4. Various shapes and magnetization patterns of PM: surface radial (left), surface parallel (middle),

and bread loaf (right).

Figure 5. Flux densitiy along the centre of the air-gap.

Figure 6. Maximal cogging torque and output torque for various shapes and magnetization patterns of PMs

versus magnet span αm.

Conclusion

Numerous parametric FEM calculations and laboratory measurements have proved that

majority of rotor design techniques are very efficient in reducing the parasitic cogging

A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors182

torque in PMSMs. If they are combined with several additional stator design techniques

(optimization of slot openings, additional notches in stator teeth), the decreasing of the

cogging torque Tcog max

could be even improved, while maintaining the output torque T0

of the PMSM at the same level. Presented ascertainments are already brought into use

in mass-production of PMSM motors for high demanding special applications.

References

[1] J. Gieras, M. Wing, Permanent Magnet Motor Technology, New York, 1997, Marcel Dekker Incorpora-

tion.

[2] N. Bianchi, S. Bolognani, Design techniques for reducing the cogging torque in surface-mounted PM

motors, IEEE Trans. Industry Application, Vol. 38, No. 5, September/October 2002, pp. 1259–1265.

[3] R. Lateb, N. Takorabet, F. M. Tabar, J. Enon, A. Sarribouete, Design technique for reducing the cogging

torque in large surface mounted magnet motors, ICEM 2004 International Conference on Electrical Ma-

chines, Proceedings ICEM 2004 CD-ROM, Krakow, Poland, 5-8 Sept 2004.

[4] M.S. Islam, S. Mir, T. Sebastian, Issues in Reducing the Cogging Torque of Mass-Produced Permanent-

Magnet Brushless CD Motor, IEEE Tran. Industry Application, Vol. 40, No. 3, May/June 2004, pp. 813-

820.

[5] M. Furlan, A. Černigoj, M. Boltežar, A coupled electromagnetic-mechanical-acoustic model of a DC

electric motor, Compel, Vol. 22, No. 4, 2003, pp. 1155-1165.

Table 1. Gathered results of rotor design techniques

Design technique Tcog max (Nm) Tcog max / T0 T0 / T0 basic mod.

Basic PMSM model (αm = 56°) 4,42 17,7% 100%

Optimal magnet span (αm = 40°) 2,10 9,4% 89%

Magnet pole shifting with magnet span (γ = 56°, αm = 50°) 0,99 4,0% 98%

Bread loaf magnet shape 0,024 0,11% 89%

A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors 183

Optimum Design of Linear Motor for

Weight Reduction Using Response Surface

Methodology

Do-Kwan HONG, Byung-Chul WOO, and Do-Hyun KANG

Korea Electrotechonolgy Research Institute, P. O. Box 20, Changwon, 641-120, Korea

Tel: +82-55-280-1395, Fax: +82-55-280-1547, E-mail: [email protected]

Abstract. This paper presents an optimum design procedure of linear motors to

reduce the weight of the machines with the constraints of thrust and detent force

using response surface methodology (RSM). RSM is well adopted to make the

analytical model of the minimum weight with constraints of thrust and detent force,

and it enables objective functions to be easily created, and a great deal of computa-

tion time can be saved. Therefore, it is expected that the proposed optimization

procedure using RSM can be easily utilized to solve the optimization problem of

the linear motors.

Introduction

In many applications were the motion is essentially linear. It is possilble to use linear

motors instead of rotary motors. Linear motors are electromagnetic devices developing

mechanical thrust without mechanical slider-crank system mechanism. Advantages of

the linear motors include low noise, reduced operating cost, and incearsed flexibility of

operation due to gearless feature [1]. The linear motors, however, have some practical

limitations. One of the major reasons of the limitations is that inherently large air gap

causes low power density. In order to increase the power density, permanent magnet

(PM) type longitudinal flux linear motors (LFLMs) can be considered for the applica-

tion of the linear motors in high power density systems. Since LFLMs can produce

high magnetic thrust and reluctance thrust with relatively small air gap. There are sev-

eral practical examples of LFLM in [2,3]. In this paper we consider the development of

a LFLM for use in linear compressor applications. Figure 1 shows the structure of the

developed LFLM. It has two important electromechanical components, a linear actua-

tor and springs for refrigeration application. For short stroke applications like in this

situation in odrer to recover the energy at the end of the displacement mechanical

springs are used. By controlling the operating frequency of the actuator around me-

chanical resonance frequency, the system has higher efficiency than conventional ro-

tary type compressors.

In order to increase the performance of the system it was mandatory to consider a

method of optimization. RSM is recently receiving attention for its modeling ability of

electromagnetic devices performance by using statistical fitting method, given that

RSM is well adopted to develop analytical models for the complex problems. With this

analytical model, an objective function with constraints can be easily created, and

computation time can be saved.



doi:10.3233/978-1-58603-895-3-184

184

This paper presents the optimum design of longitudinal flux actuator for linear

compressor using RSM. Its design goal is to reduce the weight of the machine with the

constraints of thrust and detent force with respect to the initial machine. At first step,

most influential design variables and their levels should be determined and be arranged

in a table of orthogonal array. Each response value is determined by 3D finite element

method (FEM). With the use of reduced gradient algorithm (RGA), finally, the most

desired set is determined, and the influence of each design variables on the objective

function can be obtained. The weight can be reduced by 10.09%, thrust force improved

by 3.06% and detent force improved by 4.15% of initially designed PM type LFLM.

Optimum Design for Longitudinal Flux Linear Motor (LFLM)

RSM Method

RSM seeks for the relationship between design variables and response through statisti-

cal fitting method. A polynomial approximation model is commonly used for a second-

order fitted response (u) and can be written as follow

2

0

k k k

j jj jj j ij i j

j l j i j l

u x x x xβ β β β ε

= = =

= + + + +∑ ∑ ∑ (1)

:β regression coefficients, :x design varaibles, :ε random error, :k number of de-

sign variables.

The least squares method is used to estimate unknown coefficients. Matrix nota-

tions of the fitted coefficients and the fitted response model can be written as:

1ˆ( )X X X uβ

−

′ ′= ˆ

ˆu X β= (2)

It should be evaluated at the data points. ˆ

β , where ˆ

β is the vector of the unknown

coefficients which are usually estimated to minimize the sum of the squares of the error

term. RSM method is applied in connection with FEM and the response actually repre-

sent FEM output values. Figure 2 presents the principal steps of RSM procedure.

Figure 1. PM type LFLM for compressor.

D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction 185

Figure 2. RSM.

0 10 20 30 40 50 60

0.0

0.5

1.0

1.5

2.0

2.5

Laminated steel

S23

H (kA/m)

B (T

)

Figure 3. B-H curves of the used materials.

Table 1. Specifications of LFLM analysis model

Item Unit Specification

Stator/Iron material S23

Permanent Magnet NdFe35H

Nominal air gap mm 0.5

Nominal current A 1

MMF AT(Ampere Turn) 600

Design Variables and Levels

Table 1 shows the specification of the PM type LFLM for a compressor. Figure 3

shows the B-H characteristic of the S23 material which was used for the active parts of

the LFLM. The variables represent geometrical dimentsions which are completely de-

termining the geometry of the LFLM no material variable was considered. Figure 4

shows the design variable of the PM type LFLM. Table 2 shows the design variable

D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction186

and level. The table of orthogonal array is shown in Table 3 and corresponding simula-

tion result (nominal current value 600 AT). The table of orthogonal array is determined

by considering the number of design variables and each level of them. After obtaining

the experimental data from 2D FEM, the function necessary to draw response surface is

extracted. In order to determine the equations of the response surface, several experi-

mental designs are developed to establish the approximate equation using the smallest

number of experiments. The first level of DV6 and the second level of the other design

varaibles have the values of the initially designed LFLM in Table 2.

Optimum Design Result

Table 2 represents the mixed orthogonal array, which is determined by considering the

number of the design variables and each level of them. After getting the experimental

data by 2D FEM, the function to draw response surface is extracted. In order to deter-

mine the equations of the response surface, several experimental designs are developed

to establish the approximate equation using the smallest number of experiments. The

purpose of this paper is to minimize the object function (weight) with constraints of

thrust and detent force. Table 3 and Table 4 show optimum solution and comparison

result of initial model and optimum model, respectively. The two fitted second order

polynomial of the object functions for the seven design variables are as follows.

2 2

2 2 2 2 2

1.44767 0.078 1 0.07477 2 0.01819 3 0.05701 4

0.08289 5 0.006 6 0.00346 7 0.00091 1 0.000465 2

0.00196 3 0.0024 4 0.0006 5 0.00335 6 0.00129 7

Weight DV DV DV DV

DV DV DV DV DV

DV DV DV DV DV

= + + + +

+ − + − +

+ + − + +

(3)

Figure 4. Design variable of PM type LFLM.

Table 2. Design variable and level

Design Variable

Level

DV1 DV2 DV3 DV4 DV5 DV6 DV7

1(–1) 8.449 5.95 1.9975 4.25 5.95 1.833 16.15

2 (0) 9.94 7 2.35 5 7 3.666 19

3 (1) 11.431 8.05 2.7025 5.75 8.05 5.499 21.85


2 2

2 2 2 2 2

57.8049 4.6812 1 3.2581 2 3.5414 3 0.1728 4

0.0871 5 0.196 6 0.5666 7 0.0336 1 0.0492 2

0.2469 3 0.1479 4 0.0906 5 0.2294 6 0.0434 7

Thrust

F DV DV DV DV

DV DV DV DV DV

DV DV DV DV DV

= + + + −

+ + − + −

− − + + −

(4)

2 2

2 2 2

0.38872 0.09841 1 0.03092 2 0.07204 3 0.03208 4

0.03093 5 0.12183 6 0.12523 7 0.05814 1 0.03928 4

0.03498 5 0.03957 6 0.06569 7

Detent

F DV DV DV DV

DV DV DV DV DV

DV DV DV

= + + + −

− + − + −

− + +

(5)

The adjusted coefficients of the multiple determination R2

adj for three responses are

weight (99.6%), FThrust

(99.9%) and FDetent

(98.1%). In Table 3 and 4, the optimal point

is searched to find the point of less than 10.09% of the weight, 4.15% of detent force,

and greater than 3.06% of the thrust force of the initially model. Table 4 and Table 5

show the optimum solution and the comparison result of initial model and optimum

model, respectively. Figure 5 shows the 2D flux line and flux density of the initially

model. Pareto chart of thrust force, detent force and weight are presented in Fig. 6.

Also from this figure the most sensitive design variables can be identified. The re-

sponse surface of thrust force according to change of the most influential design vari-

ables are shown in Fig. 7. Figure 8 shows an interaction plot of means for thrust force

between design variables. Figure 9 shows a comparison between initial and optimum

model. The weight can be reduced 10.09%, thrust force improved by 3.06% and detent

force improved by 4.15% of the initially designed PM type LFLM. Figure 10 shows the

detent force and thrust force profile of the optimum model by mmf.

Table 3. Table of mixed orthogonal array L18

(21

×37

)

Exp. DV1 DV2 DV3 DV4 DV5 DV6 DV7

Avg. thrust

force(N)

Max.detent

force(N)

Weight

(kg)

1 8.449 5.95 1.9975 4.25 5.95 1.833 16.15 46.72 0.33 1.16

2 8.449 7 2.35 5 7 3.666 19 53.23 0.33 1.37

3 8.449 8.05 2.7025 5.75 8.05 5.499 21.85 59.41 0.4 1.62

4 9.94 5.95 1.9975 5 7 5.499 21.85 50.60 0.37 1.36

5 9.94 7 2.35 5.75 3 1.833 16.15 58.30 0.34 1.6

6 9.94 8.05 3 4.25 5.95 3.666 19 64.42 0.46 1.41

7 11.431 5.95 2.35 4.25 8.05 3.666 21.85 58.80 0.39 1.49

8 11.431 7 2.7025 5 5.95 5.499 16.15 66.76 0.97 1.46

9 11.431 8.05 1.9975 5.75 7 1.833 19 61.65 0.36 1.66

10 8.449 5.95 2.7025 5.75 7 3.666 16.15 53.27 0.53 1.37

11 8.449 7 1.9975 4.25 8.05 5.499 19 49.92 0.38 1.37

12 8.449 8.05 2.35 5 5.95 1.833 21.85 55.72 0.25 1.37

13 9.94 5.95 2.35 5.75 5.95 5.499 19 54.53 0.46 1.35

14 9.94 7 2.7025 4.25 7 1.833 21.85 60.47 0.33 1.43

15 9.94 8.05 1.9975 5 8.05 3.666 16.15 57.85 0.49 1.59

16 11.431 5.95 2.7025 5 8.05 1.833 19 62.72 0.43 1.57

17 11.431 7 1.9975 5.75 5.95 3.666 21.85 57.81 0.33 1.49

18 11.431 8.05 2.35 4.25 7 5.499 16.15 66.71 0.91 1.54

Table 4. Optimum level and optimum size

Design variable

Level

DV1 DV2 DV3 DV4 DV5 DV6 DV7

Optimum level 0.508 –1 1 –1 –1 –1 0.4116

Optimum size 10.70 5.95 2.703 4.25 5.95 1.833 20.173


Table 5. Comparison of initial model and optimum model

Model

Weight

(kg)

Avg.

Thrust force (N)

Peak.

Detent force (N)

Initial 1.4575 57.8900 0.3594

Optimum (predicted RSM) 1.3099 60.0000 0.3609

2D FEM (optimum simulation) 1.3104 59.6664 0.3445

Variation between initial

and 2D FEM model (%)

–10.09 3.06 –4.15

(a) Flux line (b) Flux density

Figure 5. Flux density and flux line in LFLM using FEM (initial model).

(a) Thrust force (b) Weight

(c) Detent force

Figure 6. Pareto chart of thrust force(Avg.), detent force and weight.


Figure 7. Surface plot of thrust force(Avg.).

Figure 8. Interaction plot of means for thrust force (avg).

-3 -2 -1 0 1 2 3

0

10

20

30

40

50

60

optimum

initial

thrust

Initial (peak) : 0.3594 N

Optimum (avg.) : 59.6664 N

Optimum (peak) : 0.3445 N

Deten

t, thrust fo

rce (N

)

Position (mm)

MMF (AT) : 0 0 600 600

Initial (avg.) : 57.89 N

detent

-3 -2 -1 0 1 2 3

0

20

40

60

80

100

120

Deten

t, th

ru

st fo

rce (N

)

Position (mm)

MMF(AT) : 0 300 600 900 1200

Optimum model

Figure 9. Comparison between initial and optimum

model.

Figure 10. Detent and thrust force profile.


Conclusion

In this paper, an optimum design procedure is introduced to design LFLM to reduce its

weight and to improve thrust force and detent force of the initially designed LFLM

with many shape design variables, Also, Optimization design by RSM and table of

orthogonal array are presented in detail for the LFLM in this paper. The performance of

optimized LFLM is improved as compared with the initial model. Based on this

method the weight of optimized LFLM is reduced by 10%. Therefore, when this pro-

posed approach is applied, it can efficiently raise the precision of optimization and re-

duce the number of iterations of experiments in the optimization design by RSM.

References

[1] H. Lee, S. S. Jeong, C. W. Lee and H. K. Lee, “Linear Compressor for Air-Conditioner,” International

Compressor Engineering Conference at Purdue, pp. 1-7, 2004.

[2] T. Mizuno, M. Kawai, F. Tsuchiya, M. Kosugi, and H. Yamada, “An Examination for Increasing the Mo-

tor Constant of a Cylindrical Moving Magnet-Type Linear Actuator,” IEEE Trans. Magn., Vol. 41,

No. 10, pp. 3976-3978, October, 2005.

[3] P. Zheng, A. Chen, P. Thelin, W. M. Arshad, and C. Sadaranani, “Research on a Tubular Longitudinal

Flux PM Linear Generator Used for Free-Piston Energy Converter,” IEEE Trans. Magn., Vol. 43, No. 1,

pp. 447-449, January, 2007.


Analytical Evaluation of Flux-Linkages

and Electromotive Forces in Synchronous

Machines Considering Slotting, Saliency

and Saturation Effects

Antonino DI GERLANDO, Gianmaria FOGLIA and Roberto PERINI

Dipartimento di Elettrotecnica – Politecnico di Milano, Piazza Leonardo da Vinci,

32 – 20133 Milano, Italy

[antonino.digerlando, gianmaria.foglia, roberto.perini]@etec.polimi.it

Abstract. An analytical approach to the evaluation of flux linkages and e.m.f.s in

salient-pole synchronous machines is developed, capable to accurately allow for

the actual winding structure, the stator and rotor air-gap geometry (slotting and sa-

liency), under any saturated, steady-state or transient operating condition.

The method is based on a Park d-q decomposition of the stator m.m.f. distri-

bution (preserving the spatial harmonics) and on the use of FEM identified satura-

tion functions.

A relevant feature is that the self and mutual inductances, evaluated in un-

saturated conditions, are simply corrected by using the saturation functions. Sev-

eral transient FEM simulations validate the method.

Introduction

An analytical procedure for the evaluation of the air-gap field, of the inductances, of

the e.m.f.s and of the electromagnetic torque of salient-pole, three-phase synchronous

machines was previously developed, considering anisotropy, slotting and actual wind-

ing structure [1–3]: the procedure showed good accuracy features, but it was affected

by the significant limitation of the operation in unsaturated conditions.

Starting from that theory, the method is here extended to any saturated operating

condition: this generalisation is based on a d-q approach to the calculation of the air-

gap field distribution, made possible by the use of FEM identified saturation func-

tions [4] and by a d-q decomposition of the stator m.m.f. distribution [5].

In this paper, the expression of the coil and of the phase flux linkage and e.m.f. is

obtained, while the electromagnetic torque evaluation is performed in another pa-

per [6].

Based on the developed method, various simulations have been carried out, in dif-

ferent operating conditions, comparing the results with those obtained by correspond-

ing transient FEM calculations [7]: the accuracy level is investigated and discussed, for

the validation of the described analytical method.

In the model, for now the presence of the rotor damper cage is not considered.



doi:10.3233/978-1-58603-895-3-192

192

The d-q Decomposition of m.m.f. and Flux-Density Air-Gap Distributions

As more completely described and illustrated in [5], by using the Park transformation,

each stator phase current can be expressed in terms of phase d, q instantaneous compo-

nents:

( ) ( ) ( )k kd kqi t i t i t= + k = 1s, 2s, 3s. (1)

The meaning of (1) is resumed by the flow-chart of Fig. 1.

Thus, being ξ the generic angular position along the stator, the instantaneous

m.m.f. distribution ms(ξ, i

1s(t), i

2s(t), i

3s(t)) produced by the stator three-phase winding

actual structure [1] can be decomposed as follows:

( ) ( ) ( )( ) ( ) ( )

( ) ( )

s 1s 2s 3s s 1sd 2sd 3sd s 1sq 2sq 3sq

sd sq

m ,i t , i t , i t m ,i , i , i m ,i , i , i

m , t m , t .

ξ = ξ + ξ =

= ξ + ξ

(2)

As can be verified, the msd

(t) and msq

(t) distributions, produced by the d-axis and

q-axis instantaneous current terns, act along the d, q axis respectively, showing the

same stepped waveform features of the total m.m.f. [5].

Therefore, the model described by (1) and (2) represents a more general transfor-

mation than the classical Park one, usually applied to decompose sinusoidally distrib-

uted m.m.f.s only.

As shown in [5], the instantaneous saturation functions are expressed by

( ) ( ) ( )( )d d r d

t i t , i tσ = σ , ( ) ( ) ( )( )q q u qt i t , i tσ = σ , (3)

where: id and i

q are 1/ 3 times the current Park vector components i

Pd, i

Pq; i

r is the

rotor current; iu is an equivalent d-axis current corresponding to the resultant d-axis

m.m.f. mu:

( ) ( ) ( )u u f r Pd di t m N i t i t= = + α ⋅ , (4)

αPd

is the classical d-axis Potier coefficient, Nf is the field winding turn number per

pole.

Called ζ = ζ(t) the rotor angular mechanical position, mr(ξ − ζ(t), i

r(t)) the rotor

m.m.f. distribution, βs and β

r the stator slotting and rotor saliency functions respec-

Stator phase

global currents:

i1s(t), i2s(t), i3s(t)

Park

transformation

on the rotating

frame

Park

current

components:

iPd(t)

iPq(t)

Distinct anti-

transformation of the

Park components

Stator phase

d, q current

components:

i1sd(t), i2sd(t), i3sd(t)

i1sq(t), i2sq(t), i3sq(t)

Figure 1. Flow chart illustrating the d-q decomposition of the stator phase currents.

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces 193

tively [1], the air-gap flux density distribution (radial component, measured along the

circumference at half the minimum air-gap width [5]) can be expressed as follows in

saturated operation:

( )( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )

( )( ) ( )( )

o s r

d sd r r q sq

d d.ns q q.ns

d q

b , t , t g t

t m , t m t ,i t t m , t

t b , t , t t b , t , t

b , t , t b , t , t .

ξ ζ = μ ⋅β ξ ⋅β ξ − ζ ⋅

⋅ σ ⋅ ξ + ξ − ζ + σ ⋅ ξ =

= σ ⋅ ξ ζ + σ ⋅ ξ ζ =

= ξ ζ + ξ ζ

(5)

where bd.ns

(ξ, ζ(t), t) and bq.ns

(ξ, ζ(t), t) are the non-saturated d and q flux density field

distributions.

As can be observed in the first formulation of (5), the saturation functions are

originally applied to the axis m.m.f. distributions: in fact, they can be interpreted as the

factors that, along each axis, express the ratio between the distribution of the magnetic

potential difference (m.p.d.) at the air-gap and the distribution of the m.m.f.

The second line formulation of (5) shows that the d and q axis flux density distri-

butions can be evaluated as the unsaturated ones, times the corresponding saturation

functions.

The Development of the Flux-Linkage Expression in Saturated Conditions

The flux linkage ψk of the phase k includes the main flux linkage (ψ

mk) and the leakage

flux linkage (ψℓk

):

( ) ( ) ( )k mk k

t t tψ = ψ +ψ . (6)

As regards the leakage flux linkage ψℓk

, its rigorous analytical evaluation is diffi-

cult to be performed, as like as its FEM evaluation [3].

However, for the leakage model, here the well known assumptions of the classical

theory will be adopted:

− the leakage flux paths exhibit wide portions developing in air: thus, leakage

can be considered as a substantially unsaturated phenomenon;

− the leakage flux linkage is assumed to be independent on the rotor position.

Considering the typical structure of the stator winding, in which the shorted coil

pitch is usually employed, in general the following constructional properties occur:

some slots contain conductors of different phases; moreover, some end-windings of

different phases are close each others. As a consequence, the leakage flux linkage ex-

hibits self and mutual terms, all due to the stator currents only; however, considering

the three-phase structural symmetry, the phase self leakage inductances are all equal

(Lℓself

), as like as the phase mutual leakage inductances (Lℓmutual

). Therefore, the leakage

flux linkage of the phase 1s can be written as follows:

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces194

( ) ( ) ( ) ( )

( ) ( ) ( )

1s self 1s mutual 2s mutual 3s

self mutual 1s s 1s

t L i t L i t L i t

L 2 L i t L i t ,

ψ = ⋅ + ⋅ + ⋅ =

= − ⋅ ⋅ = ⋅

(7)

where the “service” equivalent leakage inductance Lℓs

has been introduced, thanks to

the link:

( ) ( ) ( )1s 2s 3si t i t i t 0+ + = , (8)

consequence of the hypothesis of the stator winding insulated neutral point. Of course,

the leakage flux linkages of the other two phases have expressions similar to (7): ψℓk=

Lℓs⋅i

k, (k = 1s, 2s, 3s).

As regards the main flux linkage ψmk

, it can be expressed as a function of the teeth

fluxes. From the expression (5) of the flux density distribution, the tooth flux in the j-th

stator tooth [2] can be expressed as follows:

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

fj

j

ij

fj fj

ij ij

j j

t

d d.ns q q.ns

d td.ns q tq.ns

, t R b , , t d

R t b , , t d t b , , t d

t , t t , t ,

ξ

ξ

ξ ξ

ξ ξ

ϕ ζ = ⋅ ⋅ ξ ζ ⋅ ξ =

⎛ ⎞= ⋅ ⋅ σ ⋅ ξ ζ ⋅ ξ + σ ⋅ ξ ζ ⋅ ξ =⎜ ⎟

⎝ ⎠

= σ ⋅ϕ ζ +σ ⋅ϕ ζ

∫

∫ ∫

(9)

where ξij and ξ

fj are the j-th tooth initial and final angular positions (along the axes of

the slots adjacent to the j-th tooth itself), ℓ the lamination stack length, R the average

radius at the minimum air-gap (along the d axis).

Equation (9) shows that also the tooth flux can be expressed as the sum of the un-

saturated d and q axis components, ϕtd.nsj

(ζ(t), t) and ϕtq.nsj

(ζ(t), t), each multiplied by

the corresponding saturation function.

The main flux linkage ψmk

of the phase k can be expressed as a function of the

tooth fluxes:

( ) ( )j

mk j jk t

t tψ = Σ Γ ⋅ϕ ; (10)

in (10) Γjk

is an integer number, called linkage coefficient [2]: the value of Γjk

ex-

presses how many times the j-th tooth flux is linked with the k-th phase; the sign of Γjk

depends on the winding direction of the phase winding coils, compared with the posi-

tive radial direction adopted for the generic tooth flux.

Thanks to (9), the main phase flux linkage can be considered as the sum of the d

and q components; moreover, each axis main flux linkage term can be expressed as the

corresponding unsaturated main flux linkage component, multiplied by the suited satu-

ration function; therefore, from (10) it follows:

( ) ( ) ( ) ( ) ( ) ( ) ( )mk mdk mqk d md.ns.k q mq.ns.k

t t t t t t tψ = ψ +ψ = σ ⋅ψ +σ ⋅ψ . (11)


On the other hand, each non-saturated main flux linkage component (ψmd.ns.k

(t) and

ψmq.ns.k

(t)) can be expressed as proportional to the corresponding current components

(i1sd

, i2sd

, i3sd

, ir and i

1sq, i

2sq, i

3sq respectively), multiplied by the non saturated self and

mutual inductance functions [2]:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

md.ns.k kr r kv vd

v

mq.ns.k kv vq

v

t L i t L i t ;

t L i t ; v, k = 1s, 2s, 3s.

ψ = ζ ⋅ + ζ ⋅

ψ = ζ ⋅

∑

∑

(12)

The expressions of ψmd.ns.k

(t) and ψmq.ns.k

(t) of (12) can be considered as the limit of

the main flux components ψmdk

(t) and ψmqk

(t) given by (11), when the saturation is neg-

ligible (i.e., when σd, σ

q → 1).

Considering (7) and (12), (6) becomes (again with: v, k = 1s, 2s, 3s):

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

k d kr r kv vd

v

q kv vq s k

v

t t L i t L i t

t L i t L i t .

⎛ ⎞

⎜ ⎟ψ = σ ⋅ ζ ⋅ + ζ ⋅ +⎜ ⎟⎝ ⎠

⎛ ⎞

⎜ ⎟+ σ ⋅ ζ ⋅ + ⋅⎜ ⎟⎝ ⎠

∑

∑

(13)

The structure of (13) confirms the previous remarks concerning the possibility to

express a saturated quantity as the product between the unsaturated one, times a satura-

tion function; however, in this case, this property is particularly important, because it

allows to evaluate the non saturated self and mutual inductance functions once and for

all, as a function of the air-gap geometry and of the rotor position only, regardless of

the current amplitude.

Considering the heaviness of the inductances and inductance derivatives calcula-

tion process [2], the cited property is crucial for an acceptable application of the ma-

chine analytical model in saturated operating conditions.

In order to illustrate the accuracy of the described flux linkage evaluation method,

Fig. 2 shows the phase flux-linkage waveforms, in ideally unsaturated and saturated

conditions, in steady-state, loaded, balanced operation, with d and q reaction compo-

nents, for the 8-pole, fractional slot winding machine considered in [4] (waveforms

evaluated analytically by (13) and by FEM transient simulations [7]). As can be seen,

the calculation accuracy in saturated conditions is similar to that obtained in unsatu-

rated ones, thus confirming the soundness of the saturation model: of course, the filter-

ing effect of the winding distribution makes the waveforms almost sinusoidal.

If the effect of saturation on more distorted quantities is to be examined, it is useful

to analyse the coil flux linkage, shown in Fig. 3: in it, the waveforms correspond to

those of Fig. 2, for the same machine and in the same loaded condition: as can be ob-

served, the agreement between analytical and FEM results remains acceptable.

The steady-state operating conditions of Figs 2 and 3 correspond to constant values

σd and σ

q in (13).

In order to evaluate the soundness of the analytical method in all the d and q axis

saturation conditions, the most suited way is to consider an holding torque test; in this


condition, all the rotor and stator currents are maintained constant (Ir, I

1s, I

2s, I

3s) and

the rotor is supposed to be driven at constant rotational speed Ω. So, the stator current

Park vector has a constant amplitude, but its d, q components Id and I

q change with the

rotor positions, in such a way to produce all the possible reaction situations (magnetiz-

ing (Id > 0), demagnetizing (I

d < 0), generating (I

q < 0) and motoring (I

q > 0) condi-

tions). Such operating condition should be considered as a virtual test, because difficult

to be actually performed in case of large rating machines. The simulated results of this

test are shown in Figs 4, 5, for the coil and phase winding flux linkage respectively: as

can be observed, the correctness of (13) is confirmed. It should be noted the important

flux linkage reduction due to saturation, that is unsymmetrical, depending on the mag-

netizing or demagnetizing reaction.

0 2 4 6 8 10 12 14 16 18 20

−6

−4

−2

ψph.ns

ψph

t [ms]

[Wb]

6

4

2

0

0 2 4 6 8 10 12 14 16 18 20

−0.20

−0.15

−0.10

−0.50

0

0.50

0.10

0.15

0.20

[Wb]

ψc

ψc.ns

t [ms]

Figure 2. Phase flux linkage at load (machine data:

[4]): ψph.ns = unsaturated operation; ψph = saturated

operation; solid lines = analytical; dotted lines =

FEM.

Figure 3. Coil flux linkage waveforms; same ma-

chine and loaded operating conditions as in Fig. 2;

solid lines = analytical results; dotted lines = FEM

results.

0 5 10 15 20

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

ψc.ht

[Wb]

t [ms]

ψc.ht.ns

0 5 10 15 20

−14

−12

−10

−8

−6

−4

−2

0

2

4

6

[Wb]

ψph.ht

ψph.ht.ns

t [ms]

Figure 4. Coil flux linkage, in ideally unsaturated and

in saturated operation, during rotation with constant ir,

is1, is2, is3 values (holding torque test operation): solid

line = analytical results; dotted line = FEM results.

Figure 5. Phase flux linkage, in ideally unsaturated

and in saturated operation, in the same conditions

of Fig. 4 (holding torque test operation): solid



The Development of the e.m.f. Expression in Saturated Conditions

By performing the time derivative of (13), and posing Ω = dζ/dt, it follows:

( )

( )

( )

k k kt ks d kr r kv vdv

q kv vq d kr r kv vdv v

q kv vq s k d kv vd kr rv v

q kv vqv

e e e e dL d i dL d i

dL d i L di dt L di dt

L di dt L di dt d dt L i L i

d dt L i v,k 1s,2s,3s.

Ω= + + = Ω⋅σ ⋅ ζ ⋅ + ζ ⋅ +

+Ω⋅σ ⋅ ζ ⋅ + σ ⋅ ⋅ + ⋅ +

+ σ ⋅ ⋅ + ⋅ + σ ⋅ ⋅ + ⋅ +

+ σ ⋅ ⋅ =

∑

∑ ∑

∑ ∑

∑

(14)

As can be noted, in addition to the two classical components (the speed e.m.f.

ekΩ

(t) and the transformer e.m.f. ekt

(t)), a third term arises (that can be called “satura-

tion” e.m.f.), eks

(t), proportional to dσd/dt and to dσ

q/dt.

It is interesting to analyze some particular operating conditions, again for coil and

phase winding waveforms.

Figures 6 and 7 refer to the steady-state, no-load, operation, with constant values

of the speed Ω and of the rotor current Ir; in this condition, (14) reduces to the follow-

ing no-load speed term:

( )o o

k k d r kr re e I ,0 dL d I

Ω= = Ω⋅σ ⋅ ζ ⋅ . (15)

The following remarks can be made:

0 5 10 15 20

−60

−40

−20

0

20

40

60

ec0

ec0.ns

[V]

t [ms]

0 5 10 15 20

−1.5

−1.0

−0.5

0

0.5

1.0

1.5

t [ms]

[kV]

eph0.ns

eph0

Figure 6. Coil e.m.f. in no-load operation: ec0.ns =

non saturated core; ec0 = saturated core; solid lines =

analytical results; dotted lines = FEM results.

Figure 7. Phase e.m.f. in no-load operation: eph0.ns =

non saturated core; eph0 = saturated core; solid lines =



− as expected, the phase winding waveform is less distorted than the coil one,

thanks to the filtering effect of the adopted winding structure (fractional slot,

with shorted pitch coils);

− the waveform analytical reproduction is accurate, both in unsaturated and in

saturated operation.

In steady state, loaded operation, the field current Ir is constant, while the stator

currents i1s

, i2s

, i3s

are balanced, sinusoidally time dependent quantities; considering that

the d, q Park current components have constant values (Id, I

q), σ

d and σ

q can be evalu-

ated once and for all; from (14), we obtain (again with v, k = 1s, 2s, 3s):

( ) ( )

( ) ( ) ( )

( )

loadk d r d kr r kv vd

v

q u q kv vq d r d kv vdv v

q u q kv vq s kv

e I , I dL d I dL d i

I , I dL d i I , I L di dt

I , I L di dt L di dt .

= Ω⋅σ ⋅ ζ ⋅ + ζ ⋅ +

+ Ω⋅σ ⋅ ζ ⋅ + σ ⋅ ⋅ +

+ σ ⋅ ⋅ + ⋅

∑

∑ ∑

∑

(16)

Figures 8 and 9 show the coil and phase winding e.m.f.s respectively, in steady

state, loaded, balanced operation, both in ideally unsaturated conditions and in satu-

rated ones; the following remarks can be proposed:

− the saturation reduces significantly the waveforms amplitude, roughly main-

taining their shape;

− a good agreement can be observed between analytically evaluated waveforms

and FEM calculated ones, both in ideally unsaturated conditions and in satu-

rated operation;

− the coil e.m.f. waveforms are highly distorted, because of the loaded opera-

tion;

− the phase winding e.m.f. waveforms are less distorted, thanks to the filtering

effect due to the winding factor;

− however, the armature reaction affects also the distortion level of the phase

e.m.f. waveform, mainly because of the third harmonic components; in fact,

the line to line e.m.f. appears more sinusoidal.

The last operating condition here considered is the holding torque test, already ex-

amined in Figs 4 and 5 as concerns the coil and phase flux linkages; here, the coil and

phase e.m.f.s are analyzed, both in ideally unsaturated operation and in saturated condi-

tions. Considering that, in this case, the time derivatives of the currents are zero,

(14) reduces to:

( )( ) ( )

( ) ( )( )

( )

ht ht htk k ks d r d kr r kv vd

v

q u q kv vqv

d kv vd kr r q kv vqv v

e e e I , I t dL d I dL d I

I t , I t dL d i

d dt L I L I d dt L I ; k, v 1s, 2s,3s

Ω= + = Ω⋅σ ⋅ ζ ⋅ + ζ ⋅ +

+Ω⋅σ ⋅ ζ ⋅ +

+ σ ⋅ ⋅ + ⋅ + σ ⋅ ⋅ =

∑

∑

∑ ∑

(17)


Figures 10 and 11 show the coil and phase winding e.m.f.s corresponding to the

holding torque test, evaluated in ideally unsaturated conditions (i.e., when σd, σ

q→ 1)

and taking into account the saturation, on the basis of (17); the following remarks are

valid:

− the analytically evaluated waveforms are fairly similar to those evaluated by

FEM transient simulations;

− the saturation has an important influence on the waveforms amplitude and

shape;

0 5 10 15 20

−80

−60

−40

−20

0

20

40

60

80

t [ms]

ec

ec.ns

[V]

0 5 10 15 20

−2.0

−1.5

−1.0

−0.5

0

0.5

1.0

1.5

2.0

t [ms]

[kV]

eph.ns

eph

Figure 8. Coil e.m.f. in loaded operation: ec.ns = non

saturated core; ec = saturated core; solid lines =


Figure 9. Phase e.m.f. in loaded operation: eph.ns =

non saturated core; eph = saturated core; solid lines =


0 5 10 15 20

−150

−100

−50

0

50

100

t [ms]

ec.ht

ec.ht.ns

[V]

0 5 10 15 20

−4

−3

−2

−1

0

1

2

3

t [ms]

eph.ht

eph.ht.ns

[kV]

Figure 10. Coil e.m.f., in ideally unsaturated and in

saturated operation, during rotation with constant ir, is1,

is2, is3 values (holding torque test operation): solid


Figure 11. Phase e.m.f., in ideally unsaturated and

in saturated operation, in the same conditions of

Fig. 9 (holding torque test operation): solid line =

analytical results; dotted line = FEM results.


− the validity of (17) is substantially confirmed, even if some local discrepan-

cies can be observed between analytical and FEM calculated waveforms, in

the saturated operation.

Conclusion

An analytical approach to the evaluation of the phase flux linkage and e.m.f. wave-

forms of salient-pole synchronous machines has been developed, able to take into ac-

count the actual structure of the stator winding, the slotting and anisotropy features

under saturated operating conditions.

The saturation has been modelled by using suited saturation functions, that are p.u.

quantities, dependent on the amplitudes of the d-q m.m.f. main sinusoidal components:

this saturation model allows to evaluate the actually saturated operating quantities as

the corresponding unsaturated ones multiplied by the saturation functions.

The most important consequence of this approach is that the self and mutual in-

ductances, and the corresponding derivatives with respect to the rotor position, can be

evaluated once and for all in unsaturated conditions, subsequently including the satura-

tion effects corresponding to the actual conditions.

Several operating situations have been analysed, comparing the waveforms calcu-

lated by the developed analytical approach with those obtained by means of corre-

sponding transient FEM simulations: in general, the agreement appears satisfactory,

showing the soundness of the developed method.

References

[1] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Calculation of the Air-Gap Magnetic Field in Salient-

Pole Three-Phase Synchronous Machines with Open Slots”, ISEF 2005 – XII Int. Symp. on Electromag.

Fields in Mechatronics, Electrical and Electronic Eng., Baiona, Spain, Sept. 15-17, 2005, Proc. on CD,

ISBN N° 84-609-7057-4, paper EE-3.14.

[2] A. Di Gerlando, G. Foglia, R. Perini: “Calculation of Self and Mutual Inductances in Salient-Pole, Three-

Phase Synchronous Machines with Open Slots”, ibidem, paper EE-3.15.

[3] A. Di Gerlando, G. Foglia, R. Perini: “E.M.F. and Torque Analytical Calculation in Salient-Pole, Three-

Phase Synchronous Machines with Open Slots”, ibidem, paper EE-3.16.

[4] A. Di Gerlando, G. Foglia, R. Perini: “FEM identification of d-q Saturation Functions of Salient-Pole

Synchronous Machines”, ISEF 2007 – XIII International Symposium on Electromagnetic Fields in

Mechatronics, Electrical and Electronic Engineering, Prague, Czech Republic, September 13-15, 2007.

[5] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Model of the Air-Gap Magnetic Field in Synchronous

Machines considering slotting, saliency and saturation effects”, ISEF 2007.

[6] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Calculation of the Electromagnetic Torque in Synchro-

nous Machines considering slotting, saliency and saturation effects”, ISEF 2007.

[7] Maxwell 2D FEM code, Version 10, Ansoft Corporation, Pittsburgh, PA, USA.


Radiation in Modeling of Induction

Heating Systems

Jerzy BARGLIKa

, Michał CZERWIŃSKIb

, Mieczysław HERINGc

and

Marcin WESOŁOWSKIc

a

Silesian University of Technology, Krasińskiego 8, 40-019 Katowice, Poland


b

The Industrial Institute of Electronics, Długa 44/50, 00-241 Warsaw, Poland


c

Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland


Abstract. The paper presents analysis of third type boundary conditions applied

for numerical simulation in more frequently used induction heating systems. A

special emphasis was put on analysis on radiation heat transfer with taking into

consideration multiple reflections phenomenon. Obtained results confirmed neces-

sity of usage of multiple reflections model for analysis of high temperature induc-

tion heating systems.

Introduction

Induction heating seems to be modern, environment-friendly industrial technology

quite well explored from theoretical and practical point of view. Usage of the induction

heating technologies leads to significant energy savings, distinct shortening the time of

heating and consequently to the growth of a total efficiency. A powerful tool for de-

signing and optimization is a suitable computer model based on a mathematical model-

ing of the task. The model is typically based on a system of non-linear second order

partial differential equations for coupled electromagnetic and temperature fields. De-

velopment of the mathematical modeling of induction heating processes as well as the

computations with required accuracy by means of professional software and sometimes

also by some user codes has reached a high level. However there are some exceptions.

One of them seem to be modeling of high temperature induction heating systems, for

instance surface induction hardening of steel bodies. During such processes a radiation

heat transfer plays an important role. The phenomenon is connected not only with high

temperatures of the workpiece, but also with a typical arrangement of the system char-

acterized by heated to high temperature charge and many surrounding elements with

distinctly lower and often various temperatures. Majority of papers on induction heat-

ing present classical approach: taking into account radiation in a simplified way only

without considering multiple reflections phenomenon. There are two reasons of the

such the approach. One of them seems to be a false idea that multiple reflections do not

influenced strongly on accuracy of calculations. The more important is the second rea-

son: lack of professional software having precise, well done algorithms for such calcu-

lations. There are of course some packages having well prepared procedures for model-



doi:10.3233/978-1-58603-895-3-202

202

ing of radiation heat transfer with multiple reflections for instance TAS (Thermal

Analysis System) but in that case software is not well prepared for electromagnetic

calculations. The paper presents analysis of coupled electromagnetic and temperature

in typical induction heating system with taking into consideration radiation heat trans-

fer with multiple reflections.

Model of Radiation Heat Transfer in a System with Many Elements

Let us consider an induction heating system with N surfaces of different temperatures

T. For such a system the energy balance for each surface is given by Eq. (1), describing

energy losses from the inductively heated charge to external surfaces [1]:

( ), 4

, , , 0

1 1

1

σ

N N

k i i i

k i k i k i i

i i ii i

P

T

S

δ ε

ϕ δ ϕ

ε ε= =

⎛ ⎞−

− ⋅ ⋅ = −⎜ ⎟

⎝ ⎠

∑ ∑ (1)

where ,k i

δ denote Kronecker delta defined as by (2):

,

1 for = 1

0 for 1

k i

k

k

δ

⎧ ⎫

= ⎨ ⎬≠⎩ ⎭

(2)

and i

ε – effective emissivity of surface i, ,k i

ϕ – view factor between two surfaces k

and i, i

P – energy losses of surface i, 0

σ = 5.67·10–8

W/(m2

·K4

) – Stefan–Boltzmann

constant, i

S – area of surface i.

Usually i

S are elementary surfaces obtained which are lines or faces of a single

elements of mesh. Main difficulty to solve the system of equations (1) is to calculate

view factors ,k i

ϕ , defined as the fraction of total radiant energy that leaves surface k

which arrives directly on surface i. Another complication is that a radiating surfaces are

a gray diffuse body, so total emissivity are less than one. Radiant energy that leaves

surface k which arrives directly on surface i is not total absorbed on it, but according to

Lambert law, is partly reflect and diffuse [1]. This effect of energy balance between all

radiating surfaces in one enclosure called a reflection effect.

Formulation of Technical Problem and Illustrative Example

Let us consider an axi-symmetric model of induction hardening system of cylindrical

steel workpiece by three-turns cylindrical inductor. Main parameters of the induction

heating system are as follows:

Workpiece:

diameter d = 100 mm; length l = 200 mm; thermal conductivity λ = 18 W/(m·K);

density ρ = 8000 kg/m3

; specific heat c = 500 J/(kg·K); effective emissivity ε =

0.8; conductivity γ = 7·106

S/m; relative magnetic permeability μr = 1;

J. Barglik et al. / Radiation in Modeling of Induction Heating Systems 203

Inductor:

number of turns n = 3; inner diameter Di = 130 mm; outer diameter D

e= 154 mm;

width of one turn w = 16 mm; gap between turns Δw = 8 mm; effective emissivity

of internal surface ε = 0.3 ÷ 0.8;

Supply source:

current density within the inductor J = 3·107

A/m2

; frequency f = 1000 Hz.

In order to determine influence of radiation heat transfer on accuracy of tempera-

ture calculations, in the first stage of computations a phenomenon of convection was

neglected. Third type boundary condition having only radiation component was ap-

plied. We modeled the radiation heat transfer in induction heating process for several

values of inductor temperature, its emissivity, distance between inductor and workpiece

and shape of conductor. These elements have strongly influence on radiation heat trans-

fer and on temperature profile in the workpiece. In order to simplify calculations

steady-state analysis of temperature field was used.

However before start the simulation it was necessary to solve also a transient prob-

lem. The obtained results (Fig. 1) show that maximal temperature differences are no-

ticed in steady state. So in order to simplify the calculations in the further part of the

paper only steady-state analysis will be provided. So the analysis could be done for that

state.

For calculations of weakly coupled electromagnetic and temperature fields Quick

Field 2D (QF) and TAS packages were use. Electromagnetic calculations were made

by means of QF (number of nodes is equal to 10428). Based upon results of specific

power density released in the workpiece taken from QF steady-state temperature field

Figure 1. Dependence of temperature of workpiece on time of heating with and without taking into consid-

eration multiple reflections.

J. Barglik et al. / Radiation in Modeling of Induction Heating Systems204

was calculated by means of TAS 3D. with 18373 nodes, 17800 elements, 17680 vol-

ume heat sources, 1720 radiation surfaces. Total power released in the workpiece is

equal to 6702 W. Some results showing temperature distribution within the surface of

the workpiece (Fig. 2) and in its longitudinal (Fig. 3) and transversal (Fig. 4) cross-

sections were presented below.

O

C

Figure 2. Steady-state temperature field in workpiece heated by cylindrical inductor.

O

C

Figure 3. Steady-state temperature field in a longitudinal cross-section of the workpiece.


Temperature distribution in steady-state along length of the workpiece surface is

presented in Fig. 5. Two cases were solved: with and without taking into consideration

multiple reflection effect. The calculations confirmed that if multiple reflection phe-

nomenon was neglected temperature of the workpiece is much lower (about 100o

C).

The dependence between maximal temperature and total emissivity of the work-

piece was shown in Fig. 6. For the case without multiple reflections (not shown in

Fig. 6) maximal temperature is equal to 1197o

C.

The calculations, that apply to inductor’s temperature influence on maximum tem-

perature of charge proves, that he’s faint and can be entirely omitted. This is true in

case of heating systems characteristic for hardening processes. In systems used in plas-

tic forming of metals processes situation could be different.

Assuming, that the workpiece transfer heat by natural convection, according to

generally accepted formulas [2], the value of convection heat transfer coefficient

changes for considered system and temperature range from α = 4.5 to 8 W/(m2

·K), with

practically constant value for temperatures bigger than 530o

C. Results of calculations

are shown in Fig. 7.

It follows from them, that participation of convection in heat transfer process is in

comparison to radiation is rather small. It proves necessity to build model for tempera-

ture calculations taking into account radiation heat transfer with multiple reflections

effect.

O

C

Figure 4. Steady-state temperature field in a longitidinal cross-section of the workpiece.


600

700

800

900

1000

1100

1200

1300

1400

0 25 50 75 100 125 150 175 200

Distance, mm

Temperature,

0

C

ε = 0.3 ε = 0.8 no reflections

Figure 5. Steady state temperature distribution on the workpiece surface of charge for two different values of

emissivity and without multiple reflection phenomenon.

1210

1220

1230

1240

1250

1260

1270

1280

1290

1300

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Emissivity, -

Temperature,

0

C

Figure 6. Maximum temperature of workpiece in function of total emissivity of inductor.


600

700

800

900

1000

1100

1200

1300

1400

0 20 40 60 80 100 120 140 160 180 200

Distance, mm

Temperature,

0

C

with convection no convection

Figure 7. Steady state temperature on the surface of charge for inductor emissivity 0.3 with and without

convection losses.

Modeling in ANSYS Environment

Further simulations were made in ANSYS environment in 2D coordinate system. They

include analysis of dependence of charges temperature in function of distance from

inductor and its geometry. ANSYS is one of the rare packages, that allow simulation of

radiation heat transfer with multiple reflection effect in both 2D and 3D coordinate

systems. Such operation is impossible for TAS, that realizes this operation in 3D only.

In purpose of objectivity in all cases the same volume power distribution in charge was

maintained, as in earlier calculations. In purpose of comparison of simulation results

obtained in different environments in Fig. 8, was shown temperature field received

from ANSYS, with the same conditions and body loads, as presented previously (ob-

tained with usage of QF+TAS packages).

Figures 9 and 10 present temperature distribution on the surface of the workpiece

for three different distances between inductor and the workpiece, with constant emis-

sivity of inductor and workpiece and with the same power distribution within the

workpiece. It follows, that value of emissivity has big influence on reflected radiation

effect. This is frequently in classical models. On mentioned reflection effect essential

influence has distance between inductor and workpiece L. The influence of the inductor

can be omitted only for big values of distance inductor-workpiece L, that is avoided

due to system efficiency.


1

MN

MX

50

235.578

421.156

606.733

792.311

977.889

1163

1287

APR 27 2007

14:23:54

NODAL SOLUTION

STEP=1

SUB =4

TIME=1

TEMP (AVG)

RSYS=0

SMN =50

SMX =1287

Figure 8. Temperature field with 15mm distance of inductor from charge and emissivity 0.3.

600

700

800

900

1000

1100

1200

1300

0 20 40 60 80 100 120 140 160 180 200 220

Distance, mm

Temperature,

0

C

no reflections L = 15 mm L = 35 mm L = 55 mm

Figure 9. Temperature distribution on the charge surface of with constant emissivity ε = 0.8 and different

distances of inductor from charge (L).

Last series of simulations concerned on influence of inductor geometry on tem-

perature field distribution in the workpiece. Results are presented in Fig. 11. Change of

the geometry in analyzed case was reduced to replacement of rectangular cross section

of the conductor into circular one. It was assumed, that the surface of conductor cross

section does not change and their geometrical centers are located in same place. The

biggest influence of inductor geometry was observed with its small values of emissiv-

ity. In this case differences reached level of about 25o

C. For bigger values of emissivity

this difference significantly decreased.


600

700

800

900

1000

1100

1200

1300

0 20 40 60 80 100 120 140 160 180 200 220

Distance, mm

Temperature,

0C

no reflections L = 15 mm L = 35 mm L = 55 mm

Figure 10. Temperature distribution on the surface of charge with constant emissivity ε = 0.3 and different

distance between inductor and charge (L).

600

700

800

900

1000

1100

1200

1300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Distance, m

Temperature,

0

C

circular turn ε = 0,8 circular turn ε = 0,3

rectangural turn ε = 0,8 rectangural turn ε = 0,3

Figure 11. Temperature on the surface of workpiece in function of emissivity and for two different shapes of

conductor.

Conclusions

Obtained results confirmed necessity of consideration of multiple reflection effects in

majority of induction heating systems. Errors caused by neglecting of multiple reflec-


tions phenomena may reach the order of 100o

C. For analyzed systems radiation takes a

crucial part in heat transfer to surroundings. Convection is significantly smaller. Induc-

tor is a kind of mirror, that reflect part of radiation and in result increasing temperature

of charge (especially in close to surface area, that is most interesting in hardening proc-

esses).

Acknowledgement

This work was financially supported by the Polish Ministry of Science and Higher

Education (Grant Projects 9T08C 04678 and 503 /G/1041/0744/006).

References

[1] Siegel R., Howell J.: Thermal radiation heat transfer. New York. McGraw-Hill 1972.

[2] Hering M.: Termokinetyka dla elektryków. Warszawa. WNT 1980.


Time-Domain Analysis

of Self-Complementary and Interleaved

Log-Periodic Antennas

A.X. LALAS, N.V. KANTARTZIS and T.D. TSIBOUKIS

Department of Electrical and Computer Engineering,

Aristotle University of Thessaloniki, GR-54124, Thessaloniki, Greece


Abstract. A systematic near- and far-field analysis of dual polarisation log-

periodic antennas is presented in this paper. The investigation of these demanding

structures is performed by means of a 3D finite-difference time-domain (FDTD)

methodology, properly tailored to tackle their geometrical details and abrupt dis-

continuities. In particular, the analysis delves into the design parameters of self-

complementary, interleaved and trapezoidal-toothed structures and conducts a

thorough examination of their radiation characteristics. Furthermore a new discre-

tisation concept involving very flat cells which enhance algorithmic performance

is introduced. Numerical verification addresses an extensive set of realistic appli-

cations with diverse parameter setups as well as instructive comparisons which in-

dicate the merits of the proposed formulation.

Introduction

Log-periodic antennas, introduced by DuHamel and Isbell [1], are typically used in

wideband systems because their structure exhibits a practically independent behaviour

with regard to operating frequency. As many applications require dual polarisation ar-

rangements, potential choices comprise two different designs, i.e. the self-

complementary (SC) and the interleaved (IL) configuration [2–5]. To this direction,

one may also add an alternative design; the trapezoidal-toothed (TT) one, which shares

similar attributes. Due to the inherent broad frequency spectrum, such antennas are

essentially encountered in radar and measuring systems, attaining high-Q resonance

properties. Amid the degrees of freedom of log-periodic antennas, one can discern their

electrically thin ground-plane-backed dielectric substrate. So, prospective bandwidth

advancement may be accomplished by increasing the substrate thickness or reducing its

dielectric constant. However, the former issue is regularly accompanied by inductive

impedance offsets and augmentation of the surface-wave effect. Thus, it becomes ap-

parent that a meticulous procedure should be followed for the design of these particular

radiators and the most significant: prior to any fabrication process to avoid expensive

construction costs and totally misleading products.

Towards the preceding deductions, the impact of design parameters on the radia-

tion characteristics of log-periodic antennas has been comprehensively explored via

several measurement attempts [6]. Moreover, various frequency-domain computational

investigations have been presented, chiefly via commercial software packages [7].

Nevertheless, to our best knowledge, only a limited amount of time-domain attempts



doi:10.3233/978-1-58603-895-3-212

212

have been discussed. Therefore, it is the goal of this paper to introduce a Finite-

Difference Time-Domain (FDTD) algorithm [8] for the efficient modelling and accu-

rate simulation of 3D self-complementary, interleaved as well as trapezoidal-toothed

antennas and derive reliable guidelines of their radiation characteristics. Special em-

phasis is drawn on the geometrical peculiarities of the devices along with their periodi-

cally-repeated attributes. Pursuing the improvement of the standard FDTD discretiza-

tion rationale, a novel flexible method, incorporating very flat cells, is devised. In this

manner, the domain is divided into more robust lattices devoid of erroneous mecha-

nisms, oscillatory vector parasites or late-time instabilities. Additionally, open bounda-

ries are truncated through diverse versions of the Perfectly Matched Layer (PML) ab-

sorber [9], which offer notable wave annihilation rates without any other non-physical

conventions. To substantiate these qualities, the proposed approach is successfully ap-

plied to various real-world configurations, concerning all three structures, and numeri-

cal outcomes are carefully compared to extract possible similarities or prominent dis-

crepancies.

Structural Description of Log-Periodic Antennas

The design parameters of a log-periodic antenna are α, β, Rmax

, Rmin

, the number of

teeth N, geometrical ratio τ, and width ratio χ, as shown in Fig. 1. Specifically, τ and χ

are given by

1

1

n n

n n

R r

R r

τ

−

+

= = and

1

n

n

r

R

χ

+

= (1)

and therefore, Rn+1

and rn are terms of geometrical progressions

min

1 1nn

R

R

τ

++

= and min

nn

r

r

τ

= (2)

Figure 1. Geometry of a log-periodic planar element.

A.X. Lalas et al. / Time-Domain Analysis of SC and IL Log-Periodic Antennas 213

During the design process, τ and χ can be acquired from

min

max

N

R

R

τ = and χ τ= (3)

given the rest of the parameters. Taking into account that the size of the antenna is

fixed, we use (3) and then (2) to resolve the dimensions of metal surfaces. This is ex-

actly the process followed throughout this paper. In particular, the self-complementary

and interleaved structures are schematically depicted in Figs 2a and 2b, respectively.

The antennas are realised through the use of two metallic sheets, properly shaped and

mounted on a dielectric board, while the feed sections are constructed by integrated

tapered-microstrip baluns. Two out of four branches are located on the upper side of

the board and two are placed at its bottom. An alternative implementation of a dual

polarisation device is the trapezoidal-toothed log-periodic radiator, whose geometry is

illustrated in Fig. 2c.

