mathematical models of hysteresis and their applications

Preface

"But I have lived, and have not lived in vain: My mind may lose its force, my blood its fire, And my frame perish even in conquering pain, But there is that within me which shall tire Torture and Time, and breathe when I expire..."

Lord Byron

This book is a greatly expanded, revised and updated version of the previous book "Mathematical Models of Hysteresis" (Springer-Verlag, 1991). This book deals with mathematical models of hysteresis nonlinearities with "nonlocal memories". The distinct feature of these nonlinearities is that their future states depend on past histories of input variations. It turns out that memories of rate-independent hysteresis nonlinearities are quite selective. Indeed, only some past input extrema (not the entire input variations) leave their marks upon the future states of rate-independent hysteresis nonlinearities. Thus, special mathematical tools are needed to describe nonlocal selective memories of such hysteresis nonlinearities. The origin of such tools can be traced back to the landmark paper of Preisach.

The first three chapters of this book are primarily concerned with Preisach-type models of hysteresis. All these models have a common generic feature: they are constructed as superpositions of the simplest hysteresis nonlinearities--rectangular loops. The discussion in these chapters is by and large centered around the following topics: various generalizations and extensions of the classical Preisach model of hysteresis (with special emphasis on vector generalizations); finding of necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by various Preisach-type models; solution of identification problems for these models, their numerical implementation and extensive experimental testing. Our exposition of Preisach-type models of hysteresis has two salient features. The first is the strong emphasis on the universality of the Preisach models and their applicability to the mathematical description of

ix

x Preface

hysteresis phenomena in various areas of science and technology. The second is the accessibility of the material in the first three chapters to a broad audience of researchers, engineers and students. This is achieved through the deliberate use of simple mathematical tools. The exception is the discussion of the identification problems for the vector Preisach models in the third chapter, where some machinery of integral equations and the theory of irreducible representations of the group of rotations are occasionally used.

The book contains three new chapters that deal with applications of the Preisach formalism to the modeling of thermal relaxations (viscosity) in hysteretic materials as well as to the modeling of superconducting hysteresis and eddy current hysteresis. In Chapter 4, Preisach models driven by stochastic inputs are used for the description of thermal relaxations in hysteretic systems. This approach explicitly accounts for the hysteretic nature of materials, their past histories and stochastic characteristics of internal thermal noise. In this sense, this approach has certain advantages over traditional thermal activation type models of viscosity. This approach also reveals the origin of universality of intermediate lnt-type asymptotics for thermal relaxations. Some results of experimental testing of thermal decay in magnetic materials are presented and the phenomenon of scaling and "data collapse" for viscosity coefficients is reported. The chapter also presents the modeling of temperature dependent hysteresis within the framework of randomly perturbed fast dynamical systems and the discussion of functional (path) integration models of hysteresis and their connections with Preisach-type models.

Chapter 5 covers the modeling of superconducting hysteresis. It starts with the discussion of the critical state (Bean) model for superconductors with ideal (sharp) resistive transitions. It is demonstrated 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. The results of extensive experimental testing of the Preisach modeling of superconducting hysteresis are reported and the remarkable accuracy of this modeling is highlighted. The case of gradual resistive transitions described by "power laws" is treated through nonlinear diffusion equations and analytical solutions of these equations are found for linear, circular and elliptical polarizations of electromagnetic fields.

Chapter 6 deals with eddy-current hysteresis in magnetically nonlinear conductors. It is demonstrated that in the case of sharp magnetic transitions (abrupt saturation), the eddy current hysteresis can be represented in terms of the Preisach model. This representation reveals the remarkable fact that nonlinear (and dynamic) eddy current hysteresis can

Preface xi

be fully characterized by its step response. Eddy current hysteresis for gradual magnetic transitions is studied by using nonlinear diffusion equations and analytical solutions of these equations are reported for linear and circular polarizations of electromagnetic fields. The developed techniques are used to study "excess" eddy current and hysteresis losses as well as rotational eddy current losses.

In this book, no attempt is made to refer to all relevant publications. For this reason, the lists of references are not exhaustive but rather suggestive. The presentation of the material in the book is largely based on the publications of the author and his collaborators.

I first heard about the Preisach model during my conversation with Professor K. M. Polivanov. This was about thirty years ago, and at that time I lived in Russia. Shortly thereafter, my interest in the Preisach model was strongly enhanced as a result of my discussions with Professors M. A. Krasnoselskii and A. Pokrovskii. When I came to the United States, my work on hysteresis modeling was encouraged by Dr. O. Manley from the U.S. Department of Energy. My research on the Preisach models has benefited from many penetrating discussions I have had with Professor D. Fredkin (University of California, San Diego). I was also fortunate to have such wonderful graduate students as G. Friedman, C. Korman, and A. Adly, who assisted me at different times in my work on hysteresis and who became important contributors in this field in their own right. I am very grateful to my collaborators Professor M. Freidlin, Drs. G. Bertotti, C. Serpico and C. Krafft for the gratifying experience I have had work- ing with them. I acknowledge with gratitude the numerous stimulating discussions I had with Professors A. Visintin, M. Brokate, J. Sprekels and P. Krejci over the past twenty years. I am very thankful to Mrs. P. Keehn who patiently, diligently and professionally typed several versions of the manuscript. In the preparation of the manuscript, I have also been assisted by my students Chun Tse and Mihai Dimian. Finally, I gratefully acknowledge the financial support for my research on hysteresis from the U.S. Department of Energy, Engineering Research Program.

Introduction

The topic being discussed in this book is mathematical models of hysteresis. Special emphasis is placed on the mathematical exposition of these models which makes them quite general and applicable to the description of hysteresis of different physical nature. There are, however, two additional reasons for this emphasis. As was pointed out by A. Ein- stein [1], "... mathematics enjoys special esteem, above all other sciences, [because] its laws are absolutely certain and indisputable . . . . "Mathemat- ics has achieved and maintained this exceptional position because its results are derived from a few (more or less self-evident) axioms by a chain of flawless reasonings. Since it is based on impeccable logic, mathematics can provide some level of security (and clarity) for natural sciences which is not attainable otherwise. For this reason, the rigorous mathematical treatment of natural sciences is highly desirable and should be attempted whenever is possible. In addition, mathematics more and more often serves as a vehicle of communication between scientists and engineers of different specializations. As a result, if some area of science is represented in a rigorous mathematical form, its accessibility is strongly enhanced. With these thoughts in mind, it is hoped that the mathematical exposition of hysteresis models undertaken here will bring much needed clarity into this area and will make it appealing to the broader audience of inquiring researchers.

This monograph has been written by an engineer for engineers. For this reason, mathematics is largely used in the book as a tool rather than a topic of interest in its own right. As a result, many mathematical subtleties of hysteresis modelling are omitted. These subtleties are by and large related to the fact that hysteresis operators are naturally defined on sets of piece-wise monotonic functions that do not form complete function spaces. This leads to the problem of continuous extension of hysteresis operators from the above sets to some complete function spaces. The reader interested in this type of mathematical problems is referred to the study by the Russian mathematicians M. Krasnoselskii and A. Pokrovskii [2] as well as to the books of A. Visintin [3] and M. Brokate and J. Sprekels [4].

xiii

xiv Introduction

u(t) ~1

-I HT

FIGURE 1

f(t)

The phenomenon of hysteresis has been with us for ages and has been attracting the attention of many investigators for a long time. The reason is that hysteresis is ubiquitous. It is encountered in many different areas of science. Examples include magnetic hysteresis, ferroelectric hysteresis, mechanical hysteresis, superconducting hysteresis, adsorption hysteresis, optical hysteresis, electron beam hysteresis, economic hysteresis, etc. However, the very meaning of hysteresis varies from one area to another, from paper to paper and from author to author. As a result, a stringent mathematical definition of hysteresis is needed in order to avoid confusion and ambiguity. Such a definition will serve a twofold purpose: first, it will be a substitute for vague notions, and, second, it will pave the road for more or less rigorous treatment of hysteresis.

We begin with the definition of scalar hysteresis and, for the sake of generality, we adopt the language of control theory. Consider a transducer (see Fig. 1) that can be characterized by an input u(t) and an output f(t). This transducer is called a hysteresis transducer (HT) if its input-output relationship is a multibranch nonlinearity for which branch-to-branch transitions occur after input extrema. This multibranch nonlinearity is shown in Fig. 2. For the most part, the case of rate-independent hysteresis nonlinearity will be discussed. The term "rate-independent" means that

f

I ~ U

FIGURE 2

Introduction xv

branches of such hysteresis nonlinearities are determined only by the past extremum values of input, while the speed (or particular manner) of input variations between extremum points has no influence on branching. This statement is illustrated by Figs. 3a, 3b and 3c. Figures 3a and 3b show two different inputs ul(t) and u2(t) that successively assume the same extremum values but vary differently between these values. Then, for a

u~(t)

~ t (a)

u2(t)

* t (b)

fl ~ U ~c)

FIGURE 3

xvi Introduction

B

- / H

FIGURE 4

rate-independent HT, these two inputs will result in the same f-u diagram (see Fig. 3c), provided that the initial state of the transducer is the same for both inputs.

The given definition of rate-independent hysteresis is consistent with existing experimental facts. Indeed, it is known in the area of magnetic hysteresis that a shape of major (or minor) loop (see Fig. 4) can be specified without referring to how fast magnetic field H varies between two extremum values, q-Hm and -Hm. This indicates that time effects are negligible and the given definition of a rate-independent hysteresis transducer is an adequate one. It is worthwhile to keep in mind that, for very fast input variations, time effects become important and the given definition of rate-independent hysteresis fails. In other words, this definition (as any other definition) has its limits of applicability to real life problems.

It is also important to stress that the notion of rate-independent hysteresis implies three distinct time scales. The first is the time scale of fast internal dynamics of the transducer. The second is the time scale on which observations (measurements) are performed. This time scale is much larger than the time scale of internal transducer dynamics so that every observation can be identified with a specific output value of the transducer. The third is the time scale of input variations. This time scale is much larger than the observation time scale so that every measurement can be associated with a specific value of input.

In the existing literature, the hysteresis phenomenon is by and large linked with the formation of hysteresis loops (looping). This may be mis- leading and create the impression that looping is the essence of hysteresis. In this respect, the given definition of hysteresis emphasizes the fact that history dependent branching constitutes the essence of hysteresis, while

Introduction xvii

looping is a particular case of branching. Indeed, looping occurs when the input varies back and forth between two consecutive extremum values, while branching takes place for arbitrary input variations.

From the given definition, it can also be concluded that scalar hysteresis can be interpreted as a nonlinearity with a memory which reveals itself through branching.

In the given definition of hysteresis, the physical meanings of the input u(t) and the outputf(t) were left unspecified. It was done deliberately, for the sake of mathematical generality. However, it is not difficult to specify the meanings of u(t) and f(t) in particular applications. For instance, in magnetism u(t) is the magnetic field and f(t) is the magnetization, in mechanics u(t) is the force andf(t) is the displacement (length), in adsorption u(t) is the gas pressure and fit) is the amount of material adsorbed. The notion of hysteresis transducer may have different interpretations as well. For instance, in magnetism the HT can be construed as an infinitesimally small volume of magnetic material, and the corresponding input-output hysteresis nonlinearity can be interpreted as a constitutive equation for this material.

All rate-independent hysteresis nonlinearities fall into two general classifications: (a) hysteresis nonlinearities with local memories, and (b) hysteresis nonlinearities with nonlocal memories. The hysteresis nonlinearities with local memories are characterized by the following property. The value of outputf(t0) at some instant of time to and the values of input u(t) at all subsequent instants of time t ~ to uniquely predetermine the value of output f(t) for all t > to. In other words, for hysteresis transducers with local memories the past exerts its influence upon the future through the current value of output. This is not the case for hysteresis transducers with nonlocal memories. For such transducers, future values of outputf(t) (t ~ to) depend not only on the current value of outputf(t0) but on past extremum values of input as well.

Typical examples of hysteresis nonlinearities with local memories are shown in Figs. 5, 6, and 7. Figure 5 shows the simplest hysteresis nonlinearity with local memory. It is specified by a major loop which is formed by ascending and descending branches. These branches are only partially reversible (their vertical sections are not reversible). This type of hysteresis nonlinearity is characteristic, for instance, of single Stoner-Wolhfarth magnetic particles [5]. For this type of hysteresis, branching occurs if extremum values of input exceed +Um or -Um.

A more complicated type of hysteresis nonlinearities with local memories is illustrated by Fig. 6. Here, there is a set of inner curves within the major loop and only one curve passes through each point in the f - u diagram. These curves are fully reversible and can be traversed in both

xv i i i Introduction

- U m

F I G U R E 5

Urn ~ U

f

U

f

U

F I G U R E 6 F I G U R E 7

directions, for a monotonically increasing and decreasing input u(t). For this type of hysteresis, branching may occur only when ascending or descending branches of major loops are reached.

A hysteresis nonlinearity with local memory that has two sets of inner curves (the ascending and descending curves) is shown in Fig. 7. This type of hysteresis was probably first described by Madelung [6] in the beginning of the century, and afterwards it was independently invented by many authors time and time again (see, for instance, [7] and [8]). For this hysteresis nonlinearity, only one curve of each set passes through each point in thef-u diagram. If the input u(t) is increased, the ascending curve

P. U

Introduction

FIGURE 8

xix

is followed; if it is decreased, the descending curve is traced. Thus, branching occurs for any input extremum. However, in general, minor loops are not formed; if u(t) varies back and forth between the same two values, the output usually exhibits a continued upward drift.

It is clear from the above examples that all hysteresis nonlinearities with local memories have the following common feature: every reachable point in the f -u diagram corresponds to a uniquely defined state. This state predetermines the behavior of HT in exactly one way for increasing u(t) and exactly one way for decreasing u(t). In other words, at any point in the f -u diagram there are only one or two curves that may represent the future behavior of HT with local memory (see Fig. 8). This is not true for hysteresis transducers with nonlocal memories. In the latter case, at any reachable point in the f -u diagram there is an infinity of curves that may represent the future behavior of the transducer (see Fig. 9). Each of these curves depends on a particular past history, namely, on a particular sequence of past extremum values of input. By analogy with the random process theory, hysteresis nonlinearities with local memories can be called Markovian hysteresis nonlinearities, while hysteresis nonlinearities with nonlocal memories are non-Markovian. It is clear that hysteresis nonlinearities with nonlocal memories are much more complicated than those with local memories.

Mathematical models of hysteresis nonlinearities with local memories have been extensively studied by using differential and algebraic equations. These models have achieved high level of sophistication that is reflected, for instance, in publications [9-12]. However, the notion of

F I G U R E 9

U

J

f

Introduction XX

F I G U R E 10

hysteresis nonlinearities with local memories is not consistent with experimental facts. For instance, it is reported in [13] that crossing and partially coincident minor loops have been experimentally observed. These loops are schematically shown in Figs. 10 and 11, respectively. The existence of crossing minor loops attached to a major loop is more or less obvious, while the presence of partially coincident minor loops is a more subtle phenomenon. The existence of crossing and partially coincident mi- nor loops clearly suggests that the states of the corresponding hysteresis

Introduction

f

U

F I G U R E 11

xxi

transducers are not uniquely specified by their inputs and outputs. Thus, hysteresis of this transducer does not have a local memory.

This book is solely concerned with mathematical models of hysteresis with nonlocal memory. The question arises, why are these models needed? The answer is that the hysteresis transducer is usually a part of a system. As a result, its input is not known beforehand, but is determined by the interaction of the transducer with the rest of the system. Since the input of HT is not predictable a priori, it is impossible to specify ahead of time the branches of hysteresis nonlinearity which will be followed in a particular regime of the system. This is the main impediment as far as self-consistent mathematical descriptions of systems with hysteresis are concerned. To overcome the difficulty mentioned above, mathematical models of hysteresis are needed. These models represent new mathematical tools that themselves (due to their structure) will detect and accumulate input extrema and will choose appropriate branches of the hysteresis nonlinearity according to the accumulated histories. Coupled together with mathematical description of the rest of the system, these models will constitute complete and self-consistent mathematical descriptions of systems with hysteresis. Without such models, the self-consistent mathematical descriptions of systems with hysteresis are virtually impossible.

We next turn to the discussion of vector hysteresis. This hysteresis can be characterized by a vector input ~(t) and vector output f(t) (see Fig. 12). Two- and three-dimensional vector inputs and vector outputs are most relevant to practical applications. That is why only two- and three- dimensional vector hysteresis models are discussed in the book. However,

xxii Introduction

VHT

FIGURE 12

the formal mathematical generalization of these models to n dimensions (n > 3) is straightforward. It is believed that such a generalization will be performed by the reader if it is needed.

The most immediate problem we face is how to define vector hysteresis in a mathematically rigorous as well as physically meaningful way. To do this, it is important to understand what constitutes in the case of vector hysteresis the essential part of past input history that affects the future variations of output. In the case of scalar rate-independent hysteresis, experiments show that only past input extrema (not the entire input variations) leave their mark upon future states of hysteresis nonlinearities. In other words, the memories of scalar hysteresis nonlinearities are quite selective. There is no experimental evidence that this is the case for vector hysteresis. As a result, we must resign ourselves to the fact that all past vector input values may affect future output variations. The past input variations can be characterized by an oriented curve L traced by the tip of the vector input ~(t) (see Fig. 13). Such a curve can be called an input "hodograph." Vector rate-independent hysteresis can be defined as a vector nonlinearity with the property that the shape of curve L and the direction of its tracing (orientation) may affect future output variations, while the speed of input hodograph tracing has no influence on future output variations. Next, we demonstrate that scalar rate-independent hysteresis can be construed as a particular case of vector rate-independent hysteresis. This case is realized when the vector input is restricted to vary along only one direction (one line). In fact, it can be successfully argued (at least in the area of magnetics) that there is no such a thing as scalar hysteresis. Whenever we talk about scalar hysteresis, we are actually dealing with some specific properties of vector hysteresis that have been observed

k y

FIGURE 13

Introduction

U'minl ;U'min2 U'max2 U'maxl

FIGURE 14

xxi i i

for vector input variations restricted to some fixed directions. It is apparent that, for unidirectional input variations ~(t) = ~u(t), input hodographs (see Fig. 14) are uniquely determined by current values of u(t) as well as by past extrema of u(t). In this sense, vector rate-independent hysteresis is reduced to scalar rate-independent hysteresis with the input u(t).

Next, we shall give another equivalent definition of rate-independent vector hysteresis in terms of input projections. This definition will be convenient in the design of mathematical models of vector hysteresis. Con- sider input projection along some arbitrary chosen direction. As the vector ~(t) traces the input hodograph, the input projection along the chosen direction may achieve extremum values at some points of this hodograph. In this sense, the extrema of input projection along the chosen direction samples certain points of the input hodograph. If the projection direction is continuously changed, then the extrema of input projections along the continuously changing direction will continuously sample all points on the input hodograph. In this way, the past extrema of input projections along all possible directions reflect the shape of input hodograph and, consequently, the past history of input variations. Thus, we arrive at the definition of vector rate-independent hysteresis as a vector nonlinearity with the property that past extrema of input projections along all possible directions may affect future output values. It is clear that mathematical models of vector hysteresis are imperative for self-consistent descriptions of systems with vector hysteresis. These models should be able to detect and store past extrema of input projections along all possible directions and choose the appropriate value of vector output according to the accumulated history.

This book deals exclusively with the mathematical models of hysteresis that are purely phenomenological in nature. Essentially, these models represent the attempt to describe and generalize experimental facts. They provide no insights into specific physical causes of hysteresis. Neverthe- less, they have been and may well continue to be powerful tools for device design. There are, however, fundamental models of hysteresis which attempt to explain experimental facts from first principles. For instance, in micromagnetics, these principles require that the equilibrium distribution of magnetization should correspond to free energy minimum. The minimized energy basically includes the exchange energy, the anisotropy

xxiv Introduction

energy, the energy of interaction with an applied field, the magnetostatic self-energy, and possibly some other terms. It turns out that there are many (at least two) different local minima of the total energy for a given applied field. Since only one of these energy minima corresponds to the thermodynamic equilibrium state, the others must be metastable. They may persist for a very long time. These persisting metastable states are responsible for the origin of hysteresis.

Although the above micromagnetic approach is fundamental in nature, its implementation encounters some intrinsic difficulties.

First, in order to carry out this approach, the detailed information of microscopic material structure is needed. Only on the basis of this information can the above-mentioned terms of minimized energy be specified. However, the detailed knowledge of material microstructure is often not available.

Second, the micromagnetic approach leads to nonlinear differential (or integrodifferential) equations which are quite complicated to solve even using sophisticated numerical techniques. In part, this is because the solution of these equations may exhibit highly irregular behavior. In- deed, domains and their walls should emerge from the micromagnetic approach. The domain walls are small regions where the direction of magnetization changes quite rapidly, from some particular direction in one domain to a different direction in an adjacent domain. In a way, these domain walls can be mathematically construed as interior layers. This suggests that micromagnetic problems may well belong to the class of sin- gularly perturbed problems. (This fact has not been appreciated enough in the existing literature). To resolve the fast variations of magnetization over the domain walls, very fine meshes are needed. But, the domain walls usually move when the applied field is changed. Thus, it is not clear a priori where the fine meshes should be located. This may seriously complicate the numerical analysis.

Finally, the detailed domain structure which can be produced by the micromagnetic approach may be irrelevant to some practical problems. This is the case, for instance, in the design of devices for which the average value of magnetization over regions with dimensions much larger than domain dimensions is of interest.

Summarizing the above discussion, it can be concluded that the phenomenological approach is more directly connected with macroscopic experimental data. For this reason, it is of a great value to device de- signers. The fundamental micromagnetic approach, on the other hand, is intimately related to material structure and, therefore, it can be useful in the design of new materials.

Introduction xxv

RefeFences 1. Einstein, A. (1983). Geometry and experience, Sidelights on Relativity, New

York: Dover Publications.

2. Krasnoselskii, M. and Pokrovskii, A. (1983). Systems with Hysteresis, Moscow: Nauka.

3. Visintin, A. (1994). Differential Models of Hysteresis, Berlin: Springer.

4. Brokate, M. and Sprekels, J. (1996). Hysteresis and Phase Transitions, Berlin: Springer.

5. Stoner, E. C. and Wolhfarth, E. P. (1948). Trans. Roy. Soc. London 240: 599.

6. Madelung, E. (1905). Ann. Physik 17: 865.

7. Everett, D. H. and Smith, F. W. (1954). Trans. Faraday Soc. 50: 187.

8. Potter, R. I. and Schmulian, R. J. (1971). IEEE Trans. Mag. 7: 873.

9. Chua, L. and Stromsmoe, K. (1970). IEEE Trans. Circuit Theory 17: 564.

10. Chua, L. and Bass, S. (1972). IEEE Trans. Circuit Theory 19: 36.

11. Boley, C. D. and Hodgdon, M. L. (1989). IEEE Trans. Mag. 25: 3922.

12. Jiles, D. C. and Thoelke, J. B. (1989). IEEE Trans. Mag. 25: 3928.

13. Barker, J. A., Schreiber, D. E., Huth, B. G. and Everett, D. H. (1985). Proc. Roy. Soc. London A386: 251.

CHAPTER 1

The Classical Preisach Model of Hysteresis

1.1 DEFINITION OF THE CLASSICAL PREISACH M O D E L

This model has a long and instructive history that can be best characterized by the following eloquent statement of J. Larmor made in his preface to the book [1]:

...scientific progress, considered historically, is not a strictly logical process, and does not proceed by syllogisms. New ideas emerge dimly into intuition, come into consciousness from nobody knows where, and become the material on which the mind operates, forging them gradually into consistent doctrine, which can be welded on to existing domains of knowledge.

This is exactly what has happened with the Preisach model. The origin of this model can be traced back to the landmark paper of F. Preisach [2] published in 1935. Preisach's approach was purely intuitive. It was based on some plausible hypotheses concerning the physical mechanisms of magnetization. For this reason, the Preisach model was first regarded as a physical model of hysteresis. It has remained primarily known in the area of magnetics where this model has been the focus of considerable research for many years. This has resulted in the further development of the Preisach model, and many improvements and valuable facts have been accumulated in this area. These developments and results are recorded (for instance) in the publications [3-12]; it is important to note that the given list of references is not complete by any standards.

Somewhat in parallel with the mentioned developments in magnetics, the Preisach model was independently invented and then extensively studied and tested for adsorption hysteresis by D. H. Everett and his collaborators [13-16]. This clearly indicated that the applications of Preisach's model were not limited only to the area of magnetics.

CHAPTER 1 The Classical Preisach Model of Hysteresis

The next decisive step in the direction of better understanding of the model was made in the 1970s and 1980s when the Russian mathematician M. Krasnoselskii and his colleagues undertook a comprehensive mathematical study of systems with hysteresis. It was then gradually realized that the Preisach model contained a new general mathematical idea. As a result, this model was separated from its physical connotation and represented in a purely mathematical form that is similar to a spectral decomposition of operators [17]. In this way a new mathematical tool has evolved that can now be used for the mathematical description of hysteresis of various physical nature. At the same time, this approach has strongly revealed the phenomenological nature of the Preisach model. This has put this model into a new perspective and has led to the clarification of many controversies that have long surrounded the Preisach model.

We next proceed to the purely mathematical description of the Preisach model. Consider an infinite set of simplest hysteresis operators }9~. Each of these operators can be represented by a rectangular loop on the input-output diagram (see Fig. 1.1). Numbers ~ and/J correspond to "up" and "down" switching values of input, respectively. It will be assumed in the sequel that c~ ~/J, which is quite natural from the physical point of view. Outputs of the above elementary hysteresis operators may assume only two values, +1 and -1 . In other words, these operators can be interpreted as two-position relays with "up" and "down" positions corresponding to ~,~u(t)= +1 and ~,~u(t)=-1, respectively.

A

+1 f

l a b

d

(x ! ~ U

F I G U R E 1.1

1.1 DEFINITION OF THE CLASSICAL PREISACH MODEL

As the input, u(t), is monotonically increased, the ascending branch abcde is followed. When the input is monotonically decreased, the descending branch edfba is traced. It is clear that the operators }3~ represent hysteresis nonlinearities with local memories (see the Introduction). Along with the set of operators }9~ consider an arbitrary weight function/~(ol, fl) that is often referred to as the Preisach function. Then, the Preisach model can be written as follows:

A

f(t) = F u ( t ) - ~>t~ #(~, fl)G~u(t) d~ dfl. (1.1)

A Here F is used for the concise notation of the Preisach hysteresis operator that is defined by the integral in (1.1).

It is apparent that the model (1.1) can be interpreted as a continuous analog of a system of parallely connected two-position relays. This interpretation is illustrated by the block diagram shown in Fig. 1.2. According to this diagram, the same input u(t) is applied to each of two-position relays. Their individual outputs are multiplied by #(c~, fl) and then integrated over all appropriate values of ~ and ft. As a result, the output, f(t), is obtained. Discrete approximation to the above block-diagram can be used as device realizations of the Preisach model (1.1).

It is clear from the above discussion that the Preisach model is constructed as a superposition of simplest hysteresis operators }9~. These operators can be construed as the main building blocks for the model (1.1). The idea that a complicated operator can be represented as a superposition of simplest operators is not entirely new and was exploited

u(t)

A

~'o,

A

A

|

|

/// f

.(a,~) |

f(t)

FIGURE 1.2


before in mathematics, particularly in the functional analysis [17]. For instance, according to the spectral decomposition theory for self-adjoint operators, any self-adjoint operator can be represented as a superposition of projection operators that are, in a way, the simplest self-adjoint operators. The above analogy shows that the Preisach model (1.1) can be interpreted from the mathematical point of view as a spectral decomposition of complicated hysteresis operator P into the simplest hysteresis operators 9~. There is also an interesting parallel between the Preisach model and wavelet transforms that are currently very popular in the area of signal processing. Indeed, all rectangular loop operators }9~ can be obtained by translating and dilating the rectangular loop operator }91,_1, that can be regarded as the "mother loop operator." Thus, the Preisach model can be viewed as a "wavelet operator transform."

The Preisach hysteresis nonlinearity (1.1) is constructed as a superposition of elementary hysteresis nonlinearities }9~ with local memories, nevertheless, it usually has a nonlocal memory. (This fact will be proved in the next section.) It is remarkable that a new qualitative property of nonlocal memory emerges as a collective property of a system having huge (infinite) number of simple and qualitatively similar components.

Having defined the Preisach model, it is appropriate to describe the directions along which the further discussion will proceed. The subsequent discussion in this chapter will be centered around the following five topics.

1. How does the Preisach model work? In other words, how does this model detect local input extrema, accumulate them and choose the appropriate branches of hysteresis nonlinearity according to the accumulated histories?

The answer to this question will reveal the mechanism of memory formation in the Preisach model. It will require the development of a special diagram technique that will constitute the mathematical foundation for the analysis of the Preisach model.

2. What experimental data are needed for the determination of the Preisach function,/~(~,/J), for a given hysteresis transducer?

This is the so-called identification problem. The solution to this problem is very important as far as the practical applications of the Preisach model are concerned.

3. What are the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the Preisach model?


The significance of this problem is that its solution will clearly establish the limits of applicability of the Preisach model.

4. How can the Preisach model be implemented numerically?

This is an important question because it seems at first that the numerical evaluation of two-dimensional integrals is required for the numerical implementation of the Preisach model. However, it will be shown in the sequel that the evaluation of the above integrals can be completely avoided.

5. How can the Preisach model be useful for the computation of hysteretic energy dissipation?

It is well known that the hysteretic energy dissipation can be easily evaluated for the case of periodic (cyclic) input variations. However, the problem of computing hysteretic energy losses for arbitrary (not necessary periodic) input variations has remained unsolved. It will be shown in this chapter (see Section 5) that the Preisach model can bring about the solution to this problem. This solution can be useful for the computation of entropy production and, in this way, may facilitate the development of irreversible thermodynamics of hysteretic media.

There is another problem which has not been mentioned above but has been extensively studied by Krasnoselskii and his colleagues. This problem is related to the fact that hysteresis nonlinearities are naturally defined on the set of continuous and piecewise monotonic inputs. How- ever, the above set of functions does not form a complete function space. This constitutes the main difficulty as far as the rigorous mathematical treatment of differential (or integral) equations with hysteresis nonlinearities is concerned. Thus, the problem of continuous extension of hysteresis operators from the above set of piecewise monotonic inputs to some complete function spaces presents itself. The essence of the problem is in finding (or description) of such complete function spaces. The solution to this problem is important because these function spaces form the natural "environment" for the rigorous mathematical study of equations with hysteresis nonlinearities. Nevertheless, this problem is more or less of purely mathematical nature. It is not directly related to practical applications, and for this reason will not be discussed in the book. The reader interested in the discussion of this problem is referred to Krasnoselskii and Pokrovskii [18] as well as more recent publications [19-24].

It is apparent from the above discussion that the Preisach model has been defined without any reference to a particular physical origin of hysteresis. This clearly reveals the phenomenological nature of the model and its mathematical generality. To further emphasize this new approach to


the Preisach model, it is instructive to compare the definition of the model given above with the definition regularly used in magnetics.

In magnetics, separate magnetic "particles" (or "domains") are introduced. These particles have rectangular hysteresis loops and they play the same role as elementary hysteresis operators }9~#. A loop typical for such particles is shown in Fig. 1.3. Here Hu and Ha are "up" and "down" switching magnetic fields, respectively, and ms is the magnetic moment (magnetization) that is identical for all particles. The notation ~(Hu, Ha) is used for the particle having the hysteresis loop shown in Fig. 1.3. The magnetic material is considered to be composed of many such particles. It is also assumed that the different particles have some distribution of reversal field Hu and Ha that can be characterized by the distribution function q~(Hu, Ha). This function plays the same role as/~(e, fl) in (1.1). It is typical to speak about statistical nature of the distribution function ~(Hu, Hd), although the author is not aware of any rigorous justification or experimental evidence for this statistical interpretation. By using the magnetic particles and their distribution function, the Preisach model is usually defined in magnetics as follows

M(t) - [ [ ~(Hu, Ha)~(Hu, Ha)H(t)dHudHa, (1.2) JJH u >~Ha

where M is the magnetization, while

~(Hu, Ha)H(t) = +ms (1.3)

+ m s -

/Hd

- m s

,Hu ~H

FIGURE 1.3


if the particle is switched up, and

~(Hu, Ha)H(t) - -ms (1.4)

if the particle is switched down. The given definition is framed in terms of magnetics and can naturally

be called the "magnetic" definition. It is obvious that the "magnetic" definition of the Preisach model is mathematically equivalent to the previous definition (1.1).

The "magnetic" definition suggests some relation of the Preisach model to certain physical realities such as magnetic particles, their distribution, etc. For this reason, the Preisach model has long been regarded as the physical model. However, there are some intrinsic difficulties with respect to this interpretation. Indeed, if the magnetic particles are to have some features of reality, then their geometric shapes and mutual spatial locations should be important and affect the value of magnetization. But, in the "magnetic" definition of the Preisach model there is no reference whatsoever to the spatial locations of the particles or to their shapes. In the mathematical definition (1.1), this difficulty does not appear because there is no need to relate the operators, 9~fi, to some real physical objects.

Another difficulty comes from the fact that many (almost all) particles in the Preisach model have asymmetrical rectangular loops (Hu :~ -Ha). Such loops are not physical for single particles. The use of asymmetrical loops is commonly justified on the ground that asymmetry is caused by the interaction between the particles, and the amount of asymmetry is a measure of the interaction. More precisely, this means that each particle "feels" not only the applied (external) magnetic field H(t), but also the interaction magnetic field that is due to the magnetization of adjacent particles. The interaction magnetic field may, of course, vary from one particle to another. These interaction magnetic fields result in shifted (asymmetrical) hysteresis loops when these loops represent the dependence of magnetization on the applied field alone.

Although the given explanation sounds plausible, it is not completely satisfactory. This explanation does not account for the fact that the interaction fields depend on particular states of magnetization, shapes, and location of the particles. This information is not represented in the Preisach model at all. In the case of purely mathematical definition (1.1) of the Preisach model, the above difficulty does not appear. This is because there is a straightforward phenomenological explanation for the need of operators 9~fi with asymmetrical loops. These operators are needed in order to describe asymmetrical minor loops exhibited by actual hysteresis nonlinearities (see Fig. 1.4). Indeed, if only operators }9~,_r with symmetrical loops were used, then the model (1.1) being constructed as a superposition


U

FIGURE 1.4

of such }9-operators would also describe only symmetrical minor loops. In other words, };-operators with symmetrical loops do not form a complete set of operators in the sense that hysteresis nonlinearities with asymmetrical minor loops cannot be represented as superpositions of }9-operators from this set.

From the discussion presented above the following conclusion can be reached. The "magnetic" definition of the Preisach model is possible and historically was first developed. However, this definition obscures the model, conceals its mathematical and phenomenological nature, and nar- rows the area of applicability of this model to the field of magnetics. The definition (1.1), on the other hand, interprets the Preisach model as a new mathematical tool whose importance may well extend beyond the area of magnetics. For this reason, the purely mathematical definition (1.1) seems to be more attractive and will be used throughout the book.

1.2 GEOMETRIC INTERPRETATION A N D M A I N PROPERTIES OF THE PREISACH MODEL

The mathematical investigation of the Preisach model is considerably facilitated by its geometric interpretation. This interpretation is based on the following simple fact. There is a one-to-one correspondence between operators G~ and points (c~, fl) of the half-plane a ~> fl (see Fig. 1.5). In other words, each point of the half-plane c~ ~> fl can be identified with only one particular };-operator whose "up" and "down" switching values are respectively equal to a and fl coordinates of the point. It is clear that this

1.2 GEOMETRIC INTERPRETATION

identification is possible because both }9-operators and the points of the half-plane c~ ~ ]~ are uniquely defined by pairs of numbers, c~ and/~.

Consider a right triangle T (see Fig. 1.5). Its hypotenuse is a part of the line c~ - /~ , while the vertex of its right angle has the coordinates c~0 and ]~0 with/~0 = -c~0. In the sequel, this triangle will be called the limiting triangle and the case when/~(c~, ~) is a finite function with a support within T will be discussed. In other words, it will be assumed that the function/~(~, ]~) is equal to zero outside the triangle T. This case covers the important class of hysteresis nonlinearities with closed major loops. At the same time, the above case will not essentially limit the generality of our future discussions.

To start the discussion, we first assume that the input u(t) at some instant of time to has the value that is less than ~0. Then, the outputs of all }9-operators which correspond to the points of the triangle T are equal to -1. In other words, all }9-operators are in the "down" position. This corresponds to the state of "negative saturation" of the hysteresis nonlinearity represented by the model (1.1).

Now, we assume that the input is monotonically increased until it reaches at time tl some maximum value Ul. As the input is being increased, all 9-operators with "up" switching values c~ less than the current input value u(t) are being turned into the "up" position. This means that their outputs become equal to +1. Geometrically, it leads to the subdivision of the triangle T into two sets: S+(t) consisting of points (~,~) for which the corresponding }9-operators are in the "up" position, and S-(t) consisting of points (~,/J) such that the corresponding }3-operators are still in the "down" position. This subdivision is made by the line c~ = u(t) (see

O~ A

(oco, Pol ,,.Jc~p /

T ~~'~

/

(Z

P

FIGURE 1.5 FIGURE 1.6

10 CHAPTER 1 The Classical Preisach Model of Hysteresis

Fig. 1.6) that moves upwards as the input is being increased. This upward motion is terminated when the input reaches the maximum value Ul. The subdivision of the triangle T into S+(t) and S-(t) for this particular instant of time is shown in Fig. 1.7.

Next, we assume that the input is monotonically decreased until it reaches at time t2 some minimum value u2. As the input is being decreased, all }3-operators with "down" switching values fl above the current input value, u(t), are being turned back into the "down" position. This changes the previous subdivision of T into positive and negative sets. Indeed, the interface L(t) between S+(t) and S-(t) has now two links, the horizontal and vertical ones. The vertical link moves from right to left and its motion is specified by the equation fl = u(t). This is illustrated by Fig. 1.8. The above motion of the vertical link is terminated when the input reaches its minimum value u2. The subdivision of the triangle T for this particular instant of time is shown in Fig. 1.9. The vertex of the interface L(t) at the above instant of time has the coordinates c~ = Ul and g = u2.

Now, we assume that the input is increased again until it reaches at time t3 some maximum value u3 that is less than Ul. Geometrically, this increase results in the formation of a new horizontal link of L(t) which moves upwards. This upward motion is terminated when the maximum u3 is reached. This is shown in Fig. 1.10.

Next, we assume that the input is decreased again until it reaches at time t4 some minimum value u4 that is above u2. Geometrically, this input variation results in the formation of a new vertical link that moves from right to left. This motion is terminated as the input reaches its min imum

s§

/

s-(t)

/

s§

ot

/

= u ( t )



(x

/

11


/

( U 1 , U 2 ) 1 [ lu~u~l

s +

/

FIGURE 1.11

value U4. As a result, a new vertex of L(t) is formed that has the coordinates c~ = u3 and fl = u4. This is illustrated by Fig. 1.11.

By generalizing the previous analysis, the following conclusion can be reached. At any instant of time, the triangle T is subdivided into two sets: S+(t) consisting of points (c~, fl) for which the corresponding }9-operators are in the "up" position, and S-(t) consisting of points (c~, fl) for which the corresponding };-operators are in the "down" position. The interface L(t) between S + (t) and S-(t) is a staircase line whose vertices have c~ and ~ coordinates coinciding respectively with local maxima and minima of input at previous instants of time. The final link of L(t) is attached to the line c~ = fl and it moves when the input is changed. This link is a horizontal


S-(t) G "~

S+(t) I

s-lo


one and it moves upwards as the input is increased (see Fig. 1.12). The final link is a vertical one and it moves from right to left as the input is decreased (see Fig. 1.13).

Thus, at any instant of time the integral in (1.1) can be subdivided into two integrals, over S + (t) and S-(t), respectively:

Since

and

A

#(o~, fl ) f ,~u( t ) dol dfl f (t) - F lz(t) - +(t)

f s tz ( ol , fl ) G f u ( t ) d o~ d fl . Jr- - ( t )

G~u(t) = +1, if (or, fl) ~ S+(t)

G ~ u ( t ) - -1 , if (o~, fl) ~ S- ( t ) ,

from (1.5) we find

(1.5)

(1.6)

(1.7)

f ( t ) - f f s tz ( ol , fl ) dot d fl - / f s tz ( ot , fl ) dot d fl . (1.8) +(t) - ( t )

From the above expression, it follows that an instantaneous value of output depends on a particular subdivision of the limiting triangle, T, into positive and negative sets S + (t) and S-(t). This subdivision is determined by a particular shape of the interface L(t). This shape, in turn, depends on the past extremum values of input because these extremum values are the coordinates of the vertices of L(t). Consequently, the past extremum val-

1.2 GEOMETRIC INTERPRETATION 13

ues of input shape the staircase interface, L(t), and in this way they leave their mark upon the future.

To make the above point perfectly clear, consider two inputs ul(t) and u2(t) with two different past histories for t < t'. This means that they had different local extrema for t < Y. It is next assumed that these inputs coincide for t >~ t'. Then according to (1.8), the outputs fl (t) andf2(t) corresponding to the above inputs are given by the formulas

fl(t) = ffs-~(t) fl)d dfl - ffsl(t) #(oe, fl)doldfl, (1.9)

f2(t) =/fs_~(t)tz(ot, fl)dotdfl- ffs2(t) #(o~,fl)dotdfl, (1.10)

where S-((t) and Sl( t ), S-f (t) and S 2 (t) are positive and negative sets of two subdivisions of T associated with ul(t) and u2(t), respectively.

The above two subdivisions are different because they correspond to two different input histories. Thus, from (1.9) and (1.10) we conclude

A(t) #f2(t) for t > t'. (1.11)

It is clear that inequality (1.11) holds even if the outputs A(t') and f2(t') are somehow the same at time t'. This means that the Preisach model (1.1) describes, in general, hysteresis nonlinearities with nonlocal memories.

The above discussion reveals the mechanism of memory formation in the Preisach model. The memory is formed as a result of two different rules for the modification of the interface L(t). Indeed, for a monotonically increasing input, we have a horizontal final link of L(t) moving upwards, while, for a monotonically decreasing input we have a vertical final link of L(t) moving from right to left. These two different rules result in the formation of the staircase interface, L(t), whose vertices have coordinates equal to past input extrema.

It is apparent from the previous analysis that the Preisach model can be defined in purely geometric terms, without any reference to the analytical definition (1.1). Indeed, the formula (1.8), along with the above two rules for the modification of L(t), can be interpreted as an independent definition of the Preisach model. This definition is fully equivalent to the previous one. However, the geometric definition may be convenient for further generalization of the Preisach model. For instance, new and more general rules for the subdivision of T into positive and negative sets, S+(t) and S-(t), may be introduced. In these rules, the links of L(t) may not necessarily be the segments of straight lines parallel to coordinate axes. Furthermore, different functions/z + (~, fl) and # - (~, fl) may be


defined on the positive and negative sets, respectively. All these modifications may result in some meaningful generalizations of the Preisach model. However, the above possibilities have not been examined enough in the existing literature.

Having described the geometric interpretation of the Preisach model, we are now well equipped for the discussion of the main properties of this model. We begin with the simplest property which expresses the fact that the output value, f+, in the state of positive saturation is equal to the minus output value, - f - , in the state of negative saturation. In the state of positive saturation, the input u(t) is more than d0 and all }9-operators are in the "up" position. Hence, according to (1.8), we find:

f+ - ~iT #(~" fl) do~ dfl. (1.12)

Similarly, in the state of negative saturation the input u(t) is less than fl0 and all }9-operators are in the "down" position. As a result, we obtain

f - - - f /T #(~' fl) d~ dfl. (1.13)

From (1.12) and (1.13), we have

f + = - f - . (1.14)

It is important to keep in mind that the saturation valuesf + and f - remain constant for any value of input u(t) above do and below rio, respectively. In other words, after ascending and descending branches merge together, they become flat. Partly for this reason, it is often said that the Preisach model does not describe reversible components of hysteresis nonlinearities. These components are regarded as being responsible for finite slopes of ascending and descending branches after they merge together. The in- ability of the Preisach model to describe the reversible components of hysteresis nonlinearities has long been viewed as a deficiency of the model. It will be shown later in the book that this deficiency along with some others can be removed by the appropriate generalization of the Preisach model.

We next proceed to the more interesting property which further elucidates the mechanism of memory formation in the Preisach model. It turns out that this model does not accumulate all past extremum values of input. Some of them can be wiped out (erased) by subsequent input variations. To make this property clear, consider a particular past history that is characterized by a finite decreasing sequence {ul,u3,u5, u7} of local input maxima and an increasing sequence {U2, U4, U6, U8} of local input minima. A typical ~-fl diagram for this kind of history is shown in Fig. 1.14. Now, we assume that the input u(t) is monotonically increased until it reaches

15

(X

/

. . . . . U 1

. . . . . . U 3 I

U 8

(X

U l



some maximum value u 9 that is above u3. This monotonic increase of input u(t) results in the formation of a horizontal final link of L(t) that moves upwards until the maximum value u9 is reached. This results in a modified ~-~ diagram shown in Fig. 1.15. It is evident that all vertices whose u-coordinates were below u9 have been wiped out. It is also clear that the wiping out of vertices is equivalent to the erasing of the history associated with these vertices. Namely, the past input maxima and minima that were respectively equal to ~- and/~-coordinates of the erased vertices have been wiped out. We have illustrated how the wiping out of vertices occurs for monotonically increasing inputs. However, it is obvious that the wiping out of vertices occurs in a similar manner for monotonically decreasing inputs as well. Thus, we can formulate the following property of the Preisach model.

WIPING-OUT PROPERTY Each local input maximum wipes out the vertices of L(t) whose u-coordinates are below this maximum, and each local minimum wipes out the vertices whose ~-coordinates are above this minimum.

The wiping-out property is asserted above in purely geometric terms. This makes this property quite transparent. However, the same property can also be described in analytical terms. The analytical formulation complements the geometric one because it is directly phrased in terms of time input variations.

Consider a particular input variation shown in Fig. 1.16 for the time interval to ~< t ~< t t. We assume that at the initial instant of time to the input value u(to) was below 1~0. This means that the initial state is the state of negative saturation. Consequently, the whole history has been written by


MI

�9 rrl I 'rll z

F I G U R E 1.16

the input variation after time to. We would like to specify explicitly local input extrema that will be stored by the Preisach model at time t'. Con- sider the global max imum of the input at the time interval [to, t']. We will use the notations M1 for this max imum and t~- for the instant of time the m a x i m u m was reached:

M1 -- max u(t), u(t +) = M1. (1.15) [t0,t']

It is clear that all previous input extrema were wiped out by this maximum. Now, consider the global m in imum of the input at the interval [t~-, t']. We will use the notation ml for this min imum and t I for the time it was reached:

ml = min u(t), u(tl) = ml. (1.16) [t~-,t']

It is apparent that all intermediate input extrema that occurred between t~- and t~- were erased by the min imum ml. Next, consider the global m a x im um of the input at the interval [ t l , t' ]. The notations M2 and t + are appropriate for this max imum and the time it occurred:

M 2 = max u(t), u(t-~) -- M2. (1.17) [t~-,t']


It is obvious that this maximum wiped out all intermediate input extrema that occurred between t I and t~-. As before, consider the global minimum of input at the time interval [t~-, t'] and the notations m2 and t 2 will be used for this minimum and the time it was achieved:

m2 -- min u(t), u(t2) -- m2. (1.18) [t+,t ,]

It is clear that this minimum wiped out all intermediate input extrema. Continuing the above line of reasoning, we can inductively introduce

the global maximum Mk and global minimum mk:

Mk -- max u(t), u(t~) = Mk (1.19) [t~-_l,t']

and mk = min u(t), u(t-~) = mk. (1.20)

[t~-,t']

Only these input extrema were accumulated by the Preisach model, while all intermediate input extrema were erased. It is natural to say that Mk and mk (k = 1, 2,. . .) form an alternating series of dominant maxima and minima.

It is evident from the above analysis that c~- and/~-coordinates of vertices of the interface L(t') are equal to Mk and mk, respectively. It is also clear that the alternating series of dominant extrema is modified with time. This means that new dominant extrema can be introduced by the time varying input, while the previous ones can be erased. In other words, Mk and mk are functions of t' as it is clearly suggested by their definitions (1.19) and (1.20).

Now, the wiping-out property can be stated in the following form.

WIPING-OUT PROPERTY Only the alternating series of dominant input extrema are stored by the Preisach model. All other input extrema are wiped out.

It is worth noting that the wiping-out property is somewhat natural and consistent with experimental facts. Indeed, experiments in the area of magnetics show the existence of major hysteresis loops whose shapes do not depend on how these loops are approached. In other words, the major hysteresis loops are well-defined. It means that any past history is wiped out by input oscillations of sufficiently large magnitude. This is in complete agreement with the wiping-out property.

Consider another characteristic property of the Preisach model that is valid for periodic input variations. Let ul (t) and u2(t) be two inputs that may have different past histories (different alternating series of dominant


- ] [_ ___

. p p


extrema). However, starting from some instant of time to, these inputs vary back and forth between the same two consecutive extremum values, u+ and u_. It can be shown that these periodic input variations result in minor hysteresis loops. Let Figs. 1.17 and 1.18 represent c~-~ diagrams for the inputs ul(t) and u2(t), respectively. As the inputs vary back and forth between u+ and u_, the final links of staircase interfaces Ll(t) and L2(t) move within the identical triangles T1 and T2. This results in periodic shape variations for Ll(t) and L2(t) which in turn produce periodic variations of the outputs, fl(t) andf2(t). This means that some minor hysteresis loops are traced in the f -u diagram for both inputs (see Fig. 1.19). The positions of these two loops with respect to the f-axis are different. This is because the above two inputs have different past histories that lead

U

2

I

1 I

U+ ~U

FIGURE 1.19


to different shapes for staircase interfaces, Ll(t) and L2(t). As a result, the values of outputs for the same values of inputs are different. This is easily seen from the formula (1.8). However, it can be proved that the above two hysteresis loops are congruent. It means that the coincidence of these loops can be achieved by the appropriate translation of these loops along the f-axis.

The proof of the congruency of the above loops is equivalent to showing that any equal increments of inputs ul(t) and u2(t) result in equal increments of outputs fl(t) and f2(t). To this end, let us assume that both inputs after achieving the same minimum value u_ are increased by the same amount: AUl = Au2 = Au. As a result of these increases, the identical triangles AT1 and AT2 are added to the positive sets S-~(t) and S~-(t) and subtracted from the negative sets S l(t) and S 2 (t) (see Figs. 1.17 and 1.18). Now, using the formula (1.8), we find that the corresponding output increments are given by

2 f f #(06/3) doe dfl, (1.21) a fl ddA T1

Af2 = 2 H #(or, fl) dot dfl. ,IdA T2

Since A T1 = A T2, we conclude

(1.22)

All = Af2. (1.23)

The equality (1.23) has been proved for the case when inputs ul(t) and u2(t) are monotonically increased by the same amount after achieving the same minimum value u_. Thus, this equality means the congruency for the ascending branches of the above minor loops. By literally repeating the previous reasoning, we can prove that the same equality (1.23) holds when the inputs ul(t) and u2(t) are monotonically decreased by the same amount Au after achieving the maximum value u+. This means that the descending branches of the above minor loops are congruent as well. Thus, we have established the following property of the Preisach model.

CONGRUENCY PROPERTY All minor hysteresis loops corresponding to back- and-forth variations of inputs between the same two consecutive extremum values are congruent.

1.2.1 Poss ib le appl icat ions of the Preisach m o d e l as a neural ne twork

We conclude this paragraph with a remark on possible applications of the Preisach model beyond the area of hysteresis modeling. It is clear from


the previous discussion that the Preisach model has the ability to detect and store the alternating series of dominant extrema of input. In other words, the Preisach model is endowed with memory. For this reason, the Preisach model might have appeal as the mathematical model of memory with some interesting properties, and its device realization (see Fig. 1.2) might be utilized as an unusual storage device. We will discuss below only a few peculiarities of this memory.

First, the mechanism of memory formation in the Preisach model is quite simple and results from the parallel connection of qualitatively similar elements (two-position cells) }~. Since this model employs very simple elements and has little structure, its memory formation can be interpreted as a "spontaneously emerged collective effect." (This kind of ter- minology is in vogue in the area of neural networks; see, for instance, [25].)

Second, storage of information is not localized. Indeed, the model (1.1) stores the information (extremum values of input) not in particular separate cells (as in the case of computer storage devices), but some ensembles of the cells }9~ participate in storage of each bit of information. As a result, if some of the cells are destroyed, the stored information might still be preserved. This opens the way for cell replenishment without affecting the currently stored information.

Third, some subsequent events may erase the information previously stored in the memory (wiping-out property). This erasing occurs if only new events make stronger impacts (characterized by larger input extrema) than previous ones.

Fourth, as it will be shown in Chapter 4, the Preisach model exhibits "fading" memory. This means that the stored information is slowly and gradually erased in the presence of small random perturbations (intrinsic noise).

The properties discussed above are somewhat similar to those being observed (or suspected) for memories in biological systems. However, it will be imprudent to speculate how far this similarity goes. Nevertheless, it may be expected that the mathematical tool (1.1) might find some interesting applications beyond the conventional area of hysteresis modeling.

1.3 IDENTIFICATION PROBLEM FOR THE PREISACH MODEL. REPRESENTATION THEOREM

Next, we proceed to the discussion of the identification problem for the Preisach model. The essence of this problem is in the determination of weight function #(~,~). The set of first-order transition curves will be

1.3 IDENTIFICATION PROBLEM 21

(X f

s-

I

; / i / I P ' U

/


used for this purpose. These curves can be defined as follows. First, the input u(t) should be decreased to the value that is less than fl0. This brings a hysteresis nonlinearly to the state of negative saturation. Next, the input is monotonically increased until it reaches some value ~'. The corresponding a-fl diagram is shown in Fig. 1.20. As the input is increased, an ascending branch of a major loop is followed (see Fig. 1.21). This branch will also be called the limiting ascending branch because usually there is no branch below it. The notation f~, will be used for the output value on this branch that corresponds to the input value u = oY. A first-order transition (reversal) curve is formed as the above monotonic increase of the input is followed by a subsequent monotonic decrease. The term "first-order" is used to emphasize the fact that each of these curves is formed after the first reversal of input. The notation f~'l~' will be used for the output value on the transition curve attached to the limiting ascending branch at the point f~,. This output value corresponds to the input value u = fl' (see Fig. 1.21). The above monotonic decrease of input modifies the previous a-fl diagram shown in Fig. 1.20. A new a-fl diagram for the instant of time when the input reaches the value fl' is illustrated by Fig. 1.22.

Now, we define the function

1 = (1.24)

This function is equal to one-half of the output increments along the first- order transition curves. The next step is to express this function in terms of the Preisach function lZ(a, fl). To this end, we compare the a-fl diagrams shown in Figs. 1.20 and 1.22. It is clear from these diagrams that the triangle T(a', fl') is added to the negative set S- and subtracted from the


s- / 0~ _ _ _ _ ~

~ T(~,~ ' )

/ ~ p'

FIGURE 1.22

positive set S + as a result of the monotonic input decrease from the value u = a ' to the value u =/~'. Using the above fact and the formula (1.8), we find that the Preisach model will match the output increments along the first-order transition curves if the function #(a,/~) satisfies the equation

r 1 6 2 -- - 2 H /~(~, j3) dc~ d]8.

d d T (~',~') (1.25)

By comparing the formulas (1.24) and (1.25), we obtain

(~',t~') (1.26)

The integral over the triangle T(~',/~') can be written as the following double integral:

F(c/, 13') = , #(G,/~) dc~ d/~. (1.27)

By differentiating the last expression twice (first with respect to ~' and then with respect to ~'), we find

# (ol',/J') = - 32F(c~" j~') . (1.28)

Invoking (1.24), the expression (1.28) can be written in another equivalent form:

1 32f~,t~, (1.29) #(c~' , /~')- 2 O~' 3fl'"

The formula (1.29) allows for a simple geometric interpretation of the function #(~', ~'). Indeed, for the first derivative off~,~, with respect to ]~'

1.3 IDEN I I[FICATION PROBLEM 23

we have

3f~,~, - tan0 (d , ~'), (1.30)

0/~'

where 0(c~',/~') is the angle be tween the axis u and the tangent to the first- order transit ion curve f~,~, at the point u = ~'.

From (1.29) and (1.30), we find

1 3 tan O(ol',/~') #(c~',/~') - ~ 3c~' " (1.31)

From (1.31), we conclude that the Preisach function /~(c~',/~') is positive if tan0(c~ ~, ]~') is a monotonical ly increasing function of c~' for any fixed/J'. The last condit ion is satisfied if all first-order transit ion curves are monotonical ly increasing functions of/~', and they do not intersect inside the major loop but merge together at the point where the descending and ascending branches of the major loop meet one another (see Fig. 1.23). 1 To secure the above merge of first-order transit ion curves, tan 0(c~',/~') should increase as a function of c~' for any fixed/~' in order to compensate for larger values off~,~,. It is wor thwhi le not ing that in the "magnet ic" definition of the Preisach model (see Section 1.1) the function #(c~,/J) is treated as a distr ibution function wi th some statistical connotation. For this reason, this function is tacitly a s sumed to be positive. In our treatment, the last p roper ty is directly related to experimental facts.

t

�9 U

F I G U R E 1.23

lit is useful to keep in mind that the descending branch of the major loop can be interpreted as a particular case of the first-order transition curve for which a t - ~0.


For positive #(a, fl), F(c~', fl') is a monotonically increasing function of c~' for any fixed fl' and a monotonically decreasing function of fl' for any fixed a'. This follows directly from (1.26).

If the Preisach function is determined from (1.29) (or (1.28)), then the expression (1.25) is satisfied. This means that the Preisach model matches the output increments along the first-order transition curves. We next show that this also implies that the first-order transition curves themselves are matched. To this end, consider the particular case when f l ' = rio. For this case, we have

f~,& = f - . (1.32)

From (1.25) and (1.32), we obtain: / ' f

f- - f ~, - - 2 l ] #(or, fl ) dot d fl . ddT (~',&)

(1.33)

Thus, the Preisach model matches the output increment, f - -f~,, along the limiting ascending branch.

For c~' = c~0, we have

f~0 = f+ . (1.34)

From (1.34) and (1.33), we find

f - - f + - - 2 f f w tz ( ol , fl ) d ol d fl , (1.35)

where T = T(c~0, rio) is the limiting triangle. Since major hysteresis loops are usually symmetric, the following

equality holds: f + = - f - . (1.36)

From (1.35) and (1.36), we derive:

f - - - f f w #(o~, fl) dot dfl. (1.37)

This means that the Preisach model matches the output value in the state of negative saturation. From the last fact and (1.33), we conclude that the model matches the limiting ascending branch. Since the limiting ascending branch and the output increments along first-order transition curves are matched, we deduce that the first-order transition curves themselves are matched.

The Preisach function,/z(c~, fl), has been found by using the first-order transition curves. These curves are attached to the limiting ascending branch, and each of them is formed when a monotonic increase along this branch is followed by a subsequent input decrease. For this reason,


these curves can be named first-order decreasing transition curves. How- ever, by almost literally repeating the previous reasoning, a similar expression for/x(c~,/~) can be found by using the first-order increasing transition curves. These curves are attached to the descending branch of the major loop. Each of these first-order increasing transition curves is formed as a monotonic decrease along the limiting descending branch is followed by a subsequent input increase. The notation f~,, will be used for the output value on the limiting descending branch. This value is achieved when the input is monotonically decreased from some value above c~0 to the value u =/~ ' . The corresponding c~-~ diagram is shown in Fig. 1.24. The notation f~,,~,, will be used for the output value on the first-order increasing transition curve that is attached to the limiting descending branch at the point f~,,. This output value corresponds to u = c~" (see Fig. 1.21). The corresponding ~-~ diagram is shown in Fig. 1.25.

By using the function

F(~",/~") - - f f tx(o~, ~) do~ d~, (1.38) J J T (~",~")

from Figs. 1.24 and 1.25 and the formula (1.8), we obtain

F(c~", ~") - ~(f~,,~,, -f~,,). (1.39)

From (1.38), as before we find

~(O/" /J ' ) -- aRE(co"/~ ' ) ' - 3c~"3~" " (1.40)

It is clear on the physical grounds (symmetry considerations) that first- order decreasing and increasing transition curves are congruent. In math-

y y

(x

/

| I

T

/

/

(~, ~")



ematical terms, this means that if

~'=-o/ and o/'=-~', (1.41)

then

f~,, = -f~, and f~,,~,, = -f~,~,. (1.42)

From (1.24), (1.39), and (1.42) we find that if (1.41) holds, then

F(d',/~") = F(d,/~'). (1.43)

By substituting (1.41) into (1.43), we obtain

F(-fl',-o/) = F(d , fl'). (1.44)

Next, substituting (1.43) and (1.41) into (1.40), we derive

a2F(cr ' ) #(- f l ' , - c~ ' ) = - Off'at,' " (1.45)

By comparing (1.45) and (1.28), we find

/z(-fl ' , -c~') = #(~' , fl'). (1.46)

The formulas (1.44) and (1.46) express the mirror symmetry of functions F(c~, fl) and/z(~ , fl) with respect to the line c~ = - f l (see Fig. 1.26). This symmetry is a consequence of the congruency of the first-order decreasing and increasing transition curves.

If the line c, = - f l is the interface between the positive and negative sets, S + (t) and S-(t), then, according to the above symmetry and (1.8), we find that the output is equal to zero:

fit) = 0. (1.47)

(z

/

'" ~ f f ) \ \

FIGURE 1.26

27

(x

/ X L

1

1.3 IDENTIFICATION PROBLEM


For this reason, the state corresponding to the above interface is called in magnetics "the demagnetized state." However, this state cannot be exactly achieved. This is because an interface L(t) is always a staircase line whose links are parallel to c~ and/~ axes. As a result, an actual staircase interface can only approximate the line c~ = - ~ . Such an approximation can be achieved, for instance, if the input u(t) is an oscillating function whose amplitude is slowly decreased to zero starting from some value above d0 (see Fig. 1.27). The corresponding staircase interface is shown in Fig. 1.28.

We next proceed to the formulation and the proof of the fundamental theorem which gives the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the Preisach model.

REPRESENTATION THEOREM The wiping-out property and the congruency property constitute the necessary and sufficient conditions for a hysteresis nonlinearity to be represented by the Preisach model on the set of piece-wise monotonic inputs.

PROOF. Necessity: Let a hysteresis transducer be representable by the Preisach model. Then, this transducer should have the same properties as the model. In particular, it should have the wiping-out and congruency properties.

Sufficiency: Consider a hysteresis transducer that has both the wiping- out property and the congruency property. It can be recalled that the wiping-out property means that the future values of output do not depend on all past extremum values of input but only on those that form an alternating series of dominant input extrema. The congruency property, on the other hand, suggests that all minor hysteresis loops formed


as a result of back-and-forth input variations between the same two consecutive input extrema are congruent. We intend to prove that the above properties imply that the hysteresis transducer can be represented by the Preisach model.

The proof of the last statement is constructive. First, it is assumed that the weight function, #(c~, ~), is found for the given transducer by matching its first-order transition curves. This can be accomplished by using the formula (1.28). This formula is equivalent to (1.25), which means that the integrals of #(c~,/7) over triangles T(~',/7') are equal to one-half of output increments, (1/2)Af = (1/2)(f~, -f~,~,), along the first-order transition curves. Next, it will be proved that if the above weight function is substi- tuted in (1.1), then the Preisach model and the given transducer will have the same input-output relationships. This statement is true for the first- order transition curves due to the very way the weight function,/~(c~,/7), is determined. The induction argument will be used to prove that the same statement holds for higher-order transition curves as well. Let us assume that the above statement is true for transition curves with number 1, 2 , . . . , k. Then, for the induction inference, we need to prove that this statement holds for a transition curve number k + 1.

Let a be a point at which the transition curve number k 4- 1 starts (see Fig. 1.29). The point, a, corresponds to some input value u = ~'. According to the induction assumption, the output values of the transducer and the Preisach model coincide at this point. Thus, it remains to be proved that the output increments along the transition curve number k 4- 1 are the same for the actual transducer and for the Preisach model. Consider an

f

U

FIGURE 1.29


/

/

(z

/


29

arbitrary input value u - fit < ~,. The output increment for the transducer will be equal to the increment o f f along some curve ab (see Fig. 1.29). Let Figs. 1.30 and 1.31 represent ~-fl diagrams for the Preisach model at the time instants when u - ~ t and u - ill, respectively. From these diagrams we find that the input decrease from ~' to fl' results in adding the triangle T(c~', fl') to the negative set, S-, and subtracting the same triangle from the positive set, S +. Using the above fact and the formula (1.8), we find that for the Preisach model the output increment along the transition curve number k + 1 is given by

-- 2 f f tt(oe, fl) do~ dfl. (1.48) af JJT (o,',~')

However, according to the way the function, tt(a, fl), is defined, the right- hand side of (1.48) is equal to the increment of the transducer output along the first-order transition curve cd (see Fig. 1.29). Thus, it remains to be shown that the output increments along the curves ab and cd are the same. It is here that the wiping-out and congruency properties will be used. The proof proceeds as follows. If starting from the point b we monotonically increased the input value from fl' back to a' , then, according to the wiping- out property, we would arrive at the same point a by moving along some curve ba (see Fig. 1.29). Indeed, the wiping-out property implies that as soon as the input exceeded the value al, the history associated with back- and-forth input variations between ~' and fit should be erased and the subsequent output variation should follow the transition curve number k. But, this would be possible if only for u - c~' we arrived back at the point a. Similarly, if starting from the point d we monotonically increased the input value from fl' to c~', then, according to the same wiping-out property,


we would arrive at the point c moving along some curve dc (see Fig. 1.29). Now, by invoking the congruency property, we conclude that the hysteresis loops bab and dcd are congruent. This is true because both loops result from back and forth input variations between the same two consecutive input extrema, a' and fl'. From the congruency of the above loops, we find that the output increments along the curves ab and cd are the same. Consequently, the output values of the transducer and the Preisach model coincide along the transition curve number k 4- 1. The last fact has been proved for any fl' ~> y' (see Fig. 1.29). However, according to the wiping- out property, the transition curve number k 4- 1 should coincide with the transition curve number k - 1 for fl' < y'. Thus, the case fl' < y' falls in the domain of the induction assumption.

In the above discussion, we have considered the transition curve number k + 1 corresponding to the monotonically decreasing input. However, the case when the transition curve number k 4- 1 is formed as a result of monotonically increasing input can be treated similarly. The only difference in the proof will be that the first-order increasing transition curves must be employed instead of the first-order decreasing transition curves used in the above reasoning.

This completes the proof of the theorem. 77

1.3.1 D i s c u s s i o n

It is easy to see that the essence of the given proof is in the reduction of higher-order transition curves to the first-order transition curves. This reduction rests on both the wiping-out property and congruency property.

The proven theorem is important because it clearly establishes the limits of applicability of the Preisach model. These limits are formulated in purely phenomenological terms, without any reference to the actual physical nature of hysteresis. This reveals the physical universality of the Preisach model. The theorem also explicitly indicates the factors which affect the accuracy of the Preisach model. Indeed, deviations from the conditions of the representation theorem may serve as the measure of model accuracy.

The above theorem allows for a simple explanation of so-called statistical instability of the Preisach model. This instability has been a very popular topic in the "magnetic" literature. It usually means that weight functions, #(a, fl), determined from different experimental data are not identical. Since these functions have been construed in magnetics as "particle distributions" (with some statistical connotation), the dependence of /z(c~, fl) on a particular experimental way of its determination has been


termed as "statistical instability." The origin of this "statistical instability" can be easily understood from the following discussion.

The formula (1.28) for/x(c~,/~) has been derived from the expression (1.26). This means that the weight function, #(~, ~), can be always determined if the integrals of/~(~,/~) over the triangles T(~', ~') are somehow experimentally found. It has been proposed before (see (1.25)) to find these integrals by matching the first-order transition curves. However, this is not the only way it can be done. Indeed, according to the formula (1.48), the same integrals can be found by matching the output increments along any particular transition curves. But, we will end up with the same values for these integrals if and only if higher-order transition curves and the first-order transition curves are congruent. According to the proof of the representation theorem, this congruency takes place if and only if the wiping-out and congruency properties are valid for hysteresis transducers. If these properties are not valid, then, by matching different transition curves, we will end up with different values for the integrals over the triangles T(~', ~') and, consequently, with different values of #(~, fi). This is exactly what had happened when the "statistical instability" was dis- covered (see [8]). Thus, from the phenomenological point of view, the origin of "statistical instability" comes from the fact that the wiping-out and congruency properties may not be exactly satisfied for actual hysteresis nonlinearities. Under these circumstances, the Preisach model cannot serve as an absolutely accurate representation for actual hysteresis nonlinearities, but it can still be used as an approximation. The quality of this approximation will depend on the extent to which the wiping-out and congruency properties are satisfied.

It has been mentioned before that the weight function,/x(~, r can be found by matching output increments along any transition curves. Thus, the question arises: "Which transition curves should be used for the above purpose?" If the wiping-out and congruency properties are valid, then it really does not matter which transition curves are used for the determination of #(~,/~). All of them will theoretically lead to the same result. However, from the practical point of view, the first-order transition curves have some clear advantages. First, it is easier to find these curves experimentally than higher-order transition curves. Second, measurements of these curves start from a well-defined state, namely, the state of negative (or positive) saturation. This is not the case for some experimental techniques described in the literature. For instance, it has been suggested in the literature to use a demagnetized state as a starting state in the experimental determination of #(~, ~). But, as it was pointed out before, the demagnetized state is not well defined for the Preisach model. This state depends on a particular way it has been prepared. As a result, some errors


and discrepancies may be introduced if the demagnetized state is used as the initial state for the experimental determination of tt(u, fl).

It is often asserted in the literature that the Preisach model describes reversal curves with zero initial slopes. This statement is based on the following reasoning. Let f~,/~, be a reversal curve which is traced for a monotonically decreasing input. This curve starts at the point f~, which corresponds to the local input maximum u = u'. The difference f~,/~, - f~, is then given by (1.25) and the current slope of this reversal curve at the point fl = fi' can be found from (1.30). By using (1.25), (1.26), (1.27), and (1.30), it is easy to derive the following expression for the initial slope

lim tan 0 (oe', fl') = 2 lim # (oe, fi') da. /~'--+~' /~'-+c~' ,

In the literature, the function it(u, fl) is usually construed as a distribution function for magnetic particles and, for this reason, it is tacitly assumed to be bounded. This assumption and the last formula result in zero initial slopes for the reversal curvesf~,/~,. This property is often regarded as an intrinsic property of the Preisach model. However, this is not true. Indeed, if the phenomenological approach is adopted, then there is no need to interpret tt as a distribution for magnetic particles and the assumption that tt is bounded can be removed. The freedom of choice of function #(~, ~) should be used in order to match as many experimental facts as possible. It is easy to show that experimentally observed nonzero initial slopes of reversal curves f~,#, can be matched by the Preisach model if we allow for delta-type (Dirac function) singularities of #(c~, ~) along the line c~ =/~. It is easy to see that these singularities can be interpreted as an additional (and fully reversible) term in the Preisach model consisting of degenerate (zero width) rectangular loop (or step) operators G,~"

; #(o~, f l ) G ~ u ( t ) du d~ + k ( ~ ) G ~ u ( t ) dot, O4)

where function it(u,/?) is now free of singularities. This form of the Preisach model leads to the following modification of

formulas (1.25)-(1.27):

"(L" ) L" f ~ , ~ , - f ~ , = - 2 f ~, #(or, fl ) d o~ d ,8 - 2 , k ( o~ ) dot.

From the last formula we derive:

tan 0 (c~',/~') -- 3f~'/~' = 2 L ~', . + 2k(e'),


ms

fo~13

(X G2

- m

w

FIGURE 1.32

and

lim tan 0 (d, ~') - 2k(0/').

Thus, the actual nonzero initial slopes of the first-order reversal curves can be matched by the appropriate choice of function k(0/).

This section will be concluded by the discussion of a simple example that illustrates the application of the formula (1.29). Consider a hysteresis nonlinearity with a local memory shown in Fig. 1.32. This hysteresis nonlinearity has a set of inner curves that are parallel to u-axis. The above hysteresis nonlinearity is completely characterized by the following parameters: ms,0 /1 ,0 /2 , w . Our goal is to find the Preisach representation (1.1) for this nonlinearity. To this end, we have to find the weight function #(0/, ~). This requires the specification of first-order transition curves f ~ and the subsequent application of the formula (1.29). It is clear from Fig. 1.32 that for any 0/such that 0/1 < 0/ < ~ the corresponding first-order transition curve f ~ consists of two parts: (a) a particular inner curve and (b) a part of the limiting descending branch if 0/1 - - W ~ ~ ~ 0/2 - - W, /J -4- W ~ 0/ < OK).

It is apparent that f ~ varies only along the second part.

34 C H A P T E R 1 The Classical Pre isach M o d e l of Hys te res i s

From the above remarks, we find

fo,~ = - m s 4- 2ms

0/2 -- 0/1 ~ ( / J -- 0/1 4- W), (1.49)

0/1- - W ~ ~ ~ 0/2 -- W , fl 4- w ~< 0/~< cx~, (1.50)

and

f ~ - const (1.51)

for all other possible values of ft. It is worthwhile to keep in mind that, depending on a particular value

of fl, the constant in (1.51) may assume one of the following two values: - m s , - m s 4- ((2ms)/(0/2 - 0/2))(0/- 0/1).

From (1.49), (1.50), and (1.51), we derive:

3fo~ 2ms - - , ( 1 . 5 2 )

O/J 0/2 -- 0/1

0 / 1 - - W ~ / J ~ 0 / 2 - - W , fl 4- w ~< 0/~< oo, (1.53)

and

= 0 (1.54)

for all other values of a and ft. The last three formulas can be writ ten in the following equivalent form:

3fo~ 2ms - , i f (0/ , f l ) ~ S2, ( 1 . 5 5 )

O/J 0/2 --0/1

and

Ofo,~ _ O, if (0/, fl) ~ s2, (1.56) 0~

where f2 is the region defined by the inequalities (1.53) (see also Fig. 1.33). From (1.55) and (1.56), we obtain:

O2fo~ 2ms

00/0/J 0/2 -- 0/1 ~ 8 ( 0 / - fl - w), (1.57)

O/1 ~ 0/ ~ 0/2, (1.58)

and (O2fot~)/(O0/Ofl) is equal to zero otherwise. From (1.29), (1.57), and (1.58), we find

ms #(0/, f l ) - ~ 8 ( 0 / - fl - w) (1.59)

0/2 -- 0/1


/

~3

/ /

35

FIGURE 1.33

for

or1 ~ ot ~ or2 (1.60)

and/~(c~,/~) is equal to zero otherwise. Thus, the weight function #(~,/~) has the support on the line c~ - /~ -

w = 0. This support is the bold segment (see Fig. 1.33) of the above line. By substituting (1.59) and (1.60) into (1.1), after simple transforma-

tions we obtain

m s f~ ~ = G,~-wu(t)dot. (1.61) fit) Or2 - - Ot l 1

This is the Preisach representation of the hysteresis nonlinearity shown in Fig. 1.32. It is left to the reader as a useful exercise to verify that the formula (1.61) indeed describes all branches of the above hysteresis nonlinearity.

It is clear from Fig. 1.33 that by varying Otl,Ot2, and w we can freely move around the support of the function tz in the half-plane ~ ~>/~. This fact and the expression (1.61) suggest that the equivalent representation for the Preisach model can be obtained as a superposition (parallel connection) of infinite number of hysteresis nonlinearities shown in Fig. 1.32. However, the representation (1.1) is preferable due to the most elementary nature of hysteresis operators }9~.

It is not accidental that in the above example the function #(c~, ]3) has a "line support." This is related to the fact that the above hysteresis nonlinearity has a local memory. It can even be proved in general that if we have


- U U

FIGURE 1.34

a hysteresis nonlinearity with a local memory that can be represented by the Preisach model, then the weight function/~(~, fl) has a support along some particular curve. This curve may intersect all possible staircase interfaces L ( t ) only once.

It was emphasized before that the Preisach model can describe (and usually does describe) hysteresis with nonlocal memories. These nonlinearities are much more complex than those with local memories. However, it would be a mistake to think that any hysteresis nonlinearity with local memory can be represented by the Preisach model. In fact, there are very simple hysteresis nonlinearities with local memories that cannot be described by the model. For instance, this is true for the hysteresis nonlinearity shown in Fig. 1.34. Indeed, let the input vary back and forth between two consecutive extremum values ~' and fl',-urn < fl ' < or' < Um.

Then, depending on the past history, the bold reversible parts of descending and ascending branches will be traced back and forth. These parts can be construed as degenerate minor loops. Since these parts are not congruent, the congruency property of the representation theorem is not satisfied. Consequently, the hysteresis nonlinearity shown in Fig. 1.34 cannot be represented by the Preisach model. However, this hysteresis nonlinearity is typical for single Stoner-Wohlfarth magnetic particles. It is known from the theory of these particles that the reversible parts of their hysteresis loops result from uniform rotations of magnetization within these particles, that is from reversible processes. Incongruence of the above parts is the reason why the Preisach model cannot describe the hysteresis loops of

1.4 NUMERICAL IMPLEMENTATION 37

single Stoner-Wohlfarth particles. For this reason, it is often asserted that the Preisach model does not fully account for reversible processes. It will be shown in the next chapter that this deficiency of the Preisach model can be removed by its appropriate generalization.

1.4 NUMERICAL IMPLEMENTATION OF THE PREISACH MODEL

The Preisach model can be numerically implemented by using the formula (1.8) for the computation of the output, fit), and the formula (1.29) for the determination of the weight function, #(ot,/~). Although the above approach is straightforward, it encounters two main difficulties. First, it requires the numerical evaluation of double integrals in (1.8). This is a time consuming procedure that may impede the use of the Preisach model in practical applications. Second, the determination of the weight function #(ot,~) by employing the formula (1.29) requires differentiations of experimentally obtained data. These differentiations may strongly amplify errors (noise) inherently present in any experimental data. It turns out that another approach can be developed for the numerical implementation of the Preisach model. This approach completely circumvents the above two difficulties. It is based on the explicit formula for the integrals in (1.8). This formula directly involves (without any differentiation) the experimental data used for the identification of/,(ot,/~). Moreover, the above formula will be a valuable tool for the theoretical investigation of the Preisach model, and it will be extensively used later in the discussion of vector models of hysteresis (see Chapter 3).

