THE FINITE ELEMENT METHOD IN MAGNETICS

MIKLÓS KUCZMANN AMÁLIA IVÁNYI THE FINITE ELEMENT METHOD IN MAGNETICS AKADÉMIAI KIADÓ, BUDAPEST

This book was sponsored by the Széchenyi István University in Győr 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 Bíró,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 1 2 Potential formulations in electromagnetic field 3 2.1 Maxwell sequationsandinterfaceconditions... 4 2.1.1 Maxwell sequations... 4 A.ThedifferentialformofMaxwell sequations... 4 B.TheintegralformofMaxwell sequations... 6 C. The constitutive relations.... 9 D.Energyofelectromagneticfields... 10 2.1.2 Interface and boundary conditions..... 11 A.Electricandmagneticfieldintensity... 12 B. Electric and magnetic flux density, current density....... 13 2.1.3 ClassificationofMaxwell sequations... 14 2.2 Magnetostaticsandeddycurrentfieldproblems... 16 2.2.1 Staticmagneticfields... 17 2.2.2 Eddycurrentfields... 18 2.3 Potential formulations of static magnetic field and eddycurrentfieldproblems... 21 2.3.1 Staticmagneticfields... 21 A. The reduced magnetic scalar potential, the Φ-formulation... 22 B. The total magnetic scalar potential, the Ψ-formulation... 30 C. The combination of the magnetic scalar potentials, the Φ Ψ-formulation... 31 D. Applying the reduced magnetic scalar potential with appropriate representation of T 0... 33 E. The magnetic vector potential, the A-formulation... 34 F. Combination of the magnetic vector potential and the magnetic scalar potential, the A Φ-formulation... 40 2.3.2 Eddycurrentfields... 43 A. The current vector potential and magnetic scalar potential, the T, Φ-formulation... 43

vi CONTENTS B. The magnetic vector potential and electric scalar potential, the A, V-formulation... 47 C. The modified magnetic vector potential, the A -formulation. 49 2.3.3 Couplingstaticmagneticandeddycurrentfields... 50 A. The gauged T, Φ Φ formulation... 51 B. The ungauged T, Φ Φ formulation... 52 C. The gauged A,V A formulation... 53 D. The ungauged A,V A formulation... 54 E. The ungauged A A formulation... 55 F. The gauged T, Φ A formulation... 55 G. The ungauged T, Φ A formulation... 57 H. The gauged T, Φ A Φ formulation... 58 I. The ungauged T, Φ A Φ formulation... 60 J. The gauged A,V Φ formulation... 61 K. The ungauged A,V Φ formulation... 62 L. The gauged A,V A Φ formulation... 63 M. The ungauged A,V A Φ formulation... 64 3 Weak formulation of nonlinear static and eddy current field problems 66 3.1 TheweightedresidualandGalerkin smethod... 66 3.1.1 Differentialequationsinelectromagneticfieldcomputation... 67 3.1.2 Theweightedresidualmethod... 69 A. Weighted residual of elliptic differential equations with scalar potential... 69 B. Weighted residual of elliptic differential equations with vector potential... 70 C.Weightedresidualofparabolicdifferentialequations... 73 3.2 Approximation of unknown functions and weighting functions..... 73 3.3 TheweakformulationwithGalerkin smethod... 75 3.3.1 The reduced magnetic scalar potential, the Φ-formulation... 76 3.3.2 Combination of magnetic scalar potentials, Φ Ψ-formulation.. 77 3.3.3 The magnetic vector potential, the A-formulation with implicit enforcementofcoulombgauge... 79 3.3.4 The magnetic vector potential, the A-formulation with a numerical technique, which is not sensitive to Coulomb gauge.. 80 3.3.5 Combination of the magnetic vector potential and the magnetic scalar potential, the A Φ-formulation... 82 3.3.6 The gauged T, Φ Φ formulation... 86 3.3.7 The ungauged T, Φ Φ formulation... 90 3.3.8 The gauged A,V A formulation... 92 3.3.9 The ungauged A,V A formulation... 96 3.3.10 The ungauged A A formulation... 99

