the finite element method in magnetics

THE FINITE ELEMENT METHOD IN MAGNETICS

´ KUCZMANN–AMALIA ´ ´ MIKLOS IVANYI

THE FINITE ELEMENT METHOD IN MAGNETICS

´ BUDAPEST ´ AKADEMIAI KIADO,

This book was sponsored by the Széchenyi István University in Gy˝or by the Pollack Mihály Faculty of Engineering, University of Pécs and by the ’Tanoda’ Foundation Pécs

Scientific Review Ao.Univ.-Prof. Dipl.-Ing. Dr.techn. Oszkár B´ıró, DSc Institut für Grundlagen und Theorie der Elektrotechnik Graz University of Technology Dr. Péter Kis, PhD Research Engineer, Furukawa Electric Institute of Technology Assistant Professor, Department of Automation and Applied Informatics Budapest University of Technology and Economics

ISBN 978 963 05 8649 8

Published by Akadémiai Kiadó Member of Wolters Kluwer Group P.O. Box 245, H-1519 Budapest, Hungary www.akademiaikiado.hu

c M. Kuczmann, A. Iványi, 2008

All rights reserved. No part of this book may be reproduced by any means or transmitted or translated into machine language without the written permission of the publisher.

Printed in Hungary

Contents 1

Introduction

2 Potential formulations in electromagnetic field 2.1 Maxwell’s equations and interface conditions . . . . . . . . . . . . . . . 2.1.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . A. The differential form of Maxwell’s equations . . . . . . . . . B. The integral form of Maxwell’s equations . . . . . . . . . . . C. The constitutive relations . . . . . . . . . . . . . . . . . . . . D. Energy of electromagnetic fields . . . . . . . . . . . . . . . . 2.1.2 Interface and boundary conditions . . . . . . . . . . . . . . . . . A. Electric and magnetic field intensity . . . . . . . . . . . . . . B. Electric and magnetic flux density, current density . . . . . . . 2.1.3 Classification of Maxwell’s equations . . . . . . . . . . . . . . . 2.2 Magnetostatics and eddy current field problems . . . . . . . . . . . . . . 2.2.1 Static magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Eddy current fields . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Potential formulations in static magnetic and eddy current field problems . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Static magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . A. The reduced magnetic scalar potential, the Φ-formulation . . . B. The total magnetic scalar potential, the Ψ-formulation . . . . . C. The combination of the magnetic scalar potentials, the Φ − Ψ-formulation . . . . . . . . . . . . . . . . . . . . . . D. Applying the reduced magnetic scalar potential with appropriate representation of T 0 . . . . . . . . . . . . . . . . . . . E. The magnetic vector potential, the A-formulation . . . . . . . F. Combination of the magnetic vector potential and the magnetic scalar potential, the A − Φ-formulation . . . . . . . . 2.3.2 Eddy current fields . . . . . . . . . . . . . . . . . . . . . . . . . A. The current vector potential and magnetic scalar potential, the T , Φ-formulation . . . . . . . . . . . . . . . . . . . .

1 3 4 4 4 6 9 10 11 12 13 14 16 17 18 21 21 22 30 31 33 34 40 43 43

vi

CONTENTS

2.3.3

B. The magnetic vector potential and electric scalar potential, the A, V -formulation . . . . . . . . . . . . . . . . . . . . C. The modified magnetic vector potential, the A⋆ -formulation . Coupling static magnetic and eddy current fields . . . . . . . . . A. The gauged T , Φ − Φ formulation . . . . . . . . . . . . . . . B. The ungauged T , Φ − Φ formulation . . . . . . . . . . . . . . C. The gauged A, V − A formulation . . . . . . . . . . . . . . . D. The ungauged A, V − A formulation . . . . . . . . . . . . . E. The ungauged A⋆ − A formulation . . . . . . . . . . . . . . . F. The gauged T , Φ − A formulation . . . . . . . . . . . . . . . G. The ungauged T , Φ − A formulation . . . . . . . . . . . . . . H. The gauged T , Φ − A − Φ formulation . . . . . . . . . . . . I. The ungauged T , Φ − A − Φ formulation . . . . . . . . . . . . J. The gauged A, V − Φ formulation . . . . . . . . . . . . . . . K. The ungauged A, V − Φ formulation . . . . . . . . . . . . . . L. The gauged A, V − A − Φ formulation . . . . . . . . . . . . M. The ungauged A, V − A − Φ formulation . . . . . . . . . . .

3 Weak formulation of nonlinear static and eddy current field problems 3.1 The weighted residual and Galerkin’s method . . . . . . . . . . . . . . . 3.1.1 Differential equations in electromagnetic field computation . . . . 3.1.2 The weighted residual method . . . . . . . . . . . . . . . . . . . A. Weighted residual of elliptic differential equations with scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . B. Weighted residual of elliptic differential equations with vector potential . . . . . . . . . . . . . . . . . . . . . . . . . C. Weighted residual of parabolic differential equations . . . . . . 3.2 Approximation of unknown functions and weighting functions . . . . . . 3.3 The weak formulation with Galerkin’s method . . . . . . . . . . . . . . . 3.3.1 The reduced magnetic scalar potential, the Φ-formulation . . . . . 3.3.2 Combination of magnetic scalar potentials, Φ − Ψ-formulation . . 3.3.3 The magnetic vector potential, the A-formulation with implicit enforcement of Coulomb gauge . . . . . . . . . . . . . . . . . . 3.3.4 The magnetic vector potential, the A-formulation with a numerical technique, which is not sensitive to Coulomb gauge . . 3.3.5 Combination of the magnetic vector potential and the magnetic scalar potential, the A − Φ-formulation . . . . . . . . . . . . . . 3.3.6 The gauged T , Φ − Φ formulation . . . . . . . . . . . . . . . . . 3.3.7 The ungauged T , Φ − Φ formulation . . . . . . . . . . . . . . . 3.3.8 The gauged A, V − A formulation . . . . . . . . . . . . . . . . 3.3.9 The ungauged A, V − A formulation . . . . . . . . . . . . . . . 3.3.10 The ungauged A⋆ − A formulation . . . . . . . . . . . . . . . .

47 49 50 51 52 53 54 55 55 57 58 60 61 62 63 64 66 66 67 69 69 70 73 73 75 76 77 79 80 82 86 90 92 96 99

vii

CONTENTS

3.3.11 3.3.12 3.3.13 3.3.14 3.3.15 3.3.16 3.3.17 3.3.18

The gauged T , Φ − A formulation . . . . . The ungauged T , Φ − A formulation . . . The gauged T , Φ − A − Φ formulation . . The ungauged T , Φ − A − Φ formulation . The gauged A, V − Φ formulation . . . . . The ungauged A, V − Φ formulation . . . The gauged A, V − A − Φ formulation . . The ungauged A, V − A − Φ formulation

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100 101 102 110 114 115 115 120

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125 125 130 131 146

5 Modeling of ferromagnetic hysteresis 5.1 Ferromagnetic materials . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Short description of magnetic materials . . . . . . . . . . . . . . 5.1.2 Ferromagnetic hysteresis . . . . . . . . . . . . . . . . . . . . . . 5.1.3 The hysteresis operator . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Preisach model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The scalar model . . . . . . . . . . . . . . . . . . . . . . . . . . A. Description of the model . . . . . . . . . . . . . . . . . . . . B. The distribution function . . . . . . . . . . . . . . . . . . . . C. The Everett function . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Pseudocode for the scalar Preisach model . . . . . . . . . . . . . 5.2.3 The vector model . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 The classical scalar Preisach model via neural networks . . . . . A. A brief introduction to neural networks . . . . . . . . . . . . . B. Neural Preisach models . . . . . . . . . . . . . . . . . . . . . 5.3 The neural network based hysteresis model . . . . . . . . . . . . . . . . 5.3.1 The scalar model . . . . . . . . . . . . . . . . . . . . . . . . . . A. Training sequence and preprocessing of measured data . . . . B. Operation of the model . . . . . . . . . . . . . . . . . . . . . 5.3.2 The isotropic vector hysteresis operator . . . . . . . . . . . . . . A. Description of the model . . . . . . . . . . . . . . . . . . . . B. The identification process . . . . . . . . . . . . . . . . . . . . C. Verification of the model . . . . . . . . . . . . . . . . . . . . 5.3.3 The anisotropic vector hysteresis operator . . . . . . . . . . . . . A. Description of the model . . . . . . . . . . . . . . . . . . . . B. Fourier expansion of the measured 2D vector Everett function . C. Fourier expansion of the measured 3D vector Everett function .

161 162 162 165 168 168 168 169 174 180 182 183 185 185 188 190 190 190 193 198 198 200 205 206 207 208 211

4 The finite element method 4.1 Fundamentals of FEM . . . . . . . . . . . . . 4.2 Approximating potentials with shape functions 4.2.1 Nodal finite elements . . . . . . . . . . 4.2.2 Edge finite elements . . . . . . . . . .

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viii

CONTENTS

5.4

D. The relation between the scalar and the vector Everett functions E. Verification of the model . . . . . . . . . . . . . . . . . . . . 5.3.4 Some properties of the vector hysteresis characteristics . . . . . . Measuring magnetic hysteresis . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 System for scalar hysteresis measurement . . . . . . . . . . . . . A. The measurement set-up using LabVIEW . . . . . . . . . . . B. Simulation of measured static curves by the scalar NN model . 5.4.2 Vector hysteresis measurement system . . . . . . . . . . . . . . . A. The measurement set-up . . . . . . . . . . . . . . . . . . . . B. Measuring results . . . . . . . . . . . . . . . . . . . . . . . .

214 217 218 223 223 223 230 231 233 235

6 The polarization method and the fixed point technique 6.1 Solution of nonlinear equations . . . . . . . . . . . . . . . . . . . . 6.2 Nonlinearity in electromagnetic field simulation . . . . . . . . . . . 6.3 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Definition of metric spaces . . . . . . . . . . . . . . . . . . 6.3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Banach fixed point theorem . . . . . . . . . . . . . . . . . 6.4 The optimal value of parameters µ and ν . . . . . . . . . . . . . . . 6.4.1 Using the inverse characteristics . . . . . . . . . . . . . . . 6.4.2 Using the direct characteristics . . . . . . . . . . . . . . . . 6.5 The applied formulation, summary . . . . . . . . . . . . . . . . . . 6.5.1 Using the inverse characteristics . . . . . . . . . . . . . . . 6.5.2 Using the direct characteristics . . . . . . . . . . . . . . . . 6.5.3 Proof of the nonexpansive property of Maxwell’s equations .

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238 238 243 245 245 246 247 248 248 251 251 251 252 253

7 Application of the finite element method 7.1 Introduction . . . . . . . . . . . . . . . . . . . . 7.2 Illustration how to select T 0 , the C-magnet . . . 7.3 Iron cube in homogeneous magnetic field . . . . 7.4 Conducting cube in homogeneous magnetic field 7.5 Steel plates around a coil, linear problem . . . . . 7.6 Steel plates around a coil, nonlinear problem . . . 7.7 Asymmetrical conductor with a hole . . . . . . . 7.8 Rotational single sheet tester . . . . . . . . . . .

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255 255 257 262 267 272 275 277 281

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8 References

292

9 Index

305

10 About the authors

309

11 About the book

310

1 Introduction The aim of this book is to present a survey of the simulation of ferromagnetic hysteresis and its importance in electromagnetic field computation. We try to focus on the problems from an electrical engineering point of view. The book is divided into six chapters. In electrical engineering practice, the simulation of devices, measuring arrangements and various electrical equipments is based on Maxwell’s equations coupled with the constitutive relations. It is very important to take into account the hysteretic behavior as well as the vector property of the magnetic field quantities. However, in some cases it is adequate to use constant permeability or single valued nonlinearity. Hysteresis characteristics must be taken into consideration when, for example, hysteresis losses in electrical machines or effects related to the remanent magnetization are to be calculated. The second chapter is a summary of Maxwell’s equations, their form in the case of nonlinear static magnetic field problems and of nonlinear eddy current field problems. Here, we give the correct description of all the possible potential formulations used in solving static magnetic field problems and eddy current field problems. Chapter 3 presents the weighted residual method, which can be used to solve the nonlinear partial differential equations obtained from Maxwell’s equations by applying potentials. Here the weighted residual method is applied to the weak formulation. The finite element method is a possible technique to solve numerically the partial differential equations formulated by using the weak form. It is noted here that the finite element method is one of the most widely used methods to solve electromagnetic field problems. In chapter 4, we summarize the finite element method from scratch, the nodal finite elements as well as the newer vector elements. This chapter is very useful for students studying applied electromagnetics in faculties of electrical engineering, too. Chapter five contains a summary of modeling and measuring ferromagnetic hysteresis and this work is based on our experiments. A short summary of the fundamental relations in magnetism and magnetic materials is also introduced. The widely used Preisach model is presented as well as neural network based scalar and vector hysteresis models developed by the authors. The method of identification of hysteresis models is also given here. Applying neural networks in the field of function approximation has the advantage of easy identification and the resulting model can approximate the measured object attractively

2

1. INTRODUCTION

with continuous output. At the same time, neural networks are black box models and there is no connection between the physical meaning of the phenomenon to be simulated and the parameters of the model. The main challenge of simulations is to predict the behavior of an arrangement. The basis of simulations is to work out a mathematical model of the given arrangement and a mathematical model of the material behavior. These models can be applied to calculate the physical quantities, which we are interested in. The sixth chapter presents the polarization method as well as the fixed point technique to solve nonlinear electromagnetic field problems. The magnetic field intensity or the magnetic flux density is split into a linear term and a nonlinear term, as defined by the polarization method. Nonlinear equations can be formulated as a fixed point equation, which is solved iteratively by using the fixed point technique. This method results in a convergent numerical tool to solve nonlinear equations, or a system of nonlinear equations, however, the parameter of the linear term must be selected in a special way. The last chapter is the collection of seven problems solved by the finite element method. The problems have been solved by using the user friendly graphical user interface and functions of COMSOL Multiphysics, which is a commercial finite element software. The authors want to express their grateful acknowledgement to Prof. Oszkár B´ıró, Prof. Imre Sebestyén, Prof. Maurizio Repetto, Prof. Carlo Ragusa for their assistance during developing the nonlinear finite element procedures, to dr. Péter Kis and dr. János Füzi for their help in developing the scalar hysteresis measurements. The authors would like to express their thanks to Prof. György Fodor, Prof. Oszkár B´ıró and Prof. Imre Sebestyén for the reading of the parts of the research and for the helpful and fruitful suggestions. The authors thank to Prof. Oszkár B´ıró, dr. Péter Kis, dr. István Standeisky for reading of this book and making reviews. Their advices helped us to prepare this book better. We would like to express our special thanks to Prof. Oszkár B´ıró for giving advices, many helps during studying the finite element method as well as the colleagues of the Group of Electromagnetic Theory, Department of Broadband Infocommunications and Electromagnetic Theory, Budapest University of Technology and Economics. The first author is grateful to dr. Péter Kis for learning together and for the useful discussions. The first author thanks the scholarship provided by the Tateyama Laboratory Hungary Ltd. during his PhD studies. This book was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences (BO/00064/06). The Laboratory of Electromagnetic Fields, Department of Telecommunications, Széchenyi István University in Gy˝or is supported by the Széchenyi István University (15-3002-51,15-3002-57), by the Department of Telecommunications of Széchenyi István University and by EPCOS Ltd. Thanks go to Prof. Gábor Borbély, the Head of Department of Telecommunications for the support during preparing this book. This book was supported by the Hungarian Scientific Research Fund (OTKA PD 73242 ELE). Last but not least, the first author would like to thank his family, his parents and his wife, L´ıvia Kuczmann-Kicsák for the daily assistance and patience. The authors want to express their appretiate thanks to the Széchenyi István University in Gy˝or, to the Pollack Mihály Faculty of Engineering, University of Pécs, and to ’Tanoda’ Foundation Pécs for sporsoring the publication. Dr. Miklós Kuczmann, PhD Associate Professor

Dr. Habil Amália Iványi, DSc Professor

2 Potential formulations in electromagnetic field The numerical analysis or the computer aided design (CAD) of an arrangement, which require electromagnetic field calculation can be characterized by the electric and magnetic field intensities and flux densities. For determination of these quantities in the electromagnetic field, one method is to find the solution of the partial differential equations of the field quantities under prescribed boundary conditions. The mathematical description of electric and magnetic fields can be formulated by the differential form of Maxwell’s equations, which are the collection of partial differential equations of the electric field intensity E, the magnetic field intensity H, the electric flux density D and the magnetic flux density B. The source of the electromagnetic field can be the electric current density J , the electric charge density ρ, the currents, voltages, etc. Constitutive relations between the above quantities are defined to take into account the macroscopic properties of the medium where the electromagnetic field has been studied. It is very important to note that the numerical analysis of electromagnetic fields results in an approximate solution of the partial differential equations expressed from Maxwell’s equations. The partial differential equations to be solved can be formulated for field quantities or for potentials and the resulting solution fulfilles the prescribed boundary conditions as well. There are several studies on the basic equations of the electromagnetic fields and on their solution based on different potential formulation [8, 16, 51, 77–79, 94, 125, 129, 169, 170, 172]. In the first part of this chapter a short summary of the basic equations, i.e. Maxwell’s equations, boundary and interface conditions can be found based on the above cited widely read books. Here, we only focus on linear and nonlinear static magnetic field problems, moreover on linear and nonlinear eddy current field problems. The set of Maxwell’s equations can be transformed into one or two partial differential equations by using potentials. Boundary and interface conditions can also be represented by potentials. The potential formulations are presented in the second part of this chapter.

4

2. POTENTIAL FORMULATIONS IN ELECTROMAGNETIC FIELD

2.1 Maxwell’s equations and interface conditions 2.1.1 Maxwell’s equations A. The differential form of Maxwell’s equations Electromagnetic field simulation problems can be characterized by the field intensities and the flux densities described by the partial differential equations derived from Maxwell’s equations under prescribed boundary conditions. Maxwell’s equations define relations for the electromagnetic field quantities and the source elements, i.e. the electric and the magnetic field intensities, the electric and the magnetic flux densities, the charge and current distribution. Expletively, the so-called constitutive relations are appended to Maxwell’s equations to describe the properties of the media. The differential form of Maxwell’s equations is as follows: ∇ × H(r, t) = J (r, t) + ∇ × E(r, t) = −

∂D(r, t) , ∂t

∂B(r, t) , ∂t

(2.1) (2.2)

∇ · B(r, t) = 0,

(2.3)

∇ · D(r, t) = ρ(r, t),

(2.4)

B(r, t) = µ0 [H(r, t) + M (r, t)], J (r, t) = σ[E(r, t) + E i (r, t)],

(2.5)

D(r, t) = ε0 E(r, t) + P (r, t). The presented symbols and variables with their name and unit in SI system can be found in Table 2.1 (µ0 = 4 π · 10−7 H/m, ε0 = 8.854 · 10−12 F/m). This is the set of partial differential equations, which can be used in the frame of any numerical field analysis tool, e.g. in the finite element method. The field quantities are depending on space r and on time t, therefore in the following a shorter notation will be used, that is H = H(r, t), J = J(r, t), D = D(r, t), E = E(r, t), B = B(r, t), ρ = ρ(r, t), M = M (r, t), E i = E i (r, t), P = P (r, t). The sources of the electromagnetic field are the electric current density J and the electric charge density ρ. The first Maxwell’s equation (2.1) covers the Ampere’s law in differential form. The first term of the right-hand side is the current density of coils (generated by electrons moving inside coils) and eddy currents inside conducting material. The second term is the so called displacement current in dielectric media, which is generated by a time-varying electric field. This equation represents that currents in conductors or eddy currents flowing in conducting materials and time-varying electric fields generate magnetic fields as it is presented in Fig. 2.1. The second Maxwell’s equation (2.2) is the differential form of Faraday’s law. This states that the time-varying magnetic field induces electric field as it is illustrated in Fig. 2.2.

5

2.1. MAXWELL’S EQUATIONS AND INTERFACE CONDITIONS

Table 2.1. Name and SI units of symbols in Maxwell’s equations Notation

Quantity

SI unit

H(r, t)

magnetic field intensity

E(r, t)

electric field intensity

A m V m

B(r, t)

magnetic flux density

T

D(r, t)

electric flux density

J (r, t)

electric current density

ρ(r, t)

electric charge density

C m2 A m2 C m3 A m V m C m2 H m S m F m

M (r, t)

magnetization

E i (r, t)

impressed electric field

P (r, t)

polarization

µ0

permeability of vacuum

σ

conductivity

ε0

permittivity of vacuum

J+

J

∂D ∂t

H

H

(a) The electric current density J generates magnetic field H

(b) The electric current density J and the displacement current density ∂D/∂t generate magnetic field H

Fig. 2.1. Explanation of the first Maxwell’s equation

In time-varying situation the first and the second Maxwell’s equations are coupled, i.e. the time-varying magnetic field generates electric field and this electric field generates eddy currents inside the conducting materials, which currents modify the source magnetic fields. Equation (2.3) states that the magnetic field is divergence-free, i.e. free magnetic charges do not exist physically and magnetic flux lines close upon themselves. This is the magnetic Gauss’ law. Equation (2.4) is the electric Gauss’ law. This means that the source of electric field is the electric charge and electric flux lines start and close upon the charge.

6


∂B ∂t

E Fig. 2.2. The time variation of the magnetic flux density ∂B/∂t generates electric field intensity E There is an important relationship between the variation of the current density and the charge density, the so-called current continuity equation (also called charge conservation law). This law is coming from the first Maxwell’s equation. Taking the divergence of (2.1), the following equation can be obtained: ∂ ∂D = ∇ · J + ∇ · D. (2.6) ∇ · (∇ × H) = ∇ · J + ∂t ∂t The left-hand side is equal to zero according to the identity ∇ · (∇ × v) ≡ 0,

(2.7)

for any vector v = v(r, t) and the second term on the right-hand side can be rewritten by applying equation (2.4). Finally the current continuity equation has the form ∇·J +

∂ρ = 0. ∂t

(2.8)

This equation means that the variation of currents and charge distribution according to space and time is depending on each other. B. The integral form of Maxwell’s equations There is an other form of Maxwell’s equations, the so-called integral form, Z I ∂D H · dl = J+ · dΓ, ∂t l Γ I Z ∂B E · dl = − · dΓ, l Γ ∂t I B · dΓ = 0, I

Γ

Γ

D · dΓ =

Z

ρ dΩ.

Ω

The constitutive relations have the same form as in (2.5).

(2.9)

(2.10) (2.11) (2.12)

7


The relationship between the integral forms (2.9) and (2.10) and the differential forms (2.1) and (2.2) can be noticed by applying Stokes’ theorem, Z I v · dl = ∇ × v · dΓ, (2.13) l

Γ

where v = v(r, t) is a space vector, which may depend on time, the vectors dl and dΓ are illustrated in Fig. 2.3. In this figure, the relation between the circulation about the loop l and the positive (outer) normal unit vector n can be seen as well. The first Maxwell’s equation (see Fig. 2.3) says that the line integral of the magnetic field intensity vector along any closed loop l is equal to the sum of currents or to the surface integral of current densities flowing across the area Γ bounded by the path l. The displacement current has the same effect as the conducting current. This is the classical form of Ampere’s law. In other words it can be stated that if a current is flowing inside a conductor, then a magnetic field is generated in the vicinity of this conductor. i2

l

i1

Γ

n dΓ dl H·dl H

Fig. 2.3. The Ampere’s law

The surface integral of source current density J 0 = σE i is equal to the current i flowing into the surface Γ, i.e. Z J 0 · dΓ = i, (2.14) Γ

where i = i(t) or i = I. The second Maxwell’s equation says that the line integral of the electric field intensity vector along any closed loop l is equal to the surface integral of the time variation of magnetic flux density vector across the area Γ bounded by the path l. This is the classical form of Faraday’s law. In other words, if the flux Φ = Φ(t) is varying with time, then a corresponding induced voltage ui = ui (t) is generated between the two poles of a coil placed inside this time varying flux (see Fig. 2.4), ui (t) = −

dΦ(t) , dt

(2.15)

8


dΦ dt

Φ

l

Γ

n dΓ

ui

·d l

dl

E

E

Fig. 2.4. The Faraday’s law because of definitions I ui (t) = E · dl,

and Φ =

l

Z

Γ

B · dΓ.

(2.16)

The relationship between the integral form (2.11) and (2.12) and the differential form (2.3) and (2.4) can be expressed by applying Gauss’ theorem, Z I v · dΓ = ∇ · v dΩ, (2.17) Γ

Ω

where v = v(r, t) is a space vector, which may depend on time, the vector dΓ and volume Ω are represented in Fig. 2.5. The third Maxwell’s equation says that the surface integral of the magnetic flux density on a closed surface Γ is equal to zero, i.e. the flux lines are closed.

n dΓ D·dΓ D Ω ρ

Fig. 2.5. The Gauss’ law


9

The fourth Maxwell’s equation declares that the surface integral of the electric flux density on a closed surface Γ is equal to the volume integral of the charge density enclosed by this surface representing the volume Ω as it is illustrated in Fig. 2.5. The flux lines are starting and closing at electric charges. The outer normal unit vector of the volume Ω is denoted by n. The volume integral of charge density is equal to the charge Q, Z ρ dΩ = Q. (2.18) Ω

C. The constitutive relations Equations in (2.5) collect the so-called constitutive relations, which – depending on the properties of the examined material – describe the relationship between field quantities. In the simplest case these relations are linear, i.e. M = χH,

E i = 0,

P = ε0 χd E,

(2.19)

where χ and χd are the magnetic and the dielectric susceptibility and B = µH,

J = σE,

D = εE,

(2.20)

where µ = µ0 (1 + χ) = µ0 µr ,

(2.21)

ε = ε0 (1 + χd ) = ε0 εr .

(2.22)

and

Here µr = 1 + χ is the relative permeability, εr = 1 + χd is the relative permittivity of the material and the conductivity σ is constant. The second equation in (2.20) is the differential form of Ohm’s law. Constitutive relations are generally nonlinear, that is the permeability, the conductivity and the permittivity depend on the appropriate field quantities, µ = µ(H, B),

σ = σ(E, J),

ε = ε(E, D).

(2.23)

This can be written in another way, B = B(H),

J = J (E),

D = D(E),

(2.24)

where B(·), J (·) and D(·) are operators. The first one of the above equations will be in focus in the following, which can be expressed by the hysteresis operator (see chapter 5) as B = B{H},

or H = B −1 {B}.

(2.25)

If the material properties are independent of space r, they are homogeneous, otherwise they are inhomogeneous, µ = µ(r),

σ = σ(r),

ε = ε(r).

(2.26)

10


Constitutive relations may depend on the frequency of excitation as well, µ = µ(f ),

σ = σ(f ),

ε = ε(f ).

(2.27)

When the constitutive parameters depend on the direction of the applied field, the materials are anisotropic, otherwise they are isotropic (the linear equations in (2.20) states for isotropic materials). In the anisotropic case the permeability, the conductivity and the permittivity are tensors, B = [µ]H,

J = [σ]E,

D = [ε]E,

(2.28)

for example 

µxx [µ] =  µyx µzx

µxy µyy µzy

 µxz µyz  . µzz

(2.29)

In the most general situation, the constitutive relations depend on all of the above variables, e.g. B = B{H, r, f }.

(2.30)

D. Energy of electromagnetic fields The energy balance equation of electromagnetic fields can be expressed from Maxwell’s equations. After multiplying the equation (2.1) by −E and the equation (2.2) by H, then summing these equations, the following expression can be obtained: H · ∇ × E − E · ∇ × H = −H ·

∂B ∂D −E·J −E· . ∂t ∂t

(2.31)

After applying the identity ∇ · (E × H) = H · ∇ × E − E · ∇ × H

(2.32)

on the left-hand side and integrating the resulting equation over a volume Ω bounded by a surface Γ and using Gauss’ theorem, it results in the energy balance equation of the electromagnetic fields, I Z Z ∂D ∂B E · J dΩ + (E × H) · dΓ = 0. (2.33) E· dΩ + +H · ∂t ∂t Ω Ω Γ The first volume integral is the rate of change of energy density of electromagnetic fields w = w(r, t) in the volume Ω, i.e. ∂w ∂D ∂B =E· +H · . ∂t ∂t ∂t The second integral can be reformulated by using the second equation in (2.5), J |J |2 − Ei · J = − Ei · J . E ·J = σ σ

(2.34)

(2.35)

11


This results in Z Z ∂B |J|2 ∂D dΩ + +H · dΩ E· ∂t ∂t Ω σ ZΩ I − E i · J dΩ + (E × H) · dΓ = 0. Ω

(2.36)

Γ

Here the second integral is the dissipated energy term, i.e. the Joule heat, the third integral contains the energy supplied by the sources and finally the last surface integral is the intensity of energy flow, i.e. the rate of change of energy crossing a unit area whose normal is oriented in the direction of the vector E × H (see Fig. 2.6(a) and Fig. 2.6(b)). This is the so-called Poynting vector, S = E × H.

(2.37) n

H

S ·dΓ

dΓ S

E Ω

S =E×H

(b) The energy crossing the area dΓ

(a) The direction of Poynting vector S

Fig. 2.6. The Poynting vector The energy density w = w(r, t) in linear and isotropic media has the form w(r, t) =

1 εE 2 + µH 2 . 2

(2.38)

In nonlinear and anisotropic media the following expressions can be used to determine the electric and the magnetic energy density: Z D Z B we = E(D) · dD, and wm = H(B) · dB, (2.39) 0

0

which depend on the shape of characteristics given by the operators in (2.24).

2.1.2 Interface and boundary conditions Field equations are valid for points in whose neighborhood the physical properties of the medium vary continuously, or in special case they are constants. Let us imagine that there is a surface bounding one medium with material parameters µ1 , σ1 and ε1 from

12


another one body with material parameters µ2 , σ2 and ε2 , i.e. there occur sharp changes in parameters. In this situation interface conditions on the electromagnetic field quantities must be fulfilled. In the case of static magnetic field problems and in eddy current field problems the open boundary is usually modeled by a sphere with a radius r → ∞. The energy crossing the area of this boundary is equal to zero, because the variation of energy of electric and magnetic field is taking place inside the bounding sphere. This condition can only be fulfilled if lim r2 (E × H) · n = 0,

(2.40)

lim r|E| = 0,

(2.41)

r→∞

i.e. r→∞

and

lim r|H| = 0.

r→∞

This means that the electric and the magnetic fields must vanish at infinity. In other words, we can say that during the solution of Maxwell’s equations, the interface and the boundary conditions have to be taken into account along the interface of the materials and at the boundary surfaces of the problem region. Here interface and boundary conditions are collected without deduction. The used notations are presented in Fig. 2.7. The interface Γ (boundary surface) is placed between two regions denoted by Ω1 and Ω2 with material parameters µ1 , σ1 , ε1 and µ2 , σ2 , ε2 , respectively. The normal unit vector n is conventionally equal to the outer normal unit vector of the medium filling the region Ω1 (here n = n1 and n = −n2 ). E 2 , D2 , H 2 , B 2 , J 2

Ω2 µ2 , σ2 , ε2 n

Γ E 1 , D1 , H 1 , B 1 , J 1

µ1 , σ1 , ε1 Ω1

Fig. 2.7. For the behavior of electric and magnetic field quantities along the interface Γ

A. Electric and magnetic field intensity The interface conditions prescribe continuity for the tangential component of the electric field intensity, n × (E 2 − E 1 ) = 0.

(2.42)


13

The surface current density K relates to the tangential component of the magnetic field intensity vector, n × (H 2 − H 1 ) = K.

(2.43)

The surface current density K is flowing on the surface tangentially to the normal vector unit n. If there is no surface current density on the interface, the tangential component of the magnetic field intensity is continuous, n × (H 2 − H 1 ) = 0.

(2.44)

If Γ denotes the bounding sphere of domain Ω1 , i.e. E 2 = 0 and H 2 = 0, moreover E = E 1 and H = H 1 , then the boundary conditions can be formulated as −n × E = 0,

or E × n = 0,

(2.45)

and −n × H = K,

or H × n = K,

(2.46)

or (if K = 0) −n × H = 0,

or

H × n = 0.

(2.47)

B. Electric and magnetic flux density, current density On the interface of different dielectric materials the normal component of the electric flux density is continuous only if there is no surface charge, ρs = 0, n · (D2 − D 1 ) = 0,

(2.48)

otherwise the normal component of the electric flux density has a jump on the interface, n · (D2 − D 1 ) = ρs .

(2.49)

On the interface of different magnetic materials, the normal component of the magnetic flux density must be continuous, n · (B 2 − B 1 ) = 0.

(2.50)

The charge conservation law yields the continuity of the normal component of the conducting current in the case of eddy current field, n · (J 2 − J 1 ) = 0,

(2.51)

or generally n · (J 2 − J 1 ) + n · is valid on the interface.

∂D 2 ∂D 1 − ∂t ∂t

=0

(2.52)

14


If Γ denotes the bounding sphere of domain Ω1 , i.e. D 2 = 0, B 2 = 0, J 2 = 0 and ∂D2 /∂t = 0, moreover D = D1 , B = B 1 and J = J 1 then the boundary conditions can be formulated as −n · D = ρs ,

or D · n = −ρs ,

(2.53)

or n · D = 0,

or D · n = 0,

(2.54)

if ρs = 0 and −n · B = 0,

or B · n = 0,

(2.55)

−n · J = 0,

or J · n = 0,

(2.56)

and

or generally −n · J − n ·

∂D = 0, ∂t

or J · n +

∂D · n = 0. ∂t

(2.57)

It can be noted that equation (2.55) can be generalized as −n · B = b,

or B · n = −b,

(2.58)

where b = b(r, t) is the charge density of fictitious magnetic surface charges. These general conditions will be rewritten in problem specific form in section 2.2.

2.1.3 Classification of Maxwell’s equations The usual classification of Maxwell’s equations is presented here. In the simplest case the time variation of the field quantities can be neglected, i.e. ∂/∂t = 0. This kind of fields is denoted as static field, when the magnetic, the electric and the current fields can be regarded independently, because there are no interactions between them. In time varying case when ∂/∂t 6= 0, the magnetic and electric fields are coupled resulting the section of eddy current fields and wave propagation of electrodynamics. (i) Static magnetic field. The time independent current density J 0 = J 0 (r) generates a time independent magnetic field intensity H = H(r) and the time independent magnetic flux density B = B(r). The equations are the following: ∇ × H = J 0,

(2.59)

∇ · B = 0,  in air,  µ0 H, in magnetically linear material, µ0 µr H, B=  µ0 (H + M ), in magnetically nonlinear medium.

(2.60) (2.61)


15

In nonlinear medium, the magnetization vector M = M (r) is depending on the magnetic field intensity vector, i.e. M = H {H}. This operator can be described by hysteresis models as it is introduced in the fifth chapter and this operator is denoted by B = B{H}. This constitutive relationship has an inverse form as well,  in air,  ν0 B, in magnetically linear material, ν0 νr B, H=  −1 B {B}, in magnetically nonlinear medium.

(2.62)

Here ν0 = 1/µ0 , νr = 1/µr are the reluctivity of vacuum and the relative reluctivity. In magnetically nonlinear medium it can be represented by an inverse hysteresis operator, H = B −1 {B}. The source current distribution is solenoidal, which is evident from the current continuity equation (2.8), ∇ · J 0 = 0.

(2.63)

This means that all current lines either close upon themselves, or start and terminate at infinity. This is the situation when magnetic field generated by current carrying coils is simulated, or the static behavior of electrical machines is the question. If J 0 = 0, then a boundary value problem should be solved, which can be used to simulate e.g. the field of magnetic poles. (ii) Static electric field. The source of electrostatic field described by the electric field intensity E = E(r) and by the flux density D = D(r) is the charge ρ = ρ(r). The field equations are as follows: ∇ × E = 0,

(2.64)

∇ · D = ρ,

(2.65)

D = εE,

or D = ε0 E + P ,

(2.66)

where P = P (r) is the polarization vector. For example, this is the situation when simulating the electrostatic field inside capacitors. Supposing static electric field is useful in high voltage applications, e.g. in transmission line problems. (iii) Currents in conducting materials. The motion of charges results in currents flowing in the conducting material, which can be expressed by time independent equations ∇ × E = 0,

(2.67)

∇ · J = 0,

(2.68)

J = σE,

or J = σ(E + E i ).

(2.69)

16


Here E i = E i (r) is the impressed electric field intensity. The term σE i can be represented by the source current density J 0 = σE i , i.e. J = J 0 + σE.

(2.70)

These equations can be used efficiently when the current distribution of a coil cannot be supposed to be uniform, but can be computed as a current flow problem. (iv) Eddy current field. In time varying case (∂/∂t 6= 0), the magnetic and electric fields are coupled, but the displacement current can be neglected in the usual frequency range, because |J | ≫ |∂D/∂t|. The equations of ’quasi-static’ field problems are as follows: ∇ × H = J, ∇×E =−

∂B , ∂t

∇ · B = 0,  in air,  µ0 H, µ0 µr H, in magnetically linear material, B=  µ0 (H + M ), in magnetically nonlinear medium, J = σE.

(2.71) (2.72) (2.73) (2.74) (2.75)

The constitutive relation of the magnetic material may have the inverse form (2.62). This approximation is applicable in all power frequency applications involving metallic structures. The analysis of power losses in electrical machines or some nondestructive testing problems and the prediction of transient problems can be simulated by this kind of equations. (v) Wave propagation. In this case all terms of Maxwell’s equations are considered without any modification. This group of equations is useful in the analysis of waveguides, cavities or resonators at high frequencies. In the following sections, we turn our attention mainly to the linear and nonlinear static magnetic field and the eddy current field problems.

2.2 Magnetostatics and eddy current field problems In the case of magnetic materials, there are two groups of problem have to be formulated, the static magnetic field and the eddy current field. The typical structure – as they can be found in electrical engineering applications – with equations, interface and boundary conditions of these problems are formulated in this section.

17

2.2. MAGNETOSTATICS AND EDDY CURRENT FIELD PROBLEMS

2.2.1 Static magnetic fields The typical structure of a static magnetic field problem can be seen in Fig. 2.8. In the case of static magnetic field, the currents flowing in coils i (represented by the source current density J 0 ) generate electric field as well as magnetic field, but the magnetic field has no interaction with the electric field. In this case Maxwell’s equations can be written as ∇ × H = J 0,

in Ω0 ∪ Ωm ,

(2.76)

∇ · B = 0, in Ω0 ∪ Ωm ,  in air, Ω0 , µ0 H, in magnetically linear material, Ωm , B = µ0 µr H,  B{H} = µo H +R, in magnetically nonlinear medium, Ωm .

(2.78)

∇ · J 0 = 0.

(2.79)

(2.77)

The last relation is defined in section 6.5.2. Taking the divergence of equation (2.76), the solenoidal property of the source current density can be obtained,

The constitutive relationship (2.78) can be used in its inverse form as well,  in air, Ω0 , ν0 B, in magnetically linear material, Ωm , H = ν0 νr B,  −1 B {B} = νo B+I, in magnetically nonlinear medium, Ωm .

(2.80)

The last relation is defined in section 6.5.1. A current excitation i (or a current density J 0 ) with given amplitude is placed into air Ω0 and symbol Ωm represents the magnetic material (which can be represented by linear or nonlinear, isotropic or anisotropic, single valued or hysteretic model). The current density can be calculated from known current as |J 0 | =

Ni , Si

(2.81)

ΓB

air

ΓH

i

µ0

n

Γm0 magnetic material

nm n0

Ωm ΓH

µr or B{·} ΓB

ΓB

H, B Γm0

Ω0 ΓB

Fig. 2.8. Structure of a static magnetic field problem

18


where N is the number of turns of the coil and Si is the cross section area of the coil. The direction of J 0 is generated by the geometry of the coil. The excitation current generates magnetic field H around the coil. Maxwell’s equations are valid in the problem region Ω = Ω0 ∪ Ωm . The problem region Ω is bounded by ∂Ω, which has two disjunct parts ΓH and ΓB , i.e. ∂Ω = ΓH ∪ ΓB . Along the dashed line in Fig. 2.8 denoted by ΓH , the tangential component of the magnetic field intensity is given by a known surface current density K. If K = 0 then ΓH represents a symmetry plane. The boundary ΓB is usually the closing boundary or a symmetry plane of the problem region, where the normal component of the magnetic flux density is vanishing. It is assumed to be known in a term b. Along the interface between the magnetic material and the air region Γm0 , the tangential component of magnetic field intensity and the normal component of the magnetic flux density are continuous. These conditions can be formulated as (see section 2.1.2) H × n = K,

or H × n = 0,

on ΓH ,

(2.82)

and B · n = −b,

or B · n = 0,

on ΓB ,

(2.83)

where n is the outer normal unit vector of the region, moreover H 0 × n0 +H m × nm = 0, and B 0 · n0 + B m · nm = 0, on Γm0 ,

(2.84)

where n0 , nm , H 0 , H m , B 0 and B m are the outer normal unit vectors of the region filled with air and with magnetic material, moreover the magnetic field intensity and the magnetic flux density in the appropriate region along the interface, respectively, and it is evident that n0 = −nm along Γm0 .

2.2.2 Eddy current fields In time varying case (∂/∂t 6= 0), the electric and the magnetic fields are coupled. Currents flowing in coils generate magnetic field in the vicinity of the coil. The time variation of this magnetic field induces electric field that causes eddy currents flowing in conducting materials. The magnetic field generated by the eddy currents modifies the effect of sources. A typical structure of an eddy current field problem can be seen in Fig. 2.9. In the eddy current free region denoted by Ωn , there is a magnetic field with equations ∇ × H = J 0, ∇ · B = 0, B = µH,

in Ωn ,

(2.85)

in Ωn , or

H = νB,

(2.86) in Ωn .

(2.87)

Sometimes it is filled with air (µ = µ0 , or ν = ν0 ). At low frequencies and with normal conducting materials, the displacement currents are small compared with the conducting currents and they can be neglected, i.e. |∂D/∂t| ≪ |J | .

(2.88)

19

2.2. MAGNETOSTATICS AND EDDY CURRENT FIELD PROBLEMS

ΓB

nonconducting region (air)

ΓHn

i(t)

µ0

n

Γnc conducting region

σ

Ωc ΓHc

µr or B{·} H e ΓE

σE

nc nn

ΓB

H, B Ωn Γnc ΓB

Fig. 2.9. Structure of an eddy current field problem Consequently, the studied electromagnetic fields can be described by the ’quasi-static’ Maxwell’s equations written in differential form as ∇ × H = J,

in Ωc ,

∂B , in Ωc , ∂t ∇ · B = 0, in Ωc , µ0 µr H, in magnetically linear material, Ωc , B= B{H} = µo H +R, in magnetically nonlinear medium, Ωc ,

∇×E =−

J = σE,

in Ωc .

(2.89) (2.90) (2.91) (2.92) (2.93)

The last relation in (2.92) is defined in section 6.5.2. Of course, the constitutive relation (2.92) can be used in its inverse form as well, in magnetically linear material, Ωc , ν0 νr B, H= (2.94) B −1 {B} = νo B+I, in magnetically nonlinear medium, Ωc . The last relation is defined in section 6.5.1. Taking the divergence of equation (2.89), the solenoidal property of the induced current density can be obtained, ∇ · J = 0,

(2.95)

meaning that eddy currents close upon themselves. A current excitation i(t) (or a source current density J 0 (t)) with given signal shape is placed into the nonconducting region Ωn (which is usually the air region) and the eddy current distribution σE is unknown in the conducting material Ωc (the corresponding magnetic field generated by eddy currents is denoted by H e in the illustration in Fig. 2.9). Region Ωc is sometimes filled with magnetic material (which can be represented by linear or nonlinear, isotropic or anisotropic, single valued or hysteretic model).

20


Static and ’quasi-static’ Maxwell’s equations are valid in the eddy current free region Ωn and in the eddy current region Ωc , respectively. The two regions are coupled through the interface Γnc . The eddy current free region Ωn is bounded by two disjunct parts ΓHn and ΓB , the eddy current region is bounded by two disjunct parts ΓHc and ΓE . Generally, the problem region Ω = Ωn ∪ Ωc is bounded by ∂Ω = ΓHn ∪ ΓB ∪ ΓHc ∪ ΓE . Along the dashed line placed in the eddy current free region in Fig. 2.9 denoted by ΓHn , the tangential component of the magnetic field intensity is given by a known surface current density K. If K = 0 then ΓHn represents a symmetry plane, where the tangential component of the magnetic field intensity is vanishing. The boundary ΓB is usually the closing boundary of the problem region, where the normal component of the magnetic flux density is vanishing. It is assumed to be known by a term b. The dashed line signed in the eddy current region ΓHc usually denotes a symmetry plane, where the tangential component of the magnetic field intensity is zero and ΓE is the boundary where the tangential component of the electric field intensity is vanishing. The surface ΓE may also model electrodes where a voltage source is connected to the conductor. Along the interface between the two disjunct region Γnc , the tangential component of magnetic field intensity and the normal component of the magnetic flux density are continuous, moreover the normal component of the induced eddy currents are equal to zero. These conditions can be formulated as (see section 2.1.2) H × n = K,

or H × n = 0,

on ΓHn ,

(2.96)

and B · n = −b,

or B · n = 0,

on ΓB ,

H × n = 0,

on ΓHc ,

(2.98)

E × n = 0,

on ΓE ,

(2.99)

(2.97)

and

and

where n is the outer normal unit vector of the region, moreover H c × nc + H n × nn = 0,

on Γnc ,

(2.100)

and B c · nc + B n · nn = 0,

on Γnc ,

(2.101)

and J · nc = 0,

on Γnc ,

(2.102)

where nn , nc , H n , H c , B n and B c are the outer normal unit vector of the region filled with air and with conducting material, moreover the magnetic field intensity and the

2.3. POTENTIAL FORMULATIONS IN STATIC MAGNETIC AND EDDY CURRENT FIELD PROBLEMS

21

magnetic flux density vectors in the appropriate region on the boundary, respectively and it is evident that nn = −nc along Γnc . The magnetic field must satisfy the initial conditions, as well B n (t = 0) = B n,0 ,

in Ωn ,

and B c (t = 0) = B c,0 ,

in Ωc .

(2.103)

As a conclusion of the boundary conditions on ΓHc and on ΓE , the normal component of the eddy current density is vanishing on the boundary ΓHc because of equations (2.98) and (2.89), while on boundary ΓE , the normal component of magnetic flux density is equal to zero coming from equations (2.99) and (2.90), i.e. J ·n= 0

on ΓHc ,

(2.104)

B·n =0

on ΓE .

(2.105)

and

As a proof, let us take the normal component of the first Maxwell’s equation (2.89). It results in the expression (∇ × H) · n = J · n, i.e. ∇ · (H × n) = J · n. The left-hand side of this equation is equal to zero on ΓHc according to (2.98), i.e. J · n = 0 on ΓHc . The equation B · n = 0 can be obtained similarly.

2.3 Potential formulations in static magnetic and eddy current field problems There are several potential formulations applicable to calculate the electromagnetic field quantities, fundamentally scalar and vector potentials can be used [8–16,18,31–33,35,40, 51,77–79,87,89,94,137,138,150–152,157,167–170,172,187,197]. The aim of potential formulations is to reduce the solution of Maxwell’s equations to the solution of different type of partial differential equations at prescribed boundary conditions. This section deals with the potential formulations of static magnetic field and eddy current field problems.

2.3.1 Static magnetic fields The general definition of static magnetic field problems can be found in section 2.2.1. The static magnetic field is defined by Maxwell’s equations (2.76), (2.77), constitutive relations in (2.78) or in (2.80), moreover the boundary and interface conditions (2.82), (2.83) and (2.84). The static magnetic field can be described by the reduced magnetic scalar potential Φ, by the total magnetic scalar potential Ψ, by the combination of these scalar potentials (Φ − Ψ-formulation), or by the magnetic vector potential A. The combination of these formulations is also valid, resulting the A−Φ-formulation or the A−A−Φ-formulation. It is noted that the most economical way is using the scalar potential, however, it results in some problems, especially when iron parts with high permeability are present.

22


In this situation the combination of the two scalar potentials can be used, but this introduces some extra conditions on the iron/air interfaces. The known rotational part of the magnetic field intensity according to coil’s current must be calculated beforehand. This potential is called impressed current vector potential T 0 . The use of magnetic vector potential is more general, however, the number of unknowns is increasing when simulating 3D problems. A. The reduced magnetic scalar potential, the Φ-formulation The magnetic field intensity vector can be split in two parts as H = T 0 + H m.

(2.106)

The curl of the so-called impressed current vector potential T 0 is equal to the source current density J 0 and H m is nonrotational, ∇ × T 0 = J 0,

and ∇ × H m = 0.

(2.107)

The first Maxwell’s equation (2.76) can be fulfilled by this decomposition, ∇ × H = ∇ × (T 0 + H m ) = ∇ × T 0 + ∇ × H m = J 0 .

(2.108)

The divergence of T 0 can be selected according to Coulomb gauge, i.e. ∇ · T 0 = 0,

(2.109)

which selection can be useful when creating the function T 0 . It is important to note that the two terms T 0 and H m can be calculated separately. The first step is the generation of the function T 0 according to the source current density J 0 satisfying the first equation in (2.107), then H m can be represented by a scalar potential. Generating the function T 0 . The first step is to determine the impressed current vector potential T 0 . There are many possibilities for the construction of the source term T 0 from the known source current density J 0 . Here the four most attractive and the most widely used solutions are formulated. The impressed field can be calculated by using the Biot–Savart’s law, T 0 can be built up by simple analytical expressions if the geometry of coils is simple, or by the solution of a partial differential equation with appropriate boundary conditions. It must be noted that T 0 is calculated in free space, i.e. µ = µ0 must be set everywhere in the problem region. (i) Applying the Biot–Savart’s law. The impressed current vector potential T 0 can be selected as the magnetic field of coils in free space H s calculated by Biot–Savart’s law (Fig. 2.10), Z 1 J 0 (r J ) × (r f − rJ ) dΩJ , (2.110) T 0 (r f ) ≡ H s (r f ) = 4π ΩJ |r f − r J |3 or (Fig. 2.11) T 0 (r f ) ≡ H s (r f ) =

I0 4π

I

lI

dl × (r f − rI ) 3

|r f − r I |

.

(2.111)


23

J 0 (r J )

dΩJ

rf − rJ

H s (rf )

rJ rf O Fig. 2.10. Magnetic field intensity H s generated by source current density I0 H s (rf )

rf − rI dl

rf

coil rI O

Fig. 2.11. Magnetic field intensity H s generated by a filamentary conductor Here rf is the field point where H s (r f ) is calculated, r J and r I are the source points where J 0 (r J ) and I0 (r I ) is given. Symbol ’O’ denotes the origin of the coordinate system. It results in a divergence-free impressed current vector potential. The source term should be calculated with a given value of current (I0 = 1A is a good choice), or a given value of current density ( |J 0 | = 1A/m2 ). Then the calculated source term T 0 can be multiplied by the actual value of current during simulation, e.g. when i(t) is sinusoidal or I = 0.2A, etc. Simple analytical formulation can be used when the geometry of the coil is simple (for example the formula H = I0 /2rπ can be used in case of one filamentary conductor in 2D simulations). However, this step can be a very time consuming task, especially in 3D situations. (ii) Creating a simple function satisfying ∇ × T 0 = J 0 . Extremely simple analytical expressions can be formulated in case of simple coil shapes as it is illustrated by a cylindrical coil in Fig. 2.12. In this case the integration can be avoided and a simple, tangentially continuous analytical expression can be built up. If T 0 has only one component in the z direction, i.e. T 0 = T0,z ez moreover T0,z = T0,z (r) or T0,z = T0,z (x, y), then source current density in the r − ϕ or in the x − y plane can be represented easily.

24


z

h

J0 r |J 0 |

J0

R′ R r

L

−|J 0 | T0,z

T0,z = |J 0 |(R′ − r)

T0,z = |J 0 |(R′ − r) |J 0 |L r

Fig. 2.12. Creating T 0 for cylindrical shaped coil In the illustration shown in Fig. 2.12, the cylindrical shaped coil is placed around the z-axis with center at the origin of a cylindrical coordinate system. The inner and outer radius of the coil is R and R′ , respectively, the height of the coil is h and the current density is J 0 = |J 0 |eϕ . The vector potential T 0 has only one component in the z direction, i.e. T 0 = T0,z ez , where  if |z| ≥ h/2 or r > R′ ,  0, if |z| < h/2 and r ≤ R, |J 0 |L, T0,z = (2.112)  |J 0 |(R′ − r), if |z| < h/2 and r > R and r ≤ R′ . In this case T 0 is equal to zero outside the coil, it is constant inside the coil where the current density is equal to zero and it is varying linearly inside the coil where the current density is not zero. The divergence of T 0 is not equal to zero.

(iii) Minimizing an appropriate functional. The following functional can be built up to find out the source term T 0 : Z h i 2 2 F {T 0 } = |∇ × T 0 − J 0 | + (∇ · T 0 ) dΩ, (2.113) Ω

which has to be minimized. This is equivalent to the solution of the following partial differential equations defined in free space: ∇ × T 0 = J 0,

in Ω,

(2.114)


∇ · T 0 = 0,

25

in Ω.

(2.115)

The boundary conditions T 0 × n = 0, T 0 · n = 0,

on ΓH ,

(2.116)

on ΓB

(2.117)

must also be satisfied (∂Ω = Γ = ΓH ∪ ΓB ). A partial differential equation with appropriate boundary conditions can be formulated for the mentioned functional (2.113) what is an advantage of this formulation. According to the principle of minimum energy, the functional (2.113) has minimum value only if the vector field T 0 satisfies the partial differential equations (2.114) and (2.115). This minimum is a stable configuration of the vector potential T 0 . Let us first extract the functional (2.113), Z h i F {T 0} = |∇ × T 0 |2−2(∇× T 0 )·J 0 +|J 0 |2+(∇ · T 0 )2 dΩ,

(2.118)

Ω

and let us introduce a small variation of the vector potential T 0 as T 0 + αδT 0 (α is a small positive number), which results in a variation of the field energy around the stable configuration of the potential T 0 , Z h |∇ × (T 0 + αδT 0 )|2 F {T 0 + αδT 0 } = Ω

2

−2[∇ × (T 0 + αδT 0 )] · J 0 + |J 0 | i +[∇ · (T 0 + αδT 0 )]2 dΩ,

i.e.

(2.119)

Z h F {T 0 +αδT 0 } = |∇ × T 0 |2 + 2α (∇ × T 0 )·(∇ × δT 0 ) Ω

+α2 |∇ × δT 0 |2 − 2(∇ × T 0 ) · J 0 − 2α (∇ × δT 0 ) · J 0 2

+ |J 0 | + (∇ · T 0 )2 + 2α(∇ · T 0 )(∇ · δT 0 ) i +α2 (∇ · δT 0 )2 dΩ.

(2.120)

This variation can be represented in Taylor series approximation as F {T 0 + αδT 0 } = F {T 0 } + αδF {T 0 , δT 0 } + α2 δ 2 F {δT 0 }. (2.121) Here δF {T 0 , δT 0 } = 2

and

Z h

(∇ × T 0 )·(∇ × δT 0 ) − (∇ × δT 0 ) · J 0 (2.122) i +(∇ · T 0 )(∇ · δT 0 ) dΩ,

Ω

26


Z h i 2 2 |∇ × δT 0 | + (∇ · δT 0 ) dΩ. δ F {δT 0 } = 2

(2.123)

Ω

In concern with the stationary concept, the first variation δF {T 0 , δT 0 } has to be disappear and the second variation δ 2 F {δT 0 } has to be positive. In this case F {T 0 } leads indeed to its minimum value. The definition of the first variation is (which must be equal to zero) ∂F {T 0 + αδT 0 } = 0, (2.124) δF {T 0 , δT 0 } = ∂α α=0

from which Z h (∇ × T 0 ) · (∇ × δT 0 ) − (∇ × δT 0 ) · J 0 Ω

i +(∇ · T 0 )(∇ · δT 0 ) dΩ = 0,

(2.125)

and it is easy to see that the second variation (2.123) is usually positive. Using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(2.126)

with u = ∇ × T 0 , v = δT 0 in the first integral and with u = J 0 , v = δT 0 in the second integral and the identity ∇ · (ϕ v) = v · ∇ϕ + ϕ∇ · v,

(2.127)

with ϕ = ∇ · T 0 , v = δT 0 in the third term results in Z δT 0 · (∇ × ∇ × T 0 − ∇ × J 0 ) dΩ Ω I + δT 0 · [(∇ × T 0 − J 0 ) × n] dΓ ZΓ I − δT 0 · (∇∇ · T 0 ) dΩ + (δT 0 · n) ∇ · T 0 dΓ = 0. Ω

(2.128)

Γ

The first boundary integral can be rewritten according to the two parts ΓH and ΓB of the boundary Γ, I δT 0 · [(∇ × T 0 − J 0 ) × n] dΓ ZΓ (2.129) δT 0 · [(∇ × T 0 − J 0 ) × n] dΓ. = ΓH ∪ΓB

On ΓH the Dirichlet type boundary condition T 0 × n = 0 must be satisfied, i.e. the tangential component of the variation is equal to zero, δT 0 × n = 0. According to the identity δT 0 · [(∇ × T 0 − J 0 ) × n] = (∇ × T 0 − J 0 ) · (n × δT 0 ) = − (∇ × T 0 − J 0 ) · (δT 0 × n),

(2.130)


27

the boundary integral on ΓH is vanishing. On the rest part ΓB the Neumann type boundary condition (∇ × T 0 − J 0 ) × n = 0 must be prescribed to vanish the surface integral term. The second boundary integral can also be rewritten in the form I Z (δT 0 · n) ∇ · T 0 dΓ. (δT 0 · n) ∇ · T 0 dΓ = Γ

(2.131)

ΓH ∪ΓB

On the part ΓB the Dirichlet type boundary condition T 0 · n = 0 must be specified, consequently δT 0 · n = 0. On the part ΓH the Neumann type boundary condition ∇ · T 0 = 0 should be applied to vanish the boundary integral term. This means that an extra boundary condition must be specified on ΓH and on ΓB . Finally, the integral equation (2.128) results in the following partial differential equation and the boundary conditions: ∇ × ∇ × T 0 − ∇∇ · T 0 = ∇ × J 0 , T 0 × n = 0,

in Ω,

on ΓH ,

(2.132) (2.133)

∇ · T 0 = 0,

on ΓH ,

(2.134)

T 0 · n = 0,

on ΓB ,

(2.135)

(∇ × T 0 ) × n = 0,

on ΓB .

(2.136)

This means that the integral equation (2.128) can be fulfilled for any variation δT 0 if the partial differential equation (2.132) and the boundary conditions defined in (2.133)–(2.136) are solved. The solution of this boundary value problem is a divergence free impressed current vector potential and it can be solved by any numerical field calculation procedure (in this book we are focusing on the Finite Element Method). This formulation results in gauged impressed current vector potential. (iv) Minimizing a functional combined with an appropriate numerical technique. The following functional can be built up too, to find out the source term T 0 : Z 2 F {T 0 } = |∇ × T 0 − J 0 | dΩ, (2.137) Ω

which has to be minimized. This is equivalent to the following partial differential equation defined in free space: ∇ × T 0 = J 0,

in Ω,

(2.138)

and the boundary conditions are T 0 × n = 0, T 0 · n = 0,

on ΓH , on ΓB .

(2.139) (2.140)

28


A partial differential equation with boundary conditions can be formulated for the mentioned functional (2.137), which can be obtained in a similar way presented in the last item, only the terms according to ∇ · T 0 = 0 are missing. The result can be formulated as ∇ × ∇ × T 0 = ∇ × J 0, T 0 × n = 0, T 0 · n = 0,

in Ω,

(2.141)

on ΓH ,

(2.142)

on ΓB .

(2.143)

The solution of this boundary value problem is a possible choice of the impressed current vector potential. It can be solved by a numerical field calculation procedure, which is not sensitive to Coulomb gauge. This formulation results in an ungauged impressed current vector potential. Finally T 0 can be regarded as known, because this quantity is calculated before the numerical simulation. Calculating the nonrotational part H m . The second step of Φ-formulation is the determination of the nonrotational part of the magnetic field intensity H m in (2.106). It can be derived from the negative gradient of a magnetic scalar potential Φ, H m = −∇Φ,

(2.144)

because of the identity ∇ × (∇ϕ) ≡ 0 for any scalar function ϕ = ϕ(r) (or ϕ = ϕ(r, t)). By this formulation the magnetic field intensity can be written as H = T 0 − ∇Φ,

(2.145)

which satisfies equation (2.76) exactly. The magnetic scalar potential Φ is usually called the reduced magnetic scalar potential, because the source term is fundamentally hidden in T 0 . Applying the linearized form of the nonlinear constitutive relation in (2.78) results in the magnetic flux density B = µo (T 0 − ∇Φ) + R.

(2.146)

The divergence of magnetic flux density is equal to zero according to (2.77). Finally the linear partial differential equation of the problem has the form ∇ · (µo ∇Φ) = ∇ · (µo T 0 ) + ∇ · R,

in

Ω,

(2.147)

which is a generalized Laplace–Poisson equation. Partial differential equation (2.147) can be obtained in another way as well. Starting from the solenoidal property of the source current density defined by the equation (2.79), J 0 can be written by the curl of the impressed current vector potential according to the identity ∇ · (∇ × v) ≡ 0 for any vector function v = v(r) (or v = v(r, t)), J 0 = ∇ × T 0.

(2.148)


29

Substituting the relation (2.148) into Maxwell’s equation (2.76) results in ∇ × H = ∇ × T0

⇒

∇ × (H − T 0 ) = 0.

(2.149)

The curl-less vector field H − T 0 can be determined by the negative gradient of the magnetic scalar potential Φ, i.e. H − T 0 = −∇Φ

⇒

H = T 0 − ∇Φ.

(2.150)

Substituting the relation of magnetic field intensity in (2.150) into (2.77) and using the linearized constitutive relation in (2.78) also leads to the partial differential equation (2.147). Boundary conditions. Let us now define the boundary and interface conditions of the Φ-formulation. The impressed current vector potential T 0 must satisfy the symmetry conditions on symmetry planes, if any. The tangential component of the impressed current vector potential must vanish on ΓH and its normal component must be equal to zero on ΓB as it was mentioned in (2.133) and in (2.135). The tangential component of the magnetic field intensity can be set to the given value as follows on ΓH : H × n = (T 0 − ∇Φ) × n = K,

on ΓH .

(2.151)

A vector H = H(r) can be written by the two orthogonal components as (see Fig. 2.13) H = (H · n) · n + n × (H × n).

(2.152)

The second term is the tangential component, i.e. n × (H × n) = n × [(T 0 − ∇Φ) × n] = n × K,

(2.153)

which can be reformulated as n × (−∇Φ × n) = n × (K − T 0 × n) ,

(2.154)

from which −∇Φ can be selected as n × (H × n) H ×n

n

H

n

H

(H · n) · n

H ×n

n × (H × n)

Fig. 2.13. Definition of the normal and tangential component of a vector H

30


−∇Φ = n × (K − T 0 × n) ,

(2.155)

because n × (K − T 0 × n) = n × {[n × (K − T 0 × n)] × n} .

(2.156)

Using the definition of the gradient of a vector field, the equation (2.155) results in the following Dirichlet boundary condition: Z (2.157) Φ = Φ0 , and Φ0 = n × (K − T 0 × n) · dl, on ΓH . l

If one applies a known source magnetic field intensity vector H 0 on the boundary ΓH , which can be written as a surface current term as H 0 × n = K, or n × K = H 0 , then Z Φ = Φ0 , and Φ0 = (H 0 − n × [T 0 × n]) · dl, on ΓH (2.158) l

is valid. Here path l is lying on ΓH and it can be arbitrary. It is noted that symmetry planes are usually equi-potential surfaces. On the rest part ΓB , setting the normal component of the magnetic flux density results in a Neumann type boundary condition, B · n = −b

⇒

(µo T 0 − µo ∇Φ + R) · n = −b,

on ΓB ,

(2.159)

since B = µo (T 0 − ∇Φ) + R. If no Dirichlet boundary conditions are prescribed, then Φ = 0 must be specified at some points. In this case the reduced magnetic scalar potential is unique. The Φ-formulation. The partial differential equation and the boundary conditions of a static magnetic field problem solved by the reduced magnetic scalar potential can be written as ∇ · (µo ∇Φ) = ∇ · (µo T 0 ) + ∇ · R, Φ = Φ0 ,

in

Ω,

on ΓH ,

(µo T 0 − µo ∇Φ + R) · n = −b,

(2.160) (2.161)

on ΓB ,

(2.162)

where Φ0 can be determined by the integral (2.157) or (2.158). B. The total magnetic scalar potential, the Ψ-formulation The Φ-formulation is satisfactory if the permeability of the medium is not very high. In this case the simulated magnetic field H = T 0 −∇Φ does not much differ from the source magnetic field T 0 . By increasing the permeability of the medium with linear constitutive relation by increasing the slope of the nonlinear characteristics (e.g. when soft magnetic materials are modeled), the resultant magnetic field H is decreasing inside the magnetic material, i.e. T 0 ≈ ∇Φ, which is a strong disadvantage in numerically point of view.


31

This is the so-called cancellation error, when two almost equal quantities are subtracted from each other. Fortunately, the source current density J 0 is usually zero in ferromagnetic parts with high value of permeability. In this case, the magnetic field intensity can be derived from the so-called total magnetic scalar potential Ψ as H = −∇Ψ,

(2.163)

since ∇ × H = 0 (J 0 = 0) inside highly permeable regions. This can be the situation of magnetic poles. However, in most practical cases, the magnetic field intensity is generated by currents flowing in coils. The source coils are usually placed somewhere around the ferromagnetic parts, but source currents are not flowing inside ferromagnetic materials. In this situation, it is possible to use the reduced magnetic scalar potential Φ in the air region and the total magnetic scalar potential Ψ inside the highly permeable region and the two formulations must be coupled through the interface between the two regions. C. The combination of the magnetic scalar potentials, the Φ − Ψ-formulation This formulation is outdated, the method presented in the next item is more modern. In this situation the problem region is subdivided into two regions ΩΦ and ΩΨ as it is illustrated in Fig. 2.14 and Ω = ΩΦ ∪ ΩΨ . The domain ΩΦ contains all coils and the permeability is usually equal to µ0 , this is the air region. Here, the magnetic field intensity is expressed as H = T 0 − ∇Φ,

in

ΩΦ .

(2.164)

Because of the same value of permeability in ΩΦ , no cancellation errors occur there. The rest region contains all the ferromagnetic bodies, but the source current density is equal to zero, that is why the magnetic field intensity can be derived by using the total magnetic scalar potential, H = −∇Ψ,

in ΩΨ .

(2.165)

The partial differential equations are as follows in the two subregions: ∇ · (µ0 ∇Φ) = ∇ · (µ0 T 0 ) , ∇ · (µo ∇Ψ) = ∇ · R,

in

Ψ = Ψ0 ,

Z

in

ΩΦ ,

(2.166)

ΩΨ .

(2.167)

The boundary parts ΓH and ΓB are subdivided into ΓHΦ , ΓHΨ , ΓBΦ and ΓBΨ . On the ΓHΦ ∪ ΓHΨ boundaries, Dirichlet type boundary conditions are specified according to the tangential component of the magnetic field intensity, H × n = K, i.e. Z (2.168) Φ = Φ0 , and Φ0 = n × (K − T 0 × n) · dl, on ΓHΦ , l

and Ψ0 =

l

or

(n × K) · dl,

on ΓHΨ ,

(2.169)

32


n ΓBΦ ΩΨ

ΩΦ

nΨ

H = −∇Ψ

H = T 0 − ∇Φ

nΦ

n

J 0 6= 0

J0 = 0 ΓBΨ

ΓΦ,Ψ

ΓHΦ

µ = µ0

µ = µ0 µr or B{·} ΓHΨ

P0 Fig. 2.14. The scheme of a static magnetic field problem with reduced and total magnetic scalar potentials

Φ = Φ0 ,

and Φ0 =

Z

l

Ψ = Ψ0 ,

and Ψ0 =

Z

l

(H 0 − n × [T 0 × n]) · dl, H 0 · dl,

on ΓHΨ .

on ΓHΦ ,

(2.170)

(2.171)

On the rest part ΓBΦ ∪ ΓBΨ , Neumann type boundary conditions can be specified according to the condition B · n = −b, (µ0 T 0 − µ0 ∇Φ) · n = −b, (−µo ∇Ψ + R) · n = −b,

on ΓBΦ ,

(2.172)

on ΓBΨ .