The starting point for the derivation of the explicit formula for fit) is the expression (1.8). It is worthwhile to remind here that the positive (S + (t)) and negative (S-(t)) sets in (1.8) are separated by the staircase interface L(t). This interface have vertices whose ot and/~ coordinates are equal to Mk and mk, respectively (see Fig. 1.35). As discussed in Section 1.2, numbers Mk and mk form the alternating series of dominant input extrema.

By adding and subtracting the integral of #(ot,/3) over S+(t), the expression (1.8) can be represented in the form:

- - [ [ #(ot, ~)dot d~ + 2 [ [ #(ot, ~)dot d~, (1.62) f(t) ,liT adS +(t)

where, as before, T is the limiting triangle. According to (1.26), we find:

f f T #(ot, fl ) dot d fl - F(oto, ]3o). (1.63)


(X

, (M k, mk_l)

s+i ,---- ~ . . . .

\ -13

i', /

FIGURE 1.35

The positive set, S+(t), can be subdivided into n trapezoids Qk (see Fig. 1.35). As a result, we have

n(t) ffs #('*'fl)d'*dfl= ffo #(oe, fi)doedfl. (1.64) +(t) k=l k(t)

It is clear that the number, n, of these trapezoids and their shapes may change with time. For this reason, n and Qk are shown in (1.64) as functions of time.

Each trapezoid Qk can be represented as a difference of two triangles, T(Mk, mk-1) and T(Mk, mk). Thus, we obtain

f f co # ( " ) d - f f r # ( ol , fl ) doe d fl k(t) (Mk,mk-1)

- f f #(oe, fl) doe dfl, ( 1.65) JJT (Mk,mk)

where, for the case of k = 1, m0 in (1.65) is naturally equal to rio. According to (1.26), we find:

F(Mk, ink-l) , (1.66) (Mk,mk-1)

and

fT a(~, t~) e~ e~ F(Mk, mk). (Mk,mk)

(1.67)


From (1.65), (1.66), and (1.67), we derive:

/Q /x(~, ~) do~ d~ - F(Mk, mk-1) -- F(Mk, mk). (1.68) k(t)

From (1.62), (1.63), (1.64), and (1.68), we obtain

n(t)

f (t) = -F(c~o, ~o) + 2 ~ [ F ( M k , mk-1) -- F(Mk, mk)]. (1.69) k=l

It is clear from Fig. 1.35 that mn is equal to the current value of input

mn = u(t). (1.70)

Consequently, the expression (1.69) can be written as

n(t)-I

f (t) - -F(~o, ~o) + 2 ~ [F(Mk, mk-1) - F(Mk, mk)] k=l

+ 2[F(Mn, m n - 1 ) - F(Mn, u(t))]. (1.71)

The last expression has been derived for monotonically decreasing input, that is, when the final link of interface L(t) is a vertical one. If the input u(t) is being monotonically increased, then the final link of L(t) is a horizontal one and the c~-/~ diagram shown in Fig. 1.35 should be slightly modified. The appropriate diagram is shown in Fig. 1.36. This diagram can be considered as a particular case of the previous one. This case is realized when

mn(t) = Mn(t) = u(t). (1.72)

(x

I I /

FIGURE 1.36


According to the definition (1.26) of F(~,/~), we find

F(Mn, mn) = F(u(t), u(t)) = 0. (1.73)

From (1.69), (1.72), and (1.73), we derive the following expression for f(t) in the case of monotonically increasing input:

n( t ) - I

f(t) -- -F(olo, ~o)§ ~ , [F(Mk, mk_l)-F(Mk, mk)]§ (1.74) k=l

The function F(~,/~) is related to experimentally measured first-order transition curves by the formula (1.24). Using this formula, expressions (1.71) and (1.74) can be written in terms of the above experimental data as follows:

n-1

f (t) - mf q- § E (fMkm k _ fMkmk_l ) § fMnu(t) --&nmn-1, (1.75) k=l

n -1

f i t ) - -f+ + ~(fMkmk--fMkmk-1) § (1.76) k=l

Here, f+ is, as before, the positive saturation value of output, and the last term in (1.74) has been transformed by using the formulas (1.24) and (1.44). Thus, we have derived the explicit expressions (1.75) and (1.76) for f(t) in terms of experimentally measured data. These expressions constitute the basis for the numerical implementation of the Preisach model.

Using the above expressions, the digital code that implements the Preisach model has been developed. This code determines the output of a hysteresis transducer from an input, a set of first-order transition curves and an input history that are all specified by the user. The code also allows the user the flexibility of specifying an analytical expression for the weight function, #(c~,/~), instead of first-order transition curves. In this case, the function F(ol,/~) defined by (1.26) is precomputed and then used in the formulas (1.71) and (1.74) for the evaluation of output values.

As usual, the code performs three main functions: preprocessing, processing, and postprocessing. During the preprocessing stage, a square mesh covering the limiting triangle T (see Fig. 1.37) is created and a dis- cretized set of first-order transition curves is entered. This set consists of mesh values of the functionf~/~. In the case when an analytical expression for #(~,/~) is specified, the mesh values of F(~, fl) are computed during the preprocessing. At the processing stage, an input history and current values of input are entered. Using these data and the formulae (1.19) and (1.20), the alternating series of dominant input extrema {Mk, mk} is first


1 /

-;-~-~-~-~-~4-;-44-;-~- ~-~-444-~ / c~" + + + + ~ + + + + + + + ~ + + + y - ~ - ~ - ~ ~ . ~ - ? - ~ - ~ . ~

-§ § § **-,-§ § § -,-,-,-,-,-§ ~-§

." ? ; . FT?,-?. ; ~ 7 ' - T - " T T

~- + + - r + + 4 ~

~-,-,-§247

FIGURE 1.37

determined and then continuously updated for each new instant of time. Using Mk, mk, and mesh values of f ~ (or F(~fl)), all terms in the formulae (1.75) and (1.76) (or (1.71) and (1.74)) are computed. This is done by first determining particular square (or triangular) cells to which points (Mk, mk), (Mk, mk-1), and (Mn, u(t)) belong, and then by computing the values o f f~ (or F(c~, fl)) at these points by means of interpolation of the mesh values o f f ~ (or F(a, fl)) at the vertices of the above cells. In the code, the interpolation is based on the following local approximation to f ~ in each square mesh cell:

fot fl "" Co 4- Clot if- C 2 fl 4- C 3 ot fl . (1.77)

The C-coefficients in (1.77) are found by matching the values of f ~ in the cell vertices. For triangular cells that are adjacent to the line c~ - fl, the linear local approximations to f ~ are used"

furl ~ Co q- Clot -Jr- C 2 fl . (1.78)

After computing all necessary values of f ~ , the current values of output are evaluated by employing the formulae (1.75) and (1.76). At the postprocessing stage, the computed output values are stored in a special file along with the corresponding input values and the values of Mk and mk. The user has the option to plot the output and input as functions of time, the path traced on the f -u plane and the set of first-order transition curves used in computations.

Below, we present some examples computed by using the above code. Figs. 1.38, 1.39, and 1.40 show the set of first-order transition curves, the initial staircase interface L(t), and the input u(t), respectively. These data


. 04

. 02

0

g

- . 02

- . 04

!

- 2500

L

- 2 0 0 0 - 1 5 0 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 I n p u t

FIGURE 1.38

2000

1000

- I 0 0 0

-2000

m

m

w-- i

I-

, i , , , , i , , , ~ l , , , , j , , , , ( , , ....

m

1

m

, I I J I I ! I ~ , I I i i ~ , 1 , , , t , l j i

- 2 0 0 0 - I 0 0 0 0 I 0 0 0 2 0 0 0

FIGURE 1.39

1 . 4 N U M E R I C A L I M P L E M E N T A T I O N 4 3

i l l | I ] l | | I l ' w l l | i | l l | i I | I | I _

2 0 0 0

1 0 0 0

0

- 1 0 0 0 _

- 2 0 0 0

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 T i m e

F I G U R E 1 . 4 0

I , , ' ' 1 ' ' ' ' i ' ' ' ' ! ' ' ' ' I ' ' ' ' I '

. 0 4

. 0 2

~ 0

- . 0 2

- . 0 4

I 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0

T i m e

F I G U R E 1 . 4 1

44

O

.04

.02

- . 0 2

- . 0 4

I

n

B


' 1 ' ' ' ' I ' ' ' ' I " " ' ' ! ' ' ' ' 1 ' '

, , 1 , , I , J I , , , I , , , , I i , i , l , ,

- 2 0 0 0 - 1 0 0 0 0 1000 2 0 0 0 I n p u t

F I G U R E 1 . 4 2

have been used to compute the outputf(t). The computed outputf(t) and the f - u path of the transducer are given in Figs. 1.41 and 1.42, respectively.

As was pointed out before, the derived formulas (1.75) and (1.76) (or (1.71) and (1.74)) are useful not only for the numerical implementation of the Preisach model but for its theoretical investigation as well. For instance, by using (1.75) and (1.76), the following proposition can easily be proven.

PROPOSITION Any f - u path of hysteresis transducer representable by the Preisach model is piecewise congruent to first-order transition curves. In other words, any f - u path of such a transducer consists of pieces, which are congruent to some particular first-order transition curves.

PROOF. Suppose that the input u(t) is being monotonically decreased from its previous maximum value u = Mn until it reaches the value u -- mn-1. Then, the formula (1.75) is valid. In this formula, all terms except fMmu(t) do not vary. Consequently, the f - u path traced during the above input variation is congruent to the first-order transition curve fMnu that is attached to the limiting ascending branch at the point fMn- As the input reaches the value mn-1, the cancellation of the last two terms in (1.75) oc-


curs. If the input is being further decreased remaining between u -- mn-1 and u = mn-2, then the formula (1.75) is modified as follows

n - 2

f ( t) - - f + + ~_, (fMkmk -- fMkmk_ 1 ) -Jr- &n-lU(t) -- fMn-lmn-2" k-1

(1.79)

From this formula, as before, we conclude that the f - u path traced during the above input variation is congruent to the first-order transition c u r v e fMn_lU that is attached to the limiting ascending branch at the point f =fMn_l. By continuing the same line of reasoning, we find that if the input is being monotonically decreased between u - mk and u -- mk-1, then the correspondingf-u path is congruent to the first-order transition curve fMku. For the case when the input is monotonically increased, the formula (1.76) is valid. From this formula, we find that if the input is monotonically increased between the values u = Mk and u = Mk-1, then the time varying part of the output is described by the t e r m f_mk_l,_u(t). Thus, the correspondingf-u path is congruent to the first-order transition curvef-mk_l,-U that is attached to the limiting ascending branch at the point f =f-mk-l" This completes the proof. [3

It is evident from the proven theorem that initial slopes of reversal curves described by the Preisach model coincide with initial slopes of experimentally measured first-order transition curves and, consequently, in general they are not equal to zero. This is consistent with our discussion of initial slopes of reversal curves presented in the previous section.

Another important application of the formula (1.75) and (1.76) is the derivation of the expressions for linear extrapolations of output corresponding to small input increments. Such extrapolative expressions can be useful for numerical analysis of systems with hysteresis by time-marching techniques. The derivation of these expressions proceeds as follows. Con- sider the case when the input has been monotonically increased for t f ~< t and this monotonic increase continues just after the time t. Then, the formula (1.76) is valid for the output before and after the time t. From this formula, by using the local Taylor expansion at time t for t' > t, we find

f (t') - f i t ) "" - ~ Of --mn-l~

fi=-u(t) [u(t') - u(t)]. (1.80)

It is convenient to represent the last formula in terms of input and output increments:

3f -mn_l~ Af = - ~ Au. (1.81)

Off fl=-u(t)


It is important to keep in mind that mn-1 in (1.81) may change with time. This occurs because of the wiping out property. It is easy to see that, at any instant of time, mn-1 in (1.81) is the largest mk in the series of Mk and mk.

If a monotonic increase of input for t' K t is followed by a monotonic input decrease for t' > t, then the input value u(t) can be treated as the last maximum, Mn, and from (1.75), we find:

n-1

f i t ) - -f+ + y] (&kmk -- &kmk+l ) "4- fu(t)u(t) -- fu(t)mn-1 k=l

(1.82)

and

n-1

f(t') = - f + + E ( f M k m k -- fMkmk_l) + fu(t)u(t') -- fu(t)mn-l" k=l

(1.83)

From (1.82) and (1.83), we obtain

f (t') - fit)=fu(t)u(t') -- fu(t)u(t). (1.84)

Using the last expression, we derive

A f ~" Ofu(t)fl fl=u(t)

Au. (1.85)

Now, we consider the case when the input has been undergoing a monotonic decrease for t I ~ t. If this decrease continues for t t > t, then the formula (1.75) is valid. From this formula as before we find

A f "" Of Mnfi fi=u(t)

Au. (1.86)

In the last formula, Mn may change with time. It is clear that at any instant of time Mn in (1.81) is the smallest Mk in the alternating series of Mk and mk.

If the monotonic decrease of input for t' K t is followed by a subsequent monotonic input increase for t t > t, then the input value u(t) can be considered as ran-1 and the formula (1.76) can be used for t' > t. From this formula, as before we derive

Of -u(t)fi

/?=u(t) Au. (1.87)

Thus, for the previously increasing input, the output can be linearly extrapolated by using the formulas (1.81) and (1.85) for Au > 0 and Au < 0, respectively. For the previously decreasing input, the output can be linearly extrapolated by using the formulae (1.86) and (1.87) for Au < 0 and


Au > 0, respectively. It is important to note that the above formulas are given in terms of slopes of experimentally measured first-order transition curves.

The formulae (1.81), (1.85), (1.86), and (1.87) give local linearizations of output increments at time t. In other words, these linearizations are fairly accurate only for small time difference t - t f. However, by employing the formulae (1.75) and (1.76), it is possible to find global linearizations of output increments which are valid for finite (not necessary small) time intervals. The simplest way to do this is to use a special set of continuous inputs u(t). Consider an arbitrary time interval [t0,t']. We define the set K[to,t,] of inputs that are continuous on this interval and have the properties: (a) any two inputs from the set assume their m a x i m u m and min imum values at the same time instances, (b) maxima and minima that constitute alternating series of dominant extrema are also assumed at the same times. For two inputs u(t) and ~(t), the last condition can be expressed as

u(t-~) = Mk, ~(t~-) --/~Ik, (1.88)

u(t-~) -- mk, ~(t~-) = mk, (1.89)

where the same notations as in Section 1.2 are used above. It is obvious that K[to,t, ] is a convex set of functions. Indeed, if the

above two properties are satisfied for inputs u(t) and ~(t), they are also satisfied for the input Xu(t) + (1 - X)~(t) where 0 ~ ~ K 1. It is also clear that if u(t) belongs to K[to,t'], then Xu(t) also belongs to K[to,t,] for any X ~ 0 and it does not belong to K[to,t'] for ~ < 0. This means that the set K[to,t,] is a cone in the Banach space of continuous functions. It is clear that the set of inputs

+ and t~- By specifying dif- K[to,t'] depends on the alternating sequence of t k ferent times t+k and t~-, we end up with different sets of inputs. For this reason, it would be more precise to adopt the notation K[to,t-~,tl ..... t+,tn,t,]. But for the sake of conciseness, we preserve the previous notation.

From the above definition of K[to,t,], it follows that any two inputs from K[to,t,] are monotonically decreased (or increased) on the same time intervals. This means that at any instant of time the same formula (1.75) (or (1.76)) can be applied to the computat ions of outputs. Thus, assuming that at time t both inputs, u(t) and ~(t), are monotonically decreased, we can use (1.75) for u(t) and the similar expression for ~(t):

n-1

f(t) = - f+ + ~ (fMk~k - &k~lk-1 ) "q- &nU(t) - & n m n " k=l

(1.90)


By subtracting (1.75) from (1.90) and retaining only linear terms of Taylor expansions, we derive

n-l( ~f~ ) gr i t ) - ~ Of~ 8Mk Jr-. 3mk -sF k=l Mk,mk Mk,mk

+ (1.91) q---~ff Mnu(t)r~Mn ---~- Mnu(t) where the following notations are used:

grit) =d~(t) - fit), 8u(t) = ~(t) - u(t), (1.92)

8Mk -- Mk -- Mk, 8mk = Fnk -- mk. (1.93)

If both inputs are monotonically increased at time t, then the formula (1.76) is applied and, as before, we derive

n-1 ~ ~Mk +. 8mk

k=l Mkmk --~ Mkmk

Of~ IlMkmk 13Mk aft3

( . G

+-SF

--mn-1 3ot _mn_l,_u(t)) 3ran-1

8u(t). (1.94) --mn-1,--u(t)

The expressions (1.91) and (1.94) represent a global linearization of output increment on the interval [t0, t']. These expressions may be useful in the analysis of periodic regimes in systems with hysteresis by perturbation techniques.

We conclude this section by discussing another example that demonstrates the usefulness of the above explicit formulas for the output fit). Consider an input u(t) that has an infinite number of oscillations as time progresses to infinity. We assume that the amplitude of these oscillations (around some value u(~) ) monotonically diminishes to zero. In other terms, this means that

lim Mk = lim mk -- u(o~) = ~. (1.95) k---> ~ k--+oo

1.5 THE PREISACH MODEL AND HYSTERETIC ENERGY LOSSES 49

The described input variations are typical for an hysteretic magnetization processes.

We are going to prove that there exists the limit for the output f i t) as time goes to infinity.

By using (1.75), we find

o o

lim f(t) = -F(ol0, rio) 4- 2 ~[F(Mk , mk-1) -- F(Mk, mk)]. (1.96) t--+ oo"

k = l

Thus, it remains to be proven that the infinite series in (1.96) converges. To this end, we introduce the following notations:

F(Mk, ink-l) -- a2k , F(Mk, mk) - - a 2 k + l (1.97)

and rewrite (1.96) as follows:

oO

lim f(t) = -F(c~0, rio) 4- 2 ~_,(--1)kak . (1.98) t--, e~ ~

k = 2

Since usually #(c~, fl) >~ 0, from (1.26) and (1.97) we find that ak > 0. More- over, since

mk-1 < mk and Mk > Mk+l, (1.99)

we conclude that

a2k = F(Mk, ink-l) > F(Mk, mk) = a 2 k + l , (1.100)

a 2 k + l = F(Mk, mk)> F(Mk+l, mk)-- a 2 k + 2 . (1.101)

In addition, from (1.95) and (1.26) we have

lira a2k = lira a2k+ 1 = F(~, ~) = 0. k ~ k ~

Thus, the expression (1.98) is the alternating series with monotonically decreasing to zero coefficients. According to the well-known theorem, this series converges. This proves that there exists the limit off(t) for t --~ cx~.

1.5 THE P R E I S A C H M O D E L A N D HYSTERETIC E N E R G Y LOSSES

A hysteresis phenomenon is associated with some energy dissipation which is often referred to as hysteretic energy losses. The problem of determining hysteretic energy losses is a classical one. It has been attracting considerable attention because the hysteresis energy loss is an important component of "core losses" occurring in almost all electromagnetic power


U

FIGURE 1.43

devices as well as in many high frequency microwave devices. For this reason, the means for accurate predictions of hysteresis losses and their reduction are important for optimal design of various equipment. The solution to the above problem has long been known for the particular case of periodic (cyclic) input variations. In magnetics, this solution is most often associated with the name of C.P. Steinmetz. This solution shows that a hysteretic energy loss per cycle is equal to an area enclosed by a loop resulting from periodic input variations (see Fig. 1.43). However, energy dissipation occurs for arbitrary (not necessary periodic) variations of input. The problem of computing hysteretic energy losses for arbitrary input variations has remained unsolved. A solution to this problem would be of both theoretical and practical importance. From the theoretical point of view, the solution to the above problem will allow for the calculation of internal entropy production that is a key point in the development of irreversible thermodynamics of hysteretic media. In magnetics, it may also allow one to separate a dissipated energy from an energy stored in magnetic field. This eventually may lead to expressions for electromagnetic forces in hysteretic media. From the practical viewpoint, the solution to the above problem may bring new experimental techniques for the measurement of hysteretic energy losses occurring for arbitrary input variations.

It should not be surprising that the expression for hysteretic energy losses has been found only for the case of periodic input variations. The reason behind this fact is that the hysteretic energy losses occurring for periodic input variations can be easily evaluated by using only the energy conservation principle; no knowledge of actual mechanisms of hysteresis


or its model is required. The situation is much more complicated when arbitrary input variations are considered. Here, the energy conservation principle alone is not sufficient, and an adequate model of hysteresis should be employed in order to arrive at the solution to the problem. It turns out that the Preisach hysteresis model is very well suited for this purpose.

In this section, the Preisach model will be used for the derivation of general expressions for hysteretic energy losses. These expressions will be given in terms of the weight function, #(a, j3), as well as in terms of experimentally measured first-order transition curves. Furthermore, a formula that relates the hysteretic energy losses occurring for arbitrary input variations to the losses occurring for certain periodic input variations will be found. This formula may result in simple techniques for the measurement of hysteretic losses occurring for arbitrary input variations. The application of the mentioned results to the irreversible thermodynamics of hysteretic media will be discussed as well.

We begin by defining the input, u(t), and the output, fit), as work variables. This means that the infinitesimal energy supplied to the transducer (media) in the form of work is given by

~ W = u d f . (1.102)

In magnetics, u is the magnetic field H, f is the magnetization M, and the formula (1.102) becomes the classical expression for the work done in magnetizing a unit volume of magnetic media:

8 W = H d M . (1.103)

Similarly, in mechanics, u is the force F, f is the specific length L, and from (1.102) we find the standard formula

8 W = F d L . (1.104)

Now, we proceed to the derivation of expressions for hysteretic energy losses. We first consider the case when a hysteresis nonlinearity is represented by a rectangular loop shown in Fig. 1.44. If a periodic variation of input is such that the whole loop is traced, then the hysteretic energy loss for one cycle, Qcycle, is equal to the area enclosed by the loop

Qcycle = 2(o~ -/J) . (1.105)

It is clear that the horizontal links of the loop are fully reversible and, for this reason, no energy losses occur as these links are traced. Thus, it can be concluded that only "switching-up" and "switching-down" result in energy losses. It can be assumed (on the physical grounds) that there is symmetry between the above two switchings. In other words,


+1

~H

-1

~U

FIGURE 1.44

these switchings are identical as far as energy losses are concerned. Con- sequently, the same energy loss occurs for each of these switchings. As a result, we conclude that the energy loss per switching, q, is given by

q = ~ - ft. (1.106)

The product/z(a, fl)}9~ can be construed as a rectangular hysteresis loop with output values equal to itz(~, fl). For this reason, switchings of such loops will result in energy losses equal to #(a, fl)(c~ - fl). In the Preisach model, any input variation is associated with switchings of some rectangular loops #(c~, fl)}9r These switchings represent irreversible processes occurring during input variations. Consequently, it is natural to equate the hysteretic energy loss occurring for some input variation to the sum of energy losses resulting from the switching of rectangular loops during this input variation. Since in the Preisach model we are dealing with continuous ensembles of rectangular loops, the above summation should be replaced by integration. Thus, if f2 denotes the region of points on c~-fl diagram for which rectangular loops were switched during some input variation, then the hysteretic energy loss, Q, for this input variation is given by

Q = I I lz(o~, fl)(o~ - fl)dotdfl. (1.107) M

JJn


This is the fundamental formula for hysteretic energy losses, and all subsequent result will follow from this expression. It is clear from the above reasoning that the derivation of the formula (1.107) rests on the following two facts: (a) for rectangular hysteresis loops hysteretic losses can be evaluated for arbitrary input variations, (b) the Preisach model represents complicated hysteresis nonlinearities as superpositions of rectangular loops. The above two facts make the Preisach model a very convenient tool for the solution of the problem at hand. However, it has to be kept in mind that the formula (1.107) cannot be applied to any hysteresis nonlinearity. It has certain limits of applicability that are the same as for the Preisach model itself.

A typical shape of the region f~ is shown in Fig. 1.45. It is clear from this figure that f2 can be always subdivided into a triangle and some trapezoids. The trapezoids, in turn, can be represented as differences of triangles. Thus, if the integral in (1.107) can be evaluated for any triangular region, then it will be easy to determine this integral for any possible shape of f2. For this reason, it makes sense to compute the values of the above integrals over various triangles. By using these values, hysteresis losses can be easily found for any input variations. In the case when f~ is a triangle, the integral (1.107) can be evaluated in terms of first-order transition curves. The derivation proceeds as follows.

Consider the function (~ -/~)F(~,/~). By differentiating this function, we find

0 2 0Ola~ [(Ol --/~)F(cG ~)] = 0F(01,0/~ ~) -- 0F(cg,0ol/~) if- (Ol --/~) 02F(~ (1.108)

I

i

/ i

/ [J" /

-L] J1 I////



By using (1.28) and (1.108), we derive:

OF(a, fl) OF(a, fl) 02 ~(~'P)(~- P) = ap - a~ a~ap [ ( a - f l ) F ( o ~ , f l ) ] . (1.109)

Let T(u+,u_) be a triangle (see Fig. 1.46) swept during the monotonic increase of input from u_ to u+. According to (1.107), the above input variation results in the hysteretic loss Q(u_, u+) that is given by

[ [ #(a, fl)(ot - fl) da dfl. (1.110) Q(u_,u+) JJT (u+,u_)

Substituting the expression (1.109) into (1.110), we find:

Q(u_, u+) - ~IT 0F(a, fl) do~ dfl (u+,u_) Off

fr aF(~,fl) - (u+,u_) ~ do~ d fl

(u+,u_) a~a# [(~ - #)F(~, #)] d~ alp.

The first integral in (1.111) can be evaluated as follows:

ffr af(~,p) (u+,u_) Og d~ d~ = fuU+(fu ~ _ _ aF(~' ~) d~) a~a-----F-

1 u+ in+ = f(o~,a,)dol - f(ol, u_)d~,

f u+ -- - f(a, ,u_) do~,

(1.111)

(1.112)

since F(a,c~) = 0. Similarly, for the second integral in (1.111) we obtain

aF(~, ~) d~ dp = F(u+, p) dp. (u+,u_ ) Ool

Finally, for the third integral in (1.111) we derive

02 (u+,u_) a~at~ [(~ -/~)F(~, ~)] d~ d/~

~u+(~flu+ 02 ) - _ a o , a ~ [(~ - ~)f(~ ,~)] d~ d~

fu u+ 3 = _ ~-fi[(u+ - fl)F(u+,fl)]dfl - - (u+ - u_)F(u+,u_). (1.114)

(1.113)


Substituting (1.112), (1.113), and (1.114) into (1.111), we find

Q(u_,u+) -- (u+ - u_)F(u+,u_)

- F(u+, ~) d~ - F(~, u_) d~. (1.115)

The last formula has two main advantages over (1.110). First, its application requires the evaluation of one-dimensional integrals. Second, this formula expresses the losses directly in terms of experimentally measured first-order transition curves.

By using the expression (1.115), the hysteretic energy losses can be evaluated for arbitrary input variations. Consider some input variation for which the region fl has the shape shown in Fig. 1.45. Then, the corresponding energy loss is given by (1.107) that can be written as follows:

k

Q = Q(Mn, mn) + E [ Q ( M n _ i , mn-i) - Q(Mn-i , mn-i-1)], (1.116) i=1

where it is assumed that mn-k-1 -- u(t) (1.117)

and each term in (1.116) can be evaluated by employing the formula (1.115).

Next, we discuss some interesting qualitative properties of energy losses occurring in hysteresis transducers described by the Preisach model. Consider a cyclic variation of input between two consecutive extremum values u_ and u+. During the monotonic increase of input from u_ to u+, the final horizontal link of L(t) sweeps the triangle T(u+,u_) (see Fig. 1.46). Consequently, the losses occurring during this monotonic increase are given by (1.110). On the other hand, during the monotonic decrease of input from u+ to u_, the final vertical link of L(t) sweeps the same triangle, and the corresponding losses Q(u+, u_) will be given by the same integral in (1.110). Thus,

Q(u_,u+) -- Q(u+,u_) (1.118)

and we obtain the following result.

For any loop, the hysteretic losses occurring along ascending and descending branches are the same.

The above result can be used to find the formula which relates a hysteretic loss occurring for arbitrary input variations to certain cyclic hysteretic losses. Suppose that the input u(t) is monotonically increased from some minimum value u_ and it reaches successively the values Ul


f

u u

I

/


and U2 with u2 > Ul (see Fig. 1.47). We are concerned with the hysteretic loss Q(Ul, u2) during the monotonic input increase between Ul and u2. For this input increase we have (see Fig. 1.48)

= T(ua, u _ ) - T(Ul, U_). (1.119)

Consequently, Q(Ul, U2) -- Q(u_,u2) - Q(u_ ,u l ) . (1.120)

Using the above result (see formula (1.118)), losses Q(u_, U2) and Q(u_, Ul) can be expressed in terms of cyclic losses

1-- 1 Q(u_,ul) -- -2 Q(u_,ul), Q(u_,u2) -- ~Q(u_, u2), (1.121)

where Q(u_,ul) is the hysteretic loss per cycle when the input is peri- odically varied between u_ and Ul; the notation Q(u_,u2) has a similar meaning.


1 Q(Ul, U2) -- ~ [ Q ( u - , u 2 ) - Q(u-,ul)] . (1.122)

The last formula expresses the loss occurring during the monotonic increase of input. By literally repeating the same line of reasoning a similar formula can be derived for the case of monotonic input decrease between Ul and u2 (Ul > u2)"

1 Q(Ul, U 2 ) - ~ [Q(u+, u2) - Q(u+, Ul)], (1.123)


where u+ is the last input maximum (see Fig. 1.48). The formulas (1.122) and (1.123) may be useful from the practical

point of view, because it is much easier to measure cyclic losses than those occurring for nonperiodic input variations.

It is instructive to show that the derived expressions for hysteretic energy losses are consistent with the classical result: a hysteretic energy loss occurring for a cyclic input variation is equal to the area enclosed by the loop resulting from this cyclic input variation.

Consider a cyclic input variation between u_ and u+. According to (1.118) and (1.115), we find that the hysteretic loss per cycle for the above input variation is given by

Q(u_,u+) -- 2Q(u_,u+)

[ fu -- 2 (u+ - u_)F(u+,u_)- F(u+,fl)dfl

fu u+ ] - F(ol, u_)do~ . (1.124)

On the other hand, the area enclosed by the corresponding loop is given by

W= $ udf. (1.125) d u _It+U_

Since

~u udf + ~u f dU= ~u d(uf)-O, (1.126) _It+U_ _It+U_ _U+bl_

we find

w - - fu f d u = - / u f dU- /u f du. (1.127) _It+u_ _U+ +It_

To evaluate the last two integrals, we shall use the formulae (1.71) and (1.74). In the case of monotonic input increase, the formula (1.74) is appropriate. This formula can be written in the form

fit) = C 4- 2F(u(t), u_), (1.128)

where u_ is used instead of mn-1 and the constant C is given by

n - 1

C = -F(oto, rio) 4- 2 ~ [F(Mk, mk-1) -- F(Mk, mk)]. k=l

(1.129)

From (1.128), we find

fu fu f du = C(u+ - u_) 4- 2 F(u, u_) du. _ld+

(1.130)


In the case of monotonic input decrease from u+ to u_, the formula (1.71) is appropriate. This formula can be rearranged as follows

f i t ) - C + 2[F(u+,u_) - F(u+,u(t))] , (1.131)

where the notation u+ is used instead of Mn, and C is the same history- dependent constant as in (1.128).

From (1.131), we obtain:

u f du - C(u_ - u+) 4- 2(u_ - u+)F(u+,u_) +U-

- 2 F ( u + , u) du. (1.132) +

Using (1.127), (1.130), and (1.132), we derive

[ /u u+ /u u+ ] W - 2 (u+ - u_)F(u+,u_) - f ( u + , u ) d u - f ( u , u _ ) d u . (1.133)

It is apparent that the expressions (1.124) and (1.133) are identical. This proves that the expressions for hysteretic energy losses derived above are consistent with the classical result (1.125).

We next discuss the applications of the above results to the irreversible thermodynamics of hysteretic media. It is clear that any hysteresis phenomenon is accompanied with energy dissipation. This means that hysteretic processes are irreversible, and consequently they fall in the domain of irreversible thermodynamics. Irreversible thermodynamics is the far- reaching extension of classical thermodynamics that describes reversible processes. For this reason, it is appropriate to begin with the brief review of the formal structure of classical thermodynamics.

Classical thermodynamics is based upon three main principles. The first principle of classical thermodynamics is the law of energy conservation. According to this principle, there exists a function of state, called the internal energy, U, of the closed system. This state function is such that its infinitesimally small change dU may occur as a result of energy exchange with the surroundings in the form of heat dQ, as well as a result of energy #W added to the system (or spent by the system) in the form of work. Mathematically, the first principle is expressed by

d U = d Q +dW, (1.134)

where a stroke is put across the symbol d in (1.134) to emphasize that the quantities//Q and # W are path-dependent infinitesimals, which are sometimes called imperfect differentials. In other words, the above quantities depend on particular path traced by the system during its transition from one equilibrium state to another.


The second principle of classical thermodynamics postulates the existence of another state function, called the entropy S. This principle also relates 8Q to the differential of entropy dS for reversible processes by the formula

dS- - #Q (1.135) T '

where T is the absolute temperature. For irreversible processes, the above equality is replaced by the in-

equality 8Q

dS > ~ . (1.136) T

The formulae (1.134) and (1.135) are often combined into one formula:

1 dS = -~ (dU - d W) (1.137)

that constitutes the mathematical foundation of classical thermodynamics. The factor 1/T in (1.137) and (1.135) can be mathematically interpreted as an integrating factor for the imperfect differential #Q.

The first and second principles of classical thermodynamics are com- plemented by the Nernst-Plank postulate which is sometimes called the third principle of thermodynamics. According to this principle, the entropy of any system vanishes at zero temperature. This principle provides some useful information concerning the asymptotic behavior of entropy. However, the bulk of phenomenological thermodynamics does not require this principle. For this reason, the above postulate cannot be compared in importance with the first and second principles of classical thermodynamics.

Classical thermodynamics has by and large been extended in two main directions. The first extension is based on the introduction of new variables describing the composition of the system. This approach has been very successful in applications of thermodynamics to chemical re- actions. It has led to the development of chemical thermodynamics. The second extension is based on the generalization of the second principle of classical thermodynamics; it has led to the development of thermodynamic theory of irreversible processes that is often called irreversible thermodynamics. This theory has been mainly developed by I. Prigogine and his collaborators.

As mentioned above, the basic difference between classical and irreversible thermodynamics lies in the way in which the second principle is stated. In irreversible thermodynamics, the second principle is formulated as follows.


1. The change in entropy dS can be split into two parts:

dS -= de S 4- di S, (1.138)

where #e S is due to the flow of entropy into the system from its surroundings, while at/S is the generation of entropy by irreversible processes within the system. The term #i S is often called internal entropy production.

2. The internal entropy production at/S is never negative. It is zero if only the system undergoes a reversible process and positive if it undergoes an irreversible process. Thus,

di S/> 0. (1.139)

3. For closed systems, the term G S is related to the energy ~e Q re- ceived in the form of heat by the formula that is similar to (1.135):

#eQ deS= T ' (1.140)

where T is the absolute temperature that is assumed to be definable (at least locally) for nonequilibrium situations.

It is also assumed in irreversible thermodynamics that the formula (1.137) holds with dS meaning the total change in entropy.

From (1.138) and (1.140), we find

TdS =c-leQ + T di S. (1.141)

The last expression allows one to relate the entropy production to the dissipated energy within the system. Indeed, from (1.137) and (1.141) we find

d U = d W +c-leQ + T di S. (1.142)

The first two terms in (1.142) have the same meaning as in (1.134) while the last term can be interpreted as the energy supplied to the system as a result of dissipating (irreversible) processes. This interpretation is especially clear for adiabatically isolated (G Q = 0) systems that undergo cyclic changes (~ at W = 0). The increase in internal energy during one cycle (fi dU) of such systems is only due to dissipating processes within the system (~ T di S). From (1.142) and the foregoing discussion, we find that the entropy production can be related to the dissipated energy di Q by the formula:

#iQ diS-= ~ . (1.143) T

It is apparent that the novel part of the above formulation of the second principle is the introduction of internal entropy production di S. However,


this new quantity is useful if only it can be evaluated (in a mathematical form) for different irreversible processes. This is the central problem of irreversible thermodynamics and it has been emphasized in many books on this subject. For instance, in the book [26, p. 90] by I. Prigogine we find: "The main feature of the thermodynamics of irreversible processes consists of the evaluation of the entropy productions . . . . " Similarly, in [27, p. 21] we read: "In thermodynamics of irreversible process, however, one of the important objectives is to relate the quantity #i S, the entropy production, to the various irreversible phenomena which may occur inside the system." Finally, in [28, p. 69] it is noted: "A central problem of irreversible thermodynamics, in fact, is the development of formulas for the entropy production di S in specific cases."

The above problem has been resolved for various irreversible processes which are caused by macroscopic non-uniformit ies of the system. Exam- ples of such processes include heat flows due to temperature gradients, diffusion flows due to density gradients, electric current flows due to potential gradients, and so forth. For the above processes, the so-called entropy balance equation (1.138) is written in local (differential) form in which the entropy production, #i S, is replaced by the entropy source. The entropy source is then found as a sum of several terms each being product of a flux characterizing a particular irreversible process and a quantity, called thermodynamic force, which is related to a particular macroscopic nonuniformity of the system (temperature gradient, for instance). In this way, many useful results have been established. The celebrated On- sanger reciprocity principle for phenomenological coefficients is the most known example of these results. However, the above developments cannot cover hysteresis phenomena. This is because hysteresis is not caused by macroscopic nonuniformities and therefore cannot be linked to gradients of some physical quantities. As a result, different approaches should be developed for the calculation of entropy production in the case of irreversible hysteretic processes. It is logical to expect that mathematical models of hysteresis may help to solve the above problem. It is shown below that by using the Preisach model and the expressions for hysteretic energy losses derived on the basis of this model, the entropy production for hysteresis processes can indeed be found.

Consider the input u(t) which is monotonically increased from its previous local minimum value u_. If the current input value, u(t), is lower than Mn, then by using (1.115), for the dissipated energy we find

fu u ]u u Q ( u _ , u ) = (u - u _ ) F ( u , u _ ) - F(u, f l )df l - F ( ~ , u ) d a . (1.144)


Employing (1.144), it is easy to conclude, that the energy dissipation, #i Q, that occurs as a result of the input increase from u to u 4- du, is given by

OQ(u_,u) diQ = du. (1.145)

3u From (1.145) and (1.143), we derive:

1 3Q(u_,u) di S = - du. (1.146)

T 3u

By differentiating (1.144) with respect to u and substituting the result into (1.146), after simple transformations we obtain

~i S = ---~-du ~u u -~uO [F(u,u_) - F(u, fl)] dfl. (1.147)

If the input is monotonically decreased from its previous local m ax im u m value, then the internal entropy product ion occurring as a result of the input decrease from u to u - du is given by

dU fuU+ O ~i S = ~ ~ [F(u+, U) -- V(~, U)] d~. (1.148)

The derivation of the last formula is similar to that for (1.147). It has been tacitly assumed in the previous derivation that the monotonic input variations are such that no previous history is wiped out. However, by using (1.116), it is easy to extend (1.147) and (1.148) to the most general case. The details of this extension are left to the reader.

Very interesting discussion of thermodynamic aspects of hysteresis can be found in [29, 30].

The presentation of the material in this chapter is largely based on the references [31-36].

References 1. Poincar6, H. (1952). Science and Hypothesis, New York: Dover.

2. Preisach, E Z. (1935). Phys. 94: 277.

3. Neel, L. (1958). C. R. Acad. Sci. Paris Sdr. I Math. 246: 2313.

4. Biorci, G. and Pescetti, D. (1958). Nuovo Cimento 7: 829.

5. Biorci, G. and Pescetti, D. (1959). J. Phys. Radium 20: 233.

6. Biorci, G. and Pescetti, D. (1966). J. Appl. Phys. 37: 425.

7. Brown, W. F. Jr. (1962). J. Appl. Phys. 33: 1308.

8. Bate, G. (1962). I. Appl. Phys. 33: 2263.

9. Woodward, J. G. and Della Torre E. (1960). J. Appl. Phys. 31: 56.

10. Della Torre, E. (1965). J. Appl. Phys. 36: 518.


11. Barker, J. A., Schreiber, D. E., Huth, B. G., and Everett, D. H. (1985). Proc. Roy. Soc. London Ser. A 386: 251.

12. Damlanian, A. and Visintin, A. (1983). C. R. Acad. Sci. Paris S~r. I Math. 297: 437.

13. Everett, D. H. and Whitton, W. I. (1952). Trans. Faraday Soc. 48: 749.

14. Everett, D. H. (1954). Trans. Faraday Soc. 50: 1077.

15. Everett, D. H. (1955). Trans. Faraday Soc. 51: 1551.

16. Enderby, J. A. (1956). Trans. Faraday Soc. 52: 106.

17. Friedman, A. (1982). Foundation of Modern Analysis, New York: Dover.


19. Brokate, M. (1989). IEEE Trans. Magnetics 25: 2922.

20. Brokate, M. and Visintin, A. (1989). J. Reine Angew. Math. 402: 1.

21. Brokate, M. and Friedman, A. (1989). SIAM J. Control Optim. 27: 697.

22. Visintin, A. (1984). Nonlinear Anal. 9: 977.

23. Visintin, A. A. (1982). Ann. Mat. Pura Appl. 131: 203.

24. Friedman, A. and Hoffmann, K.-H. (1988). SIAM J. Control Optim. 26: 42.

25. Hopfield, J. J. (1982). Proc. Nat. Acad. Sci. U.S.A. 79: 2554.

26. Prigogine, I. (1961). Introduction to Thermodynamics of Irreversible Processes, New York: Wiley.

27. deGroot, R. and Mazur, P. (1963). Non-Equilibrium thermodynamics, North- Holland: Amsterdam.

28. Vincenti, W. G. and Kruger, C. H. Jr. (1965). Introduction to Physical Gas Dynam- ics, New York: Wiley.

29. Bertotti, G. (1996). Phys. Rev. Lett. 76: 1739-1742.

30. Bertotti, G. (1998). Hysteresis in Magnetism, Boston: Academic Press.

31. Mayergoyz, I. D. (1991). Mathematical Models of Hysteresis, New York: Springer- Verlag.

32. Mayergoyz, I. D. (1986). Phys. Rev. Lett. 56: 1518-1521.

33. Mayergoyz, I. D. (1985). J. Appl. Phys. 57: 3803-3805.

34. Mayergoyz, I. D. (1986). IEEE Trans. Magnetics 22: 603-608.

35. Doong, T. and Mayergoyz, I. D. (1985). IEEE Trans. Magnetics 21: 1853-1855.

36. Mayergoyz, I. D. and Friedman, G. (1987). J. Appl. Phys. 61: 3910-3912.

CHAPTER 2

Generalized Scalar Preisach Models of Hysteresis

2.1 " M O V I N G " PREISACH M O D E L OF HYSTERESIS

The classical Preisach model of hysteresis has been discussed in detail in the previous chapter. It has repeatedly been emphasized that this model has some intrinsic limitations. The most important of them are the following:

1. The classical Preisach model describes hysteresis nonlinearities which exhibit congruency of minor loops formed for the same reversal values of input. However, many experiments show that actual hysteresis nonlinearities may substantially deviate from this property.

2. The classical Preisach model is rate-independent in nature and does not account for dynamic properties of hysteresis nonlinearities. However, for fast input variations these properties may be essential.

3. The classical Preisach model describes hysteresis nonlinearities with wiping out property. This property is tantamount to the immediate formation of hysteresis loop after only one cycle of back-and-forth variation of input between any two reversal values. However, experiments show that hysteresis loop formation may be preceded by some "stabilization process" that may require large number of cycles to achieve a stable minor loop. This process is also called in the literature "accommodation" or "reptation" process.

4. In the classical Preisach model, a scalar output exhibits hysteretic variations with respect to only one scalar input. However, in the case of magnetostrictive hysteresis, the strain is a hysteretic function of two variables: magnetic field and stress. Thus, the problem of developing the Preisach type models with two inputs presents itself. These models may find applications beyond the area of mag-

65

66 CHAPTER 2 Generalized Scalar Preisach Models of Hysteresis

netostrictive hysteresis, for instance, in the modeling of piezoelec- tric hysteresis.

5. The classical Preisach model deals only with scalar hysteresis nonlinearities. However, in many applications vector hysteresis is encountered. Properties of this hysteresis are usually quite different from scalar hysteresis properties.

To remove (or relax) the above mentioned limitations, essential generalizations of the classical Preisach model are needed. These generalizations for the case of scalar hysteresis are discussed in this chapter, while vector Preisach models are treated in Chapter 3. In this chapter, experimental testing of various scalar Preisach type models of hysteresis is presented as well. The presentation of the material in this chapter is largely based on our publications [1-12].

We begin this chapter with some interesting modification of the classical Preisach model. This modification will reveal that the Preisach model does describe to a certain extent reversible properties of hysteresis nonlinearities. This fact has been overlooked in the existing literature. Apart from the mentioned fact, this modification will be also instrumental in the further generalizations of the classical Preisach model which are discussed in subsequent sections.

The classical Preisach model has been defined as

A

fit) = Fu(t) = #(c~, ~)~,~u(t) dc~ d~, (2.1)

where T is the limiting triangle specified by inequalities/J0 K/J K c~ K c~0. This triangle is the support of the function/~(c~,/~) and it does not change with input variations.

+ We next subdivide the triangle T into three sets Su(t), Ru(t) and Su(t) (see

Fig. 2.1), that are defined as follows:

(Ol, ~) E Su+(t) if/J0 K/~ K c~ ~< u(t), (2.2)

(~,/~) ~ Ru(t) if/~0 K/~ K u(t), u(t) ~ ot ~ c~0, (2.3)

(~ , ~) E Su(t) if u(t) ~ ~ ~ c~ ~ oto. (2.4)

By using the above subdivision, we can represent (2.1) in the form

f(t) - /~Ru(t ) #(c~, ~)~,~u(t) do~ d~ +/ /s+ u(t)

I~(o~, ~)G~u(t) dot dfl

+//s #(o~,~)G~u(t)do~d~. u(t)

(2.5)

2.1 "MOVING" PREISACH MODEL OF HYSTERESIS

I

/ . . . .

,-I-

/

(X

~(X 0 /

67

FIGURE 2.1

4- Since u(t) >~ ~ for any (~, fl) ~ Su(t) , then G~u(t) = +1 and

ffs u(~176 #(o~,fl)da, dfl. (2.6) +u(t~ u+~t~

Similarly, u(t) <~ fl for any (or, fl) ~ Su+(t) and

t'[" #(ol, fl)~,o~u(t) d~ dfl = - [ [ #(~, fl) do~ dfl. (2.7) dJs u(~ J J Su(t)

By substituting (2.6) and (2.7) into (2.5), we obtain

f(t) = f ~ .(o,.~)9.~u(t)do, dfl + f/s #(a, fl)dadfl u(t) +u(t)

- f f u(~,~)d~d~. (2.8) ddS u(t)

We next find a simple expression for the last two terms in (2.8). Consider a monotonic increase of input from some value below fl0 (state of negative saturation) to some value u(t). Then the output will change along the ascending branchfu~t), and according to (2.8) we find

-- f f R # ( ~ " fl ) d ~ d fl + f f s # ( a, , fl ) d ol d fl u(t) u-t-(t)

- f f ~(~,/~) d~ d/~. diS u(t)

f u+( t ) -

(2.9)


Similarly, if we consider a monotonic decrease of input from some value above d0 (state of positive saturation) to the same value u(t), then by using (2.8) we obtain

f~t) = f fRu(t) #(~ fl) d~ dfl + f fSu+(t) lz(ot, fl) dot dfl

r r - H #(c~, fl) dc~ dfl. (2.10)

dis u(t) By summing up (2.9) and (2.10), we derive

f f s # ( ol , fl ) d o~ d fl - f f ss # ( oe , fl ) doe d fl = 1 +u(t) ~t, -2 (fu+((t) + f~t))" (2.11)

By substituting (2.11) into (2.8), we finally obtain

fit) = #(ol, f)~,~eu(t) dot dfl + -~ (fu+(t) ff-f~t))" u(t)

(2.12)

The last expression is formally equivalent to the classical one (2.1). How- ever, in this expression the integration is performed not over the fixed limiting triangle T but over the rectangle Ru(t) which changes along with input variations. For this reason, the expression (2.12) is termed here as a "moving" Preisach model. It is also clear from (2.12)that l(fu+(t ) +f~t)) represents a fully reversible component of hysteresis nonlinearity described by the classical Preisach model. In this respect, the first term in the right- hand side of (2.12) can be construed as irreversible component of the classical Preisach model. To make the last point transparent, consider some output increment Afu corresponding some input value u (see Fig. 2.2). This increment depends on a particular history of input variations and can be regarded as a measure of irreversibility of hysteresis nonlinearity for this particular history. It is clear from the diagram shown in Fig. 2.3 and the expression (2.12) that the above increment is given by

Afu = 2 f fn #(or, fl) doe dfl. (2.13)

The region of integration f2 belongs to Ru(t). This is true for any past history. This fact clearly suggests that the first term in the right-hand side of (2.12) describes irreversible processes. Thus the expression (2.12) gives the decomposition of hysteresis nonlinearity described by the classical Preisach model into irreversible and reversible components.

It is stressed in the above discussion that the expression (2.12) is mathematically equivalent to the classical definition (2.1). However, it is apparent that this equivalence holds only for input and output variations


~ U u


69

confined to the region enclosed by major hysteresis loop. Outside this region, the classical Preisach model prescribes flat saturation values for output, while the moving model (2.12) prescribes the actual experimentally observed values f,+ a n d f~t) for the states of negative and positive u(t) saturation, respectively. This is the case because for these states ascending and descending branches merge together and consequently fu~t ) =f~t)" As far as the first term in (2.12) is concerned, it is clear from (2.3) that this term vanishes. Thus, the moving model (2.12) can be regarded as a generalization of the classical model as far as the description of hysteresis nonlinearities beyond the limits of major loops is concerned.

It is instructive to consider directly the identification problem for the model (2.12) without invoking our previous discussion of this problem for the classical model (2.1). The essence of this problem is in determining the function/~ by fitting the model (2.12) to some experimental data. Sup- pose that, starting from the state of negative saturation, the input u(t) is monotonically increased until it reaches some value a. Then the input is monotonically decreased until it reaches some value ft. The corresponding diagram is shown in Fig. 2.4. By using (2.12) and this diagram, we find that the output valuers3 corresponding to the final value of output is given by

f~fl - - f fR(~,flo,3,3) (o~o,3o,o.3)

+ f~ +f3-. (2.14) 2

70

(~,Bo

CHAPTER 2 Generalized Scalar Preisach Models of Hysteresis

(%'

d ~,~)

/ 4

%


Here R(c~0, 30, c~, fl) is the rectangle whose opposite vertices are the points (c~0, rio) and (c~, fl); the same meaning has the notation R(c~, rio, 3, 3).

Now suppose that, starting from the state of positive saturation, the input u(t) is monotonically decreased until it reaches the value 3. The corresponding c~-fl diagram is shown in Fig. 2.5. From this diagram and (2.12) we find the following expression for the resulting value of output:

f• ~(~', ~') a~' a~' + f ~ - (~o,3o,3,3 )

J; +i; (2.15)

Next we introduce the function

T(oe, fl) = f ~ - fo,~, (2.16)

which is equal to output increments between the limiting descending branch and first-order transition curves. From (2.14), (2.15) and (2.16) we find

T ( o~ , 3) - 2 f f #(c~', 3') dc~' d fl ' J JR (~o,3o,ot,3 )

= 2 f-~ ( f , i . 217.

By differentiating the last expression two times, we obtain

1 32T(a,/3) /~(c~, 3 ) = - - �9 (2.18)

2 Ool Off


$+(t) -..

Rk

O~ S-(t) i I ' l l I ,,,,, ,

_ I

|

/ /

~ (Mk,mk)

71

FIGURE 2.6

By recalling the definition (2.16) of T(a, fl), from (2.18) we derive

1 32f~ (2.19) = 2

which coincides with the expression (1.29) obtained for the classical Preisach model.

Next, we show that the integration in (2.12) can be avoided and that the explicit expression for f(t) in terms of experimentally measured function T(~, fl) can be derived. To start the derivation, consider the c~-fl diagram shown in Fig. 2.6. Then, according to (2.12), we have

f(t) = / f S + ( t ) f f s #(or, fl ) d ol d fl 4- # ( ol , fl ) dot d fl - - ( t ) fu+(t) ff- f~ t )

----2 ffs+(t) #(~ fl) d~ dfl - ffRu(t) lz(ol, fl) dol dfl + fu+(t) q- fu(t) . (2.20)

2

To evaluate the integral over Ru(t) in (2.20), we consider a monotonic increase of input from the state of negative saturation to some value u(t). Then from (2.12) we find

/R f'+ f u ~ t ) - - ~ ( o ~ , ~ ) d o l d / J if- u(t) ff-f~t). (2.21)

u(t) 2


By substituting (2.21) into (2.20), we conclude

- - 2 f f # ( oe , fl ) doe d fl f(t) + f u+(t) �9 (2.22) dis +(t)

It is clear that the moving model (2.12) is equivalent to the classical Preisach model as far as description of purely hysteretic behavior is concerned. For this reason, the wiping-out property and congruency property of minor loops are valid for the moving model. According to the wiping- out property, the staircase interface L(t) in Fig. 2.6 has vertices whose and fl coordinates are equal to Mk and mk, respectively, and, as before, numbers Mk and mk form the alternating series of past dominant input extrema. Now we subdivide S + (t) into rectangles Rk and represent the integral in (2.22) as

n(t) ffs #(oe, fl)doedfl. (2.23) +(t) k = l k

It is easy to see from the expression (2.17) for the function T(~, fl) that the integrals over Rk can be represented in the form

/ /R #(~ , fl) do~ dfl - - T(Mk, mk). T(Mk+l,mk) (2.24) k

From (2.22), (2.23) and (2.24) we obtain

n(t)

f (t) - 2 E [ T ( M k + I , mk) -- T(Mk, mk)] + fu+(t). (2.25) k = l

This is the final result which expresses explicitly the output f( t) in terms of experimentally measured function T.

Up to this point, we have discussed the modeling of "counter clock- wise" hysteresis. For this type of hysteresis, the positive saturation state is achieved for a larger input value than the negative saturation state, and for any two consecutive descending and ascending branches the descending branch is above the corresponding ascending branch (see, for instance, Fig. 2.2). This type of hysteresis is typical for magnetic materials and it can be modelled by using the classical Preisach model with positive measure #(a, fl). It is apparent that the classical Preisach model with negative measure can be used for the modeling of "clock-wise" hysteresis when the positive saturation state is achieved for a smaller input value than the negative saturation state. This type of hysteresis occurs in superconductors due to their diamagnetic nature. The modeling of superconducting hysteresis is extensively discussed in Chapter 5. There exists, however, the

2.2 PREISACH MODEL OF HYSTERESIS 73

"clock-wise" hysteresis with the property that the positive saturation state is achieved for a larger input value than the negative saturation state. This is typical for hysteresis that occurs for certain front propagation problems such as nonlinear diffusion of electromagnetic fields in magnetically nonlinear conductors (see Chapter 6). The modeling of this type of hysteresis is not readily available within the framework of the classical Preisach model, however it can be easily accomplished by using the "moving" Preisach model (2.12). Indeed, the proper saturation states can be modelled by the appropriate choice of the reversible terms 1 (fu~t) ff-f~t) )' while the "clock-wise" nature of hysteresis is imposed by choosing a negative measure #(c~, ]~) in the irreversible component. In other words, the decomposition (2.12) of hysteresis nonlinearity on reversible and irreversible components makes it possible to separately control the (clock-wise or counter clock-wise) orientation of hysteresis and the location of saturation states.

2.2 PREISACH MODEL OF HYSTERESIS WITH I N P U T - D E P E N D E N T MEASURE

The Preisach model with input-dependent measure is discussed in this section. It has the following advantages over the classical model. First, the congruency property of minor loops is relaxed for this model. This results in a broader area of applicability of this model as compared with the classical model. Second, the model with input-dependent measure allows one to fit experimentally measured first- and second-order reversal curves. Since higher-order reversal curves are "sandwiched" between first- and second-order reversal curves, it is natural to expect that this model will be more accurate than the classical one.

The Preisach model with input-dependent measure can be mathematically defined as

//R f'+ f i t ) - /~(c~,/~, u( t ) )~ ,~u( t ) dc~ d~ + u(t) -}- f~t ) . (2.26)

u(t) 2

It is clear that a new feature of this model in comparison with the "moving" model (2.12) is the dependence of the distribution function/~ on the current value of input, u(t). For this reason, the model (2.26) is also termed as the "nonlinear" Preisach model. Due to the new feature mentioned above, the first term in the right-hand side of (2.26) can be construed as a partially reversible component of hysteresis nonlinearity. Indeed, for each


(Z

~ ( t ) ' ~ ~

/ ; u(t)

FIGURE 2.7

pair (c~, fl), the integrand #(c~, fl, u(t))G~u(t) is reversible for input variations between ~ and ft.

The "nonlinear" Preisach model admits a geometric interpretation which is similar to that for the classical model. In other words, it could be easily demonstrated that at any instant of time t the rectangle Ru(t) is subdivided into two sets (see Fig. 2.7): S + (t) and S-(t) consisting of points (c~, fl) for which Gnu( t )= 1 and Gnu( t )= -1 , respectively. The interface L(t) between S+(t) and S-(t) is a staircase line whose vertices have c~ and

coordinates equal (respectively) to past dominant extrema Mk and ink. Using the above geometric interpretation, the model (2.26) can be represented in the form

fit) : i/s #(ot, fl, u(t))dotdfl- Sfs #(ot, fl, u(t))dotdfl +(t) -(t) ~+

u(t) q- f~t) + . (2.27)

2

It can be shown that the following two properties are valid for the "nonlinear" model.

PROPERTY A (Wiping-Out Property) Only the alternating series of subsequent global extrema Mk and mk are stored by the "nonlinear" Preisach model.

The proof of this property is completely identical to that for the classical model.

2.2 PREISACH MODEL OF HYSTERESIS

i i i i i I i i i ' ]

i i I I

e _~ a

U_ U U+ ~U

fu'

75

e a ,k

U_ U U+


PROPERTY B (Property of Equal Vertical Chords) All minor loops resulting from back-and-forth input variations between the same two consecutive extrema have equal vertical chords (output increments)for the same input values (Fig. 2.8).

PROOF. Consider a minor loop formed as a result of back-and-forth input variations between u+ and u_ (Fig. 2.9). Let f~t and fu be output values on descending and ascending branches of this loop, respectively. These values correspond to the same value of input u from the interval u_ < u < u+. By using the expression (2.27),fu" and f~ can be computed as

f u - f ,+ u ( ~ . fl . u ) d ~ d fl - f f s '- # ( ol , fl , u ) d o~ d fl

+ fu + + f u , (2.28) 2

f " = ffs.; .(~.fl , u)d.~dfl- f f s''-/z(~,,fl, u)d~,dfl

+ fu + + f u , (2.29) 2

where the sets S~_, St_, S~ and S'5 are shown on the diagrams presented in Fig. 2.10. It is clear from these diagrams that

s'~ - s'+ = R ( u + , u , u _ ) , S'_ - S"_ = R ( u + , u , u _ ) .

From (2.28), (2.29) and (2.30), we derive

f~' - fu = 2 ~ ]z(ot, fl, u)dol dfl. JaR (u+,u,u_)

(2.30)

(2.31)


/

I- s: R,u_ u u+,

I I

u u+

/

I s:' s: I ~(~/ - R(U_,U,U+)

I I

p ' ~ p _ U U +

/

(a) (b)

FIGURE 2.10

According to (2.31), we find that for any u ~ (u+,u_) the corresponding vertical chord does not depend on a particular past history preceding the formation of minor loop. This proves that all comparable minor loops, that is the loops with the same reversal input values u+ and u_, have equal vertical chords.

It is left to the reader as a useful exercise to prove that comparable minor loops described by the nonlinear model (2.26) are not necessary congruent.

Now we turn to the identification problem of determining the distribution function #(~, fl, u) by fitting the model (2.26) to some experimental data. It turns out that for the solution of this identification problem the sets of first- and second-order reversal curves are required. These curves can be measured experimentally as follows. We first decrease the input, u(t), to such a negative value that the outputs of all operators 9~fi are equal to - 1 (state of negative saturation). Then we monotonically increase the input until it reaches some value c~. As we do this, we will follow along the ascending branch of the major loop (see Fig. 2.11). As we already know the first-order reversal curves are attached to this ascending branch and they are formed when the above monotonic increase of u(t) is followed by a subsequent monotonic decrease. The notation f + will be used for the output values on the first-order reversal curve attached to the ascending branch of the major loop at the poin t f +. The second-order reversal curves are attached to the first-order reversal curves, and they are formed when the above monotonic decrease is followed by a monotonic increase. The notation f~u will be used for the output values on the second-order re-


f

I

u

77

F I G U R E 2.11

versal curve attached to the first-order reversal curve fo, u at the point f ~ (Fig. 2.11).

Consider the function

P(o~, fl, u) = fo<u - f<~u, (2.32)

which has the physical meaning of output increments between the first- and second-order reversal curves.

It is clear from the definition of this function that

P(c~, u, u) = P(u, fl, u) - P(u , u, u) = 0. (2.33)

This proper ty will be employed later in our discussion. N o w we will try to relate the function, P(c~, fl, u), to the distribution function, #~(c~, fl, u). To this end, we will use the diagrams shown in Fig. 2.12. From these diagrams and (2.27) we conclude

ion- fi~ "(<'"~' u)~.~.- ii~ .(~..~..u)~.~. + iu+ +in + ' - 2 '

io~u: ff~ .(~..~..u)~.~.- ff~ .(~..~..u)~.~ (2.34) +

f + + f ; 2

S+ - S+ - R(a, fl, u), S_ - S_ - R(c~, fl, u). (2.35)


d s d

\

S+

/

u

-- ~ T Z ~ ( o t . f 3 S+ , ,U )

u

(a) (b)

FIGURE 2.12

By using (2.34) and (2.35), we derive

P ( o~ , fl , u) - 2 f f # ( o~ ' , fl ', u) d o~ ' d fl ' a I R (~,~,u)

f (s ) = 2 U(oe', fl', u) dfl' dot'. (2.36)

By differentiating the last expression twice, we obtain

1 32P(~, ,6, u) /,(c,, fl, u ) = . (2.37)

2 3ol3fl

By recalling the definition (2.32) of P(c~, fl, u), from (2.37) we find

1 O2f~u (2.38) ~(~ , /~ ,u ) - ~ o~ o~"

It is clear from the above derivation, that the expressions (2.37) and (2.38) are valid if fl < u < ~. If u ~< fl or u ~> c~, then we define

#(c,, fl, u) -- 0, (2.39)

which is consistent with (2.33). Thus, if the distribution function, #(c~, fl, u), is determined from (2.36)

or (2.38), then the nonlinear model (2.26) will match the increments between first- and second-order reversal curves. Next we shall show that the limiting ascending branch will be matched by this model as well. In- deed, it is easy to see that the output values fu on the limiting ascending


branch predicted by the nonlinear model (2.26) are equal to

f u - - f f R tt(d, fl',u)dddfl' +f+ +fu. (2.40) (~0,&,u) 2

According to (2.36) and (2.32), we have

ffR 1 1 (~0,e0,u) ~(~'' ~'' u)a~ a~' = ~P(~0, &, u) - -~Gou -f~oeou)

1 = ~ ( f ~ - f + ) . (2.41)

By substituting (2.41) into (2.40), we find

fu=fu +. Thus, if the function, #(~, 3, u), is determined according to (2.36) or (2.38), then the nonlinear model (2.26) fits: (a) the output increments between the first- and second-order reversal curves, (b) the ascending branch of the major loop. Since the ascending branch of the major loop can be construed as a second-order reversal curve, we conclude that the nonlinear model (2.26) fits the sets of first- and second-order reversal curves.

In the above discussion we have used second-order increasing reversal curves (see Fig. 2.13) in order to determine the distribution function,

I

FIGURE 2.13


I

I L

T - - ;, ~-u ] I I I / i

f - I / i ',

F I G U R E 2 . 1 4

/x(a, fl, u). However, by almost literally repeating the previous line of reasoning, a similar expression can be found for #(a, fl, u) by using second- order decreasing reversal curves ffi6~. One of these curves is shown in Fig. 2.14. By using this figure, we can introduce the function

P(fi,&, ~t) = f fic~ - f fi~. (2.42)

In the same way as before, we can show that

P(f i , = 2 #(ot',fl',~t)dot'dfl' (2.43) (~,~,~)

and

1 02P(~,ol,~_t) (2.44) / x ( & , f i , f i ) - 2 O & O f i "

If

f i - - ~ , ~ - - / ~ , f i - - u , (2.45)

then due to the symmetry between increasing and decreasing second- order reversal curves (see Fig. 2.14) we find

P(fi, &, fi) - P(~,/~, u). (2.46)

By using (2.45) and (2.46) in (2.44), we derive

1 32p(ot, fl, u) #(-/~, - ~ , -u ) - . (2.47)

2 Oot Off


From (2.47) and (2.37), we conclude

#(c~,/~, u) =/~(-/~, - ~ , -u) . (2.48)

The last formula can be regarded as a generalization of the mirror symmetry (1.46) previously established for the classical Preisach model. E3

We next proceed to the proof of the following important result.

REPRESENTATION THEOREM The wiping out property and the property of equal vertical chords for minor loops constitute the necessary and sufficient conditions for the representation of a hysteresis nonlinearity by the nonlinear Preisach model on the set of piece-wise monotonic inputs.

PROOF. Necessity: If a hysteresis nonlinearity is representable by the nonlinear Preisach model, then this nonlinearity should have the same properties as the model. This means that this nonlinearity should exhibit the wiping-out property and the property of equal vertical chords for comparable minor loops.

Sufficiency: Consider a hysteresis nonlinearity which has both the wiping out property and the property of equal vertical chords. For this nonlinearity we find the distribution function #(~, ~, u) by using formula (2.37) (or (2.38)). Then the nonlinear Preisach model (2.26) will match exactly the sets of all first- and second-order reversal curves. We intend to prove that this model will match all possible higher-order reversal curves as well. The proof is based on the induction argument. Let us assume that the above statement is true for all possible reversal curves up to the order k - 1. Then, for the induction inference to take place, we need to prove that the same statement holds for any reversal curve of order k. Let a be a point at which a reversal curve of order k is attached to a reversal curve of order k - 1 (see Fig. 2.15). According to the induction assumption, the output values for the actual hysteresis nonlinearity and for nonlinear Preisach model coincide at each point of the reversal curve of order k - 1. Thus, it remains to be proved that the output increments between the reversal curves of orders k and k - 1 are the same for the actual hysteresis nonlinearity and for the nonlinear model (2.26). It is here that the wiping-out property and the property of equal vertical chords will be used. According to the wiping-out property, the kth reversal curve should meet the ( k - 1)th reversal curve at the point b, which is the point of inception of the latter curve. As a result, a minor loop is formed. Consider a comparable minor loop (loop with the same reversal input values), which is attached to the limiting ascending branch (Fig. 2.15). This loop is formed by some first- and second-order transition curves. According to the property of equal

FIGURE 2.15


k

/ k-1

,

f(k) __f(k-1) = 2 f fR lz(oe', 13', u) doe' dfl', (o~,~,u)

vertical chords, this loop has the same vertical chords as the loop formed by ( k - 1)th and kth reversal curves. Consider an arbitrary value u of input such that fi < u < c~. Using the diagram shown in Fig. 2.16, it is easy to derive the following expression for the output increment

(2.49)

U

o~

(eL, 1

y_~,, RIcz,~,,,l

82

FIGURE 2.16


wherefu (k) andfu (k-l) are the output values on the kth and ( k - 1)th reversal curves, respectively, corresponding to the input value u and predicted by the model (2.26). From (2.36) we find that the right-hand side of (2.49) is equal to the output increment el between the first- and second-order transition curves. Consequently,

f(u k) _ f(k-1) =el. (2.50)

According to the property of equal vertical chords we have

el = cd. (2.51)


f(k) F(k-1) -~ u -- cd. (2.52)

Thus the nonlinear model predicts the correct output increments between the kth and (k - 1)th reversal curves. From this fact and the above induction assumption we conclude that the nonlinear model predicts the correct output values on the reversal curve of order k. This concludes the proof of the theorem. []

The proved theorem establishes the exact bounds of applicability of the nonlinear Preisach model. It is apparent that the property of equal vertical chords is more general than the congruency property. Indeed, if comparable minor loops are congruent, then they have equal vertical chords. However, if comparable minor loops have equal vertical chords, they are not necessarily congruent. This clearly shows that the nonlinear Preisach model (2.26) has a broader area of applicability than the classical Preisach model. Next we shall show that the nonlinear model (2.26) contains the classical Preisach model as a particular case. The exact statement of this fact is given by the following theorem.

REDUCTION THEOREM If all comparable minor loops of hysteresis nonlinearity are congruent, then the nonlinear Preisach model for this nonlinearity coincides with the classical Preisach model as far as purely hysteretic behavior of this nonlinearity is concerned.

PROOF. Since all comparable minor loops are congruent, these loops have equal vertical chords. Assuming also that the wiping out property holds, we can represent the mentioned hysteresis nonlinearity by the nonlinear Preisach model (2.26). We next show that, because of congruency property, the distribution function # in (2.26) does not depend on u. It is clear from


f

I

I I

I I I I

(x (x

fdpu

F I G U R E 2.17

Fig. 2.17 that the congruency proper ty results in the congruency of second- order reversal curves f~,~u, f ~ u and f~"~u. This means that the derivative

often does not depend on a Consequently, 3u

O3f~ --0. (2.53) Oot Off Ou

From (2.53) and (2.38) we find

_= 0. (2.54) 3u

Thus # does not depend on u and is only a function of a and fl:

/z(a, fl, u) = v(a, fl). (2.55)

By substi tuting (2.55) into (2.26), we obtain

/ f R f u+( t ) ff- f ~ t ) f ( t ) = v(a , f l ) G ~ u ( t ) d a d f l + . (2.56) u(t) 2

It remains to be proved that v(a, fl) in (2.56) coincides with #(a, fl) in (2.12). The proof is straightforward. By using (2.12) and (2.56) we find that for any a, fl and u such that fl ~ u ~< a we have

P(a, fl, u) - f~u - f~flu - 2 f f_ /z(~', fl') da' dfl', (2.57) JJl< (,~,~,u)


P(~,/~,u) =f~u - f ~ u - 2 / / v(~', ~') dc~' d/~'. (2.58) ddR (~,~,u)

From (2.57) and (2.58), as before, we derive

1 32p(ol, [3, u) #(a,/~) = v(~,/~) = . (2.59)

2 Oa Off Thus, it is proved that under the congruency condition, the nonlinear Preisach model (2.26) coincides with the moving Preisach model (2.12). On the other hand, the moving model coincides with the classical Preisach model as far as the description of purely hysteretic behavior is concerned. From here we conclude that under the congruency condition the nonlinear model (2.26) coincides with the classical Preisach model, and this concludes the proof of the theorem. E3

We next turn to the discussion of numerical implementation of the nonlinear model (2.26). We intend to show that double integration in (2.26) can be completely avoided and that explicit expressions for the outputf( t ) in terms of experimentally measured function P(c~, ~, u) can be derived. The starting point of our derivation is the expression (2.27) that can be modified as follows:

fit) = ffR i~(ol,~,u(t))dold~ - 2 f /s I~(o~,~,u(t))do~d~ u(t) -(t)

fu+(t) q-'f~t) + . (2.60)

2 According to (2.41), we have

f fR I~(~ ~" u(t)) d~ d~ = f ~t) - f u+(t) (2.61) u(t) 2 "


f (t) = f~t) - 2 f f I~(ol, fl, u(t)) dot dfl. (2.62) d iS -(t)

We next subdivide S-(t) into rectangles Rk (see Fig. 2.18) and represent the integral in (2.62) as

n(t)

f / s I~(ol,~,u(t))dotd~ = E f i r #(a,~,u(t))dad~. (2.63) -( t) k= l k

It is easy to see that

Rk -- a(Mk, ink, u(t)) -- R(Mk+I, mk, u(t)). (2.64)


(X

/

FIGURE 2.18


f /Rk t~(~, ~, u(t)) d~ d~

= f ~ #(ol, fl, u(t))doldfl-//R #(c~,fl, u(t))dotdfl (Mk,mk,u( t) ) (Mk+ l,mk,u( t ) )

1 = i[P(Mk, mk, u(t)) -- P(Mk+x,mk, u(t))]. (2.65)

By substituting (2.65) into (2.63) and then (2.63) into (2.62), we obtain

n(t)

fit) =f~t) + E[P(Mk+ l'mk'u(t)) -P(Mk, mk, u(t))]. (2.66) k = l

This is the final formula which expresses explicitly the output f(t) in terms of the experimentally measured function P. This formula has been used to develop a digital code which numerically implements the nonlinear Preisach model. This code computes output values by using input values, a set of second-order reversals curves and an input history which are all specified by the user. The structure of the code is very similar to the structure of the code which implements the classical Preisach model. For this reason, the detailed discussion of this code is omitted.

We next proceed to the discussion of hysteretic energy losses for hysteresis nonlinearities described by the nonlinear Preisach model (2.26). We begin with the case of elementary hysteresis nonlinearity represented


A ~(a,~ u)~u

87

~(a,~,u)

b

a

U

FIGURE 2.19

by the expression #(c~, fl, u(t))G~u(t). It follows from (2.39) that the input- output relationship for this nonlinearity is described by a loop shown in Fig. 2.19. It is clear that the branch I of this loop is fully reversible except for its part a along which the irreversible switching up occurs. Similarly, the branch 2 of the loop is fully reversible except its part b along which the irreversible switching down occurs. If a periodic variation of input is such that the whole loop is traced, then the hysteretic energy loss for one cycle, Qcycle, is equal to the area enclosed by the loop

Qcycle = 2 /~(c~, fl, u) du. (2.67)

It is clear that this energy loss occurs as a result of irreversible switchings along the vertical parts of the above loop. Consequently

Qcycle - q~ 4- q~, (2.68)

where q~ and q~ are losses occurred as a result of irreversible "up" and "down" switchings, respectively. Due to the symmetry between branches I and 2, it can be concluded that

q~ =q~. (2.69)


From (2.67), (2.68) and (2.69) we find

q~ = q~ - #(ot, fl, u) du. (2.70)

In the nonlinear Preisach model, any input variation is associated with irreversible switchings of some elementary loops #(a, fl, u(t))f,~u(t). It is natural to equate the hysteretic energy loss occurring for some input variation to the sum of energy losses resulting from the irreversible switchings of the above elementary loops. Since in the nonlinear Preisach model we deal with continuous ensembles of the elementary hysteresis loops, the above summation should be replaced by integration. Thus, if f2 denotes the region of points on ~-fl diagram for which the elementary hysteresis loops are irreversibly switched during some monotonic input variation, then the hysteretic loss, Q, for this input variation is given by

Q - f f~ ( f ~ #(o~, fl, u)du) dadfl. (2.71)

In the particular case when the distribution function/z does not depend on u, from (2.71) we obtain the expression

Q = f f~ #(c~, fl)(a - fl)dol dfl, (2.72)

which has been previously derived for the classical Preisach model (see Chapter 1).

A typical shape of the region f~ is shown in Fig. 2.20. The region f2 can be subdivided into a triangle and some trapezoids which in turn can be

FIGURE 2.20


represented as differences of triangles. Thus, if the integral (2.71) can be evaluated for any triangle region, then it will be easy to determine this integral for any possible shape of f2. In the case when f2 is a triangle, a very simple expression for the integral (2.71) in terms of experimentally measured function P can be derived. The derivation proceeds as follows. Let T(u+,u_) be a triangle (see Fig. 2.21) swept during a monotonic increase of input from some local minimum u_ to some value u+. According to (2.71), this input variation results in the hysteretic loss:

(L") Q(u_, u+) - lz(ot, fl, u) du da d~ (u+,u_)

- fui+ (fu~ (f'iz(,,~,u)du)d~)d~. (2.73)

By using Fig. 2.22, we can change the order of integration in the two interior integrals as follows:

f.'(L" ) fu'(fu u ) ~(ot, ~, u) du d~ - ~(o~, ~, u) ct~ du.

Now, by invoking the formula (2.37), from (2.74) we find

fu~ ~ ) l fu~ oaP(~ ~(o~, ~, u) du d~ = --~ oo~o~

lfu'~[OP(~ 2 _ 3a

(2.74)

3P(ot,ootu_, u) ] du. (2.75)

(x (u+,u )

~ - - -

- iii I I

u_ (X


90 CHAPTER 2 General ized Scalar Preisach Models of Hysteresis

U+

~ , _ - - / / / I I I

im �9

U_ U+

FIGURE 2.23

By substituting (2.75) into (2.73) and by changing this time the order of integration according to Fig. 2.23, we derive

1/uU+(/u~ Q( u_, u+ ) = - -i _ _ 3ol

3P(c~,oolu_, u) ] du) dot

1 /uU+(~u+IoP(~ 'u 'u ) _ 3 P ( ~ 2 _ 3o~ Oo~

l f u+ 2 [P(u+, u, u) - P(u, u, u) - P(u+, u_, u)

4- P(u, u_, u)] du. (2.76)

By using (2.33) in (2.76), we finally obtain

1 /u l + Q(u_, u + ) - ~ P(u+,u_,u)du. (2.77)

The formula (2.77) is remarkably simple if compared with a similar formula (1.115) derived for the classical Preisach model. The formula (2.77) allows one to evaluate hysteretic energy losses for arbitrary input variations. Indeed, consider some input variation for which the region ~ has the shape shown in Fig. 2.20. Then, the corresponding hysteretic energy loss is given by (2.71) which can be written as

k

Q - Q(Mn, mn) + ~-~[Q(Mn-i, mn-i) - Q(Mn-i, ran-i-I)]. (2.78) i=1

Each term in (2.78) can be evaluated by employing the formula (2.77).


The formula (2.77) admits the following geometric interpretation. Consider a minor loop formed as a result of back-and-forth input variations between two consecutive extrema u_ and u+. If a hysteresis nonlinearity is representable by the nonlinear Preisach model, then all comparable minor loops of this nonlinearity have equal vertical chords. Con- sequently, P(u+, u_,u) can be construed as a vertical chord of the above minor loop. This implies that the integral in (2.77) is equal to the area A of this loop. Thus, the expression (2.77) can be written as

A Q(u_, u+) - ~ . (2.79)

From (2.79) we immediately infer the following result.

For any loop of hysteresis nonlinearity representable by the nonlinear Preisach model, hysteretic losses occurring along ascending and descending branches are the same.