CONTENTS vii 3.3.11 The gauged T, Φ A formulation...100 3.3.12 The ungauged T, Φ A formulation...101 3.3.13 The gauged T, Φ A Φ formulation...102 3.3.14 The ungauged T, Φ A Φ formulation...110 3.3.15 The gauged A,V Φ formulation...114 3.3.16 The ungauged A,V Φ formulation...115 3.3.17 The gauged A,V A Φ formulation...115 3.3.18 The ungauged A,V A Φ formulation...120 4 The finite element method 125 4.1 Fundamentals of FEM.......125 4.2 Approximatingpotentialswithshapefunctions...130 4.2.1 Nodalfiniteelements...131 4.2.2 Edgefiniteelements...147 5 Modeling of ferromagnetic hysteresis 162 5.1 Ferromagneticmaterials...163 5.1.1 Shortdescriptionofmagneticmaterials...163 5.1.2 Ferromagnetichysteresis...166 5.1.3 Thehysteresisoperator...169 5.2 ThePreisachmodel...169 5.2.1 Thescalarmodel...169 A.Descriptionofthemodel...170 B.Thedistributionfunction...175 C.TheEverettfunction...181 5.2.2 Pseudocode for the scalar Preisach model........183 5.2.3 Thevectormodel...184 5.2.4 TheclassicalscalarPreisachmodelvianeuralnetworks...186 A. A brief introduction to neural networks........186 B.NeuralPreisachmodels...189 5.3 Theneuralnetworkbasedhysteresismodel...191 5.3.1 Thescalarmodel...191 A. Training sequence and preprocessing of measured data.... 191 B.Operationofthemodel...194 5.3.2 Theisotropicvectorhysteresisoperator...199 A.Descriptionofthemodel...199 B.Theidentificationprocess...201 C.Verificationofthemodel...206 5.3.3 Theanisotropicvectorhysteresisoperator...207 A.Descriptionofthemodel...208 B. Fourier expansion of the measured 2D vector Everett function. 209 C. Fourier expansion of the measured 3D vector Everett function. 212

viii CONTENTS D. The relation between the scalar and the vector Everett functions 215 E.Verificationofthemodel...218 5.3.4 Somepropertiesofthevectorhysteresischaracteristics...219 5.4 Measuringmagnetichysteresis...224 5.4.1 Systemforscalarhysteresismeasurement...224 A.Themeasurementset-upusingLabVIEW...224 B. Simulation of measured static curves by the scalar NN model. 231 5.4.2 Vectorhysteresismeasurementsystem...232 A.Themeasurementset-up...234 B.Measuringresults...236 6 The polarization method and the fixed point technique 239 6.1 Solutionofnonlinearequations...239 6.2 Nonlinearityinelectromagneticfieldsimulation...244 6.3 Metricspaces...246 6.3.1 Definition of metric spaces.....246 6.3.2 Definitions...247 6.3.3 Banachfixedpointtheorem...248 6.4 The optimal value of parameters μ and ν...249 6.4.1 Usingtheinversecharacteristics...249 6.4.2 Usingthedirectcharacteristics...252 6.5 Theappliedformulation,summary...252 6.5.1 Usingtheinversecharacteristics...252 6.5.2 Usingthedirectcharacteristics...253 6.5.3 Proof of the nonexpansive property of Maxwell s equations...254 7 Application of the finite element method 256 7.1 Introduction......256 7.2 Illustration how to select T 0,theC-magnet...258 7.3 Ironcubeinhomogeneousmagneticfield...263 7.4 Conducting cube in homogeneous magnetic field........268 7.5 Steel plates around a coil, linear problem......273 7.6 Steel plates around a coil, nonlinear problem....276 7.7 Asymmetrical conductor with a hole....278 7.8 Rotationalsinglesheettester...282 8 References 293 9 Index 306 10 About the authors 310 11 About the book 311

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 Bíró, 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 Bíró 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 Bíró, 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 Bíró 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 Győr 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, Lívia Kuczmann-Kicsák for the daily assistance and patience. The authors want to express their appretiate thanks to the Széchenyi István University in Győr, to the Pollack Mihály Faculty of Engineering, University of Pécs, and to Tanoda Foundation Pécs for sporsoring the publication. Dr. Miklós Kuczmann, PhD Dr. Habil Amália Iványi, DSc Associate Professor Professor