(2.173)

Interface boundary conditions must be specified along the interface between the two subregions ΩΦ and ΩΨ . This is denoted by ΓΦ,Ψ , where the tangential component of the magnetic field intensity vector and the normal component of the magnetic flux density vector have to be continuous. The condition for the tangential component of the magnetic field intensity can be written as (T 0 − ∇Φ) × nΦ + (−∇Ψ) × nΨ = 0,

on ΓΦ,Ψ ,

(2.174)

which can be defined as a Dirichlet boundary condition similarly to the boundary condition (2.157) Z (2.175) Ψ = Φ + nΦ × (T 0 × nΦ ) · dl, on ΓΦ,Ψ . l


33

Here nΦ and nΨ are the outer normal unit vectors of the appropriate subregion, and it is noted that nΨ = −nΦ . Moreover at a point P0 , Φ = Ψ must be specified to make the solution unique. The condition for the normal component of the magnetic flux density can be written as (µ0 T 0 − µ0 ∇Φ) · nΦ + (−µo ∇Ψ + R) · nΨ = 0,

on ΓΦ,Ψ ,

(2.176)

which is a Neumann type boundary condition. The collection of partial differential equations, boundary and interface conditions of the Φ − Ψ-formulation is as follows: ∇ · (µ0 ∇Φ) = ∇ · (µ0 T 0 ) , ∇ · (µo ∇Ψ) = ∇ · R,

in

in

ΩΦ ,

(2.177)

ΩΨ ,

(2.178)

Φ = Φ0 ,

on ΓHΦ ,

(2.179)

Ψ = Ψ0 ,

on ΓHΨ ,

(2.180)

(µ0 T 0 − µ∇Φ) · n = −b,

on ΓBΦ ,

(2.181)

(−µo ∇Ψ + R) · n = −b, on ΓBΨ . Z Ψ = Φ + nΦ × (T 0 × nΦ ) · dl, on ΓΦ,Ψ ,

(2.182) (2.183)

l

(µ0 T 0 − µ0 ∇Φ) · nΦ + (−µo ∇Ψ + R) · nΨ = 0,

on ΓΦ,Ψ ,

(2.184)

where Φ0 and Ψ0 can be determined by the integrals (2.168), (2.170), or by (2.169), (2.171). D. Applying the reduced magnetic scalar potential with appropriate representation of T 0 The cancellation error inside ferromagnetic materials with high permeability can be eliminated by the appropriate representation of the impressed current vector potential in numerical field analysis procedures. The rotational part of the magnetic field intensity is originally a smooth function with an infinite number of continuous derivatives, whereas ∇Φ is usually interpolated by simple functions depending on the used method, e.g. by polynomials having finite number of continuous derivatives. A so-called compatible approximation should be used to interpolate the impressed field and the unknown nonrotational part. In a finite element procedure this means that the vector field T 0 should be represented by edge elements and the scalar field Φ should be interpolated by nodal elements and the order of them must be the same. In this case, the Φ-formulation has been defined by (2.160)–(2.162) is satisfactory. It is advantageous, because the interface condition (2.183) can be difficult to implement, moreover edge element representation results in a tangentially continuous vector field.

34


E. The magnetic vector potential, the A-formulation The magnetic vector potential is defined by B = ∇ × A,

(2.185)

which satisfies (2.77) exactly, because of the identity ∇·∇×v ≡ 0 for any vector function v = v(r). Substituting the definition (2.185) into the first Maxwell’s equation (2.76) and using the linearized constitutive relation from (2.80), it leads to the partial differential equation ∇ × (νo ∇ × A) = J 0 − ∇ × I,

in Ω,

(2.186)

when using the source current density J 0 , or ∇ × (νo ∇ × A) = ∇ × T 0 − ∇ × I,

in Ω,

(2.187)

when using the impressed current vector potential T 0 to represent coils as it is introduced in (2.148). To ensure the uniqueness of the magnetic vector potential, the divergence of it can be selected according to Coulomb gauge, ∇ · A = 0.

(2.188)

This gauging is useful, because the vector potential A′ = A + ∇ϕ also satisfies (2.77) as well as (2.186) or (2.187), because of the identity ∇ × (∇ϕ) ≡ 0, where ϕ = ϕ(r) is a scalar function. This is the reason why the magnetic vector potential is not unique. Let us first study a two-dimensional problem, then a three-dimensional one. Gauging is satisfied automatically in 2D, but unfortunately it is not true in 3D. The origin of numerical problems is the lack of uniqueness of the magnetic vector potential. 2D problems. In 2D problems Coulomb gauge ∇ · A = 0 is satisfied automatically, if the source current density has only z component, the magnetic field intensity vector and the magnetic flux density vector have x and y components, i.e. J 0 =J0,z (x, y) ez , H =Hx (x, y) ex + Hy (x, y) ey ,

B = Bx (x, y) ex + By (x, y) ey .

(2.189)

In 2D, J 0 is used to represent source current density. The magnetic vector potential has only z component, A = Az (x, y) ez , because (Ax = 0, Ay = 0 and Az = Az (x, y)) ex ey ez ∂ ∂Az ∂Az ∂ 0 = ex B = ∇ × A = ∂x ∂y − ey , ∂y ∂x 0 0 Az

(2.190)

(2.191)


35

i.e. Bx (x, y) = ∂Az /∂y and By (x, y) = −∂Az /∂x. The divergence of this one component vector potential is equal to zero, ∂Az (x, y) = 0. (2.192) ∂z Now, let us define the boundary conditions of the A-formulation. On the part ΓH of the boundary, the tangential component of the magnetic field intensity vector can be set to the prescribed value by the relation ∇·A=

H ×n=K

⇒

(νo ∇ × A + I) × n = K,

on ΓH ,

(2.193)

which is a Neumann type boundary condition. The normal component of the magnetic flux density can be set as B · n = −b

⇒

(∇ × A) · n = −b,

on ΓB .

(2.194)

The left-hand side of the last formulation can be rewritten as (∇ × A) · n = ∇ · (A × n) = −b,

(2.195)

finally ∇ · (n × A) = b,

(2.196)

i.e. n × A = α,

on ΓB ,

(2.197)

where ∇ · α = b. This is a Dirichlet type boundary condition. The selection of α is not evident [16], but in many practical cases b = 0, so n × A = 0,

on ΓB

(2.198)

can be selected. Finally, the nonlinear partial differential equation and the boundary conditions of a two-dimensional static magnetic field problem, which solution satisfies Coulomb gauge can be formulated as ∇ × (νo ∇ × A) = J 0 − ∇ × I, (νo ∇ × A + I) × n = K,

n × A = α,

on ΓB .

in Ω,

on ΓH ,

(2.199) (2.200) (2.201)

3D problems. In three-dimensional problems, the uniqueness of the vector potential is not so evident and it can be prescribed by (i) implicit enforcement of Coulomb gauge (e.g. when using nodal elements in the finite element approximation), (ii) applying a numerical technique, which is not sensitive to Coulomb gauge (e.g. when using vector elements in the finite element approximation and taking care about the representation of source current density). The first method results in the gauged, the second one in the ungauged version of the A-formulation.

36


(i) Implicit enforcement of Coulomb gauge. When applying the partial differential equation (2.199) and boundary conditions (2.200) and (2.201) to solve a three-dimensional static magnetic field problem, the uniqueness of the vector potential is not prescribed, which results in numerical difficulties. The Coulomb gauge can be prescribed implicitly by the following formulations. The result of this formulation is a modification of the partial differential equation (2.199), moreover additional boundary conditions are appended to the original boundary value problem. The first step is to define the functional of the partial differential equation (2.199). Linear case is presented for simplicity, when νo = ν and I = 0, Z Z Z 1 2 K · A dΓ. (2.202) ν|∇ × A| dΩ − J 0 · A dΩ − F {A} = 2 Ω Ω ΓH According to the principle of minimum energy, this functional has minimum value only if the magnetic vector potential satisfies the partial differential equation (2.199) and the boundary conditions (2.200) and (2.201). This minimum is a stable configuration of the magnetic vector potential. Let us now modify this functional by introducing Coulomb gauge as Z Z Z 1 ν|∇ × A|2 dΩ − J 0 · A dΩ − F {A} = K · A dΓ 2 Ω Ω ΓH Z (2.203) 1 2 + ν(∇ · A) dΩ. 2 Ω This functional has minimum value only if the magnetic vector potential satisfies the partial differential equation (2.199), the boundary conditions (2.200) and (2.201), and Coulomb gauge, however, extra boundary conditions must be appended to the boundary value problem defined by the equations (2.199), (2.200) and (2.201). This procedure must be done, because we have two partial differential equations to be solved, (2.199) and ∇ · A = 0 with only one unknown A. The two partial differential equations can be formulated in one partial differential equation by the help of (2.203). The partial differential equation and boundary conditions, which minimize the functional (2.203) can be derived as follows. Let us introduce a small variation of the magnetic vector potential of A as A + αδA, which results in a variation of the field energy around the stable configuration of A, Z 1 ν|∇ × (A + αδA)|2 dΩ F {A + αδA} = 2 Ω Z Z J 0 · (A + αδA) dΩ − K · (A + αδA) dΓ (2.204) − Ω ΓH Z 1 ν[∇ · (A + αδA)]2 dΩ. + 2 Ω It is important to note that the small variation should not modify the prescribed Dirichlet boundary conditions, i.e. the vector field representing the small variation δA must satisfy homogeneous Dirichlet boundary condition.


37

The variation of the functional can be reformulated as Z Z 1 F {A + αδA} = ν|∇ × A|2 dΩ + α ν(∇ × A) · (∇ × δA) dΩ 2 Ω Ω Z α2 + ν|∇ × δA|2 dΩ 2 Ω Z Z J 0 · A dΩ − α J 0 · δA dΩ − ZΩ ZΩ (2.205) − K · A dΓ − α K · δA dΓ ΓH ΓH Z Z 1 2 + ν(∇ · A) dΩ + α ν(∇ · A)(∇ · δA) dΩ 2 Ω Ω Z α2 ν(∇ · δA)2 dΩ. + 2 Ω This variation can be represented in Taylor series approximation as F {A + αδA} = F {A} + αδF {A, δA} + α2 δ 2 F {δA}.

(2.206)

In concern with the stationary concept, the first variation δF {A, δA} has to be disappear and the second variation δ 2 F {δA} has to be positive. In this case F {A} leads indeed to its minimum value. The definition of the first variation is as follows (it must be equal to zero): ∂F {A + αδA} δF {A, δA} = = 0, (2.207) ∂α α=0 which is

δF {A, δA} = −

Z

ZΩ

Z

ν(∇ × A) · (∇ × δA) dΩ − J 0 · δA dΩ Ω Z ν(∇ · A)(∇ · δA) dΩ = 0. K · δA dΓ +

ΓH

(2.208)

Ω

It is easy to see that the second variation Z Z 1 1 δ 2 F {δA} = ν|∇ × δA|2 dΩ + ν(∇ · δA)2 dΩ 2 Ω 2 Ω

(2.209)

is usually positive, so (2.208) really states the minimum value of the functional (2.203). Let us now obtain the according partial differential equation and boundary conditions for (2.208). After using the identities ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(2.210)

∇ · (ϕv) = v · ∇ϕ + ϕ ∇ · v,

(2.211)

and

38


with u = δA, v = ν∇ × A, ϕ = ν∇ · A and v = δA, the following relation can be obtained: Z δA · [∇ × (ν∇ × A)] dΩ δF {A, δA} = IΩ Z + [δA × (ν∇ × A)] · n dΓ − J 0 · δA dΩ Γ Ω Z Z (2.212) δA · [∇(ν∇ · A)] dΩ K · δA dΓ − − Ω IΓH + ν∇ · A(δA · n) dΓ = 0. Γ

This equation can be rewritten in the form Z δF {A, δA} = δA · [∇ × (ν∇ × A) − J 0 − ∇(ν∇ · A)] dΩ Z ZΩ [(ν∇ × A) × n] · δA dΓ − K · δA dΓ + ΓH ZΓH ∪ΓB ν∇ · A(δA · n) dΓ = 0. +

(2.213)

ΓH ∪ΓB

The sum of the first and the second surface integrals on the part ΓH is equal to zero, because of the boundary condition (ν∇ × A) × n = K. The first boundary integral is equal to zero on the rest part ΓB , because δA must satisfy homogeneous Dirichlet boundary condition, i.e. n × δA = 0 on ΓB and it can be acknowledged according to the identity [(ν∇ × A) × n] · δA = [n × δA] · (ν∇ × A). The last boundary integral in (2.213) can be eliminated too by some extra boundary conditions, ν∇·A = 0 or A·n = 0 should be specified somewhere, but it must be formulated after some words about the first integral. Now, it can be seen that the partial differential equation to be solved is ∇ × (ν∇ × A) − ∇(ν∇ · A) = J 0 ,

in

Ω,

(2.214)

because in this case the first integral in (2.213) is equal to zero for any value of the variation δA. Let us take the divergence of equation (2.214), ∇ · [∇ × (ν∇ × A)] − ∇ · [∇(ν∇ · A)] = ∇ · J 0 .

(2.215)

The first term on the left-hand side and the right-hand side are obviously equal to zero, i.e. the Laplace equation −∇ · ∇(ν∇ · A) = 0,

in Ω

(2.216)

for the scalar variable ν∇ · A is left. Here the magnetic vector potential A is the only one independent variable, so only one partial differential equation should be used. Fortunately, the solution of a Laplace equation −∇ · ∇ϕ = 0 (ϕ = ϕ(r)) can be identically equal to


39

zero if the potential ϕ is prescribed on the whole boundary as a homogeneous Dirichlet type boundary condition, or a homogeneous Dirichlet type boundary condition on one part and a homogeneous Neumann type boundary condition on the rest part of the boundary. The solution of Laplace equation −∇ · ∇ϕ = 0,

in Ω

(2.217)

is identically zero in Ω if (α) ϕ = 0 on ∂Ω, or (β) ϕ = 0 on Γ1 and ∂ϕ/∂n = 0 on Γ2 and ∂Ω = Γ1 ∪ Γ2 , moreover n is the outer normal unit vector of the region Ω. Let us follow the second way. First, let us take the normal component of the partial differential equation (2.214) only on the boundary ΓH , n · [∇ × (ν∇ × A)] − n · ∇(ν∇ · A) = n · J 0 ,

on ΓH .

(2.218)

The first term on the left-hand side is equal to zero according to n · [∇ × (ν∇ × A)] = ∇ · [(ν∇ × A) × n] = ∇ · K = 0,

(2.219)

because of the current conversation law. The right-hand side is obviously zero, because the normal component of the source current density is equal to zero on the boundary ΓH . Finally, a homogeneous Neumann boundary condition −n · ∇(ν∇ · A) = −

∂ (ν∇ · A) = 0, ∂n

on ΓH

(2.220)

is specified automatically for the scalar quantity ν∇ · A. On the rest part of the boundary ΓB , homogeneous Dirichlet boundary condition must be specified, ν∇ · A = 0,

on ΓB .

(2.221)

In this way ν∇ · A ≡ 0 in the whole problem region Ω and on the boundary ΓH ∪ ΓB , i.e. the last surface integral in (2.213) can be eliminated. As a last check on the uniqueness of the vector potential, let us append a ∇ϕ term to the magnetic vector potential A as A + ∇ϕ and take the divergence of this term, ∇ · (A + ∇ϕ) = ∇ · A + ∇ · ∇ϕ = 0.

(2.222)

The first term of this equation is equal to zero according to Coulomb gauge, that is why only ∇ · ∇ϕ = 0 has to be analyzed. On the boundary part ΓB , the boundary condition n × (A + ∇ϕ) = α results in n × ∇ϕ = 0 for ϕ, i.e. ϕ is constant on ΓB . On the rest part ΓH , the normal component of ∇ϕ can be specified as ∇ϕ · n = 0. This can be obtained by the condition A · n = 0 on ΓH . Finally, it is known that if ∇ · ∇ϕ = 0 in Ω and ∇ϕ · n = ∂ϕ/∂n = 0 on ΓH , then the constant value of ϕ on ΓB can only be zero.

40


This means that ϕ = 0 in the problem region Ω and on the boundary ΓH ∪ ΓB , i.e. the magnetic vector potential is unique. Finally, the partial differential equation and the boundary conditions of the static magnetic field problem, which solution is unique according to Coulomb gauge can be written as follows: ∇ × (νo ∇ × A) − ∇(νo ∇ · A) = J 0 − ∇ × I, (νo ∇ × A + I) × n = K, A · n = 0, n × A = α, νo ∇ · A = 0,

on ΓH ,

in Ω,

(2.223) (2.224)

on ΓH ,

(2.225)

on ΓB ,

(2.226)

on ΓB .

(2.227)

This formulation is the so-called gauged A-formulation. (ii) Applying a numerical technique, which is not sensitive to Coulomb gauge. This formulation is newer than the first one. The only one rule is that the impressed current vector potential and the unknown magnetic vector potential must be approximated by the so-called vector elements in the finite element procedures. The partial differential equation and the boundary conditions of the static magnetic field problem, which solution is not sensitive to Coulomb gauge can be written as ∇ × (νo ∇ × A) = ∇ × T 0 − ∇ × I, (νo ∇ × A + I) × n = K, n × A = α,

on ΓB .

on ΓH ,

in Ω,

(2.228) (2.229) (2.230)

First, T 0 must be calculated by one of the methods presented on page 22. This is called ungauged A-formulation. F. Combination of the magnetic vector potential and the magnetic scalar potential, the A − Φ-formulation Applying the magnetic vector potential A in the whole domain of a static magnetic field problem is the general solution, however, applying the total or the reduced magnetic scalar potential Ψ or Φ (or their combination) is a more economic way. The combination of these methods results in the A − Φ-formulation as it is illustrated in Fig. 2.15. It is possible to use the magnetic vector potential only in the region filled with iron, ΩA and the reduced magnetic scalar potential in the air region ΩΦ where µ is equal to µ0 . The two potential formulations are coupled through interface conditions along the interface ΓA,Φ . The main advantage of this formulation is that the number of unknowns in the air domain can be decreased significantly, however, extra interface conditions have to be formulated. In the subregion ΩA , the partial differential equation of the magnetic vector potential is the same as in (2.223), but J 0 = 0 is true in iron. The boundary conditions on ΓHA and on ΓBA are the same as in (2.224), (2.225) and in (2.226), (2.227), respectively. The partial differential equation (2.160) and the boundary conditions (2.161), (2.162) are valid


41

n ΓBΦ ΩA

ΩΦ

nA

B =∇×A

H = T 0 − ∇Φ

nΦ

n

J 0 6= 0

J0 = 0 ΓBA

ΓA,Φ

ΓHΦ

µ = µ0

µ = µ0 µr or B{·} ΓHA

Fig. 2.15. The scheme of a static magnetic field problem with magnetic vector potential and reduced magnetic scalar potential in the subregion ΩΦ partially bounded by ΓHΦ and ΓBΦ , but R = 0 and µ = µ0 . The tangential component of the magnetic field intensity and the normal component of the magnetic flux density must be continuous on the interface between the two subregions, ΓA,Φ . Finally, the partial differential equations, the boundary and interface conditions of the static magnetic field problem, which solution is unique according to Coulomb gauge can be written as ∇ × (νo ∇ × A) − ∇(ν0 ∇ · A) = −∇ × I, ∇ · (µ0 ∇Φ) = ∇ · (µ0 T 0 ) , (νo ∇ × A + I) × n = K, A · n = 0,

ΩΦ ,

νo ∇ · A = 0,

on ΓHA ,

(2.233) (2.234)

on ΓBA ,

(2.235)

on ΓBA ,

(2.236)

on ΓHΦ ,

(µ0 T 0 − µ0 ∇Φ) · n = −b,

(2.237) on ΓBΦ ,

(2.238)

(νo ∇ × A + I) × nA + (T 0 − ∇Φ) × nΦ = 0, (∇ × A) · nA + µ0 (T 0 − ∇Φ) · nΦ = 0, A · nA = 0,

on ΓA,Φ .

(2.231) (2.232)

on ΓHA ,

n × A = α, Φ = Φ0 ,

in

in ΩA ,

on ΓA,Φ ,

on ΓA,Φ ,

(2.239) (2.240) (2.241)

42


On the ΓA,Φ part of the boundary of the subdomain ΩA (∂ΩA = ΓHA ∪ΓBA ∪ΓA,Φ ), the normal component of the magnetic vector potential must be set to zero (see equation (2.241)). Coulomb gauge ∇ · A = 0 is satisfied in the domain ΩA partially bounded by ΓHA ∪ ΓBA , but it is not true on the new boundary part ΓA,Φ . Here either the normal or the tangential component of the magnetic vector potential must be specified. According to the condition (2.239), the tangential component of the magnetic field intensity is specified by the unknown scalar potential, i.e. (νo ∇ × A + I) × nA = −(T 0 − ∇Φ) × nΦ , which is similar to the condition (2.233). The boundary part ΓA,Φ is similar to the part ΓHA , where the normal component of the magnetic vector potential has been specified. Unfortunately, the behavior of the magnetic field in the vicinity of the iron/air interface may be very strange if the finite element method is used, because of the weak coupling between the two regions represented by the equations (2.239) and (2.240). The reason of it is that the normal component of the magnetic flux density in iron and the tangential component of the magnetic field intensity in air may be very low as it is illustrated in Fig. 2.16 (here, an iron cube is placed into a homogeneous magnetic field H0 ). It is easy to see that B · n ≈ 0 on the left side of the iron cube, where H × n ≈ 0 as well and this strange behavior can not be eliminated by decreasing the size of mesh in the vicinity of iron/air interface. A possible solution of this numerical difficulty is moving the A/Φ interface from the iron/air interface into the air region, which results in the so-called A − A − Φ-formulation (Fig. 2.16). air

ΓH

ΓB

ΓH

ΓB

Φ ΓA,Φ A

iron A

H0

Fig. 2.16. Illustration for the A − A − Φ-formulation At the end of this section, the partial differential equations, the boundary and interface conditions of the static magnetic field problem in the case of ungauged version are also presented, ∇ × (νo ∇ × A) = −∇ × I, ∇ · (µ0 ∇Φ) = ∇ · (µ0 T 0 ) ,

(νo ∇ × A + I) × n = K,

in ΩA , in

ΩΦ ,

on ΓHA ,

(2.242) (2.243) (2.244)


n × A = α, Φ = Φ0 ,

43

on ΓBA ,

(2.245)

on ΓHΦ ,

(2.246)

(µ0 T 0 − µ0 ∇Φ) · n = −b,

on ΓBΦ ,

(2.247)

(νo ∇ × A + I) × nA + (T 0 − ∇Φ) × nΦ = 0, (∇ × A) · nA + µ0 (T 0 − ∇Φ) · nΦ = 0,

on ΓA,Φ ,

on ΓA,Φ .

(2.248) (2.249)

2.3.2 Eddy current fields Eddy current field is defined by the equations (2.89), (2.90), (2.91), (2.92) or (2.94) and (2.93) and the boundary conditions introduced in section 2.2.2. Two potential functions can be used in the eddy current region, either a current vector potential T or a magnetic vector potential A. The current vector potential T can be coupled with a magnetic scalar potential, here the reduced magnetic scalar potential Φ is used. The magnetic vector potential A can be coupled with an electric scalar potential, denoted by V . A. The current vector potential and magnetic scalar potential, the T , Φ-formulation The solenoidal property of the induced eddy current density (2.95) results in the possibility of applying the current vector potential T to represent the eddy current field in conducting materials, ∇·J =0

⇒

J = ∇ × T,

(2.250)

because of the identity ∇ · ∇ × v ≡ 0 for any vector function v = v(r), or v = v(r, t). Substituting this relation to the first Maxwell’s equation (2.89), i.e. ∇×H =∇×T

⇒

∇ × (H − T ) = 0

(2.251)

results in the reduced magnetic scalar potential Φ as H − T = −∇Φ

⇒

H = T − ∇Φ,

(2.252)

because ∇ × ∇ϕ ≡ 0 for any scalar function ϕ = ϕ(r) or ϕ = ϕ(r, t). The first Maxwell’s equation (2.89) has been satisfied exactly by this formulation. Applying the impressed current vector potential T 0 to represent the known source current density J 0 placed in the eddy current free region takes easier the coupling of the present formulation with the reduced magnetic scalar potential in the eddy current free region, i.e. appending T 0 to (2.252) is advantageous, H = T 0 + T − ∇Φ.

(2.253)

This can be done, since ∇ × T 0 = 0 in the eddy current region Ωc . It is noted that the reduced magnetic scalar potential is used in this formulation and we have to take care about the representation of T 0 (see point D on page 33).

44


The expression J = σE = ∇×T can be written according to the constitutive relation (2.93), from which the electric field intensity E can be expressed by the current vector potential as E=

1 ∇ × T. σ

(2.254)

Substituting this expression and the linearized form of the constitutive relation in (2.92) into Faraday’s law (2.90) results in the partial differential equation ∂Φ ∂R ∂T ∂T 0 1 (2.255) ∇ × T + µo − µo ∇ = −µo − , in Ωc . ∇× σ ∂t ∂t ∂t ∂t The magnetic Gauss’ law (2.91) can be rewritten in the form ∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R,

in

Ωc .

(2.256)

The solution of these partial differential equations with the below described boundary conditions results in two unknowns (T and Φ) of the T , Φ-formulation. Let us now define the boundary conditions. On the ΓHc part of the boundary, the tangential component of the magnetic field intensity vector must vanish, H ×n=0

⇒

(T 0 + T − ∇Φ) × n = 0,

on ΓHc .

(2.257)

The boundary part ΓHc always represents a symmetry plane, where T 0 × n ≡ 0, i.e. the Dirichlet boundary conditions T × n = 0,

and Φ = Φ0 ,

on ΓHc

(2.258)

have to be specified and Φ0 is constant. On ΓHc , the equation J · n = 0 satisfies automatically, because of J · n = (∇ × T ) · n = ∇ · (T × n),

(2.259)

and T × n = 0 is prescribed here. On the rest of boundary ΓE , the tangential component of the electric field intensity must be equal to zero, 1 E×n=0 ⇒ (2.260) ∇ × T × n = 0, on ΓE , σ and the normal component of the magnetic flux density must vanish, B·n =0

⇒

(µo T 0 + µo T − µo ∇Φ + R) · n = 0,

on ΓE .

(2.261)

These are Neumann type boundary conditions. Finally, here is the collection of partial differential equations and boundary conditions of the T , Φ-formulation, which is however, not gauged, ∂R ∂T ∂T 0 1 ∂Φ ∇ × T + µo − µo ∇ = −µo − , in Ωc , ∇× (2.262) σ ∂t ∂t ∂t ∂t


∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R, T × n = 0,

in

on ΓHc ,

Φ = Φ0 , on ΓHc , 1 ∇ × T × n = 0, σ

45 Ωc ,

(2.263) (2.264) (2.265)

on ΓE ,

(µo T 0 + µo T − µo ∇Φ + R) · n = 0,

(2.266) on ΓE .

(2.267)

The solution of the problem defined by the above equations and boundary conditions is not unique, because the divergence of the current vector potential has not specified yet. The Coulomb gauge should be used in this formulation, similarly to the gauge fixing method applied in the A-formulation (see on page 35), i.e. ∇ · T = 0,

in Ωc

(2.268)

must be specified. Let us first append a −∇ (1/σ∇ · T ) term to the left-hand side of the equation (2.262), 1 1 ∂T ∂Φ ∇×T −∇ ∇ · T +µo − µo ∇ ∇× σ σ ∂t ∂t (2.269) ∂R ∂T 0 = − µo − , in Ωc ∂t ∂t Taking the divergence of this equation and taking equation (2.263) into account results in a Laplace equation for the scalar variable 1/σ∇ · T , 1 −∇ · ∇ ∇ · T = 0. (2.270) σ The solution of a Laplace equation −∇ · ∇ϕ = 0 can be equal to zero if the potential ϕ is prescribed on the whole boundary by a homogeneous Dirichlet boundary condition, or a homogeneous Dirichlet boundary condition on one part and a homogeneous Neumann boundary condition on the rest of the boundary. After taking the normal component of the equation (2.269) on the boundary segment ΓE , a homogeneous Neumann boundary condition can be prescribed automatically 1 ∂ ∇ · T = 0, on ΓE , − (2.271) ∂n σ because the normal component of the first term in (2.269), 1 1 ∇×T ·n= ∇· ∇×T ×n ∇× σ σ

(2.272)

is equal to zero according to the boundary condition (2.266) and the normal component of the sum of the last four terms in (2.269) is equal to zero, too, because of the condition (2.267).

46


Consequently, on the rest of the boundary ΓHc , the following Neumann boundary condition: 1 ∇ · T = 0, σ

on ΓHc

(2.273)

must be specified. In this way 1/σ∇ · T ≡ 0 in the whole problem region Ωc and on the boundary ΓHc ∪ ΓE . Finally, the partial differential equations and the boundary conditions of an eddy current field problem, which solution is unique according to Coulomb gauge can be written as 1 ∂T 1 ∂Φ ∇×T −∇ ∇ · T +µo − µo ∇ ∇× σ σ ∂t ∂t (2.274) ∂R ∂T 0 = − µo − , in Ωc , ∂t ∂t ∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R, T × n = 0, Φ = Φ0 ,

in

on ΓHc ,

(2.277)

1 ∇ · T = 0, on ΓHc , σ 1 ∇ × T × n = 0, on ΓE , σ

T · n = 0,

on ΓE .

(2.275) (2.276)

on ΓHc ,

(µo T 0 + µo T − µo ∇Φ + R) · n = 0,

Ωc ,

(2.278) (2.279) on ΓE ,

(2.280) (2.281)

The last boundary condition (2.281) is introduced analogously to the boundary condition (2.225) in the A-formulation. As a last check of the uniqueness of the current vector potential, let us append a ∇ϕ term to the vector potential T as T + ∇ϕ and take the divergence of this term, ∇ · (T + ∇ϕ) = ∇ · T + ∇ · ∇ϕ = 0.

(2.282)

The first term of this equation is equal to zero according to Coulomb gauge, i.e. only ∇ · ∇ϕ = 0 has to be analyzed. On the boundary part ΓHc , the boundary condition (T + ∇ϕ) × n = 0 results in (∇ϕ) × n = 0, i.e. ϕ is constant on ΓHc . On the rest part ΓE , the normal component of ∇ϕ can be specified as ∇ϕ · n = 0. This can be obtained by introducing T · n = 0 on ΓE . Finally, it is known that ∇ · ∇ϕ = 0 in Ω and ∇ϕ · n = ∂ϕ/∂n = 0 on ΓE , the constant value of ϕ on ΓHc can only be zero. This means that ϕ = 0 in the problem region Ω and on the boundary ΓHc ∪ ΓE , i.e. the current vector potential is unique.


47

B. The magnetic vector potential and electric scalar potential, the A, V -formulation The divergence-free magnetic flux density vector can be described by the curl of the magnetic vector potential A, since ∇ · ∇ × u ≡ 0, for any vector function u = u(r), or u = u(r, t), i.e. B = ∇ × A.

(2.283)

This automatically enforces the satisfaction of magnetic Gauss’ law (2.91). Substituting expression (2.283) into Faraday’s law (2.90) results in ∂A ∂A ∂ = 0, (2.284) ⇒ ∇× E + ∇ × E = − ∇ × A = −∇ × ∂t ∂t ∂t because rotation (i.e. derivation by space) and derivation by time can be replaced. The curl-less vector field E+∂A/∂t can be derived from the so-called electric scalar potential V (∇ × ∇ϕ ≡ 0, for any scalar functions ϕ = ϕ(r), or ϕ = ϕ(r, t)), E+

∂A = −∇V, ∂t

(2.285)

and the electric field intensity vector can be described by two potentials as E=−

∂A − ∇V. ∂t

(2.286)

Substituting the relations (2.283) and (2.286) into (2.89) and using the linearized constitutive relation in (2.94) leads to the partial differential equation ∇ × (νo ∇ × A) + σ

∂A + σ∇V = −∇ × I, ∂t

in Ωc .

(2.287)

The charge conservation law (2.95) with the constitutive relation (2.93) and with the formulation (2.286) results in the second partial differential equation of this formulation, ∂A (2.288) −∇ · σ + σ∇V = 0, in Ωc . ∂t There are two unknown functions (A and V ), that is why two equations must be formulated, however, the second one is coming from taking the divergence of Ampere’s law (2.89). Now, let us define the boundary conditions of the problem. On the ΓHc part of the boundary, the tangential component of the magnetic field intensity vector must vanish, H ×n=0

⇒

(νo ∇ × A + I) × n = 0,

on ΓHc .

(2.289)

This is a Neumann boundary condition for A. The normal component of eddy currents must be equal to zero on ΓHc , which can be formulated by the Neumann boundary condition J ·n

⇒

−σ

∂A · n − σ∇V · n = 0, ∂t

on ΓHc .

(2.290)

48


On the rest part of the boundary ΓE , the tangential component of the electric field intensity must be equal to zero, ∂A (2.291) E×n=0 ⇒ − − ∇V × n = 0, on ΓE . ∂t This boundary condition can be specified by two Dirichlet boundary conditions, n × A = 0,

and V = V0 ,

on ΓE ,

(2.292)

because −A× n = n × A and V0 is constant. Here B · n = 0 satisfies explicitly, because B · n = (∇ × A) · n = ∇ · (A × n),

(2.293)

and A × n = 0 has been prescribed yet. Finally, here is the collection of equations and boundary conditions of the ungauged A, V -formulation, ∂A + σ∇V = −∇ × I, ∂t ∂A + σ∇V = 0, in Ωc , −∇ · σ ∂t ∇ × (νo ∇ × A) + σ

in Ωc ,

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A · n − σ∇V · n = 0, on ΓHc , −σ ∂t n × A = 0, on ΓE , V = V0 ,

on ΓE .

(2.294) (2.295) (2.296) (2.297) (2.298) (2.299)

The solution of the problem defined by the above equations and boundary conditions is not unique, because the divergence of the magnetic vector potential has not specified yet. The Coulomb gauge should be used in this formulation similarly to the gauge fixing method applied in A-formulation of static magnetic field problems (see on page 35), i.e. ∇ · A = 0 must be specified. First, let us append the left-hand side of the partial differential equation (2.294) by the term −∇ (νo ∇ · A), ∂A + σ∇V = −∇ × I, in Ωc . (2.300) ∂t Taking the divergence of this equation and taking the equation (2.295) into account, it results in the Laplace equation for the scalar variable νo ∇ · A, ∇ × (νo ∇ × A) − ∇ (νo ∇ · A) + σ

−∇ · ∇ (νo ∇ · A) = 0.

(2.301)

After taking the normal component of the equation (2.300) on the boundary segment ΓHc , a homogeneous Neumann boundary condition can be set up automatically ∂ (νo ∇ · A) = 0, on ΓHc , ∂n because the normal component of the first and the last terms in (2.300), −

(2.302)


49

[∇ × (νo ∇ × A + I)] · n = ∇ · [(νo ∇ × A + I) × n]

(2.303)

is equal to zero according to the boundary condition (2.296) and the normal component of the sum of the last two terms in (2.300) is equal to zero, too, because of the condition (2.297). Consequently, on the rest part of the boundary ΓE , the following Neumann type boundary condition: νo ∇ · A = 0,

on ΓE

(2.304)

must be specified. In this way νo ∇ · A ≡ 0 can be satisfied in the whole problem region Ωc and on the boundary ΓHc ∪ ΓE . Finally, the partial differential equations and the boundary conditions of an eddy current field problem, which solution is unique according to Coulomb gauge can be written as ∇ × (νo ∇ × A) − ∇ (νo ∇ · A) + σ ∂A −∇ · σ + σ∇V = 0, ∂t

∂A + σ∇V = −∇ × I, ∂t

in Ωc , (2.305)

in Ωc ,

(2.306)

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A + σ∇V · n = 0, on ΓHc , − σ ∂t A · n = 0,

n × A = 0, V = V0 ,

(2.307) (2.308)

on ΓHc ,

(2.309)

on ΓE ,

(2.310)

on ΓE ,

νo ∇ · A = 0,

(2.311)

on ΓE .

(2.312)

Here, equation (2.309) is introduced according to the proof presented on page 39. C. The modified magnetic vector potential, the A⋆ -formulation This formulation can only be applied when the ungauged version of the A, V -formulation is used. In this situation, the electric scalar potential V can be supposed to be equal to zero, if and only if, the conductivity σ is constant. The set of partial differential equations and boundary conditions are coming from the equations (2.294)–(2.299), but the partial differential equations and boundary conditions containing V have been eliminated, i.e. ∇ × (νo ∇ × A⋆ ) + σ

∂A⋆ = −∇ × I, ∂t

(νo ∇ × A⋆ + I) × n = 0, ⋆

n × A = 0,

on ΓE .

on ΓHc ,

in Ωc ,

(2.313) (2.314) (2.315)

50


2.3.3 Coupling static magnetic and eddy current fields In most eddy current field problems, the conductors carrying the eddy currents are at least partially surrounded by a nonconducting medium free of eddy currents (e.g. air) where a static magnetic field is present (Fig. 2.17). The static magnetic field is induced both by the eddy currents and by the source current of coils. That is why the potential formulations of the static magnetic field and of the eddy current field must be coupled. n ΓB J0 = 0

nc

Ωn

nn

n

J 0 6= 0 Γnc

Ωc

ΓHn

ΓE µ = µ0 µr or B{·}

µ = µ0 µr

ΓHc Fig. 2.17. The eddy current region is surrounded by nonconducting region

The static magnetic field in Ωn can be described by a magnetic scalar potential (reduced, total or the combination of them), or by a magnetic vector potential. In the first case, applying the reduced magnetic scalar potential Φ is simpler to use, however, currents of coils must be represented by an impressed current vector potential T 0 , which must be realized in a special way. Application of the magnetic vector potential is a more general way. Combination of the two formulations can also be supposed, e.g. A − Φ, or A − A − Φ method. The eddy current field in Ωc can be represented by a vector potential coupled with a scalar potential. The current vector potential T can be coupled with the reduced magnetic scalar potential Φ and the magnetic vector potential A can be coupled with the electric scalar potential V . It is important to note that the vector potentials can be either ungauged or gauged. The ungauged version has infinite number of solutions for the potentials, but electromagnetic field quantities calculated from the potentials are unique, of course, since the solution of Maxwell’s equations is unique. The gauged versions result in only one unique solution for the potentials, too. Here, the possible potential formulations are shown in the gauged and ungauged situations. The partial differential equations and the boundary conditions are based on


51

the last section, but the interface conditions between the regions with and without eddy currents are presented below. First, the basic formulations are shown, the T , Φ formulation in the eddy current region is coupled to the Φ formulation in the eddy current free region resulting the T , Φ − Φ formulation, then the A, V formulation in the eddy current region is coupled to the A formulation in the eddy current free region resulting the A, V − A formulation. The T , Φ − Φ formulation can not be used when the eddy current region is multiply connected. This problem can be solved by the T , Φ − A formulation and by the T , Φ − A − Φ formulation. The A, V − A formulation is not an economic procedure, there are three unknown functions to represent A in air, that is why the A, V − Φ and the A, V − A − Φ formulations have been developed. A. The gauged T , Φ − Φ formulation The reduced magnetic scalar potential Φ is used in the eddy current free region Ωn as well as in the eddy current region Ωc . It is a continuous scalar variable in the entire region Ωc ∪ Ωn and on the interface Γnc , too. The equations to be used are (2.160)–(2.162) and (2.274)–(2.281), however, some continuity equations on Γnc must be appended to these equations. The magnetic field intensity vector is derived as H = T 0 − ∇Φ in Ωn and it is written as H = T 0 + T − ∇Φ in Ωc . The tangential component of the magnetic field intensity can be set to be continuous on Γnc by a continuous magnetic scalar potential and by setting the tangential component of the current vector potential equal to zero by the boundary condition T × n = 0 on Γnc . It must be noted that T 0 × n is continuous. Vanishing the normal component of eddy current density on Γnc satisfies automatically, because J = ∇ × T and J · n = (∇ × T ) · n = (T × n) · ∇. The term in the last brackets has been set to zero, i.e. J · n = 0 on Γnc . The continuity of the normal component of magnetic flux density results in a Neumann type boundary condition (see (2.329)). The divergence free property of T has been defined only in the domain Ωc and on the boundary ΓHc ∪ ΓE . On the rest segment Γnc , 1/σ∇ · T = 0 must be specified as a new Neumann type interface condition. It can be introduced similarly to (2.273). The summarized equations of the gauged version are as follows (see Fig. 2.17): 1 ∂Φ 1 ∂T ∇× ∇×T −∇ ∇ · T +µo − µo ∇ σ σ ∂t ∂t (2.316) ∂R ∂T 0 − , in Ωc , = − µo ∂t ∂t ∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R, −∇ · (µ∇Φ) = −∇ · (µT 0 ) , T × n = 0, Φ = Φ0 ,

on ΓHc ,

on ΓHc ,

1 ∇ · T = 0, σ

on ΓHc ,

in Ωn ,

in

Ωc ,

(2.317) (2.318) (2.319) (2.320) (2.321)

52


Φ = Φ0 , on ΓHn , 1 ∇ × T × n = 0, σ

(2.322) on ΓE ,

(µo T 0 + µo T − µo ∇Φ + R) · n = 0, T · n = 0,

(2.323) on ΓE ,

(2.324)

on ΓE ,

(µT 0 − µ∇Φ) · n = −b,

(2.325) on ΓB ,

(2.326)

Φ is continuous on Γnc ,

(2.327)

T × nc = 0,

(2.328)

on Γnc ,

(µT 0 − µ∇Φ) · nn + (µo T 0 + µo T − µo ∇Φ + R) · nc = 0, on Γnc , (2.329) 1 ∇ · T = 0, σ

on Γnc .

(2.330)

B. The ungauged T , Φ − Φ formulation The reduced magnetic scalar potential Φ is used in the eddy current free region Ωn as well as in the eddy current region Ωc and it is a scalar continuous variable in the entire region Ωc ∪ Ωn and on the interface Γnc , too. The equations of ungauged T , Φ − Φ formulation can be built up by using (2.160)–(2.162) and (2.262)–(2.267), which must be appended by some continuity equations defined on Γnc . The interface conditions can be defined as in the gauged version, however, (2.330) is not used. The summarized equations are as follows (see Fig. 2.17): ∂Φ 1 ∂R ∂T ∂T 0 (2.331) ∇× ∇ × T + µo − µo ∇ = −µo − , in Ωc , σ ∂t ∂t ∂t ∂t ∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R, −∇ · (µ∇Φ) = −∇ · (µT 0 ) , T × n = 0, Φ = Φ0 ,

in

in Ωn ,

(2.334)

on ΓHc ,

(2.335) (2.336) on ΓE ,

(µo T 0 + µo T − µo ∇Φ + R) · n = 0, (µT 0 − µ∇Φ) · n = −b,

(2.332) (2.333)

on ΓHc ,

Φ = Φ0 , on ΓHn , 1 ∇ × T × n = 0, σ

Ωc ,

on ΓB ,

(2.337) on ΓE ,

(2.338) (2.339)

Φ is continuous on Γnc ,

(2.340)

T × nc = 0,

(2.341)

on Γnc ,

(µT 0 − µ∇Φ) · nn + (µo T 0 + µo T − µo ∇Φ + R) · nc = 0, on Γnc . (2.342)


53

Note. The T , Φ − Φ formulation can not be used in case of multiply connected region. In this situation the formulations T , Φ − A or T , Φ − A − Φ can be used. Note. The T , Φ − Φ formulation can not be useful in case of multiply connected region, however, a simple modification can be introduced in this case. The holes filled with nonconducting materials (σ = 0) can be replaced by conducting materials with very small conductivity (σ ≈ 0 comparing with the conductivity of the conducting material, e.g. σ = σc /a, where a is constant). C. The gauged A, V − A formulation The magnetic vector potential A is used in this formulation throughout the region Ωc ∪Ωn and the electric scalar potential V only in Ωc . Here, the equations (2.223)–(2.227) and (2.305)–(2.312) have to be used to prepare the formulation, but the set of these equations have to be appended by continuity equations defined on Γnc . The magnetic vector potential is continuous, meaning that the tangential and the normal component of the magnetic vector potential are continuous on Γnc . The continuity of the tangential component of the magnetic vector potential immediately enforces the continuity of the normal component of the magnetic flux density from (2.101) and (2.357), (∇ × A) · nc + (∇ × A) · nn =∇ · (A × nc ) + ∇ · (A × nn ) =∇ · (A × nc + A × nn ) = 0.

(2.343)

The continuity of the tangential component of the magnetic field intensity vector must be prescribed by an additional interface condition on Γnc (see the Neumann type condition (2.359)). It is obvious that the normal component of the eddy current density must vanish on Γnc . The divergence free property of A has been defined only in the domain Ωc ∪ Ωn bounded by ΓHc ∪ ΓE ∪ ΓHn ∪ ΓB , but not on Γnc . On the part Γnc ν∇ · A must be continuous, which can be prescribed by a Neumann type condition (see (2.361)). The summarized equations are as follows (see Fig. 2.17): ∇ × (νo ∇ × A) − ∇ (νo ∇ · A) + σ ∂A −∇ · σ + σ∇V = 0, ∂t

∂A + σ∇V = −∇ × I, ∂t

in Ωc ,

∇ × (ν∇ × A) − ∇(ν∇ · A) = J 0 ,

in

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A + σ∇V · n = 0, on ΓHc , − σ ∂t A · n = 0,

on ΓHc ,

(ν∇ × A) × n = K, A · n = 0,

on ΓHn ,

in Ωc , (2.344) (2.345)

Ωn ,

(2.346) (2.347) (2.348) (2.349)

on ΓHn ,

(2.350) (2.351)

54


n × A = 0, V = V0 ,

on ΓE ,

(2.352)

on ΓE ,

(2.353)

νo ∇ · A = 0,

on ΓE ,

(2.354)

n × A = α,

on ΓB ,

(2.355)

ν∇ · A = 0,

on ΓB ,

(2.356)

nc × A + nn × A = 0, A · nc + A · nn = 0,

on Γnc ,

(2.357)

on Γnc ,

(2.358)

(νo ∇ × A + I) × nc + (ν∇ × A) × nn = 0, ∂A + σ∇V · nc = 0, on Γnc , − σ ∂t (νo ∇ · A) nc + (ν∇ · A) nn = 0,

on Γnc ,

(2.359) (2.360)

on Γnc .

(2.361)

Note. It must be highlighted that the interface condition (2.358) must be eliminated, if the permeability is changing abruptly along the interface Γnc . In this case Γnc is an iron/air interface (remember Fig. 2.16). The condition ∇ · A = 0 on Γnc can be satisfied according to the discretization. The problem can be solved by the ungauged A, V − A formulation too, presented in the next item. D. The ungauged A, V − A formulation The magnetic vector potential A is used in this formulation in the region Ωc ∪ Ωn and the electric scalar potential V only in Ωc . The equations of ungauged A, V − A formulation can be built up by using (2.228)–(2.230) and (2.294)–(2.299), which must be appended by some continuity equations defined on Γnc . The interface conditions can be defined as in the case of gauged version, however, equations (2.349), (2.351), (2.354), (2.356), (2.358) and (2.361) are not used. Here, T 0 is used to represent the source current density J 0 . The summarized equations are as follows (see Fig. 2.17): ∂A + σ∇V = −∇ × I, ∂t ∂A + σ∇V = 0, in Ωc , −∇ · σ ∂t

∇ × (νo ∇ × A) + σ

∇ × (ν∇ × A) = ∇ × T 0 ,

in

Ωn ,

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A + σ∇V · n = 0, on ΓHc , − σ ∂t (ν∇ × A) × n = K, n × A = 0,

on ΓE ,

on ΓHn ,

in Ωc ,

(2.362) (2.363) (2.364) (2.365) (2.366) (2.367) (2.368)


V = V0 ,

55

on ΓE ,

n × A = α,

(2.369)

on ΓB ,

(2.370)

nc × A + nn × A = 0,

on Γnc ,

(2.371)

(νo ∇ × A + I) × nc + (ν∇ × A) × nn = 0, ∂A + σ∇V · nc = 0, on Γnc . − σ ∂t

on Γnc ,

(2.372) (2.373)

Note. If σ is constant in Ωc and the ungauged A, V −A formulation is used, then V = 0 can be selected. This results in the ungauged A⋆ − A formulation. Note. A more economical way of A-based formulations is using the reduced magnetic scalar potential in the air region resulting the A, V −Φ and the A, V −A−Φ formulations. E. The ungauged A⋆ − A formulation The equations of this formulation are (2.228)–(2.230) and (2.313)–(2.315) appended by the continuity equations (2.100) and (2.101) (see Fig. 2.17), ∇ × (νo ∇ × A⋆ ) + σ

∂A⋆ = −∇ × I, ∂t

∇ × (ν∇ × A) = ∇ × T 0 ,

in

(νo ∇ × A + I) × n = 0,

on ΓHc ,

⋆

(ν∇ × A) × n = K, ⋆

in Ωc ,

Ωn ,

(2.374) (2.375) (2.376)

on ΓHn ,

(2.377)

n × A = 0,

on ΓE ,

(2.378)

n × A = α,

on ΓB ,

(2.379)

⋆

nc × A + nn × A = 0,

on Γnc ,

(νo ∇ × A⋆ + I) × nc + (ν∇ × A) × nn = 0,

(2.380) on Γnc .

(2.381)

F. The gauged T , Φ − A formulation The T , Φ − Φ formulation is not capable of treating multiply connected conductors when all the eddy current free region is described by the reduced magnetic scalar potential Φ. This difficulty can be overcome by applying the magnetic vector potential in the eddy current free region, Ωn . This is called T , Φ − A formulation. In this formulation, the gauged current vector potential T with the reduced magnetic scalar potential Φ is used in the eddy current region Ωc and the gauged magnetic vector potential A is used in the region free of eddy currents, Ωn . The equations to be used are (2.274)–(2.281) and (2.223)–(2.227). Some continuity equations and boundary conditions defined on Γnc must be appended. The magnetic field intensity is derived as H = T 0 +T −∇Φ in Ωc and H = ν∇×A in Ωn , while the magnetic flux density can be written as B = µo (T 0 + T − ∇Φ) + R

56


in Ωc and B = ∇ × A in Ωn . The continuity equations (2.100) and (2.101) can be reformulated by these equations. The potential functions are not continuous on Γnc , therefore this surface acts as a boundary of the two subregions. It must be noted here that the uniqueness of a vector potential can be ensured by defining either its tangential component and its divergence or its normal component and the tangential component of its curl. The tangential component of the current vector potential can not be set to zero on the interface Γnc since this would imply that no net current can flow around the hole filled by A. That is why the other conditions must be prescribed here, i.e. T · n = 0 on Γnc . The curl of the current vector potential is the electric field intensity since E = 1/σ∇ × T , which tangential component is continuous on any surface, i.e. E c × nc + E n × nn = 0 must be satisfied. From the second Maxwell’s equation (2.90) ∂ ∂ (∇ × E) · n = − (∇ × A) · n ⇒ ∇·(E × n) =∇· − (A × n) , (2.382) ∂t ∂t from which E×n=

1 ∇×T σ

×n=−

∂ A × n. ∂t

(2.383)

This is a condition for the tangential component of the magnetic vector potential too, which uniqueness on Γnc can be ensured by prescribing its divergence. The summarized equations of the gauged version are as follows (see Fig. 2.17): ∂T ∂Φ 1 1 ∇× ∇×T −∇ ∇ · T +µo − µo ∇ σ σ ∂t ∂t (2.384) = − µo ∂T 0 /∂t − ∂R/∂t, in Ωc , ∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R, ∇ × (ν∇ × A) − ∇(ν∇ · A) = J 0 , T × n = 0, Φ = Φ0 ,

in

in

Ωn ,

on ΓHc ,

(2.389) (2.390)

A · n = 0, on ΓHn , 1 ∇ × T × n = 0, on ΓE , σ

n × A = α,

on ΓE , on ΓB ,

(2.386)

(2.388)

1 ∇ · T = 0, on ΓHc , σ (ν∇ × A) × n = K, on ΓHn ,

T · n = 0,

(2.385)

(2.387)

on ΓHc ,

(µo T 0 + µo T − µo ∇Φ + R) · n = 0,

Ωc ,

(2.391) (2.392) on ΓE ,

(2.393) (2.394) (2.395)


ν∇ · A = 0,

57

on ΓB ,

(2.396)

(T 0 + T − ∇Φ) × nc + (ν∇ × A) × nn = 0,

on Γnc ,

(µo T 0 + µo T − µo ∇Φ + R) · nc + (∇ × A) · nn = 0, T · nc = 0, on Γnc , ∂A 1 ∇ × T × nc − × nn = 0, σ ∂t

ν∇ · A = 0,

on Γnc ,

(2.397) (2.398) (2.399)

on Γnc ,

(2.400)

on Γnc .

(2.401)

G. The ungauged T , Φ − A formulation In this formulation, the ungauged current vector potential T with the reduced magnetic scalar potential Φ is used in the eddy current region Ωc and the ungauged magnetic vector potential A is used in the region free of eddy currents, Ωn . The ungauged T , Φ − A formulation can be built up by using (2.262)–(2.267) and (2.228)–(2.230), which must be appended by some continuity and boundary conditions defined on Γnc . The interface conditions can be defined as in the gauged version, however, equations (2.389), (2.391), (2.394), (2.396), (2.399) and (2.401) are not used. The impressed current vector potential T 0 is used in this formulation to represent J 0 . The summarized equations of the gauged version are as follows (see Fig. 2.17): ∂R 1 ∂T ∂T 0 ∂Φ ∇ × T + µo − µo ∇ = −µo − , in Ωc , ∇× (2.402) σ ∂t ∂t ∂t ∂t ∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R, ∇ × (ν∇ × A) = ∇ × T 0 , T × n = 0, Φ = Φ0 ,

in

in

Ωc ,

(2.403)

Ωn ,

(2.404)

on ΓHc ,

(2.405)

on ΓHc ,

(2.406)

(ν∇ × A) × n = K, on ΓHn , 1 ∇ × T × n = 0, on ΓE , σ (µo T 0 + µo T − µo ∇Φ + R) · n = 0, n × A = α,

(2.407) (2.408) on ΓE ,

(2.409)

on ΓB ,

(T 0 + T − ∇Φ) × nc + (ν∇ × A) × nn = 0,

(2.410) on Γnc ,

(µo T 0 + µo T − µo ∇Φ + R) · nc + (∇ × A) · nn = 0, 1 ∂A ∇ × T × nc − × nn = 0, on Γnc . σ ∂t

on Γnc ,

(2.411) (2.412) (2.413)

Note. A more economical way of the T , Φ − A formulations is using the reduced magnetic scalar potential in a subregion of Ωn . This is the T , Φ − A − Φ formulations.

58


H. The gauged T , Φ − A − Φ formulation This is a modification of the gauged T , Φ − A formulation. Applying the magnetic vector potential in the static magnetic field region is not economical, because it requires three scalar unknown functions (Ax , Ay and Az ). This can be reduced to one by using the reduced magnetic scalar potential in a subregion of air, which is simple connected. The current vector potential T with the reduced magnetic scalar potential Φ is used in the eddy current region Ωc , the magnetic vector potential A is used in the holes of the eddy current region, which is free of eddy currents and denoted by ΩA . The rest region is described by the reduced magnetic scalar potential Φ and this region is denoted by ΩΦ . The scheme of this formulation can be seen in Fig. 2.18. The problem region has three disjunct parts, Ω = Ωc ∪ ΩnA ∪ ΩnΦ . This means that the boundary of eddy current free region and the interface have two disjunct parts, ΓHn = ΓHA ∪ ΓHΦ , ΓB = ΓBA ∪ ΓBΦ , Γnc = ΓncA ∪ ΓncΦ . The further interface between ΩnA and ΩnΦ is ΓA,Φ . n ΓBΦ µ = µ0 ΩΦ nc J0 = 0 ΓncΦ nc

ΓHΦ

J 0 6= 0

nc ΓncΦ

nΦ

n

J 0 = 0 ΓHc

ΓA,Φ Ωc ΓE ΓncA µ = µ0 µr or B{·}

ΩA Ωc J 0 6= 0

ΓHc

ΓncA ΓE

ΓBA ΓHA Fig. 2.18. Applying A and Φ in the air region in the case of multiply connected eddy current region The equations to be used are (2.274)–(2.281), (2.223)–(2.227) and (2.160)–(2.162) in the subregion Ωc , ΩnA and ΩnΦ , respectively, which must be appended by some continuity equations and boundary conditions defined on Γnc and ΓA,Φ . The interface conditions on ΓncΦ are the same as on Γnc in the gauged T , Φ − Φ formulation, see (2.327)–(2.330). The interface conditions on ΓncA are the same as on Γnc in the gauged T , Φ − A formulation, see (2.397)–(2.401). The tangential component of the magnetic field intensity and the normal component of the magnetic flux density must be continuous on the interface ΓA,Φ , moreover this is a boundary of the region ΩnA , where A · n = 0 is prescribed because of the condition for the tangential component of the magnetic field intensity, which is a condition for (ν∇ × A) × n (see (2.439)).


59

The summarized equations of the gauged version are as follows: 1 ∂T 1 ∂Φ ∇×T −∇ ∇ · T +µo − µo ∇ ∇× σ σ ∂t ∂t

(2.414)

= − µo ∂T 0 /∂t − ∂R/∂t, in Ωc ,

∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R,

−∇ · (µ∇Φ) = −∇ · (µT 0 ) ,

Ωc ,

(2.415)

in ΩnΦ ,

∇ × (ν∇ × A) − ∇(ν∇ · A) = J 0 ,

T × n = 0,

in

in

(2.416) ΩnA ,

(2.417)

on ΓHc ,

(2.418)

Φ = Φ0 , on ΓHc , 1 ∇ · T = 0, on ΓHc , σ (ν∇ × A) × n = K, on ΓHA , A · n = 0,

ν∇ · A = 0,

(2.421) (2.422) (2.423)

on ΓE ,

(µo T 0 + µo T − µo ∇Φ + R) · n = 0, n × A = α,

(2.420)

on ΓHA ,

Φ = Φ0 , on ΓHΦ , 1 ∇ × T × n = 0, σ

T · n = 0,

(2.419)

(2.424) on ΓE ,

(2.425)

on ΓE ,

(2.426)

on ΓBA ,

(2.427)

on ΓBA ,

(2.428)

(µT 0 − µ∇Φ) · n = −b,

on ΓBΦ ,

(2.429)

Φ is continuous on ΓncΦ ,

(2.430)

T × nc = 0,

(2.431)

on ΓncΦ ,

(µT 0 − µ∇Φ) · nΦ + (µo T 0 + µo T − µo ∇Φ + R) · nc = 0, on ΓncΦ , (2.432) 1 ∇ · T = 0, on ΓncΦ , (2.433) σ (T 0 + T − ∇Φ) × nc + (ν∇ × A) × nA = 0, on ΓncA , (2.434)

(µo T 0 + µo T − µo ∇Φ + R) · nc + (∇ × A) · nA = 0, T · nc = 0, on ΓncA , ∂A 1 ∇ × T × nc − × nA = 0, σ ∂t

ν∇ · A = 0,

on ΓncA ,

on ΓncA ,

(T 0 − ∇Φ) × nΦ + (ν∇ × A) × nA = 0, (µT 0 − µ∇Φ) · nΦ + (∇ × A) · nA = 0, A · nA = 0,

on ΓA,Φ .

on ΓncA ,

(2.435)

(2.436) (2.437) (2.438)

on ΓA,Φ , on ΓA,Φ ,

(2.439) (2.440) (2.441)

60


I. The ungauged T , Φ − A − Φ formulation Applying the same notations presented in the last item, the summarized equations of the gauged version can be set up, however, J 0 is represented by T 0 . The scheme of this formulation can be seen in Fig. 2.18. The equations to be used are (2.262)–(2.267), (2.228)–(2.230) and (2.160)–(2.162) in the subregion Ωc , ΩnA and ΩnΦ , respectively. Some continuity equations and boundary conditions defined on ΓncΦ (see (2.340)–(2.342)), on ΓncA (see (2.412)–(2.413)) and on ΓA,Φ (see (2.439) and (2.440)) must be appended. Finally, the system of equations of the ungauged T , Φ − A − Φ formulation are as follows: ∂R ∂T ∂T 0 1 ∂Φ ∇ × T + µo − µo ∇ = −µo − , in Ωc , ∇× (2.442) σ ∂t ∂t ∂t ∂t ∇ · (µo T − µo ∇Φ) = −∇ · (µo T 0 ) − ∇ · R, −∇ · (µ∇Φ) = −∇ · (µT 0 ) , ∇ × (ν∇ × A) = ∇ × T 0 , T × n = 0, Φ = Φ0 ,

in

Ωc ,

(2.443)

in ΩnΦ ,

(2.444)

ΩnA ,

(2.445)

in

on ΓHc ,

(2.446)

on ΓHc ,

(ν∇ × A) × n = K,

(2.447) on ΓHA ,

Φ = Φ0 , on ΓHΦ , 1 ∇ × T × n = 0, σ

(2.449) on ΓE ,

(µo T 0 + µo T − µo ∇Φ + R) · n = 0, n × A = α,

(2.448)

(2.450) on ΓE ,

(2.451)

on ΓBA ,

(µT 0 − µ∇Φ) · n = −b,

(2.452) on ΓBΦ ,

(2.453)

Φ is continuous on ΓncΦ , T × n = 0,

(2.454)

on ΓncΦ ,

(2.455)

(µT 0 − µ∇Φ) · nn + (µo T 0 + µo T − µo ∇Φ + R) · nc = 0, on ΓncΦ , (2.456) (T 0 + T − ∇Φ) × nc + (ν∇ × A) × nA = 0,

on ΓncA ,

(µo T 0 + µo T − µo ∇Φ + R) · nc + (∇ × A) · nA = 0, ∂A 1 ∇ × T × nc − × nA = 0, on ΓncA , σ ∂t (T 0 − ∇Φ) × nΦ + (ν∇ × A) × nA = 0, (µT 0 − µ∇Φ) · nΦ + (∇ × A) · nA = 0,

on ΓA,Φ , on ΓA,Φ .

on ΓncA ,

(2.457) (2.458) (2.459) (2.460) (2.461)


61

J. The gauged A, V − Φ formulation This formulation is a modification of the A, V − A potential formulation. The magnetic vector potential in the eddy current free region Ωn has been replaced by the reduced magnetic scalar potential Φ. The two potentials A and V remain in Ωc . The number of unknown potential functions is decreased from three to one in Ωn , but it can be used only if the eddy current region Ωc is a simple connected domain. The equations to be used are (2.305)–(2.312) and (2.160)–(2.162), moreover some continuity equations and boundary conditions defined on Γnc must be appended to these equations. The tangential component of the magnetic field intensity and the normal component of the magnetic flux density must be continuous on the interface Γnc , moreover the normal component of the eddy current density must vanish there. The interface is a boundary of the A-region, too, i.e. the uniqueness of the magnetic vector potential must be satisfied on Γnc as well. It can be performed by using the condition A · n = 0, because of the condition for the tangential component of the magnetic field intensity, which is a condition for (ν∇ × A) × n (see (2.473)). The summarized equations are as follows (see Fig. 2.17): ∇ × (νo ∇ × A) − ∇ (νo ∇ · A) + σ

∂A + σ∇V = −∇ × I, in Ωc , ∂t

(2.462)

∂A + σ∇V = 0, −∇ · σ ∂t

in Ωc ,

(2.463)

−∇ · (µ∇Φ) = −∇ · (µT 0 ),

in Ωn ,

(2.464)

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A − σ + σ∇V · n = 0, on ΓHc , ∂t

(2.467)

on ΓHn ,

n × A = 0, V = V0 ,

(2.466)

on ΓHc ,

A · n = 0, Φ = Φ0 ,

(2.465)

(2.468)

on ΓE ,

(2.469)

on ΓE ,

νo ∇ · A = 0,

(2.470)

on ΓE ,

(µT 0 − µ∇Φ) · n = −b,

(2.471) on ΓB ,

(νo ∇ × A + I) × nc + (T 0 − ∇Φ) × nn = 0,

(2.472) on Γnc ,

(∇ × A) · nc + (µT 0 − µ∇Φ) · nn = 0, on Γnc , ∂A + σ∇V · nc = 0, on Γnc , − σ ∂t A · nc = 0,

on Γnc .

(2.473) (2.474) (2.475) (2.476)

62


Note. In the case of multiply connected region, the so-called gauged A, V − A − Φ formulation can be used. K. The ungauged A, V − Φ formulation This formulation is a modification of the A, V − A potential formulation, the magnetic vector potential in the eddy current free region Ωn has been replaced by the reduced magnetic scalar potential Φ. The two potentials A and V remain in Ωc . The number of unknown potential functions is decreased from three to one in Ωn , but it can be used only if the eddy current region Ωc is a simple connected domain. The equations of the ungauged A, V − Φ formulation can be built up by using the relations (2.294)–(2.299) and (2.160)–(2.162), which are completed by some continuity and boundary conditions defined on Γnc . The interface conditions can be defined as in the gauged version, however, equations (2.467), (2.471) and (2.476) are not used. The summarized equations are as follows (see Fig. 2.17): ∂A + σ∇V = −∇ × I, ∂t ∂A + σ∇V = 0, in Ωc , −∇ · σ ∂t

∇ × (νo ∇ × A) + σ

−∇ · (µ∇Φ) = −∇ · (µT 0 ),

in Ωc ,

(2.478)

in Ωn ,

(2.479)

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A − σ + σ∇V · n = 0, on ΓHc , ∂t Φ = Φ0 ,

V = V0 ,

(2.480) (2.481)

on ΓHn ,

n × A = 0,

(2.482)

on ΓE ,

(2.483)

on ΓE ,

(µT 0 − µ∇Φ) · n = −b,

(2.477)

(2.484) on ΓB ,

(νo ∇ × A + I) × nc + (T 0 − ∇Φ) × nn = 0,

(2.485) on Γnc ,

(∇ × A) · nc + (µT 0 − µ∇Φ) · nn = 0, on Γnc , ∂A − σ + σ∇V · nc = 0, on Γnc . ∂t

(2.486) (2.487) (2.488)

Note. If the conductivity of the media placed in Ωc was constant, then V = 0 could be supposed. This results in the ungauged A⋆ −Φ formulation. This is based on the equations (2.477)–(2.488) neglecting the term σ∇V in (2.477) and ignoring the equations (2.478), (2.481), (2.484) and (2.488). Note. In the case of multiply connected region, the so-called A, V − A − Φ formulation can be used.


