Stream function approach for determining optimal surface currents

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Stream function approach for determining optimal surface currents

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Stream function approach for determining optimal surface currents
Citation for published version (APA): Peeren, G. N. (2003). Stream function approach for determining optimal surface currents. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR570424
DOI: 10.6100/IR570424
Document status and date: Published: 01/01/2003
Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
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Download date: 14. Jun. 2022

Stream Function Approach for Determining
Optimal Surface Currents

Copyright c 2003 by Geran Peeren. Printed by Universiteitsdrukkerij, Eindhoven University of Technology.
CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Peeren, Gerardus, N. Stream function approach for determining optimal surface currents / by Gerardus N. Peeren. – Eindhoven : Technische Universiteit Eindhoven, 2003. Proefschrift. – ISBN 90–386–0792–X NUR 919 Subject headings : electric coils / magnetic fields / construction optimization / numerical methods 2000 Mathematics Subject Classification : 78M50, 78M25, 65M32, 35Q60, 37E35, 33E05, 33C55

Stream Function Approach for Determining
Optimal Surface Currents
PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen op dinsdag 9 december 2003 om 16.00 uur
door Gerardus Nerius Peeren
geboren te Eindhoven

Dit proefschrift is goedgekeurd door de promotoren:
prof.dr. W.H.A. Schilders en prof.dr. R.M.M. Mattheij

Aan mijn vrouw D´e, en mijn dochter Merel.

Contents

1 Introduction

1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Problem description

11

2.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Objective of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Mathematical problem description . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Maxwell’s equations for linear media . . . . . . . . . . . . . . . 16

2.3.2 Quasi-static conditions . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Series representation of the current density . . . . . . . . . . . 19

2.3.4 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.5 Mutual inductance and mutual resistance operators . . . . . . 22

2.3.6 Uniqueness condition . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.7 Analogy with electric circuits . . . . . . . . . . . . . . . . . . . 26

2.3.8 Integral expressions for the magnetic field . . . . . . . . . . . . 28

2.4 Cost functions related to efficiency . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Minimum energy cost function . . . . . . . . . . . . . . . . . . 31

2.4.2 Alternative cost functions . . . . . . . . . . . . . . . . . . . . . 33

2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.1 The electromagnetic field . . . . . . . . . . . . . . . . . . . . . 35

2.5.2 Spherical Harmonics Expansion . . . . . . . . . . . . . . . . . . 36

2.5.3 Force in a static magnetic field . . . . . . . . . . . . . . . . . . 42

2.6 Computational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

viii

Contents

3 Stream functions

47

3.1 Divergence-free vector fields . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1.1 Discretization into windings . . . . . . . . . . . . . . . . . . . . 48

3.1.2 Stream function pairs . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Two-dimensional divergence-free vector fields . . . . . . . . . . . . . . 54

3.2.1 Stream functions in R2 . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.3 Stream functions on surfaces . . . . . . . . . . . . . . . . . . . 58

3.2.4 Stream function class Ψ(S) . . . . . . . . . . . . . . . . . . . . 61

3.2.5 Derivation from stream function pair . . . . . . . . . . . . . . . 62

3.3 Computational considerations . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.1 Non-zero divergence . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.2 Discretization into windings from the stream function . . . . . 64

3.4 Axially symmetric problems . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.1 Surface representation . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.2 The stream function and surface vector function . . . . . . . . 71

3.4.3 Derived function . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4.4 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4.5 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4.6 Derived function . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5 Stream functions in electromagnetic problems . . . . . . . . . . . . . . 74

3.5.1 Mutual inductance expressed in the stream function . . . . . . 75

3.5.2 Magnetization and surface currents . . . . . . . . . . . . . . . . 75

3.5.3 Stream function as vertical magnetization . . . . . . . . . . . . 76

3.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Numerical methods

79

4.1 Discretization of the stream function . . . . . . . . . . . . . . . . . . . 79

4.1.1 Polygonal meshes . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1.2 Normalized set of basis functions . . . . . . . . . . . . . . . . . 81

4.1.3 Axially symmetric case . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Calculating the stream function . . . . . . . . . . . . . . . . . . . . . . 87

4.2.1 Initial value problem . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.2 The optimization problem . . . . . . . . . . . . . . . . . . . . . 90

4.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 92

Contents

ix

5 Computation of physical quantities

95

5.1 Triangular and quadrilateral meshes . . . . . . . . . . . . . . . . . . . 96

5.1.1 An(x0) for a triangular mesh element . . . . . . . . . . . . . . 97

5.1.2 An(x0) for a quadrilateral mesh element . . . . . . . . . . . . . 98

5.2 Axially symmetric problems . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.1 Mutual Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.2 Lorentz force in a static background field . . . . . . . . . . . . 101

5.2.3 Spherical Harmonics Expansion . . . . . . . . . . . . . . . . . . 102

5.2.4 Expressions for the magnetic field . . . . . . . . . . . . . . . . 103

5.2.5 Integration to λ . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2.6 Reduction of the function Fm to standardized functions . . . . 107

5.2.7 Singular cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Examples

117

6.1 MRI gradient coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1.1 Principle of Magnetic Resonance . . . . . . . . . . . . . . . . . 118

6.1.2 The MRI system . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1.3 Example of an X gradient coil . . . . . . . . . . . . . . . . . . . 122

6.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2 Design of a magnetizer . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2.1 Principle of permanent magnetization . . . . . . . . . . . . . . 132

6.2.2 Electric circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2.3 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2.4 Electromagnetic design . . . . . . . . . . . . . . . . . . . . . . 139

6.2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 147

7 Conclusions and Recommendations

149

A Modified Complete Elliptic integrals

153

A.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.2 Computation using the Bartky transformation . . . . . . . . . . . . . 154

A.2.1 The Bartky transformation . . . . . . . . . . . . . . . . . . . . 154

A.2.2 Application of the Bartky transformation . . . . . . . . . . . . 157

A.2.3 The case p = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.3 Computation using recurrence relations . . . . . . . . . . . . . . . . . 164
PublicationVersionStream FunctionDiscussionBartky Transformation