Integral Methods in Low-Frequency Electromagnetics

Integral Methods in Low-Frequency Electromagnetics......Page 5
Contents......Page 7
List of Figures......Page 13
List of Tables......Page 25
Preface......Page 27
Acknowledgments......Page 29
1.1 Fundamentals......Page 31
1.1.1 Maxwell's equations in integral form......Page 32
1.1.3 Constitutive relations and equation of continuity......Page 33
1.1.5 Conductors......Page 34
1.1.7 Magnetic materials......Page 35
1.1.8 Conditions on interfaces......Page 36
1.2.1 Scalar electric potential......Page 38
1.2.2 Magnetic vector potential......Page 39
1.3.1 Static electric field......Page 40
1.3.2 Static magnetic field......Page 42
1.3.3 Quasistationary electromagnetic field......Page 44
1.3.4 General electromagnetic field......Page 45
1.4 Energy and forces in electromagnetic fields......Page 46
1.4.1 Energy of electric field......Page 47
1.4.2 Energy of magnetic field......Page 48
1.4.3 Forces in electric field......Page 49
1.4.4 Forces in magnetic field......Page 53
1.5.2 Balance of power in linear electromagnetic field......Page 54
2.1 Continuous models in electromagnetism......Page 57
2.1.1 Differential models......Page 58
2.1.2 Integral and integrodifferential models......Page 61
2.2 Methods of solution of the continuous models......Page 62
2.2.3 Methods based on the stochastic approach......Page 63
2.3.1 Methods built on the basic laws of electromagnetics......Page 64
2.3.2 Methods based on various transforms......Page 65
2.3.3 Direct solution of the field equations......Page 73
2.4 Numerical methods and their classification......Page 76
2.5.1 Difference methods......Page 78
2.5.2 Weighted residual methods......Page 83
2.5.3 Variational and other related methods......Page 88
2.6 Finite element method......Page 92
2.6.1 Discretization of the definition area and selection of the approximate functions......Page 93
2.6.2 Computation of the functional and its extremization......Page 103
2.8 Important mathematical aspects of numerical methods......Page 106
2.8.1 Stability......Page 107
2.9 Numerical schemes for parabolic equations......Page 108
2.9.1 Explicit scheme......Page 109
2.9.2 Implicit scheme......Page 110
3.1 Introduction......Page 113
3.2.2 Electric field generated by a solitary filamentary conductor of infinite length......Page 114
3.2.3 Electric field of charged thin circular ring......Page 115
3.2.4 Magnetic field generated by a solitary filamentary conductor of infinite length......Page 118
3.2.5 Magnetic field of thin circular current carrying loop......Page 120
3.2.6 Electric field generated by a system of uniformly charged parallel thin filaments of infinite length......Page 123
3.2.7 Magnetic field generated by a system of currents carrying parallel filamentary conductors of infinite length......Page 126
3.3.2 Magnetic field of an infinitely long massive conductor carrying DC current......Page 127
3.3.3 Magnetic field of a massive ring of rectangular cross section......Page 131
3.4 Forces acting in the system of long massive conductors......Page 136
3.4.1 Self-inductance of a massive ring of rectangular cross section......Page 140
3.4.2 Radial force on a massive ring of rectangular cross section......Page 145
3.4.3 Cylindrical air-core coils and their parameters......Page 148
3.4.4 Electric field of an idealized thundercloud......Page 158
3.5.2 Magnetic field around a helicoidal air-core coil......Page 163
4.1 Integral versus differential models......Page 175
4.2.1 Electrostatic fields produced by charged bodies......Page 179
4.2.2 Eddy currents in linear homogeneous systems......Page 180
4.2.3 Planar and axisymmetric arrangements......Page 183
4.3.1 Electric field of a thin charged circular ring......Page 186
4.3.2 Current density in a harmonic current carrying massive hollow conductor......Page 189
4.3.3 Current density in a system consisting of a harmonic current carrying massive hollow cylindrical conductor—a coaxial shielding pipe......Page 195