63

L. The gauged A, V − A − Φ formulation The aim of this formulation is to modify the A, V − Φ potential formulation in such a way that the resulting formulation will be able to simulate multiply connected regions by introducing the magnetic vector potential in the holes placed in the eddy current region. The region of holes filled with nonconducting material is denoted by ΩnA . The two potentials A and V remain in Ωc and Φ in the remaining part of the eddy current free region ΩnΦ , i.e. Ωn = ΩnA ∪ ΩnΦ . The problem region has three disjunct parts, that is Ω = Ωc ∪ ΩnA ∪ ΩnΦ . This means that the boundaries of eddy current free region and the interface have two disjunct parts, ΓHn = ΓHA ∪ ΓHΦ , ΓB = ΓBA ∪ ΓBΦ , Γnc = ΓncA ∪ ΓncΦ . The further interface between ΩnA and ΩnΦ is ΓA,Φ . The scheme is plotted in Fig. 2.18. The equations to be used are (2.305)–(2.312), (2.223)–(2.227) and (2.160)–(2.162) in the subregion Ωc , ΩnA and ΩnΦ , respectively, which must be completed by some continuity equations and boundary conditions defined on Γnc and ΓA,Φ . The interface conditions on ΓncA are the same as on Γnc in the gauged A, V − A formulation, see (2.357)–(2.361). The interface conditions on ΓncΦ are the same as on Γnc in the gauged A, V − Φ formulation, see (2.473)–(2.476). The tangential component of the magnetic field intensity and the normal component of the magnetic flux density must be continuous on the interface ΓA,Φ , moreover this is a boundary of the region ΩnA , where A · n = 0 is prescribed because of the condition for the tangential component of the magnetic field intensity, which is a condition for (ν∇ × A) × n (see (2.514)). It is noted that the last three boundary conditions are the same as in the gauged T , Φ − A − Φ formulation. The summarized equations are as follows: ∇ × (νo ∇ × A) − ∇ (νo ∇ · A) + σ ∂A + σ∇V = 0, −∇ · σ ∂t

∂A + σ∇V = −∇ × I, in Ωc , ∂t

in Ωc ,

∇ × (ν∇ × A) − ∇ (ν∇ · A) = J 0 ,

(2.489) (2.490)

in ΩnA ,

(2.491)

in ΩnΦ ,

(2.492)

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A + σ∇V · n = 0, on ΓHc , − σ ∂t

(2.493)

−∇ · (µ∇Φ) = −∇ · (µT 0 ),

on ΓHc ,

A · n = 0,

(ν∇ × A) × n = K, on ΓHA ,

A · n = 0, Φ = Φ0 ,

on ΓHΦ ,

n × A = 0, V = V0 ,

on ΓE ,

on ΓE ,

(2.494) (2.495)

on ΓHA ,

(2.496) (2.497) (2.498) (2.499) (2.500)

64


νo ∇ · A = 0,

on ΓE ,

(2.501)

n × A = α,

on ΓBA ,

(2.502)

ν∇ · A = 0,

on ΓBA ,

(2.503)

(µT 0 − µ∇Φ) · n = −b,

on ΓBΦ ,

(2.504)

(νo ∇ × A + I) × nc + (ν∇ × A) × nA = 0,

on ΓncA ,

(νo ∇ · A)nc + (ν∇ · A)nA = 0, on ΓncA , ∂A − σ + σ∇V · nc = 0, on ΓncA , ∂t A · nc + A · nn = 0,

(2.506) (2.507)

on ΓncA ,

nc × A + nn × A = 0,

(2.508)

on ΓncA ,

(2.509)

(νo ∇ × A + I) × nc + (T 0 − ∇Φ) × nΦ = 0,

on ΓncΦ ,

(∇ × A) · nA + (µT 0 − µ∇Φ) · nΦ = 0, on ΓncΦ , ∂A + σ∇V · nc = 0, on ΓncΦ , − σ ∂t A · nc = 0,

on ΓncΦ ,

(2.510) (2.511) (2.512) (2.513)

(ν∇ × A) × nA + (T 0 − ∇Φ) × nΦ = 0, (∇ × A) · nA + (µT 0 − µ∇Φ) · nΦ = 0, A · nA = 0,

(2.505)

on ΓA,Φ , on ΓA,Φ ,

on ΓA,Φ .

(2.514) (2.515) (2.516)

M. The ungauged A, V − A − Φ formulation Applying the same notations and theory, presented in the last item, the summarized equations of the gauged version can be set up, but the impressed current vector potential T 0 is used to represent J 0 , see Fig. 2.18. The equations to be used are (2.294)–(2.299), (2.228)–(2.230) and (2.160)–(2.162) in the subregion Ωc , ΩnA and ΩnΦ , respectively, which must be completed by some continuity equations and boundary conditions defined on ΓncΦ (see (2.486)–(2.488)), on ΓncA (see (2.371)–(2.373)) and on ΓA,Φ (see (2.514) and (2.515)). Finally, the system of equations of the ungauged A, V − A − Φ formulation are as follows: ∂A + σ∇V = −∇ × I, ∂t ∂A −∇ · σ + σ∇V = 0, in Ωc , ∂t ∇ × (νo ∇ × A) + σ

∇ × (ν∇ × A) = ∇ × T 0 , −∇ · (µ∇Φ) = −∇ · (µT 0 ),

in

ΩnA ,

in ΩnΦ ,

in Ωc ,

(2.517) (2.518) (2.519) (2.520)


65

(νo ∇ × A + I) × n = 0, on ΓHc , ∂A − σ + σ∇V · n = 0, on ΓHc , ∂t (ν∇ × A) × n = K, Φ = Φ0 ,

(2.522)

on ΓHA ,

(2.523)

on ΓHΦ ,

n × A = 0, V = V0 ,

(2.521)

(2.524)

on ΓE ,

(2.525)

on ΓE ,

n × A = α,

(2.526)

on ΓBA ,

(µT 0 − µ∇Φ) · n = −b,

(2.527) on ΓBΦ ,

(2.528)

(νo ∇ × A + I) × nc + (ν∇ × A) × nA = 0, ∂A − σ + σ∇V · nc = 0, on ΓncA , ∂t nc × A + nn × A = 0,

on ΓncA ,

(2.530)

on ΓncA ,

(2.531)

(νo ∇ × A + I) × nc + (T 0 − ∇Φ) × nΦ = 0,

on ΓncΦ ,

(∇ × A) · nA + (µT 0 − µ∇Φ) · nΦ = 0, on ΓncΦ , ∂A − σ + σ∇V · nc = 0, on ΓncΦ , ∂t (ν∇ × A) × nA + (T 0 − ∇Φ) × nΦ = 0, (∇ × A) · nA + µ (T 0 − ∇Φ) · nΦ = 0,

(2.529)

on ΓA,Φ , on ΓA,Φ .

(2.532) (2.533) (2.534) (2.535) (2.536)

3 Weak formulation of nonlinear static and eddy current field problems There are several methods to solve the partial differential equations of the electromagnetic fields based on the weighted residual method [78]. The first group is based on the direct form of the weighted residual method, as the finite difference method [52, 78, 94, 176] or the classical Galerkin technique [78]. The weak form of the weighted residual method results in the variational method [77] and the finite element method [16, 17, 30, 83, 94, 124, 141, 149, 159, 166, 198, 199], finally the boundary element method [23] and Trefftz method [78] are coming from the inverse form of the weighted residual method. The numerical analysis of electromagnetic field problems with the aid of the Finite Element Method (FEM) has been one of the main directions of research in computational electromagnetics [16, 17, 30, 83, 94, 124, 141, 149, 159, 166, 198, 199]. This is the most widely used technique to approximate the solution of the partial differential equations. The basis of this extensively studied method is the weak formulation of partial differential equations, which also will be presented in this chapter, especially for magnetostatics and for eddy current fields as well as the potential formulations of these fields. FEM is presented in the next chapter.

3.1 The weighted residual and Galerkin’s method This section presents the weighted residual method, which is a family of methods for solving partial differential equations. We are focusing only on the partial differential equations obtained from the static magnetic field problems and from the eddy current field problems. The application of Galerkin’s method to solve partial differential equations is one possibility, however, it is the most widely used technique. The finite element method is based on the Galerkin’s method of the weighted residual method [8, 10, 16–18, 30, 32, 51, 77–79, 83, 94, 124, 141, 149, 159, 166, 169, 170, 172, 198, 199].

67

3.1. THE WEIGHTED RESIDUAL AND GALERKIN’S METHOD

The weighted residual method can be applied to minimizes the residual of a partial differential equation. The best approximation for the potentials can be obtained when the integral of the residual of the partial differential equation multiplied by a weighting function over the problem domain is equal to zero. The weighting function can be arbitrary, but in Galerkin’s method, the weighting functions are selected to be the same as those used for expansion of the approximate solution.

3.1.1 Differential equations in electromagnetic field computation The order of partial differential equations in electromagnetic field computation problems is two. Generally these equations have the form −∇ · [c ∇Φ(r, t)] + d

∂Φ(r, t) ∂ 2 Φ(r, t) +e = f (r, t), ∂t ∂t2

(3.1)

or ∇× [c ∇× A(r, t)] −∇[c ∇·A(r, t)]+d

∂ 2 A(r, t) ∂A(r, t) = F (r, t), (3.2) +e ∂t ∂t2

or ∇ × [c ∇ × A(r, t)] + d

∂ 2 A(r, t) ∂A(r, t) +e = F (r, t). ∂t ∂t2

(3.3)

Here c, d and e are known parameters of the medium to be analyzed, which can be linear or nonlinear. Functions f (r, t) and F (r, t) are known source terms. The unknown scalar function Φ(r, t) and the unknown vector function A(r, t) are depending on space r and time t as well. According to the parameters c, d and e, the following three groups of partial differential equations can be formulated in the analyzed domain Ω: elliptic, parabolic and hyperbolic type partial differential equation. The elliptic type partial differential equations have the form (d = e = 0) −∇ · [c ∇Φ(r, t)] = f (r, t),

in

Ω,

(3.4)

or ∇ × [c ∇ × A(r, t)] − ∇[c ∇ · A(r, t)] = F (r, t),

in Ω,

(3.5)

or ∇ × [c ∇ × A(r, t)] = F (r, t),

in Ω.

(3.6)

The first one is the Poisson equation of static fields. If f (r, t) = 0, then Laplace equation can be obtained. The parabolic type partial differential equations represent the diffusion equation (in this case e = 0), −∇ · [c ∇Φ(r, t)] + d or

∂Φ(r, t) = f (r, t), ∂t

in Ω,

(3.7)

68

3. WEAK FORMULATION OF NONLINEAR STATIC AND EDDY CURRENT FIELD PROBLEMS

∇ × [c ∇ × A(r, t)] − ∇[c ∇ · A(r, t)] + d

∂A(r, t) = F (r, t), ∂t

in Ω, (3.8)

or ∇ × [c ∇ × A(r, t)] + d

∂A(r, t) = F (r, t), ∂t

in

Ω.

(3.9)

Finally, wave equation is a hyperbolic type partial differential equation (d = 0), −∇ · [c ∇Φ(r, t)] + e

∂ 2 Φ(r, t) = f (r, t), ∂t2

in

Ω,

(3.10)

or ∇× [c ∇ × A(r, t)]−∇[c ∇·A(r, t)] + e

∂ 2 A(r, t) = F (r, t), ∂t2

in Ω, (3.11)

or ∇ × [c ∇ × A(r, t)] + e

∂ 2 A(r, t) = F (r, t), ∂t2

in Ω.

(3.12)

Only the elliptic and the parabolic partial differential equations of the scalar function and of the vector function will be presented in this book, because they represent the static magnetic field and the eddy current field problems. The following shorter notations will be used below: Φ = Φ(r, t), A = A(r, t), f = f (r, t) and F = F (r, t). The left-hand side of the above partial differential equations is usually called the differential operator of the actually used potential function. The partial differential equations are usually given in the domain Ω, which has a boundary, Γ = ∂Ω. As it was presented in the previous section, there are Dirichlet and Neumann type boundary conditions on the disjunct parts of the boundary. Here, ΓD and ΓN are the two parts of the boundary where Dirichlet and Neumann boundary conditions are prescribed, respectively and Γ = ΓD ∪ ΓN . Dirichlet boundary condition is specified on ΓD , i.e. Φ = g,

or n × A = G,

or A · n = G,

on ΓD .

(3.13)

Here, the functions g = g(r, t), G = G(r, t) and G = G(r, t) are known on the part of boundary, ΓD . Here n is the outer normal unit vector of the domain Ω. Neumann boundary condition is given on the rest ΓN , (c∇Φ) · n = c

∂Φ = h, or (c∇ × A) × n = H, ∂n or c ∇ · A = H, on ΓN .

(3.14)

Here, the functions h = h(r, t), H = H(r, t) and H = H(r, t) are known on the boundary part ΓN .

69


3.1.2 The weighted residual method A. Weighted residual of elliptic differential equations with scalar potential ˜ i.e. Φ ≃ Φ, ˜ but this The potential function Φ can be approximated by a function Φ, function does not satisfy the differential equation exactly, however, it is possible to select ˜ which satisfies Dirichlet boundary conditions exactly (see next sections). a function Φ, The weighted residual method is based on the inner product of the differential operator of the partial differential equation and a weighting function N = N (r), defined as Z N [−∇ · (c ∇Φ)] dΩ. (3.15) < N, PDE >= Ω

In this case N is a real function. Using the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v and ϕ = N and v = c ∇Φ, the inner product has the form Z Z I − N ∇ · (c ∇Φ) dΩ = c ∇Φ · ∇N dΩ − N (c∇Φ · n) dΓ. (3.16) Ω

Ω

Γ

Let us determine the following inner product in the same way: Z I Z Φ∇ · (c ∇N ) dΩ = c ∇Φ · ∇N dΩ − Φ(c∇N · n) dΓ, − Ω

Ω

(3.17)

Γ

i.e. Φ and N are replaced, then Z Z − N ∇ · (c ∇Φ) dΩ = − Φ∇ · (c ∇N ) dΩ Ω Ω I + [Φ(c∇N · n) − N (c∇Φ · n)] dΓ,

(3.18)

Γ

because the first integrals on the right in (3.16) and in (3.17) are the same. ˜ as well, This relation is valid for the approximating trial function Φ Z Z ˜ · (c ∇N ) dΩ ˜ dΩ = − − Φ∇ N ∇ · (c ∇Φ) Ω Ω I h i ˜ ˜ · n) dΓ. + Φ(c∇N · n) − N (c∇Φ

(3.19)

Γ

By subtracting equation (3.18) from (3.19) and applying the definition of elliptic partial differential equation −∇ · (c ∇Φ) = f , the following equation can be obtained: Z Z h i ˜ − f dΩ = − (Φ ˜ − Φ)∇ · (c ∇N ) dΩ N −∇ · (c ∇Φ) Ω Ω I n (3.20) o ˜ − Φ)(c∇N · n) − N [c∇(Φ ˜ − Φ) · n] dΓ. (Φ + Γ

70


Let us denote the two disjunct segments of the boundary, Z Z h i ˜ ˜ − Φ)∇ · (c ∇N ) dΩ N −∇ · (c ∇Φ) − f dΩ = − (Φ Ω Ω Z n o ˜ ˜ (Φ−Φ)(c∇N ·n)−N [c∇(Φ−Φ)·n] dΓ. +

(3.21)

ΓD ∪ΓN

The potential function Φ is prescribed on ΓD , Φ = g and the normal component of the gradient of Φ is given on ΓN , −∇Φ · n = h (see equations (3.13) and (3.14)). It is possible to select the approximation function to satisfy Dirichlet condition exactly, but N = 0,

on ΓD

(3.22)

should be specified in this case. These result in Z Z h i ˜ − f dΩ = − (Φ ˜ − Φ)∇ · (c ∇N ) dΩ N −∇ · (c ∇Φ) Ω Ω Z h Z i ˜ − Φ)(c∇N · n)dΓ − N (c∇Φ) ˜ · n − h dΓ. + (Φ ΓN

(3.23)

ΓN

Supposing that the difference between the unknown potential function Φ and the ˜ is equal to zero, i.e. the first and the second integrals on the approximation function Φ right of (3.23) disappear, it results in Z Z h i h i ˜ − f dΩ+ ˜ · n − h dΓ = 0. N −∇ · (c ∇Φ) N (c∇Φ) (3.24) Ω

ΓN

This is the sum of two terms, the first one is the inner product of the partial differential equation and the weighting function, the second term is the Neumann type boundary condition multiplied by the weighting function, integrated only on ΓN . The Dirichlet boundary condition can be specified by an appropriate selection of the approximation function on ΓD . This is called the weak formulation of elliptic differential equations with scalar potential. B. Weighted residual of elliptic differential equations with vector potential ˜ i.e. A ≃ A, ˜ but this The potential function A can be approximated by a function A, function does not satisfy the differential equation exactly, however, it is possible to select ˜ which satisfies Dirichlet boundary conditions exactly (see next sections). a function A, Let us first determine the inner product of the differential operator and a weighting function W = W (r), Z W · [∇ × (c ∇ × A) − ∇(c ∇ · A)] dΩ. (3.25) < W , PDE >= Ω

Here, W is a real function. Using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v

(3.26)


71

with u = c ∇ × A and v = W , moreover the relation ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v,

(3.27)

with ϕ = c ∇ · A and v = W , the inner product has the form Z

W · [∇ × (c ∇ × A) − ∇(c ∇ · A)] dΩ Z Z = c ∇ × A · ∇ × W dΩ + c ∇ · A ∇ · W dΩ Ω I IΩ + [(c ∇ × A) × W ] · n dΓ − c ∇ · A (W · n) dΓ. Ω

Γ

(3.28)

Γ

Let us next determine the following inner product in the same way: Z

A · [∇ × (c ∇ × W ) − ∇(c ∇ · W )] dΩ Z c ∇ × A · ∇ × W dΩ + c ∇ · A ∇ · W dΩ = Ω IΩ I + [(c ∇ × W ) × A] · n dΓ − c ∇ · W (A · n) dΓ, ZΩ Γ

(3.29)

Γ

then Z

Ω

Z

W · [∇ × (c ∇ × A) − ∇(c ∇ · A)] dΩ

A · [∇ × (c ∇ × W ) − ∇(c ∇ · W )] dΩ I IΩ + [(c ∇ × A) × W ] · n dΓ − c ∇ · A (W · n) dΓ IΓ IΓ − [(c ∇ × W ) × A] · n dΓ + c ∇ · W (A · n) dΓ.

=

Γ

(3.30)

Γ

Applying the identity (a × b) · n = (n × a) · b = −(a × n) · b in the first (a = c ∇ × A, b = W ) and in the third (a = c ∇ × W , b = A) boundary integral terms, it results in Z

=

ZΩ

W · [∇ × (c ∇ × A) − ∇(c ∇ · A)] dΩ

A · [∇ × (c ∇ × W ) − ∇(c ∇ · W )] dΩ I IΩ − [(c ∇ × A) × n] · W dΓ − c ∇ · A (W · n) dΓ IΓ IΓ + [(c ∇ × W ) × n] · A dΓ + c ∇ · W (A · n) dΓ. Γ

Γ

(3.31)

72


˜ as well, This relation is valid for the approximating trial function A Z ˜ − ∇(c ∇ · A)] ˜ dΩ W · [∇ × (c ∇ × A) Ω Z ˜ · [∇ × (c ∇ × W ) − ∇(c ∇ · W )] dΩ = A Ω I I ˜ × n] · W dΓ − c ∇ · A ˜ (W · n) dΓ − [(c ∇ × A) Γ Γ I I ˜ dΓ + c ∇ · W (A ˜ · n) dΓ. + [(c ∇ × W ) × n] · A Γ

(3.32)

Γ

By subtracting equation (3.31) from equation (3.32) and applying the definition of elliptic partial differential equation ∇ × (c ∇ × A) − ∇(c ∇ · A) = F , the following equation can be obtained: Z ˜ − ∇(c ∇ · A) ˜ − F ] dΩ W · [∇ × (c ∇ × A) Ω Z ˜ − A) · [∇ × (c ∇ × W ) − ∇(c ∇ · W )] dΩ = (A Ω I n I (3.33) o ˜ − A) (W · n) dΓ ˜ − A)] × n · W dΓ − c ∇ · (A − [c ∇ × (A IΓ I Γ ˜ ˜ − A) · n] dΓ. + [(c ∇ × W ) × n] · (A − A) dΓ + c ∇ · W [(A Γ

Γ

The boundary can be split in two disjunct segments, Γ = ΓD ∪ ΓN . Dirichlet boundary condition is prescribed only on the segment of the boundary surface ΓD and Neumann boundary condition is prescribed only on the segment ΓN . The first and the second boundary integral terms contain Neumann type boundary conditions, (c∇ × A) × n = H, or c ∇ · A = H, respectively. The last boundary integral terms contain Dirichlet type boundary conditions, n × A = G, or A · n = G. These are coming from (3.13) and (3.14). Dirichlet boundary conditions can be specified exactly by an appropriate selection of the approximation function. In this case n × W = 0,

or W · n = 0,

on ΓD

(3.34)

must be specified. Supposing that the difference between the unknown potential function A and the ˜ is zero, i.e. approximation function A Z ˜ W ) = (A ˜ − A) · [∇ × (c ∇ × W ) − ∇(c ∇ · W )] dΩ = 0, R(A, (3.35) Ω

it results in the weak formulation of the elliptic differential equations with vector potential, Z ˜ − ∇(c ∇ · A) ˜ − F ] dΩ W · [∇ × (c ∇ × A) Ω Z Z (3.36) ˜ × n − H] · W dΓ + (c ∇ · A ˜ − H) (W ·n) dΓ = 0. + [(c ∇ × A) ΓN

ΓN

3.2. APPROXIMATION OF UNKNOWN FUNCTIONS AND WEIGHTING FUNCTIONS

73

The first term is the inner product of the partial differential equation and the weighting function, the second and third terms are the Neumann type boundary conditions multiplied by the weighting function, integrated only on ΓN . Dirichlet boundary conditions can be approximated exactly on ΓD . The above formulation is valid if the gauged vector potential is used. The formulation of ungauged vector potential is as follows: Z ˜ − F ] dΩ W · [∇ × (c ∇ × A) Ω Z (3.37) ˜ × n − H] · W dΓ = 0, [(c ∇ × A) + ΓN

because the form of partial differential equation is the following: ∇ × (c ∇ × A) = F . C. Weighted residual of parabolic differential equations As it is presented in the last section, the partial differential equations of a transient eddy current field problem has some terms containing time derivative of the potentials. These terms are appended to the integrand of the weighted residual, however, additional initial conditions must be specified as Φ(t = 0) = Φ0 ,

and A(t = 0) = A0 .

(3.38)

3.2 Approximation of unknown functions and weighting functions In the previous section, we have defined the weighted residual of the static magnetic field problem and the eddy current field problem, but we have not introduced the approximating functions and the weighting functions, yet [10,16–18,30,78,83,94,124,141,149,159,166, 198, 199]. The approximating scalar and vector potential functions and the scalar and vector weighting functions are selected as

and

e t) = Φ(r, t) ≃ Φ(r, N (r) =

K X

k=1

I X i=1

Pk (r) bk ,

e Qi (r) ϕi (t), A(r, t) ≃ A(r, t) W (r) =

L X

P l (r) cl ,

J X

Qj (r) aj (t), (3.39)

j=1

(3.40)

l=1

where Qi (r), Qj (r), Pk (r) and P l (r) are the basis functions of the approximation, ϕi (t), aj (t), bk and cl are the unknown coefficients, which must be determined. This is the most general form of approximation and weighting function. According to the selection of the basis functions, the weighted residual method results in several technique.

74


The weighted residual method has three forms as it is shown in Fig. 3.1, (i) the direct form, (ii) the weak form, (iii) the inverse form. Applying the direct form of the weighted residual method, i.e. using directly e.g. (3.24), (3.36) or (3.37) and the formulations in (3.39) and (3.40) results in a system of linear equations, which solution gives the unknown coefficients ϕi (t) or aj (t) of the approximating potential functions. The finite difference method and the general Q = P, Q = P

Q 6= P, Q 6= P Direct form

Moment and finite difference methods

Bubnov–Galerkin methods

Weak form

Finite element method

General weak form

Inverse form

Trefftz method

Boundary element method

Fig. 3.1. The numerical field analysis methods

3.3. THE WEAK FORMULATION WITH GALERKIN’S METHOD

75

Bubnov–Galerkin method belong to this group. The finite difference method is a popular and widely used technique. The disadvantage of this form is that the basis functions must be differentiated twice, because the differential operator is a second order one. The weak formulation of the weighted residual method can be obtained when applying the rule of integration by parts to decrease the order of the differential operator in the inner product. This means that the order of approximating function can be just one. The finite element method or the general weak form can be derived from this group of the weighted residual method, moreover the weak formulation of static magnetic field problems and eddy current field problems is presented in the next section. In the case of finite element method, the weighting function and the basis function of the approximating function are the same. The boundary element method and the Trefftz method are based on the inverse form of partial differential equations. In this situation, the weighting function is a solution of the homogeneous partial differential equation. Further analysis of the direct form and the inverse form is out of scope of this book, since we are focusing on the weak formulation, which is the way to the Finite Element Method.

3.3 The weak formulation with Galerkin’s method The finite element method use the the weak formulation with Galerkin’s method when the basis functions of the approximating function and the weighting function are the same. Here, the weak formulations of the potential formulations according to Galerkin’s method are presented, which are appropriate in the finite element method. In the following N = N (r) denotes the scalar weighting function as well as the basis functions of approximating function and W = W (r) denotes the vector weighting function as well as the basis functions of approximating function [10, 16–18, 30, 78, 83, 94, 124, 141, 149, 159, 166, 198, 199]. Scalar potentials Φ = Φ(r), or Φ = Φ(r, t) are approximated by an expansion in terms of I elements of an entire function set Ni , e = ΦD + Φ≃Φ

I X

N i Φi ,

(3.41)

i=1

where ΦD satisfies the prescribed non homogeneous Dirichlet type boundary conditions, ΦD = g,

on ΓD ,

(3.42)

that is why the elements of the function set must prescribe homogeneous Dirichlet type boundary condition, Ni = 0,

on ΓD .

(3.43)

Finally, non homogeneous Dirichlet boundary conditions can be prescribed directly by the help of the function ΦD .

76


Vector potentials A = A(r), or A = A(r, t) are approximated by an expansion in terms of J elements of an entire function set W j , e = AD + A≃A

J X

W j Aj ,

(3.44)

j=1

where AD satisfies the prescribed non homogeneous Dirichlet boundary conditions, n × AD = G,

or AD · n = G,

on ΓD ,

(3.45)

that is why the elements of the function set must prescribe homogeneous Dirichlet type boundary condition, n × W j = 0,

or

W j · n = 0,

on ΓD .

(3.46)

Finally, non homogeneous Dirichlet type boundary conditions can be prescribed directly by the help of AD . Shape functions Ni and W j are the elements of an entire function set, which can be defined in many ways. The definition of these elements are presented in chapter 3.3.18. e Ψ, e A, e Ve (e e will denote the approximated unknown In the following Φ, v ) and T potential functions.

3.3.1 The reduced magnetic scalar potential, the Φ-formulation The weak formulation of the partial differential equation (2.160) and the Neumann type boundary condition (2.162) can be written as Z Z e e · n + R·n e e 0 · n − µo ∇Φ Nk ∇· (µo ∇Φ) dΩ + Nk b + µo T dΓ Ω Γ Z Z B (3.47) e 0 ) dΩ + e dΩ, k = 1, · · · , I, = Nk ∇· (µo T Nk ∇ · R Ω

Ω

where

Nk = 0,

on ΓH .

(3.48)

The second order derivative in the first integral can be reduced to first order by using the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

e and the Gauss’ theorem with the notations ϕ = Nk and v = µo ∇Φ I Z e dΩ + µo Nk ∇Φ e · n dΓ µo ∇Nk · ∇Φ − Ω Γ Z e 0 · n − µo ∇Φ e · n dΓ = e ·n+R + Nk b + µo T Γ I Z B e 0 · n dΓ e 0 dΩ + µo Nk T µo ∇Nk · T − I Γ ZΩ e dΩ + Nk R e · n dΓ, ∇Nk · R − Ω

Γ

(3.49)

(3.50)

77


e 0 and where the right-hand side has been modified with notations ϕ = Nk and v = µo T e too. The boundary integrals on the right-hand side are valid over the boundary of v = R, the whole problem region Γ = ΓH ∪ ΓB , i.e. Z Z e dΩ + e · n dΓ − µo ∇Nk · ∇Φ µo Nk ∇Φ Ω ΓH ∪ΓB Z e ·n+R e · n dΓ = e 0 · n − µo ∇Φ + Nk b + µo T ZΓB Z (3.51) e 0 · n dΓ e 0 dΩ + − µo Nk T µo ∇Nk · T Ω ΓH ∪ΓB Z Z e · n dΓ. e dΩ + ∇Nk · R − Nk R Ω

ΓH ∪ΓB

e 0 ·n, µo ∇Φ·n e on the part ΓB are disappearing, e and R·n It is easy to see that the terms µo T Z Z Z e e µo ∇Nk · ∇Φ dΩ + µo Nk ∇Φ · n dΓ + Nk b dΓ = − Ω Γ ΓB Z Z H e 0 · n dΓ e 0 dΩ + − µo Nk T µo ∇Nk · T (3.52) Ω ΓH Z Z e · n dΓ. e dΩ + ∇Nk · R − Nk R Ω

ΓH

On the rest part of the boundary ΓH , the shape function Nk is equal to zero because of Dirichlet type boundary condition (2.161). Finally, the following weak form can be obtained for the Φ-formulation (k = 1, · · · , I): Z Z Z Z e 0 dΩ + ∇Nk · R e dΩ + Nk b dΓ. (3.53) e dΩ = µo ∇Nk · T µo ∇Nk · ∇Φ Ω

Ω

Ω

ΓB

3.3.2 Combination of magnetic scalar potentials, Φ − Ψ-formulation

The partial differential equation (2.177) in ΩΦ , the partial differential equation (2.178) in ΩΨ , the Neumann type boundary conditions (2.181) and (2.182) on ΓBΦ ∪ ΓBΨ and the interface condition (2.184) on ΓΦ,Ψ can be summarized in the following formula: Z Z e e dΩ Nk ∇ · (µ∇Φ) dΩ + Nk ∇ · (µo ∇Ψ) ΩΦ ΩΨ Z e 0 · n − µ∇Φ e · n dΓ + Nk b + µT ΓBΦ

+

Z

ΓBΨ

+

Z

ΓΦ,Ψ

=

Z

ΩΦ

e ·n+R e · n dΓ Nk b − µo ∇Ψ

(3.54)

h i e 0 − µ∇Φ) e · nΦ + (−µo ∇Ψ e + R) e · nΨ dΓ Nk (µT

e 0 ) dΩ + Nk ∇ · (µT

Z

ΩΨ

e dΩ, Nk ∇ · R

k = 1, · · · , I.

78


Here Nk = 0,

on ΓHΦ ∪ ΓHΨ .

(3.55)

The second order derivatives in the first and in the second integrals can be reduced to first order and the right-hand side can also be manipulated by using the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

(3.56)

e then ϕ = Nk and v = µo ∇Ψ, e moreover with the notations ϕ = Nk and v = µ∇Φ, e 0 , finally ϕ = Nk and v = R e and the Gauss’ theorem ϕ = Nk and v = µT −

− +

Z

ΩΦ

Z

ΩΨ

Z

e dΩ + µ∇Nk · ∇Φ

e dΩ + µo ∇Nk · ∇Ψ

ΓBΦ

+

Z

ΓBΨ

+

Z

ΓΦ,Ψ

− −

Z

ΩΦ

Z

ΩΨ

Z

ΓHΦ ∪ΓBΦ ∪ΓΦ,Ψ

Z

e · nΦ dΓ µNk ∇Φ

ΓHΨ ∪ΓBΨ ∪ΓΦ,Ψ

e · nΨ dΓ µo Nk ∇Ψ

e 0 · nΦ − µ∇Φ e · nΦ dΓ Nk b + µT e · nΨ dΓ e · nΨ + R Nk b − µo ∇Ψ

(3.57)

h i e 0 − µ∇Φ) e · nΨ dΓ = e · nΦ + (−µo ∇Ψ e + R) Nk (µT

e 0 dΩ + µ∇Nk · T e dΩ + ∇Nk · R

Z

Z

ΓHΦ ∪ΓBΦ ∪ΓΦ,Ψ

ΓHΨ ∪ΓBΨ ∪ΓΦ,Ψ

e 0 · nΦ dΓ µNk T

e · nΨ dΓ. Nk R

Here the orientation of normal unit vectors is also indicated, i.e. nΦ is pointing outward the subregion ΩΦ and nΨ is pointing outward the subregion ΩΨ (see Fig. 2.14). e 0 · nΦ , µ∇Φ e · nΦ , µo ∇Ψ e · nΨ and R e · nΨ on ΓBΦ and on ΓBΨ are The terms µT disappearing in the boundary integral terms. All the terms on ΓΦ,Ψ have also pairs in the first, second and in the last two surface integrals. The rest parts are ΓH type boundaries where Dirichlet type boundary condition is prescribed, that is why Nk = 0 can be supposed there. Finally the weak formulation of the Φ − Ψ-formulation can be written as Z

Z e dΩ + e dΩ µ∇Nk · ∇Φ µo ∇Nk · ∇Ψ ΩΦ ΩΨ Z Z Z e 0 dΩ + e dΩ + µ∇Nk · T ∇Nk · R = ΩΦ

where k = 1, · · · , I.

ΩΨ

ΓBΦ ∪ΓBΨ

(3.58) Nk b dΓ,

79


3.3.3 The magnetic vector potential, the A-formulation with implicit enforcement of Coulomb gauge The weak formulation is built up by using the partial differential equation (2.223) and the Neumann type boundary conditions (2.224) and (2.227). The weak formulation is, Z h i e − ∇(νo ∇ · A) e dΩ W k · ∇ × (νo ∇ × A) ZΩ h i e + Ie ×n − K dΓ W k · νo ∇ × A + (3.59) Z Z ZΓH e (W k · n) dΓ = e dΩ, W k · J 0 dΩ − W k · (∇ × I) + ν0 ∇ · A ΓB

Ω

Ω

where

n × W k = 0,

on ΓB ,

(3.60)

and W k · n = 0,

on ΓH ,

(3.61)

where k = 1, · · · , J. The second order derivatives in the first integral can be reduced to first order one and the last integral can be reformulated by using the following identities: ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v,

(3.62)

∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.63)

and

e v = W k , u = νo ∇ × A e and u = Ie and the Gauss’ with the notations ϕ = νo ∇A, theorem, i.e. Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A Z ZΩ i h e × W k · n dΓ − e (W k ·n) dΓ νo ∇ × A νo ∇· A + ΓH ∪ΓB ΓH ∪ΓB Z Z (3.64) h i e (W k · n) dΓ e e + W k · νo ∇ × A + I × n − K dΓ + νo ∇· A Z Z ΓB ZΓH (Ie × W k ) · n dΓ. W k · J 0 dΩ − (∇ × W k ) · Ie dΩ − = Ω

Ω

ΓH ∪ΓB

The first surface integral can be reformulated according to h i h i e × W k · n = n × νo ∇ × A e · Wk νo ∇ × A h i e ×n . = −W k · νo ∇ × A

(3.65)

80


The last integral can be rearranged by using the identity

i.e.

e · W k = W k · (Ie × n), −(Ie × W k ) · n = −(n × I)

(3.66)

Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A ZΩ Z h i e × n dΓ − e (W k · n) dΓ W k · νo ∇ × A νo ∇· A − ΓH ∪ΓB ΓH ∪ΓB Z Z (3.67) h i e + Ie × n − K dΓ + νo ∇· A e (W k · n) dΓ + W k · νo ∇ × A Γ ΓB Z Z Z H W k · J 0 dΩ − (∇ × W k ) · Ie dΩ + = W k · (Ie × n) dΓ. Ω

Ω

ΓH ∪ΓB

Now, it is easy to see that the first and the last surface integrals are vanishing on ΓH . On the rest ΓB , the Dirichlet boundary condition (2.226) must be satisfied, i.e. n × W k = 0 and h i e × W k · n = [W k × n] · νo ∇ × A e νo ∇ × A (3.68) e . = − [n × W k ] · νo ∇ × A

This is the reason why the first and the last boundary integral terms can be eliminated. It is evident that the second boundary integral is vanishing on ΓB (because of the fourth boundary integral term) and it is equal to zero on the rest part of the boundary because of the Dirichlet boundary condition (2.225), so W k · n = 0 on ΓH . Finally, the following weak formulation can be given to the A-formulation appended by the implicit Coulomb gauge enforcement: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A Z Z ZΩ (3.69) W k · J 0 dΩ + W k · K dΓ − (∇ × W k ) · Ie dΩ, = Ω

ΓH

Ω

where k = 1, · · · , J.

3.3.4 The magnetic vector potential, the A-formulation with a numerical technique, which is not sensitive to Coulomb gauge The partial differential equation (2.228) and the Neumann boundary condition (2.229) can be summarized in Z h i e dΩ W k · ∇ × (νo ∇ × A) ZΩ h i e + Ie × n − K dΓ + W k · νo ∇ × A (3.70) Γ Z Z H e 0 dΩ − = Wk · ∇ × T W k · ∇ × Ie dΩ, k = 1, · · · , J. Ω

Ω


81

Here n × W k = 0,

on ΓB .

(3.71)

The second order derivative in the first integral can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.72)

e The right-hand side can also be simplified with the notations v = W k and u = ν∇ × A. e 0 and u = I. e After by applying the same identity with the notations v = W k , u = T using Gauss’ theorem the following equation can be obtained: Z Z h i e e ×W k ·n dΓ νo (∇ × W k ) · ∇ × A dΩ + νo ∇ × A ΓH ∪ΓB ZΩ h i e + Ie × n − K dΓ W k · νo ∇ × A + Z ZΓH e 0 dΩ + e 0 × W k · n dΓ (∇ × W k ) · T = T (3.73) Ω ΓH ∪ΓB Z − (∇ × W k ) · Ie dΩ Ω Z − Ie × W k · n dΓ. ΓH ∪ΓB

The first boundary integral can be eliminated by the procedure applied in equations above (3.65)–(3.68). The integrand of the boundary integral term remained on the right-hand side can be rewritten in the following forms: e 0 = − (n×W k )· T e 0. e 0 ·W k = (W k × n)· T e 0 × W k ·n = n × T (3.74) T

The second one is equal to zero on ΓH , because the tangential component of the impressed current vector potential is vanishing there, the last one is useful on the rest ΓB of the boundary, because of the Dirichlet type boundary condition (2.230). As a consequence of these manipulations, the boundary integral terms can be eliminated. Finally, the weak formulation of the A-formulation, which must be realized by a numerical technique, which is not sensitive to Coulomb gauge is as follows: Z Z e e 0 dΩ νo (∇ × W k ) · ∇ × A dΩ = (∇ × W k ) · T Ω Ω Z Z (3.75) W k · K dΓ − + (∇ × W k ) · Ie dΩ, k = 1, · · · , J. ΓH

Ω

e 0 such that it satisfies It is possible to select the impressed current vector potential T the Dirichlet boundary condition e 0 × n = K, T

on ΓH .

(3.76)

82


In this case the according boundary integral term on the right-hand side on the boundary part ΓH can be reformulated as Z Z e 0 × W k ) · n dΓ = (n × T e 0 ) · W k dΓ = (T ΓH ΓH Z Z (3.77) e 0 × n) · W k dΓ = − K · W k dΓ, − (T ΓH

ΓH

i.e. the weak formulation of the ungauged A-formulation can be written as, Z Z e e 0 dΩ νo (∇ × W k ) · ∇ × A dΩ = (∇ × W k ) · T Ω Ω Z (∇ × W k ) · Ie dΩ, −

(3.78)

Ω

where k = 1, · · · , J.

3.3.5 Combination of the magnetic vector potential and the magnetic scalar potential, the A − Φ-formulation There are two unknown potentials in this formulation, the magnetic vector potential A in the iron region and the reduced magnetic scalar potential Φ in the air region, that is why two equations must be realized. The first is coming from the region ΩA , the second is from the region ΩΦ , but the two equations are coupled through the interface ΓA,Φ . It is important to note that the resulting system of equations should be symmetric. The first weak formulation is based on the partial differential equation (2.231), on the Neumann boundary conditions (2.233), (2.236) and on the vector type interface condition (2.239). The integral equation is very similar to (3.59), Z h i e − ∇(νo ∇ · A) e dΩ W k · ∇ × (νo ∇ × A) Ω Z Z A h i e + Ie × n−K dΓ + e (W k · n) dΓ + W k · νo ∇× A νo ∇· A ΓHA

+

Z

ΓBA

h i e + I) e × nA + T e 0 − ∇Φ e × nΦ dΓ = W k · (νo ∇ × A

(3.79)

ΓA,Φ

− where

Z

W k · ∇ × Ie dΩ,

ΩA

n × W k = 0,

k = 1, · · · , J,

on ΓBA ,

(3.80)

and W k · n = 0,

on ΓHA ∪ ΓA,Φ .

(3.81)


83

Obtaining the weak form of this weak formulation is very similar to the manipulations starting on page 79. After applying the identities (3.62), (3.63), (3.65) and (3.66), the following equation can be obtained: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A Ω Z A h i e × n dΓ − W k · νo ∇ × A ΓHA ∪ΓBA ∪ΓA,Φ

−

+

Z

ΓHA ∪ΓBA ∪ΓA,Φ

Z

ΓHA

+

Z

ΓBA

+

Z

ΓA,Φ

−

Z

ΩA

Wk ·

h

e (W k ·n) dΓ νo ∇· A

i e + Ie × n − K dΓ νo ∇ × A

(3.82)

e (W k · n) dΓ νo ∇ · A

i h e + I) e × nA + T e 0 − ∇Φ e × nΦ dΓ = W k · (νo ∇ × A

e (∇ × W k ) · IdΩ +

Z

ΓHA ∪ΓBA ∪ΓA,Φ

W k · (Ie × n)dΓ.

The boundary of the region ΩA is ∂ΩA = ΓHA ∪ ΓBA ∪ ΓA,Φ (see Fig. 2.15). It should be noted that the normal unit vector n in the first and in the second boundary integral terms must be replaced by nA on the interface ΓA,Φ . The first boundary integral is vanishing on the part ΓHA and on ΓA,Φ according to the third and to the last boundary integral terms on the left-hand side, moreover it is equal to zero according to the Dirichlet boundary condition (2.235) on ΓBA . The second boundary integral term is vanishing on ΓBA , because of the fourth boundary integral term and it is equal to zero on the rest part ΓHA ∪ ΓA,Φ because the Dirichlet boundary conditions (2.234) and (2.241) must be satisfied there. The term in the third and in the fifth boundary integrals according to the nonlinear residual Ie are vanishing because of the same terms in the last boundary integral. The last integral is equal to zero on ΓBA since (2.235) is prescribed there. Finally, the following equation can be written: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A ZΩA e × nΦ dΓ − W k · ∇Φ ΓA,Φ

=

Z

ΓHA

−

Z

ΩA

W k · K dΓ −

Z

ΓA,Φ

e (∇ × W k ) · IdΩ.

e 0 × nΦ dΓ Wk · T

(3.83)

Unfortunately, this is not the final weak form. One more modification is necessary because of symmetry, but this will be presented after the second weak form.

84


The partial differential equation in (2.232), the Neumann type boundary condition (2.238) and the interface condition with scalar type (2.240) build the second equation, which is similar to (3.47), Z Z e 0 · n − µ∇Φ e dΩ + e · n dΓ Nk ∇ · (µ∇Φ) Nk b + µT ΩΦ

+

Z

ΓBΦ

h i e · nΦ dΓ e 0 − ∇Φ e · nA + µ T Nk (∇ × A)

(3.84)

ΓA,Φ

=

Z

ΩΦ

where

e 0 ) dΩ, Nk ∇ · (µT

Nk = 0,

on ΓHΦ ,

(3.85)

and k = 1, · · · , I. Obtaining the weak form is very similar to that on page 76. After using the manipulations (3.49), (3.50) and (3.51), but supposing linear media in ΩΦ , the following equation can be obtained: Z Z e e · n dΓ µ∇Nk · ∇Φ dΩ + µNk ∇Φ − ΩΦ

+

Z

e 0 · n − µ∇Φ e · n dΓ Nk b + µT

ΓBΦ

+

Z

ΓA,Φ

−

Z

ΩΦ

ΓHΦ ∪ΓBΦ ∪ΓA,Φ

h i e 0 − ∇Φ e · nΦ dΓ = e · nA + µ T Nk (∇ × A)

e 0 dΩ + µ∇Nk · T

Z

ΓHΦ ∪ΓBΦ ∪ΓA,Φ

(3.86)

e 0 · n dΓ. µNk T

The first surface integral is vanishing on ΓBΦ ∪ ΓA,Φ , because of the same term with opposite sign in the second and in the third surface integrals (of course n = nΦ on ΓA,Φ ). The last surface integral is vanishing on the same part of the boundary according to the second and the third boundary integrals. The first as well as the last surface integrals are equal to zero on the rest part ΓHΦ , because Nk = 0 there according to the Dirichlet boundary condition (2.237). The result is as follows: Z Z e · nA dΓ = e dΩ + − µ∇Nk · ∇Φ Nk (∇ × A) ΩΦ

−

Z

ΩΦ

e 0 dΩ − µ∇Nk · T

Z

ΓA,Φ

(3.87)

Nk b dΓ.

ΓBΦ

It can be seen that the surface integral terms on the left-hand side of equations (3.83) and (3.87) have no got the same format, however, – from the finite element point of view – they should have. The surface integral term of (3.83) can be rewritten as Z e × nΦ dΓ − W k · ∇Φ ΓA,Φ

=

Z

Z e × nA dΓ = W k · ∇Φ

ΓA,Φ

e dΓ. nA · W k × ∇Φ

ΓA,Φ

(3.88)


85

The identity ∇ × (ϕv) = ϕ∇ × v − v × ∇ϕ

(3.89)

should be applied as n · [∇ × (ϕv)] = ϕ[n · (∇ × v)] − n · (v × ∇ϕ) e and v = W k , i.e. with the notations ϕ = Φ Z Z e dΓ = e A · (∇ × W k )] dΓ nA · W k × ∇Φ Φ[n ΓA,Φ

ΓA,Φ

−

Z

C

(3.90)

(3.91)

e k · dl, ΦW

where curve C is bounding the interface between the A-region and the Φ-region. If this surface is closed (e.g. when symmetry planes are not taken into account), the curve integral can be eliminated. Otherwise, the curve C meets with boundary type of ΓB and/or ΓH (Fig. 2.15). In the first case, n × W k = 0, i.e. W k · dl = 0. In the second case Φ is given by (2.237), i.e. Φ = Φ0 . Finally, Z Z e k · dl = Φ0 W k · dl, ΦW (3.92) C

CH

where CH denotes the path lying on ΓA,Φ and meeting ΓH . The weak form the gauged A − Φ-formulation can be collected as follows: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A Ω Z A e Φ[(∇ × W k ) · nA ] dΓ + ΓA,Φ

=

Z

ΓHA

+

Z

CH

−

Z

W k · K dΓ +

Z

ΓA,Φ

Φ0 W k · dl

(3.93)

e (∇ × W k ) · IdΩ,

ΩA

where k = 1, · · · , J and Z Z e dΩ + − µ∇Nk · ∇Φ ΩΦ

−

e 0 × nA dΓ Wk · T

Z

ΩΦ


with k = 1, · · · , I.

Z

ΓA,Φ

ΓBΦ

e · nA dΓ = Nk (∇ × A)

Nk b dΓ,

(3.94)

86


3.3.6 The gauged T , Φ − Φ formulation

There are two unknown potentials in this formulation, the current vector potential T in the eddy current region Ωc and the reduced magnetic scalar potential Φ in the whole region Ωc ∪ Ωn , that is two equations must be worked out. These are coming from the partial differential equations (2.316)–(2.318) and the boundary and interface conditions (2.319)–(2.330). It is important to note that the resulting system of equations should be symmetric. The first weak formulation is based on the partial differential equation (2.316), on the Neumann type boundary conditions (2.321), (2.323) and on the interface condition (2.330), " # Z e e ∂ T 1 ∂ Φ 1 e −∇ e + µo W k · ∇× dΩ ∇× T ∇·T −µo ∇ σ σ ∂t ∂t Ωc Z Z 1 1 e e Wk · ∇ · T (W k · n) dΓ + ∇ × T × n dΓ = (3.95) + σ ΓHc ∪Γnc σ ΓE Z Z e0 e ∂T ∂R dΩ − dΩ, k = 1, · · · , J, µo W k · Wk · − ∂t ∂t Ωc Ωc where

W k · n = 0,

on ΓE ,

(3.96)

and W k × n = 0,

on ΓHc ∪ Γnc .

(3.97)

The second order derivatives in the first integral can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.98)

∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

(3.99)

e and the identity with the notations v = W k and u = 1/σ∇ × T

e and v = W k also must be used. This results in with the notations ϕ = 1/σ∇ · T Z 1 e + 1∇ · Wk ∇ · T e dΩ (∇ × W k ) · ∇ × T σ Ωc σ # Z " e e ∂T ∂Φ µo W k · dΩ − µo W k · ∇ + ∂t ∂t Ωc Z 1 1 e e ∇ × T × W k · n − ∇ · T (W k · n) dΓ (3.100) + σ σ ΓE ∪ΓHc ∪Γnc Z Z 1 1 e e + ∇ · T (W k ·n) dΓ + ∇ × T × n dΓ = Wk · σ ΓHc ∪Γnc σ ΓE Z Z e0 e ∂T ∂R µo W k · dΩ − Wk · dΩ. − ∂t ∂t Ωc Ωc


87

The first and the last boundary integral terms on the left are vanishing on the boundary part ΓE . The first integrand of the first boundary integral term can be reformulated as

1 e ∇ × T × Wk · n σ 1 e ∇×T · Wk = n× σ 1 e ×n , = − Wk · ∇×T σ

(3.101)

which is the same as the last one with opposite sign. The first term of the first boundary integral term is equal to zero on the rest part ΓHc ∪ Γnc , because of the Dirichlet type boundary and interface conditions (2.319) and (2.328), i.e. W k × n = 0,

on ΓHc ∪ Γnc .

(3.102)

The second term of the first and the second boundary integral terms are vanishing on ΓHc ∪ Γnc and the first one is equal to zero on the rest part ΓE because of the Dirichlet type boundary condition (2.325) and Wk · n = 0

(3.103)

there. Finally, the first equation of the weak form is the following:

1 1 e e (∇ × W k ) · ∇ × T + ∇ · W k ∇ · T dΩ σ Ωc σ # " Z e e ∂T ∂Φ dΩ = − µo W k · ∇ µo W k · + ∂t ∂t Ωc Z e0 ∂T − µo W k · dΩ ∂t Ωc Z e ∂R Wk · − dΩ, ∂t Ωc Z

(3.104)

where k = 1, · · · , J. The partial differential equations (2.317) and (2.318), moreover the Neumann type boundary conditions (2.324), (2.326), (2.329) can be summarized in the weak formulation formulation presented next. The time derivative of these partial differential equations and the according Neumann type boundary conditions must be performed, anyway the resulting system of equations will not be symmetric. It is noted that it is useful to multiply the partial differential equations (2.317) and (2.318) by −1.

88


After taking the time derivative, the following form can be obtained: ! e e ∂Φ ∂T − dΩ Nk ∇ · µo − µo ∇ ∂t ∂t Ωc ! Z e ∂Φ dΩ Nk ∇ · µ∇ + ∂t Ωn ! Z e e0 e e ∂R ∂T ∂T ∂Φ + · n dΓ Nk µo + µo − µo ∇ + ∂t ∂t ∂t ∂t ΓE ! Z e0 e ∂T ∂Φ ∂b +µ · n − µ∇ · n dΓ + Nk ∂t ∂t ∂t ΓB ! Z e e0 ∂Φ ∂T + ·nn dΓ Nk µ − µ∇ ∂t ∂t Γnc ! Z e e e0 e ∂T ∂Φ ∂R ∂T + ·nc dΓ Nk µo + µo − µo ∇ + ∂t ∂t ∂t ∂t Γnc ! Z e0 ∂T = dΩ Nk ∇ · µo ∂t Ωc ! Z e0 ∂T + dΩ Nk ∇ · µ ∂t Ωn Z e ∂R dΩ, Nk ∇ · + ∂t Ωc

(3.105)

Nk = 0,

(3.106)

Z

where on ΓHc ∪ ΓHn ,

and k = 1, · · · , I.

(3.107)

The first, the second and the last three integral terms can be reformulated by the use of the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

(3.108)

e /∂t, or v = µ∂ T e /∂t, v = µo ∇∂ Φ/∂t, e or with the notations ϕ = Nk and v = µo ∂ T e e e e v = µ∇∂ Φ/∂t and v = µo ∂ T 0 /∂t, or v = µ∂ T 0 /∂t and v = ∂ R/∂t.


89

The above manipulations result in the following equation: Z

Z e e ∂T ∂T dΩ − · n dΓ µo ∇Nk · Nk µo ∂t ∂t Ωc ΓE ∪ΓHc ∪Γnc Z Z e e ∂Φ ∂Φ µo ∇Nk · ∇ dΩ − µ∇Nk · ∇ dΩ − ∂t ∂t Ωc Ωn Z e ∂Φ + · n dΓ Nk µo ∇ ∂t ΓE ∪ΓHc ∪Γnc Z e ∂Φ Nk µ∇ · n dΓ + ∂t ΓB ∪ΓHn ∪Γnc ! Z e0 e e e ∂T ∂T ∂Φ ∂R + · n dΓ Nk µo + µo − µo ∇ + ∂t ∂t ∂t ∂t ΓE ! Z e0 e ∂b ∂T ∂Φ Nk + +µ · n − µ∇ · n dΓ ∂t ∂t ∂t ΓB ! Z e0 e ∂T ∂Φ Nk µ ·nn dΓ − µ∇ + ∂t ∂t Γnc ! Z e e0 e e ∂R ∂T ∂T ∂Φ ·nc dΓ = + µo − µo ∇ + Nk µo + ∂t ∂t ∂t ∂t Γnc Z Z e0 e0 ∂T ∂T dΩ − dΩ µo ∇Nk · µ∇Nk · − ∂t ∂t Ωc Ωn Z Z e0 e0 ∂T ∂T + · n dΓ + · n dΓ Nk µo Nk µ ∂t ∂t ΓE ∪ΓHc ∪Γnc ΓB ∪ΓHn ∪Γnc Z Z e e ∂R ∂R − dΩ + · n dΓ. ∇Nk · Nk ∂t ∂t Ωc ΓE ∪ΓHc ∪Γnc

(3.109)

It must be noted that the boundary of the region Ωc is ∂Ωc = ΓE ∪ ΓHc ∪ Γnc and the boundary of the region Ωn is ∂Ωn = ΓB ∪ ΓHn ∪ Γnc .

The fourth boundary integral term defined on ΓE is vanishing, because of the same terms with opposite sign in the eighth, first, second and the last boundary integral terms. The sixth and seventh boundary terms defined on Γnc are vanishing, too, because of the same terms with opposite sign in the ninth, the third, moreover the eighth, the first, the second and the last boundary integral terms. The first and the second boundary integral terms are vanishing on the rest ΓHc , because of the Dirichlet type boundary conditions (2.320), i.e. Nk = 0 on ΓHc . The last two terms in the fifth boundary integral, defined on ΓB , are vanishing according to the terms in the third and in the ninth boundary integrals. The third one is equal to zero on ΓHn because of (2.320). The last three integrals are equal to zero on the rest part ΓHc and on ΓHn as well.

90


Finally, the following weak formulation can be obtained: Z Z Z e e e ∂T ∂Φ ∂Φ µo ∇Nk · µo ∇Nk ·∇ µ∇Nk ·∇ dΩ − dΩ − dΩ = ∂t ∂t ∂t Ωc Ωc Ωn Z Z e0 e0 ∂T ∂T (3.110) − µo ∇Nk · dΩ − µ∇Nk · dΩ ∂t ∂t Ωc Ωn Z Z e ∂R ∂b − Nk dΓ − ∇Nk · dΩ. ∂t ∂t ΓB Ωc

Finally, the weak formulation of the T , Φ − Φ formulation satisfying Coulomb gauge is the following: Z 1 e + 1∇ · Wk ∇ · T e dΩ (∇ × W k ) · ∇ × T σ Ωc σ # Z " e e ∂T ∂Φ dΩ = − µo W k · ∇ µo W k · + (3.111) ∂t ∂t Ωc Z Z e0 e ∂T ∂R µo W k · Wk · − dΩ − dΩ, k = 1, · · · , J, ∂t ∂t Ωc Ωc Z −

Z Z e e e ∂T ∂Φ ∂Φ µo ∇Nk · µo ∇Nk ·∇ µ∇Nk ·∇ dΩ + dΩ + dΩ ∂t ∂t ∂t Ωc Ωc Ωn Z Z e0 e0 ∂T ∂T dΩ + µ∇Nk · dΩ = µo ∇Nk · ∂t ∂t Ωc Ωn Z Z e ∂R ∂b + Nk dΓ + ∇Nk · dΩ, k = 1, · · · , I. ∂t ∂t ΓB Ωc

(3.112)

3.3.7 The ungauged T , Φ − Φ formulation There are two unknown potentials in this formulation, too, the current vector potential T in the eddy current region Ωc and the reduced magnetic scalar potential Φ in the whole region Ωc ∪Ωn , that is two equations must be realized, coming from the partial differential equations (2.331)–(2.333) and the boundary and interface conditions (2.334)–(2.342). The first weak formulation is based on the partial differential equation (2.331) and on the Neumann boundary condition (2.337), # " Z e e ∂T ∂Φ 1 e Wk · ∇ × dΩ ∇ × T + µo − µo ∇ σ ∂t ∂t Ωc Z Z e0 1 ∂T e × n dΓ = − (3.113) + Wk · µo W k · ∇×T dΩ σ ∂t ΓE Ωc Z e ∂R Wk · dΩ. − ∂t Ωc


91

Here W k × n = 0,

on ΓHc ∪ Γnc ,

(3.114)

and k = 1, · · · , J. The second order derivatives in the first integral can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.115)

e, with the notations v = W k and u = 1/σ∇ × T Z "

# e e 1 ∂ T ∂ Φ e +µo W k · dΩ (∇×W k ) · ∇× T − µo W k ·∇ ∂t ∂t Ωc σ Z 1 e × W k · n dΓ + ∇×T σ ΓE ∪ΓHc ∪Γnc Z 1 e × n dΓ = ∇×T Wk · + σ ΓE Z Z e0 e ∂T ∂R − µo W k · Wk · dΩ − dΩ. ∂t ∂t Ωc Ωc

(3.116)

The first and the second boundary integral terms are vanishing on the boundary part ΓE , the integrand of the first boundary integral term can be reformulated as 1 e ∇ × T × Wk · n σ 1 e ∇×T · Wk (3.117) = n× σ 1 e ×n , = − Wk · ∇×T σ which is the same as the second one with opposite sign. The first boundary integral term is equal to zero on the rest part ΓHc ∪ Γnc because of the Dirichlet type boundary and interface conditions (2.334) and (2.341), i.e. W k × n = 0 on ΓHc ∪ Γnc . Finally, the first equation of the weak form is the following: # Z " e e 1 ∂T ∂Φ e dΩ = (∇×W k ) · ∇ × T +µo W k · − µo W k ·∇ ∂t ∂t Ωc σ (3.118) Z Z e0 e ∂T ∂R µo W k · Wk · − dΩ − dΩ, k = 1, · · · , J. ∂t ∂t Ωc Ωc Fortunately, the weak formulation of the partial differential equations (2.332) and (2.333) and the Neumann type boundary conditions (2.338), (2.339), (2.342) is the same as presented in the last point H.

92


The weak formulation of the ungauged T , Φ − Φ formulation is the following: # Z " e e 1 ∂ T ∂ Φ e +µo W k · dΩ = (∇×W k )· ∇ × T − µo W k ·∇ ∂t ∂t Ωc σ Z e0 ∂T (3.119) − µo W k · dΩ ∂t Ωc Z e ∂R − dΩ, Wk · ∂t Ωc and Z e e ∂T ∂Φ µo ∇Nk ·∇ dΩ + dΩ ∂t ∂t Ωc Ωc Z e ∂Φ + µ∇Nk ·∇ dΩ ∂t Ωn Z Z e0 e0 ∂T ∂T µo ∇Nk · µ∇Nk · = dΩ + dΩ ∂t ∂t Ωc Ωn Z Z e ∂b ∂R dΩ. Nk dΓ + ∇Nk · + ∂t ∂t ΓB Ωc −

Z

µo ∇Nk ·

(3.120)

This weak formulation must be realized by a numerical technique, which is not sensitive to Coulomb gauge. Here k = 1, · · · , J and k = 1, · · · , I, respectively.

3.3.8 The gauged A, V − A formulation There are two unknown potentials in this formulation, the magnetic vector potential A in the entire problem region Ωc ∪ Ωn and the electric scalar potential V defined only in the eddy current region Ωc , consequently two equations are needed. These are coming from the partial differential equations, the boundary and interface conditions given by (2.344)–(2.361). It is important to note that the resulting system of equations should be symmetric, we are going to take care about it. First of all, electric scalar potential V is replaced by the function v = v(r, t) defined by Z t v= V (τ ) dτ, (3.121) −∞

from which V =

∂v . ∂t

(3.122)

The resulting system of equations will be symmetric by this modification, and this function is approximated by ve.


93

The first weak formulation is based on the partial differential equations (2.344) and (2.346), on the Neumann type boundary conditions (2.347), (2.350), (2.354), (2.356) and on the interface conditions (2.359), (2.361), Z h i e − ∇ νo ∇ · A e dΩ W k · ∇ × νo ∇ × A Ωc ! Z e ∂e v ∂A dΩ + σ∇ Wk · σ + ∂t ∂t Ωc Z h i e − ∇ ν∇ · A e dΩ + W k · ∇ × ν∇ × A Ω Z n Z e e + νo ∇ · A(W ν∇ · A(W k · n) dΓ + k · n) dΓ ΓE ΓB Z h i e + Ie × n dΓ (3.123) + W k · νo ∇ × A +

Z

Z

ΓHc

ΓHn

Wk ·

h i e × n − K dΓ ν∇ × A

i e + Ie × nc + ν∇ × A e × nn dΓ νo ∇ × A Z ZΓnc h i e e · n ) + ν∇ · A(W · n ) dΓ = ν∇ · A(W W k · J 0 dΩ + k c k n Γnc Ωn Z e dΩ, − W k · (∇ × I) +

Wk ·

h

Ωc

where

n × W k = 0,

on ΓE ∪ ΓB ,

(3.124)

W k · n = 0,

on ΓHc ∪ ΓHn ,

(3.125)

and

and here k = 1, · · · , J. The second order derivatives in the first and in the third integrals can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v, e or u = ν∇ × A e and the identity with the notations v = W k and u = νo ∇ × A, ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

(3.126)

(3.127)

e or ϕ = ν∇ · A e and v = W k also must be used. The last with the notations ϕ = νo ∇ · A, e This results integral on the right-hand side can also be rewritten by v = W k and u = I. in

94


Z

h

i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A Ωc ! Z e ∂A ∂e v dΩ + σ∇ Wk· σ + ∂t ∂t Ωc Z h i e + ν∇ · W k ∇ · A e dΩ ν (∇ × W k )· ∇ × A + Ω Z n i h e × W k · n dΓ + νo ∇ × A ΓE ∪ΓHc ∪Γnc

+

Z

h

ΓB ∪ΓHn ∪Γnc

− − +

Z

ΓE ∪ΓHc ∪Γnc

Z

ΓE

+ +

e νo ∇ · A(W k · n) dΓ

ΓB ∪ΓHn ∪Γnc

Z

Z

ΓHc

ΓHn

Wk ·

Z

e ν∇ · A(W k · n) dΓ

e νo ∇ · A(W k · n) dΓ + Wk ·

Z

i e × W k · n dΓ ν∇ × A

h

Z

ΓB

(3.128)


i e + Ie × n dΓ νo ∇ × A

h i e × n − K dΓ ν∇ × A

h i e + Ie × nc + ν∇ × A e × nn dΓ νo ∇ × A Z ZΓnc h i e e νo ∇ · A(W W k ·J 0 dΩ + k · nc ) + ν∇ · A(W k · nn ) dΓ = Ωn Z ZΓnc (∇ × W k ) · Ie dΩ − (Ie × W k ) · n dΓ. −

+

Ωc

Wk ·

ΓE ∪ΓHc ∪Γnc

The first and the second boundary integrals are vanishing on the boundary part ΓHc ∪ Γnc and on ΓHn ∪Γnc , respectively, because of the seventh, eighth, ninth and the last boundary integral terms after using the identity h i h i e × W k · n = n × ν∇ × A e · Wk ν∇ × A h i (3.129) e ×n . = − W k · ν∇ × A

The first and the second as well as the last boundary integral terms are equal to zero on the rest parts ΓE and ΓB as well, because of the Dirichlet type boundary condition (2.352) and (2.355), i.e. W k × n = 0 on these boundaries. The third and the fourth boundary integral terms are vanishing on ΓE , on ΓB and on Γnc , because of the fifth, the sixth and the last boundary integral terms on the left-hand side. On the part ΓHc and on ΓHn these integral terms are equal to zero, too, because of Dirichlet type boundary conditions (2.349) and (2.351), i.e. W k · n = 0 there. The last boundary integral term containing


95

the nonlinear residual term Ie is vanishing, because of the same terms on the left-hand side on ΓHc ∪ Γnc and the Dirichlet type boundary condition (2.352). Here the identity e · W k = (Ie × n) · W k −(Ie × W k ) · n = −(n × I)

must be used on the right-hand side. Finally, the first equation of the weak form is the following: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A Ωc ! Z e ∂A ∂e v dΩ + σ∇ Wk · σ + ∂t ∂t Ωc Z h i e + ν∇ · W k ∇ · A e dΩ ν (∇ × W k ) · ∇ × A + Ω Z Z Z n (∇ × W k ) · Ie dΩ, W k · KdΓ − = W k · J 0 dΩ + Ωn

ΓHn

(3.130)

(3.131)

Ωc

and k = 1, · · · , J. The partial differential equation (2.345), with the Neumann type boundary condition (2.348) and the interface condition (2.360) can be summarized in the formulation presented next. Here, the conditions are multiplied by −1, ! Z e ∂A ∂e v − Nk ∇ · σ dΩ + σ∇ ∂t ∂t Ωc ! (3.132) Z e ∂e v ∂A + · n dΓ = 0, + σ∇ Nk σ ∂t ∂t ΓHc ∪Γnc where Nk = 0,

on ΓE ,

(3.133)

and k = 1, · · · , I. The first volume integral can be reformulated by the use of the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v e + σ∇∂e v /∂t, with the notations ϕ = Nk and v = σ∂ A/∂t ! Z e ∂A ∂e v ∇Nk · σ dΩ + σ∇ ∂t ∂t Ωc ! Z e ∂e v ∂A − Nk σ · n dΓ + σ∇ ∂t ∂t ΓE ∪ΓHc ∪Γnc ! Z e ∂e v ∂A · n dΓ = 0. + σ∇ + Nk σ ∂t ∂t ΓHc ∪Γnc

(3.134)

(3.135)

96


The boundary integral terms are vanishing on ΓHc ∪ Γnc and the first one is equal to zero because Nk = 0 on ΓE according to Dirichlet type boundary condition (2.353). Finally, the following weak formulation can be written: ! Z e ∂A ∂e v ∇Nk · σ dΩ = 0, k = 1, · · · , I. (3.136) + σ∇ ∂t ∂t Ωc Finally, the weak formulation of the A, V − A formulation satisfying Coulomb gauge is the following: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k )· ∇ × A Ωc ! Z e ∂e v ∂A Wk · σ dΩ + + σ∇ ∂t ∂t Ωc Z h i e + ν∇ · W k ∇ · A e dΩ (3.137) + ν (∇ × W k )· ∇ × A =

ZΩn Ωn

−

Z

Ωc

Z

Ωc

W k · J 0 dΩ +

Z

ΓHn

(∇ × W k ) · Ie dΩ, ∇Nk ·

W k · KdΓ

e ∂A ∂e v σ + σ∇ ∂t ∂t

!

dΩ = 0.

(3.138)

Here k = 1, · · · , J and k = 1, · · · , I, respectively.

