Modelling of Core Noise from Power Transformers - KTH · PDF fileModelling of Core Noise from Power Transformers ... ABB’s in-use toolchain for simulating transformer core noise

Modelling of Core Noise from Power Transformers

M O R I T Z K R E U T Z E R

Master of Science Thesis Stockholm, Sweden 2011

Modelling of Core Noise from Power Transformers

M O R I T Z K R E U T Z E R

Master’s Thesis in Numerical Analysis (30 ECTS credits) at the Scientific Computing Master Programme Royal Institute of Technology year 2011 Supervisor at CSC was Jesper Oppelstrup Examiner was Michael Hanke TRITA-CSC-E 2011:084 ISRN-KTH/CSC/E--11/084--SE ISSN-1653-5715 Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-100 44 Stockholm, Sweden URL: www.kth.se/csc

AbstractTransformer noise is a significant contribution to unwantedambient noise, especially in the vicinity of electrical trans-mission facilities. This issue is becoming more and moreimportant, be it due to an increasing distribution of elec-trical facilities and rising awareness among people regardinghealth and standard of living. Thus, it is desirable to lowerthe disturbance level caused by a power transformer, e.g.,by lowering its overall level or changing its characteristics.

Noise generated by vibrations of the transformer core,caused by magnetostriction, is a major contributor to theoverall noise radition from power transformers. Within thisthesis work a complete model for this specific type of corenoise from power transformers is developed, starting fromscratch and covering the electromagnetic behavior, the me-chanical characteristics and eventually the acoustics. Ineach of those steps, the modelling procedure is explainedand the resolution of occurring problems is presented. Fur-thermore, possible simplifications which are applied to themodel, e.g., the absence of hysteresis in the electromag-netical model, are pointed out. In the end of each step,the simulation results are verified using reference solutions.The software tool used for the analysis, as well as for pre-and post-processing, is COMSOL Multiphysics 4.2, includ-ing its interface to MATLAB and MATLAB itself.

As outcome of this work, a complete model is presentedwhich provides a convenient, flexible and efficient toolchainfor simulating this kind of noise. The model is built tocomplement the state-of-the art toolchain for simulatingthis phenomenon, which incorporates a collection of variouspieces of software and scripts.

ReferatModellering av buller hos järnkärnan i

krafttransformatorer

Transformatorbuller bidrar märkbart till oönskat omgiv-ningsoljud, särskilt i närheten av anläggningar för överfö-ring av elkraft. Fenomenet blir allt viktigare dels på grundav tillkomst av allt fler elkraftsanläggningar och dels pågrund av allmänhetens allt större medvetenhet om hälsaoch aktsamhet om levnadsstandard. Därför bör de upplev-da akustiska störningar som orsakas av krafttransforma-torer minskas genom att man minskar dess ljudnivå ellerandra karakteristika.

Buller från vibrationer hos järnkärnan orsakas av mag-netostriktion och ger stort bidrag till den totala ljudstrål-ningen från en krafttransformator. Examensarbetet utveck-lar en komplett simuleringsmodell för denna specifika typav kärn-buller, innefattande det elektro-magnetiska uppfö-randet, mekaniska karakteristika och slutligen akustiken –utstrålningen från den vibrerande järnkärnan.

För vart och ett av dessa steg gås modelleringsprocessenigenom och uppkommande problem löses. Vissa möjliga för-enklingar av modellen görs i processen, t.ex. har hysteres-effekter försummats. Simuleringsresultaten verifieras medhjälp av referensresultat från mätningar och från räkningarmed andra metoder.

Som modelleringsverktyg används COMSOL Multiphy-sics 4.2 kopplad till MATLAB. Arbetet har etablerat enintegrerad, flexibel och effektiv modell för simulering avmagnetostriktivt buller. COMSOL-modellen komplemente-rar den befintliga modell-kedjan som består av en samlingolika program och kommando-filer.

Contents

1 Introduction 11.1 Transformer core properties . . . . . . . . . . . . . . . . . . . . . . . 11.2 Noise generation in transformers . . . . . . . . . . . . . . . . . . . . 21.3 Toolchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Electromagnetics 52.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Geometry and boundary conditions . . . . . . . . . . . . . . . 72.2.2 Choice of physics interface . . . . . . . . . . . . . . . . . . . . 92.2.3 Material properties . . . . . . . . . . . . . . . . . . . . . . . . 92.2.4 Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.5 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.6 Solver settings . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.7 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Reference criteria . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 High damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.3 Medium damping . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.4 Low damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Mechanics 273.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Geometry and boundary conditions . . . . . . . . . . . . . . . 303.3.2 Choice of physics interface . . . . . . . . . . . . . . . . . . . . 313.3.3 Obtaining the magnetic flux density . . . . . . . . . . . . . . 32

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.1 Eigenmode analysis . . . . . . . . . . . . . . . . . . . . . . . 363.4.2 Displacement results . . . . . . . . . . . . . . . . . . . . . . . 38

4 Acoustics 41

4.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Geometry and boundary conditions . . . . . . . . . . . . . . . 424.2.2 Choice of physics interface . . . . . . . . . . . . . . . . . . . . 444.2.3 Extrusion of the displacement . . . . . . . . . . . . . . . . . . 44

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.2 Sound power results . . . . . . . . . . . . . . . . . . . . . . . 45

5 Conclusions and Outlook 47

Bibliography 49

List of Tables

2.1 Coefficients for the approximation of µr (cf. Equation 2.13) . . . . . . . 122.2 Coil parameters for high damping . . . . . . . . . . . . . . . . . . . . . 192.3 Coil parameters for medium damping . . . . . . . . . . . . . . . . . . . 222.4 Coil parameters for low damping . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Mechanical properties of the core . . . . . . . . . . . . . . . . . . . . . . 27

List of Figures

1.1 Geometry of the core (all lengths in mm) . . . . . . . . . . . . . . . . . 21.2 Schematic view of noise generation and its transmission path in a trans-

former [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Schematic view of a coil . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Overview of the geometry and boundary conditions for the electromag-

netics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 The two-dimensional mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 B-H-relationship for 23ZDKH90 steel . . . . . . . . . . . . . . . . . . . . 102.6 Approximated curves for µr in both rolling and cross direction . . . . . 112.7 Relative error of the analytical representation of µr in both rolling and

cross direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 Approximated B-H curves in both rolling and cross direction . . . . . . 122.9 Axisymmetric vs. plane setup . . . . . . . . . . . . . . . . . . . . . . . . 132.10 Discrete model for a three phase transformer . . . . . . . . . . . . . . . 142.11 Joint between the middle limb and the yoke (marked with a circle) . . . 162.12 By in the tree limbs (high damping) . . . . . . . . . . . . . . . . . . . . 202.13 Comparison of By at point under consideration (high damping) . . . . . 202.14 Comparison of the FFT of By at point under consideration (high damping) 212.15 By in the tree limbs (medium damping) . . . . . . . . . . . . . . . . . . 222.16 Close-up of By in the tree limbs (medium damping) . . . . . . . . . . . 232.17 Comparison of By at point under consideration (medium damping) . . . 232.18 Comparison of the FFT of By at point under consideration (medium

damping) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.19 By in the tree limbs (low damping) . . . . . . . . . . . . . . . . . . . . . 25

3.1 Realignment of magnetic domains leading to magnetostriction . . . . . . 303.2 Overview of the geometry and boundary conditions for the mechanics . 313.3 Magnetostriction curves in rolling direction . . . . . . . . . . . . . . . . 333.4 Magnetostriction curves in cross direction . . . . . . . . . . . . . . . . . 343.5 The first seven eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Overall core displacement over the frequency range . . . . . . . . . . . . 383.7 Displacement of the core at 2312 Hz . . . . . . . . . . . . . . . . . . . . 39

4.1 The three-dimensional model of the core . . . . . . . . . . . . . . . . . . 424.2 Overview of the geometry and boundary conditions for the acoustics . . 434.3 Extrusion from two to three spatial dimensions, e.g. for one single

boundary segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

List of Symbols

λMS Magnetostriction strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

λL, µL Lamé parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa

ρ Electric charge density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cm3

A Magnetic vector potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vsm

B Magnetic flux density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .T

D Electric induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cm2

E Electric field intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vm

H Magnetic field intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Am

Je External current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Am2

J Current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Am2

µ Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hm

ν Poisson ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

ρ Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kgm3

σ Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sm

A Cross section area of the coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m2

acoil Cross section area of one winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .m2

E Elasticity modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Pa

G Shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Pa

k Wave number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1m

N Number of windings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

p Acoustic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa

[σ] Mechanical stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa

[c] Tensor of elasticitiy moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa

[S] Linear strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1

Introduction

Noise pollution is an important issue in modern societies. Power transformers are asignificant contributor to this unwanted ambient noise. It is therefore desirable tooptimize the acoustic behaviour of power transformers, i.e. either to minimize thelevel of noise radiation or to optimize its characteristics in order to lower the levelof disturbance for humans.

This thesis work consists of three major parts, each covering one element of thecoupling chain which represents the creation of transformer noise. Firstly, the core’selectromagnetic characteristics are analyzed, followed by its mechanical propertiesand finally its acoustics.

