Preface |
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iii | |
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Mathematical Preliminaries |
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1 | (26) |
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1 | (1) |
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1 | (1) |
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2 | (2) |
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2 | (1) |
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Definition of the Gradient, Divergence, and Rotational |
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3 | (1) |
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4 | (3) |
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5 | (2) |
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7 | (7) |
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7 | (2) |
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9 | (2) |
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11 | (2) |
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13 | (1) |
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14 | (7) |
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14 | (3) |
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17 | (3) |
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20 | (1) |
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21 | (2) |
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Application of Operators to More than One Function |
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23 | (1) |
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Expressions in Cylindrical and Spherical Coordinates |
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24 | (3) |
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Maxwell Equations, Electrostatics, Magnetostatics and Magnetodynamic Fields |
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27 | (96) |
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27 | (1) |
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28 | (7) |
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The Electric Field Intensity E |
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30 | (1) |
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The Magnetic Field Intensity H |
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30 | (1) |
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The Magnetic Flux Density B and the Magnetic Permeability μ |
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31 | (1) |
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The Electric Flux Density D and Electric Permittivity ε |
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32 | (1) |
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The Surface Current Density J |
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33 | (1) |
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33 | (1) |
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The Electric Conductivity σ |
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34 | (1) |
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Local Form of the Equations |
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35 | (5) |
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40 | (2) |
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The Approximation to Maxwell's Equations |
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42 | (5) |
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The Integral Form of Maxwell's Equations |
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47 | (2) |
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49 | (19) |
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50 | (1) |
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50 | (1) |
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Force on an Electric Charge |
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51 | (1) |
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The Electric Scalar Potential V |
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51 | (5) |
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Nonconservative Fields: Electromotive Force |
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56 | (3) |
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Refraction of the Electric Field |
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59 | (4) |
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63 | (2) |
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Laplace's and Poisson's Equations of the Electric Field for Dielectric Media |
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65 | (3) |
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Laplace's Equation of the Electric Field for Conductive Media |
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68 | (1) |
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68 | (5) |
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Maxwell's Equations in Magnetostatics |
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70 | (1) |
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70 | (3) |
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73 | (1) |
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73 | (1) |
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73 | (4) |
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Magnetic Field Refraction |
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77 | (3) |
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Energy in the Magnetic Field |
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80 | (3) |
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83 | (24) |
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84 | (1) |
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85 | (1) |
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85 | (1) |
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85 | (3) |
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The Influence of Iron on Magnetic Circuits |
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88 | (2) |
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90 | (1) |
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General Properties of Hard Magnetic Materials |
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90 | (4) |
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The Energy Associated with a Magnet |
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94 | (6) |
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Principal Types of Permanent Magnets |
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100 | (2) |
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Dynamic Operation of Permanent Magnets |
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102 | (2) |
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Inductance and Mutual Inductance |
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104 | (1) |
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104 | (1) |
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Energy in a Linear System |
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105 | (2) |
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107 | (16) |
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Maxwell's Equations for the Magnetodynamic Field |
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108 | (4) |
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Penetration of Time-Dependent Fields in Conducting Materials |
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112 | (1) |
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113 | (1) |
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113 | (1) |
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114 | (1) |
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114 | (1) |
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Solution of the Equations |
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115 | (8) |
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Brief Presentation of the Finite Element Method |
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123 | (74) |
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123 | (1) |
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The Galerkin Method -- Basic Concepts |
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124 | (11) |
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The Establishment of the Physical Equations |
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124 | (1) |
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125 | (2) |
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Application of the Weighted Residual Method |
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127 | (4) |
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Application of the Finite Element Method and Solution |
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131 | (3) |
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134 | (1) |
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Dirichlet Boundary Condition -- Imposed Potential |
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134 | (1) |
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Neumann Condition -- Unknown Nodal Values on the Boundary |
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135 | (1) |
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A First-Order Finite Element Program |
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135 | (12) |
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Example for Use of the Finite Element Program |
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142 | (5) |
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Generalization of the Finite Element Method |
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147 | (19) |
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High-Order Finite Elements: General |
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148 | (1) |
