Eddy Currents E-Book

Sheppard J. Salon

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 390

Veröffentlichungsjahr: 2023

Beschreibung

EDDY CURRENTS 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, Modeling 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.

Leseprobe

Table of Contents

Cover

Table of Contents

Title Page

Copyright

About the Authors

Preface

Part I: Theory

1 Basic Principles of Eddy Currents

1.1 Introduction

1.2 Faraday's Law and Lenz's Law

1.3 Proximity Effect

1.4 Resistance and Reactance Limited Eddy Currents

1.5 Electromotive Force (emf) and Potential Difference

1.6 Waves, Diffusion, and the Magneto-Quasi-static Approximation

1.7 Skin Depth or Depth of Penetration

1.8 Diffusion, Heat Transfer, and Eddy Currents

1.9 The Diffusion Equation and Random Walks

1.10 Transient Magnetic Diffusion

1.11 Coupled Circuit Models for Eddy Currents

1.12 Summary

Notes

2 Conductors with Rectangular Cross Sections

2.1 Finite Plate: Resistance Limited

2.2 Infinite Plate: Reactance Limited

2.3 Finite Plate: Reactance Limited

2.4 Superposition of Eddy Losses in a Conductor

2.5 Discussion of Losses in Rectangular Plates

2.6 Eddy Currents in a Nonlinear Plate

2.7 Plate with Hysteresis and Complex Permeability

2.8 Conducting Plates with Sinusoidal Space Variation of Field

2.9 Eddy Currents in Multi-Layered Plate Geometries

2.10 Thin Wire Carrying Current Above Conducting Plates

2.11 Eddy Currents in Materials with Anisotropic Permeability

2.12 Isolated Rectangular Conductor with Axial Current Applied

2.13 Transient Diffusion Into a Solid Conducting Block

2.14 Eddy Current Modes in a Rectangular Core

2.15 Summary

Notes

3 Conductors with Circular Cross Sections

3.1 Axial Current in a Conductor with Circular Cross Section: Reactance-Limited Case

3.2 Axial Current in Composite Circular Conductors

3.3 Circular Conductor with Applied Axial Flux: Resistance-Limited Case

3.4 Circular Conductor with Applied Axial Flux: Reactance-Limited Case

3.5 Shielding with a Conducting Tube in an Axial Field

3.6 Circular Conductors with Transverse Applied Field: Resistance-Limited Case

3.7 Cylindrical Conductor with Applied Transverse Field: Reactance-Limited Case

3.8 Shielding with a Conducting Tube in a Transverse Field

3.9 Spherical Conductor in a Uniform Sinusoidally Time-Varying Field: Resistance-Limited Case

3.10 Diffusion Through Thin Cylinders

3.11 Surface Impedance Formulation for Electric Machines

3.12 Summary

Notes

Part II: Modeling

4 Formulations

4.1 Mathematical Formulations for Eddy Current Modeling

5 Finite Differences

5.1 Difference Equations

5.2 The Two-Dimensional Diffusion Equation

5.3 Time-Domain Solution of the Diffusion Equation

5.4 Equivalent Circuit Representation for Finite Difference Equations

6 Finite Elements

6.1 Finite Elements

6.2 The Variational Method

6.3 Axisymmetric Finite Element Eddy Current Formulation with Magnetic Vector Potential

7 Integral Equations

7.1 Surface Integral Equation Method for Eddy Current Analysis

7.2 Boundary Element Method for Eddy Current Analysis

7.3 Integral Equations for Three-Dimensional Eddy Currents

Part III: Applications

8 Induction Heating

8.1 Simplified Induction Heating Analysis

8.2 Coupled Eddy Current and Thermal Analysis: Induction Heating

9 Wattmeter

10 Magnetic Stirring

10.1 Introduction

10.2 Analysis

11 Electric Machines

11.1 Eddy Currents in Slot-Embedded Conductors

11.2 Solid Rotor Electric Machines

11.3 Squirrel Cage Induction Motor Analysis by the Finite Element Method

Notes

12 Transformer Losses

12.1 Foil Wound Transformer

12.2 Phase Shifting Transformers

Appendix A: Bessel Functions

Appendix B: Separation of Variables

B.1 One-Dimensional Separation of Variables in Rectangular Coordinates

B.2 Two-Dimensional Separation of Variables in Cylindrical Coordinates

Appendix C: The Error Function

Appendix D: Replacing Hollow Conducting Cylinders with Line Currents Using the Method of Images

Appendix E: Inductance of Parallel Wires

Appendix F: Shape Functions for First-Order Hexahedral Element

References

Index

End User License Agreement

List of Tables

Chapter 6

Table 6.1 Elements and node numbering for the uppermost four-element block....

