Integral Methods in Low-Frequency Electromagnetics Book

List of Figures xi

List of Tables xxiii

Preface xxv

Acknowledgments xxvii

1 Electromagnetic Fields and their Basic Characteristics 1

1.1 Fundamentals 1

1.1.1 Maxwell's equations in integral form 2

1.1.2 Maxwell's equations in differential form 3

1.1.3 Constitutive relations and equation of continuity 3

1.1.4 Media and their characteristics 4

1.1.5 Conductors 4

1.1.6 Dielectrics 5

1.1.7 Magnetic materials 5

1.1.8 Conditions on interfaces 6

1.2 Potentials 8

1.2.1 Scalar electric potential 8

1.2.2 Magnetic vector potential 9

1.2.3 Magnetic scalar potential 10

1.3 Mathematical models of electromagnetic fields 10

1.3.1 Static electric field 10

1.3.2 Static magnetic field 12

1.3.3 Quasistationary electromagnetic field 14

1.3.4 General electromagnetic field 15

1.4 Energy and forces in electromagnetic fields 16

1.4.1 Energy of electric field 17

1.4.2 Energy of magnetic field 18

1.4.3 Forces in electric field 19

1.4.4 Forces in magnetic field 23

1.5 Power balance in electromagnetic fields 24

1.5.1 Energy in electromagnetic field and its transformation 24

1.5.2 Balance of power in linear electromagnetic field 24

2 Overview of Solution Methods 27

2.1 Continuous models in electromagnetism 27

2.1.1 Differential models 28

2.1.2 Integral and integrodifferential models 31

2.2 Methods of solution of the continuous models 32

2.2.1 Analytical methods 33

2.2.2 Numerical methods 33

2.2.3 Methods based on the stochastic approach 33

2.2.4 Specific methods 34

2.3 Classification of the analytical methods 34

2.3.1 Methods built on the basic laws of electromagnetics 34

2.3.2 Methods based on various transforms 35

2.3.3 Directsolution of the field equations 43

2.4 Numerical methods and their classification 46

2.5 Differential methods 48

2.5.1 Difference methods 48

2.5.2 Weighted residual methods 53

2.5.3 Variational and other related methods 58

2.6 Finite element method 62

2.6.1 Discretization of the definition area and selection of the approximate functions 63

2.6.2 Computation of the functional and its extremization 73

2.6.3 Further prospectives 76

2.7 Integral and integrodifferential methods 76

2.8 Important mathematical aspects of numerical methods 76

2.8.1 Stability 77

2.8.2 Convergence 78

2.8.3 Accuracy 78

2.9 Numerical schemes for parabolic equations 78

2.9.1 Explicit scheme 79

2.9.2 Implicit scheme 80

3 Solution of Electromagnetic Fields by Integral Expressions 83

3.1 Introduction 83

3.2 ID integration area 84

3.2.1 Review of typical problems 84

3.2.2 Electric field generated by a solitary filamentary conductor of infinite length 84

3.2.3 Electric field of charged thin circular ring 85

3.2.4 Magnetic field generated by a solitary filamentary conductor of infinite length 88

3.2.5 Magnetic field of thin circular current carrying loop 90

3.2.6 Electric field generated by a system of uniformaly charged parallel thin filaments of infinite length 93

3.2.7 Magnetic field generated by a system of currents carrying parallel filamentary conductors of infinite length 96

3.3 2D integration area 97

3.3.1 Review of typical problems 97

3.3.2 Magnetic field of an infinitely long massive conductor carrying DC current 97

3.3.3 Magnetic field of a massive ring of rectangular cross section 101

3.4 Forces acting in the system of long massive conductors 106

3.4.1 Self-inductance of a massive ring of rectangular cross section 110

3.4.2 Radial force on a massive ring of rectangular cross section 115

3.4.3 Cylindrical air-core coils and their parameters 118

3.4.4 Electric field of an idealized thundercloud 128

3.5 3D integration area 133

3.5.1 Review of typical problems 133

3.5.2 Magnetic field around a helicoidal air-core coil 133

4 Integral and Integrodifferential Methods 145

4.1 Integral versus differential models 145

4.2 Theoretical foundations 149

4.2.1 Electrostatic fields produced by charged bodies 149

4.2.2 Eddy currents in linear homogeneous systems 150

4.2.3 Planar and axisymmetric arrangements 153

4.3 Static and harmonic problems in one dimension 156

4.3.1 Electric field of a thin charged circular ring 156

4.3.2 Current density in a harmonic current carrying massive hollow conductor 159

4.3.3 Current density in a system consisting of a harmonic current carrying massive hollow cylindrical conductor-a coaxial shielding pipe 165