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High-Order Finite Elements: Notation |
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148 | (5) |
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High-Order Finite Elements: Implementation |
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153 | (3) |
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Continuity of Finite Elements |
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156 | (1) |
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157 | (1) |
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Transformation of Quantities -- the Jacobian |
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158 | (3) |
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Evaluation of the Integrals |
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161 | (5) |
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166 | (3) |
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169 | (6) |
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First-Order Triangular Element |
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171 | (1) |
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Second-Order Triangular Element |
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172 | (1) |
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Quadrilateral Bi-linear Element |
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173 | (1) |
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Quadrilateral Quadratic Element |
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174 | (1) |
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Coupling Different Finite Elements |
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175 | (2) |
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Coupling Different Types of Finite Elements |
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175 | (2) |
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Calculation of Some Terms in the Field Equation |
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177 | (5) |
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178 | (2) |
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Evaluation of the Second Term in Eq. (3.72) |
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180 | (1) |
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Evaluation of the Third Term in Eq. (3.72) |
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181 | (1) |
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Evaluation of the Source Term |
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181 | (1) |
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A Simplified 2D Second-Order Finite Element Program |
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182 | (15) |
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182 | (2) |
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184 | (1) |
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The Finite Element Program |
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185 | (12) |
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The Finite Element Method Applied to 2D Electromagnetic Cases |
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197 | (86) |
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197 | (1) |
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197 | (16) |
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Electrostatic Fields: Dielectric Materials |
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198 | (2) |
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Stationary Currents: Conducting Materials |
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200 | (1) |
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Magnetic Fields: Scalar Potential |
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201 | (2) |
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The Magnetic Field: Vector Potential |
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203 | (8) |
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The Electric Vector Potential |
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211 | (2) |
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Application to 2D Eddy Current Problems |
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213 | (20) |
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First-Order Element in Local Coordinates |
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213 | (6) |
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The Vector Potential Equation Using Time Discretization |
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219 | (7) |
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The Complex Vector Potential Equation |
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226 | (5) |
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Structures with Moving Parts |
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231 | (2) |
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Axi-Symmetric Application |
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233 | (6) |
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The Axi-Symmetric Formulation for Vector Potential |
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236 | (3) |
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Advantages and Limitations of 2D Formulations |
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239 | (2) |
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241 | (7) |
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Method of Successive Approximation |
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241 | (1) |
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The Newton-Raphson Method |
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242 | (6) |
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Geometric Repetition of Domains |
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248 | (3) |
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249 | (1) |
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250 | (1) |
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251 | (5) |
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251 | (1) |
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252 | (1) |
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252 | (1) |
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253 | (3) |
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Voltage-Fed Electromagnetic Devices |
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256 | (4) |
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260 | (10) |
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Calculation of Electrostatic Fields |
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261 | (1) |
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Calculation of Static Currents |
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262 | (3) |
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Calculation of the Magnetic Field -- Scalar Potential |
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265 | (2) |
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Calculation of the Magnetic Field -- Vector Potential |
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267 | (3) |
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270 | (13) |
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Eddy Currents: Time Discretization |
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270 | (2) |
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Moving Conducting Piece in Front of an Electromagnet |
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272 | (4) |
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Time Step Simulation of a Voltage-Fed Device |
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276 | (3) |
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Thermal Case: Heating by Eddy Currents |
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279 | (4) |
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Coupling of Field and Electrical Circuit Equations |
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283 | (60) |
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283 | (1) |
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Electromagnetic Equations |
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283 | (8) |
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Formulation Using the Magnetic Vector Potential |
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284 | (1) |
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The Formulation in Two Dimensions |
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285 | (1) |
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285 | (1) |
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285 | (2) |
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287 | (2) |
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Equations for the Whole Domain |
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289 | (1) |
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The Finite Element Method |
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290 | (1) |
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Equations for Different Conductor Configurations |
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291 | (11) |
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Thick Conductors Connections |
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292 | (1) |
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292 | (2) |
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294 | (5) |
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Thin Conductors Connections |
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299 | (1) |
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Independent Voltage Sources |
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300 | (1) |
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Star Connection with Neutral |
