Understand the theory of eddy currents with this essential reference
Eddy currents are electrical current loops produced when a conductor passes through a magnetic field, or is otherwise subject to a change in magnetic field direction. These currents play a significant role in many industrial processes and areas of electrical engineering. Their properties and applications are therefore a subject of significant interest for electrical engineers and other professionals.
Eddy Currents: Theory, Modelling and Applications offers a comprehensive reference on eddy currents in theory and practice. It begins with an introduction to the underlying theory of eddy currents, before proceeding to both closed-form and numerical solutions, and finally describing current and future applications. The result is an essential tool for anyone whose work requires an understanding of these ubiquitous currents.
Eddy Currents readers will also find:
- Professional insights from an author team with decades of combined experience in research and industry
- Detailed treatment of methods including finite difference, finite element, and integral equation techniques
- Over 100 computer-generated figures to illustrate key points
Eddy Currents is a must-have reference for researchers and industry professionals in electrical engineering and related fields.
Table of Contents:
1 Basic Principles of Eddy Currents 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Faraday’s Law and Lenz’s Law . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Proximity Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Resistance and Reactance Limited Eddy Currents . . . . . . . . . . . . 14
1.5 Electromotive Force (emf) and Potential Difference . . . . . . . . . . . 18
1.6 Waves, Diffusion and the Magneto-Quasistatic Approximation . . . . . 29
1.7 Skin Depth or Depth of Penetration . . . . . . . . . . . . . . . . . . . . 35
1.8 Diffusion, Heat Transfer and Eddy Currents . . . . . . . . . . . . . . . . 38
1.9 The Diffusion Equation and Random Walks . . . . . . . . . . . . . . . . 42
1.10 Transient Magnetic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 44
1.11 Coupled Circuit Models for Eddy Currents . . . . . . . . . . . . . . . . 51
1.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2 Conductors with Rectangular Cross-Sections 59
2.1 Finite Plate: Resistance Limited . . . . . . . . . . . . . . . . . . . . . . 60
2.2 Infinite Plate: Reactance Limited . . . . . . . . . . . . . . . . . . . . . . 63
2.3 Finite Plate: Reactance Limited . . . . . . . . . . . . . . . . . . . . . . . 70
2.4 Superposition of Eddy Losses in a Conductor . . . . . . . . . . . . . . . 77
2.5 Discussion of Losses in Rectangular Plates . . . . . . . . . . . . . . . . 79
2.6 Eddy Currents in a Nonlinear Plate . . . . . . . . . . . . . . . . . . . . . 90
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2.7 Plate with Hysteresis and Complex Permeability . . . . . . . . . . . . . 106
2.8 Conducting Plates with Sinusoidal Space Variation of Field . . . . . . . 109
2.9 Eddy Currents in Multi-layered Plate Geometries . . . . . . . . . . . . . 122
2.10 Thin Wire Carrying Current Above Conducting Plates . . . . . . . . . . 132
2.11 Eddy currents in materials with anisotropic permeability . . . . . . . . 151
2.12 Isolated Rectangular Conductor with Axial Current Applied . . . . . . 155
2.13 Transient Diffusion into a Solid Conducting Block . . . . . . . . . . . . 158
2.14 Eddy Current Modes in a Rectangular Core . . . . . . . . . . . . . . . . 167
2.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3 Conductors with Circular Cross-Sections 175
3.1 Axial Current in a Conductor with Circular Cross-Section: Reactance
Limited Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.2 Axial Current in Composite Circular Conductors . . . . . . . . . . . . . 183
3.3 Circular Conductor with Applied Axial Flux: Resistance Limited Case . 193
3.4 Circular Conductor with Applied Axial Flux: Reactance Limited Case . 197
3.5 Shielding with a Conducting Tube in an Axial Field . . . . . . . . . . . 202
3.6 Circular Conductors with Transverse Applied Field: Resistance Limited
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
3.7 Cylindrical Conductor with Applied Transverse Field: Reactance Limited
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
3.8 Shielding with a Conducting Tube in a Transverse Field . . . . . . . . . 221
3.9 Spherical Conductor in a Uniform Sinusoidally Time Varying Field:
Resistance Limited Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
3.10 Diffusion Through Thin Cylinders . . . . . . . . . . . . . . . . . . . . . . 227
3.11 Surface Impedance Formulation for Electric Machines . . . . . . . . . . 235
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3.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
4 Formulations 245
4.1 Mathematical Formulations for Eddy Current Modeling . . . . . . . . . 246
5 Finite Differences 265
5.1 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
5.2 The Two Dimensional Diffusion Equation . . . . . . . . . . . . . . . . . 268
5.3 Time Domain Solution of the Diffusion Equation . . . . . . . . . . . . . 273
5.4 Equivalent Circuit Representation for Finite Difference Equations . . . 276
6 Finite Elements 291
6.1 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
6.2 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
6.3 Axisymmetric Finite Element Eddy Current Formulation with Magnetic
Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
7 Integral Equations 339
7.1 Surface Integral Equation Method for Eddy Current Analysis . . . . . . 340
7.2 Boundary Element Method for Eddy Current Analysis . . . . . . . . . . 345
7.3 Integral Equations for Three-Dimensional Eddy Currents . . . . . . . . 360
8 Induction Heating 367
8.1 Simplified Induction Heating Analysis . . . . . . . . . . . . . . . . . . . 367
8.2 Coupled Eddy Current and Thermal Analysis: Induction Heating . . . . 375
9 Wattmeter 385
10 Magnetic Stirring 399
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11 Electric Machines 407
11.1 Eddy Currents in Slot-Embedded Conductors . . . . . . . . . . . . . . . 407
11.2 Solid Rotor Electric Machines . . . . . . . . . . . . . . . . . . . . . . . . 443
11.3 Squirrel Cage Induction Motor Analysis by the Finite Element Method 459
12 Transformer Losses 471
12.1 Foil Wound Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
12.2 Phase Shifting Transformers . . . . . . . . . . . . . . . . . . . . . . . . . 475
A Bessel Functions 479
B Separation of Variables 481
B.1 One-Dimensional Separation of Variables in Rectangular Coordinates . 481
B.2 Two-Dimensional Separation of Variables in Cylindrical Coordinates . 484
C The Error Function 487
D Replacing Hollow Conducting Cylinders with Line Currents Using
the Method of Images 489
E Inductance of Parallel Wires 493
F Shape Functions for First Order Hexahedral Element 497
About the Author :
Sheppard J. Salon, PhD, is Professor Emeritus in the Department of Electric Power Engineering at Rensselaer Polytechnic Institute in Troy, New York, USA, and 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 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 Türkiye. 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 of industrial experience and is currently an engineering consultant.
