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Contents
1 Introduction 1
2 Vector Fields 3
2.1 Basic Operators and Equations 3
2.1.1 Vector Fields and Operators 3
2.1.2 Definition of a Vector Field 5
2.1.3 Decomposition of a Field 7
2.1.4 Scalar and Vector Potentials 8
2.1.5 Green's Theorem 9
2.1.6 Green's Formula 9
2.2 Electrostatic Field 10
2.2.1 Maxwell's Equations for Electrostatics 10
2.2.2 Electrostatic Potentials 13
2.2.3 Electrostatic Energy 15
2.2.4 Field of a Charged Plane in a Rectangular Domain 16
2.2.5 Field of a Point Charge in a Spherical Domain 16
2.2.6 Field of a Dipole 18
2.2.7 Field of a Line Charge in a Cylindrical Domain 19
2.2.8 Field of a Surface Charge on a Sphere 20
2.2.9 Energy and Forces in the Electrostatic Field 22
2.2.10 Force between the Plates of a Capacitor 24
2.2.11 Force at the Interface between Two Dielectric Materials 25
2.3 Magnetostatic Field 28
2.3.1 Maxwell's Equations for Magnetostatics 28
2.3.2 Magnetostatic Potentials 30
2.3.3 Magnetostatic Energy 32
2.3.4 Field of a Line Current in a Three-Dimensional Domain: Differential A...
2.3.5 Energy and Forces in the Magnetostatic Field 34
2.3.6 Force on an Electromagnet 37
2.3.7 Test Problems 38
2.4 Steady Conduction Field 39
2.4.1 Maxwell's Equations for Conduction Field 39
2.4.2 Potentials 40
2.4.3 Power Loss 42
2.4.4 Analytic Functions of Complex Variable 42
2.4.5 Field of a Cylindrical Conductor 42
3 Analytical Methods for Solving Boundary-Value Problems 45
3.1 Method of Green's Function 45
3.1.1 Green's Formula for Electrostatics 46
3.1.2 Green's Functions for Boundary-Value Problems 46
3.1.3 Field of a Point Charge Surrounded by a Spherical Surface at Known ...
3.1.4 Field of a Surface Dipole Distributed on a Sphere of Radius R 56
3.1.5 Green's Formula for Two-Dimensional Magnetostatics 57
3.1.6 Field of a Line Current in a Three-Dimensional Domain: Integral Appro...
3.1.7 Field of a Current-Carrying Conductor of Rectangular Cross-Section 59
3.2 Method of Images 60
3.2.1 Magnetic Field of a Line Current in a Slot 67
3.2.2 Magnetic Field of a Line AC Current over a Conducting Half-Space 70
3.3 Method of Separation of Variables 71
3.3.1 Magnetic Field of a Current Uniformly Distributed in a Slot 73
4 Numerical Methods for Solving Boundary-Value Problems 77
4.1 Variational Formulation in Two-Dimensional Magnetostatics 77
4.2 Finite Elements for Two-Dimensional Magnetostatics 80
4.2.1 Discretization of Energy Functional 80
4.2.2 Local Shape Functions in Rectangular Coordinates 82
4.2.3 Coefficient Matrix and Source Vector 84
4.2.4 From Potential to Field 85
4.2.5 Magnetic Field in a Slot Solved by the Finite Element Method 86
4.3 Finite Elements for Three-Dimensional Magnetostatics 95
4.3.1 Surface and Solid Modelling 95
4.3.2 Local Shape Functions in Rectangular Coordinates 95
4.3.3 Comparison of 2 D and 3 D Simulations of an Electromagnet 97
5 Time-Varying Electromagnetic Field 101
5.1 Maxwell's Equations in Differential Form 101
5.2 Poynting's Vector 103
5.3 Maxwell's Equations in the Frequency Domain 103
5.4 Plane Waves in an Infinite Domain 105
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Index
AAmpère's equation, 102
Analytic functions, 42
Area coordinates, 83
Automated optimal design, 2, 141
BBandwidth, 88
Bessel's function, 124
Biot-Savart's law, 34, 128, 137
Boundary conditions, 7, 13, 14, 31, 38, 42, 43, 67, 69, 70, 72-74, 78, 94, 102, 114, ...