In the beginning of this first chapter, basic information about the transformercore under consideration is provided. In the following, an overview of mechanismsof noise creation in power transformers is given and their relevance with respect tothis work is pointed out. Lastly, the current toolchain which is used by ABB for thesimulation of core noise and the toolchain utilized within this work are presented,which also gives further support to the motivitation for this work.

1.1 Transformer core properties

The transformer under investigation is a three-phase three-limb transformer com-prising a core made out of 23ZDKH90 sheet steel. Figure 1.1 shows a schematicview of the transformer core under investigation.

To approximate a circular cross section area of the core, which would be theideal design, the core is assembled from several "packages" of steel sheet with varyingsizes. Cross-section views of the yoke and the limbs, depicting the package structure,can also be seen in Figure 1.1. In order to avoid the appearance of eddy currentsaccompanied by losses each of these packages consists of multiple thin layers of sheetsteel.

Note that transformer steel is anisotropic, also in magnetic permeability (cf.Figure 2.5). In order to force the direction of the magnetic flux into the rightdirection, the core is assembled from laminations with varying rolling direction. In

1

CHAPTER 1. INTRODUCTION

60

702

700

140

120

100

(a) Schematic view of the core

Yoke140

120

100

60

Limb

9.40

9.55

9.55

39.32

(b) Cross-section views of the yoke and the limbs

Figure 1.1: Geometry of the core (all lengths in mm)

the limbs, the material is rolled in y-direction (hence, the cross direction, whichis always in-plane with respect to the sheet structure, is x). In the yoke, thosedirections are interchanged. The rolling directions are also depicted in Figure 1.1by the pale arrows in the core. In the following, quantities in rolling directionwill by a superscript "rd", whereas quantities in cross rolling direction will have asuperscript "crd".

1.2 Noise generation in transformers

Figure 1.2 depicts the noise generation mechanisms in transformers. They greyelements of this flowchart are not taken into account in this model. Transformersare electromagnetical systems which radiate noise under certain circumstances. Thereason for this are forces arising due to electromagnetical fields.

There are two major mechanisms of noise generation in transformers, namelymagnetostrictive and magnetomotive forces, the former resulting from the appear-ance of strains due to an applied magnetic field (cf. section 3.2) and the latteraccounting for the generation of a magnetic flux. The magnetomotive forces acting

2

1.3. TOOLCHAIN

Alternating electromagnetic flux

Magnetostrictive forces Magnetomotive forces

Vibration of core due to magnetostriction

Transverse relativemotion of

core laminations

Vibrationof winding

coils

Core vibration

Acoustic waves in fluid contained intransformer tank

Vibration oftransformer mounting

Structure-borneAir-borne

Tank vibration

(Tonal) noise radiation

Air-borne

Structure-borne

Figure 1.2: Schematic view of noise generation and its transmission path in a trans-former [5]

in the current-carrying windings are analyzed in [7]. Both magnetostrictive andmagnetomotive forces act in the core. Magnetomotive forces cause the laminationsof the core to strike against each other in case residual gaps between them arepresent. In contrast to this, magnetostriction causes a vibration of the entire corewith twice the line frequency. Vibrations due to magnetostriction are the major con-tributor to core noise [5]. Thus, this cause will be in the focus of this thesis. Thestructure-borne transmission of sound waves through the transformer mountings isnot taken into account. Actually, the acoustic waves in the fluid, which is containedin the transformer tank surrounding the core, caused by the core vibration finallycause the tank to vibrate which eventually radiates tonal noise. However, withinthis thesis this fluid is considered air and the tank is not taken into account. I.e.,only the noise radiation of the bare core is taken into consideration.

1.3 ToolchainIn this section, first ABB’s current toolchain used to carry out a coupled simulationfor the prediction of core noise is presented. Additionally, a motivation to change

3

CHAPTER 1. INTRODUCTION

the state of the art is given. Afterwards, the toolchain used in this work is in thefocus.

ABB’s in-use toolchain for simulating transformer core noise includes severalstages which are more or less separated from each other. ABB’s in-house softwareAce is used for the computation of the magnetic field inside the core. The resultsare processed by means of a Python1 script, which looks up several measurementtables to compute the magnetostriction strains. Additionally, this script computesforces resulting from the magnetostriction strains, as those are needed as input forthe next step. In the end, those forces are imported to Abaqus2 where eventuallythe acoustic analysis is carried out. In order to gain flexibility, convenience andmaybe performance, it is desirable to simplify this toolchain, i.e. make it use fewerdifferent tools.

The general goal of this thesis work is to determine whether the behavior of atransformer core with respect to the magnetic field inside it, magnetostriction, vi-bration and noise radiation can be predicted by means of a single software, namelyCOMSOL3 Multiphysics® 4.2. This software provides several modules which them-selves provide various physics interfaces for different applications. COMSOL Multi-physics uses the finite element method (FEM) in order to solve the arising equations.It is also used for pre- and post-processing, i.e. creating the model, generating themesh and analyzing the results. Additionally, the MATLAB interface provided byCOMSOL is utilized, which is called LiveLink™for MATLAB and therewith alsoMATLAB itself.

1http://www.python.org2http://www.simulia.com/products/abaqus_fea.html3http://www.comsol.com

4

Chapter 2

Electromagnetics

This chapter deals with the modelling of the electromagnetical behavior of thetransformer core. In addition, the developed model is verified using a referencesolution. The core’s electromagnetic characteristics constitute the first link in thecoupling chain. Thus, it is important to obtain accurate and reliable results for theelectromagnetics as they serve as input for the mechanical and finally the acousticsimulation.

At the beginning of this chapter, a brief review of the governing equations usedfor the electromagnetical analysis, i.e., Maxwell’s equatios, is given. In the followingsection, a two-dimensional model of the core is developed by means of COMSOLMultiphysics, yielding information about the distribution of the magnetic flux den-sity in the core. This section includes solutions for problems arising during the mod-elling process and information about the excitation of the modelled core. Finally,the simulation results are verified using the results of ABB’s in-house software Aceas reference. After this is done, the computed values for the magnetic flux densitycan be used as input for the computation of magnetostriction in the next chapter.

2.1 Governing equations

In this section, the main equations constituting the core’s electromagnetical char-acteristics, i.e. Maxwell’s equations, are presented. Maxwell’s equations are basedon the contributions of Ampère (2.1), Gauß (2.3, 2.4) and Faraday (2.2). The term∂D∂t is called "displacement current" and forms the major contribution of Maxwellto the set of equations. Maxwell’s equations are given as

∇×H = J + ∂D∂t

(2.1)

∇×E = −∂B∂t

(2.2)

∇ ·D = ρ (2.3)∇ ·B = 0 (2.4)

5

CHAPTER 2. ELECTROMAGNETICS

with H the magnetic field, J the current density, D the electric induction, E theelectric field and B the magnetic flux density. Additionally, the following constitu-tive equation is used:

B = µ0µrH (2.5)

with µ0 the permeability in vacuum (4π ·10−7 V sAm) and µr the relative permeability

in the material. As the analysis is carried out in the low-frequency domain, it ispossible to omit the term ∂D

∂t [6] in Equation 2.1. Ohm’s law states that

J = σE (2.6)

whith σ the conductivity. After adding an external current density Je, the relationis

J = σE + Je (2.7)

In the coils, the induced voltage Vind has to be taken into account when computingthe external current density Je. Thus, Je computes as follows:

Je = N(Vcoil + Vind)ARcoil

(2.8)

where N is the number of coil windings, A is the cross-sectional area of the coil,Vcoil the the prescribed coil voltage and Rcoil the total coil resistance given as

Rcoil =∫

A

NL

σcoilacoilAdA (2.9)

L is equal to the out-of-plane thickness in two-dimensional setups or 2πr for axiallysymmetric models. A visualization of a coil can be seen in Figure 2.1.

AN windings

acoil σcoil

Figure 2.1: Schematic view of a coil

One can introduce the magnetic vector potential A, which is defined as

B = ∇×A (2.10)

Hence, from Faraday’s law (cf. Equation 2.2) one can deduce that

E = −∂A∂t

(2.11)

6

2.2. MODELLING

Combining those relations, the general pde for the transient analysis can be givenas

σ∂A∂t

+∇×(∇×Aµ0µr

)= Je (2.12)

with Je in the coils being computed as in Equation 2.8.

2.2 ModellingThe model is investigated in two dimensions considering one single package, namelythe largest one. It sounds reasonable to assume that the modelling can be done ina straightforward way. However, some adjustments and customizations have tobe done in order to obtain a valid and reliable model. Before these measures areexplained in detail, an overview of the geometry under consideration, as well as theboundary conditions, is given.

2.2.1 Geometry and boundary conditionsFigure 2.2 depicts the basic geometry and boundary conditions of the core model.The core is surrounded by air in order to take the (small) stray field into account

Magnetic Insulation

Air

AirAir

Pointunder

conside-ration

Coil domain 1

2

Coil domain 3

4 Coil domain 5

6

Figure 2.2: Overview of the geometry and boundary conditions for the electromag-netics

and hence obtain a more precise simulation. After a relatively thin layer of air, a

7


magnetically insulating boundary condition is applied on the outer boundary whichimposes the constraint n ·H = 0, i.e. the normal component of the magnetic fieldis forced to be zero.

