Eindhoven University of Technology MASTER Design and ...as functions of the coil thickness and permanent magnet thickness. 37 5.1 A 2D thermal equivalent circuit of the C-shaped actuator

Eindhoven University of Technology

MASTER

Design and optimization of a C-shaped actuator with a soft magnetic composite core

Engelen, D.H.M.

Award date:2005

Link to publication

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https://research.tue.nl/en/studentthesis/design-and-optimization-of-a-cshaped-actuator-with-a-soft-magnetic-composite-core(26e075d8-c912-46fa-ab80-bb2b97bac570).html

Design and Optimization ofa C-shaped actuator with aSoft Magnetic Composite

Core

Dirk EngelenStudent ID: 533499

Tecnotion BV Almelo and TU/e

Supervisors:Prof.dr.ir. A.J.A Vandenput

Dr. E.A Lomonovair. P. Krechting, PDEng

28th November 2005

Abstract

A novel linear actuator has been designed under the authority of Tecnotion BVAlmelo in cooperation with Eindhoven University of Technology (TU/e). Themain goal of the research is to create an actuator with a relatively high forcedensity in comparison with existing linear actuators. The actuator is called theC-shaped actuator and will be a new type of linear motor for Tecnotion BV.

The stator of the actuator has of a C-shaped back iron. At the inside of theback iron frame a NS array of permanent magnets is fixed at each side. Therotor/mover bar consists of a core on which six coils are wound equidistantlyaround the periphery of this core. The core is made of soft magnetic composites(SMC) and is able to handle the flux from all the three sides originated fromthe permanent magnets. Therefore, this C-shaped actuator uses effectively threesides of the mover bar for the force production.

After an analytical approach, the actuator is modelled by means of a magneticequivalent circuit (MEC). This MEC model allows fast calculations to get a properunderstanding of several dimensions and its force production.

Since almost every actuator is limited by its thermal behavior, also a thermalequivalent circuit (TEC) is made. The TEC model can be used to define thecontinuous force of the C-shaped actuator and to analyze its steady state thermalbehavior.

Both models are implemented in one Mathcad script. To verify and obtainextra information a 3D finite element model is made in Maxwell 3D (by AnsoftCo). This FEM model is fully parametric. This allows the user to do a parametricsearch in 3D FEM (Optimetrix by Ansoft Co), choose the right design and dosome magnetostatic field analysis in the post processor.

To come to a proper design, it is necessary to know the specific goal of thisactuator. When this information is available and the global dimensions obtainedfrom the MEC model are inspected, one can start with the electromagnetic designroutine. This design routine consists of three steps and uses the parametric 3DFEM model. After these steps, the dimensions of the new C-shaped actuator areknown and the actuator is ready to be manufactured.

A main disadvantage of the C-shaped actuator is the high level of coggingforce in the initial design. Therefore, a few solutions are investigated with mag-netostatic 3D FEM to reduce this disturbing force.

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ii

Preface

This thesis concludes the work I have done on the final project to obtain my Masterof Science degree at the Electrical Engineering Department of the EindhovenUniversity of Technology. It mainly describes the design flow of a novel linearactuator. To have a proper design start a magnetic equivalent circuit modeland thermal model have been built. To verify the models and the design of theactuator 3D finite element software has been used. The project was formulatedand conducted under authority of Tecnotion BV in combination with TU/e.

I would like to thank prof.dr.ir.Andre Vandenput for the opportunity to makemy final project in the Electromechanics and Power Electronics group and for hissupervision.

I am grateful to dr. Elena Lomonova for her support, ideas and comments onmy thesis.

I would also like to thank ir. Peter Krechting, PDEng, of Tecnotion BV forhis clear advices and coaching efforts to finalize this project.

Furthermore I would like to thank ir. Helm Jansen for reviewing my thesisand Anton Lebedev, M.Sc., for providing the BH-curve for Somalay 500. Theyhave also supported me with using the 3D finite element package.

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iv

Contents

List of Figures ix

List of Symbols x

1 Introduction 11.1 Background, topic and goal . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline of the C-shaped actuator . . . . . . . . . . . . . . . . . . . 11.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Shape and geometry of the actuator 52.1 Principle of the C-shaped actuator . . . . . . . . . . . . . . . . . . 52.2 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Back EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Basic magnetic design . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Soft Magnetic Composites and permanent magnets 153.1 Background of soft magnetic composites . . . . . . . . . . . . . . . 153.2 Somaloy 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Permanent magnets . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Magnetic Equivalent Circuit Model 234.1 Analogies between electric and magnetic circuits . . . . . . . . . . 234.2 Magnetic equivalent circuit model of the C-shaped actuator . . . . 25

4.2.1 Flux paths in the actuator . . . . . . . . . . . . . . . . . . . 254.2.2 MEC model . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Calculation of MEC model . . . . . . . . . . . . . . . . . . . . . . 334.4 Force extraction from MEC . . . . . . . . . . . . . . . . . . . . . . 344.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Thermal analysis 395.1 Sources and materials . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Thermal equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . 415.3.1 Thermal convection resistance . . . . . . . . . . . . . . . . . 41

v

CONTENTS

5.3.2 Thermal conduction resistance in the coil . . . . . . . . . . 445.3.3 Heat sources . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Electromagnetic design 496.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Figures of merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 Continuous and peak force . . . . . . . . . . . . . . . . . . . . . . . 52

6.3.1 Continuous force . . . . . . . . . . . . . . . . . . . . . . . . 526.3.2 Peak force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4 Introduction to the design flow . . . . . . . . . . . . . . . . . . . . 536.4.1 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4.2 Optimetrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4.3 Design steps . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.5 Design flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.5.1 First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.5.2 Second step . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.5.3 Third step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.6 Conclusion and recommendation . . . . . . . . . . . . . . . . . . . 61

7 Cogging Force 637.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.2 Cogging reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.2.1 Mover extension . . . . . . . . . . . . . . . . . . . . . . . . 657.2.2 Skewing of the permanent magnets . . . . . . . . . . . . . . 657.2.3 Varying the shape of the extension . . . . . . . . . . . . . . 677.2.4 Shift of the permanent magnets . . . . . . . . . . . . . . . . 68

7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8 Conclusion and Recommendation 718.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.2 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

References 76

A Neorem permanent magnets 77

B Design Results 78B.1 Description steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.2 Step 1a results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.3 Step 1b interpolated results . . . . . . . . . . . . . . . . . . . . . . 81B.4 Step 2 results Continuous . . . . . . . . . . . . . . . . . . . . . . . 82B.5 Step 2 results Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.6 Step 3 results Continuous . . . . . . . . . . . . . . . . . . . . . . . 84B.7 Step 3 results Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

C Inductance 86

D Analytical versus FEM 87

vi

CONTENTS

E Field plots 88E.1 No current B-field . . . . . . . . . . . . . . . . . . . . . . . . . . . 89E.2 No current B-field . . . . . . . . . . . . . . . . . . . . . . . . . . . 90E.3 Continuous current B-field . . . . . . . . . . . . . . . . . . . . . . . 91E.4 Continuous current B-field . . . . . . . . . . . . . . . . . . . . . . . 92E.5 Peak current B-field . . . . . . . . . . . . . . . . . . . . . . . . . . 93E.6 Peak current B-field . . . . . . . . . . . . . . . . . . . . . . . . . . 94E.7 Peak current H-field . . . . . . . . . . . . . . . . . . . . . . . . . . 95

F Mathcad script 96

vii

List of Figures

1.1 A 3D impression of the C-shaped linear actuator. . . . . . . . . . . 2

2.1 The transformation from 3D to 2D topology by folding out theC-shaped actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Flux paths which are created by the permanent magnets and thecurrent is equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 The magnetic field (red arrow), due to the magnetomotive forceproduced by the applied current in the coils at different steps. . . 7

2.4 Axial flux torus type non-slotted surface mounted permanent mag-net motor configuration (TORUS-NS)in 2D and 3D with flux di-rections. [Aydin et al., 2004] . . . . . . . . . . . . . . . . . . . . . 8

2.5 Mover bar with the double three-phase windings. . . . . . . . . . . 102.6 Cross-section of the C-shaped actuator with one flux path. . . . . 13

3.1 Schematic picture of soft magnetic composite material. . . . . . . . 163.2 Initial Permeability vs. Frequency . . . . . . . . . . . . . . . . . . 163.3 The BH-curve for heat-treated Somaloy Somaloy 500+0.5% Kenol-

ube at different densities. . . . . . . . . . . . . . . . . . . . . . . . 173.4 a) Core loss/cycle as a function of frequency; Heat-Treated Somaloy

500 +0.5%Kenolube. b) Hysteresis loop at DC and 200Hz; Heat-Treated Somaloy 500+0.5%Kenolube, 7.47g/cm3. . . . . . . . . . 18

3.5 Demagnetization curve with recoil loop and the operation point ofthe permanent magnet. . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6 Temperature dependent demagnetization curves of a Neorem rareearth permanent magnet. . . . . . . . . . . . . . . . . . . . . . . . 20

3.7 Demagnetization curve of a rare earth permanent magnet. . . . . . 21

4.1 A flux tube in a magnetic field. . . . . . . . . . . . . . . . . . . . . 244.2 The C-shaped motor with only the bottom side. . . . . . . . . . . 254.3 The main flux paths in the C-shaped actuator. . . . . . . . . . . . 264.4 Magnetic equivalent circuit of the C-shaped actuator. . . . . . . . 274.5 Flux density distribution on the top of the permanent magnet. . . 294.6 Reluctance calculation for a trapezoidal element with a direction

of the flux perpendicular to w(x). . . . . . . . . . . . . . . . . . . . 304.7 Circular-arc straight line reluctance patterns to model the leakage

flux between two magnets. . . . . . . . . . . . . . . . . . . . . . . . 314.8 The flux distribution in the middle of the SMC core. . . . . . . . . 324.9 The flux with different current densities in 2D FEM. . . . . . . . . 35

viii

LIST OF FIGURES

4.10 Lorentz forces obtained by the MEC model and 3D FEM Maxwellas functions of the coil thickness and permanent magnet thickness. 37

5.1 A 2D thermal equivalent circuit of the C-shaped actuator. . . . . . 425.2 A flow of two media as function of Reynolds number, namely air

(Prandtl=0.72) and water (Prandtl=7.07). . . . . . . . . . . . . . . 435.3 Different heat flow directions in the coil. . . . . . . . . . . . . . . . 445.4 The 2D thermal equivalent circuit with the temperature distribu-

tion of the coil and the SMC core. . . . . . . . . . . . . . . . . . . 455.5 a) Thermal analysis of the 3D FEM model without the stator, b)

The cross-section of the bottom side of the coil in the middle. . . . 475.6 Thermal analysis of the 3D FEM model with permanent magnets

and the back iron included. . . . . . . . . . . . . . . . . . . . . . . 48

6.1 Cross-section of the C-shaped actuator in XZ-plane . . . . . . . . . 516.2 The most outer values and middle value for [Lpm, Lcoil]. The pole

pitch is varying between 15mm and 45mm as a function of Csm. . 566.3 The optimum value for the pole pitch with a given coil and perma-

nent magnet thickness. . . . . . . . . . . . . . . . . . . . . . . . . . 576.4 The steepness function, which is independent on the current. . . . 586.5 a) Continuous force and b) peak force for different configurations. . 586.6 Cfm for the continuous and peak regimes which is proportional

with the acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . 596.7 Lorentz force as a function of the pole pitch and the pitch ratio α. 606.8 Two figures of merit in design step 3. . . . . . . . . . . . . . . . . . 60

7.1 Cogging force with no extra adjustments to reduce cogging. . . . 647.2 One side of the mover bar with the extension of the SMC core. . . 657.3 The cogging force as a function of position and extension length. . 667.4 Cogging force with an extension of 6.18mm . . . . . . . . . . . . . 667.5 The cogging force as function of position and skew angle. . . . . . 677.6 The cogging force as function of position and triangle length. . . . 687.7 The cogging force as function of position and shift distance. . . . . 69

D.1 Forces obtained by the Analytical model and 3D FEM Maxwellas function of the coil thickness and permanent magnet thickness.Results obtain from section 6.5 . . . . . . . . . . . . . . . . . . . . 87

ix

List of Symbols

FEM finite element methodSMC soft magnetic compositeMEC magnetic equivalent circuitTEC thermal equivalent circuitA surface [m2]A magnetic vector potential [V s/m]Br remanent flux density [V s/m2]B,B magnetic flux density [V s/m2]B top value of the flux density [T ]B mean flux density [T ]C specific heat capacity [J/kgm2]Csm steepness divided by mass [N2/Wkg]Csv steepness divided by volume [N2/Wcm3]Cfm force divided by mass [N/kg]Cfv force divided by volume [N/cm3]D,D electric flux density [As/m2]E,E electric field strength [V/m]f frequency [Hz]F force [N ]F magnetomotive force [A]ffill fill factorhc convection heat transfer coefficient [W/m2K]Hsmc height of the SMC mover bar [mm]H,H magnetic field strength [A/m]Hc coercive magnetizing force [A/m]I,i current [A]J,J electric current density [A/m2]k correction factor for the MEC modelK machine constant [N/A]Lc length of the coil [mm]Lcoil coil thickness [mm]Lpm permanent magnet thickness [mm]Liron back iron thickness [mm]Lairgap airgap [mm]Lglue glue layer thickness [mm]M,M magnetization [A/m]N amount of windings

x

LIST OF FIGURES

Nu Nusselts numberP,P polarization [As/m2]P power [W ]P permeance [H]Q heat transfer [W ]R resistance [Ω]Re Reynolds numberR reluctance [H−1]S steepness [N2/W ]T temperature [K]v speed [m/s]V volume [m3]Wsmc width of the SMC mover bar [mm]Wpm width of the permanent magnet [mm]W energy [J ]α0 temperature coefficientα pitch ratioε emissivity of a body or surfaceλ thermal conductivity [W/mK]µ0 permeability of vacuum, per definition: 4π · 10−7[Hm−1]µr relative permeabilityρcu material density in [kg/m3]σ specific conductivity [(Ωm)−1]τcp coil pitch [mm]τmp magnet pitch [mm]τpp pole pitch [mm]φ flux in [V s]φsmc flux in SMC core [V s]ψ flux linkage [V s]

xi

LIST OF FIGURES

xii

Chapter 1

Introduction

1.1 Background, topic and goal

Permanent magnet linear actuators are used extensively in speed- and position-controlled drive systems, for example, in factory automation and as equipmentof a semiconductor manufacturer. Linear brushless permanent magnet actuatordrives can offer significant speed control and positional accuracy.

This report contains a new design for a novel linear actuator. The main ideais to allow a 3D flux in the structure and use effectively three sides of the moverfor the force production. The 3D flux is possible due to the use of soft magneticcomposite materials (SMC), which have isotropic properties. The name C-shapedactuator is considered because of the predetermined shape of the stator. Thedefinition of U-shaped actuator is not chosen on purpose, because it is alreadyreserved and widely used for ironless linear actuators.

The main goal of the research is to create an actuator with a relatively highforce density [Sahin, 2002] in comparison with existing linear actuators . The an-alytical analysis is done with basic electromagnetic design tools. The numericalanalysis is done with a magnetic equivalent circuit, a finite element software pack-age. The finite element package is developed by Ansoft Corporate and is calledMaxwell [Ansoft Co Maxwell 2D and 3D, n.a.]. The Maxwell software has twodifferent versions, namely Maxwell 2D and 3D. The 3D version can be coupledwith the Optimetrics module to perform parametric searches.

The thermal behavior is estimated with a thermal equivalent resistance circuitand with the finite element software from Ansoft Co. Later on, the models can beverified and adapted with the help of a thermal camera and temperature sensors.

The prototype and its analysis are the essential investigation steps towardsa new linear actuator product line for Tecnotion BV, Almelo. Currently, theymanufacture ironless and iron core linear actuators. The C-shaped actuator willbe an addition to the already available product types.

1.2 Outline of the C-shaped actuator

Figure 1.1 shows an impression of the C-shaped actuator that will be designed.Due to the complexity of the geometry it is wise to introduce the following defi-nitions of the actuator assembly:

1

Chapter 1. Introduction

Figure 1.1: A 3D impression of the C-shaped linear actuator.

• (Stator) The back iron consists of two vertical sides (YZ-plane) and onehorizontal part (XY-plane). The permanent magnet arrays, with the NSstructure, are fixed on the back iron. The permanent magnets are alter-nately magnetized (see figure 1.1).

• (Rotor) The mover bar consists of a rectangular soft magnetic compositecore with six coils which are wound around this core.

The flux, originated from the permanent magnets, links from one permanentmagnet to the other one via the air, coils and/or the mover. The mover bar iscovering two permanent magnets with six coils, this can be extended to twelvecoils over four permanent magnets.

1.3 Structure of the thesis

This chapter and the second one of this thesis provide an introduction to theC-shaped motor. The principles of the actuator are explained with an analyticalapproach to predict the force production.

The third chapter contains information about the soft magnetic composite andthe permanent magnets that have been used.

2

1.3. Structure of the thesis

Before the electromagnetic design flow is discussed, first two models are ex-plained in the chapters 4 and 5. Chapter 4 contains the magnetic equivalent circuit(MEC) model to estimate the magnetic properties and produced force. Chapter5 contains the thermal equivalent circuit model that will predict the steady statethermal behavior of the C-shaped actuator.

In chapter 6 the actual design flow is presented.The C-shaped motor has a rather big disturbing force in its initial design,

namely cogging force. Chapter 7 gives some options to reduce this cogging force.The thesis ends with conclusions and recommendations.

3

Chapter 1. Introduction

4

Chapter 2

Shape and geometry of theactuator

This chapter contains the description of the working principle of the C-shapedactuator. Due to the novelty and uniqueness of the proposed configuration, theC-shaped actuator will be first explained in two dimensions and later on from thethree dimensional point of view. Further, some initial calculations for propulsionforce, induced voltage and some basic magnetic calculations are treated.

2.1 Principle of the C-shaped actuator

The structure of the actuator is presented in figure 1.1, which is used for 3DFEM analysis of the C-shaped actuator. The stator consists of back iron andpermanent magnets. The permanent magnet arrays are attached to the back ironand the permanent magnets are alternately magnetized. The mover bar(rotor)will be kept in position by a linear bearing construction. The first prototype willbe attached to test setup, which is going to be built by Tecnotion BV. The actualconstruction of the bearing and suspension is beyond the scope of this thesis andwill be manufactured by Tecnotion BV.

Figure 2.1: The transformation from 3D to 2D topology by folding out the C-shaped actuator.

5

Chapter 2. Shape and geometry of the actuator

Figure 2.2: Flux paths which are created by the permanent magnets and thecurrent is equal to zero.

The mover bar can only be propulsed in the Y-direction. Six individual con-centrical coils (A, C’, B, A’, C, B’) are wound around and distributed equidistantlyalong the periphery of the mover bar (see figure 1.1) and connected to a threephase power supply that supplies the electric energy to the C-shaped actuator.

With the aim to present the operating principle, it is wise to start from a2D-view. The easiest way to make this simplification is to unwrap the structure,like is done in figure 2.1.

The whole arrangement of the C-shaped actuator is now presented as an un-rolled linear structure. The permanent magnets are now alternately magnetizedin the Z-direction. Meanwhile, some of the peculiarities of the electromagneticstructure and behavior of the actuator are missed. For example, there are lessedges in the mover bar, the flux from the permanent magnets is not going tothe same cross-section area of the mover bar and the back iron. Therefore, thisapproach is only applicable for a better understanding of the working principleof C-shaped actuator. The schematic view of the actuator depicts immediatelythe similarities between the C-shaped actuator and a classical linear synchronouspermanent magnet machine. The main flux paths of the 2D structure are shownin figure 2.2.

The propulsion force is generated as an action of the armature magnetic fluxand permanent magnet flux [Gieras and Piech, 2000]. The double three-phasewinding produces a sinusoidal or quasi-sinusoidal distribution of the magnetomo-tive force.

The three-phase current in the windings produces the magnetomotive force. Instep one, of figure 2.3, the resultant of the magnetic field, that is produced due tothe magnetomotive force, is figured as a red arrow. The currents are proportionalto the magnetomotive force and represented in the vector diagram. Accordingthe Lorentz force the mover bar in step one will go to the right.

In figure 2.3, the other sequential steps are pictured, which are shown themovement of the actuator in right direction.

The velocity of mover is proportional with the input frequency of the three-phase power supply. For this configuration the velocity vmover can be expressed

6

2.1. Principle of the C-shaped actuator

Figure 2.3: The magnetic field (red arrow), due to the magnetomotive force pro-duced by the applied current in the coils at different steps.

as:vmover = 2fsτpp (2.1)

This is the synchronous speed and is dependent on the input frequency fs andthe pole pitch length τpp.

To analyze the working principle in the real topology, the structure have tobe fold back to its original shape. Now, the flux paths of the permanent magnetsare going from the three sides of the stator into the mover bar. The flux density,in the mover bar, is the highest at the position above the space between the twopermanent magnets (see figure 2.2). This concerns the back iron too. The backiron is not that fast saturated when compared with the SMC core of the moverbar, because the surface of the cross-section of the back iron is larger and have tohandle only the flux path of its own side.

In most other standard linear synchronous machines, the core/armature ofthe rotor is made of laminated steel. This reduces the eddy current losses in thesteel. In case of the C-shaped actuator, laminated steel is not suitable as materialfor the core in the mover bar. Laminated steel can only be magnetized in thedirection of laminated structure itself. The flux which flows perpendicular to the

7


laminated steel will notice a very high magnetic resistance due to the insulationbetween steel the plates. The core in the mover bar of the C-shaped actuator hasto allow flux lines from all directions without creating a large magnetic resistancein a specific direction, in other words it have to allow a 3D flux flow. Soft magneticcomposite material can handle this 3D flux, due to its unique material properties.

The concentration of the flux in the core results in a relatively high flux densityin the mover. Therefore, this part will require a careful look, because it is theonly common flux path. The properties of the soft magnetic composite materialwill be closer inspected in chapter 3.