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300 | (1) |
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301 | (1) |
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Star Connection without Neutral Wire |
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301 | (1) |
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Connections Between Electromagnetic Devices and External Feeding Circuits |
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302 | (32) |
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Reduced Equations of Electromagnetic Devices |
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303 | (1) |
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Feeding Circuit Equations and Connection to Field Equations |
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303 | (1) |
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Calculation of Matrices G1 to G6 |
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304 | (1) |
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Circuit Topology Concepts |
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305 | (10) |
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Determination of Matrices G1 to G6 |
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315 | (9) |
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324 | (5) |
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Taking Into Account Electronic Switches in the Feeding Circuit |
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329 | (1) |
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Discretization of the Time Derivative |
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330 | (4) |
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334 | (9) |
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Simulations with Known Voltage Waveforms |
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334 | (1) |
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334 | (3) |
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Three-Phase Induction Motor |
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337 | (1) |
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Massive Conductors in Series Connection |
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337 | (2) |
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Modeling of a Static Converter-Fed Magnetic Device |
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339 | (4) |
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Movement Modeling for Electrical Machines |
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343 | (24) |
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343 | (1) |
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Methods with Non-Discretized Airgaps |
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343 | (1) |
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Methods with Discretized Airgaps |
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344 | (1) |
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344 | (5) |
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349 | (4) |
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The Skew Effect in Electrical Machines Using 2D Simulation |
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353 | (9) |
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362 | (5) |
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Three-Phase Induction Motor |
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362 | (2) |
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364 | (3) |
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Interaction Between Electromagnetic and Mechanical Forces |
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367 | (52) |
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367 | (1) |
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Methods Based on Direct Formulations |
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368 | (21) |
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Method of the Magnetic Co-Energy Variation |
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368 | (2) |
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The Maxwell Stress Tensor Method |
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370 | (13) |
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The Method Proposed by Arkkio |
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383 | (1) |
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The Method of Local Jacobian Matrix Derivation |
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384 | (2) |
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Examples of Torque Calculation |
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386 | (3) |
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Methods Based on the Force Density |
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389 | (12) |
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Preliminary Considerations |
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389 | (2) |
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Equivalent Sources Formulations |
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391 | (1) |
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392 | (1) |
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Equivalent Magnetic Charges |
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393 | (1) |
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Other Equivalent Source Distributions |
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394 | (1) |
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Formulation Based on the Energy Derivation |
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395 | (3) |
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Comparison Among the Different Methods |
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398 | (3) |
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Electrical Machine Vibrations Originated by Magnetic Forces |
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401 | (12) |
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Magnetic Force Calculation |
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402 | (1) |
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402 | (1) |
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Calculation of the Natural Response |
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403 | (1) |
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Calculation of the Forced Response Directly in Harmonic Regime |
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404 | (1) |
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Calculation of the Forced Response Using the Modal Superposition Method |
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405 | (2) |
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Example of Vibration Calculation |
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407 | (6) |
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Example of Coupling Between the Field and Circuit Equations, Including Mechanical Transients |
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413 | (6) |
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419 | (52) |
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419 | (1) |
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420 | (3) |
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423 | (7) |
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Anomalous or Excess Losses |
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430 | (3) |
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433 | (5) |
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435 | (3) |
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438 | (14) |
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438 | (3) |
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Procedure for the Numerical Implementation of the JA Method |
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441 | (2) |
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Examples of Hysteresis Loops Obtained with the JA Method |
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443 | (4) |
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Determination of the Parameters from Experimental Hysteresis Loops |
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447 | (5) |
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452 | (1) |
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The Inverse Jiles-Atherton Model |
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452 | (3) |
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452 | (2) |
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Procedure for the Numerical Implementation of the Inverse JA Method |
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454 | (1) |
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Including Iron Losses in Finite Element Calculations |
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455 | (16) |
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Hysteresis Modeling by Means of the Magnetization M Term |
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457 | (2) |
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Hysteresis Modeling by Means of a Differential Reluctivity |
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459 | (4) |
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Inclusion of Eddy Current Losses in the FE Modeling |
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463 | (2) |
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Inclusion of Anomalous Losses in the FE Modeling |
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465 | (1) |
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Examples of Iron Losses Applied to FE Calculations |
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466 | (5) |
Bibliography |
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471 | (12) |
Index |
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