By using this result we can relate hysteretic losses occurring for arbitrary input variations to certain cycle losses. This can be accomplished in exactly the same way as it was done for the classical Preisach model. In particular, the expressions (1.122) and (1.123) can be derived.

We conclude this section by the discussion of two facts which may facilitate further appreciation of the advantages of the nonlinear Preisach model over the classical one. Consider the hysteresis nonlinearities exhibited by the Stoner-Wohlfarth magnetic particles (see Fig. 2.24). It is

m

fu

- h m

T TT I I I I I I I I I I I I

hm U

FIGURE 2.24


pointed out in Chapter I that these nonlinearities cannot be described by the classical Preisach model. The reason is that degenerate minor loops traced as the input varies back and forth between c~' and/~t are not congruent. It is easy to see that the hysteresis nonlinearities exhibited by the Stoner-Wohlfarth magnetic particles can be described by the nonlinear Preisach model. First, this is clear from the fact that the property of equal vertical chords is satisfied because all minor loops are degenerate and, consequently, they have equal (zero) vertical chords. Second, the explicit representation (2.26) for the Stoner-Wohlfarth hysteresis nonlinearities can be found. Indeed, it can be shown that

#(o~, ~ u(t)) _f~t)-fU+(t)8(ol - hm, ~ + hm), (2.80) " 2

where S is the Dirac function. By substituting (2.80) into (2.26), we find

_ u(t) ~hm,_hmU(t ) _ff . (2.81) fit) ~ t ) - - ~ + ~ t ) ff-~u+(t) 2 2

It can be easily checked that the expression (2.81) indeed represents the hysteresis nonlinearities depicted in Fig. 2.24.

Finally, the following observation may help the reader to appreciate the extent to which the nonlinear Preisach model is more general than the classical one. The classical Preisach model represents hysteresis nonlinearities that, for any reversal point, have (regardless of past history) only one branch starting from this point (see Fig. 2.25a). This branch is

f f A

/ /

/

i i / / / / ," / / / i

, ///

max

U

(a)

/I i i

ii

J

I I ,, V / / ',

i m a x

(b)

FIGURE 2.25

2.3 "RESTRICTED" PREISACH MODELS OF HYSTERESIS 93

congruent to one of the first-order reversal curves. This is an apparent consequence of the congruency property of the classical model. On the other hand, the nonlinear Preisach model (2.26) describes hysteresis nonlinearities with the property that for any reversal point there are infinite possible branches starting at this point (see Fig. 2.25b). A particular realization of these branches is determined by a particular past history. This property follows, for instance, from the expression (2.66) where all terms are input-dependent. For the classical model, the similar expressions (1.71) and (1.74) have only the last terms which are input-dependent. The above observation shows that the nonlinear model is endowed with a much more general mechanism of branching than the classical model.

2.3 "RESTRICTED" PREISACH M O D E L S OF HYSTERESIS

In the previous section we have discussed the nonlinear Preisach model of hysteresis which is a far-reaching generalization of the classical Preisach model. This generalization has been achieved by assuming that the distribution function # is dependent of the current value of input u(t). Another approach to the generalization of the classical Preisach model is to assume that the function # depends on stored past extremum values of input, {Mk, mk}. This approach was briefly explored in [13]. Here, we shall follow this approach as well, however, our treatment of the model itself and the identification problem for this model will deviate appreciably from the discussion in [13].

We begin with the simplest case when the function/~ depends only on the first global maximum, M1. Thus we define new Preisach type model a s

f(t) - [[~ a(c~, fl,M1)~,~u(t) da dfl + C M 1 , (2.82) ,J ,l l M 1

where M1 is the largest input maximum since the departure from the state of negative saturation, and the support of ,(c~, ~,M1) is the triangle TM1 defined by inequalities ]~0 ~</~ ~< c~ ~< M1 (see Fig. 2.26). We shall use the following interpretation of the above model. For any fixed extremum M1, the model (2.82) will be used to describe hysteresis behavior in the region confined between an ascending branchf~t) and a first-order reversal curve fMl~ (see Fig. 2.27). For this reason, the model (2.82) can be termed as a "restricted" Preisach model of hysteresis. However, the above "restriction" does not diminish at all the region of applicability of the model. This is because M1 may assume any value between ~0 and c~0.

TM


/

jM,

94

FIGURE 2.26

f

w e

, / / L ~ ' / / / i _-. M 1

f Mll30(, fMl~

/ ~ _ _ ~ ~ ~ : : ; ; ' ~ ~ second-order reversal curve

FIGURE 2.27

The constant CM1 in (2.82) can be easily determined from the condition that the model matches the output values fill for u(t)= M1 and f_ for


u(t) = rio. Indeed, from (2.82) we easily derive

fM1 = [f~ ~(~" fl,M1) dc~ dfl + CM1, (2.83) ,,, ,, 1 M 1

f - -- -- I f [d,(oGfl,M1)dold~ q-CM1. (2 .84) s,J lM 1


f~ '~ l " q - f - CM1 = 2 ' (2.85)

which leads to the following representation for the model (2.82):

f i t ) - f/TM lz(c~,fl,M1)G~u(t)do~dfl + f~/I1 -3Fr (2.86) 1 2

Next, we shall be concerned with the solution of the identification problem. The essence of this problem is in determining the function/~(c~, fl,M1) by fitting the model (2.86) to some experimental data. It turns out that ]z(ol, ~,M1) can be found by matching second-order reversal c u r v e s fMl~c~ shown in Fig. 2.27. To do this, we introduce the function

1 F(o~, [3, M1) - - -~ ( f Ml fiO~ -- f M~ fi ). (2.87)

This function is equal to one half of the output increments along the second-order reversal curves. To relate this function to/~(c~, ~,M1), consider diagrams shown in Fig. 2.28. By using exactly the same line of

cf

(a)

cf

f w I /

(b)

FIGURE 2.28


reasoning as in the case of the classical Preisach model (see Section 3 of Chapter 1), from these diagrams and (2.86) we derive

F ( ol , fl , M1) -- f f # ( ol ', fit, M1) dol t d fl ' ddT (~,~)

) = , #(ol',/3',M1)dol' dfl'.

By differentiating (2.88) twice, we obtain

(2.88)

32F(ol, fl , M1) #(ol,/~,M1) = -

0c~ 0/~

By recalling the definition (2.87) of F(ol, fl, M1), from (2.89) we find

(2.89)

#(ol,~,M1) - - 1 3 ~vll/~~.2f A (2.90) 2 Ool 013

Thus, if the distribution function # is determined from (2.89) or (2.90), then the restricted Preisach model will match the output increments along second-order reversal c u r v e s fMlfiOt attached to the first-order reversal curve fMlfi. This implies that the limiting ascending branch f+ will be matched because this branch coincides with the second-order reversal curves fMl&~ and because the value f_ is matched by the model. From (2.87) we also find that the output increments along the first-order reversal curve fM~ will be matched because these increments are twice of F(M1,/3,M1). Since the limiting ascending branch, output increments along the first-order transition curvesfM~ and the output increment along the second-order reversal curves fMa~ will all be matched, we conclude that the second-order transition curvesfM1/~c~ themselves will be matched.

Up to this point it has been tacitly assumed that input variations are started from the state of negative saturation (u(t) ~</30). This is a natural assumption because the state of negative saturation is a well-defined state. However, another well-defined state is the state of positive saturation (u(t) f> ol0). If input variations start from this state, then it is natural to assume that the distribution function depends on the first global mini- m u m ml. This leads us to the following restricted Preisach model:

f ( t ) = [f~ ~t(ol,~,ml)G~u(t)dold/3 -}-Cml, (2.91) ,!,11 m 1

where the support of/2(ol,/3, ml) is the triangle Tm~ defined by inequalities ml ~</3 ~< ol ~< ol0 (see Fig. 2.29).

The model (2.91) can be regarded as a counterpart of the model (2.82) and our immediate goal is to establish some connections between distribution functions # and/2. To this end, we first find the cons t an t Cml in

2.3 "RESTRICTED" PREISACH MODELS OF HYSTERESIS

(X /

97

FIGURE 2.29

(2.91) by matching the output values fml and f+"

fml = - - / / T /~(ot,/J, ml) do~ dfl + Cml, (2.92) m 1

f+ = /fT ~(0/,/J, ml) do~ dfl + Cal. (2.93) ml


G+f+ Cml = 2 ' (2.94)

and the restricted model (2.91) can be represented as

f i t ) = f fr ~(o~,~,ml)~,~u(t)dotd~ +fm~ 2 +f+" (2.95) ml

To determine the distribution function ~(c~,,8,ml), we shall use the decreasing second-order reversal r fml~ fi which are attached to the increasing first-order reversal curves fmla (see Fig. 2.30). As before we can introduce the function

1 F(c~, fi, m l ) - -~(fma6 -fml6fi) (2.96)

and show that

F(c~, fi, ml) -/fT(&fi)/~(c~', 13')dot' d]3'. (2.97)


f

T

FIGURE 2.30

From (2.97), we derive

/~(~, fi, ml) - - - O 2 F ( ~ ' fi, ml) 3& 3 fi " (2.98)

By invoking (2.96) in (2.98), we obtain

2 1 3 ~l~fi (2.99)

/~(~,fi, ml) -- ~ 3~3-------7"

Next, we will utilize the symmetry between the ascending and descending second-order reversal curves. In mathematical terms, this symmetry means that if

m l = - M 1 , ~ = - / J , f i = - ~ , (2.100)

then

F(~, fi, ml) - F(~,/J, M1). (2.101)

The last formula can also be expressed as

F ( - / J , - ~ , - M 1 ) - F(~,/J, M1). (2.102)


By substituting (2.101) and (2.100) into (2.98) and taking into account (2.89), we derive

/~(~, fi, ml) = O2f(~ - 3Ol 3~ = #(oG ~,M1). (2.103)

From (2.100) and (2.103) we finally obtain

lZ(Ol, fl, M1) = ~ ( - fl, -Ol, -M1). (2.104)

The last expression can be regarded as a generalization of the mirror symmetry property (1.46) previously derived for the classical Preisach model. Indeed, if M1 = d0 and ml = fl0, (fl0 = -c~0), then the models (2.86) and (2.95), respectively, coincide with the classical Preisach model. Conse- quently, we have

~(~, t~,~0) = ~(~, 3), ~(~, 3 , -~0 ) = ~(~, t~), (2.105)

and the equality (2.104) can be rewritten as

~(~, ~) = ~ ( - t ~ , - ~ ) , (2.106)

which is the property of mirror symmetry for the classical Preisach model. The formula (2.104) also shows that, having determined the function

/z(ol, fl, M1), we have solved the identification problem for the model (2.86) and its counterpart (2.95). This suggests that the model (2.95) is not independent of the model (2.86), but rather complements the latter model. In a way, we deal here with the same model which is written in two different forms. These two different forms correspond to two different initial states (states of positive or negative saturation). For this reason, in our subsequent discussion we will not draw any distinction between the models (2.86) and (2.95). In the sequel, all results will be discussed for the model (2.86), and it will be tacitly implied that they are also valid for the model (2.95).

It is clear that the model (2.86) has almost identical structure with the classical Preisach model. For this reason, it is apparent that the wiping-out property holds for the restricted model (2.86). However, the congruency property of minor hysteresis loops undergoes some modification. This modification can be described as follows. We call minor hysteresis loops comparable if they are formed as a result of back-and-forth input variations between the same consecutive reversal values and these input variations take place some time after the same largest input maximum was achieved. The given definition of comparable minor loops implies the possibility of different past input histories between the time when the largest maximum M1 was achieved and the time when the above mentioned back-and-forth input variations commence. By using the same reasoning as in Section 2 of


Chapter 1, we can prove that all comparable minor loops described by the restricted model (2.86) are congruent. It is clear from the above statement that the congruency of minor hysteresis loops prescribed by the model (2.86) for the same input reversal values are to a certain extent history dependent. This is not the case for the classical Preisach model.

It turns out that the wiping-out property and the modified congruency property are characteristic of the restricted model (2.86) in a sense that the following result is valid.

REPRESENTATION THEOREM The wiping-out property and the congruency property of comparable minor loops constitute necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the restricted Preisach model (2.86).

The proof of this theorem is very similar to the proof of the representation theorem for the classical Preisach model. For this reason, this proof is omitted.

It is clear that the modified congruency property is more general than the congruency property of the classical Preisach model. Thus the restricted model (2.86) is more general than the classical one. In addition, the restricted model (2.86) allows one to fit experimentally measured first- and second-order reversal curves, while the classical Preisach model is able to fit only first-order reversal curves. Since higher-order reversal curves are "sandwiched" between first- and second-order reversal curves, it is natural to expect that the restricted model will be more accurate than the classical Preisach model.

We next turn to the discussion of numerical implementation of the restricted model (2.86). As before, we will show that double integration in (2.86) can be completely avoided and that an explicit expression for the output, fit), in terms of the experimentally measured function, F(c~, f l ,M1), can be derived. The derivation proceeds as follows. From (2.86) by means of simple transformations we find

f (t) -/~S~dl (t) #(o~, fl, M1) do~ dfl --//SM 1 (t) tz ( ol , fl , M1) d ol d fl

f~l +f 4-

2

= 2/ / s~ 1 (t) # ( o~ , fl , M1) d ol d fl - / f T M lz ( o~ , fl , M1) d o~ d fl 1

4- f~l +f-

(2.107)


By invoking (2.83) and (2.84), we derive

/JTM /z(ol,/J, M1) dc~ dj3 f~'~l - f - (2.108) 2 1


/(t) =2 f f s tz(c~,fl, M1)dc~d~+f_ . (2.109) ~v/1 (t)

The set S~I (t) can be subdivided into n trapezoids Qk shown in Fig. 2.31. As a result, we have

n(t)

/ f s #(ol,~,M1)doldfi - E / / Q #(ol,~M1)dc~d~. (2.110) ~,I1 (t) k=l k

Each integral in the right-hand side of (2.110) can be represented as

/ f Q # ( c~ , [3 , M l ) d ol d [3 - / f T iz ( ol , /3 , M l ) d ol d /3 k (Mk,mk-1)

- - f f #(ol, [3,M1) do~ dfl. (2.111) JJT (Mk,mk)

(X (Mk,mk) l

I I I

' , I ' ,

FIGURE 2.31


By recalling (2.88), from (2.111) we find

ffQ #(Ot, ~,M1)dotd~ = F(Mk, mk_l,M1) - F(Mk, mk, M1). k

(2.112)

By substituting (2.112) into (2.110) and then (2.110) into (2.109), we finally obtain

n(t)

d(t) - 2 y~[F(Mk, mk_l,M1) - F(Mk, mk, M1)] 4- f - . k = l

(2.113)

Thus, the formula (2.113) expresses the output, fit), explicitly in terms of experimentally measured second-order reversal curves which are related to the function F by Eq. (2.87). This formula has a two-fold advantage over the expression (2.86). First, double integration is avoided. Second, the determination of #(c~,/~,M1) through differentiation of experimentally obtained data (see (2.89) or (2.90)) is completely circumvented. This is a welcome feature because the above differentiation may amplify noise inherently present in any experimental data.

So far we have discussed the restricted Preisach model for which the distribution function # depends only on the first global extrema M1 or ml. It is natural to call it a first-order restricted Preisach model. However, further generalizations in this direction are possible. In these generalizations, the function # is assumed to be dependent on some finite sequence of past dominant extrema: l~(Ot,~,Ml, ml,M2,m2,...,Mk, mk). It is natural to call these models as high-order restricted Preisach models. To make the general idea of these models clear, consider a second-order restricted Preisach model. In this model, the function # is assumed to be dependent of M1 and ml, and the model itself is defined as

fit) =//TM #(c~, ~,Ml, ml)G~u(t)dc~d~ 4- CMlm 1, (2.114) lml

where the support of # is the triangle ZMlml specified by inequalities ml ~< /J ~< c~ ~< M1 (see Fig. 2.32).

For any fixed extrema M1 and ml, the model (2.114) will be used to describe hysteresis behavior in the region confined between the first- order reversal c u r v e fM1X and the second-order reversal c u r v e fMlmlOt (see Fig. 2.33). This explains the origin of the term "restricted" used for the name of this model. However, the described restriction does not narrow at all the region of applicability of this model. This is because M1 and ml are free parameters which may assume any values between/~0 and d0. These values depend on a particular past input history and may even change with time when the previous extrema M1 and ml are wiped out by subsequent input variations.

2.3 "RESTRICTED" PREISACH MODELS OF HYSTERESIS

/

(M1 ,ml)

Mlm

S /

Y

O~

.p

103

FIGURE 2.32

f

fMlml~p

/ / / ~ _ .~ / / " M1

FIGURE 2.33


The c o n s t a n t CMl,ml in (2.114) can be determined from the condition that the model matches the output v a l u e s f ~ andfMlml"

f~/I1 = ffTM tx(ot, f l ,Ml , ml)dotdfl q-CMlml , (2.115) lml

fMlml = -- /fTM #(ot, fl,Ml,ml)doldfl ff-CMlml. lml

From (2.115) and (2.116), we find

(2.116)

f l~ 1 -F- fMlml CMlml = 2 ' (2.117)

which leads to the following expression for the model (2.114):

f ( t ) - / fTM # (~ , f l ,Ml ,m l )~ ,~u ( t )d~d f l -Jr- fM1 -}-fMlml (2.118) lm 1 2 "

To determine the distribution function #(c~, fl ,M1, ml), the experimentally measured third-order reversal curves fM~,m~ (see Fig. 2.33) should be employed. By using these curves, we can introduce the function

1 F(o~, fl, Ml , ml ) - - -~ ( f Ml ml Ot - - f Ml ml Otfi ). (2.119)

Exactly in the same way as before it can be shown that the second-order restricted Preisach model will fit the third-order reversal curves if the function F(ot, f l ,Ml , ml) is related to the function #(ol, f l ,Ml , ml) by the equation

f F

F ( ot , fl , M l , m l ) -- U # (or', fl ', M l , m l ) dot' d fl ' . JJT (~,~)

From the last equation we easily derive

32F(ot, fl, M1, ml ) tx(oG fl, Ml , ml) = -

and 1 02dMlmlOefi

#(C~, fl, M1, ml) = -~ 3ot Off

(2.120)

(2.121)

(2.122)

By its design, the model (2.11) is valid under the condition that the initial state of hysteresis transducer is the state of negative saturation. If the initial state is the state of positive saturation, then the following counterpart of the model (2.111) can be used:

f i t ) - f f T ~z(ot, fl, f f n l , ~ l l ) ~ u ( t ) d ~ d f l + fYml +J~IM, (2.123) /~Ilrh 1 2 '


where if/1 is a global minimum and M1 is a subsequent global maximum of the input.

As before, we can establish the following relationship between the function/~ and/~:

/~(ol,/J, M1, ml) - / ~ ( - / J , - o t , - M 1 , - m l ) , (2.124)

which shows that (2.118) and (2.123) are just different forms of the same model. These two different forms correspond to two different well-defined initial states of hysteresis transducer.

It is apparent that the congruency property of minor hysteresis loops undergoes further modification (further relaxation) in the case of the second-order restricted Preisach model (2.118). The essence of this modification is that for minor loops to be congruent the corresponding inputs must vary back-and-forth between the same consecutive reversal values and also assume in the past the same values of M1 and ml. It can be proven that the wiping-out property and the above-mentioned modified congruency property of minor loops constitute necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the model (2.118). This is the so-called representation theorem for the second-order restricted Preisach model. As before, it can be shown that the following expression can be used for the numerical implementation of the model (2.118):

n(t)

fit) = 2 E[F(Mk, mk_x,Ml,ml) -- F(Mk, mk, MI,ml)] -}-fMiml. (2.125) k

The main advantage of this expression over the formula (2.118) is that it represents the output, fit), directly in terms of the experimentally measured third-order transition curves.

It is clear from the previous discussion that higher-order restricted Preisach models of hysteresis can be defined as

f ( t ) - ff_ i~(ot,~,Ml,ml,...,Mk)~,~u(t)dotd~ J J l M lml""Mk

-}- CMlml...Mk. (2.126)

It is also apparent now how higher-order transition curves can be used for the determination of the function #(c~, ~,M1, ml,. . . ,Mk). It goes without saying that by increasing the order of the restricted Preisach model we can increase the accuracy of this model. However, this increase in accuracy is amply paid for by the increase in the amount of experimental data required for the identification of higher-order models. For this reason, the

1 0 6 CHAPTER 2 Generalized Scalar Preisach Models of Hysteresis

RM 1

(X

u (a ,13) I u

R (0(,, ,u)

F I G U R E 2 . 3 4

use of restricted Preisach models of order higher than two does not seem to be practically feasible or attractive. It is instructive to note that another approach to the construction of Preisach models that can match any desired number of higher-order reversal curves has been recently proposed in [14-16].

We conclude this section with a brief discussion of "the restricted nonlinear" Preisach model of hysteresis.

It is clear from the name of this model that it combines main features of "restricted" and "nonlinear" models. The first-order restricted nonlinear (input-dependent) Preisach model can be defined as

f fR fMlu(t) + fu+(t) f(t) = tz(a, fl, u(t),M1)9~u(t)dadfl + 2 " (2.127) M1 U

where the moving support of #(ot, fl, u(t),M1) is the rectangle RMlU specified by inequalities fl0 ~< fl ~< u(t), u(t) <<, ~ <, M1 (see Fig. 2.34), and the meaning offMlu(t) is clear from Fig. 2.35.

By introducing the function

P(~, fl, u, M1) =fMl~U --fMl~u (2.128)

and by using the same reasoning as in the previous section, we can establish that

M1)-2 f f R I~(a',fl',u, M1)dol'dfl'. (2.129) (~,~,u)

u

M 1


F I G U R E 2.35

From (2.129) we can derive the following expressions:

and

1 32p(oe, fl, u, M1) #(oe, fl, u, M1) - - - , (2.130)

2 3oeOfl

1 02fMlfl~u #(oe, fl, u, M1) . . . . �9 (2.131)

2 Ooe Off

We can also derive the following explicit expression for the output, fit), of the model (2.127):

n(t)

f ( t ) - - & l u ( t ) -[- ~ _ , [ P ( M k + l , m k , u ( t ) ,M1) - P(Mk, mk, u( t ) ,M1)] . k

(2.132)

It is worthwhile to remember here that the function P(~, fl, u, M1) is directly related to experimental data and it has the meaning of output increments between the second- and third-order reversal curves.

It is easy to formulate and to prove the representation theorem for the model (2.127). This is left to the reader as a useful exercise.


2.4 D Y N A M I C P R E I S A C H M O D E L S OF H Y S T E R E S I S

In all previous sections our discussion has been centered around the classical and generalized Preisach models of hysteresis that are rate- independent in nature. The term "rate-independent" implies that in these models only past input extrema leave their mark upon the future values of output, while the speed of input and output variations has no influence on branching. The intent of this section is to relax the "rate-independence property" of Preisach type models. For this reason, new Preisach type models that are applicable to the description of dynamic hysteresis are introduced. The identification problem of fitting these models to some experimental data is then studied. Finally, some discussion is presented concerning numerical implementation of the dynamic Preisach models of hysteresis.

The main idea behind the dynamic Preisach-type models of hysteresis is to introduce the dependence of #-functions on the speed of output

variations, af This leads to the following dynamic Preisach models of hys- 37" teresis:

f(t) - ff~> l~(oe, g, df--f )~,a3u(t)doedfl, (2.133)

f(t) - # oe, g, u(t), ~ G3u(t) doe dg + u(t)

f~t) ff- fu+(t) (2.134)

The above models are "dynamic" generalizations of the classical Preisach model and the nonlinear Preisach model, respectively. Similar generalizations for the restricted Preisach models are apparent. They will not be discussed in this section because their treatment mostly parallels that of the model (2.133).

The direct utilization of the models (2.133) and (2.134) is associated with some untractable difficulties. First,/~-functions depend on the un-

known quantity, af and this complicates numerical implementations of 37, the models (2.133) and (2.134). Second, it is not clear how to pose the identification problems for these models. The above difficulties can be completely circumvented by using the power series expansions for /~-

functions with respect to ~t"

(dr) (oe,/j) _}_ ~ft/AI(C~, j~) _} - ~ , ~ , ~ = ~ 0 " " , (2.135)

dr) (oe, g, u(t)) + df (oe, ~, u(t)) + # oe, fl, u ( t ) ,~ =/~0 ~-~#1 " ' ' . (2.136)

2.4 DYNAMIC PREISACH MODELS OF HYSTERESIS 109

By retaining only the first two terms of the above expansion, we arrive at the following dynamic models:

fit) = J X ~ #o(a, ~)G~u(t) da d~

~1 (O/, ~)G~u(t) da d~,

f i t ) - / /Ru( t ) tto(ol, fl, u(t))~,~u(t) dot dfl + f~t) q-.fu+(t)

-Jl- -dtdf //Ru(t) #l (C~, ~u(t))~,~u(t) dotdfl.

(2.137)

It is clear that in the case of very slow output variations, the above models are reduced to the corresponding rate-independent hysteresis models. This means that #0-functions in (2.137) and (2.138) should coincide with the #-functions of rate-independent models (2.1) and (2.26), respectively. In other words, the #0-functions in (2.137) and (2.138) can be determined by matching first-order and second-order transition curves, respectively.

The above reasoning suggests that the models (2.137) and (2.138) can be represented in the following equivalent forms:

fit) =d~(t)+ ~ ~>~ ].tl (Or, ~)G~u(t) dot d~, (2.139)

df /fR (og, u(t))~,~u(t) do~ fit) u(t) (2.140)

where the f-terms stand for the "static" components of hysteresis nonlinearities:

J:(t)-- /~ #o(o~,~)G~u(t)dotd~, (2.141)

]:(t) - f f tto(o~, ~,u(t))~,~u(t)dc~d~ + du+(t) -+-d~t) . (2.142) u(t) 2 ,I dR

The expressions (2.139) and (2.140) are transparent from the physical point of view. They show that the instant speeds of output variations are directly proportional to the differences between instant and "static" output values.

We next turn to the identification problems of determining the #l- functions by fitting the models (2.139) and (2.140) to some experimental data. We first consider the identification problem for the model (2.139).

(2.138)


The following experiments can be used to solve this problem. Starting from the state of negative saturation, the input u(t) is monotonically increased until it reaches some value cz at t = to and it is kept constant for t ~> to. As the input is being kept constant, the output relaxes from its value f~ at t - to to its static value ?~. According to the model (2.139), this relaxation process is described by the differential equation

df + f = ~ (2.143)

where

"~cz - / I s ]Zl(OlZ,~')dccZdfl'-/~s #l(Ol',~')dolZd~', (2.144)

and S; and S + are negative and positive sets on the Preisach diagram (see Fig. 2.36) corresponding to the above input variation.

The solution to Eq. (2.143) is given by

f i t ) - (f~ -J:~)e - t + f~. (2.145)

Thus r~ has the meaning of relaxation time and can be experimentally measured.

Next, the hysteresis nonlinearity is brought back to the state of negative saturation. Starting from this state, the input is again monotonically increased until it reaches the value c~. Then the input is monotonically decreased until it reaches some value/~ at time t - t~ and it is kept constant for t > t~. As the input is being kept constant, the output relaxes from its

s~

s~

0( , J

s~ ((x

Soq3

y FIGURE 2.36 FIGURE 2.37


value f~/~ at t = t~ to its static value 3~/~. The model (2.139) yields the following differential equation for the above relaxation process:

df + f =d~/~ ' (2.146)

where according to the model (2.139) and the Preisach diagram shown in Fig. 2.37 the coefficient r~/~ is given by the expression

raft = ffS #l(ot',/3')dot'd/3'-/~s #l(ot',/3')dot'd/3'. (2.147) ;e :e

By solving (2.146), we find t

f i t ) - (f~ -K~)e ~e +K~. (2.148)

Thus, r ~ has also the meaning of relaxation time and can be experimentally measured. It is apparent, that the relaxation time r~ can be construed as the relaxation time r ~ , and, consequently, it belongs to the set {r~/~}. We next show that by knowing these relaxation times, we can find the function/,1 (ot, ~). TO this end, we introduce the function

q(ot, fl) = r~ - r~/~. (2.149)

From (2.144), (2.147) and (2.149), we find

q(ot, fl) = - 2 [[~ ptl (ot',/~') dot' dfl'. (2.150) J J l ~

where T~/~ is a triangle shown in Fig. 2.37. From (2.150), we can easily derive the expression

1 32qr /Zl (ot, /~) = ~ ----~'3ot (2.151)

By recalling (2.149), from (2.151) we find

]Zl(ot, f l) - - 132ra--------~fl. (2.152) 2 3ot Off

The solution to the identification problem for the input-dependent (nonlinear) model (2.140) can be found in a similar way. For this reason, our comments will be concise. We consider two types of relaxation processes. A first type process occurs after monotonic input increase to some value Ot and subsequent monotonic decrease of input to some value u. According to the model (2.140), the first type relaxation processes are described by the differential equation

d/+f=Ku (2.153) r~u -d7


where

run = f / s #l(Ot',fl',u)dot'dfl'- f / s #l(C~',fl',u)do~'dfl'. (2.154) =u L

From the previous differential equation, we find

f i t ) = (f~u -Y:~u)e - t +Y:~u. (2.155)

A second type relaxation process occurs when the input is first monotonically increased to c~, then decreased to fl, again monotonically increased to u, and kept constant afterwards. These relaxation processes are described by the equation

df +f=y~ u ' (2.156) r~u -~

where

r ~ u - / / S #l(Ot',fl ' ,u)do~'dfl'-f/S I~l(Ot',fl',u)dot'dfl'. (2.157) ;~u +~u

As before, we have t

d(t) = (f~u -j:~u)e ~u + f~u. (2.158)

It is clear from (2.155) and (2.158) that r~u and r~u have the physical meaning of relaxation times and can be measured experimentally. Know- ing these relaxation times, we can define the function

Q(c~, fl, u) = run -- r~u. (2.159)

From (2.154), (2.157) and (2.159), we derive

Q(c~, fl, u) = - 2 f f /,/,1 (c~', fl', u) dol' dfl'. (2.160) ddR (~,fl,u)

From (2.160), we obtain

1 32Q(o~, fl, u) #I(CGfl, U) = ~ 00~ Off " (2.161)

By invoking (2.159), from (2.161) we have the alternative expression for #1:

1 02to, flu ~I(OG fl, U)-- ~ . (2.162)

2 Oot Off Consider the relaxation processes that are symmetric to those dis-

cussed above. Suppose that we start from the state of positive saturation and reduce the input to some value fi, then increase the input to some


value & and keep it constant thereafter. Suppose also that the subsequent output variations are characterized by the relaxation times rd6. If

f i = - ~ , & = - f l , (2.163)

then due to the symmetry we have

rfi~ = r~ . (2.164)

As before, we can derive that

1 32rfi~ (2.165) ~1 (6, ~) -- 2 3& Off"

By substituting (2.164) and (2.163) into the right-hand side of (2.165) and by recalling (2.152), we obtain

1 32rotfl _ [Zl (6 , ~ ) -- 2 Off 3c~ - -#1(or,/J). (2.166)


]Zl (--~J,--or) -- --~1 (or, fl). (2.167)

Similarly, we can introduce the relaxation times rfi6~ and prove that

1 32 rd~ (2.168)

~ l ( ~ , f i , U ) - 2 36 Off"

If

f i - - - ~ , 6 = - f l , ~ = - u , (2.169)

then due to the symmetry we have

rfi~ = r~u. (2.170)

By using the same line of reasoning as before, from (2.162), (2.168), (2.169) and (2.170) we derive

/Z 1 (--fl, --0~, --U) -- --#1 (Or, fl, U). (2.171)

We next proceed to the numerical implementation of the dynamic models. The models (2.139) and (2.140) can be represented as the following differential equation:

af a(u(t)) ~ + f ( t ) = f ( t ) , (2.172)


where the hysteretic coefficient a(t) is defined by the formulas

?l(u(t)) = - f ~ #1 (a, fl)f,~flu(t) da dfl, (2.173)

?z(u(t)) = - [ L #1 (Or, ~, u(t))f ,~u(t) da dfl (2.174) d dl ' ( u(t)

for the models (2.139) and (2.140), respectively. The explicit solution to Eq. (2.172) is well-known and can be expressed

as If0 ~0 t f(~) ] f (t) = b(t) + d~ (2.175)

a(~)b(~) "

where f0 is the initial output value, and b(t) is given by the formula

( f O td~ ) b( t )=exp - a -~ " (2.176)

The expressions (2.175) and (2.176) can be efficiently used for computing f(t), if f(t) and a(t) are known. The calculation of static ;Ccomponents in the case of the classical Preisach model as well as the nonlinear Preisach model has been discussed in detail in the previous sections. Since the mathematical structure of a(t) is similar to the mathematical structure of the rate-independent Preisach models, similar explicit formulas can be derived for the numerical evaluation of a(t). In particular, it can be shown that the following formulas are valid for a(t) in the case of models (2.139) and (2.140), respectively:

n(t) ~(u(t)) = 2q(c~o, rio) - 4 ~ [ q ( M k , ink-l) -- q(Mk, ink)],

k=l (2.177)

1 ?z(u(t)) = - -~ Q(ao, rio, u(t))

n(t) - y~[Q(Mk+l,mk, u ( t ) ) - Q(Mk, mk, u(t))].

k=l (2.178)

The above formulas are convenient not only because they give explicit expressions for integrals in (2.173) and (2.174), but also because these expressions are presented in terms of experimentally measured data.

It has been tacitly assumed in the foregoing discussion that the relaxation processes used in the identification procedures are well characterized by single relaxation times. If this is not the case and several relaxation times have to be employed to describe the above relaxation processes, then


the discussed dynamic models must be generalized. The natural way to generalize these models is to use higher-order differential equations with hysteretic coefficients in order to account for several relaxation times. We demonstrate such a generalization for the second-order dynamic model:

d2f ~(1) df II (2) (u(t)) -~s + (u(t)) ~ + f (t) =;?(t), (2.179)

where

f(t) = ~ txo(c~, ~)~,~u(t) d{~ d~, (2.180) d d{~

/f~ tXl(Ol, ~)G~u(t)dold~, (2.181) ~(1) (u(t)) -- >~

/ f~ #2(c~, dc~ d/~. (2.182) ~(2) (u(t)) -- >~r [3)G~u(t)

To find IXl(~,/~) and #2(c~,/~), we shall use the same two relaxation processes as for the identification of the model (2.139). In the first process, we start from the state of negative saturation and monotonically increase input to some value c~ and keep it constant thereafter. As the input is being kept constant, the output relaxes. According to (2.179), this relaxation is described by the differential equation

a~ ) d2f a(1) d / + f _;~ (2.183) ? g + �9

A solution to this equation has the form t t

f(t) = cgl)e 41) + C~)e r(2) "q-fo/, (2.184)

where r (1) and r (2) are the roots of the characteristic equation

a~)r 2 + a(1)r + 1 = 0. (2.185)

Consequently,

a~ ) = 1 r(1)r(2), (2.186)

(1) r(2) a(1)= -r~ + (2.187)

~ 1 ) r12 ) "

The c o n s t a n t s l (1) and r (2) have the physical meaning of relaxation times and can be measured experimentally. This leads to the experimental de-

termination of a~ ) and a (1) according to (2.186) and (2.187).

116 C H A P T E R 2 G e n e r a l i z e d Scalar P r e i s a c h M o d e l s of H y s t e r e s i s

Now we consider the second relaxation process. In this case we start from the state of negative saturation, increase input to some value ot, then decrease input to some value fl and keep it constant thereafter. As the input is being kept constant, the output relaxes and this relaxation is governed by the equation

a(2) d 2 f ..(1) df + f __~, (2.188) ag + a7

whose solution is given by t t

, - . (1) r (1) . - , (2 ) r (2) ~ f(t) = ~,~e ~ + t ,~e ~e + f ~ . (2.189)

_(1) _(2) The constants ~/~ and , ~ have the meaning of relaxation times and can

_(2) and be measured experimentally. As soon as this is done, we can find u~/~ a(1)

oefl"

a(2) 1 (2.190) eft "-- _(1)_(2) '

_(1) _(2) _(1) z~fl + t~/~ (2.191) u otfl - - - _(1)_(2) "

Next we introduce the functions

(1) = a(1) _(1) (2.192) ~t~ -%t~ '

(2) _. a~) ..(1) (2.193)

which are directly related to the above-mentioned experimental data. Us- ing the same reasoning that we used many times before, we can establish the formulas

(1) = 2 I f (ot', fl') dot' dfl', (2.194) otfl J J T(ot,fl )

1"1

(2) = 2 f fT t'2 (ot', fl') dot' dfl', ~ (~,~1

from which we derive

(2.195)

,,2 (1)

1*l(ot, /J) = 1 0 q~ (2.196) 2 3ot Off'

,,2 (2) 1 o q~

1*2(ot, fl) = - - . (2.197) 2 Oot Off

2.5 PREISACH MODEL OF HYSTERESIS WITH ACCOMMODATION 117

Thus, the identification problem for the second-order model (2.179)- (2.182) is solved. Extensions to higher-order dynamic models are straightforward.

2.5 PREISACH M O D E L OF HYSTERESIS WITH A C C O M M O D A T I O N

The Preisach type models described in previous sections exhibit the wiping-out property. This property results in an immediate formation of minor hysteresis loops after only one cycle of back-and-forth input variations between any two consecutive extremum values. However, experiments show that in some cases hysteresis loop formations may be preceded by some stabilization process that may require appreciable number of cycles before a stable minor loop is achieved. This stabilization process is often called in the literature "accommodation" or "reptation" process. Sometimes this accommodation process can be appreciable and then it is important to model it. In this section we discuss a certain modification of the moving Preisach model which allows one to account for the accommodation process. Another approach to the Preisach modelling of accommodation is described in [15].

We define the Preisach model with accommodation as

- f f~ /z(~, fl,f(m))~,~u(t)dol dfl 4- r(u(t)). (2.198) f(t) J d l*~ u(t)

In the previous formula r(u(t)) stands for a fully reversible component which is represented by some single-valued function of u(t), while

d (m)- Mk f if (a, fl) ~ Rk(t), (2.199)

where Mk f are local extremum values of output f and Rk(t) are rectangular

regions formed after the extremum M f had been achieved (see Fig. 2.38). It is easy to see that the regions Rk(t) are reduced with time and can

even be completely wiped out. This fact is reflected in (2.199) by indicating

that Rk are dependent of t. It is also clear from Fig. 2.38 that M f are local output minima for even k and local maxima for odd k.

We next prove that the model (2.198) does account for the accommodation process. For the sake of clarity, we will show this for the simplest case when the initial state of a hysteresis transducer is the state of negative saturation, and the input u(t) is monotonically increased to some value u+ and is being varied back-and-forth between u+ and u_ thereafter. More general cases can be treated in a similar way.

Ro

I


(X Ro. [

R2k+l I---~ I U+

(X

/ i

Y i

118


As the input is increased to the value u+, the output reaches maximum

value M1 f which, according to (2.198) and Fig. 2.39, is equal to

M f = - f ~ # (ot, fl, Mfo) dot dfl + r(u+), (2.200) 0

where M0 f =f_ , (2.201)

and f_ is the output value in the state of negative saturation. As the input is monotonically reduced to the value u_, the output

reaches minimum value M f. According to (2.198) and Fig. 2.40, this minimum value is given by

Mr2 = ffI~l ~(a'fl 'Mf)dadfl- ffRo #(~176

+ r(u_). (2.202)

It is clear that M2 S # M0 f =f_. (2.203)

As the input is monotonically increased again to the value u+, the output

reaches some maximum value M f. According to (2.198) and Fig. 2.41, for this value we find

+ r(u+). (2.204)

2.5 PREISACH MODEL OF HYSTERESIS WITH ACCOMMODATION

Ro I

R1-

U

Ro

U+

2


119

By comparing Figs. 2.39 and 2.41 as well as formulas (2.200) and (2.204) and by taking the inequality (2.203) into account, we conclude

M3 f g= M f. (2.205)

As the input is monotonically decreased again to the value u_, the output

reaches some minimum value M4 f which according to Fig. 2.42 is given by

~(~, e,M3 ~) a~ ae - fs ~(~' e' M~ a~ ae

+ r(u_). (2.206)

By comparing formulas (2.202) and (2.206) and by taking into account that R1 - R3 and the inequality (2.205), we find

M4 f r M f. (2.207)

By continuing the same line of reasoning, we can establish that

f Mf2k+l ~k M2k_l , and

V2ik # Vaik_2 �9

(2.208)

(2.209)

The last two inequalities show that some stabilization (accommodation) process is going on before a stable minor loop is asymptotically reached. This accommodation process is illustrated by Fig. 2.43.

We next proceed to the discussion of the identification problem. It turns out that the model (2.198)-(2.199) is very general in nature and it

120

R3-


(x Ro /

I I i I I i


is not clear at this point how to solve the identification problem for this model. However, the identification problem becomes tractable if the dependence of/z-function on f(m) is factored out. In this way, we arrive at the following model:

f ( t ) - lf_ v(f(m))#(~176 + r(u(t)), (2.210) J J l ~ u(t)

where f(m) are defined in the same way as in (2.199), and/;-function is assumed to be known. Thus the identification problem consists in determining the functions r(u) and/~(r by fitting the model (2.210) to some experimental data. We begin with the determination of the function r(u). Suppose that starting from the state of negative saturation the input is monotonically increased to some value u. As a result of this monotonic increase, the output will reach some value f+ on the limiting ascending branch. According to (2.210), for this output value we have

f+ = -v(f_) f f #(c~,/~)d~d/~ + r(u). (2.211) ddR (~0,&,u)

Now consider the state of positive saturation and suppose that the input is monotonically decreased to the same value u. In this case, the limiting descending branch will be followed and some output valuefu on this branch will be reached. According to (2.210), for the above mentioned output value we find

f f #(c~,/~) dc~ d/~ 4- r(u), (2.212) fu 1) ( f q _ )

ddR (~0,&,u)

2.5 PREISACH MODEL OF HYSTERESIS WITH A C C O M M O D A T I O N 121

where as before f+ is the output value in the state of positive saturation. From (2.211) and (2.212), we derive

r(u) = v(f+)f+ + v(f_)fu. (2.213) v(f +) + v(f_)

Consider a particular case when v-function is even. In this case, since f_ = -f+, we find

v(f+)- v(f_), (2.214)

and from (2.112) we conclude

1 r(u) - -~ (fu + + fu). (2.215)

This coincides with the previous result for the moving Preisach model. We next turn to the determination of the/ ,-function in (2.210). Con-

sider first-order reversal curves f~3 which are attached to the limiting ascending branch. By fitting these curves to the model (2.210), we arrive at the following equation:

/ / ~(~', b')a~' ab' f~e l) ( fot ) ddR (~,&,3,3)

- v(f_) ~ /z(d,/3') dd dfl' + r(3). (2.216) ddR (~0,30,~,3)

From (2.216), we obtain

f~3 - fifl) - ffR #(d,/3 ') d d dfl' v(f~) (~,~o,b,~)

v(f_) ffR #(d,/3') dd dfl. (2.217) v(f~) (~o,~o,~,~)

On the other hand, by fitting the limiting descending branch f~- to the model (2.210), we find

- ~(f+) ff ~(~,, fl,) dd dfl' + r(fl). (2.218) f; ddR (~0,&,3,3)

The expression (2.218) can be transformed as follows:

f; r([3 )

ff~ ~ (~', b')a~' a~ ~(f+) (~o,eo,~,e)

+ I I /z(c~', fl') d d d3'. (2.219) J J R (~,3o,3,3)


By introducing the function

T(oe, fl) = [ [ #(d , fl') do~' dg' (2.220) J a R (~0,&,~,3)

and by using (2.217) and (2.220), we derive the following expression for T(c~,/3) in terms of experimentally measured data:

v(f~)[f; - r ( g ) ] - v(f+)[f~ e - r(g)] T(c,,fl) = . (2.221)

v(f +)[v(f,~) + v(f_)]

By knowing the function T(~, g) and by employing the same reasoning as usual, we can derive from (2.220) the following expression for #:

32T(ol, fl ) #(c~, g) = -- . (2.222)

O~ Og

For the purpose of numerical implementation of the model (2.210) we do not need to know the function/, . This is because an explicit expression for the output in terms of the function T(c~,/3) can be derived. The derivation proceeds as follows. From (2.210) and Fig. 2.38, we find

n(t)

f(t) = Z v ( G _ I ) f f R , (~ ,e )a~ae k=l 2k-1

n(t)

E v(M+k) ffI~ it(a, fl)dc~ dg. (2.223) k=0 2k

Due to the definition of T(~,/5), we have

ffR ~(~' fl) dol dfl - T(Mk+I, ink) -- T(Mk, ink). (2.224) 2k-1

Similarly, we find

ffR ~(~' fl) dol dfl - T(Mk+I, mk+l) -- T(Mk+I, mk), (2.225) 2k

where it is assumed that T(Ml, mo) =0. (2.226)

By substituting (2.224) and (2.225) into (2.223), we derive

n(t)

f ( t ) - ~ v(Mfk_l)[T(Mk+l,mk)- T(Mk, mk)] k=l

n(t)

- Z v (G)[T(Mk+l 'mk+l) - T(Mk+l,mk)]. (2.227) k=0

2.5 PREISACH MODEL OF HYSTERESIS WITH ACCOMMODATION 123

This is the final expression for the output in terms of the function T which is directly related to the experimental data by the formula (2.221).

We will conclude this section by the discussion of one sufficient condition which guarantees the convergence of the stabilization (accommodation) process. This condition is given by the following inequality:

maxlv' (Mf)[. n~x [[/~(~, fl)do~ dfl - q < 1, (2.228) Mf JJR

where v' stands for the derivative of v-function, and R is any rectangle within the limiting triangle.

The proof that the condition (2.228) guarantees the convergence of the stabilization process is given below for the simplest case. This is the same case that was discussed in the beginning of this section when we proved that the model (2.198) did account for accommodation. More general cases can be treated similarly.


0

- v(M2dk) ffR p~(ot, fl)do~dfl + r(u+), (2.229) +

where the notation R+ is used for R2k because they do not change from one cycle to another.

Similarly we have

G = v(M2fk-1)ffa_ #(ot, fl)dadfl

- v(f_) f~ tx(ot, fl)dadfl + r(u_), (2.230) JJl< 0

where R_ stands for R2k+l. From (2.229) and (2.230), we derive

f Mf2k_l M2k+X -- -- - ['(M2fk) - " ( G - s ) ]

I f #(ot, fl)doldfl, (2.231) x J Jl< +

x f/R_ #(~'/~) d~ d#. (2.232)


It is apparent that

[v(Mf2k)- v(G_2)I ~ max v ' (Mf ) I IG-G_21 , (2.233) MY Iv(Mf2k-1)- v(G-3 ) l ~ maxlv'(Md)lIG-1--G-B1" (2.234) MY

From (2.228) and (2.231)-(2.234), we derive

I G + I -- G - 1 1 ~ q l G - G - 2 I, (2.235)

]M+k -- G - 2 1 ; qIG-X -- G - 3 I" (2.236)

By combining (2.235) and (2.236), we obtain

IM2fk+X -- G - 1 1 * q2 I G - 1 -- G - 3 I' (2.237)

]G+2 -- G I ; q a I G -- G - 2 1 " (2.238)

Inequalities (2.237) and (2.238) mean that extrema {M2fk+l } {Mf2k } and form two contracting sequences. For this reason, there exist the limits:

Y: Mu d (2.239) lim M2k+l = +, k--+c~

lim M/2k = M/u_. (2.240) k---~ cx~

Consequently, the condition (2.228) guarantees the convergence of the accommodation process.

2.6 M A G N E T O S T R I C T I V E H Y S T E R E S I S A N D P R E I S A C H M O D E L S W I T H T W O I N P U T S

The essence of magnetostrictive phenomenon is the dependence of strain on magnetization. Since the magnetization can be varied by applied magnetic fields, this opens the opportunity to control the strain of magnetostrictive rods by magnetic fields. For this reason, magnetostrictive materials (especially Terfenol type materials with giant magnetostriction) are very attractive as actuator materials for many applications such as robotics, active vibration damping, micromotors, etc. However, magnetostrictive materials exhibit hysteresis, which represents a problem for fine positioning applications. If hysteretic effects of magnetostrictive materials could be predicted, then actuator controllers could be designed to correct for these effects. This would result in high precision actuators powered by

2.6 MAGNETOSTRICTIVE HYSTERESIS 125

magnetostrictive materials�9 It is clear that mathematical models for magnetostrictive hysteresis could facilitate the design of the above controllers. The Preisach modelling of magnetic hysteresis of Terfenol type materials was first attempted in [18].

It is known that magnetostrictive materials exhibit hysteretic behavior with respect to variations of two variables: magnetic field and stress. Thus, the problem of developing the Preisach type models with two inputs presents itself.

For the sake of generality, we consider a hysteresis nonlinearity that can be characterized by two inputs u(t) and v(t) and an output fit). In magnetostriction applications u(t) is the magnetic field, v(t) is the stress, while f(t) is the strain. We shall discuss the following Preisach type model with two inputs:

f i t) - / ~ /, (oe, fl, v(t))p~u(t) doe dfl

+ / ~ v(oe, fl, u(t))~,~v(t)doe dfl. (2.241)

In the above model, the dependence of the functions/, and v on v(t) and u(t), respectively, reflects the cross-coupling between two inputs.

The investigation of the model (2.241) is greatly facilitated by its equivalent representation as a model with moving supports for/,(oe, fl, v(t)) and v(oe, fl, u(t)). To arrive at this representation, we introduce the sets S+u(t), au(t), S u(t), + Sv(t), Rv(t) and S-~t ), which can be defined by the same formulas as (2.2)-(2.4). By using these sets, the expression (2.241) can be transformed as follows:

f i t ) - / /Ru( t ) i~(oe, fl, v(t))~,~u(t) doe dfl + f /R v(oe, fl, u(t))~,~v(t) doe dfl v(t)

+ / / s I~(oe, fl, v(t))doedfl- / f s #(oe, fl, v(t))doedfl +u(t) u(t)

+ / f s v(oe, fl, u(t)) doe dfl - / f s v(oe, fl, u(t)) doe dfl. +v(t) ~t)

(2.242)

We next find a simple expression for the last four terms in (2.242). Con- sider first a state of negative saturation and a subsequent monotonic increase of inputs until they reach some values u(t) and v(t) Let f,+ be �9 u(t)v(t) the resulting output value. Then, according to (2.242), we find:

fu+(t)v(t) -- _ / f a u ( t ) # ( oe , fl , v ( t ) ) doe d fl - / f R v ( oe , fl , u ( t ) ) doe d fl v(t)


+ f~s #(o~,g,v(t))do~dg- ffs v(o~,g,u(t))doldg u+(t) ;{t)

+ f~S v(ol, g,u(t))dotdfl- l~s v(ol, g,u(t))do~dfl. (2.243) +v(t) ~t)

Similarly, starting from the state of positive saturation, we can define f~t)v(t) and derive the following expression:

f~t)v(t) --" f~Ru(t) #(ol, g,v(t))dotdg + lfRv(t) v(ol, g,u(t))do~dfl

+ f~s+ #(~,g,v(t))d~dg - ffs- #(~,g,v(t))d~dg u(t) u(t)

4- f~S v(~,g,u(t))do~dfl- f~s v(o~,g,u(t))dotdg. (2.244) v(t)+ U(t)

From (2.243) and (2.244) we derive

Su+(t) u(t)

+ fs + v(ol, g,u(t))d~dg - ~s- v(~,g,u(t))d~dg. v(t) v(t)

(2.245)

By substituting (2.245) into (2.242), we arrive at the following representation of the model (2.241):

f(t) - J~R #(ol.g.v(t))9.,u(t)dotdg 4-11, v(ol.g.u(t))9..flv(t)d~dg u(t) v(t)

1 + 4- -2 G(t)v(t) 4- f~t)v(t))" (2.246)

The last expression can be regarded as a generalization of (2.12). By using the same line of reasoning as in previous sections, it can

be established that the model (2.246) has the following two characteristic properties.

WIPING-OUT PROPERTY Only the alternating series of past dominant extrema of u(t) and v(t) are stored by the model (2.246), while all other past extrema of u(t) and v(t) are wiped out.


PROPERTY OF EQUAL VERTICAL CHORDS All minor hysteretic loops corresponding to the same consecutive extremum values of u(t) for the same fixed value of v have equal vertical chords regardless of the past history of variations of u(t) and v(t). The same is true for minor hysteretic loops formed as a result of back- and-forth variations of v(t) for any fixed value of u.

It turns out that the model (2.246) has also the following distinct property.

PATH INDEPENDENCE PROPERTY Consider two points (Ul, Vl) and (u2,v2) on the u-v plane and a set of paths connecting these points and corresponding to monotonic variations of both u(t) and v(t) (see Fig. 2.44). Then, the output increment predicted by the model (2.246) does not depend on a particular monotonic path between the points (Ul, Vl) and (u2, v2).

PROOF. For any instant of time the model (2.246) can be represented as

f(t) = / / s # ( c ~ , ~ , v ( t ) ) d ~ d ~ - / / s #(~,~,v( t ) )d~d~ +(t) -(t)

+(t) -(t)

1 + if- -2 (fu(t)v(t) -ff d~t)v(t))" (2.247)

Here S+(t),S-(t),~+(t) and ~-( t ) are positive and negative sets on which Ru(t) and Rv(t) are subdivided, respectively.

(u2,v2)

(U 1 ,Vl)

FIGURE 2.44


It is clear that the above subdivision on positive and negative sets will be the same at the point (u2, v2) for all monotonic paths connecting this point with (Ul,Vl). This is because the monotonic variations of u(t) and v(t) will result in the same modification of negative and positive sets upon achieving the values u2 and v2, respectively. From here it is easy to conclude the validity of the above property.

We next proceed to the discussion of the identification problem for the model (2.246). In order to determine the functions #(c~, fl, v) and v(~, fl, u) we shall use the first-order transition curvesf~v andfu~ . As before, these curves are measured for piece-wise monotonic input variations started from the state of negative saturation. Namely, f ~ v is the output resulting from monotonic increases of two inputs to the values c~ and v, respectively, and subsequent monotonic decrease of u(t) to the value ft. The output valuefur can be determined in a similar way.

We shall define the functions

1 F(~, fl, v) = ~ ( f~v -f-~v), (2.248)

1 G(c~, fi, u) - -~ ( fu~ - f ~ ). (2.249)

By using (2.246) and the diagram technique, it is easy to show that

a I R (~,~)

= # (d , fl', v) dfl' dd , (2.250) 0

u) = f f u) J J R (~,/~)

= v (d, fl', u) dfl' dd , (2.251) 0

where R(c~, fl) is the rectangle shown in Fig. 2.45. From (2.250) and (2.251) we derive

O2F(ot, fl, v) #(c~, g, v) = , (2.252)

O~ aft

O2G(c~, fl, u) v(c~,/3, u) = . (2.253)

a~ Off As far as the numerical implementation of the model (2.246) is con-

cerned, it can be accomplished without the formulas (2.252) and (2.253)


(o~,13)

R(OC,~)

, "~

_

/

FIGURE 2.45

but by employing explicit expressions for the integrals in (2.246) in terms of the functions F(ol,/~, v) and G(~,/~, u). The derivation of these expressions proceeds as follows.

For the first integral in (2.246) we find

f/R #(ot,~,v(t))~,~u(t)do~d~ = 2 / / s /z(ol,~,v(t))dold~ u(t) +(t)

- [ f~ #(ol,/~, v(t)) do~ d~. (2.254) J Jl~, u(t)

Similarly,

[ ( v(c~, ~,u(t))~,~v(t)dol d~ - 2 ( [ v(c~,/~,u(t)) d~ d/~ d JR u(t) J J f2+(t)

- [f~ v(o~,~,u(t))dotd~. (2.255) J J l '4 v(t)

From (2.243) and (2.244), we derive

1 f+ //R -2 (u(t)v(t) - f ~t)v(t)) = - /z(cr v(t)) dot dfl u(t)

- [ l v(o~,~,u(t))do~d~. (2.256) J Jl~ v(t)

By substituting (2.254) and (2.255) into (2.246) and taking (2.256) into account, we obtain


(X S-(t)

s§

FIGURE 2.46

I f fs #(o~,fl, v(t))dotdfl + f f~ f i t ) - 2 +(t) +(t) +

+ fu(t)v(t)

v(o~, fl, u(t)) dot dfl I

(2.257)

A typical geometry for the set S + (t) is shown in Fig. 2.46. From this figure follows

n ( t )

+(t) k=l k (2.258)

By using (2.250), we find

//R Iz(~ fl" v(t)) d~ dfl - F(M~U) mk(u)" v(t)) - _V..k+l,~i /r(u) m~U)v(t)), k

(2.259)

where M~ u) and m~ u) form the alternating series of past dominant extrema of u(t). From (2.258) and (2.259) we conclude

f fs #(o~,fl, v(t))do~dfl + ( t )

n ( t )

_ X--,rF,~,~(u)- (u) ~,^/r(u). (u) m k , v(t)) - v(t))] (2.260) Z_,L V**K ~V,~k+l ~ m k ~ �9

k=l


By using the same line of reasoning as before, we can derive the expression

v(~, f l , u( t ) )d~df l +(t)

l(0 Z..,t , kX-'rG'M(V),m(V) [^/i(v) . (v) k ,u(t)) -- G m k ,u(t))] (2.261)

- - ~ , ~ , ~ k + l r �9

k=l

By substituting (2.261)and (2.260) into (2.257), we obtain

(t)

f (t) - f+ ~t^,~(u) .(u) u(t)v(t) + 2 ~ [ F ( M ~ u ) " (u) m k ,v(t)) - v(t))] --~d,~k+l ~ m k

k=l

l(0 2 X--,rGtM(V) . (v) , ,~ _ Gt^A(v) " (v) m k , u, t , j m k , u,t,jj(~] (2.262) +

Z_.rL ~, k i ~d,~k+l / �9 k= l

This is the final expression for the output in terms of experimentally measured functions F and G defined by (2.248) and (2.249), respectively.

We next formulate the following result.

REPRESENTATION THEOREM The wiping-out property, the property of equal vertical chords and the path independence property constitute the necessary and sufficient conditions for the representation of actual two input hysteresis nonlinearities by the model (2.246) on the set of piece-wise monotonic inputs.

The proof of this theorem closely parallels the proof of the representation theorem from Section 2 of this chapter. The only new element is the utilization of the path independence property. By using this property, it can be easily shown that an arbitrary simultaneous piece-wise monotonic variation of u(t) and v(t) can be reduced to two consecutive piece-wise monotonic single input variations during which the other input remains constant. Further details of the proof of the theorem are left to the reader.

In conclusion of this section, we consider one peculiar property of strain hysteresis in comparison with magnetic hysteresis. This peculiarity stems from even symmetry of strain hysteresis which exhibits butterfly- shaped major hysteretic loops with respect to variations of magnetic field (see Fig. 2.47). It is clear that the first-order transition curves f ~ v will also exhibit even symmetry. This can be expressed mathematically as

f ~ , ~ , v = f ~ v and fa, v = ~ (2.263)

if f l ' = - ~ and c~ '=-f l . (2.264)


IJv �9 [ , , I

~ a v 8 �9

//

I

i fa~v �9 8 ~ U

~ a

FIGURE 2.47

By using (2.263) and (2.264), it can be easily proven that

(2.265)

Thus, the function #(~,/J, v) is of odd symmetry with respect to the line c~ = -~ . In the case of magnetic hysteresis, the/z-function has an even symmetry with respect to the same line.

It follows from (2.265) that the function tt(~, ~,v) can not be positive everywhere on the u-l? plane. Actually, this function changes its sign across the line ~ =-1~. As a result, first-order and higher-order transition curves are not enclosed by major hysteretic loops. This property has been experimentally observed and it also distinguishes the strain hysteresis from magnetic hysteresis.

2.7 EXPERIMENTAL TESTING OF PREISACH-TYPE MODELS OF HYSTERESIS

In our previous discussions of Preisach-type models of hysteresis we consistently tried to establish the necessary and sufficient conditions under which actual hysteresis nonlinearities can be represented by these models. These conditions establish the limits of exact applicability of Preisach-type models. As a result, these conditions allow one to judge to what extent one or another Preisach-type model is applicable. These conditions also clearly reveal the phenomenological nature of Preisach-type models and their

2.7 EXPERIMENTAL TESTING OF PREISACH-TYPE MODELS 133

physical universality. However, it is unrealistic to expect that actual hysteresis nonlinearities will satisfy these conditions exactly. For instance, the experimental results presented in [19] show that neither the congruency property nor the property of equal vertical chords of comparable minor loops are satisfied for some typical materials used in magnetic recording. For this reason, the classical and "nonlinear" Preisach models as well as some other Preisach-type models cannot be expected to be absolutely accurate. However, we may try to use these models as approximate ones. Then the question arises how to judge the accuracy of these models.

It has been emphasized in this book that branching is the essence of hysteresis. Consequently, any mathematical model of hysteresis is expected to predict this branching. In other words, models of hysteresis are expected to predict higher-order reversal curves (branches) which occur after several (or many) reversals of input. This prediction should be based on some limited experimental data used for the identification of hysteresis models. Thus, the accuracy of the classical, "nonlinear" and "restricted" Preisach models can be measured by their abilities to predict higher-order reversal curves on the basis of information provided by the first- and second-order reversal curves. This is exactly the criterion which has been chosen for the experimental testing of the accuracy of the Preisach-type models.

This testing was first performed for two typical particulate materials: Co-coated y-Fe203 and iron magnetic tape materials. The experimental work consisted of two main stages. During the first stage, the first- and second-order transition curves were measured by using a vibrating sample magnetometer. Experimentally measured first-order transition curves for the above two materials are shown in Figs. 2.48 and 2.49, respectively. For each first-order transition curve there is a set of second-order transition curves which are attached to the above first-order curve. Exper- imentally measured sets of second-order transition curves are shown in Figs. 2.50 and 2.51, respectively. These first- and second-order transition curves were used for the identification of the classical, "nonlinear", and "restricted" Preisach models.

During the second stage of the performed experiments, several sets of higher-order (with the order up to six) transition curves were measured. The sequences of field reversal for these higher-order transition curves were chosen in such a way that these higher-order transition curves were more or less uniformly (and densely) distributed over the area enclosed by major hysteresis loops. This was important in order to get the information about uniform accuracy of Preisach-type models. Then, by using the developed computer software which implements the Preisach-type models, various sets of higher-order transition curves predicted by the Preisach-


.04

.02

-.02

-.04

-2000 -1500 -I000 -500 0 500 I000 1500 2000

FIGURE 2.48

type models were computed for the same sequences of field reversals as in the experiment. The comparison of the higher-order transition curves predicted by the classical Preisach model (2.1) with the experimentally measured higher-order transition curves is demonstrated in Figs. 2.52 and 2.53. Similar comparisons for the nonlinear Preisach model (2.26) and the restricted Preisach model (2.86) are shown in Figs. 2.54 and 2.55. It is evident from these figures that the actual (experimentally measured) transition curves are "sandwiched" between the predictions of the nonlinear and restricted Preisach models. This observation has prompted the idea to use an "average" (superposition) model

l f f fit)- -~ [/~(c~, fl, u(t)) + #(ot,~,M1)]~,o,~(t)doldfl

Z + u<t + +f l +f- + 4 " (2.266)

2.7 EXPERIMENTAL TESTING OF PREISACH-TYPE MODELS

.I,t-

.05 1

-.05

-.I

-3000 -2000 -I000 0 I000 2000 3000

FIGURE 2.49

The numerical implementation of this model can be carried out by using the formula

f - q- f~t) n(t) 1 + -~ E[P(Mk+l, mk, u( t ) ) - P(Mk, mk, u(t))] f(t) =

k=l

n(t) q- Z[F(Mk, mk_I,M1) - F(Mk, mk, M1)],

k=l

(2.267)

which follows from (2.66) and (2.113). It is clear that the "average" model inherits the common "wiping-out" property of the "nonlinear" and "restricted" models. Functions /x(cc, fi, u) and /2(r can be individu- ally determined by matching first- and second-order reversal curves exactly as in the case of those models. On the other hand, the "average" model further relaxes the congruency property of the "restricted" Preisach model and the property of equal vertical chords of the nonlinear model.

136 C

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Namely, the "average" model describes hysteresis nonlinearities whose minor loops have equal vertical chords only if they are formed after the same largest input maximum M1 was achieved. In other words, the property of equal vertical chords becomes history dependent. It is clear that the average model much further relaxes the congruency property of the classical Preisach model than the "nonlinear" or "restricted" models. At the same time, the average model requires the same experimental data for its identification as the two aforementioned models.

The "average" model turned out to be remarkably accurate for all used (and densely distributed) sequences of reversal values of magnetic fields. The comparison of the higher-order transition curves predicted by this model with the experimentally measured higher-order transition curves is demonstrated in Figs. 2.56 and 2.57. The above comparison suggests a higher accuracy of the "average" Preisach model. This prompted the attempt to test this model for other magnetic materials. This testing was performed for six different magnetic tape materials: Ampex 641, Ampex-797, Ampex-D1, TDK-YHS, Maxell-Beta and Memorex-3480. Sample results of the comparison of the higher-order transition curves predicted by the "average" model with the experimentally measured higher-order transition curves are demonstrated for the above materials in Figs. 2.58-2.63, respectively.

Thus, it can be concluded that the extensive experimental testing presented above has revealed a remarkable accuracy of the "average" Preisach model of hysteresis. It is believed that this accuracy is related to the fact that the average model provides the far-reaching generalization of the congruency property of the classical Preisach model. Actually, this generalization goes much further than in the cases of "nonlinear" and "restricted" Preisach models, although the average model needs the same experimental data for its identification as the two aforementioned models.

References 1. Mayergoyz, I. D. (1991). Mathematical Models of Hysteresis, Berlin: Springer-

Verlag.

2. Mayergoyz, I. D. (1986). Phys. Rev. Lett. 56: 1518-1521.

3. Mayergoyz, I. D. and Friedman, G. (1988). IEEE Trans. Mag. 24: 212-217.

4. Mayergoyz, I. D. (1988). IEEE Trans. Mag. 24: 2925-2927.

5. Mayergoyz, I. D., Friedman, G. and Salling, C. (1989). IEEE Trans. Mag. 25:

3925-3927.

6. Mayergoyz, I. D., Friedman, G. and Adly, A. A. (1990). J. Appl. Phys. 67: 5466- 5468.



8. Adly, A. A., Mayergoyz, I. D. and Bergqvist, A. (1991). J. Appl. Phys. 69: 5777- 5779.

9. Friedman, G. and Mayergoyz, I. D. (1991). J. Appl. Phys. 69: 4611-4613.

10. Adly, A. A. and Mayergoyz, I. D. (1992). IEEE Trans. Mag. 28: 2268-2270.

11. Mayergoyz, I. D. and Adly, A. A. (1992). IEEE Trans. Mag. 28: 2605-2607.

12. Mayergoyz, I. D. and Andrei, P. (2002). J. Appl. Phys. 91: 7645-7647.

13. Wiesen, K. and Charap, S. H. (1988). IEEE Trans. Mag. 24: 249-251.

14. Friedman, G. and Cha, K. (2000). Journal of Material Processing and Manufactur- ing Science 9: 70-78.

15. Friedman, G. and Cha, K. (2001). J. Appl. Phys. 89: 7236-7238.

16. Friedman, G. (2000). Physica B 275: 173-178.

17. Della Torre, E. (1999). Magnetic Hysteresis, IEEE Press.

18. Restorff, J. B., Savage, T. H., Clark, A. E. and Wum-Fogle, M. (1990). J. Appl. Phys. 67: 5016-5018.

19. Salling, C. and Schultz, S. (1988). IEEE Trans. Mag. 24: 2877-2879.

CHAPTER 3

Vector Preisach Models of Hysteresis

3.1 CLASSICAL S T O N E R - W O H L F A R T H M O D E L OF VECTOR HYSTERESIS

In magnetics, research on the modelling of scalar and vector hysteresis has been pursued along two quite distinct lines. Modelling of scalar hysteresis has been dominated by the Preisach approach. This approach can be traced back to a landmark paper [1]; it has been proved to be very successful and has won many followers. On the other hand, phenomenological modelling of vector hysteresis has long been centered around the classical Stoner-Wohlfarth (S-W) model [2]. As a result, this model has further been developed and used in the area of magnetic recording [3-5]. The attractiveness of the S-W model can be attributed to its strong appeal to physical intuition. This appeal is, in turn, based on the fact that the S-W model is designed as an ensemble of single-domain, uniaxial magnetic particles. Since these particles have some features of physical realities, the S-W model is usually regarded as a physical (not mathematical) model. Due to its popularity in magnetics, the S-W model is a natural bench- mark for comparison with other vector hysteresis models. This is the main reason why we precede our discussion of vector Preisach models by the discussion of the S-W model.

Since single-domain, uniaxial magnetic particles are the main building blocks of the S-W model, we begin with the discussion of hysteresis of these particles.

We consider a single-domain, uniaxial magnetic particle with magnetization (magnetic momentum) M which may change its orientation under the influence of an applied field but has a constant magnitude. Such a particle is now commonly called a Stoner-Wohlfarth (S-W) particle. It is clear from the symmetry consideration that the vector M of this particle lies in the plane formed by the easy axis x and the applied magnetic field H

149

150 CHAPTER 3 Vector Preisach Models of Hysteresis

M

H

~- X

FIGURE 3.1

(see Fig. 3.1). The orientat ion-dependent par t of free energy g of the S-W particle is given by

g = Ksin2 0 -/VI �9 (3.1)

where K is the anisotropy constant and 0 is the angle between the easy

axis and M. The first term in the r ight-hand side of (3.1) represents the anisotropy energy, while the second term in (3.1) is the energy of interac-

tion of magnetic m o m e n t u m M with the applied magnetic field. By using the Cartesian coordinates shown in Fig. 3.1, the expression

(3.1) can be represented as

g - K sin 2 0 - MHx cos 0 - MHy sin 0. (3.2)

Equil ibrium orientations of M correspond to minima of g, and they can be found from the equations

Og = 0, (3.3)

O0

02s 302 ~ O. (3.4)

From (3.2) and (3.3) we derive

2K sin 0 cos 0 + MHx sin 0 - MHy cos 0 = 0.

By introducing the so-called switching field c~:

2K 0 / ' - "

M '

the expression (3.5) can be rewrit ten as

c~ sin 0 cos 0 + Hx sin 0 - Hy cos 0 - 0,

(3.5)

(3.6)

(3.7)

3.1 CLASSICAL STONER-WOHLFARTH MODEL 151

which is equivalent to HK

sin 0 cos 0 =0/. (3.8)

Equation (3.7) (as well as (3.8)) is a quartic equation with respect to cos0. For this reason, this equation may have two or four real solutions. Which

of these two cases is realized depends on the applied magnetic field, H. In the first case, we have only one min imum and, consequently, only one equilibrium orientation of M. In the second case, there are two minima and this results in two equilibrium orientations of M. Thus, on H-plane there are two different regions where M has one and two equil ibrium orientations, respectively. On the boundary between the above two regions one min imum and one max imum merge together. In the case of a minimum, the inequality (3.4) holds, while for a max imum we have 02~/002 ~ 0. Consequently, on the above boundary we have

0s 02s = 0 and = O. (3.9)

30 002 The first condition in (3.9) leads to Eq. (3.8). Now, let us derive the second equation by using both conditions in (3.9). From (3.2), by using the notation (3.6), we derive

1 3g Hx Hy = 0/q - - . (3.10)

sin 0 cos 0 30 cos 0 sin 0

By differentiating (3.10), we obtain

3 ( 1 )3g 1 32g HxsinO HycosO 00 sin0 cos 0 ~ + sin0 cos 0 002 = cos2-------O + ~ " (3.11) sin 2 0

By using both conditions (3.9), from (3.11) we find

HK Hy --0. (3.12) cos 3 0 sin 3 0

Thus, on the boundary which separates regions with one and two minima, Eqs. (3.8) and (3.12) are satisfied. These are two linear equations with respect to Hx and Fly. By solving these equations, we find

Hx - -0/cos 3 0, Hy - 0/sin 3 0. (3.13)

From (3.13) we find the equation for the boundary separating the above two regions:

Hx2/3 _}_/_/2/3 = 0/2/3 (3.14) --y

This equation represents the astroid curve shown in Fig. 3.2. This astroid curve helps to visualize the solution of the quartic equation (3.7). This

152 C H A P T E R 3 Vector Pre isach M o d e l s of Hys te res i s

Hy

(Hxl'~y -~~

(X

r H x /

-(X

F I G U R E 3.2

solution can be found by using the following geometric construction [6, 7]. For a given external magnetic field H with components Hx and Hy the directions of M that satisfy Eq. (3.7) are parallel to those tangent lines to the astroid that pass through the point H (see Fig. 3.2). The proof of the above statement proceeds as follows.

..)

Let Hx and Hy be Cartesian components of H, and let Hxl and Hy~ be Cartesian coordinates of the point on the astroid at which the above mentioned tangent line touches the astroid. For the slope of this tangent line we have

m = Hyl - Hy (3.15) Hxl - H x "

which is tantamount to

Hy I - Hy = m(Hxl - Hx). (3.16)

Since the point (Hxl, Hyl) belongs to the astroid, we have

Hx2/3 /q2/3 _ c~2/3 (3.17) 1 -~- ~ ~Yl

By using implicit differentiation of (3.17) with respect to Hxl, we derive

2H-1/3 2H-1/3dHyl "3 X1 "q- 3 Yl dHx1 = 0 . ( 3 . 1 8 )

From (3.18) we obtain

aH l (H,I 1/3 m = dHx~ - - Hxl l " (3.19)


Let fl be the angle formed by the above tangent line with the easy axis of the particle. Then,

m = tan fl, (3.20)

and from (3.19) and (3.20) we find

Hyl = - tan 3 ft. (3.21) Hx I

Expressions (3.17) and (3.21) can be construed as two simultaneous equations with respect to two unknowns: Hx1 and Hy I . By solving these equations, we obtain

H x l - - -O l COS 3 ~, Hy 1 -- o~ sin 3 ft. (3.22)

By substituting (3.20) and (3.22) into (3.16), we find

cg sin 3 fl - Hy = tan fl (-c~ cos 3 fl - Hx) . (3.23)

A trivial transformation leads to

c~ (sin 3 fl cos fl + cos 3 15 sin fl) + Hx sin fl - Hy cos fl -- 0, (3.24)

which can be simplified as

c~ sin fl cos fl + Hx sin fl - Hy cos fl = 0. (3.25)

It is clear that Eqs. (3.7) and (3.25) are identical. Consequently,

fl =0 . (3.26)

This proves the validity of the above described geometric construction.

It can be shown that equilibrium orientations of M correspond to the tangent lines with smallest slopes. It is clear that, when the point H is outside the astroid, only two tangent lines, are possible, and therefore there is only one equilibrium orientation of M. When H is inside the astroid, there are four tangent lines. However, only two of these tangent lines represent equilibrium orientations of the magnetization M. Which one is realized depends on the previous history of the magnetization.

The described geometric rules allow one to compute hysteresis loops of a S-W particle for the case when the applied magnetic field is restricted to vary along one arbitrary chosen direction. Suppose that this direction is specified by the line a-a' (see Fig. 3.3) and that the magnetic field is first monotonically increased from its value H_ corresponding to the point 1 to the value H+ corresponding to the point 6 and then is monotonically decreased back to H_. The dependence of the magnetization projection along the line a-a' on the value of the magnetic field H exhibits hysteresis that is shown in Fig. 3.4. This is clear from Fig. 3.3. Indeed, as we move


Hy

6 a

H•

FIGURE 3.3

up along the line a-a ' , the equilibrium (stable) orientations of M coincide with directions of tangent lines to the right-hand side of the astroid until we reach the point 5. At this point a "switch" from the right-hand side to the left-hand side of the astroid occurs and, after that point, stable orientations of M coincide with directions of tangent lines to the latter part of the astroid. As we move down along the line a-a I from the point 6, we use the tangent lines to the left-hand side of the astroid to determine the stable direction of M. However, at point 2 a "switch" from the left-hand side to the right-hand side of the astroid occurs and, after that point, stable orientations of M coincide with directions of tangent lines to the latter part of the astroid. Thus for the points of the line a-a ' which are inside the astroid, there are two stable orientations of M which result in two different branches of the hysteresis loop shown in Fig. 3.4. It is clear from the preceding discussion that if the applied field varies along the easy axis x then a S-W particle exhibits a rectangular hysteresis loop shown in Fig. 3.5. It is also clear that if the applied magnetic field is varied along the direction perpendicular to the easy axis, then due to the symmetry there is no hysteresis effect and a S-W particle exhibits a single valued magnetization curve shown in Fig. 3.6. Thus, the shape of hysteresis loops depends on the direction along which the applied field is being varied.

It has been shown in Chapter 2 (see formula (2.81)) that the loop shown in Fig. 3.4 can be represented in terms of the rectangular 9-1oop. My former student G. Friedman found a very interesting generalization of the formula (2.81) (see reference [8]). This generalization represents the

3.1 CLASSICAL STONER-WOHLFARTH MODEL

M a-a'

/

= H a-a'

155

FIGURE 3.4

Mx

= H x

My

Hy


magnetization of a S-W particle for arbitrary (not only collinear) variations of magnetic field in terms of rectangular }9-loops. The basis for this representation is the notion that there are two distinct states (vector branches) for any S-W particle. In the first state, stable orientations of the magnetization coincide with directions of tangent lines to the left-hand side of the astroid. The notation A/l+(~(t),H(t)) will be used for the magnetization in the first state, where ~(t) is the angle formed by the applied magnetic field with the easy axis and H(t) is the magnitude of magnetic field. In the second state, the stable orientations of M coincide with directions of tangent lines to the right-hand side of the astroid. The notation

. . +

M-(~(t),H(t)) will be used for the magnetization in the second state. It


is clear that M+(~o(t),H(t)) and M-(~o(t),H(t)) can be found geometrically by using the previously described "astroid" rule or by solving the quartic equation (3.7). The rectangular loop operator G,-~ will be employed to describe switching from the first state to the second state and vice versa. The input v(t) for this operator is given by

cos~0(t) [Hx(t)2/3 § Hy(t)2/313/2 v(t) -- I cos~0(t)] (3.27)

It is clear from (3.27) and (3.14) that v(t) reaches the value c~ as the tip of H(t) crosses the right-hand side of the astroid, and v(t) reaches the value -c~ as the tip of H(t) crosses the left-hand half of the astroid. This shows that the switching of the rectangular loop }9~,_~ occurs at the same time as the switching of the S-W particle from one state to another. By using this fact, we can represent the magnetization M(t) of the S-W particle as

~l(t) = -~(t)~,~,_~v(t) § d(t), (3.28)

where 1

~(t) = ~[/~I + (~o(t),H(t)) - ~I- (~o(t),H(t))], (3.29)

d(t) = -~ IA/I+ (~o(t), H(t)) + ~l- (~o(t), H(t)) 1. (3.30)

It is clear from (3.28), (3.29) and (3.30) that/~I(t)--A/l+(~o(t),H(t)) when G,_~v( t ) - 1, and M(t)= M-(~p(t),H(t)) when G,_r -1 . Switchings of }9~,_~v(t) from 1 to - 1 and vice versa occur at the times when the tip of H(t) crosses the astroid. This proves that formulas (3.27)-(3.30) give the right representation for the magnetization of the S-W particle.