2.2 Force

The C-shaped actuator is more or less comparable with an axial flux torus typenon-slotted motor, a rotating machine as figured in 2.4(a). This motor uses onlythe two long flat sides of the windings to produce the torque, in contrast with theC-shaped actuator, which uses three sides.

stator with

airgap windings

magnets

rotor

(a) (b)

Figure 2.4: Axial flux torus type non-slotted surface mounted permanent magnetmotor configuration (TORUS-NS)in 2D and 3D with flux directions. [Aydin et al.,2004]

In the torus topology, the windings in the airgap are used for the torqueproduction [Aydin et al., 2004]. This is also applicable for the force productionin the C-shaped actuator. Both actuators have airgap-windings, therefore it ispossible to use Lorentz force to estimate the propulsion force on the windings[Reece and Preston, 2000].

The reluctance force component will be generated due to a current in combi-nation with an inductance variation per position, defined as:

Frel =12I2 dL(x)

dx(2.2)

In the torus-NS topology [Upadhyay et al., 2004] [Gieras and Wing, 2002], thereluctance force component is negligible. A 3D FEM simulation has calculate the

8

2.2. Force

self inductance per coil as function of the position. The variation, as plotted inAppendix C, is minimal and therefore not introducing an extra force componentfor smaller current densities. The inductance calculations in Maxwell are donewith low currents. It is wise to analyze this reluctance force in later stage withother condition, this is not incorporated in this thesis.

A rotating slotless axial flux permanent magnet has no cogging/cogging forcebecause of its slotless circular structure with no end-effects. The permanent mag-net will experience the same reluctance on every position and therefore the coggingforce will be negligible for the torus motor. This is definitely not applicable onthe C-shaped actuator, because the mover bar is not infinite long. In the section3.3 and chapter 7 the cogging force will be further evaluated.

The cogging force is only disturbing in moving direction if it is attached toa proper bearing system. Nevertheless, there are also other force components,due to the permanent magnets. The permanent magnets will attract the SMCcore on all sides. The permanent magnets on the horizontal part will attract theSMC core significantly to the z-direction. This detail will be important when aproper bearing has to be determined in a later stage. The attraction force onthe core due to the permanent magnets on the vertical sides is nett zero. On theother hand, the force on vertical sides of the back iron is significant and equal tothe attraction force in the z-direction on the core. This is also important frommechanical point of view, when defining the back iron.

The force acting on one thin conductor that is crossing magnetic flux lines canbe expressed by the Lorentz’ force equation, but the 3D shape and the commonflux path in the actuator makes it convenient to look at the flux linkage.

The force calculation starts from the voltage equation of one single coil withN turns:

u = iR +dψ

dtwhere ψ = Nφ (2.3)

The flux linkage ψ is position and current dependent therefore:

dψ

dt=

δψ

δy· dy

dt+

δψ

δi· di

dt(2.4)

For a linear inductance holds L = δψδi and equation 2.4 becomes:

dψ

dt=

δψ

δy· dy

dt+ L · di

dt(2.5)

Multiplying equation 2.3 with the current, the equation becomes a power balance:

P = i2R + i · δψ

δy· dy

dt+

12· dLi2

dt(2.6)

The mechanical power is as function of force F and the velocity v, where v = dydt :

Pmech = F · dy

dt= i · δψ

δy· v (2.7)

where the force of one coil can be defined as:

F = i · δψ

δy(2.8)

9


Equation 2.8 will be used to calculate the propulsion force of the C-shaped actu-ator.

The prototype is proposed as a double three-phase winding in series, like infigures 2.2 and 2.5. Three coils have the same length as one single permanentmagnet:

τpp = τcp (2.9)

In figure 2.6 can be seen how the pole pitch τpp is depending on the coil pitch

Figure 2.5: Mover bar with the double three-phase windings.

τcp and vice versa. The pole pitch τpp is therefore an important design variable.The first three coils are the phases A, C ′ and B. The next set of coils are thenegative phases, A′, C and B′ respectively, see also figure 2.2.

To predict the force contribution of the coils analytically, equation 2.8 canbe used. This equation needs the flux distribution in the mover bar. For thisanalytical approach, there is assumed that the flux is sinusoidally distributed inthe core of the mover bar and is defined as:

φsmc(y) = φsmc cos(π

τppy) (2.10)

In subsection 2.4, the flux φsmc will be defined. The current per coil can bedefined as:

Ik(y) = I sin(

π

τppy +

π

6(2k + 1) + ϕ

)(2.11)

where k is an integer and represents the coils A, C ′, B, A′, C and B′ as 0, 1, 2,3, 4 and 5 respectively. ϕ is a variable phase shift and I is the amplitude of thecurrent.

The force is dependent on the amount of flux linkage change per positionmultiplied by the current of a coil. To approach the flux linkage per coil, the

10

2.3. Back EMF

mean value of the flux linkage is determined over one coil length per coil:

ψk(y) =3

τpp

∫ y+τpp3 (k+1)

y+τpp3 (k)

Nφsmc cos(π

τppy)dy

=3ψsmc

π

[sin

(π

τppy +

π

3(k + 1)

)− sin

(π

τppy +

π

3k

)](2.12)

The derivative with respect to the position y per coil is:

dψk(y)dy

=3ψsmc

τpp

[cos

(π

τppy +

π

3(k + 1)

)− cos

(π

τppy +

π

3k

)](2.13)

The total force can be defined after combining equation 2.11 and 2.13 as follows:

Ftot =5∑

k=0

Ik(y) · dψk(y)dy

(2.14)

= I

5∑

k=0

sin(

π

τppy +

π

6(2k + 1) + ϕ

)· dψk(y)

dy

The peak current can be defined as the RMS value by I/√

2:

Ftot = Irms

5∑

k=0

√2 sin

(π

τppy +

π

6(2k + 1) + ϕ

)· dψk(y)

dy(2.15)

The sum of equation 2.15 contains the force constant Kf and can be rewritten bysome goniometric relations as:

Kf =3√

22

ψsmc

τpp

5∑

k=0

[− 1+sin(

2π

τppy +

2π

3k +

π

2)− sin(

2π

τppy +

2π

3k +

π

6)]

(2.16)

Equation 2.16 can be derived in a mathematical program like Mathematica. Theforce constant can be derived as:

Kf = 9√

2φsmc

τppN (2.17)

The force can now be described as function of the machine constant Kf :

Ftot = Kf · Irms (2.18)

2.3 Back EMF

The voltage equation 2.3 and equation 2.5 reveal that when the actuator moveswith a certain speed, the flux changes per position, this will generate a certainvoltage, back electromotive force (EMF):

ecoil =dψk(y)

dy· dy

dt= N

dφk(y)dy

· v (2.19)

11


The velocity of the mover is already defined in equation 2.1. This equation canbe combined with equation 2.13. The maximum of equation 2.13 will be 3

τppψ.

This gives for one coil a top value of:

ecoil =3

τppψsmc

3π

v (2.20)

To obtain the rms value equation 2.20 has to be multiplied by√

2. The velocityequation 2.1 can also be implemented:

Ecoil = 6√

2Nφsmc3π

fs (2.21)

Defining the EMF constant as:

KE =3√

2π

Nφsmc3

τpp(2.22)

the generated rms voltage of one coil Ecoil can be described as a function of themachine constant and velocity:

Ecoil = KE · v (2.23)

The propulsion force is proportional to the magnetic field and the phase cur-rents, see equation 2.15. The back EMF induced in the phase windings is pro-portional to the magnetic field and the speed of the mover bar, see equation 2.20.The total thrust force is the sum of the forces produced by all the phases. Onlyif the force functions are sinusoidal, balanced and have a symmetric sinusoidalcommutation of the phase currents a smooth propulsion force will be acquired.In practice this is not the case. So higher harmonics due to non linear behavior,non sinusoidally back EMF, etc. will created disturbance forces.

2.4 Basic magnetic design

To obtain the force and back EMF it is necessary to know the maximum fluxvalue φsmc in the main flux path. Analytically this can be done with Ampere’slaw: ∮

C

Hdl = NI (2.24)

By assuming that the back iron and the core in the mover bar have an infinitepermeability and neglecting the magnet leakage flux, the simple circuit equationresults:

2HpmLpm + 2HairgapLairgap + 2HcoilLcoil + 2HglueLglue = 0 (2.25)

The relative permeability of air, copper and glue is assumed to be one. Therefore,Lgap = Lairgap + Lcoil + Lglue is defined. This gives the following equation:

2HpmLpm + 2HgapLgap = 0 (2.26)

Equation 2.26 can also be written as:

Hpm = −BgapLgap

µ0Lpm(2.27)

12

2.5. Conclusion

Figure 2.6: Cross-section of the C-shaped actuator with one flux path.

The permanent magnet can be defined as:

Bpm = µ0µrpmHpm + Br (2.28)

Where Bpm, Hpm and Bgap are the permanent magnet flux density, permanentmagnet field strength and flux density in the gap. Br and µrpm are the perma-nent magnet remanence and relative permeability. Inserting 2.28 into 2.27 andassuming that there is no tangential flux density component, so Bpm = Bgap theflux density in the gap is:

Bgap =Br

1 + µrpm Lgap

Lpm

(2.29)

Bgap is the flux density of flux line that originates from the middle of one per-manent magnet and goes via the SMC core to the adjacent permanent magnetto return via the back iron, see figure 2.6. In section 2.2, there is assumed asinusoidally distributed flux density. Bgap is therefore the top of this flux densitydistribution. The mean value of Bgap can be deriving by multiplying it by 2/π.The amount of flux in the gap φgap is equal to φsmc. Therefore, the top value ofthe flux φsmc in the middle of the SMC core from three sides can be defined as:

φsmc = 3 · 2π

Bgap ·Asmc =6π

Bgapτpp

2Wcoil (2.30)

The surface Asmc is half a pole pitch multiplied by the width of the coil Wcoil.Wcoil is the mean width of this surface, because the width of the permanentmagnets is larger than the width of the mover bar, see figure 2.1.

Combining equation 2.18 and 2.30, it is comparable to the Lorentz force equa-tion.

2.5 Conclusion

This analytical approach gives a good understanding of the working principle ofthe C-shaped actuator. There are made a few assumptions to come to this basicapproach. Chapter 4 will give a closer look to the magnetic properties of thisactuator.

In Appendix D the forces obtained with the analytical function are comparedwith results of section 6.5. The analytical tool is not suitable for the design flow,because it takes not in account the amount leakage flux, the shape of flux pathsand the saturation in the SMC core. And therefore it is not accurate enough.

13


14

Chapter 3

Soft Magnetic Compositesand permanent magnets

An essential part of the proper performance of a designed actuator depends onthe right selection of the hard and soft magnetic materials.

Soft magnetic composites (SMC) are not very often used in electrical ma-chines nowadays [Andersson, n.a.]. But these materials have a good potentialfor some novel actuator designs with complex structures where the flux works in3-dimensional paths[Jack, 1998], like the C-shaped actuator.

The C-shaped actuator is a synchronous permanent magnet machine with alarge number of permanent magnets. The large path between the permanentmagnet and the mover bar brings out a high magnetic resistance. It is thereforeimportant to chose the proper permanent magnets with the right sizes to avoidproblems at a later stage.

3.1 Background of soft magnetic composites

Soft magnetic composites can be described as ferromagnetic particles surroundedby an electrical insulated film. These particles (see figure 3.1) will be hold togetherwith a binder. By compressing all these particles in a very intensive way thereappears a solid piece. This piece of material has a very low conduction due tothe overall insulating layers between all the particles. On the other hand it has arather high permeability (µmax is around 500).

The manufacturing process starts with a powder of insulated particles. Thispowder is compacted to green bodies (unsintered bodies) [Roberts et al., 2001].This green body can not be conventionally sintered with high temperatures. Hightemperatures will damage and destroy the binder, that increases the electricalconductivity, hence destroying the electromagnetic properties in alternating fields.As consequence this piece of soft magnetic composite will not have the samemechanical strength as conventional sintered components. A general rule of thumbis that less additive and higher beat treatment temperature will improve themagnetic properties but also result in lower strength.

The compressing will also introduce some stress, which has a negative effecton the magnetic properties; this stress can be removed with a sufficiently hightemperature. This heat-treatment temperature is normally set lower than 500oC.

15

Chapter 3. Soft Magnetic Composites and permanent magnets

Figure 3.1: Schematic picture of soft magnetic composite material.

Initial permeability vs. FrequencySomaloy

TM500+0.5%Kenolube

TM

0

20

40

60

80

100

120

140

160

1 10 100 1000

frequency [kHz]

Initia

l perm

eabili

ty

Heat-Treated at 500°C (932°F)

Green

Figure 3. Initial Permeability vs. Frequency, Double Impacts (Green and Heat-Treated, GD=7,45g/cm

Figure 3.2: Initial permeability vs. frequency, double impacts (green and heat-treated, GD = 7, 45g/cm3).

3.2 Somaloy 500

Somaloy 500 is a soft magnetic composite material suitable for components thatrequire high magnetic induction [Andersson, n.a.]; it is produced by Hoganas. Thematerial consists of high purity iron powder particles, coated with an inorganiccoating. Somaloy 500 can be mixed with different types of organic additives.Samples for low to medium frequency applications, which need a low conductivitymust be mixed with Kenolube and heat treated till 500oC. A relatively highstrength can be achieved without the addition of any binder additives.

The next measurements [Andersson, n.a.] are done with samples with 25 turnsof copper wire. Figure 3.2 shows that the initial permeability is relatively stablefor the heat-threated sample thus the eddy current losses are low.

The initial magnetization curve is plotted in figure 3.3 for two samples withdifferent densities in [gr/cm3]. A higher density gives a higher magnetic inductionat relatively low field strengths.

Figure 3.4(a) gives the total core loss per cycle versus the frequency. The flux

16

3.2. Somaloy 500

Figure 3.3: The BH-curve for heat-treated Somaloy Somaloy 500+0.5% Kenolubeat different densities.

density in the core will be kept on 1 Tesla for two density levels. A higher densityresults in a lower DC-loss but the density does not affect the dynamic losses verymuch, as the slopes of the curves are very similar.

The losses in the soft magnetic material consist of hysteresis losses and eddycurrent losses.

Hysteresis losses will appear when during magnetization the domain walls inthe material move. These movements will be hindered by internal stress. So theapplied magnetizing force must therefore exceed this stress force. Some energy istherefore converted into heat while moving the domain walls. The power dissi-pated per unit volume due to the hysteresis at an applied magnetic field H withfrequency f , is ph = f

∫HdB = fAhyst. So the hysteresis losses are proportional

to the frequency.

If a piece of electrical conducting material is placed in a time varying electro-magnetic field, it will be surrounded by an electric field, due to Faraday’s law;∮

Eds = −dΦdt . This electric field will introduce a certain current density which

will create resistive losses in the material. The eddy current loss pe can be writtenas pe = KeB

2maxf2, where Ke is a constant whose value depends on the type of

material.

The core losses will be dominated by the hysteresis losses for Somaloy 500,because the material is highly resistive due to coating between the particles andtherefore the eddy current losses will be very small. This is shown in figure 3.4(b).The enclosed area of the DC-loop follows the 200Hz loop closely, this reveals thelow proportion of eddy-current losses. The slope goes linear with the frequency,see figure 3.4(a), this shows also that hysteresis losses are dominating.

Somaloy 500 is used for the core in the mover bar in the C-shaped actuator.It will hold the coils and handle the common flux path created by the permanentmagnets from the three sides.

17


(a)

(b)

Figure 3.4: a) Core loss/cycle as a function of frequency; Heat-Treated Somaloy500 +0.5%Kenolube. b) Hysteresis loop at DC and 200Hz; Heat-Treated Somaloy500+0.5%Kenolube, 7.47g/cm3.

18

3.3. Permanent magnets

Figure 3.5: Demagnetization curve with recoil loop and the operation point of thepermanent magnet.

3.3 Permanent magnets

Every hard magnetic material can be described by its BH hysteresis loop. Theproperties of primary importance in the selection of a magnet are these that definethe magnitude and stability of the field that it can provide. The hysteresis loopincludes the coercivity Hc in [A/m], saturation Bsat in [T ] and remanence Br

in [T], as well as the behavior of the hysteresis loop in the second quadrant (seefigure 3.5). Permanent magnets are classified as hard magnetic materials, whichhave low permeability and high coercivity, typically > 10000A/m [Furlani, 2001].This high coercivity makes them difficult to magnetize and demagnetize, becauseonce magnetized they tend to remain magnetized. It means also that a thinnerpermanent magnet can be used to withstand the demagnetization field.

A permanent magnet can produce a magnetic field with no excitation windingand no dissipation of electric power. External energy is involved only in changingthe energy of magnetic field, not maintaining it (see also section 4.1).

The basis for evaluating of a permanent magnet is the part of the hysteresisloop (see figure 3.5), that is located in the second quadrant and called the de-magnetization curve. When a reversed magnetic field intensity is applied to amagnetized toroidal specimen, the magnetic flux density drops down to the mag-nitude located at point [Hm, Bm], the operating point of permanent magnet. Atthe moment this reversal magnetic flux density is removed, the flux density re-turns to point [0, B0], as figured in figure 3.5, according to a minor hysteresis. Dueto the applied field, the field strength has reduced the remanence. The straightline between [0, B0] and [Hm, Bm] is called the recoil line with slope µrec.

As long as the applied negative field strength does not exceed the maximumvalue at point [Hm, Bm], the permanent magnet can be regarded as being rea-sonable permanent [Gieras and Wing, 2002]. If, however, a greater negative fieldstrength H is applied, a new and lower recoil line will be established.

19


The operation point can be affected due to the change of the applied fieldand/or a change of the reluctance from the flux path originated by the permanentmagnet. A variation in the applied field will change the operation point over βas the blue lines (parallel) in figure 3.5. A change of the reluctance will vary theline in angle α, plotted with the red lines.

The energy that is stored in the permanent magnet, when operating along therecoil line, between B0 and Bm is given by [Strahan, 1998]:

Wm =∫

Vm

∫ Bm

B0

Hm · dBmdvm (3.1)

where Vm is the volume of permanent magnet in [m3].

[kA/m]

NEOREM 412a / NEOREM 512 a

[kOe]

[kG][T]

1.0

0.0

1.2

0.2

1.4

0.4

0.6

0.8

5

10

0.10

0.20

0.30

0.40

0.50

4.002.001.501.000.75

120014001600

20 15 1010 5

18002000 8001000 400 200 0600

-H

B,J

Temperature 20o

60o 80o 120o100o

Figure 3.6: Temperature dependent demagnetization curves of a Neorem rareearth permanent magnet.

The permanent magnets, that are proposed by Tecnotion, for the C-shaped ac-tuator are Neorem 412a permanent magnets (see figure 3.6 and/or Appendix A)[Neorem Magnets, n.a.]. For normal temperatures the BH-curve in the secondquadrant is assumed to be linear and equation 2.28 can be used with constantµrpm . The slope of the recoil line is also the same as the µrpm , like in figure 3.7.So now equation 3.1 becomes:

Wm = (A1 + A2)Vm (3.2)

The field energy outside the permanent magnet follows as:

Wext =12

∫

V∞−Vmag

(HB)dV (3.3)

20

3.3. Permanent magnets

Figure 3.7: Demagnetization curve of a rare earth permanent magnet.

where V∞ refers to the infinite space. During magnetization, energy is also trans-ferred to the surrounding space, so outside Vmag. When in this space, V∞−Vmag,B = µ0µrH holds and µr should not be a function of H. If the system is withoutmoving media and currents:

∫

V∞(HB)dV = 0 (3.4)

Now equation 3.3 can be written as:

Wext = −12

∫

Vmag

(HB)dV = −12(HmBm)Vm = A3Vm (3.5)

For this reason the energy-product (BH)max of a permanent magnet is usedas a quality factor.

If the temperature increases, the linear part is falling down at a certain fieldstrength. It is therefore very important to look at the operating point for aconfiguration. High currents and rising temperatures can cause irreversible de-magnetization. The actuator will not be able anymore to produce the same forceas before.

21


22

Chapter 4

Magnetic Equivalent CircuitModel

Electromagnetic actuators can be modelled in several ways. The most commonapplied method is finite element analysis, as this method can deal with a wide va-riety of electromagnetic devices and material properties. However, finite elementanalysis is a time consuming method, especially for three-dimensional geometrieslike the C-shaped actuator. Therefore, this method is not convenient for theevaluation of a large number of designs. Iron-core actuators are often modelledwith magnetic equivalent circuits. These circuits consist of a limited number ofreluctances and sources which are representing different parts of the geometry.As an advantage, these magnetic circuit networks can be solved very fast and theapproach is similar to electrical network problems. In this chapter the magneticequivalent circuit for the C-shaped actuator will be explained and validated with3D finite element calculations.

4.1 Analogies between electric and magnetic cir-cuits

A magnetic equivalent circuit of an actuator consists of a limited number of ele-ments. Each element represents a flux path or tube in the actuator. Flux tubesare defined as geometrical figures in which all lines of the flux are perpendicularto their bases and no lines of this flux cut their sides [Ostovic, 1989]. This isshown in figure 4.1.

The surfaces U1 and U2 are describing a scalar potential and are perpendicularto the flux lines φ in [V s]. The magnetic scalar potential difference between thebases is equal to the magneto-motive force drop F in [A].

