Eddy Currents

About the Authors ix Preface x Part I Theory 1 1 Basic Principles of Eddy Currents 3 1.1 Introduction 3 1.2 Faraday's Law and Lenz's Law 5 1.3 Proximity Effect 8 1.4 Resistance and Reactance Limited Eddy Currents 11 1.5 Electromotive Force (emf) and Potential Difference 14 1.6 Waves, Diffusion, and the Magneto-Quasi-static Approximation 22 1.7 Skin Depth or Depth of Penetration 27 1.8 Diffusion, Heat Transfer, and Eddy Currents 30 1.9 The Diffusion Equation and RandomWalks 32 1.10 Transient Magnetic Diffusion 34 1.11 Coupled Circuit Models for Eddy Currents 39 1.12 Summary 43 2 Conductors with Rectangular Cross Sections 45 2.1 Finite Plate: Resistance Limited 45 2.2 Infinite Plate: Reactance Limited 48 2.3 Finite Plate: Reactance Limited 53 2.4 Superposition of Eddy Losses in a Conductor 58 2.5 Discussion of Losses in Rectangular Plates 59 2.6 Eddy Currents in a Nonlinear Plate 68 2.7 Plate with Hysteresis and Complex Permeability 80 2.8 Conducting Plates with Sinusoidal Space Variation of Field 83 2.9 Eddy Currents in Multi-Layered Plate Geometries 94 2.10 Thin Wire Carrying Current Above Conducting Plates 100 2.11 Eddy Currents in Materials with Anisotropic Permeability 112 2.12 Isolated Rectangular Conductor with Axial Current Applied 115 2.13 Transient Diffusion Into a Solid Conducting Block 118 2.14 Eddy Current Modes in a Rectangular Core 125 2.15 Summary 129 3 Conductors with Circular Cross Sections 131 3.1 Axial Current in a Conductor with Circular Cross Section: Reactance-Limited Case 131 3.2 Axial Current in Composite Circular Conductors 136 3.3 Circular Conductor with Applied Axial Flux: Resistance-Limited Case 144 3.4 Circular Conductor with Applied Axial Flux: Reactance-Limited Case 146 3.5 Shielding with a Conducting Tube in an Axial Field 151 3.6 Circular Conductors with Transverse Applied Field: Resistance-Limited Case 155 3.7 Cylindrical Conductor with Applied Transverse Field: Reactance-Limited Case 157 3.8 Shielding with a Conducting Tube in a Transverse Field 165 3.9 Spherical Conductor in a Uniform Sinusoidally Time-Varying Field: Resistance-Limited Case 167 3.10 Diffusion Through Thin Cylinders 169 3.11 Surface Impedance Formulation for Electric Machines 175 3.12 Summary 181 Part II Modeling 183 4 Formulations 185 4.1 Mathematical Formulations for Eddy Current Modeling 185 5 Finite Differences 199 5.1 Difference Equations 199 5.2 The Two-Dimensional Diffusion Equation 201 5.3 Time-Domain Solution of the Diffusion Equation 205 5.4 Equivalent Circuit Representation for Finite Difference Equations 207 6 Finite Elements 219 6.1 Finite Elements 219 6.2 The Variational Method 220 6.3 Axisymmetric Finite Element Eddy Current Formulation with Magnetic Vector Potential 248 7 Integral Equations 255 7.1 Surface Integral Equation Method for Eddy Current Analysis 255 7.2 Boundary Element Method for Eddy Current Analysis 260 7.3 Integral Equations for Three-Dimensional Eddy Currents 270 Part III Applications 277 8 Induction Heating 279 8.1 Simplified Induction Heating Analysis 279 8.2 Coupled Eddy Current and Thermal Analysis: Induction Heating 285 9 Wattmeter 291 10 Magnetic Stirring 303 10.1 Introduction 303 10.2 Analysis 304 11 Electric Machines 311 11.1 Eddy Currents in Slot-Embedded Conductors 311 11.2 Solid Rotor Electric Machines 339 11.3 Squirrel Cage Induction Motor Analysis by the Finite Element Method 352 12 Transformer Losses 361 12.1 FoilWound Transformer 361 12.2 Phase Shifting Transformers 363 Appendix A Bessel Functions 367 Appendix B Separation of Variables 369 B.1 One-Dimensional Separation of Variables in Rectangular Coordinates 369 B.2 Two-Dimensional Separation of Variables in Cylindrical Coordinates 371 Appendix C The Error Function 373 Appendix D Replacing Hollow Conducting Cylinders with Line Currents Using the Method of Images 375 Appendix E Inductance of Parallel Wires 379 Appendix F Shape Functions for First-Order Hexahedral Element 381 References 383 Index 387