Accurate Analysis via the 3D FDTD Methodology

The length and width of the antennas is set to 90 mm and their height is 6.67 mm. Rmax

and Rmin

are 40 mm and 5 mm, respectively, with these values kept fixed throughout

the paper. The computational domain is divided into 144 × 144 × 44 cells with Δx =

Δy = Δz = 0.8333 mm and Δt = 1.458 ps. Furthermore, termination of the unbounded

space is attained by a 6-cell PML absorber. The two ports are excited by hard sources,

appropriately phase-shifted to ensure circular polarisation. Our interest principally fo-

cuses on the variation of α, N and the relative dielectric permittivity εr of the dielectric

board.

From near and far-field data, acquired by our simulations, we can estimate the

normalised (in dB) current distribution upon the surface of the structures and their

radiation patterns. Some indicative results are illustrated in Figs 3a to 3c. It is empha-

sised that current values outside the outline of metal surfaces have no physical mean-

ing, since they are merely a side-effect of the processing algorithm. As observed,

maximum values of current distributions appear over the edges of the metal structures

and hence the inner material is obsolete. This notification confirms previously-reported

(a) (b) (c)

Figure 2. Geometry of (a) a self-complementary, (b) an interleaved, and (c) a trapezoidal toothed structure.

A.X. Lalas et al. / Time-Domain Analysis of SC and IL Log-Periodic Antennas214

(a) (b) (c)

Figure 3. Current distribution of (a) an SC antenna when N = 5, α = 80ο

, β = 10ο

, (b) an IL antenna when

N = 4, α = 160ο

, β = 10ο

, and (c) a TT antenna when N = 5, α = 80ο

, β = 10ο

.

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

2.0028 GHz

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

2.1949 GHz

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

2.2909 GHz

(a) (b) (c)

Figure 4. Radiation pattern at x-plane (blue line for Eθ; red line for: Eφ) of (a) an SC antenna when N = 4,

α = 80ο

, β = 10ο

, (b) an IL antenna when N = 5, α = 160ο

, β = 10ο


,

β = 10ο

.

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

2.0028 GHz

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

2.1949 GHz

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

2.2909 GHz

(a) (b) (c)

Figure 5. Radiation pattern at z-plane (blue line for Eθ; red line for: Eφ) of (a) an SC antenna when N = 4,

α = 80ο

, β = 10ο

, (b) an IL antenna when N = 5, α = 160ο

, β = 10ο


,

β = 10ο

.

measurements [6] and leads to the wire structures. In addition, radiation patterns for

each antenna, over the x- and z-plane, are presented in Figs 4 and 5, respectively.


The impact of N, α variation and that of the relative dielectric permittivity εr on the

radiation characteristics is next examined. Radiation patterns for an SC structure, over

the x- and z-plane, are shown in Figs 6 and 7. More specifically in Figs 6a and 7a, an

investigation on the behaviour of N is provided. Because of the fixed size of the an-

tenna, any change of N is actually translated to a corresponding change of τ and there-

fore teeth dimensions vary when a supplementary one is inserted. As a result, teeth

resonate in different frequencies for each case but the overall performance of the an-

tenna is similar for small modifications.

On the other hand, Figs 6b and 7b give an investigation on the variation of α. As α

increases, a more bidirectional behaviour is revealed. This is because for smaller an-

gles, dimensions of the teeth are smaller too and thus can not resonate. Subsequently in

Figs 6c and 7c, an examination on the effect of relative dielectric permittivity εr is de-

picted. The attitude of the antenna is, now, more bidirectional for smaller εr. In this

context, for a substrate with a high dielectric constant, the effective length of the teeth

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

2.0028 GHz

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

2.0028 GHz

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

2.0028 GHz

(a) (b) (c)

Figure 6. Investigation (x-plane) on the variation of (a) N when α = 80ο

, β = 10ο

(blue line for N = 3; red line

for N = 4, green line for N = 5; black line for N = 6), (b) α when N = 4, β = 10ο

, (blue line for α = 20ο

, red line

for α = 40ο

, green line for α = 60ο

; black line for α = 80ο

), and (c) εr when N = 5, α = 80ο

, β = 10ο

(blue line

for εr = 2.8; red line for εr = 3.38, green line for εr = 4.4).

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

2.0028 GHz

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

2.0028 GHz

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

2.0028 GHz

(a) (b) (c)

Figure 7. Investigation (z-plane) on the variation of (a) N when α = 80ο

, β = 10ο

(blue line for N = 3; red line

for N = 4, green line for N = 5; black line for N = 6), (b) α when N = 4, β = 10ο

(blue line for α = 20ο

; red line

for α = 40ο

, green line for α = 60ο

; black line for α = 80ο

), and (c) εr when N = 5, α = 80ο

, β = 10ο

(blue line

for εr = 2.8; red line for εr = 3.38, green line for εr = 4.4).


is decreased. Thus, when increasing the value of εr, lower frequencies can not reso-

nate [10,11].

Next, some additional comparisons are conducted. Radiation patterns of an SC

structure, for different frequencies, are shown in Figs 8a and 9a. Bi-directional behav-

iour is observed in each case. Moreover, in Figs 8b and 9b a comparison of the struc-

tures presented earlier is depicted. Their behaviour is satisfactory in the sense of being

bi-directional. The differences on the amplitude of the radiation patterns are explained

taking into account the mismatching of the ports. In order to avoid them a more accu-

rately modelling of the ports is needed.

It is noteworthy to observe the performance of the antenna in the case of reducing

the distance between its two metallic surfaces. To model these intricate cases, a

new flat-cell discretization approach, namely Δx = Δy >> Δz, is developed. We exam-

ine two cases. In the first, the height of the antenna is 2.0202 mm and the FDTD do-

main is divided into 148 × 148 × 90 cells with Δx = Δy = 0.8333 mm, Δz = 0.2525 mm,

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

−30

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

−20

−10

0 dB

θ [deg]

30

150

60

120

90

90

120

60

150

30

180 0

(a) (b) (c)

Figure 8. Comparison of radiation patterns (x-plane) for different (a) frequencies, when N = 5, α = 80ο

,

β = 10ο

(blue line for f = 2.675 GHz; red line for f = 2.867 GHz, green line for f = 3.6352 GHz), (b) structures

(blue line for SC; red line for IL, green line for TT), and (c) distances of metallic surfaces at 2.0028 GHz,

when N = 5, α = 80ο

, β = 10ο

(blue line for Δz = 0.8333 mm; red line for Δz = 0.2525 mm, green line for

Δz = 0.1263 mm).

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

−30

−20

−10

0 dB

φ [deg]

30

210

60

240

90

270

120

300

150

330

180 0

(a) (b) (c)

Figure 9. Comparison of radiation patterns (z-plane) for different (a) frequencies, when N = 5, α = 80ο

,

β = 10ο

(blue line for f = 2.675 GHz; red line for f = 2.867 GHz, green line for f = 3.6352 GHz), (b) structures

(blue line for SC; red line for IL, green line for TT), and (c) distances of metallic surfaces at 2.0028 GHz,

when N = 5, α = 80ο

, β = 10ο

(blue line for Δz = 0.8333 mm; red line for Δz = 0.2525 mm, green line for

Δz = 0.1263 mm).


and Δt = 0.703 ps. In the second case, the height of the antenna is 1.0101 mm. The

FDTD domain is divided into 148 × 148 × 146 cells with Δx = Δy = 0.8333 mm, Δz =

0.1263 mm, and Δt = 0.374 ps. Again, termination of open boundaries is attained by a

6-cell PML along x- and y-axis and a larger PML setup towards z-axis. This is deemed

necessary to ensure that the thickness of the absorber will be the same at every direc-

tion in the grid.

As we can notice from Figs 8c and 9c when the distance between the two metallic

surfaces decreases, the behaviour of the antenna at a specific frequency follows a more

omni-directional pattern, basically due to loss of antenna resonance. Moreover, lower

frequencies are suppressed at the entrance of the tapered-microstrip baluns owing to the

smaller height of these waveguiding structures. Consequently, they can not excite the

metallic parts of the antenna. Overall, the preceding investigations prove that band-

width enhancement may be conducted either by augmenting the thickness of the sub-

strate or by decreasing its dielectric constant.

Conclusion

A consistent 3D FDTD technique for the rigorous analysis of log-periodic antennas has

been presented in this paper. Three different structures with respect to their periodical

features are investigated, whereas for improved precision a new flat-cell approach is

developed. Numerical validation involves a variety of simulations and comparisons

between different antenna types. Particularly, the influence of the teeth number and

spanning angle on the antenna overall performance are extensively explored and com-

bined with a study on the effect of the relative dielectric constant and the distance of

the basic metallic surfaces. Results confirm the benefits of the proposed time-domain

method and demonstrate the potential to be employed as a promising tool for log-

periodic antenna characterization. Future aspects involve a more detailed modelling of

the ports in order to obtain higher levels of accuracy.

Acknowledgement

This work was supported by the National Scolarships Foundation of Greece (IKY).

References

[1] R.H. DuHamel, and D.E. Isbell, Broadband logarithmically periodic structures, Record of 1957 IRE

National Convention, Part 1, 99, pp. 119-128.

[2] A.L. Van Hoozen, et al., Conformal log-periodic antenna assembly, US Patent 6,011,522, 4 Jan. 2000.

[3] A.L. Van Hoozen, et al., Bidirectional broadband log-periodic antenna assembly, US Patent 6,018,323,

25 Jan. 2000.

[4] D.A. Hofer, et al., Compact multipolarised broadband antenna, US Patent 5,212,494, 18 May 1993.

[5] D. Campbell, Polarised planar log-periodic antenna, US Patent 6,211,839 B1, 3 April 2001.

[6] R.H. DuHamel, and F.R. Ore, Logarithmically periodic antenna designs, Record of 1958 IRE National

Convention, Part 1, Vol. 100, pp. 139-151.

[7] K.M.P. Aghdam, R. Faraji-Dana, and J. Rashed-Mohassel, Compact dual-polarisation planar log-

periodic antennas with integrated feed circuit, IET Microw. Antennas Propag., Vol. 152, pp. 359-366,

2005.

[8] A. Taflove, and S. Hagness, (3rd ed.), Computational Electrodynamics: The Finite-Difference Time-

Domain Method, Boston: Artech House, 2005.


[9] J.P. Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic wave,

J. Comp. Phys., Vol. 127, pp. 363-379, 1996.

[10] E. Avila-Navarro, J.M. Blanes, J.A. Carrasco, C. Reig, and E.A. Navarro, A new bi-faced log periodic

printed antenna, Microw. Optical Technol. Lett., Vol. 48, pp. 402-405, 2006.

[11] B.L. Ooi, K. Chew, and M.S. Leong, Log-periodic slot antenna array, Microw. Optical Technol. Lett.,

Vol. 25, pp. 24-27, 2000.


New Spherical Resonant Actuator

Y. HASEGAWAa

, T. YAMAMOTOa

, K. HIRATAa

, Y. MITSUTAKEb

and

T. OTAb

a

Department of Adaptive Machine Systems, Osaka University, Yamadaoka, 2-1,

Suita-city, Osaka 565-0871, Japan

[email protected]

b

Advanced Technologies Development Laboratory, Matsushita Electric Works, Ltd.,

1048, Kadoma, Osaka 571-8686, Japan

[email protected]

Abstract. This paper proposes the new spherical resonant actuator. The basic con-

struction and the operating principle of the actuator are described. The torque

characteristics of the actuator are computed by the 3-D FEM analysis. The geome-

try of the mover is investigated to improve the torque characteristics and the ef-

fectiness is clarified by both of the computation and the measurement of a proto-

type. Futhermore, the dynamic characteristics of the improved model are also con-

firmed by the measurement.

1. Introduction

Recently, multi-dimensional actuators are a topic of great interest because of solution

for vibration, noise, size constraints and limitations on operating speed [1]. Particularly,

spherical actuators [2] are studied as the application to the joints and eyeballs for robots

because they can be freely rotated in every axis direction.

Due to the computer progress, the computer simulation becomes an effective tool

to design electric devices and actuators. Authors have been studying the analyzed

method employing the 3-D FEM to apply to multi-dimensional actuators with compli-

cated magnetic structure [3].

In this paper, the new spherical actuator is proposed and the torque charactersistics

are computed through the FEM analysis. The validity of the computation is verified by

the comparison with the measurement of a prototype. The resonance characteristics

around two rotation axes are confirmed through the measurement.

2. Basic Structure and Operating Principle

Figure 1 shows the basic construction of the proposed spherical resonant actuator. It

has the hybrid magentic structure [4] so that the magnetic flux by the current can not

flow through the permanent magnet because permanent magnet has large magnetic

resistance. The mover has four magnetic poles made of cross-shaped iron, permanent

magnets (Br = 1.42 T), and spherical iron cores. The stator has four spherical magnetic

poles with exciting coils of 100 turns. The air-gap between both spherical faces is



doi:10.3233/978-1-58603-895-3-220

220

0.3 mm. The mover is connected to four common resonance springs to be operated in

resonance frequency.

Figure 2 shows the cross section of x-z plane of the basic model. The magnetic flux

by permanent magnet flows along the solid line, and the mover keeps balance at the

center. When the coils are excited as shown in this figure, the flux flows along the dot-

ted line, and the flux in the air-gap becomes unbalanced, and torque is generated. The

mover can be rotated around arbitrary axis by changing the amplitude and direction of

four coil currents.

3. Static Torque of the Basic Model

Figure 3 shows the computed torque characteristics of the basic model employing

the 3-D FEM when the coil A and C are excited, and the mover is rotated from 0 to

5 degree in step of 1 degree around y-axis. The average torque constant is

13.9 × 10–2

mN·m/A. Figure 4 shows the distribution of magnetic flux vectors with the

coil excitation of 0 and 100 A. When coils are not excited, the magnetic flux by the

magnet flows around the mover ploes and the upper parts of stator poles. On the other

Figure 1. Basic construction of the proposed model.

Figure 2. Magnetic circuit of the basic model. Figure 3. Torque characteristics of the basic model.

Y. Hasegawa et al. / New Spherical Resonant Actuator 221

hand, when both of coil A and coil B are excited, the magnetic flux by the current

flows through the yoke of whole stator and the mover. It is found that the magnetic

path by the current is completely divided with the magnetic path by the magnet.

4. Static Torque of Improved Model

Figure 5 shows the improved model, which has the same structure as the basic model

mentioned above except the mover. The mover of this model has the full-scale spheri-

cal yoke in order to keep the facing area between the mover and the stator, and has the

ring-shaped permanent magnet (Br = 1.42 T) at the center. Figure 6 shows the compari-

son between the measured and the calculated torque characteristics with the coil excita-

tion of 0 and 100 A. As can be seen, both results are in good agreement. This actuator

has the stable position at the rotation angle of 0.0 degree. The computed average torque

constant is 27.4 × 10–2

mN·m/A. It is twice as large as the basic model. Figure 7 shows

the distribution of the flux density vectors, Fig. 8 shows the contours of the flux density

in the neighborhood of facing area. The magnetic flux density of the improved model

becomes stronger than that of the basic model, because the magnetic flux from whole

of the ring-shaped magnet flows into the stator poles. As a result, the average torque

constant becomes large. And the cogging torque characteristic of the improved model

shows the linearity versus rotation angle compared with the basic model.

5. Dynamic Characteristics of Improved Model

Figure 9 shows the prototype of the proposed model, which has the gimbal mechanism

to be operated in the spherical surface with 0.3 mm gap between mover and stator, and

has four resonant springs on the upside of the mover. Figures 10 and 11 show the meas-

ured frequency characteristics for x- and y- directions while it is operated at the same

time. When the input voltage of 2.4 V (peak to peak) is applied with resonance fre-

quency of 173 Hz for x-direction, the, the maximum rotation angle is 4.3 degree (peak

(a) 0AT (b) 100AT (Coil A + Coil C)

Figure 4. Distribution of flux density vectors.

Y. Hasegawa et al. / New Spherical Resonant Actuator222

to peak) and the average current is 0.4 A. On the other hand, when the same voltage is

applied with resonance frequency of 127 Hz for y-direction, the maximum rotation

(a) overall view

(b) x-z section

Figure 5. Construction of the improved model.


angle is 6.4 degree (peak to peak) and the average current is 0.5 A. The difference of

resonance frequencies for x- and y- directions is due to the inertia of the gimbal

mechanism. Figure 12 shows the trajectory of the mover when it is operated at resonant

frequencies of x- and y-directions (173 Hz and 127 Hz). As shown, it is found that this

actuator can be operated in arbitrary direction.

Figure 6. Comparison between measured and calculated torque characteristics of improved model.

(a) Basic model (b) Improved model

Figure 7. Distribution of flux density vectors.


(a) Basic model (b) Improved model

Figure 8. Contours of flux density in the neighborhood of facing area.

Figure 9. Prototype.

6. Conclusions

This paper presentedthe new spherical resonant actuator. The torque characteristics

were computed through the 3-D FEM analysis. The effect of mover geometry on the

torque characteristics was investigated. And, it was found that average torque of the

improved model was twice as large as the basic model. The validity of the computation

was verified by the comparison with the measurement of a prototype. Furthermore, the

dynamic characteristics versus rotation angle were confirmed through the measurement.

As a result, it was found that the proposed actuator was operated in spherical surface.

This research was supported in part by “Special Coordination Funds for Promoting

Science and Technology: Yuragi Project” of the Ministry of Education, Culture, Sports,

Science and Technology, Japan.


Figure 10. Measured frequency characteristics (x-di-

rection).

Figure 11. Measured frequency characteristics (y-di-

rection).

Figure 12. Trajectory of multi-motion. (x direction: 173 Hz, y direction: 127 Hz).

References

[1] A. Tanaka, M. Watada, S. Torii and D. Ebihara, “Proposal and Design of Multi-Degree-of-Freedom

Spherical Actuator”, 11th MAGDA Conference, PS2-3, pp. 169-172, 2002.

[2] E.h.M. Weck, T. Reinartz, G. Henneberger and R.W. De Doncker, “Design of a spherical motor with

three degrees of freedom”, Annals of the CIRP, Vol. 49, pp. 289-294, 2000.


[3] K. Hirata, T. Yamamoto, T. Yamaguchi, Y. Kawase and Y. Hasegawa, “Dynamic Analysis Method of

Two-Dimensional Linear Oscillatory Actuator Employing Finite Element Method”, IEEE Transaction on

Magnetics, Vol. 43, No. 4, pp.1441-1444, 2007.

[4] K. Hirata, Y. Ichii and Y. Kawase, “Novel Electromagnetic Structure with Bypass Magnetic Path for Re-

set Switch”, IEEJ Trans. IA, Vol. 125, No. 3, pp. 293-296, 2005.




C1. Electrical Machines and Transformers


Influence of the Correlated Location

of Cores of TPZ Class Protective Current

Transformers on Their Transient State

Parameters

Elzbieta LESNIEWSKA and Wieslaw JALMUZNY

Department of Applied Electrical Engineering & Instrument Transformers,

Technical University of Lodz, ul. Stefanowskiego 18/22, 90-924 Lodz, Poland


Abstract. A method of avoidance of mutual influences between secondary wind-

ings of multi-core type in a designed protective current transformer is presented in

this paper. The analysis was performed for different location of cores with secon-

dary windings. The mutual influence is determined on the basis of 3D field distri-

butions obtained by the numerical field method. Some results of computation were

compared with test results.

Introduction

Protective current transformers are very important parts of electric power systems.

They are indispensable for the proper functioning of a system, because they are ele-

ments of the protection system. There are two kind of protective current transformers;

class P to protection at steady state and TP to protection at transient state. The multi-

core type current transformer is composed of a number of cores with individual secon-

dary windings and a common primary bar in the same casing. During a transmission

line short circuit, the primary current takes on an exponential component resulting in

core saturation and a deformation of the secondary current. The cores of measuring

current transformer and protective class P should be saturated during a transmission

line short circuit. Therefore the measuring current transformer has a core without air

gaps. The TPZ class protective current transformer has a core with air gaps which

guarantee linearity of magnetic characteristic of the core at an assumed value of pri-

mary short circuit current Ipsc

. Behaviour of protective current transformer at a transient

state is very important because it influences the proper functioning of the protection

system. The mutual influence between windings can occur through the magnetic field.

The aim of research was to estimate the influence of the correlated location of

cores on current error and the phase displacement at rated state and transformation er-

rors at transient state. During the design process of the multi-core type current trans-

former it is important to predict the mutual coupling between the current transformers

and then to avoid it.


231

A Mathematical Model

The field-and-circuit method for steady state, based on the solution of Helmholtz equa-

tion for 3D electromagnetic field determines secondary voltage and compares it with

the circuit equation of the same secondary voltage.

It is impossible to solve the transient problem using the complex method, since the

wave-shapes of field quantities are considerably deformed. Joining field-and-circuit

method and space-time 3D analysis allows computing the secondary current vs. time

assuming a non-sinusoidal wave-shape of the primary current. For 3D analysis a full

set of time-dependent differential equations must be solved instead.

( )curl curl gradV

t

ν σ

∂⎛ ⎞= − −

⎜ ⎟∂⎝ ⎠

A

A (1)

div gradV

t

σ

⎛ ∂ ⎞⎛ ⎞− − =

⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠

A

0 (2)

Using the commercial software 3D based on the numerical finite element method

enables solving this equation and estimating the secondary current vs. time while the

primary current has an exponential component. The non-linear magnetic characteristic

of cores was taken into account. The boundary conditions were

0× =A n and V = 0 (3)

at the boundary of the whole system with the surrounding air. The applied time step-

ping was 0.0001 s. The mesh of 616043 elements for 3D model was result of accuracy

analysis. Further mesh refinement does not change the solution.

Computational Results

As an example, the three toroidal protective current transformers TPZ class 1200 A/1 A

with eight air gaps of δ = 3 mm were considered. The total length of air gaps is 24 mm.

The air gaps are rotated one by one at 45 degrees.

Figure 1. Multi-core type current transformer composed of three cores with individual secondary windings

and a common primary bar.

E. Lesniewska and W. Jalmuzny / Protective Current Transformers232

The neighbouring cores with secondary windings can be located in a different way.

Two border cases were considered in work; one when the cores with gaps are in the

same position and the second when the cores are turned at 22,5 degrees to each other.

The analysis was performed for rated steady state and transient state.

The current error and the phase displacement indicate the accuracy class of de-

signed current transformer. The computations of current error and phase displacement

were performed at rated state for load R = 5 Ω. The measurements were performed in

the same conditions using the type Φ5304 measuring bridge with comparator produced

by the company POCTOK. The test for the single TPZ class protective current trans-

formers 1200 A/1 A gives the following results: the current error –0.92% and the phase

displacement 398minutes.

The results of 3D analysis carried out for the individual TPZ class protective cur-

rent transformers 1200 A/1 A were: the current error –0.88% and the phase displace-

ment 346 minutes. Table 1 shows results of computation for two different positions of

cores. Both constructions give convergent results of errors. The middle protective cur-

rent transformer has somewhat better conditions if the cores are turned at 22,5 degrees

to each other.

The computations of the instantaneous error current vs. time were performed at

transient state for load R = 5 Ω, specified primary time constant Tp = 50 ms and rated

symmetrical short-circuit current factor Kssc

= 25. The primary current was equal

ip = 30 2 (cos314.16t–e

–20t

) kA.

In Fig. 5 the results of the computed instantaneous error current vs. time at a tran-

sient states are presented for three protective TPZ current transformers. All curves are

very close. For both case the magnetic field linked between current transformers is

practically negligible.

In Fig. 4 we can observe that the both constructional solution give practically neg-

ligible magnetic flux density in the neighbouring core caused by the other protective

current transformers.

a) b)

Figure 2. Cores with gaps in the same position and the cores are turned at 22.5 degrees with relation to each

other.

E. Lesniewska and W. Jalmuzny / Protective Current Transformers 233

a)

b)

Figure 3. Distribution of magnetic flux density [T] of protective current transformer at rated steady state at

the same conditions for two different positions of cores.

Table 1. The current error and the phase displacement of the TPZ current transformer at rated state Rb = 5 Ω

TPZ class protective current transformers 1200A/1A (δ = 24 mm)

gaps in the same position gaps turned at 22.5°

core: outside middle outside outside middle outside

current error ΔI [%] –0.887 –0.883 –0.880 –0.828 –0.800 –0.825

phase displacement δi [min] 342 347 342 341 334 341

Peak instantaneous alternating current errors have been determined on the basis of

obtained curves (Fig. 5)

ˆ

ˆ 100%

2

ac

ac

psc

i

I

ε

ε = ×

(4)

where Ipsc

= Kssc

Ipn

= 25 * 1200 = 30 kA, îεac

– maximum instantaneous error of the

alternating current component and is equal 9.7%, and have been determined on the ba-

sis of test is equal 11.4%. Test was carried out using the d.c. saturation method.


a)

b)

Figure 4. Distribution of magnetic flux density [T] of protective current transformers at transient state for

two different positions of cores at the same conditions and time t = 0.054 s.

-5

0

5

10

15

20

25

0 0,05 0,1 0,15 0,2

time (s)

in

stan

ten

ou

s erro

r cu

rren

t

(A

)

outside middle outside

Figure 5. Instantaneous error current vs. time for three protective current transformers (in terms of secondary

winding).


The next problem was recognizing the influence of the two neighbouring protec-

tive current transformers type 10P 45 1200 A/5 A on operation of TPZ clas protective

current transformer 1200 A/1 A. The same computation was performed, for the case

with different class of current transformer.

The computations of current error and phase displacement were performed at rated

state of the TPZ class protective current transformer for load R = 5 Ω and the 10P class

protective current transformers of the accuracy limit factor ALF = 45 for load

S = 15 kV and cosφ = 0.8.

In Fig. 6 can be observe that the cores of both P class protective current transform-

ers are more saturated but the TPZ class protective current transformer with core with

air gaps works in normal condition. Its current error and the phase displacement practi-

cally did not change.

a)

b)

Figure 6. Distribution of magnetic flux density [T] of the two P class protective current transformers and one

TPZ class protective current transformer in the middle at rated steady state a) three current transformers

together b) only the TPZ class protective.


Research shows significant difference between protective current transformers

class P and TP (Fig. 7). Nevertheless that the accuracy limit factor of them is very big

(45) and symmetrical short circuit current factor Kssc

during test is 25, P class protec-

tive current transformer cores are saturated and the current transformation are incorrect,

in opposite to TPZ class protective current transformer (Figs 8, 9). It causes by an ex-

ponential component of the primary current.

Table 2. The current error and the phase displacement of different class of current transformers at rated state

class of current

transformer:

10P

1200 A/5 A

TPZ

1200 A/1 A °

10P

1200 A/5 A

current error ΔI [%] –0.099 –0.836 –0.099

phase displacement δi [min] +7,70 +343 +7,70

a)

b)

Figure 7. Distribution of magnetic flux density [T] of the two P class protective current transformers and one

protective current transformer TPZ class in the middle at transient state at the same conditions and time

t = 0.054s a) three current transformers together b) only TPZ class protective.


-220

-170

-120

-70

-20

30

80

130

0 0,02 0,04 0,06 0,08 0,1

time (s)

seco

nd

ary cu

rren

t (A

)

TPZ P P

Figure 8. Secondary currents vs. time of TPZ and P class protective current transformers.

-5

0

5

10

15

20

25

0 0,02 0,04 0,06 0,08 0,1

time (s)

in

sta

ntan

eo

us

erro

r c

urre

nt (A

)

sigle TPZ TPZ nearby P

-50

0

50

100

150

200

250

300

350

0 0,02 0,04 0,06 0,08 0,1

time (s)

in

stan

tan

eo

us erro

r cu

rren

t

(A

)

Type P

Figure 9. Instantaneous error current vs. time for two type of protective current transformers a) a comparison

curves for separated TPZ class current transformers and working in neighbourhood of P class protective

current transformers b) for P class protective current transformers.

Conclusions

The transient state 3D analysis of protective current transformers performed with the

application of the field-and-circuit method can determine the correlated location of

cores, which guarantees a magnetically separated operation of each core with individ-

ual secondary winding.

The computing results show if correlation location of cores has an influence on

steady state and transient errors. To eliminate the mutual interactions during the design


process, the construction may be redesigned if the steady state and transient state errors

are excessive. The tests show that in these cases the TPZ class protective current trans-

formers can operate as independent devices. This means that the neighbouring current

transformers have only a slight influence on their errors.

References

[1] E. Lesniewska, Applications of the Field Analysis During Design Process of Instrument Transformers,

Transformers in Practice, Vigo Spain 2006, pp. 227-251.

[2] E. Lesniewska and J. Ziemnicki, Transient State Analysis of Protective Current Transformers at Different

Forced Primary Currents, Przegląd Elektrotechniczny 5’2006, pp. 57-60.

[3] W. Jalmuzny, D. Adamczewska, I. Borowska-Banas, Analysis of Current Difference Test Arrangement

Operation for Measuring Class TP Current Transformers, Pomiary Automatyka Kontrola (PAK),

no 10bis/2006, pp. 102-110.

[4] E. Lesniewska and W. Jalmuzny, Influence of the Number of Core Air Gaps on Transient State Parame-

ters of TPZ Class Protective Current Transformers, Compumag 2007, Aachen.

This work was supported by the Polish Ministry of Science and Higher Education (Project No. 3 T10A 004

30).


Machine with a Rotor Structure Supported

Only by Buried Magnets

Jere KOLEHMAINEN

ABB Oy, Motors, P.O. Box 633, FI-65101 Vaasa, Finland


Abstract. A buried magnet rotor structure, which is supported only by permanent

magnets, is proposed for medium speed permanent magnet machines. A machine

utilizing the construction is built, tested and compared to another machine with

traditional V-shaped poles. The machine is also simulated using Finite Element

Method and the results are compared to tested values. The obtained results demon-

strate the feasibility of the construction.

Introduction

Permanent magnet synchronous machines (PMSM) with buried magnets have been

considered in a wide range of variable speed drives. A buried magnet design has many

advantages compared to designs with surface mounted and inset magnets. With a bur-

ied magnet design flux concentration can be achieved, which induces higher air gap

flux density [1,2]. That, in turn, gives a possibility to increase torque of a machine.

The typical way of manufacture a buried PM rotor is to assemble a stack of

punched rotor disks with rectangular holes and insert magnets into these holes. The

rotor poles between the magnets are fixed to rest of the rotor structure with thin iron

bridges. The disadvantage of the supporting bridges is the leakage flux, the magnitude

of which depends on the thickness of the bridges. In low speed applications this is not

a problem, since the centrifugal forces acting on the poles are relatively small and

the bridges can be kept thin. However, as the tangential speed of the rotor surface in

medium speed applications (4000…8000 1/min) exceeds 60 m/s (corresponding to

4000 1/min in machine size IEC250) the stress in the bridges will exceed the yield

strength of the electrical steel (typically 300 MPa for grade M400-50A). The problem

can be countered by increasing the thickness of the bridges, however, this increases the

leakage flux, which in turn increases the amount of magnet material needed to get the

required torque.

However, there exists a solution with thinner bridges, where magnets are partly

used to support the pole structure [3]. In this paper we go further and study a solution

on how to get mechanically more robust rotor structures without using iron bridges. In

the solution the tensile stress is geometrically converted into compressive one and only

the magnets are used to support the pole structure. The new solution is compared to a

traditionally used solution with V-shaped poles. The comparison is done using time

stepping and static calculations using Finite Element Method (FEM) [4]. Machines

with both the rotor designs are built and tested. The machine with the new dovetail pole

design is analyzed further and results are compared to simulations.



doi:10.3233/978-1-58603-895-3-240

240

Machine Designs

An 8-pole machine with V-shaped rotor poles is used as an example for comparison to

a machine with the new dovetail design without supporting bridges. The machine has

shaft height 250 mm, nominal power 110 kW, voltage 370 V, and speed 4800 1/min.

The only difference of the two machines is in their rotor structure as it can be seen in

Fig. 1. An 8-pole machine has 8 symmetry sections in V-pole design, but in the dove-

tail design the rotor has magnets in every second pole [1].

With both designs, volume of the magnets and dimensions of the magnets seen by

stator are same. With the V shape design, total magnet width and length in one pole

are 2 × 7.3 = 14.6 mm and 52 + 52 = 104 mm and with the dovetail design, these are

1 × 14.6 = 14.6 mm and 26 + 52 + 26 = 104 mm. Length of the both rotors is 120 mm.

Magnetically, there are two major differences which affect to electrical properties of

machines. The dovetail design has not magnetic bridges between poles so the leakage

flux is reduced especially with low saturation of flux. Every second pole of the dovetail

design has a different tangential air gap length so electrical properties with high load

angles are expected to be slightly worse than with the V shape design.

Manufacturing

Two machines with both rotor types are manufactured. The general method to manu-

facture the rotor with a V shaped design is to assemble a stack of disks, compress it

using bolts and nonmagnetic end plates and shrink fit the stack on the shaft. Then, the

magnets are inserted into their holes using glue. The rotor with the dovetail design is

manufactured with a different method, which is to assemble five (one central body and

four small poles) stacks of disks, compress them using bolts and nonmagnetic end

plates and shrink fit the central body stack on the shaft. Next, the magnets are fixed to

four pole stacks using glue. Resulting poles are axially inserted to central body stack

using glue.

Figure 1. Designs with dovetail and V shaped poles with flux lines created by the flux of the magnets.

J. Kolehmainen / Machine with a Rotor Structure Supported Only by Buried Magnets 241

Results

The machines with the both designs are tested and analyzed. For all load tests, as for

our industrial cases, the direct torque control strategy with software for permanent

magnet AC machines is used with frequency converter ACS600 [5]. All electromag-

netic calculations are done with time stepping Finite Element Method [4]. In load cal-

culations, voltage source is used, the form of the voltage is sinusoidal and amplitude is

kept the same. Simulations are started with various rotor angles without initial solution

and stopped after 41 electric periods when transient oscillations have totally died away.

Constant rotor speed is used. Iron losses are calculated from the equation

( ) ( )

2 3 22

2

0

1

,

12

T

TOT h m e f

d dB dB

P k B f t k t k dt

T dt dt

σ

⎡ ⎤⎛ ⎞ ⎛ ⎞

= + +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

∫ (1)

where m

B is the maximum flux density at the node concerned, f is the frequency, σ

is the conductivity, d is the lamination thickness, h

k is the coefficient of hysteresis

loss and e

k is the coefficient of excess loss.

Open Circuit Voltage

Measured open circuit voltages of both machines, at speed 4800 1/min, are compared

to the calculated ones in the Table 1. The measuring and the calculating temperature

has been 20

o

C. In this case, the magnets have a remanence flux density of 1.1 T and

energy product 230 kJ/m

3

.

The dovetail design has 5.2% larger calculated open circuit voltage than the

V shape design. The measured open circuit voltage is 4.9% smaller than calculated for

the V shape design and it is 14.9% smaller for the dovetail design. The measured open

circuit voltages were expected to be smaller, because small rotor length and diameter

Figure 2. Manufacturing rotor with dovetail design. In the left there is a pole with magnets, in the middle

there is the rotor without the pole, and in the right the pole is inserted to rotor.

Table 1. Open Circuit Voltages

Quantity Dovetail design V-shape design

Measured voltage (V) 311.1 330.5

Calculated voltage (V) 365.5 347.6

J. Kolehmainen / Machine with a Rotor Structure Supported Only by Buried Magnets242

ratio (120/289 = 0.42) causes remarkable leakage fluxes in ends of rotors. With the

dovetail design, also axial deviations of the rotor will add leakage fluxes (the rotor

body is slightly longer than the poles). The machine with the dovetail design has also

different harmonic distribution of open circuit voltage, as can be seen in Fig. 3. The

fifth harmonic is almost same with both designs. The seventh, eleventh and thirteenth

harmonics are larger with the dovetail design while the fifth, seventeenth and nine-

teenth harmonics are larger with the V shape design. This is caused by different air gap

forms and different sizes of every second pole with dovetail design. Practically, voltage

and torque ripples are on the same level.

Calculated Electrical Properties with Different Loads

The calculated electrical properties as a function of electric load angle are compared in

Fig. 4. With the dovetail design the torque is sinusoidal; the reluctance torque is negli-

gible and the maximum torque 382 Nm with load angle 90 degrees is smaller than

torque (414 Nm) with the V shape design, because of asymmetric pole pairs. With the

V shape design, maximum torque is 426 Nm at load angle 102 degrees. Furthermore,

maximum reluctance torque is 49 Nm. In addition, the current behaves differently, be-

cause of different saturation. With the dovetail design, power factor is larger with elec-

tric load angles under 55 degrees and with higher load angles, it is slightly smaller.

However, efficiencies are slightly better with the V shape design.

Figure 3. Open voltage harmonics and torque ripples with speed 4800 1/min of dovetail and V shape de-

signs.

Figure 4. Calculated torque and current (left) and efficiency and power factor (right) as a function of electric

load angles with dovetail and V shape design.


Comparisons with Different Loads

Measured input and output powers as a function of current are compared to calculated

ones in Fig. 5. Generally, measured powers are smaller than calculated powers. The

possible reason is the same than with the case of voltages; leakage fluxes in the ends of

rotors. Difference is also bigger with the dovetail design. Same differences can be seen

in comparison of nominal load results.

Nominal Load

The measured and the calculated nominal load results of the two designs are compared

in the Table 2. Used stator winding temperature is 75

o

C. Approximated “Other Losses”

contains all other losses except friction and additional losses. Iron losses are calculated

with Eq. (1). All efficiencies have the same magnitude. With the dovetail design, the

calculated power factor is 4.1% better and measured power factor is 7.8% worse than

with the V shape design.

Torque oscillations are compared with different electric load angles in Fig. 6. The

oscillations are larger with the dovetail design and calculated load angles over 8 de-

grees. Oscillation is studied further with load angle 40 degrees in Fig. 6. Clear sixth

order torque harmonics can be seen with the dovetail design. With the V shape design,

remarkable twelfth order torque harmonic reduces total oscillation.

Figure 5. Calculated and measured input and output power with dovetail (left) and V shape (right) designs as

a function of current.

Table 2. Comparison of Nominal Load Results

Quantity

Measured

Dovetail design

Calculated

Dovetail design

Measured

V-shape design

Calculated

V-shape design

Shaft Power (kW) 110.2 110.4 111.1 110.3

Torque (Nm) 219.1 219.6 221.0 219.5

Voltage (V) 370 370 370 370

Current (A) 229.7 194.0 212.9 202.8

Efficiency 0.947 0.950 0.950 0.946

Power Factor 0.791 0.935 0.858 0.898

Total Losses (kW) 6.26 6.26 5.90 6.03

Copper Losses (kW) 1.66 1.19 1.30 1.18

Total – Copper (kW) 4.60 5.08 4.60 4.85

Iron Losses (kW) 1.50 1.27

Other Losses (kW) 3.58 3.58


Figure 6. Calculated torque oscillations as a function of electric angle (left) and torque as a function of time

with electric load angle 40 degrees (right) with dovetail and V shape designs.

Figure 7. Von Mises stress with speed 4800 1/min. The stresses are greatest in dark grey areas.

Strength of Structures

Stress Analysis

The rotor with the new dovetail design has a totally different stress distribution com-

pared to the V-pole rotor. In the V-pole rotor, all of the shear and tension stresses are in

the iron bridges whereas in the dovetail design, most of stresses are compression stress

in the magnets and shear stress near corners of magnets. Von Mises stresses in the

dovetail and in the V shape designs with are shown in Fig. 7. Computation is done us-

ing the centrifugal force associated with the speed of 4800 1/min. The largest stress in

electric steels of the dovetail design, 130 MPa, is locally in the corners of sheets. With

the V shape design, average stress in the inner bridges is 90 MPa and the largest stress

in electric steels, 200 MPa, is also locally in the corners of sheets. These values are

below the yield strength (305 MPa) of the steel. With the dovetail design, in center of

the smaller magnets the stress is 50 MPa. It is well below the maximum compressive

strength of the magnets. The calculated maximum stress in magnets is 381 MPa (lo-

cated in corners). Using magnets to compose the structure it becomes robust enough for

the speed of 4800 1/min.


Stability of Rotor with Dovetail Design

The mechanical durability test of the motors consists of two load runs with speed 4800

1/min. First, the motor is driven with different loads and operating temperatures for one

and half hours. After this, the motor was cooled over night. Finally, temperature test

has done for four and half hours. Measured vibration levels remained same thought all

tests. Visual check was done after test. The magnets remained solid. The glue seam

between inner magnets and rotor body was separated. In the sides of smaller magnets,

glue was changed its color from grey to light grey. This indicate that glue was de-

formed, not separated. Hence, stability of the rotor remained, but more tests should be

done to see whether stability remains with longer period.

Conclusion

The prototype machine with a dovetail-shaped magnet poles exhibits a significant in-

crease in mechanical stability over the conventional V-pole design. By converting

the tensile stress in the iron bridges into a compressive stress in the magnets by redes-

igning the pole geometry, a very robust construction can be achieved. The electrical

properties and the consumption of magnetic material can be kept on the same level as

in the V-pole design.

References

[1] Kolehmainen J., “Finite Element Analysis of Two PM Motors with Buried Magnets”, ICEM, Krakow,

Poland, 2004, SPRINGER MONOGRAPH “Recent Developments of Electrical Drives”, Nov. 2006.

[2] Ohnishi T., Takahashi N., “Optimal design of efficient IPM motor using finite element method”, IEEE

Trans. Magn., vol. 36, no. 5, 3537-3539, Sep. 2000.

[3] Kolehmainen J., Ikäheimo J., “Motors with Buried Magnets for Medium Speed Applications”, IEEE

Trans. Energy Convers., to be published.

[4] Flux2D software – www.cedrat.com.

[5] ACS 607-0400-5, frequency converter – www.abb.com.


FEM Study of the Rotor Slot Design

Influences on the Induction Machine

Characteristics

Joya KAPPATOU, Kostas GYFTAKIS and Athanasios SAFACAS

University of Patras, Department of Electrical and Computer Engineering,

Electromechanical Energy Conversion Laboratory, 26500 Rion Patras, Greece


Abstract. A library of parameterized Squirrel Cage Induction Machine models has

been constructed regarding the rotor slot design and is used to investigate the in-

fluences of the rotor bars shape on the machine characteristics. The slot shapes

chosen correspond to standard induction machines used for specific applications.

The calculations were conducted using the Finite Element Model (FEM) of the

machine and the torque and current waveforms against speed, as well as the field

distribution and the copper losses in the rotor bars for every model at starting have

been obtained. Useful conclusions about the influences of the slot design on the

Induction Machine behavior, mainly at starting are derived.

Introduction

As known, the rotor of Induction Machines has stronger effect than the stator on the

performance characteristics of the machine and especially on its starting performance.

On the other hand, there are specific limits of the machine variables, e.g. maximum

value for the starting current and minimum values for the starting torque, pull-up

torque and breakdown torque, depending upon its ratings. Thus, different design

classes for cage Induction Motors are set, each one suitable for different performance

requirements and specific applications. These design classes refer to the rotor design

and specially to the geometrical characteristics of the rotor slots. Furthermore the inter-

est of both the designers and the users is directed to the optimization of the machine

performance and the increase of the efficiency. For that reason several papers have

been published during the last decades on the analysis and optimization of cage Induc-

tion Motors and specially on the suitable design of the rotor slots, which is critical for

the starting performance and the shape of the torque-speed curve [1–5].

In this paper in order to compare more accurately the performance of cage Induc-

tion Machines of different rotor slot design, the 2-d Finite Element Method is used.

Four different rotor slot designs are used, which are presented schematically in Fig. 1

and they refer to standard squirrel cage motors used for specific applications. The geo-

metrical variables of the rotor cross section have been parameterized and a FEM code,

which is used to model the motor has been developed. In order to obtain a more flexi-

ble and user friendly code, a library of four parameterized models has been constructed,

which differ between them as for the shape of the rotor slots. As the shape of the slot

remains the same in each model, the slot dimensions can easily and accurately be modi-


247

fied. In all cases the stator model, the air-gap length, the stator voltage and frequency,

as well as the number of rotor slots are the same in order to compare the influences of

the rotor slots design on the operational characteristics of the machine. Besides, as the

rotor core is usually slightly saturated and also the skin effect plays an important role,

specially in the deep-bar design, the same B-H curve of the ferromagnetic material and

a suitable number of Finite Elements in the slot area are used. In all these cases the

distribution of the field in the core and the waveforms of the torque and current as

functions of the speed are calculated and useful results of the rotor slots shape effect on

the machine behavior are derived. The use of the models library results in a flexible

and more friendly code, which can be used for the application of optimization tech-

niques for every specific design type, as fewer design geometrical variables are used

than in a model of arbitrary slot shape.

FEM Analysis

The numerical analysis is based on a magnetic vector potential formulation and a

commercial package, OPERA of VF, has been used for the Finite Elements Analysis.

The model can include non-linear materials as the magnetic saturation plays an

important role, mainly in the rotor of the machine. In order to model the skin effect,

specially in the cases of deep bars and double cage a suitable mesh has been con-

structed in the bars area. In the above cases, high currents are induced in the rotor bars,

specially at starting and the skin effect must be taken into account. The stator windings

are fed from sinusoidal voltage sources, which are connected via external circuits to the

model.

A library of four parameterized models of Asynchronous machines has been con-

structed, which differ between them as for the shape and the dimensions of the cage

bars. Each design from now on will be reported with the letters a,…,d according to

Fig. 1. All models have the same stator with 36 slots and 2 pole pairs, the same value

and frequency of the supply voltage and 48 rotor slots. Although the dimensions of the

rotor bars were selected arbitrarily for the various models, some parameters which af-

fect the motor performance are computed and some useful qualitative conclusions re-

garding the influence of the bars design on the motors behaviour can be derived. These

are calculated mostly for the starting, which is critical for the machine performance.

Simulation Results

As known the design of the rotor bars affects strongly the motor performance and spe-

cially its starting. An arbitrary bar design can result to higher torque and less current

a) b) c) d)

Figure 1. Characteristic rotor slots designs.

J. Kappatou et al. / FEM Study of the Rotor Slot Design Influences248

during starting, but to much greater slip and consequently to a reduction of the effi-

ciency at nominal load. In this work the dimensions of the rotor bars were selected ar-

bitrarily for the various models and we can not achieve general conclusions. However

some parameters which affect the motor performance, mainly at starting, are computed

and some useful qualitative conclusions regarding the advantages and disadvantages of

each model can be derived.

In Fig. 2 the flux lines at starting on a cross section of the machine for the various

models of the slots, according to Fig. 1, are presented. In the same plot the current den-

sity in the bars is superimposed also. As the bar becomes deeper, Figs a and d, the leak-

age of the rotor increases and less flux is penetrating into the rotor body.

At starting the skin effect plays an important role because the skin depth is at its

smallest value. The distribution of the current density in the bars, specially in the case

of the double cage, Fig. 2d, shows clearly the skin-effect. In Fig. 3 the flux lines and

a) b)

c) d)

Figure 2. Flux lines and current density in the bars at starting for the various models of the slots according to

Fig. 1: a) bar model a, b) bar model b, c) bar model c, d) bar model d.

Figure 3. Flux lines and current density in the bars near synchronous speed for the model of the double cage,

bar model d.

J. Kappatou et al. / FEM Study of the Rotor Slot Design Influences 249

the distribution of the current density in the bars is presented for the case of the double

cage near the synchronous speed (1 Hz) for comparison with the starting, Fig. 2d.

For a better display of the skin effect, the spatial distribution of the current density

along the axis j-k of a rotor bar at starting is demonstrated in Fig. 4. In all models the

same bar has been selected for the calculations. The value of the electrical conductivity

of the rotor bars has been taken equal to 60 kS/ mm, except for the bar design c, where

a second case of less conductivity, 40 kS/mm, has been analyzed also for comparison.

The current density increases in the top of the slot, specially for the case of double

cage, Fig. 4γ, as a result of the skin-effect. In Fig. 4β the influence of the electrical

conductivity of the bars on the current density in the bar type c is demonstrated.

When a non-linear B-H curve is used in the model of the motor with double cage,

bar type d, the current density is gradually increasing along the axis of the bar compar-

ing to the linear case, essentially in the top of the slot near the air-gap.

The effect of the bar design on the waveforms of the torque and stator current

against speed is presented in Fig. 5. The motor design c, which develops the highest

starting torque is a high slip motor, while the smallest value of starting current appears

in the motor design d.

Although the dimensions of the rotor bars were selected arbitrarily for the various

models, some more motors parameters which affect the motor performance are com-

puted and some useful qualitative conclusions regarding the advantages and disadvan-

tages of each model can be derived. These are calculated at starting and are presented

in the following Table 1.

From Fig. 5 and Table 1 the following useful conclusions are obtained:

• The starting current is lower in the case of the double cage, type d, comparing

with the motor of bar type c. This can be explained from the higher value of

β γ

Figure 4. Spatial distribution of the amplitude of current density along the axis (j-k) of a rotor bar, in A/mm2

,

at starting for the different bars used: α) the type of bars a and b, β) the bar type c for two values of the con-

ductivity of the bars, γ) the bar type d for the linear and the non-linear B-H curve.


the rotor leakage and consequently the leakage reactance in the first case, as

observed in Fig. 2 c and d. Although the starting torque is greater in case c,

the ratio of starting torque per current presents higher value in the bar type d,

particularly when the magnetic saturation is taking into account in the model,

the above ratio increases more.

• The decrease of the conductivity of the bars in the bar design c, which results

in higher bar resistance, increases the starting torque and limits the starting

current. Particularly the ratio of the starting torque per ampere increases from

0.679 to 1.012.

• The mean ohmic losses per surface in the bars in the design c are much greater

than in any other case. In Table 1 also appears the resistance of a cage bar, as

calculated, for every model.

Conclusions

A library of four parameterized Squirrel Cage Induction Motors models of different

bars shapes has been constructed and an investigation of the effect of the bars design on

the characteristics of the motor has been carried out. The waveforms of torque and cur-

rent as function of speed are calculated and compared for all models. Moreover the

distribution of the field and some more parameters at starting, like the variation of the

0 200 400 600 800 1000 1200 1400 1600

0

5

10

15

20

25

30

35

0 200 400 600 800 1000 1200 1400 1600

0

5

10

15

20

25

30

35

0 200 400 600 800 1000 1200 1400 1600

0

5

10

15

20

25

30

35

0 200 400 600 800 1000 1200 1400 1600

0

5

10

15

20

25

30

35 b

d

M (N

m)

a

n (rpm)

c

0 200 400 600 800 1000 1200 1400 1600

0

10

20

30

40

50

0 200 400 600 800 1000 1200 1400 1600

0

10

20

30

40

50

0 200 400 600 800 1000 1200 1400 1600

0

10

20

30

40

50

0 200 400 600 800 1000 1200 1400 1600

0

10

20

30

40

50

a

c

n (rpm)

d

I(A)

b

a b

Figure 5. Waveforms for the various designs of the bars of: a) the torque, b) the stator current.

Table 1. Performance characteristics for the different rotor slots design at starting

Bar type according

to Fig. 1

Torque

(Nm)

Torque/Is

(Nm/A)

Amplitude of

current in the

stator (A)

Total ohmic losses

in the bars (W)/1bar

resistance (mΩ)

Mean ohmic losses

per surface in the

bars (W/mm2

)

a 12.18 0.268 45.352 5856.302/0.0938 2.8632

b 13.68 0.282 48.4 6812.75/0.09609 3.7955

c, conduc. = 60 kS/mm 24.2 0.679 35.63 8907.686/0.2354 14.944

c, conduc. = 40 kS/mm 30.212 1.012 29.8354 9202.5/0.3527 15.4385

d, linear B-H

d, non-linear

16.957

24.29

0.853

1.079

19.86

22.5

4526.206/0.425

5863.27/0.425

3.372

4.368

J. Kappatou et al. / FEM Study of the Rotor Slot Design Influences 251

current density along the axis of a bar, as well as the ohmic losses in the cage have

been computed, which give us some useful qualitative conclusions regarding the ad-

vantages and disadvantages of each model.

References

[1] M. Nurdin, M. Poloujadoff, A. Faure, “Synthesis of squirrel cage motors: A key to optimization”, IEEE

Trans. on Energy Conversion, Vol. 6, No. 2, pp. 327-335, June 1991.

[2] S. Williamson, C. McClay, “Optimization of the geometry of closed rotor slots for cage Induction mo-

tors”, IEEE Trans on Industry Applications, Vol. 32, No. 3, pp. 560-568, May/June 1996.

[3] M.R. Feyzi, H.V. Kalankesh, “Optimization of Induction motor design by using the finite element

method”, CCECE/CCGEI 2001.

[4] C. Grabner, “Investigation of Squirrel cage Induction motors with semi-closed and closed stator slots by

a transient electromechanical finite element technique”, ISEF 2005, Baiona, Spain, conference proceed-

ings record, September 2005.

[5] Min-Kyu Kim, Cheol-Gyun Lee, Hyun-Kyo Jung, “Multiobjective optimal design of three-phase Induc-

tion Motor using improved evolution strategy”, IEEE Trans on Magnetics, Vol. 34, No. 5, pp. 2980-2983,

September 1998.


Concentrated Wound Permanent Magnet

Motors with Different Pole Pair Numbers

Pia SALMINEN, Hanne JUSSILA, Markku NIEMELÄ and Juha PYRHÖNEN

Lappeenranta University of Technology, Department of Electrical Engineering,

P.O. Box 20, 53851 Lappeenranta, Finland

[email protected]

Abstract. The study addresses the torque production capabilities and losses of

concentrated wound permanent magnet machines. Different permanent magnet

synchronous motors are modelled with both semi-closed slots and open slots. With

open slots, the coils can be easily assembled to the stator, thus making it attractive

to study the performance or benefits of the concentrated wound PM machines with

open slots. Different slot and pole combinations are considered on the basis of the

finite element analysis (the Flux2D program package by Cedrat being employed in

the computations).

Introduction

This work examines the performance of different model designs for a machine with a

rated torque of 1075 Nm, a frame size of 225, and a rated speed of 400 rpm. These val-

ues represent ratings for machines used for example in the paper making industry. To

compare the machines, the electromagnetic losses are calculated to estimate the effi-

ciency of each machine. Particular attention is paid to the iron losses, Joule losses (I2

R

losses), and eddy current losses caused by the permanent magnets of concentrated

wound PM machines.

Concentrated wound machines are one type of fractional slot wound machines.

The number of slots per pole and per phase q ≤ 0.5. In a concentrated wound machine,

each coil is wound around one tooth to achieve as short end windings as possible. This

reduces the amount of copper and leads to low Joule losses by virtue of the short end

winding. It is also possible to utilize short end windings by inserting a longer stator

stack compared to the stack of integer slot wound machines in the same frame size.

Consequently, longer active parts will give more torque. Figure 1 illustrates a concen-

trated wound prototype machine, constructed within a size 225 frame. In this case, the

end windings are so short that it is possible to make the stator stack 30% longer com-

pared to the stack length of a four pole integer slot wound induction motor [1–3].

Finite Element Analysis

A set of finite element analyses (FEA) is performed to estimate the pull-out torque and

the losses of each design. The pull-out torque is the maximum torque that a motor can

sustain at synchronous operation. The finite element computations are based on Cedrat

Flux2D [4] program package using transient analysis for model concentrated wound


253

machines with different pole pair numbers, p. Figure 2 (a) shows a machine with a low

pole pair number p = 4, and 12 slots. For this machine, the number of slots per pole and

per phase q = 0.5. In Fig. 2 (b), the pole pair number is high (p = 15), there are 36 slots,

and q = 0.5. It has been noticed in previous studies [5] that with q = 0.5 the torque pro-

duction capability is higher than with q < 0.5. When q = 0.5, the cogging torque and

torque ripple may be higher than with other possible concentrated wound combina-

tions, while it is still possible to minimize torque ripples by optimising the magnet

width [6].

Semi-Closed Slots

The machines under investigation have surface mounted magnets as shown in Fig. 2.

Most of the models are constructed with semi-closed slots in the stator, except for cer-

tain special cases that have open slots. The boundary conditions for machine dimen-

sions are determined by the size 225 frame; the outer diameter 364 mm and the ma-

chine length 270 mm. The rated speed is 400 rpm and the rated torque is 1075 Nm. The

total losses of concentrated wound PM machines calculated are presented in Fig. 3.

The total losses comprise the iron losses, Joule (I2

R) losses, and eddy current

losses caused by the permanent magnets. It can be seen in Fig. 3 that the Joule losses

dominate by a large margin, but their proportion of the total losses decreases when p is

large. Consequently, if low Joule losses are an important design parameter, it is advis-

able to avoid low pole pair numbers. However, the iron losses increase as the number

of poles increase, which is also expected as the frequency increases. The proportion of

D

270 mm

Stator stack

end

windings

end

windings

Figure 1. Concentrated wound PM machine in a size 225 frame. The short end windings leave more space

inside the frame; hence, the stator stack may be designed longer in the axial direction.

a) b)

Figure 2. Concentrated wound PMSM a) with a low pole pair number and b) with a high pole pair number.