Generally, the ratio of potential difference at the ends of the flux tube and theflux through it, is a function of the geometry of the tube and the characteristicsof the medium. Mathematically, this ratio is equal to:

R =∫ l

0

dx

c(x)A(x)(4.1)

where l is the total flux tube length, A(x) is the cross-section and c(x) is a function

23

Chapter 4. Magnetic Equivalent Circuit Model

Figure 4.1: A flux tube in a magnetic field.

of the material properties. The quantity R is defined as the reluctance in magneticfields and it is analogous to resistance in electric networks. The R is a functionof field quantities and the geometry. The quantity c(x) is equal to the flux tubematerial permeability µ for magnetic fields. For electric networks c is equal to theflux tube conductivity σ. In mathematical form:

Rm =∫ l

0

dx

µ(x)A(x)[H−1] (4.2)

Rel =∫ l

0

dx

σ(x)A(x)[Ω] (4.3)

Relationships between flux density and field strength for magnetic and electricfields are, according to the Maxwell equations:

B = µH (4.4)

J = σE (4.5)

The analogy differs in one detail: to maintain a certain energy level in a mag-netic field no support from outside the field is needed. In a resistance the completeenergy must be maintained with the help of a current from a source to cover losses.The electromagnetic field can store energy, whereas in a resistance the completeenergy is irreversibly converted into heat. If all quantities in a magnetic field areconstant the stored energy is constant. Under the same circumstances the energyloss in an electric field increases with time. The flux and current are defined as:

φ =∫

BdA =F

Rm[V s] (4.6)

I =∫

JdA =V

Rel[A] (4.7)

24

4.2. Magnetic equivalent circuit model of the C-shaped actuator

All the equations give the analogy between the electric and magnetic field. Theenergy of a magnetic and electric field can be described:

Wm =∫

Fdφ [J ] (4.8)

Wel =∫

IV dt [J ] (4.9)

The potential difference for both fields is:

F =∫ l

0

Hdl [A] (4.10)

V =∫ l

0

Edl [V ] (4.11)

4.2 Magnetic equivalent circuit model of the C-shaped actuator

A magnetic equivalent circuit model has been derived for the C-shaped actuator.This model contains the leakage flux between two adjacent permanent magnetsand the non-linearity of the SMC core. This will improve the model when com-pared with the analytical model in section 2.2. The MEC model can be fitted ina later stage with results obtained by magnetostatic 3D FEM.

(a) The C-shaped actuator with only the bottomside in 3D.

(b) A schematic view of the cross-section of theC-shaped actuator with only the bottom side.

Figure 4.2: The C-shaped motor with only the bottom side.

4.2.1 Flux paths in the actuator

The stator of the C-shaped actuator, which is shown in figure 1.1, consists ofthree sides. The flux, originated from a permanent magnet (N-pole) on one of thesides, returns via the airgap, the SMC core, again the airgap, the adjacent magnet(S-pole) and the back iron. Hardly any flux returns via one of the other stator

25


sides. The only common flux path is located in the SMC mover bar. Therefore,only one of the three stator sides is modelled, like is shown in figure 4.2(a) andschematically drawn in figure 4.2(b). However, the total flux in the SMC core isthree times the flux of one side for the total actuator, which is important whensaturation is incorporated in the model. This is only applicable for a square moverbar and permanent magnets of the same size, because the amount of flux that isgenerated at one side, is approximately the same as for the other sides.

Figure 4.3 shows in detail the modelled flux paths in the C-shaped actuator.Only two permanent magnets have to be modelled to obtain the flux densitydistribution. For the modelling, the permanent magnets are cut into two piecesand the flux paths are defined from one half permanent magnet to the other halfpermanent magnet. The outer flux path φout goes through the half coils C andC ′. The inner flux path is divided into two paths. One path for modelling theleakage and the other flux path goes through the coils B and A‘, see figure 4.3.

Figure 4.3: The main flux paths in the C-shaped actuator.

4.2.2 MEC model

Figure 4.4 shows the magnetic equivalent circuit of the C-shaped actuator. Themodel consists of the reluctances for the permanent magnet, airgap, coil, gluelayer, leakage from magnet to magnet, SMC mover bar and iron. They are definedas Rpm, Rairgap, Rcoil, Rglue, Rmm, Rsmc and Riron respectively and globallydescribed as:

• The permanent magnet source consists of a magnetomotive force sourcewith an internal reluctance.

• The airgap reluctance, consisting of the reluctance of the mechanical clear-ance between the stator and the translator.

• The coil reluctance, the flux goes from the permanent magnet to the moverbar. Figure 4.2(b) shows (dashed line) that the width of the mover bar isshorter than the permanent magnet. This flux path is more like a trapezium.

26


• The reluctance of the glue layer has the same shape as the coil. The gluelayer is taken into account because its length is of the same magnitude asthe airgap. In this case the airgap is 0.5mm and glue layer is around 0.2mm.It is actually not only glue, but also an insulation layer between the coil andthe SMC mover bar.

• The leakage reluctance between two permanent magnets.

• The non-linear SMC reluctance; the total flux in the SMC stator bar is threetimes the calculated flux, because the stator has three sides with permanentmagnet arrays with the same dimensions. This total common flux path istaken into account for the reluctance calculation of non-linear soft magneticcomposite material.

• The back iron reluctance is very low, because the cross-section of the backiron is rather large in comparison with that of the mover bar. The back irondoes not saturate that fast as the SMC material does.

!" !"

# # # #

$%&'()*(

+%,- +%,-

.$ .$

/*01 ,(%2

Figure 4.4: Magnetic equivalent circuit of the C-shaped actuator.

The calculation of all the different reluctance values is explained in the nextsubsection.

27


Permanent magnet sources

For this position the SMC mover bar covers two permanent magnets. This meansthat half of the produced flux of one permanent magnet goes to the other per-manent magnet. For the modelling, the magnets are cut into two parts and theflux path is defined from one middle half magnet to the other middle half magnet[Kim et al., 2004] and [Tapia et al., 2002]. This is shown in the figures 4.3 and4.4. The permanent magnets are separated into two parts with three differentflux paths. This separation is also convenient for the force calculation in a laterstage. The outer flux path φout goes directly through the coils C and C ′. Theinner flux path φin goes through the coils A′ and B, but there is also a certainleakage flux φmm.

The permanent magnet is defined in the magnetic network as a source withinternal reluctance. The source generates the magnetomotive force Fpm and isdefined by the coercive magnetizing force Hc and the thickness of the permanentmagnet Lpm as:

Fpm = −LpmHc (4.12)

The two left sources are representing half a permanent magnet that is magne-tized in the direction of the stator and divided into two blocks [Rasmussen andRitchie, 1997a], [Rasmussen and Ritchie, 1997b]. One part is under the half coilC ′ and the other one is under coil B. This holds also for the right permanentmagnet, namely one piece is under coil A and the other under half coil C. Theouter flux path φout has no leakage and goes straight into the mover bar.

The width of the permanent magnet is defined as:

Wpm = Wsmc + 2Lglue + kin/out · Lcoil (4.13)

The factor kin/out gives the opportunity to fit the MEC model to results of mag-netostatic 3D FEM simulations for inner and the outer flux path.

The permanent magnets in the outer flux path φout have a reluctance Rpmout

with a rectangular shape with thickness Lpm (see figure 4.2(b) between the dashedlines). According to the equation 4.2 the reluctance can be expressed as:

Rpmout =Lpm

µ0µpmτpp

6 Wpm(4.14)

The pole pitch is divided by 6, then the surface is covering the half of coil C‘ orC. In a later stage the flux density can extracted in the coils C and C‘.

The inner flux path φin through the permanent magnet under the coils A‘ andB is rectangular too. The flux path cross-section is now a function of the magnetpitch and the pole pitch and is defined as:

Rpmin =Lpm

µ0µpm( τmp

2 − τpp

6 )Wpm(4.15)

This MEC model assumes an uniform distributed flux density for each per-manent magnet part, this is in practice not the case. The flux distribution ontop of the permanent magnet is shown in figure 4.5. With the help of factor kthis irregularity can be adapted and fitted with the 3D FEM predictions. For theinner flux path is kin = 0.97 and for the outer part kout = 0.8. The differencebetween these numbers can be explained by looking to the flux distribution. The

28


Figure 4.5: Flux density distribution on the top of the permanent magnet.

flux distribution at the sides, is the lowest in the middle of the top of the perma-nent magnet, which is described with outer flux path. The Mathcad script canbe found in appendix F.

Airgap reluctance

The element of the airgap reluctance Rairgap has a rectangular shape too, thedashed line in figure 4.2(b) shows the border. The flux path cross-section iscomparable with that of the permanent magnets. The length of the flux path isdifferent and therefore defined as:

Rairgapout =Lairgap

µ0τpp

6 Wpm(4.16)

Rairgapin =Lairgap

µ0(τmp

2 − τpp

6 )Wpm(4.17)

where Wpm is defined in equation 4.13. The factors are here also different forboth flux paths and defined as mentioned before: kin = 0.97 and kout = 0.8.

Coil and glue layer reluctances

The coil and glue layer reluctances should be estimated in a different way. Theshape is no longer rectangular, because the surface on the top of the permanentmagnet differs form the surface at bottom side of the mover bar. In figure 4.2(b)the borders of the dashed line in the coil are forming a trapezoidal shape.

The trapezoidal form, that characterize the flux path, is plotted in figure 4.6.The width of the trapezoidal configuration is mathematically expressed as:

w(x) = w1 +w2 − w1

hx (4.18)

where w(x) is the width of trapezium as function of x with a given height h anddepth l. The permeability µ is not a function of x. Combining equation 4.18 with

29


Figure 4.6: Reluctance calculation for a trapezoidal element with a direction ofthe flux perpendicular to w(x).

4.2, [Ostovic, 1989]:

Rtrap =1µ

∫ h

0

dx

lw(x)

=1µl

h

(w2 − w1)ln

w2

w1(4.19)

This equation can be translated to the reluctance for the coil:

Rcoilout =1

µ0µcopτpp

6

Lcoil

(−koutLcoil)ln

(Wsmc + 2Lglue

Wsmc + 2Lglue + koutLcoil

)(4.20)

Rcoilin =1

µ0µcop(τmp

2 − τpp

6 )Lcoil

(−kinLcoil)ln

(Wsmc + 2Lglue

Wsmc + 2Lglue + kinLcoil

)(4.21)

where kin = 0.97 and kout = 0.8. The glue layer is defined as:

Rglueout =1

µ0τpp

6

Lglue

(−2Lglue)ln

(Wsmc

Wsmc + 2Lglue

)(4.22)

Rgluein =1

µ0(τmp

2 − τpp

6 )Lglue

(−2Lglue)ln

(Wsmc

Wsmc + 2Lglue

)(4.23)

Leakage reluctance

To model the leakage flux between the permanent magnets a circular-arc straightline reluctance model [Lipo and Qu, 2002] has been used. This model is shownin figure 4.7 and it is only valid for modelling in a magnetic equivalent circuit.The flux lines, as shown in figure 4.7, are not representative for the actual fluxlines. Flux lines which are going from air (relatively low µr) into iron/SMC(relatively high µr) (or the other way around), will cross this medium transitionperpendicular or almost perpendicular to this transition. It is impossible for fluxlines to go slightly parallel along the mover bar under this condition.

For mathematical convenience the circular-arc straight line model is expressedthrough the permeance. The permeance has a reciprocal relationship with reluc-

30


tance defined in equation 4.24, where P is the permeance:

P =1R

[H] (4.24)

Figure 4.7: Circular-arc straight line reluctance patterns to model the leakageflux between two magnets.

The amount of the leakage flux is dependent on the distance between the twopermanent magnets, the thickness of the coil and airgap length.

The leakage permeance is interpreted as an infinite sum of differential widthpermeances, where each length is as function of x, so the length of each line willbe L(x) = τpp − τmp + πx. The sum can be written as:

Pmm =∑ µ0µcuWpmdx

L(x)(4.25)

Calculating the continuous permeance from 0 to (Lcoil + Lairgap) yields:

Pmm =∫ Lcoil+Lairgap

0

µ0µcuWpm

L(x)dx

=µ0µcuWpm

πln

(1 +

π(Lcoil + Lairgap)τpp − τmp

)(4.26)

If (Lcoil +Lairgap) > ( τmp

2 − τpp

6 ) the new integration length has to be ( τmp

2 − τpp

6 ).Otherwise the radius of the arc reaches the area of the outer flux path. Wpm ishere defined for the inner flux path, so kin = 0.97.

To incorporate effects caused by the leakage flux from one permanent magnetto another one, the factor kleak is introduced in the MEC model for improvementof the calculation precision. After a few simulations the factor kleak is chosen tobe 0.87, it gave proper results when comparing to 3D FEM. This factor 0.87 isdecreasing the amount of leakage flux, because the width of this flux tube, fromthe permanent magnet to the mover bar, looks also like a trapezium form, as

31


described previously for the coil and glue layer. In fact, the effective reluctancevalue will increase. The equation, that will be used in the MEC model, is:

Rmm =[kleak

µ0µcuWpm

πln

(1 +

π(Lc + Lairgap)τpp − τmp

)]−1

(4.27)

Increasing the factor kleak will increase the amount of leakage flux and thereforedecrease the Lorentz force.

SMC mover bar reluctance

The SMC core, which has to handle the major part of the flux, has a non-linearBH-curve like it is shown in figure 3.3. The flux paths φout and φin are going boththrough the middle of core. This introduces a certain flux density that is mainlyconcentrated in the middle of the mover bar (see figures 4.3 and 4.4). For somesetups the SMC core can reach a high level of saturation when the cross-sectionin the middle of the SMC mover bar is too narrow. This will cause an extrasignificant magnetic reluctance.

Figure 4.8: The flux distribution in the middle of the SMC core.

The change of reluctance is also obviously non-linear. An iterative procedure isimplemented to achieve a non-linear reluctance. The known BH-curve of Somaloy500 is added in the MEC model and can be interpolated with a built-in functionof Mathcad.

The script starts with the initial value of µsmc = 5 which gives an initialreluctance and flux in the SMC core. With this value of flux, the flux density canbe calculated and according to the BH-curve the field strength can be found. Anew µsmc will be defined with the field strength and the flux density:

µsmcnew =Bsmc

µ0Hnew(4.28)

The new value for µsmcnew in the SMC core will give new values for the reluctanceand flux in the core. This repetitive process will be done four times and convergeµnew to a certain value. The value can vary from µsmc = 315 to µsmc = 3 for

32

4.3. Calculation of MEC model

different SMC cores. The iterative process will always converge, because there isonly one non-linear part in this MEC model implemented and the function has aoverall rising slope.

Mathcad has no option to implement a convenient for-loop to control theconverge ratio and to stop when this converge ratio is reached.

The results obtained by the use of this function in the MEC model are verifiedwith the 3D FEM model for a reasonable range of geometric parameters.

The region where this reluctance appears has to be defined as function ofthe length and the cross-section of SMC mover bar. The length of this region isarbitrary chosen. The length of the saturation region is 0.3 times the pole pitch.The reluctance for the SMC core is defined as:

Rsmc =0.3τpp

µ0µsmcWsmcHsmc(4.29)

where µsmc will be calculated with the iteration procedure, as mentioned above.This new value can be filled in and will converge to a certain steady-state value.With this new value of the reluctance the new fluxes can be calculated. The fluxin the SMC core can determine the flux density with the next equation:

Bmov =3ksmcφsmc

WsmcHsmc(4.30)

In figure 4.8 the cross section of the middle of the mover bar is shown. The fluxdensity is not homogenously distributed in the SMC core. The bottom part ofthe core is more saturated then the side parts. To fit the flux density with 3DFEM field results a factor ksmc is introduced. The factor is put on ksmc = 0.83which gave proper results.

Iron reluctance

The iron of the stator is assumed to have an infinite permeability, because eachside of the stator has to handle only one flux path. This path has a relative largesurface of the cross-section of the iron when compared to that of the SMC core(see figure 4.2(b)).

4.3 Calculation of MEC model

All the sources and reluctances are describing the proper configuration of thepermanent magnets, coils, iron, SMC core, etc. The total magnetic equivalentcircuit gives information about all the fluxes in the model.

The calculation is done by putting the MEC circuit, as given in figure 4.4,in a 9 × 9 coefficient matrix according to the two Kirchhoff’s laws. This gives asystem of linear equations [Holmes, 1993] and [Leon, 1998]:

[Ac] · [φ] = [f ] (4.31)

33


where [Ac] is the coefficient matrix given by:

[Ac] =

−1 −1 0 0 0 0 1 0 00 0 1 1 0 0 −1 0 00 1 0 0 −1 0 0 1 00 0 −1 0 0 1 0 −1 01 0 0 0 1 0 0 0 −1

Rout −Rin 0 0 −Rpmin0 0 0 0

Rout 0 0 Rout 0 0 Rsmc 0 Riron

Rout 0 Rin 0 0 Rpmin Rsmc 0 Riron

Rout 0 Rin 0 −Rpmin0 Rsmc −Rmm 0

(4.32)where Rout and Rin are defined as:

Rout = Rpmout+ Rairgapout

+ Rcoilout+ Rglueout

(4.33)Rin = Rairgapin

+ Rcoilin+ Rgluein

(4.34)

The unknown fluxes are written in the column matrix:

[φ] =[φC‘ φB φA‘ φC φLpmin φRpmin φsmc φmm φiron

]T (4.35)

and the column vector [f ] is:

[f ] =[0 0 0 0 0 0 2Fpm 2Fpm 0

]T (4.36)

Coefficient matrix [Ac] consists of an upper part and a lower part. The upperpart, with the zeros and units, gives the Kirchhoff’s current law for this circuit.The lower part of the coefficient matrix represents the Kirchhoff’s voltage law,with all reluctances and zeros.

This matrix is implemented in Mathcad. The unknown fluxes can be cal-culated by inverting the coefficient matrix [Ac] and multiplying it as follows[φ] = [Ac]−1 · [f ]. After a few iterations (done for the nonlinear Rsmc) all thedefinitive fluxes [φ] are known and can be used for further calculations.

If all the fluxes are available, the flux density B can be derived for each fluxtube, via equation 4.6, with the relation:

B =φ

A(4.37)

where A is the cross-section in [m2] and φ the flux in [V s].

4.4 Force extraction from MEC

The forces will be calculated using the Lorentz equation. This equation can bewritten in the form:

F =∫ ∫ ∫

coil

J×B · dV (4.38)

The force is proportional to the flux that intersects the coil perpendicular to thepropulsion direction. When using Lorentz force it is assumed that flux densityproduced by the permanent magnet is left unchanged by the applied current[Molenaar, 2000]. In the real world this is not the case. In figure 4.9 is shown

34

4.4. Force extraction from MEC

(a) J = 8.3A/mm2

(b) J = 83A/mm2

Figure 4.9: The flux with different current densities in 2D FEM.

that with a very high current density the primary flux is not disturbed verysignificantly. The current density for the peak force will be 64.47A/mm2 asmaximum according to the thermal behavior, see chapter 6.3.2. The contributionof the current in the coils to the flux distribution of the permanent magnets isnot significant.

The permanent magnets are divided into several parts, as depicted in figure 4.4.This is done to extract the force per coil with the Lorentz force in an convenientway. To derive the flux density in the region of coils C and C ′, the flux has to bedivided by the area of the cross-section, see equation 4.38.

The total coil C (not the half) the cross-section is two times the cross-sectionof the half permanent magnet. The flux in the total coil is also two times theflux. Therefore, the flux density of the half coil is the same as the total coil. Dueto the symmetry of the MEC model φC = −φC‘. This can be written as:

Bout =φC

Lcurτpp

6

(4.39)

φC is the flux that flows through the half of the coil C, the length Lcur = Wsmc +2Lglue + Lcoil is the mean length of the current path. So, Bout is the mean fluxdensity, obtained from the MEC model, in coil C which is perpendicular to thecurrent in the windings. With the mean flux density, total current density JC

35


and coil thickness the Lorentz force contribution for this coil C can be calculated.This gives for coils C and C ′:

FC ‘ = − φC‘

Lcurτpp

6

· −2JC‘Lcoilτpp

6· Lcur = 2 · φC‘ · JC‘ · Lcoil (4.40)

FC =φC

Lcurτpp

6

· 2JCLcoilτpp

6· Lcur = 2 · φC · JC · Lcoil (4.41)

The flux density Bin in the inner coils A‘ and B is a bit different. The magnetpitch can vary, and therefore the amount of flux per surface that is produced bythe permanent magnet is varying too. For this flux path, where φA‘ = φB :

Bin =φA‘

Lcur

( τmp

2 − τpp

6 + (τpp−τmp)2

) (4.42)

The denominator gives the mean surface where flux is going through with thecurrent density JA‘ in coil A‘. The force, due to coils A‘ and B, can be calculatedas:

FB = φB · JB · Lcoil (4.43)FA‘ = −φA‘ · −JA‘ · Lcoil (4.44)

The coils C and C‘ are carrying the top current density, so they produce themost Lorentz force. The other currents densities are ±120o shifted, according tothe three phase system, and therefore JC = −JC‘ = J and JB = −JA‘ = 1

2 J .The forces in the coils B and A‘ are the same as in A and B‘ due to symmetry.All the forces per coil, as determined, can be added together and will produce thepropulsion force in the y-direction of one side of the C-shaped actuator. Becausethere are three effective sides and a square mover bar this force can be multipliedby three. This gives a total force of:

Flortot = 3(FA + FC‘ + FB + FA‘ + FC + FB‘) (4.45)

This can be written as function of the top current density J :

Flortot = 3 · J · Lcoil

(4φout + 2φin

)(4.46)

where φout = −2φC = 2φC‘ and φin = φA = φB = −φA‘ = −φB‘. This equationis valid for a design with identical permanent magnets on all sides and thereforehas a square mover bar.

4.5 Results

The results obtained by the MEC model are compared with a series of simulationsin magnetostatic 3D FEM. Each plot is for a given permanent magnet thicknessLpm, where the coil thickness Lcoil is varying between 1mm and 5mm. All resultsfrom the MEC are within the range of 6% difference with the FEM results and aregiven in figure 4.10. The energy error in the 3D FEM is smaller than 0.7%. Everyindividual simulation in FEM, for this energy error1, will take approximately 10

1After each iteration, the solver of Maxwell Ansoft calculates the total energy of the systemand the percentage of this energy that is caused by solution error. It then checks to see if thenumber of requested passes has been completed, or if the percent error and the change in percenterror between the last two passes match the requested values.