Table 6.2 Values of , , and S (area) for elements in the uppermost block....

Chapter 7

Table 7.1 Definition of and .

Chapter 9

Table 9.1 The torque values for different phase angles.

Chapter 11

Table 11.1 Current density and power loss comparison.

List of Illustrations

Chapter 1

Figure 1.1 Loop with flux linkage increasing and induced current direction....

Figure 1.2 Two circuits with mutual coupling. (a) Switch closes. (b) After s...

Figure 1.3 Conducting cylinder with axial magnetic flux field applied.

Figure 1.4 Parallel conductors with opposing currents.

Figure 1.5 Real part of flux density with opposing currents.

Figure 1.6 Current density with opposing currents.

Figure 1.7 Real part of flux density with currents in the same direction.

Figure 1.8 Current density with currents in the same direction.

Figure 1.9 System and load.

Figure 1.10 Losses in load vs. load resistance.

Figure 1.11 Two magnetic circuits with identical conducting plates.

Figure 1.12 Circuit with flux linkage.

Figure 1.13 Loop with open circuit.

Figure 1.14 Circuit closed through load resistor.

Figure 1.15 Equivalent circuit of Equation (1.23).

Figure 1.16 Conductor moving in a magnetic field.

Figure 1.17 Rectangular loop moving through time and space varying field.

Figure 1.18 Moving surface in a time-varying field.

Figure 1.19 Attenuation of wave in conducting material.

Figure 1.20 Magnitude and phase of current density as a function of depth.

Figure 1.21 Skin depth calculator after Knoepfel.

Figure 1.22 Temperature at various depths over a few years.

Figure 1.23 One-dimensional random walk.

Figure 1.24 Normal distribution at various times.

Figure 1.25 Infinite conductor with applied current sheet.

Figure 1.26 Current density vs. depth for different times.

Figure 1.27 Parallel flat copper bus bars.

Figure 1.28 Real and imaginary components of current density.

Figure 1.29 Magnitude of the current density.

Figure 1.30 Finite element solution to the go and return example.

Figure 1.31 Finite element solution for currents in the same direction.

Figure 1.32 Comparison of current density magnitude in one plate.

Chapter 2

Figure 2.1 Side view of lamination with path of eddy current.

Figure 2.2 Sides of eddy current loop in non-ferromagnetic conducting plate ...

Figure 2.3 Infinite conducting slab with source and induced currents.

Figure 2.4 Half-space with applied sinusoidal magnetic fields.

Figure 2.5 Resultant fields and currents in a finite-width plate.

Figure 2.6 Reduction of effective area with thickness.

Figure 2.7 Correction factor in Equation (2.44).

Figure 2.8 Superposition of eddy losses in a rectangular conductor.

Figure 2.9 Plate with tangential excitation.

Figure 2.10 Case 1: plate with current parallel.

Figure 2.11 Normalized loss density vs. normalized depth for parallel conduc...

Figure 2.12 Case 2: current perpendicular to and through the surface.

Figure 2.13 Normalized loss density vs. normalized depth for perpendicular c...

Figure 2.14 Case 3: current flowing along plate.

Figure 2.15 Loss density for a current-carrying plate vs. normalized thickne...

Figure 2.16 Parallel conducting plates carrying go-and-return current.

Figure 2.17 Normalized loss vs. normalized width for parallel rectangular co...

Figure 2.18 Normalized loss in a finite plate with uniform field applied vs....

Figure 2.19 Single conductor as the superposition of two sets of parallel co...

Figure 2.20 Normalized loss in an isolated current-carrying conductor vs. no...

Figure 2.21 Limiting curve of steel.

Figure 2.22 Making a sinusoidal wave from square waves.

Figure 2.23 Separation surface and coordinate system.

Figure 2.24 Steel cylinder with axial current and skin depth.

Figure 2.25 curve for finite element example.

Figure 2.26 Finite element results of current density at different depths.

Figure 2.27 Determination of effective permeability.

Figure 2.28 Elliptical curve.

Figure 2.29 Loss at different angles including hysteresis.

Figure 2.30 Portion of electric machine end plate subject to axial flux dens...

Figure 2.31 Eddy current contours at edges of end plate.

Figure 2.32 Loss correction factor as a function of for various .

Figure 2.33 Loss correction factor as a function of for various .