One might think that due to symmetry, it is possible to take only one half of thecore into account. In this case, a perfect magnetic conductor condition would haveto be imposed at the symmetry axis, prescribing that n×H = 0 (i.e., the tangentialcomponent of the magnetic field is set to zero). However, in the mechanical andacoustical model the whole core has to be taken into account, as no assumptionsabout symmetry can be made. Hence, as the same mesh should be used in theelectromagnetic and mechanical analysis, the full geometry has to be used also inthe electromagnetical simulation.

The coil domains are numerated from left to right, beginning with one. Notethat one physical coil is constituted by two coil domains (e.g., the left coil consistsof the coil domains one and two).

Figure 2.3: The two-dimensional mesh

The mesh can be seen in Figure 2.3. It is relatively fine in the coils and the coreand coarser in the air domain, which is reasonable because a precise computation ofthe stray field is not needed. A small mesh convergence study can be used to showthat the mesh is sufficiantly fine, i.e., the result does not change significantly whenthe mesh is getting finer. Figure 2.4 shows the the integral of the norm of the Bfield over the whole core, depending on the reciprocal of the maximum element sizeh. It can be observed that for 1/h ≥ 50 the result does not change significantly. Infact, decreasing the maximum element size h from 0.02 m to 0.01 m changes theresult by merely 2.1 %. Thus, a maximum element size h = 0.02 m yields a meshwhich is sufficiently fine. In this model, h was chosen to be 0.018 m, which is thecase when applying the predefined Element size Extra fine.

8

2.2. MODELLING

20 40 60 80 100 120 140 160 180 20089.2

89.4

89.6

89.8

90

90.2

90.4

90.6

1/h (1/m)

Dependance of solution on mesh size

Inte

gra

l of norm

(B)

over

the c

ore

Figure 2.4: Mesh convergence

2.2.2 Choice of physics interface

The Magnetic Fields interface of COMSOL’s AC/DC Module is the right choice formodelling the electromagnetic behavior of the core. It has the equations, boundaryconditions, and external currents for modelling magnetic fields, solving for the mag-netic vector potential A [1]. It also provides Multi-Turn Coil Domains which canbe excited by a voltage, thus modelling the physics of a transformer in a realsticway.

2.2.3 Material properties

The first step is to implement the properties of the core material. Note that hys-teresis is not taken into account. As the BH-curve of 23ZDKH90 steel was onlyavailable as an image file, a way had to be found to obtain a discrete table of therelationship. For this purpose, the tool Engauge Digitizer 1 has been utilized. Itis capable of creating a table based on an image depicting a curve. The resultingBH-curve can be seen in Figure 2.5. Note that the BH-curve of this sheet metal ishighly anisotropic which is typical for sheet steel. Thus, two different directions,the rolling and the cross direction, are under consideration. A highly nonlinearbehaviour can be observed including saturation. This curve is typical for ferro-magnetic materials. Furthermore, it can be seen that the material is easier to bemagnetized in the rolling direction.

1http://digitizer.sourceforge.net/

9


101 102 103 104

0.5

1

1.5

2

H (A/m)

B(T

)

rolling directioncross direction

Figure 2.5: B-H-relationship for 23ZDKH90 steel

COMSOL implements anisotropic magnetization behavior only in case a consti-tutive relation for the electromagnetics using the relative permeability µr = f(B)(rather than B = f(H)) is chosen. However, the µr derived from the BH curve (cf.Figure 2.5) is not smooth and hence not suited for being used. This can be seenespecially in the range of 0.5 T < B < 1 T. Thus, it is required to find a good (i.e.,smooth and accurate) analytical representation of µr. Several representations havebeen tested. In the end it turned out that an approximation using a combination ofGaussian functions provides good results in rolling as well as cross direction. Thegeneral relation is

µr ≈n∑

j=0aj · e

−

(|B|−bj

cj

)2

(2.13)

with n, aj , bj and cj to be determined. In order to find the values of the parameters,MATLAB’s curve fitting tool is utilized. The resulting µr curves for the rolling andcross direction can be seen in Figure 2.6. Table 2.1 summarizes the coefficients aj ,bj and cj for both representations.

The relative errors between the given and analytical representations of µr inrolling and cross direction is shown in Figure 2.7. It can be seen that the error isrelatively small. Only for relatively high values of B in rolling direction the relativeerror gets rather large. This can also be seen in Figure 2.6; here, the bending in theaccordant region is not represented very well.

The B-H-relationships deduced from the analytical representation of µr for bothrolling and cross direction can be seen in Figure 2.8.

10

2.2. MODELLING

0 1 2 3 40

2

4

6·104

B (T)

µrd r

Gaussianfrom BH curve

(a) Rolling direction

0 1 2 3 40

1,000

2,000

3,000

B (T)

µcr

dr

Gaussianfrom BH curve

(b) Cross direction

Figure 2.6: Approximated curves for µr in both rolling and cross direction

0.5 1 1.5 20

2

4

6

B (T)

erd rel


0.5 1 1.50

3 · 10−2

6 · 10−2

9 · 10−2

B (T)

ecrd rel

(b) Cross direction

Figure 2.7: Relative error of the analytical representation of µr in both rolling andcross direction

In a personal conversation on Friday, February 18 2011 at ABB Corporate Re-search in Västerås, Gunnar Russberg suggested that µr should not be dependent onthe norm of B but rather on the magnetic flux density in the accordant direction(rolling or cross). Hence, the tensor µr for the limbs is

µr =

µcrdr (|Bx|) 1 1

1 µrdr (|By|) 1

1 1 1

(2.14)

11


10−2 10−1 100 101 102 103 104 105

0

0.5

1

1.5

2

H (A/m)

B(T

)

rd, analyticalrd, measuredcrd, analyticalcrd, measured

Figure 2.8: Approximated B-H curves in both rolling and cross direction

j a b c

0 34360 1.174 0.39091 50150 0.5903 0.61892 17760 1.535 0.2564


j a b c

0 1440 1.059 0.25041 93.06 1.233 0.78472 924.2 1.25 0.1323 2286 0.7133 0.5033

(b) Cross rolling direction

Table 2.1: Coefficients for the approximation of µr (cf. Equation 2.13)

and for the yoke

µr =

µrdr (|Bx|) 1 1

1 µcrdr (|By|) 1

1 1 1

(2.15)

It is crucial to take the absolute value of the accordant component of the B field.This is due to the fact that the analytical representation used for µr (cf. Figure 2.6)is not axially symmetrical with respect to B = 0 and thus, in case the absolut valuewere not taken, µr(B) would return incorrect values for a B field which is orientedin negative x- or y-direction.

2.2.4 CoilsIn contrast to the 2D axisymmetric case, COMSOL Multiphysics does not provide astraightforward way to model cylindrical coils in 2D plane setups. In axisymmetric

12

2.2. MODELLING

models, a coil can be modelled with one single coil domain. However, this is notpossible in plane setups, where two coil domains are required to model one coil. SeeFigure 2.9 for a visualization of this issue. The two coil domains forming one coil

i

d

B,H

Figure 2.9: Axisymmetric vs. plane setup

are not connected to each other originally, i.e., the currents and voltages in both ofthem are inpedendent from each other. Hence, a way to connect them has to befound in order to find a correct model for a two-dimensional plane setup. Wheninvestigating Equation 2.8, which describes the computation of the current densityin the coils, it can be seen that only the induced voltage Vind of the coil domain itselfis taken into account. However, it is necessary to include the induced voltage ofthe coil domain itself and the induced voltage of its associated coil domain, i.e., thetwo coil domains forming the coil. This behavior can be accomplished by changingthe definition of Vind in the Equation View of the coil domain nodes in COMSOL.Originally, the expression for Vind, e.g., for the first coil domain, is

mf.Vind_1 = mf.intmtcd13(mf.Ez*N*mf.d/mf.coilDomainArea_1)

It basically computes the integral of the z-component of the electric field, multipliedwith the number of turns and the out-of-plane thickness and divided by the area ofthe coil domain, over the first coil domain. By changing the definition to

mf.Vind_1 = mf.intmtcd13(mf.Ez*N*mf.d/mf.coilDomainArea_1)-mf.intmtcd23(mf.Ez*N*mf.d/mf.coilDomainArea_2)

In the second coil domain, the signs have to be interchanged, leading to

mf.Vind_2 = mf.intmtcd13(mf.Ez*N*mf.d/mf.coilDomainArea_1)-mf.intmtcd23(mf.Ez*N*mf.d/mf.coilDomainArea_2)

the induced voltage in the second coil domain is taken into account. This connectsthe two associated coil domains one and two in the desired way, leading to correct

13


current densities in z-direction in the two domains which are in-phase, have thesame amplitude and different sign.