36

4.6. Conclusion

1 1.5 2 2.5 3 3.5 4 4.5 515

16

17

18

19

20

21

22

23

24

25

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

MEC3D FEM

(a) Lpm = 4mm

1 1.5 2 2.5 3 3.5 4 4.5 515

16

17

18

19

20

21

22

23

24

25

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

MEC3D FEM

(b) Lpm = 5mm

1 1.5 2 2.5 3 3.5 4 4.5 515

16

17

18

19

20

21

22

23

24

25

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

MEC3D FEM

(c) Lpm = 6mm

1 1.5 2 2.5 3 3.5 4 4.5 510

15

20

25

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

MEC3D FEM

(d) Lpm = 7mm

Figure 4.10: Lorentz forces obtained by the MEC model and 3D FEM Maxwellas functions of the coil thickness and permanent magnet thickness.

minutes. The time that is required for the Mathcad calculation, is only to enterthe right values for a specific design, the calculation time in negligible.

The results, as shown in 4.10, are evaluated in more detail in section 6.5.

4.6 Conclusion

A magnetic equivalent circuit model is implemented in Mathcad for the C-shapedactuator. The model has been derived and validated with magnetostatic 3D FEMpredictions. The obtained results are within 6% accuracy and therefore it is a veryuseful tool to evaluate a large number of different designs in a short period of time.It improved significantly when compared with the analytical model in section 2.2,these results are shown in appendix D.

37


38

Chapter 5

Thermal analysis

The structure of this C-shaped actuator is not very trivial or similar to otherdesigns. Therefore, a thermal model is derived. This model makes it possibleto predict the steady state temperature distribution within a certain accuracy.This predicted temperature distribution can be used to determine the maximumcontinuous current in the windings, to determine the continuous force, see section6.3.1.

5.1 Sources and materials

It is necessary to know the thermal characteristics of all the materials that aregoing to be used for the C-shaped actuator. The winding insulation has a temper-ature limit that must not be exceeded, otherwise the insulation will irreversiblybe damaged.

The main losses will be generated due to the copper losses. The wire is madeof copper, which has a certain resistance; the copper losses per winding can bedescribed as:

Pcu = RwdgI2rms [W ] (5.1)

where Rwdg is the resistance of a round copper conductor with a length l andsurface A. The resistance can be described with the equation:

Rwdg = ρcu · l

Awdg= ρcu · 3 · lmean

ffillLcoilτpp[Ω] (5.2)

The filling factor is represented as ffill, lmean is the mean length of the windingin [m]. In this case there is only one winding. The filling factor is depending onthe wire diameter, according to the proportion between the amount of insulationand copper. The method of winding is also influencing the filling factor, the coilsin the C-shaped actuator are orthocyclic wounded.

The resistivity of copper as function of the temperature is given by:

ρcu = ρ20o (1 + αo(T − 293.15)) (5.3)

in which ρ20o is the resistivity at 20oC, αo the temperature coefficient and T thetemperature in Kelvin. The coefficients for copper are ρ20o = 1.68 · 10−8Ωm andαo = 3.93 · 10−3.

39

Chapter 5. Thermal analysis

The core losses which are described in the chapter 3, will be dominated by thehysteresis losses and are dependent on the frequency. The total loss for the SMCmover bar will be around 6.5W/kg @1T with a frequency of 50Hz. The core ofthe designed actuator, in section 6.5, weights about 80 gram. The losses will bearound 0.5W in the total SMC core, when it is moving. The thermal model isbased on a standstill situation, to simulate the worst case scenario. The hysteresislosses will be assumed negligible.

5.2 Heat transfer

Heat is transferred in the C-shaped actuator by means of conduction, convectionand radiation [Remsburg, 2001]. Heat transfer due to conduction is a process inwhich the heat flows through solid, liquid, or gas, or between two media that are inintimate contact with each other such as steel, copper and insulation. Convectionis the term used to describe the transfer of thermal energy between two surfacesas a consequence of a relative velocity between them. Radiation cooling is thetransfer of heat by electromagnetic emission, primarily in the infrared spectrum.

5.2.1 Conduction

Conduction is the heat flow within a body or between two bodies. The amountof this energy flow is depending on the thermal conductivity λ(W/m ·K) of themedium. This heat transfer is described by Fourier’s law:

Qcond = −λAdT

dx[W ] (5.4)

where A is the cross-section area perpendicular available for conduction in units of[m2]. Heat flow is considered positive when the temperature is decreasing, hencethe minus sign is in equation 5.4.

The thermal characteristic can be simplified by considering the thermal prop-erties as electrical properties. The thermal resistance can be expressed as:

Rth =∆T

Qcondx

=l

λA[K/W ] (5.5)

where l the length of the heat transfer in [m] and λ the thermal conduction in[W/mK].

5.2.2 Convection

Heat removal by convection can be classified as natural and forced convection. Forthe C-shaped actuator the natural convection can be used as a cooling mechanism.If, later on, a heating problem occurs there can be experimented with forcedcooling by using a fan and/or water cooling. The heat loss is proportional to thetemperature difference between the hot object and cooling media. This can bewritten as:

Qconv = hcA(Tsurf − Tamb) [W ] (5.6)

where Qconv is in [W ], Tsurf is the temperature of the surface, Tamb is the am-bient temperature and hc is the average convection heat transfer coefficient in

40

5.3. Thermal equivalent circuit

[W/m2K]. Since hc is position and temperature dependent, convection heattransfer solutions are more complex than conduction and radiation. To avoidcomplexity, practical assumptions have been used.

5.2.3 Radiation

Radiation for a black body can be described by the Stephan-Boltzmann equation:

Qrad = εσA(T 4surf − T 4

amb) [W ] (5.7)

where σ = 5.67 · 10−8[W/m2K4] is the Stephan-Boltzmann constant. ε is theemissivity of a body or surface and is the ratio of the radiated flux emitted by abody to that emitted by a black body at the same temperature. The radiationheat transfer is very low, because the temperatures of the actuator are rather low(< 120oC) and therefore negligible.

5.3 Thermal equivalent circuit

The thermal circuit model is an analogy of an electric circuit in which the gener-ated heat in [W ] is modelled with current sources and the temperature is analogousto a voltage. The thermal resistance are expressed in [K/W ]. The thermal modelis not equipped with thermal capacitances in [J/K]; the results are not timedependent and the temperature distribution is contains the steady state values.

This model is added to the MEC model in the Mathcad script in Appendix F.Like the MEC model, the thermal circuit of the C-shaped actuator is also variablein its dimensions. The properties of the C-shaped actuator can be analyzedat its magnetostatic properties, thermal behavior and propulsion force with theMathcad script. The thermal behavior is especially important to determine thecontinuous force as discussed in section 6.3.

In order to analyze the heat transfer in the actuator, an idealized geome-try must be chosen and divided into basic elements. These elements correspondapproximately to areas, which have thermal and physical uniformity.

The division of the machine into small elements is a compromise between themodel simplicity and the accuracy. All the elements of the C-shaped actuatorare described by nodes, having an average surface temperature with respect tothe ambient. All nodes are connected to each other by conduction or convectionresistances. The thermal equivalent circuit is schematically drawn in figure 5.1.

The dashed line is the symmetry line, because the coil is symmetrical in they-z plane. The gray surface is the SMC core, which is surrounded by the windings.The thermal model is a two dimensional representation of the mover with the coil.When the coil is covering the space between the two adjacent permanent magnetsin a static situation, it has the worst cooling condition. To take the worst case,the back iron and the permanent magnets are not included.

5.3.1 Thermal convection resistance

The resistors Rctop , Rcsideand Rcbottom

are representing the convection resistances.These resistors will transport the heat to the air layer adjacent to all the threesides of the coil. Equation 5.6 describes the heat loss where Rc is defined as:

Rc =1

hcA(5.8)

41


!"#$%&

!"#$%&

!"#$ '(&

)

*

+

,

-,, ,- .

/

0

1

2

,)

34

Figure 5.1: A 2D thermal equivalent circuit of the C-shaped actuator.

where A is the surface from which heat can be transferred to the outside world.The convection heat transfer coefficient is hc in [K/W ]. The convection resistancesgive the ability to simulate the natural cooling. On the other hand, one can alsosimulate a forced cooling. The forced cooling can be realized with a cooling fan.The convection heat transfer coefficient will be higher. In the following part,the relation between an air flow and the convection heat transfer coefficient hc isexplained.

Cooling of a plane surface with a fan gives an airflow with a certain velocity.This cooler air flow will transport the heat from that surface to the surroundingair. One can use the Nusselt number to approach the the convection coefficient

42

5.3. Thermal equivalent circuit

[Sahin, 2001]:

Nu =hcL

λ(5.9)

where L is length of the plane surface and λ the specific thermal conductioncoefficient, for air 0.025W/mK.

The dimensionless Nusselt number can be found with the flow as functionof the Reynolds number (figure 5.2) [Water van de and Veerkind, 2001]. TheReynolds number is defined as:

Re =v · L

ν(5.10)

where v is the speed of the medium that flows over the surface in [m/s] and ν thekinematic viscosity, for air ν = 1.5 × 10−5m2s−1. When the Reynolds number

1 10 102 103 104 105 106 107 108

1

10

102

103

104

105

Re

Pr = 7,07

Pr = 0,72

Nu

Figure 5.2: A flow of two media as function of Reynolds number, namely air(Prandtl=0.72) and water (Prandtl=7.07).

is known, the Nusselt number can be found in figure 5.2. The convective heattransfer coefficient can be extracted with equation 5.9.

The sudden bend in the characteristic is the transition from the laminar modusto the turbulent modus; this transition will influence the cooling of the surface alot.

The top of the mover is attached to a certain suspension; this part can alsotransfer the heat due to convection to the surroundings. The effective surface ismuch bigger than the other convection surfaces. Therefore, it is decided that thesurface is multiplied by three to compensate the larger cooling surface.

43


(a) The heat flow perpendicular to the windings. Theinsulation between the wires will cause a large ther-mal resistance.

(b) The heat flow in the same direc-tion with the wires. The copper willtransport most of the heat, due to itslow thermal resistance.

Figure 5.3: Different heat flow directions in the coil.

5.3.2 Thermal conduction resistance in the coil

Although the formulation of the conduction resistance seems rather simple, agood knowledge of the material is required. One important assumption has tobe made, namely there is no interaction between the different heat flows in thedifferent coordinate directions. [Howe et al., 1993].

In figure 5.1 the coil is divided into parts with several thermal resistances.Those resistance are describing a different heat flux paths. So, when looking tofigure 5.5, the heat flux paths are divided into two different main directions. Theheat flow from the inside to the outside of the coil will notice a large thermalresistance, due to the insulation between the copper wires (see figure 5.3(a)).Rp is representing this large thermal resistance. The other heat flow is in thedirection of the winding path (see figure 5.3(b)). Copper has a relative smallthermal resistance in comparison with the insulation. This path is described withthe thermal resistances Rs.

This approach is chosen, because the thermal resistance of a single copperwire is totally different from the thermal resistance of one copper wire to anothercopper wire, due to the insulation. The specific thermal conduction coefficient ofcopper is defined as λs = 340W/mK. The heat flow between the wires, throughthe insulation, has a conduction coefficient of λp = 1W/mK. Between the wind-ings and the SMC core is glue and insulation layer. This layer has a better heatconduction than air, (λair = 0.025W/mK) and has a specific thermal conductioncoefficient λglue = 0.93W/mK. The chosen conduction coefficients of the insula-tion layers between the copper wires may be are a bit low, but they can alwaysbe increased by experimental experiences in the future.

44

5.4. Results

5.3.3 Heat sources

The only one heat source that is assumed, is dissipated energy in the copper. Theother losses are neglected because in worst case the actuator is fixed on its placeand is not moving. The amount of copper loss must be divided by two, accordingto the symmetry of the thermal equivalent model.

The copper losses are generated in the whole coil. As mentioned the coil isdivided into different sections, represented as a node with resistances Rp and Rs.In figure 5.1 all the nodes are numbered (from 0 to 12). Every node in the coilcontains a heat source, which introduces an amount of heat proportional to thevolume of that area. The heat will flow from warmer to colder places and will bedistributed over the total thermal circuit. This gives the temperature distributionwith its gradient over the circuit with respect to the ambient temperature of 25oC.

5.4 Results

The design, that is chosen in section 6.5, is being analyzed with the thermal equiv-alent model and verified with a thermal 3D FEM simulation. Both simulationsgive steady state results. The thermal equivalent circuit model has to determinethe worst case thermal behavior for a given current density. The 2D model givesa quick solution for any given configuration. After the total design procedure,one chosen configuration can be analyzed with the thermal 3D FEM software, toavoid long thermal calculations for each design.

Figure 5.4: The 2D thermal equivalent circuit with the temperature distributionof the coil and the SMC core.

45


In a standard situation, the mover bar is located in the stator. The heat canbe also transferred to the permanent magnets. The thermal equivalent model isassumed without stator to simulate the worst case. On the other hand, one mustbe careful to use the permanent magnet as a heat transfer channel, because theycan be demagnetized irreversible when becoming too hot, see section 3.3.

The heat must primary be transferred towards the cooling layer on top andalso via the outside of the coils due to convection.

The thermal 3D FEM model is the same model as used for the magnetostaticdesign procedure (chapter 6). The convection coefficient for the aluminium coolingarea is put on halu = 25[W/m2K] and it is just a thin plate at the top of the coils.The convection coefficients for the outer surface of the coils is hcoil = 10[W/m2K].The tips of the mover bar are defined on hbar = 3[W/m2K]. This convectionparameters are the same as for the thermal equivalent mode.

The conduction coefficients for the used materials are λalu = 200[W/mK],λglue = 0.92[W/mK] and λsmc = 18[W/mK]. These are the same as for the 2Dthermal equivalent circuit.

The coils have to be differently defined for the FEM model as for the thermalequivalent circuit. The heat transfer in the windings for the thermal equivalentcircuit are defined in two heat flow directions, as mentioned above, where λp =1[W/mK] and λs = 340[W/mK]. λs is the conduction coefficient of copperand λp is the conduction coefficient of the insulation. The insulation betweenthe copper wires will influence the thermal behavior significantly in the thermalequivalent circuit. The coil in the 3D FEM model is represented as a solid piecewith λcoil = 1[W/mK] dominated by the insulation in the coil.

The current density (for the configuration of the continuous force) is J =6.7[A/mm2]. The ambient temperature is Ta = 25oC. The thermal equivalentcircuit model gives a maximum temperature in the bottom of the coil of 120oC,see figure 5.4. The main heat flow goes via the aluminium cooling surface on thetop. The SMC core gets the same temperature as the coil in the middle. In thethermal equivalent circuit model, the heat transfer to tips of the SMC core is alsoincorporated.

The 3D FEM thermal simulation gives the total steady state thermal situationof the total mover bar. The current in the middle C-coils will dissipate most ofthe energy, this is the worst case. The A and B-coils will dissipate a quarterwith respect to the C-coils, according to the three-phase power supply. TheFEM results are shown in figure 5.5(a). The maximum temperature in the FEMcalculation is around 110oC. This is the temperature at the outside of the coils.The inner part of the coils will be the warmest part. A cross-section is shown infigure 5.5(b), where the maximum hot spot is around 112oC.

In normal situations the mover bar will be placed within the effective length ofthe stator. This will influence the thermal behavior, because the coils are now ableto transfer a part of the heat via the permanent magnets to the back iron. Thethermal equivalent circuit can not capture the properties of this situation. Theresults of a thermal 3D FEM simulation are shown in figure 5.6. The maximumtemperature is around 70oC for the coils. The temperature of the permanentmagnets will not exceed 40oC.

46

5.5. Conclusion

(a) (b)

Figure 5.5: a) Thermal analysis of the 3D FEM model without the stator, b) Thecross-section of the bottom side of the coil in the middle.

5.5 Conclusion

The temperature values, predicted by FEM analysis, are a bit lower than thevalues obtained by the 2D thermal equivalent circuit calculations. The maximumvalue in the coil for 3D FEM is 112oC and for the thermal equivalent model is120oC. The maximum temperature between the aluminum plate and the upperpart of the coils is around 83oC for FEM. The TEC model gives a temperatureof 87oC.

When the stator is included the thermal model, the thermal behavior sig-nificantly changes. Assuming that this simulation is correct the current for thecontinuous force can be increased. But one must be aware of the demagnetizationeffect in the permanent magnet, which can be caused by the heat (see AppendixA).

Therefore, it is wise to do some experimental thermal measurements whenthe prototype is built. These experimental results can gain extra information toadapt both models and refine the thermal predictions for later designs.

47


Figure 5.6: Thermal analysis of the 3D FEM model with permanent magnets andthe back iron included.

48

Chapter 6

Electromagnetic design

Due to the innovative geometry of the actuator, a new strategy is defined to cometo a final design for different user applications.

This chapter contains an introduction and explanation of the figures of merit(cost functions) that have been used during the design of the C-shaped actuator.The figures of merit are important criteria to compare several design options.

Based on the results of this design flow, the dimensions are determined for theprototype that is going to be built by Tecnotion BV Almelo. All the results ofthe design flow are included.

For later designs, Tecnotion has to set up a list of specifications with the clientfor each actuator. This list can be translated to the figures of merit. This willspecify the design process. The initial design dimensions can be precalculatedwith the MEC model, as described in chapter 4. The rest of the design flow isgoing to be done with a parametric search. Optimetrix (by Ansoft Co.) will beused to perform this parametric search.

6.1 Parameters

Due to the complex 3D structure of the C-shaped actuator there are a lot of dif-ferent design variables. Important parameters are the thickness of the permanentmagnets, size of the mover bar, the thickness of coils, pole and magnet pitchesand the type of the permanent magnets.

These parameters can be different for horizontal and vertical parts of theactuator structure. For example, the permanent magnets on the bottom side canbe thicker than the magnets on the vertical side. Figure 1.1 shows also an actuatorwith two different size of the permanent magnets. It means that the amount ofdesign parameters can be twice as much as mentioned.

To start with initial sizing of the C-shaped actuator, there is decided to give allthe permanent magnets the same dimensions. This implies that the parametersfor the horizontal and vertical parts are the same and the mover bar is alwaysconfigured as a square bar (see figure 6.1):

Wsmc = Hsmc (6.1)

This gives the total structure also a square shape, which is different when com-pared to most other linear actuators.

49

Chapter 6. Electromagnetic design

The C-shaped actuator, that will be designed, has to have a total width of40mm. The total height, without the mechanical suspension and cooling system,is always less than 40mm. The airgap length is fixed at 0.5mm and the glue andinsulation layer between the coils and the SMC core will be 0.2mm.

The prototype will be equipped with Neorem 412a rare earth permanent mag-nets with a the magnetic remanence Br of 1.3T . Further specifications can beviewed in section 3.3 and Appendix A.

The iron thickness of the back iron is fixed, in dialogue with Tecnotion, on4mm. It is important to mention that the back iron will have to sustain theattraction force of the magnets in all directions. So the thickness of the back ironwill be a mechanical constraint too.

After fixing several parameters there are five variable parameters left, namelythe thickness of the permanent magnets, width/height of the mover bar, thethickness of coils, the length of the pole pitch and magnet pitch. The coil andpermanent magnet dimensions will influence the operating point of the permanentmagnets. The pole pitch is strongly limited by the level of saturation in the moverbar. With the help of the MEC model and the fixed parameters, the range of the

Parameters Abbreviation Size CommentTotal width actuator Wtot 40mm fixed

Width mover bar Wsmc − variableIron thickness Liron 4mm fixedCoil thickness Lcoil − variableAirgap length Lairgap 0.5mm fixed

Glue layer Lglue 0.2mm fixedMagnet thickness Lpm − variable

Pole pitch τpp − variableMagnet pitch τmp − variable

Table 6.1: A list of all important parameters for the C-shaped actuator.

variable parameters for design flow can be achieved. The limitations of theseparameters are necessary to avoid to many simulations. An important detail isan indication for the pole pitch length. A very long pole pitch leads to a veryhigh level of saturation in SMC core and a too small pole pitch indicates a notsufficient use of the material in the actuator. All fixed and variable parametersare shown in table 6.1. As mentioned before, the total width of the actuator isfixed on 40mm. All the parameters are defined in figure 6.1. The total width ofactuator can be formulated as follows:

Wsmc + 2Lairgap + 2Lglue + 2Liron + 2Lpm + 2Lcoil = 40mm (6.2)

All the fixed values can used in equation 6.2, the width can now be defined as:

Wsmc = 30.6mm− 2Lpm − 2Lcoil (6.3)

Equation 6.3 makes Wsmc as a function of the permanent magnet thickness Lpm

and the coil thickness Lcoil. The thickness of the permanent magnet and coilscan now be freely changed.

50

6.2. Figures of merit

Figure 6.1: Cross-section of the C-shaped actuator in XZ-plane

The same can be done for the pole and magnet pitch. The pitch ratio betweenthe pole and magnet pitch is defined as:

α =τmp

τpp(6.4)

6.2 Figures of merit

When a given current density is applied to the C-shaped actuator, it will producea certain force. This amount of force is very important feature for the final design.Nevertheless, it is not the only important feature.

To deal with the large number of variable parameters and their results, it isconvenient to use values that characterize the actuator for each design. Thesevalues are known as figures of merit. They qualify the actuator by, for example:the relation between the force and its weight. By analyzing the actuator in thisway, the amount of data can be reduced, and it is possible to compare severalactuator designs based on their figures of merit.

A very useful and common used figure of merit is the steepness; it qualifies theefficiency of the actuator by dividing the squared force by the dissipated power:

S =F 2

Pdis

[N2

W

](6.5)

The steepness gives a performance number which is independent on the currentdensity:

F 2

Pdis=

K2I2

3I2Rcoil=

K2

3Rcoil

[N2

W

](6.6)

51


where K is the machine constant in [N/A]. I2Rcoil are the copper losses in [W].These losses will be dissipated in every coil, which are connected in three phaseseries. The machine constant incorporates magnetostatic properties like the fluxdensity, physical dimensions and the number of turns (see section 2.2, [Gieras andWing, 2002], [Hendershot and Miller, 1994]). The larger the machine constantthe more force will be produced with a certain current.

The copper losses have to be transported to outside world. The way of trans-portation is also dependent on the construction on the actuator. The copperlosses must be as low as possible to increase the steepness function.

The steepness can be divided by a certain volume or mass. The steepnessfunction weighted with the volume gives the opportunity to compare differentkinds of actuators with different volumes. The steepness can also be divided bytotal mass of the mover bar or total structure.