Having described the basic properties of a S-W particle, we can now proceed to the discussion of the S-W hysteresis model. This model is designed as an ensemble of S-W particles. Consider an infinite set of S-W particles with different orientations of their easy axis and different values of switching field r The notation So,~ will be used for a S-W particle whose switching field is equal to ~ and whose easy axis forms the angle 0 with the x-axis. By using this notation, the S-W model can be represented mathematically as

~l(t) - f f ~(O, ot)So,~H(t)dO do~, (3.31)

where ~(0,~) is a distribution function that should be determined by fitting the model to some experimental data.

The expression (3.31) defines the S-W model in terms of magnetic quantities such as magnetization M and magnetic field H(t). However, it


is possible to interpret the S-W model as a general mathematical model of vector hysteresis by writing this model in the form

f (t) - ~ ~(O,~)So,~F~(t) dO d~, (3.32)

where f(t) is the vector output, while ~(t) is the vector input. The output of the operator So,~ can be formally determined by using

the astroid rule or the quartic equation (3.7) in which Hx and Hy are replaced by ux(t) and uy(t), respectively. In other words, the S-W model can be defined in purely mathematical terms without using any connections of this model to some physical objects such as uniaxial, single-domain magnetic particles. Such a purely mathematical point of view of the S-W model may have two-fold advantages. First, it suggests some possibilities of using this model not only in the area of magnetics. Secondly, it may open some opportunities for further generalization of this model.

By using representation (3.28)-(3.30) for the S-W particles, the S-W model can be written in terms of };-operators. This suggests some connections between the S-W model and the Preisach-type models. In particular, Preisach-type diagrams can be used to keep track of switching of different S-W particles.

The S-W model has been known and used in magnetics for a long time. Gradually, it has been realized that this model has certain limitations. The most important of them can be summarized as follows.

Since the S-W model is designed as an ensemble (superposition) of particles (hysteresis nonlinearities) with symmetric loops, this model does not describe nonsymmetric minor loops. This limitation is often attributed to the fact that the S-W model does not account for "particle interactions." This limitation can be somewhat corrected by expanding the set of elementary hysteresis operators So,~ and by introducing the operators So,~,~ with shifted astroids (Fig. 3.7). Then, the generalized S-W model can be represented as

f (t) - / / / / ~ (O,~,~)So,~,~u(t) dO d~ d~. (3.33)

However, this generalization will require much more computational work for the numerical implementation of the S-W model. Even without this generalization, the S-W model is computationally slow. This is in partbe- cause the calculations of outputs of individual elementary operators So,~ require the solution of quartic equations associated with the astroid construction. In addition, individual outputs should be integrated over some distributions of S-W nonlinearities (particles) with respect to their easy axis directions and switching fields. This requires the evaluation of double


YT \~, \ / /

FIGURE 3.7

integrals in the case of the classical 2D S-W model (3.32). The last difficulty is magnified in the case of the generalized S-W model (3.33).

Furthermore, the identification problem of finding the distribution function ~(0,~) by fitting the S-W model to some experimental data has not been adequately addressed yet. Solutions to this problem are usually achieved by some artwork rather than by using a well established procedure.

Our research has been motivated by the desire to circumvent the limitations of the S-W model described above. To achieve this goal, we have turned to the Preisach approach and tried to extend it to the vector case. The guiding idea in our efforts has been the notion that the scalar hysteresis is a particular case of vector hysteresis. As a result, many important and characteristic properties of vector hysteresis can be exhibited in the scalar case. By exploring this notion, new vector Preisach models of hysteresis have been developed. These models have many of the desirable features of the scalar Preisach hysteresis models, and they represent a vi- able alternative to the S-W model.

3.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS A N D THEIR NUMERICAL IMPLEMENTATION

For the sake of generality, a vector hysteresis nonlineari~ will be charac-

terized below by a vector input ~(t) and a vector output f(t). In magnetic

3.2 DEFINITION OF VECTOR PREISACH MODELS 159

applications, ~(t) is the magnetic field, whilef(t) is the magnetization. The most immediate problem we face is how to define vector hysteresis in a mathematically rigorous as well as physically meaningful way. To do this, it is important to understand what constitutes in the case of vector hysteresis the essential part of input history that affects the future variations of output. In the case of scalar rate-independent hysteresis, experiments show that only past input extrema (not the entire input variations) leave their mark upon future states of hysteresis nonlinearities. In order words, the memories of scalar hysteresis nonlinearities are discrete and quite selective. There is no experimental evidence that this is the case for vector hysteresis. As a result, we must resign ourselves to the fact that all past vector input variations may affect future output values. The past input variations can be characterized by an oriented curve L traced by the tip of the vector input ~(t). Such a curve can be called an input "hodograph." Vector rate-independent hysteresis can be defined as a vector nonlinearity with the property that the shape of curve L and the direction of its tracing (orientation) may affect future output variations, while the speed of input hodograph tracing has no influence on future output variations. Next, we shall give another equivalent definition of rate-independent vector hysteresis in terms of input projections. This definition is very convenient in the design of mathematical models of vector hysteresis. Consider input projection along some arbitrary chosen direction. As the vector ~(t) traces the input hodograph, the input projection along the chosen direction may achieve extremum values at some points of this hodograph. In this sense, the extrema of input projection along the chosen direction samples certain points of the input hodograph. If the projection direction is continuously changed, then the extrema of input projections along the continuously changing direction will continuously sample all points on the input hodograph. In this way, the past extrema of input projections along all possible directions reflect the shape of input hodograph and, consequently, the past history of input variations. Thus, we arrive at the definition of vector rate- independent hysteresis as a vector nonlinearity with the property that past extrema of input projections along all possible directions may affect future output values. It is clear that mathematical models of vector hysteresis are imperative for self-consistent descriptions of systems with vector hysteresis. These models should be able to detect and store past extrema of input projections along all possible directions and choose the appropriate value of vector output according to the accumulated history.

To detect and accumulate the past extremum values of input projections along all possible directions, the scalar Preisach models (Preisach's particles) can be employed. These scalar models are continuously distributed along all possible directions (see Fig. 3.8). Thus scalar Preisach mod-


i

i s i

", ', ' / F ~ (r.u ()) " ' ~ t

- - . . " , , ,, i' r . . -

. . . . . . . . ] ? : . . . . . . . . . .

i

FIGURE 3.8

els are main building blocks for the vector model, which is constructed as a superposition of scalar models. This can be expressed mathematically in two dimensions as

f ( t) = ~_ -f P~(-f . F~(t)) d~,r, (3.34) J# I=1

and the integration in (3.34) is performed over a unit circle. Similarly, a 3D vector Preisach model can be written in the form

f ( t) = ~ -f r~(-f . ~(t))dsr, (3.35) JJ [-~ [--1

where the integration is performed over a unit sphere. The scalar Preisach models I~ are defined by

Cr(~" ~(t)) - f f v(r ~(t))d~d~ (3.36) Jdc~

for isotropic vector models, and

- / ~ v(~,/~,~)}9~ (~. ~(t))d~d~ (3.37) rrff. ~>~

for anisotropic vector models. Ideas of the construction of vector Preisach models that are somewhat

similar to those described above have been briefly mentioned (without

3.2 DEFINITION OF VECTOR PREISACH MODELS

+ - -++

F- ,,r

FIGURE 3.9

161

any analytical details) in [9, 10]. Some similarities can also be found between our definition of the vector Preisach models and a purely mathematical vector generalization of the scalar Preisach model discussed in [11].

The following proposition further elucidates the above definition of vector Preisach models.

PROPOSITION The integration in (3.35) over a unit sphere can be reduced to the integration over a unit hemisphere.

PROOF. Consider the partition of the unit sphere into two hemisphere C + and C-. For every point ~ + E C + there is a corresponding opposite point ~- ~ C- (see Fig. 3.9)

--) .q_ - - ) _

r - - - - r .

From (3.35)we find

where

rr+(~ + �9 ~(t)) =/f~>e

r%-(~-. ~(t)) =/~>~

~(~, ~,~+)9~e (~ + �9 ~(t)) a~ a~,

v(c~, g,~-)}3~ (~- . ~(t))do~ dg.

(3.38)

(3.39)

(3.40)

(3.41)


The following identity can be verified for any continuous piece-wise monotonic function v(t):

~,~v(t) = - ~,_~,_~ (-v(t)). (3.42)

Indeed, if v(t) < ~, then -v(t) > -/~ and

~,~v( t ) - - -1 , while ~_~,_~(-v(t))= +1. (3.43)

If v(t) increases and exceeds c~, then -v(t) decreases and becomes smaller than -c~. Consequently,

~,~v(t)- 1, while }9_~,_~(-v(t))=-1. (3.44)

Similarly, it can be shown that the identity (3.42) holds for any monotonic decrease of v(t).

In the last integral in (3.39) we change the variable ~- to -~+. As a result, we find

/ / c - - f -P~- (-f- " ~(t))ds_ - - / / c + ~+P'-~+ (-~ +" ~(t))ds+, (3.45)

where

P_~+(--f+ . F~(t)) - f f v(c~,~,--f+)~,~(--f + . F~(t))d~d~. (3.46) d d a

By using the identity (3.42), the last integral in (3.46) can be transformed as follows:

/ ~ v (~, ]~,-~+) 9~ (-~ +. F~(t))d~d~

= - ] ] v(c~,/~,-?+)9_~,_~(~ +. F~(t))d~d~. (3.47)

In the last integral we will change c~ to -/~ and/J to -~ , then according to (3.46) and (3.47) we obtain

/ ~ v (- /~,-c~,-~+) }3~ (7 +. ~(t))d~d~. P'_~+ ( -~+ . ~(t)) - - ~>~


f /c- -f -P~- (-f - " ~(t)) ds_

- v ( - / ~ , - ~ , - ~ + ) 9 ~ (~ +. ~( t ) )d~d~ ds+. J

(3.48)

(3.49)


By substituting (3.40) into the first integral in (3.39), we have

fc+ -f +~-~+ (-f +" ~(t)) ds+


+ ~>/~

x 9~ if+. ~(t)) d,~ d~] ds+.

By introducing a new function

from (3.51) we derive

+ ffc +A, if+. f(t) - -f F~+ FifO) ds+, +

where

" if+ ffa r~+ �9 ~(t)) = ~>~

This completes the proof.

v'(c/, fl,~+)9~/~ (~ + �9 ~(t))do~dfl.

(3.50)

(3.51)

(3.52)

(3.53)

(3.54)

[3

The proven proposit ion suggests that (3.53) and (3.54) can be regarded as an equivalent definition of the vector Preisach model. This definition will be used in our subsequent discussions and the superscript ' will be omitted. It is also clear that a similar proposit ion is valid for the two-dimensional vector Preisach model (3.34) as well. Consequently, this model can be represented as

-- rfL +~t f(t) -f F~+ (~+ . Fi(t)) dl+, (3.55) +

where L+ is a semicircle and F' is defined by (3.54). ~+ It is apparent from the above proof that particular choices of semi-

spheres and semicircles in (3.53) and (3.55), respectively, are unimportant ; all these choices will lead to equivalent vector Preisach models.

From the above proof we can also find some interesting symmetry properties of the function v'. Indeed, if in (3.52) we change ~ to - f l , fl to - ~ , and 7 + to -~ +, then we find

v ' ( - f l , - c ~ , - ~ +) - v ( - f l , - c ~ , - ~ +) + v (c~, fl,~+). (3.56)


From (3.56) and (3.52), we obtain

v'(a, fl,~+) = v ' ( - f l , - a , - ~ + ) . (3.57)

This expresses the property of mirror symmetry of the function v' with respect to the line c~ = -f t . This also extends the definition of the function v t from a unit semisphere or unit semicircle to the entire unit sphere or unit circle, respectively.

In the case of isotropic vector models, the expression (3.57) can be simplified as follows:

vt(a, fl) = v ' ( - f l , - ~ ) , (3.58)

which is similar to the symmetry property of the/z-function for the classical scalar Preisach model (see the expression (1.46)).

Up to this point, the vector Preisach models have been defined in coordinate invariant forms (3.34), (3.35), (3.53) and (3.55). However, in many applications it is more convenient to use the expressions for the vector Preisach models in spherical and polar coordinates for three and two dimensions, respectively. In the case of spherical coordinates we have

ds+ - sin 0 dO d~o, -f + = -~eo,~o, -f + �9 Ft(t) - uo,~ (t), (3.59)

where e0,~ is a unit vector along the direction specified by angles ~0 and 0, and uo,~(t) is the projection of fi(t) along the direction of e0,~.

By using (3.59), the 3D vector Preisach model (3.53)-(3.54) can be written as

-* f02rrf0~ A f ( t ) - -Jo,~I'o,~uo,~(t) sin0 dO d~o, (3.60)

where A

I'o,~uo,~(t) - v(a, fl, O,~o)G~uo,~(t)d~dfl . (3.61)

The last two formulas can be combined into one expression:

-* fo27'fo-~ (ff~ f ( t ) = -eo,~ ~ ~ v(c~, fl, 0, ~o)G~uo,~(t) d~ d f l )

x sin 0 dO d~o. (3.62)

Similarly, the 2D vector Preisach model (3.55) can be represented in polar coordinates as

73

f (t) - -GGu (t) a o, (3.63) 2

where A

I '~u~(t) - v(~, ~, ~o)G~u~(t) d~ d~. (3.64)


In the previous formulas, ~ is a unit vector along the direction specified by a polar angle r F~ is the scalar Preisach model (operator) for this direction, and u~(t) is the projection of ~(t) along the direction of ~ .

By combining (3.63) and (3.64), we have 7l"

- f (fL ) f (t) = -~ v(ot, fl, 9 ) f '~u~( t ) dot dfl d~o. (3.65)

The expressions (3.62) and (3.65) are written for anisotropic models. In the isotropic case, the function v should be independent of 0 and ~p (or of ~0 in 2D case). This leads to the following 3D and 2D isotropic vector Preisach models:

-" f02~f0~ (fL ) f i t ) - -ee,~ v(ot, f l)f ,~ue,~(t) dot dfl sin 0 dO d~o, (3.66)

yr

f (t) = -~ v(ot, f l ) f ,~u~(t) dot dfl d~o. (3.67)

In the models described above, the functions v have not yet been specified. These functions should be determined by fitting the vector models to some experimental data. This is an identification problem. It is apparent that the identification problem is the central one as far as practical applications of the above vector hysteresis models are concerned. This problem will be discussed in sufficient detail in the subsequent sections. However, it is appropriate to comment already here that the solution of the identification problem is significantly simplified by the introduction of the auxiliary function P(ot, fl,0,~p). For any fixed 0 and ~0, consider a triangle T(ot, fl) shown in Fig. 3.10. Then by definition, we have

P ( ot , fl , O , ~p ) = f f T v ( ot ' , fl ', O , ~o ) dot ' d fl ' . (~,~)

(3.68)

By using (3.68), it can easily be shown that P is related to v by the formula

32P(ot, fl, O, ~p) v(ot, fl, O, ~p) = - . (3.69)

O~ Off

Thus, if the function P is somehow determined, then the function v can be easily retrieved. However, from the computational point of view, it is more convenient to use the function P than v. This is because the double integrals with respect to ot and fl in expressions (3.62), (3.65), (3.66) and (3.67) can be explicitly expressed in terms of P, and in this way the above double integration can be completely avoided. Indeed, for any fixed direction e0,~ we can consider the corresponding ot-fl diagram. A typical example of such a diagram is shown in Fig. 3.11, where Mo,r and mo,r


/


form an alternating series of dominant maxima and minima of input projections along the direction specified by e0,~. By,..using these maxima and minima, the output of the scalar Preisach model Fo,~o,~(t) associated with the direction e0,~ can be evaluated as follows

Co,~uo,~(t) -- -P(ao, ~o) no,~(t)

4- 2 ~ [P(Mo,~,k, mo,~,k-l,0,~a)- P(Mo,~,k, mo,~,k,O,~)]. k=l (3.70)

The proof of (3.70) literally repeats the proof of the expression (1.69) for the classical Preisach model and, for this reason, it is omitted.

In the case of 2D vector Preisach model (3.63)-(3.65), a similar expression is valid for F~u~(t)"

C~u~(t) = -P(a0,/~0) n~(t)

4- 2 ~_,[P(M~,k,m~,k-l,~a) - P(M~o,k,m~,k,~)]. (3.71) k=l

By using formulas (3.70) and (3.71) the numerical implementation of 3D vector model (3.60)-(3.62) and 2D vector model (3.63)-(3.65) can be reduced to the evaluation of double and single integrals, respectively. We note here that the numerical implementation of the classical 2D Stoner- Wohlfarth model (3.32) requires the evaluation of double integrals. In this respect, numerical implementation of 2D vector Preisach model can be accomplished more efficiently than the numerical implementation of the 2D Stoner-Wohlfarth model. Another advantage of using the formulas (3.70) and (3.71) is that the function P can be directly related to experimental

3.3 SOME BASIC PROPERTIES 167

data. This will be demonstrated when we study the identification problems for the vector Preisach models.

Using the expressions (3.70) and (3.71), digital codes that implement the vector Preisach models (3.60)-(3.62) and (3.63)-(3.65) have been developed. In these codes finite meshes of directions e0,~ and ~ are used to evaluate double and single integrals in (3.60) and (3.63), respectively. For each mesh direction the integrands in (3.60) and (3.63) are computed by using (3.70) and (3.71). Some numerical examples computed by using the developed digital codes will be given in the next section.

3.3 S O M E BASIC PROPERTIES OF V E C T O R P R E I S A C H HYSTERESIS M O D E L S

In the previous section we have defined the vector Preisach models of hysteresis and discussed their numerical implementation. The purpose of this section is to study some basic properties of these models and to show that these properties are qualitatively similar to those observed in experiments.

We begin with the property of reduction of vector hysteresis to scalar hysteresis. It has been mentioned in the introduction that this property is experimentally observed when an input is restricted to vary along arbitrary chosen direction. We shall show below that a similar reduction property holds for the vector Preisach models; this property is stated more precisely below. For the sake of notational simplicity it is formulated and proven only for the 2D Preisach model (3.63)-(3.65), although it holds for the 3D Preisach model (3.60)-(3.62) as well.

REDUCTION PROPERTY OF THE VECTOR PREISACH MODEL TO THE SCALAR PREISACH MODEL Consider an input Fl(t) restricted to vary along some direction -e~o for times t >1 to. Suppose that during t >~ to, U~o(t ) = u(t) consecutively reaches values u+ and u_ (with u+ > u_) and remains thereafter within these bounds. Then, for the Preisach vector model (3.63)-(3.65), the relationship between the output projection f~ o (t) along the direction -e~o and the input u(t) exhibits the wiping-out and congruency properties. Since these properties constitute necessary and sufficient conditions for the representation of hysteresis nonlinearities by the classical scalar Preisach model, we conclude that the vector Preisach model is reduced to the scalar Preisach model.

PROOF. Without impairing the generality of our discussion, we can assume that ~0 = 0. Then, for any ~ the input projection u~(t) varies between u+ cos~ and u_ cos~. This means that for any ~0 input variations may affect ~-/~ diagrams only within the triangle T(u+ cos ~, u_ cos ~) (see


T (X

u cos~o /+

'L /

?

/ /

FIGURE 3.12

Fig. 3.12). It is also clear that all input projections u~(t) reach maximum or minimum values at the same time and that these extremum values are "cos99-multiples" of the corresponding extremum values of uo(t)= u(t). Consequently, if {Mk} and {mk} constitute an alternating series of dominant extrema of u(t), then {M~,k} and {m~,k} defined as

M~,k = Mk cos 99, m~,k = mk cos 99 (3.72)

constitute the corresponding alternating series of dominant extrema of u~(t).

From (3.63) we find 7~

f x ( t ) - ~ cos~0Gu~(t)&0. (3.73) 2

From (3.71), (3.72) and (3.73) we conclude that only the alternating series of dominant extrema Mk and mk of u(t) affect the value of fx(t). All other input extrema are wiped out. This is tantamount to the wiping-out property of the hysteretic relation betweenfx(t) and u(t).

We shall next prove that the above hysteretic relation also exhibits congruency of minor loops. Let u(1)(t) and u(2)(t) be two inputs which vary between u+ and u_ for t/> to and which may have different past histories

~ However, starting from instant of time t~, these inputs vary for to ~< t ~< t 0. ' and u' back-and-forth between the same two consecutive extrema, u+ _.

As a result of these back-and-forth input variations, some minor loops are formed. We intend to show that these minor loops are congruent. The proof of the congruency of the above loop is equivalent to showing that any equal increments of inputs u(1)(t) and u(2)(t) result in equal increments


l u§ u+cosq)

~ u ~ c o s ~ u'+cos~o

/ r I/ g / I ~ (1) M I/ V/ I ~ (2)

U C U_


of outputsfx (1) (t) andfx (2) (t). To this end, let us assume that both inputs after achieving the same value u ~_ are increased by the same amount: Au (1) =

(1) Au (2) -- Au. As a result of these increases, the identical triangles T~ and

+ (t) + t positive are sets ) T (2) added to S~, 1 and S~,2( and subtracted from the

negative sets Sf, l(t) and S~-,2(t ) (see Figs. 3.13 and 3.14). Since

A ffs r ' ~ u ~ ( t ) = v(~, fl, ~o) doe dfl - v(o~, fl, ~p) doe dfl, (3.74) + (t) -~ (t)

from (3.73) we derive 7r

Afx(1) = 2 cos (ff l , 2

(3.75)

7r

/-2 (f/T~av(Ot, fl,~o)doldfl)dcp. (3.76) Af(2) _ 2 ~ cos ~o ) 2

Since T (1) = T (2) for any ~0, we conclude that

A / ( 1 ) - - Ad (2) . (3.77)

The equality (3.77) has been proven for the case when inputs u(1)(t) and u(2)(t) are monotonically increased by the same amount after achieving the same min imum value u2. Thus this equality means the congruency for the ascending branches of the above minor loops. By literally repeating the previous reasoning, we can prove that the same equality (3.77) holds when the inputs u(1)(t) and u(2)(t) are monotonically decreased by

' This the same amount Au after achieving the same max imum value u+. implies the congruency of descending branches of the above minor loops.


Thus, the congruency property for minor loops is established. This completes the proof of the validity of the reduction property. [-1

In our discussion of the reduction property, we have proven the congruency of "scalar" minor loops described by the vector Preisach models. The last result admits the following generalization.

CONGRUENCY PROPERTY OF VECTOR MINOR LOOPS Let the tips of two inputs fi(1)(t) and fi(2)(t) trace the same closed curve for t >~ to (see Fig. 3.15). Then the tips of the corresponding outputs f(1)(t)andf (2)(t) of the vector Preisach models trace congruent closed curves for t >~ to (see Fig. 3.16). These curves may be noncollocated in space because of possibly different past input histories prior to to.

PROOF. Consider the 3D Preisach model (3.60)-(3.62). Since the tips of both inputs fi(1)(t) and fi(2)(t) trace the same closed curve, we conclude that for any direction e0,~ the corresponding ~-fl diagrams are modified with time in the identical way within the same triangles T(u+,o,~,u_,o,~). For instance, as the tips of the inputs ~(1)(t) and v(2)(t) move from the point a to the point b (see Fig. 3.15), the same regions f21,0,~ and fa2,0,~ may be added to the positive sets S + and S + and subtracted from the negative sets 0,~0,1 0,~0,2 So,~I and S~,~, 2. Since

v(~,/~, 0, ~) d~ d/~ Fo,~uo,+(t) = ~,~o (t)

- 1"/" v(u, fl, 0, ~0) dc~ dfl, (3.78) JJs o,~o(t)

we conclude that

A~ . (1) f f .o,~uo, ~ (t) = 2 v(a, fl, O, qg) da dfl, (3.79) dd~ 1,0 ,~o

A~ . (2) f f ,o,~uo,~(t ) = 2 v(a, fl, O,~o)dadfl. (3.80) dd~ 2,0 ,~o

By using (3.60), (3.79) and (3.80), we find that the corresponding output increments which connect the points A(1),B (1) and A(2),B (2) (see Fig. 3.16) are given by

~02Jr f rr ( f f~ ) a f (1) -- 2 eo,~ v(o~,fl, O,~o)dadfl sinOdOd~o, (3.81) d 0 1,0,~0

~02rrf Jr (fff~ ) Af (2) - - 2 -eo,~ v(~, fl, O, ~o) d~ dfl sin 0 dO d~o. (3.82) JO 2,0,r

3.3 SOME BASIC PROPERTIES

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Since ~1,0,~ -- ~2,0,~0 for any 0 and ~, from (3.81) and (3.82) we conclude that

Af (1)-- Af (2) . (3.83)

The equality (3.83) holds for arbitrary chosen points a and b, and this proves that the vector minor loops shown in Fig. 3.16 are congruent. E3

We next proceed to the discussion of one remarkable property which is valid for the 2D isotropic vector Preisach model (3.67).

ROTATIONAL SYMMETRY PROPERTY Consider a uniformly rotating input (that is one of constant magnitude and angular velocity)

~(t) = { ux(t) - - U m cos cot, uy = U m sin cot }. (3.84)

Then the output of the 2D isotropic Preisach model (3.67) can be represented as

f i t) =f0 +f i t ) , (3.85)

where fo does not change with time, while f (t) is a uniformly rotating vector.

PROOF. It is clear from the very definition of the uniformly rotating input ~(t) and Fig. 3.17 that

u~(t) - - U m cos(cot - ~a). (3.86)

According to the proposition proved in Section 2 of this chapter, the integration over any semicircle can be used in the definition of the 2D vector


~ ~(t)

FIGURE 3.17

Preisach model. This fact allows one to modify the definition (3.67) of the 2D isotropic Preisach model as follows:

-* lf02rr (/f~ ) f ( t ) = -~ -e~o v(ot, f l )G~u~(t) dot dfl d~o. (3.87)

It is clear from (3.86) that for all directions ~ the corresponding ot-fl diagrams are modified with time within the same triangles T = T(um,-urn) . Outside of these triangles, the ot-fl diagrams remain unchanged. These unchanged parts of ot-fl diagrams contribute to the term f0 in (3.85), while the time varying parts of ot-fl diagrams result in the time varying term

f i t ) . By using the above comment as well as (3.86) and (3.87), we find:

"~ lf02zr (ff~ ) f i t ) -- -~ -~ ~ v(ot, f l ) ~' a fi U m cos(cot - ~o ) dot d fl d ~o . (3.88)

The expression (3.88) can be represented in terms of Cartesian components as

- 1 f027r (ff~ ) f x ( t ) - ~ coscp v(ot, f l)G~UmCOS(cot-~o)dotdfl d~o, (3.89)

xf0a:r (/f~ ) )~y(t) = sin 9) v(ot, fl)G~Um cos(cot - ~0) dot dfl dcp. (3.90)

Consider some instant of time t. For this instant of time, all directions ~ can be divided into two sets such that

0 ~< cot-9) ~< Jr, (3.91)

and

Jr ~< cot-~0 ~< 2Jr. (3.92)

For the first set, all input projections u~(t) are monotonically decreasing, and this results in the ot-fl diagram shown in Fig. 3.18. For the second set, all input projections u~(t) are monotonically increasing, and this results in

3.3 SOME BASIC PROPERTIES

Ct Ct

lu oso u ~176


173

the c~-fl diagram shown in Fig. 3.19. Next we introduce new variable

0 - cot - ~0. (3.93)

It is apparent that the double integral over T in (3.89) and (3.90) is the function of 0. This justifies the following notation

G(O) - l l v(c~, ~)G~Um cos(~ot - ~0) dc~ d/~. (3.94) ddT

0 ~< 0 ~< re, (3.95)

then the diagram shown in Fig. 3.18 is valid, and from this diagram and formulas (3.68) and (3.94) we find

G(O) -- P(um, -urn) - 2P(um, Um COSO). (3.96)

If

Jr ~< 0 ~< 2re, (3.97)

then the diagram shown in Fig. 3.19 is valid, and as before we find

G(O) -- 2P(um cos0,-urn) - P(um, -urn). (3.98)

By using the change of variables (3.93), the notation (3.94), and by taking into account that

dO = -d~o, (3.99)

and

cot- 2re ~ 0 ~< cot, (3.100)


we transform (3.89) as follows:

1 i02rr J~x(t) = ~ cos(~ot-O)G(O)dO = A cos~ot + B sino~t,

where

l f02" A - -2 cosOG(O)dO,

1 io 2rr B -- ~ sin OG(O) dO.

Similarly, the expression (3.90) can be reduced to the form

d~y(t) = A sin oJt - B cos o~t.

The formulas (3.101) and (3.104) can be modified as follows:

fx(t) - vIA 2 q- B 2 cos(ogt - ~),

j~y(t)- v/A 2 -}- B 2 sin(~ot- ~),

where

(3.101)

(3.102)

(3.103)

(3.104)

(3.105)

(3.106)

B tan ~ - ~ . (3.107)

From (3.105) and (3.106), we conclude tha t f is a uniformly rotating vector

with the magnitude equal to v/A 2 -+- B 2. We next express A and B in terms of the function P and prove that the angle ~ in (3.105) and (3.106) is acute. From (3.95)-(3.98) and (3.102) we find

eli0 a = -~ cosO[P(um,-Um) - 2P(um, UmCOSO)]dO

~2rr } + cosO[2P(umcosO,-um) - P(um,-Um)]dO �9 (3.108)

In the second integral in (3.108), we will use the change of variables

0' = 0 - rr (3.109)

and take into account that

P(-um cosO, -urn) = P(um, Um COSO). (3.110)

The last formula easily follows from the symmetry property (3.58) and the definition (3.68) of the function P. By using (3.109) and (3.110), we can transform (3.108)as follows:

i0 A = - 2 cosOP(um, UmCOSO)dO. (3.111)


From (3.95)-(3.98) and (3.103), we obtain

l/J0 B = -~ sinO[P(um,-Um)- 2P(um, UmCOSO)]dO

~2rC } + sinO[2P(umcosO,-um)- P(um,-Um)]dO . (3.112)

To transform the second integral in (3.112), we will use the change of variables (3.109) and the identity (3.110). This eventually leads to the following expression for B:

f0 B -- 2P(um,-Um) - 2 sinOP(um, UmCOSO)dO. (3.113)

Thus, we have found explicit expressions (3.111) and (3.113) for A and B in terms of the function P. In the next section, it will be shown that P can be related to some experimental data. In this way we can relate A and B to the experimental data, and, consequently, find the magnitude and phase

of the uniformly rotating vector f i t ) in terms of this data. Now, we will use formulas (3.111) and (3.113) to prove that the phase

angle ~ is acute under some general conditions. The first condition can be expressed mathematically as

P(~,/~) ~ P(~,/~') if ~ ~ ~'. (3.114)

The condition (3.114) means that P(c~,/J) is a monotonically decreasing function of/~ for any fixed c~. It is clear from the definition (3.68) of P that the condition (3.114) is satisfied if v(c~,/~) is positive.


Elo ] A - - - 2 cosOP(um, UmCOSO)dO + cosOP(um, UmCOSO)dO . (3.115)

By using the change of variables 0' -- Jr - 0 in the second integral in (3.115), we derive

A -- - 2 cosO[P(um, UmCOSO) - P(um,-UmCOSO)]dO. (3.116)


A > 0. (3.117)

The second condition is

foJrSinO[P(um,-Um)-2P(um, UmCOSO)]dO > (3.118) O.


From (3.114) and (3.118), we find

B > 0. (3.119)

From (3.107), (3.117) and (3.119), we conclude that the phase angle ~ is acute. This completes the proof of the rotational symmetry property. E~

It is worthwhile noting that in spite of the nonlinear structure of the vector Preisach model (3.67) the time harmonic input (3.84) produces (up to a history-dependent constant term ;~0) a time harmonic output of the same frequency. In other words, no generation of higher order harmonics is caused by the nonlinear structure of the vector Preisach model. This remarkable fact admits the following physical explanation. The isotropic vector Preisach model (3.67) has a mathematical form which is invariant with respect to any rotation of Cartesian coordinates. The mathematical form of uniformly rotating input (3.84) is also invariant with respect to any rotation of Cartesian coordinates. Thus, on the symmetry grounds we expect that the mathematical form of the resulting output should also be invariant with respect to any rotation of Cartesian coordinates. This is possible only if the output is a uniformly rotating vector. The above discussion clearly reveals the meaning of the term "rotational symmetry property."

The rotational symmetry property has been confirmed by numerical computations. By using a digital code that implements the vector Preisach model (3.67) and that has been briefly described in the previous section, the output of the Preisach model has been computed for the input shown in Fig. 3.20. This input gradually approaches the regime of uniform rotation (3.84). The results of computations shown in Fig. 3.21 demonstrate

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that the output also gradually attains the regime of uniform rotation. The property of rotational symmetry has also been observed in the experiments. The results of these experiments are shown in Fig. 3.22. In these experiments, the applied magnetic fields were stationary, while the mag- netizable sample of Isomax material was uniformly rotated. Thus, in these experiments, the magnetization was measured in the coordinate frame uniformly rotating with respect to the sample. The component of magnetization measured along the applied field is called "parallel magnetization" while the component perpendicular to the field is named "transverse magnetization." The results shown in Fig. 3.22 clearly suggests that for large applied fields the whole magnetization of the sample moves in synchro- nism with the uniformly rotating coordinate frame. This means that we deal with uniformly rotating magnetization. As the applied field is reduced, the time constant term of magnetization appears which reveals itself (in the uniformly rotating frame) as sinusoidally changing components of magnetization.

PROPERTY OF CORRELATION BETWEEN MUTUALLY ORTHOGONAL COM-

PONENTS OF OUTPUT AND INPUT Suppose that the input ~(t) was first restricted to vary along the y-axis. It was increased from infinitely negative value to some positive value u+, and then it was decreased to zero. Some remanent value of output fr =-eyfr resulted from the above input variations. After reaching zero, the input is restricted to vary along the orthogonal x-direction (see Fig. 3.23). It is asserted that, by increasing the input in x-direction, it is possible to reduce the orthogonal remanent component of the output to any however small value.

PROOF. All directions ~ can be subdivided into sets: 7C 7C

0~<~o~<~- and -~-~<~o~<0. (3.120)

._.)

f r ~ /~(t) r X

FIGURE 3.23


~ To x (~o) u c o s ~ (~o)

u,,n u ,nq J .......


The input variations along the y-axis results in c~-fl diagrams shown in Figs. 3.24 and 3.25 for the first and second sets of directions, respectively. As the input is increased along the x-axis, the above diagrams are modified. These modifications are shown by dash lines. They result in continuous expansions of positive sets S + (t) at the expense of negative sets S~-(t).

From (3.67), we find 7[

/ (ff fv(t) = sinp 2

By introducing the notation

G(%t) = ff~>~

v(o~, fl)G~ux(t) cosp dot dfl) dp. (3.121)

v(+, fl)G#ux(t) cosp dc~ dfl,

formula (3.121) can be represented as follows:

(3.122)

Changing p to - p in the second integral in (3.123), we obtain

fy(t) = ~0 2 sin p[G(p, t) - G( -p , t)] dp. (3.124)

If UxCOSp /> c~0, then according to the diagrams shown in Figs. 3.24 and 3.25 we have

G(p,t)=G(-p,t). (3.125)

/0 ~ f0 fy(t) = sin pG(p, t) dp 4- sin pG(p, t) dp. (3.123) ~T 2


From (3.124) and (3.125), we find

y/-

f y ( t ) = ~o r c c o s ux(t)

sin q~ [G(v~, t) - G(-~p, t)] d~o. (3.126)

or0 7r Since arccos ~ --~ g as ux( t ) ~ cx~, we conclude that

f y ( t ) ~ O. (3.127)

This completes the proof of the above property. D

This property has been confirmed by numerical computations performed by using a digital code that implements the model (3.67). The results of computations are shown in Fig. 3.26 for different remanent values of the output. The property of correlation between orthogonal components of input and output has also been observed in the experiments. The results of these experiments are presented in Fig. 3.27. There is apparent qualitative similarity between the computational and experimental results. It is worthwhile noting here that the above property of orthogonal correlation has been regarded as an important "testing" property for vector hysteresis models in magnetics.

I0

-5

- I0 0

i , , J , i l , , , i , , , , i , , , , l , , , ' , l , , , , _

5

ok_

1 I000 2000 ;]000 4000 5000 riO00

FIGURE 3.26

3.4 IDENTIFICATION PROBLEM FOR ISOTROPIC MODELS

-5

- i 0

' ' ' ' I ' ' ' ' I ' ' ' ' I '

i ~ ~ ~ L I t l i 11 J i l i I i

0 .05 .I .15

FIGURE 3.27

183

3.4 I D E N T I F I C A T I O N P R O B L E M F O R I S O T R O P I C

V E C T O R P R E I S A C H M O D E L S

The essence of the identification p rob lem is in de te rmin ing the function v(~,/~) or P(c~,/~) from some exper imenta l data. It turns out that this problem can be reduced to the solution of a special integral equat ion that relates the function P(c~,/~) to some "unidirect ional" (scalar) hysteresis data. We first present the der ivat ion of this equat ion for the 2D model (3.67). The der ivat ion proceeds as follows.

Consider the projection f x( t) off ( t ) along the direction ~0 = 0. Accord- ing to Eq. (3.67), we have

yr /+ fx(t) = cose v (c~',/~') 9~,/~,u~ (t)dc~' d/~') d~0. (3.128)

N o w we restrict ~(t) to va ry along the direction ~0- 0 which means that F~(t) =-~xu(t). First, we assume that u(t) is m a d e "infinitely negat ive." Then for any c~',/J' and ~0 we have ~, /~ ,u~(t )= - 1 . Next, we assume that the input is monotonica l ly increased until it reaches some value ~. Let f~ denote the cor responding value of fx(t). As a result of the above input increase, we find that for any ~o we have }9~,/~,u~(t) - +1 if c~' < ol cos~0 and }9~,/~,u~(t)- - 1 if c~' > c~ coscp. This is shown geometr ical ly in Fig. 3.28.


4 6 s-T ( ~ ) / , s-[ (~) / ' + . I I / ;

13cos~


Consequently, for any ~ we have

/ ~ v(d, g')~,~,~,u~(t)dot' dg' ,~>/~,

=//s v(d'fl')dddfl'-f/s v(d, fl')dddfl'. (3.129) +,~ ~,~ Finally, we assume that the above input increase is followed by a subsequent monotonic decrease of u(t) until it reaches some value g. Let f~/~ denote the corresponding value of fx(t). The above input decrease modifies the previous geometric diagram that now for any ~ assumes the form shown in Fig. 3.29. According to this figure, we have

f~ ~(~',~')~,e,u~(t)a~'a~' ,>,~,

=//S v(d,g')dddfl'- //s v(d, fl')dddg'. (3.130)

Consider the function 1

F(c~, fl) = ~(fr - f ~ ) . (3.131)

By using formulas (3.68), (3.128)-(3.131), it is straightforward to show that P(c~, g) is related to F(c~, g) by the expression

7T

: cos~P(c~ cos~a)d~ F(ol, g). (3.132) COS~r YT

2

It is apparent that f~ and f~/~ can be found by measuring the first-order transition curves. Of course, the input u(t) cannot be made infinitely neg-

3.4 IDENTIFICATION PROBLEM FOR ISOTROPIC MODELS 185

ative as was assumed in the derivation. However, by making the input "sufficiently negative," a reasonable accuracy can be secured.

The formula (3.132) can be construed as the integral equation which relates P(a, fl) to the first-order transition curves. This integral equation has a peculiar structure that is revealed by the following result.

THEOREM Consider the operator Jr

fi~P = / ~ Jr cos ~oP(ol cos ~o, fl cos r dcp. 2

Monomials ak fls are eigenfunctions of the operator fi~.

(3.133)

PROOF. From (3.133)we find Jr

A~ kt~ s - f Jr cos ~0o~ k cos k ~ofl s cos s ~o d~o 2

Jr

= ock/js f gJr cosk+S+l r d~o - ~.k+l Ock/js 2

where _ (2n-1)!! J~ ~ -

Xk+s = 9 (2n)!! _ "- (2n+1)!!

i f k + s + l = 2 n ,

i f k + s + 1 - 2 n + 1 .

(3.134)

(3.135)

E]

By using the above theorem, the solution of Eq. (3.132) can be easily found when the right-hand side F(~, fl) is a polynomial

M F(~,fl)-- ~ ~ aks-(m)~ " (3.136)

m=0 k+s=m

Indeed, looking for the solution of the integral equation in the form

M P(ol, fl) ~ E -(m) k'~s = Pks ~ P , (3.137)

m=0 k+s=m

from the above theorem we find _(m)

p~m) = "ks . (3.138) Xm

By using this fact, the following algorithm for the solution of Eq. (3.132) can be suggested. We first extend the function F(u, fl) from the triangle


- a0 ~< fl ~< a ~< a0 to the square - a0 ~< a ~< a0, - a0 ~< fl ~< a0, for instance, as an even function with respect to the diagonal a - ft. Then we expand this function into the series of Chebyshev polynomials

F(a, 3) = ~ aeqTe(a)Tq(fl), (3.139) s

where the Chebyshev polynomials are defined as

1 T0(a) = 1, Te(a) = ~ cos(s arccosa). (3.140)

By assuming for the sake of notational simplicity that a0 - 1, for the expansion coefficients aeq in (3.139) we have the formula (see [12])

2e+qflf?F(~,fl)Te(r as l l v / l _a2v / l _ f l 2dad f l . (3.141)

Let teq(a, fl) be the solution of the integral equation yr

cos~oteq(acos~o,3cosqo)d~o= re(a)rq(3), (3.142) 2

then the solution of the integral equation (3.132) can be represented in the form

P(a, 3) = ~ aeqteq(a, 3). (3.143) e,q

Thus, there are two major steps in finding the solution (3.143) of the integral equation (3.132). The first step is to solve Eqs. (3.142). This can be accomplished by using the technique (3.136)-(3.138). The second step is to evaluate the expansion coefficients aeq. This can be handled by using the formula (3.141). However, for the sake of computations it is desirable to transform this formula as follows. By introducing new variables 0 and 0t:

a = cos 0, fl - cos 0', (3.144)

we find

coss cosq0' (3.145) Te(a)= 2s Tq([3)= 2q_1,

4 f ~ f ~ o' ' ' aeq= -~ F(cos0,cos )cosf, OcosqO dOdO. (3.146)

Thus the expansion coefficients aeq can be evaluated by using fast cosine- transform schemes.


There is another approach to the solution of the integral equation (3.132) that leads to a closed form expression for P(01,/3). This approach is based on the following change of variables:

x = 01 cos ~0, X =/3/01, (3.147)

which after simple transformations yields the following integral equation of the Abel type:

fo ~ x P(x, Xx) dx - 01 v/012 L X 2 ~F(01,)~01). (3.148)

Some simplification of the above equation is achieved by introducing new auxiliary functions:

01 N(x) = xP(x, Xx), R(01) - ~F(01,X01). (3.149)

Then, from (3.148) we obtain

fo ' N(x) dx = R(01). (3.150) /012 m X 2

The solution of the above equation can be found by using the following O/ trick. We multiply both sides of (3.150) by %/S2_0t2 and integrate with re-

spect to 01 from 0 to s:

fo s 01 ( fo~ N(x)dX ) da, = foS ~ dot. (3.151) V/S2 _012 J o t 2 _ X 2 V/S2 _012

By using Fig. 3.30, we find

18 ~ V / S 2 - - O r 2 JOt , 2 - - X 2

f ~ 0IN(x) dx dot v/(S2 _ 012)(012 _ X 2

foS ( fx s 01d01 ) dx. (3.152) - - N ( x ) V/(S2 _ 012)(012 _ x 2)

Next, by using simple transformations, we obtain

fx s 01d01 _ l f x S d(01 2) . (3.153) v/(S2_ 012)(012_ x 2) 2 ~ / ( s2 -x2) 2 2 _ (012 - 2 )s2+x2 2

By introducing a new variable of integration

s 2 x 2 co - - 012 q- , (3.154)

2


l (X

D. X

FIGURE 3.30

from (3.153)we derive

s 2_x 2

fx s c~dot 1 fx22s2 do) / ( S 2 - X 2 ) ( O / 2 - - X 2 ) = 2 ~(s2-x2) 2 2 --('02

s 2 _x 2

1 arcsin 2w ]~= 2 y'fr (3.155) _ ~ ~ o

2 S 2 -- 092 x2-s 2 2 I ~ = 2


/0 s 2f0s N ( x ) d x - - -- . (3.156) Jr V/S2 _ o/2

By differentiating (3.156) with respect to s and then by replacing s by c~ and ~ by s, we find

N(o~) - 2 d fo ~ sR(s) ds. (3.157) Jr doe /o12 _ S2

This is the closed form solution to the integral equation (3.150). Now, by substituting (3.149) into (3.157), we find the closed form expression for P:

_ 1 d [o~ s2F(s, Xs) ds. P(ot , Xol) (3.158)

yro/dot J0 / c ~ 2 - 8 2

The last expression is valid for any value of ~. By varying ~ from 0 to 1 and using (3.158), we can compute all required values of P. However, for actual computations, it is convenient to transform (3.158) by integration by parts. The final expression is then given by

1 fo~ F(s, 1.s) 4- s ~F(s , 1.s) P(o~ , Xot) = -- ds. (3.159)

7/" J0 V/O/2 -- S 2


Now, we turn to the three-dimensional isotropic model (3.66). Con-

sider the projection fz(t) of f i t) along the direction 0 = 0. According to (3.66), we have

fz(t) = cos0 ,>>,~, v(c~', fl')~,~,~,uo,~(t) d~' dfl'

x sin 0 dO d~a. (3.160)

We restrict ~(t) to vary along the direction 0 = 0, which means that ~(t) = -~zU(t). As before, we first assume that u(t) is made infinitely negative. Then for any ~, fl,0 and ~a we have G~uo,~(t) - - 1 . Next, we assume that the input is monotonical ly increased until it reaches some value c~. Letf~ denote the corresponding value of fz(t). As a result of the above input increase, we find that for any 0 and ~ we have G,~,uo,~(t) - +1 if c~' < c~ cos0 and ~,~,~,uo,~(t) - - 1 if c~' > c~ cos0. This is il lustrated geometrically in Fig. 3.31. Consequently, for any 0 and r we have

f f~ v(a' , fl')f,~,~,uo,~(t)da' dfl' , ~ ,

= [ /" v(ot', f l ' )do t 'd f l ' - / I " v(ot', fl')dot'dfl'. (3.161) diS +,o sss;,o

Finally, we assume that the above input increase is followed by a subsequent monotonic decrease of u(t) until it reaches some value ft. Let f~/~ denote the corresponding value of fz(t). The above ment ioned input decrease results in geometric diagrams shown in Fig. 3.32. According to this



figure, we have

ff~ v(a',fl')~,y, uo,~(t)dc~'dfl'

= H v(ot',/J') dot 'd~ ' - f f v(c~',/~') dot'd~'. (3.162) dis +,~,o J J s;,~,o

By substituting (3.161) and (3.162) into (3.160), then by subtracting one expression from another and by using (3.68) and (3.131), we derive

fo2~fo~COsOP(c~cosO, flcosO)sinOdOd~o-F(ot, fl). (3.163)

Since the integrand does not depend on r we obtain

L + cosOsinOP(olcosO,~cosO)dO - F(ot,/~). (3.164)

This is the integral equation that relates P(c~,/~) to the experimentally measured first-order transition curves. It turns out that this equation is easily solvable. By using the change of variables

x = c~ cos0, )~ =/3/~, (3.165)

Eq. (3.164) can be represented in the form

f0 ~ or2 xP(x, Xx) dx = ~n-n F(c~, X~). (3.166)

By differentiating the last equation with respect to c~, we find

1 d [c~2F(c~, X~) ] (3.167) c~P(c~, X~) = 2Jr d~

which leads to the following final expression:

P(c~,Xc~)= 1 d [c~2F(a, Xc~)]. (3.168) 2tea dc~

It is remarkable that the solution of the identification problem for 3D isotropic Preisach models of vector hysteresis turns out to be much simpler than the solution of the same problem for 2D isotropic models.

3.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR PREISACH MODELS

We shall first discuss the identification problem for the two-dimensional anisotropic model (3.65). As before, we shall use the function P(a, fl,~0)

3.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC MODELS

J FIGURE 3.33

191

defined by (3.68). We shall relate this function to the experimental data represented by the sets of first-order transition curves measured along all directions ~ . These experimental data can be characterized by the function

F(c~, fl,~p) - 2(f~ ~ - f ~ ) , (3.169) . t

which will be assumed to be known in the subsequent discussion. Of course, it is not feasible to measure first-order transition curves for all directions ~ . However, it is possible to measure these curves for some finite meshes of directions ~ and to use subsequent interpolation for computing F(~, fl, ~o) for all ~0.

Consider local polar coordinates (p, q) with polar axis directed along the vector ~r The model (3.65) in the local polar coordinates can be written as

yr ) f i t ) = ~V, v(a, fl,~p' + ~/)9~ue/(t)dadfl d~/, (3.170)

where the relationship between the angles qo, ~p~ and ~ is illustrated by Fig. 3.33.

By using the expression (3.170) and by repeating almost literally the same reasoning as in the derivation of integral equation (3.132), we find that the function P(a, fl, cp) is related to the function F(a, fl, cp) by the expression

7~

~ cos 0P(c~ cos 0,fl cos ~,~0' + 0 ) d 0 = F(oe, fl,~o'). (3.171) 2

Next we shall use the following Fourier series expansions:

F(ot, fl, ~p') - E Fn(ot, fl)e in~', (3.172)

P(a, fl, ~o) - ~ Pn(~, fl) e in~, (3.173) t l ~ - - O C


o o

V(O/,/J, 99) = y ~ Vn(CG/J) e in~. (3.174) / / - - - - - - ( 3 0

It is clear from (3.68) and (3.69) that Pn and Vn are related by the expressions:

Pn(ot, ~) = / f T Vn (c~',/~') dc~' d~', (3.175) (~,~)

a2pn(ot, ~) Vn(Ol, ~ ) "-- --

Oot O~ By substituting (3.172) and (3.173) into (3.171), we find

(3.176)

zr

Y~ ein~~ f 2_ ein~ cos ~rPn(ot cos ~, fl cos ~) d~ F I - - - - O C 2

( x )

= E Fn(o~, fl)e in~'. (3.177) n = - - o o

From (3.177) we obtain Ar

f ~ cos cos ~, 3 cos ~) = fl) ein f t ~ V n ( o l d~ Fn (ol , yr

2

(n -0 ,4-1,4-2, . . . ) . (3.178)

Since s inn~ cos qPn(o~ cos ~, ~ cos ~) is an odd function of q, from (3.178) we derive

yr

f ~ cos ~ cos cos ~,/~ cos = n fr Pn (ol f,) dfr Fn (ol , [3) yr

2

(n = 0, +1, i 2 , . . . ) . (3.179)

Thus we have obtained the infinite set of decoupled (separate) integral equations for Pn. The right-hand sides Fn(oG ~) of these equations can be computed by using fast Fourier transform (FFT) algorithms.

For the case n - 0, from (3.179) we find yr

cos ~rP0(c~ cos ~,/~ cos ~) d~k - f0(~,/J), (3.180) 2

which naturally coincides with the integral equation (3.132) for the isotropic vector Preisach model.

If n = 4-1, from (3.179) we obtain yr

COS2 ~P+I (or cos ~r, ,8 cos ~) d~r = F+l(Ot,/J). (3.181) 2

3.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC MODELS 193

Closed form solutions to integral equations (3.180) and (3.181) can be found by using the change of variables

x = a cos ~, X = f l /a (3.182)

and by reducing the above equations to the Abel type integral equations. In the case of Eq. (3.180), the result is readily available and can be expressed by the formula

1 d f ~ saFo(s, Xs) ds. P0(c~, Xc~) (3.183)

Jr c~ do~ ,Jo V/Ol 2 - s 2

In the case of Eqs. (3.181), the change of variables (3.182) leads to the integral equations

fO c~ X 2 dx or, 2 P+I (x, Xx) V/Or 2 _ X 2 = 2 F+l(Ol, Xot). (3.184)

These equations can be solved in exactly the same way as Eq. (3.148). The final result is given by the following formula

1 d f ~ s3F+l(S, Xs) ds. P+I (~, ~ ) (3.185)

Ol 2 da ,Jo v /o l 2 - s 2

Thus, if we are interested only in the first three terms of the Fourier expansion for P (this is a first-order approximation for anisotropic media), then the solution of the identification problem can be found by using the closed form expressions (3.183) and (3.185). In the case of higher order terms, the change of variables (3.182) leads to the following Abel type integral equations

f0 o x Pn(X, XX) xTn(-d) dx= ol V/Ol2 _ X 2 ~-~Fn(ol, Xol), (3.186)

where Tn are Chebyshev polynomials defined by (3.190). Unfortunately, we have not been able to find closed form solutions to these equations. However, there are many efficient numerical techniques developed for the solution of integral equations of this type (see, for instance, [12]).

The analytical solutions to integral equations (3.179) can also be found by exploiting the following property of this equation.

THEOREM Consider operators yr

AnP - cos ~ cos nOP(ol cos ~, r cos ~) dO. 2

Monomials c~ k fls are eigenfunctions of these operators.

(3.187)


PROOF. It is straightforward to check the validity of the above statement

and to get the following expressions for the eigenvalues ~(n) . ""k+s " Jr

1o" ~(n) = 2 cos nO cos k+s+l ~ dO (3.188) "~k+s

The explicit formula for the integral in (3.188) can be found in the literature (see [13]). E]

By using the above theorem, polynomial solutions to the integral equations (3.179) can be easily found. Indeed, if the right-hand sides of these equations are polynomials in the form (3.136), then the solutions to these equations will be polynomials in the form (3.137). Polynomial coefficients of the right-hand sides and the solutions will be related by the

expression (3.138) where instead of eigenvalues ).m the eigenvalues ~(n) "~k+s should be employed. By taking advantage of this fact, Chebyshev polynomial expansions of type (3.139) for Fn(ff,, ,B) can be used and the solutions can be found in the form (3.143), where the expansion coefficients can be computed by using expressions similar to (3.146).

Now, we proceed to the identification problem for three-dimensional anisotropic vector Preisach model (3.62). This problem is technically more complicated than the corresponding problem for 2D models and, for this reason, it requires a special treatment. It turns out that some facts from the theory of irreducible representations of the group of rotations of three- dimensional Euclidean space are instrumental in the treatment of this identification problem.

As before we shall use the function P(~,/~,0, ~0) defined by (3.68) and we shall relate this function to the first-order transition curves experimentally measured along all possible directions e0,~. These curves can be used to define the function

1 0, - (3.189)

It will be convenient to use spherical harmonic expansions for the functions v, P and F:

ec k

v(~ ~ ~ Vkm(~ k=O m=-k

k

P(o~, fl, O,~o) = y ~ ~_, Pkm(O~, fl)Ykm(O, qg), k=O m=-k

(3.190)

(3.191)


k F(~ "~ ~ ~ Fkm(~ ~)Ykm(O, qg) ,

k=0 m=-k (3.192)

where Wkm(O, 99) are spherical harmonics, while Vkm, Pkm and Fkm a r e corresponding expansion coefficients. It is clear from (3.68) and (3.69) that Pkm and Vkm are related by the following expressions:

Pkm(CG [3) - - / f T Vkm(O~', ~') dol' dfl', (3.193) (~,~)

32Pkm(Ol, ~) Vkm(Ol, ~) -- -- . (3.194)

O~ 03 It turns out that the identification problem can be reduced to the so-

lution of special integral equations which relate Pkm(Ol, ~) to Fkm(Ol, ~). The derivation of these equations proceeds as follows.

Consider an arbitrary direction specified by the angles 0' and ~0'. We shall use a local coordinate system xyz which is obtained from the system XYZ by Euler rotations Rz(O),Rx(O') and Rz(~ + q)') (see Fig. 3.34). The

Jr Euler angles for the inverse rotation from xyz to XYZ are equal to ~ - q)~, 0' and Jr, respectively. It is clear that the direction of the axis z coincides with the direction of e0'v'. By using local spherical coordinates ~ and gr, we can represent the model (3.62) as

f(t) fo :fo ~ v(c~,/~,~, q,0 ' , ~o')}3~ u~ r d/~)

x sin~ d~ dq. (3.195)

By using the last expression and by repeating almost literally the reasoning that was used in the derivation of the integral equation (3.163), we can show that the model (3.195) matches first-order transition curves

Z

z

o i Y

x

FIGURE 3.34


measured along the direction e0'~' if the function P satisfies the integral equation

f02~~0 ~ cos ~P(o~ cos ~, fl cos ~, ~, ~, 0', 99')sin~ d~ d~

= F(o~, fl, O', 99'). (3.196)

Next, we shall relate the function P in local coordinates to the same function in spherical coordinates 0 and 99. To achieve this, we shall use the spherical harmonic expansion (3.191) and the following facts from the theory of irreducible representations of the group of rotations (see [14-16]).

Linear combinations of spherical harmonics Wkm of the fixed order k form a linear space Hk of the irreducible representation of the group of rotations. This means that any spherical harmonic Ykm(O, 99) in coordinates 0 and 99 can be represented as a linear combination of the spherical harmonics Ykm(~, ~) of the same order k:

k Ykm(0,99,) E k ( Jr ) , -- atom, Jr ,0 ' , -~- -- 99' Wkm'(~, ~ ) , (3.197)

m'=-k

where k a mm, are the matrix elements of the irreducible representation of the rotation group in Hk. These matrix elements are functions of Euler angles that determine the rotation from the local xyz coordinate system to the original X Y Z system. These functions are sometimes called generalized spherical harmonics because they are reduced to spherical harmonics when m or m' is equal to zero.

By using (3.197) and (3.191), we find the expression for P in local coordinates

oo k k zzz ( . ) k=0 m=-km'=-k amm' Jr ,0 ' , -~ -- 99' Pkm(Ot,, fl)Ykm'(~, fr). (3.198)


oo k k

amm, Jr ,0 ' , ~ -- 99' k=0 m=-k m'=-k

~02~~0 -~ x cos~Pkm(Ot cos~, fl cos~)Ykm,(~, fr) sin~ d~ dO. (3.199)

3.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC MODELS 1 9 7

In (3.199) the integral with respect to 7t is equal to zero if m' ~ 0, and it is equal to 27r if m' = 0. Thus we obtain

cxz k 2 7 r ~ ~ k ( Jr ) -- amo yr , 0 f, -~ -- ~o'

k=0 m=-k

~0 art X cos~Pkm(Olcos~ , f lcos~)Yko(~)s in~ d~. (3.200)

It is known from the group theory (see [16]) that

am ~ re,01, 2 _ ~1 _ (_l)m 47r Ykm(Of, q~l). (3.201) 2k+ 1

It is also known (see [12]) that

~/2k + 1 Yko(~) = ~ Lk(cos ~), (3.202)

47r

where Lk are Legendre polynomials. From (3.200), (3.201) and (3.202), we derive

k F(ot, fl, O',go') --2yr ~_. ~_. (--1)mykm(O',go ')

k=0 m=-k 7g

~0 2 x Pkm(Ol cos ~, fl cos ~)Lk(COS ~) COS ~ sin ~ d~. (3.203)

By comparing (3.203) with (3.192), we find

YT

f0 r cos~, fl sin~ = ( - l~Fkm(U, fl). (3.204) m

Pkm (Ot cos ~)Ck(cos~) cos ~ d~ 2re

These are the final integral equations that relate Pkm to Fkm. These equations can be reduced to Volterra integral equations by using the following change of variables

x - c~ cos ~, X - - . fl (3.205) c~

Indeed, after simple transformations, we derive

f0 () Pkm(X, Zx)xLk x d x - ( -1) m a2 -~ -~--~ Fkm(CC, X~). (3.206)


Explicit solution to Eqs. (3.206) can be found for the cases k = 0 and k = 1. If k = 0, from (3.206) we obtain

~0 c~ c~ 2 xPoo(x, Xx) dx - ~-~n fo0(ol, Xo~). (3.207)

By differentiating (3.207) with respect to c~, we find

Poo(~,Xc~) - 1 d [c~2Foo(cG ~,Ol)]. 2Jrot dc~

(3.208)

As expected, this result coincides with the one obtained for the isotropic model (see (3.168)). If k = 1, according to (3.206) we have

~0 ~ oe3 X2plm(X, Xx) dx - (-1)m-~--~Flm(~, Xol) (m = 4-1, 0). (3.209)

By differentiating (3.209), we derive

Plm(OGXOl)- (-1)m d [c~3Flm(CGXc~) ] (m = +1,0). (3.210)

Thus, if we are interested only in the first four terms of spherical harmonic expansion for P (this is a first-order approximation for anisotropic media), then the explicit analytical solution for the identification problem is given by formulas (3.208) and (3.210).

For k > 1, Eqs. (3.206) can be solved numerically. Discretization procedures can be applied directly to Eqs. (3.206), or these equations can first be reduced by differentiation to the Volterra equation of the second kind

x ( x ) Pkm(~,X~) - Pkm(X, XX)--~L k dx

(--1) m d [ol2Fkm(OGXOl)]. 2rr~ dc~

(3.211)

In both cases we shall end up with simultaneous algebraic equations with triangular matrices which are easy to solve. However, the reduction to the second kind integral equation may be desirable as far as computational stability is concerned.

Finally, the integral equations (3.204) can be solved analytically if polynomial approximations for their right-hand sides are employed. These analytical solutions can be found in exactly the same manner as for 2D identification problems discussed before.

Up to this point, we have used scalar hysteresis data (3.169) and (3.189) measured for unidirectional variations of input ~(t) in order to solve the identification problems. However, for anisotropic media these input


variations result in vectorial data that for 2D and 3D problems can be represented in the following forms, respectively

-, 1 ( ~ _j~ ), (3.212) F(~, ~ , ~ ) -

-~ 1 (f~0~ - f ~ 0 ~ ) (3.213) r(~, ~, 0, ~) - ~

The above vectorial data account for the output components that are orthogonal to the directions of input variations. There is a natural desire to utilize these vectorial data in the identification of vector Preisach models. This can be achieved by generalizing the models themselves. The essence of generalization is in employing vectorial functions ~. This leads to the following vector Preisach model:

j~(t) = J~l~ ( f L F2(ol, fl,-f)~(~, fi(t))doldfi)dsr. (3.214)

By repeating the same line of reasoning as in Section 2 of this chapter, we can show that by means of redefining ~ the integration over a unit sphere in (3.214) can be reduced to the integration over a unit hemisphere

f(t)- /fc+ ( fL>~ F2(c~,fl,~)f,~fi(?. ~(t))d~d~)dsr. (3.215)

It can also be shown that the redefined function 9 has the following symmetry property

(~, fl, ~) = - ~ (-/3, -c~, -~) . (3.216)

By using spherical coordinates, the generalized model (3.215) can be represented as

fO2~:fo~(fL ) f ( t ) = ~ F2(ol, fl, O,g))~,~[3uo~o(t)dotdfl sinO dO dg). (3.217)

Similarly, the 2D model can be expressed in the form 7[

f_ (ff ) f(t) = F2(o~, fl, ~a)~,~u~o(t) do~ d~ dg). (3.218)

To solve the identification problems for the models (3.217) and (3.218), we as before introduce the following auxiliary functions

P(~,/3, 0, 9)) = f i t F2(c~',fi',O,9))do~'dfl', (3.219) (o~,~)

(3.220) (~,~)


These functions can be related to the experimental data (3.213) and (3.212) by the following equations, respectively

~02~~0~ P(otcos~,/Jcos~,~,~/r,0!,99!) sin~ d~ d~ -- F(oG/J,0!,~!), (3.221)

rr 2 P((x COS l~r, ~ COS 1//, (/9 ! "q- 1//) 41//~- t2(Ol, ~ , ~D!). (3.222) yr 2

By employing spherical harmonic expansions in the 3D case and Fourier expansions in the 2D case, we derive as before the following integral equations

yr

~0 ~_ * m

Pkm(Ol cos ~,/J cos ~)Lk(cOS~) sin~ d~ - (-1) -, Fkm(OG ~), (3.223)

yf

yr 2

COS n~Pn(ol cos ~,/J cos ~) d ~ - F-n(~, ~). (3.224)

By using the change of variables (3.182) and (3.205), the above integral equations can be reduced to the following forms, respectively

/o ( - ) - Pkm(X, XX)Ck x dx - ( 1)m cz ~_ Ol ~ k m ( Ol ' X Ol ) ' (3.225)

/0 ~ Tn(X) d x - 1Fn(c~,Xc~). (3.226) Pn(x, XX) V/cr _ x 2

When k and n are equal to one, the following explicit solutions of Eqs. (3.225) and (3.226) can be derived:

...>

Plm(CZ, X~) --

.->

Pl(ol, Xc~) --

(--1) m d [c~2/:lm(CGXc~)] 2rr~ dc~

1 d ~0 c~ S2Flv (s,_)~s)s 2 ds. Jrc~ dc~ ./Cr

( m - 0,+1), (3.227)

(3.228)

For other values of k and n, polynomial expansion techniques or numerical techniques can be employed for the solution of the above integral equations.

For isotropic media, the data (3.212) and (3.213) are reduced to (3.131). It can be shown that in this case ~(~,/~) = ~v(c~, fl) and thus the generalized model (3.218) is reduced to the model (3.67). The proof of the above statement is left to the reader as a useful exercise.

3.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS 2Ol

3.6 D Y N A M I C V E C T O R P R E I S A C H M O D E L S OF HYSTERESIS

The vector Preisach model of hysteresis that have so far been discussed are rate-independent in nature; they do not account for dynamic properties of vector hysteresis nonlinearities. The purpose of this section is to develop dynamic vector Preisach models of hysteresis. We shall discuss only isotropic dynamic vector models of hysteresis. We begin with 2D dynamic models; a straightforward extension to three dimensions will then follow.

The main idea of the design of the dynamic vector hysteresis models is to introduce the dependence of the function v for scalar Preisach mod-

els for all directions ~0 on the speed of output variations, dd~, along these directions. This can mathematically be expressed as

f ( t ) - f_~ -e~o fL>~ v ot, fl, --~ ~ u ~ ( t ) dot dfl d~o. (3.229)

The direct utilization of the above model is associated with some untractable difficulties that have been discussed in Section 4 of the previous chapter. These difficulties can be circumvented by using the power series

expansion of the v-function with respect to -~"

( v ot, fl, dt J = vo(ot, fl) + ~ Vl(ot, ~) + ' ' ' . (3.230)

By retaining only the first two terms of the above expansion, we arrive at the following dynamic model:

f(t) =f0(t) + ~ v 1 (ot, fllf,~u~(t) dot dfl d~, (3.231/ 2

where 7[

fo(t) - -~ vo(ot, fl)f,~u~(t) dot dfl d~o. (3.232)

It is clear that in the case of very slow input variations the second term in the right-hand side of (3.231) becomes negligible. Thus fo(t) can be construed as a rate-independent component. This means that the function v0(ot, fl) should coincide with the v-function of the rate-independent model (3.67). In other words, the function v0(ot, fl) can be determined by matching rate-independent first-order transition curves measured for unidirectional variations of the input ~(t).

202 C H A P T E R 3 Vector Pre isach M o d e l s of Hys te res i s

We next represent the model (3.231) in Cartesian coordinates. To this end, we shall use the following expressions:

e~ = ex cos ~o + e~ sin ~o, (3.233)

By substituting (3.233) and (3.234) into (3.231), after simple transformations we arrive at the following form of the model (3.231)"

d f -, -~ A ~ =f i t ) -fo(t), (3.235)

where the matrix A is given by

fi~=(hxx(fi(t)) hxy(fi(t))) (3.236) hyx(fi(t)) hyy(fi(tl) '

and the matrix entries are specified by the expressions: 7g

(iS ) COS 2 r Vl(C~, fl)~,r dc~ dfl d~p, (3.237)

7g

7I

2 __ sin2 g0 ( f f~>f i Vl(Ot, fl)~,r (3.238)

hxy(fi(t)) = ~yx(fi(t)) 7g

f= (ff ) = cos~0sin~0 Vl(C~,~)~,~u~(t)dc~d~ d~o. (3.239)

Thus the dynamic model (3.231) can be interpreted as a set of two coupled ordinary differential equations (3.235) with hysteretic coefficients (3.237)-(3.239). The expression (3.235) also suggests that the instant speed of output variations is directly proportional to the difference between instant and rate-independent output values. The last fact is transparent from the physical point of view.

We next turn to the identification problem of determining the function 1; 1(Or, fl) by fitting the model (3.235) to some experimental data. The following experiments are used to solve this problem. We restrict fi(t) to vary along the direction ~0 = 0 which means that fi(t) = -~xu(t). First, we assume that u(t) is made "infinitely negative" and then it is monotonically increased until it reaches some value c~ at t = to. Afterwards, the input is kept constant. As the input is being kept constant, the output relaxes from its valuef~ at t = to to its rate-independent valuef0~. Due to the symmetry,

df ~o = _~ df dfx dry (3.234) O t �9 ~-~ - cos ~o-~ + sin ~0 d-T"

3.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS 203

we have fo(t) = -exfo(t), f ( t ) = -~xf(t). (3.240)

Thus, according to the model (3.234)-(3.239), the above relaxation process is described by the differential equation

af - r a ~-~ =fi t ) -f0a, (3.241)

where Jr

I: "ga - - COS 2 99 Vl ( ~ * , / J ' ) d R ' d ~ '

(3.242)

and geometrical configurations of S+,~ and S~-,~ are the same as in Fig. 3.28. The solution to Eq. (3.241) is given by

t f i t ) - (fa - f o a ) e - G + foa. (3.243)

Thus ra has the meaning of relaxation time and can be experimentally measured.

Next the input u(t) is made again "infinitely negative." Then it is monotonically increased until it reaches the value c~. Afterwards, the input

' and is monotonically decreased until it reaches some value g at time t - t o it is kept constant for t > t~. As the input is being kept constant, the output relaxes from its value fa~ at t - t~ to its "static" value f0a~. Due to the symmetry expressions (3.240) hold and the model (3.235)-(3.239)yields the following differential equation for the above relaxation process:

where

af - r a ~ =fi t ) -f0a~, (3.244)

Jr /: r a f t - - COS 2 99 Vl (ol', fl') dot' d fl'

- f L+ Vl (d" g') dd dg') d~~ (3.245)

and geometry of regions S+,~ and S;,~ is the same as in Fig. 3.29. By solving (3.244), we find:

t

f i t ) = (fa~ - foa~)e ~ + foa~. (3.246)

Thus, ra/~ has the meaning of relaxation time and can be experimentally measured. We next show that by knowing relaxation times ra and ra/~ for


all possible a and fl we can determine the function Vl(O/, fl). TO this end, we introduce the functions

1 q(~, t ) - ~(r~ - r~) , (3.247)

P1 (~, t ) - ~ Vl (c~', fl') dot' dfl', (3.248) JJT (~,~)

where T(a, t ) is a triangle shown in Fig. 3.10. It is clear as before that P1 and Vl are related by the formula

O2Pl (ot,, fl) Vl(O/, t ) = - . (3 .249)

Oc~ 0/3

Thus, if the function PI(a, t ) is found, then the function Vl(a, t ) can be retrieved. However, from the computational point of view, it is more convenient to deal with the function Pl(o~, t ) rather then with vl (~, fl). This is because the double integral with respect to a and fl in expressions (3.237)- (3.239) can be explicitly expressed in terms of Pl(o~, t ) by using formulas similar to (3.71). Another advantage of using Pl(ot, t ) is that this function can be directly related to the experimental data (3.247). Indeed, by using the expressions (3.242), (3.245), (3.247), (3.248) and Figs. 3.28 and 3.29, we derive

yr f2 COS2 (PPI(a cos ~p, fl cos r dr - q(c~, fl). (3.250)

The expression (3.250) is the integral equation that relates the function PI(G, t ) to experimental data q(c~, fl). This equation is similar to the integral equation (3.132) and, consequently, the same techniques can be used for the solution of Eq. (3.250) as for Eq. (3.132). Namely, by using the change of variables

x = ~ cos~p, ;~ = fl/o~, (3.251)

Eq. (3.250) can be reduced to the following Abel type integral equation:

fO ~ X 2 dx o1, 2 PI(X, Xx) V/O/2 _ x 2 -- -~-q(c~, Xc~). (3 .252)

By using the technique discussed in Section 4 of this chapter, the following closed form solution of the above equation can be found:

_ 1 d for s3q(s, Xs) ds. P1 (3.253)

7t'ol 2 dc~ Jo J o t 2 -- S 2

We next turn to the discussion of three-dimensional dynamic vector Preisach models of hysteresis. Similar to (3.231), these models can be rep-

3.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS 205

resented in the following mathematical form

fo2 /o fit)-fo(t)+ ( 121 (ot, fl)G~uo~(t) dot dfi)

x sin 0 dO d~o, (3.254)

wheref0(t) represents a "static" component of hysteresis nonlinearities defined by the expression

* f02~f0 ~ ( / ~ ) -~ Vl(ot, fl)~,~uo~o(t)dotdfl sinOdOd~o. (3.255) fo(t) - eo~o >>,~

By using Cartesian coordinates, we have

e0e -- ex cos 9) sin 0 + ~y sin ~0 sin 0 + ez cos 0,

dfo~o dt = cos 9) sin 0 ~ + sin ~o sin 0 - ~ + cos 0 dt

(3.256)

(3.257)

By substituting (3.256) and (3.257) into (3.254), after simple transformations we find

df =)~(t) -]o(t) , (3.258)

^

where the matrix A and its entries are given by

( ?Zxx(fi(t)) ?Zxy(fi(t)) ?Zxz(fi(t)) ) A= ?Zyx(fi(tl) ~yy(fi(t)) ?Zyz(fi(tl) ,

?Zzx(fi(t)) ?Zzy(Ft(t)) ?Zzz(Ft(tl) (3.259)

f02 ?Zxx(fi(t)) - cos 2 99 sin 3 0

x ( / f ~ Vl(ot, fl)~otfluo~o(t)dotdfl)dOdcp, >>,fi

foS~fo -~ ~yy (~(t)) = sin 2 ~o sin 3 0

x ( / ~ Vl(a, fl)~'~uo~(t)dadfl)dOdcp, >~

/o2 /o ?Zzz(fi(t)) - cos 2 0 sin 0

(3.260)

(3.261)

(3.262)


/02 ~xy(Fl(t)) = ~yx(Fl(t)) = cos ~a sin ~a sin 3 0

x ( / ~ vx(ot, fl)~'~fluo~o(t)dc~dfl)dOd~p, (3.263)

~O2~fO ~ ~xz(~(t)) = ~zx(~(t)) - cos ~o cos 0 sin 2 0

x ( / ~ Vl(O~,fl)~'~fiuo~o(t)doldfl)dOd~, (3.264)

~0 2:rrfo ~ ~yz(~(t)) = ~zy(~(t)) -- sin~o cosO sin2 0

x ( / ~ Vl(Ol, fl)~'~uo~(t)dotdfl)dOd~. (3.265)

Thus the 3D dynamic vector Preisach model of hysteresis (3.254) can be interpreted as a set of three coupled ordinary differential equations (3.258) with hysteretic coefficients (3.260)-(3.265).

We next proceed to the discussion of the identification problem for the model (3.258)-(3.265). The solution to this problem is very similar to that for the model (3.235)-(3.239) except that the final form of the solution is much simpler in three dimensions than in two dimensions. The experimental data used for the identification of the model (3.258)-(3.265) is measured when the input is restricted to vary along the axis z, that is when ~(t) = -~zu(t). As before, two types of relaxation processes are considered. The relaxation processes of first type occur when the input is made "infinitely negative" and then monotonically increased to some value c~ and kept constant thereafter. The relaxation processes of second type occur when, starting from the state of negative saturation along the axis z, the input is first monotonically increased to some value ~, then monotonically decreased to some value fl and kept constant afterwards. The relaxation times r~ and r ~ of the above processes can be measured and used for computing the function q(a,/~) defined by (3.247). On the grounds of symmetry, for both types of relaxation processes we have

fo - -~z fo(t), fit) - -ez f (t). (3.266)

This results in the reduction of the model (3.258)-(3.265) to the following equation

^ a f azz (~(t)) ~ =f( t ) -fo(t). (3.267)

3.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS 207

Now, by using the same line of reasoning as in the derivation of Eq. (3.250), we can show that the function Pl(0l,/J) defined by (3.248) is related to the experimental data q(c~, 3) by the integral equation

7r

f0 Pl(ot cos0, 3 cos 0) co s2 0 sin0 dO = q(ol, 3). (3.268) 2Jr

By using the change of variables

3 x = c~ cos 0, ~. = - , (3.269) ol

the integral equation (3.268) is reduced to the form

f0 c~ 0/3 x2pI (x, Xx) dx -~ ~--~ q(a, Xcg), (3.270)

from which we derived the final expression

Pl(X, XX)= 1 d [aBq(a, Xc~)]" (3.271) 2zra 2 dc~

3.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS. EXPERIMENTAL TESTING

There are two ways in which the above vector Preisach models can be further generalized. The first way is to use generalized scalar Preisach models as the main building blocks for the construction of vector models. The second way is to generalize the notion of input projection u~(t). We begin with the first approach. To be specific, we shall use the nonlinear (input dependent) scalar Preisach models discussed in Section 2 of the previous chapter. Analysis of generalized vector Preisach models of this type is very similar to that for the "classical" vector Preisach models. For this reason, our discussion will be concise and will be centered around the description of final results, while filling in the details will be left to the reader.

We begin with 2D isotropic models that can be represented in the following mathematical form

yr

f(t) = u~o(t)

+ ~ (u(t))F~(t).

v(c~, 3, u~(t))~,~u~(t) dc~ d3) d~o

(3.272)


The numerical implementation of the above model is substantially facilitated by the use of the following function

P(c~,fl, u ) = [ f ~ v ( r (3.273) J d l~ ~flu

where R~u is a rectangle shown in Fig. 2.12. By employing the above function, we can find explicit expressions for

the double integral with respect to ~ and fl in (3.272). These expressions are similar to formula (2.66). In addition, the function P(c~, fl, u) can be directly used for the identification of the model (3.272). First- and second- order transition curves measured along any fixed direction (for instance, along the direction ~ = 0) will be utilized for the solution of the identification problem. By using these curves, the following function can be constructed

1 F(o~, fl, u) = ~(fo~u - fo,~u), (3.274)

where fo~u and fr have the same meaning as in section two of the previous chapter.