2.2.5 DampingIn order to avoid resistive losses, the resistance of the coils is preferably small inpractice. However, this leads to a system which is very lightly-damped, i.e., ittakes many cycles after startup until the system reaches a quasi-steady state whichis independent of the initial values. Damping affects the simulation results whenapplying a time-stepping scheme, which is the case in this work. Low damping cancause the mean value of the B field at a certain point to be unequal to zero. Anotherconsequence of low damping is a peak magnetic flux density which is unequal inthe outer limbs. Both those phenomena should not occur in a quasi-steady stateand are thus unwanted. However, the system has to reach that state in order tomake the results comparable to the results from Ace (which are given as a harmonicsolution) and also in order to obtain a solution which can be used as input forthe next coupling steps. See section 2.3 for several results with different dampingvalues.

In order to understand damping better, it is reasonable to create a simplifieddiscrete model on the core, not taking into account the dependance of µ on themagnetic flux density and using a highly simplified geometry. This model can easilybe analyzed concerning its damping behavior. This approach has been proposed byJesper Oppelstrup. Figure 2.10 depicts such a model. Φk denotes the magnetic flux

h

l

Φ1,H1

N turns each

Φ2,H2

Φ3,H3

i1 i2 i3

Figure 2.10: Discrete model for a three phase transformer

in each limb, Hk the magnetic field intensity in each limb, l is half the length ofthe yoke, h the height of the limbs and N the number of turns per coil. The crosssection of the core will be called S. The following relation can be found:

Bk = Φk

S(2.16)

Hk = Bk

µ(|Bk|)= Bk

µ(2.17)

14

2.2. MODELLING

Let the currents in the coils be denoted by ik. The following relations can bededuced with the rules for magnetical circuits in mind:

i1 − i2 = (H1(2l + h) −H2h )/N (2.18)i2 −i3 = ( H2h−H3(2l + h) )/N (2.19)

i1 + i2 +i3 = 0 (2.20)

The first two equations represent relations between two currents in different coils.The third equation accounts for the fact that the sum of the fluxes, and thus thesum of the currents, has to be zero. By introducing the matrices

A =

1 −1 00 1 01 1 1

(2.21)

and

C =

a −b 00 b −a0 0 0

(2.22)

with a = 2l+hN and b = h

N , one can deduce the following relations:

AI = CH (2.23)I = A−1CH (2.24)

and with the help of Ohm’s law

U = RA−1CH (2.25)

with I a matrix holding the currents, U a matrix holding the coil voltages, R thecoil resistance and H a matrix containing the magnetic field values. By constructinga matrix F containing the magnetic fluxes, one could state that

H = 1Sµ

F (2.26)

Finally, using those equations the following relation could be obtained:

U = R

SµA−1CF (2.27)

Thus, assuming that the unknown values are in F and U accounts for the load ofthe system, the stiffness matrix K can be given as

K = R

SµA−1C (2.28)

Hence, a measure for the damping can be computed by investigating the eigenvaluesof K. The higher they are, the higher the damping gets and the faster a quasi-steadystate will be reached in the time-stepping scheme.

15


In order to avoid the aforementioned problems with damping, i.e., to reach aquasi-steady state after fewer iterations, a possible solution is to change the coil’sproperties such that its damping increases, i.e., its resistance gets higher. The coilresistance is computed as given in Equation 2.9. Thus, it is reasonable to create anartificially high damping, e.g., by choosing unphysical values for the cross sectionarea per coil winding acoil (preferably small) or the coil conductivity σcoil (preferablysmall). By this, a high damping can be obtained which results in a zero mean valueof B after only a few periods. However, in this case the driving Voltage Vcoil (cf.subsection 2.2.7) has to be very high in order to reach the desired induction in thecore. Results of several analyses with different dampings can be found in section 2.3.

Another approach to solve this problem might be to increase the damping bychoosing appropriate initial values for the voltages in the coils and the magneticflux density in the core. However, this approach is not followed further within thiswork.

2.2.6 Solver settingsIt is crucial to configure the solver correctly in order to avoid convergence problems.Those problems mainly occur at at the joint between the middle limb and the yoke.This region (cf. Figure 2.11) is particularly prone to singularities due to a suddenchange in rolling direction combined with the pointy shape of the middle limb’supper part.

Figure 2.11: Joint between the middle limb and the yoke (marked with a circle)

This problem can be resolved by setting the Tolerance factor of the Fully Cou-pled solver to a value smaller than one. Setting it to 1e-2 turned out to be asensible choice. This factor represents the tolerance used for termination of theNewton iteration. The actual used tolerance for the Newton iteration computes asthe general tolerance of the Time-Dependent solver multiplied with this tolerancefactor. By tightening the tolerance, the solver is forced to compute a more accuratesolution per timestep instead of decreasing the timestep. This is especially benefi-cial for nonlinear electromagnetical problems (COMSOL Support, A. Juhasz, Emailconversation, March 14, 2011). The usedtime stepping method is BDF.

In case a comparison with a reference solution has to be carried out, furthertweaking of the solver is reasonable in order to achieve a smooth solution. Thesolver tends to increase the timestep within the linear region, leading to a rathernon-smooth solution. As the solution has to be compared to the solution of ABB’sin-house software Ace, a preferably smooth solution is desired. This can be achievedby bounding the maximum time step in the Time-Dependent solver node. Here, avalue in the order of magnitude 10−4 appears to be a reasonable choice.

16

2.3. RESULTS

2.2.7 ExcitationFinally, the excitation of the model has to be implemented. The analysis is beingcarried out in the time domain. The coils are excited by phase-shifted voltages. Anexcitation of

V 1,2coil = ±V̂ · sin(ωt+ 2/3 · π)

is used for the first coil (consisting of the coil domains 1 and 2). The phase shiftfor the second and third coil yields

V 3,4coil = ±V̂ · sin(ωt)

and

V 5,6coil = ±V̂ · sin(ωt− 2/3 · π)

The target value is a peak magnetic flux density of 1.75 T at the point underconsideration (cf. Figure 2.2). Due to the actual symmetry of the core (whichis not taken into account in the model, cf. subsection 2.2.1 for the reasons), thex-component of B is zero at this point. So, only By has to be taken into account.

2.3 ResultsIn this section, the results of the electromagnetical simulations carried out in COM-SOL Multiphysics are compared to those of Ace, which are used as a reference. Incontrast to the time-dependent approach used within this thesis, Ace performs itsanalysis in the frequency domain. In Ace, only two superharmonics are considered,namely 150 Hz and 250 Hz. Certainly one would assume that the time-dependentanalysis yields more accurate results as no simplified assumptions are taken here. Afurther difference compared to Ace is that Ace does not excite the core by voltage-driven coils as it done in this model. Instead, the total magnetic flux density isprescribed and there are no coils modelled at all. The absence of the coils in Acemakes up a significant difference to the model developed within this work. Alsohere, one would assume that the approach used in this thesis, exciting the core byactual coils meets reality better than Ace does. Anyway, as Ace computes the ref-erence solutions which are in use today, a comparison between the developed modelfor electromagnetics and Ace is made. However, the target is not to fit Ace’s valuesexactly but rather to interpret and explain the differences.

2.3.1 Reference criteriaThe results are compared by looking at the y-component of the magnetic flux densityat the point under consideration (cf. Figure 2.2) and comparing the time-dependentsignal of both solutions. As there is no point in the Ace mesh exactly meeting theideal point (which is exactly at the balance point of the core), the closest point in

17


the Ace mesh has to be taken. This is (if x = 0 denotes the axis of symmetry inthe middle) at (-0.0030505, 0).

In order to find the Ace node closest to a given point, a MATLAB functionhas been written. It reads an Ace-typical four-column text file containing the nodenumber and the x-, y- and z-position of all Ace nodes. Then it finds the Ace nodeclosest (in the 2-norm) to the given point and finally returns the node number andposition. The source code of the function can be seen in Listing 2.1.

function [ node,x,y ] = acenode( xdes,ydes )% acenode( xdes,ydes ) Returns the node number and coordinates of the

Ace node% closest to the given point

load ’geom.txt’; % load geometry informationxs=geom(:,2); % x positions of nodesys=geom(:,3); % y positions of nodes

d = [xs,ys]-repmat([xdes,ydes],length(xs),1); % compute distancevectors

dnorm = sqrt(d(:,1).^2+d(:,2).^2); % compute distance norms[~,node] = min(dnorm); % find node with minimal

distance

x=xs(node); % x position of closest nodey=ys(node); % y position of closest node

end

Listing 2.1: MATLAB function to find the Ace node closest to a given point

The results from COMSOL and Ace are usually not in phase. Thus, in orderto obtain a fair comparison, one of them has to be shifted. The MATLAB scriptwhich can be seen in Listing 2.2 uses a naïve approach to shift the Ace result untilthe error between the two results is minimal.

xdes=0; % the balance point of the core has thecoordinates (0,0) in the Ace model

ydes=0;

[node,x,y]=acenode(xdes,ydes); % find according Ace node

[tBc] = load(strcat(’acepoint.txt’)); % load COMSOL values (time and By)

%tc=tBc(1601:end,1); % take only the last two periods into accountwhen medium damping is applied

%Bc=tBc(1601:end,2);

bestshift=0;errabs=inf;

for shift=0:0.00001:0.02 % maximum shift is one period length

18

2.3. RESULTS

if(norm(Bc’-Bace(node,tc-shift)) < errabs) % a better shift has beenfounderrabs = norm(Bc’-Bace(node,tc-shift));bestshift = shift;

endend

Ba=Bace(node,tc-bestshift); % shift the Ace result

errabs = norm(Bc’-Ba) % compute the absolute errorerrrel = errabs/norm(Ba) % compute the relative error

plot(tc,Bc,tc,Ba,’r’); % plot both signalslegend(’comsol’,’ace’);

Listing 2.2: MATLAB script used to compare the results of COMSOL and Ace

2.3.2 High damping

As explained in subsection 2.2.5, damping is a big issue in the computation of theelectromagnetic field. In order to achieve a very high coil resistivity, the parametersare set as in Table 2.2 for this run. Observe the very high voltage which is needed

Parameter ValueN 1σcoil 1 S/macoil 10−8 m2

Vcoil 4.8 · 109 V

Table 2.2: Coil parameters for high damping

to obtain the desired peak flux density of 1.75 T. Figure 2.12 shows By in thethree limbs. It can be observed that the high damping yields the expected results(cf. subsection 2.2.5). A quasi-steady solution is reached after only one cycle.Additionally, the peak flux density is the same in the left and the right limb, whichalso meets reality.

