Foreword ....................................................... xi
Preface ...................................................... xiii
1 Introduction to Magnet Technology ............................ 1
1.1 Magnet Classification ................................... 1
1.2 Scientific Discoveries in High Magnetic Field ........... 3
1.3 High Field Magnets for Applications ..................... 3
1.3.1 Magnets in Energy Science ........................ 4
1.3.2 Magnets in Condensed Matter Physics .............. 4
1.3.3 Magnets in NMR and MRI ........................... 5
1.3.4 Magnets in Scientific Instruments and Industry ... 6
1.4 Structure of Magnets .................................... 7
1.4.1 Configuration of Solenoid Magnet ................. 7
1.4.2 Racetrack and Saddle-Shaped Magnets .............. 7
1.4.3 Structure of Other Complicated Magnets .......... 10
1.5 Development Trends in High Field Magnets ............... 10
1.6 Numerical Methods for Magnet Design .................... 12
1.7 Summary ................................................ 14
References .................................................. 14
2 Magnetostatic Equations for the Magnet Structure ............ 17
2.1 Basic Law of Macroscopic Electromagnetic Phenomena ..... 17
2.1.1 Biot-Savart Law ................................. 17
2.1.2 Faraday's Law ................................... 18
2.2 Mathematical Basis of Classical Electromagnetic
Theory ................................................. 20
2.2.1 Gauss's Theorem ................................. 20
2.2.2 Stokes'Theorem .................................. 20
2.2.3 Green's Theorem ................................. 21
2.2.4 Helmholtz's Theorem ............................. 21
2.3 Equations of Magnetostatic Fields ...................... 25
2.3.1 Static Magnetic Field Generated by Constant
Current in Free Space ........................... 25
2.3.2 Basic Properties of Static Magnetic Field ....... 26
2.3.3 Magnetic Media in Static Magnetic Field ......... 29
2.3.4 Boundary Conditions of Magnetostatic Field ...... 32
2.3.5 Boundary-Value Problem of Static Magnetic
Field ........................................... 34
2.3.6 Summary of Equations of Magnetostatic Problem ... 35
2.4 Summary ................................................ 37
References .................................................. 37
3 Finite Element Analysis for the Magnetostatic Field ......... 39
3.1 Introduction ........................................... 39
3.1.1 Basic Concept of the FEM ........................ 39
3.1.2 Basic Steps of the FEM .......................... 40
3.2 Functional Construction for Static Magnetic Field ...... 41
3.3 Discretization and Interpolation Function of Solution
Domain ................................................. 44
3.3.1 Principle of Selecting Subdivisions in the
Domain .......................................... 45
3.3.2 Selection of Interpolation Function ............. 45
3.3.3 Unified Expressions of Interpolation Function ... 67
3.4 Formulation of System Equations ........................ 68
3.4.1 Two-Dimensional Cartesian Coordinate System ..... 69
3.4.2 Three-Dimensional Cartesian Coordinate System ... 70
3.4.3 Axially Symmetric Scalar Potential System ....... 71
3.5 Solution of System Equation for the FEM ................ 74
3.6 Applied FEM for Magnet Design .......................... 76
3.6.1 Magnetic Field for a Superconducting Magnet
with LTS and HTS ................................ 76
3.6.2 Magnetic Field for a Superferric Dipole Magnet .. 78
3.6.3 Force Characteristics of a Superconducting
Ball in Magnetic Field .......................... 81
3.7 Summary ................................................ 87
References .................................................. 87
4 Integral Method for the Magnetostatic Field ................. 89
4.1 Integral Equation of Static Magnetic Field ............. 89
4.2 Magnetic Field from Current-Carrying Conductor ......... 91
4.2.1 Magnetic Field Generated by Rectangular
Conductor ....................................... 91
4.2.2 Magnetic Field of Arc-Shaped Winding ............ 96
4.2.3 Magnetic Field Generated by Solenoid Coil ...... 114
4.2.4 Magnetic Field of Elliptical Cross-Section
Winding ........................................ 119
4.2.5 Parallel Plane Field ........................... 122
4.2.6 Magnetic Field of Wedge-Shaped Current Block
with Triangular Cross-Section .................. 123
4.2.7 Magnetic Field of Wedge-Shaped Structure with
Rectangular Cross-Section ...................... 126
4.3 Magnetic Field with Anisotropic Magnetization ......... 128
4.3.1 Subdivision of Three-Dimensional
Ferromagnetic Media ............................ 129
4.3.2 Magnetic Field in the Cylindrical Symmetrical
System ......................................... 133
4.4 Case Studies of Complex Coil Structures ............... 139
4.4.1 Magnetic Field Distribution of
