Preface

This book covers diffusion of electromagnetic fields in magnetically nonlinear conductors and electrically nonlinear superconductors. This diffusion is described by nonlinear partial differential equations, and for this reason it is termed “nonlinear” diffusion. Nonlinear diffusion has many qualitative features that are not observed for linear diffusion, which explains why the study of nonlinear diffusion of electromagnetic fields is of significant theoretical interest. At the same time, the study of nonlinear diffusion is very important in many practical applications. Indeed, analysis of electromagnetic field diffusion in magnetically nonlinear conductors is, in a way, analysis of eddy currents in those conductors. The latter analysis is very instrumental in such diverse applications as: design of electric machines, transformers and actuators, induction heating, nondestructive testing, electromagnetic shielding, development of inductive writing heads for magnetic recording, and design of magnetic components in power electronics. On the other hand, the study of nonlinear diffusion of electromagnetic fields in superconductors is instrumental for the analysis of magnetic hysteresis in those superconductors as well as for the understanding of creep phenomena.

In spite of significant theoretical and practical interests, nonlinear diffusion of electromagnetic fields has not been extensively studied, and currently no book exists that covers this topic in depth. It is hoped that this book will bridge this gap.

The book has the following salient and novel features.

• Extensive use of analytical techniques for the solution of nonlinear partial differential equations, which describe electromagnetic field diffusion in nonlinear media.

• Simple analytical formulas for surface impedances of nonlinear and hysteretic media.

• Analytical analysis of nonlinear diffusion for linear, circular, and elliptical polarizations of electromagnetic fields.

• Novel and extensive analysis of eddy current losses in steel laminations for unidirectional and rotating magnetic fields.

• Preisach approach to the modeling of eddy current hysteresis and superconducting hysteresis.

• Extensive analytical study of nonlinear diffusion in superconductors with gradual resistive transitions (scalar and vectorial problems).

• Scalar potential formulations of nonlinear impedance boundary conditions and their finite element implementations.

The book contains five chapters and one appendix.

Chapter 1 deals with the analytical study of electromagnetic field diffusion in magnetically nonlinear conducting media in the case of linear polarization of magnetic fields. This diffusion is described by scalar nonlinear partial differential equations of the parabolic type. Discussions start with the case of abrupt magnetic transition (abrupt saturation) and proceed to the case of gradual magnetic transition (gradual saturation). For the latter case, first self-similar analytical solutions are found, which reveal that nonlinear diffusion occurs as an inward progress of almost rectangular profiles of magnetic flux density of variable height. These almost rectangular profiles of magnetic flux density represent an intrinsic feature of nonlinear diffusion in the case of sufficiently strong magnetic fields, and they occur because magnetic permeability (or differential permeability) is increased as the magnetic fields are attenuated. The analysis of the self-similar solutions suggests the idea of rectangular profile approximation of actual magnetic flux density profiles. This approximation is used to derive simple analytical expressions for the surface impedance. Chapter 1 also contains discussions of the “standing” mode of nonlinear diffusion, applications of nonlinear diffusion to circuit analysis, and the representation of eddy current, hysteresis in terms of the Preisach model. The last representation reveals the remarkable fact that nonlinear (and dynamic) eddy current hysteresis can be fully characterized by its step response.

In Chapter 2, diffusion of circularly and elliptically polarized electromagnetic fields in magnetically nonlinear conducting media is discussed. This diffusion is described by vector (rather than scalar) nonlinear partial differential equations, which naturally raises the level of mathematical difficulties. However, it is shown that these difficulties can be completely circumvented in the case of circular polarizations and isotropic media. Simple and exact analytical solutions are obtained for the above case by using power law approximations for magnetization curves. These solutions reveal the remarkable fact that there is no generation of higher-order harmonics despite nonlinear magnetic properties of conducting media. This is because of the high degree; of symmetry that exists in the case of circular polarizations and isotropic: media. Elliptical polarizations and anisotropic media are then treated as perturbations of circular polarizations and isotropic media, respectively. On the basis of this treatment, the perturbation technique is developed and simple analytical solutions of perturbed problems are found.The chapter concludes with an extensive analysis of eddy current losses in steel laminations caused by rotating magnetic fields.

Chapter 3 presents analysis of nonlinear diffusion of weak magnetic fields. In the case of weak magnetic fields, magnetic permeability (or differential permeability) is decreased as the magnetic fields are attenuated. As a result, physical features of this nonlinear diffusion are quite different from those in the case of strong magnetic fields. However, the same mathematical machinery that has been developed in the first two chapters can be used for the analysis of nonlinear diffusion of weak magnetic fields. As a result, many formal arguments and derivations presented in Chapter 3 are in essence slightly modified repetitions of what has been already discussed in the first and second chapters. These arguments and derivations are presented (albeit in concise form) for the sake of completeness of exposition.

Chapter 4 deals with nonlinear diffusion of electromagnetic fields in type-II superconductors. Phenomenologically, type-II superconductors can be treated as conductors with strongly nonlinear constitutive relations E(J). These relations are usually approximated by sharp (ideal) resistive transitions or by “power” laws (gradual resistive transitions). Discussions start with the case of ideal resistive transitions and the critical state model for superconducting hysteresis. It is shown that this model is a very particular case of the Preisach model of hysteresis and, on this basis, it is strongly advocated to use the Preisach model for the description of superconducting hysteresis. For the case of gradual resistive transitions described by the power laws, analysis of nonlinear diffusion in superconductors has many mathematical features in common with the analysis of nonlinear diffusion in magnetically nonlinear conductors. For this reason, the analytical techniques that have been developed in the first two chapters are extensively applied to the analysis of nonlinear diffusion in superconductors. Thus, our discussion of this diffusion inevitably contains some repetitions; however, it is deliberately more concise and it stresses the points that are distinct to superconductors.

In Chapter 5, nonlinear impedance boundary conditions are introduced and extensively used for the solution of nonlinear eddy current problems. These boundary conditions are based on the expressions for nonlinear surface impedances derived in the previous chapters. The main emphasis in this chapter is on scalar potential formulations of impedance boundary conditions and their finite element implementations. However, the discussion presented in the chapter is much broader than this. It encompasses such related and important topics as: a general mathematical structure of 3-D eddy current problems, calculation of source fields, analysis of eddy currents in thin nonmagnetic conducting shells, derivations of easily computable estimates for eddy current losses, and analysis of thin magnetic shells subject to static magnetic fields.

Finally, Appendix A covers the basic facts related to the Preisach model of hysteresis. This model is treated as a general mathematical tool that can be used for the description of hysteresis of various physical origins. In this way, the physical universality of the Preisach model is clearly revealed and strongly emphasized.

In the book, no attempt is made to refer to all relevant publications. For this reason, the reference lists given at the end of each chapter are not exhaustive but rather suggestive. The presentation of the material in the book is largely based on the author's publications that have appeared over the last thirty years.

In writing this book, I have been assisted by Mrs. Patricia Keehn who patiently, diligently and professionally typed several versions of the manuscript. In preparation of the manuscript, I have also been assisted by my students Chung Tse and Michael Neely. I am very grateful to these individuals for their invaluable help in my work on this book. The main part of the book was written during my sabbatical leave at the Laboratory for Physical Sciences at College Park, Maryland, and I am very thankful to Dr. Thomas Beahn for the given opportunity. My work on this book...