P. Salminen et al. / Concentrated Wound Permanent Magnet Motors254

iron losses to the machines in this study rises from 0.3 kW to 1 kW as the frequencies

increase from 27 to 140 Hz. Please note that the minimum net losses occur when p = 10

(see Fig. 3). The machine with p = 10 has about 25% lower net losses than p = 4 and

p = 21 machines.

The model parameters for the magnet material are based on the values for Neorem

495a100, the remanence flux density of which is Br = 1.05 T and the coercivity H

C =

800 kA m

–1

. The resistivity of the sintered magnets permits remarkably large eddy cur-

rent losses if the flux density may pulsate in the magnet during the motor running. This

phenomenon makes concentrated wound machines quite vulnerable to large eddy cur-

rent losses in the magnets. In integer slot wound machines with semi-closed slots the

flux density pulsation in the magnets may be negligible compared to fractional slot

machines with concentrated windings. The aim here is to investigate what kind of a

role the permanent magnet eddy current losses have at low speed (e.g. 400 rpm). Usu-

ally, a machine with geometrically wide magnets generates high eddy current losses,

while small or narrow magnets generate lower eddy current losses [7].

Figure 4 shows the eddy current losses caused by permanent magnets for several

surface-mounted PM machines. As expected, the machines with low pole pair numbers

have high eddy current losses because of their large magnets in the rotor. Geometri-

cally large surfaces are harmful with respect to the eddy current losses.

Semi-Closed Slots Compared with Open Slots

Next, the effect of the slot opening width on the losses of PM machines is examined.

The above semi-closed machines are now modified so that they have open slots. The

results obtained by the FEA are shown in Fig. 5. The results demonstrate that only the

iron losses are smaller with the open slot structures than with the semi-closed slot

structures. The Joule losses turn out to be higher with open slots than with semi-closed

slots, because more winding turns are needed in the stator slots to induce sufficient

back electromotive force in the system. With open slots, the eddy current losses caused

by permanent magnets are approximately double compared with those of motors using

Figure 3. Iron losses, Joule losses, and eddy current losses caused by permanent magnets for machines with

surface mounted magnets having semi-closed slots. The results are obtained by the FEA.

P. Salminen et al. / Concentrated Wound Permanent Magnet Motors 255

semi-closed slots. For a machine with a low pole pair number (e.g. p = 5), the eddy

current losses with semi-closed stator slots are calculated to be 400 W, while with open

slots, the losses rise to 950 W. This result suggests that the eddy current losses in the

case of concentrated wound PM machines should be evaluated carefully even at low

speeds. With high pole pair numbers, the magnet dimensions are small, and therefore

the eddy current losses are low.

The calculated pull-out (maximum synchronous operating) torques for four pairs

of model motor designs are shown in Fig. 6. Each of the designs has either open or

semi-closed stator slots. For both types, when p = 21, the available torque is relatively

0

100

200

300

400

500

0 4 8 12 16

Pole pair number, p

Losses (W

)

12 Slots

24 Slots

36 Slots

Figure 4. Eddy current losses caused by permanent magnets for machines with surface-mounted magnets

having semi-closed slots. Results are obtained by the FEA.

Figure 5. Iron losses, Joule losses and eddy current losses caused by the permanent magnets for machines

with surface mounted magnets having semi-closed slots and open slots. Results are obtained by the FEA.


small and not enough for appropriate motor operation. The machines with 16 poles and

24 poles have a high torque, and also their efficiency shows good performance. Open

slot motors achieve slightly higher torque levels than the ones with semi-closed slots,

except when p = 10; however, the efficiency with open slots is lower than with semi-

closed slots.

Conclusion

By applying finite element analytical methods, this paper examines several factors that

affect the performance and efficiency of permanent magnet electric motors. The au-

thors have, however earlier experimental data (e.g. [2]) that confirms the validity of the

calculation process. Joule heating accounts for most of the losses; however, also eddy

current effects play a significant role and must therefore be included in the analysis.

For machines with rotor surface magnets, the electromagnetic energy losses are higher

in the machines with an open-slot stator than in the machines with a semi-closed slot

structure, especially when the pole pair number is low. This is a consequence of the

flux pulsation in magnets and the large magnet surface area. It is possible to achieve a

high pull-out (maximum) torque with both open slots and semi-closed slots by using an

intermediate number of poles and slots; these designs also exhibit the lowest Joule

losses. Since Joule losses are the dominant factor in the machines (machine design), the

further studies will focus on embedded magnet motors. It is hoped that in these motors,

the Joule losses will be low even when the number of pole pairs is large.

References

[1] Cros J., Viarouge, P., Carlson, R. and Dokonal, L.V. 2004. Comparison of brushless DC motors with

concentrated windings and segmented stator. Proceedings of the International Conference on Electrical

Machines. ICEM 2004, Krakow, Poland. CD-ROM.

0

1

2

3

Semi Open Semi Open Semi Open Semi Open

Pull-out torque (p.u.)

Torque

12-slot-10-pole 24-slot-16-pole 36-slot-24-pole 36-slot-42-pole

Figure 6. Pull-out torque for machines with surface mounted magnets having semi-closed slots and open

slots. Results are obtained by the FEA.

P. Salminen et al. / Concentrated Wound Permanent Magnet Motors 257

[2] Salminen, P. 2004. Fractional slot permanent magnet synchronous motor for low speed applications. Dis-

sertation, Acta Universitatis Lappeenrantaensis 198, Lappeenranta University of Technology. 151 p.

[3] Libert, F. and Soulard, J. 2004. Investigation on Pole-Combinations for Permanent-Magnet Machines

with Concentrated Windings, Proceedings of the International Conference on Electrical Machines, ICEM

2004, Krakow, Poland.

[4] Cedrat 2007. Software solutions: Flux®. [Online] Available from http://www.cedrat.com/ [Date accessed

26.6.2007].

[5] Salminen, P., Jokinen, T. and Pyrhönen, J. 2005. The Pull-Out Torque of Fractional-slot PM-Motors with

Concentrated Winding, Electric Power Applications, IEE Proceedings, Vol. 152, Iss. 6, pp. 1440–1444.

[6] Salminen P., Pyrhönen J., Libert F., and Soulard J. 2005. Torque Ripple of Permanent Magnet Machines

with Concentrated Windings. 15–17 September 2005, International Symposium on Electromagnetic

Fields in Mechatronics, ISEF 2005, Electrical and Electronic Engineering, Baiona, Spain.

[7] Zhu, Z.Q., Ng, K., Schofield, N. and Howe, D. 2004. Improved analytical modelling of rotor eddy cur-

rent loss in brushless machines equipped with surface-mounted permanent magnets. Electric Power Ap-

plications, IEE Proceedings. Vol. 151, Issue 6, 7 Nov. 2004 Page(s):641–650.


Air-Gap Magnetic Field of the Unsaturated

Slotted Electric Machines

Ioan-Adrian VIOREL, Larisa STRETE, Vasile IANCU and Cosmina NICULA

Electrical Machines Department, Technical University of Cluj-Napoca,

Daicoviciu 15, 400020, Romania


Abstract. This paper deals with the air-gap magnetic field analytical calculation in

the case of unsaturated slotted electric machines. Different analytical estimations

of the flux density variation versus circumferential coordinate in the machine air-

gap are considered. The obtained results are compared between them and with the

two dimension finite element method (2D-FEM) calculated values, where the iron

core material nonlinearity is fully considered.

1. Introduction

The air-gap flux density provides valuable information in evaluating slotted electric

machine performance. Any method to design an electric machine, regardless its type,

requires knowledge on the air-gap magnetic flux density to calculate the main dimen-

sions, the necessary mmf, the torque (average, maximum, starting, cogging or ripple),

the back emf value and shape and the main inductances, which may be dependent on

the rotor position. It is clear by now that the common, and the most accurate way of

obtaining the air-gap flux density is based on finite element method (FEM) calculation,

but this is still time consuming even on powerful computers and it is difficult to use

FEM in iterative design optimizing procedure, or to implement FEM results directly in

on line control systems.

The electric machines’ air-gap flux density calculation was in the researchers’ at-

tention for a very long time. Important results in the domain, concerning the air-gap

magnetic field in synchronous and respectively induction machine were published in

the twenties of the last century, as were the works of Weber [1], Spooner [2], Carter [3]

and Wieseman [4]. Heller’s book, [5], represents a synthesis of Heller entire work, con-

taining also the most important other contributions published until the sixties last cen-

tury concerning the air-gap field in the induction machine. In the last years, two types

of electric machines were in the attention of the researchers, the synchronous perma-

nent magnet (PM) machine with PMs in the air-gap [6–8], and the switched reluctance

machine (SRM) [9,10]. In the case of rotating or linear transverse flux reluctance mo-

tors, the air-gap magnetic field calculation is important too and some results are given

in [11,12], but the problem is not too different from the case of SRM’s. The calculation

of the air-gap permeance of the double-slotted electric machines was also done, results

being published, as in [13–15].

Since the case of the machines with PMs on the surface of rotor and stator was

quite extensively studied recently, [6–8] for instance, in this paper will be considered


259

only the case when there are no PMs in the air-gap, or buried inside the stator or rotor

core. The stator and rotor core surfaces are considered as equipotential ones; therefore

the magnetic field in the air-gap can be analytically described by a one-dimension (1D)

model.

The air-gap flux density variation versus circumferential coordinate, calculated via

2D-FEM analysis, is, as expected, continuous and smooth, without local points where

its first derivative became zero on the entirely domain, half of the slot pitch, but at the

domain extremities. Due to this important observation, the air-gap flux density varia-

tion requires a simple analytical estimating function in the cases when only Carter’s

factor, torques or back emf have to be calculated. When main phase inductance de-

pends on the rotor position, SRM for instance, the air-gap flux density variation must

be approximated by an analytical function which gives values as closed as possible to

the actual ones, since a simple estimating function might introduce errors.

Different analytical estimations of the air-gap flux density variation versus circum-

ferential coordinate are considered in the paper. The obtained results are compared

between them and with the 2D-FEM calculated values on a simplified machine model.

The Carter’s factor, which represents a criterion for the estimation accuracy, is calcu-

lated too and compared with that obtained via 2D-FEM analysis. The best fitted estima-

tions for different cases are discussed and pertinent conclusions regarding the analytical

estimations of the air-gap flux density are presented.

2. 2D-FEM Analysis

The 2D-FEM analysis was carried on a simple model. The model’s structure contains

two slots, a coil on the stator and a non-salient rotor. For symmetry reasons the stator

length is equal to two tooth pitches. The notations contain the suffixes S, R for stator

and rotor.

The topology was parameterized in a way to make possible adequate variation of

the most important dimensions, only the tooth pitch, which is the circumferential

length, was kept constant. A linear layout was considered, but it does not affect the

generality.

In the air-gap were considered six layers in order to obtain a more homogenous

distribution of the vector magnetic potential affected by the magnetic permeability dif-

ference between the core and the air-gap domain. The flux density, calculated in the

middle of the air-gap, has a smooth and continuous variation, Figs 2 and 3.

If the ratio between the tooth width and the double of the stator yoke is small, then

the iron-core is unsaturated even if the air-gap flux density has large values, Table 1,

via 2D-FEM analysis, when t/g = 37.5, wt/2h

yS = 0.667, w

t/2h

yR = 0.333.

Based on 2D-FEM analysis obtained values, the Carter’s factor is calculated as the

ratio between the peak and the average value of the air-gap flux density,

max

/C g gav

K B B= (1)

The saturation coefficient Ksat

results as:

max max

/sat g unsat g

K B B= (2)

I.-A. Viorel et al. / Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines260

where the saturated peak air-gap flux density value Bgmax

is obtained in the tooth axis

via 2D-FEM analysis and the unsaturated air-gap flux density value is:

max 0

/ 2g unsat

B F gμ= (3)

with the following usual notations, Fig. 1, t – tooth pitch, wt – tooth width, w

s – slot

width, hyS

, hyR

– stator and rotor yoke radial length, g – air-gap radial length, F – coil

mmf in ampere turns, µ0 – air-gap permeability (4π10

–7

H/m).

The air-gap flux density average value Bgav

is given by:

/ 2

0

2

( )

t

gav gB B x dx

t

= ∫ (4)

where Bg(x) are the air-gap flux density values obtained at equidistant points via

2D-FEM analysis, the integral being numerically calculated.

Examples of the air-gap flux density variation versus the circumferential coordi-

nate x ∈ [0,t/2], calculated by FEM, are given for two values of mmf in each case, in

Figs 2 and 3. The values of the actual mmf, the pole pitch to air-gap length ratio t/g, the

slot width to double air-gap length ws/2g ratio, peak and average flux density B

gmax,

Bgav

, Carter and saturation factors KC, K

sat for all cases presented in Figs 2 and 3 are

shown in Table 2.

Two important remarks should be made considering the curves shown in Figs 2

and 3, and the values given in Table 2:

Table 1. Saturation Ksat

and Carter KC factors

Bgmax

0.8546 1.16 1.433 1.526

Ksat

1.011 1.015 1.042 1.183

Kc

1.3597 1.362 1.368 1.372

Figure 1. Machine model with nonsalient rotor.

I.-A. Viorel et al. / Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines 261

Table 2. Representative values for the curves given in Figs 2 and 3

Figure t/g ws/2g F[A] Bgmax[T] Bgav[T] KC Ksat

2200 0.855 0.628 1.36 1.011

Fig. 2 37.5 6.25

3800 1.433 1.047 1.368 1.042

1000 0.776 0.304 2.556 1.011

Fig. 3 75 25

1800 1.383 0.539 2.568 1.022

Figure 2. Flux density variation, t/g = 37.5, ws/2g =

6.25.

Figure 3. Flux density variation, t/g = 75, ws/2g =

25.

i) The air-gap flux density variation versus circumferential coordinate is repre-

sented by smooth and continuous curves.

ii) Even at important peak flux density values, the saturation is not important for

the case considered, wt/2h

y< 1 and consequently the Carter’s factors are not

dependent on the mmf.

3. Air-Gap Flux Density Variation

One of the analytical approximation of the air-gap flux density variation quite inten-

sively employed is based on the air-gap variable equivalent permenace, a method

which was first introduced in the case of induction motor [5,15] and was later extended

to other motors, SRM [16] or transverse flux motor (TFM) [12], for example.

The air-gap variable equivalent permeance can be easily defined and allows the

calculation of the air-gap magnetic field in the case of double slotted machine. Its accu-

racy depends on the accuracy of the estimation for the air-gap flux density variation

over a tooth pitch.

If Bgmin

is the minimum value of the air-gap flux density, then:

max min

2

( ) 0.5( )cosg gav g g

x

B x B B B

t

π≅ + − , [0, / 2]x t∈ (5)

The variable equivalent air-gap permeance is defined as:

max max max

( )1 2 1

( ) cosg gav g

g g g

B x B B x

P x

g B B B t g

π

⎛ ⎞Δ

= = +⎜ ⎟⎜ ⎟

⎝ ⎠

(6)


where the air-gap topology coefficient β is [6]:

max min

max max

2

g g g

g g

B B B

B B

β

− Δ

= =

2

1

0.5 1

1 ( / 2 )s

w g

β

⎛ ⎞

⎜ ⎟= −

⎜ ⎟+

⎝ ⎠

(7)

The air-gap variable equivalent permenace, function of the circumferential coordi-

nate x, comes:

1 1 2

( ) 1 cosr

C

x

P x p

g K t

π

⎛ ⎞= +

⎜ ⎟

⎝ ⎠

(8)

where the permeance coefficient pr= K

Cβ or, as proposed in [15],

4

sin

2

r C

g

p K

t

γ π

β

π β

⎛ ⎞

=⎜ ⎟

⎝ ⎠

(9)

2

4

tan( ) ln 1

2 2 2

s s s

w w w

a

g g g

γ

π

⎡ ⎤⎛ ⎞

⎢ ⎥= − + ⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦

(10)

Some equation, eventually more accurate than (5), can be developed, such as:

i) The one proposed by Weber [1]

2

max

( ) (1 2 sin )a

g g

x

B x B

t

β π

⎛ ⎞= −

⎜ ⎟

⎝ ⎠

, s

s

t w

a

w

−

= , [0, / 2]x t∈ (11)

ii) The one proposed by Heller [5]

max,

( )g g

B x B= , [0, )t

x wβ∈

max

( ) (1 cos )g g

B x B yβ β= − + , [ , / 2]t

x w tβ∈ ,

/ 2 / 2

t

t t

w

y x

t w t w

πβπ

β β

= −

− −

(12)

iii) A nonsinusoidal variation, considering a simplified flux lines topology in the

air-gap, [7]

max,

( )g g

B x B= , [0, / 2]t

x w∈

1

max,( ) 1 ( / 2)

2g g t

B x B x w

g

π

−

⎡ ⎤

= + −⎢ ⎥

⎣ ⎦

, ( / 2, / 2]t

x w t∈

(13)


iv) An exponential approximation, considering x = 0 in a slot axis, obtained

through a curve fitting procedure, given by:

max ( )

1

( )

1 10s

g g w x

B x Bβ −

=

+

, [0, / 2]x t∈ (14)

A comparison between the FEM analysis and different proposed approximations of

the air-gap flux density referred to its peak value is given in Figs 4 and 5. In Fig. 4 the

calculated values obtained in the case of t/g = 37.5, ws/2g = 6.25 and F = 3800 AT, as

in Fig. 2 and in Fig. 5 in the case of t/g = 75, ws/2g = 25 and F = 1000 AT as in Fig. 3,

are shown. From all the cases considered were chosen these two since there is an im-

portant difference between tooth pitch to air-gap length and respectively slot width to

double air-gap length ratios, which allows for a large degree of generality of the results

and conclusions.

4. Carter’s Factor

Basically, the Carter’s factor is defined by (1), the average air-gap flux density being

calculated accordingly to (4). Only the air-gap flux density variation on a half of a

tooth pitch is considered due to the symmetry against the tooth axis. Carter’s factor can

be calculated using different equations such as, [8]:

( )

1

1

1

1

/ 5 / 1C

s s

k

t w g w

−

⎡ ⎤

= −⎢ ⎥

+⎢ ⎥⎣ ⎦

(15)

1

2

2

2 1

1 tan( ) ln 1

2 4

s s s

C

s

w w wg

k a

t g w gπ

−

⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞

⎢ ⎥⎜ ⎟⎜ ⎟= − − + ⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦

(16)

Figure 4. Air-gap flux density variation, referred to

its peak value, FEM and proposed approximations,

t/g = 37.5, ws/2g = 6.25, F = 3800 A.

Figure 5. Air-gap flux density variation, referred to

its peak value, FEM and proposed approximations,

t/g = 75, ws/2g = 25, F = 1000 A.


1

3

4

1 ln 1

4

s s

C

w wg

k

t t g

π

π

−

⎡ ⎤⎛ ⎞

= − + +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

(17)

In Table 3 and 4 the calculated values of Carter’s factor for unsaturated machine,

Ksat

<1.04, based on Eqs (15)–(17) in comparison with the values obtained via 2D-FEM

analysis are given. As one can see, the differences are not that important, and this is

valid for a large number of values of the ratios t/g and ws/2g.

The adopted notations for the Carter’s factor are: KCFEM

– computed via FEM,

KCW

(0.5) – computed for Weber approximation (11) when a = 0.5, KCH

– computed for

Heller approximation (12) and KCN

, KCE

– computed for nonsinusoidal (13) and expo-

nential (14) approximations. In Tables 5 and 6 are also given the calculated values of

the air-gap topology coefficient β(7).

As it can be seen from Tables 5 and 6, all the proposed approximations, but

Heller’s, give accurate values for the Carter’s factor, very closed to the FEM based

calculations. In the case of Heller’s approximation the differences are quite important

due to the fact that the resulting average value of the air-gap flux density is too large

compared to the FEM calculations, which can be considered very close to the actual

value.

5. Conclusions

Five analytical approximations of the air-gap flux density variation versus circumferen-

tial coordinate are considered and the calculated values are compared with each other

and with the 2D-FEM analysis results, computed on a simple machine model with slots

only on the stator. The Carter’s factor for all considered approximations and for the

2D-FEM analysis are calculated too. The following remarks should be made concern-

ing the air-gap flux density analytical estimation:

i) The approximation based on the air-gap variable equivalent permeance

method is very simple but less accurate.

Table 3. Carter’s factor values, t/g = 150

ws/2g

KC

25 31.25 37.5 43.75 50

FEM 1.446 1.637 1.886 2.225 2.718

(15) 1.435 1.628 1.882 2.231 2.739

(16) 1.424 1.61 1.854 2.187 2.666

(17) 1.433 1.622 1.87 2.209 2.669

Table 4. Carter’s factor values, t/g = 37.5

ws/2g

KC

6.25 7.813 9.375 10.938 12.5

FEM 1.359 1.515 1.714 1.978 2.341

(15) 1.313 1.461 1.652 1.904 2.25

(16) 1.311 1.455 1.639 1.88 2.207

(17) 1.338 1.49 1.685 1.94 2.293


ii) Weber’s approximation is not so simple and gives better results for larger

values of the ratio between the tooth width and the slot opening. It shows an

important dependency on the air-gap topology coefficient β calculated values.

iii) Heller’s approximation depends on β too, is a function defined on two inter-

vals and gives a too large air-gap flux density average value, consequently a

too small Carter’s factor.

iv) The nonsinusoidal (13) and exponential (14) approximations are accurate and

produce values very closed to that calculated by a 2D-FEM analysis.

When only the Carter’s factor has to be calculated then one can use Eqs (15)–(17)

or can calculate it based on nonsinusoidal (13) or exponential (14) approximations.

The exponential approximation reproduces quite well the 2D-FEM computed char-

acteristic but its slope is a bit too large. The exponential or the nonsinusoidal approxi-

mations can be employed even when the main inductance variation has to be calcu-

lated.

As an overall conclusion, it must be said that there are good analytical approxima-

tions of the air-gap flux density variation and that they can be usefully implemented in

designing procedures or in on line control packages.

References

[1] C.A.M. Weber, F.W. Lee, “Harmonics due to slot openings”, A.I.E.E. Trans., vol. 43, 1924,

pp. 687-693.

[2] T. Spooner, “Tooth pulsation in rotating machines”, A.I.E.E. Trans., vol. 44, 1925, pp. 155-160.

[3] F.W. Carter, “The magnetic field of the dynamo-electric machine”, The Jour. of the I.E.E., vol. 64,

1926, pp. 1115-1138.

[4] R.W. Wiesseman, “Graphical determination of magnetic fields”, A.I.E.E. Trans., vol. 46, 1927,

pp. 141-148.

[5] B. Heller, V. Hamata, “Harmonic field effects in induction machines”, New York, Elsevier, 1977.

[6] Z.Q. Zhu, D. Howe, “Instantaneous magnetic field distribution in brushless permanent magnet dc mo-

tors, Part III: Effect of stator slotting”, IEEE Trans. on Magnetics, vol. 28, no. 1, January 1993,

pp. 143-151.

[7] A.B. Proca, A. Keyhani, A. El-Antably, W. Lu, M. Dai, “Analytical model for permanent magnet mo-

tors with surface mounted magnets”, IEEE Trans. on Energy Conversion, vol.18, no. 3, September

2003, pp. 386-391.

Table 5. Carter’s factor, t/ws = 1.5, Weber (11), Heller (12) approximations, a = 0.5

ws/2g 50 33.3 25 20 16.67 12.5

β 0.49 0.48 0.48 0.47 0.46 0.46

KCW

(0.5) 2.66 2.61 2.574 2.53 2.46 2.41

KCFEM

2.72 2.63 2.556 2.49 2.43 2.34

KCH

1.69 1.68 1.676 1.66 1.65 1.64

Table 6. Carter’s factor, t/g = 150 nonsinusoidal (13) and exponential (14) approximation

ws/2g 50 43.75 37.5 31.25 25

β 0.49 0.488 0.486 0.484 0.48

KCN

2.699 2.209 1.87 1.622 1.433

KCE

2.885 2.303 1.923 1.655 1.442

KCFEM

2.720 2.225 1.886 1.637 1.446


[8] D.C. Hanselman, “Brushless permanent-magnet motor design”, second edition, Writes’ Collective

Cranston, Rhode Island, SUA, 2003.

[9] S.A. Hossain, I. Husain, “A geometry based simplified analytical model of switched reluctance ma-

chines for real-time controller implementation”, IEEE Trans. on Power Electronics, vol.18, no. 6,

pp. 1384-1389, Nov. 2003.

[10] H.-P. Chi, R.-L. Lin, J.-F. Chen, “Simplified flux linkage model for switched reluctance motors”, IEE

Proc. – Electr. Power Appl., vol. 152, no. 3, pp. 577-583, 2005.

[11] J.-H. Chang, D.-H. Kang, I.-A. Viorel, Larisa Strete, “Transverse flux reluctance linaer motor’s analyti-

cal model based on finite element method analysis results”, IEEE Trans. on Magnetics, vol. 43, no. 4,

April 2007, pp. 1201-1204.

[12] I.-A. Viorel, G. Henneberger, R. Blissenbach, L. Löwenstein, “Transverse flux machines. Their behav-

iour, design, control and applications”, Mediamira, 2003, Cluj-Napoca, Romania.

[13] G. Qishan, Z. Jimin, “Saturated permeance of identically double-slotted magnetic structures”, IEE

Proc.-B, vol. 140, no. 5, pp. 323-328, 1993.

[14] G. Qishan, E. Andersen, G. Chun, “Airgap permeance of vernier-type, doubly-slotted magnetic struc-

tures”, IEE Proc.-B, vol. 135, no. 1, pp.17-21, 1988.

[15] I.-A. Viorel, M.M. Radulescu, “On the calculation of the variable equivalent air-gap permeance of in-

duction motors” (in Romanian), EEA-Electrotehnica, vol. 32, no. 3, 1984, pp. 108-111.

[16] I.-A. Viorel, A. Forrai, R.C. Ciorba, H.C. Hedesiu, “Switched reluctance motor performance predic-

tion”, Proc. of IEE-IEMDC, Milwaukee, USA, 1997, TBI-4.1-4.3.


Dynamic Simulation of the Transverse

Flux Reluctance Linear Motor

for Drive Systems

Ioan-Adrian VIORELa

, Larisa STRETEa

and Do-Hyun KANGb

a

Electrical Machines Department, Technical University of Cluj-Napoca,

Daicoviciu 15, 400020, Romania


b

Korean Electro-technology Research Institute, Changwon 641-120, South Korea

[email protected]

Abstract. Three different models of transverse flux reluctance linear motor

(TFRLM) are considered – a simple one which neglects the saturation, one based

on the look-up table technique and an analytical model; the last two fully consider-

ing the nonlinearities and based on finite element method (FEM) analysis. The

models are compared concerning the TFRLM dynamic regime when the motor is

fully controlled in a specific drive system.

1. Introduction

The paper’s goal is to simulate the dynamic regime of the Transverse Flux Reluctance

Linear Motor (TFRLM) when the motor is fully controlled in a specific drive system.

Three variants of models are considered – a simple linear one, neglecting the satura-

tion, a look-up table technique based on FEM results and an analytical model which

fully considers the nonlinearities and is also based on FEM results.

The machine analyzed is a basic transverse flux reluctance linear motor (TFRLM).

Some of the advantages of linear motors come from the fact that they deliver the re-

quired linear motion directly, without the need of an intermediate system to transform

rotation in translation.

TFRLM has a double salient structure and a passive mover. It has the same number

of poles on both parts on the stator and on the mover, and each of TFRLM’s phase is

constructed as an independent module, Fig. 1, [1,2]. Basically the transverse flux reluc-

tance motor (TFRM), in a rotating or linear construction, is similar to the switched re-

luctance motor (SRM) except for the armature winding which is of ring type with ho-

mopolar features. The force/torque in both cases, TFRM or SRM is produced by the

tendency of the movable part to reach a position where the inductance or the flux link-

age of the stator phase winding is maximized. Due to this similitude, equivalence be-

tween these two machines can be defined [3].

The transverse flux machine (TFM) is capable to deliver high power densities due

to its homopolar topology as it allows an increased number of poles without reducing

the MMF per pole [1].

Any attempt to describe, or to predict TFRLM dynamic behavior relays on its

mathematical model, which, as in the case of SRM, is not a simple one since both the



doi:10.3233/978-1-58603-895-3-268

268

stator and the mover have salient poles and the iron core should be quite highly satu-

rated in order to obtain good performances. The TFRLM mathematical models, or the

SRM ones which are basically similar, presented in the literature start from a simplified

idealized one [4] and continue with different models, mostly based on numerical field

calculation. Some of the models [5,6] use directly the 2D-FEM analysis results, others

only build analytical models based on 2D-FEM obtained values [7–10].

In this paper three, quite representative models are considered:

i) An analytical simplified one which does not consider the saturation effect,

which lays on the idea formulated in [4].

ii) An analytical one based on 3D-FEM analysis obtained values which com-

bines, in an original way, the models presented in [7–9].

iii) A model employing the look-up table technique, the values being obtained di-

rectly via a 3D-FEM analysis as in [5].

After a brief presentation of the sample TFRLM considered in the paper, each

model will be described and representative comparative results will be shown. Some

conclusions concerning the accuracy and versatility of the models used to study

TFRLM dynamic behavior will end up the paper.

2. TFRLM Construction

TFRLM has a double-salient topology and a passive mover, with the same number of

poles on both parts, Fig. 1. The flux is generated by the current flowing through the

coils and the thrust is based on the minimum reluctance principle; the movable part is

acting to achieve a position where the inductance and the flux linkage of the stator ex-

cited winding are maximized. The moving direction is perpendicular to the magnetic

flux.

The transverse flux reluctance linear motor is simple, robust and low cost. Due to

the latest improvements of its construction, design and control, TFRLM has become a

valuable option for variable speed linear drives.

A sample TFRLM was designed to deliver a 1000[N] thrust. The rated current is

50[A] and the number of coil turns per phase is N = 80. The motor is a long one with

Figure 1. TFRLM topology.

I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems 269

four phases and 30 pole pairs per phase. The air-gap g = 1 mm and the pole length on y

direction is wp = 20 mm. The stator and the mover core are made from the same mate-

rial, S23, which has the initial relative permeability µri

= 2023 and a saturation flux

density value of Bs = 2.1 T.

3. Mathematical Models

The machine behavior is fully described by the following equations:

ph

ph ph ph

did dx

V R i R i

dt x dt i dt

λ λ λ∂ ∂

= ⋅ + = ⋅ + +

∂ ∂

(1)

( , )f i xλ = (2)

0

i

F di

x

λ∂

=

∂

∫ (3)

2

2f

d x dx

F m c

dtdt

= + (4)

where Vph

, iph

, λ, R, F are the phase voltage, current, flux linkage, resistance and pro-

duced thrust and m, cf are the mover mass and friction coefficient.

It means that the motor behavior can be obtained by solving (1)–(4) if the parame-

ters (resistance and inductance) can be calculated or measured and if the flux linkage

dependence on the current and mover position can be defined analytically or based on

FEM analysis, respectively on tests.

The TFRLM dynamic behavior, studied by using MATLAB/Simulink environ-

ment, is described by:

2

2

1

ph ph

f

V R i

t

d x dx

F c

m dtdt

λ∂

= − ⋅

∂

⎛ ⎞= −

⎜ ⎟

⎝ ⎠

(5)

the flux linkage and the thrust being given by (2), (3) once f (i,x) is defined.

In the following, the way f (i,x) is obtained in the three cases considered is detailed.

3.1. Linear Model

In the linear model the flux leakage is neglected and the flux linkage is considered like

varying linearly with the current and mover position. It simply means that:

2

2

p

ph

m

wF

N N x i

g

μ

λ

⋅

= = ⋅

ℜ

(6)

I.-A. Viorel et al. / Dynamic Simulation of the TFRLM for Drive Systems270

where the magnetic reluctance m

ℜ is:

2

,m

p

g

Aμ

ℜ =

⋅

p p

A w x= ⋅ (7)

the core reluctance being neglected against that of the air-gap since the core relative

permeability is considered infinite; in fact it is at most greater than 103

, but smaller

than 104

.

3.2. Nonlinear Model Based on 3D FEM Analysis

TFRLM flux linkage depends on the current and on the mover position and has its

maximum value in aligned position and its minimum value in unaligned position. A

typical TFRLM phase flux linkage variation function of phase current for aligned and

unaligned mover position is shown in Fig. 3.

In Fig. 2 the following notations are made:

λ0al,

λ0un

– aligned, unaligned unsaturated flux linkage (λ0al

= L0al

.

Iph

; λ0un

= L0un

.

Iph

)

λal,

λun

– aligned, unaligned saturated flux linkage

L0al,

L0un

– aligned, unaligned unsaturated phase main inductance.

The unsaturated flux linkage in aligned λ0al

and in arbitrary mover position λ0 is:

0 0

,

al al

L iλ = ⋅ 0 0

L iλ = ⋅ (8)

where L0 is unsaturated value of the phase inductance in an arbitrary mover position.

The saturated flux linkage in the same positions is:

λal

= λ0al

/ksal

, λ = λ0/k

s(9)

and, by combining (8) and (9) results the saturated flux linkage in an arbitrary mover

position:

Figure 2. Aligned and unaligned flux linkage variation

versus phase current for TFRLM.

Figure 3. Flux linkage vs. displacement.


0r

al

sr

l

k

λ λ= ⋅ (10)

where the ratios:

0

0

0

r

al

L

l

L

= , s

sr

sal

k

k

k

= (11)

should be calculated by using aligned, average and unaligned flux linkage characteris-

tics obtained via 3D-FEM analysis [11].

Both function l0r

and ksr

should contain a cosinusoidal function dependent on the

mover displacement since the thrust produced by a phase must be zero in aligned, re-

spectively unaligned position [11].

For the sample motor considered, in Fig. 3 are given in comparison the variation of

the phase flux linkage against displacement for different currents, values calculated via

3D-FEM respectively analytical model.

In the case of the considered sample TFRLM the analytical model proposed gives

for the flux-linkage the following expression:

( )2

( , ) 0.19925 cos( ) 0.80075

0.03675 24.58 431.1

1

( ) ( )cos( )l l

i

i

i i

a i b i

λ α α

α

= ⋅ + ⋅

⋅ + ⋅ +

⋅

+

0.1385 0.0004712

( ) 3.361 0.8566i i

la i e e

− ⋅ − ⋅

= ⋅ + ⋅

0.002302 0.1375

( ) 0.1457 3.293i i

lb i e e

⋅ − ⋅

= ⋅ − ⋅

/xα π τ= ⋅ , where τ is the pole pitch.

(12)

3.3. Look up Tables Based on 3D FEM Analysis

In Figs 4 and 5 the developed thrust and phase flux linkage calculated via 3D FEM

versus mover displacement for different values of phase mmf are presented:

Figure 4. Thrust vs. displacement at different cur-

rent, 3D FEM.

Figure 5. Flux linkage vs. displacement at different

current, 3D FEM.


From these data, the extended plots covering two pole pitches and the entire do-

main for phase ampere turns were obtained, as shown in Figs 6 and 7. These extended

plots will be introduced in MATLAB/Simulink and TFRLM dynamic characteristics

will be calculated based on the look up table technique which are widely used nowa-

days and there is no reason to extend the explanations here.

4. Computed Results

From all simulations performed in MATLAB/Simulink environment some are shown

in the following, two distinct regimes are presented:

i) A steady state regime when the phase is supplied with constant current, the

rated value of 50 A.

ii) A dynamic regime when all the phases are supplied and the mover speed is

imposed at 2 m/s, the phase current being controlled function of the speed er-

ror.

The steady state regime characteristics in Figs 8 and 9 show a good agreement be-

tween the obtained and the expected results. In Fig. 8 the phase inductance variation

Figure 6. Extended thrust force vs. displacement. Figure 7. Extended flux linkage vs. displacement.

Figure 8. Phase inductance versus mover position,

constant current.

Figure 9. Phase developed thrust versus position,

constant current.


versus mover position is given when the analytical model and the look-up table tech-

nique are employed. In the case of the linear model the inductance is varying linearly

with the displacement x and therefore it is not shown.

In Fig. 9 the phase developed thrust is presented and, as expected, the values calcu-

lated via analytical model and look-up table technique are quite the same, both algo-

rithms starting from 3D-FEM analysis. In the case of the linear model the phase devel-

oped thrust is constant and has larger values.

In Figs 10, 11 and 12 the variation of the phase current, phase flux linkage and

phase developed thrust versus mover position are shown in comparison for the three

considered models in the case of a dynamic regime at constant speed and controlled

current.

A dynamic regime of entire four phases motor, at the same constant speed as be-

fore was performed and the values obtained for the resulting thrust variation function of

mover position, all three models, are presented in Fig. 13. It is clear that the look-up

table technique and the analytical model lead to quite the same results, much different

from that obtained via linear model.

Figure 10. Phase current versus mover position,

imposed speed.

Figure 11. Phase flux linkage versus mover posi-

tion, imposed speed.

Figure 12. Phase developed thrust versus mover

position, imposed speed.

Figure 13. Resulting four phase thrust, controlled

current at constant speed.


5. Conclusions

Three mathematical models for a TFRLM were considered: a linear one, a look-up ta-

ble technique based on 3D-FEM calculated values and an analytical model based also

on 3D-FEM obtained values. The models are briefly presented and TFRLM steady

state, respectively dynamic regime was simulated via MATLAB/Simulink environ-

ment.

Some results are shown and commented.

Three are the most important conclusions that occur from the study:

i) The linear model, even if it is very simple, is far of being accurate and it does

not offer pertinent information

ii) The proposed analytical model is fully validated by the fact that the values

calculated by employing it are very close to that obtained via look-up table

technique, a well known one

iii) The proposed analytical model requires the 3D-FEM calculations only for

three mover positions and consequently shorter computation time than the

look-up table technique

The overall conclusion is that a good analytical model, which is fully considering

the nonlinearities, can be a very good solution when TFRLM dynamic behavior study

has to be done.

References

[1] Viorel, I.-A., Henneberger, G., Blissenbach, R., Löwenstein, L., “Transverse flux machines. Their be-

havior, design, control and applications”, Mediamira Publishing House, Cluj, Romania, 2003.

[2] Henneberger, G., Viorel, I.-A.: “Variable reluctance electrical machines”, Shaker Verlag, Aachen, Ger-

many, 2001.

[3] Crivii, M., Viorel, I.-A., Jufer, M., Husain, I., “3D to 2D equivalence for a transverse flux reluctance

machine”, Proc of ICEM’02, Brugge, Belgium, on CD-ROM:254.pdf.

[4] Kang, D.H., “Transversalflussmaschinen mit permanenter Erregung als Linear Antriebe im schie-

nengebundened Verkehr”, Dissertation, 1997, Germany.

[5] Soares, E., Costa Branco, P.J., “Simulation of a 6/4 switched reluctance motor based on MATLAB/

Simulink environment”, IEEE Trans Aerosp. Electrom. Syst. vol. 37, no. 3, pp. 989-1009, 2001.

[6] Viorel, I.-A., et al., “A new approach to the computation of the switched reluctance motor dynamics”,

Proc of ICEM’2000, Helsinki, Finland, pp. 1605-1608.

[7] Chang, J.H., Kang, D.H., Viorel, I.-A., Larisa, S., “Transverse Flux Reluctance Linear Motor’s Ana-

lytical Model Based on Finite Element Method Analysis Results”, IEEE Trans. on Magnetics, vol. 43,

no. 4, April 2007, pp. 1201-1204.

[8] Chi, H.-P., Lin, R.-L., Chen, J.-F., “Simplified flux linkage model for switched reluctance motors”, IEE

Proc. – Electr. Powe Appl., vol. 152, no. 3, pp. 577-583, 2005.

[9] Chang, J.H., Kang, D.H., Viorel, I.-A., Tomescu, Ilinca, Strete, Larisa, “Saturated double salient reluc-

tance motors analytical model”, Proc of ICEM’06, Chaina, Greece, on CD-ROM.

[10] Viorel, I.-A., et al., “Transverse flux machine mathematical model”, Rev. Roumanie Sci. Tech., Elec-

trotechn. Energ., vol. 48, no. 2-3, pp. 369-379, 2005.

[11] Viorel, I.-A., Strete, Larisa, “Switched reluctance motor analytical flux-linkage model”, (Sent for pub-

lication).


Influence of Air Gap Diameter

to the Performance of Concentrated Wound

Permanent Magnet Motors

Pia SALMINEN, Asko PARVIAINEN, Markku NIEMELÄ and Juha PYRHÖNEN

Lappeenranta University of Technology, Department of Electrical Engineering,

P.O. Box 20, 53851 Lappeenranta, Finland

[email protected]

Abstract. The study addresses the torque production capabilities of concentrated

wound permanent magnet machines in the motor frame size 225. The main target

of the study is to investigate how much more torque can be achieved by increasing

the air gap diameter of concentrated wound machines, using the pole pair number

as a variable in the study and keeping the machine stator frame external diameter

and stack length as fixed parameters. First, the torque dependence on the pole pair

number with a constant air gap is studied.

Introduction

Concentrated wound machines belong to the group of fractional slot wound machines;

however, the number of slots per pole and per phase q ≤ 0.5. In a concentrated wound

machine, each coil is wound around one tooth. The purpose of the paper is to explicate

the selection principles of the pole pair number with concentrated wound permanent

magnet machines. The paper provides useful information about the influence of the

pole number on the performance of a motor designed to a standardized IEC motor

frame, Fig. 1.

The torque T is proportional to the product of the rotor surface average tangential

force F and the rotor radius r (T = Fr). As the rotor radius and the length of the ma-

chine are kept constant, also the rotor surface area producing torque remains equal. The

effect of the number of poles is studied. Increasing the number of the poles gives an

opportunity to increase the rotor radius, which also allows employing quite a thin yoke.

Figure 1. IEC die cast iron housing for an electric motor [1].



doi:10.3233/978-1-58603-895-3-276

276

Different structures were designed for a machine with a rated torque of 1075 Nm, a

frame size of 225 mm, and a rated speed of 400 rpm; these are values typically applied

for instance in paper machines. The machines studied here have a 364-mm stator stack

outer diameter as a mechanical constraint. In the study, the magnetic loading of the

stator yoke and the teeth as well as the current density in the stator windings were kept

constant. Also the stack length was fixed to the value of 270 mm, even though a high

pole pair number also brings a possibility of increasing the stack length, while the end

windings become smaller. The main concerns related to the use of high pole-pair num-

bers in a selected frame size are the mechanical stiffness of a thin stator yoke and the

efficiency aspects of the motor associated with the increased iron losses as the pole

number increases. Figure 2 shows three types of concentrated wound PM machines.

The flux diagrams are obtained by the finite element analysis FEA (Cedrat Flux2D) for

a) 12-slot-10-pole and b) 36-slot-30-pole machines with an air gap diameter Dδ of 254

mm, and c) 36-slot-42-pole machine with approximately 10% larger air gap diameter of

279 mm. From the magnetic saturation point of view, the stator yoke may be thinner in

machines with a higher pole number, and therefore some of the high pole number ma-

chines were also computed using air gap diameters larger than 254 mm.

Torque with a Constant Air Gap

The pull-out torque equation for a non-salient pole machine (if direct and quadrature

inductances are approximately equal) may be written as

ph PM

d

3ˆ

.

2π 2π

U Ep

T

f fL

= (1)

Rewriting the frequency f = np in Eq. (1) we get

ph PM ph PM

2 2

d d

3 3ˆ

.

2π 2π 4π

U E U Ep

T

np npL pLn

= = (2)

a) 12 slots 10 poles, q = 0.4,

D = 254 mm

b) 36 slots 30 poles, q = 0.4,

D = 254 mm

c) 36 slots 42 poles, q = 0.286,

D = 279 mm

Figure 2. Machine topologies of concentrated wound PM machines a) 12-slot-10-pole, b) 36-slot-30 pole,

and c) 36-slot-42 pole. For the machine with 12 slots and 10 poles q = Q/2pm = 12/(2·5·3) = 0.4. For the

36/30 machine q = 0.4, too and for the 36/42 machine q = 0.286.

P. Salminen et al. / Influence of Air Gap Diameter 277

In this study, the supply phase voltage Uph

, the induced phase voltage EPM

and the

speed n [1/s] were constants, and therefore the torque is inversely proportional to the

number of pole pairs p and the synchronous inductance Ld

d

1ˆ

.T

pL

∝ (3)

When the number of pole pairs p is large and the number of slots per pole and per

phase q is small, the magnetizing inductance of a PM machine is small compared with

the leakage inductance, and hence the leakage inductance may dominate. In such a

case, we may write the synchronous inductance Ld and the pull-out torque T

ˆ as [2]

pq

L

1

d∝ and

ˆ.T q∝ (4)

In other words, the pull-out torque is proportional to the number of slots per pole

and phase [2].

For concentrated wound machines, in which the coil is wound around one tooth,

the highest slots per pole and per phase number q is 0.5. In this study, this machine

type gives the highest torque values. Figure 3 shows the pull-out torques computed

with the FEA for several concentrated wound machines when q varies from 0.25 to 0.5.

The machines have the same air gap diameter of 254 mm. Other fixed parameters are

the supply phase voltage Uph

, the induced phase voltage EPM

and the speed n [1/s]. It

can be seen that machines having q = 0.25 give the smallest pull-out torques. Higher

torques are produced when q is increased. In the study, the machines with q = 0.5 give

the highest pull-out torque values of 2.1 p.u. Figure 3 indicates the validity of q when

concerning the torque production capability of concentrated wound machines. One may

investigate the machines having 8 pole pairs (in Fig. 3): As q equals to 0.25 the maxi-

mum torque is about 1 p.u., but when q is doubled to the value of 0.5 also the torque is

doubled to the value of 2 p.u. The results verify the correlation of T and q in Eq. 4.

0

0.5

1

1.5

2

2.5

0.25 0.25 0.25 0.29 0.29 0.29 0.36 0.4 0.4 0.4 0.43 0.5 0.5 0.5 0.5

24 12 8 21 7 14 11 5 15 10 7 4 12 6 8

36 18 12 36 12 24 24 12 36 24 18 12 36 18 24

Pu

ll-o

ut to

rq

ue (p

.u

.)

q

p

Q

Figure 3. Pull-out torques of concentrated wound machines with different numbers of slots per pole and per

phase q.

P. Salminen et al. / Influence of Air Gap Diameter278

With high pole pair numbers, the magnet leakage fluxes are increased to a rela-

tively high level, and thereby the inductance increases causing a reduction in the torque

production capability. Some methods to estimate the leakage fluxes are provided in [6].

In some cases of concentrated wound PM machines, the slots are unfavourably aligned

with the magnets, so that the flux available will decrease. Consequently, the flux gen-

erated by the permanent magnets in the rotor surface cannot be used completely, be-

cause there is no suitable route for the flux to travel to the stator iron. Therefore the

magnet flux leakage can be large, which can be seen in Fig. 4 illustrating the flux

routes of a 24-slot 28-pole machine. We can see that some of the flux lines do not cir-

culate around the stator slots, and hence, they are not generating torque.

For machines designed into the same frame size, the same speed and the power, it

is found out that the pull-out torque is roughly proportional to the number of slots per

pole and per phase q. This correlation may assist the motor designer to estimate the

desired pull-out torque for concentrated winding PM motors.

Torque with a Larger Air Gap Diameter

As the motor pole pair number p is increased, the magnetic loading of the stator yoke

may allow to reduce the yoke thickness. Correspondingly, one may apply larger air gap

diameters (note also that the length of the stator teeth may be reduced, because a larger

air gap diameter yields a reduced number of coil-turns. The slots will be smaller). Thus,

some high pole number machines were computed also using a larger air gap diameter.

When the stator yoke was redesigned, also the slot dimensions were varied. The slot

height was reduced and the teeth were designed thinner.

When the air gap diameter Dδ is increased by 10%, the air gap flux Φ increases in

the same proportion, and therefore the torque T may increase by 20%, as can be seen in

Eq. (5), since the torque-producing radius increases and the inductances decrease.

( ) ζcos4π

2

δBˆ

Aˆ

LDT = (5)

In Eq. (5), L is the length of the stator stack, Aˆ

is the peak value of the fundamen-

tal of the linear current density, and Bˆ

is the peak value of the air gap flux density

fundamental, and ζ the angular displacement between the air gap flux density and the

stator linear current density.

Figure 4. Flux routes of a 24-slot 28-pole machine.


Losses

The study addresses the computation of losses both analytically and by applying the

finite element analysis. Flux2D is employed in computations. The iron losses can be

calculated in a magnetic region during the analysis. The losses, computed with the

FEA, include the hysteresis losses, the Joule losses and the excess losses. In the peri-

odic state (time stepping magnetic applications over one complete period), the iron

losses are defined as

2 1.52

2

Fe h f f e

0 0

1 1 d ( ) d ( )ˆ

( )d d ,

12 d d

T T

d b t b t

P t t k b f k k k t

T T t t

σ

⎡ ⎤⎛ ⎞ ⎛ ⎞

= + +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

∫ ∫ (6)

where bˆ

is the maximum flux density in the element concerned, f the frequency, σ the

conductivity, d the lamination sheet thickness, kh the coefficient of hysteresis loss,

ke the coefficient of excess loss and k

f is the filling factor. The factors depend on

the steel material used. In the computations, magnetic steel M600-50 is used. The pa-

rameters in the computations for M600-50 are σ = 4⋅106

(1/Ω m), d = 0.5 mm, kh =

152 (Ws/T2

/m3

), ke = 2.32 (W/Ts-1)

1.5

/m3

and kf = 0.98. [3] The Joule losses may be

defined by

2

Cu ph n,P mR I= (7)

where m is the number of phases, Rph

the phase resistance and In the rated current. In

permanent magnets, the Joule losses caused by the eddy currents are evaluated in the

FEA according to

2

PM

d ,

V

P J Vρ= ∫∫ (8)

where ρ is the material resistivity, V the volume and J the current density. The friction

and bearing losses are estimated to be 20 W. Table 1 reports the results for motors with

28, 30 and with 42 poles, when the air gap diameter and the stator slot dimensions are

varied.

In Table 1, we can see that in the case of a 24-slot-28-pole machine, the pull-out

torque increased only by 5% when increasing the air gap diameter by 10%. In the case

of 36-slot-42-pole and 36-slot-30-pole machines, the air gap diameter was 10% larger,

which increases the pull-out torque by 20%. When the air gap Dδ is increased by 10%,

the air gap flux increases and therefore the torque may increase by 20%. The torque is

proportional to Dδ

2

. From the results shown in Table 1 and in Fig. 5 one may see that

the losses are lower – leading to higher efficiency – with the machines having larger air

gap diameters. Figure 5 shows how the losses are divided into Joule losses, iron losses

and eddy current losses caused by the permanent magnets. It can be seen in Fig. 5 that

with a larger air gap diameter the Joule losses were somewhat higher and the iron

losses clearly smaller than when using the smaller air gap of 254 mm.


Table 1. Computation results with different air gap diameters

Slots,

Q

Pole pair

number,

p

q Air gap

diameter

(mm)

Air gap

diameter

increase

(%)

Rated

current

(A)

Iron

losses

(W)

Freq.

(Hz)

Pull-out

torque

(p.u.)

Efficiency

at rated

load

Pull-out

torque

increase

(%)

24 28 0.286 254 0 86 540 93 1.29 0.93

24 28 0.286 274 +10% 87 420 93 1.33 0.94 +5%

36 30 0.4 254 0 92 720 100 1.58 0.93

36 30 0.4 274 +10% 94 560 100 1.86 0.93 +20%

36 42 0.286 254 0 92 1100 140 1.0 0.92

36 42 0.286 279 +10% 96 720 140 1.2 0.92 +20%

0

500

1000

1500

2000

2500

3000

3500

4000

254 mm 274 mm 254 mm 274 mm 254 mm 279 mm

Air gap diameter (mm)

Lo

sses (W

)

Eddy losses (W)

Iron losses (W)

Joule losses (W)

24-slot-28-pole

36-slot-30-pole

36-slot-42-pole

Figure 5. Joule losses, iron losses and eddy current losses caused by the permanent magnets.

Mechanical Analysis

The reduction in the stator yoke thickness may influence the mechanical rigidity of the

stator yoke. To demonstrate the mechanical behaviour, the natural frequencies of dif-

ferent designs were compared by applying analytical methods according to [4,5] as

well as an FE analysis based on the Autodesk Inventor Professional™ finite element

module. Owing to the space limitations, the analytical equations to compute the natural

frequencies are not reported here. In the analytical approach, only the vibration mode

(2,0) is of interest. Thus, the teeth and windings do not notably contribute to the stiff-

ness of the stator. With the vibration mode (2,0), the influence on the rotary inertia of

the stator core is also insignificant [4,5]. Figure 6 reports the results obtained from the

finite element analysis and from the analytical computations for 36 slot stators. In the

analysis, the slot cross-sectional area is considered to be equivalent between the de-

signs, i.e. the stator yoke thickness and the thicknesses of the stator teeth are variables.

The reported results show that the natural frequency of the stator yoke appears to

be even lower than the line frequency as the yoke thickness is reduced below 15 mm.


This may be considered the major concern for constructions such as 36-42 in the se-

lected frame size.

Conclusions

The paper analyzes the performance improvement of concentrated wound permanent

magnet machine achieved by the change of the air gap diameter. The results show that

in the selected frame size, the most attractive solution is a 36-slot 30-pole machine with

an air gap diameter of 254 mm. A slightly higher pull-out torque is achieved with the

air gap diameter of 274 mm, and the efficiency shows to slightly improve compared

with the solution with the air gap diameter of 254 mm. The analysis shows also that the

selection of 36 slots and 42 poles is questionable, because the motor performance ap-

pears to weaken significantly. The efficiency and torque production capabilities of this

machine are weaker compared to other investigated designs. Further, the line frequency

and the stator natural mechanical frequency are very close to each other causing prob-

lems in the mechanical behaviour of the structure.

(a)

0

100

200

300

400

500

600

4 6 8

10

12

14

16

18

20

22

24

26

28

30

Stotor yoke thickness [mm]

Stato

r n

atu

ral freq

uen

cy

[H

z]

254 mm

264 mm

274 mm

(b)

Figure 6. (a) The lowest shape mode for the stator at the frequency of 290 Hz. The stator studied has 36 slots

and an air gap diameter of 274 mm. (b) Natural frequencies of the stators when the yoke thickness is a vari-

able (the slot area is fixed). Curves for air gap diameters of 254 mm, 264 mm and 274 mm.


Generally, the results show that the stator yoke thickness in the frame 225 should

be 15 mm or above in order to operate on a safe regime even though the magnetic load-

ing of the yoke allowed to use a thinner yoke. Because of this limit, the practical im-

provement achieved by the change of the air gap diameter in the machine design is

quite negligible in the frame size studied.

References

[1] Komponenten für Elektromotoren. Kurt Maier Motor-Press GmbH. CD-ROM catalogue, 2007.

[2] Salminen, P., Jokinen, T., and Pyrhönen, J., 2005. The Pull-Out Torque of Fractional-slot PM-Motors

with Concentrated Winding, Electric Power Applications, IEE Proceedings. Vol. 152, Iss. 6, pp. 1440–

1444.

[3] Salminen, P., 2004. Fractional slot permanent magnet synchronous motor for low speed applications,

Dissertation, Acta Universitatis Lappeenrantaensis, ISBN, Lappeenranta, 151 p.

[4] Yang, S.J., 1981. Low-Noise Electrical Motors. Oxford, Clarendon Press, 101 p.

[5] Parviainen, A., 2005. Design of axial-flux permanent-magnet low-speed-machines – and performance

comparison between radial-flux and axial-flux machines, Dissertation, Acta Universitatis Lappeenran-

taensis, ISBN 952-214-029-5, Lappeenranta, 153 p.

[6] Qu, R., and Lipo, T.A., 2004. Analysis and Modeling of Air-Gap and Zigzag Leakage Fluxes in a Sur-

face-Mounted Permanent Magnet Machine. IEEE Transactions on Industry Applications. Vol. 40, No. 1,

pp. 121–127.


Squirrel-Cage Induction Motor

with Intercalated Rotor Slots

of Different Geometries

V. FIREŢEANU

POLITEHNICA University of Bucharest, EPM_NM Lab., Bucharest, Romania

[email protected]

Abstract. This paper deals with a particular design of squirrel-cage rotor slots and

copper bars of high power induction motors able to ensure high values of both

starting torque and breakdown torque. The innovation consists in a rotor with two

different geometries of slots, respectively of bars cross-section shape. Along the

rotor periphery, bars with cross-section of rectangular shape are followed by bars

of stepped shape cross-section and vice-versa.

Introduction

A good design of classical induction motors with respect to the electromagnetic torque

must answer two contradictory requirements, respectively high value of starting torque

and high value of breakdown torque. In case of squirrel-cage induction motors, these

characteristics are very dependent on the rotor slot geometry [1,2]. Starting from the

results presented in the reference [3] this paper studies a new configuration of the

squirrel cage able to realize a good compromise between a high value of motor starting

torque and a high value of breakdown torque.