Figure 2.34 Two-dimensional layered geometry and source.

Figure 2.35 Semi-infinite slab with filament conductor.

Figure 2.36 Imaginary component of the flux.

Figure 2.37 Real component of the flux.

Figure 2.38 Losses in the plate.

Figure 2.39 Finite thickness slab with filament conductor.

Figure 2.40 Finite thickness shield in front of a semi-infinite slab excited...

Figure 2.41 End region laminations: geometry and coordinate system.

Figure 2.42 Rectangular conductor with ac axial current applied.

Figure 2.43 Magnitude of the current density using copper at for , .

Figure 2.44 Magnitude of the current density along the axis for copper at

Figure 2.45 Magnitude of the current density along the axis for copper at

Figure 2.46 Uniform conducting block with step function of current applied t...

Figure 2.47 Field diffusion into copper block.

Figure 2.48 Current diffusion into copper block.

Figure 2.49 Location of a particular value of the field as a function of tim...

Figure 2.50 Conducting region in solenoidal field: turn-off transient.

Figure 2.51 Turn-off magnetic field transient in conducting region at differ...

Figure 2.52 Turn-off current density transient in conducting region at diffe...

Figure 2.53 Rectangular conductor with applied magnetic field.

Figure 2.54 The mode in a rectangular conductor.

Figure 2.55 Magnetic field distribution in bar soon after current is applied...

Figure 2.56 Magnetic field distribution in bar at .

Figure 2.57 Magnetic field distribution in bar near steady state at .

Chapter 3

Figure 3.1 Coordinate system defined for the long straight wire.

Figure 3.2 Current density magnitude as a function of for .

Figure 3.3 Real and imaginary components of current density as a function of...

Figure 3.4 Magnitude of current density in copper wire at three frequencies ...

Figure 3.5 Normalized resistance vs. , closed-form and high-frequency appro...

Figure 3.6 Normalized resistance vs. , closed-form and low-frequency approx...

Figure 3.7 Two layer conductor with circular cross section.

Figure 3.8 Magnitude of the magnetic vector potential.

Figure 3.9 Real and imaginary parts of for solid conductor example.

Figure 3.10 Magnitude of the current density for solid conductor example.

Figure 3.11 Magnitude of vector potential vs. radius using the two layer for...

Figure 3.12 Magnitude of vector potential vs. radius using the two layer for...

Figure 3.13 Magnitude of current density vs. radius using the two layer form...

Figure 3.14 Long cylinder with axial flux.

Figure 3.15 Long hollow cylinder with axial flux.

Figure 3.16 Long composite cylinder with axial flux.

Figure 3.17 Long cylinder with solenoidal applied field.

Figure 3.18 Magnitude, real part, and imaginary part of the current density ...

Figure 3.19 Magnitude, real part, and imaginary part of the magnetic field v...

Figure 3.20 Magnitude of the current density vs. radius for 1.0 Hz.

Figure 3.21 Magnitude of the magnetic field vs. radius for 1.0 Hz.

Figure 3.22 Shielding cylinder in axial field.

Figure 3.23 Exact and approximate formulas for and .

Figure 3.24 Exact and approximate formulas for and .

Figure 3.25 Shielding factor for and various tube thicknesses.

Figure 3.26 Long cylinder with transverse flux.

Figure 3.27 Long composite cylinder with transverse flux.

Figure 3.28 Magnitude of current density vs. , for in copper conductor.

Figure 3.29 Real and imaginary current density vs. , for in copper conduc...

Figure 3.30 Current density vs. , for in copper conductor.

Figure 3.31 Circular composite conductor with fine wires.

Figure 3.32 Conducting tube in transverse field.

Figure 3.33 Flux density in outer region.

Figure 3.34 Shielding ratio vs. frequency.

Figure 3.35 Conducting sphere with applied field.

Figure 3.36 Conducting hollow sphere with applied uniform field.

Figure 3.37 Thin conducting cylinder in applied axial field.

Figure 3.38 Magnetic field inside copper cylinder vs. time.

Figure 3.39 Surface current density vs. time.

Figure 3.40 Thin conducting cylinder with magnetic core.

Figure 3.41 Current density around the cylinder at different times.

Figure 3.42 Cylindrical electric machine geometry.

Chapter 4

Figure 4.1 Correct use of mixed scalar potential formulation.

Figure 4.2 Incorrect use of scalar potential formulation.

Figure 4.3 Closed magnetic circuit linking no current.

Figure 4.4 Scalar potential is valid with symmetry condition.