As a limit cycle solution is reached after the first cycle, i.e., within a veryacceptable runtime, it is also possible to draw a comparison to Ace after a one-cycle run. See Figure 2.13 for a comparison of both time signals at the point underconsideration. The relative error of the time-dependent COMSOL result is withrespect to the Ace result |B

acey −Bcomsol

y ||Bace

y | = 13.2%, which is fairly large.It is reasonable to investigate the frequency spectra deeper. They are shown in

Figure 2.14. The non-zero values at non-odd superharmonic frequencies are due tonumerical artifacts and do not require further regards. As mentioned above, Acetakes only two harmonics into account. This can be observed in the graph. How-ever, when looking at the FFT of the COMSOL result, it can be seen that further

19


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

By (

T)

By in the three limbs (high damping)

Middle

Left

Right

Figure 2.12: By in the tree limbs (high damping)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

By (

T)

By at the point under consideration

COMSOL

Ace

Figure 2.13: Comparison of By at point under consideration (high damping)

20

2.3. RESULTS

100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

abs(B

y)

(T)

FFT of the Ace and COMSOL results

COMSOL

Ace

Figure 2.14: Comparison of the FFT of By at point under consideration (highdamping)

harmonics show significant amplitudes. The appearance of only odd superharmon-ics is due to the fact that the function B(H) is odd. Assuming a sinusoidal inputsignal, H can in general be expressed as H = sin(ωt). Hence, having in mind an oddfunction B(H) and using H = sin(ωt) as input of this function, the resulting B(H)is also odd. When this function is expressed by means of its frequency components,i.e., the FFT of it is taken, only odd contributions have to be taken into account,as only those are also odd with respect to H = 0.

21


2.3.3 Medium dampingAs shown in subsection 2.3.2, high damping leads to good numerical behavior forthe time-stepping scheme. However, it can also be observed that the results for highdamping do not meet the Ace results very well. Hence, it is reasonable to carry outan analysis with less damping in order to see if it performs better compared to Ace.The parameters set for the coil in order to achieve a medium damping can be seenin Table 2.3. σcoil has been set to the real conductivity for copper. Due to the lower

Parameter ValueN 10σcoil 6 · 107 S/macoil 10−8 m2

Vcoil 769 V

Table 2.3: Coil parameters for medium damping

resistivity, the driving voltage Vcoil, required to reach the induction of 1.75 T, canbe lower in this case. Figure 2.15 depicts By at the center point on the three limbs.

0 0.02 0.04 0.06 0.08 0.1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

By (

T)

By in the three limbs (medium damping)

Middle

Left

Right

Figure 2.15: By in the tree limbs (medium damping)

It can be observed that a quasi-steady solution is not reached immediately. Instead,it takes a few cycles until this is done. Thus, when comparing the result to Ace it isnecessary to take away the first cycles. Furthermore, it can be seen that the peakmagnetic flux density differs between the left and the right limb. This is a severeissue as it results in a non-symmetric vibration and thus in a non-symmetric noiseradiation of the core. See Figure 2.16 for a close-up image depicting this issue.

22

2.3. RESULTS

0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075

1.66

1.68

1.7

1.72

1.74

1.76

1.78

Time (s)

By (

T)

By in the three limbs (medium damping)

Middle

Left

Right

Figure 2.16: Close-up of By in the tree limbs (medium damping)

0.16 0.165 0.17 0.175 0.18 0.185 0.19 0.195 0.2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

By (

T)

By at the point under consideration

COMSOL

Ace

Figure 2.17: Comparison of By at point under consideration (medium damping)

23


The comparison between COMSOL and Ace is shown in Figure 2.17. It can beseen that COMSOL fits the Ace results better in case the damping is not so high.In fact, the relative error adds up to |B

acey −Bcomsol

y ||Bace

y | = 5.54%, which is significantlybetter than the one obtained for the high damping. This statement receives supportby the FFT of the signals, which can be seen in Figure 2.18. They are fairly closeto each other. However, it can be observed that the 250 Hz component is nearlyabsent in the COMSOL solution. Instead, also higher, odd harmonics contribute tothe signal which is (naturally) not the case for Ace.

100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

abs(B

y)

(T)

FFT of the Ace and COMSOL results

COMSOL

Ace

Figure 2.18: Comparison of the FFT of By at point under consideration (mediumdamping)

24

2.3. RESULTS

2.3.4 Low damping

After having looked at high and medium damping, it is reasonable to also analyzelow damping. To lower the damping even more, the properties given in Table 2.4have ben applied to the coils. The signal of By in the three limbs can be seen in

Parameter ValueN 1σcoil 6 · 107 S/macoil 10−4 m2

Vcoil 45 V

Table 2.4: Coil parameters for low damping

Figure 2.19. It is obvious that severe problems occur. First of all, the mean value for

0 0.02 0.04 0.06 0.08 0.1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time (s)

By (

T)

By in the three limbs (low damping)

Middle

Left

Right

Figure 2.19: By in the tree limbs (low damping)

the B fields does not seem to tend to zero at all. It can be estimated that it wouldtake very many cycles until a quasi-steady solution, including a B field alternatingaround zero, is reached. Furthermore, the mean values for the B field in the leftand right limb seem to be the same, whereas it is different in the middle limb.Additionally, the B field in the left and right limb has positive as well as negativevalues, whereas the B field in the middle limb is purely positive. Moreover, theproblem regarding the different peak values for B in the limb described before (cf.subsection 2.3.3) persists.

Due to these reasons, especially the slow convergence of By in the middle limb

25


to a mean value of zero, a comparison with Ace can not be established for the lowdamping case within an acceptable runtime.

In the end, it can once again be stated that damping is a serious issue for themodel which would require deeper analysis in future research.

26

Chapter 3

Mechanics

In this chapter, the mechanical behavior of the core is analyzed. The mechanicalanalysis constitutes the second element in the coupling chain from electromagneticsto acoustics. The results from the previous chapter, i.e. the magnetic flux densityin the core, serve as input for the mechanical simulation. The outcome of themechanical analysis is the (harmonic) displacement of the transformer core, whichserves as input for the subsequent acoustic analysis.

First of all, an overview of the equations describing the mechanical behavior ofthe core is given. Subsequently, the main cause of core noise, magnetostriction, ispresented in some detail. After having explained the theoretical background, thefocus in the following chapter is on the modelling of the core’s mechanical behaviorin COMSOL. In the results section, an eigenmode analysis of the core is carried outin order to obtain a better understanding of the core. With this in mind, resultsfor the displacement field are presented.

The mechanical properties of the steel are given in Table 3.1. Further informa-

Property Value

Young’s modulus{ rolling direction (Erd) 120 · 109 Pa

cross (90◦) direction (Ecrd) 210 · 109 Paz-direction (Ecrd) 2.1 · 106 Pa

Poisson ratio (ν) 0.25Density (ρ) 7850 · 0.96 kg

m3

Table 3.1: Mechanical properties of the core

tion about 23ZDKH90 can be found in [8]. The factor 0.96 for density has to beintroduced in order to account for air between the laminations whose amount canbe estimated to 4%.

27

CHAPTER 3. MECHANICS

3.1 Governing equations

The differential equation which is solved is Navier’s equation, given as

−∇ · σ = FV (3.1)

with FV the volume forces and σ the mechanical stresses. The basic underlying lawof the mechanical analysis is the law of linear elasticity, known as Hooke’s law [6].It has the general form

[σ] = [c][S] (3.2)

with [σ] the mechanical stress tensor, [c] the tensor of elasticitiy moduli and [S] thelinear strain tensor. Therewith, a relation between mechanical stress and strain isgiven. In matrix notation, the general form of Hooke’s law for an isotropic materialin three spatial dimensions is given as

σ11σ22σ33σ23σ31σ12

=

λL + 2µL λL λL 0 0 0λL λL + 2µL λL 0 0 0λL λL λL + 2µL 0 0 00 0 0 µL 0 00 0 0 0 µL 00 0 0 0 0 µL

s11s22s332s232s312s12

(3.3)

where λL and µL are the Lamé parameters defined as

λL = νE

(1 + ν)(1− 2ν) (3.4)

µL = E

2(1 + ν) (3.5)

with E the elasticity modulus and ν the Poisson ratio. Note that the shear strainsand stresses are symmetric, i.e., σij = σji and sij = sji, which allows to state thematrix form in the reduced way which can be seen above.