4.4 Static and harmonic problems in two dimensions......Page 200
4.4.1 Electric field of a thin rectangular plate......Page 201
4.4.2 Electric field of a charged cylinder......Page 204
4.4.3 Harmonic currents in a long conductor of arbitrary cross section......Page 210
4.5 Static problems in three dimensions......Page 215
4.5.1 Electric field of two charged cubes......Page 216
4.6 Time-dependent eddy current problems in one dimension and two dimensions......Page 221
4.6.1 Massive conductor carrying time-dependent current......Page 222
4.6.2 Pulse current in a long conductor of rectangular profile......Page 230
4.6.3 Short-circuit effects in a three-phase system......Page 234
4.7 Static and 2D eddy current problems with motion......Page 236
4.7.1 Distribution of charge in a system of two moving conductors......Page 237
5.1 Introduction......Page 247
5.1.1 Fundamental concepts......Page 249
5.1.2 Green's functions of common differential operators......Page 254
5.2.1 Particular steps of the solution......Page 256
5.2.2 Illustrative example in one dimension......Page 257
5.3 Problems with 1D integration area......Page 260
5.3.1 Two eccentrically placed charged cylinders......Page 261
5.3.2 Magnetic field in the air gap of a rotating machine......Page 264
5.3.3 Electric field near a high-voltage three-phase line......Page 269
5.3.4 Magnetic field of a massive conductor above a ferromagnetic plate......Page 271
6.1.1 Introduction......Page 275
6.1.3 Mathematical model and its solution......Page 276
6.1.4 Illustrative example......Page 277
6.2.1 Introduction......Page 285
6.2.3 Continuous mathematical model of the problem......Page 286
6.2.4 Example of computation......Page 291
6.3.1 Introduction......Page 296
6.3.2 Formulation of the problem......Page 298
6.3.3 Continuous mathematical model......Page 299
6.3.4 Discretized model and its numerical solution......Page 303
6.3.5 Example of calculation......Page 304
7.1.1 Model problem......Page 311
7.1.2 Projection methods......Page 312
7.2 Collocation methods......Page 313
7.2.2 Optimal basis functions in one dimension......Page 315
7.2.3 Efficient assembly of the collocation matrix......Page 318
7.2.5 Transformation of points from reference to physical elements......Page 319
7.2.6 Optimal basis functions in two dimensions......Page 322
7.3 Galerkin methods......Page 323
7.4 Numerical example......Page 326
7.4.1 Basic features of the proposed higher-order technique......Page 327
7.4.2 Illustrative example......Page 328
A.1.1 Vectors......Page 331
A.1.2 Matrices......Page 334
A.1.3 Systems of linear equations......Page 336
A.1.4 Eigenvalues and eigenvectors of matrices......Page 340
A.2.1 Differential and integral operations with vectors in Cartesian coordinates......Page 341
A.2.2 Other orthogonal coordinate systems......Page 345
B.1 Bessel functions......Page 349
B.1.1 Bessel functions of the first kind......Page 350
B.1.3 Hankel functions......Page 351
B.1.5 Asymptotic forms of Bessel functions......Page 352
B.1.7 Computation of Bessel and other related functions......Page 354
B.2.2 Incomplete and complete elliptic integrals of the second kind......Page 355
B.2.3 Incomplete and complete elliptic integrals of the third kind......Page 356
B.2.4 Some other useful formulas......Page 358
B.3.1 Legendre polynomials of the first kind......Page 359
B.3.2 Chebyshev polynomials of the first kind......Page 360
C.1 Analytical calculations of some integrals over typical elements......Page 363
C.1.1 Rectangle......Page 364
C.1.2 Triangle......Page 368
C.1.3 A ring of rectangular cross section......Page 374
C.1.4 A brick......Page 375
C.2 Techniques of numerical integration......Page 376
C.2.1 Numerical integration in one dimension......Page 377
C.2.2 Numerical integration in two dimensions......Page 385
C.2.3 Numerical integration in three dimensions......Page 395
References......Page 405
Topic Index......Page 415