Figure 4.5 Vector potential and flux.

Figure 4.6 Vector potential at material boundary.

Chapter 5

Figure 5.1 Forward, backward, and central differences.

Figure 5.2 Finite difference cell and diagram.

Figure 5.3 Interface between different materials.

Figure 5.4 Boundary with normal derivative specified.

Figure 5.5 R–C circuit for the vector potential eddy current problem.

Figure 5.6 One-dimensional diffusion problem.

Figure 5.7 Equivalent circuit for the one-dimensional diffusion problem.

Figure 5.8 One-dimensional example.

Figure 5.9 Real and imaginary components of current density as a function of...

Figure 5.10 Numerical and analytical solution for magnitude of current densi...

Figure 5.11 Current density at surface and at one skin depth with sinusoidal...

Figure 5.12 Current density vs. time at three depths for a step function of ...

Chapter 6

Figure 6.1 One triangular element.

Figure 6.2 Two separate triangular elements.

Figure 6.3 Two adjacent triangular elements.

Figure 6.4 Current sheet and copper plate.

Figure 6.5 Mesh consisting of four-element, five-node blocks.

Figure 6.6 Flux density magnitude vs. depth.

Figure 6.7 Current density vs. depth.

Figure 6.8 Phase angles of flux density and current density vs. depth.

Figure 6.9 Finite depth plate example.

Figure 6.10 Comparison between FEA results and closed-form solution for flux...

Figure 6.11 Comparison between FEA results and closed-form solution for curr...

Figure 6.12 First-order triangular element.

Figure 6.13 Definitions for Equation (6.95).

Figure 6.14 Triangular finite element for equivalent circuit.

Figure 6.15 Flux network of triangular element.

Figure 6.16 Network representing the electric current.

Figure 6.17 Section of long conducting cylinder.

Figure 6.18 Exact and equivalent circuit flux density at dc.

Figure 6.19 Exact and equivalent circuit flux density at .

Figure 6.20 Exact and equivalent circuit normalized resistance vs. frequency...

Figure 6.21 Axisymmetric first-order triangle.

Chapter 7

Figure 7.1 Rectangular copper conductor with axial ac current.

Figure 7.2 Normalized current density magnitude for thin rectangular conduct...

Figure 7.3 Current density phase for thin rectangular conductor.

Figure 7.4 Current density magnitude for square conductor by integral equati...

Figure 7.5 Current density magnitude for square conductor by finite elements...

Figure 7.6 Two-dimensional region and surface.

Figure 7.7 Integrating around the singularity.

Figure 7.8 Definition of boundary terms.

Figure 7.9 Integration for a field point located on the boundary.

Figure 7.10 Boundary definitions.

Figure 7.11 Integrating around a singularity.

Figure 7.12 Real part of .

Chapter 8

Figure 8.1 Long wire parallel to conducting half-plane.

Figure 8.2 Surface current density for wire of height above a conducting p...

Figure 8.3 Incremental conducting section on surface.

Figure 8.4 Circular conductor and current distribution.

Figure 8.5 Geometry to find the image source and current distribution around...

Figure 8.6 Surface current density around conductor as a function of .

Figure 8.7 Surface current density around copper conductor of radius .

Figure 8.8 Surface current density around copper conductor by finite element...

Figure 8.9 Loss density around copper conductor of radius .

Figure 8.10 Resistivity vs. temperature.

Figure 8.11 Magnetization curves vs. temperature.

Figure 8.12 Thermal conductivity vs. temperature.

Figure 8.13 Specific heat vs. temperature.

Figure 8.14 Flow chart of computation process.

Figure 8.15 Flux distribution at showing source coils and load.

Figure 8.16 Loss density in the load at .

Figure 8.17 Temperature distribution in the load at .

Chapter 9

Figure 9.1 The 3-D model of the watt-hour meter.

Figure 9.2 The 3-D finite element mesh model of the watt-hour meter.

Figure 9.3 The current density distribution (as arrows) of the model for ...

Figure 9.4 The current density distribution (as arrows) of the model for ...

Figure 9.5 The comparison between sine and torque values.

Figure 9.6 Eddy currents in the shunt parts.

Figure 9.7 Eddy currents in the screw and aluminum parts.

Figure 9.8 After adding the light load adjustment piece.

Figure 9.9 The torque distribution over the disc.

Figure 9.10 The mesh of the model.

Figure 9.11 Flux density distribution on the disc.

Figure 9.12 The eddy current distribution on the disc.