It can be shown that the function P(c~, fl, u) is related to the experimental data (3.274) by the following integral equation

~ cos ~0P(c~ cos ~o, fl cos ~0, u cos ~0) d~0 = f(oe, fl, u). (3.275) 2

This equation is very similar to the integral equation (3.132). Thus, the same techniques can be employed for the solution of Eq. (3.275) as for the solution of Eq. (3.132). For instance, an approximate polynomial solution of the above equation can be found. The finding of this solution is based on the fact that monomials ~kflsum are eigenfunctions of the operator A:

7E P

AP = [ cos ~oP(a cos ~o, fl cos ~0, u cos ~0) d~. (3.276) d yr

2

In other words, we have

fiiak fls u m = ),.k+s+mak fls u m, (3.277)

where: YT

f 2 99)k+s+m+1 ~.k+s+m --- (COS d~. 7~

2

(3.278)

Now, by expanding the right-hand side of the integral equation (3.275) into the series of Chebyshev polynomials

F(~,/3, u) - ~ a~qmTe(oOTq(fl)Tm(u ), f.,q,m

(3.279)


we can represent the solution of the above equation in the form

P(ol, fl, u) = ~ aeqmteqm(O~, ,6, u), (3.280) e,q,m

where the polynomials teqm(a, fl, u) are mapped by operator ,4 into polynomials Te(c~)Tq(fl)Tm(u). Polynomials t~qm(Ol, fl, u) can be determined by using expressions (3.277) and (3.278). As far as expansion coefficients aeqm are concerned, they can be computed by using the formula similar to (3.146):

8 f (cos O, cos 0', cos 0") a eqm - - - ~

x cos s cos qO' cos mO" dO dO' dO". (3.281)

It is also possible to find a closed form solution to Eq. (3.275). To this end, the following change of variables is used:

x = c~ cos ~a, X = f l / a , X = u / a (3.282)

and the above equation is reduced to the Abel type integral equation

fo ~ x x P(x,)~x, x x ) d x - ~ f ( ~ , ~ , Xc~). (3.283) /cr _ x 2

Using the same reasoning as in Section 4 of this chapter, we obtain the following solution of Eq. (3.283):

1 d l ~ s2F(s, Xs, XS) ds. P(~ , Xc~, X o/) (3.284) Jr c~ dc~ ,Jo v/ol 2 - s 2

As is seen from the above discussion, function P(c~, fl, u) (and consequently function v(a, iS, u)) can be determined by matching the increments (3.274) between the first order and second order transition curves. The function ~(u) in (3.272) can be found by matching an ascending branch of a major loop. This leads to the expression

~(u) = f + + f u . (3.285) 2u

We next proceed to the discussion of 3D generalized vector Preisach model of hysteresis. These models can be represented as follows

;~(t) fazr fro ( f f R ) = -eor v(a, fl, uo~(t))G~uo~(t) da dfl sin 0 dO d~ ,l O ,J O Uog)(t )

4- ~ (u(t))~(t). (3.286)

By introducing function P defined by (3.273) and by using experimental data (3.274) the identification problem for the model (3.286) can be re-


duced to the following integral equation

/o ~ F(c~, fl, u) (3.287) - - . cos 0 sin 0 P(c~ cos 0,/5 cos 0, u cos 0) dO = 2re

The explicit solution of the above equation can be obtained by using the change of variables

x = ~ cos0, ;~ =/~/c~, X = u/c~ (3.288)

and by reducing (3.287) to the integral equation

~0 c~ Ol 2 xP(x, Xx, xx)dx = ~--~F(c~,X~, Xc~). (3.289)

Differentiation of (3.288) yields

1 d [c~2F(c~,X~, Xc~)]. (3.290) P(c~,Xc~, Xc~)= 2rrc~ d~

We next turn to generalized dynamic vector Preisach models of hysteresis. For the sake of notational simplicity, we consider only 2D models; extensions to 3D models are straight-forward. The 2D generalized dynamic Preisach models can be defined as follows

7r

3~(t) =~0(t)+ f_~_~df~ ( / / R 1;l(~,fl, u~(t))~u~(t)d~dfl)d~o, (3.291) 2 ~ u~o (t)

where f0 is a "static" component of hysteresis nonlinearity which coincides with (3.272).

By using Cartesian coordinates, the model (3.291) can be reduced to the following ordinary differential equations

A ~ =f(t) -3~0(t), (3.292)

^

where the matrix A has the following entries: 7~

Elxx(~'l(t)) - - COS2 e ~2 u~ (t)

YT

{}}. ~yy(~(t)) -- sin 2 ~0 ~2 u~ (t)

1;1 (Ol,/J, u~(t))~,~u~(t) dc~ dfl) d~o, (3.293)

1; 1 (Ol, ~, u~(t))~,~u~(t) dc~ dfl) d~o, (3.294)

7F

?~xy(ft(t)) = ?~yx(f~(t)) = / ~ ~ cos~0 sin~0 2

X ( / fR Vl(Ol, fi, u~(t))~,~u~(t)doldfi)d~o. (3.295) u~(t)


To solve the identification problem for the model (3.292)-(3.295), we restrict input to vary along the axis x. Then, on the grounds of symmetry, it is easy to conclude that

f i t ) = -Grit), fo(t) -- -~xfo(t). (3.296)

This results in the reduction of the model (3.292)-(3.295) to the following equation

a/ ?Zxx(U(t)) ~ =fi t ) - fo(t). (3.297)

We next introduce the function P1(0/, fl, u):

P1 (0/, fl, u) - [ f~ Vl (0/', fl', u) dot' dfl', (3.298) J J l~ ~/3u

and experimental data:

1 Q(0/, fl, u ) - -~(r~u - r~u), (3.299)

where r~u and r~u are relaxation times for the processes which are described in detail in Section 4 of the previous chapter. By using the expressions (3.298)-(3.299) and the same line of reasoning as in the previous section, it can be shown that the function P1(0/, fl, u) is related to the experimental data (3.299) by the following integral equation

Jr

f 2 COS 2 ~aPI(0/ ~a) de# - - Q(0/, u). (3.300) cos cos ~0, u cos Jr

2

By using the change of variables (3.282), the above equation can be reduced to the Abel type integral equation

~0 c~ X 2 o/2 PI(X, XX, x x ) d x = Q(~,x~, 0/), (3.301)

V/ 0/ 2 __ X 2 -~- X

whose solution is given by

1 d [~ s3Q(s, Xs, xs) ds. P1(0/, X0/, X0/) (3.302)

7r0/2 d o / J o v/o/2 - s 2

Finally, we shall discuss generalized anisotropic vector Preisach models. Again, for the sake of notational simplicity, we consider only 2D models. These models can be defined as

Jr

- f_=(ff ) f ( t ) = F~(0/, fl,~,u~(t))9~#u~(t)d0/d# d~ 4- ~(~(t)). (3.303) -~ u~ (t)


The experimental data defined by the function

1 F(c~,/~, u, 99) = ~ (jT~u~ - f ~ u ~ ) (3.304)

will be used for the identification of the model (3.303). As before, we introduced the auxiliary function

P(c~,/J, 99, u) - f f ~ (~',/J', 99, u) d~' d]~'. (3.305) J Jl'4 ~ u

We shall employ the following Fourier expansions:

o0 P(cr ~, u, 99) -- ~ Pn(cr u)e in~, (3.306)

n~-oo (x)

F(cr fl, u, 99) -- y ~ F-n(CG fl, u)e in~. (3.307) 1 t = - - 0 0

By using (3.304)-(3.307), it can be shown that Pn are related to Fn by the following integral equations

cos nq/Pn(ol cos O, ~ cos ~/, u cos ~/) d~/ = f n(ol, ~, u). (3.308) 2

These equations can be reduced to the Abel type integral equations

0 x 1~ o~ .. rn(-d) d x - -~-df n(ot, Xo~, Xr (3.309) Pn(x, Xx, X x) v/Ol 2 _ x 2

Various numerical techniques can be used for the solution of these equations. In particular cases when n = 4-1, the following closed-form solutions can be found:

P+l(c~,Xc~, Xc~) - 1 d ~o c~ s2C-+l(S, Xs, XS) ds. (3.310) rr c~ dc~ V/Ol 2 _ S2

As far as the function ~ (~) in (3.303) is concerned, the following expression (similar to (3.285)) can be derived:

~ (-~u) = fu+~ + fu,~ (3.311) 2

where the notations in (3.311) are self-explanatory.


Next we proceed to the discussion of the second (and, probably, most fruitful) approach to the generalization of vector Preisach models. This approach is based on the notion of generalized input projection. The corresponding vector model can be written as follows:

~f(t) f_+:r L ) - G d~, (3.312) rr/2 ~/?

where O(t) is the angle between ~(t) and the polar axis. In the case when g(O-~b) = cos(0-~b), the above model is reduced to vector Preisach models extensively studied in this chapter. This justifies the following "cosine- type" constraints on the function g(~):

7f g'(~) ~ 0 for 0 ~ ~ K -~, (3.313)

t g ( 0 ) = l , g ~ - 0 , (3.314)

= - 1 , - 4 ) = - g ( 4 ) , (3.315)

(a) First-order transition curves which are measured when the input ~(t) is restricted to vary along one, arbitrary fixed direction. By using these curves, we can introduce the function:

1 (3.316)

(b) "Rotational" experimental data measured for the case when the input is a uniformly rotating vector: ~( t )= {UmCOscot, umsincot}. It can be shown. . that. for isotropic hysteretic media, the output has the formf(t) =f0 +fl(t), where f0 does not change with time, while fl(t) is a uniformly rotating vector that lags behind the input by some angle. By using the rotational experimental data, the following function can be introduced:

R(um) = F~(t) . f l(t) , (3.317) Um

and the product I~(t)lg(O - ~) can be construed as a generalized projection of vector input ~(t) on the direction specified by the angle ~b. Function

1 g(~) = I cos~i~ sign (cos~) is an example of "cosine-type" function that satisfies the constraints (3.313)-(3.315).

In the above model, functions v(~, ~) and g(~b) are not specified in advance but rather should be determined by fitting this model to some experimental data. This is an identification problem. To perform the identification of the model, the following experimental data will be used.


that has the meaning of the projection of fl(t) on the direction of input.

Functions v(d,/i) and g(4~) will be recovered from experimental data (3.316) and (3.317). For the identification as well as computational purposes, it is convenient to introduce the function P(d, g):

=//r (~,/~)

02p(d , ~) V(d, g ) - - - - - . (3.318)

By using the same line of reasoning as before, it can be shown that model (3.312) will match the experimental data (3.316) and (3.317), if functions P(d,/~) and g(O) satisfy the equations:

yr

2 cos~P(dg(40,,Sg(q)))d4) - Y(d,/~), (3.319)

fo - 2 cos~P(um, umg(~))dq5 -- a(um). (3.320)

Since v(d, g) is nonzero only within the triangle T = {-d0 ~</~ ~< d ~< d0}, we shall require Eq. (3.319) to be satisfied within the same triangle, while Eq. (3.320) is satisfied for 0 ~< Um <~ dO. We next extend the function ./V(d,/~) from T to the square {-d0 ~< d ~< d0, --d0 ~</J ~< d0 } as an odd function with respect to the diagonal d =/~: Y'(d,/J) = --U(/~,d). Function ~'(d, g) also exhibits mirror symmetry with respect to the diagonal d = --/~: U(d,/~) = U(--g, --d). Next, we shall use the polynomial approximation for the function U(d,/~):

N

.~'(d, ~) -- E E Uks-(2n+l) dk/~s. (3.321) n=0 k+s=2n+l

Approximation (3.321) contains only odd terms because of the mentioned symmetry properties of y.(d,/~). Due to these symmetry properties, it

(2n+1) _(2n+1) is also clear that aks =--Usk . An actual polynomial approximation (3.321) can be found by using a Chebyshev polynomial series expansion of )V(d,/~) as described before. We shall also use the odd extension of R(um) from [0,d0] to [--d0,d0] and the following polynomial approximation:

N .2n+l (3.322) R(um) -- y~ R2n+lU m �9

n--0


We shall next look for P(~, fl) in the form

N p(~,, fl) _ ~ ~ Cks-(an+I) o/k'sp. (3.323)

n=0 k+s=2n+l

By substituting (3.321), (3.322) and (3.323) into (3.319) and (3.320) and by equating similar terms, we derive

Jr

,, (2n+1) f0 : q~g2n+l (2n+1) (3.324) z % cos (4) a 4 - % ,

- 2 ~ _(2n+1) fo re Cks cos~gS(qh)d~ = a2n+l. (3.325) k+s=2n+l

From (3.315) we find that the integrals in (3.325) are equal to zero if s even and

f0 J0 cosq~gS(q0d4~ = 2 cosq~gS(q~)d4~ if s is odd. (3.326)

Next, from (3.324) we obtain

a(2n+l) _(2n+l) Jr ks u0'2n+l f0 g ~g2n+l _(2n+l) -- _(2n+l) -- 2 cos (~) d~. (3.327)

Cks c0,2n+1

From (3.325), (3.326) and (3.327) we derive Jr

, (2n+1) f0 g ~g2n+l R2n+l -- -~c0,2n+l cos (~) d~

Jr

Ic(2n+1) f0 ~ -4~ ks c~ d~, (3.328)

where ~ ' stands for the summat ion over all s that are odd and less than 2n + 1 and such that k + s = 2n + 1. By using (3.327) in (3.328), we find

_(2n+1) --za0,2n+l'~ (2n+l) ~,c(2n+l)(UkS0,an+l ..2n+l ) ( a ~ ' ~ R2n+l = - 2 _ _ u0,2n+l) ] , (3.329)

which leads to ,, (2n+1)

~(2n+1) R2n+l q- za0,2n+l (3.330) c0,2n+1 ~ _ _(2n+1) _(s) " (.)(.0s) 2 ~ ' m _(2n+l) c~S)

u0.2n+l

By assuming that c 1 0,1(1)---- X, (3.331)


where X is some unknown constant, from (3.330) we derive

c(2n+1) _ lh(2n+1) (3.332) 0,2n+1 . . . . 0,2n+1"

h(2n+l) According to (3.330), constants "0,2n+1 can be recursively determined by using the formula

,, (2n+1) h(2n+l ) _ _ R2n+l if- za0,2n+l (3.333) "0,2n+1 2~_'/a(an+l))(As)) " k s

(2n+1) UOs \ a0,2n+l "0s

and the fact that h(1) = 1 "0,1 h(2n+l)

After "0,2n+1 are computed, from (3.326), (3.327) and (3.332) we find

where

f0 Jr cos~bg2n+l(~b) d~ = S2n+l (3.334) X '

..(2n+1) 82n+1 -- u0'2n+l (2n+l) " (3.335)

0,2n+1

According to (3.313)-(3.315), there exists a function x(g) which is inverse to g(~). By using this function, we shall use the following change of variables in integrals (3.334):

~b-- x(g), dc~= x'(g)dg, X (g(~)) =~b. (3.336)

By substituting (3.336) into (3.334) and integrating by parts, we transform (3.334)as follows:

_ $2n+1 (3.337) +lganl sin x(g) dg = X(2n + 1)"

We shall next make another change of variables:

g - cos !/r, dg = - sin !k dq, (3.338)

which leads to

where

~0 zr c o s 2n !/r s i n 1/rT(!/r ) dl/.r = $2n+1

X(2n + 1)' (3.339)

T (~ ) - sin X (cos ~ ). (3.340)


We shall extend T(q) from [0,rr] as a periodic odd function with half- wave symmetry. As a result, this function can be expressed as

N T(~) -- E t2n+l sin(2n 4- 1)~, (3.341)

n--0

where

2f0~ t2n+l = -- T(~) sin(2n 4- 1)~ d~. (3.342)

It is known (see [13]), that

sin(an4-1)~=singz[aancosan'~--(n--1)aan-acosan-2 1

(n--a)22n-4cos2n-4 ~ 4- 2

- ( n - 4 ) 22n-6cOs2n-6~4-''" ] 3 , (3.343)

where (~ ) - P! k!(p-k)" From (3.339), (3.342) and (3.343), we derive

1 [ $2n+1 ( n - ) 2 2 n _ 2 t2n+l = ~ 22n 2 $2n-1 2n 4- 1 1 2 n - 1

4- (n--B) 22n-482n-32 "2nnZ-3

(n-4)aan_682n_ 5 ] 1 - 3 2n Z 5 f- . . . . ;~2n+l- (3.344)

From (3.314), (3.336) and (3.340) we find

(2) T - sin ;((0) = sin -~ - 1. (3.345)

From (3.341), (3.344) and (3.345) we obtain N

K - ~(--1)ns . (3.346) n=0

Now, the identification procedure can be summarized as follows. First, we find polynomial approximations (3.321) and (3.322). Then, by using

_(2n+1) h(2n+l) coefficients Uks , R2n+l and (3.333), we recursively compute ~0,2n+1 and $2n+1. Next, by using (3.344) and (3.346), we determine ~2n+1, X and t2n+l.

_(2n+1) Finally, by using (3.341), By using (3.327) and (3.332), we compute C ks . (3.340) and (3.336), we compute T(~), x(g) and then retrieve the inverse function g(~b).


In a particular case when

g(~b) = I cos~l 1/n sign(cos~b)

the solution of the identification problem can be simplified. Indeed, in this case, the integral equation (3.319) can be written as follows:

yr

/0 F(ol, fl) = 2 cos~P(ot, cos 1/n c~,flcos 1/n qo)ddp. (3.347)

By introducing the following change of variables,

X -- Ol COS 1/n q~, X = fl/O~, (3.348)

and by using the same transformations as before, the following integral equation is obtained from (3.347):

/o ~ N(x) a(ol) - v/Ol2n _ x 2n

where the following notations are adopted:

dx, (3.349)

N(x) -- x2n-lp(x, Xx), R(cr -- (cr 2 / 2 n ) F ( o l , XoO. (3.350)

Multiplying both sides of Eq. (3.349) by 0r 2 n - 1 / v / S 2 n - Ol 2n and integrating with respect to ~ from 0 to s, and then interchanging the order of the double integration for the right-hand side, the following result could be derived:

f0 s o~2n-1 R(ol) da /S2n _ oe2n

/oS ,x (fx s Cr 2n-1 dc~ )

V/(san __ Ol2n)( O~2n __ x2n) dx

1 s s d(cc2n)

an js N(x ) (~ v/(san2xan) 2- ( cr (s2n 2 + X2n ) 2 ) dx. (3.351)

Then, by introducing the following variable of integration,

W - Ol 2n - - (S 2n -Jr-X 2n)/2, (3.352)

we obtain

~0 s o12n-1

/S2n _ cr

1loS ( ($2n-x2n)/2 ) R(ol) dot -- 2n N(x) f dw dx

d (x 2n -s an) / 2 V/( S an --2 x2n ) 2 __ W2

~f0 s 2n N(x)dx. (3.353)


By using the notations (3.350), we obtain

I f ~ nsn-lF(s, Xs)+ sn(d/ds)F(s, Xs) ds. P(o~ , )~ot ) (3.354)

yr ,Jo v /o t 2n - s 2n

From Eq. (3.354) we conclude that, no matter what value of n is chosen, the model can always match the data obtained from the "scalar" experiments. In the specific case when n = 1, the final results (3.354) coincides with the final result for the case of the identification of vector model discussed in Section 4 of this chapter.

In order to complete the identification procedure, the optimum value of the unknown n should be determined by making use of the experimental data obtained from the "rotational" experiments. One way to do this is by adopting a "least-square" approximation.

Next, we shall illustrate the above discussion by the following experimental identification and testing of the model (3.312). The identification of the model has been performed for a typical Ampex-641 (y-Fe203) magnetic tape material by using a vibrating sample magnetometer equipped with an orthogonal pair of pickup coils. As mentioned above, the identification problem has been solved by using two sets of experimental data. First, the set of first-order transition curves were measured (see Fig. 3.35).

.06

.04

,0~

o

- . 0 2

- , 0 4

- ,0B - _ L J - . ~ L . L - L ~ - 6 0 0 - -400 - 2 0 0 0 2 0 0 4 0 0 6 0 0

Maguet..ic Field [Oe]

FIGURE 3.35

220

o Q) h ~0

~0

I 0

0~

x,

O

60

40

20

CHAPTER 3 Vector Preisach Models of Hysteresis

t7 I I rr~ [,--, i i i , , i i-,,

,~ Experiment

. / , , , , -.'-- Model when n- 1 _

\ - --Model when n---3

- : x , !

- / X - %

- / A ",,, - - I g "\0 "-- _ ; / \ - . . . . . . .... -

_' ! . - i , , I I ' L I , , i l , , , I , , ,

0 2 .4 .6 .8 t

Rotating Field Amplitude [KOe]

F I G U R E 3 . 3 6

Then, the unknown function P(~, #) was found as a function of n. Next, the set of "rotational" experimental data has been measured. This has been performed by first applying a negative input field, sufficiently large to drive the sample into the state of negative saturation. Then increasing it gradually until it reaches a certain positive value where it is kept constant while sample (in-plane) rotation is introduced. This has been repeated for different positive field values and the output / input phase-lag relationship as a function of the rotating input amplitude has been obtained. It has been found that the model gives very good matching to the "rotational" phase-lag relationship when the value of n is chosen to be 3. A comparison between the computed and measured output / input phase-lag values is shown in Fig. 3.36 for two different cases of n in order to demonstrate the impact of the choice of n on the model capability to match "rotational" experimental results.

After the identification process was performed, the experimental testing of the model has been carried out in order to estimate its quantitative ability to mimic the property of correlation between mutually orthogonal components of output and input. This property has long been regarded as an important "testing" property for any vector hysteresis model in the area of magnetics. This experiment has been carried out as follows. First, the input field is restricted to the y axis and it is increased from some nega-

3.7 G E N E R A L I Z E D VECTOR PREISACH M O D E L S OF HYSTERESIS

.06

~-~ ,04

,~. ,O2

o q

�9 --.o 2 __LLU._LU_L.t._.L_...LLU...L..t.....,_-.....,_L..u_.L..L~ 0 .5 1 1.5 2 2.5

Orthogonal Field I~ [KOe]

.04

-~ .O2

r. 0 - -

0 .5 1 t .5 2 2.5 O r t h o g o n a l Fie|d Hx [KOe]

.Or ~-r--- ,T,-FrT-~--r- ]-T-r-~--FrTVT-i-7--rTr=_ ] ~ _ 0 E-- ~ _ ~ , ~ a ~ m ~ ? = _ ~ ~

- . 0 1

~i~. - . 0 2 - . 0 3

-.04 ~t l 0 .5 1 1.5 2 2.5

.02 0 r L h o 6 o n a l F i e l d H=_[KOe]

~-.02

m -.04

----.o6 ~I_L_LLLLLL~• ...... I--_LI_LL..U....d. 0 .5 l 1.5 8 2.5

Orthogonal Field If= [KOc]

E x p e r i m e n t ~ n= 1

n = 3

F I G U R E 3.37

221

tive value (sufficient to drive the sample into the negative saturation state) to some positive value Hy+, and then decreased to zero. This results in some remanent magnetization Myr. Then, the magnetic field is restricted to vary along the x axis. By increasing the field value (now Hx) gradually from zero to very high values, the orthogonal remanent component Myr is reduced and the Myr v s Hx relationship is recorded. This experiment


has been repeated for different values of Hy+ (and consequently different values of Myr). Then by using the model (3.312) the computat ional results have been obtained for the same sequences of field variations as in the experiments. Some sample comparisons between the computed and measured results of this experiment are shown in Fig. 3.37. It is clear from Fig. 3.37 that the model (3.312) gives appreciably better matching with the experimental data than the vector Preisach models discussed in previous sections.

This may be due to the presence of addit ional unknown function g(~) which can be determined from the identification process. As a result, the oppor tuni ty appears to incorporate more experimental data in the identification process, leading to a more accurate model.

The presentat ion of the material in this chapter is largely based on our publications [17-27].

References 1. Preisach, F. (1935). Zeitschriftfiir Physik 94: 227-302.

2. Stoner, E. C. and Wolhfarth, E. P. (1948). Trans. Roy. Soc. London A240: 599-642.

3. Beardsley, I. A. (1986). IEEE Trans. Mag. 22: 454-459.

4. Koehler, T. R. (1987). J. Appl. Phys. 61: 1568-1578.

5. Bertotti, G. (1998). Hysteresis in Magnetism, New York: Academic Press.

6. Slonczewski, J. C. (1956). IBM Research Memorandum, No. RM 003.111.224, Oc- tober 1.

7. Prutton, M. (1964). Thin Ferromagnetic Films, Washington, DC: Butterworth.

8. Friedman, G. (1990). J. Appl. Phys. 67: 5361-5363.


10. Barker, J. A., Schreiber, D. E., Huth, B. G. and Everett, D. H. (1985). Proc. Roy. Soc. London A386: 251.

11. Damlamian, A. and Visintin, A. (1983). Compt. Rend. Acad. Sc. Paris 297: 437.

12. Lubich, C. (1987). IMA J. Numer. Anal. 7: 97.

13. Gradstein, I. S. and Ryzkik, I. M. (1980). Tables of Integrals, Series and Products, New York: Academic Press.

14. Joshi, A. W. (1982). Elements of Group Theory for Physicists, New Delhi: Wiley.

15. Vilenkin, N. J. (1968). Special Functions and the Theory of Group Representation, Providence, Rh American Mathematical Society.

16. Lyubarski, G. Y. (1960). Applications of Group Theory in Physics, Oxford: Perga- mon Press.

17. Mayergoyz, I. D. (1991). Mathematical Models of Hysteresis, Berlin: Springer- Verlag.



19. Mayergoyz, I. D. and Friedman, G. (1987). J. Appl. Phys. 61: 4022-4024.

20. Mayergoyz, I. D. and Friedman, G. (1987). IEEE Trans. Mag. 23: 2638-2640.



23. Adly, A. A. and Mayergoyz, I. D. (1993). J. Appl. Phys. 73: 5824-5826.

24. M~ yergoyz, I. D. and Adly, A. A. (1993). IEEE Trans. Mag. 29: 2377-2379.


26. Adly, A. A. and Mayergoyz, I. D. (1997). IEEE Trans. Mag. 33: 4155-4157.

27. Adly, A. A., Mayergoyz, I. D. and Bergqvist, A. (1997). IEEE Trans. Mag. 33: 3932-3933.

CHAPTER 4

Stochastic Aspects of Hysteresis

4.1 PREISACH M O D E L WITH S T O C H A S T I C INPUT AS A M O D E L FOR VISCOSITY

It is well known that the physical origin of hysteresis is due to the multiplicity of metastable states exhibited by hysteretic materials or systems. At temporally constant external conditions, large deviations of random (thermal) perturbations may cause a hysteretic system to move from one metastable state to another. This may result in gradual (slow) changes of the output variable. This temporal loss of memory of a hysteretic system is generally referred to in the literature as "after effect", "viscosity" or "creep." This phenomenon can be of practical significance in various engineering applications where hysteresis is utilized. One important example is the magnetic storage technology where the viscosity effect is detrimental as far as the time reliability of recorded information is concerned.

Traditionally, the viscosity phenomenon has been studied by using thermal activation type models (see, for example, [1, 2]). It has been realized that these models have some intrinsic limitations. For instance, they are usually regarded "noninteracting particle" models and they are valid for hysteretic systems with energy barriers much larger than the thermal activation energy. In this section we shall explore a new approach [3-6] to the modeling of thermal relaxations based upon the Preisach model of hysteresis. In this approach random thermal agitations will be modelled by a stochastic input. We shall compare this approach with thermal activation type models and with some known experimental facts.

Consider a deterministic input u(t) which at time t = 0 assumes some value u and remains constant thereafter. In purely deterministic situation, the outputf(t) would remain constant for t/> 0 as well. However, in order to model thermal relaxations, we assume that some noise described by

225

226 CHAPTER 4 Stochastic Aspects of Hysteresis

the stochastic process Xt is superimposed on the constant input u. Conse- quently, the Preisach model is driven by the following random process:

xt - u + Xt, Xt - 0 , (4.1)

where Xt stands for the expected value of Xt. The output will also be a random process given by

J~ - ff~>~ #(a ' f l)G~xt da dfl. (4.2)

It is instructive to note that adding noise Xt to the deterministic input u(t) is mathematically equivalent to subtracting the same noise from switching thresholds ~ and ft. This is true because

(yot_xt,fl_xt)u(t)-- 9~fl(u(t) + Xt). (4.3)

Imposing noise on the switching thresholds (barriers) may be more transparent (or justifiable) from the physical point of view, while adding noise to the deterministic input makes the problem more tractable from the mathematical point of view. To simplify the problem, we shall first model the noise by a discrete-time independent identically distributed (i.i.d.) random process Xn. In this case, it can be assumed that process samples remain constant at time intervals At = tn+l - tn and undergo monotonic step changes at time tn+l. Accordingly, the Preisach model is driven by the process

Xn - u + Xn, Xn = 0. (4.4)

The output process is given by

fn - ff~>>,~ #(ol, fl)G~Xn dot dfl. (4.5)

We will be interested in the time evolution of the expected value of the output process. Since integration is a linear operation, from (4.5) we derive

f n = E{fn} - >, lz(o~,fl)E{~'a~xn}dadfl. (4.6)

Thus, the whole problem is reduced to the evaluation of the expected value, E{G~Xn}. Since G~Xn may assume only two values, +1 and -1 , we find

E{}'a~Xn} - P{G~Xn = +1} - P{G~Xn = -1}. (4.7)

It is also clear that

P{G~Xn = +1} + P{Gf lXn - -1} = 1. (4.8)

4.1 PREISACH MODEL WITH STOCHASTIC INPUT 227

By introducing the notation

P{G~Xn = +1} = q~t~(n), (4.9)

from (4.7) and (4.8) we obtain

E{G~Xn} = 2q~(n) - 1. (4.10)

We next derive the finite difference equation for q~(n). According to the total probability theorem, we have

P { G ~ X n + I = + 1 } = P { G ~ X n + I = + l l G / ~ X n = + I } P { G ~ X n = + 1 }

+ P{y'~Xn+l = +ll~'~Xn = -1}P{].'~Xn = -1}. (4.11)

It is convenient now to introduce the switching probabilities

P,~+F(n) = P{~'~Xn+l = +ll}~Xn = +1}, (4.12)

P-dF(n) = P{}"~Xn+I = +ll~'~Xn = --1}. (4.13)

+ - S i m i l a r meanings hold for the switching probabilities P~S (n) and P ~ (n). It is clear that

+ - Pr (n) + Pr (n)= 1, (4.14)

P~-~+ (n) + P~- (n) = 1, (4.15)

P{G~Xn = -1} = 1 - @~(n). (4.16)

By using (4.9) and (4.12)-(4.16), we can transform (4.11) to

+-(n))] + P-~(n). (4.17) q~(n + 1)= q~(n)[1 - (P-~-(n) 4- Po~

We next proceed to the evaluation of switching probabilities P ~ (n) and + -

P ~ (n). It is clear that these probabilities can also be defined as

P-d~-(n) = P{Xn+l > otlG~xn = -1}, (4.18)

P+~(n) = P{Xn+l < fllG~Xn = +1}. (4.19)

In general, these probabilities are difficult to evaluate because multidi- mensional conditional probability density functions are required. How- ever, the problem is significantly simplified if the noise is modelled by i.i.d, process. In this case we find

P-~d-(n) = P;+ = P{Xn+l > Ol} -- p(x) dx, (4.20)

+ - Po~ (n) = P-~- = P{Xn+l < ~} -- p(x)dx, (4.21)

oo


where p(x) is a probability density function. By using (4.20) and (4.21), the finite difference equation (4.17) can be

represented as

q~(n 4- 1) = r ~ q ~ ( n ) 4- P~-+, (4.22)

where

r ~ - 1 - (P;+ + P-~-) - p(x) dx. (4.23)

If the probability density function p(x) is strictly positive, then

0 < r ~ < 1. (4.24)

Equation (4.22) is a constant coefficient first order finite difference equation whose general solution has the form

q~(n) - Arn~ 4- B. (4.25)


B - P ; + - - P ; + (4.26) 1 - r ~ P~-+ + P~ "

From the initial condition we obtain

q~ (0)= A 4- B, (4.27)

where 1 if (~,~) ~ S+(0),

q~ (0) - 0 if (~, fl) ~ S- (0), (4.28)

and S +(0) and S-(0) are positive and negative sets on the ~-/~ diagram, respectively, at the instant of time t = 0.

From (4.25)-(4.27) we derive

P~-+ (4.29) q~(n) = [q~(0) - q~(o~)]rn~ 4- p~_+ 4- p+_,

where

P~-+ (4.30) q~fl(OO)- n--,~lim q ~ ( n ) - p ;+ 4- p_~_.


P~-+ - P~- (4.31) E{9~flXn} - 2[q~fl(0)- q~fl(cx~)]rnfl 4- p ; + 4- p~_.


We next introduce the functions

P~-+ - P~- (4.32) ;(~, ~) = p;+ + p ~ - '

J 1 if (~,/~) ~ S + (0), 0(~, ~) (4.33) / - 1 if (c~,/~) ~ S-(0).

By using (4.28), (4.32) and (4.33), we can transform (4.31) to

E{~'~Xn}- [O(ol, f l ) - ;(c~, fl)]rn~ + ;(c~, fl). (4.34)

By substituting (4.34) into (4.6), we finally find

]:n = f ~ + f f l,(ot, fl)[O(oe, fl) - ;(o~,~)]rn~do~dfl, (4.35) d ,) ot

where

~ - nlimin - f ~ > ~(~, fl)c(~, fl) d~ d~. (4.36)

It is apparent from (4.36) and (4.32) that the limiting expected value of output f ~ does not depend on the history of input variations prior to the time t - 0. In this respect, the value, f~ , bears some resemblance to anhysteretic output value. This resemblance is enhanced by the fact that f ~ = 0 if the expected value u of Xn is equal to zero, and p(x) = p(X) is an even function. This fact can be proven as follows. According to (4.20) and (4.21) we find

P~-+ = P+~-. (4.37)

By using (4.37), it is easy to check that

((r f l )= - ( ( - f l , - c~ ) . (4.38)

On the other hand, we recall that

#(~, fl) = l*(-fl,-o~). (4.39)

From (4.38) and (4.39), we conclude that/,(c~, fl)g (c~, fl) is an odd function with respect to the line a = -f t . From the last fact and (4.36), we find that

foo - 0 . (4.40)

Next, we shall compare the result (4.35) with thermal activation-type models for viscosity. For this purpose, we replace discrete time n by continuous time t and rewrite (4.35) as

ft =]:oo + f f X(ol, fl)e-~(~ (4.41) ,Idol


where

X (or,,8)-/z(o~,,8) [0(o~,,8)- ~(c~,,8)], (4.42)

1 In r~ . (4.43) ~(~ ' /~) = a-t

According to thermal activation-type models, metastable equilibrium states of hysteretic systems are separated by energy barriers EB. It is assumed that there is a continuum of these energy barriers and it is postu- lated (with some physical justification) that the viscosity phenomenon is described by the model (see [1])"

f0 f (t) =f(oo) + A g(EB)e -~(~B)t dEB, (4.44)

where g(EB) is some density of states,

X(EB) -- X0e EB/kT, (4.45)

k is Boltzmann's constant, T is the absolute temperature, while A and X0 are some constants.

It is clear by inspection that there is some similarity between our result (4.41) and the thermal activation model (4.44). Actually, our model (4.41) can be reduced to (4.44) in the particular case when only symmetrical loops (operators) G,-~ are used in the Preisach model. In this case

#(c~,/~) = 0(~,/~)~(c~ +/~), (4.46)

where ~(c~ +/~) is the Dirac delta function. By substituting (4.46) into (4.42) and (4.41), after simple transforma-

tions we can represent (4.41) as

f0 - ft = f ~ + ;~(~)e -~(~t & . (4.47)

Now, by using the change of variables

~(~) - ~(~B), ~ - ~ -1 (X(EB/), (4.48)

the expression (4.47) can be reduced to (4.40). This shows that the Preisach model of viscosity (4.41) is reduced to the thermal activation model in a very particular case. This case occurs when only symmetrical rectangular loops are used in the Preisach model. Since it is generally believed that nonsymmetrical loops in the Preisach model account for "particle interactions", the last reduction is consistent with the generally held opinion that the thermal activation model (4.44) is a "noninteracting particle" model.


We next show that under some assumptions about X (a, fl) the model (4.41) describes ln t-type variations for ft. Another (purely stochastic)justification for In t-asymptotics will be given in Section 3 of this chapter. The In t-type variations have been observed in many experiments and they are considered to be characteristic of viscosity phenomena.

The expression (4.41) can be written as follows

ft=i + f (f (4.49)

For any fixed fl, we make the change of variables from c~ to ~ in the integral with respect to a:

f (f (4.50)

We assume that the function 7r(~, fl) can be approximated as

D(fl) ~p(~, fl) ~ for ~1 < ~ < ~2 (4.51)

and zero otherwise. Then, we derive

f ~(~, fl)e -~t d~ ,~ D(fl)

Consider such instants of time t that

1

f ~2t e-V

dr. (4.52) d~ = D(fl) d~l t v

-- << t << -- . (4.53) ~2 ~1

For these instants of time, from (4.52) we derive

/ ~(~ , fl)e -~t d~ ,~ D(fl) ,, dv i t v

= dv - dv . (4.54) V J - a V

Here a is a fixed small positive number and the last two integrals exist in the sense of Cauchy principal value. Since ~lt << 1, we can use only the first two terms of Taylor expansion of e -v in the second integral. This yields

f q/(~, fl)e - ; t d~ ,.~ S ( f l ) l n t + C(fl). (4.55)


f t ~ S I n t 4-C, (4.56)

which is a famous In t-type variation for ft.


It has been observed in experiments (see, for instance, [7]) that the temperature dependence of the decay rate of ;~ is satisfactorily approximated by T 1/2 at low temperatures. In the framework of the model (4.41) this fact can be explained as follows. From (4.43) and (4.23), we find

~ (oe, r = - In p(x) dx . (4.57)

We subdivide all exponents in (4.41) into two groups: (a) exponents for such ol's and r that u - xt r [/~,oe], and (b) exponents for such c~'s and r that u - xt E [r Consider a thermal noise with small variance o -2. If U = Xt ~ [/J, O l], then

~ ~ p ( x ) d x ~ 0 as r~ ~ 0. (4.58)

This means that according to (4.57) the exponents of the first group decay very fast and they do not contribute appreciably to the "long time" decay of the integral in (4.41).

If u - YCt E [ ~ , ~ ] , then

~ ~ p ( x ) d x ~ 1 as r~ ~ 0. (4.59)

This means that the exponents of the second group decay slowly and they are by and large responsible for the "long time" decay of f t . For this reason, only these exponents are discussed below. It is apparent that

p(x) dx = 1 - p (x ) dx - p(x) dx. (4.60) o o

For small r~ the integrals in (4.60) are small quantities which we assume can be approximated as

L I p(x) dx ~, a(ol)r~, p (x ) d ~, b(~)r~. (4.61) o 0

By substituting (4.61) into (4.60) and then into (4.57), we find

~(c~,]~) ~ - l n { 1 - I a ( ~ ) + b(/J)lr~ } ~ [a(c~) + b(l~)]r~. (4.62)

It is reasonable to assume on the physical grounds that the variance is proportional to the temperature

ry 2 ,~ Z. (4.63)

From (4.62) and (4.63) we conclude that, for the exponents of the second group, the rate of decay is approximated as

~(ol, fl) ,,-, Z 1/2. (4.64)


This suggests that the overall rate of "long time" decay of ?~ may be approximated by T 1/2 at low temperatures.

It is also interesting to examine the dependence of the limiting expected value of the output, ; ~ , on the variance 0"2 of the noise. Expres- sion (4.36) is to be used for this purpose. The noise can be chosen to have normal distribution and ~ (~, ~) in (4.36) can be computed by using formulas (4.32) and (4.20) and (4.21). For illustrative purposes, Preisach function /~(c~,/J) can be chosen as

exp{ + o 4 22Uc 2]} where A is some normalization constant, Uc is the coercive value of a major hysteretic loop, while o'1 and 0"2 are parameters which are used to control the spread of the function/~(~,/~) along and around the line ~ = -/J. This Gaussian form of #(c~, ~) is often used in magnetics [8]. The computations

O " have been performed for different values of 0"1, 0"2 and the ratio X = Uc" The computations show that for arbitrary fixed values of 0"1 and 0"2 the dependence of f ~ on u is not sensitive to wide variations of 0". This is demonstrated in Figs. 4.1 and 4.2 where the computed curves f ~ vs u are plotted for the following values of X: 0.5, 0.35, 0.2, 0.15. These curves were computed for the following values of 0"1 and 0"2, respectively: 0"1 --" 0"2 = 4 0 ,

0"1 - - 80 and 0"2 -- 40. The fact t ha t ;~ is insensitive to the variance 0"2 sug-

1

oo 5

O

O 0

O

- - . 5

-1

,, ~

J u~=200 I I I I I ,! ~ , l

- 2 o o o 200 U Mean input

FIGURE 4.1


1

o

o

.<

- 2 0 0

I I I I !

I 1 I i 1 I

o 200 U Mean inpu t

FIGURE 4.2

gests that y~c~ can probably be used as another definition of anhysteretic value.

We next consider vector Preisach models driven by vector stochastic inputs as models for viscosity [9]. Below we shall only discuss the case of 2D vector Preisach model. Extensions to the 3D case will be straightforward and, for this reason, they will be omitted.

Consider a deterministic vector input ~(t) which at time t - 0 assumes some value ~ and remains constant thereafter. We assume that some zero mean vector noise described by a discrete-time random process Vn is super imposed on ~(t). As a result, the Preisach model is driven by the process

Vn = U if- ~ . (4.66)

For the output process we have yr

-- , /2 (/~ ) fn = e+ v(o~,/J, qo)},'aflVn~ dot, d~ d(p. (4.67)

From (4.67), we derive the following expression for the expected value - )

Of fn" 7r

E{fn} = f_2-d~(/~ v(o:,,[3,~)E{~'otgVn~}dotd[3)dcp. (4.68)


By modelling the noise by i.i.d, vector process with independent Cartesian components and by repeating the same line of reasoning as before, we derive

E{~'~Vn~o}- [#(c~,fl, p ) - ((o~,fl, p)]rn~ + ((o~,fl, p), (4.69)

where

+1 if (o~, fl) e S +(0), (4.70) #(ol, fl, p) = - 1 if (ol, fl) e S~-(O),

p~-+ _ p~ ~ (4.71) ((c~, fl, p) = p~_+ + p ~ f ,

P~-~+ = pe(x) dx, (4.72)

P-~ - p~o(x) dx, (4.73)

r~o = p~o (x) dx. (4.74)

In expressions (4.72)-(4.74), p~0 is a probability density function for Vn~o. This function can be expressed in terms of probability density functions for Cartesian components of Vn. Indeed, for Vn~o we have

Vnp = Xn COS 99 -}- yn sin p, (4.75)

where Xn and yn are independent Cartesian components of Vn with probability density functions px(X) and py(y), respectively. According to the well known result from the probability theory, from (4.75) we derive

1 f_l-~ ( x ) ( v - x ) px Py dx. (4.76) p~ (v) - cos p sin p ~ cos p sin p

By substituting (4.69) into (4.68) and by replacing discrete time n by continuous time t, after simple transformations we obtain

Jr

I_ (fL ) E{]y} = E{f~} + ~ Z(a, fl, p)e-~(~ dp, (4.77)

where 7~

v(ot, fl, p)((o~, fl, p)dotdfl) dp, (4.78)

x (o~,,8,,p)- v(o~, t~, ~)[~(o~,,8, e ) - ~-(,~,,8, ~)], (4.79)


1 In r~&0 (4.80) ~ ( o ~ , # , ~ 1 - a-7 "

By comparing (4.77) with (4.44), it can be concluded that the model (4.77) can be regarded as a vector generalization of the classical thermal activation type models.

For the isotropic model the symmetry property (3.58) holds. It can also be checked that for ~ = 0, p~ is an even function if px and py are even. By using these facts and the same line of reasoning as before, it can be proven that

E{f~} - 0 . (4.81)

Therefore, for ~ = 0 and regardless of the past history, the limiting expected value of output is equal to zero.

In conclusion of this section, it can be remarked that the model (4.35) (or (4.41)) has certain attractive features in comparison with the thermal activation type models (4.44). First, the model (4.35) explicitly accounts for the specific hysteretic nature of the system as well as for specific input histories. Second, stochastic characteristics of thermal noise explicitly appear in the model, whereas the thermal activation type models are formulated in purely deterministic terms. Third, thermal activation type models (4.44) are intrinsically scalar models, whereas the model (4.35) can be generalized to the vector case.

4.2 EXPERIMENTAL TESTING. SCALING A N D DATA COLLAPSE IN M A G N E T I C VISCOSITY

As discussed before, at constant external magnetic fields large deviations of internal random (thermal) perturbations may cause a hysteretic system to move from one metastable state to another. This may result in slow time variations of magnetization, which are the essence of viscosity. The described temporal variations of magnetization are usually characterized by the following intermediate "ln t" asymptotics:

M ~ Mo - S(H, T) In t, (4.82)

where S(H, T) is called the viscosity coefficient. This coefficient depends on the values of magnetic-field H, temperature T, as well as the past history of magnetic-field variations. The latter means that the viscosity coefficient is also a function of the current state of hysteretic material.

The term "intermediate asymptotics" means that formula (4.82) describes quite well the long-time behavior of magnetization. However, it does not describe properly the ultimate (at t ~ oo) value of magnetiza-

4.2 EXPERIMENTAL TESTING 237

tion. In other words, the asymptotic behavior (4.82) breaks down for very long times, at which In t diverges.

The ability of the model (4.41) to predict the viscosity coefficient (relaxation rate) has been first experimentally tested by C. E. Korman and P. Rugkwamsook [10] for y-Fe302 magnetic recording materials. By using a vibrating sample magnetometer (VSM), first-order reversal curves were measured. These experimental data were used for the identification of the Preisach model (determination of/z(r fl)). Then viscosity measurements were performed under the conditions that the sample was first brought to positive saturation and afterwards the applied field was reversed, held constant at some fixed value, and gradual decay of magnetization was observed. These magnetization decay curves (experimentally measured at constant temperature T and various fixed values of magnetic field H) were approximated by Eq. (4.82) and the viscosity coefficient as a function S(H) of magnetic field was found.

The described viscosity experiments were numerically simulated by using model (4.41) and experimentally measured first-order transition curves. The noise was assumed to be Gaussian and calculations were performed for different values of variances a 2. The calculations revealed the "ln t" intermediate asymptotics as given by formula (4.82). The slopes of these asymptotics were computed, then they were normalized and it was observed that the normalized curves S(H)/Smax computed for different variances practically collapsed into one curve. This computed universal curve is almost identical to the normalized curve S(H)/Smax found from the experimental data previously described (see Fig. 4.3). This suggests that if the variance 0 -2 of the noise is found by matching Smax, then the model (4.41) will predict the same curve S(H) as observed in experiments.

The "computational collapse" of curves S(H)/Smax prompted the idea to experimentally test the scaling and data collapse of viscosity coefficient S(H, T) for various magnetic materials, and especially for materials used in magnetic recording [11].

Our experiments have been conducted for high-coercivity thin-film CoCrPt recording media by using a vibrating sample magnetometer (VSM) with a temperature-controlled furnace (model Micro Mag 3900 of Princeton Measurements Corporation). First, the experiments were performed to clearly demonstrate that the viscosity coefficient strongly depends on the past history of magnetic-field variations. To this end, the viscosity measurements were conducted for two distinct states of magnetic material, which had almost the same values of magnetization achieved at the same value of the external magnetic field but corresponding to two different past histories of magnetic-field variations. These two distinct

0.8

0 .9

0.2

0.1

- - 0 .7 ! .it

o.6 3

0.5

,-- o .4

E :~ 0 .3

I

,.H

1 Simula t ion w~h s~ lma ,, 10 Oe 0 Smulation w~h sk:jma 15 Oe - - - x - - - S i rnulatkm with sggma = 20 Oe -- -~-- -

Experiment . - - - ~ - -

. ' ~ ;

:.:-::2 . . . . .

0 , , , ! , ' ' , , I "~ , a. - 1 0 0 0 - 9 0 0 - 8 0 0 - 7 0 0 - 6 0 0 - 5 0 0 -400 - 3 0 0 - 2 0 0 - 1 0 0 0 100 200 300 4 d

Applied Field, Oe

5~ .005

4 o . 0 0 5

,...,

3 e - 0 0 5

i!

e -

.2

. i t

o O

a

1 ~ X ~ 5

FIGURE 4.3

states and histories of their formations are shown in Fig. 4.4. Figure 4.5 presents the observed time variations of magnetization for the above two states plotted on In t scale. It is apparent from Fig. 4.5 that the above two


FIGURE 4.4

4.2 EXPERIMENTAL TESTING

x 10 -4 Magnetization vs Ln(Time) Measurement of a Hard Disk Sample _ / .. ! . . . . . . . . . . .

- 7 . 2 . . . . . . . . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- 7 . 4 - . . . . . . . i ' ~ . . . . . . : - , - ' 1

E ' ~ - 7 . 6 v r

0 . m

- 7 . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

N , ~

C

- 8 . 2

- 8 . 4

- 8 . 6 . . . . . . . . . . . . . . 0 2 4 6 8 10

Ln(Time(sec) )

FIGURE 4.5

239

magnetization variations have different (in magnitude and sign) viscosity coefficients. This clearly reveals the strong dependence of the viscosity coefficients on the past history.

In subsequent experiments, the viscosity coefficients were measured as a function of H for various temperatures T for the same (and quite simple) histories. Namely, specimens were first driven into the state of positive saturation by applying sufficiently strong external magnetic fields and then the magnetic field was gradually decreased to a desired value at which viscosity measurements were performed. Thus, the viscosity measurements were performed for the states corresponding to the points on the descending branch of the major hysteresis loops. For various fixed values of T, viscosity coefficients S(H, T) are "bell-shaped" functions of H with the maximum near (but not necessary equal to) the coercivity. These bell-shaped curves measured for two different samples are shown in Figs. 4.6 and 4.7. The samples had coercivities of about 1600 and 2600 Oe at temperature of 25 ~ respectively. The material of the second sample (with higher coercivity) had a small value of remanence (Mrt ,~ 0.85 x 103 emu/cm2). For this reason and in order to increase the signal level, the second sample was formed by stacking several layers of the same material.


x lo -~ Sample#01 Magnetic Viscosity Coefficient vs Magnetic Field

, J o ~ ' - - o T=25~ x~ :: ~ :: j X i / .S~-1 �9 �9 T=50~ ....... i . . . . . . . . . . i . . . . . . . . : . . . . . . . . . ! . .~c~.. . . . . f . .~. ; ........ 1

�9 I1~ ~ T=,0o~ :: :: ! : : 1 ~ : : ! ~ : : / ' 1 - 1 o o T = l S o ~ .... " : . . . . . . . . . : . . . . . . 1 ..... ~ i - $ i t - l l . . . . . . . . 1

I Io ~ T=200~ r ~ T I i i ~ j "~ 3 .5 I- . 1 _ _ _ _ _ . , �9 . . . . . . . . . . . . . :, . .:. ,." . . . . . . . .

" ~ z s . . . . . . . . :. . . . . . . . . . i . . . . . . . . . . . . . . . : . . . . . . . .

os ........ : . . . . . . . . . . i . . . . . . . . . . "

M a g n e t i c F ie ld H ( O e )

FIGURE 4.6

4 x 10 -s Sample#02 Magnetic Viscosity Coefficient vs Magnetic Field . . . . . . I I

o o T = 2 5 ~ ! 3.5 . . T=50Oc . . . . . . . . . ! . . . . . . . ~ a ~ ~

,', a T=7sOc : i ~ j ~ i ~ ! ~ -'~

c 3 o [] T = I O 0 ~ ' 0

i__. x . T=125Oc , #w( ~ ~ /

om2.5 v v T = 1 5 0 ~ ' o o,=,,,ocf .~'~~ 2 -..........

> 1.5 . . . . . . . . . . . . . ~ .. . . . . . . . . . . . . : . . . . . . . . . . . . . = = o

@

r ! i ,

0.5 . .

o . . ; - ' , ~ - ...... , z - o , . . . . . -4~ ~00 - 3 5 0 0 - 3 0 0 0 - 2 5 0 0 - 2 0 0 0 - 1 5 0 0

M a g n e t i c Fie ld H ( O e )

! I

, i - 1 0 0 0 - 5 0 0

FIGURE 4.7


By using the experimental data shown in Figs. 4.6 and 4.7, the hypothesis that S(H, T) admits the following scaling:

(-) S(H, T) = Smax(T)f H*(T) (4.83)

was tested. In the above formula, Smax(T) is the maximum of the viscosity coefficient as a function of T, while H*(T) is the value of the magnetic field at which Smax is achieved. The last formula suggests that S vs H curves experimentally measured for different temperatures must collapse onto one universal curve when plotted in coordinates:

S H s - Smax(t)" h = ~tH*'T-------v" (4.84)

This phenomenon of data collapse is the principal significance of scaling and its occurrence was observed for the collected experimental data. Namely, by performing the scaling described above, it was found that the curves shown in Figs. 4.6 and 4.7 practically collapsed onto the single curves shown in Figs. 4.8 and 4.9, respectively.

0.8

X

0.6 E ~D cD

0.4

0.2

o

Normalized S vs H Curve of Hard Disk Sample#01 ' ' ~ - - I I I I I ' ' - - i i 1 ...... i

o T=25~ ,= .... T=50Oc

" T= 100~ T= 150~ T=200~

i ..... j ......... 11 I ' -o'. 17 . . . . -0.9 8 -0 -0.6 -0.5

HIH"

FIGURE 4.8


0.8

i 0.6

0.4

0.2

0 - 2

Normalized S vs H Curve of Hard Disk Sample#02 ! i ! " l ' i ' i - ~ p - " !

_ ---e-- T=25~ , ,~, T ~ 5 O ~

T=75~ --e-- T=100Oc _

= T=125Oc v T=150~

- - 1 . 8 - - 1 . 6 - - 1 , 4 - - 1 . 2 - 1 - - 0 . 8 - - 0 . 6 - 0 . 4 - - 0 . 2 O

H/H*

FIGURE 4.9

In addition, the following scaling hypotheses:

Smax(Z) -- aT ~ (4.85)

H*(T) = bT ~, (4.86)

were also experimentally tested and verified with some accuracy. The results of this testing for samples 1 and 2 are shown in Figs. 4.10 and 4.11, respectively. It was found that, for thin-film recording media with coercivity below 2000 Oe, the exponent fl was fairly close to -1 . For recording media with coercivity above 2000 Oe, substantial deviations from the above value of fl were observed.

Hypotheses (4.85) and (4.86) along with the formula (4.83) lead to the following self-similar expression for the viscosity coefficient

(-) s(H, T) = aW~f ~-~ . (4.87)

The last formula (as well as formula (4.83)) reveals an interesting and peculiar mathematical structure of the viscosity coefficient as the function of two variables: H and T. The essence of this structure is that the normalized viscosity coefficient S/Smax is a function of one variable h -- H/H*(T). This structure is quite different from the traditional representation of S(H, T) in


Ln(Smax) vs Ln(T) of Hard Disk Sample

-9.9 _ o: Sample#oi l ~ "

-10 .4 ............ - ~ .... i ............. i .............. :; ............. i ............. i .............

Ln(~

7.9

7.8

7.7

7.6

7.5

~-~.7.4 C

. - - I

7.3

7.2

7.1

6"~. 6 Ln(T)

FIGURE 4.10

Ln(H*) vs Ln(T) of Hard Disk Sample . . . . . w . . . . . . . ! ! I - . . . . .

. . . . . . . . . i . . . . . . . . . . . o o Sample#011 ....... ~ ~ , o ~ ~a~o,e,o~ I

. . . . ,

. i - - - ~ _ 1 . . . . . . . . . . I 1 I i _ . .J

5.7 5.8 5.9 6 6.1 6.2 6.3

FIGURE 4.11


terms of irreversible differentiable susceptibility; the representation that is widely analyzed and discussed in the existing literature. This structure is a direct and concise reflection of experimental observations and it is not based on any ad hoc assumptions.

The experimental evidence of scaling of S(H, T) and data collapse has been presented here only for two samples. However, the described scaling and data collapse were observed for several other samples as well. Those samples had coercivities between 1600 and 2600 Oe.

Scaling and data collapse are typical for critical phenomena where their physical origin is due to the divergence of correlation length [12]. The physical origin of scaling in magnetic viscosity is not clear at this time. However, one may speculate that this scaling is somehow related to the "granular" structures of hysteretic materials which ultimately lead to multiplicity of metastable states.

In experimental studies of magnetic viscosity reported above, magnetic samples were subject to external magnetic fields that were constant in time and spatially uniform. In these experiments, the applied fields and sample magnetization were usually aligned, while demagnetizing fields were negligible. This type of experimental studies are usually performed by using magnetometers and they reveal the intrinsic scalar thermal relaxation properties of materials.

In magnetic recording, thermal relaxations of magnetization patterns recorded on hard disks are of interest. These are data-dependent thermal relaxations and they occur under quite different conditions than in the case of intrinsic thermal relaxations. First, the recording media are subject to strong local demagnetizing fields. These fields depend on recorded patterns and they vary with time due to the temporal deterioration of these patterns. Second, the demagnetizing fields are spatially nonuniform. As a result, data-dependent thermal relaxations occur with different local rates. Third, the local demagnetizing fields and magnetization are not aligned. Therefore, data-dependent thermal relaxations are inherently vectorial. Fi- nally, data-dependent thermal relaxations are measured by using giant magneto-resistive (GMR) head. Signals in these heads are generated by magnetic fields produced by virtual magnetic charges that are proportional to the divergence of magnetization (div M). It is quite possible that, due to spatially inhomogeneous and vectorial nature of thermal relaxations, very small temporal changes in local magnetization may result in appreciable magnetic charges and GMR head signals.

Data-dependent thermal relaxations are usually studied by using spin-stands. However, traditional spin-stand studies have only examined the harmonic content (mostly the fundamental harmonic) of recorded patterns. For this reason, the spatially inhomogeneous and vectorial nature

4.3 PREISACH MODEL DRIVEN BY CONTINUOUS-TIME NOISE 245

of data-dependent thermal relaxations has not been captured. It turns out that the spatial and vectorial characterization of data-dependent thermal relaxations can be performed by using the spin-stand imaging technique [13-15]. The detailed discussion of this technique is beyond the scope of this book.

4.3 PREISACH MODEL DRIVEN BY C O N T I N U O U S - T I M E NOISE. ORIGIN OF THE UNIVERSALITY OF LONG-TIME THERMAL RELAXATIONS

In Section 1, the Preisach model driven by discrete-time i.i.d, random process is used as a model for thermal relaxations in hysteretic systems. However, actual random thermal agitations are better described as continuous-time noise. From the mathematical point of view, this makes the problem quite complicated. It is shown next that these difficulties can be largely overcome by using the mathematical machinery of the "exit problem" [16].

The noise Xt in Eqs. (4.1) and (4.2) will be modeled by a (continuous time and continuous samples) diffusion process, which is a solution to the Ito stochastic differential equation [17]

dXt - b(Xt) dt + r~(Xt) dWt. (4.88)

In this equation, Wt is the Wiener process, and its formal derivative is the white noise. Formula (4.88) can be construed as a generic equation for dynamical systems driven by white noise, and trajectories of such dynamical systems can be viewed as samples of stochastic diffusion process. From the purely mathematical point of view, the Ito stochastic differential equation generates complicated diffusion processes by using the Wiener process, which is one of the simplest and most studied diffusion processes.

Now we shall return to Eq. (4.2). Since integration is a linear operation, from (4.2) we derive

)~t -- ~ /~(c~, ~)E{yo~xt} dol d~. (4.89) d .l ol

Thus, the problem is reduced to the evaluation of the expected value, E{G~xt}.

Let

q~,~(t) = Prob{G~xt - +1}. (4.90)


Since ~'ol,~Xt may assume only two values +1 and -1 , we find

E{G~xt} = 2q~,~(t)- 1. (4.91)

In this way, the problem is reduced to the calculation of q~,~(t). The last quantity can be expressed in terms of switching probabilities P-~(t) and Pk (t), which are defined as follows"

+ { k s w i t c h i n g s o f G ~ d u r i n g } Pk (t) = Prob time interval (O,t)lG~x0- +1 '

k switchings of }9r during } Pk (t) = Prob

time interval (0, t)lG~x0 - - 1

By using these switching probabilities, we derive

{E~=oP -~k(t) if }3~x0 = +1,

q~,~(t) = ~=0 P2k+l(t) if ~r = -1 .

(4.92)

(4.93)

(4.94)

The last expression is valid because occurrences of different numbers of switchings are nonintersecting (disjoint) events.

Next, we shall discuss the mechanism of switching. It is clear from Fig. 4.12 that the first switching occurs at the moment when the stochastic process xt starting from the point x0 exits the semi-infinite interval (/~, c~). Then, the second switching occurs at the moment when the process xt starting from the point x = ~ exits the semi-infinite interval ( -~,c~) . The third switching takes place at the moment when the process xt starting from the point x = c~ exits the semi-infinite interval (]~, cx~). It is apparent that the mechanism of all subsequent even switchings is identical to the mechanism of the second switching, while all subsequent odd switchings

~or =Xt

_ J3 1, i . C(, ; Xt

Xo ~L

L

i f , - , - , - r o- , - .~ f . , - j r .,-

13 •

, I . ! - " 4 " d r J - " -," - " ,4" J - " - "

I . . . . . I . . . . . . . .

FIGURE 4.12


occur in the same manner as the third switching. Thus, switchings of rectangular loops }ga~ are closely related to the exit problem for stochastic processes. This problem is one of the most studied problems in the theory of diffusion processes and the mathematical machinery developed for the solution of this problem will be utilized in the calculation of probabilities

4- G (t). The exit problems just described can be characterized by exit times rx i ,

which are random variables. In the above notation for the exit times, sub- script "x" means that process xt starts from point x, while superscripts "4-" mean that upward and downward switchings, respectively, occur at these exit times. Next, we introduce the functions

v+(t,x) = Prob{ rx i /> t},

V+ (t, x) = e(t) - v+ (t, x),

(4.95)

(4.96)

where e(t) is a unit step-function. It is clear that

Vi( t , x ) = Prob{rx i ~ t}, (4.97)

which means that V+(t,x) has the meaning of a cumulative distribution function for the random variable rx i . This, in turn, implies that

p+(t,x) = OV+(t'x) (4.98) 3t

is the probability density function for the random variable rx i . It is apparent from (4.96)-(4.98) that p+(t,x) can be easily computed if

v+(t,x) are somehow found. It turns out (and this is a well-known result from the theory of stochastic processes) that v+(t,x) is the solution to the following initial boundary value problem for the backward Kolmogorov equation:

0V + ry2(X) 02V + 0V +

Ot = 2 Ox 2 t- b(x) 3 x ' (4.99)

v (0 ,x ) - 1, v(t,c +) - 0, (4.100)

where c + are the exit points for the process Xt, which are equal to a - u0 and ]~ - u0, respectively.

Next, we shall show that switching probabilities P:~(t) can be expressed in terms of v+(t) and p+(t). Note that, according to (4.96)-(4.98), p+(t) are related to v+(t) as follows:

3 p+(t) = ~ [e( t ) - v+(t)]. (4.101)

248

~ x t

+ 1 ]. .... I I I

-1 ~ ;L ! ~L+d~,

CHAPTER 4 Stochastic Aspects of Hysteresis

- 1

I I | ! ' -=


I

t

It is clear from the very definition of v+( t ,x ) that

P:~(t) - v+(t, 0). (4.102)

It is apparent from Fig. 4.13 that the occurrence of exactly one downward switching is the union of the following disjoint elementary events: downward switching occurs in the time interval (X, X + dX) and then no upward switching occurs up to the time t. Due to the strong Markov property of Xt, the probability of this elementary event is given by

p - (~., 0)v+(t - X, fl - uo)d~.. (4.103)

Now the probability P~-(t) of exactly one downward switching can be found by integrating (4.103) from 0 to t:

P+(t) = p - (X ,O)v+( t - ,k, fl - uo)dX. (4.104)

In other words, P-((t) is the convolution of p-(t ,0) and v+(t, fl - u0)

P+(t) = p - ( t , O ) , v + ( t , fl - uo). (4.105)

By using similar reasoning, we can derive

P l ( t ) = p + ( t , O ) , v - ( t , o l - uo). (4.106)

Next, consider the probability P-~(t) of the occurrence of exactly two switchings starting from the initial state ~ x 0 = 1. According to Fig. 4.14 this occurrence can be considered as the union of the following disjoint elementary events: downward switching occurs in the time interval (;~, ~ + dk) and then exactly one upward switching occurs up to the time t. The probability of these elementary events is given by

p- (~, 0)P l ( t - ~) d~. (4.107)

Now, by integrating (4.107), we find

fo ' P-~(t) = p - ( X , O ) P l ( t - X)dX. (4.108)


From (4.106) and (4.108) we obtain

P-~(t) = p - ( t , O ) , p + ( t , fl - u o ) , v - ( t , ol - uo). (4.109)

By using the same line of reasoning, we derive

P 2 ( t ) = p + ( t , O ) , p - ( t , ot - u o ) , v + ( t , fl - uo). (4.110)

For the sake of conciseness, we introduce the notations:

p• p~(t), p+(t, fl - uo)= p+(t), p - ( t ,u - uo)= p-( t ) , (4.111)

v+( t ,O) = v:~(t), v+( t , fl - uo) -- v+( t ) , v - ( t , ~ - uo) = v - ( t ) . (4.112)

Now, by using the same line of reasoning as before and the induction argument, we can easily derive the following expressions for the switching probabilities:

2k-2 terms

P+k(t, uo) = P o ( t ) , p + ( t ) , p - ( t ) , p + ( t ) , . . . , p - ( t ) , p + ( t i ,v-(t) , (4.113)

2k terms

P2k+l(t , u o ) - p ~ ( t ) , p - ( t ) ] , p + ( t ) , . . . , p - ( t ) , p + ( t i , v+( t ) . (4.114)

By substituting (4.113) and (4.114) into (4.94), we obtain the expression for q~#(t) in terms of infinite series of iterated convolutions. These series can be reduced to geometric ones by employing Laplace transforms:

fo ~(s) = p( t )e -s t d t (Re s > 0), (4.115)

f~(s) = v( t )e -s t dt. (4.116)

It is clear that

I, (s)l < 1. (4.117)

By using these Laplace transforms, from (4.102), (4.113), and (4.114) we obtain

=

P~-k(S) --/3 0 (S)/~ + (S)V-(S)[p-(S)/9 + (S)] k- l ,

P2k+l(S)- #~?(s)~-(s)[/~-(s)#+(s)] k.


(4.118)

(4.119)

(4.120)

7,? (s)O- (s) F/o~(s) = 1 - j6-(s)~5+(s) if }~o~xo- -1 . (4.121)


A similar expression can be derived for the case }9~x0 = +1. According to (4.101)

~+ (s) - 1 - s~;+ (s). (4.122)

Thus, the problem of computing F/~ is reduced to the problem of determining fi• This can be accomplished by using the initial boundary value problem (4.99)-(4.100). The complexity of this task will depend on the nature of the stochastic process Xt, which models the noise in hysteretic systems. It is natural that the stochastic process that models the noise must be a stationary Gaussian Markov process. According to the Doob theorem [17], the only process that satisfies these requirements is the Omstein-Uhlenbeck process. This process is the solution to the following Ito stochastic differential equation:

dXt = - b X t dt + a dWt, (4.123)

where 1 /b has the meaning of the correlation time. (This means that Xt and Xt, are only significantly correlated if [t - t'[ ~< 1/b.)

The backward Kolmogorov equation for the Omstein-Uhlenbeck process has the form

0V -}- 0-2 0-2V+ 0V + = b x - - . (4.124)

Ot 2 Ox 2 Ox

This equation should be considered jointly with initial and boundary conditions (4.100). By applying the Laplace transform to (4.124) and (4.100), we arrive at the following boundary value problem for fi+(s):

0-2 d2v+(s,x) - bx d~;+(s'x~) - s~+(s,x) - -1 , (4.125)

2 dx 2 dx

~+(s,c +) =0, ~• oe)= 1/s . (4.126)

The solution to the boundary value problem (4.125)-(4.126) can be written in the form

1 ( e[X2_(c+)a]/4x2 ~)_s/b(X/~.) ) fi+(S,X)-- S 1 -- (4.127)

Z)_s/b(C+ /;O '

where l )_s /b(X/k) are parabolic cylinder functions, while

x = ~ / , / ~ .

Expressions (4.122), (4.121), and (4.127) jointly with (4.89) and (4.91) out- line the main steps of computing ft.

It is apparent that the case of continuous-time noise is computationally expensive. Some sample examples of computations for this case can be found in [18, 19].


In spite of its complexity, the case of continuous-time noise is convenient for the discussion of the origin of the universality of intermediate asymptotics

M ,-, Mo - S(H, T) lnt. (4.128)

The very fact that intermediate asymptotics (4.128) has been observed for various magnetic materials as well as for superconductors and other hysteretic systems reveals its universality. This raises the question of the physical origins of this universality. The purpose of subsequent discussion is to demonstrate that this universality can be traced back to the asymptotic temporal behavior of internal thermal noise [20]. This demonstration also suggests that ~ /~ t type relaxations may actually occur in the case of stationary Gaussian thermal noise. Our analysis also yields a simple expression for the viscosity coefficient in terms of differential susceptibility and temperature T. Experimental data supporting the theoretical analysis is also presented.

Consider a hysteretic material that was first driven into the state of positive saturation. Then, the external magnetic field was reduced to some value -H0 close to coercivity -Hc and maintained constant (in time) thereafter. Due to the inherently present thermal noise, gradual relaxations of magnetization take place. To mathematically describe these relaxations, we shall use the Preisach model:

= / / l z ( o l , fl)~'~Ht dot dfl, (4.129) Mt

Ht -- -Ho 4- ht, (4.130)

and ht is a stochastic process that models the thermal noise. It is instructive to note again that adding noise ht to -H0 is mathematically equivalent to adding noise -h t to switching thresholds a and ft. It seems natural to assume that ht is a stationary Gaussian process with decreasing in time correlations. In this case, the following temporal asymptotics is valid for ht with probability one [21]

h t - max ih~l "~ v/2R(0)lnt, (4.131) O~r~t

where R(0) is the value of the autocovariance function of ht at zero. It is this asymptotic behavior of ht that is ultimately responsible for long-time relaxations of magnetization.

Our reasoning will be based on the diagram technique for the Preisach model. According to this technique, the state of the Preisach model at the moment when the external magnetic field is reduced to -H0 is represented by the diagram shown in Fig. 4.15a. Let M0 be the magnetization in this


FIGURE 4.15

state. At subsequent instants of time, this diagram will be modified due to the time variations of noise process ht. This will lead to fluctuations AMt of magnetization

Mt = Mo - AMt. (4.132)

Consider the nature of these fluctuations during the time interval 0 ~< r ~< t. At some instant of time r0 within this interval, process hr will reach its minimum value - h t . This will enlarge the negative set S- and reduce the positive set S + by the same trapezoid f2 (see Fig. 4.15b). During the time interval r0 K r ~< t, the Preisach diagram will be further modified. However, these modifications will be confined to the right triangle [', whose right angle vertex has ~ and/7 coordinates e~ual to -H0 + h t and -H0 - h t , respectively. For small noise ht,triangle F is small and we can neglect the diagram modifications within P. Thus, we get

AMt ~ 2 Si~/x(~,/7) dc~ d/J. (4.133)

The trapezoid f2 can be represented as a difference of two right triangles: I ' ( -H0 - h m) and [ '(-H0). This means that

AMt ~ 2 ( S f c #(ol, f l ) d o e d f l - i f c /z(c~,/~) dc~ d~). (4.134) (-HO-hm) (-Ho)

It is known from the theory of the Preisach model (see Chapter 1) that

2 ~ #(~,/7) d~ d/~ - M~- - M + (H), (4.135) d i p (H)


where M~- is the positive saturation value of the magnetization, while M + (H) is the magnetization value on the descending branch of the major hysteresis loop. By using Eqs. (4.134) and (4.135), we derive

d M + A M t ~ M + ( - H o ) - M + ( - H o - hm) ~, ( - H o ) h m

d H (4.136)

By invoking formula (4.131) and the fact that d~-~(--H0) is the differential susceptibility Xd(-H0, T) evaluated at -H0 and temperature T, we transform formula (4.136) as follows:

A M t ~, v/2R(0) Xd(--H0, T)V~n t. (4.137)

Furthermore, R(0) can be interpreted as being proportional to the total energy of the thermal noise. Indeed,

F R(0) -- S(o~) do) = y T, o o

(4.138)

where S(o~) is the spectral density of noise process ht. By using Eq. (4.138) in Eq. (4.137) and taking into account Eq. (4.132), we arrive at

Me "" MO - ~ ~ X d ( - H o , T ) V ~ t, (4.139)

where X is some constant. This is the final expression for the intermediate asymptotics of ther-

mal relaxations in hysteretic systems. The immediate question is how this expression stands the test of experiment. To test the v/-~t dependence of long-time thermal relaxations, the measurements of these relaxations for thin-film CoCrPt magnetic recording materials and YBaCuO superconducting materials have been performed by using a vibrating sample magnetometer (VSM) Micro Mag 3900 of Princeton Measurements Cor- poration. The measured values of Mt were plotted on v ~ t scale and the linear dependence of Mt vs ~/~ t has been consistently observed. Some sample results of these measurements are presented in Figs. 4.16 and 4.17. To test the temperature dependence of intermediate asymptotics (4.139), we point out that for very low temperatures Xd(-H0, T) does not depend on T for magnetic materials. This means that asymptotics (4.139) predicts ~/T dependence of ~ t decay of magnetization. This prediction was made earlier (see [1]) on the basis of thermal activation model and it is consistent with the reported experimental observations (see [7]). The temperature dependence given by Eq. (4.139) has been also tested for high temperatures in the following matter. Equation (4.139) can be written in the form

Mt " Mo - S(-H0, T)~T-nnt, (4.140)


x 10 -4 CoCrPt Material Hard Disk Sample at T=25~C I

~ 0 E ~ - 1 r

._o ~ - 2

r -3

~ - 4

-5

-6

!

�9 " . . - - : . . . . . . . ~ . , - ~ ~ - ~ , _ . . . . , _ . .

I I I I

2.2 2.4 2.6 2.8 3 3.2 Sqrt(Ln(Time(sec)))

FIGURE 4.16

where the slop S(H, T) can be retrieved from time-observations of thermal relaxations for various values of H and T. Equations (4.139) and (4.140) suggest that S(H, T) = XV~Xd(H, T). By integrating that last equality with respect to H over sufficiently large range H_ K H K H+ containing many observation points, we find

fH H+ S(H, T)dH = . (4.141)

~/T [M + (H+, T) - M + (H_, T)]

If Eq. (4.139) predicts the correct temperature dependence, then X in Eq. (4.141) must remain constant. This has been verified over the temperature range of about 150~ for samples of high and low density CoCrPt recording materials. The results of this verification are presented in Fig. 4.18. It is apparent from Fig. 4.18 that ;~ remains fairly constant and exhibits slight decay with increasing T. This decay can be attributed to the fact that S(H, T) becomes very sharply peaked around coercivity with the increase in T. This makes the measurement of max S(H, T) less accurate and eventually leads to the underestimated values of the integral in Eq. (4.141).

4.3 P R E I S A C H M O D E L D R I V E N BY C O N T I N U O U S - T I M E N O I S E

E0.12 c O

~ 0 . 1 1 N

. m

r--

0.1

High T c S u p e r c o n d u c t o r S a m p l e at T = 3 0 K

0.15 !

0 . 1 4

0 . 1 3 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.09 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . .

0 . 0 8 . . . . . . . . . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . : ~ ~ . . . . . . . . . . . . . .

I

2 2.1 2 . 2 2 .3 2 .4 2 .5 S q r t ( L n ( T i m e ( s e c ) ) )

F I G U R E 4.17

255

10 . . . . . . .

= .High_Density Hard Disk Samp.I 9 .................. o LOW Density Hard Disk Sample I1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

k, 5 . . . . . . . .

o -

3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * ........................ ~

. . . . . . . . . . , . . . . . . . . . . . : . . . . . . . . . . . ~. . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i s

L~80 300 320 340 360 380 400 420 440 Temperature(K)

F I G U R E 4.18


The derivation of Eq. (4.139) is based on asymptotics (4.131). The proof of Eq. (4.131) rests on the Gaussian nature of "low probability tail" of the probability density function for ht. The Gaussian nature of stationary noise is customary justified on the grounds of the central limit theorem. This theorem holds for normal and moderately large deviations. However, very low probability tails may appreciably deviate from Gaussian tails. If the very low probability tails have asymptotics p ( x ) = e -alxlv and the process ht has sufficiently strong mixing properties, then it can be shown that Eq. (4.131) can be modified as follows:

max [hT[ "~ b(lnt) 1/". (4.142) O~r~t

By using Eq. (4.142) and literally repeating the previous line of reasoning, the following intermediate asymptotics can be derived

Mt ~ Mo - S(lnt) 1/v. (4.143)

Particularly, in the case of exponential (v = 1) tails, Eq. (4.143) leads to In t intermediate asymptotics. Experimentally, it may be quite difficult to distinguish between (lnt) 1/" and 4J-fit (or lnt) thermal relaxations. This is because In t is a slowly varying function within the time range of the validity of intermediate asymptotics.

4.4 NOISE IN HYSTERETIC SYSTEMS A N D STOCHASTIC PROCESSES O N G R A P H S

In the previous section, the Preisach model driven by a (continuous time and continuous sample) diffusion process is discussed. The mathematical treatment of this model is reduced to the analysis of random switching of ~'~Zxt which, in turn, is reduced to the "exit problem" for diffusion processes. Although the concept of random switching as an exit problem is quite transparent from the physical point view, its mathematical implementation leads to infinite series of iterated convolutions. It turns out that the mathematical treatment can be appreciably simplified by using an entirely different approach based on the theory of stochastic processes on graphs. This theory has only recently been developed [22] and applied to the study of random perturbations of Hamiltonian dynamical systems [23]. The main purpose of this section is to demonstrate that the mathematical machinery of this theory is naturally suitable for the analysis of random output processes of hysteretic systems. As a by-product of

4.4 NOISE IN HYSTERETIC SYSTEMS

11 \

/ i=-1

13 i=+1

13 12 i=-1

FIGURE 4.19

(X 1

14 // \ i=+1

257

this demonstration, we derive analytical expressions for stationary characteristics of these processes. These expressions are of interest in their own right. The discussion presented below closely follows the paper [24].