Superconducting Magnet in Space ................ 139
4.4.2 Superconducting Magnet with Very Small Stray
Magnetic Field for an Energy Storage System .... 140
4.5 Summary ............................................... 142
References ................................................. 142
5 Numerical Methods for Solenoid Coil Design ................. 145
5.1 Magnet Materials and Performance ...................... 145
5.1.1 Basic Properties of Superconducting Materials .. 146
5.1.2 Material Properties of Copper, Aluminum, and
their Alloys ................................... 153
5.2 Magnetic Field of the Superconducting Solenoid ........ 156
5.2.1 Solenoid Coils with Uniform Current Density .... 158
5.2.2 Current Density Graded by Multisolenoid Coils .. 167
5.2.3 Design of High Temperature Superconducting
Coils .......................................... 177
5.3 Design of Resistive Magnets ........................... 181
5.3.1 Resistive Magnet with Nonuniform Current
Distribution ................................... 183
5.3.2 Structure of Bitter Resistive Magnets .......... 184
5.3.3 Resistive Magnet with Iron Yoke ................ 186
5.4 Engineering Design for Superconducting Magnets ........ 186
5.4.1 10 T Cryogen-Free Superconducting Magnet ....... 186
5.4.2 Split Superconducting Magnet System with
Large Crossing Warm Bore ....................... 188
5.4.3 Superconducting Magnet with Persistent
Current Switch ................................. 192
5.4.4 Ultrahigh Field Superconducting Magnet ......... 194
5.4.5 A Bi2223 Split Pair Superconducting Magnet
for a Propulsion Experiment .................... 195
5.5 Summary ............................................... 201
References ................................................. 201
6 Series Analysis of Axially Symmetric Magnetic Field ........ 205
6.1 Laplace's Equation in Spherical Coordinates ........... 205
6.1.1 Legendre Equation and Polynomial ............... 206
6.1.2 Orthogonality of the Legendre Polynomial ....... 208
6.1.3 Associated Legendre Function and Spherical
Harmonics Yim(6,(l>) ........................... 210
6.1.4 Addition Theorem of Spherical Harmonic
Functions ...................................... 212
6.1.5 Magnetic Vector of Loop Current with Series
Expression ..................................... 214
6.1.6 Magnetic Scalar Potential of Loop Current
with Series Expression ......................... 216
6.1.7 Magnetic Field of Zonal Current with Series
Expression ..................................... 218
6.2 Series Expression of the Boundary-Value Problem ....... 223
6.2.1 Expansion of Magnetic Induction of Circular
Current Filaments .............................. 224
6.2.2 Expansion of the Magnetic Induction for
Solenoid Coils ................................. 226
6.2.3 Expansion of Magnetic Induction of Solenoid
at any Position on the z-Axis .................. 227
6.2.4 Expansion of Magnetic Fields with Multi-
Current Filaments .............................. 232
6.2.5 Expansion of Magnetic Field of Magnetization
Loop ........................................... 233
6.2.6 Calculation of Expansion Coefficients of
Arc-Type Coils ................................. 235
6.3 Magnetic Induction of Helical Coils ................... 242
6.3.1 Magnetic Field Calculation of Helical Current
Filaments ...................................... 242
6.3.2 Magnetic Induction Generated by Helical Coils .. 243
6.4 Magnetic Field of Multi-Coil Combination .............. 247
6.4.1 Configuration of Highly Homogeneous Field ...... 247
6.4.2 Determination Methods for Parameters of
Multi-Section Magnets .......................... 248
6.5 Applied Magnetic Field Series Expansion ............... 249
6.5.1 Magnetic Field for a Surgical Magnetic
Navigation System .............................. 249
6.5.2 Force of Superconducting Sphere in the
Magnetic Field ................................. 252
6.5.3 Design of Superconducting Magnet Shim Coils .... 259
6.6 Summary ............................................... 261
References ................................................. 261
7 High Field Magnet with High Homogeneity .................... 263
7.1 Definition of Magnetic Field Homogeneity .............. 263
7.2 Requirements for Magnets with High Homogeneity ........ 264
7.2.1 Large-Bore MRI Magnet System for Medical
Research and Clinical Applications ............. 264
7.2.2 Electronic Cyclotron and Focused Magnet
System ......................................... 267
7.2.3 High Homogeneity Magnet for Scientific
Instruments .................................... 267
7.2.4 Main Constraint Conditions of Inverse Problem
for High Homogeneity Magnet .................... 269
7.3 Design of High Homogeneity Magnet ..................... 271
7.3.1 Review of Inverse Problem ...................... 271
7.3.2 Continuous Current Distribution Method ......... 273
7.3.3 Solving Nonlinear Equations for the Coil