Sometimes, the acceleration can be a very important feature for a design. Theacceleration can be found by dividing the force by the mass of the mover. In thiscase, there is no mass/load attached to the mover bar.

Here are useful figures of merit:

Csm =S

Mmover

[N2

Wkg

](6.7)

Cfm =F

Mmover

[N

kg

](6.8)

Csv =S

Vmover

[N2

Wcm3

](6.9)

Cfv =F

Vmover

[N

cm3

](6.10)

With these figures of merit, one has the ability to choose a suitable setup in anefficient way. The use of figures of merit is also convenient for comparing totallydifferent already existing actuators.

6.3 Continuous and peak force

In technical data, most linear actuator companies speak about several force regimes.In this project the C-shaped actuator has to be designed for a relatively good peakand continuous forces. Both of the two force regimes differ in thermal behavior.

6.3.1 Continuous force

The continuous force regime means that the required amount of force for a longerperiod does not cause an overheated actuator. The force is proportional to theapplied current, which will generate copper losses. The copper losses are the mainlosses and will heat up the motor. By agreement with Tecnotion it is settled thatthe coils do not exceed the 120oC, the ambient temperature is defined as 25oC.

The continuous regime gives the ability for actuator designs with a thin coilto have a relative high current density when compared with a thick coil. This isobvious visible in Appendix B.4.

To calculate the steady state temperature for every design it is necessary todevelop a thermal model. This thermal model has been discussed in chapter 5.

52

6.4. Introduction to the design flow

6.3.2 Peak force

Peak force results in a high current for a short period to achieve a certain peakaction. This high current will heat up the copper of the coils in a very shorttime. If this peak current takes too long, the copper will damage the insulationlayers irreversibly. With Tecnotion, it is agreed that coils may be heated up witha maximum of 20oC/s. This number limits the time of such an action, but givesthe ability to use the actuator intensively for a short period of time.

The copper has a certain heat capacity; so when a heat is introduced, due tothe copper losses, the copper will heat up in with a certain time constant. Tocalculate the approximately heat increase with respect to the time, the followingformula is used:

∆T

∆t=

J2peak

ρcuCcuσcu(6.11)

where Jpeak is the peak current density in [A/m2]), σcu the specific conductivityin [(Ωm)−1], ρcu the density in [kg/m3] and Ccu is representing the specific heatcapacity in [J/(kgK)].

When a temperature rise of 20oC/s is the maximum, the peak current densitybecomes around 64.7A/mm2; this current density is applicable for every design.This is in contrast with the current density for the continuous force, which isdepending on the shape of the actuator.

6.4 Introduction to the design flow

6.4.1 Target

The goal of this design flow is to come to design of a C-shaped actuator with goodproperties in peak and continuous forces. This designed actuator has no specificobjective for a customer or a machine, but determines the dimensions of the firstprototype of the C-shaped actuator built by Tecnotion BV.

The amount of continuous force is very dependent on the current density, be-cause the current density is much influenced by the steady state thermal behaviorof the actuator. The current density for the peak force is equal for every design,namely 64.7A/mm2. Therefore, these design criteria are pointing to different op-tima. To make a trade-off between these optima the steepness function (equation6.6) and other figures of merit can be advised.

6.4.2 Optimetrix

All the parameters for the dimensions of C-shaped actuator are shown in table6.1. Some of the parameters are fixed and constant during the total design flow,other are variable parameters.

The C-shaped actuator is drawn and configured in Maxwell 3D [Ansoft CoMaxwell 2D and 3D, n.a.]. During the drawing session it is possible to record allthe drawing instructions in a macro language and interact with these macro lan-guage. If the completed macro file is executed, Maxwell 3D will redraw the totalstructure. All the initial instructions in the macro file are containing constantvalues. These values can be changed to parameters. All the implemented para-meters can be as functions of other parameters or can be defined with a value.When the drawing procedure is completed and the materials are assigned, the

53


nominal project can be included in an Optimetrix project. Optimetrix uses theparameters from the macro file for performing parametric searches. Every singleparameter can be assigned with a value or sweep, for example the coil thicknesscan be put to 2mm and the permanent magnet thickness have to sweep from 3mmto 5mm in 4 steps, Optimetrix will generate a list with four different setups. Thesimulation process can be started. Optimetrix gives the information about theparameters per setup to Maxwell 3D, which will redraw the structure accordingto the given parameters and simulates the setup. After the four simulations thisparametric search is completed and can be analyzed with the post-processor.

The energy error for the C-shaped actuator is put on less than 0.7% for everysimulation. The amount of tetrahedral was mostly around the 80.000 elements.

6.4.3 Design steps

The design routine consists of three main steps. Before starting with the de-sign steps, it is necessary to investigate the proposed C-shaped actuator withthe magnetic equivalent circuit model. This gives, in a short period of time, arough indication for the pole pitch length, coil thickness and permanent magnetthickness. The coil consist only of one winding for each design in this stage.

All the results of the design steps are in Appendix B and the parameters aredefined in table 6.1. The next enumaration summarizes the three steps:

Step 1 In this step, the matching pole pitch τpp will be determined for eachset of [Lcoil, Lpm]. The criterium for the optimum pole pitch of each setis the steepness function divided by the mass of the mover (see equation6.7). To avoid a large number of simulations, this procedure is equippedwith a function that only needs five sets [Lcoil, Lpm] with their matchingpole pitches. The unknown data will be derived by doing an interpolationbased on these five sets. The pitch ratio α will be fixed on 0.85, the currentdensity is fixed on 7A/mm2.

Step 2 The results from step 1 contain a series of [Lcoil, Lpm] with their matchingpole pitches τpp, therefore [Lcoil, Lpm, τpp] are fixed parameters for step two.Step 2 determines for each set the continuous force, peak force and steepnessfunction. The two forces and the steepness function can be used to obtainthe other figures of merits, as discussed in section 6.2. All these data canbe used to select the proper set of [Lcoil, Lpm, τpp]. This selection procedurecan be simplified with 3D plots. The horizontal axis of this plot contains afigure of merit, while the vertical axes are containing the coil thickness Lcoil

and the permanent magnet thickness Lpm.

Step 3 The selected design of step 2 consists of one set of [Lcoil, Lpm, τpp], whichis going to be used for step 3. The pitch ratio α was fixed on 0.85 in thefirst two steps. This pitch ratio α is now going to be varied, together withthe pole pitch. The pole pitch variations will be not very large. This stepneeds also the continuous force, peak force and the steepness function todetermine the final design.

After step 3 the design procedure is ready and therefore all the variable parameters(see table 6.1) are known. In Appendix B a sort of manual is implemented togetherwith the results. The results are listed in different tables per step, all the used

54

6.5. Design flow

figures of merit are also listed. The design procedure for the prototype is describedin the next section. The design is according to its target, as described in section6.1.

6.5 Design flow

This section contains the design flow for the prototype. Each subsection containa step.

6.5.1 First step

The first step, to start the design flow, is to determine which pole pitch belongs toa combination of magnet thickness and coil thickness, according to the steepnessdivided by the weight of the mover bar. This optima will be found by sweepingthe pole pitch length for a few fixed sets of [Lpm, Lcoil]. The pitch ratio α (seeequation 6.4) is fixed on 0.85. When using the steepness, the current density canbe fixed on 7A/mm2 for every design.

With the magnetic equivalent model is decided to vary the coil thickness be-tween 1mm and 5mm and the permanent magnet thickness will be swept between3mm and 7mm. The pole pitch will be varied between 15mm and 45mm. Nowthere are three parameters which can be changed. If every parameter sweepneeds five steps, there are needed 53 = 125 simulations. This will take to muchsimulation time. To avoid this, only five sets [Lpm, Lcoil] will be evaluated on itsoptimum pole pitch, namely [Lpm, Lcoil] = [(3, 1), (3, 5), (7, 1), (7, 5), (5, 3)]. Thesevalues are the most outer values and the middle value for the parameters. Allthese five sets will be evaluated with a parametric search where the pole pitch isvarying. Figure 6.2 shows the results of the sweep of the five sets. This figureshows an optimum for each plot, the results are also listed in a table in AppendixB.2. The matching pole pitches for every set of coil thickness and permanentmagnet are shown in table 6.2:

Lpm(mm) Lcoil(mm) τpp(mm)7 5 213 5 335 3 277 1 253 1 24

Table 6.2: For every set [Lpm, Lcoil] the matching pole pitch.

The steepness function is a function of magnetic properties and the copperlosses. Varying the permanent magnet thickness or increasing the pole pitch willinfluence significantly the mass of the mover bar. Therefore it is convenient todivide the steepness function by the mass of the mover bar.

After the simulations, only the five sets are having an matching pole pitch. Aninterpolation function (see Appendix B.3) derives the missing data. This resultsin figure 6.3. The results from table 6.2 are visible in the corners of the plot andin the middle, the rest is interpolated data.

55


15 20 25 30 35 40 45100

150

200

250

300

350

400

450

500

pole pitch [mm]

Csm

[N2 /W

kg]

(a) Lpm = 7mm and Lcoil = 5mm

15 20 25 30 35 40 45150

160

170

180

190

200

210

220

230

pole pitch [mm]

Csm

[N2 /W

kg]

(b) Lpm = 3mm and Lcoil = 5mm

15 20 25 30 35 40 45150

200

250

300

350

400

pole pitch [mm]

Csm

[N2 /W

kg]

(c) Lpm = 7mm and Lcoil = 1mm

15 20 25 30 35 40 45135

140

145

150

155

160

165

170

175

180

pole pitch [mm]

Csm

[N2 /W

kg]

(d) Lpm = 3mm and Lcoil = 1mm

15 20 25 30 35 40 45300

320

340

360

380

400

420

440

460

pole pitch [mm]

Csm

[N2 /W

kg]

(e) Lpm = 5mm and Lcoil = 3mm

Figure 6.2: The most outer values and middle value for [Lpm, Lcoil]. The polepitch is varying between 15mm and 45mm as a function of Csm.

The pole pitch in the corners of the plot is quite different from the middle.The C-shaped actuator with a large coil thickness and a small permanent magnetthickness has a large pole pitch. This compensates the relative small amountof flux due to the small permanent magnets in combination with a large coilthickness.

A large coil thickness and a large permanent magnet thickness will lead toa smaller pole pitch, because of the small cross section of the SMC core. The

56

6.5. Design flow

3 3.5 4 4.5 5 5.5 6 6.5 7 12

34

520

22

24

26

28

30

32

34

Lcoil

(mm)L

pm (mm)

pole

pitc

h (m

m)

Figure 6.3: The optimum value for the pole pitch with a given coil and permanentmagnet thickness.

combination of a small permanent magnet with a small coil thickness leads alsoto an increasing pole pitch.

Step 1 needs a value set of 35 magnetostatic 3D FEM simulations. In thefuture it is considerable to do step 1 totally with the MEC model. This willreduce the simulation time significantly.

6.5.2 Second step

The results from step one give for a combination of permanent magnet thicknessand coil thickness the matching pole pitch, see Appendixes B.3 and B.4. Now it ispossible to extract for each set of [Lcoil, Lpm, τpp] the steepness value, continuousforce and peak force as function of [Lpm, Lcoil].

The current density for the continuous force has to be obtained from thermalequivalent circuit model (see section 5.3). The table in Appendix B.4 shows thedifference between the current density J per coil thickness. A small coil thicknesscan withstand a higher current density J(A/mm2), because of its better thermalbehavior. On the other hand, the current I(A) is depending on the coil thicknessand pole pitch. Therefore, the obtained pole pitch and the given coil thicknessare also influencing the amount of force produced by the C-shaped actuator inthe continuous and peak regimes.

Figure 6.4 shows the steepness function. This function is independent on thecurrent density. There is an optimum around the middle of this plot.

The resistance of the coil is increasing when the permanent magnet thicknessis decreasing, because the length of a winding is decreasing. The machine constantK will increase when the permanent magnet thickness is increasing. These twoeffects will cause the optimum in a certain point.

The continuous force current density J [A/mm2] of the thin coil is higher thanthat of thick coil due to its thermal behavior (see section 5. The flux density isalso higher for thin coils. But the total current I[A] is much lower for a thin coil

57


34

56

7 12

34

5

35

40

45

50

55

60

65

70

75

80

85

Lcoil

(mm)Lpm

(mm)

S(N

2 /W)

Figure 6.4: The steepness function, which is independent on the current.

34

56

7 1 1.5 2 2.5 3 3.5 4 4.5 5

10

15

20

25

Lcoil

(mm)

Continuos force

Lpm

(mm)

Flo

r (N

)

(a)

34

56

7 12

34

580

100

120

140

160

180

200

220

240

260

Lcoil

(mm)

Peak force

Lpm

(mm)

Flo

r (N

)

(b)

Figure 6.5: a) Continuous force and b) peak force for different configurations.

than that of a thicker coil.Figure 6.5(a) shows a clear optimum around the middle. There is also one

sudden rise in the corner for small permanent magnets and thick coils. Thispeak is caused by the obtained pole pitch in step one, which was quite long incomparison with the rest (see figure 6.3). The steepness function (see figure 6.4)is very low for this setup.

Figure 6.5(b) shows the peak force as a function of the coil thickness andthe permanent magnet thickness. The peak force regime has not a very clearoptimum, because the applied current density is constant 64.7A/mm2. The totalcurrent will increase with an increasing coil thickness. The obtained pole pitch,from step one, is also influencing the peak force.

The figures 6.5(a) and 6.5(b) are showing the produced force for each setup.Each setup has also a different mass, because the pole pitch, coil thickness andthe width of the SMC core are varying. To analyze the C-shaped actuator on its

58

6.5. Design flow

acceleration, the force will be divided by the mass of mover bar, see equation 6.8.The figures 6.6(a) and 6.6(b) are showing the continuous and peak accelerationas function of the coil thickness and the permanent magnet thickness.

34

56

7

11.522.533.544.5580

100

120

140

160

180

200

Lpm

(mm)Lcoil

(mm)

Continuous

Cfm

(m

/s2 )

(a) The acceleration Cfm in the continuousregime in (N/kg).

3

4

5

6

7

1

2

3

4

5400

600

800

1000

1200

1400

1600

Lpm

(mm)L

coil (mm)

Peak mode

Cfm

(m

/s2 )

(b) The acceleration Cfm in the peak regime in(N/kg).

Figure 6.6: Cfm for the continuous and peak regimes which is proportional withthe acceleration.

It is clear that for the steepness function (see figure 6.4) an optimum can befound between 2mm and 4mm for Lcoil and between 4mm and 6mm for Lpm.The continuous force (figure 6.5(a)) gives Lcoil = 2.5mm and Lpm = 4.0mm. Thecontinuous acceleration (figure 6.6(a)) has also an optimum for the coil thicknessof Lcoil = 2.5mm. The peak force and acceleration (figures 6.5(b) and 6.6(b)) aredemanding a large coil and permanent magnet thickness.

When looking to the steepness function and combining the other figures, withTecnotion is decided to go for a C-shaped actuator with Lcoil = 3.5mm andLpm = 5mm. The pole pitch will be τpp = 26.8mm.

6.5.3 Third step

The third step is to figure out how the performance index looks when changingthe pitch ratio α and pole pitch τpp. The coil and permanent magnet thicknessnumbers are obtained from step two and will be fixed. Step two gave a pole pitchof 26.8mm. This size is going to be analyzed in close region around this 26.8mm.The pitch ratio α is for both previous steps being fixed on 0.85. For the finaldesign, this ratio is going to be closer inspected.

The volume of a permanent magnet is proportional to the price, so it is in-teresting to have a closer look at the cost/volume-index in comparison to theperformance/force increase. Figure 6.7 shows an increase of the continuous forcewith respect to the pole pitch. The force is increasing with the pole pitch andtherefore the volume of the permanent magnet.

The increasing of the flux by increasing the pole pitch will create more fluxdensity in the SMC core and will lead to a high level of saturation.

To analyze the C-shaped actuator without the influences of the current, thesteepness function can be advised. The steepness function, in plot 6.8(a), is di-vided by the mass of the mover, because the mass of the mover bar is proportional

59


0.60.7

0.80.9

1

2224

2628

3017

18

19

20

21

22

23

24

25

26

27

ratiopolepitch(mm)

Flo

r(N)

Figure 6.7: Lorentz force as a function of the pole pitch and the pitch ratio α.

to the pole pitch.

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

2324

2526

2728

2930

350

400

450

500

550

pitch ratiopolepitch (mm)

Csm

(N

2 /Wkg

)

(a) Steepness divided by the mass of the moverbar as function of the pole pitch and the pitchratio α.

0.7

0.8

0.9

1 23 24 25 26 27 28 29 30

27

28

29

30

31

32

33

polepitch (mm)pitch ratio

Svo

l pm (

N2 /W

kg)

(b) Steepness divided by the volume of a perma-nent magnet as function of the pole pitch and thepitch ratio.

Figure 6.8: Two figures of merit in design step 3.

There is a certain optimum in figure 6.8(a) around the values between 26mmand 27mm for the pole pitch. The optimum for the pitch ratio α goes to one.So, it is hard to determine, from this plot, what is the most suitable pitch ratiofor the definitive design. When looking to the costs for the permanent magnetand its performance rise in figure 6.8(b), where the steepness is weighted with thevolume of the permanent magnet, a maximum can be found for a pitch ratio of0.85 and a pole pitch length of 26mm. The real optimum for the pole pitch is atrade-off between these two figures.

The field plots are available in Appendix E. All the definitive dimensions aresummarized in table 6.3, including the forces obtained by FEM and MEC.

60

6.6. Conclusion and recommendation

Parameter Abbreviation Size CommentWidth mover bar Wsmc 13.6mm dependent

Height Wsmc Hsmc 13.6mm dependentIron thickness Liron 4mm fixedCoil thickness Lcoil 3.5mm variableAirgap length Lairgap 0.5mm fixed

Glue layer Lglue 0.2mm fixedMagnet thickness Lpm 5 variable

Pole pitch τpp 26.8mm variableMagnet pitch τmp 22.78mm variablePitch ratio α 0.85 variableCont. force FcontMEC

22.87N MECCont. force FcontF EM 22.99N FEMPeak force FpeakMEC

218.44N MECPeak force FpeakF EM

208.87N FEMAttraction force Fattr. 160N FEM

Table 6.3: A list of all important parameters for the C-shaped actuator.

6.6 Conclusion and recommendation

In this chapter several design criteria were discussed. In this project the C-shapedactuator had to perform well based on the steepness, continuous and peak forcecriteria. The continuous force will be around 23N and the peak force is 210N(see table 6.3). The attraction force, due to permanent magnets is 160N .

In the future it is possible to use this design routine for the C-shaped actuatorfor every specified application. Tecnotion BV has to specify, with the client,design requirements (force, acceleration, mass, etc.) for the C-shaped actuator.

For this specification it is convenient to introduce a weight function for eachdesign step like:

Cw =N∑

i=1

ki · Pi

max(Pi)(6.12)

where i is the number of a specific figure of merit, k is the weight function andP is a specific figure of merit value. To scale all the figures of merits each tablehave to be divided by the maximum value of that table, max(Pi)

The weight factor k (between 0 and 1), for each i, can be specified in advanceby Tecnotion and/or a customer. This gives the ability to come to an optimumsolution that is defined in order of importance per step. Every figure of merit,numbered by i, is weight over the maximum of the whole set of figures of merits.Now, every figure of merit becomes a number between zero and one, and can berated with the factor k.

The current is limited by the thermal behavior when applying the peak current.A too high current can cause a demagnetization effect, locally or more, whatwill damage the permanent magnet in an irreversible way (see section 3.3). Tocheck the demagnetization effect, due to the applied current and/or the thermalbehavior, the post-processor of the 3D FEM package can provide the field results(like the flux density and field strength). The demagnetization level is generallyanalyzed with help of a field strength plot.

61


62

Chapter 7

Cogging Force

Cogging is the oscillatory force or torque caused by the tendency of the rotor lineup with the stator in a particular direction, where the permeance of the magneticcircuit ”seen” by the magnets is maximized [Hendershot and Miller, 1994].

7.1 Introduction

The cogging force in a permanent magnet linear machines is caused mainly [Bianchiet al., 2005] by two phenomena. The first one arises from the interaction betweenthe permanent magnets and the finite length of the mover bar. This interactionis often called an ”end effect” or detent force. It can be minimized adapting asuitable mover bar length or modifying the shape of the tips of mover bar. Thisphenomenon is prevalent in the C-shaped linear actuator.

The second phenomenon is due to the interaction between the permanentmagnets and the armature slots. It is called ”slotting effects” and is presented inslotted permanent magnet actuators. This last effect will not be discussed furtherbecause of the slotless structure of the C-shaped actuator.

The cogging force exists even when there is no current. Cogging/detent forceis a magnetic disturbance force and depends on the relative position of the ro-tor/mover bar with respect to the permanent magnets and it is independent ofthe armature current. The mean value of cogging is zero. If the cogging force issignificant, a position dependent signal (feed forward control) has to be added tothe force command to overcome this disturbing force.

In figure 7.1 the cogging force as function of the position of the C-shapedactuator is plotted, these results are obtained from magnetostatic 3D FEM perposition. The pole pitch of the C-shaped actuator is 26.8mm as determined insection 6.5. There is no current applied and the mover bar is positioned on 15different positions separated over one pole pitch. The force is calculated with thevirtual work method with an energy error less than 0.7%. It is obvious that thecogging force is significantly presented and will cause a lot of unwanted behaviorlike force ripples. This simulation is done with a SMC core with the length oftwice the pole pitch.

The cogging force, as shown in figure 7.1, is unacceptable for this actuator.The continuous force of the actuator is predicted around 23N and the peak forceis around 210N for this C-shaped actuator.

63

Chapter 7. Cogging Force

0 6.7 13.4 20.1 26.8-40

-30

-20

-10

0

10

20

30

40

moverposition [mm]

Cog

gin

forc

e [N

] in

y-di

rect

ion

Figure 7.1: Cogging force with no extra adjustments to reduce cogging.