Figure 9.13 The mesh of the model including rotating air gap.

Figure 9.14 Braking torque vs. time = .

Figure 9.15 Braking torque vs. time .

Chapter 10

Figure 10.1 Geometry of liquid metal stirrer.

Figure 10.2 Magnetic field components vs. .

Figure 10.3 Current density components vs. .

Figure 10.4 Magnitude of the force density vs. .

Chapter 11

Figure 11.1 Source and eddy currents and cross-slot flux of a conducto...

Figure 11.2 Finite element solution of flux lines in a trapezoidal slot for ...

Figure 11.3 Conductor of width in slot of width .

Figure 11.4 Current density vs. slot depth for , , , .

Figure 11.5 Normalized loss vs. frequency for , , .

Figure 11.6 Conductor divided into strands.

Figure 11.7 Loss function vs. ratio of conductor height to effective skin de...

Figure 11.8 Geometry of rectangular hollow conductor.

Figure 11.9 Current density vs. depth in slot, for limiting case of solid co...

Figure 11.10 Current density vs. depth in slot for hollow conductor, , ,

Figure 11.11 Rectangular slot with uniform current.

Figure 11.12 Cross-slot flux density for current in bottom conductor.

Figure 11.13 Cross-slot flux density for current in top conductor.

Figure 11.14 Squirrel cage rotor conductor divided into five sections.

Figure 11.15 Equivalent circuit for five section slot.

Figure 11.16 Geometry of induction motor slot.

Figure 11.17 Geometry of five section model.

Figure 11.18 Equivalent circuit for the five section slot model (impedances ...

Figure 11.19 Vector potential equivalent circuit.

Figure 11.20 Magnitude of the current in each of five finite difference cell...

Figure 11.21 Non-uniform slot divided into slices.

Figure 11.22 Flux linking loop formed by slice and .

Figure 11.23 Flux produced by three sources.

Figure 11.24 Current density vs. depth for rectangular slot, 25-layer approx...

Figure 11.25 Slot with trapezoidal cross section.

Figure 11.26 Current density in 25 layers of trapezoidal slot shown in Figur...

Figure 11.27 Induction motor model.

Figure 11.28 Eddy current density variation in the solid rotor with rotor ra...

Figure 11.29 Mesh model of the motor with the injected current density.

Figure 11.30 FEM results of the phase currents.

Figure 11.31 Eddy currents for different slips.

Figure 11.32 Flux density in the air gap for different slip values.

Figure 11.33 Air-gap flux density and its first harmonic with FEM.

Figure 11.34 Comparison of the analytical and numerical results of the flux ...

Figure 11.35 Equivalent circuit showing an ideal transformer.

Figure 11.36 Equivalent circuit of induction motor with rotor quantities ref...

Figure 11.37 The curve for the rotor.

Figure 11.38 The curve for the stator.

Figure 11.39 The geometry of the motor.

Figure 11.40 Enlargement of the motor geometry.

Figure 11.41 The finite element mesh of the motor.

Figure 11.42 Enlargement of the air-gap region.

Figure 11.43 Equi-flux lines for rated load laminated rotor.

Chapter 12

Figure 12.1 Foil wound transformer schematic.

Figure 12.2 Magnitude of eddy current density for and copper.

Figure 12.3 Scaled total current magnitude.

Figure 12.4 Flux map of two winding transformer.

Figure 12.5 3-Phase, 36 pulse phase-shifting rectifier transformer.

Figure 12.6 Real part of the flux distribution in transformer.

Figure 12.7 Imaginary part of the flux distribution in transformer.

Appendix D

Figure D.1 Hollow cylinders and equivalent line currents.

Figure D.2 Cylindrical coordinate system and infinitely long current in .

Figure D.3 Coordinates of the equivalent current filaments.

Appendix E

Figure E.1 Flux linkage of parallel wires.

Appendix F

Figure F.1 First-order hexahedral element.