3.3.9 The ungauged A, V − A formulation

There are two unknown potentials in this formulation, the magnetic vector potential A in the entire problem region Ωc ∪ Ωn and the electric scalar potential V defined only in the eddy current region Ωc , consequently two equations are needed. These are coming from the partial differential equations, the boundary and interface conditions given by (2.362)–(2.373). It is important to note that the resulting system of equations should be symmetric, we are going to take care about it. The electric scalar potential V is replaced and approximated by v = v(r, t) defined by (3.122) and ve, respectively. The impressed current vector potential T 0 has the property that J 0 , in Ωn , (3.139) ∇ × T0 = 0, in Ωc , that is why the term ∇ × T 0 can be appended to the right-hand side of (2.362). In this case it is easier to obtain the weak formulation. The first weak formulation is based on the partial differential equations (2.362) and (2.364), on the Neumann type boundary conditions (2.365), (2.367) and on the interface condition (2.372),

97


Z

Ωc

+ +

Z

Z h i e dΩ + W k · ∇ × νo ∇ × A

Ωc

h i e dΩ W k · ∇ × ν∇ × A

ZΩn

ΓHc

+

Z

ΓHn

+ =

Wk ·

Z

ZΓnc

Wk ·

h

e ∂A ∂e v σ + σ∇ ∂t ∂t


h i e × n − K dΓ W k · ν∇ × A

!

dΩ

(3.140)

h i e + Ie × nc + ν∇ × A e × nn dΓ νo ∇ × A Z e 0 dΩ − Wk · ∇ × T W k · ∇ × Ie dΩ,

Wk ·

Ωc ∪Ωn

Ωc

where n × W k = 0 on ΓE ∪ ΓB , and k = 1, · · · , J. The second order derivative in the first and in the third integrals can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.141)

e or u = ν∇ × A, e moreover the integrals with the notation v = W k and u = νo ∇ × A, e 0 and on the right-hand side can also be reformulated by the notation v = W k , u = T e finally u = I, ! Z Z e ∂A ∂e v e νo (∇ × W k ) · ∇ × A dΩ + Wk · σ dΩ + σ∇ ∂t ∂t Ωc Ωc Z e dΩ ν (∇ × W k ) · ∇ × A + Ω Z n h i e ×W k ·n dΓ νo ∇ × A + +

Z

ΓE ∪ΓHc ∪Γnc

ΓB ∪ΓHn ∪Γnc

+

Z

ΓHc

+

Z

ΓHn

+ =

Z

ZΓnc

Wk ·

+

h

h i e + Ie × nc + ν∇ × A e × nn dΓ νo ∇ × A Z e 0 dΩ + e 0 ×W k )·ndΓ (∇ ×W k )· T (T

Wk ·

ΓB ∪ΓHn ∪Γnc

−

Z

Ωc


h i e × n − K dΓ W k · ν∇ × A

Ωc ∪Ωn

Z

h i e ×W k ·n dΓ ν∇ × A

ΓE ∪ΓHc ∪Γnc

e 0 ×W k )·ndΓ (T

e (∇ × W k ) · IdΩ −

Z

ΓE ∪ΓHc ∪Γnc

(Ie × W k ) · n dΓ

(3.142)

98


can be obtained. The first and the second boundary integral terms are vanishing on the boundary part ΓHc ∪ Γnc and on ΓHn ∪ Γnc , respectively, because of the third, fourth and fifth boundary integral terms after using the identity i h h i e × W k · n = −W k · ν∇ × A e ×n . ν∇ × A (3.143)

The first and the second boundary integral terms are equal to zero on the rest parts ΓE and ΓB , because of the Dirichlet type boundary condition (2.368) and (2.370), W k × n = 0 on these boundaries. The first and the second boundary integral terms on the right-hand side vanish on Γnc , because n = nc in the first and n = nn in the second boundary integral on the right-hand side, moreover nc = −nn . On the rest parts ΓHc and ΓHn , e 0 × n ≡ 0, because these are symmetry planes where the tangential component of T the magnetic field intensity is equal to zero and the impressed current vector potential must satisfy this homogeneous Dirichlet type condition, moreover, on ΓE and ΓB , the Dirichlet type boundary condition (2.368) and (2.370) must satisfy, i.e. W k × n = 0 on these boundaries. The last boundary integral term on the right-hand side is vanishing, too, because of the same terms on the left-hand side and W k × n = 0 on ΓE and e · W k = (Ie × n) · W k . −(Ie × W k ) · n = −(n × I)

(3.144)

Finally, the first equation of the weak form is the following: Z

Ωc

e dΩ + νo (∇ × W k ) · ∇ × A

Z

Ωc

Wk ·

e ∂A ∂e v σ + σ∇ ∂t ∂t

Z e dΩ = e 0 dΩ ν (∇ × W k ) · ∇ × A (∇ × W k ) · T Ωc ∪Ωn Ωn Z Z (∇ × W k ) · Ie dΓ. W k · K dΓ − + +

Z

ΓHn

!

dΩ (3.145)

Ωc

It is possible to select the impressed current vector potential T 0 such that it satisfies the Dirichlet type boundary condition T 0 × n = K,

on ΓHn .

(3.146)

In this case the first boundary integral term on the right-hand side of (3.142) on the boundary part ΓHn can be reformulated as Z Z e 0 × W k ) · n dΓ = e 0 ) · W k dΓ = (T (n × T ΓHn

−

Z

ΓHn

ΓHn

e 0 × n) · W k dΓ = − (T

Z

ΓHn

(3.147)

K · W k dΓ,

i.e. the boundary integral term on the right-hand side of (3.145) can be neglected. The weak formulation of the partial differential equation (2.363), the Neumann type boundary condition (2.366) and the interface condition (2.373) can be obtained in the same way presented in the last item, see the equations (3.132)–(3.136).


99

Finally, the weak formulation of the ungauged A, V −A formulation is the following: ! Z Z e ∂A ∂e v e dΩ νo (∇ × W k ) · ∇ × A dΩ + Wk · σ + σ∇ ∂t ∂t Ωc Ωc Z e dΩ (3.148) + ν (∇ × W k ) · ∇ × A =

ZΩn

Ωc ∪Ωn

Z

Ωc

e 0 dΩ − (∇ × W k ) · T

∇Nk ·

e ∂A ∂e v σ + σ∇ ∂t ∂t

!

Z

Ωc

(∇ × W k ) · Ie dΓ,

dΩ = 0.

(3.149)

This weak formulation must be realized by a numerical technique, which is not sensitive to Coulomb gauge, moreover k = 1, · · · , J and k = 1, · · · , I, respectively.

3.3.10 The ungauged A⋆ − A formulation The only one unknown potential in this formulation is the magnetic vector potential A in the entire problem region Ωc ∪ Ωn , but it is usually denoted by A⋆ in the eddy current region Ωc . The weak formulation can be obtained from the partial differential equations, the boundary and interface conditions given by (2.374)–(2.380). The impressed current vector potential T 0 has the property shown in (3.139), that is why the term ∇ × T 0 can be appended to the right-hand side of (2.374). In this case it is easier to obtain the weak formulation. The weak formulation is based on the partial differential equations (2.374) and (2.375), on the Neumann type boundary conditions (2.376), (2.377) and on the interface condition (2.381), Z Z i h e⋆ ⋆ ∂A e W k · ∇ × νo ∇ × A σW k · dΩ dΩ + ∂t Ωc ZΩc h i e dΩ W k · ∇ × ν∇ × A + ZΩn h i e ⋆ + Ie × n dΓ + W k · νo ∇ × A ΓH (3.150) Z c h i e × n − K dΓ + W k · ν∇ × A ΓHn

+

=

Z

ZΓnc

Ωc ∪Ωn

Here

h i e ⋆ + Ie × nc + ν∇ × A e × nn dΓ ν∇ × A Z e e dΩ. W k · ∇ × T 0 dΩ − W k · (∇ × I)

Wk ·

n × W k = 0,

Ωc

on ΓE ∪ ΓB ,

(3.151)

100


and k = 1, · · · , J. The first, the third and the last two integrals can be reformulated as it was presented in the last item. Finally, the weak formulation of the ungauged A⋆ − A formulation is the following: Z Z e⋆ ⋆ ∂A e νo (∇ × W k ) · ∇ × A dΩ + σW k · dΩ ∂t Ωc Z ZΩc e dΩ = e 0 dΩ (3.152) ν (∇ × W k ) · ∇ × A (∇ × W k ) · T + ZΩn − (∇ × W k ) · Ie dΓ,

Ωc ∪Ωn

Ωc

and k = 1, · · · , J. This weak formulation must be realized by a numerical technique, which is not sensitive to Coulomb gauge.

3.3.11 The gauged T , Φ − A formulation

The current vector potential T with the reduced magnetic scalar potential Φ is used in the eddy current region Ωc and the magnetic vector potential A is used in the region free of eddy currents, Ωn . There are three unknown potentials, T , Φ and A, which can be calculated by the following weak formulation obtained from the gauged version of the T , Φ − A − Φ formulation (see on page 102): Z 1 1 e e (∇ × W k ) · ∇ × T + ∇ · W k ∇ · T dΩ σ Ωc σ # " Z e e ∂T ∂Φ dΩ − µo W k · ∇ µo W k · + ∂t ∂t Ωc ! Z e ∂A (3.153) − × nA dΓ = Wk · ∂t Γnc Z e0 ∂T µo W k · dΩ − ∂t Ωc Z e ∂R − dΩ, k = 1, · · · , J, Wk · ∂t Ωc Z Z e e ∂T ∂Φ − µo ∇Nk · dΩ + µo ∇Nk ·∇ dΩ ∂t ∂t Ωc Ωc ! Z e ∂A Nk ∇ × − · nA dΓ ∂t Γnc (3.154) Z e0 ∂T = µo ∇Nk · dΩ ∂t Ωc Z e ∂R ∇Nk · dΩ, k = 1, · · · , I, + ∂t Ωc


"

! # e e ∂A ∂A − ν (∇ × W k ) · ∇ × + ν∇ · W k ∇ · dΩ ∂t ∂t Ωn Z Z e e ∂T ∂Φ · (W k × nA ) dΓ − (∇ × W k ) · nA dΓ = − ∂t Γ Γnc ∂t Z Z nc ∂J 0 ∂K Wk · dΩ − dΓ Wk · − ∂t ∂t Ωn ΓHn ! Z Z e e0 ∂Φ ∂T Wk · × nA dΓ + W k · dl, k = 1, · · · , J. − ∂t Γnc C ∂t Z

101

(3.155)

3.3.12 The ungauged T , Φ − A formulation

The current vector potential T with the reduced magnetic scalar potential Φ is used in the eddy current region Ωc and the magnetic vector potential A is used in the region free of eddy currents, Ωn . There are three unknown potentials, T , Φ and A, which can be calculated by the following weak formulation coming from the ungauged version of the T , Φ − A − Φ formulation (see on page 110): # Z " e e 1 ∂T ∂Φ e dΩ (∇ × W k ) · ∇ × T + µo W k · − µo W k · ∇ ∂t ∂t Ωc σ ! Z Z e e0 ∂T ∂A (3.156) − × nA dΓ = − dΩ Wk · µo W k · ∂t ∂t Γnc Ωc Z e ∂R dΩ, Wk · − ∂t Ωc Z e e ∂T ∂Φ dΩ + dΩ µo ∇Nk ·∇ ∂t ∂t Ωc Ωc ! Z e ∂A − · nA dΓ Nk ∇ × ∂t Γnc Z Z e0 e ∂T ∂R = µo ∇Nk · ∇Nk · dΩ + dΓ, ∂t ∂t Ωc Ωc ! Z Z e e ∂T ∂A dΩ − · (W k × nA ) dΓ ν (∇ × W k ) · ∇ × − ∂t Ωn Γnc ∂t Z e ∂Φ (∇ × W k ) · nA dΓ = − Γnc ∂t Z Z e0 e ∂T ∂Φ dΩ − W k · dl, (∇ × W k ) · − ∂t Ωn C ∂t −

Z

µo ∇Nk ·

where k = 1, · · · , J, k = 1, · · · , I and k = 1, · · · , J, respectively.

(3.157)

(3.158)

102


3.3.13 The gauged T , Φ − A − Φ formulation

The current vector potential T with the reduced magnetic scalar potential Φ is used in the eddy current region Ωc , the magnetic vector potential A is used in the holes, ΩnA and the reduced magnetic scalar potential Φ is used in the rest region free of eddy currents, ΩnΦ . There are three unknown potentials, T , Φ and A, so three equations must be set up. It is important to note that the resulting system of equations should be symmetric. That is the reason why the time derivative of partial differential equations (2.415), (2.416) and (2.417) and the according Neumann type boundary and interface conditions must be taken. This will be highlighted while obtaining the formulation. The first weak formulation can be obtained from the partial differential equations (2.414) and the Neumann type boundary and interface conditions (2.420), (2.424), (2.433) and (2.437), Z 1 e −∇ 1∇·T e Wk · ∇ × ∇×T dΩ σ σ Ωc # " Z e e ∂T ∂Φ dΩ − µo ∇ W k · µo + ∂t ∂t Ωc Z 1 e (W k · n) dΓ + ∇·T ΓHc ∪ΓncΦ σ Z (3.159) 1 e + Wk · ∇ × T × n dΓ σ ΓE # " Z e ∂ A 1 e × nc − Wk · + ∇×T × nA dΓ = σ ∂t ΓncA Z Z e0 e ∂T ∂R − µo W k · Wk · dΩ − dΩ, k = 1, · · · , J, ∂t ∂t Ωc Ωc where W k × n = 0,

on ΓHc ∪ ΓncΦ ,

(3.160)

and W k · n = 0,

on ΓE ∪ ΓncA .

(3.161)

The second order derivatives in the first two integrals can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v, e and the identity with the notations v = W k and u = 1/σ∇ × T ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

e and v = W k also must be used. This results in with the notations ϕ = 1/σ∇ · T

(3.162)

(3.163)


1 e + 1∇ · Wk ∇ · T e dΩ (∇ × W k ) · ∇ × T σ Ωc σ # Z " e e ∂T ∂Φ dΩ − µo W k · ∇ µo W k · + ∂t ∂t Ωc Z 1 e × W k · n dΓ + ∇×T σ ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ Z 1 e (W k · n) dΓ ∇·T − σ ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ Z Z 1 1 e e + Wk · ∇ · T (W k · n) dΓ + ∇× T × n dΓ σ ΓHc ∪ΓncΦ σ ΓE # " Z e ∂A 1 e + ∇× T × nc − × nA dΓ = Wk · σ ∂t ΓncA Z Z e0 e ∂T ∂R − dΩ − dΩ. µo W k · Wk · ∂t ∂t Ωc Ωc Z

103

(3.164)

The boundary integral terms on the boundary part ΓE are vanishing as well as the same term on the boundary term ΓncA , because the first boundary integral can be reformulated as 1 1 e × W k · n = −W k · e ×n . ∇×T ∇×T (3.165) σ σ

The first first boundary integral is equal to zero on the rest part ΓHc ∪ ΓncΦ because of the Dirichlet type boundary and interface conditions (2.418) and (2.431), i.e. W k × n = 0 on ΓHc ∪ ΓncΦ . The second boundary integral term is vanishing on ΓHc ∪ ΓncΦ because of the third boundary integral term, as well as on the rest part ΓE ∪ ΓncA because of the Dirichlet type boundary and interface conditions (2.426) and (2.436), W k · n = 0 there. Finally, the first equation of the weak form is the following: Z 1 e + 1∇ · Wk ∇ · T e dΩ (∇ × W k ) · ∇ × T σ Ωc σ # Z " e e ∂T ∂Φ µo W k · dΩ − µo W k · ∇ + ∂t ∂t Ωc ! (3.166) Z Z e0 e ∂T ∂A − × nA dΓ = − dΩ µo W k · Wk · ∂t ∂t ΓncA Ωc Z e ∂R Wk · dΩ, − ∂t Ωc where k = 1, · · · , J. The second weak equation of this potential formulation is coming from the time derivative of the partial differential equations (2.415) and (2.416) and the Neumann type

104


boundary and interface conditions (2.425), (2.429), (2.432), (2.435) and (2.440). It is noted that it is useful to multiply the partial differential equations (2.415) and (2.416) by −1. After taking the time derivative, the following formula can be obtained: ! ! Z Z e e ∂Φ ∂T − dΩ + dΩ Nk ∇ · µo Nk ∇ · µo ∇ ∂t ∂t Ωc Ωc ! Z e ∂Φ + dΩ Nk ∇ · µ∇ ∂t ΩnΦ ! Z e0 e e e ∂T ∂T ∂Φ ∂R Nk µo + · n dΓ + µo − µo ∇ + ∂t ∂t ∂t ∂t ΓE ! Z e0 e ∂T ∂Φ ∂b +µ · n − µ∇ · n dΓ Nk + ∂t ∂t ∂t ΓBΦ ! Z e e0 ∂Φ ∂T + · nΦ dΓ − µ∇ Nk µ ∂t ∂t ΓncΦ ! Z e ∂R e e0 e ∂Φ ∂T ∂T (3.167) + · nc dΓ + µo − µo ∇ + Nk µo ∂t ∂t ∂t ∂t ΓncΦ ! Z e ∂A + · nA dΓ Nk ∇ × ∂t ΓncA ! Z e e e0 e ∂Φ ∂R ∂T ∂T + · nc dΓ + µo − µo ∇ + Nk µo ∂t ∂t ∂t ∂t ΓncA " # ! ! Z e e0 e ∂A ∂T ∂Φ ∇× + · nA + µ · nΦ dΓ − µ∇ Nk ∂t ∂t ∂t ΓA,Φ ! Z e0 ∂T Nk ∇ · µo = dΩ ∂t Ωc ! Z Z e0 e ∂T ∂R dΩ + dΩ, Nk ∇ · Nk ∇ · µ + ∂t ∂t ΩnΦ Ωc where Nk = 0,

on ΓHc ∪ ΓHΦ ,

(3.168)

and k = 1, · · · , I. The first, the second, the third on the left and the three integral terms on the right can be reformulated by the use of the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

(3.169)

e /∂t, or v = µo ∇∂ Φ/∂t, e e with the notations ϕ = Nk and v = µo ∂ T or v = µ∇∂ Φ/∂t e e e and v = µo ∂ T 0 /∂t, or v = µ∂ T 0 /∂t, or v = ∂ R/∂t.


105

Finally, the following equation can be obtained: Z e e ∂T ∂T µo ∇Nk · dΩ − · n dΓ Nk µo ∂t ∂t Ωc ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ Z Z e e ∂Φ ∂Φ − dΩ − dΩ µo ∇Nk · ∇ µ∇Nk · ∇ ∂t ∂t Ωc ΩnΦ Z e ∂Φ + · n dΓ Nk µo ∇ ∂t ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ Z e ∂Φ Nk µ∇ · n dΓ + ∂t ΓBΦ ∪ΓHΦ ∪ΓncΦ ∪ΓA,Φ ! Z e e0 e e ∂Φ ∂R ∂T ∂T + · n dΓ + µo − µo ∇ + Nk µo ∂t ∂t ∂t ∂t ΓE ! Z e0 e ∂b ∂T ∂Φ + +µ · n − µ∇ · n dΓ Nk ∂t ∂t ∂t ΓBΦ ! Z e0 e ∂T ∂Φ + · nΦ dΓ − µ∇ Nk µ ∂t ∂t ΓncΦ ! Z e0 e e e ∂T ∂T ∂Φ ∂R + · nc dΓ + µo − µo ∇ + Nk µo ∂t ∂t ∂t ∂t ΓncΦ ! Z e ∂A + Nk ∇ × · nA dΓ ∂t ΓncA ! Z e e0 e e ∂R ∂T ∂T ∂Φ Nk µo · nc dΓ + µo − µo ∇ + + ∂t ∂t ∂t ∂t ΓncA " ! ! # Z e e e0 ∂A ∂Φ ∂T Nk ∇× · nA + µ · nΦ dΓ = − µ∇ + ∂t ∂t ∂t ΓA,Φ ! ! Z Z e0 e0 ∂T ∂T dΩ − dΩ − (∇Nk ) · µo (∇Nk ) · µ ∂t ∂t Ωc ΩnΦ Z e0 ∂T Nk µo · n dΓ + ∂t ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ Z e0 ∂T + · n dΓ Nk µ ∂t ΓBΦ ∪ΓHΦ ∪ΓncΦ ∪ΓA,Φ Z Z e e ∂R ∂R − (∇Nk ) · Nk dΓ + · n dΓ. ∂t ∂t Ωc ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ Z

(3.170)

It must be noted that the boundary of the region Ωc is ∂Ωc = ΓE ∪ ΓHc ∪ ΓncΦ ∪ ΓncA and the boundary of the region Ωn is ∂Ωn = ΓB ∪ ΓHn ∪ ΓncΦ ∪ ΓncA .

106


The fourth boundary integral term is vanishing on ΓE because of the same terms with opposite sign in the first on the right and on the left, second on the left and the last on the right boundary integrals. The second and the third fractions in the fifth boundary integrals are vanishing according to the second one on the right and the third one on the left on ΓB Φ . The interface integral term on ΓncΦ is vanishing according to the same terms with opposite sign in the first two on the right and the first, second and third boundary integral terms on the left. The ninth boundary integral and the second term in the tenth integral are vanishing, because of the terms in the first, second, third and the first two on the right. The e ∂ R/∂t terms are vanishing, too, on ΓncΦ ∪ ΓncA , because of the same terms on the left and on the right. Dirichlet type boundary condition (see (2.419) and (2.423)) is prescribed on ΓHc and on ΓHΦ , where Nk = 0, i.e. the according boundary terms are vanishing on ΓHc ∪ ΓHΦ . Finally, the following weak formulation can be obtained: Z Z Z e e e ∂T ∂Φ ∂Φ µo ∇Nk · dΩ + µo ∇Nk ·∇ dΩ + dΩ µ∇Nk ·∇ − ∂t ∂t ∂t Ωc Ωc ΩnΦ ! Z e ∂A · nA dΓ Nk ∇ × − ∂t ΓncA ∪ΓA,Φ (3.171) Z Z Z e0 e0 ∂b ∂T ∂T µo ∇Nk · µ∇Nk · Nk dΓ dΩ + dΩ + = ∂t ∂t ∂t Ωc ΩnΦ ΓBΦ Z e ∂R + ∇Nk · dΩ, k = 1, · · · , I. ∂t Ωc

This equation has been multiplied by −1 again, because of symmetry conditions. The last weak equation of this potential formulation is coming from the time derivative of the partial differential equation (2.417), the Neumann boundary and interface conditions (2.421), (2.428), (2.434), (2.438) and (2.439), " ! !# Z e e ∂A ∂A − ∇ ν∇ · dΩ W k · ∇ × ν∇ × ∂t ∂t ΩnA " ! # Z Z e e ∂A ∂K ∂A + W k · ν∇× × n− dΓ + (W k ·n) dΓ ν∇· ∂t ∂t ΓHA ΓBA ∪ΓncA ∂t # " ! ! Z e0 e e e ∂T ∂A ∂T ∂Φ + W k· × nc + ν∇ × × nA dΓ (3.172) + −∇ ∂t ∂t ∂t ∂t ΓncA # " ! ! Z e e0 e ∂A ∂T ∂Φ + W k· −∇ × nΦ + ν∇ × × nA dΓ ∂t ∂t ∂t ΓA,Φ Z ∂J 0 dΩ. = Wk· ∂t ΩnA

Here W k × n = 0, and

on ΓBA ,

(3.173)


W k · n = 0,

on ΓHA ∪ ΓA,Φ ,

107

(3.174)

moreover k = 1, · · · , J. The second order derivatives in the first integral can be reduced to first order one by using the following identities: ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v,

(3.175)

∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.176)

and

e v = W k and u = ν∇ × A, e i.e. with the notations ϕ = ν∇A, ! # " Z e e ∂A ∂A + ν∇ · W k ∇ · dΩ ν (∇ × W k ) · ∇ × ∂t ∂t ΩnA " ! # Z e ∂A + ν∇ × × W k · n dΓ ∂t ΓHA ∪ΓBA ∪ΓncA ∪ΓA,Φ Z e ∂A (W k · n) dΓ ν∇ · − ∂t ΓHA ∪ΓBA ∪ΓncA ∪ΓA,Φ " ! # Z e ∂A ∂K ν∇ × ×n− dΓ Wk · + ∂t ∂t ΓHA Z e ∂A (W k · n) dΓ ν∇ · + ∂t ΓBA ∪ΓncA # " ! ! Z e e0 e e ∂A ∂T ∂T ∂Φ + W k· × nc + ν∇× × nA dΓ + −∇ ∂t ∂t ∂t ∂t ΓncA # " ! ! Z e e0 e ∂A ∂T ∂Φ Wk · + × nΦ + ν∇ × × nA dΓ −∇ ∂t ∂t ∂t ΓA,Φ Z ∂J 0 = dΩ. Wk · ∂t ΩnA The first surface integral can be reformulated according to ! ! # " # " e e ∂A ∂A × W k · n = −W k · ×n , ν∇ × ν∇ × ∂t ∂t

(3.177)

(3.178)

and it is vanishing on ΓHA ∪ ΓncA ∪ ΓA,Φ according to the third, fifth and sixth integrals. On the rest part, ΓBA it is equal to zero because of the Dirichlet type boundary condition (2.427) and W k × n = 0 here. The second boundary integral term is vanishing on ΓBA ∪ ΓncA and it is equal to zero on the rest part ΓHA ∪ ΓA,Φ , because of the fourth boundary integral term and the Dirichlet type conditions (2.422), (2.441), respectively.

108


The following weak formulation can be obtained after the above manipulations: ! # " Z e e ∂A ∂A + ν∇ · W k ∇ · dΩ − ν (∇ × W k ) · ∇ × ∂t ∂t ΩnA Z Z e e ∂Φ ∂T − · (W k × nA ) dΓ − (∇ × W k ) · nA dΓ = ΓncA ∂t ΓncA ∪ΓA,Φ ∂t (3.179) Z Z ∂J 0 ∂K dΩ − dΓ Wk · Wk · − ∂t ∂t ΩnA ΓHA ! Z Z e e0 ∂Φ ∂T − Wk · × nA dΓ + W k · dl, ∂t ΓncA ∪ΓA,Φ C ∂t where k = 1, · · · , J. Some notes must be appended here about the boundary integral terms. First of all, the integral equation has been multiplied by −1, because of symmetry e 0 /∂t) × nc on ΓncA and W k · (∂ T e 0 /∂t) × nΦ on ΓA,Φ conditions. The terms W k · (∂ T have been put to the right-hand side and notations nA = −nc and nA = −nΦ have been e /∂t) × nc on Γnc is manipulated as used. Because of symmetry, the term W k · (∂ T A ! ! e e e ∂T ∂T ∂T Wk · (3.180) × nc = −W k · × nA = · (W k × nA ) . ∂t ∂t ∂t e e × nc ), or W k · (∇∂ Φ/∂t × nΦ ) on ΓncA ∪ ΓA,Φ can be The third term W k · (∇∂ Φ/∂t reformulated as ! ! Z Z e e ∂Φ ∂Φ Wk · ∇ nA · W k × ∇ dΓ, (3.181) × nA dΓ = ∂t ∂t ΓncA ∪ΓA,Φ ΓncA ∪ΓA,Φ after using nA = −nc and nA = −nΦ . The identity ∇ × (ϕv) = ϕ∇ × v − v × ∇ϕ

(3.182)

must be applied as n · [∇ × (ϕv)] = ϕ[n · (∇ × v)] − n · (v × ∇ϕ) e with the notations ϕ = ∂ Φ/∂t and v = W k , i.e. ! Z e ∂Φ nA · W k × ∇ dΓ ∂t ΓncA ∪ΓA,Φ Z Z e e ∂Φ ∂Φ [nA · (∇ × W k )] dΓ − W k · dl, = ΓncA ∪ΓA,Φ ∂t C ∂t

(3.183)

(3.184)

where curve C is bounding the interface between the A-region and the region where Φ can be found, i.e. Ωc ∪ ΩnΦ . If this surface is closed (e.g. when symmetry planes are not taken into account), the curve integral can be eliminated. Otherwise, the curve C meets


109

with boundary type of ΓBA ∪ ΓE and/or ΓHc ∪ ΓHA . In the first case, n × W k = 0 (see (2.427)), i.e. W k · dl = 0. In the second case Φ is given by (2.419), i.e. Φ = Φ0 . Finally, Z Z e Φ0 W k · dl, ΦW k · dl = (3.185) C

CH

where CH denotes the path lying on ΓA,Φ and meeting ΓH . The weak equations of the gauged T , Φ − A − Φ formulation are: Z 1 e + 1∇ · Wk ∇ · T e dΩ (∇ × W k ) · ∇ × T σ Ωc σ # Z " e e ∂T ∂Φ µo W k · − µo W k · ∇ dΩ + ∂t ∂t Ωc ! Z e ∂A − × nA dΓ = Wk · ∂t ΓncA Z Z e0 e ∂T ∂R dΩ − dΩ, µo W k · Wk · − ∂t ∂t Ωc Ωc −

Z

Ωc

µo ∇Nk ·

e ∂T dΩ + ∂t

Z

Ω

µo ∇Nk ·∇

e ∂Φ dΩ ∂t

(3.186)

c ! Z e e ∂Φ ∂A · nA dΓ dΩ − Nk ∇ × µ∇Nk ·∇ + ∂t ∂t ΩnΦ ΓncA ∪ΓA,Φ (3.187) Z Z e0 e0 ∂T ∂T = µo ∇Nk · dΩ + dΩ µ∇Nk · ∂t ∂t Ωc ΩnΦ Z Z e ∂R ∂b + dΩ, ∇Nk · Nk dΓ + ∂t ∂t ΓBΦ Ωc ! # " Z e e ∂A ∂A + ν∇ · W k ∇ · dΩ − ν (∇ × W k ) · ∇ × ∂t ∂t ΩnA Z Z e e ∂Φ ∂T · (W k × nA ) dΓ − (∇ × W k ) · nA dΓ = − ΓncA ∂t ΓncA ∪ΓA,Φ ∂t (3.188) Z Z ∂J 0 ∂K dΩ − dΓ Wk · Wk · − ∂t ∂t ΩnA ΓHA ! Z Z e e0 ∂Φ ∂T Wk · × nA dΓ + W k · dl. − ∂t ΓncA ∪ΓA,Φ C ∂t

Z

Here k = 1, · · · , J, k = 1, · · · , I and k = 1, · · · , J, respectively.

110


3.3.14 The ungauged T , Φ − A − Φ formulation

The current vector potential T with the reduced magnetic scalar potential Φ is used in the eddy current region Ωc , the magnetic vector potential A is used in the holes, ΩnA and the reduced magnetic scalar potential Φ is used in the rest region free of eddy currents, ΩnΦ . There are three unknown potentials, T , Φ and A, so three equations must be set up. It is important to note that the resulting system of equations should be symmetric. That is the reason why the time derivative of the partial differential equations (2.443), (2.444) and (2.445) and the according Neumann type boundary and interface conditions are taken. This will be highlighted while obtaining the formulation. The first weak formulation is coming from the partial differential equations (2.442) and the Neumann type boundary and interface conditions (2.450) and (2.459), # " Z e e ∂T 1 ∂Φ e dΩ ∇ × T + µo − µo ∇ Wk · ∇ × σ ∂t ∂t Ωc Z 1 e × n dΓ Wk · ∇×T + σ ΓE " # (3.189) Z e ∂A 1 e + ∇ × T × nc − × nA dΓ = Wk · σ ∂t ΓncA Z Z e0 e ∂T ∂R µo W k · dΩ − Wk · dΩ, − ∂t ∂t Ωc Ωc

where

W k × n = 0,

on ΓHc ∪ ΓncΦ ,

(3.190)

and k = 1, · · · , J. The second order derivative in the first integral can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.191)

e , i.e. with the notations v = W k and u = 1/σ∇ × T # Z " e e 1 ∂T ∂Φ e dΩ (∇ × W k ) · ∇ × T + µo W k · − µo W k · ∇ ∂t ∂t Ωc σ Z 1 e × W k ·n dΓ + ∇×T σ ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ Z 1 e + W k· ∇ × T × n dΓ (3.192) σ ΓE " # Z e 1 e × nc − ∂ A × nA dΓ = ∇×T Wk · + σ ∂t ΓncA Z Z e0 e ∂T ∂R dΩ − dΩ. µo W k · Wk · − ∂t ∂t Ωc Ωc

111


The boundary integral terms on the boundary part ΓE are vanishing as well as the same term on the boundary term ΓncA , because the first term of the first boundary integral can be reformulated as 1 e × Wk · n ∇×T σ 1 e ∇×T · Wk (3.193) = n× σ 1 e ×n . = − Wk · ∇×T σ

The first boundary integral is equal to zero on the rest part ΓHc ∪ ΓncΦ because of the Dirichlet type boundary and interface conditions (2.446) and (2.455), i.e. W k × n = 0 on ΓHc ∪ ΓncΦ . Finally, the first equation of the weak form is the following: # Z " e e 1 ∂T ∂Φ e dΩ (∇ × W k ) · ∇ × T + µo W k · − µo W k · ∇ ∂t ∂t Ωc σ ! Z Z e e0 ∂T ∂A (3.194) − × nA dΓ = − µo W k · dΩ Wk · ∂t ∂t ΓncA Ωc Z e ∂R dΩ, Wk · − ∂t Ωc

where k = 1, · · · , J. The second weak equation is coming from the time derivative of the partial differential equations (2.443) and (2.444) and the Neumann type boundary and interface conditions (2.451), (2.453), (2.456), (2.458) and (2.461). It is noted that it is useful to multiply the partial differential equations (2.443) and (2.444) by −1. The same form can be obtained as in the gauged T , Φ − A − Φ formulation, see (3.167), so the weak formulation is the following: −

Z

Ωc

µo ∇Nk ·

e ∂T dΩ + ∂t

Z

Ωc

µo ∇Nk ·∇

e ∂Φ dΩ + ∂t

Z

Ωn

µ∇Nk ·∇

e ∂Φ dΩ ∂t

Φ ! e ∂A − · nA dΓ Nk ∇ × ∂t ΓncA ∪ΓA,Φ Z Z Z e0 e0 ∂T ∂T ∂b = dΩ + dΩ + µ∇Nk · Nk dΓ µo ∇Nk · ∂t ∂t ∂t Ωc ΩnΦ ΓBΦ Z e ∂R + ∇Nk · dΩ, ∂t Ωc

Z

where k = 1, · · · , I.

(3.195)

112


The last weak equation of this potential formulation is coming from the time derivative of the partial differential equations (2.445) and the Neumann type boundary and interface conditions (2.448), (2.457) and (2.460), " !# Z e ∂A dΩ W k · ∇ × ν∇ × ∂t ΩnA " ! # Z e ∂A ∂K ν∇ × ×n− dΓ + Wk · ∂t ∂t ΓHA " ! ! # Z e e 0 ∂T e e ∂A ∂T ∂Φ + × nc + ν∇ × × nA dΓ + −∇ W k· (3.196) ∂t ∂t ∂t ∂t ΓncA # " ! ! Z e e0 e ∂A ∂T ∂Φ Wk · × nΦ + ν∇ × × nA dΓ −∇ + ∂t ∂t ∂t ΓA,Φ ! Z e0 ∂T = Wk · ∇ × dΩ, k = 1, · · · , J. ∂t ΩnA Here W k × n = 0 on ΓBA is used. The second order derivative in the first integral can be reduced to first order one and the integral on the right-hand side can be reformulated by using the following identity: ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.197)

e e 0 /∂t i.e. and u = ∂ T with the notations v = W k , u = ν∇ × (∂ A/∂t) Z

ΩnA

ν (∇ × W k ) ·

e ∂A ∇× ∂t "

!

dΩ

# ! e ∂A ν∇ × × W k · n dΓ + ∂t ΓHA ∪ΓBA ∪ΓncA ∪ΓA,Φ ! # " Z e ∂A ∂K Wk · ×n− dΓ + ν∇ × ∂t ∂t ΓHA " ! ! # Z e 0 ∂T e e e ∂A ∂T ∂Φ + × nc + ν∇× × nA dΓ + −∇ W k· ∂t ∂t ∂t ∂t ΓncA ! ! # " Z e e e0 ∂Φ ∂A ∂T × nΦ + ν∇ × × nA dΓ −∇ Wk · + ∂t ∂t ∂t ΓA,Φ Z e0 ∂T dΩ (∇ × W k ) · = ∂t ΩnA ! Z e0 ∂T + × W k · ndΓ. ∂t ΓHA ∪ΓBA ∪ΓncA ∪ΓA,Φ Z

(3.198)


The first surface integral can be reformulated according to ! ! # " # " e e ∂A ∂A × W k · n = −W k · ×n , ν∇ × ν∇ × ∂t ∂t

113

(3.199)

and it is vanishing on ΓHA ∪ ΓncA ∪ ΓA,Φ according to the second, third and fourth integral terms. On the rest part, ΓBA , it is equal to zero because of the Dirichlet type boundary condition (2.452) and W k × n = 0 here. The last boundary integral term can be reformulated as ! ! e0 e0 ∂T ∂T (3.200) × W k · nA = −W k · × nA . ∂t ∂t This is the reason why the last boundary integral term is vanishing on ΓncA ∪ ΓA,Φ , because of the same terms in the third and fourth boundary integral terms and n = −nc and n = −nΦ . The last boundary integral term is equal to zero on ΓBA as well, since (2.452) is a Dirichlet type boundary condition, i.e. W k × n = 0 here. According to the time derivative of (3.146) and (3.147), the surface current density K can be eliminated from the formulation and the last boundary integral term is vanishing on the rest part ΓHA . The following weak formulation can be obtained after the above manipulations: ! Z e ∂A − dΩ ν (∇ × W k ) · ∇ × ∂t ΩnA Z Z e e ∂Φ ∂T · (W k × nA ) dΓ − (∇ × W k ) · nA dΓ = (3.201) − ΓncA ∂t ΓncA ∪ΓA,Φ ∂t Z Z e e0 ∂Φ ∂T dΩ + W k · dl. (∇ × W k ) · − ∂t ΩnA CH ∂t Some notes must be appended here about the boundary integral terms. First of all, the equation has been multiplied by −1, because of symmetry conditions, moreover notations e /∂t) × nc on ΓncA nA = −nc and nA = −nΦ have been used. The term W k · (∂ T e e as well as the third term W k · (∇∂ Φ/∂t × nc ), or W k · (∇∂ Φ/∂t × nΦ ) on ΓncA ∪ ΓA,Φ have been reformulated as it is presented in the last item, see the manipulations (3.180)–(3.185). Finally, the weak formulation of the ungauged T , Φ − A − Φ formulation can be summarized as # Z " e e 1 ∂ T ∂ Φ e + µo W k · dΩ (∇ × W k ) · ∇ × T − µo W k · ∇ ∂t ∂t Ωc σ ! Z Z e e0 ∂A ∂T (3.202) − × nA dΓ = − µo W k · dΩ Wk · ∂t ∂t ΓncA Ωc Z e ∂R Wk · dΩ, − ∂t Ωc

114


Z

Z Z e e e ∂T ∂Φ ∂Φ − µo ∇Nk · µo ∇Nk ·∇ dΩ + dΩ + dΩ µ∇Nk ·∇ ∂t ∂t ∂t Ωc Ωc ΩnΦ ! Z e ∂A Nk ∇ × · nA dΓ − ∂t ΓncA ∪ΓA,Φ (3.203) Z Z Z e0 e0 ∂b ∂T ∂T dΩ + dΩ + µo ∇Nk · µ∇Nk · Nk dΓ = ∂t ∂t ∂t Ωc ΩnΦ ΓBΦ Z e ∂R + ∇Nk · dΩ, ∂t Ωc ! Z e ∂A − dΩ ν (∇ × W k ) · ∇ × ∂t ΩnA Z Z e e ∂Φ ∂T − · (W k × nA ) dΓ − (∇ × W k ) · nA dΓ = (3.204) ΓncA ∂t ΓncA ∪ΓA,Φ ∂t Z Z e0 e ∂T ∂Φ − dΩ + W k · dl. (∇ × W k ) · ∂t ΩnA CH ∂t

Here k = 1, · · · , J, k = 1, · · · , I and k = 1, · · · , J, respectively.

3.3.15 The gauged A, V − Φ formulation

The magnetic vector potential A and the electric scalar potential V are used in the eddy current region Ωc and the reduced magnetic scalar potential Φ is used in the region free of eddy currents, Ωn . The three unknown potentials, A, V and Φ can be obtained from the weak formulation of the more general gauged A, V − A − Φ method (see on page 115), Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇ × W k ) · ∇ × A Ωc ! Z Z e ∂e v ∂A e (∇ × W k ) · nA dΓ Φ W k· σ + σ∇ + dΩ + (3.205) ∂t ∂t Ωc Γnc Z Z Z e Φ0 W k · dl − W k · (T 0 × nA )dΓ + (∇ × W k ) · Ie dΩ, = Γnc

Z

C

Ωc

!

e ∂A ∂e v dΩ = 0, + σ∇ ∂t ∂t Ωc Z Z e dΩ + e · nA dΓ = − µ∇Nk · ∇Φ Nk (∇ × A) Ωn Γnc Z Z e 0 dΩ − − µ∇Nk · T Nk b dΓ. Ωn

∇Nk ·

σ

ΓB

Here k = 1, · · · , J, k = 1, · · · , I and k = 1, · · · , I, respectively.

(3.206)

(3.207)


115

3.3.16 The ungauged A, V − Φ formulation

The magnetic vector potential A with the electric scalar potential V is used in the eddy current region Ωc and the reduced magnetic scalar potential Φ is used in the region free of eddy currents, Ωn . The three unknown potentials, A, V and Φ, can be obtained by reducing the weak formulation of the more general ungauged A, V − A − Φ method (see on page 120), Z e dΩ νo (∇ × W k ) · ∇ × A Ωc ! Z e ∂A ∂e v dΩ + σ∇ Wk · σ + ∂t ∂t Ωc Z e (∇ × W k ) · nA dΓ (3.208) + Φ ΓncΦ

=

−

Z

ZC

Φ0 W k · dl

Ωc

Z

(∇ × W k ) · Ie dΩ,

Ωc

− +

∇Nk ·

Z

Ωn

Z

Γnc

− −

Z

Ωn

Z

e ∂A ∂e v σ + σ∇ ∂t ∂t

!

dΩ = 0,

(3.209)

e dΩ µ∇Nk · ∇Φ

e · nA dΓ = Nk (∇ × A)

(3.210)

e 0 dΩ µ∇Nk · T Nk b dΓ.

ΓB

Here k = 1, · · · , J, k = 1, · · · , I and k = 1, · · · , I, respectively.

3.3.17 The gauged A, V − A − Φ formulation

The magnetic vector potential A with the electric scalar potential V is used in the eddy current region Ωc , the magnetic vector potential A is used in the holes, ΩnA and the reduced magnetic scalar potential Φ is used in the rest region free of eddy currents, ΩnΦ . There are three unknown potentials, A, V and Φ, that is three equations must be set up. It is important to note that the resulting system of equations should be symmetric, we are going to take care about it.

116


The first weak formulation is based on the partial differential equations (2.489) and (2.491), on the Neumann type boundary conditions (2.493), (2.496), (2.501), (2.503) and on the interface conditions (2.505), (2.506), (2.510), (2.514) Z h i e − ∇ νo ∇ · A e dΩ W k · ∇ × νo ∇ × A Ωc ! Z e ∂A ∂e v W k· σ + σ∇ dΩ + ∂t ∂t Ωc Z h i e − ∇ ν∇ · A e dΩ + W k · ∇ × ν∇ × A ΩnA

+

Z

ΓE

+

Z

e νo ∇ · A(W k · n) dΓ +

ΓHc

+

Z

Wk ·

ΓHA

+

Z

ΓncA

+

Z

ΓncA

+ + =

Z

h

Wk · Wk · h

ΓncΦ

ΓA,Φ

Wk ·

Z

Ωn

ΓBA



h

i e × n − K dΓ ν∇ × A

h

(3.211)

i e + Ie × nc + ν∇ × A e × nA dΓ νo ∇ × A

i e e νo ∇ · A(W k · nc ) + ν∇ · A(W k · nA ) dΓ

Wk ·

Z

Z

h i e + Ie × nc + T e 0 − ∇Φ e × nΦ dΓ νo ∇ × A

i h e × nA + T e 0 − ∇Φ e × nΦ dΓ ν∇ × A

W k · J 0 dΩ −

where W k × n = 0,

Z

Ωc

e dΩ, W k · (∇ × I)

on ΓE ∪ ΓBA ,

(3.212)

and W k · n = 0,

on ΓHc ∪ ΓncΦ ∪ ΓHA ∪ ΓA,Φ ,

(3.213)

moreover k = 1, · · · , J. The second order derivatives in the first and in the third integrals can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v, e or u = ν∇ × A e and the identity with the notations v = W k and u = νo ∇ × A, ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

(3.214)

(3.215)


117

e or ϕ = ν∇ · A e and v = W k also must be used. The last with the notations ϕ = νo ∇ · A, integral on the right-hand side can also be modified by using the notations v = W k and e This results in u = I. Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇×W k )· ∇× A Ωc ! Z e ∂A ∂e v dΩ W k· σ + σ∇ + ∂t ∂t Ωc Z i h e + ν∇ · W k ∇ · A e dΩ + ν (∇ × W k )· ∇ × A ΩnA

+

Z

h

ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ

+

Z

ΓBA ∪ΓHA ∪ΓncA ∪ΓA,Φ

− − +

Z

Z


ΓE

+

Z

Z

Wk ·

ΓHA

+

Z

ΓncA

+

Z

ΓncA

+ + =

Z

Wk · Wk · h

ΓA,Φ

Wk ·

Z

Z

e ν∇ · A(W k · n) dΓ Z

ΓBA



h

(3.216)

i e × n − K dΓ ν∇ × A

h


i e e νo ∇ · A(W k · nc ) + ν∇ · A(W k · nA ) dΓ

ΓncΦ

Ωn

−

h

Wk ·

Z

i e × W k · n dΓ ν∇ × A

e νo ∇ · A(W k · n) dΓ +

ΓHc

+

h

e νo ∇ · A(W k · n) dΓ


Z

i e × W k · n dΓ νo ∇ × A

h i e 0 − ∇Φ e + Ie × nc + T e × nΦ dΓ νo ∇ × A

h i e × nA + T e × nΦ dΓ e 0 − ∇Φ ν∇ × A

W k · J 0 dΩ −

Z


Ωc

(∇ × W k ) · Ie dΓ

(Ie × W k ) · n dΓ.

The first and the second boundary integrals are vanishing on ΓHc ∪ ΓncA ∪ ΓncΦ and on ΓHA ∪ ΓncA ∪ ΓA,Φ , respectively, because of the seventh, the first term of the eighth, the ninth, the first term of eleventh and the twelfth boundary integrals after using

118


h i h i e ×n . e × W k · n = −W k · ν∇ × A ν∇ × A

(3.217)

The first and the second boundary integral terms are equal to zero on the rest parts ΓE and ΓBA as well, because of the Dirichlet type boundary conditions (2.499) and (2.502), i.e. W k × n = 0 on these boundaries. The third and the fourth boundary integral terms are vanishing on ΓE ∪ ΓncA , on ΓBA ∪ ΓncA , because of the fifth, the sixth and the tenth boundary integral terms. On the rest part ΓHc ∪ ΓncΦ and on ΓHA ∪ ΓA,Φ these integral terms are equal to zero, too, because of Dirichlet type boundary conditions (2.495), (2.513), (2.497) and (2.516), i.e. W k · n = 0 there. The last boundary integral term on the right is vanishing on ΓHc ∪ ΓncA ∪ ΓncΦ , because of the same terms on the left-hand side and the identity −(Ie × W k ) · n = (Ie × n) · W k must be used. It is equal to zero on the rest part ΓE , because of the Dirichlet type boundary condition (2.499). e can be The last two interface integral terms on the left-hand side containing ∇Φ reformulated in the same way as in the A − Φ formulation on page 84, from (3.88) to (3.92). Finally, the first equation of the weak form is the following: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇×W k )· ∇× A Ωc ! Z e ∂A ∂e v dΩ + σ∇ W k· σ + ∂t ∂t Ωc Z h i e + ν∇ · W k ∇ · A e dΩ ν (∇ × W k )· ∇ × A + Ωn (3.218) Z Z A e (∇ × W k ) · nA dΓ = Φ W k · J 0 dΩ + ΓncΦ ∪ΓA,Φ

+

Z

ΓHA

+

Z

C

W k · KdΓ +

Φ0 W k · dl −

Z

Z

Ωc

ΓncΦ ∪ΓA,Φ

ΩnA

e 0 × nA )dΓ W k · (T

(∇ × W k ) · Ie dΩ.

Here, all the normal unit vectors are changed to nA , pointing out from the regions where A can be found (Ωc ∪ ΩnA ) and k = 1, · · · , J. The partial differential equation (2.490), the Neumann type boundary and interface conditions (2.494), (2.507) and (2.512) can be summarized in the following formulation. Here, the conditions are multiplied by −1, ! Z e ∂A ∂e v − dΩ + σ∇ Nk ∇ · σ ∂t ∂t Ωc ! (3.219) Z e ∂e v ∂A · n dΓ = 0, + σ∇ + Nk σ ∂t ∂t ΓHc ∪ΓncA ΓncΦ where k = 1, · · · , I. The first integral terms can be reformulated by the use of the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v

e + σ∇∂e v /∂t, with the notations ϕ = Nk and v = σ∂ A/∂t

(3.220)

119


Z

Ωc

∇Nk ·

e ∂A ∂e v σ + σ∇ ∂t ∂t

!

dΩ

! e ∂e v ∂A · n dΓ + σ∇ Nk σ − ∂t ∂t ΓE ∪ΓHc ∪ΓncA ∪ΓncΦ ! Z e ∂A ∂e v + · n dΓ = 0. + σ∇ Nk σ ∂t ∂t ΓHc ∪ΓncA ∪ΓncΦ Z

(3.221)

The boundary integral terms are vanishing on ΓHc ∪ΓncA ∪ΓncΦ and the first one is equal to zero because Nk = 0 on ΓE according to Dirichlet type boundary condition (2.500). Finally, the following weak formulation can be obtained: ! Z e ∂A ∂e v dΩ = 0, k = 1, · · · , I. (3.222) ∇Nk · σ + σ∇ ∂t ∂t Ωc The partial differential equation (2.492), the Neumann type boundary condition (2.504) and the interface conditions (2.511) and (2.515) build the third weak formulation, Z Z e dΩ + e · n dΓ e 0 · n − µ∇Φ Nk ∇ · (µ∇Φ) Nk b + µT ΩnΦ

+

Z

ΓBΦ

h i e · n+ Aµ T e 0 − ∇Φ e · nΦ dΓ Nk (∇ × A)

(3.223)

ΓncΦ ∪ΓA,Φ

=

Z

ΩnΦ

e 0 ) dΩ, Nk ∇ · (µT

where k = 1, · · · , I. The first integral terms can be reformulated by the use of the identity ∇ · (ϕv) = v · ∇ϕ + ϕ∇ · v e with the notations ϕ = Nk and v = µ∇Φ, Z Z e dΩ + − µ∇Nk · ∇Φ ΩnΦ

+

Z

ΓBΦ

+

Z

ΓHΦ ∪ΓBΦ ∪ΓncΦ ∪ΓA,Φ

e 0 · n − µ∇Φ e · n dΓ Nk b + µT

ΓncΦ ∪ΓA,Φ

−

Z

ΩnΦ

(3.224)

e · n dΓ µNk ∇Φ

h i e 0 − ∇Φ e · nΦ dΓ = e · nA + µ T Nk (∇ × A)

e 0 dΩ + µ∇Nk · T

Z

ΓHΦ ∪ΓBΦ ∪ΓncΦ ∪ΓA,Φ

e 0 · n dΓ. µNk T

(3.225)

The first surface integral is vanishing on the part ΓBΦ ∪ ΓncΦ ∪ ΓA,Φ of the boundary because of the same term with opposite sign in the second and in the third surface integrals (of course n = nΦ on ΓncΦ ∪ ΓA,Φ ). The last surface integral is vanishing on the same part of the boundary according to the second and the third boundary integrals. The first as

120


well as the last surface integrals are equal to zero on the rest part ΓHΦ , because Nk = 0 there according to the Dirichlet type boundary condition (2.498). The result of this math is as follows: Z Z e · nA dΓ = e dΩ + µ∇Nk · ∇Φ Nk (∇ × A) − ΩnΦ

−

Z

ΩnΦ


Z

ΓncΦ ∪ΓA,Φ

(3.226)

Nk b dΓ,

k = 1, · · · , I.

ΓBΦ

Finally, the three equations of the weak formulation of the gauged A, V − A − Φ method is the following: Z h i e + νo ∇ · W k ∇ · A e dΩ νo (∇×W k )· ∇× A Ωc ! Z e ∂A ∂e v dΩ W k· σ + σ∇ + ∂t ∂t Ωc Z h i e + ν∇ · W k ∇ · A e dΩ ν (∇ × W k )· ∇ × A + ΩnA

+

Z

ΓncΦ ∪ΓA,Φ

=

Z

ΩnA

+

Z

W k · J 0 dΩ +

ΓncΦ ∪ΓA,Φ

−

Z

Ωc

Z

e (∇ × W k ) · nA dΓ Φ Z

ΓHA

W k · KdΓ

e 0 × nA )dΓ + W k · (T

(∇ × W k ) · Ie dΩ,

e ∂A ∂e v ∇Nk · σ + σ∇ ∂t ∂t Ωc Z Z e dΩ + µ∇Nk · ∇Φ − ΩnΦ

−

Z

ΩnΦ


Z

(3.227)

!

Z

C

Φ0 W k · dl

dΩ = 0,

ΓncΦ ∪ΓA,Φ

e · nA dΓ = Nk (∇ × A)

(3.228)

(3.229)

Nk b dΓ.

ΓBΦ

Here k = 1, · · · , J, k = 1, · · · , I, k = 1, · · · , I, respectively.

3.3.18 The ungauged A, V − A − Φ formulation The magnetic vector potential A with the electric scalar potential V is used in the eddy current region Ωc , the magnetic vector potential A is used in the holes, ΩnA and the reduced magnetic scalar potential Φ is used in the rest region free of eddy currents, ΩnΦ .


121

There are three unknown potentials, A, V and Φ, so three equations must be set up. It is important to note that the resulting system of equations should be symmetric. The source current density J 0 is represented by the impressed current vector potential T 0. The first weak formulation is based on the partial differential equations (2.517) and (2.519), on the Neumann type boundary conditions (2.521), (2.523) and on the interface conditions (2.529), (2.532), (2.535). The term ∇×T 0 has been appended to the right-hand side of the partial differential equation (2.517), similarly to (2.519), according to the definition (3.139). The weak formulation is as follows: Z h i e dΩ W k · ∇ × νo ∇ × A Ωc ! Z e ∂A ∂e v + σ∇ dΩ Wk · σ + ∂t ∂t Ωc Z h i e dΩ W k · ∇ × ν∇ × A + ΩnA

+

Z

ΓHc

+

Z

Wk ·

ΓHA

+

Z

ΓncA

+ + =

Z

Wk · Wk ·

ΓncΦ

Wk ·

ΓA,Φ

Wk ·

Z

Z

Ωc ∪ΩnA

− where

h

Z

Ωc


h

i e × n − K dΓ ν∇ × A

h


(3.230)

h i e + Ie × nc + T e × nΦ dΓ e 0 − ∇Φ νo ∇ × A

h i e × nA + T e 0 − ∇Φ e × nΦ dΓ ν∇ × A

e 0 )dΩ W k · (∇ × T

e W k · (∇ × I)dΩ,

W k × n = 0,

on ΓE ∪ ΓBA ,

(3.231)

and k = 1, · · · , J. The second order derivatives in the first and in the third integrals can be reduced to first order one by using the identity ∇ · (u × v) = v · ∇ × u − u · ∇ × v,

(3.232)

e or u = ν∇ × A. e The integrals on the with the notations v = W k and u = νo ∇ × A, e e This results right-hand side can also be reformulated by v = W k , u = T 0 and u = I. in

122


Z

Ωc

+

Z

Z e dΩ + νo (∇ × W k ) · ∇ × A

Z


+

Z

h


+

Z

ΓHc

+

Z

Wk ·

ΓHA

+

Z

ΓncA

+ + =

Z

Wk · Wk ·

ΓncΦ

ΓA,Φ

Wk ·

Z

Ωc ∪ΩnA

+

h

Wk ·

Z

Z

Z

h

Z

Ωc

dΩ

i e × W k · n dΓ νo ∇ × A

h

i e × W k · n dΓ ν∇ × A

i e × n − K dΓ ν∇ × A

h


(3.233)

i h e + Ie × nc + T e 0 − ∇Φ e × nΦ dΓ νo ∇ × A

h i e 0 − ∇Φ e × nA + T e × nΦ dΓ ν∇ × A

e 0 dΩ (∇ × W k ) · T

e 0 × W k ) · n dΓ (T


−

!



+

Wk ·

e dΩ ν (∇ × W k ) · ∇ × A

ΩnA

+

Ωc

e ∂A ∂e v σ + σ∇ ∂t ∂t

e 0 × W k ) · n dΓ (T

(∇ × W k ) · Ie dΓ −

Z


(Ie × W k ) · n dΓ.

The first and the second boundary integral terms are vanishing on the boundary part ΓHc ∪ ΓncA ∪ ΓncΦ and on ΓHA ∪ ΓncA ∪ ΓA,Φ , respectively, because of the third, fourth, fifth and the first term of the sixth and of the seventh boundary integral terms after using the identity h h i i e × W k · n = −W k · ν∇ × A e ×n . ν∇ × A (3.234)

The first and the second boundary integral terms are equal to zero on the rest parts ΓE and ΓBA as well, because of the Dirichlet type boundary condition (2.525) and (2.527), i.e. W k × n = 0 on these boundaries. The first and the second boundary integral terms on the right-hand side are vanishing on ΓncΦ and on ΓA,Φ according to the sixth and the seventh terms and that e 0 × nΦ ) = nΦ · (W k × T e 0 ) = −nΦ · (T e 0 × W k ), W k · (T

(3.235)


123

and −nΦ = nc and −nΦ = nA in the first and in the second case, respectively. On ΓncA , they compensate each others, because the normal unit vectors have opposite direction. Dirichlet type boundary conditions (2.525) and (2.527) are satisfied on ΓE and on ΓB , that is why these two surface integrals are equal to zero on these parts of the boundary. On the rest segments, the tangential component of the impressed current vector potential e 0 = 0, however, it is useful to apply (3.146), because the must be equal to zero, n × T surface current term K can be eliminated on the left-hand side. The surface integral terms containing the nonlinear residual term Ie are vanishing on ΓHc ∪ΓncA ∪ΓncΦ , because of the same terms on the left-hand side.On ΓE , W k × n = 0 according to the Dirichlet type boundary condition (2.525). e can be reformulated in the same way as The interface integral terms containing ∇Φ in the A − Φ formulation on page 84, from (3.88) to (3.92) and see Fig. 2.18. Finally, the first equation of the weak form is the following: Z e dΩ νo (∇ × W k ) · ∇ × A Ωc ! Z e ∂e v ∂A W k· σ + dΩ + σ∇ ∂t ∂t Ωc Z e dΩ + ν (∇ × W k ) · ∇ × A Ωn (3.236) Z A e Φ (∇ × W k ) · nA dΓ + ΓncΦ ∪ΓA,Φ

=

Z

Ωc ∪ΩnA

+

Z

C

e 0 dΩ (∇ × W k ) · T

Φ0 W k · dl −

Z

Ωc

(∇ × W k ) · Ie dΩ,

and k = 1, · · · , J. Here, all the normal unit vectors are changed to nA , pointing out from the regions where A can be found (Ωc ∪ ΩnA ). The partial differential equation (2.518), the Neumann type boundary and interface conditions (2.522), (2.530) and (2.534) can be summarized in the following formulation. Here, the conditions are multiplied by −1, ! Z e ∂A ∂e v − Nk ∇ · σ dΩ + σ∇ ∂t ∂t Ωc ! (3.237) Z e ∂A ∂e v + · n dΓ = 0, Nk σ + σ∇ ∂t ∂t ΓHc ∪ΓncA ΓncΦ which is the same as in the gauged A, V − A − Φ formulation presented in the last item. The weak formulation is repeated here, ! Z e ∂A ∂e v ∇Nk · σ dΩ = 0, k = 1, · · · , I. (3.238) + σ∇ ∂t ∂t Ωc

124


The partial differential equation (2.520), the Neumann boundary condition (2.528) and the interface conditions (2.533) and (2.536) build the third weak formulation, Z Z e 0 · n − µ∇Φ e dΩ + e · n dΓ Nk ∇ · (µ∇Φ) Nk b + µT ΩnΦ

+

Z

ΓBΦ

h i e · nA + µ T e 0 − ∇Φ e · nΦ dΓ Nk (∇ × A)

(3.239)

ΓncΦ ∪ΓA,Φ

=

Z

ΩnΦ

e 0 ) dΩ. Nk ∇ · (µT

Obtaining the weak formulation can be realized in the same way as it was presented in the last item, in the gauged A, V − A − Φ formulation, Z Z e dΩ + e · nA dΓ = − Nk (∇ × A) µ∇Nk · ∇Φ ΩnΦ

−

Z

ΩnΦ


Z

ΓncΦ ∪ΓA,Φ

(3.240)

Nk b dΓ,

k = 1, · · · , I.

ΓBΦ

Finally, the three equations of the weak formulation of the ungauged A, V − A − Φ method is the following: ! Z Z e ∂A ∂e v e νo (∇ × W k ) · ∇ × A dΩ + W k· σ dΩ + σ∇ ∂t ∂t Ωc Ωc Z e dΩ + ν (∇ × W k ) · ∇ × A Ωn (3.241) Z Z A e 0 dΩ e (∇ × W k ) · nA dΓ = Φ (∇ × W k ) · T + ΓncΦ ∪ΓA,Φ

+

Z

C

Φ0 W k · dl −

Z

Z

Ωc

ΩnA

(∇ × W k ) · Ie dΩ,

k = 1, · · · , J,

dΩ = 0,

k = 1, · · · , I,

e ∂A ∂e v ∇Nk · σ + σ∇ ∂t ∂t Ωc Z Z e dΩ + µ∇Nk · ∇Φ − ΩnΦ

−

Z

ΩnΦ


Z

!

ΓncΦ ∪ΓA,Φ

ΓBΦ

e · nA dΓ = Nk (∇ × A)

Nk b dΓ,

k = 1, · · · , I.

(3.242)

(3.243)

4 The finite element method By using scalar and vector potentials, Maxwell’s equations can be transformed into partial differential equations as it is introduced in section 2.3. Generally, the partial differential equations can be solved by numerical methods [8,10,16,17,30,51,77–79,83,94,124,141, 149,159,166,169,170,172,198,199]. One of these numerical methods is the finite element method, which is based on the weak formulation of the partial differential equations, as it is presented in the previous chapter. The basis of numerical techniques is to reduce the partial differential equations to algebraic ones whose solution gives an approximation of the unknown potentials and electromagnetic field quantities. This reduction can be done by discretizing the partial differential equations in time if necessary and in space. The potential functions, the approximation method and the generated mesh distinguish the numerical field solvers. This section summarize the finite element method as a CAD technique in electrical engineering to obtain the electromagnetic field quantities in the case of static magnetic field and eddy current field problems. Here, we show how to discretize the analyzed domain with finite elements, how to approximate potential functions with nodal and vector shape functions and how to build up the system of equations, which solution obtain the unknown potentials.

4.1 Fundamentals of FEM The Finite Element Method (FEM) is the most popular and the most flexible numerical technique to determine the approximate solution of the partial differential equations in engineering [8, 10, 16, 17, 30, 51, 77–79, 83, 94, 124, 141, 149, 159, 166, 169, 170, 172, 198, 199]. For example, commercially available FEM software package is COMSOL Multiphysics, which is able to solve one, two and three-dimensional problems [30, 190] (see chapter 7). A free mesh generator with a built-in CAD engine and post-processor is Gmsh [191]. The main steps of simulation with FEM are illustrated in Fig. 4.1. Firstly, in the model specification phase, the model of the real life problem, which simulation require electromagnetic field calculations must be set up, i.e. we have to find out the partial

126

4. THE FINITE ELEMENT METHOD

differential equations, which must be solved with prescribed boundary and continuity conditions. We have to find out, whether it is a linear or a nonlinear problem and how the characteristics look like. After selecting potentials, the weak formulation of these partial differential equations must be worked out as well. It is depending on the problem, of course, but the chosen mathematical model of the arrangement should be adequate to calculate electromagnetic field quantities in the given accuracy. The geometry of the problem must be defined by a CAD software tool, e.g. by using a user friendly interface, see e.g. Fig. 4.2. The next step is the preprocessing task. Here we have to give the values of different parameters, such as the material properties, i.e. the constitutive relations, the excitation signal and the others. The geometry can be simplified according to symmetries or axial symmetries. The geometry of the problem must be discretized by a FEM mesh. The fundamental idea of FEM is to divide the problem region to be analyzed into smaller finite elements with given shape. A finite element can be e.g. triangle (Fig. 4.3(a)) or quadrangle (Fig. 4.3(b)) in 2D, e.g. tetrahedron (Fig. 4.4(a)) or hexahedra (Fig. 4.4(b)) in 3D. A triangle has three vertices 1, 2 and 3, here numbered counter-clockwise and has 3 edges.

Specify model Preprocessing

FEM mesh generation Computation Nonlinear cycle / Time step

Optimizing or modifying model

Data

Equations of one finite element

Assembling equations

Solver

Postprocessing

Fig. 4.1. Steps of simulation by FEM

127

4.1. FUNDAMENTALS OF FEM

Fig. 4.2. COMSOL Multiphysics, a CAD environment to solve electromagnetic field problems

The quadrangle element has 4 nodes and 4 edges. A tetrahedral element has 4 vertices and 6 edges and a hexahedral element has 8 nodes and 12 edges (the usual numbering of nodes is shown in the illustrations). There are some simple rules, how to generate a mesh. Neither overlapping nor holes are allowed in the generated finite element mesh. If material interface are included in the problem region, the configuration of mesh must be adapted to these boundaries, i.e. interfaces coincide with finite element interfaces. y

y

3

3 4

edge

edge

2 1

vertex (a) Triangular element

2 x

1

vertex (b) Quadrangle element

Fig. 4.3. Typical finite elements in the two-dimensional x − y plane

x

128


z

z 4

y

8 y

5 6 3

1

7 3

4 1

edge

2

edge

vertex

2

x

vertex

x

(a) Tetrahedral element

(b) Hexahedral element

Fig. 4.4. Typical finite elements in 3D

FEM mesh, as two illustrations, generated by COMSOL Multiphysics [190] can be seen in Fig. 4.5 and in Fig. 4.6. The first 2D illustration (Fig. 4.5) shows the mesh of a horseshoe-shaped permanent magnet. The two ends are pre-magnetized in different directions. The second illustration (Fig. 4.6) presents a model of a micro-scale square inductor, used for LC bandpass filters in microelectromechanical systems. The model geometry consists of the spiral-shaped inductor and the air surrounding it (the mesh in air is not shown). The outer dimensions of the model geometry are around 0.3 mm. These illustrations are cited from the Model Documentation of COMSOL Multiphysics.

Fig. 4.5. COMSOL model of a permanent magnet, geometry is meshed by triangles

The next step in FEM simulations is solving the problem. The FEM equations, based on the weak formulations, must be set up in the level of one finite element, then these equations must be assembled through the FEM mesh. Assembling means that the global system of equations is built up, which solution is the approximation of the introduced

4.1. FUNDAMENTALS OF FEM

129

Fig. 4.6. COMSOL model of a micro-scale square inductor, geometry is discretized by tetrahedral shape finite elements

potential. The obtained global system of algebraic equations is linear, or nonlinear but linearized, depending on the medium to be analyzed. Then this global system of equations must be solved by a solver. The computation may contain iteration if the constitutive equations are nonlinear. This is the situation when simulating ferromagnetic materials with nonlinear characteristics. Iteration means that the system of equations must be set up and must be solved step by step until convergence is reached. If the problem is time dependent, then the solution must be worked out at every discrete time instant. The result of computations is the approximated potential value in the FEM mesh. Any electromagnetic field quantity (e.g. magnetic field intensity, or magnetic flux density, etc.) can be calculated by using the potentials at the postprocessing stage. Capacitance, inductance, energy, force and other quantities can also be calculated. The postprocessing give a chance to modify the geometry, the material parameters or the FEM mesh to get more accurate result. The COMSOL Multiphysics [190] has been used to show two examples about postprocessing. The pattern of the magnetic field around the permanent magnet is well known through experiments (see Fig. 4.7). Figure 4.8 shows the electric potential in the inductor and the magnetic flux lines. The thickness of the flow lines represents the magnitude of the magnetic flux.