However, as described above, the steel has different properties in rolling andcross rolling direction (which is in a 90 degree angle from rolling direction), i.e., it isorthotropic. Hence, its stiffness depends on the properties Ei, νij and Gij [4]. Usingthis and assuming x as 1-, y as 2- and z as 3-direction, Hooke’s law transforms to:

σxx

σyy

σzz

σyz

σxz

σxy

=

c11 c12 c13 0 0 0c12 c22 c23 0 0 0c13 c23 c33 0 0 00 0 0 c44 0 00 0 0 0 c55 00 0 0 0 0 c66

sxx

syy

szz

2syz

2sxz

2sxy

(3.6)

28

3.2. MAGNETOSTRICTION

with

c11 = 1aE2

x(Ezν2yz − Ey) (3.7)

c12 = −1aExEy(Ezνyzνxz + Eyνxy) (3.8)

c13 = −1aExEyEy(νxyνyz + νxz) (3.9)

c22 = 1aE2

y(Ezν2xz − Ex) (3.10)

c23 = −1aEyEz(Eyνxyνxz + Exνyz) (3.11)

c33 = 1aEyEz(Eyν

2xy − Ex) (3.12)

c44 = Gyz (3.13)c55 = Gxz (3.14)c66 = Gxy (3.15)a = EyEzν

2xz − ExEy + 2νxyνyzνxzEyEz + ExEzν

2yz + E2

yν2xy (3.16)

G denotes the shear modulus which is computed as

G = E

2(1 + ν) (3.17)

For the computation of the shear modulus between two directions, the mean valueof E between those two directions is used.

Mechanical simulations in two dimensions can be carried out in two ways, eachapplying a different simplification to the model. In the so-called plane strain case,strains in out-of-plane direction (i.e., 3-direction) can be neglected, i.e., s31 = s32 =s33 = 0. It can be applied for setups where the third dimension is very largecompared to the other ones. In contrast to this, a very small third dimension givesrise to a so-called plane stress setup, where stresses in out-of-plane direction areneglected. Thus, σ31 = σ32 = σ33 = 0. In this model, plane stress is a reasonablechoice.

3.2 MagnetostrictionThe major contributor to the overall core noise is the core’s magnetostriction. Theterm magnetostriction stands for the appearance of a strain λMS due to a change inits magnetization, which is caused by an applied magnetic field H. This effect is dueto the alignment of the (initially unaligned) Weiss domains in the direction of anapplied outer magnetic field. Figure 3.1 visualizes this effect. Note that the magne-tostriction strain is independent of the orientation of the magnetic field. Therefore,the main magnetostriction frequency is double the frequency of the magnetic fieldand at the same time double the line frequency.

29


λMS

H=0

H=H1>0

H=-H1

Figure 3.1: Realignment of magnetic domains leading to magnetostriction

In order to account for magnetostriction, Hooke’s law as given in Equation 3.2has to be adapted. This can be accomplished by substracting the magnetostrictionstrain λMS from the strain present in Hooke’s law:

[σ] = [c]([S]− λMS) (3.18)

Back-coupling, i.e. the magnetostriction influencing the magnetic field inside thecore, is not taken into account within this work. This allows to carry out themechanical simulation independently of the electromagnetical part.

3.3 Modelling

In this section the basic modelling steps are presented. After a short overview ofthe geometry and boundary conditions, the transition from the first to the secondelement in the coupling chain, i.e., the transition from the electromagnetical to themechanical simulation is explained. Afterwards the way how to circumvent themissing magnetostriction interface in COMSOL is described.

3.3.1 Geometry and boundary conditions

Figure 3.2 depicts the basic geometry and boundary conditions for the mechanicalpart of the model. In the mechanical simulation, basically the same mesh is usedas for the electromagnetics (cf. Figure 2.3). However, the surrounding air can beomitted. Additionally, a simplification can be applied by taking the coils away.All those parts can be excluded by just deselecting the according domains in themodel. There is no need to set up a new model for the mechanical simulation. Thereal model core is attached to a flexible rope. In the simulation model this settingresults in free boundary conditions all around the core, which means that there areno constraints and no loads acting on the boundary [4].

As the elasticity matrix depends on the rolling direction, two different piezoelec-tric material models have to be defined, one for the yoke and one for the limbs.

30

3.3. MODELLING

Free

Figure 3.2: Overview of the geometry and boundary conditions for the mechanics

3.3.2 Choice of physics interfaceCOMSOL does not provide a physics interface for modelling magnetostriction.Hence, a way has to be found how to model magnetostriction using the existingmodules. As the basic characteristics and equations of magnetostriction are similarto those of piezoelectricity, using the Piezoelectric Devices interface of the StructuralMechanics Module and adapting it in order to model magnetostiriction appears tobe a reasonable choice.

The relevant equation solved for piezoelectric devices is

[σ]− [σ0] = [c]([S]− [S0])− [e]T E. (3.19)

The subindices 0 signify initial values of stress or strain, respectively. [e] is thepiezoelectric coupling tensor and E the electric field. It is evident how to changeEquation 3.19 in order to make it fit Equation 3.18. By setting the piezoelectriccoupling tensor [e] and the initial stress tensor [σ0] to zero and replacing the initialstrain [S0] by the magnetostriction strain λMS, the piezoelectric equation transformsdirectly into the required equation for magnetostriction. Only setting [e] to zerowould result in a singular mass matrix, causing the solver to abort. Thus, it isnecessary to manually exclude the variable for the electric potential from the studyby disabling it in the Dependent Variable node.

The relation between the magnetic flux density and the magnetostriction strainis different between rolling and cross direction. The magnetostriction values ofthe material are given as measurements for the basic frequency and the first nineharmonics. Summing up those relations, the magnetostriction can be described as

λMS,rd2·ffund,...,20·ffund

= f(Brdffund) (3.20)

λMS,crd2·ffund,...,20·ffund

= f(Bcrdffund) (3.21)

where ffund denotes the fundamental frequency of the magnetic flux in the core.In the simplest case, i.e., for a base frequency of the magnetic flux of 50 Hz, themagnetostriction strain values are given for the frequencies 100, 200, ... 1000 Hz.

31


Assuming that Bffund is already available in COMSOL (the procedure to achievethis is explained in subsection 3.3.3), the only step left is the integration of themagnetostriction to COMSOL. The measured magnetostriction strain is availabefor the basic frequency (100 Hz) and the first nine harmonics and differs betweencross and rolling direction. For each of those frequencies, data for the amplitude(given in micro-scale) as well as the phase angle is available. Note that ampl(λMS)and phase(λMS) for all those frequencies only depend on the amplitude of the 50 Hzcomponent of the magnetic flux density. In order to be able to use the measuredvalues of λMS,rd and λMS,crd in COMSOL in the most convenient way they eachhave to be available in a text file with the following format:

% Bffund no. harmonic lamba ampl lambda phase0 1 ampl(λMS

2·ffund(0)) phase(λMS

2·ffund(0))

......

......

Bffund,max 1 ampl(λMS2·ffund

(Bffund,max)) phase(λMS2·ffund

(Bffund,max))0 2 ampl(λMS

4·ffund(0)) phase(λMS

4·ffund(0))

......

......

Bffund,max 10 ampl(λMS20·ffund

(Bffund,max)) phase(λMS20·ffund

(Bffund,max))

An interpolation function λMS,ampl(Bffund , no.harm) as well as an interpolation func-tion λMS,phase(Bffund , no.harm), defining the magnetostriction strain in dependanceon the fundamental component of B and the number of the harmonic (where 1denotes the fundamental frequency) for each rolling and cross direction has to bedefined. As underlying data, the text files containing the magnetostriction valuesare chosen. Now, given the interpolation functions for λMS and Bffund , the initialstrain at each node (x, y) can be defined as (e.g., for the yoke and only consideringthe fundamental frequency component)

[S0] =

λMS,rd(Bx,ffund(x, y), 1) 0 00 λMS,crd(By,ffund(x, y), 1) 00 0 0

(3.22)

Figure 3.3 and Figure 3.4 show the measured λ-B-curves of the steel in rollingand cross direction, respectively, including all given frequencies.

Especially in cross direction, the dominance of the fundamental frequency withrespect to the overall strain can be seen. In rolling direction, this characteristic isalso observable but not as clear as in cross direction, though.

3.3.3 Obtaining the magnetic flux density

As described before, the analysis for the electromagnetics has been carried out intime domain for several reasons. In contrast to this, the mechanical part of themodel is analyzed in frequency domain. The material’s magnetostriction values are

32

3.3. MODELLING

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Bffund

(T)

λM

S,r

d *

10

6Magnetostriction strains in rolling direction

1

2

3

4

5

6

7

8

9

10

Figure 3.3: Magnetostriction curves in rolling direction

given as measurements, depending on the magnetic flux density’s main frequency(i.e., normally 50 Hz) component, denoted by Bffund . Hence, the basic procedure togo from the time-dependent electromagnetical to the frequency-dependent mechan-ical simulation has to be as follows:

1. Compute time-dependent solution for the electromagnetics

2. Compute the Fourier transform of this solution

3. Derive the main frequency component of the magnetic flux density at eachnode

4. Use this as an input for the mechanical simulation in the frequency domain

However, when it comes to computing a Fourier transform the capabilities of COM-SOL are limited. The only possibility is to use the FFT for post-processing, i.e., tocompute the FFT of a time-dependent solution and display it in the Results view.There is no way to compute the FFT of a time-dependent signal and use this as aninput value of another, though.