Guide

Cover

Title Page

Copyright

About the Authors

Preface

Table of Contents

Begin Reading

Appendix A: Bessel Functions

Appendix B: Separation of Variables

Appendix C: The Error Function

Appendix D: Replacing Hollow Conducting Cylinders with Line Currents Using the Method of Images

Appendix E: Inductance of Parallel Wires

Appendix F: Shape Functions for First-Order Hexahedral Element

References

Index

End User License Agreement

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IEEE Press445 Hoes LanePiscataway, NJ 08854

IEEE Press Editorial BoardSarah Spurgeon, Editor in Chief

Jón Atli BenediktssonAnjan BoseJames DuncanAmin MoenessDesineni Subbaram Naidu

Behzad RazaviJim LykeHai LiBrian Johnson

Jeffrey ReedDiomidis SpinellisAdam DrobotTom RobertazziAhmet Murat Tekalp

Eddy Currents

Theory, Modeling, and Applications

Sheppard J. Salon

Rensselaer Polytechnic InstituteSchenectady, NY, USA

M. V. K. Chari

Rensselaer Polytechnic InstituteBurnt Hills, NY, USA

Lale T. Ergene

Istanbul Technical UniversityIsanbul, Turkey

David Burow

Kingsport, TN, USA

Mark DeBortoli

Clifton Park, NY, USA

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Library of Congress Cataloging-in-Publication Data:

Names: Salon, S. J. (Sheppard Joel), 1948- author.

Title: Eddy currents : theory, modeling and applications / Sheppard J. Salon [and four others].

Description: Hoboken, NJ : Wiley-IEEE Press, [2024] | Includes bibliographical references and index.

Identifiers: LCCN 2023040651 (print) | LCCN 2023040652 (ebook) | ISBN 9781119866695 (cloth) | ISBN 9781119866701 (adobe pdf) | ISBN 9781119866718 (epub)

Subjects: LCSH: Eddy currents (Electric)

Classification: LCC TK2211 .S25 2024 (print) | LCC TK2211 (ebook) | DDC 621.31/042--dc23/eng/20230906

LC record available at https://lccn.loc.gov/2023040651

LC ebook record available at https://lccn.loc.gov/2023040652

Cover Design: WileyCover Image: © Sheppard J. Salon (with edits by David Burow)

About the Authors

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 an 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 in Electric Power Engineering at Rensselaer Polytechnic Institute. He has over 30 years of industrial experience in the design, analysis, and testing of electrical equipment. He has taught graduate engineering courses and authored a number of technical papers and publications. He is currently an engineering consultant.

Preface

It has been over 50 years since the classic monographs of Eddy Currents by Jiří Lammeraner and Miloš Štafl and The Analysis of Eddy Currents by R. L. Stoll. Since that time there have been great advances both in eddy current computation and eddy current applications. Modern numerical methods are now commonplace in the analysis of eddy currents. We can now solve three-dimensional eddy current problems with complicated boundaries, non-linear, and anisotropic materials. Further, the continued importance of high efficiency electrical devices has made the study of eddy currents and the need for accuracy in loss evaluation more important.

While these former works are excellent in their treatment of many important eddy current problems and are well referenced in this book, they are written for an audience that is already well-versed with electromagnetic phenomena and low-frequency applications.

We hope this book will be accessible to people looking for a place to start in the study and applications of eddy currents. We begin with a very basic introduction to the principles on which eddy current analysis is based (Faraday's law, Ampere's law, and Kirchhoff's laws), and refer back to these ideas throughout the book. We have included a lot of tutorial information as well as dozens of worked out numerical examples. Each problem is followed by a discussion of the results and how the basic principles of eddy currents can be seen in the solution.

We also hope that this work will be useful as a reference for experienced engineers working in the field. We include many examples of closed form and analytical solutions as well as numerical methods and approximation methods for many practical applications. The basic ideas of the numerical modeling are presented along with examples of their use and methods of interpreting and checking the results. Numerical methods are used as well in the applications section, in which we attempt to analyze problems that have both analytical and numerical solutions.

We would like to acknowledge Philippe Wendling for his invaluable help. All of the authors also want to express their gratitude to family and friends for their patience, understanding, encouragement, and support during the writing process.

Sheppard J. Salon

Schenectady, NY

April 2023

Part ITheory

1Basic Principles of Eddy Currents

1.1 Introduction

The discovery of eddy currents is usually attributed to French physicist Leon Foucault. In 1855, Foucault measured the force on a rotating conducting disk. He found that it took a greater force to rotate the disk when the disk was in a magnetic field produced by an electromagnet. He also noticed that the disk was heated when spun through a magnetic field. These observations, that eddy currents can produce force and torque and also heating, are still major applications of eddy currents. The production of force or torque by eddy currents is key to the operation of induction motors and induction generators, eddy current brakes, eddy current magnetic bearings, liquid metal stirring, and electromagnetic metal forming, to name a few applications. The forces produced by eddy currents may be a hindrance. In such cases, methods of eddy current mitigation will be necessary. Eddy current heating also has many applications such as induction heating for metal treatment and induction cooktops. There are also many applications in which eddy currents result in unwanted losses, limiting the efficiency of devices and requiring more thermal management.