Fig. 4.7. COMSOL solution of the static magnetic field generated by a permanent magnet

130


Fig. 4.8. Electric potential in the device and magnetic flux lines around the device, the problem has been solved by COMSOL

4.2 Approximating potentials with shape functions It has been shown in section 2.3 that the introduced potential function can be scalar valued (e.g. the magnetic scalar potential Φ, or the electric scalar potential V ), or vector valued (e.g. the current vector potential T , or the magnetic vector potential A). The scalar potential functions can be approximated by so-called nodal shape functions and the vector potential functions can be approximated by either nodal or so-called vector shape functions, also called edge shape functions. Generally, a shape function is a simple continuous polynomial function defined in a finite element and it is depending on the type of the used finite element. Shape functions have the following general properties [124]: (i) Each shape function is defined in the entire problem region; (ii) Each scalar shape function corresponds to just one nodal point and each vector shape function corresponds to just one edge; (iii) Each scalar shape function is nonzero over just those finite elements that contain its nodal point and equals to zero over all other elements. Each vector shape function in nonzero over just those finite elements that contain its edge and equals to zero over all other elements; (iv) The scalar shape function has a value unity at its nodal point and zero at all other nodal points. The line integral of a vector shape function is equal to one along its edge and the line integral of it is equal to zero along the other edges; (v) The shape functions are linearly independent, i.e. no shape function equals a linear combination of the other shape functions. The accuracy of solution obtained by FEM can be increased in three ways. The first one is increasing the number of finite elements, i.e. decreasing the element size. It is called h-FEM. The second way is to increase the degree of polynomials building up a shape function (e.g. using Lagrange or Legendre interpolation functions). This

131

4.2. APPROXIMATING POTENTIALS WITH SHAPE FUNCTIONS

is the so-called p-FEM. The mixture of these methods results in hp-FEM. Potentials approximated by h-version or p-version are assigned in the indices of the potentials.

4.2.1 Nodal finite elements Scalar potential functions can be represented by a linear combination of shape functions associated with nodes of the finite element mesh. Within a finite element, a scalar potential function Φ = Φ(r, t) is approximated by Φ≃

m X

N i Φi ,

(4.1)

i=1

where Ni = Ni (r) and Φi = Φi (t) are the nodal shape functions and the value of potential function corresponding to the ith node, respectively. The number of degrees of freedom is m = 2 in 1D problems, m = 3 in a 2D problem using triangular FEM mesh and m = 4 in a 3D arrangement meshed by tetrahedral elements and the shape functions are linear. The nodal shape functions can be defined by the relation 1, at the node i, Ni = (4.2) 0, at other nodes. (i) In 1D, the linear shape functions can be build up by x2 − x x − x1 N1 = , and N2 = , (4.3) x2 − x1 x2 − x1 where x1 and x2 are the coordinates of the boundaries of one finite element. The linear shape functions are plotted in Fig. 4.9. It is easy to control the equation (4.2). N1 (x)

N2 (x)

1

x1

x2

Fig. 4.9. The 1D linear shape functions N1 (x) and N2 (x) If the values of the potential are known in the two boundary points x1 and x2 , then the potential can be determined easily inside the finite element x1 ≤ x ≤ x2 as (see Fig. 4.10) x2 − x x − x1 Φ1 + Φ2 . (4.4) Φ = N 1 Φ1 + N 2 Φ2 = x2 − x1 x2 − x1 Of course, it is valid in the other finite elements as well, e.g. if x2 ≤ x ≤ x3 , then x3 − x x − x2 Φ2 + Φ3 , (4.5) Φ = N 1 Φ2 + N 2 Φ3 = x3 − x2 x3 − x2 and N1 , N2 are shifted to the second finite element.

132


The scalar potential is continuous in the whole 1D region. It is noted here that the accuracy of approximation can be increased by decreasing the length of the elements, especially where the rate of change of the solution is large, e.g. between x3 and x4 in Fig. 4.10. Here, the mesh can be very fine and higher order approximation can results in better solution. Φ1

Φ4 Φ2 N1 (x)

N2 (x)

Φ3

1 x1

x2

x3

x4

Fig. 4.10. Known potential values are approximated by linear functions

One way to build up higher order shape functions is using Lagrange interpolation functions, defined by the formula Ni (x) =

m Y

j=1, j6=i

x − xj . xi − xj

(4.6)

The order is m − 1 and Ni (x) is equal to one in the node i and equal to zero in all the other nodes. Here, second and third order approximations are shown. The second order approximation can be defined by 3 quadratic shape functions (i.e. m = 3 in (4.1), see Fig. 4.11), N1 =

(x − x2 )(x − x3 ) , (x1 − x2 )(x1 − x3 )

(4.7)

N3 (x)

N1 (x)

N2 (x)

1

x1

x3

x2

Fig. 4.11. The 1D quadratic shape functions N1 (x), N2 (x) and N3 (x)

133


N2 = N3 =

(x − x1 )(x − x3 ) , (x2 − x1 )(x2 − x3 )

(4.8)

(x − x1 )(x − x2 ) , (x3 − x1 )(x3 − x2 )

(4.9)

and the new point x3 is placed in the center of the element, x3 =

x1 + x2 . 2

(4.10)

The third order approximation can be defined by 4 cubic shape functions (m = 4 in (4.1), see Fig. 4.12), N1 =

(x − x2 )(x − x3 )(x − x4 ) , (x1 − x2 )(x1 − x3 )(x1 − x4 )

(4.11)

N2 =

(x − x1 )(x − x3 )(x − x4 ) , (x2 − x1 )(x2 − x3 )(x2 − x4 )

(4.12)

N3 = N4 =

(x − x1 )(x − x2 )(x − x4 ) , (x3 − x1 )(x3 − x2 )(x3 − x4 )

(4.13)

(x − x1 )(x − x2 )(x − x3 ) , (x4 − x1 )(x4 − x2 )(x4 − x3 )

(4.14)

1(x1 + x2 ) , 3

(4.15)

and the new points x3 and x4 are placed inside the element as x3 =

x4 =

2(x1 + x2 ) . 3 N3 (x)

N4 (x) N2 (x)

1

N1 (x)

x1

x3

x4

x2

Fig. 4.12. The 1D cubic shape functions N1 (x), N2 (x), N3 (x) and N4 (x) With this technique, the interpolation functions of any order can be defined and the equation (4.2) can be controlled. Figure 4.13 shows the higher order approximation of the potential plotted in Fig. 4.10. This illustration shows the applicability of higher order functions. (ii) 2D linear shape functions can be built up as follows when using a finite element mesh with triangular finite elements. Linear basis functions can be introduced by using

134


Φ1

Φ4 Φ2

Φ3

x1

x2

x3

x4

Fig. 4.13. Known potential values are approximated by quadratic functions the so-called barycentric coordinate system in a triangle as follows. The area of a triangle is denoted by ∆ and it can be calculated as 1 x1 y1 1 (4.16) ∆ = 1 x2 y2 , 2 1 x3 y3 where (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) are the coordinates of the three nodes of the triangle in the global coordinate system building an anticlockwise sequence. The area functions (see Fig. 4.14) of a given point inside the triangle with coordinates (x, y) can be calculated as 1 x y 1 x1 y1 1 x1 y1 1 1 1 ∆1 = 1 x2 y2 , ∆2 = 1 x y , ∆3 = 1 x2 y2 , (4.17) 2 2 2 1 x3 y3 1 x y 1 x3 y3

i.e. ∆1 = ∆1 (x, y), ∆2 = ∆2 (x, y) and ∆3 = ∆3 (x, y) are depending on the coordinates x and y. The barycentric coordinates Li = Li (x, y) can be defined by the above area functions as ∆i Li = , i = 1, 2, 3. (4.18) ∆ Three linear shape functions Ni = Ni (x, y) can be described as Ni = L i ,

i = 1, 2, 3.

(4.19)

The shape function Ni is equal to 1 at the ith node of the triangle and it is equal to zero at the other two nodes, because ∆i is equal to ∆ at node i and it is equal to zero at the other two nodes. That is why the relation (4.2) is satisfied. It is obvious that the three shape functions are linearly independent. The linear shape functions Ni (i = 1, 2, 3) vary linearly over the triangle, because the fraction ∆i /∆ measures the perpendicular distance of the point (x, y) toward the vertex opposite to node i as it is illustrated in Fig. 4.15 and the linear shape function is constant along such a line. The three linear shape functions are shown in Fig. 4.16.

135


(x3 , y3 )

∆1

∆2

(x, y)

∆3

(x2 , y2 )

(x1 , y1 ) Fig. 4.14. The area function of a triangle ∆1 /∆ = 0 ∆1 /∆ = 0.25

(x3 , y3 )

∆1 /∆ = 0.5

∆1

∆1 /∆ = 0.75 ∆1 /∆ = 1

(x, y)

(x2 , y2 ) (x1 , y1 )

Fig. 4.15. Fraction ∆i /∆ measures the perpendicular distance of the point (x, y) toward the vertex opposite to node i (here i = 1) If the potential at the nodes is known, then a linear approximation of the potential function can be represented by (4.1). The derivative of a first order approximation is zeroth order, i.e. constant. The magnetic field intensity H, or the magnetic flux density B are constant within a triangle, if these are obtained from a first order approximation by H = −∇Φ, or B = ∇ × A. This may results in inaccurate solution. This is the reason why higher order approximations are studied. Here, only the second and the third order approximations are shown. Higher order shape functions can also be built up by using the barycentric coordinates L1 , L2 and L3 introduced above in (4.18) [83].

136


(x3 , y3 )

(x3 , y3 ) 1

1

(x2 , y2 ) (x1 , y1 )

(x2 , y2 ) (x1 , y1 )

(a) N1 (x, y)

(b) N2 (x, y)

1

(x3 , y3 )

(x2 , y2 ) (x1 , y1 ) (c) N3 (x, y)

Fig. 4.16. The 2D linear shape functions N1 (x, y), N2 (x, y) and N3 (x, y) A polynomial of order n must contain all possible terms xp y q , 0 ≤ p + q ≤ n, as it is presented by Pascal’s triangle, 1 x

y

x2

xy

x3

x2 y

y2 xy 2

y3

···

The first row contains the only one term of the zeroth order polynomials, the second, third and fourth rows contain the terms of the first, second and third order polynomials. Pascal’s triangle can be used to generate the elements of a polynomial with given order.

137


Such a polynomial contains m=

(n + 1)(n + 2) 2

(4.20)

elements altogether, i.e. m = 1, m = 3, m = 6 and m = 10 in the case of zeroth, first, second and third order polynomials. It means that m coefficients must be expressed, finally m points must be placed within a triangle. Pascal’s triangle can be continued, of course. The interpolation function of order n can be constructed as n Ni = PIn (L1 ) PJn (L2 ) PK (L3 ),

where

I + J + K = n,

(4.21)

and the integers I, J and K label the nodes within the triangle, resulting in a numbering scheme. Figure 4.17, Fig. 4.18 and Fig. 4.19 illustrate the numbering scheme of the first, the second and the third order approximations. It is noted that points must be inserted not only the edges, but inside the triangle, if n > 2. n (L3 ) are defined as The polynomials PIn (L1 ), PJn (L2 ) and PK I−1 Y

I−1 n L1 − p 1 Y (n L1 − p), = I −p I! p=0 p=0

PIn (L1 ) =

J−1 Y

PJn (L2 ) =

p=0

n PK (L3 ) =

J−1 1 Y n L2 − p (n L2 − p), = J −p J! p=0

K−1 Y p=0

K−1 1 Y n L3 − p = (n L3 − p), K −p K! p=0

if

I > 0,

if

J > 0,

if

K > 0,

(4.22)

(4.23)

(4.24)

and as a definition P0n = 1.

(4.25)

If n = 1, then m = 3, i.e. (see Fig. 4.17) N1 = P11 (L1 ) P01 (L2 ) P01 (L3 ) = L1 ,

(4.26)

N2 = P01 (L1 ) P11 (L2 ) P01 (L3 ) = L2 ,

(4.27)

P01 (L1 ) P01 (L2 ) P11 (L3 )

(4.28)

N3 =

= L3 ,

since P11 (Li ) =

1−1 Y

1 Li − p = Li , 1−p p=0

(4.29)

as it was mentioned in (4.19). If n = 2, then m = 6, i.e. (see Fig. 4.18) N1 = P22 (L1 ) P02 (L2 ) P02 (L3 ) = L1 (2 L1 − 1),

(4.30)

138


3 : (0, 0, 1)

2 : (0, 1, 0) 1 : (1, 0, 0) Fig. 4.17. Numbering scheme for linear element, n = 1 3 : (0, 0, 2) 5 : (0, 1, 1) 6 : (1, 0, 1)

2 : (0, 2, 0) 4 : (1, 1, 0) 1 : (2, 0, 0) Fig. 4.18. Numbering scheme for quadratic element, n = 2 3 : (0, 0, 3) 7 : (0, 1, 2)

8 : (1, 0, 2)

10 : (1, 1, 1)

6 : (0, 2, 1)

9 : (2, 0, 1) 2 : (0, 3, 0) 5 : (1, 2, 0) 1 : (3, 0, 0)

4 : (2, 1, 0)

Fig. 4.19. Numbering scheme for cubic element, n = 3 N2 = P02 (L1 ) P22 (L2 ) P02 (L3 ) = L2 (2 L2 − 1),

(4.31)

N3 =

(4.32)

P02 (L1 ) P02 (L2 ) P22 (L3 )

= L3 (2 L3 − 1),

N4 = P12 (L1 ) P12 (L2 ) P02 (L3 ) = 4 L1 L2 ,

(4.33)

139


N5 = P02 (L1 ) P12 (L2 ) P12 (L3 ) = 4 L2 L3 ,

(4.34)

N6 = P12 (L1 ) P02 (L2 ) P12 (L3 ) = 4 L1 L3 ,

(4.35)

because P12 (Li ) =

1−1 Y

2 Li − p = 2 Li , 1−p p=0

(4.36)

and P22 (Li ) =

2−1 Y

2 Li 2 Li − 1 2 Li − p = = Li (2 Li − 1). 2 − p 2 1 p=0

(4.37)

Figure 4.20 shows the shape functions N1 and N4 . The other shape functions look like these, N2 and N3 are the same as N1 , moreover N5 and N6 are the same as N4 , but they must be rotated to the corresponding nodes. (x3 , y3 )

(x3 , y3 )

1

1

(x2 , y2 ) (x1 , y1 )

(x2 , y2 ) (x1 , y1 )

(a) N1 (x, y)

(b) N4 (x, y)

Fig. 4.20. The 2D quadratic shape functions N1 (x, y) and N4 (x, y) Finally, if n = 3, m = 10, shape functions can be constructed as (see Fig. 4.21) N1 = P33 (L1 ) P03 (L2 ) P03 (L3 ) =

1 L1 (3 L1 − 1)(3 L1 − 2), 2

1 L2 (3 L2 − 1)(3 L2 − 2), 2 1 N3 = P03 (L1 ) P03 (L2 ) P33 (L3 ) = L3 (3 L3 − 1)(3 L3 − 2), 2 9 N4 = P23 (L1 ) P13 (L2 ) P03 (L3 ) = L1 (3 L1 − 1)L2 , 2 9 N5 = P13 (L1 ) P23 (L2 ) P03 (L3 ) = L2 (3 L2 − 1)L1 , 2 N2 = P03 (L1 ) P33 (L2 ) P03 (L3 ) =

(4.38) (4.39) (4.40) (4.41) (4.42)

140


9 L2 (3 L2 − 1)L3 , 2 9 N7 = P03 (L1 ) P13 (L2 ) P23 (L3 ) = L3 (3 L3 − 1)L2 , 2 9 N8 = P13 (L1 ) P03 (L2 ) P23 (L3 ) = L3 (3 L3 − 1)L1 , 2 9 N9 = P23 (L1 ) P03 (L2 ) P13 (L3 ) = L1 (3 L1 − 1)L3 , 2 N6 = P03 (L1 ) P23 (L2 ) P13 (L3 ) =

(4.43) (4.44) (4.45) (4.46)

N10 = P13 (L1 ) P13 (L2 ) P13 (L3 ) = 27 L1 L2 L3 ,

(4.47)

because P13 (Li ) =

P23 (Li ) =

P33 (Li ) =

1−1 Y

3 Li − p = 3 Li , 1−p p=0

(4.48)

2−1 Y

3 Li 3 Li − 1 3 3 Li − p = = Li (3 Li − 1), 2 − p 2 1 2 p=0

(4.49)

3−1 Y

3 Li − p 3 Li 3 Li − 1 3 Li − 2 = 3−p 3 2 1 p=0

(4.50) 1 = Li (3 Li − 1)(3 Li − 2). 2 These functions satisfy the condition (4.2). Figure 4.21 shows the shape functions N1 and N5 , as examples. The other shape functions look like these, N2 and N3 are the same as N1 , N4 , N6 , N7 , N8 and N9 look like N5 , but they must be imagined at the corresponding nodes. The shape function N10 is equal to one at the center of mass of the triangle and equal to zero on the other nine nodes. (x3 , y3 )

(x3 , y3 )

1

1

(x2 , y2 ) (x1 , y1 )

(x2 , y2 ) (x1 , y1 )

(a) N1 (x, y)

(b) N5 (x, y)

Fig. 4.21. The 2D cubic shape functions N1 (x, y) and N5 (x, y)


141

The scalar potential along any edge of a triangle is the linear combination of the values defined in the points of this edge (see Fig. 4.17, Fig. 4.18, Fig. 4.19), so that if two triangles share the same vertice, the potential will be continuous across the interface element boundary. This means that the approximate solution is continuous everywhere, however, its normal derivate is not. It is easy to see that the 1D shape functions are the same as the functions along the edges of a triangle. (iii) 3D linear shape functions can be worked out as follows when using tetrahedral finite elements. Linear basis functions can be introduced again by using the barycentric coordinate system. The volume of a tetrahedron is denoted by V and it can be expressed as x4 − x1 y4 − y1 z4 − z1 1 (4.51) V = x4 − x2 y4 − y2 z4 − z2 , 6 x4 − x3 y4 − y3 z4 − z3 where (x1 , y1 , z1 ), (x2 , y2 , z2 ), (x3 , y3 , z3 ) and (x4 , y4 , z4 ) are the coordinates of the four nodes of the tetrahedron as shown in Fig. 4.22. The volume functions according to a given point inside the tetrahedron with coordinates (x, y, z) can be calculated as x4 − x y4 − y z4 − z 1 (4.52) V1 = x4 − x2 y4 − y2 z4 − z2 , 6 x4 − x3 y4 − y3 z4 − z3 x − x1 y4 − y1 z4 − z1 1 4 V2 = x4 − x y4 − y z4 − z , (4.53) 6 x4 − x3 y4 − y3 z4 − z3 x − x1 y4 − y1 z4 − z1 1 4 V3 = x4 − x2 y4 − y2 z4 − z2 , (4.54) 6 x4 − x y4 − y z4 − z x − x1 y − y1 z − z1 1 (4.55) V4 = x − x2 y − y2 z − z2 . 6 x−x y−y z−z 3 3 3 The barycentric coordinates Li = Li (x, y, z) of a tetrahedron can be formulated as Li =

Vi , V

i = 1, 2, 3, 4.

(4.56)

Four linear shape functions Ni = Ni (x, y, z) correspondingly to the four nodes are Ni = L i ,

i = 1, 2, 3, 4.

(4.57)

A shape function Ni is equal to 1 at the ith node of the tetrahedron, moreover it is equal to zero at the other three nodes and it is varying linearly within the tetrahedron, because the fraction Vi /V measures the perpendicular distance of the point (x, y, z) toward the facet opposite to node i as it is illustrated in Fig. 4.23 and the linear shape function is constant along such a surface. That is why the relation (4.2) is satisfied. It is obvious that the four shape functions are linearly independent.

142


(x4 , y4 , z4 ) (x, y, z)

(x3 , y3 , z3 ) (x1 , y1 , z1 )

V1

(x4 , y4 , z4 )

(x2 , y2 , z2 ) (x, y, z)

V2

(x4 , y4 , z4 ) (x, y, z)

(x3 , y3 , z3 ) (x1 , y1 , z1 )

V3

(x3 , y3 , z3 )

(x1 , y1 , z1 )

(x2 , y2 , z2 ) (x4 , y4 , z4 ) (x, y, z)

V4

(x2 , y2 , z2 ) (x4 , y4 , z4 ) (x, y, z)

(x3 , y3 , z3 ) (x1 , y1 , z1 )

(x3 , y3 , z3 )

(x1 , y1 , z1 )

(x2 , y2 , z2 )

(x2 , y2 , z2 )

Fig. 4.22. The volume functions in a tetrahedron

The higher order shape functions can be worked out similarly as it was mentioned in the case of triangular elements [83]. The barycentric coordinates L1 , L2 , L3 and L4 can be used. A polynomial of order n must contain all possible terms xp y q z r , 0 ≤ p+q +r ≤ n

143


and a polynomial contains m=

(n + 1)(n + 2)(n + 3) 6

(4.58)

elements altogether, i.e. m = 1, m = 4, m = 10 and m = 20 in the case of zeroth, first, second and third order polynomials. It means that m coefficients must be expressed and m points must be placed within a tetrahedron. (x4 , y4 , z4 ) (x, y, z) Vi /V = 1

(x3 , y3 , z3 ) Vi /V = 0.75

(x1 , y1 , z1 )

Vi /V = 0.5 Vi /V = 0.25 Vi /V = 0

(x2 , y2 , z2 ) Fig. 4.23. Fraction Vi /V measures the perpendicular distance of the point (x, y, z) toward the facet opposite to node i (here i = 3) The interpolation function of order n can be constructed as n (L3 ) PLn (L4 ), Ni = PIn (L1 ) PJn (L2 ) PK

where I + J + K + L = n,

(4.59)

where the integers I, J, K and L label the nodes within the tetrahedra, resulting in a numbering scheme. Figure 4.24, Fig. 4.25 and Fig. 4.26 illustrate the numbering scheme of the first, the second and the third order approximations. n (L3 ) and PLn (L4 ) are defined in the same way The polynomials PIn (L1 ), PJn (L2 ), PK as it was presented in the 2D situation, see definitions (4.22)–(4.25). If n = 1, then m = 4, i.e. (see Fig. 4.24) N1 = P11 (L1 ) P01 (L2 ) P01 (L3 ) P01 (L4 ) = L1 ,

(4.60)

N2 = P01 (L1 ) P11 (L2 ) P01 (L3 ) P01 (L4 ) = L2 ,

(4.61)

P01 (L1 ) P01 (L2 ) P11 (L3 ) P01 (L4 )

= L3 ,

(4.62)

N4 = P01 (L1 ) P01 (L2 ) P01 (L3 ) P11 (L4 ) = L4 ,

(4.63)

N3 =

since (4.29) as it was mentioned in (4.57). If n = 2, then m = 10, i.e. (see Fig. 4.25) N1 = P22 (L1 ) P02 (L2 ) P02 (L3 ) P02 (L4 ) = L1 (2 L1 − 1),

(4.64)

144


N2 = P02 (L1 ) P22 (L2 ) P02 (L3 ) P02 (L4 ) = L2 (2 L2 − 1),

(4.65)

N3 = P02 (L1 ) P02 (L2 ) P22 (L3 ) P02 (L4 ) = L3 (2 L3 − 1),

(4.66)

N4 =

(4.67)

P02 (L1 ) P02 (L2 ) P02 (L3 ) P22 (L4 )

= L4 (2 L4 − 1),

N5 = P12 (L1 ) P12 (L2 ) P02 (L3 ) P02 (L4 ) = 4 L1 L2 ,

(4.68)

N6 =

P02 (L1 ) P12 (L2 ) P12 (L3 ) P02 (L4 )

= 4 L2 L3 ,

(4.69)

N7 =

P12 (L1 ) P02 (L2 ) P12 (L3 ) P02 (L4 )

= 4 L1 L3 ,

(4.70)

N8 = P12 (L1 ) P02 (L2 ) P02 (L3 ) P12 (L4 ) = 4 L1 L4 ,

(4.71)

N9 =

P02 (L1 ) P02 (L2 ) P12 (L3 ) P12 (L4 )

= 4 L3 L4 ,

N10 = P02 (L1 ) P12 (L2 ) P02 (L3 ) P12 (L4 ) = 4 L2 L4 ,

(4.72) (4.73)

because (4.36) and (4.37). Finally, if n = 3, m = 20 shape functions can be constructed as (see Fig. 4.26) N1 = P33 (L1 ) P03 (L2 ) P03 (L3 ) P03 (L4 ) =

1 L1 (3 L1 − 1)(3 L1 − 2), 2

1 L2 (3 L2 − 1)(3 L2 − 2), 2 1 N3 = P03 (L1 ) P03 (L2 ) P33 (L3 ) P03 (L4 ) = L3 (3 L3 − 1)(3 L3 − 2), 2 1 N4 = P03 (L1 ) P03 (L2 ) P03 (L3 ) P33 (L4 ) = L4 (3 L4 − 1)(3 L4 − 2), 2 9 N5 = P23 (L1 ) P13 (L2 ) P03 (L3 ) P03 (L4 ) = L1 (3 L1 − 1)L2 , 2 9 N6 = P13 (L1 ) P23 (L2 ) P03 (L3 ) P03 (L4 ) = L2 (3 L2 − 1)L1 , 2 9 N7 = P03 (L1 ) P23 (L2 ) P13 (L3 ) P03 (L4 ) = L2 (3 L2 − 1)L3 , 2 9 N8 = P03 (L1 ) P13 (L2 ) P23 (L3 ) P03 (L4 ) = L3 (3 L3 − 1)L2 , 2 9 N9 = P13 (L1 ) P03 (L2 ) P23 (L3 ) P03 (L4 ) = L3 (3 L3 − 1)L1 , 2 9 N10 = P23 (L1 ) P03 (L2 ) P13 (L3 ) P03 (L4 ) = L1 (3 L1 − 1)L3 , 2 9 N11 = P13 (L1 ) P03 (L2 ) P03 (L3 ) P23 (L4 ) = L4 (3 L4 − 1)L1 , 2 9 N12 = P23 (L1 ) P03 (L2 ) P03 (L3 ) P13 (L4 ) = L1 (3 L1 − 1)L4 , 2 9 N13 = P03 (L1 ) P13 (L2 ) P03 (L3 ) P23 (L4 ) = L4 (3 L4 − 1)L2 , 2 N2 = P03 (L1 ) P33 (L2 ) P03 (L3 ) P03 (L4 ) =

(4.74) (4.75) (4.76) (4.77) (4.78) (4.79) (4.80) (4.81) (4.82) (4.83) (4.84) (4.85) (4.86)

145


9 L2 (3 L2 − 1)L4 , 2 9 = P03 (L1 ) P03 (L2 ) P13 (L3 ) P23 (L4 ) = L4 (3 L4 − 1)L3 , 2 9 = P03 (L1 ) P03 (L2 ) P23 (L3 ) P13 (L4 ) = L3 (3 L3 − 1)L4 , 2

N14 = P03 (L1 ) P23 (L2 ) P03 (L3 ) P13 (L4 ) =

(4.87)

N15

(4.88)

N16

(4.89)

N17 = P13 (L1 ) P13 (L2 ) P13 (L3 ) P03 (L4 ) = 27 L1 L2 L3 ,

(4.90)

N18 = P13 (L1 ) P13 (L2 ) P03 (L3 ) P13 (L4 ) = 27 L1 L2 L4 ,

(4.91)

N19 =

P13 (L1 ) P03 (L2 ) P13 (L3 ) P13 (L4 )

= 27 L1 L3 L4 ,

(4.92)

N20 =

P03 (L1 ) P13 (L2 ) P13 (L3 ) P13 (L4 )

= 27 L2 L3 L4 ,

(4.93)

because of the equations (4.48), (4.49) and (4.50). The scalar potential along any edge of a tetrahedron is the linear combination of the values defined on the points of the given edge, so that if two tetrahedra share the same facet, the potential will be continuous across this interface. This means that the approximate solution is continuous everywhere, however, its normal derivate is not. If potentials at the nodes are known, then a linear approximation of the potential function can be represented by (4.1). The sum of all nodal shape functions is equal to 1, hence the sum of their gradient is zero, m X i=1

Ni = 1,

and

m X i=1

∇Ni = 0.

(4.94)

This means that the maximal number of linearly independent nodal basis functions is m and the maximal number of linearly independent gradients of the nodal basis functions is m − 1, i.e. shape functions are linearly independent but their gradients are not. 4 : (0, 0, 0, 1)

3 : (0, 0, 1, 0) 1 : (1, 0, 0, 0)

2 : (0, 1, 0, 0) Fig. 4.24. Numbering scheme for linear element, n = 1

146


4 : (0, 0, 0, 2) 9 : (0, 0, 1, 1)

8 : (1, 0, 0, 1)

7 : (1, 0, 1, 0)

3 : (0, 0, 2, 0) 10 : (0, 1, 0, 1)

1 : (2, 0, 0, 0)

6 : (0, 1, 1, 0)

5 : (1, 1, 0, 0) 2 : (0, 2, 0, 0)

Fig. 4.25. Numbering scheme for quadratic element, n = 2 4 : (0, 0, 0, 3) 16 : (0, 0, 1, 2)

12 : (1, 0, 0, 2)

15 : (0, 0, 2, 1) 19 : (1, 0, 1, 1) 9 : (1, 0, 2, 0) 3 : (0, 0, 3, 0)

14 : (0, 1, 0, 2)

11 : (2, 0, 0, 1) 18 : (1, 1, 0, 1)

10 : (2, 0, 1, 0) 13 : (0, 2, 0, 1)

1 : (3, 0, 0, 0)

20 : (0, 1, 1, 1) 8 : (0, 1, 2, 0)

5 : (2, 1, 0, 0) 7 : (0, 2, 1, 0) 6 : (1, 2, 0, 0) 17 : (1, 1, 1, 0) 2 : (0, 3, 0, 0)

Fig. 4.26. Numbering scheme for cubic element, n = 3

4.2.2 Edge finite elements Vector potentials can be represented either by nodal shape functions or by so-called edge shape functions [64, 83, 123, 184, 194–196]. Edge shape functions are also called vector shape functions. The natural approach is to treat the vector field T = T (r, t) as two or three coupled scalar fields Tx = Tx (r, t), Ty = Ty (r, t) and Tz = Tz (r, t), i.e. T = T x ex + T y ey ,

(4.95)

T = T x ex + T y ey + T z ez

(4.96)

and


147

in 2D and in 3D situations, respectively, ex , ey and ez are the orthogonal unit vectors in the x − y and in the x − y − z plane. Nodal shape functions can be used in this case as well, as it was presented for scalar potentials in the previous section, however, each node has two or three unknowns. Nodal shape functions can be applied to approximate the scalar components of the vector field T . For example in 3D, T can be approximated as T ≃ =

m X i=1 m X

Ni (Tx,i ex + Ty,i ey + Tz,i ez ) Ni Tx,i ex +

i=1

m X i=1

Ni Ty,i ey +

m X

(4.97) Ni Tz,i ez .

i=1

Here Ni = Ni (r) are the usual nodal shape functions defined by (4.2) and Tx,i = Tx,i (t), Ty,i = Ty,i (t), Tz,i = Tz,i (t) are the values of components of the approximated vector potential at node i. The number of degrees of freedom is 2m in a 2D problem using triangular mesh and 3m in a 3D arrangement meshed by tetrahedral elements. Nodal shape functions are used to approximate gauged vector potentials, which was the first in the history of finite element method in electromagnetics. Unfortunately, there are some problems when the usual nodal based finite elements are used to interpolate vector potentials. The lack of enforcement of the divergence condition (lack of gauging) results in a system of algebraic equations, which has infinite number of solution and the application of iterative solvers sometimes fails. We have to take care about the Coulomb gauge. There are problems on the iron/air interface when using the magnetic vector potential approximated by nodal elements and extra interface conditions must be set up to solve this problem. Fortunately, vector shape functions have been developed in the last decades, which application in static and eddy current field problems is more and more popular, because of their advantages. The use of edge shape functions solves the problems described above. Some illustrations and examples are shown in chapter 7, which aims to compare not only the different potential functions, but the performance of the nodal and edge representation of vector potentials. It will be shown that the divergence of vector shape functions is equal to zero, that is why, gauging is satisfied automatically. The ungauged potential functions are approximated by vector elements. Vector shape functions are usually called edge shape functions, because they are associated to the edges of the FEM mesh. Vector shape functions are more and more popular in wave problems, too. Instead of scalar shape functions, vector shape functions (or edge shape functions) W i = W i (r) can be applied to approximate a vector potential T , T ≃

k X

W i Ti ,

(4.98)

i=1

where Ti = Ti (t) is the line integral of the vector potential T along the edge i. First order vector shape functions are defined by the line integral Z 1, along edge i, W i · dl = (4.99) 0, along other edges, l

148


i.e. the line integral of the vector shape function W i along the ith edge is equal to one. In other words, the vector shape function W i has tangential component only along the ith edge and it has only normal component along the other edges, because W i · dl is equal to zero only if the vectors W i and dl are perpendicular to each others and |W i ||dl| > 0. Moreover, in 3D case, the vector shape function W i has zero tangential component along every facet of the 3D finite element, which not share the edge i. If two triangles share the same vertices, the tangential component of the approximated vector potential will be continuous across the interface element boundary. This is true in 3D case as well, moreover, if two tetrahedra share the same facet, the tangential component of the vector potential will be continuous across this interface. This means that the tangential component of the approximate solution is continuous everywhere, however, its normal component is not. In the words of equations, according to the definition (4.99), the line integral of the vector potential along the mth edge is equal to Tm , i.e. ! Z Z k k Z X X W i Ti · dl = T · dl = (W i Ti ) · dl lm lm i=1 i=1 lm (4.100) Z = Tm

lm

W m · dl = Tm .

That is why, edge shape functions are also called tangentially continuous shape functions. The gradients of the nodal shape functions are in the function space spanned by the edge basis functions, that is ∇Nj = where

Pk

k X

2 i=1 cji

k X i=1

cji W i ,

i=1

j = 1, · · · , m − 1,

(4.101)

> 0. Taking the curl of each equation in (4.101) results in

cji ∇ × W i = 0,

j = 1, · · · , m − 1,

(4.102)

because ∇ × (∇ϕ) ≡ 0. This shows that the maximal number of linearly independent curls of the edge basis functions is k − (m − 1). The interdependence of the curls of the edge basis functions means that an ungauged formulation leads to a singular, positive semidefinit finite element curl-curl matrix. Singular systems can be solved by iterative methods, if the right-hand side of the system of equations is consistent. We took care about it when obtaining the weak formulations of the ungauged version of potentials, because excitation current density has been taken into account by the use of impressed current vector potential, T 0 . The vector function w ij = Li ∇Lj − Lj ∇Li

(4.103)

will be applied to construct the edge shape functions, because it can be used in functions, which satisfies (4.99) and (4.100). In 2D, Li (i = 1, 2, 3) are the barycentric coordinates

149


of the triangle defined by (4.18). In 3D, Li (i = 1, 2, 3, 4) are the barycentric coordinates of the tetrahedron defined by (4.56). According to the notations in (4.103), the edges of a finite element are pointing from node i to node j, as it can be seen in Fig. 4.27 and in Fig. 4.28. The vector field wij has the following important properties, which proofs the use of vector function w ij as vector shape function. (i) Let eij is a unit vector pointing from node i to node j, then eij · w ij =

1 , lij

(4.104)

where lij is the length of edge {i, j}. This means that wij has constant tangential component along the edge {i, j}.

Since Li and Lj are linear functions that vary from node i to node j from 1 to 0 and from 0 to 1, respectively, we have eij · ∇Li = −1/lij and eij · ∇Lj = 1/lij , finally eij · w ij = Li

1 1 Li + Lj 1 + Lj = = , lij lij lij lij

(4.105)

because Li + Lj = 1 along the edge {i, j}. See, for example Fig. 4.16 and let i = 1, j = 2, so N1 = L1 is decreasing along edge {1, 2} and N2 = L2 is increasing along the same edge. See also Fig. 4.9, from which it is easy to see the gradients eij · ∇Li = −1/lij and eij · ∇Lj = 1/lij . (ii) In 2D, the function Li varies linearly from node i to the opposite edge {j, k} (see e.g. N1 in Fig. 4.16, i = 1, j = 2, k = 3), i.e. the vector field ∇Li is perpendicular to this edge, but Li is zero there, that is why w ij is perpendicular to the edge {j, k}, wij = −Lj ∇Li ,

on the edge

{j, k},

(4.106)

and the length of this vector is decreasing from node j to k according to Lj . On the other hand, the function Lj varies linearly from node j to the opposite edge {k, i} (x3 , y3 )

l2 l3

(x2 , y2 ) l1 (x1 , y1 ) Fig. 4.27. The definition of edges with local directions of the triangular finite element

150


(x4 , y4 , z4 ) l6

l4 l3 (x1 , y1 , z1 )

l5 l1

(x3 , y3 , z3 ) l2

(x2 , y2 , z2 ) Fig. 4.28. The definition of edges with local directions of the tetrahedral finite element (see N2 in Fig. 4.16), i.e. ∇Lj is perpendicular to this edge, but Lj is zero there and wij is perpendicular to the edge {k, i}, wij = Li ∇Lj ,

on the edge {k, i},

(4.107)

and the length of this vector is decreasing from node i to k according to Li . This with item (i) means that the vector function w ij has tangential component only on the edge {i, j} and it is perpendicular to the other edges.

In 3D, this is valid to the whole triangular facet with the bounding edges opposite to a node, see e.g. Fig. 4.23. (iii) The vector field wij is divergence-free, ∇ · wij =∇·(Li ∇Lj −Lj ∇Li ) = ∇·(Li ∇Lj ) − ∇ · (Lj ∇Li ) =∇Li ·∇Lj + Li ∇·∇Lj − ∇Lj ·∇Li − Lj ∇·∇Li = 0,

(4.108)

by using the identity ∇ · (ϕ v) = ∇ϕ · v + ϕ∇ · v

(4.109)

with the notations ϕ = Li , v = ∇Lj in the second and ϕ = Lj , v = ∇Li in the last term. The barycentric coordinates are linear functions of the coordinates and their gradient is constant, which divergence is equal to zero, i.e. the second and fourth terms are vanishing. The first and the third terms are equal, finally, ∇ · w ij = 0. (iv) The vector field wij has constant curl, ∇×wij =∇×(Li ∇Lj −Lj ∇Li ) = ∇·(Li ∇Lj )−∇·(Lj ∇Li ) =Li ∇×∇Lj −∇Lj ×∇Li −Lj ∇×∇Li + ∇Li ×∇Lj =2∇Li × ∇Lj ,

by using the identity

(4.110)


∇ × (ϕ v) = ϕ∇ × v − v × ∇ϕ

151

(4.111)

with the notations ϕ = Li , v = ∇Lj in the first and ϕ = Lj , v = ∇Li in the second term. The first and the third terms are equal to zero because of the identity ∇ × ∇ϕ ≡ 0 for any function ϕ. The second term can be reformulated by a × b = −b × a, finally, the result is constant, because the gradients of the barycentric coordinates are constant. First, the edge shape functions defined on triangles based on (4.103) are collected. The basic 2D vector shape functions W i can be constructed by using the first order nodal shape functions, W 1 = l1 (N1 ∇N2 − N2 ∇N1 )δ1 ,

(4.112)

W 2 = l2 (N2 ∇N3 − N3 ∇N2 )δ2 ,

(4.113)

W 3 = l3 (N3 ∇N1 − N1 ∇N3 )δ3 .

(4.114)

Here li (Fig. 4.27) denotes the length of the ith edges of the triangle and it is used to normalize the edge shape function according to (4.104). The edge basis function W i (i = 1, 2, 3) has tangential component only along the ith edge and it is perpendicular to the other two edges as represented in Fig. 4.29(a)-4.29(c). It is easy to see that an edge shape function has magnitude and direction. The value of δi is equal to ±1, depending on whether the local direction of the edge is the same as the global direction or opposite (see Fig. 4.27 for local direction). This set of vector functions is called zeroth order vector shape functions. If the approximation of the vector function T is known along the edges of the mesh, then (4.98) can be used to interpolate the function anywhere and in linear case K = 3. Higher order vector shape functions can be constructed by using the vector function wij defined by (4.103), too. This vector function must be multiplied by a complete interpolatory polynomial, which results in the higher order vector shape functions. First and second order polynomials will be used to build up first and second order vector shape functions. Here, we follow [83], the method is as follows. First of all, an indexing sequence must be set up, which is similar to the method used to build up the scalar shape functions, because the higher order vector shape functions are based on the Lagrange polynomials and (4.103). In the case of first order approximation, the numbering scheme of the third order scalar interpolation can be used and the points are shown in Fig. 4.30, Fig. 4.31 and Fig. 4.32 must be used to represent first order vector shape functions associated to the edge {1, 2}, {2, 3} and {3, 1}, respectively. In the case of second order approximation, the numbering scheme of the fourth order scalar interpolation can be used and the interpolation points shown in Fig. 4.33, Fig. 4.34 and Fig. 4.35 must be used to represent second order vector shape functions associated to the edge {1, 2}, {2, 3} and {3, 1}, respectively. The interpolation points have been selected in this special way, because the interpolation of field vectors along vertices has been avoided, i.e. the points have been shifted inside the triangle and the indexing scheme of order n + 2 is used to represent the vector interpolation of order n. This is called global numbering and denoted by (I, J, K) on the ’big’ triangle, local numbering means the numbering scheme with the real order (i, j, k) defined over the ’small’ triangle.

152


3 2.5 l

2

3

y

l

2

1.5 1 l

1

0.5 0

0.5

1 x

1.5

2

(a) The edge shape function W 1

3 2.5 l3

2 y

l

2

1.5 1 l1

0.5 0

0.5

1 x

1.5

2

(b) The edge shape function W 2

3 2.5 l

2

3

y

l

2

1.5 1 0.5 0

l1 0.5

1 x

1.5

2

(c) The edge shape function W 3

Fig. 4.29. The 2D edge shape functions

153


(0, 0, 1)

(0, 0, 3) (0, 1, 2)

(1, 0, 2)

(0, 1, 0)

(0, 2, 1)

(1, 1, 1)

(1, 0, 0)

(2, 0, 1) (0, 3, 0) (1, 2, 0) (3, 0, 0)

(2, 1, 0)

Fig. 4.30. Numbering scheme for the first order vector element associated with w12 (0, 0, 1)

(0, 0, 3) (0, 1, 2)

(1, 0, 2)

(0, 1, 0)

(0, 2, 1) (2, 0, 1)

(1, 0, 0)

(1, 1, 1) (0, 3, 0) (1, 2, 0)

(3, 0, 0)

(2, 1, 0)

Fig. 4.31. Numbering scheme for the first order vector element associated with w23 (0, 0, 1)

(0, 0, 3) (0, 1, 2)

(1, 0, 2)

(0, 1, 0)

(0, 2, 1) (2, 0, 1)

(1, 0, 0)

(1, 1, 1) (0, 3, 0) (1, 2, 0)

(3, 0, 0)

(2, 1, 0)

Fig. 4.32. Numbering scheme for the first order vector element associated with w31

154


(0, 0, 4) (0, 1, 3)

(1, 0, 3)

(3, 0, 1)

(4, 0, 0)

(2, 2, 0)

110 200

211

(3, 1, 0)

020

(0, 3, 1)

121

011

101

(0, 2, 2)

112

(2, 0, 2)

002

(1, 3, 0)

(0, 4, 0)

Fig. 4.33. Numbering scheme for the second order vector element associated with w12 (0, 0, 4) (0, 1, 3)

(1, 0, 3)

(0, 3, 1) 121

(3, 0, 1)

(4, 0, 0)

(0, 2, 2)

112

(2, 0, 2)

211

(3, 1, 0)

(2, 2, 0)

(1, 3, 0)

(0, 4, 0)

Fig. 4.34. Numbering scheme for the second order vector element associated with w23 (0, 0, 4) (0, 1, 3)

(1, 0, 3)

(3, 0, 1)

(4, 0, 0)

(0, 2, 2)

112

(2, 0, 2)

(0, 3, 1) 121 211

(3, 1, 0)

(2, 2, 0)

(1, 3, 0)

(0, 4, 0)

Fig. 4.35. Numbering scheme for the second order vector element associated with w31

It is noted here that the COMSOL Multiphysics software uses this kind of vector shape functions, however, n = 0, n = 1 and n = 2 are named as linear, quadratic and cubic vector shape functions.

155


The vector function wab (associated to the edge pointing from node a to node b) can be multiplied by the Lagrange polynomials as W IJK = αIJK Pin (l1 ) Pjn (l2 ) Pkn (l3 ) wab , ab ab

(4.115)

where n is the order of approximation and the integers i, j and k satisfy i + j + k = n (see the small triangles in Fig. 4.30-Fig. 4.35). 0 (·) = 1 and If n = 0, the basic vector shape functions can be obtained, because Pm IJK α can be selected as the length of the appropriate edge, lab , since αab is a normalizing factor. The barycentric coordinates l1 , l2 and l3 are imagined in the small triangles. The transformation between local and global numbering is as follows: i = I − 1, i = I,

j = J − 1,

j = J − 1,

i = I − 1,

j = J,

k = K,

k = K − 1,

k = K − 1,

on the edge {1, 2},

(4.116)

on the edge

(4.117)

on the edge

{2, 3},

{3, 1}.

(4.118)

The relation between the barycentric coordinates of the small and the big triangles is as follows: 1 1 n+2 n+2 n+2 , l2 = L 2 − , l3 = L 3 − . (4.119) l1 = L 1 n n n n n Using these relations, (4.115) can be written as (let here {ab} = {23} for simplicity) n+2 n+2 1 IJK IJK n n W 23 = α23 PI L1 PJ−1 L2 − n n n (4.120) 1 n+2 n w23 . − PK−1 L3 n n According to (4.22), Lagrange polynomials can be reformulated as I−1 n+2 n+2 1 Y n L1 PIn L1 −p = n I! p=0 n I−1 1 Y [(n + 2)L1 − p] = PIn+2 (L1 ) , = I! p=0

(4.121) if

I > 0,

and J−2 Y n+2 1 1 1 n+2 n PJ−1 L2 n L2 = −p − − n n (J − 1)! p=0 n n =

J−2 Y 1 [(n + 2)L2 − 1 − p] (J − 1)! p=0 J−2 Y

1 (J − 1)! p=0 n+2 =PJ−1 L2 −

=

(n + 2) L2 −

1 n+2

,

if

1 n+2

J > 1.

−p

(4.122)

156


The so-called shifted Silvester polynomials can be used to simplify the relations above [83], J−1 Y 1 1 n+2 [(n + 2)L2 − p] . (4.123) SJn+2 (L2 ) = PJ−1 L2 − = n+2 (J − 1)! p=0 Finally, the higher order vector shape functions can be formulated as follows by using the Lagrange and Silvester polynomials: n+2 W IJK = αIJK SIn+2 (L1 ) SJn+2 (L2 ) PK (L3 ) w 12 , 12 12

(4.124)

n+2 = αIJK PIn+2 (L1 ) SJn+2 (L2 ) SK (L3 ) w 23 , W IJK 23 23

(4.125)

W IJK 31

(4.126)

=

αIJK 31

n+2 SIn+2 (L1 ) PJn+2 (L2 ) SK (L3 ) w 31 .

The parameter denoted by α is a normalization factor, which must have the value such that the line integral of vector shape function W IJKL is equal to 1 on the edge pointing ab from node a to node b. Here, P0n (·) = 1,

and S1n (·) = 1.

(4.127)

The number of vector basis functions is k = (n + 1)(n + 3).

(4.128)

There is one shape function associated with the introduced interpolation nodes on the edges. It means 3(n + 1) basis functions. There are three basis functions for an interior interpolation point, because every interpolation point inside the triangle is used to build all the vector shape functions in the three edges. Since a surface vector has only two degrees of freedom, these three basis functions are not independent and one of them must be discarded. This results in n(n + 1) interior basis functions. In total, the number of shape functions is 3(n + 1) + n(n + 1) = (n + 3)(n + 1). In the case of first order approximation n = 0 and k = 3. In the case of second order approximation n = 1 and K = 8. In the case of third order approximation n = 2 and k = 15 and so on. The first order vector shape functions are as follows (presented in Fig. 4.36) from (4.124)–(4.127): 120 3 3 3 120 W 1 = W 120 12 = α12 S1 (L1 ) S2 (L2 ) P0 (L3 ) w 12 = α12 (3 L2 − 1) w 12 , (4.129) 210 3 3 3 210 W 2 = W 210 12 = α12 S2 (L1 ) S1 (L2 ) P0 (L3 ) w 12 = α12 (3 L1 − 1) w 12 , (4.130) 111 3 3 3 111 W 3 = W 111 12 = α12 S1 (L1 ) S1 (L2 ) P1 (L3 ) w 12 = α12 3 L3 w 12 ,

(4.131)

W 4 = W 012 23 = W 5 = W 021 23 = W 6 = W 102 31 = W 7 = W 201 31 = W 8 = W 111 31 =

(4.134)

α012 23 α021 23 α102 31 α201 31 α111 31

P03 (L1 ) S13 (L2 ) S23 (L3 ) w 23 = P03 (L1 ) S23 (L2 ) S13 (L3 ) w 23 =

α012 23 (3 L3 α021 23 (3 L2

S13 (L1 ) P03 (L2 ) S23 (L3 ) w 31 = S23 (L1 ) P03 (L2 ) S13 (L3 ) w 31 = S13 (L1 ) P13 (L2 ) S13 (L3 ) w 31 =

α102 31 (3 L3 − 1) w 31 , α201 31 (3 L1 − 1) w 31 , α111 31 3 L2 w 31 .

− 1) w23 , (4.132) − 1) w23 , (4.133) (4.135) (4.136)

157


3

3

2.5 2

2.5 l

2

1.5

l

2

1.5

1

1 l1

0.5 0

0.5

1 x

(a) W 1 =

1.5

l1

0.5 0

2

W 120 12

0.5

1 x

(b) W 2 =

3

1.5

2

W 210 12

3

2.5

2.5 l

l

2

3

3

l2

y

y

l2 1.5

1.5

1

1 l1

0.5 0

0.5

1 x

(c) W 4 =

1.5

l1

0.5 0

2

W 012 23

0.5

3

2.5

2.5 l3

2

l2

y

1.5

1.5

W 021 23

l3

2 l2

y

1 x

(d) W 5 =

3

2

3

y

y

l

2

l

2

3

1.5

1

1 l1

0.5 0

0.5

1 x

1.5

l1

0.5 0

2

(e) W 6 = W 102 31

0.5

1 x

1.5

2

(f) W 7 = W 201 31

3

3 l2

2.5 l

l

2

3

l

y

3

y

2

2.5

1.5 1 0.5 0

2

1.5 1 l1

l1 0.5

1 x

1.5

(g) W 3 = W 111 12

2

0.5 0

0.5

1 x

1.5

2

(h) W 8 = W 111 31

Fig. 4.36. First order vector shape functions, n = 1, k = 8 The second order vector shape functions are as follows using (4.124)–(4.127): 310 4 4 4 W 1 = W 310 12 = α12 S3 (L1 ) S1 (L2 ) P0 (L3 ) w 12 1 (4 L1 − 1)(4 L1 − 2) w12 , = α310 12 2

(4.137)

158

4. THE FINITE ELEMENT METHOD 220 4 4 4 W 2 = W 220 12 = α12 S2 (L1 ) S2 (L2 ) P0 (L3 ) w 12

= α220 12 (4 L1 − 1)(4 L2 − 1) w 12 ,

130 4 4 4 W 3 = W 130 12 = α12 S1 (L1 ) S3 (L2 ) P0 (L3 ) w 12 1 = α130 (4 L2 − 1)(4 L2 − 2) w12 , 12 2 211 4 4 4 W 4 = W 211 12 = α12 S2 (L1 ) S1 (L2 ) P1 (L3 ) w 12

= α211 12 (4 L1 − 1)4 L3 w 12 ,

121 4 4 4 W 5 = W 121 12 = α12 S1 (L1 ) S2 (L2 ) P1 (L3 ) w 12

= α121 12 (4 L2 − 1)4 L3 w 12 ,

031 4 4 4 W 6 = W 031 23 = α23 P0 (L1 ) S3 (L2 ) S1 (L3 ) w 23 1 = α031 (4 L2 − 1)(4 L2 − 2) w23 , 23 2 022 4 4 4 W 7 = W 022 23 = α23 P0 (L1 ) S2 (L2 ) S2 (L3 ) w 23

= α022 23 (4 L2 − 1)(4 L3 − 1) w 23 ,

013 4 4 4 W 8 = W 013 23 = α23 P0 (L1 ) S1 (L2 ) S3 (L3 ) w 23 1 = α013 (4 L3 − 1)(4 L3 − 2) w23 , 23 2 121 4 4 4 W 9 = W 121 23 = α23 P1 (L1 ) S2 (L2 ) S1 (L3 ) w 23

= α121 23 (4 L1 )(4 L2 − 1) w 23 ,

112 4 4 4 W 10 = W 112 23 = α23 P1 (L1 ) S1 (L2 ) S2 (L3 ) w 23

= α112 23 (4 L1 )(4 L3 − 1) w 23 ,

103 4 4 4 W 11 = W 103 31 = α31 S1 (L1 ) P0 (L2 ) S3 (L3 ) w 31 1 = α103 (4 L3 − 1)(4 L3 − 2) w31 , 31 2 202 4 4 4 W 12 = W 202 31 = α31 S2 (L1 ) P0 (L2 ) S2 (L3 ) w 31

= α202 31 (4 L1 − 1)(4 L3 − 1) w 31 ,

301 4 4 4 W 13 = W 301 31 = α31 S3 (L1 ) P0 (L2 ) S1 (L3 ) w 31 1 = α301 (4 L1 − 1)(4 L1 − 2) w31 , 31 2 112 4 4 4 W 14 = W 112 31 = α31 S1 (L1 ) P1 (L2 ) S2 (L3 ) w 31

= α112 31 (4 L2 )(4 L3 − 1) w 31 ,

211 4 4 4 W 15 = W 211 31 = α31 S2 (L1 ) P1 (L2 ) S1 (L3 ) w 31

= α211 31 (4 L2 )(4 L1 − 1) w 31 .

(4.138)

(4.139)

(4.140)

(4.141)

(4.142)

(4.143)

(4.144)

(4.145)

(4.146)

(4.147)

(4.148)

(4.149)

(4.150)

(4.151)

The second order shape functions associated to the edge {1, 2} can be seen in Fig. 4.37. The edge shape functions associated to the other two edges can be imagined as the rotation

159


3

3

2.5 2

2.5 l

2

1.5

l

2

1.5

1

1 l1

0.5 0

0.5

1 x

(a) W 1 =

1.5

l1

0.5 0

2

W 310 12

0.5

1 x

(b) W 2 =

3

1.5

2

W 220 12

3

2.5

2.5 l

l

2

3

3

l2

y

y

l2 1.5

1.5

1 0.5 0

3

y

y

l

2

l

2

3

1 l1 0.5

1 x

(c) W 3 = 3

1.5

l1

0.5 0

2

W 130 12

0.5

1 x

(d) W 4 =

1.5

2

W 211 12

3

l

3

l

2.5

2.5

2

y

2

y

2 1.5

l2

l3 1.5

1

1 l

0.5 0

1

l1 0.5

1 x

1.5

2

(e) W 5 = W 121 12

0.5 0

0.5

1 x

1.5

2

(f) W 10 = W 112 23

Fig. 4.37. Second order vector shape functions, n = 2, K = 15 of the vector functions shown in Fig. 4.37. For example, W 10 = W 112 23 is also plotted, , but along the edge {2, 3}. which is the same as W 5 = W 121 12 The vector shape functions in 3D can be constructed as the extension of the above presented 2D realization. Three-dimensional zeroth order edge shape functions can be constructed as [83], W 1 = l1 (N1 ∇N2 − N2 ∇N1 )δ1 ,

(4.152)

W 2 = l2 (N2 ∇N3 − N3 ∇N2 )δ2 ,

(4.153)

W 3 = l3 (N3 ∇N1 − N1 ∇N3 )δ3 ,

(4.154)

W 4 = l4 (N1 ∇N4 − N4 ∇N1 )δ4 ,

(4.155)

W 5 = l5 (N2 ∇N4 − N4 ∇N2 )δ5 ,

(4.156)

160


W 6 = l6 (N3 ∇N4 − N4 ∇N3 )δ6 .

(4.157)

Here li (Fig. 4.28) is the length of the edges and it is used to normalize the edge shape function according to (4.104). The value of δi is also equal to ±1 depending on whether the local direction of the edge is the same as the global direction or opposite. The edge definition employed in my analysis can be seen in Fig. 4.28. If the approximation of the vector function T is known along the edges of the mesh, then (4.98) can be used to interpolate the function anywhere and in linear case k = 6. To construct higher order vector basis functions, the points of interpolation polynomials are arranged in a pyramid format to build an applicable numbering scheme (I, J, K, L) and I, J, K, L = 0, 1, · · · , n + 2, where n is the order of the element. The illustration of numbering scheme in 3D is not easy, but it can be construct as follows. Let us imagine the same numbering scheme on the triangular facets of the tetrahedron as in Fig. 4.33-Fig. 4.35 and the integers I, J, K and L can be set up according to the facets. The vector shape functions of order n are given as [83], n+2 = αIJKL SIn+2 (L1 ) SJn+2 (L2 ) PK (L3 ) PLn+2 (L4 ) w 12 , W IJKL 12 12

(4.158)

n+2 W IJKL = αIJKL PIn+2 (L1 ) SJn+2 (L2 ) SK (L3 ) PLn+2 (L4 ) w 23 , 23 23

(4.159)

n+2 = αIJKL SIn+2 (L1 ) PJn+2 (L2 ) SK (L3 ) PLn+2 (L4 ) w 31 , W IJKL 31 31

(4.160)

n+2 W IJKL = αIJKL SIn+2 (L1 ) PJn+2 (L2 ) PK (L3 ) SLn+2 (L4 ) w 14 , 14 14

(4.161)

W IJKL 24

n+2 PIn+2 (L1 ) SJn+2 (L2 ) PK (L3 ) SLn+2 (L4 ) w 24 ,

(4.162)

n+2 = αIJKL PIn+2 (L1 ) PJn+2 (L2 ) SK W IJKL (L3 ) SLn+2 (L4 ) w 34 . 34 34

(4.163)

=

αIJKL 24

The parameters denoted by α are normalization factors, which must have the value such that the line integral of the vector shape function W IJKL is equal to 1 on the edge ab pointing from node a to node b. The number of edge shape functions when defining the nth order family is k=

(n + 1)(n + 3)(n + 4) . 2

(4.164)

For each interpolation point on the edge, there is one corresponding vector shape function, which means 6(n + 1) functions. For each interpolation point on the face of a tetrahedron there are three vector functions, but one of them is depending on the other two and it must be discarded, finally there are 4n(n + 1) vector shape functions defined on the four facets. For each interpolation points inside the element there are six basis functions. A 3D vector has only three degree of freedom, that is why three vector basis functions must be discarded resulting in n(n − 1)(n + 1)/2 vector basis functions. Totally, there are 6(n + 1) + 4n(n + 1) + n(n − 1)(n + 1)/2 = (n + 1)(n + 3)(n + 4)/2 vector shape functions. There are k = 6, k = 20 and k = 45 shape functions for n = 0, n = 1 and n = 2, respectively. As an example, the following vector shape functions can be set up when n = 1, 1200 0210 0120 1020 2010 2001 1002 0201 0102 W 2100 12 , W 12 , W 23 , W 23 , W 31 , W 31 , W 14 , W 14 , W 24 , W 24 , 0021 0012 1110 1110 1011 1011 0111 0111 1101 1101 W 34 , W 34 , W 12 , W 31 , W 14 , W 34 , W 23 , W 24 , W 12 , W 24 .

5 Modeling of ferromagnetic hysteresis In this chapter a short summary of the fundamental relations in magnetism and magnetic materials focusing mainly on ferromagnetic materials is presented. This introductory chapter is based on the well known books [7, 22, 27, 76, 77, 82, 96–100, 128, 153, 178]. Although, an increasing number of papers can be found in the literature, only the most relevant ones have been cited in the following sections, which are the basis of this work in the field of ferromagnetic hysteresis modeling. The simulation of electrical engineering applications containing ferromagnetic parts requires fast and accurate hysteresis models, because the partial differential equations derived from Maxwell’s equations must be coupled with this kind of nonlinearity by applying numerical techniques. This requirement results in developing more and more complex and consequently more and more accurate hysteresis models. However, the main goal – from engineering point of view – is the applicability in numerical field analysis and the easy identification task of the hysteresis models. The hysteresis models can be fit to measured curves by using identification procedures. One main family of hysteresis models is called macroscopic models, which handle the process of magnetism by developing differential equation formulation or statistical description of the material property. The other family is the so-called microscopic models, based on energy contributions on very small dimensions. The typical energy terms are demagnetisation, anisotropic, exchange, external field energy, etc. A minimization procedure must be used to find the energy minimum of the sum of energy terms taken into account. The microscopic models have precise physical justifications but the computation time of these models is very large, i.e. they can not be applied in electrical engineering simulations. A search of international journals and conference proceedings (for example in IEEE Transactions on Magnetics, Journal of Applied Physics, Physica B, COMPEL, JSAEM Studies in Appl. Electromagnetics and Mechanics, Preisach Memorial Book, International Journal of Applied Electromagnetics and Mechanics, Journal of Electrical Engineering, Proceedings of SMM-Soft Magnetic Materials Conference, HMM-Hysteresis Modelling and Micromagnetism Conference, International IGTE Symposium on Numerical Field

162

5. MODELING OF FERROMAGNETIC HYSTERESIS

Calculation in Electrical Engineering, CEFC-The Conference on Electromagnetic Field Computation, COMPUMAG-Conference on the Computation of Electromagnetic Fields, OPTIM-Conference on Optimization of Electrical and Electronic Equipments, ISTET-Int. Symposium on Theoretical Electrical Engineering, ISEF-International Symposium on Electromagnetic Fields in Electrical Engineering) show that the most popular hysteresis models are the Preisach model, the Jiles-Atherton model and the Stoner-Wohlfarth model. There are some new mathematical models as well without any physical meaning. The classical Preisach model and its modifications are the most widely used hysteresis models. The introduction, the identification and the properties of this kind of models can be found in this chapter, because this model is the basis of the simulation techniques based on modern, attractive and popular procedures such as neural networks, fuzzy systems, genetic algorithms and hybrid methods as it can be read in an increasing number of papers in the last years. These are mathematical models of hysteresis. The Preisach model and the Jiles-Atherton model are so-called scalar models. This means that the magnetic field intensity H (the input of the model) and the magnetic flux density B (the output of the model) are supposed to be parallel to each other. Let us think about a toroidal shape core, where this assumption is adequate. In this case, the magnetic field intensity and the magnetic flux density can be represented by their scalar value in the direction of azimuthal axis of the toroid, i.e. their direction is known (H = H(r, t), and B = B(r, t), where r and t are space vector and time, respectively). In general, these field quantities are vectors, e.g. the magnetic field intensity is H = H(r, t) and the magnetic flux density is B = B(r, t). The dependence of direction of these vectors is more complex. This is the reason why the modeling of vector hysteresis properties is very important in the simulation of two-dimensional or three-dimensional arrangements. Applying vector hysteresis models in electromagnetic field calculation software results in more accurate results. This requirement has started an extensive study and research in this field. The Stoner-Wohlfarth model is a vector model of a magnetic particles. The classical Preisach model and the Jiles-Atherton model can be extended to realize vectorial behavior as well. The measurement of the hysteresis characteristics and the vectorial properties aims the identification of different scalar and vector hysteresis models. The scalar hysteresis characteristics can be measured by using a toroidal shape core, the vector hysteresis properties can be measured by applying an Epstein frame or a single sheet tester. This task is in focus of research nowadays, because most laboratories use different type of yokes and different sensing methods and there is no standard measurement system yet.

5.1 Ferromagnetic materials 5.1.1 Short description of magnetic materials The possible origin of magnetism is the motion of electrons in different energy levels in the atomic structures of materials. Three types of motion can be associated with the magnetic behavior of materials: orbital motion of electrons around nucleus, electron spin and nuclear spin.

5.1. FERROMAGNETIC MATERIALS

163

A simple atomic model can be built as a magnetic moment of elementary current loop, which simulates the motions in the atomic structures. This is a macroscopic description of the phenomena. The magnetic field produced by the elementary loop current Ii enclosing the surface dsi can be represented by the magnetic moment mi , mi = Ii dsi .

(5.1)

The magnetic moment can be associated with the magnetic dipole. The vectorial sum of these individual magnetic moments in a volume △V containing n numbers of magnetic moments results the magnetic moment m, i.e. m=

n X

mi ,

(5.2)

i=1

and the volume density of the magnetic moments characterizes the magnetization vector M as ! n 1 X 1 (5.3) mi . M = lim m = lim △V →0 △V △V →0 △V i=1 The magnetic field quantity called magnetic induction B can be introduced by the torque τ acting on the magnetic dipole of moment m in the presence of B, τ = m × B,

(5.4)

where × denotes the cross product of the vectors m and B. This means that the magnetic induction tries to align the dipole so that the magnetic moment lies parallel to the magnetic induction, where the energy of the dipole has minimum value. The energy of the magnetic dipole is defined as w = −m · B.

(5.5)

Here · denotes the scalar product of the vectors m and B. The magnetic induction B consists of two components, the magnetic induction of the free space µ0 H, the other component is contributed from the magnetization of the material µ0 M , i.e. B = µ0 (H + M ),

(5.6)

where µ0 is the permeability of vacuum, µ0 = 4π 10−7 H/m and H is called magnetic field intensity. The relation between the magnetic field intensity H and the magnetization M depends on the properties of the material placed into a magnetic field, and this relationship is called the hysteresis characteristics. Mathematically, it can be represented by the hysteresis operator, M = H {H}.

(5.7)

164


The magnetic induction B can be expressed by substituting this operator into (5.6) as B = µ0 (H + H {H}),

(5.8)

which results in the operator form B = B{H}.

(5.9)

Generally, the operators H {·} and B{·} represent nonlinear and multivalued relations. However, in some cases a linear function can be used to express the magnetization, M = χH,

(5.10)

where χ is the magnetic susceptibility. Substituting this relation into (5.6), it results in the form B = µ0 (1 + χ)H = µ0 µr H,

(5.11)

where µr is the relative permeability, µr = 1 + χ. The susceptibility χ is not necessarily a constant, it can be a function of the applied magnetic field intensity, M = χ(H)H,

(5.12)

moreover this can be a tensor too, M = [χ]H,

(5.13)

representing the variation of the magnetization with the direction of the applied field. In the case of anisotropic materials it has the form      χxx χxy χxz Hx Mx  My  =  χyx χyy χyz   Hy  . (5.14) Mz χzx χzy χzz Hz

The special case    Mx χx  My  =  0 Mz 0

0 χy 0

  Hx 0 0   Hy  Hz χz

(5.15)

represents anisotropic materials too, but the characteristics in the orthogonal directions are not depending on each others. Isotropic description can be realized if χx = χy = χz . In general the elements of the susceptibility tensor [χ] are functions of the applied magnetic field H. More generally, susceptibility is depending on the frequency as well. The dimensions of the susceptibility tensor [χ] is 3 × 3 in three-dimensional case, 2 × 2 in two-dimensional case and 1 × 1 in scalar simulations.


