Eddy Currents

Sheppard J. Salon, PhD, is Professor Emeritus in the Department of Electrical, Computer and Systems Engineering at Rensselaer Polytechnic Institute in Troy, New York, USA, and a founder of the Magsoft Corporation. He has published on many electrical engineering subjects and his awards and honors include an IEEE Life Fellowship and the IEEE 2004 Nicola Tesla Award. M. V. K. Chari, PhD, now retired, was a Research Professor at Rensselaer Polytechnic Institute in Troy, New York, USA. He is a former Technical Leader at General Electric, an IEEE Life Fellow, and recipient of the 1993 Nicola Tesla Award. He has published extensively on electrical engineering subjects. Lale T. Ergene, PhD, is a Full Professor in the Electrical Engineering Department at Istanbul Technical University, Turkey. She was an adjunct professor at Rensselaer Polytechnic Institute in Troy, New York, USA and worked at MAGSOFT Corporation as a consulting engineer. She is IEEE Senior Member and advisory board member of the Scientific and Technological Research Council of Turkey. She has published widely on electrical engineering subjects. David Burow, PhD, is Owner and Head Programmer at Genfo, Inc., a company that provides custom programming for Macintosh, Windows, and Linux operating systems. He was a Postdoctoral Researcher at Rensselaer Polytechnic Institute and has published several papers on electrical engineering subjects. Mark DeBortoli, PhD, received a doctoral degree at Rensselaer Polytechnic Institute. He has over 30 years of industrial experience and is currently an engineering consultant.

• About the Authors ixPreface xPart I Theory 11 Basic Principles of Eddy Currents 31.1 Introduction 31.2 Faraday's Law and Lenz's Law 51.3 Proximity Effect 81.4 Resistance and Reactance Limited Eddy Currents 111.5 Electromotive Force (emf) and Potential Difference 141.6 Waves, Diffusion, and the Magneto-Quasi-static Approximation 221.7 Skin Depth or Depth of Penetration 271.8 Diffusion, Heat Transfer, and Eddy Currents 301.9 The Diffusion Equation and RandomWalks 321.10 Transient Magnetic Diffusion 341.11 Coupled Circuit Models for Eddy Currents 391.12 Summary 432 Conductors with Rectangular Cross Sections 452.1 Finite Plate: Resistance Limited 452.2 Infinite Plate: Reactance Limited 482.3 Finite Plate: Reactance Limited 532.4 Superposition of Eddy Losses in a Conductor 582.5 Discussion of Losses in Rectangular Plates 592.6 Eddy Currents in a Nonlinear Plate 682.7 Plate with Hysteresis and Complex Permeability 802.8 Conducting Plates with Sinusoidal Space Variation of Field 832.9 Eddy Currents in Multi-Layered Plate Geometries 942.10 Thin Wire Carrying Current Above Conducting Plates 1002.11 Eddy Currents in Materials with Anisotropic Permeability 1122.12 Isolated Rectangular Conductor with Axial Current Applied 1152.13 Transient Diffusion Into a Solid Conducting Block 1182.14 Eddy Current Modes in a Rectangular Core 1252.15 Summary 1293 Conductors with Circular Cross Sections 1313.1 Axial Current in a Conductor with Circular Cross Section: Reactance-Limited Case 1313.2 Axial Current in Composite Circular Conductors 1363.3 Circular Conductor with Applied Axial Flux: Resistance-Limited Case 1443.4 Circular Conductor with Applied Axial Flux: Reactance-Limited Case 1463.5 Shielding with a Conducting Tube in an Axial Field 1513.6 Circular Conductors with Transverse Applied Field: Resistance-Limited Case 1553.7 Cylindrical Conductor with Applied Transverse Field: Reactance-Limited Case 1573.8 Shielding with a Conducting Tube in a Transverse Field 1653.9 Spherical Conductor in a Uniform Sinusoidally Time-Varying Field: Resistance-Limited Case 1673.10 Diffusion Through Thin Cylinders 1693.11 Surface Impedance Formulation for Electric Machines 1753.12 Summary 181Part II Modeling 1834 Formulations 1854.1 Mathematical Formulations for Eddy Current Modeling 1855 Finite Differences 1995.1 Difference Equations 1995.2 The Two-Dimensional Diffusion Equation 2015.3 Time-Domain Solution of the Diffusion Equation 2055.4 Equivalent Circuit Representation for Finite Difference Equations 2076 Finite Elements 2196.1 Finite Elements 2196.2 The Variational Method 2206.3 Axisymmetric Finite Element Eddy Current Formulation with Magnetic Vector Potential 2487 Integral Equations 2557.1 Surface Integral Equation Method for Eddy Current Analysis 2557.2 Boundary Element Method for Eddy Current Analysis 2607.3 Integral Equations for Three-Dimensional Eddy Currents 270Part III Applications 2778 Induction Heating 2798.1 Simplified Induction Heating Analysis 2798.2 Coupled Eddy Current and Thermal Analysis: Induction Heating 2859 Wattmeter 29110 Magnetic Stirring 30310.1 Introduction 30310.2 Analysis 30411 Electric Machines 31111.1 Eddy Currents in Slot-Embedded Conductors 31111.2 Solid Rotor Electric Machines 33911.3 Squirrel Cage Induction Motor Analysis by the Finite Element Method 35212 Transformer Losses 36112.1 FoilWound Transformer 36112.2 Phase Shifting Transformers 363Appendix A Bessel Functions 367Appendix B Separation of Variables 369B.1 One-Dimensional Separation of Variables in Rectangular Coordinates 369B.2 Two-Dimensional Separation of Variables in Cylindrical Coordinates 371Appendix C The Error Function 373Appendix D Replacing Hollow Conducting Cylinders with Line Currents Using the Method of Images 375Appendix E Inductance of Parallel Wires 379Appendix F Shape Functions for First-Order Hexahedral Element 381References 383Index 387