This discussion is based on the following simple fact. The output it - yo~xt is a random binary process. This process is not Markovian. How-

ever, the two component process y t - (itt) is Markovian. THis is because the rectangular loop operators describe hysteresis with local memory. This means that joint specifications of current values of input and output uniquely define the states of this hysteresis.

The two component process yt is defined on the four edge graph shown in Fig. 4.19. The binary process it assumes constant values on each edge Ik of the above graph. This justifies the following concise notation for the transition probability density:

P(t, ~Jl~/O) l~*Ik -- p(k) (t, xl~o). (4.144)

It is obvious that

p(1) = p for x ~ fl, p(4) = p for x ~ a,

p(2) jr_/)(3) = p for x ~ [fl,~], (4.145)

where p is the transition probability density of the process Xt, which is assumed to be known.

According to the theory of Markovian processes, the following equality is valid for p(k).

4S, 4S, f 8t dx = ~ (LS) p(k) ax. k=l k=l

(4.146)

Here, 1 2 02 0

L= -~a (X)~x2 + b(x)--Ox

is the generator of the semigroup of the process xt, while f is a function that is continuous on the entire graph and sufficiently smooth inside the edges and satisfies certain "gluing" (interface) conditions at c~


and 3. These interface conditions follow from the Markovian nature of the process yt o n the entire graph [22]. In our case, the process yt " s p e n d s zero time" at the graph vertices. In this situation, the interface conditions can be written as follows [22]"

afi Xkj-~x - 0, Xkj ~7 0. (4.147)

k oj

Here, fik --flIk, summation is performed over all edges connected to a graph vertex Oj, while the derivatives are taken along the edges in out- ward directions with respect to Oj. It is known [22] that constants Xkj are (roughly speaking) proportional to the probabilities that the process will "move" from vertex Oj along the edges Ik. It is clear that in our case there is zero probability that the process yt will move from the vertex 3 along the edge/3, while the random motions along the edges I1 and 12 are equally probable. A similar assertion is valid for the vertex c~. As a result, we arrive at the following interface conditions:

3fI1] = dfI2 [ dfI 3 dfI4 [ (4.148) 31x ax =-37x '

while the values of the derivatives dfI3/dx]~ and dj~ 2/dX[ol are entirely arbitrary. It is understood that differentiation in (4.148) is performed in the direction of increasing values of x.

By integrating by parts in the equality (4.146) and by taking into account the interface conditions (4.148) and the choice inherent inf, dfi3/dx[~ and dfI2/dxl~, one finds that the transition probability density p(3) satisfies the forward Kolmogorov equation

3p (3) 1 3 2 3 3t = 2 3x 2 (r - -~x (b(x)p(3)) (4.149)

and the following boundary conditions:

p(3)]3 -- 0, p(g)]a = P[c~. (4.150)

It is also tacitly understood that the standard &type initial condition is imposed at Y0.

A similar initial boundary value problem can be stated for p(2). How- ever , p(2) can also be found by using formula (4.145).

The solution to the initial-boundary value problem (4.149)-(4.150) can be found in terms of parabolic cylinder functions and their Laplace transforms in the case when xt is the Ornstein-Uhlenbeck process (dxt = -b(xt - xo) dt + r~ dWt) with expected value x0.

4.4 NOISE IN HYSTERETIC SYSTEMS 259

(3) Simpler analytical results can be obtained for stationary densities Pst

and p(2) In this case, we have to deal with the following boundary value st �9 problem for the ordinary differential equation

1 d 2 (o.2(x)p~3))_ d (3) 2 dx 2 -~x ( b ( x ) p s t ) =0, (4.151)

(3) /9(3), Pst (fl)=0, st [,Cr (4.152)

Although the analytical solution to the above boundary value problem can be written out for any stationary diffusion process xt, below we present this solution only for the case of the Ornstein-Uhlenbeck process:

p(3) st (X) Pst(X)~O(X,~, fl), _(2) [1 /~)], "= Pst (X)---- Pst(X) -- ~O(X,~, (4.153)

where

~/ b e_b(x_xo)2 /o.2 Pst(X) = (4.154)

~O(X, Cr = f ; eb(y-x~ dy (4.155) f ; eb(y_xo)2 /0.2 dy"

If we consider the probability current

j k (x ) __ 0 ,2 dp(k)(x) - Y d x - b ( x - x o ) p ( k ) ( x ) ,

then it is easy to conclude from formulas (4.153), (4.154), and (4.155) that h(x)--J4(x) = 0, while J2(x)=-J3(x) # 0. Thus, there exists a nonzero probability current circulating in the loop formed by edges I3 and I2. The existence of the circulating loop current can be considered as the manifestation of the lack of detailed balance in the two component process yt. The existence of this circulating current can also be traced to energy losses associated with the random switchings of rectangular loop 19~/~. This dissipated energy is extracted from noise, which is the only source of energy present in the discussed problem. The situation here is somewhat analogous to one observed for stochastic resonance, where a feeble deterministic signal alone cannot affect switchings. These switchings are assisted by the internal noise and they are accompanied by the extraction of energy from the noise [25].

It is instructive to compute the expected value it of the binary output process and its variance r~. 2 It is clear that It"

-Zt - Est(Yot~xt) - Pst ~ ( i t - 1 ) - P~F ( i t - - -1), (4.156)


which is equivalent to

it = 2Psf ( i t - 1 ) - 1. (4.157)

It is apparent that

PTt ~ (it = 1) - p~) (x) dx + pst(X) dx, (4.158)

which leads to

it = 2 Pst (x) dx -4- Pst(X) dx - 1. (4.159)

When it is computed, r~. 2 can be calculated as follows: It

r~. 2 - 1 - ~ t (4.160) It

Calculations are substantially facilitated by the observation that

fo x 4~- erfi(x), e y2 d y - - - ~ (4.161)

x2 t 3 2 ; - x 2 ) (4.162) e -x2 erfi(x)dx- - ~ 2F2 1,1; ~,

where erfi(x) and 2F2 are "imaginary error function" and "generalized hy- pergeometric function," respectively.

By using the formulas presented above, the expected value it and variance a 2 have been computed as functions of the expected ("bias")

It value x0 of the input process xt for the case of hysteretic nonlinearities represented by symmetric rectangular loops 9~,_~. The results of calculations are shown in Figs. 4.20 and 4.21, respectively. These results are plotted for normalized values of input bias v - xo/o~ and normalized values of switching thresholds ~ = c~/X, where ~2= ry2/b is the variance of the stationary distribution of the input Ornstein-Uhlenbeck process xt. The dependence of it on x0 can be interpreted in magnetics as "anhysteretic" magnetization curve (see Chapter 1). This anhysteretic curve depends on the noise variance. This dependence is especially appreciable when the noise standard deviation X is comparable with the switching threshold value c~.

Next, we shall apply the obtained results to the case of hysteresis loop with "curved" ascending and descending branches (Fig. 4.22). This type of hysteresis loop is exhibited, for instance, by Stoner-Wohlfarth particles (see Chapter 3) when the applied magnetic field is restricted to vary along one direction. Suppose that the loop shown in Fig. 4.22 is driven by the Ornstein-Uhlenbeck process xt and we are interested in the stationary

4.4 N O I S E IN HYSTERETIC SYSTEMS 2 6 1

o . �9 . . . . . . . . . . i . . . . . . . . . . .

o . 6 . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . .

0.4 . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . ~ 0.5 " -

0 2 : ~ 1,0

" - " i �9 1 . 5

" ~ 0 . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . 0 , -

L M i - - 2 . 5

: - - - 3.0 - o . 2 . . . . . . . . . . . i . . . . . . . . . . . .

- 0 . 4 . . . . . . . . . . . ! . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . .

- 0 . 6 . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . ! . . . . . . . . . . . ,

- o . s . . . . . . . . . . . . . . . . . ! . . . . . . . . . . . .

-1 - 2 -1 .5 -1 -0 .5 0 0.5 1 1.5 2

v

F I G U R E 4.20

! i ! ! ! ~ o.~

! : : . . . . . 2.0 0 . 8 . . . . . . . . . . . : . . . . . . . . . . . . i. . . . . . . . i . . . . . . . . . i . . . . . . . . . . . . 2 . 5 -

! ! i i . - . a . o

0 7 . . . . . . . . . . . i . . . . . . . . . . . : i . . . . . . . . :: . . . . . . . . . . : . - ~ . o

0 .6 . . . . . . . . . . i . . . . . . . :: . . . . . . . . . . . " . . . . . . . . . . . : . . . . . . . . . .

--'*" 0.5 . . . . . . . . . . . . . . . . . .

o . 4 . . . . . . . . . . . . . . . . . ! . . . . . . . . . . . . ! . . . . . . . . .

o . 3 . . . . . . . ;. . . . . . . . . . . . . i . . . . . . . . .

0.2 -- -:: . . . . . . . . . . . . . . . . . . . . . . .

O.

-2 -1 .5 -1 -0 .5 0 0.5 1 1.5 2

v

F I G U R E 4.21


Yt

A

~'~ (X X t

FIGURE 4.22

distribution of the output process tt. The process yt is not a binary one. However, it admits the following representation in terms of the binary process it -- Gflxt:

f+(Xt) --f-(Xt) f+(Xt) ff-f-(Xt) it + , (4.163) y t - - 2 2

where the meaning off + (Xt) and f - (Xt ) is clear from Fig. 4.22. By using formula (4.163) and the appropriate change of variables,

we obtain the following expression for the stationary distribution density ~+ (y) of the process yt along the descending branch:

~s+t (y) = P (3) (g+ (y)) dR + (Y) st dy (4.164)

Here, g+(y) is the inverse of the function y =f+(x). The last formula is (3) valid for/~ < x < c~. For x > c~, density Pst should be replaced by pst. In

a similar way, the density ~s~ (Y) along the ascending branch can be computed.

The results obtained for }3~ operators can also be used to compute the stationary characteristics of the output processy~ of the Preisach model driven by the stochastic process xt:

?tst -- f ~ ~(c~, ~)Est(G~xt) dol d~, (4.165)

where Est(G~xt) can be evaluated by using expressions (4.153)-(4.162). If it is desired to evaluate the stationary value of the second moment

Est(ft 2) of the output processJ~ and its variance, the following integral must


be evaluated:

Est(f2)=/f~ ff~ l~>& 2~>/~2

Est(Yoll~lXt ~9r Xt)

X ~(Oll,/J1)~(Cr doll d/J1 dol2 d/J2, (4.166)

where Est(.) stands for stationary expected value. To compute Est(yo~l~lXt~'ot2~2xt), we consider the three component

Markovian process

~ , - ,2 ,

Xt

where i~ = YOll~l x t and it 2 = p~2~2xt. Depending on the relation between o~1, /~1, c~2 and/~2, this process is defined on graphs shown in Fig. 4.23. By using

11 \

,1/

11 \

13 16

/ 14

12 (8) Is

15

17

2

I, (b)

~2 ~14 o~ 1

~1 ~2 17 /

16

(c)

FIGURE 4.23


the same line of reasoning as before, one can easily arrive at the following (k)(x expressions for the stationary d e n s i t i e s Pst(~.t)]Ik = Pst . )"

In the first case (Fig. 4.23a) when two rectangular loops do not overlap (/J1 < O~1 </J2 < O/2), w e have

p(k) p(3), st - pst for k = 1,4,7, st [ x ) - ps t (X)~(x , a l , f l l ) , (4.167)

p ~ ) (X) -= Ps t (X)qg(X , 0/2, f12), p(2) (3), (5) p(6) (4.168) st - - Pst - - Pst Pst - - Pst - - st �9

In the second case (Fig. 4.23b) when two rectangular loops completely overlap (ill </J2 < or2 < Otl), w e have

p( k) (3) st - - Pst for k = 1,8, Pst ( x ) --- P s t ( X ) ( f l ( x , o t l , / J 1 ) ,

p(2) (3), (5) (3) st - - P s t - Pst Pst ( X ) - Pst (X)qg(X, Ot2,/J2),

p(4) (3) (5), (7) (2) st - - Pst - - Pst Pst ( X ) - - Pst (X)qg( X'c~2'j~2)' (6) _ (2) __ P(7)

Pst ~ Pst st �9

(4.169)

(4.170)

(4.171)

Finally, in the third case (Fig. 4.23c) when two rectangular loops partially overlap (ill < f12 < ~1 < C~2), we have

p( k) _(2) [1 /J1)], (4.172) st - - Pst for k - 7, Pst (X) - - P s t ( X ) -- qg(X, a l , p(3) _(2) _(4)

st - - Pst - Pst ' Pst ( X ) - P s t ( X ) ( f l ( X , ~ 2 , j~2), (4.173)

p(5) _(2) _(4) (6) _(4) (4.174) st = P s t - - Pst - - Pst " Pst ~ Pst - - Pst "

It is worthwhile noting that in the last case there is no graph edge corresponding to i~ - - 1 and i 2 - +1 because these simultaneous values of i~ and it 2 are not consistent with the definition of rectangular loops operators

}~O~lfl 1 and yO~2/J 2 . By using the above expressions for Pst-(k)" E s t ( ~ a l f l l X l ~ a 2 f l 2 x t )

can be computed. As an example, consider a particular case of the Preisach model when

all rectangular };-loops are symmetric: ~,~,-~xt - ~'~xt. Mathematically, this case is obtained when the "weight" function #(a, fl) has the form

tt(~, fl) = ~(~)~(a + fl), (4.175)

and the Preisach model is reduced to

f0 ~176 f t = ~(ot)~,~xtdot. (4.176)

For many magnetic materials, the weight function tx(c~,fl) is usually narrowly peaked around the line oe = - f t . For these materials, formulas (4.175) and (4.176) can be regarded as fairly good approximations. At


the same time, the calculations are considerably simplified because, in the case of symmetric loops, any two loops completely overlap. As a result, one has to deal with the three component Markovian process Zt defined only on the graphs shown in Fig. 4.23b.

For the case of model (4.176), formulas (4.165) and (4.166) can be written as follows:

t st ~0 ~~ -- ~ (ot)Est(},'otxt) dot, (4.177)

/o ~ (/o" ) Est(ft 2) -- 2 ~(ot) ~(ot')Est(ftotxt~o~!xt)dot , dot. (4.178)

To compute Est(f'otxt), formulas (4.153)-(4.162)can be used. To evaluate Est(GxtG,xt) in (4.178), we first remark that

Est(~'~xt~'~,xt) = 2P(Gxt}'~,xt-- 1 ) - 1. (4.179)

To find P(Gxt~,o~,xt), one has to integrate the appropriate (k) Pst (x) over those edges of the graph shown in Fig. 4.23b on which ~'~xt and ~'~,xt have the same signs. This leads to the formula

Est(~'olXt~'ot'Xt)

(5) p(3) cx~ = 2 Pst (X) dx + st (x) dx + Pst(X) dx

Og! !

i_" f_-" I_-" ] (6) (2) q- Pst (x) dx + Pst (x) dx + pst(X) dx - 1. (4.180)

Ol t Ol O 0

The integrals in the last expression can be evaluated by using formulas (4.169)-(4.171) and taking advantage of relations (4.161) and (4.162). Some sample results of calculations are shown in Figs. 4.24 and 4.25 for ?t st and Est(ft2), respectively. In these calculations, it was assumed that ~(ot) = 1 and ?t st and Est(ft 2) were computed as functions of normalized values of input bias v = xo/oto for various normalized values of &o = oto/X, where X 2 is the variance of the stationary distribution of xt. The values of?t st and Est(f 2) have also been normalized by ~ and X 2, respectively. It is worthwhile to note that, as evident from Fig. 4.25, the asymptotic values of Est(ft 2) are equal to &~ and coincide with asymptotic values of (ftst) 2. This guarantees zero asymptotic values for variance rY 2

it" The described formalism of stochastic processes on graphs can be fur-

ther extended to compute higher order moments of the output process y~. This extension is more or less straightforward in the case of the model (4.176). In this case, the relevant multicomponent Markovian processes are defined on the graph shown in Fig. 4.26 and, by using the

2 6 6 C H A P T E R 4 S t o c h a s t i c A s p e c t s o f H y s t e r e s i s

31 ! ! !

I " t " : : + o .

+ ~ ' " �9 ,i+ Q ,i ,I, ,i

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " / " " ; . . . . . . . , . . , . , i . . : . . . . . . . . . : . . . . . . . . . . . .

i_ i i i ! ,," ~ , " ! ! ... i i i i / �9 : j . . . - - - ' t - - -

i i i !i. t : i i : : : : 1 1 : " .

: : : . - , , - , - " .

0 . . . . . . . . . . . . : . . . . . . . . . . . ' d , . . . . . . . . . . . . . . . : . . . . . . . . . . . . : . . . . . . . . . . . . : . . . . . . . . . . . .

: : : ,' "I ! ! ? i

�9 . ~ ~ . /

~ ~ + ~ o �9 �9 . =

-2 .I ' . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~o �9 . ~.

. ~ . . . . . . . �9

-2 -1.5 -1 -0.5 0 0.5 1

v

I - S - 0'5 I I 1 . 0 .

1,5 - - 2.0

2.5 3.0

I

1.5 2

F I G U R E 4 . 2 4

v I .u

. . . . . 4, ~ . . . . . ! , , , I - 05

"% I I ~ 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " l . . . . . . . �9 . . . . . . . ~ ~ 1 . 5 �9

/ 2.0 2.5

t I 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . t I i i

�9 ""..,.!, ~ i : . : . ' - . . . . . . . . . . . . . . . . . . . . . . . . . . . / . ,~ I I /

I I I �9 -~...'~.. . . i . . . / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

�9 ~ I I I 4 ~ : . . . . . . . . ~ ! i ! ~ . . . . . . . - . " ~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

! i i: \ - - t t - - - ~ . ! . l / :i ! i

2 .-. .'T. . .-. .-. . ~ -. .- . . . . ~ F : . . . . . . . i . . . . . . . . . U.!.kt. . . . . . . . . i . . . . . . . -: . . . . .--.-- -..-:. . .-..-..-..---T[ i i . l~iil . i i

I : ,,, : : ,.: : : , - . - ~ ~ - - - ~ . , - . . ~ . ~ - ~ c ~

o i . . . . ] . . . . i . . . . | . . . . i . . . . i . . . . . . . . . ; . . . . -2 -1.5 -1 -0.5 0 0.5 1.5 2

V

F I G U R E 4 . 2 5

4.5 ANALYSIS OF SPECTRAL NOISE DENSITY

2k+1

1 ,

FIGURE 4.26

I1 /

267

same line of reasoning as before, we can derive the following explicit expression for p(2k+l) st (X) for edges I2k+1" k

p(2k+l) st (X) -- Pst(X) I - I 99(X'Olj" --~J)" (4.181)

j=l

Similar expressions can be derived for other graph edges.

4.5 A N A L Y S I S O F S P E C T R A L N O I S E D E N S I T Y OF H Y S T E R E T I C S Y S T E M S D R I V E N B Y

S T O C H A S T I C P R O C E S S E S

The behavior of hysteretic systems (materials) is usually affected by the internal noise that causes thermal relaxations in these systems. This implies that outputs of hysteretic systems are stochastic processes. In this section, a method for the calculation of spectral density of such output processes is presented. The discussion is centered on the case of a rectangular hysteretic loop driven by a diffusion stochastic process. This case is important because complex hysteretic systems can be modeled through the Preisach formalism as weighted superpositions of rectangular loops. The problem of calculation of the spectral density of the output process is of considerable mathematical complexity because this process is not Markovian in nature. This difficulty is overcome by using the mathematical machinery of stochastic processes on graph. In addition, the so-called "effective" distribution function is introduced that appreciably simplifies the calculation of spectral density. As a result, remarkably simple explicit expressions for


the spectral density in terms of parabolic cylinder functions are derived for the case when the input stochastic process is the Ornstein-Uhlenbeck process.

Consider a rectangular loop hysteretic nonlinearity driven by a diffusion stochastic process xt"

it - - G ~ x t . (4.182)

The input stochastic process (noise) Xt leads to random switching of rectangular loop G~- Thus, the output it is a random binary process. The main goal of this section is to compute the spectral density of it. The process it is not Markovian. The efficient way to study the random process it is to consider as before the two component Markovian process yt:

( i t ) (4.183) yt - xt "

which is defined on the four edge graph (see Fig. 4.19). Process yt can be analyzed by using the mathematical theory of Markovian processes on graphs. This approach will be pursued in our calculations of the spectral density of it.

To start these calculations, we shall first define the autocovariance matrix for yt. Consider random vectors

( i ) and y 0 = y ' = ( i ' ) y r - y = x x' ' (4.184)

generated by the process yt at times t = r and t - 0, respectively. Then, the autocovariance matrix C(r) can be defined as follows:

C ( r ) - ((y - (y))(y'- (y,))T), (4.185)

where the symbol "()" is used for the notation of average value, while superscript "T" denotes a transposed vector. By using the joint probability density function p(i,x, r;i',x',O) and stationary distribution function ps(i,x) of the process yt, the autocovariance matrix can be computed as follows:

C ( r ) - f / ~ ~ y ( y ' ) r p ( i , x , r ; i ' , x ' , O ) d x d x ' i i'

- ( / ~ y p s ( i , x ) d x ) ( / ~ ( y ' ) T p s ( i ' , x ' ) d x ' ) . (4.186) i i'

Since the process yt is Markovian, its joint probability function can be represented as the product of the transition probability function

4.5 ANALYSIS OF SPECTRAL NOISE DENSITY 269

p(i,x, rii',x',O) and the stationary probability function. This leads to the following simplification of formula (4.186):

"C(r)- f f ~ ~y(y ' )T[p( i ,x , rli',x',O) - ps(i,x)]ps(i',x')dxdx'. (4.187) J J

i i t

To further simplify the calculations, the following two-component "effective" distribution function is introduced:

(gl( i ,x ,r)) g(i,x, r) - g2(i,x, r)

f y~y'[p(i,x, rli',x',O)- ps(i,x)]ps(i',x')dx'. (4.188) i t

If this "effective" distribution function is somehow found, then the autocovariance matrix can be computed as follows:

C ( r ) - f ~ ygT(i,x, r)dx. (4.189) i

Next, we shall derive the initial-boundary value problem for g(i,x, r) on the graph shown in Fig. 4.19. On each edge of this graph, the transition probability density function satisfies the forward Kolmogorov equation

Op(i, x, r Ii', x', 0) + ~xp (i, x, rli',x',O) - 0, (4.190)

Or where Lx is the second-order differential operator whose structure is determined by the nature of the process xt. The transition probability density function satisfies the obvious initial condition

p(i,x, Oli',x',O) = Sii , r ~ ( x - x ' ) (4.191)

and certain boundary conditions at graph vertices x = ~ and x = ]~. These "vertex" boundary conditions are discussed in the previous section and they express the continuity of the transition probability function when the transition from one graph edge to another through a vertex occurs without switching of the rectangular loop G~, and zero boundary condition is imposed on the third graph edge connected to this vertex. In addition, the probability current is conserved at each vertex.

The stationary probability function satisfies the equation

Lxps(i, x) - 0 (4.192)

on each edge of the graph and the "vertex" boundary conditions. By using the formulas (4.190)-(4.192) and the definition (4.188) of g(i,x, r), the


following initial-boundary-value problem on the graph can be derived for the "effective" distribution function: g(i, x, r) satisfies the equation

3g(i,x,r) A Or + Lxg(i,x, r) - 0 (4.193)

on each graph edge, the following initial condition

g(i,x, r)lT=0- ( y - (y))ps(i,x) (4.194)

and the "vertex" boundary conditions. It is well-known that the matrix of spectral density is related to the

autocovariance matrix through the Fourier transform:

S"(~o)- C(r)e -j~r dr (4.195) O0

The element 811(O9) of the matrix S(~o) can be construed as the spectral density of the output process it. By introducing the Fourier transform of the "effective" distribution function

G ( i , x , ~ ) - ( G l ( i ' x ' ~ ~o ~ G2(i,x,o~) - g(i,x, r)e -j~T dr, (4.196)

and using formulas (4.189) and (4.195), we find:

S l l ( O ~ ) - 2 R e ( / ~ i G l ( i , x , oJ)dx). (4.197) i

Thus, if Gl(i,x,~) is found, then the spectral density of the process it can be computed. To find GI(i, x, ~o), we perform Fourier transformation of the initial-boundary-value problem (4.193)-(4.194). As a result, we arrive at the following boundary-value problem for Gl(i,x,~): it satisfies the following equation

j~Gl(i,x,o~) + L'*'xGx(i,x,~) -- (i - (i))ps(i,x) (4.198)

on each graph edge and the "vertex" boundary conditions. Thus, by solving the boundary value problem (4.198) and by using formula (4.197), the spectral density of the output process it can be computed for any input process xt. Analytical results can be obtained when xt is the Ornstein- Uhlenbeck process. This process is very appealing as a noise model because of its stationary and Gaussian nature. In the case of Ornstein- Uhlenbeck process the operator Lx has the form:

A 0 .2 d2Gl(i,x,~) d LxGl(i,x,o~)- - - ~ dx 2 + b-~x[(X- xo)Gl(i,x,~)], (4.199)

where x0 is the expected value of xt. Formulas (4.198)-(4.199) lead to linear second-order, inhomogeneous differential equation whose solution has

4.5 ANALYSIS OF SPECTRAL NOISE DENSITY 271

two distinct components: particular solution of the inhomogeneous equation and general solution of the homogeneous equation:

J c~ 0-2 d2G 0 d 2 dx 2 ~ b-~x [(x - xo)G ~ = 0. (4.200)

It is apparent from formula (4.192) that a particular solution of the inhomogeneous equation has the form - j ( i - {i))ps(i, x). Because the particular solution is purely imaginary, it does not contribute to the spectral density computed in accordance with formula (4.197). Thus, G1 in formula (4.197) can be replaced by G O that satisfies Eq. (4.200) and can be expressed as follows:

GO= J [ o-2d2GO b d 0)] ---s 2 dx 2 -~x ((X - xo)G �9 (4.201)

By substituting expression (4.201) into formula (4.197), performing integration and using the "vertex" boundary condition, the following formula can be derived for the spectral density:

$11(co) [ ] 20-2co Im ~ dGO (1,~+,co)- ~ dGO (1,--c~ ,co)+ -~x (1,/~+,co) .

(4.202)

The solution of Eq. (4.200) can be expressed in terms of parabolic cylinder functions. This fact along with formula (4.202) leads to the following analytical expression for the spectral density

t] Sll(CO) = 20-2 Im bk(cO)-~-x (oe, co ) + Ck(~O)-~-x (/~,~o) , (4.203) k = l

where functions Zk(X, CO) are related to the parabolic cylinder functions yk(x, co) by the formula:

( ~ 2 b )b(x--xo)2 Zx(X, CO) = Yk -~-g(X -- Xo),~o e -~2 (4.204)

and coefficients bk and Xk are found from the "vertex" boundary conditions. The functions Yl and y2 are the parabolic cylinder functions that vanish at +oe and - ~ , respectively.

The sample results of computation of the spectral density are shown in Fig. 4.27.

272

3 . 5 -

S.(o9 I

3

2 .5

2 "~

1.5-

1-

0.5 i_~t~~

\

\ \

~ ~o


( : z=b= (~= l

x =0 .0 0

. . . . . . x =0.2 o

. . . . x =0.4 o

c x =0 .6 0

= X =0 .8 0

^ x =1 .0 v 0

.. x =1.2

0.5 1 1.5 2 2 .5 CO

FIGURE 4.27

4.6 M O D E L I N G OF T E M P E R A T U R E D E P E N D E N T H Y S T E R E S I S W I T H I N T H E F R A M E W O R K O F R A N D O M L Y P E R T U R B E D F A S T D Y N A M I C A L S Y S T E M S

In the previous sections of this chapter, the theory of stochastic processes is used to study noise induced thermal relaxations in hysteretic systems within the framework of the Preisach model of hysteresis. In this and the next sections, the theory of stochastic processes will be used to construct novel models of hysteresis. We begin in this section with the discussion of models for temperature (noise) sensitive rate-independent hysteresis.

It has been experimentally observed that hysteretic properties of materials are sensitive to temperature, i.e. to the level of internal thermal noise. At a given temperature, various materials exhibit rate-independent deterministic (that is fully reproducible) hysteresis. However, many hysteretic properties such as coercivity, saturation magnetization, loop shape and others change appreciably with temperature. Usually, temperature dependence of hysteresis is taken into account within the framework of Preisach model in ad-hoc ways without any efforts to understand how

4.6 MODELING OF TEMPERATURE DEPENDENT HYSTERESIS 273

sensitivity to noise and deterministic properties of rate-independent hysteresis can be compatible and coexist. In the following discussion, the modeling of temperature (noise) dependent deterministic hysteresis is developed within the framework of randomly perturbed fast dynamical systems. By using the mathematical machinery of the large deviation theory, it is demonstrated that, for specific interplay of the fast system dynamics and the noise strength, randomly perturbed fast dynamical systems may exhibit rate-independent deterministic hysteresis sensitive to noise. The theory of large deviations of randomly perturbed dynamical systems is extensively developed in [26]. The subsequent discussion closely follows the papers [27, 28].

To emphasize the sensitivity of deterministic hysteresis to noise, the discussion is deliberately centered around the case of dynamical systems that under purely deterministic conditions do not exhibit hysteresis. It is demonstrated that noise induced deterministic hysteresis may occur in such systems. This phenomenon is of interest in its own right, and it clearly reveals sensitivity of deterministic hysteresis to noise. First, the case of dynamical systems with two attractors is studied. Then, the case of dynamical systems with many attractors is discussed and the mathematical machinery for identifying the attractors through which such systems evolve in time is presented.

In the case of temperature (noise) sensitive rate-independent deterministic hysteresis one deals with four distinct time scales. The first is the time scale of fast internal dynamics of hysteretic system. The second is the time scale on which observations (measurements) are performed. This time scale is much larger than the time scale of systems dynamics so that every observation can be identified with a specific stable state of the system. The third is the time scale of input variations. This time scale is much larger than the observation time scale so that every measurement can be associated with a specific value of input. In this way, every measurement establishes the connection between a specific input value and the respective state of hysteretic system. Finally, the largest time scale is the time scale on which thermal relaxations occur. It is demonstrated in the paper that for a given noise level all attractors can be subdivided in two distinct groups. The first group consists of attractors that participate in fast switchings that occur on the first time scale. These switchings lead to deterministic hysteresis. The second group consists of attractors that participate in slow switchings that occur on the fourth time scale. These switchings are known as thermal relaxations. As temperature (noise level) is changed, the "boundary" between these two groups is shifted and this results in temperature dependent deterministic hysteresis.


To start the technical discussion, let us consider one of the possible mechanisms that leads to the occurrence of rate-independent hysteresis. Suppose that the state of a system is defined by a set of interior parameters X - (X1,... , X n) ~ R n, and the evolution of the system is governed by the equations:

r~x~t = b(Xt, gt), Xo E R n, (4.205)

where gt is an exterior parameter-input, while b(x,g) is a sufficiently smooth vector field in R n+l. It is further assumed that 8 KK 1, which implies the fast dynamics of interior parameters in time. The case when this dynamics is appreciably faster than the temporal variations of input gt is discussed below. Let us also assume that the system (4.205) with a "frozen" input gt = g has a finite number s of asymptotically stable equilibriums Kl(g),...,K~($)(g) and that any trajectory of this system is attracted to one of Ki(g) (with possible exception of trajectories belonging to separatrix surfaces). An observation (measurement) Yt performed on the system can be mathematically interpreted as a certain functional of interior parameters Xt. Since there are no instantaneous observations, the above functional will include some integration (averaging) over some time h. If h is small on the time scale of input variations and large on the time scale of Xt-variations, then the relation between Yt and gt will be essentially rate-independent.

When input gt is changed in time, the vector field b(x,g) may have bifurcations that lead to hysteresis. A typical example of this situation for 1D case is shown in Fig. 4.28a, where curve g is defined by equation b(x,g) = 0, and it is assumed that b(x,g) < 0 and b(x,g) > 0 above and below this curve, respectively. It is apparent that the vector field b(x,g)

1 T ..!~ ~ X

~)

Z- / _2- /- / (b)

FIGURE 4.28


has one stable equilibrium for gt ~ (gl,g2) and two stable equilibria for gt E (gl,g2). If the initial value of input go > g2, then the evolution of the system for decreasing gt essentially follows the left branch MBA of the curve g. The state of the system is switched to point /9 on the right branch of g when gt reaches (from above) the value of gl, and the system follows the right branch (DN) towards N as the input is further decreased. When the variation of input is reversed, the state of the system moves upward along the right branch of g until the switch to the left branch occurs at point C. It is apparent that the evolution of this system results in rate- independent hysteretic relation between observation Yt and input gt.

Next, consider the case when the field b(x,g) has no bifurcations. The 1D example of this situation is shown in Fig. 4.28b, where the set of g of solutions of equation b ( x , g ) - 0 consists of three disjoint curves x = yo(g),x = yl(g) ,x = y2(g) with yl(g) < yo(g) < y2(g). It is assumed that points of x = }I1(g) and x = Y2(g) correspond to stable equilibriums, while points of Y0(g) are unstable equilibriums. It is apparent that, for sufficiently slow input variations, the system will essentially move only along one curve (Yl(g) or Y2(g)) and no hysteretic relation between Yt and gt can be observed. It will be demonstrated below that small random perturbations may result in observable deterministic hysteresis between Yt and gt although the original deterministic system has no bifurcations. Hysteresis appears due to bifurcations in the randomly perturbed system. First, the case of hysteresis induced by noise in systems with two asymptotically stable equilibriums will be discussed. Then, the case of many equilibriums will be treated.

Consider small white-noise perturbations of dynamical system (4.205)

8X~ -- b(X~, gt) + ~/-d r~ (X~) IfVt. (4.206)

Here, Wt is the vector Wiener process in R n, while cr (x) is a nondegen- erate square matrix of order n with bounded Lipshitz continuous entires, while e is a small parameter.

We next assume that the following relations for small parameters and e hold

e ,350 , l i m e S - 1 ] l n 3 l = c > 0 . (4.207)

By introducing the time r - } e ' = s and by fixing gt - g , the stochastic , differential equation (4.206) can be written as follows

(4.208)


Next, we shall introduce the (normalized) action functional defined as follows [26]: 1LT

Sg, T(Cp) • ~ [rr-l(cps)(~bs -- b(99s,g))l 2 ds, (4.209)

if ~0 ~ CO,T is absolutely continuous, and Sg0 T is infinite for all other functions from CO, T. The action functional is t~e main mathematical tool of the theory of large deviations. It is, roughly speaking, a measure of the "difficulty" for a sample of stochastic process )~'g to pass through a small neighborhood of function ~0t. This functional determines the "most probable" transition route between equilibrium points as well as the asymptotics of switching (transition) times between these equilibriums.

By using this functional, functions V12(g) and V21(g) can be introduced:

W12(g)- inf{Sgo, T(~0)" ~PO = Yl(g), ~PT = Y2(g), T > 0}, (4.210)

V21(g)=inf{sg,TOp)" r = y2(g), ~pT--YI(g),T>O}. (4.211)

When the vector field b(x,g) is potential: b(x,g) - -VxU(x,g) (which is always true for 1D case) and rr is the unit matrix, then the above functions can be expressed as the following potential differences: V12(g)= 2[U(yo(g),g)- U(yl(g),g)] and V21(g)- 2[U(yo(g),g)- U(y2(g),g)].

Finally, let us introduce the sets:

G12 = {g: V12(X) < c, W12(g) < W21 (g) }, (4.212)

G21 -- {g: W21 (g) < c, W21(g) < V12(X)}, (4.213)

where c is defined in (4.207). It is known from the theory of large deviations [26] that for the process

Xgr 'e' the transition times between equilibrium points ~'i(g) and }q(g) (i,7 {1,2}, and i ~ 7) have (in the sense of logarithmic equivalence) the expo-

1 8~ nential asymptotics exp{ F Vi7(g)} for $ 0. This means that the transition 8 times Ti7 for the process X~ 'g have the asymptotics 8 exp{ ~ Vi7(g)}. By tak-

ing into account relation (4.207), we find

} vi7(g) In �89 --81 T/7 ~ ~ exp Vi7 (g) --[- (4.214)

It follows from the last expression that Ti7 tends to infinity if V//(g) > c, and it tends to zero if Vi7(g ) < c. Thus, it can be concluded that if the input gt is monotonically increased and the state of the system moves upwards along the curve y2(g), it is switched to the curve )'l(g) as soon as the boundary of the set G21 is crossed. Similarly, if the variation of the input is


reversed and the system tracks the curve }'l(g), it is switched to the curve y2(g) as soon as the boundary of the set G12 is crossed. It is apparent that the above switchings result in a rate-independent hysteretic relation between observation Yt and input gt. It is also clear that the input switching ("coercive") values as well as the shape (width) of the corresponding hysteresis loop are determined by constant c in (4.207) and, consequently, by the interplay of fast system dynamics and the noise strength. It is worthwhile to remark that trajectory X~ of randomly perturbed system (4.206) is close to Yl(gt) or Y2(gt) in some integral norms that filter out "short-lived" excursions of stochastic process X~. Such integral norms naturally appear in functionals that relate observation Yt to the state X~ of the system due to some finite time of any observation.

Now, consider the case when for any fixed gt = g the dynamical system (4.205) has an arbitrary finite number s of asymptotically stable equilibriums Kl(g),K2(g),. . . ,K~(g)(g). Let us suppose that as the input is changed with time, these equilibriums trace disjoint curves }'l(g), Y2(g), �9 �9 y~(~)(g). It is apparent that, for sufficiently slow input variations, the deterministic system essentially moves along one of those curves and no hysteresis is observed. The situation may change when our dynamical system is randomly perturbed and its motion is governed by Eq. (4.206). In this case, the presence of noise may result in switching from one y-curve to another and, in this way, produce hysteretic behavior of the system. The immediate task is to describe the mathematical machinery that allows one to determine the switching values of input and to iden- tify the y-curves involved in switchings. This mathematical machinery is based on the notion of the action functional (4.209) that is used to define the set of functions

Vii(g)-- inf{Sg0,T(~a): ~0 ---- Yi(g), ~(T)= yj(g)}. (4.215)

It turns out that the time sequence (order) of noise induced switchings in randomly perturbed dynamical system (4.206) is in a way not random and completely governed by deterministic functions Vij(g). To fully describe the time sequence of noise induced switchings, it is necessary to introduce the hierarchy of cycles [26] defined by functions Vq(g) on the set of L of equilibrium points and the notion of/-graphs defined on L.

Each point i from L is considered as rank 0 cycle and the exit rate ei from this cycle is given by the formula

ei - min Vij. (4.216) j: j#i

The "next" state to i is defined as the state k = Jl(i), where the above minimum is reached. The transition i --~ k is the most probable (with probability close to one). It is assumed below that our dynamical system (4.205)


is generic in the sense that the above minimum as well as all similar minima considered in the subsequent discussion are achieved only at one equilibrium state. A cycle rank I is defined as an ordered subset of v equilibrium points from L with the property that for any point i of this subset we have J~(i)= i. It is clear that the ordering on rank 1 cycles is introduced by the transition operation Jl(i). To introduce such characteristics as the main state of a cycle, the stationary distribution rate on a cycle and the exit rate from a cycle, the concept of / -graphs [26] is very instrumental.

Consider an arbitrary subset A of L and some equilibrium point i ~ A. A set of directed edges (arrows) connecting some points j ~ A is called an /-graph qi if one edge starts from any point of this graph j E A (j # i), and from any point of this graph there is a directed path to i along the edges of the graph. The set of all /-graphs is denoted as Qi(A).

Now, the main state of rank I cycle C is defined as the state j* = M(C) where the following minimum is reached

X = rain min ~ Vmn. (4.217) j~C gEQj(C) (m-.n)~q

The stationary distribution rate mc(i) of the rank 1 cycle C is defined by the formula

mc(i) = min ~ Wmn - ,K. (4.218) qEQi(C) (m -+ n)eq

Finally, the exit rate ec from the cycle C is given by the expression

ec(i) = min [mc(i) 4- Vij]. (4.219) i~C,j~C

For any rank I cycle C(P) there is such i* C(P) and such j* belonging to the rank I cycle C (s) = J2(C (p)) that the above minimum is reached. In this sense, the cycle C (s) is the "next" to the cycle C (p), and the transition C(P) C (s) is the most probable (with probability close to one) among all possible transitions between C (p) and other rank I cycles. A cycle rank 2 is defined as an ordered subset of ( rank I cycles with the property that for any rank

1 cycle C(P) of this subset J2 ~ (C(P)) = C(p). It is clear that the ordering on

rank 2 cycles is introduced by the transition operator J2(C(P)). For rank 2 cycles, we can introduce main states, stationary distribution rates and exit rates by using formulas (4.217), (4.218), and (4.219), respectively. The only qualification is that C in the above mentioned formulas is a rank 2 cycle. By literally repeating the same line of reasoning, we can introduce cycles of higher ranks until we reach some rank at which there exists only one

4.6 MODELING OF TEMPERATURE DEPENDENT HYSTERESIS

FIGURE 4.29

279

cycle consisting of all equilibrium points. For this cycle the exit rate is defined as infinite. The described hierarchy of cycles is illustrated by Fig. 4.29 where four rank 1 cycles, two rank 2 cycles and one rank 3 cycle are depicted.

Now, the important notion of observable equilibrium states of dynamical systems (4.206) can be introduced. Consider an arbitrary value g of input and some equilibrium state i that exists for that input value. For this state, we can introduce the ordered sequence of cycles Ck of all possible ranks that describes the sequence of the most probable (with probability close to one) transitions initiated from i. The above sequence of cycles generates the sequence of exit rates ek given by formula (4.219). Let ej be the smallest exit rate from the above sequence such that

c < ej, (4.220)

where c is the constant from (4.207). Let j* be the main state of the cycle Cj. Then, by using asymptotics similar to (4.214), it can be shown that a very fast (theoretically instantaneous) transition from the equilibrium state i to the equilibrium state j* is induced by the noise. This transition occurs via very fast intermediate transitions between equilibrium states that form cycles Ck with k < j. In this sense, the equilibrium state i as well as the intermediate states participating in the transition are not observable, while the state j* is observable. It is clear that the observability of the state is determined by the value of constant c in (4.207) and, consequently, by the interplay of fast system dynamics and the noise strength. It is also clear that for any instant of time there is a finite number of observable equilibrium states Jn ( n - 1,2,...,s A particular observable state, occupied by the system at a given instant of time, is determined by the past


history of system dynamics. Suppose that the system (4.206) is in some observable state Jn and suppose (for the sake of being specific) that the input is monotonically decreased. If this input decrease does not violate inequality (4.220) and the condition that Jn is the main state of the cycle Cj, then the state of the system will move along the },-curve to which the

~

observable state In belongs. However, as soon as the input gt reaches the value at which the state In is no longer observable, the system is switched

~

to another observable state ]m belonging to another },-curve. The new ob- ~

servable state ]m is determined by using the same algorithm as above with the only difference that the ordered sequence of cycles Ck describes now

.~

the sequence of the most probably transitions initiated from In" It is this change in the initial state of transitions that leads to branching and hysteresis phenomena. Indeed, if after switching to the state ]m input is somewhat further decreased and then reversed, then for the monotonically increasing input the system (4.207) may not switch back to the observable

.~ ~ .~

state .In at the same input value as the switching from In to ]m occurred. This is because the equilibrium state ]m may loose its "observable" status at a different input value than it happened for the state In" It is apparent from the above discussion that inequality (4.220) subdivides all equilibrium states (attractors) in two distinct groups: equilibrium states that participate in fast switchings that lead to deterministic hysteresis and equilibrium states that form cycles Ck with k > j. These equilibrium states participate in slow switchings that are usually known as thermal relaxations.

In the previous discussion, the emphasis has been made on the noise induced hysteresis in dynamical systems that under purely deterministic conditions do not exhibit hysteresis. However, it is clear from the presented reasoning that in dynamical systems, which under purely deterministic conditions do exhibit hysteresis, the presence of noise may substantially alter the structure of hysteresis. For instance, in 1D dynamical systems (4.205), which exhibit hysteresis due to bifurcations between two existing asymptotically stable equilibrium states, the presence of noise may appreciably alter switching input values and the shape of hysteresis loop. In dynamical systems (4.205), which exhibit hysteresis due to bifurcations between many existing asymptotically stable equilibrium states, the presence of noise may make only some of those states actually observable and, in this way, appreciably change the structure of hysteresis. It is apparent from the presented discussion that the effect of small noise on hysteresis in dynamical systems is very pronounced for fast systems. Since fast systems are typical for rate-independent hysteresis and since some small noise is always inherently present in physical systems, the study of rate-independent hysteresis in fast dynamical systems without noise may be inadequate.

4.7 FUNCTIONAL (PATH) INTEGRATION MODELS OF HYSTERESIS 281

The presented discussion has been based on relation (4.207) between small parameters 8 and E that characterize the fast dynamics and the noise strength, respectively. The question is to what extent this relation is natural. By using results from [29], it can be demonstrated that for other relations between 8 and e deterministic hysteresis either does not exist (strong noise) or it is insensitive to noise (very weak noise). Under the condition (4.207), hysteresis still exists in a deterministic sense and it is quite sensitive to noise. This is consistent with many experimental facts which demonstrate that deterministic hysteresis is temperature sensitive. In this sense, the condition (4.207) can be regarded as natural for fast dynamical systems employed to describe noise sensitive deterministic hysteresis.

Finally, it is important to note that usually observation Yt is a scalar or a vector whose dimension is much smaller than the dimension of the state vector Xt. For this reason, the observation (Yt) vs input (gt) relation gives a "reduced" description of the dynamical system. This reduced description may often lead to the situation where many different state vectors Xt result in the same observation Yt. This, in turn, may be the origin of nonlocal memory of hysteretic relation between Yt and gt.

4.7 F U N C T I O N A L ( P A T H ) I N T E G R A T I O N M O D E L S

OF HYSTERESIS

The Preisach model is designed as a continuous superposition of the simplest rectangular loop operators }9~. These operators can be construed as elementary building blocks of the Preisach model. A natural way to generalize the Preisach model is to consider more sophisticated elementary hysteresis operators and to design hysteresis models as continuous superpositions of such elementary operators. In this section we pursue this approach and consider functional (path) integration models of hysteresis that are designed as superpositions of elementary hysteresis operators generated by continuous functions. A physical interpretation of the path integration models as well as their various connections with the classical Preisach model are presented. The discussion in this section follows (to a certain extent) the paper [30].

Consider a continuous function g(x) on some closed interval [x_,x+] that satisfies the condition

u_ ~ g(x_) ~ g(x) ~ g(x+) - u+. (4.221)

Such a function will be called a generating function, while x_ and x+ can be termed as lower and upper saturation values, respectively. An ele-

282

U ,

U0


k

g(x) _ .

i

i

i

X - x o x +

X

F I G U R E 4 .30

mentary hysteresis operator 9gu(t) can be associated with each generating function by traversing its upper or lower envelopes (see Fig. 4.30). This can be done as follows. Suppose that at time to the input u(t) assumes some extremum value u0 and

uo - g(xo). (4.222)

If u0 is some min imum value, then for the subsequent monotonic increase of input the upper envelope

gu + (x) = maxg(x) (4.223) [x0,x]

is traversed. On the other hand, if u0 is some maximum value, then for the subsequent monotonic decrease of the input the lower envelope

guo (x) - rrfn g(x) (4.224) [x,x0]

is traversed. This means that for monotonic input variations the elementary hysteresis operator ~,gu(t) is defined as follows:

x+(t) if u(t) is monotonically increased, ~,gu(t) - (4.225)

x- ( t ) if u(t) is monotonically decreased.

Here x+(t) and x- ( t ) are the solutions of the following equations, respectively:

g+0 ( x+ (t)) - u(t), (4.226)

guo (x-( t)) = u(t). (4.227)

Since upper g+0 (x) and lower gu0 (x) envelopes usually have "horizontal" parts parallel to x-axis, solutions of Eqs. (4.226) and (4.227) may not be


u 1

Ut

U2

U j j ...... x

X= X 2 X! X 1

FIGURE 4.31

unique for some values of u(t). This difficulty can be removed by using minimal and maximal solutions of Eqs. (4.226) and (4.227), respectively:

x+(t) - min{x: gu+(X)- u(t)}, (4.228)

x-(t) = max{x: guo(X)- u(t)}. (4.229)

The elementary operator ~,gu(t) has been so far defined for monotonic input variations. This definition can be extended to the case of piece-wise monotonic inputs by consecutively applying the definition (4.225) for each time interval of monotonic variation of u(t). The ambiguity of choosing x0 in Eq. (4.222) can be removed if it is agreed that the evolution is started from the state x_ (or x+) of negative (or positive) saturation. The given definition of elementary hysteresis operator ~,gu(t) is illustrated by Fig. 4.31. It is clear that elementary operator ~gU(t) is rate-independent. This is because the output value x(t) depends only on the current value of input u(t) and the past history of input variations but does not depend on the rate of input variations. It is also clear that the operator ~,gU(t) has local memory. This is because the simultaneous specification of output and input uniquely defines the state of elementary hysteretic nonlinearity 9gu(t). Finally, it is clear that the operator ~,gu(t) exhibits "wiping-out" property (see Section 2 of Chapter 1). In a way, the wiping out property can be regarded as a consequence of local memory. It is important to note that not all parts of g(x) are accessible. For instance, part A shown in Fig. 4.31 is not accessible. This part of g(x) will not be traversed for any input variations. In this sense, the same elementary hysteresis operator ~,gU(t) is defined on the equivalence class of functions g(x) with the same accessible parts.


Now, consider some set G of generating functions g(x) and some measure #(g) defined on this set. Then, the functional (path) integration model of hysteresis can be formally defined as follows:

f i t ) - / 6 ~,gu(t) dl~(g). (4.230)

The above model is quite general. Its structure depends on the measure /~(g) introduced on the set G. Next, we demonstrate that the classical Preisach model is a particular case of the path integration model (4.230). To this end, consider the subset Gp of G that consists of functions with two vertical parts (x_ = -1 ,u K ~) and (x+ = 1,u ~/~) separated by inacces- sible parts X(x) with/~ < X(x) < c~ (see Fig. 4.32). It is clear that for such functions

~,gu(t) = G~u(t) for g(x) ~ A~, (4.231)

where A ~ is the equivalence class of functions from Gp that have the same values of ~ and/~. Now, consider the measure/~(g) that is concentrated on the subset Gp. Since Gp is the union of nonintersecting equivalence classes A~ , the model (4.230) can be written as follows:

f (t) - /~>,~ (/A~ ~,gu(t) d#~(g)) dot d~, (4.232)

where #~ (g ) are the measures on the equivalence classes A ~ induced by the measure #(g) on G,.

By using (4.231) in (4.232), we obtain

/a~ ~,gu(t) dl~(g) -- ( /a~ d l~(g)) ~,~u(t) = l~(ol, ~)~,~u(t), (4.233)

where the following notation is introduced

~(c~, fl) = f d#~(g). (4.234) d A

By substituting (4.233) into (4.234), we end up with the classical Preisach model

f (t) - 11~>~/~(~' ~)G~u(t) dot d~. (4.235)

In the case when vertical parts in Fig. 4.32 are replaced by curved paths (see Fig. 4.33), the corresponding elementary hysteresis operator ~,gu(t) can be represented as follows:

v+(ut) - v-(ut) ,, v+(ut) if- v-(ut) x(t)- 2 y~flut + 2 ' (4.236)

where v+(u) and v-(u) are inverse of g+(x) and g-(x), respectively.


-1

"2

u

g+(x) . . . . . . . I P , - - -

<

"~X

F I G U R E 4.32 F I G U R E 4.33

By choosing measure/~(g) concentrated on the set of functions shown in Fig. 4.33 and literally repeating the same line of reasoning as before, it can be demonstrated that the path integration model (4.230) is reduced to the Preisach model with the input dependent measure (see Section 2, Chapter 2).

It has been demonstrated above that the path integration model (4.230) is reduced to the classical Preisach model if the measure/~(g) is concentrated on functions g(x) such that ~gu(t) - G~u(t). Below, it will be shown that this reduction is also possible in the cases when the measure/~(g) is concentrated on functions g(x) such that ~,gu(t) =/= G~u(t). This will further emphasize the generality of the classical Preisach model.

In general, it is not immediately obvious how to generate measure on the functional set G and how to carry out functional integration in (4.230), in other words, how to compute output fit). It turns out that the above difficulties can be appreciably circumvented if the set G is interpreted as a set of samples of a stochastic diffusion process, that is the Markovian process with continuous samples generated by the Ito stochastic differential equation

dgx = b(gx, X) dx 4- r~(gx, X) dWx. (4.237)

Here x must be construed as fictitious "stochastic" time that must not be confused with real physical time t.

In the above case, the measure/~(g) is the stochastic measure that, in principle, can be generated by using the transition probability density function for the process defined by Eq. (4.237). This is the consequence of the Markovian nature of the process gx. However, there is no need of doing this because, as it will be demonstrated below, the output f i t) can


11.1

U t

u_

U

X

X_ X t X 1

U l

1.1. t

u_

X X t X 1

FIGURE 4.34

be interpreted as an average level-crossing (stochastic) time. As a result, the mathematical machinery developed for the solution of level-crossing (exit) problems can be extensively used for the output calculations of functional integration type models (4.230).

First, we shall discuss the meaning of ~gu(t) when the generating function g(x) is a sample of diffusion stochastic process. This meaning is different for monotonically increasing and monotonically decreasing inputs. To illustrate this, let us consider Figs. 4.34 where a particular sample g(x) of diffusion stochastic process that starts from the negative saturation value u_ is shown. It is clear from this figure and the definition of ~,gu(t) that for monotonically increasing input u(t) the elementary operator ~,gu(t) has the meaning of the first level-crossing time, where the level is equal to the current value of input u(t) This time is a random variable and, consequently, the output of the functional integration model (4.230)


is equal to the average value ofthefirst level-crossing time. This is true for any value of input until u(t) reaches some maximum value Ul. It is clear from Figs. 4.34 and the definition of ~,gu(t) that for the subsequent monotonic decrease of input u(t) the elementary operator ~,gu(t) has the meaning of the last level-crossing time, where the level is equal to the current input value u(t). More precisely, this is the last time of crossing the level u(t) before the first time of crossing the level Ul. This last level-crossing time is also a random variable. Thus, for monotonically decreasing input, the output of the functional integration model (4.230) is equal to the average value of the last level-crossing time. It is this difference in the meaning of output values of the path integration model (4.230) for monotonically increasing and decreasing inputs u(t) that results in hysteresis.

Next, we shall present mathematical formalism that supports the statements outlined above. Let T(o~,x~lfl, x~) be the notation for the probability density of the first crossing time x~ of the level c~ under the condition that the sample of stochastic process crossed the level fl < c~ at the time x~. It is clear that

fx ~ T(ot, x,~lfi, x~) dx~ = 1. (4.238)

It is apparent that the probability densities p~(x~) of x~ and p~(x~) of x~ are related by the expression

f0 x~ po,(xo,) - T(c~,xo, lfl, x~)p~(x~)dx~. (4.239)

Consider the set G of all samples of the diffusion process gx that satisfy the condition (4.221). Let us analyze mathematically how the output of the path integration model (4.230) changes when the input ut is increased from u_ to some maximum value Ul and then is decreased to some minimum value u2. During the monotonic input increase, the elementary operator Xt -- ygUt has the meaning of the first time of crossing the level Ut.

By using the notations p~(x~)= pu_(X-) and p~(x~)= Put(Xt), according to (4.239) we find

fo Xt Put(Xt) -- T(ut, xt[u_,x_)pu_(x_)dx_. (4.240)

The probability density function pu_ (x_) must be chosen as a part of the characterization of the initial state of lower saturation. For instance, it can be chosen as 6 ( x - x_). After that, Eq. (4.240) permits one to compute the unknown probability density Put(Xt) provided that the first time level-crossing problem has been preliminary solved and the function T(ut, xtlu_,x_) has been found. By using the probability density Put(Xt),


the output value of the path integration model (4.230) can be computed as follows:

f ( u t ) - xtput(xt)dxt . (4:.241)

Formulas (4.240) and (4.241) can be used to compute Put(Xt) and f (u t ) for all values of ut between u_ and Ul. In this way, the ascending branch of the major loop can be computed. For ut = Ul, we have

fo xl Pul(Xl) -- r(ul,XllU_,x_)pu_(x_)dx_. (4.242)

Next, consider the monotonic decrease of input from /-/1 to u2. For this input variation, the elementary operator xt = ~'gut has the meaning of the last time of crossing the level ut before the level Ul is reached for the first time. This means that probability density Put(Xt) of the last level-crossing time xt satisfies the integral equation

f0 xl pul(Xl) -- T(Ul,XllUt, xt)Put(xt)dxt. (4.243)

This integral equation can be (in principle) solved for any value of ut between Ul and u2. In this way, Put(Xt) can be found and used in formula (4.241) for the calculation of the output value f (u t ) along the descending branch attached to the previous ascending branch at the point ut = Ul. After Pu2 (X2) is found by solving integral equation

fo xl Pul (Xl) -- T(Ul, XllU2,x2)Pu2(X2) dx2, (4.244)

it can be used in the formulas

~0 x t Put(Xt) = T(ut, xtlu2,x2)Pu2(x2)dx2, (4.245)

fo X3 Pu3(X3) = Z(u3,x31u2,x2)Pu2(x2)dx2 (4.246)

for the computations of Put(Xt) and Pu3(X3) for the third hysteresis branch when input u(t) is monotonically increased from u2 to u3. Similarly, Pu3 (X3) can be used in the integral equations

~0 x3 Pu3(X3) -- T(u3,x31ut, xt)Put(xt)dxt, (4.247)

~0 x3 Pu3(X3) -- T(u3,x31u4,x4)Pu4(x4)dx4 (4.248)

for the computations of Put(Xt) and Pu4 (X4) for the fourth branch of hysteresis loop when input u(t) is monotonically decreased from u3 to u4.


It is clear that the computations described above can be recursively used to find any branch of hysteresis described by the path integration model (4.230) with stochastic measure.

It is important to note that in the case when u3 = ul the probability densities Pu3(X3) and Pul (xl) coincide. This directly follows from the coincidence of Eqs. (4.244) and (4.246) for the above case. The coincidence of Pu3(X3) with Pul(Xl) for u3 = Ul implies the validity of the "wiping-out" property for the path integration model. This fact can also be deduced from the validity of "wiping-out" property for each elementary hysteresis operator ~,gu(t).

Now, consider a particular case when the stochastic process gx is homogeneous (translationally invariant) with respect to "stochastic time" x. Such a process is described by the Ito stochastic differential equation

dgx = b(gx) dx + ~(gx) dWx. (4.249)

Due to the translational invariance, the conditional first time level- crossing probability density T(c~, x~ Ifl, x~) has the property

T(ol, x~lfl,x~) = T(o~,x~ - x~lfl, O). (4.250)

Next, we shall use this property to compute the expressions for ascending f+(ut) and descending f - (u t ) branches of hysteresis loops formed when the input u(t) is monotonically increased from some minimum value U2k to some maximum value U2k+l and then is monotonically decreased back to U2k. For the calculation off+(ut) , formulas similar to (4.241) and (4.245) are appropriate. This leads to the expression

/o f+ (Ut) -" xtP+ut (Xt) dxt

~cxD l~oXt ) -- xt T(ut, xt - X2k[U2k, O)Pu2k(X2k)dX2k dxt. (4.251)

By using Fig. 4.35, the above double integral can be transformed as follows:

f + (ut) = xtZ(ut, xt - X2klU2k, O) dxt Pu2k (X2k) dX2k 2k

) = ( X t - X2k)Z(ut, x t - X2klU2k, O)dxt Pu2k(X2k)dX2k 2k

) if- T(ut, x t - X2klU2k, O)dxt X2kPu2k(X2k)dX2k. (4.252) 2k


Z -- Xt -- X2k, (4.253)


X t

X2k

X2k

FIGURE 4.35

from (4.252) we find

) f+ (Ut) = T(ut, zlU2k, O) dz X2kPu2k (X2k) dx2k

) -t- zT(ut, ZlU2k, O) dz Pu2k (X2k) dX2k. (4.254)

By using formulas (4.238) and (4.241) in the first integral and the normalization condition

fO cx~ Pu2k -" (4.255) (X2k) dx2k 1

in the second integral, from (4.254) we find

f+(ut)--f(u2k) + F(ut, uak), where

~0 ~176 .T'(Ut, U2k) -- zT(ut, zlu2k, O) dz.

(4.256)

It is clear from (4.256) that the current value of output on the ascending branch is determined only by the current value of input ut and the last minimum value U2k. We shall next establish a similar result for the descending branch f - (u t ) of the hysteresis loop. According to (4.241) and (4.247) we have

f(U2k+l)

/o - /o ~

= fo ~

X2k+lPu2k+l (X2k+l) dx2k+l

X2k+l (~0 X2k+l

X2k+l (~0 X2k+l

T(U2k+I,X2k+I lUt, Xt)Pu t (Xt) dxt) dx2k+ 1

Z(u2k+l,X2k+l -- xtlut, O)pu t (xt) dxt) dx2k+l. (4.258)

(4.257)

4.7 F U N C T I O N A L (PATH) I N T E G R A T I O N M O D E L S OF HYSTERESIS

X t

x 2k+ 1 = x t / / 7 " //

/ / / /

/

.~ X2k+l

F I G U R E 4.36

291

By using Fig. 4.36, the last double integral can be transformed as follows:

f(U2k+l)

/0 t/xt \

X2k§ Z(Uak+l,X2k+l -- xt]ut, O) dx2k+l)p ~ (xt) dxt. /

(4.259)

which leads to

f - (Ut) = d ( U 2 k + l ) - .~(U2k+l, ut). (4.263)

Formulas (4.256) and (4.263) show that the shapes of generic ascending and descending branches of a minor hysteresis loop are the same regardless of the past input history. The past history is reflected in the values of f(u2k) andf(u2k+l). In other words, the minor hysteresis loops corresponding to different past histories are congruent. Since the wiping-out property and the congruency of minor loops (formed for the same back- and-forth


Z = X2k+l -- Xt, (4.260)

from formula (4.259) we derive

f (U2k+l ) -- zT(U2k+l,Zlut, O)dz Put(xt)dxt

-Jr- T(u2k+l,Zlut, O)dz XtPut(xt)dxt. (4.261)

Now, by using the same reasoning that was used to simplify formula (4.254), we obtain

f(Uak+l) = .~'(U2k+l, Ut) + f-(Ut), (4 .262)


input variations) represent the necessary and sufficient conditions for the description of hysteresis by the Preisach model, we conclude that the path integration model (4.230) with the stochastic measure corresponding to the homogeneous diffusion process is equivalent to the Preisach model.

It is clear from the previous discussion that the output calculations for the path integration model can be performed if the function T(o~,x~l~,x~) is known. This function can be computed by solving the exit problem for the stochastic process defined by Eq. (4.237). This, in turn, requires the solution of initial-boundary value problems for the backward Kolmogorov equation similar to those discussed in Section 3 of this chapter. In particular, by using the mathematical machinery of the exit problem, the closed form expressions can be derived for the weight function/z(~,/~) of the Preisach model which is equivalent to the path integration model (4.230) with the stochastic measure generated by the process (4.249). Below, we present the final results; the mathematical details of the derivation can be found in the paper [30]. Consider the function

~ ( g ) - e x p - 2 rig' o.2(g,) '

(4.264)

then the function ~(~,/~) from (4.257) and #(c~,/~) can be computed as follows:

where

2 ag f'(Ot, fl)-- G ffl (fflg~(gt)dg')(fgOl ~(gt)dgt)o.2(g)~(g ) , (4.265)

#(or, fl) - K2 ~ U(c~, fl), (4.266)

ffl o/

K~ - 7r(g) dg. (4.267)

As examples, consider the following cases:

(a) gx is the Wiener process (b = 0,r~ = 1). Then ~ ( g ) = 1 ,K~ = ~ - /~ and

1 /z(0t,/J) -- ~, f'(0t, fl) - ~(ot - fl)2. (4.268)

Thus, the Preisach weight function is simply a constant and all hysteresis branches are parabolic (see Fig. 4.37).

1 0), (b) gx is the diffusion process with constant drift [ b ( g ) - ~ (X > a = 1]. In this case we have

~(g) --e -g/X, (4.269)


K~3 = e -~/x _ e-~/x, (4.270)

~-3 coth(~2-~) - 1 2z /z(oe, fl) = s inh2 , ,_ a, [ ~x-~) , (4.271)

~ ~ ) 1 ] oe_ fl)coth( 2x ~-(oe,3) =4[( 2x (4.272)

Typical hysteresis branches computed for this case are shown in Fig. 4.38.

(c) gx is the Omste in-Uhlenbeck process (b(g) = _ Z a = 1) In this X /

case, ~(g) - e g2/x, (4.273)

and/z(c~, fl) and 3c(cz, fl) are obtained by inserting the last expression in the formulas (4.265)-(4.267).

The described functional (path) integration model (4.230) admits the following physical interpretation. It is known that hysteresis is due to the existence of multiple metastable states in the system free energy F(X) (the temperature dependence is tacitly understood), which means that the system may be t rapped in individual metastable states for long times.

f 0

~

-4

-8

,,, - - . .

/

)

!

-4 -2 0 2 4

U

FIGURE 4.37

f 0

-4

8 ~

-8 -4 -2 0 2 4

CHAPTER 4 Stochastic Aspects of Hysteresis 294

U

FIGURE 4.38

Consider a simple case where the state variable X is a scalar quantity and the relevant free energy in the presence of the external magnetic field H is ~ ( X ; H ) = F(X) - HX. The metastable states available to the system are represented by ~-minima with respect to X for which 3~ /3X = 0, 3~ /3X 2 > 0. When H is changed with time, the number and the properties of these minima are modified by the variation of the term -HX. The consequence is that previously stable states are made unstable by the field action and the system moves to other metastable states through a sequence of (Barkhausen) jumps. Because the condition 3~/OX = 0 is equivalent to H = OF~ OX, one can analyze the problem by using the field representation shown in Fig. 4.39. The response of the system, expressed in terms of H(X), is obtained by traversing the upper and lower envelopes of OF~ OX for increasing and decreasing H, respectively. From the physical viewpoint, this construction amounts to assuming that the system, once made unstable by the action of the external field, jumps to the nearest available energy minimum, which means that one excludes dynamic effects that could aid the system to reach more distant minima. It is clear from the above description that OF/OX and H are similar to the generating functional g(x) and input u(t), respectively, within the framework of the function (path) integration model (4.230). The functional integra-


X

. . . . . I

FIGURE 4.39

tion model itself can be interpreted as the average hysteresis response of a statistical ensemble of independent (elementary) systems evolving in random free energy landscape. This interpretation can be of importance in applications where randomness due to structural disorders plays a key role in the appearance of hysteretic effects. A particularly important example is the motion of magnetic domain walls in ferromagnets, where various forms of structural disorder (point defects, dislocations, gain boundaries, etc.) are responsible for the random character of OF~ 3X. There are classical papers in the literature [31, 32] where the domain wall picture has been applied to the prediction of coercivity and magnetization curve shapes, starting from some assumption about the properties of F(X). Equations (4.264)-(4.267) provide a general solution for the case where the process OF/OX is Markovian, continuous, and homogeneous. In particular, the proven equivalence of Markovian disorder to the Preisach model gives a sound statistical interpretation of the latter.

References 1. Charap, S. H. (1988). J. Appl. Phys. 63: 2054.

2. Street, R. and Wooley, J. C. (1949). Proc. Phys. Soc. A 62: 562.


3. Mayergoyz, I. D. (1991). Mathematical Models of Hysteresis, Berlin: Springer- Verlag.

4. Mayergoyz, I. D. and Korman, C. E. (1991). J. Appl. Phys. 69: 2128-2134.

5. Mayergoyz, I. D. and Korman, C. E. (1991). IEEE Trans. Mag. 27: 4766-4768.

6. Mayergoyz, I. D. and Korman, C. E. (1994). IEEE Trans. Mag. 30: 4368-4370.

7. Tobin, V. M., Shultz, S., Chan, C. H. and Oseroff, S. B. (1984). IEEE Trans. Mag. 24: 2880-2882.

8. Barker, J. A., Schreiber, D. E., Huth, B. G. and Everett, D. H. (1985). Proc. R. Soc. London A 386: 251.

9. Friedman, G. and Mayergoyz, I. D. (1992). IEEE Trans. Mag. 28: 2262-2264.

10. Korman, C. E. and Rugkwamsook, P. (1997). IEEE Trans. Mag. 33: 4176-4178.

11. Mayergoyz, I. D., Adly, A. A., Korman, C., Huang, M. W. and Krafft, C. (1999). J. Appl. Phys. 85(8): 4358-4360.

12. Goldenfeld, N. (1992). Lectures on Phase Transmitions and the Renormalization Group, Reading, MA: Addison-Wesley.

13. Mayergoyz, I. D., Serpico, C., Krafft, C. and Tse, C. (2000). J. Appl. Phys. 87: 6824-6826.

14. Mayergoyz, I. D., Tse, C., Krafft, C. and Gomez, R. D. (2001). J. Appl. Phys. 89: 6991-6993.

15. Tse, C., Mircea, D. I., Mayergoyz, I. D., Andrei, P. and Krafft, C. (2002). J. Appl. Phys. 91: 8846--8848.

16. Mayergoyz, I. D. and Korman, C. E. (1994). J. Appl. Phys. 75: 5478--5480.

17. Gardiner, C. W. (1983). Handbook of Stochastic Methods, Berlin: Springer-Verlag.

18. Korman, C. E. and Mayergoyz, I. D. (1996). IEEE Trans. Mag. 32: 4204-4209.

19. Korman, C. E. and Mayergoyz, I. D. (1997). Physica B (Condensed Matter) 233: 381-389.

20. Feidlin, M. I. and Mayergoyz, I. D. (2000). Physica B (Condensed Matter) 87: 5511-5513.

21. Cramer, H. and Leadbetter, M. (1967). Stationary and Related Stochastic Processes, New York: Wiley.

22. Freidlin, M. I. and Wentzell, A. D. (1993). Ann. Prob. 24: 2215.

23. Freidlin, M. I. (1996). Markov Processes and Differential Equations: Asymptotic Problems, Berlin: Birkh~iuser-Berlin.

24. Freidlin, M. I., Mayergoyz, I. D. and Pfeiffer, R. (2000). Physical Review E 62: 1850-1855.

25. Gammaitoni, L., H~inggi, P., Jung, P. and Marchesone, F. (1998). Review of Mod- ern Physics 70: 223.

26. Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, Berlin: Springer.


27. Freidlin, M. I. and Mayergoyz, I. D. (2001). Physica B (Condensed Matter) 306: 15-20.

28. Freidlin, M. I. and Mayergoyz, I. D. (2002). J. Appl. Phys. 91: 7640-7642.

29. Freidlin, M. I. (2001). Stochastic and Dynamics 1: 261-281.

30. Bertotti, G., Mayergoyz, I. D., Basso, Y. and Magni, A. (1999). Physical Review E 60: 1428-1440.

31. Ne61, L. (1942). Cah. Phys. 12: 1.

32. Krommuller, H. (1997). Magnetic Hysteresis in Novel Magnetic Materials, Dor- drecht: Kluwer, 85.

CHAPTER 5

Superconducting Hysteresis

5.1 S U P E R C O N D U C T O R S WITH SHARP RESISTIVE T R A N S I T I O N S

It is well known that high field (hard) type-II superconductors are actually not ideal conductors of electric current. It is also known that these superconductors exhibit magnetic hysteresis. Finite resistivity and magnetic hysteresis in these superconductors appear because the motion of flux filaments is pinned by defects such as voids, normal inclusions, dislocations, grain boundaries, and compositional variations. This pinning results in the multiplicity of metastable states, which manifest themselves in hysteresis. When the flux filaments depin by thermal activation or because a current density exceeds some critical value, their motion induces an electric field. As a result, superconductors exhibit "current-voltage" laws E(J), which are strongly nonlinear. Thus, the very phenomenon (pinning) that makes type-II superconductors useful in practical applications is also responsible for their magnetic hysteresis and nonzero resistivity.

From the point of view of phenomenological electrodynamics, type-II superconductors can be treated as electrically nonlinear conductors, and the process of electromagnetic field penetration in such superconductors is the process of nonlinear diffusion. Analysis of nonlinear diffusion in type-II superconductors is of practical and theoretical importance because it can be useful for the evaluation of magnetic hysteresis in these superconductors as well as for the study of creep phenomena.

We begin with the case of a sharp (ideal) resistive transition shown in Fig. 5.1. This transition implies that persistent currents up to a critical current density Jc are always induced in superconductors. We consider nonlinear diffusion of linearly polarized electromagnetic fields in a lamination (slab) of thickness A. At first, it may seem natural to use the scalar nonlinear diffusion equation

32E OJ(E) az 2 --/~0 0-----7--' J(E) = Jc signE, (5.1)

299

300 CHAPTER 5 Superconducting Hysteresis

c

-Jc

"E

FIGURE 5.1

in order to perform the analysis. However, since the magnetic field at the slab boundary is usually specified, a simpler way to solve the problem at hand is to base our analysis on the equation

curl H = J, (5.2)

which in our one-dimensional case can be written as dH dz -- - l c . (5.3)

Since the critical current density Jc is constant, the last equation implies linear profiles of the magnetic field within the slab.

The sharp (ideal) resistive transition (see Fig. 5.1) along with formula (5.3) form the basis for the critical state model for magnetic hysteresis of type-II superconductors. This model was first proposed by C. P. Bean [1, 2], (see also [3]) and then it was further generalized in [4] to take into account the dependence of critical current density on the magnetic field. The critical state type models have been tested experimentally and have proved to be fairly accurate for simple specimen geometries (plane slabs, circular cross-section cylinders). It has also been realized that the critical state type models have some intrinsic limitations. First, these models do not take into account actual gradual resistive transitions in type-II superconductors. Second, even under the assumption of ideal resistive transitions, these models lead to explicit analytical results only for very simple specimen geometries.

Next, we shall briefly describe some basic facts concerning the critical state (Bean) model for superconducting hysteresis. Then, we shall demonstrate that the critical state type models are particular cases of the Preisach model of hysteresis. By using this fact, we shall try to make the case for the Preisach model as an efficient tool for the description of superconducting hysteresis.

Consider a plane superconducting slab subject to an external time- varying magnetic field Ho(t). We will be interested in the B vs H0 relation.

5.1 SUPERCONDUCTORS WITH RESISTIVE TRANSITIONS 301

Here, B is an average magnetic flux density that is defined as

A

B = I~oA f ~a H(z) dz, (5.4) 2

and H(z) is the magnetic field within the slab. In practice, B and H0 are quantities that are experimentally measured

and it is their relation that exhibits hysteresis. It follows from formula (5.4) that in order to compute B for any H0, we

have to find a magnetic field profile (magnetic field distribution) within the superconducting slab. This is exactly what we shall do next.

Suppose that no magnetic field was present prior to the instant of time to. It is assumed that for times t > to, the external magnetic field Ho(t) is monotonically increased until it reaches some maximum value Hm. The monotonic increase in the external magnetic field induces persisting electric currents of density Jc. According to formula (5.3), this results in the formation of linear profiles of the magnetic field shown in Fig. 5.2. The corresponding distribution of persisting electric currents is shown in Fig. 5.3. It is easy to see that the instantaneous depth of penetration of the magnetic field is given by

Ho(t) z o ( t ) - (5.5)

lc

It is also clear that

zS zo(t) ~< ~-, (5.6)

_A 2

H~

..Ho(t); z I

, i \ A !~ ~_, 2

Zo(t)

J

- \ ' I

Zo(t) A

Zo(t)

l -~ Z



H

A /---k /------k

/ k\

A

Z

F I G U R E 5.4

if

Jc2A = H*" Ho(t) ~ (5.7)

By using Fig. 5.2 and formulas (5.4), (5.5), and (5.7), we find the average value of the magnetic flux density:

I~oHo(t)zo(t) /~0(H0(t)) 2 B(t) = A = 2H* " (5.8)

Suppose now that after achieving the maximum value, Hm, the external magnetic field is monotonically decreased to zero. As soon as the maximum value Hm is achieved, the motion of the previous linear profile is terminated and a new moving linear profile of magnetic field is formed. Due to the previously induced persisting currents, the previous profile stays still and is partially wiped out by the motion of the new profile. The distribution of the magnetic field within the slab at the instant of time when the external magnetic field is reduced to zero is shown in Fig. 5.4. This figure shows that there is nonzero (positive) average magnetic flux density, which is given by

;'o H2 - > 0. ( 5 . 9 )

4H* This clearly suggests that the B vs H0 relation exhibits hysteresis. We next demonstrate the validity of this statement by computing the hysteresis loop for the case of back-and-forth variation of the external magnetic field between -Hm and +Hm. For the sake of simplicity of our computations, we shall assume that

Hm ~ H*. (5.10)

5.1 SUPERCONDUCTORS WITH RESISTIVE TRANSITIONS

~ \ \ \ \

I

H

5

Ho(t) ~ Z

303

FIGURE 5.5

We first consider the half-period when the external magnetic field is monotonically decreased. A typical magnetic field distribution for this half-period is shown in Fig. 5.5. For the penetration depths z0 and 8, shown in this figure, we have

Hm Hm - Ho (t) 8 - J c ' z0(t) = 2Jc " (5.11)

By using Fig. 5.5 and formula (5.11), we find the increment AB of the average magnetic flux density:

2/z0 ( H m - Ho)zo(t) ( H m - H0) 2 AB = A " 2 = #0 4H* " (5.12)

This leads to the following expression for the average magnetic flux density on the descending branch of the hysteresis loop:

B = B m - A B - tz~ p ,o(Hm- H0) 2 2H* 4H* " (5.13)

Consider now the half-period during which the external magnetic field is monotonically increased from - H m to +Hm. A typical magnetic field distribution for this half-period is shown in Fig. 5.6. By using this figure, as before we find

AB = I~o(Hm + H0) 2 4H* ' (5.14)

304

H

CHAPTER 5 Superconducting Hysteresis

-H Zo(t ) \ \ \ \ \

--1 Ho(t) = Z

FIGURE 5.6

and

B = -Bm if- AB = tz~ Izo(Hm q-H0) 2 (5.15) 2H* ~- 4H* "

The expressions (5.13) and (5.15) can be combined into one formula:

H (Hm T HO) 2 ] B = +/.to 2H* 4H* ' (5.16)

where the upper signs correspond to the descending branch of the loop, while the lower signs correspond to the ascending branch.