165

According to the magnetic susceptibility, the magnetic materials can be classified as follows. (i) Diamagnetic materials. The magnetic susceptibility is negative and small (e.g. |χ| = 10−7 − 10−5 ) resulting in weak magnetic properties and it is independent of temperature. (ii) Paramagnetic materials. The magnetic susceptibility is positive and small (the order is χ = 10−5 − 10−3 ), but it is inversely proportional to the temperature. (iii) Antiferromagnetic materials. These materials have also small positive susceptibility, which is depending on the temperature. Above the so-called Neel temperature the susceptibility is inversely proportional to the temperature. (iv) Ferromagnetic materials. There are small regions, which are called domains inside the ferromagnetic materials where the magnetic dipoles have strong interaction. The large number of magnetic moments in the domains have the same directed magnetization inside a domain and this results in much higher magnetization. The magnetic properties of the ferromagnetic materials can be represented by the well known hysteresis characteristics. Above the Curie temperature the materials loose their ferromagnetic properties (e.g. TC = 770 ◦C for pure iron). (v) Ferrimagnetic materials. Theoretically these materials are between the last two groups. In case of H = 0, these materials also have spontaneous magnetization, but above Curie temperature the materials behave as the paramagnetic ones, the susceptibility decreases with increase of the temperature. Here, magnetic field quantities are denoted as space vectors. As it has been introduced at the beginning of this chapter, this is the general form. In some cases – depending on the problem to be simulated – these quantities can be supposed to be scalar ones. In the following, the scalar hysteresis characteristics are the basis of the study, which can be generalized in two dimensions and in three dimensions in space. The ferromagnetic materials is in the focus in the following sections as one of the most important class of magnetic materials in the electrical engineering applications.

5.1.2 Ferromagnetic hysteresis The typical shape of the scalar hysteresis characteristics of ferromagnetic materials can be seen in Fig. 5.1. Starting from demagnetized state (H = 0, M = 0, B = 0) and applying monotonously increasing external magnetic field intensity on the ferromagnetic material the magnetization is increasing according to the first magnetization curve. Finally it reaches the saturation magnetization Ms , where the magnetic field intensity and the magnetic flux density are Hm and Bm , respectively. Subscript m denotes the maximum value of magnetic field intensity and magnetic flux density according to saturation. The initial part of the first magnetization curve is reversible, because increasing and decreasing the magnetic field intensity the magnetization comes back to the origin. Here domains have reversible rotation from a stable state. The starting slope can be expressed

166


by the initial susceptibility χin and magnetization is proportional to the magnetic field intensity, M = χin H, where χin

(5.16)

∂M = . ∂H H=0,M=0

(5.17)

This initial region is linear, thus B = µ0 (1 + χin )H = µin H,

(5.18)

and µin is the initial permeability. Beyond the initial range the domain walls have irreversible displacement from one stable point to another one. This can be proved by the Barkhausen effect. Further increase of magnetic field intensity results irreversible domain rotations, finally the saturation state is reached. There is no further increase of magnetization because the magnetic dipoles are almost parallel to each other and to the direction of the applied magnetic field. The magnetic induction can increase with the increase of the magnetic field intensity, B = µ0 (H + Ms ) = µ0 H + µ0 Ms ,

(5.19)

which is the equation of a straight line with the slope µ0 . Decreasing the magnetic field intensity from the saturation state results in decreasing magnetization, but this variation is not reversible as can be seen in Fig. 5.1. Decreasing the magnetic field intensity to zero, the magnetization has a finite value, which is called the remanent magnetization, Mr , and Br = µ0 Mr

(5.20)

is the remanent induction. 1

1 Remanence

Remanence Saturation

B/B m

M/Ms

Saturation

0

Coercive field

0

First magnetization curve

−1 −1

0 H/H

m

First magnetization curve

1

−1 −1

0 H/H

1

m

Fig. 5.1. The normalized stationary hysteresis loop of ferromagnetic materials denoted by M = H {H} and B = B{H}

167


Further decreasing the magnetic field intensity results in decreasing magnetization and magnetic induction. The state B = 0 can be reached meaning that the ferromagnetic material can be demagnetized starting from saturation state by applying the so-called coercive field H = −Hc . Here magnetization is equal to the value of coercive field, i.e. M = Hc (M > 0). In the coordinate where the magnetization is equal to zero, the magnetic induction has the value B = µ0 H (B < 0). Further decreasing the magnetic field intensity leads to negative saturation state. Finally, increasing the magnetic field intensity to positive saturation close the major hysteresis loop. The major hysteresis loop is symmetric. Along the hysteresis loop minor loops can be generated by increasing and decreasing the applied magnetic field intensity as it is presented in Fig. 5.2. The shape of these minor loops (and of course the whole hysteresis curve) can be very different from material to material. 1

M/Ms

0.5

0

−0.5

−1 −1

−0.5

0 H/H

0.5

1

m

Fig. 5.2. Minor loops

It is evident that the susceptibility is a function of the applied magnetic field intensity in the case of ferromagnetic materials when hysteresis is taken into account, i.e. M = χ(H)H,

(5.21)

where χ(H) = χdif f =

∂M ∂H

(5.22)

is the so-called differential susceptibility. The magnetic induction can also be expressed in a similar way, B = µ0 (H + M ) = µ0 (1 + χ(H)) H = µ(H)H, where

(5.23)

168


µ(H) =

∂B = µ0 (1 + χ(H)) , ∂H

(5.24)

i.e. the permeability is a function of the magnetic field intensity.

5.1.3 The hysteresis operator In engineering applications when the simulated equipment contains ferromagnetic parts, the ferromagnetic hysteresis has to be handled by hysteresis models. The model is a mathematical representation of the physical phenomena and it can be described by the general hysteresis operator as ˙ M (t) = H {H(t), M (t), H(t), M˙ (t), x[t, H(t), M (t), · · · ]}.

(5.25)

This general form evaluates the magnetization as the function of the magnetic field intensity H(t), the magnetization M (t), the time variation of the previous two field quantities and other state variables x[t, H(t), M (t), · · · ] determining the behavior of the model, e.g. the prehistory of the material, temperature dependence, stress dependence and so on. The input of the simplest hysteresis model is only the magnetic field intensity (Fig. 5.3), M (t) = H {H(t)},

and B(t) = B{H(t)},

(5.26)

which can be expanded to simulate more and more general physical phenomenon. H(t)

M (t)

H {·}

H(t)

B{·}

B(t)

Fig. 5.3. The hysteresis operator

5.2 The Preisach model 5.2.1 The scalar model The scalar hysteresis model can be introduced in special cases when the magnetic field intensity vector and the magnetization vector are aligned in the same direction specified by the direction of the excitation field. A typical example to illustrate this assumption is the variation of magnetic field quantities inside a toroidal shape core magnetized by a coil wounded on the core. This arrangement can be used to measure the scalar hysteresis characteristics of magnetic materials. Another classical problem is the simulation of the magnetic infinite half space. It is supposed that the magnetic field intensity and the magnetization have one component and their relationship can be simulated efficiently by the scalar hysteresis model. Other introductory studies of the classical Preisach model can also be read in [6, 24, 36, 47–50, 54–60, 76, 84, 85, 95, 128, 132, 140, 145, 178–182, 185, 188, 189].

169

5.2. THE PREISACH MODEL

A. Description of the model The classical Preisach model is based on the domain theory, i.e. ferromagnetic materials consist of many elementary interacting domains. The Preisach model assumes that each domain can be represented by a rectangular elementary hysteresis loop characterized by the so-called switching fields α and β, where the normalized magnetization jumps from −1 to +1 and jumps down from +1 to −1, respectively, so α > β. Switching fields can be normalized, i.e. α and β can vary in the interval [−1, · · · , +1]. This elementary hysteresis loop can be seen in Fig. 5.4 and it can be characterized by the hysteresis operator γ b(α, β), which behaves as follows. By increasing the magnetic field intensity H, the elementary hysteresis operator with switching field α ≤ H/Hm turns to the magnetization state M/Ms = +1. By decreasing the magnetic field intensity, the rectangular operator whose magnetization is positive remains in this position until the magnetic field is decreased to the value of β < H/Hm . If the normalized magnetic field intensity is decreasing below β, then the hysteresis operator flips to the other position characterized by the value M/Ms = −1. Here Ms and Hm are the magnetization in the saturation state and the corresponding magnetic field intensity, respectively. The operator can also be defined by the width of operator and the interaction field (or mean field) as well, hc =

α−β , 2

hm =

α+β . 2

(5.27)

M/Ms

hc

hc

+1

H/Hm β

α

-1 hm

Fig. 5.4. Magnetization curve of an elementary hysteresis operator, γ b(α, β)

The Preisach triangle and the staircase line are the most important results of works followed by the first results of [4,47–50,95,126,132,154,182,189]. The Preisach triangle (Fig. 5.5) is defined in the plane of switching fields α and β if α ≥ β and α ∈ [−1, ..., 1], β ∈ [−1, ..., 1]. The elementary hysteresis operators γ b(α, β) are defined only in this region and they are turned up or turned down by the normalized magnetic field intensity b(α, β) = ±H/Hm . This results in two separate regions on the Preisach triangle where γ −1 and γ b(α, β) = +1. The interface between these regions is represented and stored

170


by the staircase line L(t) whose coordinates are α and β in coincidence with the local maxima and minima of the magnetic field intensity at previous simulation time, i.e. the shape of the interface L(t) is depending on the past extreme values of the magnetic field intensity holding the prehistory of the magnetic material. The staircase line is moving from left to right if the magnetic field intensity is increasing and is moving from top to down if the magnetic field intensity is decreasing. m

β

=

β

h

+1

α

L(t)

γ b = −1

α

-1 +1

j

-1

hc

i

γ b = +1

Fig. 5.5. The Preisach triangle and the staircase line It is assumed that different particles with rectangular hysteresis operator b γ (α, β) have a distribution on the plane of switching fields, which can be expressed by the distribution function (also called weight function) µ(α, β). The magnetization M (t) at time t can be expressed by the expected value of this distribution ZZ µ(α, β) γb(α, β) H(t)/Hm dα dβ, (5.28) M (t) = Ms α≥β

where the distribution function µ(α, β) weights the effect of elementary operators γ b(α, β) and H(t) is the magnetic field intensity. The input of the model is the magnetic field intensity, which is normalized by Hm , the output of the model is the magnetization, which is the value of double integral in (5.28) denormalized by Ms . The distribution function should be normalized as ZZ µ(α, β) dα dβ = 1, (5.29) α≥β

because all hysteresis operator turn to ±1 in saturation state and M must be equal to Ms . In a computer realization, equation (5.28) can be approximated by the weighted sum of elementary operators as M (t) ≃ Ms

N +1 X

N +1 X

i=1 j=N +2−i

µ(αi , βj ) γ b(αi , βj ) H(t)/Hm ,

(5.30)

171


where N is the number of subdivisions in the Preisach triangle. Each point on the half space α ≥ β corresponds to one elementary hysteresis operator only. Here N +1 X

N +1 X

µ(αi , βj ) = 1

(5.31)

i=1 j=N +2−i

must be fulfilled. The simulation of a magnetization process using the above theory can be demonstrated as follows. In demagnetized state (H = 0, M = 0 and B = 0) the positive and the negative magnetized particles keep balance, i.e. the area where γ b(α, β) = −1 and the area where γ b(α, β) = +1 are equal. With increasing the magnetic field intensity the staircase line is moving from left to right and the area of positive magnetized particles is increasing as well. That is why the magnetization is increasing as it is plotted in Fig. 5.6. Above the saturation state, the hysteron assumes positive value over the whole Preisach triangle and M/Ms = 1. 1

1

0.5

γ=−1

M/Ms

β

0.5

0

−0.5

−1 −1

−0.5

γ=+1

−0.5

0

0 α

0.5

1

−1 −1

−0.5

0 H/Hm

0.5

1

Fig. 5.6. The Preisach triangle for the first magnetization curve By decreasing the magnetic field intensity the switching down value of β becomes lower and more domains turn off and align to the opposite directed magnetization. This is the reason of decreasing magnetization. The staircase line is moving from top to down as it is presented in Fig. 5.7, where the remanent state is illustrated. Further decreasing the magnetic field intensity results in decreasing magnetization value. Figure 5.8 shows the simulation of starting a minor loop. The magnetic field intensity is increasing, so the staircase line is moving from left to right, which results in increasing magnetization. Further continuing the magnetization as presented in Fig. 5.9 and in Fig. 5.10 the variation of magnetization can be studied while simulating a higher order minor loop. The classical scalar Preisach model is unable to describe some phenomena, such as the noncongruency [76, 84, 128] and the accommodation [76, 128, 179].

172


1

1

0.5

γ=−1

M/Ms

β

0.5

0

−0.5

−1 −1

−0.5

γ=+1

−0.5

0

0 α

0.5

−1 −1

1

−0.5

0 H/Hm

0.5

1

Fig. 5.7. The Preisach triangle and the upper branch starting from the first magnetization curve and the remanent magnetization

1

1

0.5

γ=−1

M/Ms

β

0.5

0

−0.5

−1 −1

−0.5

γ=+1

−0.5

0

0 α

0.5

1

−1 −1

−0.5

0 H/Hm

0.5

1

Fig. 5.8. Generating a second order minor loop

Minor loops simulated by the Preisach model of hysteresis are congruent, i.e. a minor loop cycling between two fixed values of the applied magnetic field has the same size and shape. The moving model [76,140] and the product model [76,85] can solve this problem. The moving model [140] contains the classical Preisach model with a feedback, which modifies the actual value of the applied magnetic field intensity by a term of δM , i.e. Hk = Hk + δMk−1

(5.32)

at the discrete time of simulation tk and δ is a small positive number (Fig. 5.11). In the product model [85], the rate of change of magnetization with respect to the applied field ∂M/∂H is expressed by a Preisach-like integral formula multiplied by a function R(M ).

173


1

1

0.5

0.5 M/M

β

s

γ=−1

0

−0.5

−1 −1

−0.5

γ=+1

−0.5

0

0 α

0.5

−1 −1

1

−0.5

0 H/Hm

0.5

1

0 H/Hm

0.5

1

Fig. 5.9. Generating a third order minor loop 1

1

0.5

0.5 M/M

β

s

γ=−1

0

−0.5

−1 −1

−0.5

γ=+1

−0.5

0

0 α

0.5

−1 −1

1

−0.5

Fig. 5.10. The hysteresis curve and the Preisach triangle when closing the third order minor loop H

P

H {·}

M

δ

Fig. 5.11. Block representation of the moving model Accommodation of a loop means that it takes many cycles for a minor hysteresis loop to stabilize and it continues until an equilibrium is reached asymptotically. The moving model and the product model also can fulfill this property. Accommodation cannot be simulated by the classical Preisach model, because a minor loop closes at the same point (H, M ), where it was started.

174


B. The distribution function There are some usual and typical forms of the distribution function µ(α, β), which can be found in the literature. The most important ones are listed below. (i) The two-dimensional Gaussian distribution functions can be used efficiently for analytical approximation of the distribution function. The following formula with four parameters (a, b, c and d) can be applied to fit the model to a given major loop [7, 55]:  2 (α+β−d)2  − (α−β−c) − 10a 10b , if α + β ≤ 0, e (5.33) µ(α, β) = 2 (α+β+d)2 −  e− (α−β−c) a 10 10b , if α + β > 0.

The parameters of distribution function (5.33) can be fitted to the hysteresis loops of the 27ZDKH95 magnetic material measured by a single sheet tester. Measurements have been performed at the Bioelectricity and Magnetism Laboratory, Vienna University of Technology [54, 144]. The measurements have been accomplished on a single sheet tester consisting of a U-shaped vertical yoke. The laser scribed, grain oriented specimen with thickness 0.27 mm is produced by Nippon Steel, which dimensions are 500×500 mm2 . The anisotropic magnetic material is separately exposed to sinusoidal waveform of an alternating flux density, first in the rolling direction than in the transverse direction. The measured data consisting of 512 H − B pairs in every curve plotted in Fig. 5.12(a) and in Fig. 5.12(b). Here 3 numbered loops are shown. Two Preisach models have been fitted, one to simulate the hysteresis characteristics in the rolling direction and another one in the transverse direction. The identification process (shown in Fig. 5.13) is based on the objective function v u 512 X 1 u t (BP,i − BM,i )2 + fP , (5.34) fo = 512 i=1 2

2 3

2

1

3

1

1

2

B [T]

B [T]

1

0

−1

−2 −100

0

−1

−50

0 H [A/m]

50

100

(a) Hysteresis loops in the rolling direction

−2 −700

−350

0 H [A/m]

350

700

(b) Hysteresis loops in the transverse direction

Fig. 5.12. Hysteresis loops of the 27ZDKH95 material

175


where BP,i is the magnetic flux density simulated by the Preisach model, BM,i is the measured magnetic flux density and fP = fP (△Hc , △Br ) is a penalty function to minimize the difference between the simulated and the measured coercive magnetic field intensity and the simulated and the measured remanent magnetic flux density, and fo has been minimized by nonlinear least-squares method starting from random values of the parameters selected from the intervals shown in Table 5.1. The table shows the value of parameters determined by the minimization algorithm.

BP

Preisach model

Modification of parameters H

Measurements

Minimization algorithm

BM

Fig. 5.13. The identification procedure

Table 5.1. Parameters of the Preisach distribution function (5.33) fitted to measured characteristics of Fig. 5.12(a) and Fig. 5.12(b)

a b c d

Rolling direction interval value [−3, +3] −0.84683 [−3, +3] −0.54231 [−2, +2] 0.42961 [0, +2] 0.94599

Transverse direction interval value [−2, 0] −0.20796 [−3, 0] −1.07260 [−1, 0] −0.61348 [−1, 0] −0.60119

Comparisons between hysteresis curves simulated by the identified Preisach model and measurements are plotted in Fig. 5.14. The simulation error can be expressed by the relative difference △B = 100

BM − BP . Bs

(5.35)

It is plotted in Fig. 5.15 for hysteresis loops 1, 2 and 3 in the rolling and in the transverse directions and it shows the simulation error of the direct hysteresis model, whose input and output are H and M or B, respectively. The simulation error also can be expressed as △H = 100

HM − H P . Hm

(5.36)

176


2

1.5

Measured Simulated

Measured Simulated

0.75 B [T]

B [T]

1

0

−1

−2 −100 2

−0.75

−50

0 H [A/m]

50

−1.5 −700

100

B [T]

B [T]

700

0 H [A/m]

350

700

0 H [A/m]

350

700

0

−0.75

−50

0 H [A/m]

50

−1.5 −700

100

−350

1.5

Measured Simulated

Measured Simulated

0.75 B [T]

1 B [T]

350

0.75

−1

0

−1

−2 −100

0 H [A/m]

Measured Simulated

0

2

−350

1.5

Measured Simulated

1

−2 −100

0

0

−0.75

−50

0 H [A/m]

50

100

−1.5 −700

−350

Fig. 5.14. Comparison between simulation and measurements in the rolling direction and in the transverse direction Here Hm is the maximum value of magnetic field intensity when magnetization curve is saturating. This relative difference can be seen in Fig. 5.16 for hysteresis loops 1, 2 and 3 in the rolling and in the transverse direction, which shows the simulation error of the inverse hysteresis model, when the input of the model is B and the output is H.

177


50

20

Descending Branch Ascending Branch

10 ∆ B [%]

∆ B [%]

25

0

−25

−50 −100

−50

0 H [A/m]

50

−20 −700

100

∆ B [%]

∆ B [%]

0 H [A/m]

350

700


10

0

−25

0

−10

−50

50

0 H [A/m]

50

−20 −700

100

−350

20


0 H [A/m]

350

700


10 ∆ B [%]

25 ∆ B [%]

−350

20


25

0

−25

−50 −100

0

−10

50

−50 −100


0

−10

−50

0 H [A/m]

50

100

−20 −700

−350

0 H [A/m]

350

700

Fig. 5.15. The difference in B between the measured and the simulated hysteresis loops in the rolling direction and in the transverse direction To compare the differences between measurements and simulation, the hysteresis losses w in unit volume H from the enclosed area of the hysteresis characteristics can also be determined as w = B H dB. The difference between the measured and the simulated hysteresis loops with respect to the losses (wM and wP , respectively) can be determined as

178


△w = 100

wM − wP , wM

(5.37)

and this can be found in Table 5.2 for the three analyzed hysteresis loops.

40

20


10 ∆ H [%]

∆ H [%]

20

0

−20

−40 −2

−1

0 B [T]

1

−20 −1.5

2

∆ H [%]

∆ H [%]

0 B [T]

0.75

1.5


10

0

−20

0

−10

−1

0 B [T]

40

1

−20 −1.5

2

−0.75

0 B [T]

20


0.75

1.5


10 ∆ H [%]

20 ∆ H [%]

−0.75

20


20

0

−20

−40 −2

0

−10

40

−40 −2


0

−10

−1

0 B [T]

1

2

−20 −1.5

−0.75

0 B [T]

0.75

1.5

Fig. 5.16. The difference in H between the measured and the simulated hysteresis loops in the rolling direction and in the transverse direction

179


Table 5.2. Error in hysteresis loss between measured and simulated loops Loop number 1 2 3

△w

Rolling direction 8.78 % 4.98 % 4.72 %

Transverse direction 22.98 % 24.54 % 5.21 %

The difference of hysteresis losses between the measured and the simulated major loops is less than 10 % (4.72 % and 5.21 % for the major hysteresis loop of the rolling and of the transverse directions, respectively). This is important, because the so-called excess losses associated to the micro-eddy current losses generated by the local rotation of the domains cannot be approached by Maxwell’s equations. The excess losses are depending on the magnetic domain rotation, on the microstructural inhomogeneity and so on. (ii) The following distribution function is based on the property that the concentric minor loops are symmetrical. The Preisach distribution function can be expressed as a product of two one-dimensional distributions [25, 173], i.e. µ(α, β) = f (−α)f (β).

(5.38)

In the paper [175] the following expression is proposed to represent the one-dimensional distribution: f (x) =

ae−bx , (1 + ce−bx )2

(5.39)

which has the main advantage that the double integration of the distribution function, i.e. the Everett function (see in the next section) can be expressed in closed analytical form as bβ

bα

(1+ce )(c+e ) 2 bβ bα bβ bα a2 (c −1)(e −e )−(c + e )(1+ce )log (1+cebα )(c+ebβ ) E(α, β) = 2 , b (c2 − 1)2 (c + ebβ )(1 + cebα )

(5.40)

and this results in a fast generation of hysteresis model. (iii) The other distribution function, which can be applied is the Lorentzian distribution, µ(α, β) =

1+

N

2 ( α−a σa )

2 1 + ( β+a σa )

,

(5.41)

where N is a normalization factor, a and σ are two unknown parameters, which must be determined in the identification process [70, 71]. (iv) The combination of Gaussian and Lorentzian distribution is a good chance to accurately approximate the measured Preisach distribution [58]. The Preisach function in the rolling direction of grain oriented sheets and in both directions of nonoriented ones can be approximated by the expression [58] µ(α, β) =

d c , + [1 + a2 (α + b)2 ][1 + a2 (β − b)2 ] enh2m ep(hc +q)2

(5.42)

180


while the Preisach distribution function in the transverse direction of grain oriented sheets can be approximated by [58] c [1 + a2 (α + b)2 ][1 + a2 (β + b)2 ] c + [1 + a2 (α − b)2 ][1 + a2 (β − b)2 ] d + nh2 p(h +q)2 , e me c

µ(α, β) =

(5.43)

where hm =

α+β 2

(5.44)

is the interaction field, and hc =

α−β 2

(5.45)

is the coercive field, moreover a, b, c, d, n, p and q are parameters, which can be determined by fitting the resulting Preisach distribution function or the major hysteresis loop with mean square error minimization algorithm. C. The Everett function Using the Everett function leads to a very favorable implementation of the Preisach model [76, 128]. The discrete Everett table can be determined from the measured first order reversal curves at very low frequency (f < 1 Hz) avoiding the computation of the integral in (5.28) or the sum in (5.30), which are time consuming operations. Applying Everett function results in a comfortable simulation technique, because it is easy to approximate the Everett function by using measurements, E(α, β) =

1 (Mα − Mα,β ), 2Ms

(5.46)

where Mα is a reversal magnetization point in the major hysteresis loop corresponding to the magnetic field intensity α and Mα,β is the value of magnetization in a reversal curve starting from the reversal point (α, Mα ), when the magnetic field intensity is equal to β. A reversal curve with given points can be seen in Fig. 5.17. After measurements, the plane α − β can be normalized by Hm . The Preisach distribution function can be expressed from the Everett function as [128] µ(α, β) =

∂ 2 E(α, β) , ∂α ∂β

while the Everett function can be determined from the distribution function ZZ E(α, β) = µ(α′ , β ′ ) dα′ dβ ′ . α≥β

(5.47)

(5.48)

181


1

M/Ms

0.5

Mα,β

0

β

Mα

α

−0.5

−1 −1

−0.5

0 H/Hm

0.5

1

Fig. 5.17. A first order reversal curve to measure the Everett function

1

1

0.75

0.75 µ(α ,β)

E(α,β)

The main disadvantage of (5.47) is that the differentiation must be performed numerically from the measured Everett function, which amplifies the measurement noise. An example – generated by the Preisach model – for a calculated Everett function E(α, β) and the corresponding distribution function µ(α, β) are plotted in Fig. 5.18(a) and in Fig. 5.18(b), respectively.

0.5 0.25

0.5 0.25

0 1

0 1 1

0.5 β

1

0.5

0.5

0

0

−0.5

−0.5 −1 −1

β

α

0.5

0

0

−0.5

−0.5 −1 −1

(a) The Everett function

α

(b) The Preisach distribution function

Fig. 5.18. The Everett function and the Preisach distribution function Knowing the Everett function, the following expression can be used to determine the magnetization [128, 140]: ! n X M = Ms −E(α0 , β0 ) + 2 [E(αk , βk−1 ) − E(αk , βk )] , (5.49) k=1

182


where αk and βk are the increasing and the decreasing sequences of the normalized magnetic field intensity applied to the sample and stored by using the staircase line, moreover α0 and β0 denote the first turning point in the first magnetization curve (e.g. α0 = 1, β0 = −1 in saturation state) and n is the number of stairs in staircase line. The Everett function can be expressed using symmetric minor loops as well via the equation (5.49) [116].

5.2.2 Pseudocode for the scalar Preisach model A possible implementation of the scalar Preisach model using the Everett function is the following. Here staircase line is represented by the matrix α0 α1 · · · αn , (5.50) Lk = β 0 β 1 · · · βn i.e. Lk is a matrix with two rows and the number of columns is changing. Input : Hk , Hk−1 , Bk−1 , Lk−1 , Inck−1 , Hmax , Mmax , Bmax , Output : Bk ,

∂B ∂H k ,

µ0 = 4π · 10−7 .

1) Normalization of input: hk =

Hk Hmax ,

hk−1 =

Hk−1 Hmax ;

2) Setting up the staircase line Lk = Lk−1 , n = number of columns in Lk , a) if |hk | ≥ 1 then Lk = [ ] (empty),

b) else if hk < hk−1 then (magnetic field is decreasing) i) if n = 0 and not Inck−1 then Inck = false, ii) else if Inck−1 then · if n = 0 then Lk = [Lk , [|hk−1 |; −|hk−1 |]], · else Lk = [Lk , [hk−1 ; Lk (2, n)]], iii) Inck = false, iv) Delete columns of Lk if hk ≤ Lk (2, :),

c) else if hk > hk−1 then (magnetic field is increasing)

i) if n = 0 and Inck−1 then Inck = true, ii) else if not Inck−1 then · if n = 0 then Lk = [Lk , [|hk−1 |; −|hk−1 |]], · else Lk = [Lk , [Lk (1, n); hk−1 ]], iii) Inck = true, iv) Delete columns of Lk if hk ≥ Lk (1, :),

183


d) n = number of columns in Lk ; 3) Using the equations: Mk = −E(α0 , β0 ) + 2

n X i=1

∂M ∂E(α, β) , =2 ∂H ∂α α=hk ∂E(α, β) ∂M =2 , ∂H ∂β

[E(αi , βi−1 ) − E(αi , βi )] ,

(5.51)

if hk > hk−1 , (5.52) if hk < hk−1 ,

β=hk

4) and Bk = µ0 (Hk Hmax + Mk Mmax ). M ∂B max = µ0 1 + ∂M 5) ∂H ∂H k Hmax . k

5.2.3 The vector model The scalar hysteresis can be regarded as a special case of more sophisticated phenomenon where the magnetic field intensity, the magnetization and the induction are vectors. It is supposed that the magnetic field intensity vector H is restricted to vary along a specified direction and the output of the hysteresis model is the magnetization in the same direction. In this case the scalar hysteresis characteristics can be obtained. This situation can be imagined inside a toroidal shape core excited by a coil. If the simulated arrangement has a more complicated shape, e.g. it contains corners, then the vector hysteresis model has to be applied, because the magnetic field intensity vector, the magnetization vector and the induction vector are not parallel to each others, especially close to sharp corners. This can be imagined for example, in the armature core of rotating machines, or in the case of yokes of transformers. The research in this field resulted in the vector Preisach model [19, 39, 101, 114, 119– 121, 126–128, 130, 131, 133, 144, 156, 174, 183]. The vector Preisach model proposed by Mayergoyz is built up as a superposition of continuously distributed scalar Preisach models [128]. In two-dimensional case the magnetization vector can be expressed as Z π/2 eϕ H {Hϕ } dϕ, (5.53) M= −π/2

where Mϕ = H {Hϕ } is the scalar magnetization in the direction eϕ , i.e. the expression Z π/2 M= eϕ −π/2



 × Ms,ϕ

ZZ

α≥β



 µ2D (α, β) γb(α, β) [H · eϕ / Hm,ϕ ]dα dβ  dϕ

(5.54)

184


results in the magnetization vector when applying scalar Preisach model to represent H {·}. Here µ2D (α, β) is the vector Preisach distribution function, Ms,ϕ and Hm,ϕ is the saturation magnetization and the corresponding magnetic field intensity in the direction ϕ. The input and the output of the model are supposed to be normalized and denormalized here. In 3D, the following integral can be obtained to determine the magnetization vector: Z π/2 Z π/2 M= eϕ,ϑ H {Hϕ,ϑ } dϕ dϑ, (5.55) −π/2

−π/2

where Mϕ,ϑ = H {Hϕ,ϑ } is the scalar magnetization in direction eϕ,ϑ . The expression of magnetization vector in three dimensions can be evaluated by substituting the scalar magnetization (5.28) into the equation (5.55), i.e. Z π/2 Z π/2 eϕ,ϑ M= −π/2



−π/2

 ×Ms,ϕ,ϑ

ZZ

α≥β



(5.56)

 µ3D (α, β) γ b(α, β) [H · eϕ,ϑ /Hm,ϕ,ϑ ] dα dβdϕ dϑ,

where µ3D (α, β) is the 3D vector Preisach distribution function, Ms,ϕ,ϑ and Hm,ϕ,ϑ is the saturation magnetization and the corresponding magnetic field intensity in direction ϕ, ϑ. The identification process can be worked out by applying either the Everett function or the Preisach distribution function both for isotropic and anisotropic case. In the isotropic case the same Everett function or the same distribution function, i.e. the same hysteresis characteristics can be measured in all directions. This is not true in anisotropic case, measured data are depending the direction [19, 101, 128, 155, 156, 174]. The identification process of the two-dimensional model is based on the integral equation [128] Z π/2 F (α, β) = cos ϕ E(α cos ϕ, β cos ϕ) dϕ, (5.57) −π/2

where F (α, β) and E(α, β) are the measured scalar Everett function and the unknown vector Everett function. The identification process of the two-dimensional model based on the Preisach distribution function can also be applied [128], Z π/2 cos3 ϕ µ2D (α cos ϕ, β cos ϕ) dϕ, (5.58) µ(α, β) = −π/2

where µ(α, β) is the measured scalar distribution function, and µ2D (α, β) is the unknown vector distribution function. These integral equations can be solved only numerically, because no analytical solution is available [101, 128, 156]. These equations will be repeated and extensively studied in sections 5.3.2 and 5.3.3. There an identification procedure can also be read.

185


5.2.4 The classical scalar Preisach model via neural networks A. A brief introduction to neural networks A short introductory section about neural networks must be inserted here to understand the use of this technique in the field of hysteresis modeling. A more detailed description can be found e.g. in [28, 38, 41], where the structure, the training methods and a wide range of applications can be found. Artificial neural networks (NNs) are parallel, information-processing systems, which are implemented by hardware (like Hopfield-type NN, HNN, Cellular Neural Network, CNN) or software (simulation tools, e.g. NN Toolbox of the software package MATLAB and so on). Applying the technique of NNs is a quite new and challenging field of research motivated by neuroscience, the study of the human brain and the nervous system. The brain’s capacity of self-accommodation, powerful thinking, remembering and problem solving capabilities have inspired many scientists to attempt computer modeling of its operation. There is an increasing number of different types of NNs, like feedforward NNs, Adaptive Resonance Theory (ART) networks, Cerebellar Model Articulation Controllers (CMAC), Principal Component Analysis (PCA) networks, CNN, HNN, and so on. In the field of modeling the group of feedforward-type NNs can be used efficiently in the simulation of magnetic hysteresis. Feedforward NNs can be applied as an alternative mathematical tool for universal function approximation. This application is based on Kolmogorov–Arnold theorem, which is the following [41]. For each positive integer n, there exist 2n + 1 continuous functions Φ1 , Φ2 , · · · , Φ2n+1 , mapping [0, 1] into the real line and having the additional property that for any continuous function f of n real variables, there is a continuous function Ψ of one variable on [0, n] into the real line such that ! 2n+1 n X X Ψ Φq (xp ) , (5.59) f (x1 , x2 , · · · , xn ) = q=1

p=1

for all values of x1 , x2 , · · · , xn in [0, 1]. The formulation (5.59) can be represented by feedforward type NNs with at least two hidden layers with nonlinear activation function. The output of an individual neuron (processing element, PE) y is the output of the so-called activation function (transfer function) ψ(s), i.e. y = ψ(s) (see Fig. 5.19). The input of this transfer function is the linear combination of I inputs, x = [x1 , x2 , · · · , xI ]T and the bias b of the PE (Fig. 5.19), s=

I X

wi xi + b = wT x + b,

(5.60)

i=1

where vector w contains the weighting parameters of a neuron, w = [w1 , w2 , · · · , wI ]T . The transfer functions are generally nonlinear, continuous and differentiable functions. There are some widely used activation functions (e.g. hard limit, linear, saturating linear,

186


radial basis function and so on), but the bipolar sigmoidal function (hyperbolic tangent function) is the most widely used one, ψ(s) =

1 − e−2s . 1 + e−2s

(5.61)

This function transfers its input space s ∈ R into the interval [−1, · · · , +1]. A feedforward NN can be built by using such PEs grouping them into layers as it can be seen in Fig. 5.19. The inputs of the neurons in layer i are the weighted sum of the outputs of the neurons in layer i − 1. In this case, a multi input single output (MISO) system with L layers has been shown for simplicity. input layer

x1

hidden layers W

(1)

P

x2

W(i)

x1

w

xI b

···

···

···

x3

W(L) t

···

xI △W

output d

···

x2

y ψ

Training algorithm

+ P

–

λ

Fig. 5.19. Structure and training of feedforward type NNs

The relation (5.59) can be realized by a NN with two layers and the number of neurons is n and 2n + 1 in the first and in the second layer, respectively. The functions Ψ and Φ are selected to be sigmoidal one. After selecting the activation functions of the individual neurons and the number of layers and neurons in the layers, the only degree of freedom left is setting the value of weights and biases of the NN. Weights and biases of neurons in a layer are collected in the weight matrix W. Training is an iterative and convergent algorithm, which modifies the weights and the biases until a given value of a suitably defined error ε (e.g. mean square error, MSE, sum squared error, SSE, etc.) between the desired output value (target) t and the output d of the NN according to the given input vector x is reached. This training method is based on the training sequence, τ (N ) = {(xk , tk ), k = 1, ..., N },

(5.62)

which may be a set of measured data, that should be approximated by the NN, i.e. tk = f (xk ), k = 1, ..., N.

(5.63)

187


Function f (·) represents the measured functional relationship between the input-output patterns and N denotes the number of measured data pairs. According to the training sequence τ (N ) , the aim of training is to find the optimal weight coefficients and biases of the network denoted by the matrix W∗ , i.e. to minimize the function W∗ : min W

N 1 X 2 (tk − N (xk , W)) , N

(5.64)

k=1

where dk = N (xk , W) is the function approximated by the NN. Once training is finished, NN – with given number of neurons and given structure – is able to approximate the function f (x) under the given value of error function ε, i.e. ∃W∗ : kN (x, W∗ ) − f (x)k < ǫ.

(5.65)

A trained NN is able to generalize the trained information. This means that not only the discrete set of the training sequence can be represented, but other input-output data set can be approximated, which are not included in the training sequence, because the NN is a continuous function approximator. Training can be performed by any optimization technique. The aim of optimization is to minimize an error function ε = ε(λ) depending on the individual values λ = t − d, ∂ε/∂W = ∇ε → 0.

(5.66)

Weights are adapted in every iteration step k as Wk+1 = Wk + △Wk .

(5.67)

The value of △Wk can be formulated in many ways. The backpropagation (BP) training algorithm is the most widely used one, but the faster Levenberg–Marquardt iteration technique as a modification of the BP method is preferred with the adaptation rule △W = −(JT J + αI)−1 JT e,

(5.68)

where J is the Jacobian matrix formed by the derivatives of each error to each weight, I is the unit matrix, α is the Levenberg parameter and e is the error vector [192]. The operation of a MISO feedforward NN model can be expressed mathematically as   ! X (L) X (L−1) X (1) Wi ψ  Wij ··· ψ N (x, W) = ψ  Wkm xm  , (5.69) i

j

m

where ψ(·) denotes the transfer functions of the neurons. Using the chain rule, the derivatives of the formulation (5.69) with respect to any input of the NN can be expressed in analytical form. This is the general form of feedforward type NNs; although not all models share all of the characteristics. Individual neurons can have numerous inputs and can send data to numerous other PEs. There is such a large variety of NNs.

188


B. Neural Preisach models The memory storage mechanism of the ferromagnetic materials can be described by means of the Preisach triangle. The idea of staircase line can be realized by applying NNs, too. A discrete set of play operators Ψi = Ψ(t, hic )

(5.70)

can be used to simulate the memory mechanism of magnetic materials [160], moreover a feedforward NN can be trained to store the experimentally measured data set B = B{Ψ(t, hic )}.

(5.71)

This hibrid technique results in a neuro-Preisach model. The block diagram of this model can be seen in Fig. 5.20. The authors have considered a NN with I = 20 (i = 0, ..., I − 1) inputs and one hidden layer with 40 neurons. Each neuron employs a bipolar sigmoidal activation function. The training data set has contained 50 H − B pairs in a nested loop. Five first order reversal curves have been measured in one magnetic sample and seven in another sample. They have demonstrated the good quality of the predictions of the model by comparing simulation results and measured symmetrical minor loops. This model has also been applied to simulate noisy and incomplete measurement data [186]. It is very difficult to identify the classical Preisach model when the reversal curves have been loaded with noise, but NNs are able to filter the noise during the learning process. Preprocessing of noisy measured data is not required in this case.

Ψ0

Ψi

discrete set of Play operators

ΨI−1

B ···

β

···

H

h

hc

m

α

neural network

Fig. 5.20. Implementation via feedforward NN of the system with Preisach memory

A possible way to represent the main features of the classical scalar Preisach model applying NNs can be found in [1]. The block representation of the Preisach model has been realized by a NN structure with bipolar sigmoidal activation function, if the hysteresis operators are represented by elementary rectangular loops γ b(α, β) and the distribution function µ(α, β) has been characterized by the weights of the network as illustrated in Fig. 5.21. This means that a large number of inputs (few hundred) and consequently a large number of unknown weights are required for the model, so the

189


authors have grouped the rectangular operators into a smaller, but more sophisticated set of 20 hysteresis operators having different widths and 10 neurons in one hidden layer. The measured data set contains 304 H − B pairs in the form of a decreasing oscillating sequence. After training, higher order minor loops can be demonstrated and compared with measured data. NN

H

B ···

···

··· hysteresis operators

Fig. 5.21. Operator NN realization of the classical Preisach model

The approximation of the Preisach distribution function or the Everett function can be realized by a feedforward NN with 2 inputs (α and β) and one output (µ(α, β) or E(α, β)) [34]. The output of a NN having continuous and differentiable activation functions can be differentiated with respect to its input and the differential susceptibility χdif f = ∂M/∂H is a continuous function of H, which can be expressed in analytical form. Applying the Newton–Raphson iteration technique in numerical field calculation problems leads to faster convergence, but this technique requires the differential susceptibility. It is also shown in the paper [34] that the integral form (5.28), as well as the energy losses can also be worked out by an analytical expression when NNs are used. It gives a computationally efficient model. A vectorial realization based on NN technique has been investigated and performed in [2]. After some mathematical formulations of the basic equations of the anisotropic 2D vector Preisach model and applying Fourier expansion, the vector Preisach model can be identified by a two layer feedforward type NN. The input layer consists of twice as many nodes as the number of elementary hysteresis operators necessary to construct a reasonably accurate hysteresis model. Only 36 rectangular elementary hysteresis operators have been used in the model. The identification of the model is based on 706 H − M pairs. Measured first order reversal curves have been compared with predictions of the model.

190


5.3 The neural network based hysteresis model 5.3.1 The scalar model This section presents a NN based scalar hysteresis model constructed on the function approximation ability of the feedforward type NN. A knowledge-based system represents the behavior of the hysteresis characteristics. Finally a hibrid model is introduced as a mathematical representation of the hysteresis characteristics [101, 109–113, 117, 118]. A. Training sequence and preprocessing of measured data When using NNs for function approximation, it may be difficult to take into account the multivalued property of the hysteresis characteristics. To overcome this problem, a variable denoted by ξ associated with the measured first order reversal curves can be introduced. The first magnetization curve and a set of the first order reversal curves have to be measured to build this model. Systematically generated ascending or descending external magnetic fields with the relation α−1 α+1 + sin(2πf t + π/2) (5.72) H(t) = Hm 2 2 can be applied to measure the first order reversal curves. Here Hm is the maximum value of the applied magnetic field intensity, f is the frequency of excitation, α = k/n and k ∈ [n, −n] is an integer. The number of generated reversal curves is 2n + 1. These measurements have to be performed at very low frequency (f < 1 Hz) to decrease the effect of eddy currents and other dynamic properties of the material under test. Time variation of the magnetic field intensity generated by the function (5.72) in case of n = 8 can be seen in Fig. 5.22(a). Here f = 1 Hz. The resulting magnetization and the hysteresis characteristics with the first order reversal curves are plotted in Fig. 5.22(b) and in Fig. 5.22(c), respectively. Similar results can be obtained, when a triangular waveform is applied. It is noted that before measuring a reversal curve, it is advisable to generate some major loops to get a loop stable after accommodation. After measurements, it is useful to work with relative quantities, H/Hm and M/Ms or B/Bm , so that the input and the output of the model can run in the interval [−1, 1], where the subscript s denotes the saturation state and m is the appropriate maximum value. If a new variable ξ is added to the measured and normalized first order reversal curves, multivalued characteristics can be represented by a single valued, two-dimensional surface with two independent variables H and ξ. It is enough to measure either the ascending or the descending branches because of the symmetry of the magnetic hysteresis characteristics. The upgrade part of hysteresis characteristics can be described by a positive real parameter (asc)

ξ (asc) = 1 −

1 + Htp /Hm , 2

(5.73)

191

1

1

0.5

0.5 M/M s

H/Hm

5.3. THE NEURAL NETWORK BASED HYSTERESIS MODEL

0

−0.5

−1 0

0

−0.5

4

8 t [s]

12

−1 0

16

(a) Magnetic field intensity

4

8 t [s]

12

16

(b) Magnetization

1

M/M

s

0.5

0

−0.5

−1 −1

−0.5

0 H/H

0.5

1

m

(c) Hysteresis loop with reversal curves

Fig. 5.22. Generating a sequence of first order reversal curves

and the downgrade part can be approximated by a negative real parameter (desc)

/Hm 1 + Htp . (5.74) 2 These parameters can be calculated for a given transition curve starting from a turning point Htp and ξ (asc) ∈ [0, 1], ξ (desc) ∈ [−1, 0]. The values ±1 represent the major curve, e.g. −1 denotes the descending curve of the measured major loop. This kind of preprocessing can be applied on any shape of hysteresis characteristics as it can be seen in Fig. 5.23(a)-5.23(d). After preprocessing, the aim is to find the approximating nonlinear functions, which can be realized by feedforward type NNs trained by the Levenberg–Marquardt BP method. ξ (desc) = −

192


1 1 0.5 M/M

M/Ms

s

0.5

0

0 −0.5 −1 0 −0.25

−0.5

−1 −1

−0.5

0 H/Hm

0.5

1 0.5

−0.5 −0.75 ξ

1

0 −0.5

H/Hm

−1 −1

(a) First order reversal curves for classical type of hysteresis

(b) Preprocessed reversal curves

1 1 0.5 M/M

M/Ms

s

0.5

0

0 −0.5 −1 0 −0.25

−0.5

−1 −1

−0.5

0 H/Hm

0.5

1

(c) First order reversal curves for pinning type of hysteresis

1 0.5

−0.5 ξ −0.75

0 −0.5 −1 −1

H/Hm

(d) Preprocessed reversal curves

Fig. 5.23. Preprocessing can be performed on any shape of characteristics (only the descencing branches have been used)

Two NNs have to be used, one to approximate the first magnetization curve and another one for the preprocessed first order reversal branches. The first magnetization curve (about 40 data pairs) can be approximated by a NN with 8 neurons in one hidden layer (with one input, H and one output, M or B) and the preprocessed first order transition curves (about 500–800 data pairs give accurate result depending on the shape of the characteristics) can be simulated by a NN with 2 inputs (H and ξ) and one output (M or B) built by 7, 11 and 6 PEs in three hidden layers. The structure of the NNs and the number of hidden layers and neurons have been set after some trials. The transfer function of the neurons is selected to be a bipolar sigmoid function.

193


The training of the two NNs took about 20 minutes on a Celeron 566 MHz computer (192 Mbyte RAM) using the Neural Network Toolbox of MATLAB. When applying a training sequence described above to identify the first magnetization curve and the preprocessed first order reversal curves, the stopping criterion M SE = 10−6 can be reached by not more than 500 and 1200 training epochs in the two NNs. The developed measurement system, the identification results and some comparison between measured and simulated curves can be found in section 5.4. B. Operation of the model As it has been stated in the previous section, this NN based hysteresis model consists of two trained NNs. The first magnetization curve and the first order reversal branches are stored in NNs, but a memory mechanism must be realized by an additional algorithm based on heuristics. The block representation of the NN based scalar hysteresis model can be seen in Fig. 5.24 and described below.

Hk

Knowledge-base about hysteresis characteristics

TDL Hk−1

If-then type rules for neural network selection

ξ MEMORY

...

Hk

...

Hk

Correction, selection

M = H {H}, B = B{H}

Mk , Bk

η

Fig. 5.24. The block representation of the scalar model

In general, the actual value of magnetization (or the magnetic induction) is depending on the actual value of the magnetic field intensity and the prehistory of the magnetic material. The normalized magnetization or the normalized magnetic flux density at the simulation step k yielded by the NN based model is constructed on the actual value of the normalized magnetic field intensity Hk /Hm , the appropriate value of the parameter ξ and a set of turning points stored in the memory of the model. The knowledge-base is the main part of the implemented NN model. It contains ’if-then’ type rules about the hysteresis phenomena and about the properties of hysteresis characteristics. This block controls the other blocks of the model. The operation of this model is based on a set of turning points in the ascending and in the descending branches (asc) (desc) and Htp , respectively. Turning points are stored in the memory denoted by Htp of the model, that is an array with the elements [Htp , Mtp , ξ]T or [Htp , Btp , ξ]T , denoted by MATRIX(asc) and MATRIX(desc) for the ascending and the descending curves

194


(see a detailed example at the end of this section). Turning points can be detected by the evaluation of a sequence of {Hk−1 , Hk } generated by a tapped delay line (TDL). After detecting a turning point Htp = HST ART = Hk−1

(5.75)

and storing it in the last column of the appropriate array, the aim is to select a transition curve with the right value of ξ for the detected turning point (Htp , Mtp ) or (Htp , Btp ), where Mtp = MST ART = Mk−1 ,

or Btp = BST ART = Bk−1 .

(5.76)

This task can be solved by the regula-falsi method. There are some basic rules implemented in the knowledge-base as ’if-then’ type rules. First, a simple rule about the behavior of the minor loops has been determined: they must be closed as illustrated in Fig. 5.25(a). Therefore the accommodation phenomena cannot be simulated yet. After detecting a turning point, the algorithm for a general minor loop can be summarized as follows. (i) If MATRIX(desc) has more than one column and the normalized magnetic field intensity Hk is increasing at the k th simulation step, i.e. Hk > Hk−1 , then the (desc) actual minor loop must be closed at the last stored value of HGOAL = Htp , (desc) which can be found in the last column of MATRIX . (ii) If MATRIX(asc) has more than one column and the normalized magnetic field intensity Hk is decreasing at the k simulation step, i.e. Hk < Hk−1 , then the actual (asc) minor loop must be closed at the last stored value of HGOAL = Htp , which can (asc) be found in the last column of MATRIX . (iii) The value of the normalized magnetization Mk yielded by the NN model at HGOAL (desc) must be equal to MGOAL = Mtp stored in the last column of MATRIX(desc) . (iv) The value of the normalized magnetization Mk yielded by the NN model at HGOAL (asc) must be equal to MGOAL = Mtp stored in the last column of MATRIX(asc) . The last two rules are the condition for closing a minor loop (and these are true for magnetic induction as well). The data sets of reached turning points where the actual minor loop is closing must be cleared from the memory by deleting the last columns of the arrays. The first column of arrays in the memory containing the parameters of the normalized major loop (i.e. [+1, +1, −1]T and [−1, −1, +1]T) cannot be cleared. The memory mechanism is very similar to the staircase line of the classical Preisach model. The results of simulation have highlighted that the value of normalized magnetization (or induction) yielded by the relevant NN and the adequate value of the normalized magnetization (or induction) chosen from the appropriate column of the memory at the value HGOAL are not equal. Therefore a correction function η = η(H, M ) (η = η(H, B)) must be introduced to eliminate this deviation while closing a minor loop. The value

195

1

1

0.5

0.5 M/Ms

M/Ms


0

−0.5

−1 −1

0

−0.5

−0.5

0 H/Hm

0.5

1

(a) Minor loops are closed

−1 −1

−0.5

0 H/Hm

0.5

1

(b) Symmetric minor loops

Fig. 5.25. Rules of minor loops in the knowledge-base of the model

of η can be calculated as the difference between the required value of magnetization (induction) and the response of the actual NN at the value of the magnetic field intensity HGOAL when an actual minor loop is closed: η = MGOAL − M (N N) (HGOAL , ξST ART ) ,

(5.77)

where ξST ART is the value of the parameter ξ at the actual value of HST ART . The parameter η has to be calculated beforehand when opening a minor loop and must be applied increasing linearly during the magnetization process from zero (when the minor loop is opening) to the given value η (when the minor loop is closing). An additional precept must be used as formulated in the following rule when symmetric hysteresis loops are assumed (see Fig. 5.25(b)). After the detection of a turning point and an ascending (descending) curve has been chosen, furthermore MATRIX(desc) (MATRIX(asc) ) has only one column, then HGOAL = −HST ART ,

and MGOAL = −MST ART .

(5.78)

If MATRIX(desc) or MATRIX(asc) has been reduced to a single column array after the wiping out process, then saturation is assumed: HGOAL = 1, MGOAL = 1,

or HGOAL = −1, MGOAL = −1,

(5.79)

depending on the direction of the magnetic field intensity. These two rules are also valid if the output of the model is the magnetic flux density. Illustration. The following illustration shows the above mentioned rules. It is supposed that the magnetic material is in demagnetized state, i.e. H = 0, M = 0 and B = 0. Starting the magnetization from demagnetized state with increasing the magnetic field intensity, the magnetization is increasing as well and the NN approximating the first

196


magnetization curve results the output of the model (Fig. 5.26(a)). The output of the model is equal to M = Ms = 1 above the saturation. Decreasing the magnetic field intensity – as it can be seen in Fig. 5.26(a) –, the NN approximating the preprocessed first order reversal curves must be chosen after calculating the appropriate value of ξ. The coordinate of the turning point has to be stored in the memory of model,     1 0.5 −1 −0.5 (desc) (asc) = 1 =  −1 −0.95  , 0.95  , MATRIX MATRIX −1 −0.75 1 0.75 i.e. HGOAL = −0.5, MGOAL = −0.95. The simulated hysteresis loop is a symmetric minor loop. Generating a minor loop by increasing the magnetic field intensity as shown in Fig. 5.26(b) results the following modification of the memory:   1 0.5 0.95  , MATRIX(desc) =  1 −1 −0.75   −1 −0.5 0 MATRIX(asc) =  −1 −0.95 0.47  , 1 0.75 0.51 i.e. HGOAL = 0.5, MGOAL = 0.95 and ξ = 0.51. Figure 5.26(c) shows the closing process of a minor loop. In this case   1 0.5 0.2 (desc) MATRIX = 1 0.95 0.7  , −1 −0.75 −0.67   −1 −0.5 0 (asc) MATRIX =  −1 −0.95 0.47  , 1 0.75 0.51

i.e. HGOAL = 0, MGOAL = 0.47 and ξ = −0.67 represents the available H − M pairs. After closing a simple minor loop, the last column of the arrays must be cleared, i.e.   1 0.5 0.95  , MATRIX(desc) =  1 −1 −0.75   −1 −0.5 MATRIX(asc) =  −1 −0.95  , 1 0.75

and HGOAL = −0.5, MGOAL = −0.95, ξ = −0.75 again. The resulting hysteresis curve can be seen in Fig. 5.26(d). Further decreasing the magnetic field intensity below HGOAL clear the memory of model and HGOAL = −1, MGOAL = −1, but ξ is the same (Fig. 5.26(e)). Further continuing the magnetization process the whole characteristics can be realized (Fig. 5.26(f)).

197

1

1

0.5

0.5 M/Ms

M/Ms


0

−0.5

0

−0.5

−1 −1

−0.5

0 H/Hm

0.5

−1 −1

1

1

1

0.5

0.5

0

−0.5

0.5

1

0

−0.5

−1 −1

−0.5

0 H/Hm

0.5

−1 −1

1

(c) Closing the minor loop

−0.5

0 H/Hm

0.5

1

0.5

1

(d) Symmetric loop

1

1

0.5

0.5 M/Ms

M/Ms

0 H/Hm

(b) Opening a minor loop

M/Ms

M/Ms

(a) The first magnetization curve and starting a symmetric loop

−0.5

0

−0.5

0

−0.5

−1 −1

−0.5

0 H/Hm

0.5

(e) Clearing the memory

1

−1 −1

−0.5

0 H/Hm

(f) Symmetric loop with higher area

Fig. 5.26. Illustration of the operation of the NN model of hysteresis

198


5.3.2 The isotropic vector hysteresis operator In this section the isotropic vector generalization of the scalar NN based hysteresis model and an approproate identification task is presented [101, 114, 119–121]. A. Description of the model The vector NN based model of the magnetic hysteresis is constructed as a superposition of scalar NN based models. The magnetization vector M can be expressed in two dimensions as Z π/2 Z π/2 eϕ Mϕ dϕ = eϕ H {Hϕ } dϕ, (5.80) M= −π/2

−π/2

where Mϕ = H {Hϕ } is the scalar magnetization in the direction eϕ and the projected magnetic field intensity is Hϕ = |H| cos(ϑH − ϕ)

(5.81)

with the direction of the magnetic field intensity vector H, denoted by ϑH . In the case of isotropic vector hysteresis characteristics, the same scalar hysteresis operator H {·} can be used in all directions. In computer realization, it is useful to discretize the interval ϕ ∈ [−π/2, π/2] with respect to (see Fig. 5.27) ϕi = −

π (i − 1)π , + 2 n′

i = 1, · · · , n′ ,

(5.82)

where n′ is the number of directions. The magnetization vector is the vectorial sum of the scalar magnetization yielded by the individual scalar hysteresis models, i.e. ′

M∼ =

n X

′

eϕ i M ϕ i =

i=1

n X i=1

eϕi H {Hϕi }.

(5.83)

In this expression the constant △ϕ = π/n′ has been skipped. In 3D, the following expression can be obtained to determine the magnetization vector: Z M=

π/2

−π/2

Z

π/2

Z eϕ,ϑ Mϕ,ϑ dϕ dϑ =

−π/2

π/2

−π/2

Z

π/2

eϕ,ϑ H {Hϕ,ϑ } dϕ dϑ, (5.84)

−π/2

where Mϕ,ϑ = H {Hϕ,ϑ } is the scalar magnetization in the direction eϕ,ϑ (Fig. 5.28) and Hϕ,ϑ is the projected magnetic field intensity, Hϕ,ϑ = [a1 a2 a3 ]ϕ,ϑ [Hx Hy Hz ]T .

(5.85)

The directions are given as a = a 1 ex + a 2 ey + a 3 ez ,

(5.86)

199


y

..

.

n′

H

ey ϑH

x

ex eϕ i

Hϕ i

ϕi

..

.

i

2 1 Fig. 5.27. The definition of directions in the 2D model z

a y ez

eϕ,ϑ ey

ex

ϑ

ϕ

x (a) The definition of directions in the 3D model

(b) The icosahedron

Fig. 5.28. Directions of the 3D model are generated by the icosahedron

200


|a| = 1 and these are corresponding to the given directions allocated by the angles ϕ and ϑ as shown in Fig. 5.28(a) and generated by an icosahedron (see Fig. 5.28(b)). The angles ϕ and ϑ are measured from the x-axis and from the x − y plane, respectively. Any kind of regular 3D formation can be used, like polyhedrons, e.g. cube, octahedron, dodecahedron and so on. The applied scalar hysteresis operator is also unique for all directions. The integral formula (5.84) can be realized by a numerical sum in the discrete form ′

M∼ =

n X

′

eϕi ,ϑi Mϕi ,ϑi =

i=1

n X i=1

eϕi ,ϑi H {Hϕi ,ϑi },

(5.87)

where n′ is the number of directions in the 3D space. B. The identification process The NN based vector hysteresis model is constructed on scalar hysteresis characteristics, which are determined by the first order reversal curves. The reversal curves are associated with the Everett function, therefore the identification process is based on expressions related to the Everett function. In case of isotropic magnetic materials, it is evident that the hysteresis characteristics in all directions of the 2D or 3D plane are the same, that is the measured Everett function F (α, β) is unique for every direction [128]. Henceforth the hysteresis characteristics measured in the x direction will be supposed to be known. After measuring the Everett function in direction x, the following condition can be obtained between the measured ϕ-independent scalar Everett function F (α, β) and the unknown 2D ϕ-independent vector Everett function E(α, β) [128]: Z π/2 cos ϕ E(α cos ϕ, β cos ϕ) dϕ, (5.88) F (α, β) = −π/2

which means that the hysteresis curve in the x direction is the projected sum of the individual hysteresis curves distributed along all the possible directions ϕ. This expression can be written by skipping the constant △ϕ = π/n′ in a discretized form as ′

F (αk , βl ) ∼ =

n X

cos ϕi E(αk cos ϕi , βl cos ϕi ).

(5.89)

i=1

Here the Everett functions F (α, β) and E(α, β) have been divided into (N +1)×(N +1) segments as 2 2 N +1 N +1 1 1 k− − , βl = −l + , (5.90) αk = N 2 N N 2 N and k, l = 1, · · · , N +1. This results the so-called Everett tables F (αk , βl ) and E(αk , βl ). In this case α ∈ [−1, · · · , +1], β ∈ [+1, · · · , −1] and N must be an even number. In the 3D case, the expression (5.89) can also be applied, but ϕi is then the spatial angle between the directions generated by the icosahedron and the x-axis.


201

Expression (5.89) can only be solved numerically during the identification process, which is divided into two main parts. In the first step, the part of the Everett function can be identified where β = 0 and along the line α = −β. In the second step, the other parts can be calculated step by step by using values from the first step and some points from the second step. a. The first step: starting the identification process. A part of the Everett function is independent of one of the two variables α and β along the line α ≥ 0 and β = 0, i.e. k = N/2 + 1, · · · , N + 1 and l = N/2 + 1, therefore (5.89) can be simplified as ′

F (αk , 0) ∼ =

n X

cos ϕi E(αk cos ϕi , 0).

(5.91)

i=1

This expression can be divided into two parts, the first n′1 points are known, but the other n′2 points contain the unknown value of the vector Everett function E(αk , 0), ′

F (αk , 0) ∼ =

n1 X

cos ϕi1 E(αk cos ϕi1 , 0)

i1 =1

(5.92)

′

+

n2 X

cos ϕi2 E(αk cos ϕi2 , 0).

i2 =1

When αj−1 < αk cos ϕi1 ≤ αj ,

and αj−1 < αj ≤ αk−1 ,

(5.93)

and assuming linear interpolation between known values of the vector Everett function (αj−1 , E(αj−1 , 0)) and (αj , E(αj , 0)), the following formula – illustrated in Fig. 5.29 – can be applied to interpolate E(αk cos ϕi1 , 0): E(αj , 0) − E(αj−1 , 0) E(αk cos ϕi1 , 0) − E(αj−1 , 0) = , αj − αj−1 αk cos ϕi1 − αj−1

(5.94)

from which E(αk cos ϕi1 , 0) =E(αj−1 , 0) αk cos ϕi1 − αj−1 [E(αj , 0) − E(αj−1 , 0)] . + αj − αj−1

(5.95)

This expression contains known values of the vector Everett function calculated in the previous steps. The number of known points increases with the index k. When αk−1 < αk cos ϕi2 ≤ αk ,

(5.96)

then a similar expression can be used, E(αk cos ϕi2 , 0) − E(αk−1 , 0) E(αk , 0) − E(αk−1 , 0) = , αk − αk−1 αk cos ϕi2 − αk−1 from which

(5.97)

202


E(αj , 0)

E(αj−1 , 0)

αj−1 1 ...

αk

αk−1

αj αk cos ϕi1

E(αk , 0)

E(αk−1 , 0)

. . . n′1

...

α

1 . . . n′2 αk cos ϕi2

Fig. 5.29. Illustration for the first step of the identification, when α ≥ 0 and β = 0

E(αk cos ϕi2 , 0) =E(αk−1 , 0) αk cos ϕi2 − αk−1 [E(αk , 0) − E(αk−1 , 0)] . + αk − αk−1

(5.98)

This contains the only one unknown value of the vector Everett function E(αk , 0). The following expression can be obtained by substituting the relations (5.95) and (5.98) into the equation (5.92): ′

F (αk , 0) =

n1 X

i1 =1

cos ϕi1 E(αj−1 , 0)

αk cos ϕi1 − αj−1 + E(αj , 0) − E(αj−1 , 0) αj − αj−1 +E(αk−1 , 0) (1 + bk )c1 − ak c2 " +E(αk , 0) ak c2 − bk c1 ,

where ak =

αk , αk − αk−1

c1 =

n2 X

bk =

αk−1 , αk − αk−1

c2 =

n2 X

(5.99)

(5.100)

and ′

i2 =1

′

cos ϕi2 ,

cos2 ϕi2 ,

(5.101)

i2 =1

and E(αk , 0) can be expressed. A similar mathematical formulation can be obtained along the line α = −β. In this case, the indices k and l are in the interval [N/2 + 1, · · · , N + 1]. According to the linear interpolation between two points, the following expressions can be obtained after some mathematical manipulations. If αj−1 = βj−1 < αk cos ϕi1 = βl cos ϕi1 ≤ αj = βj , and

(5.102)


αj−1 = βj−1 < αj = βj ≤ αk−1 = βk−1 ,

203

(5.103)

then the known value of the vector Everett function can be expressed as E(αk cos ϕi1 , βl cos ϕi1 ) = E(αj−1 , βj−1 ) q p 2 α2k cos2 ϕi1 + βl2 cos2 ϕi1 − α2j−1 + βj−1 q q + 2 α2j + βj2 − α2j−1 + βj−1 × E(αj , βj ) − E(αj−1 , βj−1 ) .

If

αk−1 = βk−1 < αk cos ϕi2 = βl cos ϕi2 ≤ αk = βk ,

(5.104)

(5.105)

then the unknown value of the vector Everett function can be expressed as E(αk cos ϕi2 , βl cos ϕi2 ) = E(αk−1 , βl−1 ) q p 2 α2k cos2 ϕi2 + βl2 cos2 ϕi2 − α2k−1 + βl−1 q + p 2 α2k + βl2 − α2k−1 + βl−1 × E(αk , βl ) − E(αk−1 , βl−1 ) .

(5.106)

Here E(αk , βl ) is the only unknown and it can be expressed after substituting the relations (5.104) and (5.106) into (5.89). This step generates some points in the vector Everett function and gives an initiative step to determine the other values (see Fig. 5.30). b. The second step: the other points on the vector Everett plane. The plane α − β of vector Everett function can be divided into triangles and a simple linear interpolation can be applied on them to calculate the values on Everett functions corresponding to the coordinates (αk cos ϕ, βl cos ϕ). In this case, the Everett function can also be divided into two parts. The first n′1 points are known, but the other n′2 points contain the only one unknown value of the vector Everett function E(αk , βl ) (Fig. 5.30), ′

F (αk , βl ) ∼ =

n1 X

cos ϕi1 E(αk cos ϕi1 , βl cos ϕi1 )

i1 =1

(5.107)

′

+

n2 X

cos ϕi2 E(αk cos ϕi2 , βl cos ϕi2 ).

i2 =1

If α > 0, and β < 0, i.e. k = l + 1, · · · , N + 1, and l = N/2 + 2, · · · , N + 1,

(5.108)

204


(α1 , β1 )

(αk−1 , βl )

...

1111 0000 0000 1111 00001111 1111 0000 0000 1111 0000 00001111 1111 0000 1111

0

..

(αk , βl−1 )

.

Unknown

(α2 , β2 )

(α3 , β3 )

..

0 Known

.

β

0 0

(αk , βl )

111 000 00 00011 111 00 11 00 11

0 0 0

... α

...

.. .

..

...

.

...

..

.

0

...

0

Fig. 5.30. Illustration for the second step of the identification process (values denoted by squares are known from the first step) and the point (αk cos ϕi1 , βl cos ϕi1 ) is on a triangle with known nodal values, then the equation of a plane defined with the following points: (αk−1 , βl , E1 ), (αk , βl−1 , E2 ) and (αk−1 , βl−1 , E3 ), can be applied as αk cos ϕi1 − αk−1 βl cos ϕi1 − βl E − E1 αk − αk−1 βl−1 − βl E2 − E1 = 0, (5.109) 0 βl−1 − βl E3 − E1

where E = E(αk cos ϕi1 , βl cos ϕi1 ), and E1 = E(αk−1 , βl ), E2 = E(αk , βl−1 ) and E3 = E(αk−1 , βl−1 ). From this relation E(αk cos ϕi1 , βl cos ϕi1 ) can be expressed as follow: (βl cos ϕi1 − βl )(E3 − E1 ) βl−1 − βl (αk cos ϕi1 − αk−1 )(βl−1 − βl )(E3 − E2 ) − . (αk − αk−1 )(βl−1 − βl )

E(αk cos ϕi1 , βl cos ϕi1 ) = E1 +

(5.110)

Varying these three points, the equation of any other triangles can be evaluated and required values can be calculated.

205


If point (αk cos ϕi2 , βl cos ϕi2 ) is on a triangle containing the unknown value of the vector Everett function E(αk , βl ), then the equation of a plane with the three points (αk−1 , βl , E1 ), (αk , βl , E2 ) and (αk , βl−1 , E3 ) can be applied as αk cos ϕi2 − αk−1 βl cos ϕi2 − βl E − E1 αk − αk−1 0 E2 − E1 = 0, (5.111) αk − αk−1 βl−1 − βl E3 − E1

where E = E(αk cos ϕi2 , βl cos ϕi2 ), and E1 = E(αk−1 , βl ), E2 = E(αk , βl ) and E3 = E(αk , βl−1 ). From this equation E(αk cos ϕi2 , βl cos ϕi2 ) can be expressed as E(αk cos ϕi2 , βl cos ϕi2 ) = E1 (βl−1 − βl )(αk cos ϕi2 − αk−1 )(E2 − E1 ) (αk − αk−1 )(βl−1 − βl ) (αk − αk−1 )(βl cos ϕi2 − βl )(E2 − E3 ) . − (αk − αk−1 )(βl−1 − βl ) +

(5.112)

This equation contains the unknown value E2 . Substituting (5.110) and (5.112) into (5.107) E2 can be expressed for all indices k and l. If α > 0 and β > 0 (k = N + 3 − l, · · · , N + 1, l = N/2, · · · , 1), then a similar procedure can be applied, but the shape of the triangles is different. The Everett function is symmetrical with respect to the line α = −β, so it is enough to identify only the half of the vector Everett function. C. Verification of the model It is assumed that the measured Everett function is known in the x direction as it is plotted in Fig. 5.31. According to the above mentioned process the identified 2D and 3D vector Everett functions can be seen in Fig. 5.32(a) and Fig. 5.32(b).