Cogging effect, as mentioned before, is caused by a permeance variation. Thisis the permeance of the flux path ”seen” by the permanent magnets. In otherwords, the operation point of the permanent magnet is varying when the moveris changing its position. This changing of the operation point implies an energyexchange, so energy is stored or gained from permanent magnets, which is causinga force ripple. This force can be defined as energy changing per positions:

Fm = −δWm

δy(7.1)

where Fm is, in this case, the cogging force in [N ]. The energy in this permanentmagnet can also be written as [Ackermann et al., 1992]:

Wm =HmBm

2Vm =

B2m

2µ0µpmVm (7.2)

The volume Vm of the permanent magnets in [m3] is constant for each design.On the other hand, the operating point and therefore Bm will change with theposition. This will introduce the cogging force.

7.2 Cogging reduction

This section contains some attempts to reduce the cogging force. All the simula-tions are done with the 3D FEM magnetostatic in combination with Optimetrix.To reduce the cogging force the SMC core is changed in length, the permanentmagnets have been skewed, the shape of the mover bar at the tip is changed andthe permanent magnets are shifted. Every method will be discussed individually.

64

7.2. Cogging reduction

7.2.1 Mover extension

The mover extension is defined as the extra length of the SMC core. In figure7.2 the extension length is defined on one side of the mover bar. The length ofthe SMC core is initial two times the pole pitch to hold the coils and handle theflux from the permanent magnets. The 3D FEM model has an extra parameterto increase the mover extension length. Varying the extension gives the followingresults as shown in figure 7.3.

Figure 7.2: One side of the mover bar with the extension of the SMC core.

When there is no extension, the results are the same as in figure 7.1, thecogging force has a peak value around 32N . It is clear that the top is decreasingwith increasing the length of the extension till around 6.7mm. This reductionof the cogging occurs always around τpp/4 for the extension length. This area isexamined in more detail in figure 7.3(b).

The minimum, if looking at the cogging force, is around 6.18mm for the ex-tended length of the SMC core. This gives a cogging force with a top value of5N .

The specific extension of 6.18mm is simulated individually, the results areplotted in figure 7.4. This simulation is also done over one pole pitch, the coggingforce is again sinusoidal, but the frequency is doubled when compared with noextended length of the SMC core, see figure 7.1. Due to the increasing of the ex-tension length the permeance variation is doubled per pole pitch, but the amountvariation is decreased. This reduces the cogging force significantly, from 32N to5N .

7.2.2 Skewing of the permanent magnets

Skewing is a well known method for minimizing torque ripple in conventionalrotary machines. Skewing has a positive effect on the torque ripple as well as the

65


0

5

10

15

051015202530-50

-40

-30

-20

-10

0

10

20

30

40

50

Extension [mm]

moverposition Y [mm]

Cog

ging

For

ce [N

]

(a) Mover extension variation in from 0mm to 15mm

55.5

66.5

77.5

05

1015

2025

30-15

-10

-5

0

5

10

15

Extension [mm]Moverposition [mm]

Cog

ging

For

ce [N

]

(b) A detailed plot, the extension varies between 5mm and7.5mm

Figure 7.3: The cogging force as a function of position and extension length.

0 6.7 13.4 20.1 26.8-6

-4

-2

0

2

4

6

moverposition [mm]

Cog

gin

forc

e [N

] in

y-di

rect

ion

Figure 7.4: Cogging force with an extension of 6.18mm

average torque produced by a conventional rotary machine. The same principlecan be applied to linear motors.

This method is also going to be analyzed in FEM 3D. In figure 7.5(a), it isfigured out how the skewing is defined for the C-shaped actuator. The extensionis put on 6.18mm as determined in the section 7.2.1.

A rotation point is located in the middle of every permanent magnet. Theskew angle is implemented as an extra parameter in the 3D FEM model. For eachangle Optimetrix will simulate 15 different position for the mover bar, separatedover one pole pitch.

66


(a) The skew angle for the bottom side.

05

1015

2025

30

05

1015

20−6

−4

−2

0

2

4

6

Moverposition (mm)Skew ratio (degrees)

Fvi

rTO

Ty

(b) The cogging force as function of the position and posi-tion.

Figure 7.5: The cogging force as function of position and skew angle.

After adapting the skew angle, all the permanent magnets have to be adjustedin the corners of the back iron to avoid overlap in that region. With the help ofthe parametric search, the cogging force can be analyzed based on its effects fordifferent skewing angles per position. The results are plotted in figure 7.5(b).

The skew angle is varied from 0o to 20o for every permanent magnet. Whenthe angle is around 13o the corner of a permanent magnet is in line with theopposite adjacent corner of the other permanent magnet.

The cogging force reduces not very significantly when the angle is increased.The explanation for these results is the relative clearance between the permanentmagnets and the SMC core. Therefore, the flux distribution between the moverbar and permanent magnet is already smooth.

Cogging reduction by means of skewing of the permanent magnets is only use-ful with a C-shaped actuator with a slotted mover bar configuration, like standardiron core linear actuators.

7.2.3 Varying the shape of the extension

Another way to reduce the cogging force is to change the shape of the mover bartips. The shape is defined in 7.6(a). The key idea is to use existing disturbanceharmonics to neutralize the overall disturbance. In figure 7.4 at 6.18mm theextension length gives a sinusoidally force pattern. This pattern can be used toreduce the cogging. The extension of 6.18mm gives the SMC core a total lengthof 2 ·τpp +2 ·6.18mm and gives two periods of cogging force in one pole pitch. Thetotal length must be stay the same, but upper and lower parts must be shifted

67


half a period. With this shape the force patterns at the top and bottom are inanti phase.

In the next simulation the mover tip is extended with the triangle shape. Theresults of the parametric search are shown in figure 7.6. The shape is more orless the same as plotted in figure 7.3. The peak value for the cogging force isat minimum around a triangle with length of 7mm. The cogging force reductionis for this triangle length around the 5N . This will not reduce the cogging forcesignificantly and therefore not useful for this kind of actuator. There are analyzed

(a) The triangles at both mover bar tips.

0

5

10

15

051015202530−25

−20

−15

−10

−5

0

5

10

15

20

25

triangle(mm)moverposition(mm)

Fco

g(N)

(b) The cogging force as function of the position and trianglelength.

Figure 7.6: The cogging force as function of position and triangle length.

other dimensions for the extension and triangle length, but all of those attemptsgave no significant reduce of the cogging.

7.2.4 Shift of the permanent magnets

Another way to reduce the cogging is to shift the vertical permanent magnetsin the moving direction, like in figure 7.7(a). The permanent magnets on thevertical part are shifted in the opposite direction. The SMC core is extendedwith 6.18mm, like is determined in section 7.2.1.

68


(a) The shift distance in the C-shaped actuator.

0 0.5 1 1.5 2 2.5 3 3.5 4

0

10

20

30−6

−4

−2

0

2

4

6

shift distance [mm]moverposition Y [mm]

Cog

ging

For

ce [N

]

(b) The cogging force as function of the shift distance andthe position.

Figure 7.7: The cogging force as function of position and shift distance.

This shift is implemented as parameter in the 3D FEM model for a parametricsearch, which gave the results as shown in figure 7.7(b). Due to the applicationof the virtual work method and to attempt to achieve an acceptable calculationtime the plot does not look very smooth, but the trend is clearly visible.

The cogging force is obviously decreasing for an increasing shift distance, ashift distance of 4mm gives a peak cogging force of around 2.2N . A closer look tothe disturbing force pattern shows the same two sinusoidal periods over one polepitch period which are increasing with the shift.

Shifting of the permanent magnets, as proposed, will also influence the propul-sion force. This is not tested with the 3D FEM package due to lack the of researchtime. A solution to reduce this effect, is to use special diagonal wound coils aroundthe mover bar. The effective length will also increase and therefore it could alsohave a positive influence on the propulsion force. Increasing of the effective lengthwill also increase the copper losses. The diagonal parts of the winding can be lo-cated on the vertical or horizontal sides.

Shifting of the permanent magnets has a positive effect on the cogging force.To improve this method the horizontal permanent magnets can be cut into twoparts and also be shifted in the same direction as the nearest vertical permanentmagnet. This option is not tested yet.

69


7.3 Conclusion

The most suitable solution of the cogging problem is, at this time, a shift of thepermanent magnets with an enlarged SMC core. It is interesting to analyze theinfluence on the propulsion force without changing the mover bar and the coilconfiguration.

Skewing the permanent magnets and changing the mover bar tips gave nosufficient results for the C-shaped actuator.

70

Chapter 8

Conclusion andRecommendation

8.1 Conclusions

• The working principle is explained and a comparison is made with the Torus-NS motor. To start the modelling, first, an analytical model is described.This gives a better understanding of the C-shaped actuator operating prin-ciple.

• The soft magnetic composite allows the actuator to use 3D flux paths.Therefore, it enables the actuator to use the three sides of the winding.It saturates faster when compared to electrical steel and has a lower perme-ability. The relatively large distance between the permanent magnets andthe SMC core introduces also a great magnetic resistance and will reducethe flux density originated by the permanent magnet.

• The magnetic equivalent circuit model predicts the generated force for anysetup of the C-shaped actuator. It does not require a significant calculationtime and the continuous force prediction results are within the 6% whencompared with the 3D FEM force predictions.

• To determine the continuous force, one must be able to analyze the thermalbehavior for a single setup with a given current. This can be done withthe 2D thermal equivalent circuit and the 3D FEM software. These modelsgive a good indication, but have to be further adapted based on practicalexperience and measurements.

• The design routine gives for a specified application the suitable dimensions.It takes three design steps to complete this routine. These design stepsrequire a lot of parametric 3D FEM simulations, but Optimetrix gives theability to put all desired setups in a list and do a long term simulationprocedure. Every single simulation takes about 10 minutes within an energyerror of less than 0.7%.

• This design routine is only applicable with the help of a 3D FEM package.When such a package is not available, the performed routine gives a good

71

Chapter 8. Conclusion and Recommendation

global trend which can be used in combination with the MEC model.

• Optimetrix requires a parametric model created in 3D FEM. The developedparametric model gives the user the freedom to simulate every demandedsetup of the C-shaped actuator in the 3D FEM package. After simulating,a setup can be analyzed using the force and field plots (flux density, fieldstrength, etc.) can be evaluated with post processor.

• Cogging force is a big disadvantage of the proposed C-shaped actuator. Theeasiest way to reduce the cogging effect is to enlarge the SMC core. Thedesigned actuator requires a mover extension up to 6.18mm. A furtherreduction can be achieved by shifting the vertical permanent magnets inthe opposite way along the moving direction.

8.2 Recommendation

• In this thesis everything is analyzed, designed and evaluated based on amagnetostatic approach. When the prototype is built, the dynamic mea-surements are very interesting for further steps for designing the next C-shaped actuator. A 3D FEM transient analysis is not an option, due to therequired accuracy and memory problems.

• Another topology can be evaluated to increase the force with the same shapeof stator by doubling the amount of coils. The mover bar will be as twiceas long holding 12 coils. The cogging will be the same as for the structurewith 6 coils.

• The thermal properties of the thermal equivalent circuit model and FEMmodel have to be adjusted with the help of experimental data obtained fromthe first built prototype.

• For the next steps, it could be recommended to extend the thermal equiv-alent circuit model with the stator, to obtain the temperatures in the per-manent magnets.

• The TEC and MEC models, made in Mathcad (see Appendix F), have to bewritten in a Matlab script. Matlab can perform large extensive parametricsearches based on the MEC and TEC models in a very short period of time.This is not possible in Mathcad.

• To provide a much faster total design time, step 1 (as described in section6.5) must be completely executed by the magnetic equivalent circuit model.This is only possible when the total Mathcad model is described in a Matlabscript.

• A further research on a C-shaped actuator could be associated with a topol-ogy with slots in the mover bar to increase the flux density and thereforethe force.

• In chapter 4 is noticed that the edges of the permanent magnet not signif-icantly contribute to the flux production. It is possible to reduce this partof the permanent magnets without loosing performance.

72

8.2. Recommendation

• It is wise to run a few FEM simulations to examine the performance im-provement by using a Halbach permanent magnet array and compare thiswith the current permanent magnet NS-structure.

• Chapter 7 of the thesis contains a few attempts to reduce the cogging force.This research can be further completed by searching for other options.

73

Chapter 8. Conclusion and Recommendation

74

References

Ackermann B, Janssen JHH, Sottek R and van Steen RI. New technique forreducing cogging torque in a class of brushlessDC motors. Electric Power Ap-plications, IEE Proceedings B, 139(4), 315–320, 1992.

Andersson O. High velocity compaction of soft magnetic composite. Hganas, n.a.Hoganas AB, Sweden.

Ansoft Co Maxwell 2D and 3D. http://www.ansoft.com. n.a. Pittsburgh PA15219, USA.

Aydin M, Huang S and Lipo TA. Axial flux permanent magnet disc machines: areview. Symposium on Power Electronics, 2004.

Bianchi N, Bolognani S and Cappello ADF. Reduction of cogging force in PMlinear motors by pole-shifting. Electric Power Applications, IEE Proceedings,152(3), 703 – 709, 2005.

Furlani E. Permanent Magnet and Electromechanical Devices. Academic Press,2001. ISBN 0-12-269951-3.

Gieras J and Wing M. Permanent Magnet Actuator Technology. Marcel Dekker,second edition, 2002. ISBN 0-8247-0739-7.

Gieras JF and Piech ZJ. Linear Synchronous Motors, Transportation and Au-tomation Systems, volume 1. CRC Press, 2000. ISBN 0-8493-1859-9.

Hendershot JR and Miller TJE. Design of brushless permanent-magnet actuators.Monographs in Electrical and Electronic Engineering. Magna physics publica-tions, Oxford science publucations, 1994. ISBN 0-19-859389-9.

Holmes P. Elektrische Netwerken. Addison Wesley, 1993. ISBN 90-6789-359-5.

Howe D, Liu ZJ, Mellor PH and Jenkins MK. Thermal analysis of permanentmagnet machines. Electrical Machines and Drives, Sixth International Confer-ence, (376), 359 – 364, 1993.

Jack A. Experience with using soft magnetic composites for electricalmachines.University, New Magnetic Materials - Bonded Iron, Lamination Steels, SinteredIron and Permanent Magnets(Digest NMo. 1998/259), 1998. Newcastle uponTyne Univ.

Kim JK, Joo SW, Hahn SC, Hong JP, Kang DH and Koo DH. Static Charac-teristics of Linear BLDC Motor Using Equivalent Magnetic Circuit and FiniteElement Method. Transactions on Magnetics, 40(2), 742–745, 2004.

75

REFERENCES

Leon SJ. Linear Algebra with applications. Prentice Hall, 5 edition, 1998. ISBN0-13-849308-1.

Lipo TA and Qu R. Analysis and modeling of airgap and zigzag leakage fluxes ina surface mounted-PM machine. IEEE, Vol. 40(1), 121–127, 2002.

Molenaar A. A novel planar magnetic bearing and motor configuration applied ina position stage. Ph.D. thesis, University of technology Delft, 2000.

Neorem Magnets. http://www.neorem.fi. n.a. 28400 ULVILA, FINLAND.

Ostovic V. Dynamics of saturated electric machines. Berlin, Springer, 1989. ISBN3-540-97079-7.

Rasmussen CB and Ritchie E. Improved Reluctance MMF Network for Calculationof Magnetic Field Distribution in Surface Mounted Permanent Magnet Motors.Electromotion Magazine, 4(1-2), 55–61, 1997a.

Rasmussen CB and Ritchie E. A magnetic equivalent circuit approach for pre-dicting PM motor performance. Industry Applications Conference, 1, 10 – 17,1997b. Thirty-Second IAS Annual Meeting.

Reece ABJ and Preston TW. Finite Element Methods in Electrical Power Engi-neering. Monographs in Electrical and Electronic Engineering. Oxford Sciencepublications, 2000. ISBN 0-19-856504-6.

Remsburg R. Thermal Design of Electronic Equipment. CRC Press, 2001. ISBN0-8493-0082-7.

Roberts G, Davidson A, Gair S and Hajto J. An overview of the powder processingof soft magnetic composites. Engineering science and education journal, pages237–240, 2001.

Sahin F. Design and development of a high-speed axial-flux permanent-magnetmachine. Ph.D. thesis, TU/e, 2001.

Sahin F. High Force Density Linear Electric Actuator. WIPO, (WO 2004/038899),2002. Patent.

Strahan RJ. Energy conversion by nonlinear permanent magnet machine. IEEproc Electr. Power Appl., Vol. 145(No. 3), 193–195, 1998.

Tapia JA, Leonardi F and Lipo TA. A Design Procedure for a PM Machine withExtended Field Weakening Capability. Industry Applications Conference, 3(37),1928–1935, 2002.

Upadhyay PR, Rajagopal KR and Singh BP. Effect of Armature Reaction on thePerformance of an Axial-Field Permanent-Magnet Brushless DC Motor UsingFE Method. IEEE TRANSACTIONS ON MAGNETICS, 40(4), 2023–2025,2004.

Water van de W and Veerkind A. Technische Warmteleer. (Syllabus), TU/eEindhoven, 2001.

76

Appendix A

Neorem permanent magnets

[kA/m]

Nominal Values at 20oC

NEOREM 412a / NEOREM 512 aNEOREM 412a / NEOREM 512 a

Br 1.30 T 13.0 kG

Coercivity

BH

c1000 kA/m 12.6 kOe

JH

c 1120 kA/m 14.1 kOe

BHmax

320 kJ/m3

40 MGOe

[kOe]

[kG][T]

1.0

0.0

1.2

0.2

1.4

0.4

0.6

0.8

5

10

0.10

0.20

0.30

0.40

0.50

4.002.001.501.000.75

120014001600

20 15 1010 5

18002000 8001000 400 200 0600

-H

B,J

Temperature 20o

60o 80o 120o100o

77

Appendix B

Design Results

B.1 Description steps

All the parameters are shown in the next table, where X is a unknown parameter:

Parameter Abbreviation Size CommentWidth mover bar Wsmc X variable

Height Wsmc Hsmc X equals Wsmc

Iron thickness Liron 4mm fixedCoil thickness Lcoil X variableAirgap length Lairgap 0.5mm fixed

Glue layer Lglue 0.2mm fixedMagnet thickness Lpm X variable

Pole pitch τpp 26.8mm variableMagnet pitch τmp 22.78mm variablePitch ratio α 0.85 variable

• Step 1 Two main parameters of the design flow are Lcoil and Lpm. Chang-ing these parameters will change the width of SMC core. The width of coreis equal to the height of the core, because the permanent magnets are thesame size for the C-shaped actuator.

The total width of actuator can be formulated as follows:

Wsmc + 2Lairgap + 2Lglue + 2Liron + 2Lpm + 2Lcoil = 40mm (B.1)

The width of the SMC mover bar, with the fixed values (see table B.1), canbe expressed as:

Wsmc = 30.6mm− 2Lpm − 2Lcoil (B.2)

Wmsc is now dependent on the combinations of Lpm and Lcoil.

Changing the width of the SMC core the amount of flux density is alsochanging. This is also applicable for the pole pitch length τpp, increasingτpp means increasing the flux density in the SMC core.

78

B.1. Description steps

The first step will give, after a few simulation, for every set [Lpm, Lcoil] thematching pole pitch length.

To avoid a large number of simulations, seven simulations are done for fivecombinations of [Lpm, Lcoil]. These combinations are defined as [Lpm, Lcoil] =[(3, 1), (3, 5), (7, 1), (7, 5), (5, 3)]. After the simulations there are five combi-nation with the optimum pole pitch lengths. The optimum is determinedwith equation B.5; steepness divided by the mass of the mover bar. Therange for the pole pitch length is, for this project, varying from 15mm to45mm in seven steps, see Appendix B.2. An optimum pole pitch can bechosen to determine from every set of [Lpm, Lcoil].

These five optimums can give to the table that is given in step 1b, seeAppendix B.3. The red numbers are the obtained optimums. The excelsheet will interpolate a the values in between. This reduces the number ofsimulations significantly.

The results can be used in step 2.

• Step 2 The second step will process the results of the first step. The onlyvarying parameter is the current density. To obtain the continuous force,the thermal model is needed. This model can gives the steady state thermalbehavior for a given current density. The current density for the peak forcecan be derived with:

∆T

∆t=

J2peak

ρcuCcuσcu(B.3)

When obtained all the current densities for both continuous and peak forces,the parametric search can be started. After these calculations one can decidewhich setup is most suitable for a given situation or precondition. A fewfigures of merit that can be used:

S =F 2

Pdis=

K2

3Rcoil

[N2

W

](B.4)

Csm =S

Mmover

[N2

Wkg

](B.5)

Cfm =F

Mmover

[N

kg

](B.6)

Csv =S

Vmover

[N2

Wcm3

](B.7)

Cfv =F

Vmover

[N

cm3

](B.8)

• Step 3 The chosen setup from second step will be used in the third step todetermine the right pole pitch and pitch ratio α.

All the results of the mentioned steps are represented in tables:

79

Chapter B. Design Results

B.2 Step 1a results

setu

pp

ole

pitc

hm

agne

tpL

_p

mL

_co

ilW

_sm

cJ(

A/m

m2

)I(

A)

KF

_lo

rR

spP

cuM

_m

ov(

gr)

Vo

l_C

(cm

3)

NS

(N2

/W)

Csv

Csm

Cfm

Cfv

K_

wd

gvo

lpm

S/v

olp

mC

f/vo

lpm

115

12.7

53

122

.60

735

.00

0.19

6.50

4.46

E-0

41.

6415

7.30

48.7

547

25.7

60.

5316

3.76

41.3

20.

138.

730.

9626

.94

6.80

220

17.0

03

122

.60

746

.67

0.19

8.94

3.35

E-0

42.

1920

9.74

65.0

063

36.5

80.

5617

4.42

42.6

50.

1412

.08

1.28

28.6

97.

023

2521

.25

31

22.6

07

58.3

30.

1911

.27

2.68

E-0

42.

7326

2.17

81.2

578

46.5

00.

5717

7.37

43.0

10.

1415

.08

1.59

29.1

87.

074

3025

.50

31

22.6

07

70.0

00.

1913

.33

2.23

E-0

43.

2831

4.61

97.5

094

54.1

80.