Thus, it is necessary to employ the MATLAB interface of COMSOL. There arebasically two ways how to incorporate MATLAB to a COMSOL model: to call anexternal MATLAB function from whithin the COMSOL desktop or to execute thewhole model as a MATLAB script and change the source code. Both of these ways

33


0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Bffund

(T)

λM

S,c

rd *

10

6

Magnetostriction strains in cross direction

1

2

3

4

5

6

7

8

9

10

Figure 3.4: Magnetostriction curves in cross direction

turned out to be not applicable. The main reason is that the time-dependent signalper node is not being saved by COMSOL as a whole but in smaller chunks.

The desired behaviour can only be achieved by applying a workaround incorpo-rating MATLAB. In this case there is no way to run the whole simulation in onego, though. The main idea looks as follows:

1. Export the model as a MATLAB script

2. Run the electromagnetical part of the model script in MATLAB

3. Evaluate the variables Bx and By for the magnetic flux densities in x- andy-direction using the COMSOL command mpheval (cf. [3]). The accordingcode can be seen in Listing 3.1).addpath(’/home/mo/programme/COMSOL42/mli/’); % add COMSOL pathmphstart; % connect to COMSOL

servermodel=core_2d_only_electromagnetics; % execute model

script

pd = mpheval(model,’mf.Bx’,’solnum’,1); % evaulate first time step[pd_pu,~,~] = unique(pd.p’,’rows’); % locations of unique nodesnopoints = length(pd_pu); % number of unique nodesN = model.sol(’sol1’).getDefaultSolnum; % number of time steps

34

3.3. MODELLING

bxt=zeros(nopoints,N); % bxt holds the time-dependent solution of mf.Bx at each node

byt=zeros(nopoints,N); % analog

for k=1:N; % evaluate per time step to avoid out of memoryerrorspd_bx=mpheval(model,’mf.Bx’,’solnum’,k); % evaulate Bxpd_by=mpheval(model,’mf.By’,’solnum’,k); % evaulate By[~,i,~] = unique(pd_bx.p’,’rows’); % find indices of unique

nodesbxt(:,k) = pd_bx.d1(i); % save Bx for this time stepbyt(:,k) = pd_by.d1(i); % save By for this time step

end

bx50 = zeros(nopoints,4); % this matrix holds the values forBx50

by50 = zeros(nopoints,4); % By50 analog

bx50(:,1:2) = pd_pu; % first two columns holds the (unique)nodes

by50(:,1:2) = pd_pu;

for i=1:size(bx50,1) % iterate over all nodes[bx50(i,3) bx50(i,4)] = fundcomp(bxt(i,:),1); % compute

amplitude and phase of fundamental component of Bx at nodeand save to Bx50

[by50(i,3) by50(i,4)] = fundcomp(byt(i,:),1); % analog in y-direction

end

dlmwrite(’bx50.dat’,bx50,’ ’); % write text file for Bx50dlmwrite(’by50.dat’,by50,’ ’); % write text file for By50

Listing 3.1: MATLAB script to compute magnetostriction

To prevent out of memory errors when invoking mpheval, the function hasto be called separately for each timestep. As mpheval internally iteratesover the elements and not over the nodes, most of the nodes appear severaltimes in the solution it returns. In order to fix this, i.e., in order to take eachnode only once, the indices of the unique nodes have to be found with theMATLAB command unique and only those nodes have to been taken intoaccount. The function fundcomp will be described in the next item.

4. Compute the component of the fundamental frequency Bx,ffund and By,ffund ofthe time-dependent signals Bx(t)/By(t) by means of FFT (cf. Listing 3.2).

function [ampl, phase] = fundcomp(b,n)% fundcomp(b,n) Finds the amplitudes of the fundamental component

and% the first n-1 superharmonics

N = length(b); % get the number of data points

35


X = fft(b)*2/N; % normalize the datap = (angle(X)); % phase angleX = X(1:ceil(N/2)); % take only the first half of the spectrum

[~, locs] = findpeaks(abs(X)) % find peaksampl = abs(X(locs(1:n))); % take the amplitudes first n peaksphase = p(locs(1:n))*180/pi; % compute phase information

end

Listing 3.2: MATLAB function to get the main frequency component

5. For each node save the node position and Bx,ffund to a three-column text file,the node position and By,ffund to a second one.

6. In COMSOL, define two interpolation functions, one for Bx,ffund and one forBy,ffund . Load the accordant files saved by MATLAB for both functions.

Then, the magnetostriction strain per node can be computed by looking up thevalue of the accordant interpolation function Bx,ffund or By,ffund at the node. Incase the mesh is the same for both electromagnetics and mechanics, calling it aninterpolation function is a bit irritating because Bx,ffund/By,ffund is just looked upat the exact node positions; there is actually no interpolation but just a look-up ofthe values.

3.4 ResultsIn this section, the results of the mechanical simulation are presented. Based onthe magnetic flux density in the core, the displacement field resulting from magne-tostriction is computed. In the first subsection, an analysis of the core’s eigenmodesand eigenfrequencies will be done. Afterwards, a frequency sweep for the core’sharmonic displacement field will be analyzed, also with regard to the previouslycomputed eigenmodes. As frequency range, 500 Hz ≤ f ≤ 2500 Hz is chosen.

3.4.1 Eigenmode analysisIn order to obtain a better understanding of the core’s properties and to be ableto interpret the displacement results more easily, an eigenmode analysis can becarried out. This can easily be accomplished in COMSOL, e.g., by using the SolidMechanics interface. Apart from the rigid body modes, there are seven eigenmodeswith according eigenfrequencies which are smaller than the upper bound of theconsidered frequency range, 2500 Hz.

36

3.4. RESULTS

(a) Eigenmode at 763 Hz (b) Eigenmode at 1150 Hz

(c) Eigenmode at 1244 Hz (d) Eigenmode at 1312 Hz

(e) Eigenmode at 1963 Hz (f) Eigenmode at 2312 Hz

(g) Eigenmode at 2372 Hz

Figure 3.5: The first seven eigenmodes

37


3.4.2 Displacement resultsFor analyzing the core’s displacement, a frequency sweep is performed. However, asa simplification the result of the electromagnetic analysis, serving as the input forthe mechanical analysis, will stay the same. The frequency is only varied for themechanical analysis. Figure 3.6 shows the integration of the displacement field overthe whole core over the frequency range. In this simulation, only the fundamentalfrequency is taken into account for the magnetostriction strain. It can be observed

500 1,000 1,500 2,000 2,50010−9

10−8

10−7

10−6

10−5

10−4

f (Hz)

∫ core

√ real

(u)2

+re

al(v

)2(m

3 )

Figure 3.6: Overall core displacement over the frequency range

that the peaks of total displacement over the frequency range meet the previouslycomputed eigenfrequencies. The highest peak in total displacement can be seen at2312 Hz, which corresponds to a frequency of the magnetic field of 1158 Hz. Thedisplacement of the core at this specific frequency can be seen in Figure 3.7.

38

3.4. RESULTS

Figure 3.7: Displacement of the core at 2312 Hz

39

Chapter 4

Acoustics

In this chapter, the focus is on the third and last element of the coupling chain,namely the acoustics. The basis for the acoustic simulation is given by the displace-ment field computed in chapter 3. As the main goal of this work is to compute thesound radiation of power transformer cores, the results of this chapter form thisthesis’ final outcome.

4.1 Governing equationsThe acoustic study is in general governed by the following inhomogeneous Helmholtzequation ([2]):

∇ · −1ρ

(∇pt − q)−k2

eqρ

= Q (4.1)

which is solved for the pressure. pt = p + pb denotes the total acoustic pressurewhich is constituted by the acoustic pressure p and a background pressure field pbwhich is zero in this simulation. k2

eq =(

ωc

)2 is the wave number, ρ denotes thedensity and c the speed of sound. q accounts for a possible dipole source whereasQ can model a monopole source. Here, both of them are zero. Hence, the resultingequation which is solved is

∇ · −1ρ

(∇p)−k2

eqρ

= 0 (4.2)

4.2 ModellingThe acoustic analysis should be carried out in three spatial dimensions. Thus, anown, distinct model has to be created for this modelling step. The model repre-sents the inner package of the core. This model together with its boundary con-ditions is described in subsection 4.2.1. For the transition from the preceding,two-dimensional mechanical analysis to the three-dimensional acoustical analysis,

41

CHAPTER 4. ACOUSTICS

an extrusion coupling variable is used. The principle of extrusion couplings in COM-SOL is explained in subsection 4.2.3. The three-dimensional model can be seen inFigure 4.1.