1

F(α,β)

0.75 0.5 0.25 0 1 1

0.5 0.5

0 β

0

−0.5

−0.5 −1 −1

α

Fig. 5.31. The measured Everett function in the x direction, F (α, β)


0.1

0.15

0.075

0.1 E(α,β)

E(α,β)

206

0.05

0.05

0.025

0

0 1

−0.05 1 1

0.5 β

1

0.5

0.5

0

0

−0.5

−0.5 −1 −1

α

0.5

0 β

0

−0.5

(a) 2D

−0.5 −1 −1

α

(b) 3D

Fig. 5.32. The identified vector Everett function, E(α, β) Identification has been performed by the method described above, 20 directions and 9 directions have been used in the 2D and in the 3D model, respectively. The measured scalar Everett function can be approximated with a relative error below 0.3% and 0.4% by the 2D and the 3D models by substituting into the condition (5.89). The resulting vector Everett functions – as highlighted in the figures – do not have the usual shape, meaning that the first order reversal curves also do not have the usual descending shape, but they can be approximated easily by the scalar NN based hysteresis model. The vector Everett function of the 2D model has smaller maximum value, because the number of directions is higher. Increasing the number of directions results in decreasing maximum value of the identified vector Everett function. First order reversal curves can be derived from the identified vector Everett function and NNs can be trained as well as the vector models can be built by using the trained networks. A comparison between measured and simulated first order reversal curves by the identified 2D and 3D models can be seen in Fig. 5.33 and Fig. 5.34. Measured curves are denoted by points, simulated curves are denoted by dotted line. Some other verification and comparisons with the anisotropic model are presented in section 5.3.4.

5.3.3 The anisotropic vector hysteresis operator In this section the anisotropic vector generalization of the scalar NN based hysteresis model and an appropriate identification task is presented [101, 114, 119–121, 156]. First of all the description of the model is introduced. The identification task is based on two steps, the first one is to use the Fourier expansion of the measured Everett functions, then applying a similar identification process as described in the last section.

207


1

Mx/Ms

0.5

0

−0.5

−1 −1

−0.5

0 H x/Hm

0.5

1

Fig. 5.33. Identification results for isotropic case using the 2D model (measured curves are denoted by points, simulated curves are denoted by dotted line) 1

Mx/Ms

0.5

0

−0.5

−1 −1

−0.5

0 H x/Hm

0.5

1

Fig. 5.34. Identification results for isotropic case using the 3D model (measured curves are denoted by points, simulated curves are denoted by dotted line)

A. Description of the model The anisotropic vector NN based model of hysteresis is constructed as a superposition of scalar NN based models in the directions eϕ like the isotropic vector model, but the scalar hysteresis characteristics are not the same in different direction. The magnetization vector M can be expressed in two dimensions as (see Fig. 5.27) M=

Z

π/2 −π/2

eϕ Hϕ {Hϕ } dϕ.

(5.113)

208


In three dimensions a similar expression can be obtained (see Fig. 5.28(a)), Z π/2 Z π/2 M= eϕ,ϑ Hϕ,ϑ {Hϕ,ϑ } dϕ dϑ. −π/2

(5.114)

−π/2

The main problem of the identification task is the angular dependence of hysteresis characteristics. Here, the aim is to apply the same identification process with minor modifications as used in the isotropic case presented in the last section. The Fourier expansion of the Everett surfaces can be used efficiently. B. Fourier expansion of the measured 2D vector Everett function It is assumed that the Everett function can be measured in any direction associated with the polar angle ϕ. This is called the scalar characterization of the vector hysteresis property, since the characteristics are not the same in different directions. In this case, the following condition can be obtained between the measured ϕ-dependent scalar Everett function F (α, β, ϕ) and the unknown ϕ-dependent vector Everett function E(α, β, ϕ): Z π/2 cos ψ E(α cos ψ, β cos ψ, ψ + ϕ) dψ. (5.115) F (α, β, ϕ) = −π/2

As it has been shown in the last section the Everett functions F (α, β) and E(α, β) have been divided into (N + 1) × (N + 1) segments resulting in the Everett tables F (αk , βl ) and E(αk , βl ). This integral can be derived in a discretized form by skipping the constant △ϕ = π/n′ as ′

F (αk , βl , ϕj ) ∼ =

n X

cos ψi E(αk cos ψi , βl cos ψi , ϕi ),

(5.116)

i=1

where ψi = ϕi − ϕj is the angle between the direction of the measured scalar Everett function in the direction eϕj and the unknown vector Everett function defined in the direction eϕi and j = 1, · · · , n′ . The identification process is based on n′ directions selected as ϕi = −

π (i − 1)π , + 2 n′

i = 1, · · · , n′ ,

(5.117)

i.e. ϕi ∈ [−π/2, · · · , π/2] and n′ is the number of directions, where the scalar hysteresis characteristics have been measured. The identification aims to calculate the unknown vector Everett functions E(αk , βl , ϕj ) in the same directions. Assuming that the magnetic material has uniaxial anisotropy (see in Fig. 5.35), the measured scalar Everett function must be π-periodic and it is an even function with respect to ϕ, if ϕ = 0 represents the rolling direction, anyway ϕ = π/2 is perpendicular to the rolling direction (also called transverse direction), i.e. F (αk , βl , ϕ + π) = F (αk , βl , ϕ), and

(5.118)

209


td B rd

Fig. 5.35. The rotating magnetic flux density and the rolling direction (rd) and the transverse direction (td), here the value of magnetic flux in saturation state can be seen

F (αk , βl , −ϕ) = F (αk , βl , ϕ).

(5.119)

It is supposed that there are n + 1 measured Everett functions F (αk , βl , ϕj ) in the interval ϕj ∈ [0, · · · , π/2], ϕj = j π/2n, j = 0, · · · , n and n′ is chosen as n′ = 2n as plotted in Fig. 5.36. y

n

..

j

. ϕj 0

x

Fig. 5.36. The directions of the measured Everett functions F (α, β, ϕ)

In the following experiments n + 1 = 10 scalar hysteresis characteristics have been assumed, i.e. ϕ0 = 0◦ , ϕ1 = 10◦ , · · · , ϕ9 = 90◦ , generated by the elliptical interpolation function F (α, β, ϕ) = Fx2 cos2 ϕ + Fy2 sin2 ϕ,

(5.120)

where Fx = F (α, β, 0) and Fy = F (α, β, π/2) are the assumed normalized Everett functions in the rolling and in the transverse directions, respectively and they are plotted in Fig. 5.37(a) and Fig. 5.37(b). Material with uniaxial anisotropy can be simulated by this assumption.


1

1

0.75

0.75 Fy(α,β)

Fx(α,β)

210

0.5

0.5

0.25

0.25

0 1

0 1 1

0.5 β

1

0.5

0.5

0

0

−0.5

−0.5 −1 −1

α

(a) ϕ = 0

0.5

0 β

0

−0.5

−0.5 −1 −1

α

(b) ϕ = π/2

Fig. 5.37. The measured Everett functions in the rolling and in the transverse direction to build the 2D anisotropic vector hysteresis model in (5.120) Due to the periodicity of the measured Everett function, it can be approximated by its Fourier series, since it is difficult to take into account the angular dependence of the measured Everett function during the identification process, but the Fourier expansion is an accurate method to handle this phenomenon. Since F (α, β, ϕ) is an even function with respect to ϕ, only the real part of the complex Fourier series must be calculated (to make the equations below shorter, the following notation will be used: F (ϕj ) = F (αk , βl , ϕj )), X F (ϕj ) ≃ Fm cos(2mϕj ), (5.121) m

where ω = 2π/π = 2 is the angular frequency. Fourier coefficients Fm = Fm (αk , βl ) must be calculated. The subscript m is the number of Fourier coefficients and it can be maximum n. The Fourier coefficients have been calculated by their definition. The zeroth order component can be calculated as Z Z 1 π/2 2 π/2 F (ϕ) dϕ = F (ϕ) dϕ. (5.122) F0 = π −π/2 π 0 Integration can be performed numerically by the well known trapezoidal rule,   n−1 π X ∆ϕ  F0 ≃ F (ϕj ) . +2 F (0) + F π 2 j=1 The higher order harmonics can be calculated as Z Z 2 π/2 4 π/2 F (ϕ)e−jmωϕ dϕ = F (ϕ) cos(2mϕ) dϕ, Fm = π −π/2 π 0

and applying numerical integration it has the form

(5.123)

(5.124)

211


Fm

  n−1 π X 2∆ϕ  ≃ F (ϕj ) cos(2mϕj ) . +2 F (0) + (−1)m F π 2 j=1

(5.125)

1

0.3

0.75

0.2 F1(α,β)

F0(α,β)

Applying this approach, the angular dependence of the measured Everett function can be handled, since the Fourier coefficients Fm are independent of the polar angle. By applying the zeroth (Fig. 5.38(a)) and only the first order (Fig. 5.38(b)) Fourier coefficients, the measured Everett functions in the directions of measurements can be approximated with a relative error as small as about 4% (this is the maximum value).

0.5

0.1

0.25

0

0 1

−0.1 1 1

0.5 β

1

0.5

0.5

0

0

−0.5

−0.5 −1 −1

α

(a) The zeroth order harmonic component, F0

0.5

0 β

0

−0.5

−0.5 −1 −1

α

(b) The first order harmonic component, F1

Fig. 5.38. The Fourier coefficients of the measured 2D Everett surfaces

Increasing the number of directions of the measured scalar Everett functions results in better approximation between the measurement directions. In this case, the relative difference between measured and simulated Everett functions is smaller than 6%. C. Fourier expansion of the measured 3D vector Everett function The 3D anisotropic vector hysteresis model is a theoretical generalization of the 2D anisotropic vector model, since there exists no well applicable method and measurement equipment to measure the 3D vector hysteresis property. The model proposed in the following is the simplest one, but the method can be generalized. It is assumed that scalar Everett function can be measured in any directions represented by the angles ϕ and ϑ measured from the x-axis and from the x − y plane, respectively. In this case the following condition (in discretized form) can be obtained between the measured scalar Everett function F (αk , βl , ϕj1 , ϑj2 ) and the unknown vector Everett function E(αk , βl , ϕi1 , ϑi2 ):

212


′

F (αk , βl , ϕj1 , ϑj2 ) ∼ =

′

n X m X

cos ψi1 ,i2

i1 =1 i2 =1

(5.126)

× E(αk cos ψi1 ,i2 , βl cos ψi1 ,i2 , ϕi1 , ϑi2 ), where ψi1 ,i2 is the spatial angle between the direction of the measured Everett function with direction ϕj1 and ϑj2 and the vector Everett function in the given direction signed by the angles ϕi1 and ϑi2 . The following properties can be assumed during the development of the model and the identification procedure as the simplest assumptions (generalization of the assumptions of the 2D anisotropic model). (i) There are (n + 1) × (m + 1) measured scalar Everett functions, i.e. F (ϕj1 , ϑj2 ) = F (αk , βl , ϕj1 , ϑj2 )

(5.127)

in the interval ϕ ∈ [0, · · · , π/2], and ϕj1 = j1 π/2n and j1 = 0, · · · , n, moreover ϑ ∈ [0, · · · , π/2], and ϑj2 = j2 π/2m and j2 = 0, · · · , m. The identification is constructed on n′ = 2n and m′ = 2m directions. (ii) The measured scalar Everett function is π-periodic with respect to angles ϕ and ϑ, i.e. F (ϕ + π, ϑ) = F (ϕ, ϑ),

(5.128)

F (ϕ, ϑ + π) = F (ϕ, ϑ).

(5.129)

(iii) The measured scalar Everett function is symmetrical with respect to the x-axis and the x − y plane, i.e. F (−ϕ, ϑ) = F (ϕ, ϑ),

(5.130)

F (ϕ, −ϑ) = F (ϕ, ϑ).

(5.131)

In the following experiments 5×5 measured scalar hysteresis characteristics have been assumed, generated by the elliptical interpolation function F (α, β, ϕ, ϑ) = cos2 ϑ (Fx2 cos2 ϕ + Fy2 sin2 ϕ) + Fz2 sin2 ϑ,

(5.132)

where Fx = F (α, β, 0, 0), Fy = F (α, β, π/2, 0) and Fz = F (α, β, 0, π/2) are the assumed normalized Everett functions in the x, y and z directions. These are plotted in Fig. 5.39(a)-5.39(c). The measured Everett function can also be approximated by its Fourier series due to the assumed periodicity. The Everett function is an even function with respect to the angles ϕ and ϑ, i.e. only the real part of the complex Fourier series can be calculated as XX Fn1 m1 cos(2n1 ϕj1 ) cos(2m1 ϑj2 ), F (ϕj1 , ϑj2 ) ≃ (5.133) n1 m1

213

1

1

0.75

0.75 Fy(α,β)

Fx(α,β)


0.5 0.25

0.5 0.25

0 1

0 1 1

0.5 β

1

0.5

0.5

0

0

−0.5

−0.5 −1 −1

0.5

0

0

−0.5

β

α

−0.5 −1 −1

(a) The Everett function in the x direction

α

(b) The Everett function in the y direction

1

Fz(α,β)

0.75 0.5 0.25 0 1 1

0.5 0.5

0 β

0

−0.5

−0.5

α

−1 −1

(c) The Everett function in the z direction

Fig. 5.39. The measured Everett functions to build the 3D anisotropic model in (5.132)

where Fn1 m1 = Fn1 m1 (αk , βl ) are the Fourier coefficients. The maximum value of n1 and m1 can be n and m. The zeroth order component F00 can be calculated by the integral F00

1 = 2 π

Z

π/2

−π/2

Z

π/2

4 F (ϕ, ϑ) dϕ dϑ = 2 π −π/2

Z

0

Z π/2 π/2

F (ϕ, ϑ) dϕ dϑ,

(5.134)

0

which can be calculated numerically by the following expression applying the trapezoidal rule:

214


F00

( π π π π ∆ϕ∆ϑ ≃ , 0 + F , + F F (0, 0) + F 0, π2 2 2 2 2 +2

j1

+2

h π i F (ϕj1 , 0) + F ϕj1 , 2 =1

n−1 X

m−1 Xh

F (0, ϑj2 ) + F

j2 =1

π 2

, ϑj2

i

(5.135) +4

n−1 X m−1 X

)

F (ϕj1 , ϑj2 ) .

j1 =1 j2 =1

The harmonic components can be calculated by the definition as well, i.e. Fn1 m1

2 = 2 π 8 = 2 π

Z

π/2

−π/2

Z

π/2

0

Z

Z

π/2

F (ϕ, ϑ) e−jn1 ωϕ e−jm1 ωϑ dϕ dϑ

−π/2

(5.136)

π/2

F (ϕ, ϑ) cos(2n1 ϕ) cos(2m1 ϑ) dϕ dϑ,

0

and numerically Fn1 m1

( π ∆ϕ∆ϑ F (0, 0) + F 0, (−1)n1 ≃2 π2 2 π π π +F (−1)n1 +m1 , 0 (−1)m1 + F , 2 2 2 n−1 i Xh π +2 (−1)m1 cos(2n1 ϕj1 ) F (ϕj1 , 0) + F ϕj1 , 2 j =1 1

+2

m−1 Xh

F (0, ϑj2 ) + F

j2 =1

+4

n−1 X m−1 X

j1 =1 j2 =1

π 2

(5.137)

i , ϑj2 (−1)n1 cos(2m1 ϑj2 ) )

F (ϕj1 , ϑj2 ) cos(2n1 ϕj1 ) cos(2m1 ϑj2 ) .

The Fourier coefficients Fn1 m1 are independent of the angles ϕ and ϑ. By applying the zeroth F00 and only the first order Fourier coefficients F01 , F10 and F11 , the measured Everett functions along the directions of the measurements can be approximated with a relative error as small as about maximum 5 − 6%. The calculated Fourier coefficients are plotted in Fig. 5.40(a)-5.40(d). D. The relation between the scalar and the vector Everett functions The angular dependence of measured scalar Everett function can be handled by applying the Fourier series as presented in the last two sections, but a relation between the scalar and the vector Fourier coefficients must be developed. This relation gives the basic expression of the identification procedure. Authors of [156] used the distribution function of the classical Preisach model. Here the Everett function is shown, because this is the basis of the NN vector hysteresis model.

215

1

0.2

0.75

0.1 F01(α,β)

F00(α,β)


0.5 0.25

−0.1

0 1

−0.2 1 1

0.5 −0.5

0.04

0.04 F11(α,β)

0.08

0

0

−0.04

−0.04

−0.08 1

−0.08 1 1

0.5

1

0.5

0.5 0

−0.5

−0.5 −1 −1

α

(b) The first order harmonic component, F01

0.08

β

−0.5 −1 −1

(a) The zeroth order harmonic component, F00

0

0

−0.5

β

α

−1 −1

0.5

0

0

−0.5

β

1

0.5

0.5

0

F10(α,β)

0

α

(c) The first order harmonic component, F10

0.5

0 β

0

−0.5

−0.5 −1 −1

α

(d) The first order harmonic component, F11

Fig. 5.40. The Fourier coefficients of the measured 3D Everett surfaces

The identification task of the 2D model is based on the integral equation F (α, β, ϕ) =

Z

π/2

cos ψE(α cos ψ, β cos ψ, ψ + ϕ) dψ,

(5.138)

−π/2

where the measured scalar Everett function along an angle ϕ, F (α, β, ϕ) must be equal to the projected sum of the vector Everett functions E(α, β, ψ + ϕ). Unfortunately, the unknown vector Everett function is placed on the right-hand side of this integral equation. This problem can be solved as it is presented in the following. The 2D vector Everett function denoted by E(ϕ) = E(α, β, ϕ) for short also depends on the polar angle ϕ and it can also be approximated by its Fourier series by assuming the

216


same conditions as used for the measured Everett function F (α, β, ϕ), i.e. E(ϕ) ≃

X

Em cos(2mϕ).

(5.139)

m

Substituting the Fourier series (5.121) and (5.139) into (5.138), the following equation can be obtained: Z π/2 X Fm (α, β) cos(2mϕ) = cos ψ −π/2 m (5.140) X × Em (α cos ψ, β cos ψ) cos(2m(ψ + ϕ)) dψ, m

and this equation can be rewritten in the form, X Fm (α, β) cos(2mϕ) m π/2

=

Z

cos ψ

−π/2

−

Z

X

Em (α cos ψ, β cos ψ) cos(2mψ) cos(2mϕ) dψ

(5.141)

m

π/2

cos ψ

−π/2

X

Em (α cos ψ, β cos ψ) sin(2mψ) sin(2mϕ) dψ,

m

by applying the identity cos(2m(ψ + ϕ)) = cos(2mψ) cos(2mϕ) − sin(2mψ) sin(2mϕ).

(5.142)

The second term is equal to zero, because the integrand is an odd function with respect to ψ, resulting the following expression: X

Fm (α, β) cos(2mϕ) =

m

XZ m

π/2

cos ψ cos(2mψ) −π/2

(5.143)

× Em (α cos ψ, β cos ψ) dψ cos(2mϕ). The identification procedure is then based on the following integral equation, which links the mth harmonic component of the unknown vector Everett function to the corresponding scalar one, i.e. Fm (α, β) =

Z

π/2

cos ψ cos(2mψ)Em (α cos ψ, β cos ψ) dψ.

(5.144)

−π/2

In this case, the identification procedure developed for the isotropic vector model can also be applied, but the term cos(2mψ) must be appended on the right. Identification results of the zeroth and the first order harmonics of the 2D vector Everett function can be seen in Fig. 5.41(a) and Fig. 5.41(b).

217


In the 3D case, a similar mathematical formulation can be used, which leads to the following equation. The basis of the identification task is: ′

Fn1 m1 (α, β) ∼ =

n X

cos ψi cos(2n1 ϕi ) cos(2m1 ϑi )

(5.145)

i=1

× En1 m1 (α cos ψi , β cos ψi ), where the angle ψi is the spatial angle between a direction generated by the icosahedron and the x-axis. In this case, equation (5.126) should be rewritten in a simplified form, F (αk , βl , ϕj , ϑj ) ∼ =

′ nX ·m′

cos ψi E(αk cos ψi , βl cos ψi , ϕi , ϑi ).

(5.146)

i=1

0.12

0.06

0.08

0.04 E1(α,β)

E0(α,β)

The four identified Fourier series E00 , E01 , E10 and E11 of the vector Everett function are plotted in Fig. 5.42(a)-5.42(d).

0.04

0.02

0

0

−0.04 1

−0.02 1 1

0.5 β

1

0.5

0.5

0

0

−0.5

−0.5 −1 −1

α

(a) The zeroth order harmonic component, E0

0.5

0 β

0

−0.5

−0.5 −1 −1

α

(b) The first order harmonic component, E1

Fig. 5.41. The identified 2D vector Fourier coefficients

E. Verification of the model From the Fourier coefficients of the vector Everett function, the Everett functions in all directions (e.g. 9 directions generated by the icosahedron) as well as the first order reversal curves can be calculated, then NNs can be trained. It is obvious that different NNs must be trained in different directions. Simulation results – trivial conditions in the directions of the x-, of the y- and of the z-axis – for the first order reversal curves obtained from the identified Everett functions in two dimensions and in three dimensions can be seen in Fig. 5.43(a), Fig. 5.43(b) and Fig. 5.43(c)-5.43(e).


0.12

0.03

0.08

0.02 E01(α,β)

E00(α,β)

218

0.04 0

0

−0.04 1

−0.01 1 1

0.5 −0.5

0.01

0.01 E11(α,β)

0.02

0

0

−0.01

−0.01

−0.02 1

−0.02 1 1

0.5

1

0.5

0.5 0

−0.5

−0.5 −1 −1

α

(b) The first order harmonic component, E01

0.02

β

−0.5 −1 −1

(a) The zeroth order harmonic component, E00

0

0

−0.5

β

α

−1 −1

0.5

0

0

−0.5

β

1

0.5

0.5

0

E10(α,β)

0.01

α

(c) The first order harmonic component, E10

0.5

0 β

0

−0.5

−0.5 −1 −1

α

(d) The first order harmonic component, E11

Fig. 5.42. The identified 3D vector Fourier coefficients There are 18 directions and 9 directions in the 2D and in the 3D model, respectively. The experimental results are denoted by points, the hysteresis characteristics given by the NN based vector model are denoted by dotted line. Increasing the number of directions results in a better approximation, but at the same time in a slower model. If the 3D vector hysteresis model is used in a finite element procedure, then a compromise has to be made.

5.3.4 Some properties of the vector hysteresis characteristics In this section, some cases, which can characterize the behavior of the vector model prescribed above will be presented.

219


0.5

0.5 My/Ms

1

Mx/Ms

1

0

−0.5

−1 −1

0

−0.5

−0.5

0 H x/Hm

0.5

−1 −1

1

(a) Comparison in the rolling direction

0.5

0.5 My/Ms

1

Mx/Ms

0 H y/Hm

0.5

1

(b) Comparison in the transverse direction

1

0

−0.5

−1 −1

−0.5

0

−0.5

−0.5

0 H x/Hm

0.5

−1 −1

1

(c) Comparison in the x direction

−0.5

0 H y/Hm

0.5

(d) Comparison in the y direction

1

Mz/Ms

0.5

0

−0.5

−1 −1

−0.5

0 H z/Hm

0.5

1

(e) Comparison in the z direction

Fig. 5.43. The resulting identification in 2D and in 3D, anisotropic case (measured: ·, simulated: dotted)

1

220


Applying an anticlockwise rotational magnetic field intensity with different amplitude, the output of the 2D vector isotropic and 2D anisotropic models have been plotted in Fig. 5.44(a), 5.44(c) and in Fig. 5.44(b), 5.44(d). 1.2

1.2

H M

0.6 Hy /Hm, My/Ms

Hy /Hm, My/Ms

0.6

H M

0

−0.6

0

−0.6

−1.2 −1.2

−0.6

0 0.6 H x/Hm, Mx /Ms

1.2

−1.2 −1.2

−0.6

(a) Under saturation 3

H M

H M

1.5 Hy /Hm, My/Ms

Hy /Hm, My/Ms

1.2

(b) Under saturation 3

1.5

0

−1.5

−3 −3


0

−1.5

−1.5


(c) In saturation state

3

−3 −3

−1.5


3

(d) In saturation state

Fig. 5.44. Simulated H and M loci of the rotational process in the isotropic ((a) and (c)) and in the anisotropic ((b) and (d)) case

The specimen is magnetized to a given value in the direction of the x-axis and the magnetic field intensity is then rotated keeping its magnitude constant. The magnetization vector has some delay in comparison with the phase of the magnetic field intensity vector, especially in the case, when the magnetic field intensity is lower then in the saturation state. Increasing the magnitude of the magnetic field intensity results in decreasing the phase delay between the magnetization and the magnetic field intensity vectors. Higher above the value Hm they rotate together. In the figures yielded by the anisotropic model, the anisotropic behavior can easily be seen, because the amplitude of magnetization in the y-axis is smaller than in the x-axis and the delay between the magnetic field intensity and magnetization vectors is larger.

221


1.6

1.6

H M

0.8 Hy /Hm, My/Ms

Hy /Hm, My/Ms

0.8

0

−0.8

−1.6 −1.6

0

−0.8

−0.8


1.6

(a) Vectors of H and M when ϑH = 60◦

−1.6 −1.6

0.5 Mx/Ms

0.5 Mx/Ms

1

0

−0.5

1.6

0

−0.8

0 H x/Hm

0.8

−1 −1.6

1.6

0.5

0.5 My/Ms

1

0

−0.5

−0.8

0 H x/Hm

0.8

1.6

(d) The hysteresis characteristics in the x-axis

1

Mx/Ms


−0.5

(c) The hysteresis characteristics in the x-axis

−1 −1.6

−0.8

(b) Vectors of H and M when ϑH = 60◦

1

−1 −1.6

H M

0

−0.5

−0.8

0 H x/Hm

0.8

1.6

(e) The hysteresis characteristics in the y-axis

−1 −1.6

−0.8

0 H y/Hm

0.8

1.6

(f) The hysteresis characteristics in the y-axis

Fig. 5.45. The simulated H and M loci of the linear excitation in the isotropic ((a), (c) and (e)) and in the anisotropic ((b), (d) and (f)) case

222


In the second test case, the magnetic field intensity vector has applied in a linear variation (ϑH = 60◦ ). In isotropic case, the magnetic field intensity and the magnetization vectors are parallel to each others as plotted in Fig. 5.45(a), 5.45(c), 5.45(e), but in anisotropic case (see Fig. 5.45(b), 5.45(d), 5.45(f)) there is a phase delay between them. Because of the anisotropic behavior the magnetization vector tries to be close to the x-axis. The projected hysteresis characteristics in the x-axis and in the y-axis are also presented. The hysteresis characteristics in the x-axis and in the y-axis are similar in the isotropic case, but the shape of the characteristics is very different in the anisotropic case. Anisotropic behavior induced by the remanent magnetization can be observed as well. Let us suppose that the magnetic field intensity is first increased in the y direction to a given value and then it is decreased to zero. This process results in a remanent magnetization in the y direction. After reaching the remanent magnetization, the magnetic field intensity is increased in the x direction. The orthogonal remanent magnetization can be reduced as presented in the Fig. 5.46(a). In the 3D case, a similar magnetization process can be applied with similar simulation results, e.g. the magnetic field intensity is increased and then decreased in the x direction resulting remanent magnetization in the x direction, then it is increased and then decreased in the z direction and finally increased in the y direction. This process can be seen in Fig. 5.46(b). H M

1.2

H M

1.2

Hz/Hm, Mz /Ms

Hy /Hm, My/Ms

0.6

0

0.9 0.6 0.3 0 0

−0.6

0.3

0 0.3

0.6

−1.2 −1.2

−0.6


(a) 2D model

1.2

0.6

Hy/Hm, M y /Ms 0.9 1.2 1.2

0.9 Hx/Hm, M x /Ms

(b) 3D model

Fig. 5.46. The induced anisotropy in isotropic material

Using a large number of directions results better approximation, but the calculations may be time consuming. Smaller number of directions gives higher error between the measured and the simulated curves, however, the model is faster. To make a compromise, the presented model with given number of directions can be used in Computer Aided Design software tools (e.g. in the Finite Element Method) with a given accuracy.

5.4. MEASURING MAGNETIC HYSTERESIS

223

5.4 Measuring magnetic hysteresis 5.4.1 System for scalar hysteresis measurement A. The measurement set-up using LabVIEW Ferromagnetic hysteresis measurement has been performed in the Magnetic Laboratory of the Department of Broadband Infocommunications and Electromagnetic Theory, Budapest University of Technology and Economics and in the Laboratory of Electromagnetic Fields, Department of Telecommunications, Széchenyi István University, Gy˝or by a computer controlled automated measuring system [91–93, 103, 106, 108, 115, 146–148, 171]. Generation of excitation and measurements are carried out by using the user-friendly environment of LabVIEW. LabVIEW is an application software to perform sophisticated display and analysis capabilities required for virtual instrumentation. This can be used to create any test measurement system by combining different hardware and software components. Data acquisition and waveform generation can be carried out by virtual software instruments and tools and the measurement can be realized by plug-in hardware connected to the computer via special National Instruments driver software [74]. An example, how to build up a waveform with LabVIEW blocks can be seen in Fig. 5.47 [74]. The graphical user interface (GUI) of the developed software is illustrated in Fig. 5.48, data and graphs in the window are presented below. The investigated ferromagnetic sample has toroidal shape, it can be seen in Fig. 5.49. It is supposed that both the magnetic field intensity with its scalar value H and the magnetic flux density B are uniform inside the material and directed along the azimuthal axis of the toroidal core. This advantageous property results in a standard and one of the simplest measurement set up to measure the scalar hysteresis characteristics of ferromagnetic materials. In this special case the axial component of the magnetic field intensity vector and of the magnetic flux density vector are supposed to represent the relationship between these two quantities. This results in scalar hysteresis characteristics. A schematic view of the measurement set-up can be seen in Fig. 5.50. The magnetic field intensity H inside the analyzed specimen has been generated by the current i(t) flowing in the primary coil of the core. The primary coil has Np turns. The time variation of the magnetic field intensity results in a time varying magnetic flux Φ(t)

Fig. 5.47. Waveform generation with LabVIEW blocks

224


Fig. 5.48. GUI of the developed measurement system

inside the specimen, finally the effect of this flux can be measured by the secondary coil wound on the core. The secondary coil has Ns turns, which output is the induced voltage u(t) and it has no load. Essentially, this measurement set up is a simple transformer.

225


waveform of current

Fig. 5.49. The toroidal shape core

u(t) S

Ns Np

NI-DAQ card

LabVIEW on PC generator

F i(t)

Fig. 5.50. Schematic view of the scalar hysteresis measurement set-up The primary coil is excited by the current i(t) supplied by the power generator. The generator has an output where a voltage proportional to the current can be picked up. The voltage proportional to the excitation current and the induced voltage of the secondary coil have been measured by the installed NI-DAQ card (National Instruments Data Acquisition Card, here NI PCI-6251 has been used).

226


The measured magnetic field intensity is proportional to the exciting current and can be calculated as Np i(t) H(t) ≃ , (5.147) l where Np = 134 is the number of turns of the excitation (primary) coil, i(t) is the excitation current and l = 157 mm is the equivalent magnetic length of the toroidal core. The magnetic flux density B can be calculated from the measured induced voltage of the measuring (secondary) coil u(t) as Z t 1 u(τ ) dτ, (5.148) B(t) = B0 + S Ns 0

where B0 is an integration constant, S = 320 mm2 is the cross-section of the core and Ns = 248 is the number of turns of the measuring coil. The measured signals, the calculated waveform of the magnetic field intensity and of the magnetic flux density and the resulted hysteresis characteristics are visualized in the user interface (see Fig. 5.48). Data acquisition and generation of excitation current can be performed simultaneously by using the intermediate virtual instruments of LabVIEW [74]. Two Analog Inputs (AI) and one Analog Output (AO) channels must be selected to measure the induced voltage and the waveform of current, moreover to generate the waveform of current, respectively. The channels have a Physical Channel name as well, like AI1, or AO4. The minimum and the maximum limit of the input and of the output range can also be set. This range is divided into 216 segments, because the applied DAQ card has a 16 bit resolution, i.e. the accuracy of measurements can be controlled. The next step is to reset the device then setting up different sampling parameters. The samples per channel variable, which is equal to the number of samples per period multiplied by the number of periods defining how much samples must be measured during one measuring period in one channel. The sampling rate variable is equal to the number of samples per channel multiplied by the frequency. These two parameters are the sampling information of the device. Then the signal generation and the measurement stage are performed. Error messages can be obtained in the case of any problems during the measurement. Measured curves are saved to a file, for further use of the measured data using other software or for a report. The waveform, the amplitude and the frequency of the excitation current, the number of measured periods and of measured samples in one period and other parameters (Np , Ns , S, l) can be set through the graphical user interface. Computer-aided measurements can only be performed by sampling using A/D and D/A converters (acquisition) [74]. This can be carried out through general DAQ devices, GPIB interfaces, RS-232 instruments and so on. There may be a weak point of the measurement process if very steep signals must be measured (e.g. induced voltage in the secondary coil), but it can be eliminated by the appropriate choice of the sampling rate. The main advantage of applying computer controlled measurement is the data analysis, such as curve fitting, frequency response, filtering, Fourier analysis and many other numerical operations realized by a software. A photo about the measurement system can be seen in Fig. 5.51.


227

Fig. 5.51. Photo about the measurement system Unfortunately, the measured signals are sometimes noisy. This noise is coming from the environment of the measurement system: from the computer, from the generator, from the net current and so on. It is very difficult to eliminate such a noise, however, the effect of this noise can be decreased by shielding the input of DAQ card. This noise has a great effect especially in the case of small exciting currents and as it can not be eliminated and it can not be decreased effectively by setting the input range of the DAQ card. The noise results in some offset in the integrated signal (5.148), which is difficult to eliminate in the time domain. That is why a Fourier analysis of the measured periodic signals has been performed and its implementation is shown in Fig. 5.52 [103, 108].

Fig. 5.52. The FFT algorithm with digital filtering implemented in LabVIEW

228


0.014

0.014

0.0105

0.0105

|U (jw)|

0.007

i

i

|U (jw)|

First, the measured signals have been Fourier transformed by the FFT block (Fast Fourier Transform) of LabVIEW, then the undesired harmonics have been eliminated by a digital filter. This method results in an attractive noise removal technique. The applied filtering technique is very simple, the unwanted harmonics have been multiplied by zero, finally the filtered spectrums have been transformed back to the time domain. The DC component of the induced voltage has also been filtered. The number of harmonic components to be used N can be set in the GUI (Graphical User Interface). The first period has been thrown out, because it contains some transients. According to experiments, the stationary state can be reached from the second period and N = 50 is usually enough. Figure 5.53(a) and Fig. 5.53(b) show a typical amplitude spectrum of the induced voltage. The number of used periods is 6 and the number of samples per period is 22222.

0.0035

0 0

0.007

0.0035

2.2222 4.4444 6.6666 8.8888 11.11113.3332

k

(a) The whole spectrum

x 10

0 0

50

100

150

200

k

4

(b) A magnified plot, k = 0, · · · , 200

Fig. 5.53. The amplitude spectrum of the measured induced voltage

The peaks for further use can be selected by a simple for loop (Fig. 5.52), and the other components associated to noise are much smaller (in the range of about 10−5 ) and they have been eliminated. The accuracy of the proposed filtering technique can be studied in Fig. 5.54(a) and in Fig. 5.54(b). The hysteresis curve is moving and it is very noisy without filtering. The mentioned method can eliminate any noise coming from measurements and the filtered hysteresis characteristics are not suffering from noise anymore. The magnetic field intensity is proportional to the applied current in the primary coil. The inverse of it is controlling the waveform of the magnetic flux density. Here, a simple feedback controller is proposed [106, 108]: 1. Measure the anhysteretic curve before controlling the flux. 2. Generate an initial waveform of the magnetic field intensity (i.e. the current) by using the inverse of the actual anhysteretic curve and the desired waveform of the magnetic flux density.

229


3. Start with the proposed initial waveform of current. 4. Measure the time variation of the magnetic field intensity and the magnetic flux density, i.e. the hysteresis characteristics. Use the proposed filtering method as well. 5. Compare the measured and the desired waveform of the magnetic flux density, i.e. build up the time variation of the error, which must be minimized. 6. Multiply the error function by a small constant value, which results in a control signal. 7. Modify the actual waveform of current by the controller signal sample by sample. 8. Repeat the sequence from step 4, while a given value of error is reached.

1

1

0.5

0.5 B [T]

B [T]

This algorithm converges in 8–10 steps. The proportional constant can be modified by another method during the algorithm, however, it is constant in this study.

0

−0.5

−1 −1000

0

−0.5

−500

0 H [A/m]

500

1000

(a) Before filtering

−1 −1000

−500

0 H [A/m]

500

1000

(b) After filtering

Fig. 5.54. The measured hysteresis loop

By this procedure the minor loops in Fig. 5.55(a) and in Fig. 5.55(b) are measured when f = 5 Hz. Figure 5.56(a) and Fig. 5.56(b) show distorted flux pattern with the frequency of f = 5 Hz and f = 50 Hz, with the waveform [106, 108] B(t) =0.463 cos(ωt + 36.33◦) + 0.234 cos(3ωt − 89.6◦ ) +0.141 cos(5ωt + 148.03◦),

(5.149)

B(t) =0.67 cos(ωt + 16.6◦) + 0.072 cos(17ωt + 163.8◦) +0.034 cos(19ωt − 177◦ ),

(5.150)

and

in the first and second case, respectively.


1.5

1.5

0.75

0.75 B [T]

B [T]

230

0

−0.75

0

−0.75

−1.5 −2000

−1000

0 H [A/m]

1000

−1.5 −2500

2000

(a) Measured minor loops along a loop

−1250

0 H [A/m]

1250

2500

(b) Measured second order minor loops

1.5

1.5

0.75

0.75 B [T]

B [T]

Fig. 5.55. Measured minor loops

0

−0.75

0

−0.75

−1.5 −1

−0.5

0 H [A/m]

0.5

(a) Flux waveform of (5.149)

1 x 10

4

−1.5 −1

−0.5

0 H [A/m]

0.5

1 x 10

4

(b) Flux waveform of (5.150)

Fig. 5.56. Measured higher order minor loops B. Simulation of measured static curves by the scalar NN model The measured first order descending curves (containing 46 branches) generated by the excitation waveform (5.72) are plotted in Fig. 5.57(a). In this case n = 20, but there are some measured curves near the coercive field where the characteristics are very steep and there is no information enough about the variation of the curves in the vicinity of coercive field. After applying the proposed preprocessing technique, a 2D surface can be obtained as shown in Fig. 5.57(b). The training sequence contained 1296 measured points. The measurement was performed at f = 0.2 Hz frequency on a toroidal shape structural steel made of material C19 [115]. In this case, only technical saturation can be described by the values Hm = 1.433 · 104 A/m and Bs = 1.688 T. Higher magnetic field intensity could not be generated by the applied power supply.

231


2

1 0.5 B/Bs

B[T]

1

0

0 −0.5 −1 0 −0.25

−1

−2 −1.5

−0.75

0 H[A/m]

0.75

x 104

1.5

(a) The measured first order reversal curves

1 0.5

−0.5 −0.75 ξ

0 −0.5 −1 −1

H/Hm

(b) The normalized and preprocessed first order reversal curves

Fig. 5.57. The measured and the preprocessed training sequence

It is very important to verify a NN based model by comparing the simulated and the measured data, which are not in the training sequence. Symmetric minor loops have been measured and the comparison between measurements and simulations can be seen in Fig. 5.58. The maximum value of the calculated error Bm − Bs △B = 100 Bm,s

(5.151)

is 35%, 10% and 8% respectively, where Bm and Bs are the measured and the simulated induction, respectively and Bm,s = 1.688 T is the measured induction in the saturation state. The maximum difference can be located near the coercive field as highlighted in the figures. The measured major loop and the measured first order reversal curves (i.e. the training sequence) can be approximated with an error value less than maximum 4 − 5%. To compare the measured and the simulated characteristics, the hysteresis losses from the enclosed area of the hysteresis loops are determined for 20 measured and simulated symmetric minor loops. The calculated relative error in the losses with respect to the measured curves is plotted in Fig. 5.59.

5.4.2 Vector hysteresis measurement system The aim of vector hysteresis measurement is to measure and to study the vector properties of ferromagnetic materials in case of alternating and rotational flux. Several techniques have been developed to achieve the required data, mainly the Epstein frame and the single sheet tester. The aim of measurements is to determine the power losses and the variation of the permeability of the magnetic material and of course to construct accurate vector hysteresis models, which can be built into electromagnetic field calculation software [39, 44–46, 61, 62, 72, 77, 81, 86, 122, 161–165, 193].

232


1.6

Simulated Measured

B [T]

0.8

0

−0.8

−1.6 −800

1.6

−400

0 H [A/m]

400

800

0 H [A/m]

2000

4000

0 H [A/m]

3000

6000

Simulated Measured

B [T]

0.8

0

−0.8

−1.6 −4000 1.6

−2000

Simulated Measured

B [T]

0.8

0

−0.8

−1.6 −6000

−3000

Fig. 5.58. Comparisons between the simulated and the measured symmetric minor loops, Hmax = 718 A/m, Hmax = 3580 A/m and Hmax = 5750 A/m

233


25

20

∆Wm [%]

15

10

5

0 0

2

4

6

8

10 12 Minor loop

14

16

18

20

Fig. 5.59. The error in the hysteresis area between the measured and the simulated characteristics A. The measurement set-up The measuring arrangement of the 2D vector magnetic hysteresis can be seen in Fig. 5.60 [122]. The orthogonal magnetic field intensity vector H has been generated simultaneously by the currents of two orthogonal exciting coils with 2 × 160 turns supplied by two KIKUSUI PBX 20-20 power supplies, which can follow any analogous control signals represented by its discrete sets generated by the mathematical functions of LabVIEW. The LabVIEW generates analog output on the DAQ card through the card-driver, which is located inside of computer. This analog signal is followed by the above device of power supply. The appropriate combination of the excitations can create any kind of magnetic field intensity vector. There are two coils orthogonal to each other (called B-coils) slipped into holes of the magnetic material to measure the two components of the induction vector B inside the ferromagnetic material and two other coils (called H-coils) superimposed on each other to detect the two orthogonal components of the magnetic field intensity vector H on the surface of the material (the tangential component of the magnetic field intensity is continuous on the surface). The B-coils have NB = 15 turns and the area of them is SB = 103 mm2 , the H-coils have NH = 320 turns on a 20 mm× 20 mm flan (the area of the inner coil isSH = 8.55 mm2 and the area of the outer coil is SH = 33.44 mm2 ). The induced voltages uH (t) and uB (t) on the four measuring coils can be acquired directly and the components of the magnetic field intensity vector and the induction vector can be calculated as H(t) = H0 +

1 µ0 SH NH

Z

0

t

uH (τ ) dτ,

(5.152)

234

5


20

85

excitation

B-coils H-coils

y

yoke x

Fig. 5.60. The 2D vector hysteresis measuring arrangement

and

B(t) = B0 +

1 SB N B

Z

t

uB (τ ) dτ,

(5.153)

0

where µ0 is the vacuum permeability, moreover H0 and B0 are integration constants. There is no air gap in the specimen, it is a compact arrangement. The computer communicates with the measuring environment through the NI-DAQ BNC-2090 panel, which wired directly to the cards inside of the computer. The data acquisition and the generation of excitation currents are happening simultaneously. The controlling of measurement and post-processing of measured data are performed in LabVIEW environment. Components of the magnetic flux density and magnetic field intensity vectors are calculated from measured data during the post-processing phase, which are stored in files. The most important parameters (amplitude, phase, frequency and signal type of the exciting current, number of periods to be measured, number of scanning per period, scanning rate, limits of channels of DAQ, parameters of the coils and so on) can be manipulated through the graphical user interface.

235


B. Measuring results The signals of measuring coils are very small and buried with noise of the DAQ card. Especially the measurement of the signal of H-coils is very difficult, because the measured voltages are in the range of just some millivolts. After integration and subtracting this effect, signals of the magnetic field intensity and induction are symmetrical. The major problem of the measurement is that the magnetic field intensity vector is not sinusoidal, however, the exciting current is sinusoidal, so an iterative method must be performed during measurement process (see Fig. 5.61). It means that some feedback must be applied according to the actually measured hysteresis curves to modify the waveform of the exciting currents to make the components of the magnetic field intensity vector or the induction vector sinusoidal. In this case a simple successive approximation technique has been applied, so that the waveform of current has been modified according to the difference of the actual induction (or magnetic field intensity) and the reference signal. After some iteration the prescribed waveform can be reached with as small error as the user want to reach. Reference signal, Bref

Bref

i (t)

Measured B(t)

D i (t) Calculating correction

B-Bref

Fig. 5.61. The iterative process As a first measurement result, the trace of the magnetic field intensity vector has been examined under clockwise rotational and arbitrary alternating flux density. Fig. 5.62(a) and Fig. 5.62(b) show the changes of the H and the B loci with increasing the magnetic flux density in the interval [0.25 T, · · · , 1.5 T] under rotational flux density. Direction x is the rolling direction, the perpendicular y direction is the transverse direction of the material according to the rolling and manufacturing process. The B trace is almost circular supervised by the iterative algorithm, the ideal circular shape is denoted by points in the figure. At low flux density (under 0.25 T) the H vector trace is almost elliptical, but at higher flux densities two hard axes appear as it can be seen in the figure, the trace of the H vector is extended in about 6 degree measured from the x and y directions. They are almost perpendicular to each other. There are also two easy axes as it can be seen in Fig. 5.62(a), but they are not in the bisectrix of the two hard axes. The main cause of this phenomena is the remanent magnetization of the material arising the rotational field of the magnetization process. An

236


3

x 10 4

2

1

By [T]

Hy [A/m]

1.5

0

−1.5

−3 −3

0

−1

−1.5

0 Hx [A/m]

1.5

3 x 10 4

−2 −2

−1

(a) The resulted H loci

0 Bx [T]

1

2

(b) The resulted B loci

Fig. 5.62. Magnetic field intensity and magnetic flux density vector under rotational flux density (o − 0.25 T, − 0.5 T, ∇ − 0.75 T, △ − 1 T, × − 1.25 T, ∗ − 1.5 T) example for the time variation of the magnetic flux density vector and the magnetic field intensity vector can be studied in Fig. 5.63. Figure 5.64 shows the resulted current shapes after the iteration process, while the initial shape of the exciting currents is sinusoidal. 3−1.6

Bx [T]

0

0.8

1.5

Hy [A/m]

1.61.6

0.8

0

0

−1.5

−3 −3

By [T]

x 10

−0.8

4

−0.8

−1.5

0 Hx [A/m]

1.5

−1.6 3 x 10 4

Fig. 5.63. Time variation of the H and B loci under rotational flux density A linear polarized flux (the x and y projection of the B vector is sinusoidal) is generated at directions 0◦ , 22.5◦ , 45◦ , 67.5◦ , 90◦ , 112.5◦, 135◦ and 157.5◦. Results are plotted in Fig. 5.65(a) and in Fig. 5.65(b). The resulted magnetic field intensity vectors are not the same in all the given directions. A hard axis can be found when the angle of magnetic flux density vector is 157.5◦, the maximum value of the magnetic field intensity

237


8

i y(t)

i(t) [A]

4

0

i x(t) −4

−8

0

0.5

1 t [s]

1.5

2

Fig. 5.64. Waveform of exciting currents, ix (t) and iy (t) after iteration

1500

0.5

B y [T]

1

H y[A/m]

3000

0

−0.5

−1500

−3000 −3000

0

−1500

0 H x [A/m]

1500

(a) The resulted H vector

3000

−1 −1

−0.5

0 B x [T]

0.5

1

(b) The resulted B vector

Fig. 5.65. Magnetic field intensity and magnetic flux density vector under alternating flux density vector is about 3000 A/m in this direction and about 2000 A/m perpendicular to this axis. Measurements are almost symmetrical to this direction. After dividing the x − y plane into more direction this easy axis can be detected more precisely as well as the hard axis of the material.

6 The polarization method and the fixed point technique This short chapter, based on the publications [20, 26, 37, 65–69, 73, 90, 91, 101, 142, 143] presents a possible method to handle nonlinear characteristics of ferromagnetic materials in electromagnetic field computation and a technique to solve the resulting system of nonlinear equations. According to the so-called polarization method, the output of the nonlinear model is split in two parts, a linear part and a nonlinear part. The system of nonlinear equations can be solved by the fixed point iteration technique. First, the general form of nonlinear equations is shortly presented, then a simple illustrative example is solved in the first part of this chapter. In the second part, the most important definitions and formulations are shown, which are important to understand the introduction of the polarization technique and the application of the fixed point based iteration technique.

6.1 Solution of nonlinear equations A nonlinear equation F (x) = 0,

(6.1)

or a system of nonlinear equations F(x) = 0 (here F is a vector valued function) generally can be solved by iterative methods. Iteration means a repeated sequence of operations, which must lead to the solution of the nonlinear equation. There are two main groups of methods, the successive approximation based methods and the techniques based on Newton’s method. Here, only the first family will be analyzed, the so-called fixed point iteration scheme is introduced to solve nonlinear electromagnetic field problems. The successive approximation method is based on the following form of nonlinear equations: x = f (x),

(6.2)

239

6.1. SOLUTION OF NONLINEAR EQUATIONS

which is usually called the fixed point form of nonlinear equations (the system of nonlinear equations can be represented by the vector valued function x = f(x)). Let us follow the iteration sequence illustrated in Fig. 6.1. Here, the fixed point equation x = f (x) is solved. A simple function y = f (x), as well as the function y = x are presented here. The point denoted by c, where c = f (c), is called fixed point, which is the solution of the nonlinear equation and it is the common point of the functions y = f (x) and y = x. The iteration can be started by any arbitrary point x0 , y0 = f (x0 ) can be calculated by using the function f (x). Value y0 is the input of the second step of iteration, i.e. x1 = y0 = f (x0 ) as it is illustrated by the linear function y = x. This results in y1 = f (x1 ) and x2 = y1 = f (x1 ). This iteration scheme can be generalized as xn+1 = f (xn ),

n = 0, 1, · · · .

(6.3)

The sequence shown in Fig. 6.1 leads to the solution of the nonlinear equation x = f (x), which is called the fixed point, because its coordinates are fixed, they do not change if n → ∞. This iteration is convergent, meaning that the distance |xn+1 − xn | is decreasing by increasing the index n. The condition |xn+1 − xn | < ε

(6.4)

can be used to stop the iteration, where ε is a small positive real number. The example shown in Fig. 6.2 shows an iteration, which is not convergent. This nonlinear equation can not be solved by the above presented iteration. The following can be summarized after this illustration. Let us denote the set of the input (argument) and of the output (value) of the function f by I and O, respectively. Convergent iteration can be realized only if O ⊂ I and function f only has solution in I, y

y=x

y = f (x)

c = f (c)

c · ·x·

2

x1

x0

Fig. 6.1. Convergent iteration to a fixed point

x

240

6. THE POLARIZATION METHOD AND THE FIXED POINT TECHNIQUE

y y = f (x)

y=x

c = f (c)

c x0

x1

x

Fig. 6.2. Iteration does not converge to the fixed point if it is continuous in I. Figure 6.1 illustrates that the rate of change of function y = f (x) should not exceed the rate of change of the function y = x, i.e. df (x) dx df (x) < (6.5) dx dx ⇒ dx < 1.

The iteration scheme (6.3) is convergent in this case. Moreover, it can be seen in Fig. 6.1, that the smaller the rate of change of function f , the smaller the number of iterations needed to reach the fixed point. In practical situations, the derivative of function f can not be calculated analytically, or it is a time consuming task. In this case the formulation |f (x) − f (y)| 0 there exists N such that d(x, xn ) < ε,

if

n ≥ N.

(6.35)

Cauchy sequence. Let X be a metric space with metric d. A sequence given as {x1 , x2 , · · · , xn } in X is said to be a Cauchy sequence if for each ε > 0 there exists N such that n ≥ N and m ≥ N implies d(xn , nm ) < ε. It is evident that a convergent sequence is Cauchy sequence.

(6.36)

247

6.3. METRIC SPACES

Complete metric space. A metric space X is said to be complete if each Cauchy sequence in X is convergent to a point in X. A subset of X (denoted by A ⊂ X) is said to be complete if each Cauchy sequence in A is convergent to a point in A. The definition of complete metric spaces is very important in Banach fixed point theorem. Before defining this theorem, some useful definitions are repeated here. Let f be a mapping from a metric space X to itself, i.e. f : X → X (it can be defined on a subset of the metric space X, of course). The mapping f is usually an operator or a function defined on the metric space X. Nonexpansive mapping. The mapping f is said to be nonexpansive, if ||f (x) − f (y)|| ≤ ||x − y||,

for all x, y ∈ X.

(6.37)

Lipschitzian mapping. The mapping f is said to be L-Lipschitzian, if ||f (x) − f (y)|| ≤ L||x − y||,

for all x, y ∈ X.

(6.38)

where L > 0 is a positive constant, called Lipschitz constant. Contraction mapping. The mapping f is said to be Banach contraction, if f is L-Lipschitzian, moreover Lipschitz constant is lower than 1 (denoted by q), ||f (x) − f (y)|| ≤ q||x − y||,

for all x, y ∈ X.

(6.39)

6.3.3 Banach fixed point theorem The Banach fixed point theorem (contraction mapping principle) was formulated by Stefan Banach (1892–1945) in 1922 [91]. The fixed point of a function f : X → X is a point x ∈ X such that f (x) = x.

(6.40)

Banach fixed point theorem is as follows. Let X be a complete metric space. Let f : X → X be a contraction mapping from metric space X to itself with contractivity coefficient q. Let x0 ∈ X be an arbitrary starting point of the iteration xn+1 = f (xn ),

n ≥ 0.

(6.41)

Mapping f has a unique fixed point c and the sequence xn converges to c, i.e. f (c) = c,

(6.42)

d(c, xn ) ≤ q n d(c, x0 ),

(6.43)

and

or equivalently d(c, xn+1 ) ≤

q d(xn+1 , xn ), 1−q

or d(c, xn ) ≤

qn d(x1 , x0 ). 1−q

(6.44)

248


The last inequality is an applicable estimate of the error in xn if q is known. The speed of convergence is depending on the value of the contractivity coefficient q. The smaller the value of q the faster the convergence, i.e. the aim is to find as small contractivity coefficient as possible.

6.4 The optimal value of parameters µ and ν 6.4.1 Using the inverse characteristics The fixed point iteration (6.25) is convergent if the mapping φβ {·} is contraction, i.e. (6.39) must be satisfied in every point of the domain under investigation, ||φβ {β1 } − φβ {β2 }|| ≤ q||β 1 − β 2 ||,

for all β 1 , β 2 ,

and 0 < q < 1. Here || · || means the square of the inner product, sZ sZ √ ||x|| = < x, x > = x · x dΩ = |x|2 dΩ. Ω

(6.45)

(6.46)

Ω

The relation (6.45) is difficult to use, a simpler relation can be formulated, as it is proved in the following. Fortunately, the set of Maxwell’s equations is nonexpansive, i.e. ||M {β 1 } − M {β2 }|| ≤ ||β1 − β 2 ||,

for all β 1 , β 2 .

(6.47)

Moreover, the parameter µ can be selected such a way that the operator (6.17) is a contraction mapping, ||fB {B 1 } − fB {B 2 }|| ≤ q||B 1 − B 2 ||,

for all B 1 , B 2 ,

(6.48)

where 0 < q < 1 must be satisfied. Relation (6.48) together with (6.47) is equivalent with (6.45) and this results in the condition for the parameter µ. Here, B 1 = M {β1 } and B 2 = M {β2 } and the right-hand side of relation (6.48) can be reformulated by using (6.47), ||fB {B 1 } − fB {B 2 }|| ≤q||B 1 − B 2 || = q||M {β1 } − M {β2 }|| ≤q||β1 − β 2 ||.

(6.49)

The left-hand side of (6.48) can also be rewritten as ||fB {M {β1 }} − fB {M {β2 }}|| ≤ q||β 1 − β2 ||,

(6.50)

which is the same as the contraction mapping (6.45) by using (6.25). This means that the mapping φβ {·} is a contraction if the mapping fB {·} defined by (6.17) is a contraction. Finally, the parameter µ can be selected so that fB {·} is a contraction. A possible solution is shown in the following. The relation (6.48) can be rewritten as

i.e.

||fB {B 1 } − fB {B 2 }|| ≤ q, ||B 1 − B 2 ||

for all B 1 , B 2 ,

(6.51)

249

6.4. THE OPTIMAL VALUE OF PARAMETERS µ AND ν

||∆β|| ||β 1 − β 2 || = ≤ q, ||B 1 − B 2 || ||∆B||

for all B 1 , B 2 ,

(6.52)

by using the notations ∆β = β 1 − β2 and ∆B = B 1 − B 2 . The integral in (6.46) can be omitted, if inequality (6.52) is true in every point of the domain Ω. These quantities can be decomposed into their three orthogonal components, ∆β = ∆βx ex + ∆βy ey + ∆βz ez ,

(6.53)

∆B = ∆Bx ex + ∆By ey + ∆Bz ez ,

(6.54)

and

where ex , ey and ez are the three orthogonal unit vectors. The inequality (6.52) can be rewritten as ∆βx2 + ∆βy2 + ∆βz2 ∆β · ∆β ≤ q2 . = ∆B · ∆B ∆Bx2 + ∆By2 + ∆Bz2

(6.55)

The upper bound of this fraction is ∆βx2 /∆Bx2 , or ∆βy2 /∆By2 , or ∆βz2 /∆Bz2 . The upper bound ∆βx2 /∆Bx2 can be rewritten as

∆βx ∆Bx

2

=

∆Bx − µ∆Hx ∆Bx

2

=

2 µ 1− , µ∆x

(6.56)

where µ∆x is the differential permeability, µ∆x =

∆Bx Bx,1 − Bx,2 = ∆Hx Hx,1 − Hx,2

(6.57)

which has two extreme values along the hysteresis curve µmin =

min

Hx,1 ,Hx,2

Bx,1 − Bx,2 , Hx,1 − Hx,2

and µmax =

max

Hx,1 ,Hx,2

Bx,1 − Bx,2 . Hx,1 − Hx,2

(6.58)

Here µmin and µmax are the minimum and maximum slope of the hysteresis characteristics. The inequality µmin ≤ µ ≤ µmax

(6.59)

is obvious, from which the inequalities 1−

µ µmin

< 0,

and 1 −

µ µmax

moreover 2 2 µ µ 1− ≤ 1− , µ∆x µmin are valid.

> 0,

(6.60)

and

2 2 µ µ 1− ≤ 1− (6.61) µ∆x µmax

250


These expressions result in 2 2 µ µ 1− ≤ 1− ≤ q2 , µ∆x µmin

and

2 2 µ µ 1− ≤ 1− ≤ q2 , µ∆x µmax (6.62)

i.e. − 1−

µ µmin

≤ q,

and 1 −

µ µmax

≤ q.

(6.63)

The minus sign on the left of the first inequality is coming from (6.60). From the last expressions, the inequalities µmin (1 + q) ≥ µ,

and µmax (1 − q) ≤ µ

(6.64)

are coming, which can be written as 0 < µmax (1 − q) ≤ µ ≤ µmin (1 + q) < 2µmin ,

(6.65)

because 0 < q < 1. This inequality occurs surely, if µmax (1 − q) ≤ µmin (1 + q),

(6.66)

µmax − µmin ≤ q < 1. µmax + µmin

(6.67)

hence

The least value of q is q=

µmax − µmin , µmax + µmin

(6.68)

and the corresponding optimal value of permeability is obtained from one of the equations in (6.64) by substituting the least value of q in (6.68), µo =

2 µmax µmin . µmax + µmin

(6.69)

The subscript o refers to the optimal value of permeability. The same formulations can be done to obtain the optimal value of permeability used in the y and z directions, too. If the three characteristics are the same, then µ is constant. This is the situation, when isotropic hysteresis model is used. Generally, the permeability is a tensor with three constants in the diagonal. Finally, in this case the inverse type hysteresis model must be applied whose input and output are the magnetic flux density and the magnetic field intensity, respectively, H = B −1 {B}. Equations (6.15) and (6.16) can be rewritten as B = µo H + β, and

where

µo =

2 µmax µmin , µmax + µmin

(6.70)

251

6.5. THE APPLIED FORMULATION, SUMMARY

β = B − µo B −1 {B}.

(6.71)

The value of µmin and µmax is defined in (6.58). In this case, the sequence (6.25) can be used in the nonlinear electromagnetic field computation. However, we use its inverse, summarized in section 6.5.1.

6.4.2 Using the direct characteristics The fixed point iteration (6.26) is convergent if the mapping φη {·} is contraction mapping, i.e. (6.39) must be satisfied, ||φη {η1 } − φη {η2 }|| ≤ q||η 1 − η 2 ||,

for all η 1 , η 2 ,

(6.72)

and 0 < q < 1. It is easy to see that this relation is very similar to (6.45). That is why, the optimal value of reluctivity in (6.20) can be obtained in the same way as it was presented in the previous section, however, β, B and µ must be changed to η, H and ν, respectively. Finally, in this case the direct type hysteresis model must be applied, which input and output are the magnetic field intensity and the magnetic flux density, respectively, B = B{H}. Equations (6.18) and (6.19) can be rewritten as H = νo B + η,

where νo =

2 νmax νmin , νmax + νmin

(6.73)

and η = H − νo B{H}.

(6.74)

The value of νmin and νmax are the minimum and the maximum slope of the inverse hysteresis characteristics. In this case, the sequence (6.26) can be used in the nonlinear electromagnetic field computation. The subscript o refers to the optimal value of reluctivity.

6.5 The applied formulation, summary 6.5.1 Using the inverse characteristics It may be better to use the reluctivity ν when applying the inverse hysteresis model, because ν(B) = ∂H/∂B. Let us now multiply the relation (6.70) by νo = 1/µo , νo B = H + νo β,

(6.75)

H = νo B − νo β.

(6.76)

i.e.

Here, we are going to denote the second term −νo β by I, H = νo B + I, and from (6.70),

(6.77)

252


νo =

νmax + νmin , 2

(6.78)

where νmax and νmin are the maximum and minimum slope of the inverse hysteresis characteristics. It is evident that νmax = 1/µmin and νmin = 1/µmax . This formulation is more convenient when obtaining the partial differential equations of electromagnetic field problems, i.e. the magnetic field intensity vector is defined by the sum of two terms, H = νo B + I. There is a condition obtained from (6.65), which must be satisfied, νo >

νmax . 2

(6.79)

The nonlinear iteration can be summarized as follows. The iteration can be started by an arbitrary value of I (0) , then in the nth iteration step (n > 0), (i) the magnetic flux density B (n) can be calculated by solving the partial differential equations obtained from Maxwell’s equations and using I (n−1) , in other words, B (n) = M {I (n−1) }, (ii) the magnetic field intensity H (n) can be calculated by applying the inverse type hysteresis model, H (n) = B −1 {B (n) }, (iii) the nonlinear residual term can be updated by using the magnetic flux density and magnetic field intensity, I (n) = H (n) − νo B (n) = B −1 {B (n) } − νo B (n) ,

(6.80)

(iv) and the sequence defined by steps (i)–(iii) must be repeated until convergence. Convergence criterion can be ||I (n) − I (n−1) || < ε,

(6.81)

where ε is a small positive real number. Convergence criterion can be defined by using the magnetic field intensity or the magnetic flux density, as well.

6.5.2 Using the direct characteristics It may be better to use the permeability µ when applying the direct hysteresis model, because µ(H) = ∂B/∂H. Let us now multiply the relation (6.73) by µo = 1/νo , µo H = B + µo η,

(6.82)

B = µo B − µo η.

(6.83)

i.e.

Here, we are going to denote the second term −µo η by R, B = µo H + R, and from (6.73),

(6.84)

253

6.5. THE APPLIED FORMULATION, SUMMARY

µo =

µmax + µmin , 2

(6.85)

where µmax and µmin are the maximum and minimum slope of the direct hysteresis characteristics. It is evident that µmax = 1/νmin and µmin = 1/νmax . This formulation is more convenient when obtaining the partial differential equations of electromagnetic field problems, i.e. the magnetic flux density vector is defined by the sum of two terms, B = µo H + R. There is a condition obtained from the ν-version of (6.65), which must be satisfied, µo >

µmax . 2

(6.86)

The nonlinear iteration can be summarized as follows. The iteration can be started by an arbitrary value of R(0) , then in the nth iteration step (n > 0), (i) the magnetic field intensity H (n) can be calculated by solving the partial differential equations obtained from Maxwell’s equations and using R(n−1) , in other words, H (n) = M {R(n−1) }, (ii) the magnetic flux density B (n) can be calculated by applying the inverse hysteresis model, B (n) = B{H (n) }, (iii) the nonlinear residual term can be updated by using the magnetic field intensity and magnetic flux density, R(n) = B (n) − µo H (n) = B{H (n) } − µo H (n) ,

(6.87)

(iv) and the sequence defined by steps (i)–(iii) must be repeated until convergence. Convergence criterion can be ||R(n) − R(n−1) || < ε,

(6.88)

where ε is a small positive real number. Convergence criterion can be defined by using the magnetic field intensity or the magnetic flux density, as well.

6.5.3 Proof of the nonexpansive property of Maxwell’s equations Theorem (6.47) is reformulated by using the notations in the formulation B = µH + R, ||M {R1 } − M {R2 }|| ≤ ||R1 − R2 ||,

for all R1 , R2 .

(6.89)

Let (H 1 , B 1 , R1 ), (H 2 , B 2 , R2 ) be two fields and polarizations having the same boundary conditions and same current densities J . The solution of Maxwell’s equations is unique, that is why the difference of these fields H d = H 1 − H 2 , B d = B 1 − B 2 verifies Z B d · H d dΩ = 0. (6.90) = Ω

254


This inner product can be reformulated as = − = 0,

(6.91)

where ν = 1/µ and Rd = R1 − R2 . It can be rewritten as = .

(6.92)

According to Schwartz inequality |< x, y >| ≤ ||x|| ||y||

(6.93)

the upper bound of the right-hand side of (6.92) can be predicted, ≤ ||B d || ||νRd ||.

(6.94)

It is obvious from (6.91) that the value of inner product is always positive because is always positive. Finally ≡ ||B d || ||νB d || ≤ ||B d || ||νRd ||,

(6.95)

i.e. the inequality ||νB d || ≤ ||νRd ||

(6.96)

is true. It can be written as ||ν(B 1 − B 2 )|| ≤ ||ν(R1 − R2 )||.

(6.97)

Parameter ν can be omitted if it is constant. Generally ν is tensor. This theorem can be proved by applying the formula H = νB + I, too, in a similar way.