Boundary-value problem, 7, 10, 14, 48, 77
CCapacitance, 25
Cauchy's problem, 136
Cauchy-Riemann's equations, 42, 43
Charge continuity, 102
Charge-free, 12, 25, 108
Charged sphere 20-21, 54-57, 63, 64
Charge density, 11, 19, 20, 24, 56
Coefficient matrix, 82, 84, 86, 87, 97
Co-energy, 16, 22, 25, 26, 32, 34, 37, 79
Complex variable, 42
Condition number, 89, 139
Conducting half-space, 60, 70
Conducting plane, 70, 115-121
Conduction
- current, 101
- field, 39, 40, 42
Conductivity, 39, 102, 107, 115, 130
Conductor, 11, 34, 42, 43, 57-59, 69, 73, 110, 115-117, 119-125, 137, 138
Constitutive law, 10, 12, 28, 29
Constrained optimization, 144
Constraint, 153-159
Convection current, 101
Coulomb's gauge, 113
Coulomb's method, 23, 25, 27
Curl, 3, 5, 7, 8, 41, 85, 114, 161
Curl-free field, 7
Current density, 35, 39, 42, 58, 60, 79, 85, 87, 101, 102, 106, 107, 113, 115, 116, 1...
Cylindrical
- conductor, 42, 43, 59, 122, 123
- coordinates, 3, 19, 33, 57, 77, 122, 165
DDelta function, 16
Design variables, 143, 144, 147
Deterministic, 146, 147
Diagonal dominance, 88
Differential approach, 33
Dirac's distribution, 16, 17, 33
Direct problems, 135
Dirichlet's problem, 10, 14, 51, 52
Displacement current, 101, 107, 113
Divergence, 3, 5, 8, 9, 14, 36, 164
Divergence-free, 6, 7
Domain, 4, 5, 7, 10-17
Double layer, 56
Dual potentials, 113-115
Dynamic optimization, 156-158
EElectric dipole, 18
Electromagnet, 37, 97-99
Electromagnetic
- field, 101-103, 107, 110, 113, 114, 125, 161
- wave, 105, 110
Electrostatic
- energy, 15, 23
- field, 10, 15, 19, 22
- images, 60-64
Energy and forces, 22-24, 34-36
Energy functional, 78-80
Error norm, 92
Euler's equation, 79
Evolution strategy, 147
FFaraday's equation, 102
Far field, 112
Field
- domain, 46, 72, 149
- intensity, 10, 28, 39, 41, 101, 106, 127
- point, 8, 45, 46, 57, 59, 110, 163, 164
Finite elements, 80, 95
Flux
- density, 10, 15, 28, 32, 35, 65-67
- surface, 4
Force, 22-27, 34-38, 98, 99, 102
Fredholm's equation, 137, 138
Frequency domain, 103, 104, 106, 107, 110, 111, 121, 123
GGalerkin's method, 79
Galilean transformation, 127, 132
Gauge, 14, 30, 41, 108-111, 113, 114
Gauss's electric equation, 102
Gauss's magnetic equation, 102
Generation, 147
Genetic algorithm, 146
Global coefficient matrix, 87
Global minimum, 146
Global shape function, 80, 81, 84
Global source vector, 82, 84, 85, 94, 97
Gradient, 14, 30, 41, 97, 108, 114, 145, 146
Gradient-based method, 145, 146
Gradient-free method, 145, 146
Green's first identity, 9
Green's formula, 9, 10, 45, 46, 56, 57,
Green's function, 45, 46, 49-51, 54, 57, 59, 137, 138
Grid, 80-82, 84-86, 88, 91-95, 144
HHadamard's condition, 135, 136
Harmonic, 13, 30, 41, 105, 113, 118
Helmholtz's equations, 104
Homogeneous medium, 12, 29, 40, 107, 108
IIdentification, 135, 141, 150-153
Ill-posed problems, 135-137