On the basis of the previous discussion, the essence of the Bean model can now be summarized as follows. Each reversal of the magnetic field Ho(t) at the boundary of the superconducting slab results in the formation of a linear profile of the magnetic field. This profile extends inward into the superconductor until another reversal value of the magnetic field at the boundary is reached. At this point, the motion of the previous profile is terminated and a new moving linear profile is formed. Due to the previously induced persisting currents, the previous linear profiles stay still and they represent past history, which leaves its mark upon future values of average magnetic flux density. These persisting linear profiles of the magnetic field may be partially or completely wiped out by new moving profiles.

Next, we shall establish the connection between the critical state (Bean) model for superconducting hysteresis and the Preisach model [5]. To do this, we shall establish that the wiping-out property and congru-


ency property hold for the Bean model. Indeed, a moving linear profile of the magnetic field will wipe out those persisting linear profiles if they correspond to the previous extremum values of H0(t), which are exceeded by a new extremum value. In this way, the effect of those previous extremum values of Ho(t) on the future average values of magnetic flux density B will be completely eliminated. This means that the wiping-out property holds. It can also be shown that the congruency property of minor loops corresponding to the same reversal values of Ho(t) holds as well. In-

deed, consider two variations of external magnetic field H~ 1) (t) and H~ 2) (t). Suppose that these external fields may have different past histories, but starting from some instant of time to they vary back-and-forth between the same reversal values. It is apparent from the previous description of the Bean model that these back-and-forth variations will affect in the identical way the same surface layers of superconductors. Consequently, these variations will result in equal increments of B, which is tantamotmt to the congruency of the corresponding minor loops.

In the case of generalized critical state models [4], the linear profiles of the magnetic field within superconductors are replaced by curved profiles. However, the creation and motion of these profiles are basically governed by the same rules as in the case of the Bean model. As a result, the previous reasoning holds, and, consequently, the wiping-out property and the congruency property are valid for the generalized critical state models as well. It was established in Chapter 1 that the wiping-out property and congruency property constitute necessary and sufficient conditions for the representation of actual hysteresis nonlinearity by the Preisach model. Thus, we conclude that the Bean model and generalized critical state models are particular cases of the Preisach model:

B(t) = / / #(c~, fl)f,~Ho(t) do~ dfl. (5.17) dd~

It is instructive to find such a function/z(~, fl) for which the Preisach model coincides with the Bean model. To do this, consider a "major" loop formed when the external magnetic field varies back-and-forth between +Hm and -Hm. Consider first-order transition curves B~ attached to the ascending branch of the previously mentioned loop. We recall that the curves B~ are formed when, after reaching the value -Hm, the external magnetic field is monotonically increased to the value c~ and subsequently monotonically decreased to the value ft. Depending on particular values of c~ and fl, we may have three typical field distributions shown in Figs. 5.7, 5.8, and 5.9. We will use these figures to evaluate the function

1 F(c~, fl) - ~(B~ - B~Z). (5.18)

306

H

/ \ \

~ Z


\

\ \

H

/ / / ~_~" -"Z


Figure 5.7 is valid under the condition

Hm q-c~ ~ 2H*.

From this figure we find

F(o~,fl) = 8H*

Figure 5.8 holds when

Hm q-Ol ~/ 2H*,

By using this figure, we derive

F ( ~ , ~ ) =

~0(o t _ / j ) 2

ol -- fl K 2H*.

a0(~ - 3) 2

8H*

/ \ / / \ \

/ / / \ \ \

/ / / / \ \ \ t

Z

(5.19)

(5.20)

(5.21)

(5.22)

FIGURE 5.9


Finally, the d is t r ibut ion of the magne t ic field s h o w n in Fig. 5.9 occurs w h e n

Hm + ol >1 2H* and c, - fl ~> 2H*. (5.23)

F rom Fig. 5.9, we obtain

/~0 (c~ - fl - H*). (5.24)

The express ions (5.20), (5.22), and (5.24) can be combined into one formula:

{ a0~/~-v-? )2 if 0 < CZ -- fl ~< 2H*, I~1 ~< Hm, 131 <~ Hm,

F(o~, fl) - -~(c~ - fl - H*) if o~ - fl I> 2H*, ]c~] ~< Hm, [fl] ~ Hm.

(5.25)

By us ing fo rmula (5.25) as well as the fo rmula (see Chap t e r 1)

a2F(cr fl) /z(c,, f l ) = - ~ , (5.26)

Ooe O3

we find

{ ~0 if 0 < c~ - fl ~ 2H*, I c*I ~< Hm, If l l~ Hm, g-~ (5.27) /z(c,, fl) = 0 otherwise.

The t rapezo ida l s u p p o r t of /z(a , fl) g iven by (5.27) is i l lus t ra ted in Fig. 5.10. Thus, it has been s h o w n that the critical state mode l for superconduc t -

ing hysteresis is a very par t icu lar case of the Preisach model . This result

(Hm,_ Hm ) (X,

I

FIGURE 5.10


has been established for one-dimensional flux distributions and specimens of simple shapes (plane slabs). For these cases, explicit analytical expressions for magnetic field distributions within the superconductors are readily available, and they have been instrumental in the discussion just presented.

Next, we shall demonstrate that the critical state model is a particular case of the Preisach model for specimens of arbitrary shapes and complex flux distributions [6]. For these specimens, analytical machinery for the calculation of magnetic fields within the superconductors does not exist. Nevertheless, it will be shown next that the superconducting hysteresis (as described by the critical state model) still exhibits the wiping-out property and the congruency property of minor hysteresis loops.

To start the discussion, consider a superconducting cylinder of arbitrary cross-section subject to a uniform external field B0(t) whose direction does not change with time and lies in the plane of superconductor cross- section (Fig. 5.11). We will choose this direction as the direction of axis x. As the time-varying flux enters the superconductor, it induces screening (shielding) currents of density • The distribution of these superconducting screening currents is such that they create the magnetic field, which at any instant of time completely compensates for the change in the external field B0(t). Mathematically, this can be expressed as follows:

8Bo(t) 4- Bi(t) = 0. (5.28)

FIGURE 5.11


Here 3Bo(t) is the change in B0(t), while Bi(t) is the field created by superconducting screening currents, and equality (5.28) holds in the region interior to these currents.

It is clear that 8Bo(t) ~ 0 when Bo(t) is monotonically increased, and 8Bo(t) ~ 0 when Bo(t) is monotonically decreased. By using this fact and (5.28), it can be concluded that there is a reversal in the direction (polarity) of superconducting screening currents as Bo(t) goes through its maximum or minimum values.

With these facts in mind, consider how the distribution of superconducting currents is generically modified in time by temporal variations of the external magnetic field. Suppose that, starting from zero value, the external field is monotonically increased until it reaches its maximum value M1 at some time t = t~-. This monotonic variation of Bo(t) induces a surface layer of superconducting screening currents. The interior boundary of this current layer extends inwards as Bo(t) is increased [see Fig. 5.12a], and at any instant of time this boundary is uniquely determined by the instantaneous values of Bo(t). Next, we suppose that this monotonic increase is followed by a monotonic decrease until Bo(t) reaches its min imum value ml at some time t = t 1. For the time being it is assumed that Imll < M1. As soon as the maximum value M1 is achieved, the inward progress of the previous current layer is terminated and a new surface current layer of reversed polarity (direction) is induced [see Fig. 5.12b]. This new current layer creates field Bi(t), which compensates for monotonic decrease in Bo(t) in the region interior to this current layer. For this reason, it is clear that the interior boundary of the new current layer extends inwards as Bo(t) is monotonically decreased. It is also clear that this boundary is uniquely determined by the instantaneous value of 8Bo(t), and, consequently, by the instantaneous value of Bo(t) for any specific (given) value of M1. Now suppose that the monotonic decrease is followed by a monotonic increase until Bo(t) reaches its new maximum value M2 at some time t = t~-. For the time being, it is assumed that M2 < Iml]. As soon as the minimum value ml is achieved, the inward progress of the second layer of superconducting screening currents is terminated and a new surface current layer of reversed polarity is introduced to counteract the monotonic increase of the external field [see Fig. 5.12c]. This current layer progresses inwards until the maximum value M2 is achieved; at this point the inward progress of the current layer is terminated. As before, the instantaneous position of the interior boundary of this layer is uniquely determined by the instantaneous value of 8Bo(t), and, consequently, by the instantaneous value of Bo(t) for a specific (given) value of ml.

Thus, it can be concluded that at any instant of time there exist several (many) layers of persisting superconducting currents [see Fig. 5.12d].


FIGURE 5.12

These persisting currents have opposite polarities (directions) in adjacent layers. The interior boundaries S~ and S~- of all layers (except the last one) remain still and they are uniquely determined by the past extremum values Mk and mk of B0(t), respectively. The last induced current layer extends inward as the external field changes in time monotonically.

The magnetic moment M of the superconductor is related to the distribution of the superconducting screening currents as follows:

M(t) - fs [r x j(t)] ds, (5.29)


where the integration is performed over the superconductor cross-section. In general, this magnetic moment has x and y components. According

to (5.29), these components are given by the expressions

Mx(t) - ~s yj(t) ds,

My(t) = - f xj(t) ds. Js

(5.30)

(5.31)

It is clear that if the superconductor cross-section is symmetric with respect to the x-axis, then only the x component of the magnetic moment is present. In the absence of this symmetry, two components of the magnetic moments exist.

It is apparent from the previous discussion that the instantaneous values of Mx(t) and My(t) depend not only on the current instantaneous value of the external field Bo(t) but on the past extremum values of Bo(t) as well. This is because the overall distribution of persisting superconducting currents depends on the past extrema of Bo(t). Thus, it can be concluded that relationships Mx(t) vs Bo(t) and My(t) vs Bo(t) exhibit discrete memories that are characteristic and intrinsic of the rate-independent hysteresis. It is worthwhile to note that it is the hysteretic relationship Mx(t) vs Bo(t) that is typically measured in experiments by using, for instance, a vibrating sample magnetometer (VSM) with one pair of pickup coils. By using a VSM equipped with two pairs of orthogonal pickup coils, the hysteretic relation between My(t) and Bo(t) can be measured as well.

It is important to stress here that the origin of rate independence of superconducting hysteresis can be traced back to the assumption of ideal (sharp) resistive transitions. This connection is especially apparent for superconducting specimens of simple shapes (plane slabs). For such specimens, the explicit and single-valued relations between the increments of the external field and the location of inward boundaries of superconducting layers can be found by resorting only to Amp6re's Law.

It is clear from the presented discussion that a newly induced and inward-extending layer of superconducting currents will wipe out (replace) some layers of persisting superconducting currents if they correspond to the previous extremum values of B0(t), which are exceeded by a new extremum value. In this way, the effect of those previous extremum values of Bo(t) on the overall future current distributions will be completely eliminated. According to formulas (5.30) and (5.31), the effect of those past extremum values of the external magnetic field on the magnetic moment will be eliminated as well. This is the wiping-out property of the superconducting hysteresis as described by the critical state model.


Next, we proceed with the discussion of the congruency property.

Consider two distinct variations of the external field, B~l)(t) and B~2)(t). Suppose that these two external fields have different past histories and,

consequently, different sequences of local past extrema, {M~l),m~ 1)} and

{M~ 2), m~2)}. However, starting from some instant of time they vary back- and-forth between the same reversal values. It is apparent from the description of the critical state model and expressions (5.30) and (5.31) that these two identical back-and-forth variations of the external field will result in the formation of two minor loops for the hysteretic relation Mx(t) vs Bo(t) [or My(t) vs B0(t)]. It is also apparent from the same description of the critical state model that these two back-and-forth variations of the external field will affect in the identical way the same surface layers of a superconductor. Unaffected layers of the persistent superconducting currents will

be different because of different past histories of B~ 1) (t)and B~ 2)(t). Accord- ing to (5.30) and (5.31), these unaffected layers of persistent currents result in constant-in-time ("background") components of the magnetic moment. Consequently, it can be concluded that the same incremental variations of

B~l)(t) and B~2)(t) will result in equal increments of Mx (and My). This is tantamount to the congruency of the corresponding minor loops. Thus, the congruency property is established for the superconducting hysteresis as described by the critical state model.

It has been previously established that the wiping-out property and the congruency property constitute the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the Preisach model. Thus, the description of the superconducting hysteresis by the crib ical state model is equivalent to the description of the same hysteresis by the Preisach model.

The question can be immediately asked, "What is to be gained from this result?" The answer to this question can be stated as follows. There is no readily available analytical machinery for the calculation of the interior boundaries of superconducting current layers for specimens of arbitrary shapes. For this reason, the critical state model does not lead to mathematically explicit results. The application of the Preisach model allows one to circumvent these difficulties by using some experimental data. Namely, for any superconducting specimen, the "first-order transition" curves can be measured and used for the identification of the Preisach model for the given specimen. By using these curves, complete prediction of hysteretic behavior of the specimen can be given at least at the same level of accuracy and physical legitimacy as in the case of the critical state model. In particular, cyclic and "ramp" losses can be explicitly expressed in terms of the first-order transition curves (see Chapter 1).


As an aside, we point out that the presented discussion can also be useful whenever numerical implementation of the Bean model is attempted. Indeed, the numerical implementation of the Bean model can be appreciably simplified by computing only the "first-order transition" curves and then by using these curves for the prediction of hysteretic behavior for arbitrary piecewise monotonic variations of the external field. The latter is possible because, whenever the congruency and wiping-out properties are valid, all hysteretic data can be compressed (collapsed) into the "first-order transition" curves.

5.2 EXPERIMENTAL T E S T I N G OF THE P R E I S A C H M O D E L I N G OF S U P E R C O N D U C T I N G HYSTERESIS

After it has been realized [5] that the critical state (Bean) model is a particular case of the classical Preisach model, several attempts have been made to test the accuracy of Preisach modeling of superconducting hysteresis. First, experimental testings of the congruency and wiping-out properties for type-II superconductors has been carried out by G. Friedman, L. Liu, and J. S. Kouvel [7]. In the reported experiments, two superconducting samples were used. One was a high temperature superconductor Ba0.575K0.425BiO3, while the other was niobium (Nb). The hysteresis loops of these superconductors are shown in Figs. 5.13 and 5.14, respectively. The wiping-out property was checked by observing closure of minor loops at the end of the first cyclic variation of the magnetic field. To examine the congruency property, minor hysteresis loops were compared for identical cyclic variations of the magnetic field with different prior histories. The results of this comparison for the BaKBiO3 sample and the Nb sample are shown in Figs. 5.15 and 5.16, respectively. These figures suggest that the congruency property is fairly accurate for these superconductors.

More extensive experimental testings of the accuracy of the Preisach modeling of superconducting hysteresis have been reported in [8]. In these experiments, higher-order reversed curves predicted by the Preisach model were compared with actual higher-order reversal curves measured for the same past extremum values of the external magnetic field as used in Preisach predictions. This comparison is the basis for the assessment of the accuracy of the Preisach model because the history dependent branching is the phenomenological essence of hysteresis.

The testing was performed for YBa2Cu3Ox superconducting samples by using a vibrating sample magnetometer (VSM) equipped with a cryo- stat (model MicroMag 3900 of Princeton Measurements Corporation). The

314 C H A P T E R 5 Supe rconduc t ing Hysteres i s

- ~ I "-.. -I

- ~

"x 0 . . . . .

%.a~,

-1o

-,o- T -- 4.2 K I_ X.~,~/- -I,

-800 -400 0 400 800

H (oe)

F I G U R E 5.13

40

'~" 2O

0

v

- 2 0

- 4 0

- 8 0 - 6000 - 4 0 0 0 - 2 0 0 0 0 2000 4000 8000

A(o~)

F I G U R E 5.14


tO

@3

0

v -1o

-20

- Ba a75K.425BiO s

T =4.2K

m..

, I , I , I ~ _ 1 _ _ ~ , ! , I , - 500 - 4 0 0 - 300 - 2 0 0 - I O0 0 t O0 200

H (o~)

20 I from 1000 O e -

"..

" ~'- "'-o I0

-~~ o - "~.

"-" K 4 ~ a B i O a : ~ - 1 o B a . a T s .

T = 4 . 2 K

-:~0 - t , [ . . . . [ . . , ] , [ , [ , ] , [ ,

- 7 0 0 - 6 0 0 - 5 0 0 - 4 0 0 - 3 0 0 - 2 0 0 - 1 0 0 0 tO0

H (o~)

I0

% o

0 - t o -

-20 -

, I , I , I , ! , l , , , -200 - 150 - 100 -50 0 50 100 t50 200

H (Oe)

F I G U R E 5 . 1 5


20

iO t9

0 0

-10

v ,~ -2O

-3O

-40~

Nb T=4.2K

f rom %

from 0 0 e

-50 ~-- - tO00 -800 -600 -400 -200 0

n (o~)

._. I0

0 O'

-I0

-2O

_30 !

-40

ZFC

Nb

T = 4.2K

-5O 0 tO0 200 300 400 500 800

H (o~)

-10

e ) - 20

~ -~0

~ -4o

- 5 0

- 6 0 400

Nb

_ T=4.2K

, I , i , I , i , i , 500 600 700 800 900 tO00

H (Oe)

FIGURE 5.16


specimens were sintered disk shaped samples about 4 mm in diameter and 2 mm in thickness. These samples were procured from Angstrom Sciences, Inc. The experiments were conducted in the wide range of temperatures (varying from 14 to 80 K). In these experiments, the first-order reversal curves were measured for each temperature. These curves were used for the identification of the Preisach model as discussed in Chap- ter 1. Then, higher-order reversal curves (up to the eighth order) were measured at each temperature for various sequences of reversal values of the applied magnetic field, that is, for various past histories. These measured higher-order reversal curves were compared with the predictions of those curves by the Preisach model computed for the same past histories as in the experiments. Sample results of these comparisons are shown in Figs. 5.17, 5.18, and 5.19 for temperatures of 14, 30, and 60 K, respectively. These sample results of the comparison between the experimental data and the Preisach model predictions are representatives of what we have observed for other temperatures. The above figures demonstrate the remarkable accuracy of the classical Preisach model in predicting various branches of superconducting hysteresis for various past histories and in the wide range of temperatures. Since history dependent branching is the essence of phenomenological manifestation of hysteresis, the above comparison suggests that the Preisach model may have a remarkable prediction power as far as the description of superconducting hysteresis is concerned. This comparison also suggests that the set of first-order reversal curves may eventually emerge as the standard experimental data that can be used for the complete phenomenological characterization of superconducting hysteresis. These first-order reversal curves can be useful not only for the prediction of branching but for calculation of cyclic and "ramp" losses as well.

As an aside, it is worth noting that there is mounting experimental and theoretical evidence that the classical Preisach model may be much more accurate for the description of superconducting hysteresis than for the description of hysteresis of magnetic materials. This is quite ironic because historically the Preisach model was first developed as a model for magnetic hysteresis and was first phrased in purely magnetic terms. This irony supports the point of view that it is beneficial to consider the Preisach model as a general mathematical tool whose usefulness extends far beyond the area of modeling of magnetic hysteresis.

The attempt has been made (see [9]) to extend the testing to the case of vector Preisach models of hysteresis and to examine their ability to mimic vectorial hysteretic behavior of type-II superconductors. It is worthwhile to mention that experimental data on vector superconducting hysteresis

0.2


YBa2Cu30 x Sample @ 14 K . . . . . ~ i , ' I Measurement I a 0"21 Simul tion

0.05

3"

~ o ~

-0.05

-0.1

-0.15

-0.2 - -0.5 0 0.5

H (Oe) x 104

0.15

YBa2Cu30 x Sample @ 14 K __

Measurement - - Simulation

0.1

0.2

0.05

o

-0"05 I

-0,1 f -0.15

-0.2 -1

318

-0'.5 ; 0'.5 1 H (Oe) x 10 4

(b)

I - - Measurement Simulation

0,15

0.1

0.05

-0.05

-0.1

-0 ,15

-0.2

YBa2Cu30 x Sample @ 14 K

-0.5 0 0.5 I H (Oe) x 10 4

(a)

FIGURE 5.17


Y B a 2 C u 3 0 x Sample @ 30 K

0.2 , . , �9 , . . . . . .

l 0.05

- 0 . 0 5

-0 .1

- 0 . 1 5

- 0 . 2 -" - 0 . 5 0 0.5 1

H (Oe) x 104

0.2

319

Y B a 2 C u 3 0 X Sample @ 30 K , . . . , . . . . . . . . . . . . . . . |

/ % I ' - - - Measuremen t l

0.15

0.1

0.05

-0.05 -0 .1

- 0 . 1 5

- 0 . 2 - 0 . 5 0 0.5 1

H (Oe) x 104

(b)


Olf -0"05 I

-0.1 I - 0 . 1 5

- 0 . 2 ; -015 . . . . . . .

Measuremen t [ Simulation. J

0.5 0 0.5 1 H (Oe) x 10 4

FIGURE 5.18

0.08

0.06

0.04

0.02

E 0 ,,..,

-0.02

-0.04

-0.06

-0.08

YBa2Cu30 x Sample @ 60 K . . . . .


YBa2Cu30 x Sample @ 60 K ._ _ L

i Measurement Simulation

-0 .5 0 0.5 1 H (Oe) x 10 =

(a)

0,08

0.06

0.04

0.02

3"

-0.02

-0.04

-0 .06

-0 .08

Measurement

-- Simulation

-0 .5 0 0.5 1 H (Oe) x 104

(b)

0.08

0.06

0.04

0.02

-0 D2

-0 .04

-0 .06

-0 .08


Measurement Simulation

- 0 , 5 0 H (Oe)

0.5 I x 10 4

320

FIGURE 5.19


is very scarce. For this reason, the experimental data presented below is of interest in its own right.

The testing has been performed for the following vector Preisach model (see Chapter 3, Section 7):

y[

- L (fL ) M(t)= e0 v(ot, fl)~,~(iH(t)lg(O-4)(t)))do~d fl dO. (5.32)

Here/~I(t) is the magnetization, e0 is a unit vector along the direction specified by a polar angle 0, Gt~ are (as before) operators represented by rectangular loops with o~ and fl being "up" and "down" switching values, respectively, ~(t) is an angle formed by a polar axis and the magnetic field H(t).

In the above model, functions v(ol, fl) and g(O - ~ ) are not specified in advance but rather should be determined by fitting this model to some experimental data. This is an identification problem. It is apparent that the expression iH(t)lg(O -dp(t)) can be construed as a generalized projection of H(t) on the direction specified by e0. Indeed, this expression is reduced to the conventional projection in the case when g(O -q~) = cos(0 -q~). This suggests that we look for functions g(O - ~) in the form:

1

g(O - 4))- sign[cos(0 - q~)] icos(0 - ~)i ~" (5.33)

For this class of models, identification of g is reduced to the determination of n. More general classes of Preisach models that are not constrained by the assumption (5.33) are discussed in Section 7 of Chapter 3.

To perform the identification of the model (5.32)-(5.33), the following experimental data has been used:

(a) First-order reversal curves measured when the magnetic field H(t) is restricted to vary along one, arbitrary fixed direction. These curves are attached to the ascending (or descending) branch of the hysteresis loop and they are traced after the first reversal (extremum) value of H(t) (see Fig. 5.20). By using these curves, the following function can be introduced:

1 F(o~, fl) - ~ (M,~ - M~t~), (5.34)

(b) "Rotational" experimental data measured for the case when the --)

sample is subject to a uniformly rotating magnetic field: H(t)= -exHm cos o)t +-~yHmsincot. For isotropic superconducting media,

the magnetization M(t) has the form: M(t) = Mo + Hi(t), where M0 does not change with time and depends only on the past history,


\ -Y\:C

M

" H

FIGURE 5.20

while M(t) is a uniformly rotating vector that lags behind H(t) by some X. This lag angle depends on Hm:

X =f(Hm), (5.35)

and this experimentally measured relation has been used, along with (5.34) for the identification of the model (5.32)-(5.33). The identification procedure is outlined in Section 7 of Chapter 3.

The vector Preisach model (5.32) was tested for high-Tc YBa2Cu3Ox superconducting samples by using a vibrating sample magnetometer (VSM model MicroMag 3900 of Princeton Measurements Corporation) that has vectorial measurement capabilities and it is equipped with a cryo- stat. The superconducting specimens were the same sintered disk shaped samples about 4 mm in diameter and 2 mm in thickness.

First, it was verified that the specimens have isotropic magnetic properties. This was done by performing "rotational" experiments when the samples were subject to uniformly rotating magnetic fields. It was found that the time varying component of the magnetization was a uniformly rotating vector. This clearly suggested that the superconducting samples had isotropic hysteretic properties.

To perform the identification of the Preisach model, the set of first- order reversal curves and the lag angle as a function of Hm were measured. These two sets of experimental data as well as the accuracy of the identification procedure are illustrated for T = 40 K in Figs. 5.21 and 5.22, respectively. The best fit of the experimentally measured curve shown in Fig. 5.22 was obtained for n = 2. It is worthwhile to point out that the relation (5.35) for superconductors is qualitatively quite different from the lag angle versus Hm relation observed for magnetic hysteretic materials (see

0.5

0.4 Measured

- - Computed

0.3

0,2

.~. 0,1

-0.1

-0.2

-0.3

-0.4


-0.5 -5000 0 5000

Hx (Oe)

FIGURE 5.21

323

-140 . . . . . .

-145

-150

~ _155 i

" 0

~-160 ._1

0 o

-165 r- n

-170

-175

/ Z

I-- Computed Measured I

7

I -180 500

i / i i i i i i

t000 1500 2000 2500 3000 3500 4000 4500 5000 Rotating Field Magnitude (Oe)

FIGURE 5.22


H m

FIGURE 5.23

Fig. 5.23). This difference can be traced back to different physical origins of hysteresis in superconductors and magnetic materials.

After the identification was performed, the ability of model (5.32) to predict the correlation between mutually orthogonal components of magnetization and magnetic field was tested. This correlation has long been regarded as an important "testing" property for vector hysteresis models. The reason is that the cross-correlation data between orthogonal components of M(t) and H(t) is qualitatively quite different from scalar and rotational data used for the identification of the model (5.32).

The testing was carried out as follows. First, the magnetic field was restricted to vary along the y-axis. It was increased from the "saturation" negative value to some positive value Hy and then decreased to zero. This resulted in some remanent magnetization Myr. Then, the magnetic field was restricted to vary along the x-axis and the curve My vs Hx was measured. These curves were measured for various values of Myr, that is for various values of Hy. By using the identified Preisach model (5.32), the computations of magnetization were performed for the same sequences of field variations as in the described experiment. The comparison between these computations and the measured data is shown in Fig. 5.24. This comparison suggests that the vector Preisach model (5.32) mimics the measured data with reasonable accuracy. This accuracy is remarkable for sufficiently high values of Hy and it deteriorates when Hy is close to zero.

It is well-known that at constant (in time) external magnetic fields, the flux filaments in type-II superconductors can be depinned by thermal activation. This may result in slow (very gradual) time variations of magnetization of superconducting samples, which is the essence of viscosity (or creep). The described gradual temporal variations of magnetization can be usually characterized by the following intermediate "ln t" asymptotics:

M(t) ~, Mo - S(H, T)lnt, (5.36)


. . . . -- ' ' ' i - ; - . y . r ~ o o , I - " - HY; 1~176176 Or,, .....

0.1 ~ Solid Lines: Measured "t

0.05 ~ ~ Dash Lines: Computed i

> , ,

=E

-0,05

-o.1

00 t i I i Hx (Oe)

0 , 1 5 . . . . . . . . . . . . . . - ; - .~=,:~ooo~ II

I -"- Hy=750Oe II

o.1 ~ Solid Lines: Measured ]

0 t o 0

:E

-0.05

-o.1

i i i i i i i i i

0.1

0.05

�9 0

-0.05

500 1000 1500 2ooo 2soo 3oo0 ~ 4oo0 4soo 5000 Hx (Oe)

. . . . . . . -J,-- ~ , , ~ , o o . I'-B- Hy= 250 Oe

Solid Lines: Measured ~ ~ ~ i ~ Dash Lines: Computed

i , , i i i i i i i | i

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Hx (Oe)

325

FIGURE 5.24


where S(H, T) is called the viscosity (creep) coefficient. This coefficient depends on the values of applied magnetic field H, temperature T as well as the past history of magnetic field variations. The latter means that the viscosity coefficient is also a function of the current state of hysteretic material.

The term "intermediate asymptotics" means that formula (5.36) describes quite well the long-time behavior of magnetization. However, it does not describe properly the ultimate (at t ~ oo) value of magnetization. In other words, the asymptotic behavior (5.36) breaks down for very long times when In t diverges.

The phenomenon of creep in type-II superconductors is very similar to magnetic viscosity of hysteretic magnetic materials, where intermediate asymptotics (5.36) has been observed as well. In the case of magnetic materials, it has been found (see Section 2 of Chapter 4) that "bell-shaped" S vs H curves measured for different temperatures T collapse onto one "universal" curve as a result of appropriate scaling. This prompted the idea to experimentally investigate the scaling and data collapse for the creep coefficient S(H, T) for type-II superconductors [10].

Our experiments were conducted for high-Tc YBa2Cu3Ox superconducting samples by using a vibrating sample magnetometer (VSM model MicroMag 3900 of Princeton Measurements Corporation). The superconducting samples were the same sintered disk shaped specimens about 4 mm in diameter and 2 mm in thickness. Hysteresis loops of these samples measured for various temperatures are shown in Fig. 5.25. These hysteresis loops are quite different from those observed for magnetic materials. First, hysteresis loops for magnetic and superconducting materials have different "orientation". This difference is attributed to the diamagnetic nature of superconductors. The second and more striking difference is that ascending and descending branches of hysteresis loops for high-Tc superconductors are not monotonic.

The viscosity (creep) experiments were performed in the wide range of temperatures (varying from 25 K to 75 K). In these experiments the magnitude of the external magnetic field was first increased to some maximum value, then it was slowly decreased to some desired value and kept constant thereafter while slow temporal variations of magnetization were recorded. Typical examples of measured temporal variations of magneti- zations plotted on ln t scale are shown in Fig. 5.26. By using such plots, the values of viscosity (creep) coefficient S(H, T) were extracted for various values of the fixed magnetic field at the selected temperatures. These measured S vs H curves for different selected temperatures are shown in Fig. 5.27. It is apparent that these curves are appreciably different from those "bell-shaped" curves observed for magnetic viscosity. First, S vs H


0,8

0.6

0,4

E 0.2

.~ o

~ -0 ,2

-0 .4

-0 .6

-0.8

I - - t = 2 5 K

" - ~ _ . _ ~

-0,5 0 0.5 1 Applied Magnetic Field H (Oe) x 10 r

0.25

0.2

0.15

0.1

0.05

P~ o ,.m

~ -o,1

.-0.15

-0.2

-0.25

/ 1

I- T=50K

4

-0.5 0 0.5 Applied Magnetic Field H (Oe) x 10 4

0.15

0.1

~ 0.05

--g o

c

~ -0.05

-0.1

, ,

I T=75 K 1

j

-0.5 0 0.5 Applied Magnetic Field H (Oe) x 104

3 2 7

F I G U R E 5 . 2 5

328

0.08

0.06

. . 0.04

E O v 0.02

C

O m

N

| 0 c

-0.02

-0.04

-0.0f

CHAPTER 5 Superconduct ing Hysteresis

I ! . . . . . . . ~ = ~I �9

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H . ~ . ~ . ......

~ ...........

H:IO00 Oe

" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H - _ ~ ~ . . .

~._~.--r~ Oe

............................................................................. H ~ ' I N

I~-,-~O Oe' , , , | . . . . . .

2 3 4 5 6 Ln(Time(sec))

FIGURE 5.26

0.018- -

0.016

0.014

~ o.o12

~ o.oI

O .

~ 0.008

'~ O.IX)G

o .oo2

" ' ' ' ' . . . . . . ' ' l - - ' T - ~ 6 K I

I - T=65 K I I - - T=60 I~ I - - - T=55 I~ I - * - T=50 K~ I - T=45 K I I -o- T=40 I ~ - T=30 KI

-0.8 -0 .6 -0 .4 -0.2 0 0.2 0.4 0,6 0.8 1 Magnetic Field H (Oe) x 10'

FIGURE 5.27


curves for superconductors are quite asymmetrical. Second, each of these curves has two maxima: main maxima and "satellite" maxima. This more complex structure of S vs H curves can be attributed to non-monotonic nature of ascending and descending branches of hysteresis loops for type-II superconductors.

By using the experimental data shown in Fig. 5.27, the hypothesis that S(H, T) admits the following scaling

S(H, T) - Smax(T)f (H,(HT) ) (5.37)

was tested. In the above formula, Smax(T) is the global maximum of the viscosity coefficient as a function of T, while H*(T) is the value of the magnetic field at which Smax(T) is achieved. The last formula suggest that S vs H curves experimentally measured for different temperatures must collapse onto one universal curve when plotted in coordinates:

S H s = Smax(Z)' h = H*(Z)" (5.38)

This phenomenon of data collapse is the principal significance of scaling and its occurrence was observed for the collected experimental data. Namely, by performing scaling described above, it was found that curves shown in Fig. 5.27 by and large collapse onto the single curve (see Fig. 5.28).

In addition, the following scaling hypothesis

Smax(Z) =aT ~ (5.39)

was also experimentally tested and verified with some accuracy. The results of this testing are shown in Fig. 5.29.

Hypothesis (5.39) along with the formula (5.37) lead to the following self-similar expression for the creep coefficient:

S(T)=aT~ ). (5.40)

The last formula (as well as formula (5.37)) reveals an interesting and peculiar structure of the creep coefficient as the function of two variables: H and T. The essence of this structure is that the normalized creep coefficient s = S/Smax(T) is a function of one variable h = H/H*(T).

Scaling and data collapse are typical for (but not limited to) critical phenomena. There, the physical origin of data collapse can be traced back to the divergence of correlation length near critical points. The physical origin of scaling in magnetic viscosity (creep) of type-II superconductors is not clear at this time. One may speculate that this scaling is related to "granular" structures of high-Tc superconductors.


1 -

0.9

0.8

0.7

o.o

r

09

0.4

0.3

0.2

0,1

0 '5 ' -20 -1 -10

. . . . . . : i i I ! . . . . . !

1-=75 K . . . . T=65K

T---B0 K T=55 K T=50 K T=45 K T=40K

--~ T~O K T-=25 K

! . . . . I , , I I

-5 0 5 10 15

Hil l

FIGURE 5.28

-4i I

............... ! i ..... ....... i ........... .............. i - 6 . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . ! . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . i . . . . . . . . . . . . . .

-8.5' '

,I . . . . . i

-~,2 3.4 3.6 3.8 4 4.2 4.4

Log(T)

FIGURE 5.29

5.3 NONLINEAR DIFFUSION IN SUPERCONDUCTORS 331

5.3 NONLINEAR DIFFUSION IN SUPERCONDUCTORS WITH GRADUAL RESISTIVE TRANSITIONS (LINEAR POLARIZATION)

In the first section of this chapter, nonlinear diffusion in superconductors with sharp (ideal) resistive transitions was discussed. However, actual resistive transitions are gradual and it is customary to describe them by the following power law:

E = �9 signl (n > 1), (5.41)

where E is electric field, ] is current density, and k is some parameter that coordinates the dimensions of both sides in the last expression.

The exponent "n" is a measure of the sharpness of the resistive transition and it may vary in the range 7-1000. Initially, the power law was regarded only as a reasonable empirical description of the resistive transition. Recently, there has been a considerable research effort to justify this law theoretically. As a result, models based on Josephson-junction coupling [11], "sausaging" [12], and spatial distribution of critical current [13] have been proposed. However, the most plausible explanation for the power law came from the thermal activation theory [14-16]. According to this theory, the electric field E induced by thermally activated drift of flux filaments (vortices) can be written in the form of the Arrhenius law:

E - Ec exp[ -U(J ) / kBr] , (5.42)

where U(J) is a current-dependent flux creep potential barrier, which sup- posedly vanishes at some critical current Jc; Ec is an electric field at J = Jc. If a logarithmic dependence of activation barrier U on current J

U(J)= Uc l n ( ~ ) (5.43)

is assumed, then from formula (5.42) we readily obtain the power law (5.41) for the resistive transition.

Whatever the theoretical rationale may be behind the power law, this law has been observed in numerous experiments. For this reason, in our subsequent discussions, this law will be used as a constitutive relation for hard superconductors. By using this constitutive relation and Maxwell's equations, it is easy to show that nonlinear diffusion of linearly polarized electromagnetic fields for monotonically increasing boundary conditions is described by the following nonlinear partial differential equation:

32J n 8J (5.44) 3Z 2 = t'tOkn O---t"


J(O,t)

~ p > l

m, t

FIGURE 5.30

We shall first consider the solution of this equation for the following boundary and initial conditions:

/(0, t ) - ctP (t >1 0, p > 0), (5.45)

J(z, 0 ) - 0 (z > 0). (5.46)

It may seem at first that these boundary conditions are of a very specific nature. However, it can be remarked that these boundary conditions do describe a wide class of monotonically increasing functions as p varies from 0 to cx~ (see Fig. 5.30). It will be shown below that for all these boundary conditions the profile of electric current density as a function of z remains practically the same. This observation will suggest using the same profile of electric current density for arbitrary monotonically increasing boundary conditions. This will lead to very simple analytical solutions.

The initial boundary value problem (5.44)-(5.46) can be reduced to the boundary value problem for an ordinary differential equation. This reduction is based on the dimensional analysis of Eqs. (5.44) and (5.45). This analysis leads to the conclusion that the following variable is dimensionless:

z ( = , (5.47)

tmv/k-n l~ol c n-1

where p ( n - 1) + 1

m = . (5.48) 2

By using this dimensionless variable, we look for the self-similar solution of initial boundary value problem (5.44)-(5.46) in the form:

l(z, t) = ctPf (( ), (5.49)

where f ( ( ) is a dimensionless function of (. By substituting formula (5.49) into Eq. (5.44), after simple transformations we end up with the following


ordinary differential equation:

d2f n df d---- i + m; -~ - pf = O. (5.50)

It is apparent that J(z, t) given by expression (5.49) will satisfy boundary and initial conditions (5.45) and (5.46) if f satisfies the boundary conditions:

riO) = 1, (5.51)

f ( ~ ) = 0. (5.52)

Thus, the initial boundary value problem (5.44)-(5.46) is reduced to the boundary value problem (5.50)-(5.52) for the ordinary differential equation (5.50). It can be proven that this nonlinear differential equation has the following group property: if f ( ( ) is a solution to Eq. (5.50), then

F(~ ) = X-2/(n-1)f (X~ ) (5.53)

is also a solution to this equation for any constant ~. This property can be utilized as follows. Suppose we have solution f ( ( ) to Eq. (5.50), which satisfies the boundary condition (5.52), however,

f(O) = a :7/: 1.

Then, by using X = a (n-1)/2, we find that

1 (n-1)/2 f (r = a f (a ~)

(5.54)

is the solution to Eq. (5.50), which satisfies (5.52) as well as the boundary condition (5.51). Thus, we can first find a solution to Eq. (5.50) subject to boundary condition (5.52), then, by using transformation (5.53), we can map this solution into the solution that also satisfies the boundary condition (5.51).

It can be shown that a solution to Eq. (5.50) satisfying the boundary condition (5.52) has the form:

_ ] b(1 - ~)1/(n-1)[1 if- b1(1 - ~') + b2(1 - ~)2 _}_...] f(~') ! 0

i f 0 ~ ~ < l ,

if~ >1.

(5.56)

By substituting formula (5.56) into Eq. (5.50), after simple but lengthy transformations, we find

b I m ( n - 1 ) ] 1/(n-1) = , ( 5 . 5 7 )

n

(5.55)


p ( n - 1 ) - m b l - - ~m~--(~-I - 1-) ' (5.58)

1 4- lbl [ (2n- 1)(3n- 2 ) - 4n] b2 = - b l . (5.59)

3 (2n - 1)

It is clear that

riO) = b(1 4- bl 4- b2 4-...) # 1. (5.60)

This difficulty is overcome by using transformation (5.53) with

X - [b(1 4- bl 4- b2 4 - " .)](n-I)/2. (5.61)

This leads to the following solution of the boundary value problem (5.50)- (5.52):

- / (1 - ~.~)1/(n-1)1+b1(1-x~')+b2(1-X~')2+'"1+bl+b2+... if 0 K X~ < 1, f ( ; ) (5.62) / 0 ifX~ > 1.

The last expression can be simplified by exploiting the fact that the exponent n in the power law is usually greater than 7. This simplification can be accomplished by using the following inequalities for bl and b2, which can be easily derived from Eqs. (5.48), (5.58), and (5.59):

1 Ibll ~ 2 n ( n - 1)' (5.63)

1 1 Ib21 ~ 6 ( n - 1)(2n- 1)n 4- 8 ( n - 1)n 2" (5.64)

From the above inequalities, for n ~ 7 we find

Ibll ~ 0.012, Ib21 ~ 0.00075. (5.65)

This suggests the following simplification of solution (5.62):

- / (1 - v / m ( n - 1)/n~) 1/(n-1) if 0 ~ ~ < v / n / m ( n - 1), f ( ( ) (5.66) / 0 if ~ > v / n / m ( n - 1).

By substituting formula (5.66) into expression (5.49) and taking into account Eq. (5.47), we end up with the following analytical expression for the current density:

z )l/(n-1) l(z, t) -- ctP (1 - ~ if z ~ dt m, (5.67)

0 if z ~ dt m,

where

d = v/(ncn-1)/[#oknm(n - 1)]. (5.68)


The brief examination of self-similar solutions (5.67) leads to the following conclusion: the profile of electric current density J(z,t) remains approximately the same in spite of wide-ranging variations of boundary conditions (5.45) (see Fig. 5.30). For typical values of n (usually n ~> 7), this profile is very close to a rectangular one. This suggests that the actual profile of electric current density will be close to a rectangular one for other boundary conditions as well. Thus, we arrive at the following generalization of the critical state model.

Current density J(z,t) has a rectangular profile with the height equal to the instantaneous value Jo(t) of electric current density on the boundary of the superconductor (see Fig. 5.31). Magnetic field H(z,t) has a linear profile with a slope determined by the instantaneous value of Jo(t).

To better appreciate this generalization, we recall that in the critical state model the current has a rectangular profile of constant (in time) height, while the magnetic field has a linear profile with constant (in time) slope.

For the zero front of the current profile we have

Ho(t) zo(t) = ~ . (5.69)

lo(t) However, Ho(t) and Jo(t) are not simultaneously known. For this reason, we intend to find Jo(t) in terms of Ho(t). To this end, we multiply Eq. (5.44) by z and integrate from 0 to zo(t) with respect to z and from 0 to t with respect to t. After some simple but lengthy transformations that are similar

J(z,t) i

t 2 r"

t 3 r"

if, t1

Zo(t 1 ) Zo(t 2) zo(t 3) (a)

"-Z

H(z,t)

t 3

t 2

- - - , . ~ Z

Zo(h) Zo(t2)Zo%) (b)

F I G U R E 5.31


to the derivation of (5.82) below, we arrive at the following equation

f zo(t) ~0 t t~ok n zJ(z , t )dz-- lrd(r)dr. (5.70)

dO By using in the last equation the rectangular profile approximation for J(z, t), we obtain

lz~ fot -~ Jo(t)z~(t)= J~(r)dr. (5.71)

By substituting formula (5.69) into Eq. (5.71), we find

d [Ha(t)] l~(t). (5.72)

2 dt I Jo(t) J

By introducing a new variable

H2(t) (5.73) y(t) = /0(t) '

we can represent formula (5.72) as the following differential equation with respect to y (t):

dy n+l __ 2(n + 1) ~2n( t)~0 (5.74) dt #ok n "

By integrating Eq. (5.74) and by using Eq. (5.73), we arrive at the following expression for J0(t):

Jo(t) = H2(t) . (5.75) {[(2(n 4- 1)/]zokn)] fo H~ n('c)d'~}l/(n+l)

By substituting formula (5.75) into Eq. (5.69), we find the following expression for zero front zo(t) in terms of the magnetic field, H0(t), at the boundary of the superconductor:

[ fO ] 1/(n+1) 1 2(n 4- 1) tH~n(.c)d.c z o ( t ) - Ho(t) #ok n (5.76)

Up to this point, nonlinear diffusion of electromagnetic fields in semi- infinite superconducting half-space has been discussed. However, the above results can be directly extended to the case of a slab of finite thickness A. This can be done due to the finite speed of propagation of zero front zo(t). As a result, if zo(t) < -~, nonlinear diffusion at both sides of a superconducting slab occurs in the same way as in the case of superconducting half-space. This is illustrated by Figs. 5.32a and 5.32b.

5.3 NONLINEAR DIFFUSION IN SUPERCONDUCTORS

J, H /t3

/t2

A / 2

.L ~Z

(tl < t2< t3)

,L

A p 2

FIGURE 5.32

337

I I

I

\ A ~Z

After the instant of time t*, when two fronts meet at the middle of the slab, formula (5.75) is not valid anymore. To find the appropriate formula for Jo(t) in the case t > t*, we shall again use the first moment relation for the nonlinear diffusion equation. However, this moment relation should be somewhat modified (in comparison with (5.70)). To find this modification, we start with the nonlinear diffusion equation:

02E Ol (5.77) O Z 2 = ['tO O---t"

We multiply both sides of this equation by z and integrate with respect to A . z from the boundary z = 0 to z -- ~.

js 32E /o ~- 3Jdz. (5.78) Z -Ez 2 ct z = ~ o z a~

Next, we shall transform the first integral in formula (5.78) by using integration by parts:

A A

/0 "~ 32E zOEIo / o ~ O E d z = A O E ( 2 ) Z-~z 2 dz = O---z - 3---z 2 Oz ,t + Eo(t). (5.79)

From equation curl E - -/~0 ~-t, we find

(2) OE t - -~o t (5.80) a z " - ~ ' "


By using the last equation in formula (5.79), as well as the power law (5.41), we obtain

A

fo g 32E /xoAOH(A ) J~(t) (5.81) Z-~z2dZ= 2 8t ~ , t -t kn �9

By substituting formula (5.81) into the moment relation (5.78), we derive A

J~(t) [d'~ A ) 0 lot 2 Ot 2" t = i~ok n-~ zJ(z, t) dz. (5.82)

Since it is assumed (in our generalization of the critical state model) that

l(z, t)= Jo(t) 0 <~ z < ~ ,

we can transform formula (5.82) as follows:

J~(t) I~~ 3H ( 2 t) tx~ dJ~ 2 8t ' = 8 d----t-"

Relation (5.83) implies that

which yields

2 H o ( t ) - H ~ , t = lo(t),

(5.83)

(5.84)

(5.85)

dJo(t) 4r0 dHo(t) ro dt + J~(t) = A " dt ' (5.87)

txo A 2 k n ~0 = ~ . (5.88)

8

Thus, in order to find Jo(t) for t/> t*, the solution to differential equation (5.75) subject to the initial condition

lo(t*) =l; (5.89)

must be found. Here J0 is the value of the current density immediately prior to the instant t*, and this value can be computed by using formula (5.75).

where

By substituting the last formula into expression (5.84), we arrive at the following ordinary differential equation for J0(t):

OH ( 2 ) A dJo(t) dHo(t) (5.86) -~ ,t = 2 dt t d----t--"


As an example, consider an important case when

Ho(t) = H0 - const (t > 0). (5.90)

For this case, Eq. (5.87) is reduced to

dlo(t) ro d-----~=-J~(t). (5.91)

This equation can be integrated by employing the following separation of variables:

dJo dt - (5.92)

J~ r0'

which leads to

where

1 ( )nl J0(t) = t + t ' (5.93)

and t' is determined from initial condition (5.89). It is interesting to note that formula (5.93) coincides with the long-

time (intermediate) asymptotics found in [16] (see also [14, 15]). These asymptotics are used to describe the phenomenon of flux creep in superconductors. To better appreciate this, we shall rewrite formula (5.93) in the form:

Jo(t) exp{ - 1 In t + t ' } = . ( 5 . 9 5 ) n - 1 r~

By assuming that

1 t + t t n>~l , t>>t' , ~ l n ( (1 , (5.96)

n - 1 r~

and by using only two terms of the Taylor expansion in formula (5.95), we find

1 t Jo(t) ~ 1 - ~ In -7- (5.97)

n - 1 r 0

This is the well-known logarithmic intermediate asymptotics, which is typical for creep phenomena. Thus, it can be concluded that long-time solutions to the nonlinear diffusion equation (5.44) are instrumental for the description of creep. The idea of using nonlinear diffusion equations for

r~---- r0 _ /z0A2k n n - 1 - 8 ( n - 1)' (5.94)

340

/t~ it2

it3

A/' 2


Hi

,k,A = z "

I

~'ll t ~ ~ r

, ~ t a t~ ,

, . t

A"'r \A 2

=Z

( t~< t2<t3)

F I G U R E 5.33

the description of flux creep can be traced back to the landmark papers of P. W. Anderson and Y. B. Kim [17] and M. R. Beasley, R. Labusch, and W. W. Webb [18].

Typical distributions of the electric current density and the magnetic field computed by using the described generalization of the critical state model for the case Ho(t) = H0 = const are shown in Figs. 5.33a and 5.33b.

Next, we intend to show that electromagnetic field diffusion in superconductors with gradual resistive transitions may exhibit a peculiar (anomalous) mode that does not exist in superconductors with ideal resistive transition. This is a standing mode. In the case of this mode, the electromagnetic field on a superconductor boundary increases with time, while the region occupied by the electromagnetic field does not expand. We shall first find the condition for the existence of this mode by using the "rectangular profile" approximation for the current density. Then we shall derive the exact expressions for the standing mode through the analytical solution of the nonlinear diffusion equation, that is, without resorting to the rectangular profile approximation. Finally, we shall compare these two results.

To start the discussion, we turn to Eq. (5.70) and try to find such a monotonically increasing boundary condition Jo(t) for which the zero front, z0(t), stands still. To this end, we assume that zo(t) = z0 = const, and, by differentiating both sides of (5.70), we arrive at

l~(t) = lz~ ~ (5.98) 2 dt "


The last expression can be t ransformed as follows:

2 dt = dJo(t) lzOknz 2 l~(t)

By integrating both parts of (5.99), we obtain

2 lzoknz~

From (5.100), we derive

Jo(t) =

~ t - 1 [J~-n(o)-j~-n(t)] n - 1

2(n-1) (ll-n(o) - lzoknz 2 t) ~

The last expression can be represented in the form:

(5.99)

where

(5.100)

(5.101)

c Jo(t) = 1 , (5.102)

(to - t)n-1

lzOknz2 )nll #~ (5.103) c = 2 ( n - l ) ' t 0 = 2 ( n - 1 ) "

Thus, we have established that, if the current density on the boundary of superconduct ing half-space varies with time according to the expressions (5.102)-(5.103), then the zero front, z0(t), of the current density stands still dur ing the time interval 0 ~< t < to. In other words, during this time interval the electromagnetic field diffusion exhibits a s tanding mode. This mode is illustrated by Fig. 5.34.

J(z,t) t 3

f

t 2

f-

t 1

f

v

z o z

FIGURE 5.34


It is desirable to express the boundary condition for the standing mode in terms of magnetic field Ho(t) at the superconductor boundary. This can be easily accomplished by using (5.102) and Amp6re's Law, which lead to:

CZo Ho(t) = 1 �9 (5.104)

(to - t) n-1

Our previous derivation has been based on the rectangular profile approximation for the electric current density. Next, we shall derive the expressions for the standing mode solution without resorting to the above approximation, but instead through analytical solution of the nonlinear diffusion equation (5.44). It is remarkable that the standing mode solution can be obtained by using the method of separation of variables. Actually, this is the only "short-time" solution that can be obtained by this method. According to the method of separation of variables, we look for the solution of Eq. (5.44) in the form:

J(z, t) = ~p(z)~p(t). (5.105)

By substituting (5.105) into (5.104), after simple transformations we derive

1 da99n(z) tzO kn d~( t ) = (5.106)

~p(z) dz 2 ~zn(t) d t "

This means that ].tO kn d~r(t) q/n(t) dt

= X, (5.107)

1 d 2~0 n(z) = X, (5.108)

qg(Z) dz 2

where X is some constant. By integrating Eq. (5.107), we easily obtain

1 [ O(t) = ( n - 1))~(t0 - t) ' (5.109)

where to is a constant of integration. Equation (5.108) is somewhat more complicated than Eq. (5.107) and

its integration is somewhat more involved. To integrate Eq. (5.108), we introduce the following auxiliary functions"

dO(z) ~ ( z ) = O(z), R(z) = d----z-" (5.110)


dR d20 d2~o n

dz dz 2 dz 2

1 = X~p(z) = X0 ~ (z). (5.111)


On the other hand,

dR dR dO dR I d(R 2) dz = dO " dz = R d---O=2 dO " (5.112)

By equating the right-hand sides of (5.111) and (5.112), we obtain

d(R 2) 1 = 2X0 ~ (5.113)

dO

By integrating Eq. (5.113), we find

n+l R(z) = 2 n ~ [ 0 ( z ) ] 2 n . (5.114)

n + l

In formula (5.114), a constant of integration was set to zero. This can be justified on the physical grounds. Indeed, the magnetic field should vanish at the zero front, that is, at the same point where J(z, t) vanishes. By using (5.105) and (5.110), it can be shown that the magnetic field and J(z,t)

1 are proportional to R(z) and 0 ~ (z), respectively. This means that these two functions should vanish simultaneously. This is only possible if the integration constant in (5.114) is set to zero.

Next, by using (5.110) in (5.114), we find

dO(z) = ~ 2n [ ] n+12_ff. dz _ n + 1X-0(z)- (5.115)

By integrating (5.115), we derive

[0(z)]nn-t __ ( n - 1) 2 1)nX(Z0 -- z) 2, (5.116)

2(n +

where z0 is a constant of integration. From (5.110) and (5.116), we obtain

1 [ ( n - 1)2X 1 ~-~-1

~0(z) = 2(n 4- 1)n (z0 - z) 2 . (5.117)

Now, by substituting (5.109) and (5.117) into (5.105), we find the following analytical (and exact) solution of nonlinear diffusion:

1 j (z , t ) = [ (n - X)lz~176 - z)2] n-1

2(n 4-1)n(t0 - t) " (5.118)

It is remarkable that, as a result of substitution, the "separation" constant X cancels out, and the solution (5.118) does not depend on X at all.


The obtained solution (5.118) can be physically interpreted as follows. Suppose that at time t - 0 the electric current density satisfies the following initial condition:

(n_1)#okn(zo_z)2 J(z, O) - [ 2(n+l)nto ] if 0 <, Z <, ZO, (5.119)

0 if Z ~> Z0.

Suppose also that the current density satisfies the following boundary condition during the time interval 0 ~< t < to:

1 I -- I n-1

(n 1)l~oknz 2 Jo(t) = / ( o , t ) - 2(n 4-1)n(t0 - t) " (5.120)

Then, according to (5.118), the exact solution to the initial boundary value problem (5.119)-(5.120) for the nonlinear diffusion equation (5.44) can be written as follows:

(n_1)#okn(zo_z)2 J(z , t ) - [ 2(n+l)n(to-t) ] if 0 <, z <~ zo, (5.121)

0 if z > z0.

This solution is illustrated by Fig. 5.35 and it is apparent that it has the physical meaning of the standing mode. It is also clear from formula (5.121) (and Fig. 5.35) that the above solution has the following self- similarity property: the profiles of electric current density for different instants of time can be obtained from one another by dilation (or contraction) along the J-axis. In other words, these profiles remain similar to one another. This suggests that solution (5.121) can be derived by using the dimensionality analysis. However, we shall not delve further into this matter.

J(z,t) t 3

Z

FIGURE 5.35


From the practical point of view, it is desirable to express the boundary condition (5.120) for the standing mode in terms of magnetic field Ho(t) on the superconductor boundary. According to Amp6re's Law, we have

Ho(t) - fo z~ l(z, t) dz. (5.122)

By substituting (5.121) into (5.122) and performing integration, we obtain

1

I I n-1 Ho(t) = n - 1 ( n - 1)txoknz~ n + I z~ 2(n + 1)n(to - t) " (5.123)

It is also instructive to compare the above exact standing mode solution with the standing mode expressions derived on the basis of rectangular profile approximation. First, it is clear from formula (5.121) (and Fig. 5.35) that, for sufficiently large n, the actual current density profiles for the standing mode are almost rectangular. Second, it is apparent that the boundary condition (5.120) can be written in the form (5.102) with c and to defined as follows:

1

I I n-1 (n 1)l~oknz 2 c = 2(n 4- 1)n ' (5.124)

( n - 1)txoknz~j~-n(o) to = . (5.125)

2(n 4- 1)n

By comparing (5.124)-(5.125) with (5.103), it can be observed that for sufficiently large n these expressions are practically identical. Thus, the rectangular profile approximation is fairly accurate as far as the prediction of the standing mode diffusion is concerned. This brings further credence to the rectangular profile approximation.

The origin of the standing mode can be elucidated on physical grounds as follows. Under the boundary condition (5.120), the electromagnetic energy entering the superconducting material at any instant of time is just enough to affect the almost uniform increase in electric current density in the region (0 ~< z ~< z0) already occupied by the field but insufficient to affect the further diffusion of the field in the material.

In the discussion presented above, the method of separation of variables has been used in order to find the "short-time" solution, which describes the standing mode of nonlinear diffusion. It turns out that the same method of separation of variables can be used in order to study "long-time" solutions, which describe the phenomenon of flux creep. As has been demonstrated by E. H. Brandt [15], these "long-time" solutions

Z

Bo t,

CHAPTER 5 Superconducting Hysteresis 346

FIGURE 5.36

can be found in quite general situations. In our presentation below, we closely follow the paper [15] of E. H. Brandt.

Consider a long superconducting cylinder of an arbitrary cross- section subject to uniform magnetic field B0 directed along the y-axis (see Fig. 5.36). This magnetic field induces currents J in the superconductor and these currents are directed along the z-axis. The vector magnetic potential A is also directed along the z-axis and it is given by the following formula:

A(r , t )= /~0 [ j ( r , , t ) ln l r_r , ld2r , +A0(r), (5.126) 2rr ds

where A0 is the vector magnetic potential of the external field:

A0(r) - -xBo. (5.127)

0B and c u r i a = B we conclude that From equations curl E - --~-

E(r, t) = - ~ A ( r , t). (5.128)

Combining formulas (5.126), (5.127), and (5.128), we derive

~s O Bo E(r, t ) = #0 3l(r',t) lnlr_ r~ I d2 i -4- x - - . (5.129) Ot Ot

Now we consider "long-time" solutions when the external magnetic field is maintained constant:

OBo B0 = const, = 0. (5.130)

Ot


By subst i tut ing (5.130) into Eq. (5.129) and taking into account power law (5.41), we arrive at the fol lowing nonl inear integro-differential equation:

1 E(r, t) -- ~lz~ ~s 3E~(r',t).ot l n [ r - r~l d2r ~. (5.131)

We look for a nonzero solution of this equat ion in the form:

E(r, t) = ~0(r) . (5.132)

By subst i tut ing formula (5.132) into Eq. (5.131), we find that the above equat ion is satisfied if

Or = - - 1, (5.133)

n

which yields

n = 1 - n" (5.134)

This means that solution (5.132) takes the form:

( t t lnn E(r, t) - ~(r) , (5.135)

and Eq. (5.131) is reduced to the fol lowing nonl inear integral equation:

-#0k ~ 1 r ! ~0(r) = 27cr(n- 1) ~ b~ l n l r - l d21" (5.136)

As n >> 1 and r can be chosen sufficiently large, Eq. (5.136) can be solved numerical ly by using contraction m a p p i n g iterations [15].

Hav ing compu ted ~0(r), we can find the electric current density:

1 l ( t ) T=-~ J(r,t) = k[~0(r)] 5 - sign~0(r). (5.137)

r

To investigate the intermediate asymptot ical behavior of the current density, we shall employ the formula

1 ( t ) l-n { 1 t }

- - exp - In - . (5.138) r n - 1 r

Since n >> 1, there are a lways such times that

1 In t - << 1. (5.139) n - 1 r


For such times, we can retain only two terms of the Taylor expansion of the exponent in formula (5.138). This leads to the following intermediate logarithmic asymptotics:

1[ 1 t] /(r, t) ~ kilo(r)] ~ 1 - ~ In sign ~o(r), (5.140)

n - 1

which is typical of creep phenomena.

5.4 NONLINEAR DIFFUSION IN ISOTROPIC SUPERCONDUCTORS WITH GRADUAL RESISTIVE TRANSITIONS (CIRCULAR POLARIZATION)

In the previous sections of this chapter, nonlinear diffusion of linearly polarized electromagnetic fields was discussed. Our analysis was based on the solution of scalar nonlinear diffusion equations. Now we turn to the discussion of more complicated and challenging problems, where electromagnetic fields are not linearly polarized. This will require the solution of vector nonlinear diffusion equations. We begin with the simplest case when electromagnetic fields are circularly polarized and superconducting media are assumed to be electrically isotropic. We shall still use the power law as the constitutive relation for the isotropic superconducting media. In the vector case, the power law (5.41) can be written as follows:

~x(~x, n~-~ (j~2 + n)~x, (5.141)

~(~x, ~ - ~ (~2 + ~ ) ~ , (5.142)

where

cr (~/E 2 if- E~) -o'([E[)-k[E[ 1-1. (5.143)

By using the above formulas, the problem of nonlinear diffusion of circularly polarized electromagnetic fields in the superconducting half-space z ~ 0 can be framed as the following boundary value problem: find the time periodic solution of the coupled nonlinear diffusion equations

~2~x ~ [o(j~2+~)~x] ~5144~ Oz 2 = a o ~

~2~y ~[~(~2+~)~y] ~5145~ Oz2 = aos~

5.4 NONLINEAR DIFFUSION IN ISOTROPIC SUPERCONDUCTORS 349

subject to the following boundary conditions:

Ex(0, t) - Em cos(cot 4- 00),

Ey(O,t) = E m sin(cot 4- 0o),

EK(~ , t) = Ey(~, t) = 0.

(5.146)

(5.147)

(5.148)

It is apparent that the mathematical structure of nonlinear differential equations (5.144)-(5.145)and boundary conditions (5.146)-(5.148)is invariant with respect to rotations of the x- and y-axes around axis z. This suggests that the solution of the boundary value problem (5.144)-(5.148) should also be invariant with respect to the above rotations. The latter implies that the electric field is circularly polarized everywhere within the superconducting media:

Ex(z, t) - E(z) cos[cot 4- 0(z)], (5.149)

Ey(z, t) - E(z) sin[cot 4- 0(z)]. (5.150)

Now we formally demonstrate that the circularly polarized solution (5.149)-(5.150) is consistent with the mathematical structure of the boundary value problems (5.144)-(5.148). First, it is clear from the above formulas that

[E(z,t)[ - ~/E2(z, t) 4- E~(z,t) - E(z). (5.151)

This means that the magnitude of the electric field, as well as the conductivity rr(IEI), do not change with time at every point of the superconducting media.

Next, we shall write formulas (5.149) and (5.150) in the phasor form:

F:x(Z) = E(z)e j~ (5.152)

It is apparent that

and

F:y(z) -- - j E (z)eJ~ (z). (5.153)

IE(z , t ) ] - [Ex(z)[ : (5.154)

(I E<z, t)I) -- (l x<Z)I) -- (I I). <5.155) Now, by using phasors (5.152) and (5.153) as well as the formula (5.155), we can exactly transform the boundary value problem (5.144)-(5.148) into the following boundary value problems for f~x(Z) and f~y(z), respectively:

aa~x dz 2 - jcottorr (IEKI)Ex, (5.156)

Ex(0) - Em, Ex(oO) - 0, (5.157)


and

where

d2Ey dz 2 = jo~#oa (] f~y]) Ey, (5.158)

Ey(O) = - jEm, F:y(CX~) -- O, (5.159)

F:m - Em ejO~ (5.160)

The above exact transformation can be construed as a mathematical proof that the circular polarization of the incident electromagnetic field is preserved everywhere within the superconducting media. This also proves that there are no higher-order time-harmonics of the electric field anywhere within the media despite its nonlinear properties.

From the purely mathematical point of view, the achieved simplification of the boundary value problem (5.144)-(5.148) is quite remarkable. First, partial differential equations (5.144)-(5.145) are exactly reduced to the ordinary differential equations (5.156) and (5.158), respectively. Sec- ond, these ordinary differential equations are completely decoupled. Fi- nally, these decoupled equations have identical mathematical structures. As a result, the same solution technique can be applied to both of them.

To solve Eqs. (5.156) and (5.158), we shall first slightly transform them. According to formulas (5.143) and (5.155), we have

a (]/~x[) - k]Ex] 1-1, (5.161)

a(IEy[) = k]F:y] 1-1, (5.162) 1 1

am m~ rY (]Em ]) = k]Em ]~- . (5.163)

From the last three expressions, we derive

EK 1-1 o'(]Ex]) -- am ~ , (5.164)

1_1

o'(IEy])--rym ~ �9 (5.165)

By substituting formulas (5.164) and (5.165) into Eqs. (5.156) and (5.158), respectively, we find

d2Ex E,K 1-1 dz 2 - ja~#orrm -~m Ex, (5.166)


d2Ey dz 2 = jmtxOO'm Ey

Ym

1_ 1 Ey. (5.167)

The solution to Eq. (5.166), subject to the boundary conditions (5.157), can be sought in the form:

where

( z)~ FZx(Z)-- Em 1 - ~ , (5.168)

= ~' 4- jd ' . (5.169)

Here, z0,c~' and ~,I are parameters, which will be appropriately chosen to guarantee that Fzx(Z) given by formula (5.168) satisfies Eq. (5.166).

It is important to keep in mind that expression (5.168) is an abbreviated form of the solution. In other words, it is tacitly understood that this form is valid for 0 ~< z ~< z0, while for z ~> z0 the solution is equal to zero.

From formulas (5.168) and (5.169), we derive

I ~zx(z)~zml- ( l - Z ) ~ ' ~ , (5.170)

which leads to 1-1 o/(n-1) ( z ) n

/~m = 1 - ~ . (5.171)

By substituting formulas (5.168) and (5.171) into Eq. (5.166), we find

CO(Or -- 1)Em (1 - ~0) a-2 ^ ( ~0) Or ~ --jcolzoamz~Em 1 - . (5.172)

It is clear that equality (5.172) will hold, if the following two conditions are satisfied:

c~l(n- 1) 2 = - - , (5.173)

n and

c~(a- 1)=jcolzoamZ~. (5.174)

From the first condition, we immediately find

2n c~' = ~ . (5.175)

n - 1

The second condition (5.174) can be construed as a characteristic equation, which can be used for the determination of c~" and z0. This is an equation


in terms of complex variable a. It can be reduced to the following two real equations:

c~'(a' - 1 ) - (Or 2 - - 0, (5.176)

c~" (2c~' - 1) -- o~#OamZ 2. (5.177)

By using formulas (5.175) and (5.176), we derive

,, v/2n(n + 1) = . (5.178)

n - 1

By substituting expressions (5.175) and (5.178) into (5.177), we arrive at the following expression for z0"

[2n(n + 1)(3n + 1)2] 1 z0 = . (5.179)

( n - 1)~/o~/Z0am

Formulas (5.168), (5.169), (5.175), (5.178), and (5.179) completely define the phasor F:x(Z) as the solution of the boundary value problem (5.156)- (5.157). The boundary value problem (5.158)-(5.159) has the same mathematical form (structure) as the boundary value problem (5.156)-(5.157). For this reason, the solution to the boundary value problem (5.158)-(5.159) can be written as follows"

Ey(z) -jEm (1 - z~) ~ = , ( 5 . 1 8 0 )

where, as before, a and z0 are given by formulas (5.169), (5.175), (5.178), and (5.179).

Expressions (5.168) and (5.180) can be converted from the phasor forms into the time-domain forms. This yields

2n

Ex(z,t)=Em(1-~O) n-1

[ v /an(n+l) ( ~0)] x cos ~ot + 00 + In 1 - (5.181) r / - 1

2n ( Z) l Ey(z,t)= Em 1 - ~

[ v /2n(n+l) ( ~}] x sin o~t + 00 + In 1 - . (5.182)

n - 1

The above formulas give the exact analytical solution to the boundary value problem (5.144)-(5.148) for coupled nonlinear diffusion equations.


This is a highly symmetric solution, which is invariant (up to a choice of initial phase 00) with respect to rotations of axes x and y.

By using the last two formulas along with expressions (5.141)-(5.143), we obtain the following relations for the electric current densities:

2

Jx(z , t ) -Jm(1-~oo) n-1

[ v /Sn(n+l) ( z ) ] x cos cot + 00 + In 1 - (5.183)

n - 1 ~ ' 2

Jy ( z , t )=Jm(1 -~o ) n-1

[ v/Sn(n+X) ( z~)] x sin cot + 00 + In 1 - (5.184) n - 1

where

Jm = crmEm. (5.185)

From the above relations, we find 2

]J(z) I -- Jmx(Z) -- Jmy(Z) - Jm ( 1 - ~0 ) n-1 �9 (5.186)

We see that for n >> 1 profile of IJ(z)l is almost rectangular. A typical plot of this profile is shown in Fig. 5.37. We can also observe the logarithmic variation of phase with respect to z. As a result, for any fixed time t, electric current densities Jx and Jy (as well as electric fields Ex and Ey) have infinite numbers of zeros (infinite numbers of oscillations) in the interval O<,z <~zo.

IJI

Z o ~ Z

F I G U R E 5.37


Up to this point, it has been assumed that the electric field components are specified on the boundary of the superconducting half-space z ~ 0. However, in applications it is more convenient to specify the boundary values of the magnetic field components. For this reason, we shall express the above solutions (5.181)-(5.182) in terms of the magnetic field at the boundary. To do this, we shall invoke the equations

dEx_ _ -j~ol~of-ty, (5.187) dz

dz = Ja~#~ (5.188)

as well as formulas (5.168) and (5.180). As a result, we obtain

f-Ix(Z) /2/m (1 - z~) ~-1 - , (5.189)

where

I-Iy(Z)~--j~-tm(1 ~o) ~-1 - , ( 5 . 1 9 0 )

/2/m- c~ Era. (5.191) co#0z0

If/2/m is given, then the last relation can be construed as a nonlinear equation for/~m. This equation is nonlinear, because z0 depends on r~m, which is a nonlinear function of Era. To make this nonlinearity manifest, we use formula (5.179)in (5.191)and derive

Hm --[o~[~ am n - 1 Em. (5.192) w#0 [2n(n 4- 1)(3n 4- 1)2]�88

Now, by recalling formula (5.163), we transform the last equation as follows:

la[/ k n - 1 E 2 nl+n Hm V (5.193) o~/~0 [2n(n 4- 1)(3n 4- 1)2] 1

From expressions (5.169), (5.175), and (5.178), we find

Ic~l = v/2n(3n + 1). (5.194) n - 1

By substituting formula (5.194) into (5.193), we end up with the nonlinear equation for Era:

) 1 Hm-- k 2n ~ l+n

n + l Em2n " (5.195)


Consequently,

and

n n

Em = ' 2n

a n

H n + l (5.196) m ,

l - -F/

2n l_n (n + 1t 2(-G- ~ Hm n+12(1-n) rYm = k~X-1 (~o/z0) ~-~ . (5.197)

2n

Thus, for any given H m we can find am from formula (5.197), and then z0 from formula (5.179). Having found z0, we can use formulas (5.189), (5.190), and (5.191) for calculations of H x ( z ) , H y ( z ) , and Em.

Next, we consider the surface impedance of the superconduct ing half- space. This impedance is defined as follows:

G(o) ~y(o) 77 =/2/y(0 ) -- /2/x(0 ) (5.198)

From formulas (5.168), (5.190), (5.191), and (5.198), we find

j~oaozo r /= ~ . (5.199)

O/

By using the polar form of the impedance

17 -- ]~ [e j~~ (5.200)

from Eq. (5.199), we derive

wttozo I ~ I - ~ , (5.201)

0/f

tan ~0 = ~-;. (5.202)

By invoking formulas (5.179) and (5.194), from (5.201) we find 1

( n + 1 ) ~ /totto (5.203) [r /[- 2n V rrm "

Similarly, from formulas (5.175), (5.178), and (5.202), we arrive at

~ / 2 n tan ~o = n + i" (5.204)

The last two formulas are remarkably simple. In the part icular case of n = 1 (linear media), these formulas lead to the wel l -known expressions:

~/tottO Jr Irl[ = , ~0 = - . (5.205)

V rrm 4


It is important to note that the magnitude of the surface impedance is field dependent. This is clearly seen from formula (5.197). In contrast, the phase ~p of the surface impedance is not field dependent. It is determined only by the sharpness of the resistive transition. Figures 5.38 and 5.39 show,

respectively, the dependence of Jr/i//o~0 and tan~p on the exponent n, V am

which is the natural measure of the sharpness of the resistive transition.

Iill ~ O ~ o ,

(~m 1.00

0.98

y 0 . 9 0 ~

0 . 8 8 ~

0.86 . . . . . , . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . .

10 20 30 40

FIGURE 5.38

n

55

54

53

52

51 . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . .

10 20 30 40 n

FIGURE 5.39


In another limiting case n = ~ (sharp transition), from formulas (5.203) and (5.204) we derive

~ tort0 tan = ~Y~. (5.206) ~a

Now we turn back to formulas (5.189) and (5.190) and convert them into the t ime-domain form. As a result, we obtain

n+l

Hx(Z,t)--Sm(1-~o)n-1

I v / 2 n ( n + l ) ( z ~ ) ] x cos tot + In 1 - (5.207) n - 1

n+l ( Z ) n-1

Hy(z,t)=Hm 1- G

x sin tot + In 1 - (5.208) n - 1 ~0 '

where, for the sake of simplicity, it is assumed that the initial phase of the magnetic field at the boundary is equal to zero.

Next, we shall show that, in the limiting case of sharp resistive transitions (n - cx~), the last two expressions are reduced to those that were asserted by C. Bean in the paper [19]. To this end, from formula (5.191), we find

Hm = Jm. ( 5 . 2 0 9 ) co~OCrmZO

From the last equation as well as formulas (5.179) and (5.194), we obtain 1

( an )~Jm (5.210) * J t o ~ 0 C r m - - n + 1 H---~"

By substituting the last expression into formula (5.179), we arrive at

v/(n 4- 1)(3n 4- 1) Hm z0 - ~ . (5.211) n-1 Jm

In the case of sharp resistive transitions,

lm =Jc, (5.212)

and the ratio

Hm Hm = = (5.213)

Jm Jc


can be construed as the field dependent (Bean) penetration depth, which we dealt with for linear polarizations of magnetic fields (see formula (5.11)). By using expression (5.213) in formula (5.211), in the limiting case of n ---> ~ , we obtain

z0 = x/38. (5.214)

Thus, the penetration depth in the case of the circular polarization is ~/3 times larger than in the case of linear polarization.

Finally, by substituting formula (5.214) into expressions (5.207) and (5.208) and letting n go to infinity, we derive

t z l [ t zt] Sx(z,t)- Sm 1 - ~ - ~ cos cot 4- x/21n 1 - ~ - ~ , (5.215)

z z Hy(z,t) = Hm(1- -~) sin[o~tq- x/21n(1- -~t ]. (5.216)

The last expressions are identical to those published in the paper [19]. The results discussed up to this point were obtained for the supercon-

ducting half-space. However, they can be easily extended to the case of a superconducting slab of finite thickness A if the penetration depth z0 satisfies the following inequality:

A z0 K 2" (5.217)

Under condition (5.217), nonlinear diffusion of electromagnetic fields at each side of the superconducting slab occurs in the same way as in the case of the superconducting half-space. As a result, we obtain the following formulas:

/2/m (1 _ -~ +z)a-1 -Go

o

/2/m (1_ -}-z~ot-1

if --~ E z E --~ + z0,

i f + z0 z - z0 ,

if -~ - z 0 ~ z ~ -~,

(5.218)

-jHm(1- ~+z~a-1 -To,

G(z)= o -jHm(1- -~-z~a-1

-To,

i f - @ ~< z <~ --~ + z0,

if --~ 4- z0 ~ z ~ -~ - z0,

A if -~ - zo ~< z ~< ~-.

(5.219)

Similar generalizations can be given for the electric field and current density. It would be interesting to find the solution to the nonlinear diffusion

a problem in the case when z0 > ~.

5.5 CASE OF ELLIPTICAL POLARIZATIONS 359

5.5 N O N L I N E A R DIFFUSION IN THE CASE OF ELLIPTICAL POLARIZATIONS A N D A N I S O T R O P I C M E D I A

In this section, we shall use the perturbation technique in order to extend the results from the previous section to more complicated situations.

To start the discussion, consider a plane electromagnetic wave penetrating the superconducting half-space z > 0. The magnetic field at the boundary of this half-space is specified as follows:

Hx(O,t)- Hm[cos(cot 4- y) 4- efx(t)], (5.220)

Hy(O,t) = Hm[sin(cot 4- y) 4- Ely(t)],

where e is some small parameter, while fx(t) and fy(t) are given periodic functions of time with the period 2~. (z)

It is apparent that this plane wave can be construed as a perturbation of the circularly polarized plane wave. By using the Maxwell equations, we find that the distribution of electric field in the half-space z > 0 is governed by the following coupled nonlinear partial differential equations:

32Ex 3Jx(Ex, Ey) (5.221) OZ 2 = ~ 0 3t '

02 Ey OJy(Ex, Ey) (5.222) 3z 2 = #o 3t "

subject to the boundary conditions

8Ex (0, t ) - -#oHm[cOCOS(Cot 4- y)4- G(t)],

3z

3Ey (0,t)= -#oHm[cosin(cot 4- y ) - eft(t)], 3z

(5.223)

(5.224)

Ex(cX~) = Ey(cx~) = O, (5.225)

where functions Jx(Ex, Ey) and Jy(Ex, Ey) are specified by formulas (5.141), (5.142), and (5.143).