5617

2.21

42.3

80.

1417

.90

1.91

28.3

36.

975

3529

.75

31

22.6

07

81.6

70.

1915

.47

1.91

E-0

43.

8336

7.04

113.

7511

062

.57

0.55

170.

4742

.16

0.14

20.8

42.

2328

.04

6.94

640

34.0

03

122

.60

793

.33

0.18

17.2

51.

67E

-04

4.37

419.

4813

0.00

126

68.0

60.

5216

2.25

41.1

30.

1323

.29

2.55

26.6

96.

777

4538

.25

31

22.6

07

105.

000.

1717

.86

1.49

E-0

44.

9247

1.91

146.

2514

164

.86

0.44

137.

4437

.86

0.12

23.9

92.

8722

.61

6.23

815

12.7

57

114

.60

735

.00

0.14

5.05

2.94

E-0

41.

0869

.85

42.7

547

23.6

00.

5533

7.78

72.2

40.

126.

781.

5215

.55

3.33

920

17.0

07

114

.60

746

.67

0.15

7.17

2.20

E-0

41.

4493

.14

57.0

063

35.7

00.

6338

3.27

76.9

50.

139.

682.

0217

.65

3.54

1025

21.2

57

114

.60

758

.33

0.16

9.05

1.76

E-0

41.

8011

6.42

71.2

578

45.5

50.

6439

1.26

77.7

40.

1312

.10

2.53

18.0

13.

5811

3025

.50

71

14.6

07

70.0

00.

1510

.53

1.47

E-0

42.

1613

9.71

85.5

094

51.3

70.

6036

7.71

75.3

70.

1214

.14

3.03

16.9

33.

4712

3529

.75

71

14.6

07

81.6

70.

1210

.19

1.26

E-0

42.

5216

2.99

99.7

511

041

.25

0.41

253.

1162

.53

0.10

13.7

33.

5411

.65

2.88

1340

34.0

07

114

.60

793

.33

0.11

10.2

61.

10E

-04

2.88

186.

2811

4.00

126

36.6

00.

3219

6.48

55.0

90.

0913

.85

4.05

9.05

2.54

1445

38.2

57

114

.60

710

5.00

0.10

10.4

09.

79E

-05

3.24

209.

5612

8.25

141

33.3

90.

2615

9.31

49.6

10.

0813

.96

4.55

7.33

2.28

1515

12.7

57

56.

607

175.

000.

058.

294.

02E

-05

3.69

51.2

742

.75

236

18.6

10.

4436

3.02

161.

660.

1911

.18

1.52

12.2

75.

4616

2017

.00

75

6.60

723

3.33

0.05

12.7

13.

01E

-05

4.92

68.3

757

.00

315

32.8

00.

5847

9.85

185.

860.

2217

.15

2.02

16.2

26.

2817

2521

.25

75

6.60

729

1.67

0.05

15.0

32.

41E

-05

6.15

85.4

671

.25

394

36.7

20.

5242

9.72

175.

880.

2120

.30

2.53

14.5

25.

9418

3025

.50

75

6.60

735

0.00

0.04

14.6

12.

01E

-05

7.38

102.

5585

.50

472

28.9

20.

3428

1.97

142.

470.

1719

.70

3.03

9.53

4.81

1935

29.7

57

56.

607

408.

330.

0414

.95

1.72

E-0

58.

6111

9.64

99.7

555

125

.93

0.26

216.

7712

4.92

0.15

20.1

73.

547.

334.

2220

4034

.00

75

6.60

746

6.67

0.03

15.0

01.

51E

-05

9.84

136.

7311

4.00

630

22.8

40.

2016

7.07

109.

670.

1320

.24

4.05

5.65

3.71

2145

38.2

57

56.

607

525.

000.

0314

.80

1.34

E-0

511

.07

153.

8212

8.25

709

19.7

90.

1512

8.66

96.2

40.

1219

.99

4.55

4.35

3.25

2215

12.7

53

514

.60

717

5.00

0.07

11.4

97.

07E

-05

6.50

127.

6148

.75

236

20.3

30.

4215

9.30

90.0

60.

2415

.50

0.96

21.2

612

.02

2320

17.0

03

514

.60

723

3.33

0.07

16.9

65.

30E

-05

8.66

170.

1465

.00

315

33.2

10.

5119

5.20

99.6

90.

2622

.90

1.28

26.0

513

.30

2425

21.2

53

514

.60

729

1.67

0.08

22.1

84.

24E

-05

10.8

321

2.68

81.2

539

445

.42

0.56

213.

5810

4.28

0.27

29.9

61.

5928

.50

13.9

225

3025

.50

35

14.6

07

350.

000.

0827

.35

3.54

E-0

512

.99

255.

2197

.50

472

57.5

90.

5922

5.65

107.

180.

2836

.89

1.91

30.1

114

.30

2635

29.7

53

514

.60

740

8.33

0.08

32.0

23.

03E

-05

15.1

629

7.75

113.

7555

167

.62

0.59

227.

1110

7.53

0.28

43.2

02.

2330

.31

14.3

527

4034

.00

35

14.6

07

466.

670.

0734

.95

2.65

E-0

517

.32

340.

2813

0.00

630

70.5

20.

5420

7.23

102.

720.

2747

.19

2.55

27.6

513

.71

2845

38.2

53

514

.60

752

5.00

0.07

38.0

82.

36E

-05

19.4

938

2.82

146.

2570

974

.42

0.51

194.

3999

.48

0.26

51.4

32.

8725

.94

13.2

829

1512

.75

53

14.6

07

105.

000.

1010

.69

1.08

E-0

43.

5796

.38

45.7

514

132

.04

0.70

332.

4111

0.93

0.23

14.3

61.

3423

.93

7.99

3020

17.0

05

314

.60

714

0.00

0.11

15.7

58.

09E

-05

4.76

128.

5161

.00

189

52.1

50.

8540

5.78

122.

560.

2621

.26

1.79

29.2

18.

8231

2521

.25

53

14.6

07

175.

000.

1220

.76

6.47

E-0

55.

9516

0.64

76.2

523

672

.50

0.95

451.

3112

9.25

0.27

28.0

02.

2332

.49

9.31

3230

25.5

05

314

.60

721

0.00

0.12

24.9

55.

39E

-05

7.14

192.

7691

.50

283

87.2

70.

9545

2.75

129.

460.

2733

.63

2.68

32.6

09.

3233

3529

.75

53

14.6

07

245.

000.

1228

.91

4.62

E-0

58.

3222

4.89

106.

7533

110

0.39

0.94

446.

4012

8.55

0.27

39.0

63.

1232

.14

9.25

3440

34.0

05

314

.60

728

0.00

0.11

30.2

34.

04E

-05

9.51

257.

0212

2.00

378

96.0

70.

7937

3.80

117.

630.

2540

.81

3.57

26.9

18.

4735

4538

.25

53

14.6

07

315.

000.

1030

.88

3.60

E-0

510

.70

289.

1513

7.25

425

89.0

80.

6530

8.09

106.

790.

2241

.66

4.02

22.1

87.

69

80

B.3. Step 1b interpolated results

B.3 Step 1b interpolated results

Function to interpolate results5 21.000 24.783 26.675 28.642 33.0004 24.406 25.194 26.640 28.113 28.784

L_coil 3 25.316 25.735 27.000 26.980 27.1842 25.488 25.758 26.074 26.077 25.9661 25.000 25.629 25.825 25.589 24.000

7.00 6.00 5.00 4.00 3.00L_pm

5 21.000 24.783 26.675 28.642 33.0004.5 22.70276 24.98854 26.65735 28.3777 30.891914 24.406 25.194 26.640 28.113 28.784

L_coil 3.5 24.86078 25.46435 26.82008 27.54643 27.983893 25.316 25.735 27.000 26.980 27.184

2.5 25.40182 25.74643 26.53706 26.5285 26.574932 25.488 25.758 26.074 26.077 25.966

1.5 25.2438 25.69373 25.9498 25.83288 24.982951 25.000 25.629 25.825 25.589 24.000

7.00 6.00 5.00 4.00 3.00L_pm

81


B.4 Step 2 results Continuousse

tup

pole

pitc

hm

agne

tpL_

coil

L_pm

W_s

mc

J(A

/mm

^2)

I(A)

KF

_lor

Rsp

Pcu

M_m

ov(g

r)V

ol_C

(cm

^3)

NS

(N^2

/W)

Csv

Csm

Cfm

Cfv

K_w

dgvo

lpm

S/v

olpm

Cf/v

olpm

diam

eter

124

20.4

13

22.6

13.3

106.

401.

96E

-01

20.8

42.

79E

-04

9.47

251.

6978

7545

.85

0.59

182.

1882

.81

0.27

14.6

91.

5329

.97

13.6

20.

352

25.6

21.7

61

420

.613

.25

113.

071.

97E

-01

22.3

12.

39E

-04

9.17

225.

5580

.64

8054

.25

0.67

240.

5498

.90

0.28

15.7

82.

0027

.10

11.1

40.

353

25.8

21.9

31

518

.613

.311

4.38

1.88

E-0

121

.55

2.15

E-0

48.

4418

7.82

78.6

981

54.9

90.

7029

2.78

114.

720.

2715

.26

2.30

23.8

89.

360.

354

25.6

21.7

61

616

.613

.35

113.

921.

73E

-01

19.7

51.

94E

-04

7.57

150.

9275

.52

8051

.52

0.68

341.

3913

0.85

0.26

13.8

72.

4820

.77

7.96

0.35

525

21.2

51

714

.613

.35

111.

251.

57E

-01

17.4

81.

76E

-04

6.54

116.

4271

.25

7846

.70

0.66

401.

1015

0.12

0.25

12.2

52.

5318

.47

6.91

0.35

625

21.2

51.

53

21.6

10.7

133.

751.

72E

-01

23.0

71.

74E

-04

9.33

255.

4481

.25

118

57.0

40.

7022

3.30

90.3

20.

2820

.35

1.59

35.7

914

.48

0.35

725

.821

.93

1.5

419

.610

.713

8.03

1.75

E-0

124

.16

1.54

E-0

48.

7922

0.96

81.2

712

266

.45

0.82

300.

7310

9.35

0.30

21.3

62.

0232

.94

11.9

80.

358

25.9

22.0

151.

55

17.6

10.7

138.

571.

68E

-01

23.2

41.

38E

-04

7.97

182.

7878

.995

122

67.7

60.

8637

0.76

127.

150.

2920

.46

2.31

29.3

210

.05

0.35

925

.721

.845

1.5

615

.610

.813

8.78

1.56

E-0

121

.61

1.25

E-0

47.

2014

6.38

75.8

1512

164

.85

0.86

443.

0414

7.61

0.28

18.8

42.

4926

.04

8.68

0.35

1025

.221

.42

1.5

713

.610

.913

7.34

1.42

E-0

119

.51

1.12

E-0

46.

3311

2.90

71.8

211

960

.09

0.84

532.

2017

2.78

0.27

16.9

02.

5523

.57

7.65

0.35

1126

22.1

23

20.6

9.1

157.

731.

53E

-01

24.1

41.

22E

-04

9.11

258.

8284

.516

363

.98

0.76

247.

2093

.28

0.29

24.9

51.

6638

.60

14.5

70.

3512

26.1

22.1

852

418

.69.

1515

9.21

1.55

E-0

124

.63

1.11

E-0

48.

4121

7.27

82.2

1516

472

.10

0.88

331.

8411

3.35

0.30

25.3

72.

0435

.32

12.0

70.

3513

26.1

22.1

852

516

.69.

216

0.08

1.50

E-0

123

.98

9.97

E-0

57.

6617

8.53

79.6

0516

475

.09

0.94

420.

5913

4.34

0.30

24.5

72.

3332

.23

10.3

00.

3514

25.8

21.9

32

614

.69.

2515

9.10

1.40

E-0

122

.35

8.97

E-0

56.

8114

1.95

76.1

116

273

.32

0.96

516.

4915

7.44

0.29

22.7

62.

5029

.33

8.94

0.35

1525

.521

.675

27

12.6

9.3

158.

101.

24E

-01

19.5

97.

95E

-05

5.96

109.

9072

.675

160

64.3

30.

8958

5.32

178.

230.

2719

.82

2.58

24.9

47.

590.

3516

26.6

22.6

12.

53

19.6

8.05

178.

441.

36E

-01

24.3

39.

28E

-05

8.87

257.

9686

.45

209

66.7

40.

7725

8.70

94.3

00.

2828

.49

1.70

39.3

514

.35

0.35

1726

.522

.525

2.5

417

.68.

117

8.88

1.40

E-0

124

.97

8.45

E-0

58.

1121

4.40

83.4

7520

876

.84

0.92

358.

3911

6.46

0.30

29.0

42.

0737

.08

12.0

50.

3518

26.5

22.5

252.

55

15.6

8.1

178.

881.

35E

-01

24.2

37.

59E

-05

7.28

175.

6980

.825

208

80.5

81.

0045

8.66

137.

910.

3028

.17

2.37

34.0

710

.24

0.35

1925

.721

.845

2.5

613

.68.

217

5.62

1.26

E-0

122

.10

6.93

E-0

56.

4213

6.59

75.8

1520

276

.09

1.00

557.

1016

1.77

0.29

25.4

12.

4930

.56

8.87

0.35

2025

.421

.59

2.5

711

.68.

2517

4.63

1.12

E-0

119

.53

6.11

E-0

55.

5910

5.30

72.3

920

068

.18

0.94

647.

4918

5.46

0.27

22.3

72.

5726

.54

7.60

0.35

2127

.223

.12

33

18.6

7.25

197.

201.

22E

-01

24.0

37.

35E

-05

8.58

256.

9688

.425

767

.35

0.76

262.

1093

.53

0.27

31.3

21.

7338

.84

13.8

60.

3522

2722

.95

34

16.6

7.3

197.

101.

26E

-01

24.8

36.

70E

-05

7.81

212.

3185

.05

255

78.9

80.

9337

1.99

116.

960.

2932

.13

2.11

37.4

011

.76

0.35

2327

22.9

53

514

.67.

3519

8.45

1.21

E-0

124

.05

5.99

E-0

57.

0817

3.49

82.3

525

581

.68

0.99

470.

8013

8.61

0.29

30.9

02.

4133

.89

9.98

0.35

2425

.721

.845

36

12.6

7.4

190.

181.

09E

-01

20.7

55.

55E

-05

6.03

131.

9375

.815

243

71.4

60.

9454

1.64

157.

280.

2726

.51

2.49

28.7

08.

330.

3525

25.3

21.5

053

710

.67.

4518

8.49

9.87

E-0

218

.60

4.89

E-0

55.

2110

0.89

72.1

0523

966

.41

0.92

658.

2218

4.33

0.26

23.5

82.

5625

.95

7.27

0.35

2628

23.8

3.5

317

.66.

6521

7.23

1.09

E-0

123

.73

5.94

E-0

58.

4225

7.68

9130

866

.92

0.74

259.

7292

.10

0.26

33.6

51.

7937

.49

13.2

90.

3527

27.5

23.3

753.

54

15.6

6.7

214.

961.

13E

-01

24.3

95.

46E

-05

7.56

210.

1686

.625

303

78.6

60.

9137

4.28

116.

070.

2834

.38

2.15

36.5

811

.34

0.35

2826

.822

.78

3.5

513

.66.

7521

1.05

1.09

E-0

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70.9

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644.

9018

1.68

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23.7

22.

5224

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6.89

0.35

3128

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43

16.6

6.15

236.

169.

76E

-02

23.0

44.

91E

-05

8.21

258.

1893

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364

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0.69

250.

5089

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0.25

35.4

11.

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3732

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46

10.6

6.3

211.

689.

06E

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83.

90E

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5.25

120.

7174

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317

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30.

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890.

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90.

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55

11.6

5.85

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298.

96E

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20.9

83.

61E

-05

5.94

156.

2081

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378

74.1

50.

9147

4.73

134.

340.

2633

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31.1

28.

810.

3539

2521

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4.5

69.

65.

922

1.25

7.85

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217

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3.34

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54.

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5.69

73.7

535

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27.7

82.

4225

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0.35

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66

204.

306.

84E

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13.9

63.

12E

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3.91

80.6

164

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20.

7761

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230.

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46.

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3541

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53

14.6

5.4

297.

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14E

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24.1

73.

21E

-05

8.51

280.

7310

7.25

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68.6

80.

6424

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86.0

90.

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54

12.6

5.45

259.

788.

37E

-02

21.7

43.

31E

-05

6.70

200.

6590

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450

70.5

40.

7835

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330.

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56

8.6

5.6

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477.

17E

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16.5

92.

89E

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4.65

110.

8973

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30.

8153

4.15

149.

630.

2327

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56.

900.

3545

2117

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6.6

5.7

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505.

72E

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11.4

22.

87E

-05

3.43

71.7

859

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331

38.0

50.

6453

0.12

159.

070.

1918

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2.12

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35

82

B.5. Step 2 results Peak

B.5 Step 2 results Peak

setu

ppo

lepi

tch

mag

netp

L_co

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pmW

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cJ(

A/m

m^2

)I(A

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orR

spP

cuM

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m^3

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S(N

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svC

smC

fmC

fvK

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volp

mS

/vol

pmC

f/vol

pmdi

amet

er

124

20.4

13

22.6

64.4

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5.8

1.96

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110

1.02

2.79

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422

2.64

251.

778

7545

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0.59

182.

1140

1.36

1.30

14.6

91.

5329

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66.0

20.

352

25.6

21.7

61

420

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550.

21.

92E

-01

105.

542.

39E

-04

217.

1722

5.5

80.6

480

51.2

90.

6422

7.39

467.

921.

3115

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2.00

25.6

252

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0.35

325

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15

18.6

64.4

7455

4.5

1.84

E-0

110

2.20

2.15

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419

8.40

187.

878

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8152

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0.67

280.

2754

4.10

1.30

14.9

32.

3022

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44.3

80.

354

25.6

21.7

61

616

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550.

21.

72E

-01

94.8

31.

94E

-04

176.

5515

0.9

75.5

280

50.9

40.

6733

7.49

628.

331.

2613

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2.48

20.5

338

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0.35

525

21.2

51

714

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537.

31.

53E

-01

82.2

11.

76E

-04

152.

5811

6.4

71.2

578

44.3

00.

6238

0.51

706.

161.

1511

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2.53

17.5

232

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0.35

625

21.2

51.

53

21.6

64.4

7480

5.9

1.73

E-0

113

9.15

1.74

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433

8.83

255.

481

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118

57.1

40.

7022

3.71

544.

741.

7120

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1.59

35.8

687

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0.35

725

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1.5

419

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831.

71.

71E

-01

141.

981.

54E

-04

318.

9722

1.0

81.2

712

263

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0.78

286.

0364

2.58

1.75

20.8

32.

0231

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70.3

70.

358

25.9

22.0

151.

55

17.6

64.4

7483

4.9

1.65

E-0

113

7.60

1.38

E-0

428

9.39

182.

878

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122

65.4

30.

8335

7.99

752.

861.

7420

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2.31

28.3

159

.53

0.35

925

.721

.845

1.5

615

.664

.474

828.

51.

52E

-01

125.

681.

25E

-04

256.

5714

6.4

75.8

1512

161

.56

0.81

420.

5585

8.56

1.66

18.3

52.

4924

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50.4

70.

3510

25.2

21.4

21.

57

13.6

64.4

7481

2.4

1.36

E-0

111

0.12

1.12

E-0

422

1.59

112.

971

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119

54.7

20.

7648

4.69

975.

321.

5316

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2.55

21.4

743

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0.35

1126

22.1

23

20.6

64.4

7411

17.5

1.51

E-0

116

8.26

1.22

E-0

445

7.32

258.

884

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361

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0.73

239.

1965

0.10

1.99

24.5

41.

6637

.35

101.

510.

3512

26.1

22.1

852

418

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1121

.81.

54E

-01

172.

651.

11E

-04

417.

6621

7.3

82.2

1516

471

.37

0.87

328.

4979

4.65

2.10

25.2

42.

0434

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84.5

90.

3513

26.1

22.1

852

516

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1121

.81.

46E

-01

163.

269.

97E

-05

376.

2517

8.5

79.6

0516

470

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0.89

396.

8291

4.49

2.05

23.8

72.

3330

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70.0

90.

3514

25.8

21.9

32

614

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1109

.01.

36E

-01

150.

368.

97E

-05

330.

9814

2.0

76.1

116

268

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0.90

481.

1710

59.2

11.

9821

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2.50

27.3

260

.14

0.35

1525

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27

12.6

64.4

7410

96.1

1.20

E-0

113

2.01

7.95

E-0

528

6.67

109.

972

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160

60.7

90.

8455

3.14

1201

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1.82

19.2

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5823

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51.1

80.

3516

26.6

22.6

12.

53

19.6

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29.2

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119

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8.81

258.

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209

64.8

40.

7525

1.35

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472.

2228

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1.70

38.2

411

3.25

0.35

1726

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2.5

417

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1423

.81.

37E

-01

194.

438.

45E

-05

514.

1121

4.4

83.4

7520

873

.53

0.88

342.

9690

6.84

2.33

28.4

02.

0735

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93.8

20.

3518

26.5

22.5

252.

55

15.6

64.4

7414

23.8

1.34

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119

1.11

7.59

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546

1.55

175.

780

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208

79.1

30.

9845

0.41

1087

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2.36

27.9

22.

3733

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80.8

00.

3519

25.7

21.8

452.

56

13.6

64.4

7413

80.8

1.23

E-0

116

9.85

6.93

E-0

539

6.64

136.

675

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202

72.7

30.

9653

2.51

1243

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2.24

24.8

52.

4929

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68.2

00.

3520

25.4

21.5

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57

11.6

64.4

7413

64.7

1.04

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114

1.74

6.11

E-0

534

1.63

105.

372

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200

58.8

00.

8155

8.42

1345

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1.96

20.7

72.

5722

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55.1

70.

3521

27.2

23.1

23

318

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1753

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20E

-01

210.

377.

35E

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678.

3125

7.0

88.4

257

65.2

50.

7425

3.91

818.

692.