Initial conditions, 113, 124, 130, 136
Inner domain, 50
Insulating medium, 10, 12, 105, 125
Integral approach, 21, 57, 137
Interface, 11, 12, 25, 27, 35, 61, 85
Inverse problems, 135-141
Irrotational, 7, 11, 12, 29, 40
Isotropic, 10, 11, 13, 15, 28, 32, 39, 113
KKelvin's transformation, 46
Kernel, 137, 138
Kronecker's index, 53
LLaminar current density, 123, 125
Laplace's equation, 13, 14, 31, 41, 71-73, 136
Laplace's law, 58, 112
Laplacian, 4, 47-49
Least-square solution, 139-141
Line
- charge, 19, 21
- current, 33, 57, 58, 64-70
Local coefficient matrix, 84, 86, 87
Local error, 91
Local minimum, 144
Local shape function, 82, 84, 95, 96
Local source vector, 85
Lorentz's equation, 102
Lorentz's gauge, 109, 111, 113
Lorentzian transformation, 132
MMagnet, 30
Magnetic
- diffusion, 129, 131, 157
- dipole, 28
- pole, 30, 150, 153, 156, 159
Magnetostatic
- energy, 32
- field, 28-38
- images, 64-67
Maxwell's equations, 10, 28, 39, 101, 103, 106, 113, 127, 161
Mechanical effect, 22
Medium, 10-12, 15, 28, 29, 32, 39, 40, 61, 101, 102, 105-108, 113, 125
Mesh, 95, 97, 98, 151, 155, 160
Method of images, 60-70
Minimization, 79, 143, 144, 146, 148, 151
Moment, 18, 56
Monopole, 19
Motional effect, 102
Multiobjective optimization, 145
Multiply-connected domain, 14
Mutation, 147
NNear field, 112
Neumann's problem, 10, 14
Newton-Raphson method, 31, 151
Nodal
- current, 82, 89
- potential, 82
Node, 80-82, 84-89, 91-94
Non-deterministic, 146, 147
Non-dominated solutions, 145, 159, 160
Non-homogeneous medium, 12, 29, 40
Non-linear, 15, 32, 131, 146, 150, 151, 153
Norm, 119, 139, 161
Normal versor, 5, 13, 23, 31, 35, 37, 46, 164
OObjective function, 143, 144, 146-149, 153-157
Observer, 10, 28, 67, 125-131
Optimization, 143-145, 152-159
Ordinary differential equation, 72, 123
Oscillating dipole, 110-112
Outer domain, 52
Over-determined system, 138, 139
PPareto front, 145, 159, 160
Pareto optimality, 159
Partial differential equation, 72
Penalty term, 144, 154
Penetration depth, 118, 123, 125
Permanent magnet, 28-30,
Permeability, 28, 29, 31, 35, 37, 38, 57, 64, 65, 67, 68, 70, 71, 73, 77, 86, 97, 102...
Permittivity, 10, 11, 24, 26, 46, 60, 61, 102, 107, 113
Phasor, 104, 105, 107, 110-112, 118, 121
Plane wave, 105, 106
Point charge, 16, 18, 21, 54, 57, 60-63, 110, 127
Poisson's equation, 13, 45, 46, 72, 73, 79
Potential, 8, 13, 16-19, 31, 34, 40-42
Power loss, 42, 121
Poynting's vector, 103, 104, 106, 112, 121
Poynting's theorem, 103
RRadial vector, 61, 112
Rectangular coordinates, 3, 4, 24, 30, 35, 41, 67, 71, 77, 80, 82, 95
Reference frame, 3, 125
Reluctivity, 28, 150
Retarded potentials, 110
Ritz's method, 79
Rotational, 34
SSample-and-rank method, 159
Scalar potential, 8, 10, 14, 31, 108, 109, 111, 113, 114
Selection, 147
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