Design ......................................... 277
7.3.4 Combined Linear and Nonlinear Method for
Inverse Problem ................................ 279
7.3.5 Regularization Methodfor Inverse Problem ....... 281
7.3.6 Ferromagnetic Shielding of Superconducting
Coil ........................................... 284
7.3.7 Solving the Magnet Structure by the Fredholm
Equation ....................................... 286
7.3.8 Nonlinear Optimization with Preset Coil
Number ......................................... 287
7.4 Design Example of High Homogeneity Magnet ............. 290
7.4.1 Active-Shield Cylindrical Magnet ............... 290
7.4.2 Openness of MRIMagnet .......................... 301
7.4.3 Short-Length Active-Shield MRI Magnet .......... 302
7.5 Design of High Field and High Homogeneity Magnet ...... 305
7.5.1 Minimum Volume Method .......................... 305
7.5.2 One-Step Nonlinear Optimal Method .............. 307
7.6 Engineering Designs and Applications .................. 309
7.7 Summary ............................................... 317
References ................................................. 318
8 Permanent Magnets and their Applications ................... 321
8.1 Introduction to Magnetic Materials .................... 321
8.1.1 Basic Parameters of Magnetism .................. 321
8.1.2 Progress in Magnetic Materials ................. 322
8.2 Classification and Characteristics of Permanent
Magnets ............................................... 324
8.2.1 Selection of Permanent Materials ............... 324
8.2.2 Selection of Soft Magnetic Materials ........... 326
8.3 Permanent Magnet Structure Design ..................... 331
8.3.1 Magnetic Circuit Design of Permanent Magnet .... 331
8.3.2 Numerical Methods of Permanent Magnet Design ... 334
8.4 Design of Magnet for Engineering Applications ......... 341
8.4.1 MRI Permanent Magnets .......................... 341
8.4.2 AMS with Permanent Magnet ...................... 349
8.4.3 Structure of Six-Pole Permanent Magnet ......... 354
8.4.4 Magnetic Resonance Imaging Logging ............. 354
8.4.5 Q&A Vacuum Birefringence Experimental Magnet ... 359
8.4.6 Permanent Magnets for Magnetic Resonance
Molecular Imaging .............................. 362
8.5 Summary ............................................... 364
References ................................................. 365
9 Shimming Magnetic Field .................................... 367
9.1 Magnetostatic Principle for Shimming Magnetic Field ... 367
9.2 Design Method for Active Shimming Coil ................ 372
9.2.1 Axial Shim Design .............................. 372
9.2.2 Radial Coil Design ............................. 382
9.2.3 Shim Design by Arbitrary Current Distribution .. 397
9.2.4 Target-Field Method for MRI Shim Coils ......... 400
9.3 Current Calculation for Active Shim Coils ............. 411
9.4 Passive Shimming Design Method ........................ 414
9.4.1 Magnetic Field Produced by Magnetic Material ... 415
9.4.2 Mathematical Optimization Model ................ 416
9.5 Summary ............................................... 420
References ................................................. 420
10 Electromechanical Effects and Forces on the Magnet ......... 423
10.1 Magnetostatic Electromechanical Effects on the
Solenoid .............................................. 423
10.1.1 Analytical Methodfor the Stress Problem in
a Solenoid ..................................... 423
10.1.2 Semi-Analytical Method for the Stress in
a Solenoid ..................................... 425
10.2 Averaged Model of the Magnet .......................... 435
10.2.1 Basic Theory of the FEM ........................ 435
10.2.2 Averaged Model for FEM ......................... 436
10.2.3 Stress Solution for a High Field Magnet ........ АЪ1
10.2.4 Equivalent Elastic Material of Magnet .......... 443
10.3 Detailed FEM for the Ultrahigh Field Solenoid ......... 445
10.3.1 Establishment of the Detailed FEM .............. 445
10.3.2 Mesh Construction in the Detailed Model ........ 453
10.3.3 Analysis Method of the Detailed Model .......... 456
10.3.4 Equivalent Treatment of Electromagnetic Force
Loading ........................................ 456
10.3.5 Finite Element Equation of Detailed FEM ........ 458
10.4 Mutual Inductance and Force Calculations .............. 459
10.5 Detailed Model for Electromechanical Stress Analysis .. 462
10.5.1 Electromagnetic Stress Analysis of 11.75 T
NMR Magnet ..................................... 462
10.5.2 Stress Analysis of a 19 T Insert ............... 466
10.5.3 Stress Analysis of' a 9.4 Т/800 mm MRIMagnet ... 470
10.6 Summary ............................................... 472
References ................................................. 473
Index ......................................................... 477
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