We will use a very broad definition of eddy currents in this treatment. We include all electric currents induced by time-varying magnetic fields and/or relative motion between conductors and magnetic fields. This also includes the redistribution of currents due to the self-field of conductors excited with external sources.

All currents induced by a time-varying magnetic field can be thought of as eddy currents, but in this book, we are mainly dealing with applications in which the current density is nonuniform in the conductor. Whether we speak of eddy currents, skin effect, or proximity effect, we are speaking of the same physical phenomenon and this is described by the same set of equations.

In this chapter, we will develop the ideas necessary to understand the eddy current phenomenon. We will begin with a very basic introduction to Faraday's law and Lenz's law. This will give us a qualitative understanding of some of the important concepts of eddy current analysis such as skin effect and proximity effect. We then discuss the concept of resistance and reactance limited eddy currents in Section 1.4. We then give a more formal introduction to Faraday's law, emf, and potential difference and discuss the different ways voltage can be produced in a conductor. In eddy current analysis, we will be solving the diffusion equation. From electromagnetic theory, this means that we are making a quasi-static approximation. The full set of Maxwell's equations will result in the wave equation for time-varying phenomena. In Section 1.6, we will justify this approximation and study its implications.

Eddy currents can sometimes be rather difficult to visualize. There are other physical phenomena such as particle diffusion and heat transfer, that are described by the same mathematics, the diffusion equation. We will derive the expressions for eddy currents formally from Maxwell's equations in Section 2.2, but making the analogy to these other areas can help clarify some of the physics. We will first look at particle diffusion by means of random walks in Section 1.9. This will introduce the diffusion equation and some of its classic solutions. In Section 1.10 we discuss the electromagnetic diffusion problem in the time domain and the analogy is made to the particle diffusion results. We then look at the concept of skin depth or depth of penetration in electromagnetic applications for steady-state sinusoidal excitation. This concept is one of the key results of eddy current analysis and will be derived formally from Maxwell's equations in Section 2.2. As an example of skin depth and sinusoidal excitation, we turn to another branch of physics, heat transfer. Heat conduction and heat storage are described by the same set of equations as electromagnetic diffusion. We use an example of heat conduction to illustrate the concepts introduced about skin depth. Another method of understanding and computing eddy currents that we will find useful is the use of magnetically coupled circuits to model the eddy currents. This idea is introduced in Section 1.11.

With this introduction, we then present a number of applications and techniques for eddy current analysis as well as several eddy current applications of interest today. Chapters 2 and 3 present the analysis of several eddy current applications of practical interest, mainly in closed form, for rectangular conductors and for conductors with circular cross-sections, respectively. In Part II of the book, we focus on modern numerical techniques to study eddy currents. First, we introduce the most common mathematical formulations for eddy currents and the different variables used for these models. We then present the formulations for finite difference, finite element, and integral equations, with several examples that refer back to the results found in Chapters 2 and 3. In Part III of the book, we consider a number of practical applications such as electric machines, transformers, induction heating, and liquid metal stirring.

1.2 Faraday's Law and Lenz's Law

We will present a more formal introduction of Faraday's law below in Section 1.5. In this section, we will introduce Faraday's law and Lenz's law in non-mathematics terms to introduce some of basic ideas of eddy currents. Faraday's law is one of Maxwell's equations. It states that there is a voltage induced in a circuit equal to the negative time rate-of-change of the flux linking the circuit. We refer to this induced voltage as the emf (electromotive force). The “negative” sign is referred to as Lenz's law but is really part of Faraday's law. Lenz's law tells us the voltage induced in the circuit will be in a direction to circulate current that will produce flux which will oppose the change in flux linking the circuit. For example, if the flux linkage is increasing, the voltage will circulate current in a direction to decrease the flux linkage.

Consider the case illustrated in Figure 1.1. The figure shows a loop with an external source of magnetic flux that is varying in time in such a way as to increase the flux linking the loop. According to Lenz's law, we expect induced current in the sense shown in the figure to oppose the increase of flux linkage. Note that it is not the direction of the flux that is important but the time rate of change of the flux linking the circuit. With flux in the same direction as shown in the figure, if the flux were decreasing, the current direction would reverse.

In another example, consider the two coplanar loops in Figure 1.2. One of the loops contains a dc source. When the switch is closed, a current will flow counter-clockwise in the powered loop. This will cause flux from the first circuit to enter the plane of circuit 2 in the direction shown. We expect current in circuit 2 to circulate in the counter-clockwise direction to counter this flux; that is, this current will oppose the change of flux linkage from the powered circuit.