4.4 Static and harmonic problems in two dimensions 170

4.4.1 Electric field of a thin rectangular plate 171

4.4.2 Electric field of a charged cylinder 174

4.4.3 Harmonic currents in a long conductor of arbitrary cross section 180

4.5 Static problems in three dimensions 185

4.5.1 Electric field of two charged cubes 186

4.5.2 Electric field of two charged plates 191

4.6 Time-dependent eddy current problems in one dimension and two dimensions 191

4.6.1 Massive conductor carrying time-dependent current 192

4.6.2 Pulse current in a long conductor of rectangular profile 200

4.6.3 Short-circuit effects in a three-phase system 204

4.7 Static and 2D eddy current problems with motion 206

4.7.1 Distribution of charge in a system of two moving conductors 207

5 Indirect Solution of Electromagnetic Fields by the Boundary Element Method 217

5.1 Introduction 217

5.1.1 Fundamental concepts 219

5.1.2 Green's functions of common differential operators 224

5.2 BEM-based solution of differential equations 226

5.2.1 Particular steps of the solution 226

5.2.2 Illustrative example in one dimension 227

5.2.3 Multidimensional problems 230

5.3 Problems with 1D integration area 230

5.3.1 Two eccentrically placed charged cylinders 231

5.3.2 Magnetic field in the air gap of a rotating machine 234

5.3.3 Electric field near a high-voltage three-phase line 239

5.3.4 Magnetic field of a massive conductor above a ferromagnetic plate 241

6 Integral Equations in Solution of Selected Coupled Problems 245

6.1 Continual induction heating of nonferrous cylindrical bodies 245

6.1.1 Introduction 245

6.1.2 Formulation of the technical problem 246

6.1.3 Mathematical model and its solution 246

6.1.4 Illustrative example 247

6.1.5 Conclusion 255

6.2 Induction heating of a long nonmagnetic cylindrical billet rotating in a uniform magnetic field 255

6.2.1 Introduction 255

6.2.2 Formulation of the technical problem 256

6.2.3 Continuous mathematical model of the problem 256

6.2.4 Example of computation 261

6.2.5 Conclusion 266

6.3 Pulsed Induction Accelerator 266

6.3.1 Introduction 266

6.3.2 Formulation of the problem 268

6.3.3 Continuous mathematical model 269

6.3.4 Discretized model and its numerical solution 273

6.3.5 Example of calculation 274

7 Numerical Methods for Integral Equations 281

7.1 Introduction 281

7.1.1 Model problem 281

7.1.2 Projection methods 282

7.2 Collocation methods 283

7.2.1 Optimal collocation points in one dimension 285

7.2.2 Optimal basis functions in one dimension 285

7.2.3 Efficient assembly of the collocation matrix 288

7.2.4 Optimal collocation points in two dimensions 289

7.2.5 Transformation of points from reference to physical elements 289

7.2.6 Optimal basis functions in two dimensions 292

7.2.7 Efficient assembly of the collocation matrix 293

7.3 Galerkin methods 296

7.3.1 Schur complement method for partially orthonormal basis 296

7.4 Numerical example 296

7.4.1 Basic features of the proposed higher-order technique 297

7.4.2 Illustrative example 298

Appendix A Basic Mathematical Tools 301

A.1 Vectors, matrices, and systems of linear equations 301

A.1.1 Vectors 301

A.1.2 Matrices 304

A.1.3 Systems of linear equations 306

A.1.4 Eigenvalues and eigenvectors of matrices 310

A.2 Vector analysis 311

A.2.1 Differential and integral operations with vectors in Cartesian coordinates 311

A.2.2 Other orthogonal coordinate systems 315

Appendix B Special Functions 319

B.1 Bessel functions 319

B.1.1 Bessel functions of the first kind 320

B.1.2 Bessel functions of the second kind 321

B.1.3 Hankel functions 321

B.1.4 Modified Bessel functions 322

B.1.5 Asymptotic forms of Bessel functions 322

B.1.6 Some other useful relations 324

B.1.7 Computation of Bessel and other related functions 324

B.2 Elliptic integrals 325

B.2.1 Incomplete and complete elliptic integrals of the first kind 325

B.2.2 Incomplete and complete elliptic integrals of the second kind 325

B.2.3 Incomplete and complete elliptic integrals of the third kind 326

B.2.4 Some other useful formulas 328

B.3 Special polynomials 329

B.3.1 Legendre polynomials of the first kind 329

B.3.2 Chebyshev polynomials of the first kind 330

Appendix C Integration Techniques 333

C.1 Analytical calculations of some integrals over typical elements 333

C.1.1 Rectangle 334

C.1.2 Triangle 338

C.1.3 A ring of rectangular cross section 344

C.1.4 A brick 345

C.2 Techniques of numerical integration 346

C.2.1 Numerical integration in one dimension 347

C.2.2 Numerical integration in two dimensions 355

C.2.3 Numerical integration in three dimensions 365

References 375

Topic Index 385