7 Application of the finite element method 7.1 Introduction This chapter contains some illustrative examples, which have been solved on the basis of the previous methods by the help of COMSOL Multiphysics [30, 190]. This commercial finite element software has the advantage that the weak formulation of a problem can be inserted easily and it is not necessary to implement mesh generators, solvers and postprocessors. The user can focus on the problem and on the weak formulation, can design the geometry of the problem under test, can insert the weak form, the boundary and interface conditions according to the applied potential formulation. It is available to select nodal or edge finite elements. Using higher order approximation is also possible, COMSOL Multiphysics uses the scalar shape functions with the order from one to five and vector shape functions with the order from one to three. The mesh generation is performed automatically. Many solvers are implemented in COMSOL Multiphysics, so all of them can be tested and the best one, the most applicable to the problem can be selected. The postprocessor is perfect, simulated data and electromagnetic field quantities obtained from the potentials can also be simulated and plotted easily. These properties are very useful from the point of view of research, because the time consuming programming task can be eliminated. Seven problems are shown in the following. The first 2D magnetostatic problem illustrates how to select the impressed current vector potential, T 0 when applying magnetic scalar potential formulations. The problem contains a C-shaped magnet and the magnetic field is generated by a coil. Here, we show the false solution according to the cancellation error, the correct solution by the help of the two scalar potentials (total and reduced), finally the edge element representation of the impressed current vector potential will be proposed. The reference solution is obtained by the magnetic vector potential. Edge element representation of T 0 is very advantageous and useful in the other formulations based on the ungauged magnetic vector potential, too.

256

7. APPLICATION OF THE FINITE ELEMENT METHOD

The second 3D problem containing an iron cube placed in a homogeneous magnetic field illustrates the effect of gauging on the magnetic vector potential A. The problem has been solved without and with Coulomb gauge and A is approximated by nodal finite elements. Unfortunately, the solution of the gauged version is not correct on the iron/air interfaces, that is why two potentials are used, one in the air region and an other one in the iron part and only the tangential component is prescribed to be continuous on the iron/air interfaces. This modification can be implemented easily by COMSOL Multiphysics. Edge element representation of ungauged A is much better, because it results in less computation time. We will show the A − Φ and A − A − Φ formulations as well. The aim of these methods is to reduce the degree of freedom. The problem is solved by the reduced magnetic scalar potential, too, as a reference solution, because no currents are present. The third problem is a modification of the second one. It contains an iron cube made of conducting material placed in a homogeneous magnetic field, too. The problem illustrates the effect of mesh generation on skin effect problems. The problem has been solved by using the Coulomb gauged A, V − A and T , Φ − Φ formulations and by the ungauged version of them. The gauged vector potentials are approximated by nodal finite elements, the ungauged vector potentials are approximated by vector finite elements. The fourth problem is a Benchmark problem containing an excitation coil and steel plates around it. First, the problem is supposed to be linear, then nonlinear with saturating characteristics, but hysteresis is neglected. The problem has been solved by nodal and edge element representation of the magnetic vector potential and by reduced magnetic scalar potential combined with the edge element representation of the impressed current vector potential. The magnetic flux density inside the steel plates simulated by the reduced magnetic scalar potential is a little larger than the result calculated by the magnetic vector potential formulations. However, simulated results are close to measured ones. After linear simulations, the nonlinear problem has been solved by the fixed point technique. The next one is a Benchmark problem again. It consists of a plate made of aluminum containing a hole in it and an excitation coil generates time varying magnetic field. The effect of eddy currents are taken into account by applying different potential formulations. The problem contains a multiply connected region and the behavior of formulations are studied. The time varying problem has been solved in frequency domain and simulated results are compared with measured ones. The last problem is the analysis of a vector hysteresis measurement system, which is under construction in the Laboratory of Electromagnetic Fields, at the Department of Telecommunications, Széchenyi István University, Gy˝or. The simulations have been implemented in the time domain because of the hysteresis characteristics and we show a time stepping scheme with the fixed point technique. Hysteresis is taken into account inside the specimen by a 2D isotropic vector hysteresis model shown in chapter 5.

257

7.2. ILLUSTRATION HOW TO SELECT T 0 , THE C-MAGNET

7.2 Illustration how to select T 0 , the C-magnet The first problem focuses on how to represent the impressed current vector potential T 0 to eliminate cancellation error on the interface between iron and air when applying the magnetic scalar potential formulations (the Φ-formulation, the Ψ-formulation and the Φ − Ψ-formulation) [13, 16, 150, 167, 168, 187]. A simple 2D example has been chosen for illustration: a C-shaped yoke is magnetized by a coil as it can be seen in Fig. 7.1. The yoke is made of iron and a linear characteristics have been supposed with permeability µ = 1000µ0. The coil has only two turns placed symmetrically to the x-axis and the value of DC current is I = 1 A (x+I = x−I = 0, y+I = ±105 mm, y−I = ±35 mm). y

GB r m0

30 mm

+I 160 mm

-I GH

40 mm

-I iron m

x

30 mm

+I 160 mm

Fig. 7.1. C-shaped yoke magnetized by a coil

Only the half of geometry has been analyzed because of symmetry along the line y = 0. The tangential component of magnetic field intensity is vanishing here, i.e. the line y = 0 is a ΓH type boundary. Artificial far boundary is supposed at the distance r = 800 mm, where B · n = 0. It is denoted by ΓB . The value of r, i.e. the place of the artificial far boundary can be determined after some trials, the boundary condition along ΓB should not modify the magnetic field in the problem region. First of all, the problem has been solved by using the magnetic vector potential A as a reference solution, because the problem is 2D and the partial differential equation (2.199) with boundary conditions (2.200) and (2.201) can be used. Here K = 0 and α = 0. The magnetic flux density B inside and around the yoke can be seen in Fig. 7.2(a) and in Fig. 7.2(b). The system of linear equations has been solved by the direct solver UMFPACK [30, 190]. It is easy to see that the magnetic field intensity is perpendicular to the line y = 0, i.e. to the boundary ΓH and the magnetic flux lines follow the shape of the yoke.


200

100

150

75 y [mm]

y [mm]

258

100

50

0 −100

50

25

−50

0 x [mm]

50

0 0

100

(a) The magnetic flux density inside and around the yoke

25

50 x [mm]

75

100

(b) A magnified plot

Fig. 7.2. The magnetic flux density simulated by using the magnetic vector potential A as a reference solution The absolute value of magnetic flux density along the line x = −65, · · · , 65 mm, y = 65 mm (along the center of upper leg of the yoke) can be seen in Fig. 7.3. This result is used as a reference solution in the following. The impressed current vector potential has two components and it can be determined easily by using the well known relation of the magnetic field intensity according to one infinite long filamentary conductor H = I/2Rπ [51, 169], i.e. T 0 = T0,x ex + T0,y ey ,

(7.1)

where 3

x 10

−4

|B| [T]

2.25

1.5

0.75

0 −66

−33

0 x [mm]

33

66

Fig. 7.3. The absolute value of magnetic flux density along the line x = −65, · · · , 65 mm, y = 65 mm calculated by the magnetic vector potential

259


T0,x

y − 0.105 I y + 0.105 · 2 = + 2π x + (y − 0.105)2 x2 + (y + 0.105)2 y − 0.035 I y + 0.035 · 2 − + , 2π x + (y − 0.035)2 x2 + (y + 0.035)2

(7.2)

I x x + · 2 2π x + (y − 0.105)2 x2 + (y + 0.105)2 I x x − + . · 2 2π x + (y − 0.035)2 x2 + (y + 0.035)2

T0,y = −

(7.3)

The impressed current vector potential can be calculated by analytical expression in nodes of the finite element mesh and the reduced scalar potential can be used in the simulations. It is noted that there is no node where the filamentary conductors are placed. The partial differential equation (2.160) with the boundary conditions (2.161) and (2.162) can be applied to solve the problem. Here Φ0 = 0 since K = 0 and the tangential component of T 0 is equal to zero along ΓH , T 0 × n = 0, moreover b = 0 on ΓB . The formulation is very simple, however, the solution is catastrophic in the iron region as it can be seen in Fig. 7.4(a) (here first order elements are used). The absolute value of magnetic flux along the line x = −65, · · · , 65 mm, y = 65 mm calculated by the reduced magnetic scalar potential can be seen in Fig. 7.4(b). −3

200

3

100

50

0 −100

Coarse mesh Very fine mesh 3rd order element

2.25 |B| [T]

y [mm]

150

x 10

1.5

0.75

−50

0 x [mm]

50

100

(a) The simulated magnetic flux density inside and around the yoke

0 −66

−33

0 x [mm]

33

66

(b) The absolute value of magnetic flux along the line x = −65, · · · , 65 mm, y = 65 mm

Fig. 7.4. The distribution of magnetic flux density vector is catastrophic when using the reduced magnetic scalar potential Φ The accuracy of solution can not be increased very well by increasing the number of finite elements, however, applying higher order approximation results in better distribution of the magnetic flux density. Here, the coarse mesh consists of 782 triangles with 419 unknowns, the very fine mesh consists of 50048 triangles with 25241 unknowns and the

260


approximation is linear. The geometry discretized by 3128 triangles when third order approximation is applied and the number of unknowns is 14239. Let us now compare the value of impressed current vector potential T 0 and the gradient of the reduced magnetic scalar potential ∇Φ. It can be seen in Fig. 7.5 for different value of the relative permeability µr of the iron yoke. The difference between the two quantities is decreasing by increasing the relative permeability, i.e. |H| = |T 0 − ∇Φ| is decreasing inside the C-shaped yoke. The impressed current vector potential is expressed by analytical formulation, while the magnetic scalar potential is approximated by first order polynomial. The difference field can be very inaccurate if they are almost the same, which results in numerical difficulties. This problem is usually referred as cancellation error. This is the reason why the accuracy can be increased by increasing the order of approximating polynomials, however, this is not a general rule (the impressed current vector potential is very simple in this situation). 16 T0 Φ , µ =10 r

T0, Φ [A/m]

12

Φ , µ =100 r

Φ , µ =1000 r

8

4

0 −66

−33

0 x [mm]

33

66

Fig. 7.5. Comparison of the impressed current vector potential and the gradient of the reduced magnetic scalar potential There are two widely used solutions for this problem. The first is a combination of the reduced and the total scalar magnetic potential, the second is applying edge elements for the representation of impressed field. The origin of cancellation error is the presence of impressed current vector potential inside regions with high permeability. In iron regions the total scalar potential Ψ without T 0 should be used and the reduced scalar potential Φ with T 0 should represent the magnetic field in the air region. In this case the partial differential equations (2.177) and (2.178) must be solved satisfying the boundary conditions (2.179), (2.180), (2.181) and (2.182), moreover the interface conditions (2.183) and (2.184). In this example, Φ0 = Ψ0 = 0, because K = 0 and T 0 × n = 0, furthermore b = 0. The line integral in (2.183) can now be expressed by analytical formula, however, this is not general. Here a coarse mesh with 782 elements and 419 unknowns has been used again. Figure 7.6 shows the comparison between results simulated by the reduced scalar potential

261


formulation with third order approximation and the two potential approach with linear and second order approximation. The first order approximation of Φ − Ψ-formulation is much better than the linear one in Fig. 7.4(b), however, the mesh is the same. The result calculated by the second order elements (Φ − Ψ-formulation) is almost the same as the third order one (Φ-formulation). 4

x 10

−4

rd

Φ − 3 order element, fine st

Φ,Ψ − 1 order element, coarse nd

Φ,Ψ − 2

|B| [T]

3

order element, coarse

2

1

0 −66

−33

0 x [mm]

33

66

Fig. 7.6. Cancellation error can be eliminated by the Φ − Ψ-formulation (the first and the last curves are practically the same)

4

x 10

−4 nd

Φ,Ψ − 2

order element, coarse

T0 vector − 1st order element T0 vector − 2nd order element

3

rd

|B| [T]

T0 vector − 3 order element

2

1

0 −66

−33

0 x [mm]

33

66

Fig. 7.7. Cancellation error can be eliminated by applying edge elements in the approximation of impressed field (the last two curves are practically the same)

262


If the impressed current vector potential and the gradient of the reduced scalar potential are in the same function space, then the cancellation error can be eliminated as it is presented in Fig. 7.7. Here the result of second order approximation simulated by the Φ − Ψ-formulation has been compared by the first, the second and the third order vector approximation of impressed field. The impressed current vector potential T 0 has been represented by edge elements and the reduced scalar potential with the same order has been applied in the simulations. The second and the third order approximations are almost the same, however, the first order one is also accurate. The difference between the magnetic flux density simulated by the magnetic vector potential and by the last method is maximum 3 · 10−6 , i.e. the relative error is 1.8 %.

7.3 Iron cube in homogeneous magnetic field The problem is a very simple structure, an iron cube is situated in a homogeneous magnetic field [150]. The linear model can be seen in Fig. 7.8 (B 0 = (1 T)ez , µ = 1000µ0).

z B0

m0 iron m

y x

20 mm

100 mm

Fig. 7.8. Iron cube in a homogeneous magnetic field The three planes x = 0, y = 0 and z = 0 are symmetry planes. On the planes x = 0 and y = 0 the normal component of the magnetic flux density is vanishing, B · n = 0, i.e. they are ΓB type boundaries, the plane z = 0 is a ΓH boundary, where the tangential component of the magnetic field intensity is equal to zero, H × n = 0. Artificial far boundaries at x = ±50 mm, y = ±50 mm and z = ±50 mm are assumed, where the cube does not affect the magnetic field. The first and the second boundaries are ΓB type where the source magnetic flux is given by boundary conditions defined by potentials, the artificial far boundaries placed at z = ±50 mm are ΓH type, where H × n = 0. These simplifications result in the analysis of the eighth of the whole problem, x ≥ 0, y ≥ 0 and z ≥ 0 (see Fig. 7.9).

263

7.3. IRON CUBE IN HOMOGENEOUS MAGNETIC FIELD

z

GB

GH

y

GH GB x

Fig. 7.9. The analyzed region, the type of boundaries is also indicated

z [mm]

z [mm]

The aim of simulations is to compare the different potential formulations, the gauged and the ungauged magnetic vector potential approximated by nodal elements, the magnetic scalar potential, the combination of the gauged magnetic vector potential and the magnetic scalar potential, finally the magnetic vector potential approximated by vector elements. First, the effect of gauging on the magnetic vector potential A has been studied. The vector field of magnetic vector potential is not unique if Coulomb gauge is not used, but the magnetic field calculated from A is unique and of course, it is correct. The behavior of the ungauged magnetic vector potential is very strange as it is presented in Fig. 7.10(a). It is a very important experiment that the number of iterations of iterative solvers needed for the solution can be very high, because of the ill-conditioned mass matrix. The solvers sometimes do not converge in the case of ungauged vector potential (the conjugate gradient method with algebraic multigrid preconditioner has been applied in the simulations). Coulomb gauge ∇ · A = 0 can be easily satisfied by the modified

y [mm]

x [mm]

(a) The ungauged magnetic vector potential

y [mm]

x [mm]

(b) The gauged magnetic vector potential

Fig. 7.10. The effect of gauging on the magnetic vector potential

264


partial differential equation (2.223). The gauged magnetic vector potential as well as the boundary conditions according to (2.224)–(2.227) can be seen in Fig. 7.10(b). The number of iterations of the solver is decreasing by introducing Coulomb gauge (Fig. 7.11). The resulting magnetic flux density and the gauged magnetic vector potential can be seen in Fig. 7.12. The results seem to be accurate, moreover the number of iterations is decreasing. However, let us take a closer look at the solution along the line y = 0, z = 0 (Fig. 7.13(a)) and y = 10mm, z = 0 (Fig. 7.13(b)). The gauged solution is not the same as the ungauged one, especially in the second case. The problem is coming from the abrupt change of the permeability along the iron/air interface.

Relative tolerance

10

10

10

10

10

2

0

without gauging −2

−4

with gauging

−6

0

50 100 150 Number of iterations

200

z [mm]

Fig. 7.11. The number of iterations is decreasing by introducing gauging

y [mm]

x [mm]

Fig. 7.12. Magnetic flux density as vectors and the gauged magnetic vector potential in iron as streamlines

265

7.3. IRON CUBE IN HOMOGENEOUS MAGNETIC FIELD

The problem can be solved by the following modifications: (i) The solution is correct if gauging is not used, however, solving the system of the resulting equations can be very difficult, so this is not an advantageous way; (ii) Applying nodal finite elements to approximate the magnetic vector potential results in continuous normal and tangential components of the magnetic vector potential. Allowing the jump of the normal component of the magnetic vector potential can solve efficiently the problem. This means that a magnetic vector potential A1 has to be used inside iron and a magnetic vector potential A2 has to be used in the air

5

on

A − Gauge off A − Gauge on Ψ A1,A2

3.75 Bz [T]

A1,A2,Ψ

2.5

1.25

0 0

12.5

25 x [mm]

37.5

50

(a) Along the line y = 0, z = 0

6 A − Gauge off A − Gauge on Ψ A1,A2

on

4.5

z

B [T]

A1,A2,Ψ

3

1.5

0 0

12.5

25 x [mm]

37.5

50

(b) Along the line y = 10mm, z = 0

Fig. 7.13. Distribution of the magnetic flux density

266


region. There are two unknown vector functions along the iron/air interface, but A × n must be continuous; (iii) Using edge elements to represent unknown vector potentials. The problem can be solved easily by the reduced magnetic scalar potential Ψ as well. The main advantage of using magnetic scalar potential is the few number of unknowns, because it can be decreased from three per node to only one per node. A very advantageous formulation can be built up by applying the magnetic vector potential inside iron and the magnetic scalar potential in the air region (A−Ψ formulation). Unfortunately, the behavior of the magnetic field is very strange on the iron/air interface, which is caused by the weak coupling of the normal component of the magnetic flux density and of the tangential component of the magnetic field intensity along the iron/air interface. If the A/Ψ interface is moved from the iron/air interface into the air region, the problem disappears resulting the A − A − Ψ formulation. Comparison between the results of the two potential formulations can be seen in Fig. 7.14. The disadvantage of this formulation is the high number of iterations and the strange behavior of error during the iterations, but this can be eliminated by an appropriate solver, as it is presented in Fig. 7.15.

Ψ

z [mm]

z [mm]

Ψ

A

y [mm]

A

x [mm]

(a) By A − Ψ-formulation

y [mm]

A

x [mm]

(b) By A − A − Ψ-formulation

Fig. 7.14. Magnetic flux density calculated by using Φ in the air region R The value of flux inside the cube over the surface z = 0, Γ Bz dΓ has been computed and compared. The reference result of paper [150] is 0.417 mVs. The number of elements in the tetrahedral mesh, the number of equations, the number of iterations (conjugate gradient method with algebraic multigrid preconditioner) and the resulting flux are shown in Table. 7.1. The magnetic flux calculated by the magnetic vector formulation is a little lower, the magnetic flux calculated by using the magnetic scalar potential is a little higher when using the same mesh. Comparison between the performance and results of different formulations is collected in Table. 7.1.

267

7.4. CONDUCTING CUBE IN HOMOGENEOUS MAGNETIC FIELD

10

Relative tolerance

10

4

2

AMG

10 10 10

0

−2

−4

GMG

10

−6

0

40 80 120 Number of iterations

160

Fig. 7.15. The performance of iteration is sensitive for the preconditioner when applying magnetic scalar potentail in the air region (AMG-Algebraic Multigrid, GMG-Geometric Multigrid) Table 7.1. Comparison between the performance and results of different formulations A-formulation No. of No. of No. of Flux FEs eqs. iter. [mVs] 146 915 11 0.339 12216 20 0.399 2552 39 0.411 16838 73812 103034 434256 88 0.414

A1 − A2 -formulation No. of No. of Flux eqs. iter. [mVs] 972 11 0.339 12309 21 0.399 74139 40 0.411 435087 96 0.414

Ψ-formulation No. of No. of Flux eqs. iter. [mVs] 306 5 0.346 3919 10 0.438 24990 12 0.423 145989 16 0.419

7.4 Conducting cube in homogeneous magnetic field The problem is a modification of the previous one, i.e. a conducting cube made of iron is situated in a homogeneous magnetic field resulting a linear eddy current field problem. The linear model can be seen in Fig. 7.8, too (B 0 = (1 T)ez , µ = 1000µ0, but the conductivity of the cube is σ = 2 · 106 S/m). The problem has been solved in the time domain and the simulated frequencies are f = 0.1 Hz, f = 1 Hz, f = 10 Hz and f = 100 Hz, and the skin depth is δ = 35.59 mm, δ = 11.25 mm, δ = 3.559 mm and δ = 1.125 mm, respectively, calculated by the relation [51, 79, 169] r 1 δ= . (7.4) πf µσ The number of calculated periods is 3 and every periods are divided into 72 time instants, i.e. ω∆t = 5◦ , where ω is the angular frequency of the external magnetic field and ∆t is the time step.

268


The problem has been solved by the A, V − A and by the T , Φ − Φ formulations and the vector potentials have been approximated by nodal and vector shape functions of second order. In case of nodal approximation, the A1 , V − A2 formulation is used, i.e. the magnetic vector potential A1 is used in the cube and A2 in the air region, similarly to that presented in the last example. The aim was to compare the results obtained by these four formulations. The finite element mesh of the problem can be seen in Fig. 7.16 in case of f = 0.1 Hz and f = 1 Hz. Here, only the discretized cube is shown. The variation of the z component of magnetic flux density inside the cube is almost linear, because the frequency is small enough. The four methods result in practically the same magnetic flux density as it is plotted in Fig. 7.17.

Fig. 7.16. The FEM mesh of the eighth of the problem when f = 0.1 Hz and f = 1 Hz (the mesh in air is not shown)

(a) Magnetic flux inside the cube, f = 0.1 Hz

(b) Magnetic flux inside the cube, f = 1 Hz

Fig. 7.17. Magnetic flux density calculated by using the four potential formulations along the line x ∈ [0, · · · , 10]mm, y = 10 mm, z = 0 mm


269

Here, the z component of the magnetic flux density is shown in the third period, when the external flux density is decreasing from +1 T to −1 T (curves are denoted by the integer k and curves are plotted at the time instants tk = (2.25 + k/18)/f ). In small frequency, the eddy currents have very small effect. It is our experience that worse results can be obtained at higher frequency when using the same mesh. The effect of eddy currents is higher and higher when increasing the frequency of the excitation and it is observable as the so-called skin effect [51, 79, 169], i.e. the magnetic flux density and the magnetic field intensity are crowded out from the interior parts of the cube. When the frequency is high enough, the magnetic field can almost be equal to zero in the central part of the cube. The changing (i.e. the gradient) of the magnetic field can be very small inside the cube and it can be very fast close to the surface when the frequency is higher and higher. The information, i.e. the change of magnetic field is crowded out to the surface, that is why finer and finer mesh close to the surface and as coarse mesh as possible in the interior space should be generated. Increasing the degree of the approximating functions is not the only way to solve this problem, the mesh must also be modified. After some trials, the finite element mesh plotted in Fig. 7.18 and in Fig. 7.19 are generated to reach accurate results at f = 10 Hz and f = 100 Hz, respectively. The mesh is finer close to the surface and it is coarser inside the cube. This modification has solved the problem of worse results as it can be seen in Fig. 7.20 and in Fig. 7.21 when the frequency is f = 10 Hz and f = 100 Hz, respectively. The figures show the difference between the solutions calculated by using the A, V − A and the T , Φ − Φ formulations. The magnetic flux density simulated by the magnetic vector potential is first order, because the magnetic vector potential is second order. However the magnetic flux density calculated by the second order current vector potential and the magnetic scalar potential is smoother than the one calculated by the magnetic vector potential. Even so, the results are comparable.

Fig. 7.18. The FEM mesh of the eighth of the problem when f = 10 Hz (only the cube is shown)

270


Fig. 7.19. The FEM mesh of the eighth of the problem when f = 100 Hz (only the cube is shown)

(a) Magnetic flux inside cube, calculated by the A, V − A formulation

(b) Magnetic flux inside cube, calculated by the T , Φ − Φ formulation

Fig. 7.20. The magnetic flux density at f = 10 Hz along the line x ∈ [0, · · · , 10]mm, y = 10 mm, z = 0 mm The magnetic field is almost equal to zero when the distance measured from the surface is higher than 5δ [51, 79, 169]. It can be seen easily in Fig. 7.21. As a consequence of the above simulations, we can say that the magnetic flux density simulated by nodal or vector representation of the unknown potentials leads to the same results, however, this is not true for simulation time. The computation time is smaller in

271


(a) Magnetic flux inside cube, calculated by the A, V − A formulation

(b) Magnetic flux inside cube, calculated by the T , Φ − Φ formulation

Fig. 7.21. The magnetic flux density at f = 100 Hz along the line x ∈ [0, · · · , 10]mm, y = 10 mm, z = 0 mm the case of vector representation of the magnetic vector potential and in the case of nodal approximation of the current vector potential at low frequency, but in the case of vector representation of the current vector potential in higher frequency. Results are presented in Table 7.2 and in Table 7.3. Here, DOF means the number of degree of freedom, i.e. the number of unknowns. The number of steps means the average number of steps of the whole time stepping scheme. It is important to note that the ungauged A, V − A formulation can be reformulated by using V = 0 as A⋆ − A formulation. The performance of this formulation is very bad, the number of step is 1022, 1014 and 928 in the case of f = 0.1 Hz, f = 1 Hz and f = 10 Hz, moreover the convergence was very weak when f = 100 Hz. The mesh was the same as it is presented in Table 7.2.

Table 7.2. Computational costs of using the magnetic vector potential Formulation A, V − A, nodal A, V − A, vector A, V − A, nodal A, V − A, vector A, V − A, nodal A, V − A, vector A, V − A, nodal A, V − A, vector

Frequency [Hz] 0.1 0.1 1 1 10 10 100 100

DOF 20180 25324 20180 25324 59579 76356 105008 136843

No. of steps 105 50 112 50 219 72 495 90

272


Table 7.3. Computational costs of using the current vector potential Formulation T , Φ − Φ, nodal T , Φ − Φ, vector T , Φ − Φ, nodal T , Φ − Φ, vector T , Φ − Φ, nodal T , Φ − Φ, vector T , Φ − Φ, nodal T , Φ − Φ, vector

Frequency [Hz] 0.1 0.1 1 1 10 10 100 100

DOF 15622 20456 15622 20456 49458 66773 86400 117297

No. of steps 17 32 19 34 53 49 383 98

7.5 Steel plates around a coil, linear problem This problem is a modification of Problem No. 10 of the TEAM Workshops [40, 135, 150]. The original task is to compute the transient eddy currents in saturated steel plates around racetrack-shaped coil excited by exponentially rised current. Here, the problem is simplified, the current as well as the permeability of steel are constant and static magnetic field has been simulated. The problem has been solved by the ungauged A-formulation represented by edge finite elements first, then by the gauged A-formulation approximated by nodal FEM to study the effect of gauging. Here the generalized minimum residual method (GMRES) with algebraic multigrid preconditioner has been applied to solve the linear system of equations. The FEM mesh of the eighth of the problem can be seen in Fig. 7.22 (it is enough to simulate the eighth of the problem because of symmetry). The convergence of the ungauged version is very bad, but gauging can speed up the solution of the system

Fig. 7.22. The FEM mesh of the eighth of the problem

273

7.5. STEEL PLATES AROUND A COIL, LINEAR PROBLEM

of linear equations (e.g. a mesh with 76662 tetrahedral elements and second order FEM approximation results in 328116 equations, which can not be solved without gauging – normalized residual was only 0.13 after 500 steps and it did not decrease –, however, it is 274 steps with gauging when the error limit is 10−6 ). Unfortunately, the magnetic flux density calculated from the gauged magnetic vector potential is not correct, especially in the vicinity of iron/air interface, because of the continuous normal component of the magnetic vector potential. Generating finer and finer mesh does not solve this problem. If the jump of the normal component of magnetic vector potential is allowed, the problem can be eliminated, however, the number of unknowns is increasing a bit as well as the number of iterations (in this case, the number of unknowns is 341109 and the number of iterations is 780). The magnetic flux density vectors inside the plate and inside the central plate can be seen in Fig. 7.23 and in Fig. 7.24.

80

80

60

60 z [mm]

z [mm]

Fig. 7.23. The magnetic flux is driven by the plates (current density inside coil can also be seen)

40

20

0

40

20

0

7.5

15 22.5 30 y [mm] (a) Continuous magnetic vector potential

0

0

7.5

15 22.5 30 y [mm] (b) Normal component of A is not continuous

Fig. 7.24. Magnetic flux density vector calculated by using the gauged magnetic vector potential approximated by nodal finite elements

274


The magnetic flux should be almost constant inside the specimen, Fig. 7.25 shows the results simulated by different A-formulations using the same mesh. It must be noted that the solution without gauge fix has been solved by the direct solver SPOOLES. 3 Gauge fix off Gauge fix on Gauge fix on, finer mesh Gauge fix on, normal jump

z

B [T]

2.25

1.5

0.75

0 0

7.5

15 y [mm]

22.5

30

Fig. 7.25. The z component of the magnetic flux density, Bz , along the line x = 0, y = 0, · · · , 30 mm, z = 0 3 T0,Φ

2.25

z

B [T]

A

1.5

T0,Φ T0,Φ, finer mesh

0.75

T0,Φ, 3rd order A, nodal T0−A, vector T0−A, vector, finer mesh

0 0

7.5

15 y [mm]

22.5

30

Fig. 7.26. The z component of the magnetic flux density, Bz , along the line x = 0, y = 0, · · · , 30 mm, z = 0 simulated by different formulations The problem has been solved by the ungauged version of the magnetic vector potential formulation, too. In this case, the coil’s current must be represented by the impressed current vector potential T 0 approximated by edge elements. Edge elements are used to represent the magnetic vector potential, too, which results in larger value of the magnetic

275

7.6. STEEL PLATES AROUND A COIL, NONLINEAR PROBLEM

flux density (see Fig. 7.26). The number of unknowns is 95397 and 290123 for a coarse and a fine mesh. In the first situation 55 and 112 steps needed to calculate the ungauged impressed field and the magnetic vector potential, respectively. In the case of fine mesh 93 and 349 steps needed. The ungauged version results in a faster solution than the gauged one. Next, the problem has been solved by the reduced magnetic scalar potential Φ. The coil’s current has been represented by T 0 approximated by edge elements, the magnetic scalar potential has been approximated by nodal elements. Figure 7.26 shows the results simulated by different formulations. It can be seen that the magnetic flux density inside the magnetic material is larger than the result obtained by the magnetic vector potential represented by nodal elements. The result calculated by the reduced scalar potential formulation can be smaller and smaller by increasing the number of unknowns or the order of approximation (the number of unknowns here is 14710 and 173163 for a coarse and a fine mesh). It must be noted that the order of T 0 and of Φ must be the same.

7.6 Steel plates around a coil, nonlinear problem The modified version of Problem No. 10 of the TEAM Workshops have been solved when the nonlinearity of the steel plates is taken into account. Measured data is known from the paper [40, 135, 150]. The measured magnetization curve of steel plates can be seen in Fig. 7.27. This curve can be modeled by simple functions, like the inverse tangent function, however, here a neural network has been trained to represent the measured data. 2

B [T]

1.5

1

0.5

0 0

2000

4000 H [A/m]

6000

8000

Fig. 7.27. The measured magnetization curve of steel plates The problem has been solved by first order reduced magnetic scalar potential Φ and by first order ungauged magnetic vector potential. The problem region is discretized by a mesh used in the last example, too, it consists of 76662 elements. The number of unknowns is 95397 for T 0 , 14710 for Φ and 95397 for A.

276


The magnetic field intensity vector H can be calculated directly when the reduced magnetic scalar potential Φ is used, obtained from the finite element method, H = T 0 − ∇Φ.

(7.5)

In this case, the direct model, B = B{H} can be used to calculate the magnetic flux density, then B = µo H + R

⇒

R = B − µo H

(7.6)

is used to update the residual term R. After it, the updated finite element equations must be solved again and the procedure described here has to be repeated until convergence. The magnetic flux density vector B can be calculated when the magnetic vector potential A is used, obtained from the finite element method, B = ∇ × A.

(7.7)

In this case, the inverse model, H = B −1 {B} can be used to calculate the actual magnetic field intensity, then the formula H = νo B + I

⇒

I = H − νo B

(7.8)

is used to update the residual term I. After it, the updated finite element equations must be solved again and the procedure described here has to be repeated until convergence. The magnetic field quantities as well as the residual terms are defined on the points of the mesh, where the scalar potentials are.

2

Bz [T]

1.5

1

0.5 T0, Φ A

0 0

7.5

15 y [mm]

22.5

30

Fig. 7.28. The z component of the magnetic flux density, Bz , along the line x = 0, y = 0, · · · , 30 mm, z = 0 simulated by different formulations

277

7.7. ASYMMETRICAL CONDUCTOR WITH A HOLE

Updating the nonlinear residual terms R, or I means that the right-hand side of the linearized, assembled system of equations is changing from iteration to iteration. The difference between two iteration steps (i.e. ∆R, ∆I, ∆H, or ∆B) is decreasing during the convergence algorithm. The iteration can be stopped if a predefined error limit is reached. The fixed point iteration scheme with 154 and 143 steps found the solution by using the Φ and the A-formulation, respectively. The z component of the resulting magnetic flux density along the line x = 0, y = 0, · · · , 30 mm, z = 0 can be seen in Fig. 7.28. The averaged value of measured data is Bz = 1.67 T [150]. Simulated results are Bz = 1.72 T, moreover Bz = 1.68 T by the Φ-formulation and by the A-formulation, respectively.

7.7 Asymmetrical conductor with a hole Benchmark problem No. 7 of the TEAM workshop consists of an asymmetrical conductor with a hole [53, 75, 136, 177], see Fig. 7.29. The conductor is made of aluminum. The source of magnetic field is the sinusoidal current flowing in the racetrack-shaped coil placed above the plate eccentrically. The conductivity of the plate is σ = 3.526 · 107 S/m, the maximum ampere-turn is 2742 AT (the ampere-turn (AT) is the unit of magnetomotive force, represented by a direct current of one ampere flowing in a single-turn loop in vacuum), the frequencies are f = 50 Hz and f = 200 Hz. The amplitude of the source current density of the coil can be calculated by the formula |J | = 2742/(0.1 · 0.025) A/m2, where the denominator is the cross-section area of the coil. The impressed current vector potential T 0 must be used to represent the current of coil in the case of ungauged A-formulation and Φ-formulation. The problem is a linear quasi-steady state eddy current field problem with multiply connected eddy current region, because there is a hole inside the conducting material. Paper [53] is a summary of the measurement system and the comparison between 25 solutions with different methods and formulations and the measurements [3, 5, 12, 21, 42, 43, 63, 88, 134, 139, 158]. Here, the problem has been solved by the formulations introduced previously. The problem has been discretized by tetrahedral finite elements. The air region has been closed by an artificial far boundary as it can be seen in Fig. 7.30. The radius of the bounding sphere has been determined after some trials. Figure 7.31 shows the discretization of the plate and the coil. After some trials, the third order approximation on a mesh with 8604 elements has been selected. The weak formulations have been transformed to the frequency domain by replacing the operator ∂/∂t by the multiplier jω, i.e. the resulting system of equations is complex. The weak form of the gauged A, V − A formulation (3.137)–(3.138) in the frequency domain can be summarized as Z h i e + ν∇ · W ∇ · A e dΩ ν (∇ × W )· ∇ × A Z ZΩc ∪Ωn Z (7.9) e + σe W · σA v dΩ = W · J 0 dΩ + W · KdΓ, +jω Ωc

Ωn

ΓHn

278


y

25 mm

294 mm s m0 i(t) 150 mm 108 mm

18 mm

s=0 m0

50 mm 25 mm

hole x

18 mm z air

100 mm

s=0 m0 hole

30 mm 19 mm

x

Fig. 7.29. A plate beneath a coil

jω

Z

Ωc

e + σe ∇N · σ A v dΩ = 0.

(7.10)

The other formulations in the frequency domain can be obtained similarly. It is noted that the problem based on nodal finite elements has been solved by the GMRES (Generalized Minimum Residual Method) solver with the Incomplete LU preconditioner . If the vector functions have been approximated by vector elements, the GMRES solver with SSOR (Symmetric Successive Over-Relaxation) preconditioner has been applied. Our experiment is that the other solvers of COMSOL Multiphysics are very slow in this case [29]. The reference solution is obtained by the A, V − A-formulation. The eddy currents have a path around the hole filled with air, as it can be seen in Fig. 7.32 and in Fig. 7.33, for f = 50 Hz and f = 200 Hz, respectively. The use of the reduced magnetic scalar potential Φ in the eddy current free region results in false solution, as it is illustrated in Fig. 7.34. This is the situation when applying the A, V −Φ-formulation or the T , Φ−Φ-formulation aiming the reduction of the degree of freedom.

7.7. ASYMMETRICAL CONDUCTOR WITH A HOLE

279

Fig. 7.30. Finite element mesh of the problem

Fig. 7.31. Finite element mesh of the plate and its surround The problem of the A, V − Φ-formulation can be solved by applying the A, V − A − Φ-formulation, when the magnetic vector potential A is introduced in the hole, too. The false solution of the T , Φ − Φ-formulation can be corrected by employing the T , Φ − A-formulation or the T , Φ − A − Φ-formulation, when the magnetic vector potential A is introduced in the whole air region, or only in the hole, respectively. The other possibility is filling the hole with a conducting material with very low conductivity, i.e. the T , Φ − Φ-formulation can be used without any modification. The conductivity of air region in the hole can be determined by trial and error method, i.e. σair = σaluminum /c, where c is a constant. Increasing c results in a model, which is closer and closer to the original one, however, the system of linear equations may become ill-conditioned and the number of iterations may increase. The nodal A, V −A-formulation as well as the nodal A, V −A−Φ-formulation can be sped up by using edge element representation of the unknown magnetic vector potential.

280


Fig. 7.32. Real part of eddy current field inside the aluminum plate at f = 50 Hz

Fig. 7.33. Real part of eddy current field inside the aluminum plate at f = 200 Hz The convergence of the gauged A, V − A − Φ-formulation is extremely slow. According to [16], the reason of this inferior performance is the only Neumann type boundary conditions (2.507) and (2.512) on ΓncA and on ΓncΦ , respectively, of the electric scalar potential V , since, there is no plane of symmetry. However, the performance of the ungauged A, V − A − Φ-formulation is much better. The fastest solver can be obtained by using the nodal T , Φ − A − Φ-formulation, however, the performance of the nodal T , Φ − Φ-formulation (the hole is filled with conducting media) is also satisfactory. It is our experience that the use of edge elements in the T , Φ-based methods slow down the solvers. In the case of edge element representation of the magnetic vector potential, V = 0 can be supposed resulting the A⋆ −A-formulation or the A⋆ −A−Φ-formulation. These methods have the advantage that the number of unknowns is decreasing, however, the solution time is increasing. Comparison of measured and simulated results in Fig. 7.35 and in Fig. 7.36 show

7.8. ROTATIONAL SINGLE SHEET TESTER

281

Fig. 7.34. False eddy current distribution inside the aluminum plate at f = 50 Hz obtained by the A, V − Φ-formulation or by the T , Φ − Φ-formulation (real part) that the solution of different potential formulations is practically identical and close to measurements. The solution is complex, following [53], the quantity q C = sign(Cr ) Cr2 + Ci2 (7.11)

has been determined, where Cr and Ci are the real and the imaginary parts of the complex quantity, respectively (e.g. of the magnetic flux density) and sign(Cr ) is the sign of the real part. Finally, C can be plotted and analyzed (Table. 7.4, where CPU time means the results of an AMD Athlon 64 X2 Dual Core Processor 4600+ 2.41GHz computer with 4GByte RAM).

7.8 Rotational single sheet tester The example closing this chapter contains an ongoing research, which aims to build up a vector hysteresis measurement apparatus. The CAD design of the arrangement is based on the finite element method. The rotational single sheet tester is one of the possible measurement arrangements of the two-dimensional vector hysteresis properties. In this case, the specimen has round shape and this is called RRSST system [61, 80, 81, 102, 104, 105, 107]. The RRSST system is an induction motor, which rotor has been removed and the round-shaped specimen has been installed in this place. The magnetic field inside the specimen can be generated by a special two phase winding excited by two independent current generators. The two orthogonal components of the magnetic field intensity vector and of the magnetic flux density vector inside the specimen can be measured by a sensor system. The tangential component of the magnetic field intensity can be measured by two coils placed onto the surface of the specimen (H-coils), the magnetic flux density inside the specimen can be measured by two coils slipped into holes of the specimen (B-coils). The block diagram can be seen in Fig. 7.37.

282


Table 7.4. Comparison between the performance and results of different formulations Method A, V − A, nodal A, V − A, nodal A, V − A, vector A, V − A, vector A, V − A − Φ, nodal A, V − A − Φ, nodal A, V − A − Φ, vector A, V − A − Φ, vector T , Φ − Φ, nodal T , Φ − Φ, nodal T , Φ − Φ, vector T , Φ − Φ, vector T , Φ − A − Φ, nodal T , Φ − A − Φ, nodal T , Φ − A − Φ, vector T , Φ − A − Φ, vector A⋆ − A, vector A⋆ − A, vector ⋆ A − A − Φ, vector A⋆ − A − Φ, vector

Frequency [Hz] 50 200 50 200 50 200 50 200 50 200 50 200 50 200 50 200 50 200 50 200

DOF 122020 122020 132344 132344 55554 55554 58377 58377 53204 53204 55949 55949 53422 53422 55999 55999 128952 128952 54147 54147

Iteration 532 857 132 132 980 9995 185 180 50 60 216 277 47 44 2353 1092 405 301 436 328

CPU time [sec] 1924 2454 616 616 1450 9919 259 255 319 345 278 355 340 325 2409 1171 1598 1213 456 384

The stator core is made of laminated iron, that is eddy currents have been neglected there. Eddy currents have been taken into account only inside the specimen. According to the preliminary studies of the problem [102, 104, 105, 107], the magnetic field intensity is much smaller inside the stator core than inside the specimen, so a linear characteristics have been supposed inside the stator core (µr = 4000) and a nonlinear hysteretic one inside the specimen. The hysteresis characteristics of the material have been simulated by the isotropic neural network based vector model. The x component of the applied hysteresis characteristics can be seen in Fig. 7.38. Here, the terms µo Hx and Rx are also presented. The arrangement has been simulated by the gauged T , Φ − Φ-formulation. The x − y plane of the arrangement has been discretized by triangular mesh (Fig. 7.39), which has been extruded in the z direction resulting in prism elements. The mesh consists of 94512 prism elements. The impressed current vector potential T 0 has been represented by edge elements (see Fig. 7.40), the unknown current vector potential T and the unknown reduced magnetic scalar potential Φ have been approximated by nodal elements. The number of unknowns is 68893 and 5800 vector hysteresis models with 20 scalar models per one vector model, i.e. the full number of scalar models is 116000. There are 3 layers inside the specimen.

283

9

6

6 Bz [mT]

9

3

z

B [mT]


Measured, y=72mm Nodal Vector Measured, y=144m Nodal Vector

0

−3 0

75

150 x [mm]

225

Bz [mT]

6

z

B [mT]

6

−3 0


75

150 x [mm]

225

6 Bz [mT]

6


0

−3 0

75

150 x [mm]

225

−3 0

6 Bz [mT]

6

z

0

−3 0

75

150 x [mm]

225

300

(g) T , Φ − A − Φ-formulation, f = 50 Hz

150 x [mm]

225

300


75

150 x [mm]

225

300

(f) T , Φ − Φ-formulation, f = 200 Hz 9


75

3

9

3

300


0

300

(e) T , Φ − Φ-formulation, f = 50 Hz

225

(d) A, V − A − Φ-formulation, f = 200 Hz 9

z

B [mT]

−3 0

9

3

150 x [mm]

3

0

300

(c) A, V − A − Φ-formulation, f = 50 Hz

75

(b) A, V −A-formulation, f = 200 Hz 9

0

B [mT]

−3 0

9

3


0

300

(a) A, V − A-formulation, f = 50 Hz

3

3 Measured, y=72mm Nodal Vector Measured, y=144m Nodal Vector

0

−3 0

75

150 x [mm]

225

300

(h) T , Φ − A − Φ-formulation, f = 200 Hz

Fig. 7.35. The z component of the magnetic flux density along x = 0, · · · , 288 mm, y = 72 mm and y = 144 mm, z = 34 mm, simulated by different formulations

284


6

2

x 10

5

2

J [A/m ]

0

y

y

0 Measured, z=19mm Nodal Vector Measured, z=0mm Nodal Vector

−1

−2 0

75

150 x [mm]

225

−5 0

300

x 10

5

150 x [mm]

225

300

x 10

6

2

J [A/m ]

2.5

2

J [A/m ]

1

0

y

y


−1

−2 0

75

150 x [mm]

225

−5 0

300

75

150 x [mm]

225

300

(d) A, V − A − Φ-formulation, f = 200 Hz

6

2

Measured, z=19mm Nodal Vector Measured, z=0mm Nodal Vector

−2.5

(c) A, V − A − Φ-formulation, f = 50 Hz x 10

5

x 10

6

2.5 2

2

J [A/m ]

1 J [A/m ]

75

(b) A, V −A-formulation, f = 200 Hz

6

2


−2.5

(a) A, V − A-formulation, f = 50 Hz

0

y

y


−1

−2 0

75

150 x [mm]

225

−5 0

300

75

150 x [mm]

225

300

(f) T , Φ − Φ-formulation, f = 200 Hz

6

2


−2.5

(e) T , Φ − Φ-formulation, f = 50 Hz x 10

5

x 10

6

2.5 2

2

J [A/m ]

1 J [A/m ]

6

2.5

2

J [A/m ]

1

x 10

0

y

y

0

−1

−2 0


75

150 x [mm]

−2.5

225

300

(g) T , Φ − A − Φ-formulation, f = 50 Hz

−5 0


75

150 x [mm]

225

300

(h) T , Φ − A − Φ-formulation, f = 200 Hz

Fig. 7.36. The y component of the eddy current density, Jy , along the two lines x = 0, · · · , 288 mm, y = 72 mm, z = 0 mm and z = 19 mm, simulated by different formulations

285


waveform of currents

measured signals

RRSST

sensors y

NI-DAQ card

LabVIEW on PC

x

specimen

generators

(thickness: 0.5 mm)

i x (t)

i y (t)

Fig. 7.37. The block diagram of the measurement system 4 µoHx

2 Bx [T]

Rx

0

−2 µmaxHx

−4 −2000

−1000

0 1000 Hx [A/m]

2000

Fig. 7.38. The hysteresis characteristics and the terms µo Hx and Rx The system of the nonlinear ordinary differential equations (3.111) and (3.112) must be solved in the time domain, because of the hysteresis characteristics and it has been solved in the (n + 1)th time step by the following scheme for the general variable a: a(n+1) − a(n) ∂a(n+1) ∂a(n) =θ + (1 − θ) , ∆t ∂t ∂t

(7.12)

where ∆t is the time step of the time discretization and θ ∈ [0, · · · , 1] is a parameter [77].

286


Fig. 7.39. The 2D mesh of the arrangement, which is extruded in z direction First, the equations (3.111) and (3.112) must be multiplied by θ∆t and it is supposed in the time step (n + 1), Z θ∆t θ∆t (n+1) (n+1) (∇ × W k ) · ∇ × T ∇ · Wk ∇ · T + dΩ σ σ Ωc " # Z ∂T (n+1) ∂Φ(n+1) µo θ∆tW k · dΩ = − µo θ∆tW k · ∇ + (7.13) ∂t ∂t Ωc Z Z (n+1) ∂T 0 ∂R(n+1) dΩ − dΩ, µo θ∆tW k · θ∆tW k · − ∂t ∂t Ωc Ωc

Fig. 7.40. The x component of the impressed current vector potential T 0

287


! ∂T (n+1) − dΩ (∇Nk ) · µo θ∆t ∂t Ωc Z ∂Φ(n+1) dΩ (∇Nk ) · µo θ∆t∇ + ∂t Ωc ∪Ωn ! Z (n+1) ∂T 0 (∇Nk ) · µo θ∆t dΩ = ∂t Ωc ∪Ωn ! Z ∂R(n+1) dΩ + (∇Nk ) · θ∆t ∂t Ωc Z ∂b(n+1) + Nk θ∆t dΓ. ∂t ΓB Z

(7.14)

Next, equations (3.111)–(3.112) are multiplied by (1 − θ)∆t in the time step (n), Z

Ωc

(1 − θ)∆t (∇ × W k ) · ∇ × T (n) dΩ σ (1 − θ)∆t ∇ · W k ∇ · T (n) dΩ σ

Ωc

µo (1 − θ)∆tW k ·

Ωc

µo (1 − θ)∆tW k · ∇

Ωc

µo (1 − θ)∆tW k ·

Ωc

(1 − θ)∆tW k ·

Ωc

+ + − − −

Z

Z

Z

Z

Z

∂T (n) dΩ ∂t ∂Φ(n) dΩ = ∂t

(7.15)

(n)

∂T 0 ∂t

dΩ

∂R(n) dΩ, ∂t

and ! ∂T (n) dΩ (∇Nk ) · µo (1 − θ)∆t − ∂t Ωc Z ∂Φ(n) (∇Nk ) · µo (1 − θ)∆t∇ + dΩ ∂t Ωc ∪Ωn ! Z (n) ∂T 0 dΩ (∇Nk ) · µo (1 − θ)∆t = ∂t Ωc ∪Ωn ! Z ∂R(n) + dΩ (∇Nk ) · (1 − θ)∆t ∂t Ωc Z ∂b(n) Nk (1 − θ)∆t dΓ. + ∂t ΓB Z

(7.16)

288


Finally, the equations (7.13), (7.15) and (7.14), (7.16) must be added using (7.12), Z

θ∆t (∇ × W k ) · ∇ × T (n+1) dΩ σ Ω Z c θ∆t + ∇ · W k ∇ · T (n+1) dΩ Ωc σ Z h i µo W k · T (n+1) − µo W k · ∇Φ(n+1) dΩ = + ZΩc (n+1) (n) − T0 µo W k · T 0 dΩ − Ωc Z W k · R(n+1) − R(n) dΩ − Ω Z c (1 − θ)∆t − (∇ × W k ) · ∇ × T (n) dΩ σ Ω Z c (1 − θ)∆t ∇ · W k ∇ · T (n) dΩ − σ Ωc Z h i µo W k · T (n) − µo W k · ∇Φ(n) dΩ, +

(7.17)

Ωc

and − + =

Z

Ωc

Z

(∇Nk ) · µo T (n+1) dΩ

ZΩc ∪Ωn

(∇Nk ) · µo ∇Φ(n+1) dΩ

(n+1) (n) − T0 (∇Nk ) · µo T 0 dΩ

ZΩc ∪Ωn (∇Nk ) · R(n+1) − R(n) dΩ + ZΩc + Nk b(n+1) − b(n) dΓ Γ Z B − (∇Nk ) · µo T (n) dΩ Ω Z c (∇Nk ) · µo ∇Φ(n) dΩ. +

(7.18)

Ωc ∪Ωn

Different schemes can be realized by selecting θ, if θ = 1, θ = 0, θ = 1/2 or θ = 2/3, the scheme is the backward Euler scheme, the forward Euler scheme, the Crank–Nicolson scheme or the Galerkin scheme, respectively. Here, the Galerkin scheme has been used. The other formulations can be formulated in a similar way. The problem is nonlinear, which must be solved iteratively. The applied fixed point based algorithm is the following.


289

1. Starting from demagnetized state, H = 0, B = 0. Initialize variables, T (0) = 0, Φ(0) = 0, R(0) = 0, n = 0, k = 1 (k is the index of fixed point iteration, for short, it is denoted only if necessary). Calculate µo using the x component of the isotropic vector hysteresis characteristics; 2. Solve the equations (7.17) and (7.18) in the time step (n + 1); 3. Calculate the magnetic field intensity inside the specimen in the time step (n + 1) (n+1) by H (n+1) = T 0 + T (n+1) − ∇Φ(n+1) ; 4. Calculate the magnetic flux density by the direct isotropic vector hysteresis model, B (n+1) = B{H (n+1) }; (n+1)

5. Update the residual term Rk+1 = B (n+1) − µo H (n+1) , which is the new value of R(n+1) in the equations (7.17) and (7.18); 6. Repeat from step 2 until the procedure is not convergent and k ← k + 1. The criteria of stopping the sequence 2–5 is

(n+1) (n+1) (7.19)

Rk+1 − Rk

< εR , or

or

(n+1) (n+1)

< εH ,

H k+1 − H k

(n+1) (n+1)

< εB ,

B k+1 − B k

(7.20)

(7.21)

where εR , εH and εB are predefined error limits, e.g. εR = 10−6 and k·k is a norm. Here, the following norm has been used: v u Nh u 1 X (n+1) (n+1) t (7.22) H k+1,i − H k,i , Nh i=1

where Nh is the length of vector storing the values of H, moreover k is the index of fixed point iteration. In other words, the iteration is stopping if two subsequent solution are close enough to each others. If time step (n + 1) is convergent, then the next time step must be iterated. Every data is modified by the iterated R(n+1) in time step (n + 1). The independent excitation currents have been prescribed by the functions ix (t) = Ix sin(ωt + α),

(7.23)

iy (t) = Iy sin(ωt + β).

(7.24)

and

290


The amplitude and the phase of currents define the polar angle and the amplitude of the magnetic field intensity vector or of the magnetic flux density vector. Controlling the flux or the magnetic field can be worked out by an iterative feedback algorithm. Here the relationship between the currents and the magnetic field intensity is studied at f = 5 Hz, f = 50 Hz and f = 500 Hz. The conductivity of the specimen is σ = 2 · 106 S/m, the value of µo is about 4000µ0 and the peak value of currents is 1 A. The thickness of the sample is 0.5 mm. In Fig. 7.41, the magnetic field intensity has been increased in the x direction then it has been rotated in the counter clockwise direction and only the stationary state of the two orthogonal components of the magnetic field intensity at point x = y = z = 0 is plotted.

400

200 Hy

y

H [A/m], H [A/m]

5Hz 50Hz 500Hz

0

x

Hx

−200

−400 1

10

19 Sample

28

36

2

Hy/200 [A/m], By [T]

H

B

1

0

−1

−2 −2

−1 0 1 Hx/200 [A/m], Bx [T]

2

Fig. 7.41. Variation of x and y components of the magnetic field intensity vector and the loci of the magnetic field intensity and the magnetic flux density at the point x=y=z=0

291


The effect of eddy currents can be sensed at f = 500 Hz, because the magnetic field has some phase shift and its amplitude becomes smaller, however, the result according to f = 50 Hz seems to be almost the same as the results of f = 5 Hz calculations. Figure 7.41 shows the rotating magnetic field intensity vector and the magnetic flux density vector when f = 5 Hz. The effect of hysteresis can be seen in the figure, because the magnetic flux density has some delay, which moreover means that the material is not in the saturation state. Figure 7.42 shows the time variation of the x and y components of the magnetic field intensity at some points inside the specimen (f = 50 Hz). The coordinates are given in the legend of the figure in mm and z = 0. 400 0,0 20,0 38,0 0,20 0,38 14,14 27,27

0

y

H [A/m]

200

−200

−400 −400

−200

0 Hx [A/m]

200

400

Fig. 7.42. Loci of magnetic field inside the specimen, rotating field

Figure 7.43 shows the loci of the magnetic field intensity vector in linear excitation when ix (t) = iy (t). The coordinates are given again in the legend of the figure in mm and z = 0. 800

0

y

H [A/m]

400

0,0 20,0 38,0 0,20 0,38 14,14 27,27

−400 −800 −800

−400

0 400 H [A/m]

800

x

Fig. 7.43. Loci of magnetic field inside the specimen, linear field

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INDEX

Modeling ∆m curves using the IEEE Trans. on Magn., 31:1809–1812,

[186] C. Visone, C. Serpico, I. D. Mayergoyz, M. W. Huang, and A. A. Adly. Neural-Preisach-type models and their application to the identification of magnetic hysteresis from noisy data. Physica B, 275:223–227, 2000. [187] J. P. Webb and B. Forghani. A single scalar potential method for 3D magnetostatics using edge element. IEEE Trans. on Magn., 25:4126–4128, 1989. [188] K. Wiesen and S. H. Charap. A better scalar Preisach algorithm. IEEE Trans. on Magn., 24:2491–2493, 1988. [189] J. G. Woodward and E. Della Torre. Particle interaction in magnetic recording tapes. J. of Appl. Phys, 31:56–62, 1960. [190] www.comsol.com. [191] www.geuz.org/gmsh. [192] www.mathworks.com. [193] T. Yamamoto and Y. Ohya. Single sheet tester for measuring core losses and permeabilities in a silicon iron steel. IEEE Trans. on Magn., 10:157–159, 1974. [194] T. V. Yioultsis, N. V. Kantartzis, C. S. Antonopoulos, and T. D. Tsiboukis. A fully explicit Whitney element - time domain scheme with higher order vector finite elements for three-dimensional high frequency problems. IEEE Trans. on Magn., 34:3288–3291, 1998. [195] T. V. Yioultsis and T. D. Tsiboukis. Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions. IEEE Trans. on Magn., 32:1389–1392, 1996. [196] T. V. Yioultsis and T. D. Tsiboukis. Development and implementation of second and third order vector finite elements in various 3-D electromagnetic field problems. IEEE Trans. on Magn., 33:1812–1815, 1997. [197] A. K. Ziarani and A. Konrad. Galerkin’s method and the variational procedure. IEEE Trans. on Magn., 38:190–199, 2002. [198] O. C. Zienkiewicz and R. Taylor. The Finite Element Method. McGraw-Hill, Maidenhead, 1991. [199] W. B. J. Zimmerman. Multiphysics Modelling with Finite Element Method. World Scientific Publishing Co., 2006.

Index Accommodation, 171

Compatible approximation, 33

Ampere’s law, 4, 7

Complete metric space, 247

Ampere-turn, 277

COMSOL Multiphysics, 125, 154, 255

Anisotropic magnetic material, 174

Constitutive relation, 9, 19, 243

Anisotropic materials, 164

Constitutive relationship, 15

Antiferromagnetic materials, 165

Contraction, 240

Area functions, 134

Contraction mapping, 243, 247

Artificial far boundary, 257, 262

Contraction mapping principle, 247

Assembling, 128

Contractive mapping, 245

Banach fixed point theorem, 247 Banach’s theorem, 245 Barkhausen effect, 166 Barycentric coordinate system, 134, 141 Barycentric coordinates, 135, 141, 148 Biot–Savart’s law, 22 Boundary conditions, 13, 14

Convergent sequence, 246 Coulomb gauge, 22, 34–36, 40, 41, 45, 46, 48, 49, 263 implicit enforcement, 35 Curie temperature, 165 Current continuity equation, 6, 15 Current vector potential, 43, 50, 51, 55, 57, 58

Cancellation error, 31, 33, 257, 260, 262

impressed, 22, 34

Cauchy sequence, 246

impressed, gauged, 27

Charge conservation law, 6

impressed, ungauged, 28

Classical Preisach model, 168

Currents in conducting materials, 15

distribution function, 170 interaction field, 169

Data Acquisition, 225

rectangular elementary hysteresis loop, Demagnetized state, 165, 171, 195 Diamagnetic materials, 165 169 staircase line, 169

Dirichlet boundary condition, 68, 75

switching fields, 169, 170

Dissipated energy, 11

Coercive field, 167

Domain, 165, 169

306

INDEX

Eddy current, 4, 43, 50

h-FEM, 130

Eddy current field, 16

Hexahedra, 126

Edge element, 33, 266

hp-FEM, 131

Edge shape functions, 130, 146, 147

Hysteresis characteristics, 163

Electric scalar potential, 43, 50, 53, 54, 92

scalar, 165

Elementary current loop, 163

Hysteresis losses, 177

Elementary hysteresis operator, 169

Hysteresis model

Energy balance equation, 10 Energy density, 11 Everett function, 180, 200

direct, 175, 252 inverse, 176, 251 Hysteresis operator, 15, 168

scalar, 200 vector, 200

Identification process anisotrop model, 206

Faraday’s law, 4, 7, 44, 47 Feedback controller, 228 FEM mesh, 126 Ferrimagnetic materials, 165 Ferromagnetic hysteresis measurement, 223

isotrop model, 201 Impressed current vector potential, 43, 258 infinite long filamentary conductor, 258 Induction remanent, 166

Ferromagnetic materials, 165

Inner product, 69

FFT, 228

Interface conditions, 12

Finite Element Method, 125

Isotropic magnetic materials, 200

First magnetization curve, 165

Isotropic materials, 164

First order descending curves, 230 First order reversal curves, 190, 200 Fixed point algorithm, 238, 243, 245, 288

Joule heat, 11 Knowledge-base, 193

Fixed point iteration, 238 Fourier analysis, 226

LabVIEW, 223

Fourier expansion, 206

Lagrange interpolation functions, 132

Fourier series, 210

Lagrange polynomials, 155

Frequency domain, 277

Laplace equation, 38, 45, 48

Functional, 25, 36

Laplace–Poisson equation, 28 Lipschitz constant, 247

Galerkin’s method, 66, 75 Gauss’ law, 5, 44, 47

Lipschitzian mapping, 247 Lorentzian distribution, 179

Gauss’ theorem, 8 Gaussian distribution function, 174

Magnetic dipole, 163

307

INDEX

energy, 163

layer, 186

Magnetic field intensity, 163

Levenberg–Marquardt iteration, 187, 191

Magnetic induction, 163, 164

neuron, 185

Magnetic moment, 163

training, 186

Magnetic scalar potential, 28, 43, 50, 257

training sequence, 186

Φ − Ψ-formulation, 33

transfer function, 185

reduced, 21, 28, 40, 50–52, 55, 57, 58,

universal function approximation, 185

61, 62, 82, 266 total, 21, 31 Magnetic susceptibility, 164

Newton’s method, 238 NN based hysteresis model, 193 Nodal elements, 35

Magnetic vector potential, 21, 34, 40, 43, Nodal shape functions, 130, 131 47, 50, 53–55, 57, 58, 61–63, 82 Magnetization remanent, 166

Noncongruency, 171 Nonexpansive, 247 Nonexpansive mapping, 247

Magnetization vector, 15, 163

Nonlinear equations, 238

Maxwell’s equations, 3, 17

Nonlinear partial differential equations, 244

classification, 14 differential form, 4 integral form, 6 quasi-static, 19 Metric space, 245 Minor loop, 167, 171, 194 Moving model, 172 Multiply connected conductor, 55 Multiply connected region, 51, 53, 62, 63 Neel temperature, 165

Objective function, 174 p-FEM, 131 Paramagnetic materials, 165 Partial differential equations elliptic, 67 hyperbolic, 68 parabolic, 67 Pascal’s triangle, 136 Permeability

Neumann boundary condition, 68

initial, 166

Neural network, 185

of vacuum, 163

activation function, 185

optimal value, 250

approximate, 187

relative, 164

backpropagation, 187

Picard–Banach theorem, 245

bipolar sigmoidal function, 186

Polarization method, 238, 243–245

feedforward, 185

Postprocessing, 129

hidden layers, 192

Poynting vector, 11

Kolmogorov–Arnold theorem, 185

Preconditioner, 272, 278

308

INDEX

Preisach distribution function, 180

Tangentially continuous shape functions, 148

Preisach model, 180

Tangentially continuous vector field, 33

implementation, 182

Taylor series, 25, 37

Preisach triangle, 169, 171

Tetrahedron, 126

Preprocessing, 126, 191

Time stepping scheme

Principle of minimum energy, 36

backward Euler, 288

Product model, 172

Crank–Nicolson, 288 forward Euler, 288

Quadrangle, 126 Reluctivity optimal value, 251 Remanent state, 171 Rolling direction, 174, 179, 208

Galerkin, 288 toroidal core, 223 Transverse direction, 174, 180, 208 Triangle, 126 Triangular finite elements, 133 Turning point, 193

Rotational single sheet tester, 281 RRSST system, 281

Uniaxial anisotropy, 208, 209

Saturation state, 165, 167, 169, 196

Variation, 25, 36

Scalar hysteresis characteristics, 168

first, 26, 37

Scalar hysteresis model, 168

second, 26, 37

Silvester polynomials, 156

Vector element, 35, 40

Skin depth, 267

Vector hysteresis measurement, 231

Solenoidal property, 15, 43

Vector hysteresis model, 183

Solver

Vector NN based model

conjugate gradient method, 263

anisotrop, 207

generalized minimum residual method,

isotrop, 198

272, 278 SPOOLES, 274 UMFPACK, 257 Static electric field, 15 Static magnetic field, 14, 16, 21 Successive approximation method, 238 Susceptibility differential, 167 initial, 166 Symmetric minor loop, 196

Vector Preisach model, 183 Vector shape functions, 130, 146, 147, 159 Wave propagation, 16 Weak formulation, 70 Weighted residual direct form, 74 inverse form, 75 weak form, 75 Weighted residual method, 66

ABOUT THE AUTHORS Dr. Miklós Kuczmann, PhD Associate Professor of Electrical Engineering has graduated at the Budapest University of Technology and Economics, Faculty of Electrical Engineering and Informatics in 2000. He has got the degree of PhD of Technical Sciences in 2005 under the supervisorship of Prof. Amália Iványi at the Department of Electromagnetic Fields. ´ He works for the Department of Telecommunications, Jedlik Anyos Institute of Informatics, Electrical and Mechanical Engineering, Faculty of Engineering Sciences, Széchenyi István University, Gy˝or, where he is the head of Laboratory of Electromagnetic Fields. He is the lecturer of ’Theoretical Electrotechnics’ and ’Signals and Systems’. His research fields are: simulation and measurements of ferromagnetic materials, coupling the nonlinearity and the Finite Element Method, Computer Aided Design of electromagnetic field problems. He is the author of the lecture book Signals and Systems, 2006, and about 100 scientific papers and conference presentations. He is the owner of the ’János Bolyai Research Scholarship’ of the Hungarian Academy of Sciences. Dr. Habil Amália Iványi, DSc Professor of Informatics has graduated at the Technical University of Budapest, Faculty of Electrical Engineering. She has got the degree of Candidate of Sciences in 1991, the degree of PhD in 1997 at Budapest University of Technology and Economics, and the degree of Doctor of Sciences in 1999 from the Hungarian Academy of Sciences. She worked for the Department of Electromagnetic Theory, Faculty of Electrical Engineering and Informatics, Budapest University of Technology and Economics, and from 2003 she is full professor at the Department of Information Technology, Pollack Mihály Faculty of Engineering, University of Pécs. She is Honorary Professor at the Transylvania University of Brasov, Romania. She had the Széchenyi Professorship from 2000–2003. She has worked out the R-functions method in numerical analysis of electromagnetic fields and several research on hysteresis modeling, she founded high-level laboratory on hysteresis research. She is the author of more then 300 scientific papers and conference presentations, the author and co-author of lecture books. She is the editor of Conference Proceedings, and the author of three books: Hysteresis Models in Electromagnetic Computation, 1997, Magnetic Field Computation with R-Functions, 1998, Continuous and Discrete Simulations in Electromagnetic Fields, 2003, she is the Editor of Preisach Memorial Book, Hysteresis Models in Mathematics, Physics and Engineering, 2005.

ABOUT THE BOOK The aims of the book are to present a survey of the simulation of ferromagnetic hysteresis and its importance in electromagnetic field computation from electrical engineering point of view, and the implementation of hysteresis models in the Finite Element Method. The book starts with a summary of Maxwell’s equations, their form in case of nonlinear static magnetic field problems and nonlinear eddy current field problems. The description of the possible potential formulations used in solving static magnetic field and eddy current field problems is presented. Finite Element Method is a possible technique to solve numerically the partial differential equations formulated by using the weak form. The review of the Finite Element Method is followed by the study of nodal finite elements as well as the newer vector elements. The modeling and measuring aspects of ferromagnetic hysteresis based on the authors’ experiments are discussed next. A short summary of the fundamentals in magnetism and magnetic materials is also introduced. The Preisach model is presented as well as the neural network based scalar and vector hysteresis models developed by the authors. The method of identification of hysteresis models is also given. One chapter presents the polarization method as well as the fixed-point technique to solve nonlinear electromagnetic field problems. Nonlinear equations obtained from Maxwell’s equations can be formulated as a fixed-point equation. This method results in a convergent numerical tool to solve nonlinear equations. The book is closed by a collection of seven problems solved by the nonlinear Finite Element Method applying the user-friendly graphical user interface and functions of COMSOL Multiphysics.

the finite element method in magnetics

the finite element method in magnetics

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