Covers the practical numerical method for the analysis and design of magnets Extensively covers the magnet design and computation aspects from theories to practical applications, emphasizing design methods of practical structures such as superconducting, electromagnetic and permanent magnet for use in various scientific instruments, industrial processing, biomedicine and special electrical equipments. The computations cover a wide range of numerical techniques and analytical derivation to efficiently provide solutions to complicated problems that are often encountered in practice, where simple analytical calculations are no longer adequate. Chapters include: Introduction of Magnet Technology, Magnetostatic Equation for the Magnet Structure, Finite Element Analysis for Magnetostatic Field, Integral Method for Magnetostatic Field, Numerical Method of Solenoid Coils Design, Series Analysis of Axially Symmetric Magnetic Field, Magnets with High Magnetic Field and High Homogeneity, Permanent Magnet and its Applications, Shimming Magnetic Field with Optimal Magnet Structure, Electro-Mechanical Effect of Magnet, Calculation Formula of Magnetic Field Vector/Recurrence Formula of Legendre Function/Magnetization Equation in Cylindrical Coordinate System. Presents a clear introduction to magnet technology, followed by basic theories, numerical analysis and practical applications Emphasizes the latest developments in magnet design, including MRI systems, and provides comprehensive numerical techniques that provide solutions to practical problems Covers the most recent developments on magnet design and application as well as emphasizing latest computation techniques for optimising and characterising the magnetostatic structure design Essential reading for Researchers in Magnets and Applications, Materials Scientists, Structural Engineers, Graduate students in Electrical Engineering.

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 ofWedge-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 Ylm(u,f) 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 Method for 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 MRI Magnet 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 Method for 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 437

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 T/800 mm MRI Magnet 470

10.6 Summary 472

References 473

Index 477