A three-phase squirrel-cage induction motor with rated power Pn = 500 kW, syn-

chronous speed 750 rpm, supplied at 6 kV, 50 Hz is studied. Two shapes in Fig. 1 of

the rotor bars cross-section, rectangular shape and respectively stepped shape are con-

sidered. Independently of the value of h2/a

2 and h

1/h

2 parameters, the rotor bars have

the same cross-section area.

Simulation Results Analysis

Starting Torque, Breakdown Torque and Energetic Parameters

In case of rectangular shape of bar cross-sections, Fig. 1a, the increase of ratio h2/a

2

between the bars height and thickness ensures the increase of per unit starting torque

(Mp/M

n), Fig. 2, and the decrease of per unit breakdown torque (M

max/M

n).

The two electromagnetic torque – slip characteristics (M-s) in Fig. 3, correspond to

the lower (h2/a

2)min

and upper (h2/a

2)max

values which characterize the rectangular bars

cross-section geometry, Fig. 1a, considered in this study. These values correspond to an

almost square bar, with a2 = 15 mm and h

2 = 14.92 mm, respectively, a deep rectangu-

lar bar, with a2 = 4 mm and h

2 = 55.93 mm.



doi:10.3233/978-1-58603-895-3-284

284

a) b)

Figure 1. Slot geometries and different shapes of rotor bars cross-section a) rectangular shape; b) stepped

shape.

Figure 2. Starting torque (Mp/Mn) and breakdown torque (Mmax/Mn) versus ratio h2/a2.

In case of stepped shape bars, Fig. 1b, the height h1 of the upper bar step is the pa-

rameter of bar cross-section geometry and h2 is the total height of the stepped bar. The

upper step of the bar has the thickness, a1 = 3 mm and the lower step has square shape.

The dependence of starting torque Mp on the ratio (h

1/h

2) in Fig. 4 relieves an op-

timal shape of stepped bars for the value (h1/h

2)opt

= 18/31.03 = 0.58. After this value,

both starting torque and breakdown torque decrease.

V. Fireteanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries 285

Figure 3. Electromagnetic torque versus slip for rectangular bars.

Figure 4. Starting torque (Mp/Mn) and breakdown torque (Mmax/Mn versus (hl/h2) for stepped shape bars.

The (M-s) characteristics in Fig. 5 correspond to the minimum value (h1/h

2)min

=

10/23.92 = 0.418 and to the optimal value (h1/h

2)opt

. The comparison of these curves

with those in Fig. 3 shows the following:

− if the almost square shape bars, with the ratio (h2/a

2)min

= 14.92/15 = 0.994 are

replaced with bars of stepped shape cross-section with (hl/h

2)min

= 10/23.92 =

0.418, the starting torque increases with 58.4% and to the breakdown torque

diminishes with 41.3%;

− if the deep rectangular bars with (h2/a

2)max

= 55.92/4 = 13.98 are replaced with

stepped bars of optimal cross-section, with (hl/h

2)opt

= 0.58, the starting torque

increases with 32.6% and the breakdown torque increases also with 9.2%.

Both starting torque and breakdown torque increase when changing the deep

rectangular shape bars with optimal stepped shape bars.

V. Fireteanu / Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries286

The high value of the starting torque in Fig. 5, corresponding to the ratio

(h1/h

2)opt

= 18/31.03 = 0.58 in case of stepped shape bars, and the high value of break-

down torque corresponding to the value (h2/a

2)min

= 14.92/15 in case of rectangular

bars, suggest the idea of an innovative squirrel-cage, where the (h1/h

2)opt

bars are inter-

calated with (h2/a

2)min

bars. It results the rotor geometry in Fig. 6, called squirrel-cage

with intercalated slots of different geometries.

The (M-s) characteristic for the new rotor geometry in Fig. 7 and the values of

starting and breakdown electromagnetic torques, Table 1, reflect a compromise be-

tween high values of starting torque, which can be obtained with stepped shape bars –

green colour in Figs 6 and 7, and high values of breakdown torque that characterise the

rectangular bars of almost square shape – magneta colour in the same figures.

Figure 5. Electromagnetic torque versus slip for stepped shape bars.

Figure 6. Squirrel-cage rotor with intercalated bars.


In case of the new rotor with intercalated bars the starting torque increases with

91.6% and the breakdown torque decreasees with only 21.6% with respect the rotor

variant with almost square bars ((h2/a

2)min

). In comparison with with rotor variant with

stepped bars with (h1/h

2)opt

, the starting torque of the new rotor variant decreases with

26.6% but the breakdown torque increases with 45.3%.

As waited, the starting current, the rotor Joule losses and the motor energetic pa-

rameters – efficiency and power factor, Table 1, have values between those obtained

when all rotor slots have only one of the two geometries in Fig. 6.

The magnetic field lines for rated load operation of the motor with intercalated

bars and the chart of current density in rotor bars at motor start-up are presented in

Fig. 8, respectively Fig. 9.

For rated load operation, the Joule effect generates 0.704 kW in each rectangular

bar and 0.783 kW in each stepped shape bar of the new rotor variant. When the rotor

contains only rectangular bars with (h2/a

2)min

, this power is 0.745 kW/bar and when the

rotor contains only stepped bars with (h1/h

2)opt

, 0.749 kW/bar.

For motor start-up, the Joule power is 5.85 kW in each rectangular bar and

11.09 kW in each stepped shape bar of the new rotor variant. When the rotor contains

only rectangular bars with (h2/a

2)min

this power is 4.93 kW/bar and when the rotor con-

tains only stepped bars with (h1/h

2)opt

, this power is 12.7 kW/bar.

Figure 7. Electromagnetic torque versus slip for intercalated rotor bars.

Table 1. Motor characteristics

Rotor bars geometry Mp/Mn Mmax/Mn Ip/In Pj2/Pn η [%] cosϕ

Rectangular bars with (h2/a2)min 0.545 2.958 6.101 1.164 95.43 0.859

Stepped bars with (h1/h2)opt 1.423 1.597 4.187 1.278 95.15 0.804

Intercalated bars:

rectangular bars (h2/a2)min +

stepped bars(h1/h2)opt

1.044 2.320 5.167 1.216 95.30 0.833


Figure 8. Magnetic field lines for motor rated load operation: rotor with intercalated bars, rectangular bars

(h2/a2)min + stepped bars(h1/h2) opt.

Figure 9. Chart of current density in rotor bars for motor start-up: intercalated bars, rectangular bars

(h2/a2)min + stepped bars(h1/h2) opt.

Time Variation and Harmonics of Electromagnetic Torque at Rated Load

The time variation of electromagnetic torque for rated load operation of the new motor

with intercalated bars is presented in Fig. 10, where the numerical values corresponds

to the half on the motor, which was considered for the electromagnetic field computa-

tion domain. The mean electromagnetic torque of the motor, Fig. 10 a), has the value

3247.24 × 2 = 6494.5 Nm. The most important harmonics of the electromagnetic

torque, Fig. 10 b), with amplitudes of around 30 Nm have the frequency 300 Hz and

respectively 700 Hz.


a)

b)

Figure 10. Time variation and harmonics of the electromagnetic torque at motor rated load.

Transient Time Variation of Motor Velocity

The three curves in Fig. 11 ilustrate the transient variation of velocity for motor no-

load start in cases: rotor with rectangular bars with (h2/a

2)min

, rotor with stepped bars

with (h1/h

2)opt

and the new rotor with intercalated bars. The time of motor start in 6.30

seconds in the first case, 5.08 seconds in the second case and 5.18 seconds in the third

case.

The three curves in Fig. 12 show the transient variation of velocity when double of

rated load is applied after no-load start of the motor in three previously defined cases.

Since in the second case – stepped bars with (h1/h

2)opt

, the breakdown torque is not high

enough, when apply a charge double with respect rated one, the motor is not able to

reach a new steady state operation regime, with a lower value of the rotor speed that

hapens in the first and the third case of rotor bar cross-section geometry.


Rotor with rectangular bars with (h2/a2)min

Rotor with stepped bars with (h1/h2)opt

New rotor with intercalated bars

Figure 11. Transient time variation of velocity at motor start-up.


Rotor with rectangular bars with (h2/a2)min

Rotor with stepped bars with (h1/h2)opt

New rotor with intercalated bars

Figure 12. Transient time variation of velocity at motor start-up

References

[1] M. Brojboiu, Concerning the influence of the rotor bar geometry on the induction motor performances,

Proc. of 5th TELSIKS’01 International Conference, Sept. 2001.

[2] J.L. Kirtley Jr., Designing Squirrel Cage Rotor Slots with High Conductivity, Proc. of ICEM’04 Confer-

ence, Sept. 2004.


[3] O.A. Turcanu, T. Tudorache, V. Fireteanu, Influence of Squirrel-Cage Bar Cross-Section Geometry on

Induction Motor Performances, Proc. of SPEEDAM’06, May 2006.

[4] V. Fireteanu, T. Tudorache, O.A. Turcanu, Optimal Design of Rotor Slot Geometry of Squirrel-Cage

Type Induction Motors, Proc. of IEMDC’06 Conferrence, May 2007.


Analysis and Performance of a Hybrid

Excitation Single-Phase Synchronous

Generator

Nobuyuki NAOEa

, Akiyuki MINAMIDE a

and Kazuya TAKEMATAb

a

Kanazawa Technical College 2-270 Hisayasu Kanazawa, Ishikawa Japan 921-8601

b

Kanazawa Institute of Technology 7-1 Oogigaoka Nonoichi,

Ishikawa Japan 921-8501

Corresponding author e-mail: [email protected]

Abstract. This paper describes a hybrid excitation-type single-phase synchronous

generator. The generator has a permanent magnet and wound fields on the same

shaft. The finite-element method (FEM) is applied to analyze the no-load charac-

teristic of the generator. The test results obtained on a 0.5-kVA prototype machine

demonstrate its capability for field regulation.

1. Introduction

Among the many advantages of a permanent magnet (PM) synchronous generators are

their simple rotor structures and high efficiency. However, because the PM flux is con-

stant, the air gap flux is difficult to control. In wound rotor synchronous generators, on

the other hand, the air gap flux is controlled by the field current. Therefore, a synchro-

nous generator with features of both PM and wound-rotor synchronous generators

would display enhanced practicality. A hybrid excitation-type synchronous machine

with both PM poles and excitation poles on the rotor has been studied [1–6].

The authors propose a hybrid excitation-type single-phase synchronous generator

(HESSG) with a two-part rotor that has both PM and wound rotors, retaining a conven-

tional stator armature winding. The generator contains the features of both PM and

wound-rotor synchronous machines. The PM part is highly efficient, and the wound

rotor provides field regulation and allows field-strengthening operation. In this paper,

the finite element analysis of the HESSG is described, and its basic characteristics are

demonstrated.

2. Structures

The structure of the HESSG is shown in Fig. 1. Unlike other types of synchronous ma-

chines, the proposed HESSG has one PM and one wound field separately mounted on

the same shaft. A 0.5-kVA, 60-Hz, two-pole prototype machine was developed. The

field winding is wound around the rotor for easy production. In the PM part, which is

5-mm thick, Nd-Fe-B permanent magnets are embedded inside the core. The rotor is a

salient pole without damper windings. The volume of the PM rotor part is within 50%



doi:10.3233/978-1-58603-895-3-294

294

of the volume of the rotor core. The volume of the wound rotor part is within 50% of

the volume of the rotor core. The stator of the prototype machine is similar to that of a

conventional synchronous machine. The magnetic circuits of the PM and the wound

rotors are independent of each other. Moreover, the flux produced by the field current

does not pass through the PM because it has a large degree of reluctance.

3. Performance

The finite element (FE) analysis was carried out using software Maxwell 2D provided

by ANSOFT. Through FE analysis, the no-load characteristics of the generator were

examined. The two-dimensional FE analysis was adopted. To determine the effective-

ness of the proposed model, we used software Maxwell 2D. The magnetic circuits of

PM and wound rotors are independent of each other. Moreover, the flux produced by

the field current does not pass the PM because it has a large reluctance. Therefore, the

EMFs of PM and wound rotors are simply added.

Figure 2 shows the cross-section configuration with the no-load PM rotor

magnetic flux distribution obtained from the FE analysis. Figure 3 shows the

computed EMF of the phase voltage with the PM rotor. Figure 4 shows the

cross-section configuration with no-load wound rotor magnetic flux distribu-

tion obtained from FE analysis. Figure 5 shows the computed EMF of phase

voltage with the wound rotor. The measured terminal voltage is compared with

the calculated one as shown in Fig. 6. It can be seen that the calculated results

agree with the measured results. The voltage regulation of the resistive load is

shown in Fig. 7. As is evident in Fig. 7, the armature EMF can be controlled by

adjusting the field current If.

PM

STATOR

FIELD WINDING

ARMATURE WINDING

SHAFT

ROTOR

Figure 1. Structure of the HESSG.

N. Naoe et al. / Analysis and Performance of an HESSG 295

Figure 2. Cross section and magnetic flux distribution in the PM rotor.

Figure 3. Computed EMF of the phase voltage in the PM rotor.

Figure 4. Cross section and magnetic flux distribution in the wound rotor.

N. Naoe et al. / Analysis and Performance of an HESSG296

Figure 5. Computed EMF of the phase voltage in the wound rotor.

Figure 6. Calculated and measured terminal voltages.

Ia[A]

Vla[V]

If[A]

without If

with If

Vla

If

Resistive load

Figure 7. Measured terminal voltage and field current waveforms.

N. Naoe et al. / Analysis and Performance of an HESSG 297

The measured terminal voltage and the field current waveforms are shown at a re-

sistive load in Fig. 8. The field current has confirmed double harmonic EMF for the

negative-phase-sequence field of the armature reaction. The field current of double

harmonic EMF can be used for the excitation of the field winding.

4. Conclution

The FE analyses of the HESSG with a two-part rotor are described in this paper. A 0.5-

kVA prototype machine was built, and its basic characteristics were demonstrated. The

FE analyses of the HESSG carried out a no-load EMF. The computed EMF clearly

agrees with the measured one. The test results clearly show that the proposed HESSG

has the capacity for field regulation. The field current of double harmonic EMF can be

used the excitation of the field winding.

References

[1] X. Luo and T.A. Lipo, “A synchronous/permanent magnet hybrid ac machine”, IEEE – International

Electric Machines and Drives Conference ’99, pp. 19-21,1999.

[2] J.A. Tapia, F. Leonardi and T.A. Lipo, “Consequent-Pole Permanent-Magnet Machine with Extended

Field-Weakening capability”, IEEE Tran. IA Vol. 39, No. 6, pp. 1704-1709, 2003.

[3] Y. Amara, J. Lucidarme, M. Gabsi, M. Lecrivain A. Ahmed and A.D. Akemakou, “A New Topology of

Hybrid Synchronous Machine”, IEEE Tran. IA Vol. 37, No. 5, pp. 1273-1281, 2001.

[4] N. Naoe and T. Fukami, “Trial Production of a Hybrid Excitation Type Synchnous Machine”, IEEE – In-

ternational Electric Machines and Drives Conference’01, pp. 545-547, 2001.

[5] K. Matsuuchi, T. Fukami, N. Naoe, R. Hanaoka, S. Takada and A. Miyamoto, “Performance Prediction

of a Hybrid-Excitation Synchronou-Machine with Axially Arranged Excitation Poles and Permanet-

Magnet Poles”, Trans. IEE Japan, Vol. 123-D, No. 11, pp. 1345-1350, 2003.

[6] N. Naoe, T. Fukami, A. Minamide and K. Takemata, “Steady-State Performance of a Hybrid Excitation

single-Phase Synchronous Generator”, XVII International Conference on Electrical Machines, PSA2-28,

p. 5, 2006, CD-ROM.

0.000 0.005 0.010 0.015 0.020

-100

-80

-60

-40

-20

0

20

40

60

80

100

-10

-8

-6

-4

-2

0

2

4

6

8

10

Time[s]

Term

inal V

oltage[V

]

Teminal Voltage

Field Current

Field C

urrent[A

]

Figure 8. Measured terminal voltage and field current.

N. Naoe et al. / Analysis and Performance of an HESSG298

Numerical Calculation of Eddy Current

Losses in Permanent Magnets of BLDC

Machine

Damijan MILJAVECa

and Bogomir ZIDARIČb

a

University of Ljubljana, Faculty of electrical engineering,

Trzaska 25, 1000 Ljubljana, Slovenia


b

TECES, Maribor, Slovenia

Abstract. The aim of the paper is the numerical calculation and analyze of eddy

currents in permanent magnets (PM) of brush-less direct current (BLDC) ma-

chines. The main source of these induced eddy currents is in reluctance variation

due to stator geometry. The study is carried out on outer rotor BLDC machine. The

3-D time-stepping finite-element analyze is used to clarify this phenomenon. The

induced eddy-current distribution in the PM and the resulting power losses are cal-

culated. Also, the study of this parasitic effect is curried out by 2D time-stepping

finite-element analyze. The induced eddy current losses in PM due to different sta-

tor geometries are calculated and analyzed.

Introduction

The main object of presented study is the outer rotor BLDC machine (Fig. 1). The per-

manent magnets (PM) are placed around the inner surface of rotor. Inside of rotor is

laminated stator. The rare-earth permanent magnets are not laminated and are electri-

cally good conductors. Any kind of magnetic flux density change induces the eddy

currents and with them power losses. These losses are shown in supplementary heating

of PM. With this, the temperature dependant working characteristic of PM (second

quadrant of B-H curve) is moving toward coordinate origin point.

The main study is based on induced eddy currents in PM just due to stator geome-

try. Stator slot openings are causing the magnetic flux density changes in permanent

magnets (Fig. 1) while they rotate around the stator. In such a study, the stator wind-

ings must not be connected to outer source and further more, no currents have to flow

in these windings. To achieve these working conditions the BLDC machine must work

in generator mode with no-load.

3D Numerical Analyze of Eddy Current Losses in Permanent Magnets

The outer diameter of studied ten-pole surface-mounted-PM BLDC machine is

160 mm. Other geometric parameters of the machine: the air-gap length is 1mm, the

thickness of the PM is 5 mm, the stator diameter is 126 mm, stator slot opening is


299

7 mm and the stack length of stator is 40 mm. Figure 1 shows in a 3-D view analysis

model.

Using the 3-D time-stepping finite-element analyzes [1] all upper described work-

ing conditions were achieved.

The stator ferromagnetic material was presented with its B-H curve (M600-35A).

The main concern of this study was to analyze the induced eddy currents in permanent

magnets so the conductivity of stator and rotor iron was set to zero. The FeNdB perma-

nent magnets were described with their magnetic and electric characteristics

(Br = 1.24 T, H

c = 900 kA/m, ρ = 0.15∗10

−5

Ωm).

In form of color scale the magnetic flux density distribution in PM at 4000 rpm is

shown in Fig. 2. The changes of magnetic flux density in PM’s (Fig. 2) rotate with

Permanent magnets

Outer rotor

Stator

Figure 1. BLDC machine geometry with finite elements.

Figure 2. Magnetic flux density distribution in PM at 4000 rpm with stator in shown position.

D. Miljavec and B. Zidaric / Numerical Calculation of Eddy Current Losses300

4000 rpm and cause the induction of eddy currents. They are presented in Fig. 3. The

presentation of these currents (Fig. 3) stands at certain moment of time.

The values of induced losses mainly depend on the speed of rotation, stator geome-

try, PM geometry and its conductivity. The induced power losses are dissipated in all

magnets and this means additional heating of magnets. The power losses for presented

geometry of BLDC machine (Fig. 1) in function of speed are presented in Fig. 4.

The penetration of induced eddy currents into PM presented in form of cross sec-

tion slices are shown in Fig. 5. The form and depth of penetration into PM mainly de-

Figure 3. Induced eddy current density in PM at 4000 rpm with stator in shown position.

Speed (rpm)

Eddy current loss in m

agnets (W

)

250

200

150

100

50

0

1000 2000 3000 4000 600050000

Figure 4. Eddy current losses in all PM calculated with the 3-D time-stepped finite-element analyzes.

D. Miljavec and B. Zidaric / Numerical Calculation of Eddy Current Losses 301

pends on stator slot opening and on rotor rotational speed. Both parameters cause the

magnetic flux density changes which are penetrating into PM.

Figure 6 shows the eddy current density lines in second and third PM (from left

side in Fig. 3) at 4000 rpm. These lines present the points in PM with equal eddy cur-

rent densities. Higher density of these lines means higher value of induced eddy cur-

Figure 5. Penetration of induced eddy current density (axial component) into second PM (from left side in

Fig. 3) at 4000 rpm.

Figure 6. Eddy current density lines in second and third PM (from left side in Fig. 3) at 4000 rpm.


rents. The vectors of eddy current density are always tangential to the presented eddy

current density lines.

One way to reduce the induced power losses is to axially segment the PM [3–7].

Figure 7 shows the eddy current density lines in PM when they are segmented in three

levels. The result shown in Fig. 6 is valid at 4000 rpm and at presented stator position.

The same situation is presented in a form of color palette, but for 4 axial segments per

PM.

Figure 7. Eddy current density lines in PM (magnets are axial segmented into 3 pieces) at 4000 rpm.

Figure 8. Induced eddy current density in PM (magnets are axial segmented into 4 pieces) at 4000 rpm.


The study was curried out for different number of axial PM segments. The eddy

current losses depending on number of PM segments at 2000 rpm and at 4000 rpm are

shown Fig. 9.

2-D Numerical Analyze of Eddy Current Losses in Permanent Magnets

On the base of the 2D finite element method with time-stepped finite-element analyze

[1,2], in the introduction described working conditions were achieved with BLDC ma-

chine model shown in Fig. 10.

Normally, the eddy currents are not included in 2D finite element calculations. To

have them taken into account, the PM were presented to FE analyze as solid conductors

with their electric and magnetic properties. Using solid conductors in calculation of

eddy currents the end-effect is neglected, because the 2-D FEM assumes that solid

conductors are infinitely long and currents flow only in axial direction.

1 2 3 4 5

0

20

40

60

80

100

120

Number of segments per magnet

Peddy

, n = 2000 rpm

Peddy

, n = 4000 rpm

Ed

dy cu

rren

t lo

ss in

m

ag

nets (W

)

Figure 9. Power losses in PM depending on number of PM’s axial segments at 2000 rpm and 4000 rpm.

Rotor

Stator

Permanent magnets

Slot openings

Figure 10. 2-D outer rotor BLDC machine geometry with finite elements.


The Fig. 11 shows the induced eddy current density distribution on PM magnet

cross-section as consequence of magnetic flux density changes only due to slot opening

at rotor speed of 4000 rpm.

The eddy current losses in PM, calculated with described 2-D FEM analyze, as a

function of the rotor speed and different stator slot openings are presented in Fig. 12.

Good agreements between results from 2-D (Fig. 12, 7 mm slot opening) and 3-D

(Fig. 4) analyses are mainly due to relatively thick rotor beck iron. This thickness al-

lows the PM’s magnetic flux to distribute itself in the rotor beck iron in a quite the

same manner, so in 2-D and 3-D analyses.

magnet

boundary

Figure 11. 3D view of induced eddy current density distribution in PM due to slot opening at 4000 rpm.

0 1000 2000 3000 4000 6000 70005000

Eddy current loss in m

agnets (W

)

250

200

150

100

50

0

Stator slot openings:

7 mm original

6 mm

5 mm

4 mm

Speed (rpm)

Figure 12. Eddy current losses versus speed and stator slot opening.


Conclusion

The eddy current losses in permanent magnets (PM) of outer rotor BLDC machine has

been analyzed both 3-D and by 2-D time stepping finite-element method. The eddy

current losses in PM have been induced only by reluctance variation due to stator air

gap side surface. It has been shown that the reduction of these induced losses can be

achieved by axial segmentation of PM and by narrowing the stator slot openings.

References

[1] FLUX2D, software for electromagnetic design from CEDRAT, 2006.

[2] D. Maga, R. Hartansky, “Numerical solutions (Numericke riesenia)”, University of Defence, Brno,

Czech republic, 2006, ISBN 80-7231-130-1.

[3] Yacine Amara, Jiabin Wang, David Howe, “Analytical Prediction of Eddy-Current Loss in Modular Tu-

bular Permanent-Magnet Machines”, IEEE Trans. On Energy Conversion, vol. 20, No. 4, December

2005.

[4] Hiroaki Toda, Zhenping Xia, Jiabin Wang, Kais Atallah, David Howe, “Rotor Eddy-Current Loss in

Permanent Magnet Brushless Machines”, IEEE Trans. On Magnetics, vol. 40, No. 4, July 2004.

[5] W.N. Fu, Z.L. Liu, “Estimation of Eddy-Current Loss in Permanent Magnets of Electric Motors Using

Network-Field Coupled Multislice Time-Stepping Finite Element Method”, IEEE Trans. On Magnetics,

vol. 38, No. 2, March 2002.

[6] D. Nedeljkovic, R. Fiser, V. Ambrozic, “Time-optimal magnetization of inductors with permanent mag-

net cores”, Inf. MIDEM, 2004, Vol. 34, No. 1, pp. 32-36.

[7] Kinjiro Yoshida, Yasuhiro Hita, Katsumi Kesamaru, “Eddy-Current Loss Analysis in PM of Surface-

Mounted-PM SM for Electric Vehicles”, IEEE Trans. On Magnetics, vol. 36, No. 4, July 2000.


Analysis of High Frequency Power

Transformer Windings for Leakage

Inductance Calculation

Mauricio Valencia FERREIRA DA LUZa

and Patrick DULARb

a

GRUCAD/EEL/CTC, C.P. 476, 88040-900, Florianópolis, SC, Brazil


b

University of Liège, Institut Montefiore, Sart Tilman, B28, BE-4000, Liège, Belgium


Abstract. This paper deals with the analysis of high frequency power transformer

windings. The current and magnetic flux densities are determined by the finite

element method. The considered system is a pot core transformer, to which the ax-

isymmetric magnetic vector potential formulation is applied. The leakage induc-

tance obtained by the finite element method is compared with the analytical one

for different types of winding. The contribution of this paper is focused on the de-

velopment of the analytical equations to calculate the leakage inductance.

Introduction

In the last two decades energy supply systems have undergone some major changes,

although this is not always visible from outside. Power electronics plays a growing role

in the grid through power flow conditioners and converters of electricity. High fre-

quency power transformers are the main components of the modern power electronics.

The investigation of high frequency power transformers for switching mode power

supply application is very important for power electronic area.

Winding structure of high frequency transformers is the major factor to determine

the performance of the transformer. The coil winding is often the most delicate part of

a finite element method (FEM) model, since it is used within a loss computation and

coupled to a thermal FEM model. This part of the device has significant temperature

dependence. In general, two types of coil windings are distinguished: stranded coil

windings and massive or solid conductor coil windings. The fundamental distinction

between both types is the occurrence of significant internal eddy currents, being a fre-

quency determined phenomenon. Therefore, it is possible that the same coil construc-

tion is represented as stranded for relatively low frequencies and as solid for the higher

frequencies.

When a transformer transfers energy, large currents flow through its windings and

cause Joule losses. Depending on the skin depth relative to the winding strand size,

internal skin and proximity effects occur, increasing the losses. Part of the leakage flux

passes through conductive parts, including tank, inducing eddy currents, and thus addi-

tional losses.

The magnetic flux distribution and the currents flowing through the windings are

two important characteristics to be considered for high frequency transformer design.


307

Not all the magnetic flux created by primary winding of transformer follows the mag-

netic circuit and links the other windings. In addition to the mutual flux, which does

link both of the windings, there is the leakage flux that leaks from the core and returns

through the air and consequently causes imperfect coupling. Therefore, the leakage

inductance is a very important factor for transformer design; it can cause overvoltage in

power switch at turn-off action.

It is important to make a very clear distinction between leakage field (or flux) and

stray field (or flux). The leakage is formed by the flux that links one winding and does

not link the other winding. It can be measured as a voltage drop at the transformer ter-

minals. The leakage flux does not necessarily escape the transformer. Stray fields nec-

essarily escape the transformer. The stray flux can link one or two of the windings.

Stray flux can exist in the air without adding to the leakage inductance. The stray flux

is not measured as a voltage drop at the terminals. It can be measured with a coil in the

neighborhood of the transformer. A portion of the leakage flux can also be stray flux

when it escapes the transformer boundaries.

Magnetic Vector Potential Formulation and Types of Coil Windings

A bounded domain Ω of the two or three-dimensional Euclidean space is considered.

Its boundary is denoted Γ. The equations characterising the magnetodynamic problem

in Ω are [1]:

jh = curl , bet

curl ∂−= , 0 div =b , (1a-b-c)

rbhb +μ= , ej σ= , (2a-b)

where h is the magnetic field, b is the magnetic flux density, br is the permanent mag-

net remanent flux density, e is the electric field, j is the electric current density, includ-

ing source currents js in Ω

s and eddy currents in Ω

c (both Ω

s and Ω

c are included in

Ω), μ is the magnetic permeability and σ is the electric conductivity.

The boundary conditions are defined on complementary parts Γh and Γ

e, which

can be non-connected, of Γ,

0

h

=×

Γ

hn , 0 .

e

=

Γ

bn , 0

e

=×

Γ

en , (3a-b-c)

where n is the unit normal vector exterior to Ω. Furthermore, global conditions on volt-

ages or currents in inductors can be considered [1].

The a-formulation, with a magnetic vector potential a and an electric scalar poten-

tial v, is obtained from the weak form of the Ampère equation (1a) and (2a-b) [1], i.e.

,0)' ,()' ,grad v ( )' , (

',)' curl, ()' curl, urlc (

s

h

scct

sr

=−σ+∂σ

+>×<+ν−ν

ΩΩΩ

ΓΩΩ

ajaaa

ahnabaa

),(F'a

Ω∈∀a

(4)

M.V. Ferreira da Luz and P. Dular / Analysis of High Frequency Power Transformer Windings308

where s

hn× is a constraint on the magnetic field associated with boundary Γh of the

domain Ω and μ=ν /1 is the magnetic reluctivity. Fa(Ω) denotes the function space

defined on Ω which contains the basis and test functions for both vector potentials a

and a'. (. , .)Ω

and <. , .>Γ denote a volume integral in Ω and a surface integral on Γ of

products of scalar or vector fields. Using edge finite elements for a, a gauge condition

associated with a tree of edges is generally applied.

In FEM model, two types of coil windings are distinguished [2]: Stranded coil

windings: the coils are made of strands having a radius smaller than the skin depth. It is

not required to discretize every individual conductor and the entire coil cross-section is

meshed as a whole. The stranded winding circuit model corrects resistance and losses

by the filling factor. The losses due to the leakage flux have to be considered in the

model by an external calculation. Massive or solid conductor coil windings: when the

skin effect is significant, the individual conductors have to be modelled and meshed

separately so eddy currents can be included. This implies that the elements cannot be

larger than the skin depth at the outskirts. This is the case with thick wire coils and foil

(sheet) windings. It depends on the relative sizes of the conductor and the conductor’s

insulation whether this individual model is merely a set of touching conductive regions

or a detailed conductor/insulation composite. All kinds of Joule and eddy current losses

are automatically included in the winding model.

Analytical Prediction of Leakage Inductance

The leakage inductance is a very important factor for transformer design. It can be es-

timated via analytical methods. For example, the leakage inductance for winding ar-

rangements can be calculated by [3]:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+μ= ∑∑

g2

w2

1o1h

3

h

M

1

b

l

N L ,

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+μ= ∑∑

g2

w2

2o2h

3

h

M

1

b

l

N L ,

(5a-b)

where L1 and L

2 are the primary and secondary leakage inductances, respectively. N

1 is

the number of primary turns, N2 is the number of secondary turns, M is the number of

section interfaces, lw is the mean length, h is the height and b is the breadth of the wind-

ing. Figure 1 shows the winding arrangements for calculation of leakage inductance.

Application

In order to demonstrate and validate the proposed method, we consider a pot core trans-

former. It is a single-phase transformer with single layer, a primary winding located on

the top section of three copper wires and a secondary winding arranged at the bottom

M.V. Ferreira da Luz and P. Dular / Analysis of High Frequency Power Transformer Windings 309

half section. Figure 2(a) shows the axisymmetric studied domain and the 2D mesh.

Figures 2(b)–(e) show the magnetic flux distribution.

The primary winding and the secondary winding are totally separated. At the oper-

ating frequency of 1 kHz, the magnetic flux mostly goes through the core structure and

couples both windings. At operating frequencies of 10 kHz and 100 kHz, the phenom-

ena of leakage flux happen and the majority of flux goes out from the magnetic core

into the gap between the primary and secondary windings. In the last picture, at the

operating frequency of 1 MHz, almost all the flux goes out of the core and only a very

low flux couples the primary and secondary windings. This leakage flux generates eddy

Figure 1. Winding arrangements for calculation of leakage inductance.

(a) (b) f = 1 kHz (c) f = 10 kHz

(d) f = 100 kHz (e) f = 1 MHz

Figure 2. (a) Mesh 2D and axisymmetric studied domain. (b), (c), (d) and (e) Zoom of the magnetic flux

distribution of an axisymmetric pot core transformer.


currents flowing inside the primary winding and the top section of the secondary wind-

ing. Therefore, no magnetic flux can penetrate the secondary winding as the operating

frequency further increases. The phenomenon of leakage flux inside a high frequency

operated transformer can be explained by the eddy current flowing in the windings.

According to these numerical simulations, the transformer can be defined not working

well the operating of 10 kHz or above.

Figure 3 shows an axisymmetric pot core transformer with (a) separated windings

and (b) interweaving windings. The interweaving windings have a winding structure

with the most number of section interfaces. If the number of section interfaces in-

creases, then, the leakage inductance decreases. The relationship between them is that

leakage inductance is inversely proportional to the square of the number of section in-

terfaces of the winding. Each wire of primary winding is separated by secondary wire,

as shown in Fig. 3. The leakage inductance of this winding structure is much less than

the leakage inductance of the separated winding structure.

The comparison of the leakage inductance of these transformers (Fig. 3) is shown

in Table 1.

Conclusions

In this paper, the leakage inductance was numerically calculated with a magnetody-

namic formulation and with adapted techniques for considering stranded and massive

conductors. The comparison of the results between the analytical equation and the FEM

simulation was satisfactory. An axisymmetric pot core transformer with separated

windings and interweaving windings was analyzed. The leakage inductance of the in-

terweaving winding structure is very small compared with the separated winding struc-

tures. This structure can increase the operating frequency of the transformer. However,

Figure 3. (a) Pot core transformer with separated winding and (b) pot core transformer with interweaving

windings.

Table 1. Leakage inductance of two winding structures at the frequency of 1 MHz

Winding Structures Leakage inductance –

analytical equation

Leakage inductance –

MEF

Primary 226.504 μH 229.684 μHSeparated

windings Secondary 226.516 μH 229.963 μH

Primary 3.712 μH 4.828 μHInterweaving

windings Secondary 3.713 μH 4.831 μH

M.V. Ferreira da Luz and P. Dular / Analysis of High Frequency Power Transformer Windings 311

it is not easy to wound circular copper wires in a perfect square arrangement by hand or

by machine. Therefore new structure of windings or new structure of whole transform-

ers is urgently needed for modern power electronics.

References

[1] M.V. Ferreira da Luz, P. Dular, N. Sadowski, C. Geuzaine, J.P.A. Bastos, “Analysis of a permanent

magnet generator with dual formulations using periodicity conditions and moving band”, IEEE Transac-

tions on Magnetics, Vol. 38, No. 2, pp. 961-964, 2002.

[2] J.P.A. Bastos, N. Sadowski, “Electromagnetic modeling by finite elements”. Marcel Dekker, Inc, New

York, USA, 2003.

[3] P.L. Dowell, “Effect of eddy currents in transformer windings”, Proc. IEE, Vol. 113, No. 8, pp. 1387-

1394, 1966.


Influence of the Stator Slot Opening

Configuration on the Performance of an

Axial-Flux Induction Motor

Asko PARVIAINENa

and Mikko VALTONENb

a

AXCO-Motors Oy, Laserkatu 6, 53850 Lappeenranta, Finland

b

Lappeenranta University of Technology, Skinnarilankatu 34,

53850 Lappeenranta, Finland


Abstract. Axial-flux induction machines are found to be an attractive solution in

some specific industrial applications. However, the rotor construction of an axial-

flux induction motor is sensitive to harmonic losses caused by permenace harmon-

ics and time harmonics in the supply current. The paper concentrates on studying

the possibility of improving the machine efficiency by adjusting the stator slot

opening configuration.

Introduction

Axial-flux induction machines are found to be an alternative solution to radial flux ma-

chines in several industrial applications. However, because of the limited number of

machines manufactured in the past, their design principles are not very well known and

documented. In particular for medium speed, i.e., machines with a rated speed of

4000 min–1

–10 000 min–1

, solid-rotor-core axial flux machines may be considered a

totally new machine topology. The target of this work is to find out whether the stator

slot opening configuration has an influence on the performance of a solid-rotor-core

axial flux induction motor. It was assumed that rotor losses, caused by the stator per-

menace harmonics, will be reduced if a totally enclosed slot opening is employed. Fig-

ure 1 shows the typical loss density distribution, caused by the stator slot openings on

the rotor surface. Further, it was expected that a change in the stator leakage inductance

will reduce the losses caused by the time harmonics in the frequency converter use.

Based on these assumptions, three different slot opening configurations were studied

by FE analysis and by measurements for the actual machines. The paper is a continua-

tion for the work commenced by the authors in [1] and [2].

Computation Model

The performance characteristics of the axial-flux solid-rotor-core induction motor were

evaluated by using a two-dimensional, non-linear, time-stepping FEA, i.e., the mag-

netic saturation, skin effect and the movement of the rotor with respect to the stator

were taken into account. The analysis was carried out by applying FLUX2D™ finite


313

element software. Most of the spatial harmonics in the air gap field are generated by

the variation of the air gap permeance caused by the stator slot opening. In the electri-

cally conducting ferromagnetic rotor material, the losses caused by eddy currents may

be significant. The power loss in the solid-rotor-core materials can be computed as

2

dV

P Vρ= ∫∫∫ J (1)

where ρ is the resistivity of the material, J the current density, and V the volume of the

rotor conducting material

Slot Opening Configurations

Figure 2 illustrates the slot opening configurations analyzed in this study. The structure

(a) is a conventional semi-open slot configuration, while the structure (b) depicts a

conventional semi-open slot opening, but the slot opening is filled with semi-magnetic

slot wedge material. The structure (c) is a totally enclosed slot opening configuration.

The corresponding stator was manufactured by adjusting the punch line parameters in

the stator manufacture so that there remained a narrow steel lamination bridge in the

slot opening towards the air-gap.

Figure 1. Loss density distribution on the rotor surface caused by the stator slot opening.

Figure 2. Slot opening shapes.

A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration314

Prototype Machines

Three different equal stators were manufactured corresponding to the slot opening

shapes (a)–(c). The corresponding machines are 45 kW/6000 min–1

single-sided axial

flux machines.

Test Setup

For each stator, an input-output-based load test was performed. The stator configura-

tions were measured using the same housing and the same rotor in order for the test

setups to be equal between the designs. The machine air-gap length and the parameters

of the frequency converter were also kept the same in the tests. Figure 4 provides the

test configuration employed.

Figure 3. Stator with open slot.

Figure 4. Test setup.

A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration 315

Results

Figures 5 and 6 provide the results of the test runs. A comparison with the computed

values is also provided in Fig. 6.

0,7

0,74

0,78

0,82

0,86

0,9

0,94

25 30 35 40 45 50 55 60 65 70 75

Torque [Nm]

Efficiency

semi-closed slot

totally enclosed slot

open slot

Figure 5. Efficiency of the motor as a function of torque. The motor rated torque is 75 Nm.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

enclosed slot semi-closed open slot enclosed slot semi-closed open slot

Slot opening shapes

Po

wer facto

r / E

fficien

cy

0

30

60

90

120

150

Cu

rren

t [A

]

cosfii measured cosfii calculated

efficiency measured efficiency calculated

current measured current calculated

Figure 6. Effects of the slot opening shapes on the efficiency, power factor, and current at the nominal point

of the motor.

A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration316

Discussion

The measurement results show some unexpected behavior: according to the measure-

ments, a totally enclosed slot opening yields a poorer performance in terms of motor

efficiency than semi-closed slot opening. Also the effect of the totally enclosed slot on

the motor power factor is considerable. The weakest performance, in terms of effi-

ciency, is for the open-slot stator; this is an expected result. Considering the stator

manufacturing issues, a totally closed slot causes some major problems in the winding

in serial production. Therefore, the practical relevance and applicability of this configu-

ration are questionable.

Conclusions

In this paper, the influence of the slot opening shape on the performance of a solid-

rotor-core medium-speed axial flux induction motor is studied. The results from the

computations and measurements show some interesting results. In the studied slot

opening configurations, the totally enclosed slot opening shows no benefits over the

semi-closed slot provided with semi-magnetic slot wedge material. When the obtained

result is combined to the time-consuming winding process of the stator with a totally

enclosed slot openings, this slot opening shape cannot be recommended in machines

that are aimed to be manufactured for industrial purposes.

References

[1] M. Valtonen, A. Parviainen and J. Pyrhönen, Determination of the Rotor Losses in an Inverter Supplied

Axial-Flux Solid-Rotor Core Induction Motor by Using 2D FEM, ICEMS 2006 CD-Proceeding, 2006.

[2] M. Valtonen, A. Parviainen and J. Pyrhönen, Inverter Switching Frequency Effects on the Rotor Losses

of an Axial-Flux Solid-Rotor Core Induction Motor, PowerEng 2007 CD-Proceeding, 2007.

A. Parviainen and M. Valtonen / Influence of the Stator Slot Opening Configuration 317

Characteristics of Special Linear Induction

Motor for LRV

Nobuo FUJIIa

, Kentaro SAKATAa

and Takeshi MIZUMAb

a

Kyushu University /Dept. of Electrical and Electronic Systems Eng., Japan

[email protected]

b

National Traffic Safety & Environment Laboratory, Japan

[email protected]

Abstract. The special linear motor with function of linear transformer is studied

analytically by using a special integral equation method (IEM) and a finite element

method (FEM). The basic configuration is a wound-secondary type of linear in-

duction motor (LIM), and comprises the primary winding of single-phase concen-

trated coil and the secondary winding with two-phase distributed coil. The com-

puted results by IEM and FEM agree well in the flux density and thrust respec-

tively. The estimate of thrust by IEM is easy, especially for dynamic analysis. The

thrust of apparatus has about 2.1 kN/m and an enough value for the propulsion of

LRV.

Introduction

For a future public transportation which is in harmony with environment, a new type of

light rail vehicle (LRV) is hoped. The overhead-wireless and non-contact power collec-

tion for on-board power is the key to the development. The linear induction motor

(LIM) will be one of the key devices [1–4].

The authors have proposed a new apparatus with functions of propulsion and non-

contact power collection [5]. The apparatus changes to the transformer and the LIM

respectively by only the signal of converter connected to the onboard winding. The

basic configuration is a wound-secondary type of LIM. In standstill condition in the

section of guide way with the ground winding, the batteries on board are charged by

the non-contact power collection. For the acceleration and the deceleration of vehicle in

the section with ground winding, the vehicle is driven by the LIM operation using the

ground commercial power. In the section without the ground winding, the vehicle is

run by the power supplied from the onboard battery source.

In the paper, the characteristics of linear motor are studied analytically by using

the special integral equation method (IEM) and the finite element method (FEM).

Basic Configuration and Analytical Model

Figure 1 shows the basic configuration. Figure 2 shows the dimensions of model, in

which the real secondary winding has the 16 poles. In the analytical model, the length

of longitudinal (x-) direction is extended infinity as the periodic model of Fig. 3 is

adopted. The primary winding set on the ground is the concentrated winding of single-



doi:10.3233/978-1-58603-895-3-318

318

phase current as shown in Fig. 4(a), which is supplied from commercial power source

with a fixed frequency. The secondary winding equipped on the vehicle is the double-

layer winding as shown in Fig. 4(b) and two-phase arrangement for linear motor opera-

tion, and is changed to the single-phase arrangement for the transformer operation. The

secondary current and frequency are controlled by an onboard inverter. The secondary

frequency is the slip-frequency for the vehicle speed with a slip. The slip power is

charged to the onboard battery. The thrust for propulsion force of vehicle is obtained by

the interaction between the alternative flux of primary winding and the shifting mag-

netic field of secondary winding.

The special integral equation method (IEM) named ELF/MAGIC is used for the

analysis, which has no mesh in the air region. To check the validity the FEM analysis is

also used. The analytical model is shown in Figs 3 and 4. The model is for a pair of

pole and the periodic method is used. In the design, one coil of the primary winding has

z

x

Secondary

(Vehicle)

Primary

(Ground)

b

+

-

a−

a

b−

Onboard converter (Two-phase inverter operation) Linear motor

Figure 1. Special wound type of linear induction motor and onboard converter for LRV.

a a-a-a -b -b b b

1035

1400=x

2 34

300

Secondaryz

y

x

12

100

20

11

30

Primary

Figure 2. Dimensions of analytical model.

(a) Primary coil (b) Secondary coil

Figure 3. Model for IEM or FEM. Figure 4. Coil shape for IEM or FEM.

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV 319

32 turns with rated current of 113 A, and one coil of the secondary winding is 8 turns

with rated current of 110 A.

Flux Density in Air Gap

Figures 5 and 6 show the x- and z-component air-gap flux density respectively. The

position of z = 1 mm means 1mm height from the surface of primary core, and

z = 11 mm means 1mm height from the surface of secondary core. The shape of flux

density distribution is varied by the position in the air gap, because the slot pitch and

the phase between currents of the primary and the secondary are different respectively.

The solid line represents the value computed by IEM, in which the tooth of 11 mm

width is divided into two in the direction of x, and the broken line represents the value

computed by FEM, in which the tooth width is divided into 11. There is little difference

between two computed results.

Figure 7 shows the distributions of flux density with time at middle of air gap

(z = 6 mm). Although the shifting magnetic field generates by the secondary winding,

the point of zero flux is present because the influence of alternative magnetic field of

primary winding with single-phase current is large.

-0.1 0 0.1

-0.4

-0.2

0

0.2

0.4

z = 1 mm

x-position x(m)

x-co

mp

on

en

t

flu

x d

en

sity

Bx

(T

) IEM

FEM

-0.1 0 0.1

-0.4

-0.2

0

0.2

0.4

x-position x(m)

z = 11 mmx-com

ponent

flux density Bx

(T

)

IEM

FEM

Figure 5. Distribution of x-component flux density in air gap.

-0.1 0 0.1

-1.0

-0.5

0

0.5

1.0

z = 1 mm

x-position x(m)

z-com

ponent

flux density Bz

(T

)

IEM

FEM

-0.1 0 0.1

-1.0

-0.5

0

0.5

1.0

x-position x(m)

z = 11 mm

z-co

mp

on

en

t

flu

x d

en

sity

Bz

(T

)

IEM

FEM

Figure 6. Distribution of z-component flux density in air gap.

-0.1 0 0.1

-0.20

0

0.20

x-position x(m)

x-co

mpo

nen

t

flux

den

sity

Bx (T

)

ωt = 0, 30, 60, ..., 180°z = 6 mm

-0.1 0 0.1

-1.0

-0.5

0

0.5

1.0

x-position x(m)

z-com

ponent

flux density Bz (T

)

ωt = 0, 30, 60, ..., 180°z = 6 mm

(a) x-component (b) z-component

Figure 7. Flux density distribution with time at standstill.

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV320

Thrust at Standstill Condition

Figure 8 shows the computed thrust distribution at near surface of primary member and

surface of secondary member respectively. Although there are large ripple in the

distributions, the computed results of IEM and FEM are agree well. Figure 9 is the

thrust distribution at the middle position of air gap. The 2D current sheet model means

the two-dimensional electromagnetic theoretical method with smooth cores, in which

Carter’s coefficient for both cores is used to correct the real air gap. At this position, it

is suitable to estimate the spatial average thrust as the ripple is small. The computed

value for the 2 divided model of IEM is almost equal to that for 11 divided model of

FEM. That is, the convergence of force in the IEM is very well compared to FEM. At

standstill condition, the thrust distribution doesn’t move in longitudinal direction and

only the magnitude varies with time, as shown in Fig. 10. The whole thrust in a pair of

pole length changes with time as shown in Fig. 11. The value of current sheet model is

about 13% smaller than that of IEM as shown in Fig. 12.

-0.1 0 0.1

-50

0

50

100

z = 1 mm

x-position x(m)

Thrustfx (kN

/m

)

IEM

FEM

-0.1 0 0.1

-50

0

50

100

x-position x(m)

z = 11 mm

Thrustfx (kN

/m

)

IEM

FEM

Figure 8. Thrust distribution.

-0.1 0 0.1

0

20

40

x-position x(m)

z = 6 mm IEM

FEM

2D current sheet

Thrustfx

(kN

/m

)

-0.1 0 0.1

-20

0

20

40

x-position x(m)

Th

ru

stfx

(k

N/m

)ωt = 0, .... , 90°

ωt = 90, .... , 180°z = 6 mm

Figure 9. Comparison among thrust distributions of

three types of analytical methods.

Figure 10. Thrust distribution with time at standstill.

90 180

2

4

6

0

Th

ru

stfx

(kN

/m

)

Time ω t (deg.)

IEM

FEM

-180 -90 90 180

-2

-1

1

2

0

2D current sheet model

IEM

FEM

ThrustFx

(kN/m)

Phase ϕ (deg.)

δ = 90°

Figure 11. Thrust (whole in model) with time. Figure 12. Comparison among thrust (time average)

of three types of analytical methods.


Thrust in Moving Condition

As the linear motor has the special winding configuration, the dynamic characteristics

are studied by using IEM, which is especially convenient because there is no need the

meshing in air region. Figure 13 shows the distribution of z-component flux density

with time when the vehicle with secondary winding is running at speed of 21.6 km/h.

As the flux by primary winding is alternative at fixed position and the flux by secon-

dary winding with moving speed of 21.6 km/h is shifting field with 28.6 Hz, the wave

of flux distribution varies at the observed position. The distribution observed from the

primary member varies in waveform for the moving effect as shown in Fig. 13(a). The

flux observed from the secondary moves and varies as shown in Fig. 13(b), which is

much different from Fig. 7(b) and represents the effect of moving clearly. On the thrust

shown in Fig. 14, the effect of moving appears. Figure 15 shows the variation with time

in the whole thrust for a pair of pole length. For the moving time of °= 8401tω the ve-

hicle moves by the distance of 0.28 m which is the length of two poles. Although there

is difference between the thrusts at 21.6 km/h and standstill, those averages are almost

equal. The spatial and time average thrust is about 2.1 kN/m and an enough value for

practical use.

Conclusions

1. The validity of computed flux density in the air gap and the thrust respectively

is confirmed from the agreement of results of IEM and FEM analysis.

2. The thrust of proposed apparatus has about 2.1 kN/m and an enough value for

the propulsion of LRV.

-0.1 0 0.1

-1.0

-0.5

0

0.5

1.0

x-position x(m)

z-co

mp

on

en

t

flu

x d

en

sity

Bz

(T

)ωt = 0, 30, 60, ..., 180°

z = 6 mm

-0.1 0 0.1

-1.0

-0.5

0

0.5

1.0

x-position x(m)

z-co

mp

on

en

t

flu

x d

en

sity

Bz

(T

)

ωt = 0, 30, 60, ..., 180°z = 6 mm

(a) Observed on the primary member (b) Observed on the secondary member

Figure 13. z-component flux density distribution with time at moving speed of 21.6 km/h.

-0.1 0 0.1

-20

0

20

40

x-position x(m)

Thrustfx

(k

N/m

)

ωt = 0, .... , 90°

ωt = 90, .... , 180°z = 6 mm

180 360 540 720

2

4

6

0

Th

ru

stfx

(k

N/m

) v =0, v =21.6km/s

Time ω1t (deg.)

Figure 14. Thrust distribution with time at moving

speed of 21.6 km/h.

Figure 15. Comparison between thrusts at standstill

and moving speed of 21.6 km/h.

N. Fujii et al. / Characteristics of Special Linear Induction Motor for LRV322

3. The convergence of solution of IEM is well for a division of elements in a

three-dimensional analysis.

4. The estimate of thrust (force) by IEM is easier compared with the FEM, espe-

cially for dynamic analysis.

References

[1] B. Yang, M. Henke, H. Grotstollen, “Pitch Analysis and Control Design for the Linear Motor of a Rail-

way Carriage”, IEEE IAS Annual Meeting (IAS2001), pp. 2360-2365, 2001, Chicago.

[2] B. Yang, H. Grotstollen, “Application, Calculation and Analysis of the Doubly Fed Longstator Linear

Motor for the Wheel-on-Rail NBP Test Track”, EPE-PEMC 2002, 2002 Dubrovnik, Croatia.

[3] R. Shindoh, T. Mizuma, “Evaluation of Air Suspended LIM Driven Transit System”, Proc. of 6th

Int.

Sympo. on Magnetic Suspension Technology, pp. 345-348, Oct. 2001, Turin, Italy.

[4] N. Fujii, I. Hirata, K. Kawamura, K. Nishimura, “Investigation of High Performance Drive of Linear

Motor Train for Urban Transit”, Trans. IEE of Japan, vol. 114-D, pp. 910-917, Sep. 1994 (in Japanese).

[5] N. Fujii, T. Mizuma, “Device with Functions of Linear Motor and Non-contact Power Collector for

Wireless Drive”, Trans. IEE of Japan, vol. 126-D, No. 8, pp. 1113-1118, Aug. 2006 (in Japanese).


Electromagnetic Computations in the End

Zone of Power Turbogenerator

M. ROYTGARTSa

, Yu. VARLAMOVb

and А. SMIRNOVa

a

OJSC “Power Machines” Branch ELECTROSILA

Telephone: +7 (812) 387 47 88, Fax: +7 (812) 388 18 14,


b

St.-Petersburg State Polytechnic University

Telephone: +7 (812) 348 91 77, Fax: +7 (812) 388 18 14,


Abstract. The mathematical model and computations results of electromagnetic

fields, eddy currents and losses in the construction of the power turbogenerators

end zone are presented. Effect of the stator and rotor size relations, geometry of

the skewed core end part, shape, sizes and arrangement of stator core screens and

pressure plate by the method of numerical experiment are investigated. The results

of numerical modeling in designs of powerful turbogenerators are implemented.

Computational and test data are compared.

1. Introduction

At designing and exploitation modern high loaded turbogenerators inevitably there is a

problem of limitation of heating and increase of use reliability of end zone. The heating

of structural element is determined both efficiency of ventilating, and intensity of allo-

cated losses dependent on affecting electromagnetic fields, characteristics of used ma-

terials, features of design [1,2].

Aim of this paper consists in numerical calculation and analysis of electromagnetic

fields, eddy currents and losses in the most loaded end zone of non-salient pole syn-

chronous machines in steady state mode of operation. This problem is essentially three-

dimensional in view of complicated geometry of a field sources – currents in stator and

rotor windings, and also composite configuration of computational area including end

surfaces of the stator and a rotor cores, screens, casing, end shields. The full three-

dimensional analysis of a field is time-consuming even at using of modern computer

technology, therefore in practice the introducing of the justified assumptions is expedi-

ent.

2. The Mathematical Model

Application of Fourier series for harmonic presentation of currents and fields in the

direction of electric machine rotation made it possible to combine analytical approach



doi:10.3233/978-1-58603-895-3-324

324

to the problem in the direction of angular coordinate with numerical analysis in the

plane of rotational axis. In this case geometry of conductors, configuration and elec-

tromagnetic properties of the core, end plate and shield of the stator, rotor geometry,

space-time law of variation of currents in the stator and rotor windings have been taken

into account. The computation model covers the end zone and a part of active length of

the stator and rotor adjacent to it (Figs 1, 2).

The field sources are densities of the rotor and stator currents in the singled out

sub-areas of the slot and end portions of the winding, including the linear, bent parts

and heads. Two-layer type of the winding, shortening of the stator winding pitch, bend-

ing of the end portions, phase shift of currents in the bars and in the rotor winding have

been taken into account. Amplitudes of current densities were determined by the for-

mulas well-known in the electric machine theory with the only difference that the

winding factors were assigned by the distribution factors and the shortening of pitch

varying along the length of the end portions was automatically taken into account at

computation of the field in the end zone. The analytical representation of the field

sources as rotating waves as well as uniformity or periodicity of the properties of the

machine construction in the direction of rotation made it possible to obtain (in the

quasi-linear statement of problem) the results being also in the form of superposed ro-

tating waves, as follows:

)](exp[),,(),,,(xm

vtjzrXtzrX ψϕωϕ +−= (1)

where Xm

(r, z) – complex amplitude, r, z, ϕ – cylindrical coordinates, v – harmonic

number, ψx – starting phase.