Figure 4.1: The three-dimensional model of the core

4.2.1 Geometry and boundary conditionsThe geometry and boundary conditions of the acoustical model can be seen in Fig-ure 4.2. For the acoustical simulation, only the air surrounding the core is taken intoaccount. The excitation is modelled by Normal acceleration boundary conditionson the boundaries where the core matches the air. This boundary condition uses agiven normal acceleration an and imposes the following condition:

− n ·(− 1ρ

(∇pt − q))

= an (4.3)

Taking into account the simplifications described in section 4.1, the boundary con-dition takes the following form:

− n ·(− 1ρ

(∇p))

= an (4.4)

The values for the normal acceleration an can be deduced from the preceding me-chanical simulation. The displacement field computed by it is denoted by (u,v),

42

4.2. MODELLING

Air

AirAir

PML

Normal acceleration

Figure 4.2: Overview of the geometry and boundary conditions for the acoustics

where u denotes the displacement in x-direction and v stands for the displacementin y-direction. For example, on boundaries which are oriented in x-direction, an

can be expressed as an = ±vtt. A negative value for an accounts for an outwardacceleration, whereas a positive an models an inward acceleration. Hence, the signof an depends on whether the surface normal is oriented in positive or negativex-direction. In the first case, the sign has to be negative, as a positive displacementvalue on such a surface creates an outward acceleration. In the latter case it isanalog. This can easily be modelled in COMSOL by using the built-in variablesnx and ny, which represent the surface normal in vectors in x- or y-direction, re-spectively. As the analysis is carried out in frequency domain, time-derivatives canbe exchanged by multiplications with jω, resulting in an = ∓ω2v for boundaries inx-direction and an = ∓ω2u for those in y-direction. The core is surrounded by twospheres. The inner sphere represents the control volume, on whose surface the soundpower is measured. The shell created by the outer sphere implements a perfectlymatched layer (PML) of spherical type. A PML is not a boundary condition butrather an auxiliary domain absorbing the incident wave without producing any re-flections. This is accomplished by the introduction of a complex-valued coordinatetransformation with the additional requirement of not affecting the wave impedance[2]. In this model, a PML of spherical type is used. As the core is placed exactly inthe center of the spherical domain, the center coordinate of the PML can be set tothe balance point of the core.

According to [2], an appropriate value for the thickness of the PML is onewavelength of the smallest frequency. The wavelength λ can be computed as

λ = cs/f (4.5)

with cs the speed of sound and f the minimum frequency.

43

CHAPTER 4. ACOUSTICS

4.2.2 Choice of physics interface

For modelling the acoustics, the Pressure Acoustics Interface of COMSOL’s Acous-tics Module is used. It has the equations, boundary conditions, and sources formodelling acoustics, solving for the sound pressure [2]. It provides the possibility tomodel perfectly matched layers which is necessary for the model. Furthermore, itenables the definition of a Normal Acceleration boundary condition, which makes iteasy and convenient to couple the displacement results of the mechanical simulationto the acoustical model.

4.2.3 Extrusion of the displacement

As the mechanical model has been simulated in two dimensions, its relevant result,i.e., the displacement at the core’s boundaries, has to be extruded in order toallow it to couple with the three-dimensional acoustic model. For this purpose,

Figure 4.3: Extrusion from two to three spatial dimensions, e.g. for one singleboundary segment

a linear extrusion coupling operator, coupling the computed displacement on thecore’s boundary from a line in two dimension to a surface in three dimensions, canbe utilized. The principle of extrusion, e.g. for the outer boundary of the loweryoke, can be seen in Figure 4.3. By this, the displacement value of a single pointon the boundary in 2D is extruded in z-direction.

4.3 ResultsThe acoustical results form the final outcome of the model and the target whichhas to be optimized.

4.3.1 Measure

The measure which is used for the sound radiation is the sound power of the trans-former core. It is computed by the integral of the normal sound intensity along acontrol surface. In this model, the control surface is formed by the inner sphere.As the computed acoustical intensity is only available in cartesian coordinates, it

44

4.3. RESULTS

cannot be used directly for the calculation of the intensity. Its normal componentcan be obtained by computing the scalar product of the intensity vector with thesurface normal. The surface normal nr on a spherical surface can be computed by

nr = x− r0|x− r0|

(4.6)

where r0 denotes the center of the sphere which is the balance point of the core in

this model. x =

xyz

is the location of the point where the normal vector should

be computed. The normal component Ir of the acoustical intensity I =

Ix

Iy

Iz

canbe computed as

Ir = nr · I (4.7)

In order to obtain the sound power of the core, Ir has to be integrated over a surface(in this case it is spherical) surrounding the core.

4.3.2 Sound power resultsFor the computation of the sound power, a frequency sweep similar to the mechanicalanalysis is performed. The speed of sound in air is cs = 340 m/s and the minimumfrequency is f = 500 Hz. Hence, λ and thus the thickness of the PML yieldsλ = 340/500 m = 0.68 m.

As stated above, the acoustic analysis is carried out in three spatial dimensions.Due to the fact that the hardware used for modelling within this work (Intel Core2 Duo with 3 GB of memory) is too weak to perform the incidental calculations,it is not possible to present sound power results at this point. However, the modelis capable of computing the core’s sound power if the used hardware is powerfulenough.

45

Chapter 5

Conclusions and Outlook

The main goal of this work was to create a complete model for core noise frompower transformers in COMSOL Multiphyiscs. It had to be found out whetherit is possible to replace the current in-use toolchain of ABB, which incorporatesseveral tools, by only one single tool, namely COMSOL. Different simplifications,for example the disregard of hysteresis, the absence of bolts or joints in the coreand purely two-dimensional results for the electromagnetic and mechanical analysishave been taken.

It is possible to create a complete model, including all three coupling steps (elec-tromagnetics, mechanics, acoustics), in COMSOL. Either there are given interfaceswhich provide the capabilities to model the physics, which is the case for the electro-magnetics and acoustics, or it is possible to adjust a given interface in case there isno predefined interface, as it is the case for magnetostriction where the PiezoelectricDevices module has been utilized.

A special property of this model is that it follows a mixed approach in terms ofspatial dimensionality (electromagnetics and mechanics in 2D, acoustics in 3D) aswell as in terms of analysis approaches (electromagnetics in time-domain, mechanicsand acoustics in frequency-domain). The change in dimensionality can easily bemodelled in COMSOL. However, the change from a time- to a frequency-dependentanalysis requires a work-around which entails the utilization of MATLAB becauseFFT results in COMSOL are available only for post-processing purposes and notfor further computation. Due to this fact, the model cannot be analyzed entirely inCOMSOL, but MATLAB has to be employed when going from the electromagneticsto the mechanics.

In future work, several enhancements of the model can be made. The mostevident improvement is the abandonment of the several simplifications which weretaken. For example, by implementing a hysteresis model in COMSOL the results forthe electromagnetic simulation could become more accurate and closer to reality.Furthermore, the core could be modelled in more detail, e.g., taking the jointsand bolts into account. Another enhancement would be to create a fully three-dimensional model of the core for all analysis steps, not just the acoustics. In the

47

CHAPTER 5. CONCLUSIONS AND OUTLOOK

end one could even consider taking the sheet structure into account in order toobtain even better results.

This thesis considers a simple model core consisting only of one single package.In order to obtain pratice-oriented results, it is desired to create a model of thecomplete core, taking into account all packages. Finally, one also would have toadd the fluid which is surrounding the tank and finally the tank itself. A workwhere this has been taken into account is [7].

As stated above, numerical damping forms a big issue in the time-steppingscheme which is used for the electromagnetical analysis. It may be desirable toanalyze, understand and finally be able to control damping better. A promisingapproach to this is the use of correct initial data.

A big improvement to the model would be if it were possible to run the entiresimulation inside COMSOL. However, accomplishing this depends on the develop-ment of COMSOL, i.e. whether it will be possible in a future version to computethe FFT of a time-dependent signal at every node and use this for successive studysteps.

48

Bibliography

[1] COMSOL. AC/DC Module User’s Guide 4.2.

[2] COMSOL. Acoustics Module User’s Guide 4.2.

[3] COMSOL. LiveLink™for MATLAB® User’s Guide 4.2.

[4] COMSOL. Structural Mechanics Module User’s Guide 4.2.

[5] Tom Hilgert, Lieven Vandevelde, and Jan Melkebeek. Comparison of Magne-tostriction Models for Use in Calculations of Vibrations of Magnetic Cores. IEEETransactions on Magnetics, 44, June 2008.

[6] Manfred Kaltenbacher. Numerical Simulation of Mechatronic Sensors and Ac-tuators. Springer, 2nd edition, 2007.

[7] Mustafa Kavasoglu. Load Controlled Noise of Power Transformers: 3D Mod-elling of Interior and Exterior Sound Pressure Field. Master’s thesis, KTHComputer Science and Communication, 2010.

[8] Nippon Steel Corporation, http://www.nsc.co.jp/en/product/sheet/pdf/DE304.pdf.Electrical Steel Sheets.

49

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ISSN-1653-5715

www.kth.se

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Modelling of Core Noise from Power Transformers - KTH · PDF fileModelling of Core Noise from Power Transformers ... ABB’s in-use toolchain for simulating transformer core noise