After some time, the system has reached steady state with dc current in loop 1 and no current in loop 2. If we now open the switch, interrupting the current in loop 1, current will be induced in loop 2 to maintain the steady-state flux linkage. Current will be induced in the clockwise direction in this case.

Figure 1.1 Loop with flux linkage increasing and induced current direction.

Figure 1.2 Two circuits with mutual coupling. (a) Switch closes. (b) After steady state is reached, open switch in loop 1.

We can now apply these ideas to the case of induced current in a circular wire. Referring to Figure 1.3, we have a conductor of circular cross section with time varying magnetic flux produced by an external source, in the axial direction.

Let us assume that the source flux density is uniform over the cross section and is varying sinusoidally with time. By symmetry, in this cylindrical geometry, we expect the eddy currents to follow circular trajectories and the eddy currents circulating to oppose the change in source flux. If we divide the conductor into concentric layers as shown in the figure, we can make the following argument. The induced voltage from the source in each circuit will be

Figure 1.3 Conducting cylinder with axial magnetic flux field applied.

(1.1)

where is the root-mean-square (RMS) value of the flux density, is the cross-sectional area enclosed by each of the loops, and is the angular frequency of the source. This tells us that the emf in the outer circuits is greater than the emf of the inner circuits due to the increased surface area and therefore the increased flux linking the current path. The current circulating in the outer circuit, loop 1, now produces a reaction flux which, by Lenz's law, opposes the change in source flux. This reduces the total flux in the interior of the cylinder, which is the sum of the source and reaction flux. Circuit 2, with a smaller emf, will also have current circulating in a direction to oppose the change in source flux. This will cause a further reduction in the net flux inside circuit 2. Note that the reaction field from circuit 2 does not pass through the conductor area of loop one, but only inside the path of loop 2. This continues for loops 3 to . As we move inward, therefore, the source emf is smaller and the effects of the reaction field are greater. The problem is complicated by the fact that each loop has a resistance and inductance, and therefore there will be a phase shift in the currents. We will deal with this issue in Section 1.7. For our purposes here, it is enough to note that we will have higher current and flux in the outer rings and lower current and flux in the inner loops. There is a characteristic length called the depth of penetration or skin depth, which we will treat more formally in Chapter 2, that describes a distance from the surface in which “most” of the current and flux is contained. We will see that this characteristic depth depends on the material properties (conductivity and permeability) and the frequency of excitation.

1.3 Proximity Effect

In the previous example, we considered the nonuniform current distribution in a circular conductor due to the self-field of the conductor. The term proximity effect is often used to describe the situation in which the emf produced by near-by sources influences the current distribution in the conductor. Referring to Figure 1.4, we can apply Faraday's law and Lenz's law to illustrate this effect. The figure depicts two parallel conductors with opposing currents, a go-and-return circuit.

Considering the current on the left, in the direction, we see that the field produced by conductor 1 in conductor 2, is increasing in the direction assuming that the current in conductor 1 is increasing. By the application of Lenz's law, the emf will tend to circulate current in conductor 2 such that the current on the left side of conductor 2 is in the direction, while the current on the right side is in the direction. This is a circulating current, going down one side of the conductor and returning on the other side. Integrating this circulating current density over the surface of the conductor will result in zero total current. This circulating current will subtract from the load current () on the right side and add on the left side. By applying this argument to conductor 1 and considering the field produced by conductor 2, we find the currents add on the right side of conductor 1 and subtract on the left side. The conclusion is that the currents crowd to the inside surfaces when the load currents are in the opposite direction.

If the load currents were in the same direction, we will find the currents crowd to the outside surfaces. In practice, if we are dealing with ac currents, the addition and subtraction of the circulating current is not a direct arithmetic addition or subtraction. Depending on the impedance of the circulating current path, there will be a phase shift that must be included. Figures 1.5 and 1.6 show the results of a finite element analysis of the two conductor problem for the case of opposing current direction. Figures 1.7 and 1.8 show the results of a finite element analysis of the two conductor problem for the case of load currents in the same direction. We can see that the results conform to our expectations on the nonuniformity of the current produced by the external fields.

Figure 1.4 Parallel conductors with opposing currents.

Figure 1.5 Real part of flux density with opposing currents.

Figure 1.6 Current density with opposing currents.

We will discuss the losses in conductors in detail in Chapters 2 and 3