Scalar magnetic potential um

out of eddy currents zone is defined by the following

equation:

mmm

Hdivurvugraddiv02

22

22)/( μμμ =− (2)

Figure 1. Computation model of turbogenerator

160 MW with air cooling.

Figure 2. Computation model of turbogenerator

800 MW with water cooling.

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator 325

where μ is the medium magnetic permeability, H0m

is the current vector function ampli-

tude, index «2» with differential operator means differentiation in the (r, z) plane.

In the eddy currents zone Maxwell’s equations are directly integrated expressed

through electric field strength E:

0

1

=+ EjrotErot ωγ

μ

(3)

EroteE

r

jv

rotE2

][ +×=ϕ

(4)

where ω – circular frequency, γ – medium conductivity, eφ– unit vector in the direction

of rotation.

For subdomain adjoint boundaries classic boundary conditions of continuity of

magnetic intensity tangent components and normal component of magnetic flux density

as well as non-linear surface impedance boundary conditions, which simplify signifi-

cantly the solution with sharply expressed surface effect, have been applied [3,4].

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

∂

∂

−

−

=⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

−

=

∂

∂

m

mmm

u

r

vu

j

Z

n

u

j

Z

n

u

2

2

2

2

0

2

2

0τωμωμ

(5)

where Z is a wave impedance of conducting ferromagnetic body thickness and curva-

ture radius of which are much larger than electromagnetic field penetration depth, n –

normal to the boundary surface, directed inside of conducting medium.

With the help of the impedance boundary conditions the geometrical dimensions,

shape and electromagnetic properties of the turbogenerator ferromagnetic housings and

end shields were taken into account [5,6].

For simulation of non-uniform magnetic properties of the stator core along and

across the rolled steel, for account of insulation intervals between the stacked core

laminations as well as for account of magnetic properties of the slot and tooth zone, the

equivalent anisotropic magnetic permeability has been used. The averaging was made

under condition of keeping the magnetic flux. When non-linear case was considered,

for each of iteration the average magnitude of the magnetic permeability tensor has

been re-computed in each point of the computation area.

The average value of magnetic permeability tensor components are obtained as

( )

( )[ ]

( )[ ]zzzzze

ze

rre

k1//kk1

k1//kk1

1k1

−−+=

−−+=

−+=

μμμ

μμμ

μμ

ϕ

ϕϕϕϕϕ

ϕ

, (6)

where kφ

and kzare iron factors in direction of rotation and axial coordinate.

M. Roytgarts et al. / Electromagnetic Computations in the End Zone of Power Turbogenerator326

3. Results of Calculations

The computations of the end zone for air-, water- and hydrogen-cooled turbogenerators

63 to 1000 MW have been done according to the programs developed. The scalar po-

tential and three components of the function of current in the area free of eddy currents

as well as three components of field strength, eddy currents and losses in the nonmag-

netic end plate, electromagnetic shield, fan screen, housing and end shields have been

determined.

3.1. Load Changing

Using developed mathematical model the computations of the turbogenerator end

zones at no-load, sustained short-circuit as well as resistive-inductive and resistive-

capacity load were made. Phase shift of magnetic excitation field and armature reaction

field vectors is changed with load. Losses in the end zone are changed correspondingly.

The results of calculations (p.u.) of additional losses in the end zone structural compo-

nents of the air cooled turbogenerator at rated resistive-inductive, resistive and resis-

tive-capacity load according to the capability curve are given in Table 1.

Essential increase of losses in the resistive-capacity load mode confirms that nu-

merical analysis of the end zone is required for successful design of powerful high-

utilized turbogenerators.

3.2. Phase Shift Between Current Layers of Stator Winding

In two-layer windings, one side of coil lays in the upper layer, another – in the lower.

Due to the shortening of pitch lower side of coils are shifted respectively upper ones,

phases of current load rotating waves are shifted correspondingly. If coils begin in the

upper layer, current wave of lower layer remains behind, if coils begin in lower layer,

current wave of upper layer remains behind. Due to the presence of steel tooth cores

with high magnetic permeability in the machine active length, for machine operation it

doesn’t practically matter which coil side is a right or which is left one. In the end zone

of the electrical machine the value of additional losses is determined by the total mag-

netic field essentially depending from the current layer being the next to the design

component. As one can see from vector diagram and calculation results (Figs 3, 4),

having air gap induction constant, losses in the stator end zone reduce by 10% when

stator winding lower layer wave is lagging, losses in the fan shield reduces when stator

winding lower layer wave is leading.

Table 1. Losses in the end zone of loaded turbogenerator

Power factor Screen end Whole screen Plate end Whole plate End shield Housing

0.85 lag 1 1 1 1 1 1

1.0 1.78 1.37 1.32 1.12 0.58 0.6

0.95 lead 2.48 1.83 1.74 1.38 0.4 0.44


A. Stator winding lower layer wave is leading. B. Stator winding lower layer wave is legging.

Figure 3. Magnetic field strength for turbogenerator 160 MW at rated load.


4. Test Results

The numerical computation results were compared with the experimental data. When

conducting the experimental tests using JSC Electrosila test rig the end zone of tur-

bogenerators was equipped with three-component induction measuring coils,

Rogovsky’s sensors and temperature detectors.

Indications of the temperature detectors depend, to a considerable degree, on cool-

ing rate in the area of the sensors installed while the induction measuring coils and

Rogovsky’s sensors make it possible to measure directly the field strength, eddy cur-

rents and to determine the density of power losses released. Figures 5, 6 present the

numerical calculations results and experimental values of magnetic field induction on

the cylindrical surface of the end plate with water cooling tubes for 800 MW com-

pletely water cooled turbogenerator.

A X

Direction of

rotation

A X

A

AA

ZZ

ZZ BB

BB XX

XX CC

CC YY

YY

A AA

ZZ BB

ZZ BB

XX CC

XX CC

YY

YY

A A

Direction of

rotation

Ia

Ia2

Ia1

If

E

Ia

Eδ

(1-β)π

Iδ

Figure 4. Winding design and current vector diagram.

Figure 5. Radial component of induction on horisontal part of end plate. Turbogenerator 800 MW at rated

load. o – experimental results.


5. Conclusions

The developed technique of calculation provides the analysis of an electromagnetic

condition of the end zone of turbogenerators in all operational conditions, allows se-

lecting of electromagnetic loads, efficiency of cooling, shape, sizes and placement of

structural components, including stator and rotor cores with windings, screen, pressure

plate, housing and end shields.

Mathematical modeling of powerful turbogenerators has shown essential increase

of losses in stator end zone when resistive-inductive load is changing to resistive-

capacity one.

Selection of the housing and end shields material, geometry and electromagnetic

loading should be done taking into account the losses released and cooling system effi-

ciency.

On the basis of on numerical calculations and study of winding design and vector

diagram it is shown that when the beginnings of coils are laid in the upper winding

layer, losses in the stator end zone are reduced.

Numerical experiment has shown essential increase of external magnetic leakage

fields and concerned additional losses in saturated turbogenerators has been established

when using anisotropic steel with radial direction of rolling in stator core.

The comparison of the numerical and analytical computations for the test problems

as well as numerical computations and results obtained during the experimental studies

of electromagnetic fields in the end zone of powerful turbogenerators shows the cor-

rectness of obtained assumptions.

Now technique has improved by the consideration of inhomogeneous properties of

a design in direction of rotation. Eddy currents and losses in the end part of stator core

are determined for real geometry of steel segments.

Figure 6. Axial component of induction on horisontal part of end plate. Turbogenerator 800 MW at no-load

conditions. Î – experimental results.


References

[1] Sameh R. Salem and Menoj R. Shah. Electromagnetic design practices of turbo-generator end region, In-

ternational Conference on Electrical Machines, ICEM 2002, 25–28 August, Brugge-Belgium, 2002. Con-

ference record.

[2] R.D. Stancheva and I.I. Iatcheva. Numerical determination of operating chart of large turbine generator,

International Conference on Electrical Machines, ICEM’2002, 25–28 August, Brugge-Belgium. Confer-

ence record.

[3] M. Roytgarts, V. Chechurin. Electromagnetic field computation at the strongly expressed skin effect. – In

book: Methods and means of boundary problem solution, L.: LPI, 1981, pages 68-76.

[4] V. Chechurin, Yu. Varlamov, M. Roytgarts. Surface impedance for electromagnetic field computing in

large turbogenerators. International Conference on Electrical Machines, Helsinki, 28–30 Aug. 2000,

ICEM 2000 Proceedings Vol. 2, pp. 1035-1037.

[5] V. Chechurin, I. Kadi-Ogly, M. Roytgarts, Yu. Varlanov. Computation of Electromagnetic Field in the

End Zone of Loaded Turbogenerator. IEMDC 2003. Proceedings of the International on Electrical Ma-

chines and Drives Conference, Madison, Wisconsin, USA, June 1–4, 2003.

[6] Yu. Varlamov, M. Roytgarts, V. Chechurin. Numerical electromagnetic analysis of the end zone of

power turbogenerators. In book: Problems of creation and exploitation of new types of power equipment.

Russian Academy of Sciences, Department of electrical power problems, Issue 6, St.-Petersburg, 2004,

pp. 60-78 (In Russian).




C2. Actuators and Special Devices


The Impact of Magnetic Circuit Saturation

on Properties of Specially Designed

Induction Motor for Polymerization

Reactor

Andrzej POPENDA and Andrzej RUSEK

Technical University of Czestochowa, Al. Armii Krajowej 17,

42-200 Czestochowa, PL

[email protected]

Abstract. The impact of magnetic circuit saturation on properties of specially de-

signed induction motor driving the mixer of polymerization reactor is studied in the

paper. The frequency characteristic of Butterworth’s low-pass filter was proposed

to represent a nonlinear magnetization function in mathematical model of induc-

tion machine. Digital simulation of polymerization reactor drive has been made on

the basis of presented mathematical model of specially designed induction motor

and the examples of transient responses and trajectories are shown.

Description of Driving Unit

The polymerization reactors play an important role in production of polyethylene. The

drives for polymerization reactors work under extraordinary conditions because of ne-

cessity of keeping a constant temperature in reactor chamber and ethylene atmosphere

and working pressure of 2800 × 105

Pa [1]. A driving motor adapted for vertical work has

specific dimensions because of socket fixing in upper part of reactor chamber. Supply

systems are often damaged due to exceptional working conditions, including feed of the

motor via specially designed electrodes. A non-typical driving set is characterized by the

following extremely difficult operating conditions: (1) location of 55 kW-

induction motor in closed tubular seat with diameter of 302 millimeters and total length

of 919 millimeters, (2) impossibility of application of both ventilation in the motor and

standard power supply due to location of working motor directly in reactor chamber

with the pressure of 2800 × 105

Pa, (3) vertical single-point suspension of working mo-

tor together with the mixer; upper bearing aligns the rotor in stator, (4) the work of

vertically suspended motor with application of self-aligning non-lubricate slide bearing

containing the large-size rings made of sintered carbides, etc.

The extreme working conditions of motors working in polymerization reactor

chambers resulted in the necessity of developing the new prototypes of specially

designed induction motors being more resistant.


335

Parameters of Specially Designed Induction Motor

The geometrical dimensions of rotor and stator slots of specially designed induction

motor made as a prototype in the frame of purposeful grant no. Nr 6 T10 2003C/06105

are given in Fig. 1 [1]. This figure depicts cross-sections of both slots.

Additional calculations made on the basis of considered slot dimensions and meas-

urements of both a non-loaded motor and motor with blocked rotor allow estimating the

substitute diagram parameters of induction motor as well as the operating parameters of

induction motor [1]. The following values of magnetization reactance related to unsatu-

rated magnetic circuit and magnetization reactance at nominal voltage supplying the

motor were assumed for calculations: 10.7 Ω and 5.34 Ω.

The one-sided displacement of current from squirrel-cage rotor bars occurs during

starting the motor. This phenomenon is known as a skin effect [2]. The depth σ of cur-

rent penetration into rotor bar is measured from external end of slot and determines the

working surface of bar cross-section for passage of current at given f2 frequency of current

in rotor bars. Therefore, the rotor resistance depends on σ. The skin effect disappears

during the work of motor with nominal rotational speed.

Approximation of Magnetization Curve for Induction Motor

The frequency characteristic of Butterworth’s low-pass filter was proposed in order to

represent a nonlinear magnetization function [3]. In dependence describing Butter-

worth’s low-pass filter the longitudinal flux ψ in main magnetic circuit of motor (main

flux) instead of frequency is taken as an argument. A few modifications of Butter-

worth’s polynomial allow obtaining the optional gradient of non-loaded motor curve in

a range of main flux proportionality to magnetizing current im:

( )( )

ψ

ψ ψ 1

m

q

mn

n

L b

k

La

= =

+

, ψ , ψm m n mn mn

L i L i= = , (1)

where: Lm is magnetization inductance relating to main magnetic circuit of motor, L

mn

is magnetization inductance at nominal voltage supplying the motor, ψ is longitudinal

flux in main magnetic circuit of motor, ψn is longitudinal flux at nominal voltage supplying

the motor. The following values of coefficients in (1) derived from estimated parameters of

motor substitute diagram: a = 3, b = 2, q = 4.

Figure 1. The cross-sections of rotor and stator slots.

A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation336

Assuming that ratio of any longitudinal flux and nominal longitudinal flux

is approximately equal to ratio of respective voltages the following dependence

may be derived:

( )1 ψ ψ ψ

1 1

ψ ψ ψ ψ

q q

m

mn N N N N N

i U U

a a

i k b bU U

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟= = + ≈ +⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠. (2)

The proposed approximation and corresponding to that a non-loaded motor curve

are depicted in Fig. 2. The linearization of non-loaded motor curve through the nominal

saturation point is also shown.

Despite the fact that there are some differences between approximated and meas-

ured curves, the proposed representation is enough precise. The difference between

experimentally obtained and approximated magnetization reactance does not exceed

few percents in the range of magnetizing current variability from zero to rated value of

motor current [3]. A simple representation of the nonlinear magnetization curve was

achieved at the cost of the abovementioned differences.

The Supply System

The schematic diagram of supply system for induction motor is depicted in Fig. 3. The

induction motor drives the mixer in polymerization reactor chamber. The supply system

for induction motor consists of switchhouse P17 segmented into two fields. The field

no. 13/14 allows directly connecting the motor to the grid and it consists of the circuit

breaker and the fuses and contactor. The field no. 25/26 allows supplying the motor via

Figure 2. Approximation of magnetization curve and corresponding to that a non-loaded motor curve.

Figure 3. The schematic diagram of supply system for drive of mixer in polymerization chamber.

A. Popenda and A. Rusek / The Impact of Magnetic Circuit Saturation 337

frequency converter and contains: the circuit breaker with fuses and frequency converter

and contactor that is located behind the converter.

In the design of supply systems for electrical machines and electrical equipments it

should not be omitted that mentioned machines and equipments exert influence on power

grid and local electric power systems. The drops and variations of mains voltage as well

as the harmonics generated by electric machines and other disadvantageous phenomena

from power quality point of view occur as a result of this influence. The author of [4]

pays attention to mentioned phenomena and he proposes number of prognosis analyses

concerning power demand for individual consumers or groups of consumers as well as

determined areas connected with work of selected sets of local consumers in power sys-

tems.

The Vector Equations of Induction Machine

The equations of induction machine mathematical model without derivation of mag-

netization inductance Lm are more advantageous in case of taking into consideration the

magnetic circuit saturation. The zero value of rotor voltage 0r

u = should be taken

into consideration in a case of squirrel-cage motor. The equations in stationary coordi-

nate system 0αβ using spatial vectors [5,6] are given as follows:

ψs s s

s

d

R i u

dt

=− + , ψ ω ψr b mr

r r

d

R i jp

dt

=− + (3)

where: α β

ψ ψ ψs s

s

j= + ,α β

ψ ψ ψr r

r

j= + ,α βs ss

i i ji= + ,α βr rr

i i ji= + are flux and

current vectors of both stator and rotor, α βs ss

u u ju= + is stator voltage vector, Rs, R

r are

stator and rotor windings resistances, ωm is angular velocity of rotor, p

b is number of cou-

ple pairs. The Eqs (3) should be completed with following flux-current dependencies

(4) and equation of motion (5):

2

ψ ψr m

s r

s

s r m

L L

i

L L L

−

=

−

, 2

ψ ψ

s m

r s

r

s r m

L L

i

L L L

−

=

−

, (4)

( )1

ω

θm e t m

m

d

M M M

dt

= − − , ( )*

ψe b s

s

M p Im i= ⋅ (5)

where: ( )σ

1s ms

L j L L= + + , ( )σ

1r mr

L j L L= + + ,α βm m m

L L jL= + are self-

inductance vectors of both stator and rotor windings and magnetization inductance vec-

tor, θm is moment of rotor inertia, M

e is output torque of motor, M

m is load torque of

motor, Mt is moment of friction in lower slide bearing of motor. The components of

magnetization inductance vector Lm are as follows:


( )α

4

α

2

3 ψ ψ 1

mn

m

n

L

L =

+

,

( )β

4

β

2

3 ψ ψ 1

mn

m

n

L

L =

+

, (6)

where: α α σ α

ψ ψs s s

L i= − ,β β σ β

ψ ψs s s

L i= − are components of main flux vector. The

following auxiliary variables are defined in order to obtain the appropriate results of

numerical analysis:

α

2 3a s

i i= , β α

1 2 1 6b s s

i i i= − , β α

1 2 1 6c s s

i i i=− − , (7)

2 2

α βs s s

I i i= + , n = 9,55ωm, (8)

where: ia, i

b, i

c are phase currents of motor, |I

s| is absolute value of stator current vector,

n is rotational speed of rotor.

Examples of Transient Responses and Trajectories

A digital simulation of polymerization reactor drive has been made on the basis of pre-

sented mathematical model of specially designed induction motor. The examples of

transient responses and trajectories for selected working conditions of drive are shown

in Figs 4–9. The magnetic circuit saturation is taken (a) and not taken (b) into consid-

eration.

Connection of Motor to the Power Grid

a)

0,0 0,1 0,2 0,3 0,4 0,5

0

400

800

1200

1600

t [s]

n [rp

m]

0

400

800

1200

1600

M

e

[N

m]

-500

-250

0

250

500

i

a

[A

]

b)

0,0 0,1 0,2 0,3 0,4 0,5

0

400

800

1200

1600

t [s]

n [rp

m]

0

400

800

1200

1600

Me

[N

m]

-500

-250

0

250

500

ia

[A

]

Figure 4. Transient responses of motor including time functions of stator phase current ia, output torque M

e

and rotational speed n.


Starting the Motor Supplied via Frequency Converter Applying Constant Ratio E/f

Method

a)

0,0 0,2 0,4 0,6 0,8

0

400

800

1200

1600

t [s]

n [rp

m]

0

200

400

600

Me

[N

m]

-400

-200

0

200

400

ia

[A

]

b)

0,0 0,2 0,4 0,6 0,8

0

400

800

1200

1600

t [s]

n [rpm

]

0

200

400

600

M

e

[N

m]

-400

-200

0

200

400

i

a

[A

]

Figure 7. Transient responses of motor including time functions of stator phase current ia, output torque M

e

and rotational speed n.

a)

0 300 600

0

400

800

1200

1600

Me

[N

m]

|Is

| [A]

b)

0 300 600

0

400

800

1200

1600

Me

[N

m]

|Is

| [A]

Figure 5. Trajectories of motor output torque as a function of absolute value of current vector.

a)

-600 -300 0 300 600

-600

-300

0

300

600

isβ

[A

]

isα

[A]

b)

-600 -300 0 300 600

-600

-300

0

300

600

i

sβ

[A

]

isα

[A]

Figure 6. Trajectories of current vector on phase plane.


Conclusion

The mathematical model together with mathematical and numerical studies of drive for

polymerization reactor with specially designed induction motor is presented in the pa-

per. The mathematical model considers number of real phenomena e. g. saturation of

motor magnetic circuit, the skin effect occurring in bars of squirrel-cage rotor, the mo-

ment of friction in lower slide bearing of motor as a function of rotational speed of ro-

tor and dependence of load torque resulting from polymerization process.

The saturation of main magnetic circuit of motor causes additional distortions of

both transient phase currents and transient output torque (Figs 4a, 6a, 7a, 9a). The

higher harmonics occur in mentioned variables as a result of a nonlinear magnetization

curve taking into consideration. The saturation of main magnetic circuit taken and not

taken into consideration causes minimal differences between extreme values of both

output torque and absolute value of current vector during starting the motor (Figs 5, 8).

The differences may depend on assumed initial conditions. Significant increase of mag-

netizing current when nominal value of supply voltage is exceeded is additional disad-

vantageous of magnetic circuit saturation.

a)

0 200 400

0

250

500

750

M

e

[N

m]

|Is

| [A]

b)

0 200 400

0

250

500

750

Me

[N

m]

|Is

| [A]

Figure 8. Trajectories of motor output torque as a function of absolute value of current vector.

a)

-400 0 400

-400

-200

0

200

400

i

sβ

[A

]

isα

[A]

b)

-400 0 400

-400

-200

0

200

400

isβ

[A

]

isα

[A]

Figure 9. Trajectories of current vector on phase plane.


References

[1] A. Rusek, A. Popenda, Transient states of polymerizer drive including real load of specially designed in-

duction motor, 17th Int. Conf. on Electrical Machines, Chania, Crete Island, Greece, (CD ROM), 2006.

[2] F. Kohlrausch, Praktische Physik (in German), Bd. 2. Stuttgart, Teubner 1968.

[3] A. Popenda, The mathematical model of induction machine with variable mutual inductance, Int. Conf.

PCIM Proceedings (CD ROM), Nuremberg Germany, June 2006.

[4] T. Popławski, Application of the Takagi-Sugeno (TS) fuzzy logic model for load curves prediction in the

local power system, III-rd International Scientific Symposium Elektroenergetika 2005, Stara Lesna Slo-

vak Republic, 2005.

[5] K.P. Kovacs, J. Racz, Transiente Vorgange in Wechselstrommaschinen (in German), Ung. Akad. D. Wiss.,

Budapest 1959.

[6] J.P. Kopylow, Elektromechaniczeskije preobrazowatieli energii (in Russian), Energia, Moscow 1973.


Electromagnetic Design of Variable-

Reluctance Transducer for Linear Position

Sensing

J. CORDA and S.M. JAMIL

School of Electronic and Electrical Engineering

University of Leeds

Leeds LS2 9JT

United Kingdom


Abstract. The linear position transducer based on variable reluctance principle is a

contactless device distinguished by a simple construction which allows accom-

plishment of a large sensing range without adverse effects on accuracy and sensi-

tivity. The goal in the electromagnetic design of the transducer is to achieve its

output directly proportional to position, which is of particular relevance for precise

real-time position feedback control of linear drives under dynamic conditions.

Such a requirement is resolved through a combined consideration of magnetic and

electrical arrangements, which includes magnetic circuit optimisation and investi-

gation of the effects of core losses and motional e.m.f. on the transducer output.

Introduction

Linear servo drive systems usually include a position transducer which is most com-

monly of Linear Variable Differential Transformers type – LVDT [1]. In linear systems

with long displacement this contactless transducer has major drawbacks due to an in-

crease of the absolute error with measuring range and a reduced sensitivity expressed in

terms of output voltage per unit length. Other types of contactless transducers [2–4]

which alleviate the above drawbacks are however a major cost component of a linear

drive system.

To overcome the above drawbacks, a research has been carried out on an alterna-

tive contactless linear position transducer of variable-reluctance (VR) type [5,6]. Such

transducers rely on repetitive inductance variations of its sensors with position, which

are produced by changes of the reluctance between magnetic saliencies of the outer

part, here referred to as the sensors, and the inner part having a form of transversely

slotted rod, here referred to as the mover. These transducers were originally developed

for the use in electronic commutation of phases of the linear tubular switched reluc-

tance motor (LTSRM) [7] where it is essential for the mover to be unrestricted in terms

of displacement. In other words, when the transducer is used in conjunction with

LTSRM, the mover of the latter interacts with the transducer. The transducer design

can be adapted for the use with other types of linear motors with a toothed ferromag-

netic structure on the mover but without permanent magnets.


343

To provide precise continuous position sensing under dynamic conditions of op-

eration of LTSRM, it is desirable that the transducer output is directly proportional to

the position. So, the objective of the electromagnetic design presented here is to ac-

complish the output of the VR type transducer similar to the one available from a linear

encoder where the output varies with displacement in a linear fashion.

Transducer Construction and Operating Principles

The linear position transducer of VR type shown in Fig. 1 comprises two identical sen-

sors with E-shaped magnetic cores interacting with a slotted ferromagnetic shaft. Each

E-core is formed of low-loss laminations assembled into a thin stack. The central sali-

ency of the core, holding a primary coil, has a width of a full mover pitch, λ. Each of

the two outer saliencies holds a secondary coil and has a width of λ/2. The slots be-

tween the saliencies are also λ/2 wide. Thus, when one of the outer saliencies is in full

alignment with the mover tooth, the other one is in full misalignment. The two E-core

sensors, diametrically opposite to each other and mutually shifted along the shaft axis

by λ/4, are held by a nonmagnetic shell supported on the slotted mover by means of

slide bearings.

Provided that the radial gap between the salient surfaces of the E-core and the shaft

is small compared to the tooth width and that the excitation is below saturation level,

the flux of the central saliency produced by an excitation current is nearly independent

of the mover position and it is equal to the sum of the two position dependent fluxes of

the outer saliencies plus a small amount of flux leakage. The interaction between the

fields of the two sensors is insignificant (Fig. 2) because the excitation of their prima-

ries is arranged so to produce magnetic polarities of identical senses.

Variations of the two secondary flux linkages of each E-core against displacement

are in anti-phase. When considering all four outer saliencies at a given excitation cur-

rent in the primaries of two sensors, the four secondary flux linkage waveforms against

displacement and hence the waveforms of mutual inductances between each secondary

and the corresponding primary, hereinafter referred to as the ‘mutual inductance pro-

file’, are spatially shifted from one another by λ/4, as illustrated in Fig. 3.

When the primary coils are fed with sinusoidal current of constant amplitude and

carrier frequency, the amplitude of each secondary voltage is modulated with the corre-

sponding variation of mutual inductance with displacement (Fig. 4). The transducer

output (Fig. 5) is formed from the envelope differences of induced secondary voltages

and therefore the waveform of the output voltage variation with position is principally

determined by the mutual inductance profile.

Figure 1. VR transducer with E-core sensors. Figure 2. Axial magnetic field plot.

J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing344

The requirement for a linear output addresses, at first, the need to optimise propor-

tions of the magnetic circuit so that the mutual inductance profile has a high level of

linearity over the rising and falling segments in the waveform. These segments are in-

dicated in Fig. 3 and are hereinafter referred to as ‘λ/4-segments’. Secondly, the effects

of eddy currents and motional e.m.f., which have adverse impact on linearity and dy-

namic error, need to be minimised.

Magnetic Circuit Optimisation

At a given geometry of the mover, which was selected while optimising the design of

the LTSRM, the main parameters affecting the transducer’s mutual inductance profile

are the airgap length between the sensor and the mover, and the ratio of the sensor’s

pole to slot widths. The reduction of the airgap increases the span of mutual inductance

variation, i.e. the transducer sensitivity, but at the same time it enhances the effect of

mechanical imperfections on the waveform profile and increases the cost of manufac-

turing. For the mover diameter of 40 mm, the airgap of 0.2 mm is considered as an op-

timum.

The effect of changing the sensor pole/slot width ratio on the mutual inductance

profile was examined by magnetostatic field analysis using FE field solver and the re-

sults are summarised in Table 1. The example is related to the mover pitch length of

10 mm, a tooth/slot ratio of 4/6 and a slot depth of 6 mm. The parameter values were

constrained with regard to achieve a segment of linear variation of at least λ/4 length on

Position [mm]

Secondary voltage [v]

Figure 3. Flux-linkage (mutual inductance) vs. position

waveforms.

Figure 4. Modulated waveform of induced

secondary voltage.

λ/4

Output signal

Position

V24

V31

V42

V13

λ/4 λ/4

Figure 5. Output voltage vs. position waveform.

J. Corda and S.M. Jamil / Electromagnetic Design of VR Transducer for Linear Position Sensing 345

both rising and falling part of mutual inductance waveform, which ensures that any

displacement is correlated with a repetitive linear segment. The criterion used for se-

lecting the best λ/4-segment of the mutual inductance waveform was the minimal

nonlinearity, which is expressed as a percentage ratio of the maximum deviation from

the linear least-square fit line to the span of variation over the λ/4-segment.

Equivalent Circuit Model

Figure 7 illustrates an approximate equivalent circuit representing one E-core sensor

when the primary is subjected to a current-fed excitation of constant amplitude. The

core losses caused by eddy-currents and hysteresis are coarsely represented by a resis-

tive branch.

Under a current-fed sinusoidal excitation the primary current is unaffected by

changes of the circuit parameters.

In an ideal case without core losses (r = ∞), the secondary voltage is given by

tIMvms

ωω cos= (1)

and its magnitude is modulated by the variation of mutual inductance in a direct linear

relationship

msIMV ω=

(2)

If the core losses are taken into account, the instantaneous secondary voltage equa-

tion is

Table 1. Effects of sensors proportions at fixed mover tooth/slot ratio 4/6 and λ = 10

Sensor

pole/slot

width ratio

Span of mut.

ind. variation

[mH]

Maximum

deviation

[mH]

Nonlinearity

[%]

5.0/5.0 2.29 0.006 0.26

4.0/6.0 2.10 0.008 0.38

3.5/6.5 2.06 0.011 0.53

3.0/7.0 2.02 0.018 0.89

i iμ

iw

vS

r

vS

M2

M1

Figure 6. Variation of mutual inductances against

displacement for optimised E-core sensor.

Figure 7. Equivalent circuit of one E-core sensor.


)sin(

r

v

tI

dt

d

Mvs

ms−= ω (3)

and the relationship between the secondary voltage magnitude and the mutual induc-

tance has a nonlinear form given by

2

)/(1 rM

IM

Vm

s

ω

ω

+

= (4)

So, when the mutual inductance rises linearly with displacement the gradient of

secondary voltage envelope will not remain constant but will decrease progressively.

This undesirable effect becomes more enhanced at higher excitation frequencies. The

resistance representing the core losses rises with frequency but at a progressively

slower rate. For the prototype E-core sensor the estimated values of r at 1, 2, 5 and

10 kHz are respectively 0.75, 1.3, 1.9 and 2 kΩ, and the nonlinearities calculated on the

basis of Eq. 4 are 0.1, 0.2, 0.4 and 1.5%. Nonetheless, a higher excitation frequency

increases the secondary voltages which is desirable from a viewpoint of the reduction

of the noise-to-signal ratio. In addition, as discussed below, a higher excitation fre-

quency is required if the transducer is to be used at higher running speeds.

The model considered above assumed that the induced e.m.f. was caused simply

by the time-varying current excitation. Beside this transformational e.m.f. there is an-

other e.m.f. component caused by the mutual inductance variation during motion.

Likewise the core losses, this component e.m.f. (hereinafter referred to as ‘the motional

e.m.f.’) distorts the simple linear relationship (2) between the secondary voltage and

mutual inductance.

A simple estimate of the ratio of amplitudes of the motional (undesirable) to the

transformational (desirable) e.m.f. can be made if these e.m.f.s are considered in isola-

tion of the core losses. The amplitudes of the two e.m.f.s are given by (dM /dt)Im and

ωMIm respectively, and their ratio is

dp

dM

Mf

f

dp

dM

ME

Ew

transform

motional

⋅⋅⋅=⋅⋅=

1

2

1

.

π

λ

ω

υ

(5)

where dM /dp is the gradient of linear λ/4-segment, υ denotes the mover speed and fw

/f

is the ratio between the frequency of the envelope waveform and the excitation fre-

quency.

So, the percentage error caused by the motional e.m.f. is directly proportional to

the ratio fw

/f . For the transducer having mutual inductance profile shown in Fig. 5, the

e.m.f. ratio is not greater than 0.65 fw

/f. So, for instance, at the speed of 0.1 m/sec i.e.

fw = 10Hz, the error caused by the motional e.m.f. at the excitation frequency of 1kHz

is not larger than 0.65%. If the excitation frequency is increased to 5 kHz, the error is

not larger than 0.13%.

Under running conditions the transducer error is caused by effects of both the core

losses and motional e.m.f., and its minimisation requires a compromise in selecting the

excitation frequency.


Prototype Transducer

An experimental prototype transducer was designed in conjunction with a 4-phase

LTSRM and it interacts with the mover which is constructed as a shaft made of mild

steel with a 40 mm diameter, a 6 mm slot depth, a 10 mm pitch length, and the

tooth/slot ratio of 4/6. The magnetic core of each sensor was made of low-loss lami-

nated steel and has the outer dimensions of 30 × 13 mm, the stack thickness of 5 mm

and was fixed at 0.2 mm from the shaft. The central and outer saliencies are 10 and

5 mm wide respectively. The primary and secondary coils have 200 turns each. The

sensors’ primaries are connected in series and fed by a sinusoidal current provided

from a linear amplifier.

The amplitude variations of each full-wave rectified secondary voltage are tracked

by a sample-and-hold circuit. The transducer output is formed from linear λ/4-segments

which are extracted from the four differential voltage waveforms using a comparative-

logic circuit [5].

The transducer was tested first statically at various excitation frequencies. The am-

plitude of excitation current was accordingly adjusted to ensure that the amplifier oper-

ated in the linear region and that there was no magnetic saturation in the sensors. At

excitation frequency of 1kHz the nonlinearity on each λ/4-segment was within 0.35%.

At 5 kHz, the effect of core losses was still low causing a relatively small increase of

nonlinearity to 0.5%. However, when the frequency was increased to 10 kHz, the im-

pact of core losses became apparent, and the nonlinearity rose to 4.5%.

The action of the sample-and-hold circuit and the associated low-pass filter in the

processing unit generates a slight shift between the electronically derived envelope and

the actual secondary voltage envelope. The error caused by this shift is speed depend-

ent and can be assessed on the basis of positional resolution of the output from sample-

and-hold circuit. The positional resolution is given by 0.5υ/f and at excitation fre-

quency of 5 kHz and speed of 0.1 m/s the absolute error due to the resolution is not

larger than 0.01 mm which corresponds to 0.4% of a λ/4-segment. When combined

with the error caused by the motional e.m.f., assessed in the previous section, the total

dynamic error is not larger than 0.53%.

Figure 8 shows the measured output from the prototype transducer at operating

speed of 0.1 m/s and excitation frequency of 5 kHz.

Figure 8. Output signal measured at the speed of 0.1 m/s and excitation frequency of 5 kHz.


The important feature of the transducer is that the position error occurring over in-

dividual λ/4-segments are not accumulative, which allows accomplishment of a large

sensing range without adverse effects on accuracy and sensitivity.

Conclusions

The paper has presented the electromagnetic design of a linear position transducer

based on repetitive reluctance variations between saliencies of two longitudinally

shifted E-core sensors and a transversely slotted ferromagnetic shaft which in the stud-

ied case represented the mover of a linear tubular switched reluctance motor.

The transducer with an optimised magnetic circuit is distinguished by the linear

segments in the cycle of inductance variation against position, where each segment

covers one quarter of the shaft pitch. The inductance variations were utilised to give the

transducer output linearly proportional with position. This was achieved by applying a

current-fed sinusoidal excitation to the primary coils, and by using a processing unit for

tracking the amplitude variations of the sensors’ induced voltages and for the identifi-

cation of the envelope linear segments.

Exciting the transducer sensors with a high frequency enables more accurate track-

ing of the amplitude variation of secondary voltages, a better sensitivity and a relative

reduction of the undesirable impact of the motional e.m.f., but because of the increase

of core losses it has an adverse impact on the profile of the secondary voltage envelope.

A prototype transducer was accomplished for a linear drive application with a

1000-mm long displacement at a speed range up to 0.1 m/s. Its absolute position error

is not larger than 0.026 mm which corresponds to a non-linearity of 1.03% occurring

over a 2.5-mm long repetitive segment. (The position error occurring over individual

λ/4-segments are not accumulative.) This is a substantial improvement compared to a

typical LVDT transducer with a non-linearity of 0.5%, which at a full stroke of

±250 mm is equivalent to a position error of 1.25 mm.

References

[1] E.O. Doebelin, “Measurement Systems – Application and Design”, McGraw-Hill, 4th

edition, 1990.

[2] Y. Kano, S. Hasebe, C. Huang and T. Yamada, “New Type Linear Position Differential Transformer Po-

sition Transducer”, IEEE Transaction on Instrumentation and Measurement, Vol. 38, No. 2, April 1989,

pp. 407-409.

[3] Y. Kano, S. Hasebe, C. Huang, T. Yamada and M. Inubuse, “Linear Position Detector with Rod Shape

Electromagnet”, IEEE Transaction on Magnetics, Vol. 26, No. 5, September 1990, pp. 2023-25.

[4] A.E. Bennemann and R.L. Hollis, “Magnetic and Optical – Fluorescence Position Sensing for Plannar

Linear Motors”, IEEE International Conference on Intelligent Robots, Vol. 3, 1995, pp. 101-107.

[5] J. Corda, J.K. Al-Tayie and P. Slater, “Contactless linear position transducer based on reluctance varia-

tion”, IEE Proceedings – Electric Power Applications, Vol. 146, No. 6, Nov. 1999, pp. 151-158.

[6] J. Corda and J.K. Al-Tayie, “Enhanced performance variable reluctance transducer for linear position

sensing”, IEE Proceedings – Electric Power Applications, Vol. 150, No. 5, Sept. 2003, pp. 623-628.

[7] J. Corda and E. Skopljak, “Linear switched reluctance actuator”, IEE Publication No. 376, International

Conference on Electrical Machines and Drives, Oxford, 1993, pp. 535-539.


The Influence of the Matrix Movement in a

High Gradient Magnetic Filter on the

Critical Temperature Distribution in the

Superconducting Coil

Antoni CIEŚLA and Bartłomiej GARDA

AGH University of Science and Technology, al. Mickiewicza 30,

31-059 Kraków, Poland


Abstract. The subject of the paper is an analysis of dynamical state of work of the

High Gradient Magnetic Separator. HGMS requires high values of the magnetic

field in big spaces. Superconducting coils are used for that field excitement. Sepa-

rator works in two states. First state – static – magnetic filtration, and dynamic –

exchanging the matrix. Matrix is made of ferromagnetic wool. So it influences on

the field excited by the coil. During dynamical state, the matrix is taking out the

filter and it affects on the field distribution in the coil windings. Unfortunately the

level of temperature required to maintain the superconductivity properties locally

might be dramatically lowered and it could be the cause of the serious accidents.

Magnet designers have to take into consideration also the dynamical state of the

magnetic filter.

1. High Gradient Magnetic Separator

In this paper a prototype superconducting magnet is considered (Fig. 1). The winding is

made of a composite multifibre conductor with a NbTi superconductor placed in a cop-

per matrix and cooled by liquid helium [1]. This electromagnet provides the source of

magnetic field in a High Gradient Magnetic Separator (HGMS). HGMS is a device that

can separate a solid particles and slurries. Effectiveness of that action is proportional to

the value of the magnetic field and “nonhomegenity” of the field in the working space

of that filter (high gradient methods). That strong magnetic field is usually excited by a

superconducting coil. High nonhomegenity is made using a matrix which is usually

made of the stainless magnetic wool. The magnetic fraction of the feed passes through

the matrix and is attached to the ferromagnetic elements. The non-magnetic particles

are collected out side of the matrix. So while that matrix is filled with those magnetic

particles it has to be removed from the working space of the filter (Fig. 2). Because the

field is excited by the superconducting winding it is not easy to turn it off and exchange

the matrix without the field. Superconducting coils works in the closed circuit. Such a

coil has gathered the big energy amount While matrix is taken out the filter first some

force appears due to the energy change [2] and it also influences on the field distribu-

tion in the coil windings. That field is linked with the temperature distribution of the



doi:10.3233/978-1-58603-895-3-350

350

coil what could be the cause of quench. In the paper authors present an analysis of

critical temperature distribution change while moving out the matrix from the filter.

2. Material Properties

The winding is to be made of a composite multifibre conductor with NbTi. To

simplify the calculations, the critical surface of the superconductor has been semi-

linearized. On Fig. 3 authors present this critical surface. It is the surface that binds

three critical values magnetic field, current density and the temperature. All those val-

ues must not be exceeded. All the superconducting properties was calculated using a

reduced state model widely described in [3].

z [m]

water

cryostat

matrix

canister

with slurry

of the filter

ferromagnetic

of the canal

filling

superconductorfiltration

product

Figure 1. High Gradient Magnetic Filter.

B0, zo

Bm

High Gradient

Magnetic Filter

Bn,

b

Le

Superconducting Solenoid

a1

a2

z

direction of matrix movement

Figure 2. Removing of the matrix from the working space of the coil.

T

B

J

Bc0

Tc0

Jc;t

Bp

Tp

Figure 3. Semi-linearized critical surface of superconductor.

gwindin

A. Ciesla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter 351

To obtain semi-linearized critical surface first one have to find the relation be-

tween the field, current and temperature:

1 2 3

( , )

cT B J B Jα α α= + + (1)

where:

( )

T

c

ptcc

cpccp

c

c

T

BJB

TBBBT

B

T

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ −−

−=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

0

;0

000

0

0

3

2

1

;

)(

;

α

α

α

(2)

and Tc0

– is the critical temperature for B and J = 0, Bc0

– critical magnetic field induc-

tion for J and T = 0, and Jc;t

is usually given in the table as the function of Bp for fixed

temperature Tp. Usually all those values are provided by the superconductor producers.

In our case authors used for model description an NbTi wire produced by Vac-

uumschmelze GmbH from Hanau with the superconducting properties presented in

Table 1.

Coil designers usually construct magnets minimizing the superconductor volume

fixing field value and the shape of that field in the working space. When the coil is cal-

culated it has to be proved that the all values are lower than the critical surface, if not,

our possible solution must be recalculated. The field is to be excited in the area where

the ferromagnetic wool is placed (matrix). That wool is described by so called “packing

factor” defined as the ratio of value of the ferromagnetic wool and the total value of the

matrix. In our model authors applied an pacing factor on the level of 7% what is the

average value of the real matrix models (3÷15%). Another problem is that ferromag-

netic wool has nonlinear magnetic properties. Special model was used which compares

an ferromagnetic wool magnetic properties and pacing factor of the matrix. On the

Fig. 4 authors present an μr as an function of the field B.

Table 1. Superconductor characteristic Jc = f(B) for fixed temperature Tc0 = 4,2 K

B [T] 3 4 5 6 7 8

Jc [A/m2

] 2.588 2.235 1.882 1.544 1.147 0.735

B [T]

0 1 2 3 4

μr

1,0

1,1

1,2

1,3

1,4

1,5

ε =0.15

ε =0.10

ε =0.05

Figure 4. Magnetic relative permeability in the function of magnetic induction.

A. Ciesla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter352

Magnet designers calculate the coil when the matrix is placed directly in the center

of the working space. All the critical values are fulfilled and the superconducting wind-

ing is in “no danger” of the quench. But unfortunately when the matrix is being moved

out of the windings local unstable points might appear and what is worst the supercon-

ductor might lose its superconducting properties.

3. Simulation of the Dynamical State

Authors made some simulations with the matrix movements calculating the critical

temperature distribution for different position of the matrix. Using FEM model and

having calculated an field distribution and applying Eq. (1) authors could calculate

critical temperature distribution on the cross-section of the coil. Having that data au-

thors made some animation. Figure 5 presents the some important situation. Figure 5a

presents the situation when matrix is being placed in the center of the coil (static state

of work of the filter). Figures 5 from b to f shows distribution during the dynamical

state of work of the filter. It can be seen that position of the matrix has the great influ-

ence on the critical temperature distribution. Starting from Fig. c to Fig. e critical tem-

perature is lower than temperature of the liquid helium so the superconductivity can not

be maintain using liquid helium as cooling factor.

On Fig. 6 it can be seen the influence of the matrix movement on the minimal criti-

cal temperature in the whole area of the coil. Critical temperature is divided by the

4.2 K (the temperature of liquid helium), so when the locally temperature is lower then

1 it means that at that point we could have problem to maintain that low temperature

and the superconductive state might be lost. From the figure it can be seen that during

static state of the filter the maximum critical temperature is on the level of 1,2÷1,3 of

the liquid helium, but if the matrix is being moved the critical temperature is being

lowered rapidly, and the dangerous quench situation appears. Above dangerous situa-

tion might appear when middle of the matrix is between 0,12÷0,34 m and the critical

temperature is lower then 4,2 K.

4. Conclusions

Movement of matrix made of the ferromagnetic wool has an great influence on critical

temperature in the coil windings. Unwanted accidents might happen during dynamical

state of the work of the filter. The problem is getting harder when the shape of the coil

cross-section is not simply rectangular. There are two solutions to avoid the problems.

First is to lower the current in the coil just before the matrix is being taken out the fil-

ter. It is a little problematic action because of big energy change. Otherwise one can

lower the input current in the coil, but then also field value in the filter is lowered fur-

thermore the filtering effectiveness is decreased dramatically.


a) Tmin rel = 1,24, z0 = 0 m b) Tmin rel = 1,01, z0 = 0,11 m

c) Tmin rel = 0,96, z0 = 0,12 m d) Tmin rel = 0,79, z0 = 0,21 m

e) Tmin rel = 0,95, z0 = 0,32 m f) Tmin rel = 1,16, z0 = 0,5 m

Figure 5. Influence of the matrix movement on the critical temperature. Pictures made for some different

positions of the matrix.

A. Ciesla and B. Garda / Matrix Movement in a High Gradient Magnetic Filter354

0,70

0,80

0,90

1,00

1,10

1,20

1,30

0 0,1 0,2 0,3 0,4 0,5 0,6

z [m]

Trel

[/]

Liquid Helium Line 4,2K

Line of the border of the coil

Figure 6. Influence of the position of the middle of the matrix z0 (Fig. 2.) on the minimal critical temperature

of superconductor of the coil.

Acknowledgement

The work presented in this paper was supported by the Polish State Committee for Sci-

entific Research, Warsaw, in the frame of project Internal Research (“Badania własne”)

contract No. BG10.10.120.561/III/9/p.

References

[1] A. Cieśla, Superconductor Electromagnet as a DC Machine (the Constructional and Exploitations Peculi-

arities), Proceeding of 43rd ISCT University of Ilmenau, September 21–24, 1998.

[2] A. Cieśla, B. Garda, Analysis of the force acting on the matrix of a superconducting filter in high mag-

netic field, proceedings of ISEF 2001, pp. 247-250, September 20–22, 2001.

[3] M.A. Green, Calculating the Jc, B, T surface for Niobum Titanum using a reduced state model, IEEE

Transactions on Magnetics, Vol. 25, No. 2, March, pp. 2119-2122; 1989.


Macro- and Microscopic Approach to the

Problem of Distribution of Magnetic Field

in the Working Space of the Separator

Antoni CIEŚLA

AGH – University of Science and Technology,

Department of Electrical Engineering, al. Mickiewicza 30,

PL 30 – 059 Kraków, Poland, Tel: +48 12 617 39 86,


Abstract. The subject of this paper is field distribution in a magnetic separator

containing ferromagnetic matrix where the magnetic field is generated by a dc su-

perconducting magnet. To develop invariable conditions for the extraction of par-

ticles from the slurry in a filter matrix, it is necessary to create a homogenous

magnetic field within the working space of the device. The source of the field is

usually a solenoidal coil winding with superconducting wire and, in order to

achieve the design objective of field uniformity, various configurations have been

considered using optimisation techniques (macroscopic model). Also distribution

of the magnetic field around single ferromagnetic filament of the steel wool placed

in homogenous field of the separator is considered (microscopic model). Analysis

of the single filament is essential to describe particles movement in the matrix.

Some simulation results are presented and the most promising solutions are high-

lighted.

Principle of Magnetic Separation in Matrix Separator

When fine particles are dispersed in air, water, sea water, oil, organic solvents, etc.,

their separation or filtration by using a magnetic force is called magnetic separation.

(Fig. 1).

Figure 1. High Gradient Magnetic Separator.



doi:10.3233/978-1-58603-895-3-356

356

In the magnetic field generated by this superconducting solenoidal winding, a ma-

trix is positioned in which particles from the slurry flowing through a separator are

extracted. The matrix is a canister filled with gradient-generating elements such as

chips or ferromagnetic wool, to which particles of specific magnetic properties are at-

tracted. The separator matrix should be placed in a homogenous magnetic field to de-

velop consistent conditions during the technological process (the filtration of sus-

pended solids). Therefore, the shape and dimensions of the superconducting winding

that generates a magnetic field of required homogeneity must be carefully considered.

Distribution of Magnetic Field in the Working Space of the Separator

A circular solenoid of rectangular cross-section (Fig. 2) is the most common coil shape

used in magnetic separation. The shaded region is where a cylindrical HGMS matrix is

located. The coil is characterized by the parameters, collected in Table 1.

The basic relationships that may be applied to the solenoid winding are [1]:

( )0 0 1 0

μ α,βB a JK= (1)

( )2 2

0

2

α α β

α,β β ln

1 1 β

K

+ +=

+ +

(2)

( )3 2

1

2 β α 1V aπ= − (3)

solenoid

matrix

B0 B

n

za1

a2

Bm

b

Lm

Figure 2. Configuration of the matrix and solenoid of the separator.

Table 1. Parameters characterizing coil windings

GEOMETRIC PARAMETERS ELECTRIC PARAMETERS

2 a

1

– inside diameter of solenoid,

2 a

2

– outside diameter of solenoid,

2 b – length of solenoid,

Le – length of matrix.

The above parameters are interconnected through

the following relationships:

2

1

α;

a

a

=

1

β;

b

a

=

1

β ;e

e

L

a

=

1

β.e

b L

a

−=Δ

J – averaged density of current in solenoid,

B

0

– magnetic flux density in the geometric centre

of the solenoid,

Bm

– maximum value of flux density on the winding

external surface, in its middle plane,

Bn

– value of flux density on the end of matrix.

A. Ciesla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field 357

( ) ( )2

3

1

α,β β α 1

2

V

v

aπ

= = − (4)

where V is the volume of the winding.

To determine the shape of the winding, the relationships (2) and (4) should be ap-

plied. In Eqs (2) and (4), four variables occur: v, K0, α, and β; two of them being inde-

pendent ones. Figure 3 presents computed distributions of the relative value of the z

component of magnetic flux density on the symmetry axis as shown in Fig. 2.

In this case an analytical expression for the magnetic field distribution on the

symmetry axis can be achieved:

( )( )

( )( )

( )

( )

2 22 2

2 2 2 2

2 22 2

1 1 1 1

ln ln

2

a a b z a a b zI

H b z b z

a a b z a a b z

⎛ ⎞⎟⎜ + + − + + + ⎟⎜ ⎟⎜= − + + ⎟⎜ ⎟⎜ ⎟⎜ + + − + + + ⎟⎟⎜⎝ ⎠

(5)

The analysed magnetic field distribution is for the value of z = ±150 mm and is re-

lated to the length of the matrix (2 Le= 300 mm).

The author [1] proposes considering other possibilities of windings for magnetic

field excitation in HGMS, which promises improved homogeneity of the magnetic field

in the working space of the separator. Two of the possible improved designs are pre-

sented in Fig. 4. Field distributions for the designs of Fig. 4 have been predicted nu-

merically using finite elements modelling.

For convenience, when comparing the results, the inhomogeneity factor ε has been

introduced and defined as maxz

z ave

B

B

ε= where B

z max is the maximum value of B

z, and

Bz ave

is the mean (average) value of Bz on the desired length, in our case – length of the

matrix. Figures 5 and 6 present distribution graphs of a relative value of z component

of magnetic flux density on the symmetry axis for z = ±150 mm for the two proposed

new designs (Figs 5 and 6 should be compared with Fig. 3).

The author proposes the shaping of the magnetic field using the construction from

Fig. 4b, where the coil is divided into small sections with different current density. This

z [mm]

-150 -125 -100 -75 -50 -25 0 25 50 75 100 125 150

Bz

/B

zmax

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

b = 100 mm

b = 160 mm

b = 200 mm

b = 250 mm

MATRIX

a)

b

a

1

c1

c

2

z

b)

J3

J2

J1

J2

J3

d d d3

2

bz

Figure 3. Distribution of Bz (z component of magnetic

flux density) for r = 0 for the case depicted by Fig. 2.

Figure 4. Proposed designs: a) coil with addi-

tional turns of the winding, b) coil with variable

current density in the cross-section (J1, J

2, J

3);

( ) ( )1 1 1 2 1 2

,

e e

L a c L a cβ β= − = −.

A. Ciesla / Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field358

solution leads to both technical and economical considerations. The technical aspect

consists in a good usage of superconductor. From the economical point of view, the

variable cross-section method makes it possible to minimize the volume of the super-

conductor used. However, there may be a problem of supplying different currents to

different sections, as in the case of using the superconductor this would require using

several current leads. This could cause an increase in liquid helium evaporation from

the cryostat.

Motion of the Particles in Matrix Separator

The separator matrix in which magnetic separation occurs is placed in the magnetic

field being induced by solenoidal winding described in the previous chapter.

Figure 7 shows matrix of the High Gradient Magnetic Separator [2].

As Fig. 7b indicates, there are a few arrangements possible between the directions

of magnetic field and particle flow velocity with respect to the ferromagnetic wire.

Three of the configurations are shown in Fig. 8 [3]. Since it is assumed that the fibre is

z [mm]

-150 -125 -100 -75 -50 -25 0 25 50 75 100 125 150

Bz

/ B

z m

ax

0,75

0,80

0,85

0,90

0,95

1,00

b = 250 [mm]

b = 200 [mm]

b = 160 [mm]

b = 100 [mm]

MATRIX

z [mm]

-150 -125 -100 -75 -50 -25 0 25 50 75 100 125 150

Bz

/B

z m

ax

0,75

0,80

0,85

0,90

0,95

1,00

b = 100 [mm]

J2/J

1=0,9524; J

3/J

1=1,9048

b = 160 [mm]

J2/J

1=1,013; J

3/J

1=1,430

b = 200 [mm]

J2/J

1=1,057; J

3/J

1=1,134

b = 250 [mm]

J2/J

1=1,042; J

3/J

1=1,063

MATRIX

Figure 5. Distribution of Bz (z component of the mag-

netic flux density) for r = 0 for the design from Fig. 4a;

(β1 = 5 and β

2 = 3,75).

Figure 6. Distribution of Bz (z component of the

magnetic flux density) for r = 0 for the design

from Fig. 4b.

a)

b) c)

Figure 7. Matrix of the separator (a), matrix filling with stainless steel magnetic wool (wool seen with a

magnifier) (b), one ferromagnetic filament of the steel wool treatment as a collector (c).


always orthogonal to the external magnetic field, the fluid flow can be either parallel

(longitudinal (L) orientation) or perpendicular (transverse (T) orientation) to the direc-

tion of magnetic field. The third alternative configuration, when the initial fluid flow is

parallel to the wire axis is known as the axial (A) configuration. Analysis of the particle

movement around the collector will concern configuration L and T.

The particles in the suspension that flows through the matrix are under the influ-

ence of the magnetic force and move in the direction of the collector (which is a single

fibre of ferromagnetic wool). The distribution of the magnetic field around the collector

shown in Fig. 9 is described by he following equation [4]:

0 0 2 2

0

1 cos 1 1 sin 1

for 1

for

c c

r

a a

a

k

a

k

K K

H A H

r r

b

r

R

b

H H r

R

θθ θ

⎡ ⎤⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎢ ⎥⎟ ⎟= + − −⎜ ⎜⎟ ⎟⎢ ⎥⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

< <

= < <∞

(6)

where:

( ) ( ) ( )0

1 1 , 1 / 1 ,

and .

c c

wk wo a k

A K K

r r R

ε ν ν

ν μ μ

= − = − +

= =

ε0 – the packing factor of the matrix, µ

wk, µ

wo– relative magnetic susceptibility of the

collector and medium, respectively.

Motion of a single particle in nonhomogeneous magnetic field in the vicinity of a

collector is analyzed (see Fig. 9).

Svoboda [3] showed the complexity of the process of particles deposition on a col-

lector and proposed another formulation of the problem. Accepting the above, we are

going to limit our considerations only to such particle size for which the magnetic in-

teraction is decisive. Under these conditions the following equation is assumed to be

valid:

c

c

dv

m F

dt

=Σ

(7)

collector

Rk

particle b

H0

v0 Fg

Fs

x

y

Figure 8. Representation of three geometrical configura-

tions.

Figure 9. Particle with radius b in the magnetic

field around the collector with radius Rk.


where F

Σ is the sum of forces affecting particles in the matrix.

The complete equation of the particle motion including most important forces oc-

curring in the matrix is as follows [4]:

( ) ( )c

0 0 c 0

v 1

g 6 v v grad H B

2c c c c c c

d

V V b V

dt

ρ ρ ρ πη χ⎛ ⎞⎟⎜= − + − + ⋅ ⎟⎜ ⎟⎟⎜⎝ ⎠

(8)

Particles affected by the magnetic force move towards the collectors and settle on

their surface. Particles outside the capture zone that is determined by the border trajec-

tory, will not be captured by the collector. Deposition takes place up to moment when

the balance of holding magnetic force and shear force is achieved.