3830

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1.73

37.6

312

1.32

0.35

2227

22.9

53

416

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1740

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23E

-01

213.

256.

70E

-05

609.

0521

2.3

85.0

525

574

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0.88

351.

6910

04.4

42.

5131

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2.11

35.3

610

1.00

0.35

2327

22.9

53

514

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1740

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16E

-01

202.

185.

99E

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544.

7917

3.5

82.3

525

575

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0.91

432.

5011

65.3

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4629

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2.41

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36

12.6

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57.0

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117

9.53

5.55

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545

7.39

131.

975

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243

70.4

70.

9353

4.10

1360

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2.37

26.3

32.

4928

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72.0

90.

3525

25.3

21.5

053

710

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1631

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94E

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145.

844.

89E

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390.

0610

0.9

72.1

0523

954

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0.76

540.

4514

45.5

02.

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2.56

21.3

156

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0.35

2628

23.8

3.5

317

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2106

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08E

-01

227.

745.

94E

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791.

0225

7.7

9130

865

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0.72

254.

4688

3.82

2.50

33.3

01.

7936

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127.

590.

3527

27.5

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753.

54

15.6

64.4

7420

68.5

1.09

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122

4.52

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570

0.53

210.

286

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303

71.9

60.

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2.41

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32.8

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400.

3528

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55

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120

8.87

4.99

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560

8.28

166.

981

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295

71.7

20.

8842

9.76

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56

11.6

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7.96

126.

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281

65.0

40.

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4.36

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3530

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57

9.6

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6.87

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57.3

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2.52

20.5

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293

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3.91

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71.

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129.

920.

3532

28.1

23.8

854

414

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91E

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239.

364.

52E

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9820

8.7

88.5

1535

472

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0.82

347.

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46.9

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4.85

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4.35

160.

581

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69.4

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8643

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3729

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90.5

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25.2

21.4

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610

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2166

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39E

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181.

723.

90E

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549.

4112

0.7

74.3

431

760

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0.81

497.

9315

05.4

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2.47

17.3

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0.35

3630

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4.5

315

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2988

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60E

-02

256.

923.

94E

-05

1055

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269.

810

0.42

543

862

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0.62

231.

7995

2.11

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37.6

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420.

3537

28.4

24.1

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54

13.6

64.4

7427

46.6

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223

6.11

3.84

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586

8.56

205.

089

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52.

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4.5

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664

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87E

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166.

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586.

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5.7

73.7

535

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0.64

406.

6814

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2.42

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4.5

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664

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2195

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83E

-02

127.

883.

12E

-05

451.

1180

.664

.695

322

36.2

50.

5644

9.71

1586

.35

1.98

18.7

62.

3015

.79

55.7

00.

3541

3328

.05

53

14.6

64.4

7435

46.1

7.17

E-0

225

4.10

3.21

E-0

512

12.5

228

0.7

107.

2552

053

.25

0.50

189.

6790

5.11

2.37

37.2

62.

1025

.31

120.

780.

3542

28.6

24.3

15

412

.664

.474

3073

.37.

77E

-02

238.

783.

31E

-05

937.

4020

0.7

90.0

945

060

.82

0.68

303.

1211

89.9

92.

6534

.96

2.24

27.1

910

6.76

0.35

4326

.722

.695

55

10.6

64.4

7428

69.1

7.10

E-0

220

3.60

3.11

E-0

576

9.21

151.

481

.435

420

53.8

90.

6635

5.94

1344

.75

2.50

29.8

12.

3822

.62

85.4

40.

3544

24.8

21.0

85

68.

664

.474

2664

.96.

02E

-02

160.

312.

89E

-05

616.

0911

0.9

73.1

639

041

.71

0.57

376.

1514

45.6

22.

1923

.46

2.40

17.3

666

.71

0.35

4521

17.8

55

76.

664

.474

2256

.64.

90E

-02

110.

582.

87E

-05

438.

3971

.859

.85

331

27.8

90.

4738

8.54

1540

.41

1.85

16.2

22.

1213

.13

52.0

60.

35

83


B.6 Step 3 results Continuous

setu

pp

ole

pitc

hra

tiom

agne

tpL

_co

ilL

_p

mW

_sm

cJ(

A/m

m2

)I(

A)

KF

_lo

rR

spP

cuM

_m

ov(

gr)

Vo

l_C

(cm

3)

NS

(N2

/W)

Csv

Csm

Cfm

Cfv

K_

wd

gvo

lpm

S/v

olp

mC

f/vo

lpm

dia

me

ter

123

.00

0.70

16.1

3.5

513

.66.

7518

1.1

9.71

E-0

217

.59

5.81

E-0

55.

7214

3.23

70.1

525

354

.07

0.77

377.

5112

2.80

0.25

24.5

71.

6931

.99

10.4

00.

352

24.7

50.

7017

.325

3.5

513

.66.

7519

4.9

9.77

E-0

219

.05

5.40

E-0

56.

1615

4.13

75.4

927

358

.93

0.78

382.

3312

3.59

0.25

26.6

81.

8232

.39

10.4

70.

353

26.5

00.

7018

.55

3.5

513

.66.

7420

8.4

9.94

E-0

220

.71

5.05

E-0

56.

5716

5.03

80.8

329

265

.27

0.81

395.

5312

5.51

0.26

29.0

31.

9533

.51

10.6

30.

354

28.2

50.

7019

.775

3.5

513

.66.

7322

1.8

9.96

E-0

222

.09

4.73

E-0

56.

9917

5.92

86.1

631

169

.83

0.81

396.

9312

5.55

0.26

30.9

72.

0833

.63

10.6

40.

355

30.0

00.

7021

3.5

513

.66.

7223

5.2

9.88

E-0

223

.23

4.46

E-0

57.

4018

6.82

91.5

033

172

.97

0.80

390.

6112

4.36

0.25

32.7

02.

2133

.09

10.5

40.

356

23.0

00.

7717

.767

53.

55

13.6

6.75

181.

11.

03E

-01

18.5

95.

81E

-05

5.72

143.

2370

.15

253

60.4

30.

8642

1.88

129.

820.

2725

.97

1.87

32.3

99.

970.

357

24.7

50.

7719

.119

375

3.5

513

.66.

7519

4.9

1.04

E-0

120

.24

5.40

E-0

56.

1615

4.13

75.4

927

366

.55

0.88

431.

8013

1.34

0.27

28.3

52.

0133

.15

10.0

80.

358

26.5

00.

7720

.471

253.

55

13.6

6.74

208.

41.

04E

-01

21.7

05.

05E

-05

6.57

165.

0380

.83

292

71.6

20.

8943

3.97

131.

470.

2730

.40

2.15

33.3

210

.09

0.35

928

.25

0.77

21.8

2312

53.

55

13.6

6.73

221.

81.

05E

-01

23.3

94.

73E

-05

6.99

175.

9286

.16

311

78.3

30.

9144

5.27

132.

980.

2732

.80

2.29

34.1

910

.21

0.35

1030

.00

0.77

23.1

753.

55

13.6

6.72

235.

21.

05E

-01

24.7

74.

46E

-05

7.40

186.

8291

.50

331

82.9

60.

9144

4.05

132.

600.

2734

.86

2.43

34.0

910

.18

0.35

1123

.00

0.85

19.4

353.

55

13.6

6.75

181.

11.

06E

-01

19.2

35.

81E

-05

5.72

143.

2370

.15

253

64.6

00.

9245

0.99

134.

230.

2726

.85

2.04

31.6

59.

420.

3512

24.7

50.

8520

.913

753.

55

13.6

6.75

194.

91.

08E

-01

21.1

25.

40E

-05

6.16

154.

1375

.49

273

72.4

40.

9646

9.98

137.

020.

2829

.58

2.20

32.9

99.

620.

3513

26.5

00.

8522

.392

53.

55

13.6

6.74

208.

41.

10E

-01

22.9

35.

05E

-05

6.57

165.

0380

.83

292

79.9

70.

9948

4.56

138.

920.

2832

.13

2.35

34.0

19.

750.

3514

28.2

50.

8523

.871

253.

55

13.6

6.73

221.

81.

11E

-01

24.6

64.

73E

-05

6.99

175.

9286

.16

311

87.0

21.

0149

4.64

140.

150.

2934

.57

2.51

34.7

29.

840.

3515

30.0

00.

8525

.35

3.5

513

.66.

7223

5.2

1.09

E-0

125

.72

4.46

E-0

57.

4018

6.82

91.5

033

189

.46

0.98

478.

8613

7.69

0.28

36.2

02.

6633

.61

9.66

0.35

1623

.00

0.92

21.1

025

3.5

513

.66.

7518

1.1

1.11

E-0

120

.02

5.81

E-0

55.

7214

3.23

70.1

525

370

.07

1.00

489.

2413

9.80

0.29

27.9

72.

2231

.63

9.04

0.35

1724

.75

0.92

22.7

0812

53.

55

13.6

6.75

194.

91.

11E

-01

21.5

95.

40E

-05

6.16

154.

1375

.49

273

75.7

21.

0049

1.29

140.

100.

2930

.24

2.38

31.7

69.

060.

3518

26.5

00.

9224

.313

753.

55

13.6

6.74

208.

41.

14E

-01

23.7

45.

05E

-05

6.57

165.

0380

.83

292

85.7

41.

0651

9.56

143.

850.

2933

.27

2.55

33.5

99.

300.

3519

28.2

50.

9225

.919

375

3.5

513

.66.

7322

1.8

1.13

E-0

124

.98

4.73

E-0

56.

9917

5.92

86.1

631

189

.33

1.04

507.

7614

2.00

0.29

35.0

32.

7232

.82

9.18

0.35

2030

.00

0.92

27.5

253.

55

13.6

6.72

235.

21.

13E

-01

26.6

04.

46E

-05

7.40

186.

8291

.50

331

95.6

91.

0551

2.19

142.

410.

2937

.44

2.89

33.1

19.

210.

3521

23.0

00.

9922

.77

3.5

513

.66.

7518

1.1

1.11

E-0

120

.16

5.81

E-0

55.

7214

3.23

70.1

525

371

.01

1.01

495.

7914

0.73

0.29

28.1

62.

3929

.70

8.43

0.35

2224

.75

0.99

24.5

025

3.5

513

.66.

7519

4.9

1.13

E-0

121

.97

5.40

E-0

56.

1615

4.13

75.4

927

378

.42

1.04

508.

8114

2.57

0.29

30.7

82.

5730

.48

8.54

0.35

2326

.50

0.99

26.2

353.

55

13.6

6.74

208.

41.

15E

-01

23.9

15.

05E

-05

6.57

165.

0380

.83

292

86.9

61.

0852

6.93

144.

870.

3033

.50

2.75

31.5

78.

680.

3524

28.2

50.

9927

.967

53.

55

13.6

6.73

221.

81.

14E

-01

25.2

14.

73E

-05

6.99

175.

9286

.16

311

90.9

51.

0651

6.99

143.

280.

2935

.34

2.94

30.9

78.

580.

3525

30.0

00.

9929

.73.

55

13.6

6.72

235.

21.

15E

-01

26.9

64.

46E

-05

7.40

186.

8291

.50

331

98.2

41.

0752

5.86

144.

290.

2937

.94

3.12

31.5

08.

640.

35

84

B.7. Step 3 results Peak

B.7 Step 3 results Peak

setu

pp

ole

pitc

hra

tiom

agne

tpL

_co

ilL

_p

mW

_sm

cJ(

A/m

m2

)I(

A)

KF

_lo

rR

spP

cuM

_m

ov(

gr)

Vo

l_C

(cm

3)

NS

(N2

/W)

Csv

Csm

Cfm

Cfv

K_

wd

gvo

lpm

S/v

olp

mC

f/vo

lpm

dia

me

ter

123

.00

0.70

16.1

03.

55

13.6

64.4

7417

309.

51E

-02

164.

525.

81E

-05

522.

0314

3.23

70.1

525

351

.85

0.74

361.

9911

48.6

22.

3524

.06

1.69

30.6

797

.32

0.35

224

.75

0.70

17.3

33.

55

13.6

64.4

7418

629.

65E

-02

179.

625.

40E

-05

561.

7515

4.13

75.4

927

357

.43

0.76

372.

6311

65.3

92.

3826

.34

1.82

31.5

798

.74

0.35

326

.50

0.70

18.5

53.

55

13.6

64.4

7419

939.

59E

-02

191.

155.

05E

-05

601.

4716

5.03

80.8

329

260

.75

0.75

368.

1211

58.3

12.

3628

.00

1.95

31.1

998

.14

0.35

428

.25

0.70

19.7

83.

55

13.6

64.4

7421

259.

61E

-02

204.

174.

73E

-05

641.

1917

5.92

86.1

631

165

.01

0.75

369.

5611

60.5

72.

3729

.88

2.08

31.3

198

.33

0.35

530

.00

0.70

21.0

03.

55

13.6

64.4

7422

579.

40E

-02

212.

214.

46E

-05

680.

9118

6.82

91.5

033

166

.14

0.72

354.

0011

35.8

82.

3231

.13

2.21

29.9

996

.24

0.35

623

.00

0.77

17.7

73.

55

13.6

64.4

7417

301.

02E

-01

176.

115.

81E

-05

522.

0314

3.23

70.1

525

359

.41

0.85

414.

8212

29.5

92.

5125

.75

1.87

31.8

594

.40

0.35

724

.75

0.77

19.1

23.

55

13.6

64.4

7418

621.

00E

-01

186.

955.

40E

-05

561.

7515

4.13

75.4

927

362

.22

0.82

403.

6812

12.9

62.

4827

.41

2.01

30.9

993

.13

0.35

826

.50

0.77

20.4

73.

55

13.6

64.4

7419

931.

02E

-01

202.

505.

05E

-05

601.

4716

5.03

80.8

329

268

.18

0.84

413.

1412

27.0

92.

5129

.66

2.15

31.7

294

.21

0.35

928

.25

0.77

21.8

23.

55

13.6

64.4

7421

251.

01E

-01

214.

704.

73E

-05

641.

1917

5.92

86.1

631

171

.89

0.83

408.

6612

20.4

22.

4931

.42

2.29

31.3

793

.70

0.35

1030

.00

0.77

23.1

83.

55

13.6

64.4

7422

579.

79E

-02

220.

894.

46E

-05

680.

9118

6.82

91.5

033

171

.66

0.78

383.

5511

82.3

42.

4132

.40

2.43

29.4

590

.77

0.35

1123

.00

0.85

19.4

43.

55

13.6

64.4

7417

301.

06E

-01

183.

155.

81E

-05

522.

0314

3.23

70.1

525

364

.26

0.92

448.

6212

78.7

02.

6126

.78

2.04

31.4

989

.75

0.35

1224

.75

0.85

20.9

13.

55

13.6

64.4

7418

621.

06E

-01

197.

855.

40E

-05

561.

7515

4.13

75.4

927

369

.69

0.92

452.

1312

83.7

02.

6229

.01

2.20

31.7

390

.10

0.35

1326

.50

0.85

22.3

93.

55

13.6

64.4

7419

931.

08E

-01

214.

795.

05E

-05

601.

4716

5.03

80.8

329

276

.70

0.95

464.

7713

01.5

22.

6631

.46

2.35

32.6

291

.35

0.35

1428

.25

0.85

23.8

73.

55

13.6

64.4

7421

251.

06E

-01

225.

604.

73E

-05

641.

1917

5.92

86.1

631

179

.38

0.92

451.

2112

82.3

92.

6233

.02

2.51

31.6

790

.01

0.35

1530

.00

0.85

25.3

53.

55

13.6

64.4

7422

571.

03E

-01

231.

374.

46E

-05

680.

9118

6.82

91.5

033

178

.62

0.86

420.

8212

38.4

52.

5333

.94

2.66

29.5

486

.92

0.35

1623

.00

0.92

21.1

03.

55

13.6

64.4

7417

301.

08E

-01

187.

355.

81E

-05

522.

0314

3.23

70.1

525

367

.24

0.96

469.

4313

08.0

32.

6727

.40

2.22

30.3

484

.55

0.35

1724

.75

0.92

22.7

13.

55

13.6

64.4

7418

621.

10E

-01

204.

525.

40E

-05

561.

7515

4.13

75.4

927

374

.46

0.99

483.

1313

26.9

72.

7129

.99

2.38

31.2

385

.78

0.35

1826

.50

0.92

24.3

13.

55

13.6

64.4

7419

931.

10E

-01

219.

225.

05E

-05

601.

4716

5.03

80.8

329

279

.90

0.99

484.

1713

28.3

92.

7132

.11

2.55

31.3

085

.87

0.35

1928

.25

0.92

25.9

23.

55

13.6

64.4

7421

251.

09E

-01

231.

024.

73E

-05

641.

1917

5.92

86.1

631

183

.24

0.97

473.

1513

13.2

02.

6833

.81

2.72

30.5

984

.89

0.35

2030

.00

0.92

27.5

33.

55

13.6

64.4

7422

571.

07E

-01

241.

774.

46E

-05

680.

9118

6.82

91.5

033

185

.84

0.94

459.

5012

94.1

12.

6435

.46

2.89

29.7

083

.65

0.35

2123

.00

0.99

22.7

73.

55

13.6

64.4

7417

301.

09E

-01

189.

065.

81E

-05

522.

0314

3.23

70.1

525

368

.47

0.98

478.

0613

19.9

92.

7027

.65

2.39

28.6

479

.08

0.35

2224

.75

0.99

24.5

03.

55

13.6

64.4

7418

621.

11E

-01

205.

885.

40E

-05

561.

7515

4.13

75.4

927

375

.45

1.00

489.

5513

35.7

62.

7330

.19

2.57

29.3

380

.02

0.35

2326

.50

0.99

26.2

43.

55

13.6

64.4

7419

931.

11E

-01

221.

925.

05E

-05

601.

4716

5.03

80.8

329

281

.88

1.01

496.

1713

44.7

62.

7532

.51

2.75

29.7

280

.56

0.35

2428

.25

0.99

27.9

73.

55

13.6

64.4

7421

251.

12E

-01

237.

904.

73E

-05

641.

1917

5.92

86.1

631

188

.27

1.02

501.

7213

52.2

62.

7634

.82

2.94

30.0

681

.01

0.35

2530

.00

0.99

29.7

03.

55

13.6

64.4

7422

571.

08E

-01

243.

784.

46E

-05

680.

9118

6.82

91.5

033

187

.28

0.95

467.

1813

04.8

82.

6635

.76

3.12

27.9

978

.17

0.35

85

Appendix C

Inductance

The simulated C-shaped actuator per coil one winding is specified as follows: The

Parameter Abbreviation Size CommentTotal width actuator Wtot 40mm fixed

Width/Height mover bar Wsmc 13.6mm dependentIron thickness Liron 4mm fixedCoil thickness Lcoil 3.5mm variableAirgap length Lairgap 0.5mm fixed


Pole pitch τpp 26.8mm variablePitch ratio α 0.85 variable

Energy error < 0.5% FEM# Tetrahedra 72.000 FEM

mean inductance of one coil is around 1.8µH and not significantly changing perposition.

0 3.35 6.7 10.05 13.4 16.75 20.1 23.45 26.81.806

1.808

1.81

1.812

1.814

1.816

1.818

1.82x 10

−8

Moverposition (mm)

L coil(H

)

Self Induction of coil A, C‘,B

86

Appendix D

Analytical versus FEM

1 1.5 2 2.5 3 3.5 4 4.5 515

16

17

18

19

20

21

22

23

24

25

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

Analytical3D FEM

(a) Lpm = 4mm

1 1.5 2 2.5 3 3.5 4 4.5 515

16

17

18

19

20

21

22

23

24

25

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

Analytical3D FEM

(b) Lpm = 5mm

1 1.5 2 2.5 3 3.5 4 4.5 515

16

17

18

19

20

21

22

23

24

25

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

Analytical3D FEM

(c) Lpm = 6mm

1 1.5 2 2.5 3 3.5 4 4.5 510

11

12

13

14

15

16

17

18

19

20

Coil thickness [mm]

For

ce [N

] in

y−di

rect

ion

Analytical3D FEM

(d) Lpm = 7mm

Figure D.1: Forces obtained by the Analytical model and 3D FEM Maxwell asfunction of the coil thickness and permanent magnet thickness. Results obtainfrom section 6.5

87

Appendix E

Field plots

This Appendix contains the field plots of the designed C-shaped actuator in sec-tion 6.5.

The dimensions and further specification are given in the next table:

Parameter Abbreviation Size CommentWidth mover bar Wsmc 13.6mm dependent

Height Wsmc Hsmc 13.6mm dependentIron thickness Liron 4mm fixedCoil thickness Lcoil 3.5mm variableAirgap length Lairgap 0.5mm fixed


Pole pitch τpp 26.8mm variableMagnet pitch τmp 22.78mm variablePitch ratio α 0.85 variableCont. force FcontMEC 22.87N MECCont. force FcontF EM 22.99N FEMPeak force FpeakMEC

218.44N MECPeak force FpeakF EM

208.87N FEMAttraction force Fattr. 160N FEM

The field plots are obtained from the post-processor in Maxwell 3D. The nexttable shown the order of the field plots.

Regime Field PositionNo current B-field cross-section in the middle of mover bar in the xz planeNo current B-field cross-section in the middle of mover bar in the yz planeCont. force B-field cross-section in the middle of mover bar in the xz planeCont. force B-field cross-section in the middle of mover bar in the yz planePeak force B-field cross-section in the middle of mover bar in the xz planePeak force B-field cross-section in the middle of mover bar in the yz planePeak force H-field cross-section in the middle of mover bar in the yz plane

88

E.1. No current B-field

E.1 No current B-field

89

Chapter E. Field plots

E.2 No current B-field

90

E.3. Continuous current B-field

E.3 Continuous current B-field

91


E.4 Continuous current B-field

92

E.5. Peak current B-field

E.5 Peak current B-field

93


E.6 Peak current B-field

94

E.7. Peak current H-field

E.7 Peak current H-field

95

Appendix F

Mathcad script

96

Documents

Eindhoven University of Technology MASTER Design and ...as functions of the coil thickness and permanent magnet thickness. 37 5.1 A 2D thermal equivalent circuit of the C-shaped actuator