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) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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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 Physicalmodel ............................... 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 Computationalissues............................ 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 Fourierseries ............................ 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 Polygonalmeshes.......................... 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 Axiallyb symmetric problems . 99 5.2.1 Mutualb 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 SummaryandDiscussion ......................... 114 6 Examples 117 6.1 MRIgradientcoils ............................. 118 6.1.1 Principle of Magnetic Resonance . 118 6.1.2 TheMRIsystem .......................... 119 6.1.3 Example of an X gradient coil . 122 6.1.4 Results ............................... 127 6.2 Designofamagnetizer........................... 131 6.2.1 Principle of permanent magnetization . 132 6.2.2 Electric circuit . 133 6.2.3 Designparameters ......................... 136 6.2.4 Electromagnetic design . 139 6.2.5 Notes ................................ 145 6.3 SummaryandDiscussion ......................... 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 x Contents B Logarithmic Complete Elliptic Integral 169 B.1 Generalapproach.............................. 169 B.1.1 Lell(kc, a) near kc = 0, a =0 ................... 170 B.1.2 Lell(kc, a) for large a ........................ 171 B.2 Expressions for the derivatives . 171 B.2.1 Derivatives of Mell(kc, a)...................... 172 B.2.2 Derivatives of Nell(kc,y)...................... 174 C Vector algebra 175 Glossary 177 Symbols...................................... 178 Topological terminology . 179 Bibliography 181 Index 185 Summary 189 Samenvatting 193 Curriculum Vitae 197 Chapter 1 Introduction 1.1 Background Ever since the first intelligent humans created tools to make their life easier, the opti- mal shape of such a tool posed an interesting problem. Even without any knowledge of the fundamental physical laws, our forefathers were able to get a intuitive feeling for the optimal shape. For example, a fist axe (see Figure 1.1) has to have a ’sharp’ (drawing by D. Stapert/H.R. Roelink, B.A.I. Groningen) Figure 1.1: Fist axe, over 45,000 years old edge to maximize its ability to cleave wood, related to maximization of the local pres- sure. The hull of a vessel has to have a certain shape in order to minimize the force 2 Chapter 1. Introduction needed to propel it and to keep the vessel stable, related to the complex physical laws in fluid dynamics and mechanics (see Figure 1.2). In short, we humans have a very (source: Vasily N. Khramushin, Saint-Petersburg — Yuzhno-Sakhalinsk) Figure 1.2: Phoenician’s sea trade vessel, about 2500 years ago useful intuitive understanding of many physical laws, and in addition have the rather astonishing capability of ’inverting’ these laws to find the optimal shape of objects among the infinite number of possible shapes. When physical science began to evolve, physical laws were discovered and formulated, as for instance Archimedes’ law and Newton’s three laws of motion. These laws, and the related mathematics — which was often also developed at the same time — gave an insight why the shape of the ob- ject was as it should be, and often this insight could be used to improve its efficiency. Also physical laws were discovered that were less intuitive, that is, remotely or not related to phenomena that an average person would experience in his everyday life. An example are the various laws related to electromagnetism as formulated by a/o Faraday, Amp`ere and Volt, and later unified by Maxwell. Electromagnetism is the focus area of this thesis, and Maxwell’s equations will be discussed in more detail in Section 2.3.1. Although these laws incorporate the rather abstract concept of ’fields’, fortunately skilled persons are able to develop an intuitive feeling for many electro- magnetic phenomena, perhaps because analogies exist with common phenomena like streaming fluid.