Subsequent steps of a grain movement in the vicinity of the ferromagnetic element

of the matrix – collecting grains of particular magnetic properties – are presented in

Fig. 10.

Figure 11 shows simulation of the particles trajectory for one collector (for con-

figuration L from Fig. 8) and for five collectors (for configuration L and T from Fig. 8).

Conclusions

The high force separation capabilities of superconducting magnets and their application

for the most difficult separation problems of paramagnetic or low susceptibility materi-

als are now recognized. Recent advances in superconducting technology mean that

technology, once limited to the laboratory research, can be successfully used on a large

scale.

Theoretical model presented in this paper makes it possible to determine basic

variables that characterize extraction particles from slurry in High Gradient Magnetic

Separator.

Figure 10. Steps of particles’ movement in the vicinity of the collector in the matrix separator.


a) b)

c) d)

e) f)

Figure 11. Results of the particles trajectories for different conditions: one collector (configuration L from)

(a, b); five collectors (configuration L) (c, d); five collectors (configuration T ) (e, f).

Acknowledgement

The work presented in this paper was supported by the Polish State Committee for Sci-

entific Research, Warsaw, in the frame of the project No 4 T12A 027 28 (2005–2007).

References

[1] A. Cieśla, B. Garda, J. Sykulski: Shaping of Magnetic Field Distribution in a High Gradient Magnetic

Filter. Archiwum Elektrotechniki, Vol. LI, No 4, pp. 403–415, 2002.

[2] A. Cieśla: Use of the superconductor magnet to the magnetic separation. Some selected problems of ex-

ploitation. Int. Journal of Applied Electromagnetics and Mechanics 19 (2004), pp. 327–331, IOS Press.

[3] J. Svoboda: Magnetic Methods for the Treatment of Minerals. Elsevier Science Publishers B.V., 1987.

[4] Mayuree Natenapit, Wirat Sanglek: Capture radius of magnetic particles in random cylindrical matrices

in high gradient magnetic separation, Journal of Applied Physics, Vol. 85, No 2, pp. 660–664, 1999.


Electric Field Exposure Near the Poles of a

MV Line

D. DESIDERI, A. MASCHIO and E. POLI

Università di Padova, Dipartimento di Ingegneria Elettrica, via Gradenigo 6/a,

Padova, Italy


Abstract. In this work the electric field on the head of a human body under a me-

dium voltage line is studied, close to, at a short distance from, and far from a rein-

forced concrete pole. Two models of the body have been studied numerically: a

more complete one and another simplified. The numerical data obtained with both

models were found equivalent, in order to compute the field on the top of the head.

The simplified configuration has been experimentally validated. The results show

a reduction of the field, close to the pole, compared with the value generally calcu-

lated neglecting the poles.

Introduction

The distribution of the electric field on a human body (“body” in the following) stand-

ing under a power transmission line is basic for the calculations of the induced current

density, subjected to recommendations and standards giving limits for human exposure

to electromagnetic fields.

The calculations are generally performed on a conductive surface, representing a

body, by using numerical techniques such as Finite Element Method, Boundary Ele-

ment, and Charge Simulation Method [1–4].

The computations are especially important on the top of the head, where the field

is maximum [4]: when doing such computations, the ambient field prior to the presence

of the body (“unperturbed field” in the following) is generally assumed to be vertical

and uniform [2–4]. Moreover, the presence of the poles is generally neglected.

Close to a reinforced concrete pole (“pole” in the following) of a medium voltage

(MV) line, these assumptions on the field are no longer valid: moreover, near to the

pole, the unperturbed field can be much higher than that in the midpoint of the span [5].

The aim of this work is to investigate the variation of the field on the head of a

body when approaching the pole of a MV line.

Numerical Analysis

Aim of the numerical analysis was to investigate whether two different models of the

body, the first one closer to the actual body shape and the second one more suitable as

regards the construction of a simple, cheap and easily manageable experimental set-up,

gave equivalent results for the evaluation of the electric field on the head of the body.


363

The Model

A MV overhead line supported by poles with a pole-top-pin construction has been con-

sidered. The line model is shown in Fig. 1, where both Cartesian (x, y, z) and cylindri-

cal (r, θ, z) coordinates are indicated. Three straight cylindrical conductors, with the

same diameter of 0.01 m, whose axes are parallel to each other and to the ground, rep-

resent the three wires. In agreement with typical spacings used for the analyzed con-

struction [6], the axes of the two lateral conductors are as high as a = 10 m, while the

axis of the central top conductor is at a + b = 10.8 m over the ground: the distance 2c

between the lower conductors is 1.52 m. The pole is assumed to be a cylindrical con-

ductor at zero potential, with its vertical axis on the z axis: its height is a = 10 m and

the radius of its cross-section is 0.17 m. The xz vertical plane containing the axis of the

central overhead conductor is at y = 0.

In a first representation (case A, Fig. 2-a), a model of the body complete silhouette

has been implemented. It is composed by two parts, both at zero potential: i) a vertical

conducting cylinder put on the ground, with a diameter of 0.30 m and a height of

1.55 m; ii) a conducting sphere with a diameter of 0.20 m, placed on the cylinder, with

the centre on the axis of the cylinder. The chosen dimensions are very close to the

range reported in [3]. A second simple model (case B, Fig. 2-b) was also used: a con-

ducting sphere, at zero potential, with a diameter of 0.25 m, with the top placed at

1.75 m, i.e. at the same height of the model of case A.

The analyses performed in the following refer only to a symmetrical configuration,

i.e. with the axis of the cylinder and the centre of the spheres on the xz plane at y = 0.

Figure 1. MV line with a pole-top-pin construction.

Figure 2. Numerical models of the body: (a) complete model (case A); (b) simple model (case B).

D. Desideri et al. / Electric Field Exposure Near the Poles of a MV Line364

The Charge Simulation Method

The electric field around the MV power line was computed by using the charge simula-

tion method (CSM). In this method conductors and their surface charges are replaced

by simulating charges placed inside the equipotential contours of the conductors [7–8].

The presence of the ground plane is taken into account by a corresponding set of image

charges. The magnitude of the simulating charges is initially unknown but is readily

calculated when potentials are imposed on a number of points of the contours. Then

potentials and electric fields can be computed everywhere. Furthermore, the accuracy

of the method is tested by calculating potentials on control points of the conductors.

A section of the MV line, extending 30 m apart from the pole on both sides, was

considered. For the representation of the overhead conductors and of the pole, seg-

ments with a constant charge density were used.

The equipotential contour of each overhead conductor was simulated by means of

55 couples of segments. The segments of each couple are on a xy plane, parallel to the

axis of the conductor, on opposite sides, one twentieth of the radius apart from it along

y (Fig. 3-a). The couples differ one from another for their length: the start point of each

segment varies from couple to couple, while the end point is, for all the couples, the

end of the conductor.

The pole was simulated by 69 sets of 10 segments, regularly placed all around the

axis of the pole, on a circumference with a radius of one twentieth of the radius of the

pole (Fig. 3-b). The lengths of the segments of the same set are equal, while they are

different from set to set; all the segments end at the top of the pole. It is worth noting

that the presence of the body induces, on the surface of the pole, charges that are higher

on the side where the body is closer to the pole, i.e. the charge density can vary

strongly with the angle θ around the pole (Fig. 1). Thus, the above-stated charge con-

figuration has been adopted for the pole, in order to correctly reproduce the equipoten-

tial contour of the surface of the pole.

For the body representation, rings with a constant charge density along the circum-

ference and charged points were used. Inside the spheres (case A and case B), 7 rings

with different radii were placed on xy planes: the major radius was for the ring located

in the middle of the sphere. Moreover, 2 charged points were placed near the top and

the bottom of the spheres. On each ring, distributed along the circumference, 15

charged points were added. Therefore each sphere has been replaced by 7 rings and 107

charged points.

Figure 3. Charge configuration: (a) one couple of charged segments inside the equipotential surface of an

overhead conductor, (b) one set of 10 segments inside the equipotential surface of the pole.

D. Desideri et al. / Electric Field Exposure Near the Poles of a MV Line 365

Inside the cylinder (case A), 34 rings of the same radius were placed on xy planes.

Furthermore, for each ring, distributed along the circumference, 9 charged points were

added. Therefore the cylinder has been replaced by 34 rings and 306 charged points.

Finally, the body induces on the overhead conductors a very low charge, and there-

fore the above mentioned representation with simple couples of charged segments has

been used.

The accuracy was tested. With 1 kV imposed on the central phase conductor and

–0.5 kV on the other ones, on the control points a deviation, on the potential, lower

than 2 V was found: on the pole and on the top of the head a difference of less than

1 mV was computed.

It is worth noting that the charge simulation method was chosen in this study be-

cause it allows to compute the electric field with a small number of simulating charges:

for the calculations, the largest matrix to be solved was only of 1434 × 1434. Conse-

quently, each computation run requires about 70 seconds on a PC with 504 MB RAM,

1.4 GHz Intel Celeron M.

Electric Field Computation

The way to compute, by the CSM, the rms value of the three components of the electric

field is simple. It is well known that a sinusoidal electric field ( )E P,t

can be expressed

as the combination of two terms:

( )E P,t

=1 2

( ) cos( ) ( ) ( )E P t E P sen tω ω+

(1)

and the rms values of the three components result:

2 2

1 2

( )

2 2

i i

i

E E

E P = + (2)

with i indicating one of the coordinates (x, y, z). In a simple way, the calculation of

1

( )E P

and of 2

( )E P

is done respectively setting the potentials of the three overhead

conductors at t1 = 0 and at t

2 = π/(2ω). Any choice of the potential configuration at

t1 = 0 is valid: different values of the potential configuration result in different expres-

sions for 1

( )E P

and 2

( )E P

.

Since the MV power line is fed by a set of balanced three-phase voltages, to take

profit of the symmetry of the whole system (the pole-top-pin configuration, the models

used for the body and their location on the plane y = 0), the following two configura-

tions have been chosen. The first one is with the central top conductor at maximum

potential, and the two lateral conductors at one half (and opposite) of the maximum; the

second one, shifted of π/(2ω) in time, is with the central top conductor at zero potential,

and the two lateral conductors subjected to potentials, with opposite sign, equal in

magnitude to the maximum multiplied by 3 /2. With this choice, it can be noticed

that, in the plane y = 0, the x and z components of the electric field, with the values

indicated for the second configuration, are zero.


A line voltage of 20 kV rms has been considered. In Fig. 4 the vertical component

Ez of the electric field on the top of the head, at y = 0, against the distance of the center

of the body from the axis of the pole, is reported, both for case A (continuous line) and

for case B (dashed line).

Experimental Validation

The case B configuration has been realized. A hollow sphere, in brass, with a diameter

of 0.25 m, bonded to the ground by a thin copper thread, has been used, with a wooden

tripod as supporting structure. The top of the sphere has been placed at 1.75 m over the

ground, under the central conductor of the MV line, in three different measuring posi-

tions (with the centre at 0.32 m, 1.14 m and 6.67 m from the axis of the pole). The elec-

tric field has been measured by using a PMM EHP-50C, usually located on the top of

the sphere, except near to the pole where it was located also by the sphere’s side. Fig-

ure 5 shows the experimental set-up placed on-site.

In Table 1, with reference to the vertical component of the electric field (Ez) on the

top of the head, the numerical data obtained with the two depicted representations (case

A and case B) are reported and compared with the experimental values. The values

computed with the simple sphere are in good agreement with those obtained with the

complete model (a difference less than 8%). Moreover the numerical analysis relative

to the case B configuration has been experimentally validated: the difference between

numerical and experimental data is of about 20%. In Table 1 it is also apparent the

large drop of Ez on the top of the head when shifting the body from 6.67 m to 1.14 m

and, even more, to 0.32 m. It can be seen that the Ez variation with the distance is the

same for both measured and computed fields.

Finally, in the position at 0.32 m, on the side of the sphere opposite to the pole, the

measured electric field parallel to the overhead conductors was 266 V/m, in agreement

with the numerical data. The field is higher than on the top: that happens close to the

pole, due to the local enhancement of the unperturbed field observed in [5].

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10

Ez

[V/m]

x [m]

Figure 4. Vertical component of the electric field (Ez) at y = 0 on the top of the head for: case A (continuous

line) and case B (dashed line).


Conclusions

A comparison between two different models of the body, centered on the vertical plane

of symmetry along the line, has been performed in this work.

A first conclusion is that, when only the evaluation of the field on the top of the

head is required, a simple sphere can be used, instead of choosing among the different

shapes or sizes of the model of the body used in literature. This solution is much easier

to be handled in the numerical analysis and to be implemented as experimental appara-

tus.

A second conclusion is that the presence of the pole strongly reduces the field on

the top of the head near the supporting structure. Close to the pole, a severe overestima-

tion of the field results, when the line supporting structures are neglected in the compu-

tations.

References

[1] E. Poli, Magnetic and electric field computation in proximity of power-line towers and supported bus-

bar, CIGRE 33-96 (WG-07) IWD 19, 1996.

[2] O. Bottauscio, R. Conti, Magnetically and electrically induced currents in human body models by ELF

electromagnetic fields, 10th ISH, pp. 5-8, 1997.

Figure 5. Case B configuration: on-site experimental set-up.

Table 1. Ezon the top of the body: numerical results (case A and case B) and experimental data

position case A case B measured

at 0.32 m 144 V/m 132 V/m 168 V/m

at 1.14 m 350 V/m 330 V/m 412 V/m

at 6.67 m 681 V/m 650 V/m 789 V/m


[3] A.B. Mahdy, H.I. Anis, Field exposure modeling using charge simulation, 5th ISH, paper 33.20, 1987.

[4] CIGRE WG 36-01, Electric and magnetic fields produced by transmission systems. Description of phe-

nomena - practical guide for calculation, Paris 1980.

[5] D. Desideri, A. Maschio, E. Poli, Environmental analysis of the electric field due to MV overhead lines

supported by concrete poles, 4th International Workshop on Biological Effects of EMFs, pp. 876-883,

2006.

[6] Standard Handbook for Electrical engineers, 10th ed., D. G. Fink and J. M. Carroll Eds., McGraw-Hill,

1968.

[7] N. H. Malik, “A review of the charge simulation method and its applications,” IEEE Trans. Electr. Insu-

lation, vol. 24, pp. 3-20, Feb. 1989.

[8] E. Poli, “The use of image charges in the charge simulation method: a parallel-plane dielectric plate cov-

ering a conductor,” IEEE Trans. Magn., vol. 28, pp. 1076-1079, Mar. 1992.


Study on High Efficiency Switched

Reluctance Drive for Centrifugal Pumping

System

Jian LI, Junho CHA and Yunhyun CHO

Depart of Electrical Engineering, Dong-A University, 840 Hadan 2 dong, Saha-gu,

Pusan, 604-714, Korea


Abstract. Variable speed drives are widely used in pump system for the energy ef-

ficiency potential. In this paper a switched reluctance motor and variable speed

drive is designed. Considering pumping system’s requirements, ‘base speed’ and

rated torque is carefully decided to give a high efficiency in wide speed range. Ro-

tor’s shape is also studied to give good dynamic performance. The motor’s geome-

try was optimized for high efficiency.

1. Introduction

In last decades, the pumping system driven by variable frequency controlled induction

motors are widely used in industry. But switched reluctance motor has advantages such

as high efficiency, high reliability, and low cost construction. More importantly, sub-

merged pumps are in slender shape and SRMs can be much more easily manufactured

in slender shape than induction motors. So the demand of SRM application in pumping

systems is getting stronger.

Variable Speed Drive (VSD) converts everyday pumps into sensor-controlled

pumps that automatically adjust speed in real time to changes in pressure of pumping

system. As more flow is required, the pump motor speeds up to provide smooth and

consistent performance. The pump’s operating characteristics is show in Fig. 1. In a

constant pressure piping system, system curve changes from 1 to 2 when large flow

rate is required, so the motor’s speed increases from 3

n to 2.n In our project, the

pump’s speed range is from 2000 rpm to 3600 rpm. When design a switched reluctance

motor, the rated torque should first satisfies pump’s shaft torque. It’s important to de-

cide the ‘base speed’ since motor’s dynamic performance is much better below base

speed [1] but higher efficiency can be obtained above ‘base speed’. These two cases are

shown in Fig. 2. Another problem is that switched reluctance motor’s efficiency gets

lower when operating with smaller load, and usually this is common in variable speed

driven pumps.

2. Preliminary Design and Flux Linkage Analysis

Design of switched reluctance motor is a complicate procedure due to its nonlinearities

because of core saturation and salient poles. Many guides and considerations were



doi:10.3233/978-1-58603-895-3-370

370

given in literature [1,2], but the flux-linkage was first further studied here to give a high

efficiency design. Firstly, the rated power, operating speed range are derived from the

application’s requirements. These data together with dimensional constraints are used

to derive cross shape detentions using equations from [3]. The motor studied in this

paper has 6 stator poles and 4 rotor poles and the dimensions of core are given in Ta-

ble 1.

The well-known expression for inductance calculation will be used:

2

phT

L

R

= (1)

where ph

T is the number of the turns of the excited phase, R is the reluctance of the

magnetic circuit in which the inductance is calculated. Since it is a complicated mag-

netic circuit, consisting of parts, such as stator pole, stator yoke, air gap, rotor pole,

rotor yoke, the total reluctance will be the sum of single reluctance along the magnetic

Figure 1. Pump performance curves.

Figure 2. Motor and pump torque curves.

J. Li et al. / Study on High Efficiency Switched Reluctance Drive for Centrifugal Pumping System 371

path. The reluctance will be calculated by means of geometrical dimensions and mag-

netic permeability:

l Hl

R

S BSμ

= = (2)

where l is the length of the magnetic path, S is the area which is penetrated by the mag-

netic flux, magnetic permeability μ is given by the values of B and H in the B/H curve

of the lamination material.

According to the operation modes, the angle between the unaligned and aligned

positions which is the equal to half the rotor pole pitch can be divided into three re-

gions: a) fully unaligned to start of the pole overlap, b) starting of the pole overlap to

full pole overlap, and c) full pole overlap to fully aligned condition of stator and rotor

poles. Each region is then modeled by several flux tubes [4] (as shown in Fig. 3), the

inductance of which is calculated at given phase current.

Assume a certain flux density in the stator pole, and then flux densities in other

parts of the machine such as rotor pole, rotor back iron, stator yoke, and air gap are

derived as the areas of cross sections of these parts for assumed flux paths are obtained

from the machine geometry and the assumed stator pole flux density. From the flux

densities in various parts of the machine and the flux density vs. magnetic field inten-

Table 1. Specifications of 6/4 SRM for 2.2 kW centrifugal pump

Item Quantity

Core material Non-oriented silicon steel 500N60

Stator outer diameter (mm) 150

Rotor outer diameter (mm) 80

Shaft diameter(mm) 24

Stack length (mm) 80

Stator yoke width (mm) 12.1

Rotor yoke width (mm) 13.2

Stator pole arc (degrees) 30

Rotor pole arc (degrees) 32

Air-gap length (mm) 0.4

DC voltage (V) 310

Rated Torque (N.m) 5.8

Rated Speed (rpm) 3600

Figure 3. Magnetic flux tubes to calculate the inductance at unaligned position.

J. Li et al. / Study on High Efficiency Switched Reluctance Drive for Centrifugal Pumping System372

sity characteristics of the lamination material, corresponding magnetic field intensities

are obtained. Given the magnetic field intensities and length of the flux path in each

tube, their product gives the magneto motive force (mmf). The mmfs for various parts

are likewise obtained, and for the magnetic equivalent circuit and stator excitation Am-

pere’s circuital law is applied. If error between the applied stator mmf and that given

equivalently by various parts of the machine reveals a discrepancy, then that error is

used to adjust the assumed flux density in the stator pole and entire iteration continues

until the error is reduced to a set tolerance value.

Once the phase inductance ( , )L iθ is known, it can be multiplied with the phase

current ( )i θ to get the phase flux linkage ( , )iλ θ as shown in Fig. 4, whereθ is the angle

of displacement of the rotor from the fully unaligned position.

The energy conversion can be modelled by the flux linkage information. In order

to maximize the power density at a fixed inverter volt-ampere rating, the SRM must be

designed with a sufficiently narrow air gap [5].

After obtaining the flux-linkage from the analytical method, the curves can be de-

scribed by linear and polynomial fit for unaligned and aligned position:

( ) *u ui A iλ = (3)

5 4 3 2 1

5 4 3 2 1

( ) * * * * *a

i B i B i B i B i B i Aλ = + + + + + (4)

Equation (4) is a linear equation whereu

λ is the flux linkage at unaligned position

andu

A is linear coefficient, inductance for this position actually. Equation (5) is a five

order polynomial which accurately determines the nonlinear relationship between flux

linkage and current.

Define

5 4 3 2

5 4 3 2 1

( ) ( ) ( )

* * * * ( )*

a u

u

F i i i

B i B i B i B i B A i A

λ λ= −

= + + + + − +

(5)

Figure 4. Flux-linkage versus current and rotor position.


And take derivative of it

( )

0

dF i

di

=

(6)

By solving the above equation we can get the currentp

I at which co-energy incre-

mentc

WΔ is maximum for the same current increment iΔ as shown in Fig. 2. This

value is defined as maximum co-energy increment current (MEIC), which is the opti-

mum operating point of the motor. The increment of coenergy was shown in Fig. 5 (b).

When the motor is under deep saturation, the increment of co-energy does not in-

crease but decreases. This is because magnetic circuit at aligned position is under deep

saturation and the flux- linkage only increases slightly but at unaligned position mag-

netic circuit is far from saturation so flux-linkage increases linearly with current. Exci-

tation current should be at or a little larger than MEIC to take full use of co-energy

increment when we design the motor. The advantage of this criterion is verified in the

following sections.

After the cross section of motor was designed, the saturation effect is the similar

with same winding turns while stack length varies. This is illustrated by the increment

of co-energy vs. current curves at various stack lengths in Fig. 3. The MEICs are al-

most constant in spite of stack length caused by saturation effects under same electric

loading.

The dimension of cross section is given in Table 1 and the winding per pole has 80

turns. The MEIC 13.5p

I A= is derived from Eqs (5) and (6). Comparing the data in

Table 2, the stack length 90 mm with winding of 80 turns is mostly suitable to the rated torque 5.8 N.m.

3. Modeling and Analysis of Switched Reluctance Drive

When considering the performance of the designed motor, there are many criteria such

as efficiency, maximum output power, thermal effect and so on among which effi-

ciency seems to be mostly critical especially in the continuous operating apparatus.

ΔWC

Ι

P

Δi

ΔWC

ΔWC

ΔWC

λ

λ

u

λ

a

ΔiΔi

Flu

x link

age

W

b

Current [A]

Δi 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Current (A)

Incr

emen

t o

f co

-en

erg

y (J

)

(a) (b)

Figure 5. (a) Variation of co-energy increment with respect of exciting current. (b) the increment of the co-

energy according to current evaluated by the proposed method.


Dynamic system is simulated by a computer program or a standard modeling package,

such as Spice and Simulink, which usually ignore the effects of the multi-phase cur-

rents on the saturation, mutual inductance effects, electromagnetic transient phenome-

non and its accuracy depends on the fitting of the static magnetization.

A more accurate model was developed using FE method coupling with electrical

circuit as shown in Fig. 7. The control algorithm and inverter are modelled in Mat-

lab/Simulink environment and FEM model of motor is developed in commercial soft-

ware package Flux 2D. The two models are connected by a coupling block supported

by Flux 2D. In the control algorithm, the turn on angle is set to give enough current

rising time for p

I before overlap position and conducting angle is constant with 30°.

The current and torque waveforms from simulation results are shown in Fig. 8.

Accurate efficiency estimation needs more investigation on losses. The power

losses in the electromechanical devices are mainly of three types: iron losses, copper

losses and the mechanical losses. Copper losses can be easily calculated using phase

current waveforms from simulation results (7),

2

0

1

( )

T

Cu s

P mR i t dt

T

= ∫ (7)

where m is number of phases ands

R is resistance of winding. The iron loss is calculated

by LS model introduced in [7] which is based on dynamic B-H characteristics associ-

ated with transient finite element simulation. It is more accurate compared with the

method in [6].

0 5 10 15 20 25 30 35

0.0

0.1

0.2

0.3

0.4

0.5

Increment of Co-energy

(J)

70mm

80mm

90mm

100mm

Current (A)

Figure 6. Increment of co-energy with respect of the variation of current for different stack length.

Table 2. SpeciMotor stack length and power at MEIC

Turns per

Pole

MEIC (A)

Saturate

Current (A)

Co-energy

(W)

Torque

c

T (N.m)

60 18.1 9.03 3.1 5.81

70 15.4 7.74 3.01 5.75

80 13.5 6.52 3.08 5.78

90 12.1 5.48 3.13 5.90


Figure 7. Dynamic FEM simulation coupling with external circuit.

Figure 8. Phase currents and torque from simulation.


The motor in Table 1 was analyzed by the above method and the efficiency of mo-

tor is 89.7%. Then the prototype of the designed motor was manufactured and tested.

The motor and inverter are shown in Fig. 9. The mechanical losses were obtained by

turning off the inverter at rated speed and the efficiency achieves 88.2% which is ex-

cluding mechanical losses. This is a little lower than simulation result because of the

losses in the inverter.

4. Conclusions

The design and analysis of high efficiency switched reluctance motor for pumping sys-

tem have been proposed in this paper. The maximum co-energy increment current

(MEIC) was defined after modeling and analyzing the flux linkage curves. The motor’s

geometry was optimized according to the mechanical requirements of pumping system.

then dynamic FEM simulation which can estimate the iron loss using a more accurate

LS models was used to analyze both the motor and drive. And the performance was

verified by experiments.

References

[1] M.N. Anwar, Iqbal Husain, and Arthur V. Radum, “A Comprehensive Design Methodology for Switched

Reluctance Machines”, IEEE Trans. Industrial Electronics, vol. 37, No. 6, February 2001.

[2] T.J.E. Miller, “Optimal Design of Switched Reluctance Motors” IEEE Trans. Industrial Electronics, vol.

49, No. 1, February 2002.

[3] R. Krishnan, Switched Reluctance Motor Drives: Modeling, Simulation, Analysis, Design, and Applica-tions. Boca Raton, CRC: 2001, pp. 79–120.

[4] N.K. Shen and K.R. Rajagopal, “Calculation of the flux-Linkage characteristics of a switched reluctance

motor by flux tube method” IEEE Trans. Magnet, vol. 41, No. 10, October 2005.

[5] Arthur V. Radun, “Design Considerations for the Switched Reluctance Motor”, IEEE Trans. Industrial Applications, vol. 31, No. 5, September/October 1995.

[6] Peter N. Materu and Ramu Krishnan, “Estimation of Switched Reluctance Motor Losses,” IEEE Trans-actions on Industry Applications, vol. 28, No. 3, pp. 668–679, May/June 1992.

[7] FLUX® 8.10 2D Application New features.

Figure 9.




C3. Special Applications


Power Quality Effects on Ferroresonance

Luca BARBIERI, Sonia LEVA, Vincenzo MAUGERI and Adriano P. MORANDO

Politecnico di Milano, Dipartimento di Elettrotecnica

Piazza L. da Vinci, 32-20133 Milano (Italy)

luca1.barbieri; [email protected]

sonia.leva; [email protected]

Abstract. Ferroresonance is a complex phenomenon very dangerous for electric

power system. In this paper, the power quality effect on ferroresonant circuit is for

the first time investigated by using nonlinear dynamics theory. In particular, the

software package AUTO is used to determine the period solutions of the differen-

tial equation describing a typical voltage transformer circuit, supplied by distorted

voltage, in which ferroresonance can occur.

Introduction

Ferroresonance is a complex phenomenon – due to the interaction between a non linear

inductance and a capacitance – very dangerous for electric power system. Research

involving in transformers has been conducted over the last 80 years. The word ferrore-

sonance is first seen in the literature in 1920 [1], although papers on resonance in trans-

formers appears as early as 1907 [2]. Early analysis was done using graphical method,

which appeared in American literature as early as 1938 [3]. More exacting and detailed

works was done later by Hayashi in the 1950s [4]. Several works focuses on improve-

ment of system and transformer models used with software such as EMTP [5].

Today the term ferroresonance is firmly established in the power system engineer's

vocabulary and is used to not only describe the jump to higher current fundamental

frequency state but also bifurcations to subharmonic, quasi-periodic and even chaotic

oscillations in any circuit a non linear inductor. The connection of ferroresonance to

nonlinear dynamics and chaos was established in 1988 and published in 1992 [6]. The

application of nonlinear dynamics and chaotic theory in studying of a ferroresonant

circuits is first seen in literature in 1990 [7] and in 1994 [8].

In case of ferroresonance, remnant magnetization of core, voltage at the time

switching, and amount of charge on the capacitance are all initial conditions which

determine the steady state response of the non linear dynamical system. Even with mi-

nutely small differences in initial condition, it is possible that subsequent initiations of

ferroresonance may result in very different voltage waveforms. Bifurcation theory has

proved to be the adequate mathematical framework for the study of non linear dynami-

cal systems. This theory implies the calculation of a solution of system of ODE with

respect to a parameter. The critical values of this parameter, where the type or number

of solution of system changes, are called the bifurcation values. Different approaches

exist to solve the system, amongst them the Galerkin method, which transforms the

problem into frequency domain [9]. In this present work the software package AUTO

[10] was used.


381

There is an increasing interest in Power Quality (PQ) topics among both customers

and utilities. The effects of the disturbances, mainly harmonics, interharmonics and

unbalances, have been deeply investigated and in particular their negative effects have

been highlighted in the technical literature in this field. Power losses, thermic effects,

degradation of the electric insulation, interferences with electronic, control systems and

telecommunication lines, voltage fluctuations and flicker are the most common subjects

of interesting and well-established studies in the PQ bibliography.

This paper investigates – for the first time – the influence of these disturbances that

could decrease the quality level of the system on ferroresonance circuit. Our attention

will be focused on single-phase electrical network in which is present a Voltage Trans-

former (VT). Figure 1 shows the typical VT circuit arrangement where CCB

is the cir-

cuit breaker capacitance and CBB

is the total bus bar capacitance to earth. Ferroreso-

nance conditions occurred upon opening the circuit breaker with DS1 closed and DS

2

open, leading to failure of the transformer primary winding [11].

System Configuration

The basic ferroresonance equivalent circuit [11] subject of the present study is sche-

matically depicted in Fig. 2. The resistor R represents transformer core losses, Csh

is

total phase-to-earth capacitance of the circuit and Cs

is the circuit breaker grading ca-

pacitance.

The φ-i characteristic of the transformer (Fig. 2) is simulated by the following sev-

enth-order polynomial:

7

i a bφ φ= + (1)

BB

C

CB

C

( )e t

1

DS2

DS

circuit breaker

VT

Figure 1. Typical Voltage Transformer circuit arrangement.

( )e tR

sh

C

s

C

( )d

v t

dt

φ

=

( )i t

φ

i

Figure 2. Basic ferroresonance circuit.

L. Barbieri et al. / Power Quality Effects on Ferroresonance382

where a = 3.42, b = 0.41, i is the current in p.u. value and φ is the flux in the trans-

former core in p.u. value.

The time behaviour of the ferroresonant circuit is described by the differential

equation:

( ) ( )

( ) ( )

2 ( cos cos )s

n n

sh s sh s sh s

Cdv v i

E t t

dt C C R C C C C

d

v

dt

ω ω αω ω

φ

= + − −

+ + +

=

(2)

where E is the RMS of the fundamental harmonic supply phase voltage, α is the weight

of the nth

harmonic components supply phase voltage and ω is the angular supply fre-

quency (ωn=nω).

The software package AUTO computes and continuates the solutions of systems of

algebraic and autonomous differential equations. The computation of periodical solu-

tions can be treated as boundary value problem. A periodically forced system of order k

can be transformed into an autonomous system by adding a stable non linear oscillator

with the desired pulsation ωn:

( )

( )

2 2

2 2

n

n

x x y x x y

y y x y x y

ω

ω

= + − +

= − − +

(3)

the solution of these equations are: x = sin(ωnt) and x = cos(ω

nt). By coupling (3) to the

system (2), a sixth-order system is formed that can be solved by AUTO.

Simulation Results

The system defined in (2) presents three different oscillation modes (three different

periodical attractors): a normal response plus two possible types of ferroresonance con-

ditions. Csh

is taken as bifurcation parameter. All the results are obtained by detailed

numerical simulation, and preliminary analyses are based on mathematical theories and

method of [12]. At first, the supply voltage is supposed sinusoidal then a distortion

harmonic component is considered. In the following analysis, instead of using actual

values of circuit parameter E, ω, Cs, C

sh, etc., system equations are made dimensionless

by employing per unit value.

Sinusoidal Supply Voltage

In this first case n = 0 and α = 0 in (2) and (3). First of all, the bifurcation diagram is

given in Fig. 3 by using AUTO [10]. There are seven types of bifurcations and three

different branch of solutions, one for each attractors.

The branch A is obtained from the continuations of the normal sinusoidal response

(Fig. 4). At Csh

*

= 0.03 nF a Limit Point (LP) bifurcation is detected. In this point the

stable periodic solution (Fig. 3 full line) coalesces and obliterates with unstable peri-

odic solution (Fig. 3 dashed line).

L. Barbieri et al. / Power Quality Effects on Ferroresonance 383

From the continuations of fundamental frequency ferroresonance are not detected

bifurcations (Fig. 3 attractor B). The solution remains stable for all range of Csh

value.

Operation in ferroresonance region is demonstrated by the high amplitude of trans-

former voltage waveform and by the presence of high frequency harmonic components

in the voltage waveform power density spectrum (Fig. 5).

The branch C is obtained from the continuations of the subharmonic ferroreso-

nance (Fig. 6), the resulting waveform is still periodic, but with a period twice the pe-

riod of the supply cycle. We found four limit point bifurcation and two period doubling

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

Csh [nF]

max ( flux ) [p.u.]

LP

B

C

A

-0.05 0 0.05 0.1 0.15 0.2

2.85

2.9

2.95

3

3.05

3.1

Csh [nF]

max ( flux ) [p.u.]

PD

C

D

Figure 3. Bifurcation diagram at Cs = 0.5 nF, R = 225 MΩ (full line for stable solution, dashes line for un-

stable solution).

12.66 12.67 12.68 12.69 12.7

-0.2

-0.1

0

0.1

0.2

time [s]

v [p

.u.]

12.66 12.67 12.68 12.69 12.7

-0.2

-0.1

0

0.1

0.2

time [s]

ph

i [p

.u

.]

Figure 4. Normal sine wave response. Cs = 0.5 nF, R = 225 MΩ, C

sh = 3 nF, E = 1 p.u.

12.66 12.67 12.68 12.69 12.7

-3

-2

-1

0

1

2

3

time [s]

v [p.u

]

12.66 12.67 12.68 12.69 12.7

-4

-2

0

2

4

time [s]

ph

i [p.u

.]

Figure 5. Fundamental frequency ferroresonance. Cs = 0.5 nF, R = 225 MΩ, C

sh = 3 nF, E = 1 p.u.

PD


bifurcation (PD). In this point the branch D (Fig. 3) of periodic-doubled solution

emerges, and the original branch of stable periodic continues as a branch of unstable

solution at the post-bifurcation.

From Fig. 3 it is also possible note that the bifurcation diagram can be subdivided

in two different region based on Csh

value. In the first region, where Csh

< 0.03 nF, only

ferroresonance oscillations can occur. In the second region, for Csh

> 0.03 nF, both the

normal and the ferroresonant oscillation can occur. In such cases the steady state oscil-

lation mode will be determined by initial conditions.

Distortion Supply Voltage

For what concern the presence of harmonic components, different cases changing the

harmonic order (n = 2 and n = 7) and the weight of the considered harmonic (α = 0.01,

0.1, 0.5) have been analyzed. For each one has been calculated the bifurcation diagram

by using AUTO.

In Figs 7 and 8 the n = 7, α = 0.1 case is shown. By analyzing the obtained results

it is possible to note that the bifurcation diagram is substantially equal to the one ob-

tained in the sinusoidal case. Also in this case there are seven types of bifurcations and

three different branch of solutions, one for each attractors. The three A, B and C

branches have the same meaning that in the sinusoidal case:

12.62 12.64 12.66 12.68

-3

-2

-1

0

1

2

3

time [s]

v [p

.u

.]

12.62 12.64 12.66 12.68

-3

-2

-1

0

1

2

3

time [s]

ph

i [p

.u

.]

Figure 6. Subharmonic ferroresonance. Cs = 0.5 nF, R = 225 MΩ, C

sh = 0.19 nF, E = 1 p.u.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

Csh [nF]

max

( flu

x ) [p

.u

.]

LP

B

C

A

0.03 0.031 0.032 0.033 0.034 0.035

0

0.1

0.2

0.3

0.4

0.5

0.6

Csh* [nF]

α

(a) (b)

Figure 7. (a) Bifurcation diagram at Cs = 0.5 nF, R = 225 MΩ (full line for stable solution, dashes line for

unstable solution); (b) Bifurcation diagram in two-parameter space Cs-α.


− A represents the normal wave response: the voltage v has the same waveform

of the power supply, v and φ have low amplitude.

− B represents the fundamental frequency ferroresonance: v and φ have high

amplitude with the same period of the power supply.

− C represents the subharmonic ferroresonance: v and φ have high amplitude but

with a period twice the period of the power supply.

Furthermore, the flux and voltage waveforms (see Fig. 8) – distorted for the pres-

ence of harmonic components – have amplitude only a few different from the previous

waveforms.

12.66 12.665 12.67 12.675 12.68 12.685 12.69 12.695 12.7

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

time [s]

v [p

.u

.]

12.66 12.665 12.67 12.675 12.68 12.685 12.69 12.695 12.7

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

time [s]

flux [p.u.]

(a)

12.66 12.665 12.67 12.675 12.68 12.685 12.69 12.695 12.7

-3

-2

-1

0

1

2

3

time [s]

v [p.u.]

12.66 12.665 12.67 12.675 12.68 12.685 12.69 12.695 12.7

-4

-3

-2

-1

0

1

2

3

4

time [s]

flu

x [p.u.]

(b)

12.62 12.63 12.64 12.65 12.66 12.67 12.68 12.69

-3

-2

-1

0

1

2

3

time [s]

v [p.u.]

12.62 12.63 12.64 12.65 12.66 12.67 12.68 12.69

-3

-2

-1

0

1

2

3

time [s]

flu

x [p.u.]

(c)

Figure 8. (a) Normal wave response Cs = 0.5 nF, R = 225 MΩ, C

sh = 3 nF, E = 1 p.u.; (b) Fundamental

frequency ferroresonance. Cs = 0.5 nF, R = 225 MΩ, C

sh = 3 nF, E = 1 p.u.; (c) Subharmonic ferroreso-

nance. Cs = 0.5 nF, R = 225 MΩ, C

sh = 0.19 nF, E = 1 p.u.


As AUTO has the possibility of freeing a second parameter, the bifurcation point

in a two-parameter space may be determinate. This can be applied to the continuation

of LP represented in Fig. 7a: the goal is analyze the behaviour of the capacitance value

Csh

*

as a function of α between 0 and 0.6. The result of the continuation of this LP is

shown in Fig. 7b. Also in this case it is possible to note that the presence of harmonic

components does not give big difference.

It is possible conclude that concerning on the ferroresonance phenomenon the

presence of harmonic components does not give any contribution.

Conclusions

The influence of the PQ-disturbances decrease the quality level of the voltage and cur-

rent of the system but on the point of view of ferroresonance phenomenon, the har-

monic components change only a few the bifurcation diagram. Furthermore, by con-

firming the limited influence of the harmonic components on the ferroresonance, the

Csh

value that individuates the two different working region characterised by only fer-

roresonance oscillations and normal/ferroresonant oscillation change in imperceptible

way as a function of the harmonic components RMS value.

In the near future this work will be extended to the study the PQ effect on the bi-

furcation analysis of three-phase ferroresonant circuit. Besides more attention will be

spent to the accurate modelisation of voltage transformer and high power transformer.

References

[1] P. Boucherot, Existence de Deux Régime en Ferrorésonance, R.G.E., pp. 827-828, December 10, 1920.

[2] J. Bethenod, Sur le Transformateur et Résonance, L’Eclairae Electrique, pp. 289-296, November 30,

1907.

[3] P.H. Odessey and E.Weber, Critical Condition in Ferroresonance, Trans. AIEE, vol. 57, pp. 444-452,

1938.

[4] C. Hayashi, Nonlinear Oscillations in Physical System, McGraw-Hill Book Company, New York, NY,

1964.

[5] D.L. Stuehm, B.A. Mork and D.D. Mairs, Five-Legged Core Transformer Equivalent Circuit, IEEE

Trans. Power Delivery, vol. 4, no. 3, pp. 1786-1793, July, 1989.

[6] B.A. Mork, Ferroresonance and Chaos . Observation and Simulation of Ferroresonance in a Five-

Legend Core Distribution Transformer, Ph.D. Thesis, North Dakota State University, May, 1992.

[7] C. Kieny, Application of the Bifurcation Theory in Studying and Understanding the Global Behavior of

a Ferroresonant Electric Power Circuit, IEEE Trans. Power Delivery, vol. 6, no. 2, pp. 866-872, April,

1990.

[8] B.A. Mork and D.L. Stuehm, Application of Nonlinear Dynamics and Chaos to Ferroresonance in Dis-

tribution Systems, IEEE Trans. Power Delivery, vol. 9, no. 2, pp. 1009-1017, April, 1994.

[9] C. Kieny, A. Sbai, Ferroresonance Study Using Galerkin Method with Pseudo-Archlength Continuation

Method, IEEE Trans. Power Delivery, vol. 6, no. 4, pp. 1841-1847, 1991.

[10] E.J: Doedel and J.P. Kernévez, AUTO: Software for Continuation and Bifurcation Problems in Ordi-

nary Differential Equations, Applied Mathematics Report, Californian Institute of Technologies, 1986.

[11] Z. Emin, B.A.T. Al Zahawi, D.W. Auckland, Y.K. Tong: Ferroresonance in Electromagnetic Voltage

Transformer: A Study Based on Nonlinear Dynamics, IEE Proc. Gener. Transm. Distrib., vol. 144,

no. 4, pp. 383-387, July, 1997.

[12] Y.A. Kuznetsov, Elements of applied bifurcation theory, Springer, New York, 1995.


FEM Computation of Flashover Condition

for a Sphere Spark Gap and for a Special

Three-Electrode Spark Gap Design

Matjaž GABER and Mladen TRLEP

University of Maribor, Faculty of electrical engineering and computer science,

Smetanova 17, SI-2000, Slovenia

[email protected]

Abstract. Electrical discharges in gases represent a complex problem in high-

voltage techniques. The paper presents the findings which have been used to build

a model of flashover in gases designed for use in engineering practice. The model

is based on the electrostatic FEM calculation and on the value of the flashover

electrical field intensity in air. The model has been used to calculate the flashover

for a two-electrode sphere spark gap and a special three-electrode spark gap.

Introduction

In classical theory Townsend [1,2] defined the physical picture of flashover by (1),

where α, β, γ and d represent the first Townsend ionization coefficient, the electron

attachment coefficient, the second Townsend ionization coefficient and the distance

between the electrodes. Unfortunately, (1) applies for gases of low pressure only be-

cause some coefficients cannot be defined at normal air pressure [3].

[ ]exp( ) 1 1d

α

γ α β

α β

− − =

−

(1)

As a consequence, it has become necessary to search for new approaches to elec-

trical discharge in gases of pressures near the normal air pressure. These attempts pro-

duced a new theory that uses numerical mathematics and advanced computers, and has

lately yielded some significant results [4–7]. However, the use of such models for the

calculation of flashover is very demanding, so it is limited to scientific research pur-

poses. In engineering practice, the condition for flashover is determined by using the

flashover electrical field intensity condition 3 MV/mb

E = only. This means that the

flashover takes place if the value of the electrical field intensity exceeds the flashover

electrical field intensity at any point of the calculation. This condition often turns out to

be insufficient, so we decided to find the criteria that would determine the flashover in

air more precisely and be more suitable for engineering practice.

The conditions in air vary due to the changes in air humidity, temperature and

pressure. As a result, the mean free path of electron 5

10 me

λ

−

≈ and the ionization

energy of the average molecule in air 18

5 10 Jion

W

−

≈ ⋅ are slightly changed. Thus the



doi:10.3233/978-1-58603-895-3-388

388

flashover electrical field intensity is 3 MV/m.b

E ≈ We decided to test this condition on

some examples of standard two-electrode sphere spark gaps.

Computational Determination of Flashover Voltage for a Two-Electrode Spark

Gap

We selected some standard sizes of sphere spark gaps from [8], modeled them and used

the obtained models in our calculations. The diameters of the selected spheres were

2 cm, 10 cm, 50 cm and 200 cm.

Having in mind the speed of computations, we decided to use triangles of the first

order-degree for the FEM mesh. The mesh contained 20000 to 40000 of elements, de-

pending on the distance between the electrodes. The size of finite elements in the space

between the electrodes is proportional to the size of the electrodes, considering the dis-

tance between them. This approach simplifies the evaluation of the results obtained

with different electrode diameters.

Since in our computation we set the relative voltage of the low voltage (LV) elec-

trode to 0 V and the relative voltage of the high voltage (HV) electrode to 1 V as the

boundary condition on the electrode surfaces, we could use (2) to calculate the flash-

over voltage Ub. The maximum electrical field intensity E

max has been determined from

the FEM calculation as the electrical field intensity in the finite element in which the

calculated electrical field intensity is the greatest. As the flashover electrical field in-

tensity Eb is not precisely defined, we made calculations for three assumed flashover

electrical field intensity values of Eb: 2.8 MV/m, 3 MV/m and 3.2 MV/m.

max

b

b

E

U

E

= (2)

The calculated values were compared with the values defined in the standard for

voltage measurements with sphere spark gaps [9]. In addition, we determined the per-

centage of discrepancy between the flashover voltage Ub and the standard value for the

flashover voltage for electrical field intensity 3 MV/m.b

E =

Figure 1 shows how the flashover voltage depends on the distance between two

electrodes with a diameter of 2 cm. We can see that the calculated values of the flash-

over voltage are up to 50% lower than the values defined by the standard. The discrep-

ancy decreases slightly with the increasing distance between the electrodes. The per-

centage of discrepancy of the calculated values shown in Fig. 1 would not change sub-

stantially even the 3% measuring uncertainty defined in the standard was considered.

In Fig. 2 the same dependence is shown for a sphere spark gap with a diameter of

10 cm. Calculations were made only for some standard distances between the elec-

trodes. The discrepancy between the calculated values and the values defined in the

standard is about 10%. The calculated flashover voltage values are again lower than the

standard values, and the discrepancy again decreases with the increasing distance be-

tween the electrodes.

In Fig. 3 we can see the conditions in case of a sphere spark gap with a diameter of

50 cm. In the part where the accuracy of the standard value is 3%, the calculated values

deviate from the standard values for less than 2%. In the part where the standard val-

M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap 389

ues are defined at a 5% accuracy, the calculated values deviate up to 10%. In the case

of spheres with a diameter of 50 cm, the calculated values exceed the standard values,

and the discrepancy increases with the increasing distance between the electrodes.

Figure 1. Flashover voltage depending on the distance between the electrodes, sphere diameter is 2 cm.



M. Gaber and M. Trlep / FEM Computation of Flashover Condition for a Sphere Spark Gap390

Figure 4 shows the conditions for a sphere diameter of 200 cm. Again the calcu-

lated values exceed the standard values. In this case the discrepancy reaches up to 30%,

and it increases with the increasing distance between the electrodes.

The discrepancy between computational results and the standard values is surpris-

ingly great. One of the observations derived from computational results is that the val-

ues of flashover voltage obtained with smaller spheres are always smaller than the val-

ues defined in the standard. Also, we can observe that the calculated flashover voltages

obtained with large spheres always exceed the standard values. Further observations

reveal that the discrepancy decreases with the increasing distance between small elec-

trodes, but in contrast, it increases with the increasing distance in the case of large

spheres. It should be noted that the number of finite elements varies little in the compu-

tations, so the elements are a hundred times larger if the spark gap is a hundred times

bigger (linear dimension increment). At the same time it holds for the latter that the

size of finite elements grows if the distance between the electrodes increases. Having in

mind these findings, we made a dense-mesh calculation for the sphere with a diameter

of 200 cm, which confirmed our assumption that the obtained flashover voltage would

be lower. These findings lead to the conclusion that there exists an optimal surface

(volume) where the average value of the electrical field intensity and the flashover

electrical field intensity are equal, so the effective flashover can take place. If we look

at the results we can see that the optimal surface is equal to the size of the finite ele-

ment whose greatest electrical field intensity has been reached with a sphere with a

diameter of 50 cm and a distance of 140 mm between the electrodes.

When we are dealing with small spheres and short distances between them the

flashover strongly depends on the presence of free electrons in air. In such cases the

electrical field is non-homogeneous. As a consequence, ionization can take place only

in a small volume (short distances and small diameters) and only a small number of

new free electron generations are available. This explains why the number of all elec-

trons depends so much on the number of the initial free electrons in case of flashover.

Greater ionization volumes yield a greater number of generations, so the number of all

electrons depends less on the number of the initial free electrons. In our calculation the

number of the initial free electrons was not considered, but we assumed equal concen-

tration for all cases.



Flashover Voltage of a Three-Electrode Spark Gap

The same condition as applied for calculating the flashover voltage was used to analyze

the operation of a special three-electrode spark gap (Fig. 5) used in a current impulse

generator [9]. If the trigger electrode is properly placed, this kind of spark gap has a

wider bandwidth (3) than a spark gap with the trigger electrode placed in the main elec-

trode [10]. ΔUW

, UW max

, UW min

, Ub, U

b1 and U

b2 represent the bandwidth, the maximum

working voltage, the minimum working voltage, the flashover voltage between the

main electrodes, the flashover voltage between the upper main electrode and the trigger

electrode, and the flashover voltage between the lower main electrode and the trigger

electrode. It turns out that the widest bandwidth is obtained if Ub1

= 2Ub2

[11], which

actually is the equipotential line.

max min 2

max

W W b b

W

W b

U U U U

U

U U

− −

Δ = = (3)

We took the basic dimensions for the three-electrode spark gap model from [10]

and increased them linearly until the diameter of the main electrode was 10 cm and the

distance between the main electrodes 27.8 mm. Calculations were made for an elec-

trode with a sharp inner edge (Fig. 6, left) and an electrode with a rounded inner edge

(Fig. 6, right). The obtained results were similar, the only difference was observed in

the size of the bore in the electrode: the size of the bore in the electrode with a rounded

edge is smaller (10 to 60 mm) than the bore in the electrode with a sharp edge (40 to

100 mm). Therefore, the conditions for the sharp-edged electrode will not be presented,

except in Fig. 7. The impact of the shape of electrode edges on the conditions in the

spark gap is shown in Fig. 7, where the diameter of the bore in the rounded electrode is

Figure 5. Three-electrode spark gap: to the left the picture of the spark gap (1 – HV main electrode, 2 – LV

main electrode, 3 – trigger electrode), to the right the results of computation.

Figure 6. Shapes of trigger electrode edges.


33.3 mm and the bore in the sharp-edged electrode 88.9 mm. Despite the difference in

the size of bores, a relatively good agreement of curves is observed. It is a consequence

of high electrical field values on sharp edges.

When the electrode with a rounded edge is placed in the middle of the distance be-

tween the main electrodes and the trigger voltage is 50% of the flashover voltage, we

can see that at a certain bore size the flashover voltage stops changing with the distance

(Fig. 8). In this case the flashover voltage is approximately 70 kV. If we look up in

Fig. 2 we can see that the calculated flashover voltage of the spark gap without the

trigger electrode is app. 70 kV. The conclusion is that in this case it is not reasonable to

increase the bore diameter over 45 mm. If we did so, we would get a direct flashover

between the main electrodes. Provided that the amplitude of negative voltage signal is

high enough, the spark gap can nevertheless be used in certain cases, but the shrinkage

of the effective bandwidth makes such use senseless. Figure 9 also shows the condi-

tions in case of a rounded trigger electrode placed in the middle of the distance between

the main electrodes. Here we can see that it is the best to use a trigger electrode with a

bore diameter of about 45 mm because in this case the greatest possible part of flash-

over voltages is covered, which means 32 kV out of 70 kV at a bore diameter of

Figure 7. Flashover voltage depending on the percentage of trigger electrode voltage, for rounded and sharp-

edged electrodes.

Figure 8. Flashover voltage depending on trigger electrode bore diameter for different voltages on trigger

electrode.


44.4 mm. It should be noted that with respect to Fig. 2 the flashover voltages for the

given case are about 10% lower than the expected measured values.

The data presented in Fig. 9 are not suitable for laboratory use, so we recalculated

the percentage record of the trigger electrode voltage into the actual trigger electrode

voltage. The results are presented in Fig. 10.

In all the cases calculated so far the trigger electrode was placed in the middle of

the distance between the main electrodes. Figure 11 presents the results for the case

where the trigger electrode is shifted 4.6 mm downwards. The peaks of curves are at

different values of the trigger electrode voltages, which is explained by the fact that the

position of the trigger electrode changes in relation to the equipotential line while its

bore is increasing.

Conclusion

The condition for flashover in air can be simply acquired with FEM-based electrostatic

calculations where the size of finite elements has to be adjusted to the size of the spark

Figure 9. Flashover voltage depending on percentage of flashover voltage on trigger electrode for different

trigger electrode bore diameters.

Figure 10. Flashover voltage depending on the actual trigger electrode voltage for different trigger electrode

bore diameters.


gap. If the size of finite elements in the critical region is appropriate, the results can be

corrected by comparison with the standard. In addition to the presence of the initial free

electrons, the flashover depends on the pressure, humidity and temperature of the gas,

which is included in the standard in the calculation of correction factors. The presented

results will be confirmed by measurements.

References

[1] M. J. Druyvesteyn, F. M. Penning, “The Mechanism of electrical discharges in gases of low pressure,”

Reviews of modern physics, Vol. 12, No. 2, pp. 104-120, April 1940.

[2] A. Pedersen, “Criteria for spark breakdown in sulfur hexafluoride”, IEEE Transactions on power appa-

ratus and systems, Vol. PAS-89, No. 8, Nov./Dec. 1970.

[3] J. M. Meek, “A theory of spark discharge,” Physical review, Volume 57, April 1940.

[4] A. Fiala, L. C. Pitchford, J. P. Boeuf, “Two-dimensional, hybrid model of low-pressure glow dis-

charges,” Physical review E Vol. 49, No. 6, June 1994.

[5] J. P. Boeuf, “Characteristics of dusty nonthermal plasma from a particle-in-cell Monte Carlo simula-

tion,” Physical review A, Vol. 46, No. 12, Dec. 1992.

[6] Yu. V. Serdyuk, A. Larsson, S. M. Gubanski, M. Akyuz, “The propagation of positive streamers in a

weak and uniform background electric field,” J. Phys. D: Appl. Phys.3,4 pp. 614-623, 2001.

[7] M. Akyuz, A. Larrson, V. Cooray, G. Strandberg, “3D simulations of streamer branching in air,” Jour-

nal of electrostatics, No. 59, pp. 115-141, 2003.

[8] IEC 60052, “Voltage measurement by means of standard air gaps,” Geneva, november 2002.

[9] M. Gaber, J. Pihler, “Design of current impulse generator,” Electrotechnical Review, Vol. 73(1),

pp. 53-58, 2006.

[10] P. Osmokrović, N. Arsić, Z. Lazarević, D. Kušić, “Numerical and experimental design of three-

electrode spark gap for synthetic test circuit,” IEEE Transactions on Power Delivery, Vol. 9, No. 3,

July 1994, pp. 1444-1450.

[11] J. Dams, K. F. Geibig, P. Osmokrović, A. J. Schwab, “Computer aided design of three-electrode spark

gaps,” 4th

ISH, Athens 1983.

Figure 11. Flashover voltage depending on the percentage of trigger electrode voltage for different diameters

of trigger electrode.


Recent Developments in Magnetic Sensing

Barbaros YAMAN

a

, Sadık SEHIT

b

and Ozge SAHIN

c

Dokuz Eylul University, Department of Electrical and Electronics Eng.

35160 Buca, Izmir-Turkey

a

[email protected], b

[email protected], c

[email protected]

Abstract. Measurement systems must have some properties such as sensitivity,

accuracy, stability and low power consumption. New sensor types are continu-

ously being developed according to new needs of applications besides improving

measurement systems. In this study, recently developed magnetic sensors are

searched. Fundamentals of new magnetic sensor types, their advantages and appli-

cation areas are presented and compared in a table to form a guide for magnetic

sensor users.

Introduction

Magnetic sensors sense magnetic field and generate electrical output according to its

input. Basically, sensing capability and application areas of the sensor is used to clas-

sify the sensors. In industry, generally magnetic sensors are used in control and meas-

urement systems, especially current sensing and position sensing. Indeed these sensors

can not be used for special applications because of their sensitivity, size, power con-

sumption and stability. At recent years, with development of the silicon technology,

new sensor types are developed to be used at scientific researches and contribute to the

new developed technologies.

SQUID

Superconductor materials have a property as resistance decreases when temperature

became lower. SQUID (Superconducting Quantum Interferometer Devices) is the most

sensitive type of the magnetic sensors. SQUID magnetometers have an ultimate com-

bination of field and spatial resolution [1]. Unsurpassed magnetic field sensitivity and

wide bandwidth of the SQUID sensors allows creating a large variety of measurement

systems with unique resolution for different applications [2]. A SQUID can be realized

as:

• It can measure magnetic flux on the order of one flux quantum. A flux quan-

tum can be visualized as the magnetic flux of the Earth’s magnetic field

(0.5 Gauss = 0.5 × 10

–4

Tesla) through a single human red blood cell (diame-

ter about 7 microns).

• It can measure extremely tiny magnetic fields. The energy associated with the

smallest detectable change in a second, about 10

–32

Joules, is about equivalent

to the work required to raise a single electron 1 millimeter in the Earth’ gravi-

tational field.



doi:10.3233/978-1-58603-895-3-396

396

A SQUID is essentially a magnetic flux to voltage transducer (Fig. 1). SQUIDs are

amazingly versatile, being able to measure any physical quantity that can be converted

to flux. DC SQUIDs are structures involving two Josephson junctions connected by a

low inductance superconducting loop. The basic phenomena governing the operation of

the SQUID devices are flux quantization in a superconducting loop and the Josephson

Effect. SQUID sensors have been used in various applications like SQUID magne-

tometer for physical property measurements of small samples, non-destructive evalua-

tion (NDE) of sub-surface defects, bio-magnetic evaluation of human heart and brain,

geomagnetic prospecting, detection of gravity waves etc. [1,2].

High-Resolution, Chip-Size Magnetic Sensor Arrays

Arrays of very small sensing device in a single chip are not new phenomena. This

technology was used at the late of 70’s in the auto-focus mechanism of the 35mm cam-

eras. Photodiodes were used at these systems. But in deed used in magnetic sensors in

single chip array is a new phenomena. These nano-technology arrays of micron-sized

magnetic sensors and sensor spacing on a single chip can be used to detect very small

magnetic fields with very high spatial resolution [2]. The older magnetic sensor tech-

nologies as Hall-effect and Anisotropic Magnetoresisitive (AMR) were not able to be

applied in these applications either due to size, high power consumption or low sensi-

tivity issues. Giant magneto resistive (GMR) and Spin-Dependent Tunneling (SDT)

makes it possible the production of these devices [3]. This kind of the devices are used

at the credit cards, magnetic biosensors, magnetic imaging and application which are

needed measurement of very small magnetic field and to detect the change in magnetic

field. This kind of integrating technique reduces the effect of the noise and simplifies

the sensor/signal processing interface [3].

The development of Giant Magnetoresistive (GMR) and Spin Dependent Tunnel-

ing (SDT) materials has opened up a new era of miniature, solid-state, magnetic sen-

sors. These deposited, multi-layer materials exhibit large changes in resistance in the

presence of a magnetic field. Their thin film nature allows the fabrication of extremely

small sensors using traditional photolithography techniques from the semiconductor

industry. Since these sensitive films can be deposited on semiconductor wafers, inte-

grated sensors can be manufactured that incorporate both sensing elements and signal

Figure 1. Schematic of SQUID.

B. Yaman et al. / Recent Developments in Magnetic Sensing 397

processing electronics same chip. On-chip integration is especially important for sin-

gle-chip sensor arrays. A single-chip, sensor array with a large number of sensors re-

quires a correspondingly large number of off-chip connections unless on-chip multi-

plexing is utilized. The area of the bonding pads can dominate the chip area unless on-

chip electronics are used [2,3].

MEMS Magnetometers

There is a military requirement to measure the magnetic bearing accurately to detect

targets and navigational needs.

MEMS-based magnetometers are based on the Lorentz force to cause movement of

a resonant MEMS device. The change in amplitude of the resonance caused by the

Lorentz force is sensed capacitively and related to the magnetic force which caused it

[5]. The famous Lorentz equation

F = q E + q v x B (1)

defines the force on a charged particle (charge q) moving at a velocity v through a re-

gion where there is an electric field (E) and magnetic field (B) present. The magnetic

force is perpendicular to both the local magnetic field and the particle’s direction of

motion. No magnetic force is exerted on a stationary charged particle. The flow of an

electric current down a conducting wire is ultimately due to the movement of electri-

cally charged particles (in most cases, electrons) along the wire. Thus if we make a

MEMS structure and pass a current down a conducting beam, in the presence of a

magnetic field the Lorentz force will cause the free structure to move in a perpendicu-

lar direction [5].

This basic principle has been be applied to a magnetometer designed using MEMS

techniques with appropriate electronic control. It consists of a shuttle mass suspended

by two straight suspension arms, with differential capacitive pick-off to detect the posi-

tion, and electrostatic actuators to force the device into resonance (Fig. 2). There is a

positive feedback loop between the pickoff and electrostatic actuator. To measure the

Figure 2. Schematic of MEMS resonant magnetometer.

B. Yaman et al. / Recent Developments in Magnetic Sensing398

magnetic field an alternating current is passed down the suspension at the same fre-

quency as the resonance. In a closed loop mode the electrostatic force could be adjusted

to null the magnetic force. For the device to operate efficiently it has to be packaged in

a vacuum to achieve a high Q [5,6].

MEMS technology can improve magnetic sensors by minimizing the effect of

noise.

The most successful MEMS products exploit one or more of the following charac-

teristics:

− advantageous scaling properties for improved device or system performance,

− batch fabrication to reduce size and hopefully cost [5],

− circuit integration to improve performance [2].

Spring Specialized LVDT Sensor

An LVDT is a device that produces an electrical output proportional to the displace-

ment of a separate movable core. The spring specialized LVDT measures the variations

of the magnetic field to determine the change of spring position. In this device, there is

no contact between movable coil and coil structure. This gives the LVDT an infinite

life. The infinite mechanical life is also important in high-reliability mechanisms and

systems found in aircraft, missiles, space vehicles and critical industrial equipment

[8,9]. It consists of a primary coil and two secondary coils symmetrically spaced on a

cylindrical form [8].

A free-moving, rodshaped magnetic core inside the coil assembly provides a path

for the magnetic flux linking the coils (Fig. 3). When the primary coil is energized by

an external AC source, voltages are induced in the two secondary coils. These are con-

nected series opposing so the two voltages are of opposite polarity. Therefore, the net

output of the transducer is the difference between these voltages, which is zero when

the core is at the center or null position [9]. When the core is moved from the null posi-

tion, the induced voltage in the coil toward which the core is moved increases, while

the induced voltage in the opposite coil decreases. This action produces a differential

voltage output that varies linearly with changes in core position. The phase of this out-

put voltage changes abruptly by 180° as the core is moved from one side of null to the

other [9].

Figure 3. LVDT structure.


PLCD (Permanent Magnet Linear Contact-Less Displacement)

PLCD-Displacement Sensors (Permanentmagnetic Linear Contactless Displacement

Sensors) basically consist of a special soft magnetic core surrounded by a coil, wound

around its entire length. On each end of the core there is a second short coil [10].

A permanent magnet guided close to the sensor causes localized magnetic satura-

tion and thereby a virtual division of the core. The position of the saturated area along

the sensor axis can be determinated by the coil system. The sensor is supplied by an

external electronic module or by an integrated circuit which also produce the output

signal. The output signal is linearly dependent on the position of the magnet. The signal

can be conditioned by a Standard electronic module, which gives either a current out-

put (4–20 mA) or a voltage output (0–5 V or 0–10 V) [10].

The most significant property of Displacement Sensors PLCD is their operation

without any mechanical connection between driving magnet and sensor element. This

property enables the driving of the sensor through walls made of nonmagnetic materi-

als. The applications, for which the advantages of this quality are very significant, is

the indication of the position of a piston in hydraulik or pneumatic systems. The per-

manent magnet is simply fixed on the piston and drives the PLCD sensor through the

cylinder from e.g. aluminum or brass [8].

PLCD sensors have the following properties:

• Continuous, contact less, linear displacement measurement,

• No mechanical connection between driving magnet and sensor,

• Magnetic operation adaptable to individual applications,

• Control through partitions of non-ferromagnetic materials,

• Large mounting tolerances [8].

Fiber Optic Magnetometers

Some of the principal reasons for the popularity of optical fiber based sensor systems

are [2] small size, light weight, immunity to electromagnetic interference (EMI), pas-

sive (all dielectric) composition, high temperature performance, large bandwidth,

higher sensitivity as compared to existing techniques, and multiplexing capabilities.

Moreover, the widespread use of optical fiber communication devices in the telecom-

munication industry has resulted in a substantial reduction in optical fiber sensor cost

As a result, optical fiber sensors have been developed for a variety of applications in

industry, medicine, defense and research. Some of these applications include gyro-

scopes for automotive navigation systems strain sensors for smart structures and for the

measurement of various physical and electrical parameters like temperature, pressure,

liquid level, acceleration, voltage and current in process control applications. Small,

fiber optic-based magnetic field sensors could be used for many applications which are

inaccessible to electrical based sensors. A variety of optic fiber based magnetic field

sensors have already been developed. But these, in general, are relatively large or have

low sensitivity when compared to small electrical based semiconductor Hall effect sen-

sors or magnetoresistive sensors commonly in use.


Conclusion

Magnetic field measurement has become an essential procedure in many industrial,

scientific and defense projects today. Weather satellites routinely map the earth’s mag-

netic field for scientific information. Airplanes and ships use the earth’s magnetic field

to compute direction as well as altitude, in case of airplanes. Measuring magnetic field

is one of the primary methods of estimating high voltages and currents in industrial

environments. Magnetic field detection is of much interest to defense scientists, par-

ticularly in the area of minefield and submarine detection. All the above reasons have

made the development of reliable, rugged, extremely sensitive magnetic field sensors,

highly essential.

This article includes information on recently developed magnetic sensor types.

One of them is SQUID, the most sensitive magnetic sensor, is used in very low mag-

netic field detection. Another type is GMR, development version of the AMR. GMR

produced by using silicon technology reduces size. Other types PLCD and LVDT re-

move the effect of the contact and reduce the mechanical problems and effective using

range. MEMS has advantages as scaling properties for improved device or system per-

formance; batch fabrication to reduce size and hopefully cost. MEMS were developed

in 2001. These sensor types are compared in Table 1.

Table 1. Development and applications of magnetic sensors

Magnetic Sensor Development Year Application

SQUID 1970– Laboratory instrumentation, biomedical applications (mag-

netoencephalography, magnetocardiology, etc), military

applications (for submarine and mine detection), in geo-

physics (from prospecting for oil and minerals to earth-

quake prediction) and in non-destructive evaluations. [2]

GMR 1988 Imaging of magnetic media

Magnetic bioassay

Non destructive testing

Magnetic couplers

Hard Disk Heads

LVDT-PLCD 1990– A spring specialized LVDT sensor is used at [10].

Automotive:

Shock Absorbers

Suspension

Brake systems

Industrial automation

Professional balances

Dynamometers

Load cells

Vibrations monitoring

Transportation and aerospace [8]

PLCD is used for indication of the position of a piston in

hydraulic or pneumatic systems.

FIBER OPTIC 1991 Measuring low level magnetic fields and strain produced in

a magnetostrictive material in the presence of an external

magnetic field.

MEMS 2001 MEMS technologies include many microsensors (e.g.,

inertial sensors, pressure sensors, magnetometers, chemical

sensors, etc.), microactuators (e.g., micromirrors, microre-

lays, microvalves, micropumps, etc.), and microsystems

(e.g., chemical analysis systems, sensorfeedback- con-

trolled actuators, etc.) [2]


These sensors are generally used in specific systems. In future they may be of

more common use. But development of materials technology is important parameter

for development of these sensors since materials restrict the performance of devices.

References

[1] SQUIDs–Highly sensitive magnetic sensors, Materials Science Division, Indira Gandhi Centre for

Atomic Research.

[2] James Lenz and Alan S. Edelstein, Magnetic Sensors and Their Applications, IEEE Sensors Journal,

Vol. 6, N. 3, June 2006.

[3] Dr. Carl H. Smith and Robert W. Schneider, Chip-Size Magnetic Sensor Arrays, NVE Corporation,

Eden Prairie, MN.

[4] J. Moreland, J. Kitching, P.D.D. Schwindt, S. Knappe, L. Liew, V. Shah, V. Gerginov,Y.-J. Wang and

L. Hollberg, Chip Scale Atomic Magnetometers, Time and Frequency Division National Institute of

Standards and Technology Boulder, Colorado.

[5] D.O. King, K.M. Brunson, A.L. McClelland, R.J.T. Bunyan, MEMS Magnetometers and Gradiometers

for Magnetic Compasses with Bearing Correction, IEE seminar on MEMS Sensor Technologies, April

2005.

[6] Jack W. Judy and Nosang Myung, Magnetic Materials for MEMS, University of California, Los Ange-

les, CA, USA.

[7] http://www.hitachigst.com/hdd/research/recording_head/headprocessing/index.html.

[8] Derek Weber & Enrico Giorgione, New Applications for Magnetic Based Sensors, Plcdand Lvdt Prin-

ciples and Applications, Inprox Sensors.

[9] Schaevitz, LVDT Technology, LVDT Functional Advantages and Operation Principles. http://

www.schaevitz.com/.

[10] PLCD Displacement Sensor for Industrial Applications Tyco Electronics. http://www.

tycoelectronics.com.


Modelling of Open Magnetic Shields’

Operation to Limit Magnetic Field of

High-Current Lines

R. GOLEMAN, A. WAC-WŁODARCZYK, T. GIŻEWSKI and

D. CZERWIŃSKI

Institute of Electrical Engineering and Electrotechnologies, Faculty of Electrical

Engineering and Computer Science, Lublin University of Technology, 20-618 Lublin,

38A Nadbystrzycka St.,Poland


Abstract. The paper presents the analysis of magnetic field around high-current

line (50 Hz) and the influence of the shields on spatial distribution of the field. The

comparison of the systems with open ferromagnetic single shield for each line and

the systems with common shield has been made. Open shields are characterised by

the air gap located above the conductor or current line. Ferromagnetic shields are

made of transformer plate M117-30P 5. The calculations have been made for the

shields of defined angles i.e. 90°, 120° and 150° and various thickness.

Introduction

In our surroundings human beings have been influenced by various kinds of elec-

tric and magnetic fields of different frequencies. Such fields at controlled values of the

intensity, frequency and time of treatment may have positive impact on human organ-

isms or they may be used in diagnostics. Low-magnetic fields are neutral for people but

in most cases high-magnetic fields are dangerous. Apart from biological influence on

living organisms, artificially generated magnetic and electric fields have hazardous

impact on a wide range of sensible technical devices that cause disturbances or even

make their operation impossible. The protection against the influence of electromag-

netic field is difficult in particular in case of lower intensity and low frequency level.

Magnetic field generated by transformers and high-current lines rarely exceed the

safety level however it may disturb the operation of other technical devices e.g. it may

interact on the deflection of the monitors situated in the areas of considerable concen-

tration of field intensity. Such interaction is characterised by lines distortion and vibra-

tion of the image displayed by the monitor which make the operation difficult. Shield-

ing buses are usually applied to reduce the influence of the magnetic field generated by

currents in the buses on the surroundings. The paper presents the analysis of magnetic

field around high-current line and the influence of the shield on spatial field distribu-

tion [1–5]. Open ferromagnetic shield systems [5], individual for each bus (Fig. 1) and

the systems with one common shield for all buses (Fig. 2) were compared. Open shield

are characterised by the air gap above the conductor or current line. Ferromagnetic

shields are formed by M117-30P5 transformer sheets. The calculations were carried out

for the shields of characteristic angles like 90°, 120°, 150° and different thickness.


403

Bus Shielding System with Open Magnetic Shield

The system of current buses made of aluminium was selected for the analysis (Fig. 3).

The assumption was made that the current intensity in the bus is 3 kA.

Vector potential in the system with the shield can be described by the following

equations:

in current bus

JA =⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

rot

1

rot

0μ

(1)

in air

0rot

1

rot

0

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

A

μ

(2)

Figure 1. Section of tested bus-shield system, shielding of particular phases.

Figure 2. Section of tested bus-shield system, common bus shielding for three phases.

Figure 3. Schematic diagram of analysed system with magnetic shield: 1–current buses, 2–shield.

R. Goleman et al. / Modelling of Open Magnetic Shields’ Operation404

in magnetic shield

0rot

1

rot

0

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

A

r

μμ

(3)

where: A – vector potential, J – current density vector, r

μμ ,

0– magnetic permeability

of vacuum and relative permeability of the shield, γ – conductivity, ω – pulsation.

Current density in conductive medium, in two-dimensional field is described by

the following relation

V

t

gradγγ −

∂

∂

−=

A

J (4)

Assuming that the value of scalar potential V is constant in cross-section of the

conductor and taking the relation (4) into consideration as well as the following relation

VU grad⋅−= l (5)

voltage between conductor’s ends is

s

A

d∫∂

∂

+=

s

t

RiRU γ (6)

where: R – conductor’s resistance determined at direct current, l – conductor’s length,

s – conductor’s section.

According to II Kirchhoff’s law for current circuits the following equation can be

written

L3L1

L1 L3 L obc L1 L3 L( )( ) ( d d )

s s

U U R R i i R γ

t t

∂∂− = + − + −

∂ ∂∫ ∫

AA

s s (7)

L3L2

L2 L3 L obc L2 L3 L( )( ) ( d d )

s s

U U R R i i R γ

t t

∂∂− = + − + −

∂ ∂∫ ∫

AA

s s (8)

Currents in buses fulfil the equation

03LL2L1=++ iii (9)

At the assumption that current line and the shields are long enough, 2D analysis

would be sufficient. The equations (1–9) were numerically solved with the application

of FLUX 2D package. Cyclic boundary conditions were taken for the calculation of

triple-phase model of shielded buses. They enable to calculate vector potential beyond

the region limited by boundary curve taking into account its decay into infinity. Com-

putation results have been presented as maps of magnetic flux density (Figs 4, 5) and

charts of maximum values of magnetic flux density (Figs 6–10).

R. Goleman et al. / Modelling of Open Magnetic Shields’ Operation 405

Figure 4. Distribution of magnetic flux density around buses with single magnetic shields for ω t = 90°.

Figure 5. Distribution of magnetic flux density around buses with common magnetic shield for ω t = 90°.

0 1000 2000 3000

x, mm

0

0.001

0.002

0.003

0.004

0.005

0.006

B, T

alpha=90

alpha=120

alpha=150

Figure 6. Distributions of maximum value of magnetic flux density in plane A (100 mm) of current line with

single magnetic shields of different values of the angle α = 90°, α = 120°, α = 150° and thickness of the wall:

5 mm.


The charts of magnetic flux density were determined in two planes located under

the shield marked as A in Figs 4 and 5 distant from conductors’ axis of 100 mm and

1000 mm respectively.

Conclusions

In the paper the influence of open magnetic shields on magnetic field distributions

around three phase buses was investigated. Open magnetic shields can be used to limit

0 1000 2000 3000

x, mm

0

0.0001

0.0002

0.0003

0.0004

B, T

alpha=90

alpha=120

alpha=150

Figure 7. Distributions of maximum values of magnetic flux density in plane A (1000 mm) of current line

with single magnetic shields of different values of the angle α = 90°, α = 120°, α = 150° and thickness of the

wall: 5 mm.

0 1000 2000 3000

x, mm

0

0.001

0.002

0.003

0.004

B, T

alpha=90

alpha=120

alpha=150

Figure 8. Distributions of maximum values of magnetic flux density in plane A (100 mm) of current line

with common magnetic shield of different values of the angle α = 90°, α = 120°, α = 150° and thickness of

the wall: 5 mm.


magnetic field only in this part which is directly in front of the shield. In such systems

the intensity of the field in the area above the shield increases and the highest values

occur at the edges of the shield.

The distribution of magnetic flux density under the shield depends on shield type

and the distance from its surface. In the system with separate shields for each bus in the

protected zone close to the surface of the shield the highest value of magnetic flux den-

sity was noticed under central bus however under remaining buses the values are lower

of about 25%. Negligible increase of magnetic flux density appears also under shields’

edges of extreme buses.

0 1000 2000 3000

x, mm

0

0.002

0.004

0.006

0.008

0.01

B, T

without shield

single shields

common shield

Figure 9. The comparison of the distributions of maximum values of magnetic flux density in plane A

(100 mm) in the systems: non-shielded with individual shields and with common shield (α = 120°, thickness

of shields’ wall: 5 mm).

0 1000 2000 3000

x, mm

0

0.0001

0.0002

0.0003

0.0004

0.0005

B, T

without shield

single shields

common shield

Figure 10. The comparison of the distributions of maximum values of magnetic flux density in plane A

(1000 mm) in the systems: non-shielded with individual magnetic shields and common shield (α = 120°,

thickness of shields’ wall: 5 mm).


In the system with common shield, in the area close to the surface of the shield the

highest value of magnetic flux density was noticed under shield’s edges whereas under

central bus the system is well shielded. Along with the increase of the distance from

conductor’s axis the distribution of magnetic flux density under the shields tends to be

more steady, the value of magnetic flux density increases and shielding efficiency is

lower. The change in the length of shield’s gap which corresponds to the increase of

the shield’s angle from 90° to 150°, has reasonable impact on shielding efficiency.

In case of individual shield it results in increased shielding efficiency of about 8%

at the points located under the shield in the symmetry axis. This influence is noticeable

in case of common shield. Assumed change of shield’s angle corresponds to the in-

crease of the shielding efficiency of about 30% at the plane points mentioned above.

The change in magnetic shield’s thickness in the range from 5 to 10 mm does not

influence magnetic flux density distributions and shielding efficiency since in consid-

ered models the shields were not subject to saturation. The simulations revealed that

the system with common magnetic shield has better shielding efficiency. It’s shielding

coefficient at the points located in symmetry plane of the system and located

25–925 mm from the shield varies from 0,12 to 0,65 and is lower of about 0,2–0,3

compared to the system with individual shields.

References

[1] K. Bednarek, R. Nawrowski, A. Tomczewski, Electromagnetic compatibility in the neighbourhood of

high-current lines, X International Symposium on Electromagnetic Fields in Electrical Engineering, ISEF

2001, pp. 423-428.

[2] O. Bottauscio, M. Chiampi, D. Chiarabaglio, M. Zucca, Use of grain-oriented materials in low-frequency

magnetic shielding, Journal of Magnetism and Magnetic Materials, Vol. 215-216, pp. 130-132, 2000.

[3] R. Goleman, J. Szponder, Reduction of low frequency weak magnetic fields using laminar screens,

X International Symposium on Electromagnetic Fields in Electrical Engineering, pp. 267-270, 2001.

[4] T. Rikitake, Magnetic and Electromagnetic Shielding, Terra Scientific Publishing Company, Tokyo

1987.

[5] K. Wassef, V.V.Varadan, K.K.Varadan, Magnetic field shielding concepts for power transmission lines,

IEEE Trans. Magnetics, Vol. 34, No. 3, pp. 649-654, 1998.


Selected Problems of the Flux Pinning in

HTc Superconductors

J. SOSNOWSKI

Electrotechnical Institute, 04-703 Warsaw, Pozaryskiego 28, Poland

Abstract. The applications of the HTc superconductors in electricity depends

among other on the current capability of the superconducting materials, especially

in the form of wires and tapes. In the paper is investigated the influence of the

electromagnetic interaction of the vortices with the nano-sized pinning centers on

the critical current of HTc superconductors. The capturing interaction is consid-

ered taking into account the change in the free energy density arising during the

motion of pinned vortex against it’s equilibrium position. The influence of the

pinning interaction on magnetic induction distribution and flux trapped is consid-

ered too for ceramic HTc superconductors.

Introduction

High temperature oxide superconductors (HTs) were discovered more than twenty

years ago. Now it is time therefore for numerous applications of these materials in elec-

tric devices. Unique electromagnetic phenomena appearing in them should be taken

into account before constructing the superconducting device. In the paper is investi-

gated the influence of the nano-sized pinning centers on the critical current of the HTc

superconductors. It is considered too related mechanism of the flux trapping in these

materials, which giant value allows to treat HTc superconductors as promising perma-

nent magnets very useful in magnetic levitation process, as magnetic bearing, motors

etc. We mention that word record of remanent magnetic moment reached just for HTc

superconducting YBaCuO macromolecule attains the giant value of 17 T, even in not

so low temperature of 29 K. This temperature is presently frequently obtained both

using cryogenic liquids – hydrogen or without any liquids with help of cryocoolers.

Critical Current Analysis

As follows from theoretical analysis especially based on the Ginzburg-Landau theory

[1] superconducting state is characterized by the order parameter, which means that this

state is energetically more favor than normal one. It denotes further that normal phase

enhances the energy of system and therefore should be minimized. It is in fact the base

of the proposed pinning interaction model. The movement of the captured on nano-

sized pinning center vortex with normal core of the radius equal to the temperature

dependent coherence length ξ(T), at zero temperature ξ0, enhances the normal state

energy of the superconductor. From the other side Lorentz force acting on the pinned

vortex during current flow and elasticity forces tear off the vortices from the pinning

centers, which leads then to their movement and dissipation effects. Critical current



doi:10.3233/978-1-58603-895-3-410

410

density is reached, while the potential barrier height against capturing the vortices van-

ishes.

During the movement of the vortex on the distance x against the pinning center of

the size d, as it is shown in the Fig. 1, the normal part energy of the superconductor

with captured vortex increases initially according to the relation 1 valid for x < xc:

( )

22

2 2

2

arcsin 1

2 2 2 2

o cH l d d d

U x dx

μ ξ

πξ ξ

ξ ξ

⎡ ⎤

⎛ ⎞

⎢ ⎥= + − − −

⎜ ⎟⎢ ⎥⎝ ⎠

⎣ ⎦

(1)

while for larger declinations, it is for x > xc according to the following expression:

( )

2

2 2

3

arcsin 1

2 2

o cH l x x x

U x

μ ξ π

ξ ξ ξ

⎡ ⎤

⎛ ⎞

⎢ ⎥= + + −

⎜ ⎟⎢ ⎥⎝ ⎠

⎣ ⎦

(2)

Critical value of the declination parameter xc is defined here by the formula:

2

1 ( )

2

c

d

x ξ

ξ

= − (3)

The pinning potential barrier which should be passed by the vortex during its

movement on the distance x for both cases of x smaller or larger than xc is given now

respectively by the expressions:

( )

2

2

2

o c

H ldx

U x

μ

Δ = (4)

Figure 1. Scheme of the vortex interaction at T = 0 with the nano-sized pinning center.

J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors 411

( )

2 22 2

3

arcsin arcsin 1 1

2 2 2 2 2

o c

H l x d x x d d

U x

⎡ ⎤⎛ ⎞ ⎛ ⎞

⎢ ⎥Δ = − + + − + −⎜ ⎟ ⎜ ⎟

⎢ ⎥⎝ ⎠ ⎝ ⎠

⎣ ⎦

μ ξ π

ξ ξ ξ ξ ξ ξ

(5)

For determination of the total energy balance the potential of the Lorentz force as

well as the elasticity energy of the vortex lattice should be taken into account. Captur-

ing of the vortices by the nano-sized pinning centers causes the shift of the vortex from

it’s equilibrium position in the regular vortex array, thus leading to an increase in the

elasticity energy of the magnetic structure of the vortex lattice. This effect is the func-

tion of the declination of the individual vortex from it’s equilibrium position in the lat-

tice. We have taken it into account by assuming that the enhancement of the vortex

elasticity energy is proportional then to the square of the length of the vortex deflection

from the equilibrium site., with coefficient of proportionality expressed by the value of

the parameter α

( )2

2

2

2

( )s

el

a

c x

U x

l

−

= = −

πξ ξ

α ξ (6)

Parameter cs is the corresponding elasticity shear modulus, and l

a≈ l denotes the

length on which the magnetic flux lines of vortices are distorted. This model leads

therefore finally to the relation describing the potential barrier for flux creep process

ΔU in the function of reduced current density I = j/jC, where j

C is defined as transport

current density j for which potential barrier disappears.

( ) ( )2 2

2 2 2

1 1 2

2

o c

H l

U i z i i

μ ξαξΔ = + − − − (7)

Parameter z appearing in Eq. (7) is defined according to the relation:

2

2

arcsin 1 1 arcsin

2 2 2

d d d

z i i i

ξ ξ ξ

⎛ ⎞⎟⎜ ⎟= + − − − −⎜ ⎟⎜ ⎟⎜⎝ ⎠ (8)

Hc is critical magnetic thermodynamic field, l pinning center thickness. Parameter

α describes the elasticity energy of the vortex lattice, as was stated previously. Let’s

insert now the expressions 7–8 into the constitutive relation describing generated elec-

tric field in the flux creep process just in the function of the potential barrier height ΔU,

what allows us to predict the form of the current -voltage characteristics:

0

exp 1 exp

B C B

U j U

E B a

k T j k T

ω

⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞Δ Δ⎟ ⎟⎜ ⎜⎢ ⎥⎢ ⎥⎟ ⎟=− − + − −⎜ ⎜⎟ ⎟⎢ ⎥⎢ ⎥⎜ ⎜ ⎟⎟⎟ ⎟⎜⎜ ⎝ ⎠⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

(9)

The dependence of the real critical current i.e. filling the electric field criterion on

material’s parameters is determined in this way. An example of the results of the

J. Sosnowski / Selected Problems of the Flux Pinning in HTc Superconductors412

calculations the influence of the pinning centers dimensions on the critical current in

arbitrary units shown in Fig. 2 indicates that for too small pinning centers the capturing

of vortices is not effective, while for larger dimensions critical current related to indi-

vidual vortex – pinning center interaction saturates. Analogous calculations have been

performed concerning the influence of the elasticity parameter of the vortex lattice α

on the critical current, while results shown in the Fig. 3 also indicate on the importance

of this effect for the critical current density, what should be interesting for supercon-

ducting tape technology. Numerical data presented in the Fig. 3 suggest that rigid vor-

tex lattice does not allow for movement of the individual vortex, which stay in their

equilibrium position in vortex lattice and therefore flux pinning and critical current

density appropriately decrease for such case in an approximation of low concentration

nano-sized pinning centers.

Flux Trapping Analysis

Described previously pinning interaction influences too the magnetic induction distri-

bution and then the flux trapped magnitude. In the case of the free movement of the

vortices the magnetization is reversible and therefore none remanent moment arises.

0

2

4

6

8

0 0.5 1 1.5 2 2.5 3d/ξ

Jc(a.u)

B=1 T2T

5T

10T

T=10 K

Figure 2. Influence of the nano-sized pinning centers dimensions d/ξ in reduced units on the critical current

versus applied magnetic field.

0

5

10

15

20

25

0 5 10 15 20

α (105

J/m2

)

Jc(a.u)

B=1T

B=5T

B=10T

Figure 3. Influence of the vortex lattice elasticity parameter α on the critical current density versus applied

magnetic induction.


However for HTc superconductors anomalous giant value of the flux trapped is ob-

served, which allows to classify these materials as potentially very promising perma-

nent magnets. Analysis of this problem is in this chapter performed. Set of Equations

has been obtained, describing the dependence of the flux trapped Ftr on the amplitude

of external magnetic induction in the cycle 0 → Bm

→ 0, taking into account the ce-

ramic structure of HTc superconductors. Schematic view of magnetic induction profile

in this case is shown in Fig. 4. Tooth-like shape of magnetic induction follows just

from the approximation that in ceramic materials the superconducting grains are im-

mersed in the matrix characterized by worse superconducting properties, which effect

can be expressed too in the language of an existence weak Josephson’s junctions. Such

shape of magnetic induction profiles reflects then appearance of large intra-granular

currents and weak Josephson’s currents flowing between grains.

As it was mentioned in the introduction the flux trapping value of the high tem-

perature superconductors has essential meaning from the technical point of view. In the

present chapter it will be performed an analysis of this material parameter by consider-

ing the nature of the generation flux trapped, taking into account especial features of

the HTc superconductors. Performed analysis should be helpful therefore for finding

necessary conditions for optimizing the magnitude of this very important parameter.

In Fig. 4 it is drawn the variation of the magnetic induction profile inside the su-

perconducting ceramics, taking into account the existence of their granular structure,

which leads to different critical current values inside the grains and in surrounding ma-

trix. As it is shown here for increasing magnetic field trapped in grains magnetic flux

reduces total flux trapped value, while in decreasing one adds to the total trapped flux.

Four regions have been considered in dependence on the maximal magnetic induction

amplitude Bm in the magnetic field cycle, according to the profiles presented in the

Fig. 4. First one has the place for maximal external magnetic field smaller than the first

penetration field which is the superposition of the first critical magnetic field Bc1

/μ0 and

magnetic potential barrier Δ. In this region magnetic induction does not penetrate gen-

erally inside superconductor and clearly the trapped magnetic flux Ftr vanishes. For

higher amplitudes of increasing external magnetic field magnetic induction begins to

penetrate the superconducting material and flux trapped per unit volume of the super-

conducting ceramic arises. This region of non-total magnetic induction penetration is

given by the condition:

Figure 4. The magnetic induction profile in the magnetic field cycle 0 → Bm → 0. The influence of the

granular structure of the ceramic superconductors on magnetic induction distribution is shown in this picture.


1

0e

Bγ> > (10)

where it has been introduced definition Be

1

= Be – B

c1 – Δ. Parameter γ, which de-

scribes the value of the full penetration of magnetic induction inside the superconduct-

ing cylindrical sample is equal to:

0 c

j Rγ μ= (11)

Trapped magnetic flux normalized to the sample volume is described by the inte-

gral over the superconducting cylindrical sample from the magnetic flux profile in the

cross section presented in the Fig. 4 and for the magnetic field range determined by

Eq. (10), for time stable magnetic induction distribution is given by Eq. (12). In real

cases magnetic induction profiles according to flux creep process are varying during

the relaxation effects, which also will be in the paper discussed.

⎥

⎦

⎤

⎢

⎣

⎡

+−=sg

ee

tr

nB

BB

F

22

)(1

2

21

γ

γ

(12)

In Eq. (12) it has been taken into account explicitely, according to the previous

considerations existence of the superconducting grains immersed inside the surround-

ing matrix. The grains concentration is described by the parameter n, while parameter

Bsg

is normalized to the grain’s cross-section and describes trapped flux in individual

cylindrical grain. It is determined therefore by the relation.

1

3

g

sg g

B Bc

γ= + (13)

Bc1g

is the first critical field, while parameter γg describing penetration of magnetic

induction into grains, is defined as 0g cg g

j Rγ μ= , where the radius of the supercon-

ducting grain is Rg, while critical current density inside grains j

cg. Relations 12–13 were

obtained in an approximation of the linear magnetic induction profiles, known fre-

quently as Bean’s approach. For larger values of the magnetic induction amplitude, it is

for the range 1

2e

Bγ γ≥ ≥ flux trapped is equal to:

3

1 2 12

1

2

2 ( ) 2 1

1

2 4 3 2 2

sge e

tr e

nB B B

F B

γ γγ

γγ

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎟⎜⎟ ⎟⎜ ⎜ ⎟⎜⎟ ⎟= − − + − − ⎟⎜ ⎜⎜⎟ ⎟ ⎟⎜ ⎜⎜⎟ ⎟⎟ ⎟⎜ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎟⎜⎜ ⎟⎝ ⎠

(14)

In the last case of the maximal magnetic field defined as 1

2e

B γ≥ , flux trapped

reaches constant value

3

tr sg

F nB

γ= + . Above model of flux trapping allows to deter-

mine remanent moment of the high temperature superconductors, very important pa-

rameter for describing the magnetic properties of ceramic superconducting material. An

example of the computer calculations of the influence on flux trapped of filling ceramic


material with superconducting grains is shown in Fig. 5 and indicates on relevance of

this effect.

References

[1] J. Sosnowski, Superconductivity and applications, Warsaw, Poland, Book Publisher of Electrotechnical

Institute, 2003, pp. 1-200, in polish language.

Figure 5. The influence of the filling the HTc superconductor with the superconducting grains (parameter n

value) on the square root of the flux trapped in the magnetic induction cycle 0 → Bm → 0. Numbers at curves

indicate value of the parameter n, defined at Eq. (12).


The Effect of the Direction of Incident

Light on the Frequency Response of p-i-n

Photodiodes

Jorge Manuel Torres PEREIRA

Institute of Telecommunications, Department of Electrical and

Computer Engineering, Instituto Superior Técnico, Technical University of Lisbon,

Av. Rovisco Pais, 1049-001 Lisboa, Portugal


Abstract. This paper investigates the effects of the direction of the incident light

on the transit time limited frequency response of p-i-n photodiodes. The simulation

model starts from dividing the absorption region into any desired number of layers

and, for each layer, the continuity equations are solved assuming that the carriers’

drift velocities are constant. The frequency response of the multilayer structure is

calculated from the response of each layer using matrix algebra. The bandwidth is

seen to increase when the light is incident on the p side. In this case the use of con-

stant saturation velocities underestimates the device’s bandwidth.

Introduction

In optical communication systems photodetectors are key elements on the receiver’s

side. Photodiodes have been widely used because they are easy to fabricate and have

good frequency response. The basic photodiode structure is of the p-i-n type and con-

sists in a double heterojunction, which is known to have some advantages over the cor-

responding homojunction, namely lower dark current and noise. For the long-

wavelength range, i.e. 1.3–1.6 μm, the most important semiconductor materials are InP,

for the n- and p-regions, and one element of the InGaAsP family, for the i-region. The

choice of semiconductor for the i-region takes into account the wavelength of the radia-

tion to be detected and makes sure that the semiconductor is lattice matched to InP.

Both of these conditions are met by the ternary In0.53

Ga0.47

As, which is commonly used

for the absorption region [1].

The optical system’s performance may be determined by the photodetector’s fre-

quency response and therefore it is necessary to have a very detailed knowledge of its

behaviour in the frequency domain. The frequency response of p-i-n photodiodes has

been investigated by several authors using different numerical techniques and assump-

tions [2–5]. Analytical expressions for the frequency response of Si p-i-n photodiodes

have been presented by assuming constant values for the carrier velocities [2]. This

approximation is not valid for InGaAs/InP photodiodes with large absorption region’s

width and/or small reverse bias voltage because the dependence of the carrier’s drift

velocity on the electric field should be taken into account. An analytical solution for the

frequency response of InGaAs p-i-n- detectors, that includes the influence of the elec-

tric field, may be expressed in terms of a frequency response function [3]. Numerical


417

approaches involving a finite element calculation [4], and the Ramo’s theorem [5] have

also been implemented. A simple and powerful numerical technique that looks at the

device as a sequence of spatially uniform layers, each one with analytical solutions, has

been developed to obtain the frequency response of multilayer structures [6]. This

technique has been applied to several types of devices and situations [7,8] and is used

in this paper to investigate the effects of the direction of the incident light on the transit

time limited frequency response of n+

InP/n–

InGaAs/p+

InP photodiodes. The influence

of the absorption layer width and bias voltage on the frequency response is also consid-

ered.

Device Structure and Modelling

The p-i-n structure under study is shown in Fig. 1. The contact layers are of highly

doped InP semiconductor. Between the contacts there is an absorption layer of

In0.53

Ga0.47

As with width a

. The light is assumed to be absorbed only in the InGaAs

layer. Under normal bias conditions the electric field, which extends throughout the

absorption region, is responsible for the charge transport of the optical generated elec-

tron-hole pairs. In general the carriers’ drift velocities depend on the electric field,

which is likely to change spatially in that region. By dividing the absorption region into

a certain number of layers enables us to assign to each layer a constant value for the

electric field. These discrete values of the electric field are then used to obtain the car-

riers’ drift velocities in each layer where they are assumed to be constant. This ap-

proach may be used for any known electric field profile, the accuracy depending on the

number and size of layers.

The numerical model starts by solving, for each layer and direction of light, the

continuity equations which, in this case, have analytical solutions.

In the frequency domain the electron and hole current densities for the i

th

layer,

Jin

(x,ω) and Jip

(x,ω) respectively, obey the following equations [8]:

( , )

( , ) ( , )

v

in

in i

in

d J xi

J x G x

dx

= +

ωω

ω ω (1)

a

n+ n- p+

InP InP

In0.53

Ga0.47

As

x0

Φ

xi-1 x

i

ith

layer

i

Φ'

Figure 1. Schematic of the p-i-n structure used in this work. The figure shows the light incident on the n+

(Φ)

and on the p+

side (Φ').

J.M.T. Pereira / The Effect of the Direction of Incident Light418

( , )

( , ) ( , )

v

ip

ip i

ip

d J xi

J x G x

dx

= − +

ωω

ω ω (2)

where v , vin ip

are the electron and hole drift velocities respectively and ( , )i

G x ω re-

fers to the electron-hole generation rate due to optical absorption in the ith

layer given

by

( )1 1

( , )x

i i iG x q e x x x

−

−

= ≤ ≤

α

ω αφ , (3)

for light incident on the n-side, and given by

( )( )

1 1( , )

ax

i i iG x q e x x x

− −

−= ≤ ≤

α

ω αφ (4)

for light incident on the p-side. The parameter α is the absorption coefficient, q is the

magnitude of the electron charge and 1φ is the amplitude of the sinusoidal input optical

flux component.

The ith

layer may then be represented by a set of linear response coefficients

, ,i i i

T S R

and Di which are used to compute the corresponding response coefficients

for the multilayer structure. The quantities ,i i

T S

are related to the electron and hole

current densities through the equations

1

1

( ) ( )

( ) ( )

ip i ip i

i i

in i in i

J x J x

T S

J x J x

−

−

⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(5)

ipp ipn ip

i i

inp inn in

T T S

T and S

T T S

⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦

(6)

The quantities i

R

and Diare obtained from the equation

1

1

( )

( )

( )

ip iT

i i i

in i

J x

p R D

J x

−

−

⎡ ⎤

= +⎢ ⎥

⎢ ⎥⎣ ⎦

ω (7)

( ) ( , ) ( , )

i

i in ipp J x J x dx⎡ ⎤= +

⎣ ⎦∫

ω ω ω . (8)

The quantity Di is a scalar whereas

T

i ip inR R R⎡ ⎤=

⎣ ⎦

.

The frequency response I(ω) may then be expressed as [6]:

( ) ( ) ( )n n nn a

I D R S T= − ω (9)

J.M.T. Pereira / The Effect of the Direction of Incident Light 419

where D, Rn, S

n, T

nn are the coefficients for the whole structure which may be obtained

following a simple set of rules

1, 1

1, 1 1

1, 1

1, 1 1

i i i i

i i i i i

T T T

i i i i i

T

i i i i i i

T T T

S S T S

R R T R

D R S D D

+ +

+ + +

+ +

+ + +

=

= +

= +

= + +

(10)

where the subscripts (i), (i+1) and (i+1,i) refer to the ith

layer, (i+1)th

layer and the un-

ion of the two layers respectively.

For the ith

layer and light incident on the n-side, , ,

i i iT S R

and Di are given by [8]:

0

0

ip

in

i

i

i

e

T

e

−

⎡ ⎤

⎢ ⎥=

⎢ ⎥⎣ ⎦

ωτ

ωτ

( )

( )

ip

i i

in

f i

R

f i

⎡ ⎤

⎢ ⎥=

⎢ ⎥−⎣ ⎦

ωτ

ωτ

(11)

( )

( )

1

i

ip ix

i i

in i

f i

S q e

f i

αωτ α

αφ

ωτ α

−

⎡ ⎤−

⎢ ⎥=

⎢ ⎥− − −⎣ ⎦

(12)

( ) ( )

( ) ( )

( )2

1

α

α ωτ

αφ

α ωτ

α ωτ

α ωτ

−⎡ − −

= −⎢

+⎣

⎤−

⎥−

− ⎥

⎦

i i

i inx

i i

i in

i ip

i ip

f f i

D q e

i

f f i

i

(13)

with / v , / vip i ip in i in

τ τ= = and θ

(θ) (1 ) / θf e

−

= − . When the light is incident from

the p-side the expressions for andi iT R

do not change but Si

and Di

are given by:

( )

( )

( )

1

a i

ip ix

i i

in i

f i

S q e

f i

− −

⎡ ⎤+

⎢ ⎥=

⎢ ⎥− − +⎣ ⎦

α

ωτ α

αφ

ωτ α

(14)

( ) ( )

( ) ( )

( )2

1

a i i

i inx

i i

i in

i ip

i ip

f f i

D q e e

i

f f i

i

− − −

⎡ − − −

= +⎢

− +⎢⎣

⎤− −

⎥+

+ ⎥

⎦

α α

α ωτ

αφ

α ωτ

α ωτ

α ωτ

(15)

In this work a linear electric field profile is assumed and may be expressed as [9],


2

2

( ) ( )d d

d

aa

U U U

E x x U U

⎛ ⎞−

= + >⎜ ⎟

⎝ ⎠

(16)

with

2

2

d a

d

n

qN

U

ε

=

. U

d is called the punchthrough voltage, U is the reverse bias volt-

age, Nd is the residual donor concentration in the absorption region and ε

n is the In-

GaAs dielectric constant.

The electron and hole drift velocities in the In0.53

Ga0.47

As are calculated, for each

value of the electric field, from two empirical expressions that show very good agree-

ment with experimental results [4]:

( ) ( )

( )

v ( ) v / 1

v ( ) v tanh / v

n n n

p p p p

E E E E

E E

γ γ

μ β β

μ

= + +

=

(17)

withn

μ = 1.05 m2

V–1

s–1

;p

μ = 0.042 m2

V–1

s–1

; vn

= 6 × 104

m/s; vpl

= 4.8 × 104

m/s;

β = 7.4 × 10–15

(m/V)2.5

; γ = 2.5 and the electric field E expressed in (V/m). Figure 2

shows the electron and hole drift velocities as a function of the electric field for

In0.53

Ga0.47

As.

Results and Discussion

In the calculations we used α = 1.15 μm–1

(λ = 1.3 μm) and ε = 14.1ε0 for the ternary

compound [1]. The results were obtained for an absorption region divided into 100

layers. However, due to the electric field profile and the carriers’ drift velocity type of

dependence on the electric field, the results seem to stabilize for a number of layers as

low as 10.

Drift velo

city

(×

105

m/s)

4

2

0

0 5 10

Electric field (MV/m)

electron

hole

In0.53

Ga0.47

As

Figure 2. Electron and hole drift velocities as a function of the electric field for

In0.53

Ga0.47

As.


Figure 3 shows the computed frequency response of two structures of the type de-

scribed before with a

= 1.5 μm and 3 μm, impurity concentration Nd = 10

21

m–3

, and

bias voltage U = 10 V. For both directions of the incident light it is seen that shorter

devices have a better frequency response which is mainly due to a decrease in the carri-

ers’ transit time. However, for each structure, better frequency response is obtained

when the incident light is on the p+

side. This result agrees with other published results

and may be explained in terms of a larger electron-hole optical generation rate closer to

the p+

contact so that it is the electrons which must travel across the absorption region

[2]. As it is seen from Fig. 2 the drift velocity is higher for electrons than for holes.

In Fig. 4, the computed frequency response of a structure with a

= 3 μm,

Nd = 10

21

m–3

, and bias voltage U = 6 V is compared to the results obtained using the

carriers’ saturation velocity values. When using the carriers’ saturation velocity the

device’s bandwidth is seen to be overestimated or underestimated depending whether

the light is incident from the n+

side or p+

side respectively. These results confirm the

Figure 3. Computed frequency response of two structures with a

= 1.5 μm and 3 μm.

Figure 4. Computed frequency response of one structure with a

= 3 μm for U = 6 V and when the satura-

tion velocities are used.


type of dominant carrier in each case and may be explained in terms of the velocity-

field relationship for the electrons and holes. In fact the electron’s average velocity in

the absorption region is higher than its saturation velocity whereas the hole’s average

velocity is lower than its saturation velocity.

Figure 5 shows the bandwidth as a function of a

for U = 6 V and U = 10 V when

the light is incident on the n- or p-side. The results show that, for a fixed a

, the band-

width is always higher when light is incident on the p-side than when light is incident

on the n-side. Furthermore, when the bias voltage changes from 6 V to 10 V, the band-

width increases for light incident on the n-side and it decreases for light incident on the

p-side. We may then conclude that the electron’s transit time determines the bandwidth

when light is incident on the p-side and that the hole’s transit time determines the

bandwidth when light is incident on the n-side.

Conclusions

A numerical model has been implemented to investigate the effect of the direction of

light on the frequency response of p-i-n photodiodes. This model may be applied to

devices with an arbitrary electric field profile in the absorption region and non-uniform

illumination. Better bandwidths are obtained when the light is incident on the p side.

Bandwidth values obtained with the saturation velocity values are underestimated and

overestimated for light incident on the p and n side respectively.

References

[1] J.E. Bowers and C.A. Burrus, “Ultrawide-band long-wavelength p-i-n photodetector”, J. Lightwave

Technol., vol. 5, pp. 1339-1350, 1987.

[2] G. Lucovsky, R.F. Schwarz, R.B. Emmons, “Transit-Time Considerations in p-i-n diodes”, J.Appl. Phys.,

35(3), pp. 622-628, 1964.

[3] R. Sabella, S. Merli, “Analysis of InGaAs p-i-n photodiode frequency response”, IEEE J. Quantum Elec-

tron., vol. 29, pp. 906-916, 1993.

1.5 2.0 2.5 3.0

25.0

21.0

17.0

13.0

9.0

5.0

( m)a

μ

f3dB

(G

Hz)

Light incident on the p-side

Light incident on

the n-side

21 3

10 md

N

−

=

6 VU =

10 VU =

6 VU =

Figure 5. Computed bandwidth as a function of a

for U = 6 V and U = 10 V.


[4] M. Dentan, B. de Cremoux, “Numerical simulation of the nonlinear response of a p-i-n photodiode under

high illumination”, J. Lightwave Technol., vol. 8, nº 8, pp. 1137-1144, 1990.

[5] Z. Šušnjar, Z. Djurić, M. Smiljanić, Ž. Lazić, “Numerical calculation of photodetector response time us-

ing Ramo’s theorem”, Proceed. MIEL’95, vol. 2, NIS, Serbia, pp. 717-720, 1995.

[6] J.N. Hollenhorst, “Frequency response theory for multilayer photodiodes”, J. Lightwave Technol., vol. 8,

nº 4, pp. 531-537, 1990.

[7] J.M. Torres Pereira, “Frequency-Response Simulation Analysis of InGaAs/InP SAM-APD Devices”, Mi-

crowave Opt. Technol. Lett., 48 (4), pp. 712-717, 2006.

[8] Jorge Manuel Torres Pereira, “Modeling the frequency response of p+

InP/n-

InGaAs/n+

InP photodiodes

with an arbitrary electric field profile”, EPNC’2006, Maribor, Slovenia, vol. 1, pp. 181-182, 2006.

[9] J.B. Radunovic, D.M. Gvozdic, “Nonstationary and nonlinear response of a p-i-n photodiode made of

two-valley semiconductor”, IEEE Trans. Electron Devices, vol. 40, nº 7, pp. 1238-1244, 1993.


3-D Finite Element Mesh Optimization

Based on a Bacterial Chemotaxis

Algorithm

S. COCOa

, A. LAUDANIa

, F. RIGANTI FULGINEIb

and A. SALVINIb

a

DIEES, Università di Catania, V. A. Doria 6, I 95125 Catania, Italy

b

DEA, Università di Roma 3, Via della Vasca Navale, 84, 00146 Roma, Italy

asalvini@uniroma3

Abstract. An approach based on a bacterial chemotaxis algorithm dedicated to the

3-D finite element mesh optimization is presented. By starting from an assigned

mesh, the ‘bacterial chemotaxis algorithm’ modifies the initial configuration by

changing the spatial coordinates of specific mesh nodes (or possibly by canceling

them) while all node connections are kept fixed. This approach guarantees an ap-

preciable reduction of the number of elements preserving a high quality of the

mesh. This is a fundamental rule for reducing computational costs and preserving

the accuracy of the numerical results. The approach has been validated on the

meshing of a 3-D Helix structure of a TWT.

1. Introduction

Dedicated 3-D numerical simulators are necessary to develop suitable electromagnetic

analysis for a correct design of electronic and electric devices. Moreover, the Finite

Element (FE) approach is the most employed method implemented. The key of success

of the FE is mainly due to its capability to give accurate solutions for several different

kinds of electromagnetic problems, but also for its vocation for analyzing 3-D complex

geometries. To obtain accurate results with an acceptable computational cost, it is nec-

essary to provide mesh refinement and adaptive mesh generation in critical spatial re-

gions of the device to be studied. In particular 3-D electromagnetic analyses, the gen-

eration of the finest mesh, able to guarantee accurate results, is a very difficult task due

to the complexity of the device shape (e.g. the helix slow wave structure in a Traveling

Wave Tube (TWT)). Usually, the mesh is generated by assembling together elementary

small blocks (primitives) having simple shape. In other words, the overall mesh is ob-

tained by applying a special “connect-it” function to all the interface elements between

adjacent primitives. Unfortunately, this way to operate may produce a low quality

mesh. This aspect is due to the non-optimal choice of the initial shape of the connecting

elements. Consequently, a refinement procedure is necessary in order to increase the

mesh quality [1].

The aim of this paper is to present an approach for the 3-D finite element mesh op-

timization based on Bacterial Chemotaxis Algorithm (BCA) [2]. The choice of this

specific optimization algorithms can be justified by the particular nature of the problem


425

under analysis as described in [3]. By starting from an assigned mesh, the BCA modi-

fies the initial configuration by moving the specific mesh nodes (or possibly by cancel-

ing them), while all node connections are kept fixed. This approach guarantees an ap-

preciable reduction of the number of elements preserving a high quality of the mesh.

The paper is structured as follows: in the Section 2 the main aspects of mesh opti-

mization are presented; in the Section 3 the Bacterial Chemotazis Algorithm and its

application to mesh optimization is discussed; two experiments for the approach valida-

tion are discussed in Section 4; authors’conclusions follow in Section 5.

2. Mesh Optimization

The main motivation of the optimization of a Finite Element mesh is to improve the

quality of a poorly shaped elements in order to reduce computation time, also by de-

creasing the number of elements of the original mesh. Two different approaches can be

carried out to optimizing a mesh, local and global. A local approach search a optimal

discretization in the proximity of an assigned mesh node (e.g. Laplacian smoothing),

while a global mesh optimization is aimed to improve mesh quality for the entire do-

main. Unfortunately the local approach is not able to guarantee an overall mesh en-

hancement, thus a global approach, although its more expensive computational cost, is

preferable. In addition there are two main problems in the mesh optimization: the for-

mer is that the optimal mesh does not exist for 3D domain, since it is impossible to fill

an arbitrary space with regular tetrahedra; the second is the choice of a metric for the

evaluation of quality of a 3D unstructured mesh is an open problem. In fact it is possi-

ble to qualify a mesh according to “geometrical metrics”, related to interpolation error

of numerical method, or to “condition number metrics”, related to condition number of

the discretized FE matrices. In our optimization approach we have adopted geometric

metrics, which will briefly discuss in the next paragraph, since the Bacteria Chemotaxis

behaviour can be easily connected to geometrical properties of the mesh.

Tetrahedron Shape Measures and Fitness Functions

Tetrahedron Shape Measures (TSM) are usually defined in terms of the ratio of two

characteristic sizes of the tetrahedron (the volume of the tetrahedron, the length of

longest or smallest edge, the radius, the area or the volume of the insphere or circum-

sphere, etc) and usually vanish with the volume of tetrahedron and achieve the maxi-

mum possible quality of 1 only for regular elements [4,5].

Among the different shape measures the usually adopted are:

− the Radius Ratio ρ

3inscribed

circumscribed

ρρ

ρ= (1)

where inscribed

ρ and circumscribed

ρ are the radius of the inscribed sphere and circumscribed

sphere respectively.

− The Mean Ratio η

S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm426

3 2

6

2

1

12 9

i

i

V

l

η

=

=

∑

(2)

where V is the volume of the tetrahedron and liis the length of the i-th edge of the tet-

rahedron.

− The Edge Ratio σ defined as the ratio between the smallest edge over the

largest edge of the tetrahedron.

− The Aspect Ratio γ

1 6

2 6

max

inscribed

i

i

l

ργ

< <

= (3)

where inscribed

ρ is the radius of the inscribed sphere and liis the length of the i-th edge of

the tetrahedron. Clearly in order to evaluate the overall quality of a mesh we should

refer to a statistical parameter like the minimum, the average, the maximum of the

choosen shape measures of the tetrahedra. Among these quantities the average value is

the most important. Infact the maximum is pratically useless since reaches the 1.0

value, while the minimu is less important because should be erroneus evaluate the qual-

ity of a mesh by considering just the worst tetrahedra, which often belong to the

boundary and cannot successfully be modify during optiomization. In addition as dis-

cussed in literature [4,5] there is an equivalence among the tetrahedron shape measures

and conseguently a mesh optimized by using as fitness function one of the previuos

defined tetrahedron shape measure is close to be optimal for the other tetrahedron

shape measures. In the following and in the examples we adopt as fitness function for

our test the Edge Ratio σ , however very similar results have been found by using all

the other TSMs.

3. BCA

The BCA [2] takes inspiration by the motion of particular micro-organisms (bacteria)

like in natural life. The bacterium activity is due to the different chemical properties

encountered into its habitat. This behavior is called bacterial chemotaxis. In particular,

a bacterium is sensitive both to the gradient of the nutritive substance concentrations

and to the gradient of harmful substances. Thus, the bacterium motion mechanism is

due to the evolution of the bacterium position for achieving survival or, simply, a better

life conditions. Thus, all scientific observations of a particular bacterium specie behav-

ior (e.g. Escherichia coli and Salmonella typhimurium), can be effort for heuristic evo-

lutionary computation (virtual bacterium). Thus, from the knowledge of the chemical

mechanisms governing the bacterial chemotaxis, mathematical abstractions and imple-

mentation of numerical algorithms has been made for optimization problems [2,3].

A mathematical description of a bacterium motion can be developed by the deter-

mination of suitable probabilistic distributions referred both to the motion duration and

to the velocity vector (speed and direction) of the bacterium. The bacterium motion is

performed in a n-dimensional hyperspace in which its velocity vector is made of n

speeds and n directions components.

S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm 427

In the present implemented BCA for mesh optimization, the virtual bacterium mo-

tion follows the next listed rules: a) the bacterium population consists of an assigned

number individuals; b) each bacterium follows its own path, each consisting of a se-

quence of rectilinear trajectories; c) each path segment is identified in a n-dimension

space which dimension is depending on the number of parameters to be optimized; d)

the bacteria speeds and directions are updated for each calculation step according to the

Error Index of solutions; e) new speeds and directions are generated by means of a

Gaussian probabilistic functions; f) the path duration τ is assumed to be constant

The “BCA” optimizes the initial configuration of the primitive blocks by changing

the assigned coordinates of specific nodes without changing the connections which do

not belong to the boundaries. On the other hand, the BCA can cancel some

nodes/elements. Following this approach we obtain a mesh having an higher quality

and also a lower number of elements, i.e. it is possible to achieve accurate results with

a low computational cost.

4. Approach Validation

The BCA mesh optimizer has been tested by using several finite element model. Here-

after two examples are illustrated in order to validate the approach. The first example

presented regards a very simple cilindrical geometry. In particular, by starting from a

mesh made of 420 points, 1710 tetrahedra, 2483 edges, and having an average TSM

(edge ratio σ ) of 0.31, an optimized mesh is obtained after 61 iterations, consisting of

281 points, 893 tetrahedra, 1431 edges, and having an average TSM (edge ratio) of

0.44. The enhancement of the overall quality was of more than 40%, while the number

of elements, points and edges is reduced of about 35% with respect the initial one. Fig-

ure 1 shows the intial mesh of the cylinder and the otpimized one. Figure 2 shows the

histogram of element TSM for the two meshes before and after BCA optimization. The

initial mesh have an high percentage (about 75%) of tetrahedra having a TSM of 0.3.

Figure 1. Visualization of the initial mesh (left) and of the optimized one (right).


It is worth noticing that the high percentage of element having a TSM quality of

0.3 in the optimized mesh is relative to elements belonging to the boundary.

The second test here presented regards the optimization of an helix slow wave

structure of a TWT [6]. The original mesh has 5064 points, 27324 tetrahedra, 33335

edges, an average TSM (edge ratio) of 0.135. The optimized mesh, after 88 iterations,

has 1747 points, 8633 tetrahedra, 10951 edges, an average TSM (edge ratio) of 0.361.

The enhancement of the overall quality was of more than 150%, while the number of

elements, points and edges is reduced to 1/3 of the initial one. Figure 3 shows the opti-

mized mesh obtained. Obviously, we do not show the initial mesh because it is made of

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

tetrahedron shape measure

% o

f el

emen

ts

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50


% o

f el

emen

ts

Figure 2. Histogram of the percentage of elements having an assigned shape measure for the two mesh of

helix, before and after the optimization.

Figure 3. Visualization of the optimized mesh of a TWT Helix: the number of elements is about 1/3 of initial

mesh.

S. Coco et al. / 3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm 429

a high number of elements which makes the mesh not intelligible. In Fig. 4 the histo-

gram of element TSMs for the two meshes before and after optimization is presented,

showing the enhancement of the elements quality achieved by BCA optimization.

5. Conclusions

The presented approach based on a bacterial chemotaxis algorithm dedicated to the 3-D

finite element mesh optimization seems to be an attractive method. By starting from an

assigned mesh, the ‘bacterial chemotaxis algorithm’ modifies the initial configuration

by changing the spatial coordinates of specific mesh nodes (or possibly by canceling

them) while all node connections are kept fixed. This approach guarantees an apprecia-

ble reduction of the number of elements preserving a high quality of the mesh. This is a

fundamental rule for reducing computational costs and preserving the accuracy of the

numerical results.

References

[1] S. Coco, A. Laudani, S. Pulvirenti e M. Sergi, An object-orientated 3D finite element determinis-

tic/neural mesh generator, in Software for Electrical Engineering Analysis and Design V, Wessex Insti-

tute of Technology, United Kingdom, pp. 27-36. 2001.

[2] S.D. Muller, J. Marchetto, S. Airaghi, P. Kournoutsakos, “Optimization Based on Bacterial Chemotaxis”,

IEEE Trans. on Evolutionary Computation, Vol. 6, No. 1, February 2002, pp. 16-29.

[3] F. Riganti Fulginei, A. Salvini, Comparative Analysis between Modern Heuristics and Hybrid Algo-

rithms, COMPEL, The International Journal for Computation and Mathematics in Electrical and Elec-

tronics Engineering, MCB University Press, March 2007, Vol. 26, No. 2, pp. 264-273.

[4] J. Dompierre, P. Labbe, F. Guibault and R. Camarero, Proposal of Benchmarks for 3D Unstructured Tet-

rahedral Mesh Optimization, 7th International Meshing Roundtable, Dearborn, MI, pp. 459-478, October

1998.

[5] A. Liu, B. Joe, Relationship between tetrahedron shape measures, BIT 34, pp. 268–287, 1994.

[6] S. Coco, A. Laudani, G. Pollicino, R. Dionisio, R. Martorana, A FE tool for the electromagnetic analysis

of slow-wave helicoidal structures in Traveling Wave Tubes, IEEE Transactions on Magnetic, Vol. 43,

Issue 4, pp. 1793-1796, April 2007.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50


% o

f el

emen

ts

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25


% o

f el

emen

ts

Figure 4. Histogram of the percentage of elements having an assigned shape measure for the two mesh of

helix, before (left) and after (right) the optimization.


Electro-Quasistatic High Voltage Field

Simulation of Large Scale 3D Insulator

Structures Including 2D Models for

Conductive Pollution Layers

Daniel WEIDAa

, Thorsten STEINMETZa

, Markus CLEMENSa

, Jens SEIFERTb

and Volker HINRICHSENc

a

Helmut-Schmidt-University, University of the Federal Armed Forces Hamburg,

22043 Hamburg, Germany

b

Lapp Insulators GmbH, 95632 Wunsiedel, Germany

c

High-Voltage Laboratory, Technische Universität Darmstadt, D-64283

Darmstadt, Germany

Abstract. Simulations of large scale 3D insulator structures covered by conductive

pollution layers are presented. These conductive pollution layers on the surfaces

are thin with respect to the device dimensions. The 3D geometric modeling of

these layers is expensive in time and leads to ill-conditioned linear systems of

equations due to large aspect ratios. Alternatively, in this paper they are modeled

as 2D surface layers coupled to the 3D geometric model in the electro-quasistatic

field simulations. Numerical results for realistic 3D high voltage insulator struc-

tures covered by both 3D and 2D pollution layers are presented.

1. Introduction

High voltage insulator structures are essential for electric power transmissions, where

the maximum electric field stress is a crucial design criterion. If the field stress on the

surface of the structure is too high, the corrosion of the insulator material due to elec-

tric discharge phenomena is intensified, thus reducing its lifetime. During operation,

the insulators structures which are exposed to environmental influences are contami-

nated by dirt. For the analysis of the field stress, the pollution layers covering the insu-

lator structure have to be taken into account. These layers are naturally not included in

the CAD design data of the insulators. Hence, for conventional 3D simulations, they

have to be provided additionally, which is expensive especially for curved surfaces and

– due to the necessarily required finer mesh resolutions – leads to more degrees of free-

dom (DoF) in the discrete models. Whereas the electric conductivity of the insulator

and the ambient air vanishes, a perfect electric conductivity could be used as modeling

assumption for the simulation of the electric field in the pollution layers. Perfect con-

ductive material behavior can be considered by projection techniques applied to Kry-

lov-subspace methods [7]. However, with this approach, the conductive current through

the pollution layer can not be calculated which prevents the computation of the Ohmic

losses eventually required for coupled or subsequent thermodynamic simulations. In

this paper, these thin conductive dirt layers positioned on the outer surface of the insu-


431

lator are modeled as 2D surfaces, assuming, that the conductive currents normal to the

insulator surfaces are negligible. These 2D surfaces coupled into the 3D model save

expensive further geometric modeling, they help to avoid additional difficulties in the

mesh generation and the introduction of additional, effectively inessential DoF for the

thin dirt layers.

In order to cope with both capacitive and resistive material behavior, the simula-

tions are performed under the electro-quasistatic (EQS) assumption [1–3], i.e. the

omission of the magnetic induction term in Faraday’s law. Thus, a scalar potential

function φ(r, t) allows computing the resulting irrotational electric field intensities. The

EQS differential equation for the electric scalar potential in the time domain reads

( ) ( )( )div grad , 0t

r tκ ε ϕ− + ∂ = . (1)

Here, the electric conductivity is denoted by κ, the electric permittivity by ε. Simu-

lations of contaminated insulators have already been reported e.g. in [3], using 3D dirt

layers only. In this paper 2D pollution layers with a finite electric conductivity are in-

troduced and the simulation results are compared to the 3D approach.

2. Description of the Model Problem and Discrete Formulation

The application of Green’s first integral theorem to the weak form of the EQS differen-

tial equation (1)

( )( )div grad d 0t

v Vκ ε ϕ

Ω

− + ∂ =∫ (2)

yields

d

grad grad d grad grad d d

d

v V v V v

t n

ϕκ ϕ ε ϕ

Ω Ω ∂Ω

∂⋅ + ⋅ = Γ

∂∫ ∫ ∫ . (3)

At this point the solution function φ and the test function v in (3) are elements of

the infinite dimensional function spaces ( )Ω

1

H and ( )Ω

1

0

H , respectively. Applying

Galerkin FEM techniques, φ and v can be approximated by finite nodal shape func-

tions. Using Whitney-FEM [5], the geometrical discretization (3) reads

d

0

dt

κ ε+ =C Φ P Φ . (4)

Here, Φ is the vector of the electric nodal scalar potential. Since the pollution lay-

ers p

Γ are very thin compared to the lateral dimensions, the tangential component of the

total current density (electric + displacement current density) in the pollution layers

D. Weida et al. / Electro-Quasistatic High Voltage Field Simulation432

outweighs its normal component. Considering these anisotropic material behavior,

equation (3) is rewritten as

d

grad grad d grad grad d

d

d

grad grad d grad grad d d ,

dp p

S S S S S S

v V v V

t

v v v

t n

κ ϕ ε ϕ

ϕ

κ ϕ ε ϕ

Ω Ω

Γ Γ ∂Ω

⋅ + ⋅

∂+ ⋅ Γ+ ⋅ Γ = Γ

∂

∫ ∫

∫ ∫ ∫

(5)

where S

ε and S

κ are the electric surface permittivity and conductivity and S

grad the

surface gradient. The discrete formulation of (5) reads

( ) ( ),2 ,2

d

0

d

D D

t

κ κ ε ε+ + + =C C Φ P P Φ , (6)

which is an ordinary differential equation (ODE) in the time domain. The matrices κ

C

and ε

P are assembled element-wise for each volume element, the matrices ,2Dκ

C and

,2DεP for each pollution layer surface element via a strongly modified version of the

FEMSTER C++ class library [6] with nodal shape functions based on Sylvester-

Lagrange interpolation polynomials.

3. Electro-Quasistatic 3D Simulations

Transient EQS simulations of high voltage insulators for applications on overhead

transmission lines with 21 sheds covered by 3D dirt layers and 2D dirt layers are

performed. The electric conductivity and permittivity of the 1mm thick pollution

layersis set to κ = 0.03 A/Vm and ε = 8.854e-12 As/Vm (κS = 3.0e-5 A/m and

εS = 8.854e-15 As/V), while the insulator and the ambient air feature zero conductivity.

A time-harmonic electrode voltage of 105 kV with a frequency of 50 Hz is applied.

The time integration of (6) is performed by an implicit Singly-Diagonal-Runge-Kutta

SDIRK3(2) method with four internal stages [4] where the resulting linear system of

equations are solved by a preconditioned conjugate-gradient iterative solver (PCG).

The preconditioning is carried out by an algebraic multigrid method.

3D vs. 2D Pollution Layer

In the first example, the top sides of the sheds of a high voltage insulator are half-

covered by conductive pollution layers. These are modeled both as thin 3D volumes

and as 2D surfaces, respectively. On the electrodes of the insulator that are assumed to

be perfect conductors, the voltage profile depicted in Fig. 1 is applied. The insulator in

Fig. 2 is embedded in a vacuum box serving as computational domain. Homogeneous

Neumann conditions are assigned to the outer surfaces.

The simulation results show a nearly identical distribution of the scalar potential as

well as the electric field intensity. However, in addition to avoiding a more complicated

D. Weida et al. / Electro-Quasistatic High Voltage Field Simulation 433

geometric modeling of the pollution layer, the following simulation is more efficient

using the 2D pollution layers than the 3D dirt layers. In the table below, the number of

PCG steps and the solution time required for the four stages of a specific time step so-

lution shown in Fig. 3 are listed.

Type of pollution layer Number of DoFs Total number of PCG steps Total solve time

3D 386.157 471 543 sec

2D 291.093 290 271 sec

Entire Pylon with Insulator Covered by 2D Pollution Layers

The first example above shows the efficiency and reliability of the 2D pollution layer

models. As next model problem the entire pylon of an overhead transmission line is

simulated. The configuration shown in Fig. 4 consists of a cylindrical pylon, a traverse,

a 21 shed insulator and the central phase cable. The outer surfaces of its surrounding air

Figure 1. Ramped time-harmonic sinus voltage impressed on the insulators electrodes.

Figure 2. CAD Model of the insulator with sheds’ upside half-polluted.


region are set to 0 V except the surfaces normal to the cable on which homogeneous

Neumann conditions are defined. The top sides of the sheds of the high voltage insula-

tor are covered by conductive pollution layers.

The simulation results in Fig. 5 show the effect of the traverse, the insulator and

the corona ring on the radial field of the cable. The effect of the conductive pollution

layers is obvious in Fig. 6. While the electric field magnitude is smaller at the pollution

layers compared to the simulation without pollution layers, it is increasing between the

pollution layer and the next shed.

The consideration of additional pollution layers results in increased numerical

costs as shown in the table below. The number of PCG steps needed to solve the stage

systems of equations more than octuplicates for the model problem with 2D pollution

layers. However, based on the numerical results achieved for the first example, the use

of 3D pollution layers is assumed to raise the numerical costs even more. Particularly,

if the more complex modeling, the essential mesh refinement and the larger aspect ra-

tios of the 3D pollution layers are taken into account, the approach containing 2D pol-

lution layers turns out to be more efficient.

Figure 3. Comparison of 3D EQS simulations with 3D (left) and 2D (right) pollution layers, respectively.

The Finite-Element mesh, the scalar potential Φ and the electric field magnitude |E| are shown.

Figure 4. Geometry of the entire electricity pylon: the top sides of the insulator sheds are polluted.


4. Conclusion

EQS simulations of high voltage insulator structures covered by thin pollution layers

were presented. Modeling these layers as 2D surfaces instead of 3D volume bodies

resulted in both a smaller number of DoF and a more economical modeling process,

respectively. Thus, the efficiency of the simulation scheme was clearly increased. Nu-

merical results which compared the scalar potential and electric field distributions of

the particular approaches as well as the efficiency of the proposed method were shown.

pollution layer Number of DoFs Total number of PCG steps Total solve time

None 706.007 28 129,96 sec

2D 706.007 230 508,30 sec

Figure 5. 3D EQS simulation with 2D pollution layers. The scalar potential Φ and the electric field magni-

tude |E| are shown.

Figure 6. Electric field magnitude |E| along the rod of the insulator. Even with the effect of the pollution

layer the EPRI (Electric Power Research Institute) norm (|E| < 4.5 kV/cm) is still fulfilled.


References

[1] R.M. Matias, A. Raizer, Calculation of Electric Field Created by Transmission Lines, by 3D-FE Method

Using Complex Electric Scalar Potential, ACES Journal, Vol. 12(1), pp. 56-60, 1997.

[2] K. Preis, FEM Numerical Computation of Transient Quasistatic Electric Fields, Proceedings of the 4th

International Conference on Computation of Electromagnetics (CEM) 2002.

[3] U.v. Rienen, M. Clemens, T. Weiland, Simulation of Low-frequency Fields on High-Voltage Insulators

with Light Contaminations, IEEE Transactions on Magnetics, Vol. 32(3), pp. 816-819, 1996.

[4] T. Steinmetz, M. Helias, G. Wimmer, L.O. Fichte, M. Clemens, Electro-Quasistatic Field Simulations

Based on a Discrete Electromagnetism Formulation, IEEE Transactions on Magnetics, Vol. 42(4),

pp. 755-758, 2006.

[5] A. Bossavit and L. Kettunen, Yee-like schemes on staggered cellular grids: A synthesis between FIT and

FEM approaches, IEEE Transactions on Magnetics, Vol. 36(4), pp. 861-867, 2000.

[6] P. Castillo, J. Koning, R. Rieben, M. Stowell, D. White, Discrete Differential Forms: A Novel Methodol-

ogy for Robust Computational Electromagnetics, 2003.

[7] M. Clemens M. Wilke, G. Benderskaya, H. De Gersem, W. Koch, and T. Weiland: Transient electro-

quasi-static adaptive simulation schemes, IEEE Trans. Magn., Vol. 40(2), pp. 1294-1297, 2004.


Electromagnetic Aspects of Data

Transmission

Liliana BYCZKOWSKA-LIPIŃSKAa

and Sławomir WIAKb

a

Technical University of Lodz Institute Computer Science, ul.Wólczańska 215, Poland

[email protected]

b

Technical University of Lodz Institute of Mechatronics and Information Systems,

ul. Sefanowskiego 18/22, Poland

[email protected]

Abstract. The aim of this paper is to analyze the problems connected with trans-

mission of information and the application of electromagnetic wave in data trans-

mission. The issue of electromagnetic compatibility is discussed. The problem of

electromagnetic radiation as a threat to natural environment is presented.

1. Electromagnetic Phenomena in Data Transmission

Telecommunication is a branch of science and technology that deals with transmission

of information from a transmitter to one or more receivers (either humans or technical

appliances appropriately adapted to this purpose) [2,4,5].

In practice, contemporary telecommunication is teleinformatics, as it integrates

two technical disciplines:

− telecommunication, which deals with transmission of information,

− informatics, which deals with data processing.

The integration of these two disciplines is obvious. However, it is not complete

and does not proceed simultaneously in all countries. Currently it concerns:

• developing new structures of data networks integrating data transmission ser-

vices into one technical form capable of gaining access to processing systems

and commutation between them;

• developing new algorithms improving the quality and quantity of data trans-

ferred, creating computer networks, integrating access times and processing

power, as well as development and integration of teleinformatic systems.

The process of information transfer from a transmitter to a receiver in a telecom-

munication network comprises the following stages:

1. Processing, which involves the adaptation of information to transfer condi-

tions. This refers mainly to the change of original information into electro-

magnetic signal and appropriate processing of this signal in order to improve

the quality and quantity of information transferred (modulation, analog-digital

processing, coding) and information retrieval (demodulation, digital-analog

processing, decoding).



doi:10.3233/978-1-58603-895-3-438

438

2. Teletransmission from one point of telecommunication network to another.

Depending on the frequency range, an appropriate transmission channel is

used, i.e. copper wire, optical fiber or wireless.

3. Commutation and telecommutation dealing with joining and disjoining the

elements of the transfer channel of information carrying signals. This stage of

the process comprises also designing, manufacturing, installation and exploi-

tation of telecommutation devices and the issues of movement in telecommu-

nication networks.

In telecommunication, it is the electromagnetic wave that is used as a carrier of

original information. Figure 1 shows electromagnetic spectrum [6–10].

The majority of contemporary telecommunication systems is based on broad fre-

quency electromagnetic phenomenon.

The following electromagnetic wave ranges can be distinguished (Fig. 1):

− electromagnetic waves used in wireless transmission, commonly known as ra-

dio waves,

− electromagnetic waves used in copper wire transmission (electric current),

− electromagnetic waves used in optical fiber transmission (optical infrared,

visible, and ultraviolet radiation).

This is an example of how your paper is to be prepared according to the instruc-

tions. When numbering equations enclose the number in parentheses and place it flush

with the right hand margin as shown below.

Figure 1. Electromagnetic spectrum [10,12].

L. Byczkowska-Lipinska and S. Wiak / Electromagnetic Aspects of Data Transmission 439

Table 1 shows frequencies and wavelengths of electromagnetic radiation used for

information transfer.

Since the properties of electromagnetic waves used in wireless transmission (radio

range) are very diverse, the division into subranges was introduced. At present, deci-

mal division of radio waves introduced by Consultative Committee on International

Radio (CCIR) and compatible with the Radio Regulations is used (Table 2).

The decimal classification is totally formal and does not reflect the properties of

particular wave subranges. Therefore, the traditional division is often used, as illus-

trated in Table 3.

Wavelength λ is expressed with (1)

,

f

c

=λ (1)

where: f – frequency, c = 3⋅108

m/s – the speed of electromagnetic wave propagation in

vacuum.

Table 1. Frequencies and wavelengths of electromagnetic radiation

Range Wavelength λ,

[m]

Frequency f,

[Hz]

Radio waves 108

...10–4

3...3⋅1012

Optical radiation 10–4

...10–8

3⋅1012

...3⋅1016

X-rays 10–8

...10–11

3⋅1016

...3⋅1019

Gamma rays < 10–11

> 3⋅1019

Table 2. Subranges of radio waves

Designation Symbol Wavelength Frequency

Decamegametric 105

– 104

km 3 – 30 Hz

Megametric 104

– 103

km 30 – 300 Hz

Hectokilometric 1000 – 100km 300 – 3000 Hz

Myriametric VLF 100 – 10 Km 3– 30 kHz

Kilometric LF 10 – 1 Km 30 – 300 kHz

Hectokilometric MF 1000 – 100 m 300 – 3000 kHz

Decametric HF 100 – 10 m 3 – 30 MHz

Metric VHF 10 – 1 m 30 – 300 MHz

Decimetric UHF 100 – 10 cm 300 – 3000 MHz

Centimetric SHF 10 – 1 cm 3 – 30 GHz

Milimetric EHF 10 – 1 mm 30 – 300 GHz

Decimilimetric 1 – 0,1 mm 300 – 3000 GHz

Table 3. Traditional division of electromagnetic wave ranges

Designation Wavelength Frequency

Very long ≥ 20 km ≤ 15 kHz

Long 20 – 3 km 15 – 100 kHz

Medium 3000 – 200 m 100 – 1500 kHz

Intermediate 200 – 100 m 1,5 – 3 MHz

Short 100 – 10 m 3 – 30 MHz

Ultrashort 10 – 1 m 30 – 300 MHz

Microwaves ≤ 1m ≥ 300 MHz

L. Byczkowska-Lipinska and S. Wiak / Electromagnetic Aspects of Data Transmission440

2. Transmitters and Receivers Of Electromagnetic Wave

Contemporary data transmission systems are based to a large extent on the application

of electromagnetic wave to wireless information transfer. This refers mainly to inter-

continental communication links via satellite.

Antenna is a part of a transmitter or receiver, designed to send or receive electro-

magnetic waves [5]. Depending on the radiation characteristics, three antenna types

may be distinguished, namely omnidirectional, one-directional and two-directional

[12].

The properties of an antenna are characterized by the following parameters:

• gain – a ratio that indicates how much energy an antenna radiates in a given

direction compared to the isotropic antenna and expressed in dBi,

• beamwidth – antenna’s radiation characterized by the so called lobes, i.e. the

main lobe, and smaller side and rear lobes (Fig. 2). Beamwidth is the angle

within which the intensity of radiation is one-half the maximal intensity

(Fig. 3),

• front-to-back ratio – the ratio of power radiated to the front of an antenna ver-

sus the amount of power radiated to the back of an antenna (Fig. 4),

Figure 2. Characteristics of antenna’s radiation.

Figure 3. Beamwidth.

main

side

lobe

rear

lobe


• polarization – depends on the characteristics of electromagnetic field vector;

there are two types of polarization:

− linear polarization – depending on the direction of electric field vector, it

can be either horizontal or vertical,

− circular polarization – depending on the direction in which the electric

field vector rotates, it can be left-hand-circular or right-hand-circular.

• impedance – the ratio between the voltage and current intensity at antenna’s

terminal; it is a very important parameter, because its mismatch with the

transmitter can result in some fraction of the wave’s energy reflecting back to

the source,

• VSWR (voltage standing wave ratio) – characterizes the energy reflecting

back to the source as a result of the mismatch between the antenna and the

transmitter,

• capacity – the frequency range, within which antenna maintains its parame-

ters, e.g. radiation characteristics and input impedance.

3. Technologies Used in Data Transmission

The first technologies used for data transmission networks were wire technologies,

used in corporate networks. With time, there appeared new solutions using the existing

phone lines, wireless transmission and even the existing energetic networks.

According to the type of transmission channel and the standards used in transmis-

sion of digital signals, data transmission networks (computer networks) can be divided

into several groups [1,4,5,7–9,11].

1. Wired networks built from scratch:

a) HAVI (Home Audio Video Interoperability) – a standard which allows all

manner digital consumer electronics and home appliances to interoperate

by means of one controlling appliance;

b) Ethernet 10Base–T (using UTP cat. 5);

Figure 4. Front-to-back ratio.


c) IEEE 1394 Protocol (freeware) – international hardware and software

standard that integrates entertainment, communication and computation

systems, thanks to which data may be transferred at 100, 200 or

400 Mb/s;

d) UpnP (Universal Plug and Play) – allows devices to exchange data under

the control of a controlling device in the network.

2. Wire networks using existing cables:

a) Home PNA (Home Phone Line Alliance) – computer network standards

designed to operate over existing phone lines,

b) PLC (Power Line Communication) broadband network using outdoor

electrical cables,

c) PLC (Power Line Communication) narrowband network using indoor

electrical cables.

3. Wireless LANs:

a) Home RF (Home Radio Frequency) – a standard providing high effi-

ciency wireless connectivity for multimedia computer applications ac-

cording to three versions of SWAP (Shared Wireless Access Protocol);

b) Bluetooth – a set of standards and products that enable devices to find

each other and connect seamlessly over a short range. This technology is

a universal method of access to existing data networks,

c) IEEE 802.11 WLAN – a Wireless LAN transmission standard that offers

two network connection methods:

− ad-hoc that requires no server and no access point,

− client/server that requires a central server, coordinating all stations

in the network.

d) IrDA (Infrared Data Association) – infrared digital signals transmission

protocol.

In wired networks, the connection of the elements with network wiring system is

stationary. The change of localization of a particular device requires new configuration

of cable connection. Wireless networks provide full mobility of the workstation within

the network.

4. Typology of Wireless Networks

A great variety of wireless transmission technologies and devices employed for creat-

ing wireless LAN helps to create different kinds of data transmission networks:

− fully wireless;

− wireless with an access to wired network;

− wireless bridges, used for connecting distant segments of wired networks or

connecting individual WLANs.

The above types can be mixed. An example of a fully wireless network is illus-

trated in Fig. 5.


Individual devices belonging to such a network work in a “peer-to-peer” system,

which enables them to exchange resources. They are also capable of moving within the

scope of the network, and a potential loss of a direct connection between two devices

does not have to mean transmission breakdown, because indirect information transfer is

possible.

Wireless networks with an access to the resources of wired network are the most

common type of wireless networks, e.g., LAN. Communication between the two seg-

ments (wired and wireless) is held by specialized mediating circuits, called AP (Access

Points). They enable the cooperation with Ethernet and Token Ring, in both cases cre-

ating a mixed network (Fig. 6).

5. Electromagnetic Compatibility

The increase in the number of electronic systems and devices used in various fields of

human activity results in greater electromagnetic interference.

Electromagnetic compatibility the capability of an object (device, installation or

system) to be operated in its intended operational environment without causing elec-

tromagnetic interference [14].

The sources of electromagnetic phenomena may be objects emitting electromag-

netic waves both intentionally (television and radio stations) and unintentionally (hair

dryer).

Electromagnetic compatibility is connected with the following notions:

1. Emission of disturbances,

2. Immunity to disturbances.

Every piece of electrical equipment that is in operation can induce disturbances of

various levels and different character, one of them being the propagation of electro-

magnetic waves.

The issue of limiting the influence of unwanted signals should be taken into con-

sideration in the process of designing, constructing and manufacturing of devices and

systems that use electromagnetic field (teleinformatics). These problems should be

dealt with in the earliest phase of the design process. Usually, changes introduced in

stacja docelowa

zasięg stacji

Figure 5. An example of wireless station with multistage transmission [6].

destination

range


the later stages are not very efficient. Moreover, the number of solutions available de-

creases with time, and the later they are introduced, the more costly they get.

The phenomena occurring in natural environment and human activity result in the

emergence of unwanted random that in turn cause disturbances. They occur every-

where, both in technical devices and in natural environment, and their character and

intensity depend on many factors.

Every device is susceptible to electromagnetic interference and can be a source of

disturbance for other device, as well as its own elements. The disturbances can be of

either interior or exterior origin. If the source is inside a given device, then interior

disturbances occur. If the source is outside a given object, exterior disturbances take

place. The sources can be divided into low and high frequency sources, with the high

frequency sources being most difficult both from the theoretical and the practical point

of view.

Classification of sources depends on the criteria, which may include: origin, physi-

cal and biological phenomena, and mathematical description of a signal.

Classification of disturbances according to different criteria:

punkty dostępu

sieć Ethernet

sieć

Token

Ring

punkt dostępu

Figure 6. Examples of mixed networks, a) wireless segment and Ethernet, b) wireless segment and Token

Ring [6].

a)

b)

assess points

Ethernet

access point


1. According to the origin:

− natural (cosmic and terrestrial),

− human-caused.

2. According to the character of physical phenomena:

− vibroacoustic (vibrations, acoustic oscillations),

− biological, connected with natural environment both animate and inani-

mate (changes in temperature, humidity, pressure, the presence of fungi,

molds and dust),

− electromagnetic (noises emitted by electronic elements and circuits, sig-

nals from radio and television transmitters, switching and ignition sig-

nals, signals from energetic lines, light devices and other devices used at

home and in industry).

3. According to mathematical description:

− deterministic disturbances (can be defined in terms of mathematical de-

pendencies),

− stochastic disturbances (random; they cannot be defined in terms of

mathematical dependencies, as the signal observation results are different

each time. They are dealt with by means of probability theory).

Figure 7 illustrates an example scheme of disturbances in a television receiver.

In practice, electromagnetic compatibility includes two stages. First, the emission

channel of disturbing signals is determined, their values are evaluated and compared

with the acceptable values. Second, their reliability level in the environment of elec-

tromagnetic interference is determined.

While designing and operating electronic systems and devices, one should allow

for different types of disturbance, as they can influence substantially the way a given

system or device works.

A tmo s p h e ric

d is c h a rg es

O th e r

b ro a d c as t in g

s ta t io ns

E le c t r ic

e q u ip me n t in

me c h a n ic a l

v e h ic le s

V ib ro a c o u s t ic

p h e n o me n a

E le c t ro s ta t ic

d is c h a rg es

En e rg e t ic l in e s

a n d su bs ta t io ns

Ra d io a n d te le v is io n

re c e iv e rs

E le c t r ic a l d o me s t ic

e q u ip me n t

A tmo s p h e ric

p re c ip ita t io n (a g a in s t

a n te n na ’s c o ns t ru ct io n )

Figure 7. Sources of signals disturbing the work of a television receiver.


Some phenomena are strongly connected with each other and it is difficult to sepa-

rate them. What is important is the intensity of disturbances compared to the proper

signals of an appliance.

To deal with the increasing wave of electromagnetic disturbances, as well as for

commercial reasons, the European Union has passed a law that prohibits marketing and

installation of devices which do not fulfill the protection requirements.

6. The Influence of Electromagnetic Field on Living Organisms

The role of electromagnetic field in data transmission has become a subject of increas-

ing interest among ecologists, who examine how electromagnetic fields (EMF) emitted

by telecommunication devices, in particular those using wireless technology, can affect

humans. Potential health hazards resulting from mobile phone usage (wireless trans-

mission of information) concern not only the consumers and producers of mobile

phones, but also governmental and non-governmental organizations responsible for

citizens’ health. The exposure to EMF emitted by mobile telephony is below the ac-

ceptable norms. It is important to remember that these norms were developed on the

basis of the predicted thermal effects and do not allow for other effects of electromag-

netic emission.

Since mobile phones emit EMF in close proximity to the head, it is the influence of

EMF on central nervous system that arouses most of the concerns. Thus, most experi-

mental research in this field is concerned with the reaction of nervous system.

EEG tests did not give unequivocal results either. Part of them did not indicate any

changes in brain activity due to the exposure to EMF [14] and some of the works re-

port an increase of beta1 and delta waves 15 min after EMF exposure finished [15].

The examination of ulnar and facial nerves’ conduction did not reveal any vital

disorders either during or after the exposure to 900 MHz EMF of a mobile phone [14].

Hormone tests conducted on people who were exposed to mobile phone EMF for 4

weeks (2 h/day, 5 days a week), did not reveal any change in the level of pituitary

gland, corticotrophin (ACTH), thyrotrophic (TSH), growth hormone (GH), prolactin

(PRL), Latinizing hormone (LH) and follicle stimulating hormone (FSH). It was only

noticed that the concentration of TSH fell, but still maintained an acceptable level [15].

While examining circadian rhythm of melatonin secretion, no remarkable changes were

noticed

7. Summary

It is the electromagnetic phenomenon in all frequency ranges (depending on the trans-

mission channel) that carries data. A vital problem connected with the development of

all-type electric devices and teleinformatics is the influence of electromagnetic fields

on natural environment and living organisms.

The producers of mobile phones, scientists and technologists admit that the ra-

diation of a mobile phone’s antenna penetrates through the head and neck, which in-

creases brain temperature. Contradicting opinions about the influence of this phenome-

non on health have been the subject of hot debates all over the world. While in opera-

tion, mobile phone is the only device with a radiation source placed in close proximity

to the head. The issue of electromagnetic threats posed by telecommunication devices


is very problematic and should be approached with reason. Further research in the field

is needed in order to determine the actual influence of electromagnetic field on the en-

vironment.

References

[1] Bing B.: Wireless local area networks. New York John Wiley & Sons, Inc. 2002.

[2] Byczkowska-Lipińska L., Mandzij B.: Aspekty informatyczne w telekomunikacji. Łódź, ŁTN, 2004,

(in Polish).

[3] Byczkowska-Lipińska L., Cegielski M.: Architektury systemu informatycznego dla przedsiębiorstw

małej i średniej wielkości. Zeszyty naukowe seria: Technologie informacyjne nr 4, Gdańsk, 2004, (in

Polish).

[4] Derfler F.J.: Poznaj sieci komputerowe. W. MIKOM, 2003, (in Polish).

[5] Hallberg B.: Sieci komputerowe – kurs podstawowy. W. „Edition 2000”, 2001, (in Polish).

[6] Nowicki K., Woźniak J.: Przewodowe i bezprzewodowe sieci LAN. OWPW, Warszawa 2003,

(in Polish).

[7] Ogletree T.: Rozbudowa i naprawa sieci. W. Helion, Gliwice 2002, (in Polish).

[8] Plumley S.: Sieci komputerowe w domu i biurze. W. Helion, Gliwice 2004, (in Polish).

[9] Sprtack M.: Sieci komputerowe. Księga eksperta. W. Helion, Gliwice 2005, (in Polish).

[10] Stojmenović I.: Handbook of Wireless Networks and Mobile Computing. John Wiley & Sons, Inc. New

York 2002.

[11] Wajda K.:Wybrane zagadnienia z budowy i eksploatacji sieci korporacyjnych.WPT, 2002, (in Polish).

[12] Wesołowski K.: Systemy radiokomunikacji ruchomej. WKiŁ, Warszawa 2002, (in Polish).

[13] Zieliński B.: Bezprzewodowe sieci komputerowe. W. Helion, Gliwice 2005, (in Polish).

[14] Hasse L., Krakowski Z., Spiralski L., Kołodziejski J., Konczakowska A.: „Zakłócenia w aparaturze

elektronicznej” Radioelektronik Sp. Z o.o., W-awa 1995, (in Polish).

[15] Badania Kompatybilności Elektromagnetycznej: http://www.delta.poznań.pl./info/kompatybil.htm (in

Polish).

[16] Di Barba P., Savini A., Wiak S.: Field Models in Electricity and Magnetism. Springer, 2007, mono-

graph, under publishing procedure.


Application of the Magnetic Field

Distribution in Diagnostic Method of

Special Construction Wheel Traction

Motors

Zygmunt SZYMAŃSKI

Institute of Electical Engineering and Automation in Mines, Silesian University of

Technology Gliwice, 44-100 Gliwice, str. Akademicka 2, Poland

Email: [email protected]

Abstract. The paper presents a review of a special construction traction motors

applied in electric and hybrid vehicle. Mathematical model of the magnetic fields

distribution in traction wheel motor. Dynamic model of PMSM traction motor

based on 2D magnetic flux distribution are also presented in the paper. ANSYS,

JMAG and FEMM computer programs were applied in calculation of magnetic

fields distribution for different faults state of the motors. On the base of magnetic

field distributions were analyzed different failures situations and method of limita-

tion their negative effects. Some laboratory experiments was realized for the trac-

tion motor.

Introduction

Modern drive system of the road traction vehicle should ensure: environmental safety,

high reliability and economical speed control in specific duty circumstances. A signifi-

cant improvement in economical and power indexes can be achieved by: application of

new design drive motors (wheel induction motor, permanent magnet motor, or hybrid

drive system which contained petrol and electric motor), application of modern voltage

converters controlled by microprocessor systems and optimum control of machines and

electric vehicles. Special construction wheel traction motors: double rotor motor and

cutting magnetic core motor, are presented in the paper. Modern construction of the

permanent magnet wheel traction motor is described in the paper. Mathematical model

of the magnetic fields distribution in wheel traction motor (equivalent scheme and fi-

nite element method) are presented in the paper. Most occur failure in traction motor

include: broken rotor bars and end ring connectors, stator faults, eccentricity and bear-

ing faults. The consequences of such faults are: unbalanced phase voltage and line

currents, torque pulsations, mechanical vibrations and excessive heating. Ones of the

non-invasive technique for the traction motor faults diagnosis is monitoring and proc-

essing of the stator currents, to detect spectrum harmonics in motor characteristics for

the various types of faults. ANSYS, JMAG and FEMM computer programs were


449

applied in analysis of magnetic fields distribution in traction motors with various types

of faults. On the base of magnetic field distributions were calculated basic parameters

of different type traction motors. Some results of computer calculation are presented in

paper. Results of calculation were partially verified in laboratory experiments.

Wheel Traction Motors

For electric vehicle applications, wheels motor drives represented a new attractive pos-

sible solution for their lightness and compactness. The wheels are directly driven by the

electric motor and the gears are not necessary anymore. For axial flux motor applied in

electric vehicle must be realized: high power/weight and torque/weight ratio, high effi-

ciency and suitable shape to match constrain space. In the axial flux motors family, the

axial flux PM (permanent magnet), axial flux induction motor, and cutting magnetic

circuit motor are potential solution [1,3]. Axial flux induction motor Trim has one sta-

tor core with two polyphase windings and two rotors with two different shafts which

may rotate independently. All the three magnetic cores (the stator and the two rotors)

are in the form of discs with slots for the stator windings and the rotor cages. In this

case, the motor can not be mounted inside the wheels but between them. Two identical

three phase windings are connected in series in such a way, that the stator current flows

in the same direction in any back to back stator slot. There is one main flux which links

the stator windings and the two rotor cages. The motor has a small stator yoke which

reduces the iron core cost and the iron losses, but long ends windings which results in

copper losses. Induction traction motor with twin rotor axial flux is a physical combi-

nation of two motors into one in such a way, that their magnetic circuits are no longer

independent. The rotor is on different shafts and may rotate independently. Different

solution is permanent magnet wheel traction motor. The outer rotor may be designed to

have permanent magnet NdFeB magnets, with high remanence (1, 1 T) and coercity

(1275 kA/m.) on the inside surface, and concentrated windings on the stator. In motor

applied in electric vehicle drive systems must be concerned problems: rise temperature

to avoid demagnetization effects, exposition to road shocks, spills, dust and particles,

and force demands on the wheel and tire. High number of poles reduced the torque

ripple and yields a smaller magnetic yoke, decreasing volume and weight. Figure 1 shows a 2D drawing of the FCAFPM (field controlled axial flux permanent magnet) motor, and Fig. 2 left and right side a rotor of that motor [1]. The stator structure is formed by two strip wound or tape wound stator rings, circumferentially wound DC field winding and two sets of 3-phase AC windings. The stator core is divided into two sections in order to place the DC field winding in between, which makes it possible to vary the net air gap flux. The stator core has also slots to accommodate the two sets of 3 phase AC windings which are connected in parallel. The rotor is divided into two

parts. The upper permanent magnets of the left rotor, which are mounted on every other

pole, are magnetized as N-poles and aligned with the upper permanent magnets of the

right rotor which are magnetized as S-poles. The lower section of the pole is a non-

magnet pole of iron core. Similarly, the magnets in the lower sections of the poles are

magnetized as S-poles which are again located on every other pole and aligned with the

N-pole side magnets.

Z. Szymanski / Application of the Magnetic Field Distribution in Diagnostic Method450

Mathematical Model of Traction Motor

For analysis of magnetic field distribution in the wheel traction motors are applied two

methods: equivalent scheme method and finite element method. In equivalent scheme

method total magnetic circuits is divided for particular elements contain: stator and

rotor core, air gap, stator and rotor teeth and slot and yokes. The slots being skewed will produce a small MMF component in the radial direction, but this MMF will en-

Figure 1. Scheme of FCAFOPM motor [1].

Figure 2. Rotor core of the wheel PM motor [1].

Z. Szymanski / Application of the Magnetic Field Distribution in Diagnostic Method 451

counter high reluctance and very little radial flux will eventuate, and so this effect is also ignored. This analysis is similar to a single rotor machine. Concerning of boundary and border conditions, ensured a flux continuous at all points in the machine system, differential equation described of magnetic field distributions is presented in (1) [4]:

( )0

a gas ra a tsa tra

s r s sa

e e e e e

gC FF F H H rT T H H

pμα α α α α− = + + + −

∂∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

( )( )

0

r raa

s ra ga s tsa r tra s ra

rSgC

F F B T H T H H Hp

ο ν ν ν ν ν ν

ν

μ ν− = + + + −

( )( )

0

r rbb

s rb gb s tsb r trb s rb

rSgC

F F B T H T H H Hp

ο ν ν ν ν ν ν

ν

μ ν− = + + + −

(1)

where:

( ) ( )( )1 3

2

S

ν

ν

− +

=,

,

1

2

s zs

s

tz

a

rπ

= −

1

2

r zr

r

tz

a

rπ

= −

, 1

2

r zs

sr

tz

a

rπ

= −

Full analysis of magnet field distribution, in double rotor induction motor, realized

equivalent scheme method is presented in [3,4]. In finite element method total mag-

netic circuit is divided for digitized elements. Mathematical model of FEA field distri-

bution analysis described equations (2)–(5) [2]. The magnetic scalar potential is used to

describe the magnetic field in non-conducting materials.

( )2

div grad =0

grad grad =0.

μ

μ μ

− Ψ

Ψ⋅ + ∇ Ψ (2)

The magnetic field strength can be expressed as the sum of three components:

= e

+ +s m

H H H H (3)

where: Hs – magnetic field obtained as a result of the source’s current flow in the air,

Hm – magnetic field existing in ferromagnetic material in the surroundings,

He

– magnetic field existing of some conducting material in the surroundings.

Using the formula Biot-Savart law for the Hs component becomes:

2

0

V

1

= = dv

4 rμ π

⌠⎮⎮⌡

×r

s

J 1

H B . (4)

The Hm, H

e components can be described with equation (6):

= grad + − Φm e

H H . (5)


The magnetic vector potential can describe the electromagnetic field in the con-

ducting area. The magnetic vector potential is defined by: rot =B A , and after

evaluation conducted to the equations (6):

( )

1 1

rot rot - grad div = grad

t

div γgradV div γ 0

t

V

∂γ

μ μ ∂

∂

∂

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜ ⎟⎜⎟ ⎟ − + ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎟⎜ ⎜ ⎜⎟ ⎟⎜ ⎜ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞⎟⎜+ =⎟⎜ ⎟⎟⎜⎝ ⎠

A

A A

A

(6)

For obtaining the unambiguous solution of the equations (3)–(7) in the whole area,

the equations (2)–(6) have to be completed by consideration in equations of conditions

of continuity of solution at borders of areas described by different potentials. The

Galerkin procedure and the weighted residual method are used to transform the partial

differential equation to a digitized set of non-linear algebraic equations. Boundary con-

ditions and interface conditions will also be embedded in these equations.

Diagnostic Model of the Traction Motor

Magnetic flux distribution are applied in detection of the failure work state in electrical

or mechanical part of wheel vehicle. Most occur failure in traction motor include: bro-

ken rotor bars and end ring connectors, stator faults, eccentricity and bearing faults.

Analysis of the influence broken bars on traction motor parameters and magnetic fields

distributions are presented in the paper. This analysis used a transient eddy current,

accounting for the rotation rotor, with considerations of non-linear characteristics of the

magnetic materials. Analysis is based on magnetic vector potential formulation (7):

( )s

A

A J

t

μν ρ ρ σ= −

∂∇× ∇×

∂

(7)

where: μA – magnetic potential, μ – magnetic permeability, Js

– current density,

σ – electric conductivity.

Stator windings are fed from sinusoidal voltage source Us, which are connected

from series stator resistance Rsand leakage inductance L

s.Solving the equation (8) line

current is obtained.

s

ss

I

U RI L

t t

μφ

+ +

∂∂=

∂ ∂

s s

with

A dl and dSJI

μφ

−

− − −

= =∫ ∫∫

(8)

ANSYS, JMAG and FEMM computer programs were applied in calculation of

magnetic fields distribution, for different faults state of the motors. Results of computer

calculations are presented in Figs (3–6).


Analysis of magnetic field distribution were realised for different types of wheel

traction motors: double rotor induction motor, outer permanent magnet axial motor,

surface permanent magnet motor. ANSYS, JMAG and FEMM computer program were

applied in calculation of magnetic fields distribution. Each simulation cycle was carried

out in the following steps: specification of motor geometry, specification of material

properties, specification of boundary condition and excitation sources, generating the

solution and graphical postprocessing and analysis. Analysis was performed for two

work conditions of the motor: no-load and rated load torque. Computer program enable

calculation of different kind of faults: different number broken rotor bars, short-circuit

of stator winding. Results of computer calculation are applied to calculation of basic

parameters of the different constructions motors, and diagnostic of technical states of

the traction motor. In Fig. 3 presented a magnetic field distribution in supplied of single

phase winding a stator traction motor, in Fig. 4 are presented application of ANSYS

computer program for calculation of magnetic field distribution. Figure 5 presents a

flux density distribution and lines flux in healthy traction motor. Figure 6 present a

distribution of radial component of the flux density outside the stator traction motor for

healthy and for 5 broken bars of the rotor, and in Fig. 7 presented a torque characteris-

Figure 3. Magnetic field distribution in single phase

stator winding supply.

Figure 4. FEM analysis of stator magnetic field.

Figure 5. Flux density distribution and lines flux in healthy traction motor [2].


tics for 5 broken bars. Figure 8 presents a characteristic of potential distribution in the

middle of the air gap of the traction motor for different fault cases [4]. Analysis of

computer calculations realized in different computer program, enable elaborate an

original diagnostic an monitoring procedures, which make possible realization of diag-

nostics systems with prediction of traction motors faults states. Results of application

Diagprzem computer program will be presented in next papers.

The state of all components is monitored and recorded to Digital Fault Recorders

(DFR) [4,5], while the electrical values of every sensors, of the wheel vehicle drive

system terminal are measured by installed current and voltage transformers. Mechani-

cal and kinetics values (speed, temperature and strength) are measured by intelligent

Figure 6. Distribution of radial component of the flux density outside the stator traction motor for healthy

and for 5 broken bars of the rotor.

Figure 7. Traction motor torque characteristics for 5 broken bars.

Figure 8. Potential distribution in the middle of the air gap of the traction motor.


sensors and transducers. From the operator perspective an alarm situation arises when a

monitored value exceeds a predefined upper or lower limit, activating a sound or light

alert on control panel. An expert operator would handle this situation by first checking

the control panel indications, trying then to locate the faulted area, according to the

theoretical state of the switching equipment and the current values of the measurement

points. ANSYS, JMAG and FEMM computer programs were applied in calculation of

magnetic fields distribution for different faults state of the motors.

A sophisticated fault diagnosis and monitoring system can detect similar contradic-

tions and point out the optimal restoration sequence. The proposed expert system, use a

dedicated module for the topology and state estimation of the wheel vehicle. Some di-

agnostics algorithms are realized for qualification of failure work state a traction motor

and also of the whole wheel vehicle. Results of computer simulations of the different

failure state of the traction motor are presented in the paper. Results of calculations

were partially verified in laboratory experiments.

Conclusion

For analysis of magnetic field distribution can applied different method of analysis:

equivalent scheme, finite element and finite boundary method. Knowledge of magnetic

field distribution in steady state and in transient state of the motor enables precisely

computation of their parameters. FEA in 2D and 3D enables considerate a saturation of

magnetic circuit. From the power capability and the principal dimensions, the axial flux

PM and induction motors can be mounted into the wheels to realize the driving strat-

egy. The axial flux interior PM and induction motor seem to be best compromise in

terms of power/weight ratio, efficiency, compactness capability characteristics. Analy-

sis of computer calculations realized in different computer program, enable elaborate

an original diagnostic an monitoring procedures, which make possible realization of

diagnostics systems with prediction of traction motors faults states

References

[1] M. Aydin, J. Yao, S. Huang, T.A. Lipo: Design consideration and experimental results od an axial flux

PM motor with field control. Proceeding of ICEM’04, Cracow, Poland, 2004r, pp. 764-770.

[2] C. Bocaletti, C. Bruzzese: A procedure for squirrel cage induction motor phase model parameters identi-

fication and accurate rotor faults simulation: mathematical aspect. Proceedings of ISEF’05. September,

Vigo, Spain 2005.

[3] D. Platt, B.H. Smith: Twin rotor drive for an electric vehicles. IEEE Proceedings-B, vol. 140, no. 2,

March 1993r, pp. 131-138.

[4] Z. Szymański: Analysis of the field distribution in special construction traction wheel motor in electric

and hybrid vehicles. Proceedings of ISEF’05, September 15–17, Baiona, Spain.

[5] Z. Szymański: Application of the artificial intelligence methods in diagnostics of mine machine drive

system. Proceedings of 20th World Mining Congress, 7–11 November, Tehran, Iran.


Author Index

Albert, J. 16

Apanasewicz, S. 8, 58

Arsov, L. 167

Barbieri, L. 381

Barglik, J. 202

Bednarek, K. 85

Binder, A. 116

Buchau, A. 16, 93

Byczkowska-Lipińska, L. 438

Cardoso, J.R. 158

Cecílio, J. 144

Černigoj, A. 179

Cha, J. 370

Cho, Y. 370

Cieśla, A. 350, 356

Clemens, M. 431

Coco, S. 425

Corda, J. 343

Cundeva, S. 167

Cundeva-Blajer, M. 167

Czerwiński, D. 403

Czerwiński, M. 202

de Oliveira, O.L. 158

Deak, Cs. 116

Dems, M. 130

Desideri, D. 363

di Gerlando, A. 192

di Napoli, A. 151

Dolezel, I. v

Duchesne, S. 137

Dular, P. 307

Faktorová, D. 21

Ferreira da Luz, M.V. 307

Fireţeanu, V. 284

Fišer, R. 179

Foggia, A. 175

Foglia, G. 192

Frenner, K. 80

Fujii, N. 318

Fujiwara, K. 47

Gaber, M. 388

Garda, B. 350

Gašparin, L. 179

Gawrylczyk, K.M. 64

Gawrys, P. 34

Giżewski, T. 403

Goleman, R. 403

Gorenc, D. 108

Gyftakis, K. 247

Hafla, W. 16, 80, 93

Hameyer, K. 39, 101

Hasegawa, Y. 220

Hering, M. 202

Herranz Gracia, M. 39

Hinrichsen, V. 431

Hirata, K. 220

Hong, D.-K. 184

Iancu, V. 259

Ida, K. 124

Irimie, D. 116

Ishihara, Y. 47

Jalmuzny, W. 231

Jamil, S.M. 343

Jussila, H. 253

Kang, D.-H. 184, 268

Kantartzis, N.V. 212

Kappatou, J. 247

Kawase, Y. 124

Kitagawa, W. 47

Kobylanski, L. 175

Kolehmainen, J. 240

Komęza, K. 130

Krawczyk, A. v

Kucharski, D. 26

Lalas, A.X. 212

Laudani, A. 425

Lecointe, J.-Ph. 137

Lemos Antunes, C. 144

Lesniewska, E. 231

Leva, S. 381

Li, J. 370

Lidozzi, A. 151

Lopez-Fernandez, X.M. 53

Marusic, I. 108

Maschio, A. 363

Maugeri, V. 381

Advanced Computer Techniques in Applied ElectromagneticsS. Wiak et al. (Eds.)IOS Press, 2008© 2008 The authors and IOS Press. All rights reserved.

457

Miljavec, D. 299

Minamide, A. 294

Mipo, J.-C. 175

Mitsutake, Y. 220

Mizuma, T. 318

Morando, A.P. 381

Naoe, N. 294

Napieralska-Juszczak, E. 137

Nicula, C. 259

Niedbała, R. 26

Niemelä, M. 253, 276

Ota, T. 220

Parviainen, A. 276, 313

Pawłowski, S. 8

Pereira, J.M.T. 417

Perez, J. 72

Perini, R. 192

Petrovic, Lj. 116

Pineda, M. 72

Poli, E. 363

Popenda, A. 335

Przybylski, M. 34

Puche, R. 72

Purcarea, C. 116

Putek, P. 64

Pyrhönen, J. 253, 276

Reichert, K. 116

Riganti Fulginei, F. 425

Roger Folch, J. 72

Roger, D. 137

Roytgarts, M. 324

Rucker, W.M. 16, 80, 93

Rusek, A. 335

Safacas, A. 247

Sahin, O. 396

Sakata, K. 318

Salminen, P. 253, 276

Salvini, A. 425

Sartori, C.A.F. 158

Scheiblich, C. 80

Sehit, S. 396

Seifert, J. 431

Serrao, V. 151

Shiota, H. 124

Slusarek, B. 34

Smirnov, А. 324

Solero, L. 151

Sosnowski, J. 410

Soto Rodriguez, A. 53

Souto Revenga, D. 53

Steinmetz, T. 431

Stepien, S. 101

Strete, L. 259, 268

Szymanski, G. 101

Szymański, Z. 449

Takemata, K. 294

Tamto, Y. 175

Todaka, T. 47

Trlep, M. 388

Tsiboukis, T.D. 212

Turowski, J. 53

Valente, H. 144

Valtonen, M. 313

Varlamov, Yu. 324

Viorel, I.-A. 259, 268

Wac-Włodarczyk, A. 403

Weida, D. 431

Wesołowski, M. 26, 202

Wiak, S. v, 438

Woo, B.-C. 184

Yamagiwa, A. 124

Yamaguchi, T. 124

Yamamoto, T. 220

Yaman, B. 396

Zakrzewski, K. 3

Zidarič, B. 299

458



Documents

Advanced Computer Techniques in Applied Electromagnetics