Next, we shall look for the periodic solution of the boundary value problem (5.221)-(5.225) in the following form:

Ex(z, t) - E~ t) 4- eex(z, t), (5.226)

0 (z, t) 4- e@(z, t). Ey(z,t) = Ey (5.227)

By substituting expressions (5.226) and (5.227) into Eqs. (5.221) and (5.222) and boundary conditions (5.223)-(5.225) and by equating the terms of like


powers of e, we arrive at the following boundary value problems for E 0, E~ and ex,@:

and

32E o 3Jx(E ~ E~) 32E~ 3Jy(E ~ E~) 8 Z 2 - - ~ 0 8t " 8z 2 = tto 8t ' (5.228)

E 0 3 -x (0, t) = -calzoHm cos(cat q- y), 3z

(5.229)

OE o 3z

~ ( 0 , t) =-cat, toHm sin(cat + F), (5.230)

o E~ = Ey(oo)= O, (5.231)

[ 1 32ex O 3Jx (E0, E~)ex + (E ~ E~)@ , (5.232)

32@ 3131Y oJy E~)@], (5.233) Oz 2 = t~0 ~ G (E~ E~)eK + -~y (E ~

3ex (0, t) -- -ttoHmf~(t), (5.234) 3z

0@ (0, t ) - ttoHmf~(t), (5.235) 3z

ex(e~, t) = @(cx~, t) = 0. (5.236)

The boundary value problem (5.228)-(5.231) describes the penetration of circularly polarized plane wave into the superconducting half-space. The solution to this problem has been found in the previous section. For the case when the initial phase, y, is such that the initial phase of E ~ on the boundary (z = 0) is equal to zero, this solution is given by the following expressions:

z cos(o~t+o(z)) (5.237) EO(z,t)=Em 1 - ~ 0

( )o z E~(z,t)-- Em 1 - ~ sin(cat + O(z)), (5.238)

4v/2n(n + 1)(3n + 1)2 1 1 zo - , am=kErn , (5.239)

~/calzOCrm(n - 1)

( z ) O(z) = oe" In 1 - ~ , (5.240)


a , 2n a , , v/2n(n + 1) (5.241) n - l ' - n - 1 '

and Em can be found from the nonlinear equation:

Hm = lot' § i~"] Em. (5.242) w~OZO

By substituting (5.237) and (5.238) into Eqs. (5.232) and (5.233) and by using expressions (5.141)-(5.143), after simple but somewhat lengthy transformations we arrive at the following equations for ex and @:

32ex(Z , t) ( Z ) -2 3Z 2 - - #Oam 1 - -~0

3 [ { l + n 1-ncos(2~ot+20(z) )}ex(z , t ) x 0--7 2n + 2n

1 - n sin(2~ot + 20(z))@(z, t)l, (5.243) + 2n 3

32ey(z, t ) ( z ) -2 3z 2 = lzo 1 - -~o

3 11 - n sin(2~ot + 20(z))ex(Z, t) x 3--7 2n

{ l + n X - n (2o~t+aO(z))}@(z,t)] (5.244) + 2n 2----7- cos

Equations (5.243) and (5.244) are coupled linear partial differential equations of parabolic type with variable in time and space coefficients. We would like to find the periodic solutions of these equations subject to the boundary conditions (5.234)-(5.236). To this end, we introduce new complex valued state variables:

�9 (z, t) = ex(Z, t) + j@(z, t), (5.245)

!/r(z, t) -- ex(z, t) - j@(z, t). (5.246)

By using these state variables, and some simple transformations, we can represent Eqs. (5.243) and (5.244) in the following form:

02~ 1 - n ( 1 _ z~)-2 3z 2 = lzOr~m 2n

3 I I + n ~ + ( 1 z~) jaee" " ] -- - eJ2~~ ~ , (5.247) x o t 1 - n


1 n(1 2 8z 2 = I~OCrm 2n

3 [ l + n~r + ( l - - ~o )-J2~"e-J2~tcI) ]. (5.248) x8~ 1 - -n

Assuming that functions fx(t) and fy(t) in boundary conditions (5.234) and (5.235) are functions of half-wave symmetry (the case that is usually of most practical interest), we conclude that ex(z,t) and @(z,t) will also be the functions of half-wave symmetry. For this reason, we will use the following Fourier series for ~(z, t) and ~(z, t):

o o

~(Z, t) -- y ] ~2k+l(Z) ej(2k+l)c~ (5.249) k=-~

o o

@(z, t) - ~ ~r2k+l(z)e j(2k+l)c~ (5.250) k=-~

It is clear from (5.245), (5.246), (5.249), and (5.250) that

~k+l (Z) -- ~r_2k_ 1 (Z), (5.251)

~r2~k+ 1 (Z) -- (D_2k_ 1 (Z), (5.252)

where the superscript " , " means a complex conjugate quantity. By substituting (5.249) and (5.250) into (5.247) and (5.248) and by

equating the terms with the same exponents, after simple transformations we derive

( Z - ~ ) 2 d2~ak+l [ ( Z ) jaa" ] 1 - -d~z2 = jX2k+l a~2k+l if- 1 -- ~0 q2k-1 , (5.253)

( Z ) 2da~ak-1 I ( Z ) -j2~'' ] 1 - G ~ -~jXak-1 a~2k-1 if- 1 - G ~2k+1 , (5.254)

(k = 0 ,+1,•

where we have introduced the following notations:

l + n 1 - n ~ . a = 1 - n ' X2k+l -- (2k + 1)o~#0Crm 2n (5.255)

Thus, we have reduced the problem of integration of partial differential equations (5.247)-(5.248) to the solution of an infinite set of ordinary differential equations with respect to Fourier coefficients ~2k+1 and ~r2k_ 1. These simultaneous equations are only coupled in pairs. It allows one to solve each pair of these coupled equations separately. After ~2k+1 and


~r2k_ 1 are found, we can compute aa(z,t) and O(z,t) and then ex(z,t) and @(z,t). Another simplification is that according to (5.251) it suffices to solve coupled equations (5.253) and (5.254) only for nonnegative values of k.

We shall seek a solution of the coupled equations (5.253) and (5.254) in the form:

cI)2k+l (Z) -- a2k+l (1-- ~O ) t, (5.256)

~2k-1 B2k-l (1-- ~O ) fl-j2~" - . (5.257)

By substituting (5.256) and (5.257) into (5.253) and (5.254), we end up wi th the following simultaneous homogeneous equations with respect to A2k+l and B2k-1:

(f12 _ fl _ jX2k+lZ2a)aak+l _ jX2k+lZ2Bak_l = O, (5.258)

�9 2 --JX2k-lZoA2k+l

4- [(fl -- j2ol") 2 -- (fl --j2cr (5.259)

The above homogeneous equations have nonzero solution for A2k+l and B2k-1 if and only if the corresponding determinant is equal to zero. This yields the following characteristic equation for t :

(f12 _ fl _ jX2k+lZ2a)[(fl _ j2c~,,) 2 _ (fl _ j2c~") - jX2k_xz2a] (5.260)

-Jr- X2k+lX2k_l z4 = O. From expressions (5.255) and (5.239), we find

(2k 4- 1)(3n 4- 1)(n 4- 1 ) / n 4- 1 X2k+lZ~a (5.261)

(n - 1) 2 V 2n '

X2k+lX2k-lZ~ = (4k2 - 1)(3n 4- 1)2(n 4- 1) , /n __+ 1 (5.262) ( n - l ) 2 V 2n "

From the last two expressions we conclude that the coefficients of the characteristic equation (5.260) depend on the exponent, n, of the power law and k. Consequently, the roots of this equation also depend only on n and k. It can be proven that the above characteristic equation has two roots ~l(n, k) and fl2(n, k) with positive real parts. After these roots are found, the solution of coupled equations (5.253) and (5.254) can be represented in the


form:

A (1) ( 1 _ Z ) fll A (2) (1_ Z) t2 (I)2k+l (Z) = ""2k+l ~00 q-" ~2k+l ~00 1 (5.263)

R(1) (l Z) fil-j2c~" R ( 2 ) ( l Z ~ f12-j2a" - - , (5.264) ~2k-1 (Z) -- "-'2k-1 ~00 -}- "-'2k-1 ZO J

where for the sake of notational simplicity we have omitted the dependence of fll and ~2 o n n and k.

From boundary conditions (5.234)-(5.235) and expressions (5.245)-

(5.246), we obtain the following equations for A (1) A (2) R(1) and 2k+1' 2k+1' "-'2k-1 B (2) .

2k-1

A (1) a A(2) t �9 l ~1"-2k+l "4-P2r "ff]f'x,2k+l]' (5.265)

(i l l - J2c~") B(1)2k-1 -ff ( ~ J 2 - j2c~") B~k )_ 1

-- - -Z0"0 Hm [ G , 2 k - 1 -- Jf~c,2k-1]' (5.266)

�9 2 (1) ; . z2B(1) (~2 ~1 ]Xak+lzoa)aak+X 1 -- -- - -JX2k+l 0 2k- ~ 0 ' (5.267)

�9 2 ( 2 ) ; . Z2n(2) (~2 ~2 ]Xak+lzOa)Aak+l = 01 -- -- -- JX2k+l 0D2k_l (5.268)

wherefx~,2k+l andf;,2k+ 1 are complex Fourier coefficients off~ andre. By solving simultaneous equations (5.265)-(5.268), we can find co-

efficients A(1) A(2) n(1) and n(2) Then, by using (5.263), (5.264), "12k-1' " ~2k+1' "-'2k-1 "-'2k-1" (5.249)-(5.250) and (5.245)-(5.246), we can determine perturbations ex(z, t) and @(z,t), which in turn can be used in (5.226)-(5.227) to compute the total electric field.

Consider the particular case when

fx(t) = cos~ot, fy(t) = sin~ot. (5.269)

This case corresponds to elliptical polarization of the incident field. It is easy to see that in this case the right-hand sides of Eqs. (5.265) and (5.266) are equal to zero for all k except k = 1. This means that only first and third harmonics are not equal to zero. We have reached this conclusion because we have considered only first-order perturbations with respect to e. If we consider higher-order perturbations with respect to e, we shall recover higher-order harmonics of electric field.

So far, we have dealt with isotropic superconducting media. Now, we proceed to the discussion of nonlinear diffusion in anisotropic media. The first question to be addressed is how the power law that describes gradual

5.5 CASE OF ELLIPTICAL POLARIZATIONS 3 6 5

resistive transitions can be generalized to the case of anisotropic media. A reasonable generalization of the power law is given by the following formulas:

1_ 1

Jx(Ex, Ey) = (1 + e)kEx(v/(1 + e)E2x + (1 - e )E~)~ ,

Jy(Ex, Ey) - ( 1 - e)kEy(v/(1 + e)E 2 + ( 1 - e)E~) 1-1,

(5.270)

(5.271)

where e is some relatively small parameter, which accounts for anisotrop- icity of media. It is clear that the superconductor properties enter into Eqs. (5.270) and (5.271) through parameters n, e, and k.

In the limiting case of e = 0, expressions (5.270) and (5.271) are reduced to

- kE -~-lEx, (5.272) jx(~ = ( ~ / ) 1 1 ~,,-,x, Ey) kEx E2+E~ -~-

-- kE 1-1Ey, (5.273) 1_ 1

_ +

which are constitutive relations for isotropic superconducting media with gradual resistive transitions described by the power law.

Thus, the anisotropic media with constitutive relations (5.270) and (5.271) can be mathematically treated as perturbations of isotropic media described by the power law. This suggests that the perturbation technique can be very instrumental in the mathematical analysis of nonlinear diffusion in anisotropic media with constitutive relations (5.270) and (5.271).

Formulas (5.270) and (5.271) lead to power law-type resistive transitions along the x- and y-axis:

1 Jx(Ex) = kx]Ex] ~ sign Ex, (5.274)

1 Jy(Ey) = kyIEy]-~ sign Ey, (5.275)

l+n with kx = k(1 + e) ~ and ky - k(1 - e) a n .

In the limiting case of n = cx~, expressions (5.274) and (5.275) describe ideal ("sharp") resistive transitions with critical currents jc = (1 + e)k and J~ - (1 - e)k. It is also important to note that the Jacobian matrix for J(E) defined by Eqs. (5.270) and (5.271) is symmetric. This guarantees the absence of local cyclic (hysteretic type) losses.

Now consider a plane circularly polarized electromagnetic wave penetrating the superconducting half-space z > 0. The magnetic field on the boundary of this half-space is specified as follows:

Hx(0, t) = Hm cos(~ot + }I), (5.276)

3 6 6 CHAPTER 5 Superconducting Hysteresis

Hy(0, t )= H m sin(cot 4- y). (5.277)

By using the Maxwell equations, it is easy to find that the distribution of electric field in the half-space z > 0 satisfies the following coupled nonlinear partial differential equations

32Ex 3z 2 m ~o

32Ey 3Z 2 - - ],t o

subject to the boundary conditions:

OJx(Ex, Ey) 3t

OJy(Ex, Ey) 3t "

(5.278)

(5.279)

OEx ~z (O, t) -- -ttocoHm cos(cot + y),

(5.280) 3Ey (0,t) -- -ttocoHm sin(cot + y), 3z

Ex(cX)) = Ey(cx)) = 0. (5.281)

Next, by using the perturbation technique, we shall look for the solution of the boundary value problem (5.278)-(5.281) in the form:

Ex(z, t) - E~ t) + eex(z, t), (5.282)

Ey(z, t) = E~(z, t) 4- eey(z, t). (5.283)

We shall also use the following e-expansions for constitutive relations (5.270) and (5.271):

JK(EK, Ey) = J~ (EK, Ey)

[ 1 - n E2-E~] 4-eJ~ Ey) 14- 2n " E 2 4 - ' " ' (5.284)

ly(Ex, Ey) = J (EK, Ey)

-eI~(Ex'Ey)[ 1 - I - n a n " E2 - . . . , (5.285)

where J~ Ey) and J~(Ex, Ey) are defined by expressions (5.272) and

(5.273), respectively, while E - ~/Ex 2 4- E~.

By substituting expressions (5.284)-(5.285)into Eqs. (5.278)-(5.279) and boundary conditions (5.280)-(5.281), and equating the terms of like powers of e, we end up with the following boundary value problems for


E ~ E~ and ex,@"

O2EO ojO(EO, E~) O2E~ 0 0 OI'y(EK, 3z 2 = leO 3t ' 3z 2 = leO 3t

E 0 O -x (0, t) -- -o)leoHm cos(o)t 4- y), Oz

E~ (0, t) = -wleoHm sin(wt 4- y), 3 3z

=o, = Ey

and

32ex 3 ( 3 J 0 3J 0 ) 3z--- T - leo -~ -~x (EO" E~)ex 4- -~y (E O, E~)ey

(5.286)

(5.287)

(5.288)

[Jx ( ( 0 2 )1 3 o o 1 - n EK) -(E~)2 (5.289) = leo-~ (E x, E~) 1 4- an " (E0) 2 '

02ey 0 ( OJ~ oJ~ ) O z---- T - le o -~ \ -~x ( E ~ " E ~ ) ex 4- -~y ( E ~ , E ~ ) @

[J~ ( - ( ~ (5.290) 3 1 n Ex) = - l e 0 ~ (E~ 1 - an " (E0) 2 '

8ex (0, t) = 8ey Oz -O-~-z (0't) = 0, ex(oo, t) = ev(oc, t) = O.

The boundary value problem (5.286)-(5.288) describes the diffusion of circularly polarized electromagnetic wave in the isotropic superconducting half-space z > 0. The solution to this problem has been found in the previous section. For the case when the initial phase y in (5.287) is such that the initial phase of E ~ on the boundary (z = 0) is equal to zero, this solution is given by formulas (5.237)-(5.242).

By substituting (5.237) and (5.238) into Eqs. (5.289) and (5.290) and by using expressions (5.272) and (5.273), after straightforward but somewhat lengthy transformations we derive the following equations for ex and @:

(5.291)

32ex ( z ) - 2 O [ ( l + n 1 - n ) OZ 2 leOrYm 1--~0 ~ e x an + an cosa[o)t+O(z)]

1 - n sina[o)t 4- 0(z)]] +ev 2n


( z ) n2~-1 3 [3n4-1 = lZ O rYm E m 1- -~0 ~ L 4 n cos(cot 4- O(z))

4- 1-n ]

4n cos 3(cot 4- 0(z)) , (5.292)

O2ey ( ~0)-23 [ 1 - n Oz 2 lXO~m 1 - -~ ex 2n sin2[cot 4- 0(z)]

14-n 4- ey 2n 2n

cos 2[cot 4- O(z)]) ]

( z)~--- lO[3n4-1 = -#OrymEm 1 - G -~ 4n sin(cot 4- O(z))

_ ~14n- n sin 3 (cot 4- 0 (z)) ] (5.293)

To simplify the above equations, we introduced new state variables:

q~(z, t) -- ex(z, t) + j@(z, t),

7t(z, t) = ex(z, t) - jey(z, t).

(5.294)

(5.295)

By using these state variables, we can transform Eqs. (5.292) and (5.293) as follows:

32 ~ 1 - n . . . . ( Zo ) -2 3 [14- n ~ 4- (1 z ) J2~ " 1 3z 2 2n I~oam 1 -~ 1 - n ~o eJ2wt~

( --~0) 2 [ ( ) -jOt" = lZOamEm 1 - z ~ 0 3n + 1 1 - z e_JoJt -~ 4n ~o

4- 1 nt, l 4n ~ eJ3c~ ' (5.296)

3 2 ~ 1 - n ( -~o) 3Z 2 2----~/Z0am 1-- -23114-n~4-(1-zoo)-J2r176 1 n

Z 0 m mll t 2

e j~ot 4n zo/

4- 1 n(, zt'3 ] 4n ~o e-J3~ " (5.297)


By looking for the solution of Eqs. (5.296) and (5.287) in terms of Fourier series:

(x) ~(z,t) = ~ ~2k+l(Z) ej(2k+l)~

k=-oo (5.298)

OK) 1/r(z,t) = ~ llr2k+l(Z) ej(2k+l)~~

k=-oo (5.299)

it can be shown that only ~3, ~-1,1/rl, and l/r_ 3 are not equal to zero. For ~3 and l/r I the following coupled (ordinary differential equations) can be derived:

( Z ) 2d2~3 I ( z ) J 2 ~ " 1 1 - ~o dz 2 jX3 aq~3 + 1 - G �9 ~1

( ~ 0 0 ) 2n q-j3~ -- j~gEm 1 - z ~-1 ,

z2 [ ( 1 - ~ ) d2 !/r1 ( 1 - z ) -j2c~'' dz----g- - jx1 a ~ l q- -~0 ~31

( z) ~+~'' -- j v lEm 1 - ~00 ' (5.301)

where

(5.300)

1 - n l + n = , , (5.302) X2k+l (2k -+- 1)oo#0r7 m 2n a = 1 - n

1 - n 3 n + 1 ~'3 = 3r 4n vl = ~o/z0erm 4n (5.303)

The solution of Eqs. (5.300)-(5.301) should be subject to the boundary conditions

d~3 d~l (0) dz (0) = ~ = 0, ~3(0<)) = l/r 1(OO) = 0, (5.304)

which follow from the boundary conditions (5.291). Similar ODEs can be derived for ~0-1 and ~-3. However, this can be

avoided because ~0-1 and l/r I as well as ~-3 and 993 are complex conjugate. The particular solution of ODEs (5.300) and (5.301) has the form:

( ~ 0 0 Z ) X3 ' 1/r~p) ( ~ 0 0 Z ) ~'1 �9 o )~p (z)= C3 1 - (z) = C1 1 - , (5.305)


where 2n 2n X 3 = -Jr- j3o~', X1 = -ff jc~". (5.306)

n - 1 n - 1 Coefficients C3 and C1 satisfy the following simultaneous equations:

[XB(X3- 1)-jxsaz~]C3-jx3z~C1 =j;sz2Em, (5.307)

-jxIz2C3 + [XI(X1 - 1)-jxiaz2]Cx =jvlZ2Em. (5.308)

It is clear from (5.239), (5.302), (5.303), and (5.306) that the coefficients in Eqs. (5.307)-(5.308) depend only on n. This opens the opportunity to compute the ratios C1 ~Era and C3 ~Era as functions of n.

It has been shown before that the solution of homogeneous ODEs corresponding to (5.300)-(5.301) has the form:

( Z ) fl ~r~ h) (Z t f l - Ja~ q~h)(z)=A 1 - G ' (z)=B 1 - G , (5.309)

where fl is the solution of the following characteristic equation:

( f12 fl _jxsaz2)[(fl _ j2cr (fl - jack")- jx laz 2]

-Jr- XBXlZ 4 = O. (5.310)

It can be shown that the above characteristic equation has two roots fll and /J2 with positive real parts. By using these roots and expressions (5.305) and (5.309), the solution of Eqs. (5.300)-(5.301) can be written as follows:

( Z ) / ~ 1 ( Z ( Z ~ (5.311) CI)3(Z) a l 1 - G q-a2 1 - ~2 X3 - - m if-C3 1 - ,

z0 ] z0 ]

,,.-,, 0 . 5 .

ii E

I..IJ

�9 " o 0 . 4 .

o

~ 0.3

�9 ~ 0.2

~ 0.1

~ 0.0 0.0 0.2 0.4 ~Z0 0.6 0.8 1.0

FIGURE 5.40

371

.I--4

6 i0.

.,--Im(l~2)

2i

8,

046,

d) /Re(131) im(~l) ,.-,

2

" . . . . . . . . . . . . . . " . . . . . . . . . n 0 i0 15 20 25 30 D

5.5 CASE OF ELLIPTICAL POLARIZATIONS

~_ /Re(~2) , , , , i . . . . ~ . . . . i , , , : . . . . : - : - : ; n 0 5 i0 15 20 25 30

FIGURE 5.41

2

1 . 5 '

0 . 7 9 5 4 9 5

. . . . ~ ' ' ~ o - " - ~ s . . . . ~o . . . . ~s . . . . ~o ~ n

Im( )

o.7~+s++ _",C3, R~t-~-m/

. . . . .~ . . . . ~ o . . . . i s . . . . :~o . . . . :~s . . . . 3 0 ~ n

FIGURE 5.42


0.1157420.1- . . . .

- 0 . 1

-0.129171-

I m( ~---21 m ) /

' ' ' - ~ ' ' ' '~o . . . . ~s . . . . ]o . . . . is . . . . ]o = n

0.070674~ o

0.989855

. . . . .~ . . . . . . . . . . . . . 2.o _ . . 2.5 . . . . 3.0 ~ \ . . . . . . . . . . . n

Re(EA---~2 ~)

Im( ~-~2 m ) ______~___ , /

FIGURE 5.43

~I (Z) = BI I I - ~o ) ~1-j2~

( z) l q - C 1 1 - ~ 0 .

Z 1 ]~2-j2o~" + B 2 1 - ~

(5.312)

The unknown coefficients A2, A2, B1, and B2 can be found from the boundary conditions (5.304) at z - 0 and from the fact that expressions (5.309) should satisfy homogeneous ODEs corresponding to Eqs. (5.300)-(5.301). This yields the following simultaneous equations for the above coefficients:

/~1A1 Jr- ~J2A2 - - - X 3 C 3 , (5.313)

( ~ 1 - - j2o~")B1 + (~J2 - j2c~")B2 = - X I C 1 , (5.314)

(~2 _ ~1 -- jx3az~)al -- jx3zgB1 = O, (5.315)


(~2 _ ~2 - - jx3az2)a2 - - jX3Z2B2 = O. (5.316)

Again, it is easy to see that the coefficients of characteristic equation (5.310) as well as the coefficients of s imultaneous equations (5.313)-(5.316) depend only on n. This allows one to compute the roots ~1 and/~2 as well as the ratios A1/Em, A2/Em, B1/Em and B2/Em as functions of n. In the limiting case of n - cx~ (ideal resistive transition-critical state model), one can compute specific numerical values of the above quantities. These val-

3 9~/-2 ues are as follows: ~1 = 2 + j~/2, /~2 = 1.921 + j3.699, C1/Em -" -~ - j ,

C 3 / E m = j ~ 6 2 , a l / E m = -0.129 +j0.116, a 2 / E m = 0.071 - j0.990, el/Em =

-0.043 +j0.039, B2/Em = -1.899 +j0.513. By using these values all desired quantities can be found. For instance, the magni tudes of the first and third harmonics el and e3 of the perturbat ion can be computed as the functions

O.

D - - S X X - I o . . . . ~'s . . . . do . . . . ds . . . . ~0 ~ n

x x ~ Re(= B-2) / I - m

- 0 . 0 4 3 0 5 7 _ i J . . . . . . .

:c&65[EC---

Im(B2) / r-m

o .... } .... ~o .... ~5 .... {o .... {s .... ~o~ n

\ - 1 . 8 9 8 9 6 7

\

Re( E )

FIGURE 5.44


0 . 5 G) II N

~I 0.4. .

(D

c 0.3-

E c

o 0.22 ,i..., .

. Q

(:D 0.1- GL

-- IGI

zl%l J

. . . . , . . . . , . . . . , . . . . , . . . . , . . . . , ~ n 5 10 15 20 25 30

F I G U R E 5.45

of z. The results of these computations are shown in Fig. 5.40. For gradual resistive transitions (finite n), the roots /J1 and ]~2 as well as all the mentioned coefficients have been computed as functions of n. The results of these computations are presented in Figs. 5.41 through 5.44. Finally, Fig. 5.45 shows the dependence of lell and le31 on n at z = 0.

The presented analysis can be extended to the case of nonlinear diffusion of elliptically polarized electromagnetic fields in anisotropic superconducting media described by constitutive relations (5.270)-(5.271). In this case, the perturbation technique with respect to two small parameters can be employed. One small parameter is involved in the constitutive relations, while another is used in the boundary conditions. Mathemat- ical details of this perturbation technique are almost identical to those presented below. For this reason, the discussion of this perturbation technique is omitted here. The results presented in the last three sections have been published in [20-24].

References 1. Bean, C. P. (1962). Phys. Rev. Lett. 8: 250-252.

2. Bean, C. P. (1964). Rev. Modern Phys. 36: 31-39.

3. London, H. (1963). Phys. Lett. 6: 162-165.

4. Kim, Y. B., Hempstead, C. F. and Strand, A. R. (1962). Phys. Rev. Lett. 9: 306-308.

5. Mayergoyz, I. D. and Keim, T. A. (1990). J. Appl. Phys. 67: 5466-5468.


7. Friedman, G., Liu, L. and Kouvel, J. S. (1994). J. Appl. Phys. 75: 5683-5685.


8. Mayergoyz, I. D., Adly, A. A., Huang, M. W. and Krafft, C. (2000). J. Appl. Phys. 87(9): 5552-5554.

9. Mayergoyz, I. D., Adly, A. A., Huang, M. W. and Krafft, C. (2000). IEEE Trans. Mag. 36(5): 3505-3507.

10. Mayergoyz, I. D., Adly, A. A., Huang, M. W. and Krafft, C. (2000). IEEE Trans. Mag. 36(5): 3208-3210.

11. Nikols, C. S. and Clarke, D. R. (1991). Acta Metall. Mater. 39: 995.

12. Elkin, J. W. (1987). Cryogenics 2: 603.

13. Plummer, C. J. G. and Evetts, J. E. (1987). IEEE Trans. Mag. 23: 1179-1182.

14. Brandt, E. H. and Gurevich, A. (1996). Phys. Rev. Lett. 76: 1723-1726.

15. Brandt, E. H. (1996). Phys. Rev. Lett. 76: 4030-4033.

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17. Anderson, P. W. and Kim, Y. B. (1964) Rev. Modern Phys. 36: 39-43.

18. Beasley, M. R., Labusch, R. and Webb, W. W. (1969). Phys. Rev. 181: 682-700.

19. Bean, C. P. (1970). J. Appl. Phys. 41: 2482-2483.




23. Mayergoyz, I. D. and Neely, M. (1996). J. Appl. Phys. 79: 6602-6604.

24. Mayergoyz, I. D. and Neely, M. (1997). J. Appl. Phys. 81: 4234-4236.

CHAPTER 6

Eddy Current Hysteresis. Core Losses

6.1 EDDY CURRENT ANALYSIS IN THE CASE OF ABRUPT (SHARP) M A G N E T I C T R A N S I T I O N S

It is well known that the analysis of eddy currents in conducting and magnetically nonlinear media (see Fig. 6.1) requires the solution of the following nonlinear diffusion equation:

02H OB(H) = r ~ ~ . (6.1)

3Z 2 3t

Here: H and B are magnetic field and magnetic flux density, respectively; r~ is the conductivity of media, while B(H) stands for nonlinear constitutive relationship between B and H.

The analytical solution of nonlinear diffusion equation (6.1) encounters formidable mathematical difficulties. In the past, these difficulties were fully circumvented only for the case of very simple magnetic nonlin-

y

0

FIGURE 6.1

377

378 CHAPTER 6 Eddy Current Hysteresis. Core Losses

~H

FIGURE 6.2

earities describing abrupt (sharp) magnetic transitions. Such a transition for nonhysteretic media is shown in Fig. 6.2. It can be mathematically represented by the following expression:

B(H) -- Bm sign(H), (6.2)

where, as usual, sign(H) is defined by

/ 1 if H > 0, (6.3) sign(H) / -1 if H < 0 .

The development of the analytical technique for the solution of nonlinear diffusion problems with constitutive relation (6.2) can be traced back to the landmark paper of W. Wolman and H. Kaden [1] published about seventy years ago. This technique was afterwards independently rediscovered and further extended by V. Arkad'ev [2] in Russia and by W. MacLean [3], H. M. McConnell [4], and P. Agarwal [5] in the United States. This technique is traditionally derived by using integral forms of Maxwell's equations (such as Amp6re's Law and Faraday's Law of electromagnetic induction) rather than by directly solving the nonlinear diffusion equation (6.1). Below, we deviate from this tradition and give a simple derivation of this technique based upon the solution of Eq. (6.1). To this end, we shall first modify this equation by introducing shifted magnetic flux density of b(H) defined as follows:

b(H) = B(H) + Bm = 2BIns(H), (6.4)

where s(H) is the unit step function

1 if H > 0 , s ( H ) - 0 if H < 0 . (6.5)

In terms of b(H), the nonlinear diffusion equation (6.1) takes the form

32H 3b(H)

3z 2 3t (6.6)

6.1 CASE OF ABRUPT (SHARP) MAGNETIC TRANSITIONS

B b

3 7 9

_ _ _ t z~ Zo(t)

=Z

(a) (b)

F I G U R E 6.3

= Z

We consider the solution of this equation for the following initial and boundary conditions:

H(z,O) = O, (6.7)

B(z, O) = -Bm or b(z, 0) = 0, (6.8)

H(O,t) = Ho(t) > 0. (6.9)

It is clear that magnetic flux density B and shifted flux density b have spatial distributions as shown in Figs. 6.3a and 6.3b, respectively. Indeed, as the magnetic field Ho(t) is increased at the boundary z = 0, this increase extends inside the media causing B and b to switch from -Bin to Bm and from 0 to 2Bm, respectively. The distributions of B(z) and b(z) will be fully described if we find the expression for the front zo(t) in terms of H0(t), Bin, and r~. Indeed, if zo(t) is known, then

I Bm if z < z0(t), (6.10) B(z,t) = -Bin if z > z0(t), !

and 1 2Bm if z < z0(t), (6.11) b(z,t)

0 if z > zo(t). /

To find z0(t), we shall represent the nonlinear diffusion equation (6.6) as two coupled first-order partial differential equations:

~ w = -r~b(H), (6.12)

3z OH 3w

= - - - . (6.13) 3z 3t

It is easy to see that partial differential equations (6.12) and (6.13) are formally equivalent to Eq. (6.6). Indeed, by formally differentiating Eq. (6.12)


with respect to t and Eq. (6.13) with respect to z and then subtracting the results, we arrive at Eq. (6.6). However, Eqs. (6.12) and (6.13) have some mathematical advantages over Eq. (6.6). First, Eq. (6.6) contains the time derivative of the discontinuous function b(H) and, for this reason, this equation is not rigorously defined (in a classical sense) for abrupt magnetic transitions. Equations (6.12) and (6.13) do not contain the derivative of discontinuous functions and retain mathematical sense for abrupt magnetic transitions. Actually, a solution to nonlinear diffusion equation (6.6) can be defined as a solution to coupled equations (6.12) and (6.13). Sec- ond and more importantly, coupled equations (6.12) and (6.13) are easy to solve. Indeed, from the definition of b(H), we have

3w _ J -2r~Bm if z < zo(t), (6.14)

3z | 0 if z > zo(t).

Because function w(z,t) is defined by Eqs. (6.12) and (6.13) up to a constant, from expression (6.14) we find that w(z, t) is linear with respect to z when 0 ~ z ~ zo(t) and it can be assumed to be equal to zero when z ~ zo(t)"

w(z,t) = / w(0, t ) [1 - z--~t)] if z K z0(t),

i 0 if z ~ zo(t). (6.15)

w(0,t) It is clear from (6.15) that the slope of w(z,t) is equal to -z-G~ for 0 ~ z zo(t). According to Eq. (6.14), the same slope is equal to -2crBm. Thus

and

w(0, t)

zo(t) = 2r~Bm, (6.16)

w(0, t)-- 2r~Bmzo(t).

By using expression (6.16) in formula (6.15), we find

w ( O , t ) - 2crBmz if z ~ zo(t), w(z, t ) - 0 if z >1 zo(t).

From the last relation, we obtain

Ow(z,t) { ~w(0,t) __ ~ if Z ~ Zo(t),

Ot if Z ~ Zo(t).

By substituting expression (6.19) into Eq. (6.13), we arrive at

dw(O,t) OH (z, t ) - ~ if z ~ z0(t),

3z 0 if z ~ zo(t).

(6.17)

(6.18)

(6.19)

(6.20)

6.1 CASE OF ABRUPT (SHARP) MAGNETIC TRANSITIONS 381

This means that at every instant of time H(z, t ) has a constant negative slope with respect to z for 0 <<, z <<, zo(t) and the zero slope for z >~ zo(t). The latter is consistent with the fact that H(z, t) = 0 for z >~ zo(t). Thus

H(z, t ) = / H0(t)[1 - z0-~] if z <~ zo(t), (6.21) [ 0 if z >~ zo(t).

By comparing expressions (6.20) and (6.21), we find

Ho(t) dw(O,t) = ~ . (6.22)

zo(t) dt

According to formula (6.17), we have

dw(O,t) dzo(t) = 2 c r B m ~ . (6.23)

dt dt By substituting the last relation into (6.22), we obtain

dzo(t) (6.24) Ho(t) = 2crBmzo(t) dt "

o r

dz2(t) Ho(t) = r r B m ~ . (6.25)

dt

By integrating Eq. (6.25) and taking into account that z0(0) - 0, we finally arrive at

zo(t) = ( fo S ~ dr ) l/2 crBm " (6.26)

Expression (6.26) together with (6.21) fully describe the solution of nonlinear diffusion equation (6.1) in the case of abrupt magnetic transitions. By using this solution as well as the expressions

OH J -- 3 z ' j = erE, (6.27)

we can derive the following formulas for the induced (eddy) current density j and electric field E:

H0(t) if z ~ z0(t), j ( z , t ) = zG--(ff (6.28)

0 if z >~ zo(t),

H0(t) if Z <<, Zo(t), E(z, t) = rrzo( t )

0 if z >f zo(t). (6.29)

At first, it may seem that formula (6.29) is in contradiction with the continuity condition for tangential components of electric fields. How- ever, this continuity is valid only for stationary boundaries. In the case


of moving boundaries, the above continuity condition is replaced by (see J. A. Kong [6])

x (E + - E - ) - - ( ~ . v ) (B + - B - ) ,

where ~ is a unit normal to a moving boundary, v is its local velocity, while E +, E-, B +, and B- are the vector values of electric field and magnetic flux density on two sides of the moving boundary, respectively.

For our problem, the last boundary condition yields

E(zo(t)) = 2 B m ~ dzo(t)

dt

which, according to formula (6.24), leads to

E(zo(t)) = Ho(t) crzo(t)

The last formula is consistent with Eq. (6.29). Spatial distributions of H(z, t) and j(z, t) are shown in Figs. 6.4a and

6.4b, respectively. It is clear that positive rectangular fronts of B, j, E and linear front of H move inside the medium as long as Ho(t) remains positive. As soon as Ho(t) reaches zero value and then becomes negative, the above motion is terminated and rectangular B- and j-fronts and linear H- fronts of opposite polarity are formed and they continuously move inside the conducting medium. By literally repeating the same line of reasoning as before, it can be shown that the same expression (6.26) is valid for a new zero front, z0(t), with only one correction: a minus sign appears in front of the integral.

Now, we can consider the important case when the magnetic field at the boundary is sinusoidal:

Ho(t) = Hm sin ~ot. (6.30)

a ~

H~ ~

1, Zo(t) =z

j(t)

FIGURE 6.4

v Zo(t) =Z


IB lz~(t)

_B .__l ~ Z

! am'

-am

B

~ m /

Zo(t) I + T z0(~)

l (0<t<{) (T<t<T)

FIGURE 6.5

~ Z

383

It is clear that, during the positive half-cycle, the positive rectangular front of B propagates inside the medium (see Fig. 6.5a). This inward motion of z-~(t) is terminated at t - ~. During the negative half-cycle, the negative rectangular front of B is formed and it moves inside the medium (see Fig. 6.5b). At t = T, this inward motion of z o (t) completely wipes out the positive rectangular wave of B. During subsequent cycles, the situation repeats itself.

Next, we want to find the relation between electric and magnetic fields at the boundary z = 0. We consider only the positive half-cycle; for the negative half-cycle this relation remains the same. By combining formulas (6.29) and (6.26), we obtain

Eo(t) = E(O,t) = Ho(t) ( Bm )1/2 r~zo(t) = No(t) r~ ~o . (6.31)

By substituting (6.30) into (6.31), performing integration, and introducing the notation

Bm ! ~ m - - Hm (6.32)

we arrive at

Eo(t)- Hm~/~ m sin cot . (6.33) ,/1 - cos o)t

Thus, we can see that the electric field, E0(t), at the boundary is not purely sinusoidal and contains higher-order harmonics. This generation of higher-order harmonics can be attributed to the nonlinear magnetic properties of the conducting media. It is interesting to point out that this


is not always the case, and it will be shown in this chapter that for circular polarization of electromagnetic fields there is no generation of higher- order harmonics despite the nonlinearity of media.

By using expression (6.33), we can find the first harmonic E~l)(t) of the electric field at the boundary

E~l)(t)- Hm~co~ m (a cos cot + bsincot), (6.34)

where coefficients a and b are given by the following integrals:

4 fo } sincotcoscot dt_2_ fo :r sin~" cos~" d~, (6.35) a - ~ ~/1 - cos cot Jr ~/1 - cos

4 f0 } sin2 cot dt -2- fo Jr sin2{ d~'. (6.36) b = ~ ,/1 - cos cot Jr ,/1 - cos

By performing integration in (6.35) and (6.36), we arrive at

4~/2 8~/2 ~ ~ a-- 3zr ' b - 3zr (6.37)

Expression (6.34) can also be written in the following form:

= & 2 § b 2 Hm~/corm sin(cot + ~0). (6.38) E~l)(t) V (9"

According to (6.37), we find a

v/a 2 § b2= 1.34, tan~0 = ~ = 0.5, (6.39)

which leads to the expression

-(1.34),/co/zm Hm sin(cot + 26~ (6.40) E~l)(t) V o"

The last formula can be rewritten in the phasor form

E~1)=(1.S4~co~mejJ.-Tr)Ho, (6.41)

where the symbol . . . . . is used for the notation of phasors, while j = ~L-1. The last expression can be represented in terms of surface im-

pedance r/:

~ 1 ) __ , H 0 , (6.42)

where

1.34./cot~m j~ - - e6.77. (6.43)


It is instructive to recall that in the case of linear conducting media the surface impedance is given by

~(~) -- ~ a # eJ-~. (6.44)

By using expressions (6.26), (6.30), and (6.32), the penetration depth zo(T/2) of an electromagnetic field in magnetically nonlinear conducting media can be found:

( T ) ~ 2 (6.45) z~ 2 -- r

The last expression has the same "appearance" as the classical formula for the penetration depth, 8, in linear conducting media:

tory/z (6.46)

However, in spite of formal similarities, there are two essential differences between formulas (6.45) and (6.46). First, formula (6.45) gives a complete penetration depth; there is no time-varying electromagnetic field beyond zo(T/2), that is, for z > zo(T/2). On the other hand, formula (6.46) gives a distance at which the electromagnetic field is attenuated only to e -1 times its value at the boundary. Second, in formula (6.46)/z is constant and the penetration depth is field independent, while in expression (6.45) #m is inversely proportional to Hm (see (6.32)), which makes the penetration depth field dependent. The last remark is also valid as far as comparison of expressions (6.43) and (6.44) for surface impedances is concerned. In the case of linear conducting media, the surface impedance (6.44) is field independent, while for magnetically nonlinear conducting media the surface impedance (6.43) is a function of Hm.

It is also important to stress that the surface impedance for nonlinear conducting media is defined as the ratio of first harmonic phasors of electric and magnetic fields. For this reason, the value of the surface impedance may depend on the boundary conditions for H. To illustrate this fact as well as to appreciate the range of possible variations of r/, consider the case when the magnetic field at the boundary varies with time as follows:

/ ~ (sin o~t - �89 sin 2o~t) if 0 ~< t ~< T, Ho(t) (6.47) / 1~ (sin ~ot 4- I sin 2~ot) if T K t <~ T.

Here, 1.3 is the maximum value of sin o~t 4- I sin2~ot; consequently, Hm has the meaning of the peak value of Ho(t).


This boundary condition is chosen because it leads to the sinusoidal electric field Eo(t) at the boundary. To demonstrate this, we substitute (6.47) into (6.26) and after integration we obtain

Hm z~(t) - 2.6--~aBm (1 T coscot), (6.48)

where the superscripts "4-" and " - " correspond to positive and negative half-cycles, respectively.

Now, by using expressions (6.47) and (6.48) in formula (6.29) and taking into account the definition (6.32) of #m, we end up with

E0(t) = 1.24~c~ m Hm sincot. (6.49)

Next, in order to find the surface impedance r/ that corresponds to

the boundary condition (6.47), we determine the first harmonic H~l)(t) of H0(t):

H~l)(t) = l~(acoscot 4- bsincot), (6.50)

where coefficients a and b are given by the following integrals:

2 Z T Ho(t) a -- ~ 1.3 Hm cos cot dt, (6.51)

2 Z T 3H0(t) sincotdt. (6.52) b=-~ 1. Hm

By substituting expression (6.47) into formulas (6.51) and (6.52) and by performing integration, we obtain

4 - - - - 1 a = 3yr b = 1. (6.53)

By using these values of a and b, we transform expression (6.50) as follows:

Hm H~)_ )~1 ( t ) - ~ sin(cot - ~), (6.54)

where 4

tan ~0 -- 3zr -- 0.424, ~0 -- 23 ~ (6.55)

Now, by transforming expressions (6.49) and (6.54) into phasor forms, we compute the surface impedance

E0 = 1.47./co/Zm eJ 7.:83 ~ -- ~1) V r r

(6.56)


By employing expression (6.48), we can find the penetration depth, zo(T/2), in the case of sinusoidal variation of the electric field on the boundary:

/

zo(T/2) = ~/ 1_57 . (6.57) V COo'k6m

Comparison of expressions (6.43) and (6.45) with expressions (6.56) and (6.57) is suggestive of to what extent the surface impedance and the penetration depth may depend on a particular time variation of the magnetic field on the boundary.

The results of the previous analysis can be extended to the practically important case of magnetically nonlinear conducting laminations. Such laminations are used in many applications. For instance, steel laminations are stacked together to form magnetic cores of transformers, electric ma- chines, and various actuators. Laminated permalloy heads as well as thin film heads are widely used in magnetic recording. In all these designs, magnetic laminations are employed for flux-guiding purposes. For this reason, it is desirable that cross-sections of magnetic laminations are utilized effectively. To check this, distributions of magnetic flux density over lamination cross-sections can be computed by using the previously derived expressions. Indeed, during an initial stage of positive half-cycle, magnetic fields penetrate from both sides of the laminations in the same way as in the case of semi-infinite half-space (see Fig. 6.6a). The motion of the positive front z-~(t) can be determined by using formula (6.26) if the magnetic field Ho(t) on the boundary of the lamination is known. This is usually the case when the current through the coil, which creates the magnetic flux, is known. When the voltage applied to the coil is known, then the boundary value Eo(t) of the electric field can be determined. By using E0(t), the motion of the zero front can be found according to the formula

Eo(t) -- 2Bm dz-~ (t) dt ' (6.58)

which leads to 1Lt z-~(t)- ~ Eo(r)dr. (6.59)

We note that formula (6.58) is easily derived from expressions (6.20), (6.23), and (6.27).

At the instant of time ta when

a z~- (tA) = ~-, (6.60)

388

1

CHAPTER 6 Eddy Current Hysteresis. Core Losses

lZo(t) ] ~ 2

~ Z

(0<t< tA) (a)

Bm

(ta___ t < T )

(b)

"~Z

I A 2

A ~Z

( T < t < T + t A ) (T+tA< t<T )

(c) (d)

FIGURE 6.6

~ Z

(where A is the lamination thickness), the two positive fronts are merged together (see Fig. 6.6b) and the distribution of magnetic flux density over a lamination cross-section is uniform. It remains this way during the rest of the positive half-cycle. With the commencement of the negative half-cycle, negative fronts of magnetic flux density are formed and they penetrate from both sides of the lamination (see Fig. 6.6c). At the instant of time T + t a these negative fronts are merged together (see Fig. 6.6d) and the distribution of magnetic flux density remains uniform during the rest of the negative half-cycle. At subsequent cycles, the situation repeats itself. It is clear from the above discussion that the lamination cross-section will be effectively utilized if ta is substantially smaller than T/2. The validity of this fact can be evaluated for every particular case by using formula (6.60) along with expression (6.26) or (6.59).

The analytical technique just presented can be generalized to the case when abrupt magnetic transitions are described by a rectangular hystere-


-H H c ~H

FIGURE 6.7

389

sis loop as shown in Fig. 6.7. Again, we begin with the case when the initial value of the magnetic field throughout conducting media is equal to zero, while the initial value of magnetic flux density is equal to -Bin. Suppose that the magnetic field Ho(t) at the boundary is increased. Un- til this field reaches the coercive value, Hc, nothing happens. As soon as Ho(t) exceeds the coercive value He, the rectangular front of magnetic flux density is formed and it moves inside the medium. To compute the zero front z0(t), we introduce the shifted magnetic field h(z, t):

h(z, t) = H(z, t) - Hc, (6.61)

and rewrite the nonlinear diffusion equation (6.6) as follows:

02h Ob(h) = c r y . (6.62)

3Z 2 3t

Now, by literally repeating the same line of reasoning as before, we can derive the following expression for zo(t)"

zo(t) - ( fttc h~ dr ) 1/2 rrBm " (6.63)

Here tc is the time when Ho(tc) - Hc, (6.64)

while ho(t) is the boundary value of the magnetic field, h(0,t). As far as the distributions of h(z, t), j(z,t), and E(z, t) are concerned, the same formulas (6.21), (6.28), and (6.29) are valid, however, Ho(t) in these formulas must be replaced by ho(t).

The propagation of the positive rectangular front of magnetic flux density will continue until the magnetic field at the boundary is reduced back to its coercive value Hc (or ho(t) is reduced to zero). As the magnetic field at the boundary is reduced from Hc to -Hc, nothing happens and


induced eddy currents and electric fields are equal to zero. As soon as the magnetic field on the boundary is reduced below - H c , the negative front of magnetic flux density is formed and it moves inside the medium. The motion of this negative front can be determined by using the same formula (6.63) with the following corrections: (a) the minus sign appears in front of the integral in (6.63), (b) ho(r) is defined as Ho(t) + H o and (c) the time tc is determined from the equation Ho(tc) = - H c .

Next, consider the example when the magnetic field at the boundary is sinusoidal and given by expression (6.30). We want to find the surface impedance r/. To this end, we shall first find the electric field Eo(t) at the boundary. It is clear from the previous discussion that

Eo(t) - 0 if 0 ~ t <, to (6.65)

where 1

tc - - arcsin ~o Hmm '

(6.66)

and

Eo(t) = ho( t ) ( Bm )1/2 Cr fttc ho(z') dz"

T if tc <, t <, -~ - to (6.67)

T T Eo(t) = 0 if ~ - tc <, t <~ ~ . (6.68)

Similar expressions can be written for the negative half-cycle. By substituting formula (6.30) into (6.67), by performing integration

and taking into account the definition of h(t),/Zm, and tc, we derive

~co~m sin cot - sin cotc

Eo(t) = Hm v/cos cote - cos cot - (cot - cote)sin cote (6.69)

The first harmonic of Eo(t) can be written in the form (6.34), where coefficients a and b are determined by the following integrals:

2 [:~-~c (sin ( - sin (c) cos ( d( a = -- ! , (6.70)

zr j ~c v/COS (c - cos ( - (( - (c) sin (c

2 [,~r-~c (sin ( - sin (c) sin ( d( b (6.71)

zr j ;c v/cos (c - cos ( - (( - (c) sin (c

and

(c - arcsin ( Hc Hmm)" (6.72)


In terms of a and b, the surface impedance r/is given by

rl -- v/a2 -+- b2 V/cO~mry ej~'

where

(6.73)

a tango - ~. (6.74)

By using formulas (6.70) and (6.71), the values of v/a 2 -+-b 2, tan ~o, and ~o have been computed as functions of ~'c. The results of the computations are shown in Figs. 6.8a, 6.8b, and 6.8c. By using these figures, the dependence of surface impedance r/on the coercitivity Hc can be evaluated.

In the previous discussion, we dealt with nonlinear diffusion of electromagnetic fields in conductors with plane (flat) boundaries. Now, we shall extend our study to the case of nonlinear diffusion in a cylinder. This study will shed some light on how the curvature of conducting boundaries may affect the process of nonlinear diffusion.

Consider an infinite conducting cylinder of radius R (see Fig. 6.9) subject to time-varying uniform magnetic field Ho(t) whose direction is parallel to the cylinder axis. We assume that this cylinder is magnetically homogeneous with constitutive relation described by Eq. (6.2). In other words, we shall consider the case of abrupt (sharp) magnetic transition.

Suppose that initial values of magnetic flux density and magnetic field are equal to -Bm and 0, respectively. Next, suppose that the magnetic field at the boundary is increased from its zero value and remains positive. As the magnetic field at the boundary is increased, this increase extends inside the conducting cylinder causing the transition of magnetic flux density from -Bin to -FBm. As a result, a rectangular front of magnetic flux density is formed and it moves from the boundary of the cylinder toward its axis (see Fig. 6.10).

We intend to derive the expression for the radial coordinate, r0(t), of this front in terms of r~, Bm, R and the magnetic field, H0(t), at the boundary. To this end, we shall exploit the circular symmetry of the problem. According to this symmetry, electric field lines and lines of electric current density are circular ones. They exist only for r ~ ro(t). Consider an electric field line Lr of radius r and let us apply the law of electromagnetic induction to this line:

~ L E . d l - d~(r, t) (6.75) m ~

r dt '

where ~(r, t) is the magnetic flux that links Lr.

x~a2 -[- b 2


0.8

0.6

0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0

tan()

0.50

0.40

0.30

0.20

0.10

0.00

0.0 0.5 1.0 1.5 2.0

30

25

" 2O ra~

15 0/)

10

5

0 0.0

i 0.5 1.0 1.5 2.0

(a)

(b)

392

FIGURE 6.8


R Lr

ro(t) B

ro(t) "R

-Bm I FIGURE 6.9 FIGURE 6.10

393

= r

Due to circular symmetry, we have

~L E. dl E(r , t ) . 2J r r . (6.76) / ,

By using Fig. 6.10, it is easy to see that the following expression is valid for flux ~(r, t):

�9 (r, t ) - Bmzr(r 2 - r2(t)) - BmJr~( t ) , (6.77)

which can be further reduced to the form

�9 (r, t) = BmTf (r 2 - 2r~(t)). (6.78)

From the last formula, we find

d ~ ( r , t ) dr2(t) - d--------~ = 2 J r B m d----~" (6.79)

By substituting expressions (6.76) and (6.79) into formula (6.75), we end up with

E ( r , t ) - Bm dr2(t_~) (6.80) r dt "

By using the last expression, we obtain the following equation for electric current density:

crBm dr2(t) j(r, t) = rr E(r, t) - - - . (6.81)

r dt

Now, we recall that OH(r , t )

= ~ . (6.82) j ( r , t ) - Or


By integrating the last expression from ro(t) to R and taking into account that H(ro(t), t) - 0, we obtain

Ho(t) - H(R, t) - - j(r, t) dr. (6.83) (t)

By substituting formula (6.81) into (6.83) and performing the integration, we arrive at

H~ =r~Bm(ln r ~ ) d~(t---~)dt " (6.84)

It can be shown that

l n ( ~ ) dr~(t)dt

Indeed,

d [r2(t) ln( dt

d l n ( ~ ) ] I dr~(t) - ~ [ ~ ( t ) ~. ~-~ .

~ ) ] - l n ( ~ ) d r ~ ( t ) dro(t) dt + ro(t) d-----t-

which justifies equality (6.85).

= ln (~ )d r2 ( t )d t I dr2(t) q- - - ~ 1 2 dt

(6.85)

(6.86)

This is a nonlinear equation for ro(t). It is convenient to transform this equation as follows:

2 fo Ho(r)dr _ r~(t___)) In ~(t) r~(t) + 1. (6.88) tTR2Bm R 2 R 2 R 2

We shall next introduce the variable

and the function

X(t)- r2(t) R2 , (6.89)

F(X) = X(lnX - 1)+ 1. (6.90)

By using the above function, Eq. (6.88) can be represented in the form

2 fo Ho(r)dr r y a 2 B m - F ( ~ , ( t ) ) . (6.91)

f0 Ho(r)dr _ ~(t) In ro(t) ~(t) - - a 2 r~Bm R - 2 " (6.87)

By substituting this equality into formula (6.84), then performing integration from 0 to t and taking into account that r0(0) = R, we obtain


F(k) 1.0

0.8

0.6

0.4

0.2

0.0

�9 . . , , . , , , . , | �9 . . , , , ,

0.0 0.2 0.4 0.6 0.8 1.0

FIGURE 6.11

395

This form is convenient for the graphical solution of Eq. (6.88). Indeed, the function, F(k), can be precomputed and its graph can be constructed (see Fig. 6.11). Then, for every instant of time, the left-hand side of Eq. (6.91) can be evaluated and plotted along the vertical axis in Fig. 6.11. Finally, by drawing the horizontal line until it intersects with the graph of the function F(k) and the vertical line until it intersects with the horizontal axis, we determine the value of k(t) corresponding to the left-hand side of Eq. (1.306). By using this value and formula (6.89), we find ro(t).

Equation (6.91) can also be solved numerically by using, for instance, the Newton iterative technique. This equation can also be solved analytically, albeit approximately. The analytical solution is based on the following approximation of F(X):

(X - 1) 4 + (X - 1)2 F(k) ~ . (6.92)

2 To appreciate the accuracy of approximation (6.92), the graphs of function F(k) and its approximation are plotted in Fig. 6.12. This figure suggests that approximation (6.92) is quite accurate. By substituting formula (6.92) into Eq. (6.91), we end up with the quadratic equation for ( k - 1) 2. By solving this equation and taking into account expression (6.89) for k, we arrive at the following approximate formula for r0(t):

I j 1 ro(t) ~, R 1 - --~ +

4 fo Ho(r)dr + . (6.93)

r~ R2 Bm

The value of ro(t) given by formula (6.93) can be used as an initial guess in Newton iterations. However, this may be unnecessary, because it is quite

3 9 6 CHAPTER 6 Eddy Current Hysteresis. Core Losses

1.0

0 8

0 6

0 4

0.2

0.0

' ' ' , �9 , , i �9 �9 �9 , �9 �9 �9 i �9 , �9

_

0.0 0.2 0.4 0.6 0.8 1.0

FIGURE 6.12

conceivable that the accuracy of formula (6.93) is higher than the accuracy of abrupt magnetic transition assumption, which was used in the derivation of Eq. (6.88).

Having determined the radial coordinate, r0(t), of the front, we can fully describe the distribution of magnetic flux density. We can also find the electric field by invoking formula (6.80). However, there is some in- convenience in using this formula. This is because this formula requires the differentiation of r0(t), which is found graphically or numerically. This difficulty can be circumvented by recalling formula (6.84) and expressing dr~ at as follows:

dr 2 Ho(t)

dt aBm In ro(t) " (6.94) R

By substituting the last expression into Eq. (6.80), we end up with

Ho(t) E(r,t) - (6.95)

ar ln ro(t) " R

The last formula does not contain the derivative of ro(t) and, for this reason, it is convenient for calculations. By using formula (6.95), we can also find an important relationship between electric and magnetic fields at the cylinder boundary:

Ho(t) E o ( t ) - a R l n r0(t) ' (6.96)

R

where the notation E(R, t ) = Eo(t) has been used. Up to this point, we have discussed the situation when the magnetic

field Ho(t) at the boundary is assumed to be positive. If the magnetic field

6.2 EDDY CURRENT HYSTERESIS

B

FIGURE 6.13

~F

397

Ho(t) is reduced to zero and then becomes negative, the motion of the "positive" rectangular front of magnetic flux density is terminated and the "negative" rectangular front is formed and it moves from the cylinder boundary toward its axis (see Fig. 6.13). By repeating the same line of reasoning that was used in the derivation of formula (6.88), we arrive at the similar expression:

- 2 fo Ho(r)dr r2(t) r~(t) r2(t) o.R2Bm -- R---- T- In R2 R2 }- 1. (6.97)

This equation can be used for the determination of the radial coordinate ro(t) of the "negative" rectangular front of magnetic flux density in the same way as we have used Eq. (6.88).

The previous discussion can be easily extended to the case when the abrupt (sharp) magnetic transition is described by the rectangular hysteresis loop shown in Fig. 6.7. In this case, the "positive" rectangular fronts are formed and they move inward when Ho(t) - Hc is positive. The "negative" rectangular fronts are formed and they move inward when Ho(t) + He is negative. The radial coordinates of "positive" and "negative" fronts can be determined by solving modified equations (6.88) and (6.97), respectively. Modification of these equations consists in the replacement of the

integral ~0 Ho(r)dr by integrals ~o(Ho(r) - Hc)dr and ~o(Ho(t) 4- Hc)dr, correspondingly. When Ho(t) is between -Hc and +Hc, everything is still and there is no movement of rectangular fronts.

6.2 E D D Y C U R R E N T H Y S T E R E S I S A N D T H E P R E I S A C H M O D E L

To explain the phenomenon of eddy current hysteresis, consider, as an example, a time variation of magnetic field Ho(t) shown in Fig. 6.14. It will

398

Ho(t)


~ t

FIGURE 6.14

be assumed that the following inequalities are valid:

fO tl ftx,2 Ho(r ) dr > - Ho(r ) dr > Ho(r ) dr > . . .

ftl 6 > - H o ( r ) d r .

(6.98)

It will also be assumed that magnetic properties of conducting material are described by rectangular magnetization curves (abrupt magnetic transitions) and that the initial value of magnetic flux density is equal to - B m .

According to the previous assumptions, we conclude that during the time interval 0 < t K tl a "positive" rectangular front of magnetic flux density is formed and moves toward the axis of a cylinder (see Fig. 6.15a). The radial coordinate ro(t) of this front can be determined by solving Eq. (6.88) or by using the approximate formula (6.93). At time t = tl, the motion of the positive rectangular front is terminated and a negative rectangular front of magnetic flux density is formed. During the time interval tl < t K t2, the latter front moves toward the cylinder axis (see Fig. 6.15b). The radial coordinate of this front can be determined by solving nonlinear equation (6.97). At time t = t2, the motion of this negative rectangular front is terminated and a new positive rectangular front of magnetic flux density is formed. This front moves toward the cylinder axis during the time interval t2 < t ~ t3, and the distribution of magnetic flux density for this time interval is shown in Fig. 6.15c. At subsequent time intervals (t3 < t ~ t4, t4 < t ~ t5, and t5 < t <~ t6), new negative and positive rectangular fronts of magnetic flux density are formed, and the distribution of magnetic flux density at time t = t6 looks like the one shown in Fig. 6.15d.

It is important to note that radial coordinates r~ k) of still (motionless) rec-


1 -B~ /r~

~ r ~ r

~B

-Bm,

(b)

B~

B B

=r R

_Om!lU U FIGURE 6.15

=r

399

tangular fronts form a monotonically increasing sequence

(1) r~2) r~4) (6) r 0 < <r~ 3)< <r~ 5 )<r 0 < . . . < R . (6.99)

This follows directly from inequality (6.98) and Eqs. (6.88) and (6.97). In other words, the past time variations of magnetic field Ho(t) at the conductor boundary leave their mark upon future magnetic flux distributions over the conductor cross-section. This suggests that there is a hysteretic relation between the magnetic flux through the conductor cross-section and the magnetic field at the boundary. To clearly understand this hysteretic relation, we introduce the magnetic flux

~0 R �9 (t) - 2re B(r, t)r dr, (6.100)

and the function

~0 t wo(t) = H o ( r ) d r . (6.101)


CI)

2

3

FIGURE 6.16

Next, we shall plot ~(t) versus wo(t). It is evident from Eq. (6.88) and Fig. 6.15a that as wo(t) is monotonically increased during the time interval 0 < t < tl, the flux ~(t) is also monotonically increased starting from its initial value - - C I ) m - - -rcR2Bm . Thus, the branch "1" is traced in Fig. 6.16 dur ing the above time interval. For the time interval tl < t ~ t2, the magnetic field, H0(t), is negative and the function wo(t) is monotonically decreased. It is clear from Fig. 6.15b that for the same interval the magnetic flux is monotonically decreased as well. As a result, the branch "2" is traced in Fig. 6.16. During the time interval t2 < t ~ t3, the magnetic field Ho(t) is positive and the function wo(t) is monotonically increased. It is obvious from Fig. 6.15c that for the same time interval the magnetic flux is monotonically increased as well. This results in the branch "3" in Fig. 6.16. By using the same line of reasoning, it is easy to see that new branches "4," "5," and "6" will be formed during the time intervals t3 < t ~< t4, t4 < t ~ t5 and t5 < t K t6. Thus, the relation between ~(t) and wo(t) is a mult ibranch nonlinearity. It is also clear that branch-to-branch transitions occur after each ext remum value of wo(t). Indeed, the function wo(t) assumes its (local) max imum values at times t = tl, t = t3, t -- t5, and its (local) min imum values at times t = t2, t -- t4, and t = t5, and at all these time instants transitions to new branches occur. It is important to stress that the ~(t) vs wo(t) relation is rate independent. This means that the value of ~(t) depends on the past ext remum values of wo(t) as well as the current value of w0(t), however, it does not depend on the rate of time variations of wo(t). The last statement is obvious from the fact that ~(t) is fully determined by radial coordinates of rectangular fronts of magnetic flux density, and these "front" coordinates depend only on the values of wo(t) and do not depend on the rate of its time variations.


. . . . . . . .

I

FIGURE 6.17

401

It is apparent that the branching described above occurs inside some major ("limiting") hysteresis loop shown in Fig. 6.17. This major loop is formed when for two subsequent monotonic variations of wo(t) the corresponding fronts of magnetic flux density reach the cylinder axis. Beyond the major loop magnetic flux ~(t) may assume only two values: +~m or - ~ m .

The major hysteresis loop as well as the branching inside this loop exhibit some asymmetry. This asymmetry can be completely removed by redefining the function wo(t) as

Wm (6.102) ~Vo(t) -- Wo(t) 2 "

where Wm is specified in Fig. 6.17. In our subsequent discussion, it will be tacitly assumed that the above shifting of wo(t) is performed when it is needed.

Now, we can summarize our previous discussion by stating that the essence of eddy current hysteresis is the multibranch rate independent nonlinear relation between the magnetic flux ~(t) and the function wo(t) = fo Ho(r)dr . It is interesting to explore the use of Preisach models of hysteresis for the description of eddy current hysteresis [7-9].

The Preisach model is given by the following expression:

f ( t ) = f f /~(c~, ~)~,~u(t) dol d~, (6.103) dda

where u(t) is a physical quantity called input, f i t ) is a physical quantity called output, }9~ are rectangular loop operators with a and/~ being "up" and "down" switching values, respectively, while/~(a,/~) is a weight function. Details related to formula (6.103) can be found in Chapter 1.


The Preisach model (6.103) describes a rate-independent hysteretic relation between input u(t) and output fit). In Chapter 1 the theorem is proven, which states that wiping-out and congruency properties constitute the necessary and sufficient conditions for a hysteretic nonlinearity to be represented by the Preisach model. The last theorem is very instrumental in establishing the connection between eddy current hysteresis and the Preisach model. To do this, we recall that in the case of eddy current hysteresis, �9 (t) vs wo(t) relation is a rate-independent hysteretic nonlinearity. Now, we shall demonstrate that this nonlinearity exhibits wiping-out and congruency properties. Indeed, each time wo(t) is monotonically increased (or decreased), a rectangular front of magnetic flux density is formed and it moves toward the cylinder axis. This moving front will wipe out those previous rectangular fronts of magnetic flux density if they correspond to the previous extremum values of w0(t), which are exceeded by its new extremum value. In this way, the effect of those previous extremum values of wo(t) on the future values of magnetic flux ~(t) is completely eliminated. This means that the wiping-out property holds. Next, we shall demonstrate the validity of the congruency property. Consider two vari-

(1) W~2) w~l)(t) and w~ 2) ations of wo(t): w 0 (t) and (t). Suppose that (t) have different past histories (different past extrema) but, starting from some instant of time, they vary monotonically back-and-forth between the same reversal (extremum) values. It is apparent from the mechanism of nonlinear diffusion described at the beginning of this section that these back-

and-forth variations of w~l)(t) and w~2)(t) will affect in the identical way the same surface layers of the conducting cylinder. Consequently, these variations will result in equal increments of magnetic flux ~(t), which is tantamount to the congruency of the corresponding minor loops. Since the wiping-out and congruency properties are established for the ~(t) vs wo(t) relation, this relation can be represented by the Preisach model. Thus, by taking formula (6.101) into account, we find

~(t) - ff~ ~>/~ #(~ ( f0 t Ho(r) dr) da d~. (6.104)

By using the following relations between flux ~(t) and voltage v(t) as well as between magnetic field Ho(t) and current i(t):

lf0t �9 (t) = ~ v(r) dr + ~0, (6.105)

N Ho(t) = -;-i(t), (6.106)

6.2 EDDY CURRENT HYSTERESIS 403

expression (6.104) can be written in the form

lf0t /~ ( N f 0 t ) v(r) dr + ~o - >~/3 lz(ot, fl)G[3 fir)dr dc~ dfl. (6.107)

The last expression can be interpreted as a terminal voltage-current relation for a coil placed around a conducting magnetic cylinder. It is important to note that this terminal relation is valid for arbitrary time variations of current and voltage.

Formula (6.104) (as well as (6.107)) has been derived for a conducting magnetic cylinder of circular cross-section. However, this formula can be generalized for a conducting magnetic cylinder of "arbitrary" cross- section. For such a cylinder, the nonlinear diffusion equation has the form

32H 32H 3B(H) ~ - ~ (6.108) 3X 2 +- ~ - - cr 3t '

where for the case of abrupt magnetic transitions

B(H(t)) - - B m signH(t). (6.109)

In Eq. (6.108), x and y are coordinates in the cylinder cross-section plane, while the magnetic field is always normal to this plane.

Let us now assume that the initial value of the magnetic flux density in the cylinder is equal to -Bin. Let us also assume that Ho(t) varies with time as shown in Fig. 6.14. By using the same line of reasoning as before, we conclude that positive rectangular fronts of magnetic flux density are formed and moved inwards for odd time intervals (t2k < t < t2k+l), while negative fronts are formed and moved inwards for even time intervals (t2k+l < t < t2k+2). Next, we shall transform nonlinear diffusion equations (6.108)-(6.109) into rate independent forms for odd and even time intervals. To this end, we introduce the function

W2k4.14- __ H(r) dr H(t) = k

4- 0W2k4"10t )" (6.110)

By integrating Eq. (6.108) with respect to time from t2k to t and by using formula (6.109), we derive

+ ] V W2k+l -- cr Bm sign 0W2k+l Ot

(6.111)

The last equation is valid within the region + ~2k+l(t) occupied by a newly

formed positive front. In this region, function W2k4.14- is monotonically in- +

3W2k+1 creased with time and, consequently, sign ( 0t ) -- 1. In the same region,


we also have B(t2k)- - -Bm. As a result, Eq. (6.111) takes the form of the Poisson equation:

V2W2k++ 1 -- 2crBm. (6.112)

The solution of the last equation is subject to the following boundary conditions:

+ + W2k+l (t)lL = w0,2k+l (t) = Ho(r) dr, (6.113)

k

w~+ l(t)lL-~k+~(t ) = 0, (6.114)

+ 8W2k+l = 0, (6.115)

31; L + 2k+1 (t)

+ where v is a normal to the moving boundary L2k+l(t) of the region

+ ~ 2 k + 1 (t) .

Boundary conditions (6.114) and (6.115) at the moving boundary L + 2k+l(t) follow from the fact that magnetic field and tangential component of electric field are equal to zero at the points of + L2k+l (t) for the time interval t2k < r < t, that is, before the arrival of the positive front.

During even time intervals, Ho(t) < 0 and negative fronts of the magnetic flux density are formed and they extend inwards with time. By introducing the function

W2k ~--- S(r ) dr, (6.116) k-1

and by literally repeating the same line of reasoning as before, we end up with the following boundary value problem:

V2W2k = -2r~Bm, (6.117)

sj w2~(t)lL = WO,2k(t) -- Ho(r) dr, (6.118) k-1

W2k(t)lLak(t) -- O, (6.119)

aWRk I - 0 . (6.120) OV L2k(t)

It is interesting to note that nonlinear diffusion equation (6.108) is transformed into linear Poisson equations (6.112) and (6.117). However, nonlinearity of the problem did not disappear; it is present in boundary conditions (6.114)-(6.115) and (6.119)-(6.120), which should be satisfied


+ at moving boundaries L2k+l(t) and L2k(t ), respectively. Locations of these boundaries are not known beforehand and should be determined from the fact that zero Dirichlet and Neumann boundary conditions must be simultaneously satisfied at these boundaries. In other words, formulas (6.112)- (6.115) and (6.117)-(6.120) define boundary value problems with moving boundaries, and these moving boundaries are the source of nonlinearity.

The following properties can be inferred by inspecting boundary- value problems (6.112)-(6.115) and (6.117)-(6.120).

RATE INDEPENDENCE PROPERTY Boundary value problems (6.112)-(6.115) and (6.117)-(6.120) are rate independent because there are no time derivatives in the formulations of these boundary value problems. Consequently, the instantaneous positions and shapes of moving boundaries L-~k+l (t) and L2k(t ) are determined by instantaneous boundary values of W~,2k+l(t) and w~,2k(t ), respectively.

SYMMETRY PROPERTY Boundary value problems (6.112)-(6.115) and (6.117)- (6.120) have identical (up to a sign) mathematical structures. This suggests that,

+ if Iw~,2kl -- Iw0,2k+l I, then the corresponding boundaries L2k and L2k+l + are identical. In other words, there is complete symmetry between inward motions of positive and negative fronts.

Now we introduce the function

f0 t wo(t) -- No(r) dr. (6.121)

It is clear that function wo(t) is a sum of the appropriate functions W~2k(t ) and W~2k(t ). It is also clear that wo(t) achieves local maxima at t = t2k+l and local minima at t = t2k. Next, we intend to show that q~(t) vs wo(t) is a rate-independent hysteretic relation. The rate independence of the above relation directly follows from the previously stated rate independence property. It is also true that ~(t) vs wo(t) is a hysteretic relation. Indeed, the current value of ~(t) depends not only on the current value of wo(t) but on the past extremum values of wo(t) as well. This is because the past extremum values of wo(t) determine the final locations and shapes of positive and negative rectangular fronts of B that were generated in the past. These past and motionless rectangular fronts affect current values of ~(t). It is also apparent that there are reversals of ~(t) at extremum values of wo(t). In other words, new branches of �9 vs wo relation are formed after local extrema of wo(t). The previous discussion clearly suggests that �9 vs wo is a rate independent hysteretic relation. Next, we shall demonstrate that this hysteretic relation exhibits the wiping-out and congruency


properties. Indeed, every monotonic increase (or decrease) of wo(t) results in the formation of a positive (or negative) rectangular front of the magnetic flux density, which extends inwards. This moving front will wipe out those previous rectangular fronts if they correspond to those previous extremum values of w0(t), which are exceeded by a new extremum value of wo(t). In this way, the effect of those previous extremum values of wo(t) on the future values of magnetic flux ~(t) is completely eliminated. This means that the wiping-out property holds. Now we shall demonstrate the validity of the congruency property. Consider two different boundary

conditions: w~l)(t) and w~2)(t). Suppose that w~l)(t) and w~2)(t) have different past histories (different past extrema) but, starting from some instant of time, they vary monotonically back-and-forth between the same two extremum (reversal) values. It is apparent that these back-and-forth varia-

tions of w~l)(t) and w~2)(t) will affect in the identical way the same surface layers of the conducting cylinder. Consequently, those variations will result in equal increments of the magnetic flux, which is tantamount to the congruency of the corresponding minor loops. Since the wiping-out and congruency properties constitute necessary and sufficient conditions for applicability of the Preisach model, we conclude that the �9 vs wo relation can be represented by the Preisach model. As a result, we arrive at the following representation of eddy current hysteresis:

ff~ (fot ) �9 (t) = )~ lz(ot, fl)G~ Ho(r)dr do~ dfl, (6.122)

which is valid for cylinders of arbitrary cross-sections. It is worthwhile to stress two remarkable points related to the above

result. First, memory effects and dynamic effects of eddy current hysteresis are clearly separated. The memory effects are taken into account by the structure of the Preisach model, while the dynamic effects are accounted for by the nature of the input (fo Ho(r)dr) to this model. Second, the last formula suggests that the Preisach model can be useful for the description of hysteresis exhibited by spatially distributed systems. This is in contrast with the traditionally held point of view that the Preisach model describes only local hysteretic effects in magnetic materials.

Next, we turn to the discussion of properties of function #(c~, fl) in formula (6.122). By using the symmetry property, it can be inferred that the same increments of wo(t), which occurred after different extremum values of w0(t), result in the same increments of ~(t). This fact implies that the integral

F(~, fl) = [ [ #(~', fl') d~' dfl' (6.123) ,liT (~,[3 )


over a triangle T(~, ]~), defined by inequalities c~' < c~, ~' > ]~, c~' - /5 ' ~ 0, does not depend on c~ and ~ separately but rather on the difference c~ - ]~. In other words, the value of the above integral is invariant with respect to parallel translations of the triangle T(c~, ]~) along the line c~ =/~. This is only possible if

/~(c~, ]~)=/~(~ - /J) . (6.124)

This means that function/~ assumes constant values along the lines c~ - ]~ = const. By using this fact, it can be easily observed that function/~ can be found by measuring only the ascending (or descending) branch of the major loop of �9 vs w0 hysteretic nonlinearity. It can also be seen that any path traversed on the (w0, ~) plane is piecewise congruent to the ascending branch of the major loop. (See Fig. 6.16.) Thus, �9 vs w0 hysteretic nonlinearity is completely characterized by the ascending branch of the major loop. This branch can be found experimentally by measuring the step response of eddy current hysteresis. Indeed, by assuming initial condition B(O) -- -Bin, we apply the field Ho(t) - s(t), where s(t) is the unit step function. We can then measure flux ~s(t), which corresponds to wo(t) = t. By excluding time t, we find the function ~s(W0), which describes the ascending branch of the major loop. Thus, we arrive at the remarkable conclusion that nonlinear (and dynamic) eddy current hysteresis can be fully characterized by a step response.

It is clear from the previous discussions that the ascending branch of the major loop can also be experimentally found by measuring response �9 (t) to any monotonically increasing function w0 = X(t), that is, to any positive and sufficiently large current i(t). Indeed, for any monotonically increasing function w 0 - X(t), we can find the inverse function t - x-l(w0). By substituting the latter function into response ~(t), we find the ascending branch ~(x-l(w0)) of the major loop. By using this branch, we can predict eddy current hysteresis for arbitrary time variations of current i(t).

Boundary value problems (6.112)-(6.115) and (6.117)-(6.120) can be used for a very elegant derivation of the formula for the front zo(t) in the case of plane boundary, that is, in the 1D case. In that case, the boundary value problem (6.112)-(6.115) is reduced to:

d2w = 2r~Bm if 0 < z < z0(t), (6.125)

dz 2

w(0, t)-- w0(t), (6.126)

w(zo(t),t) =0, (6.127)


dw(z, t) I -- 0. (6.128)

dz zo(t)

The solution to Eq. (6.125) that satisfies the boundary conditions (6.126) and (6.128) has the form

W(Z,t)-- crBmz 2 - 2r~Bmzzo(t) 4- wo(t). (6.129)

To find z0(t), the boundary condition (6.127) is used, which leads to

-r~Bmz2(t) 4- wo(t) - O. (6.130)

The last expression yields

_ /wo( t ) ( foHo(r)dr) 1 Zo(t) V crBm - - ~ ' (6.131)

which is identical to formula (6.26). To find H(z, t), we differentiate both sides of Eq. (6.129) with respect

to time and recall (6.110), which yields

H(z, t ) - -2r~Bmzddl t) 4- Ho(t). (6.132)

From formula (6.131) we find

dzo(t) Ho(t) = (6.133)

dt 2crBmzo(t)" By substituting the last expression into formula (6.132), we arrive at

( zt H(z,t)=Ho(t) l-z-- ~ , (6.134)

which is consistent with formula (6.21). By using formula (6.131), we can derive the expression for the ascend-

ing branch of the major loop of eddy current hysteresis in the case of a magnetically nonlinear conducting lamination. In the above case, we have

= -~m 4- 2Bmzo(t). (6.135)

By substituting formula (6.131) in the last expression, we find 1

cI)=-cI) m 4-2|Bmw~ ~ ~ ~ , (6.136) [ . . ] rY

which is the equation for the ascending branch. It is clear from this equation that this branch has a vertical (infinite) initial slope, that is, the slope at w0 - 0 (see Fig. 6.16). For this reason, weight function #(c~,/~) is singular and must be understood as a distribution. Although we have arrived at


~r

I

I I I

-Om Om

FIGURE 6.18

CI)

409

this conclusion for the case of lamination, it is of a general nature. This is because at initial stages (i.e., for small penetration depths) nonlinear diffusion in conducting bodies with curvilinear boundaries occurs almost (i.e., asymptotically) in the same way as in the case of plane (flat) boundaries. As a result, the ascending branches always have vertical initial slopes and functions/~(~, fl) are always singular.

The above difficulty can be completely circumvented if we consider the inverse w0 vs �9 hysteretic relation. This relation is shown in Fig. 6.18. It can be mathematically shown that this inverse hysteretic relation can also be represented by the Preisach model

ff+ w o C t ) - >>,~ (6.137)

with the weight function v(~, fl), which has the following property:

v(~, f l )= v(~ - fl). (6.138)

The last property suggests that function v(~, fl) can be fully determined by using the ascending branch of the major loop. In the case of lamination, this branch can be found analytically. Indeed, by using formula (6.136), we derive

O" [O+( I )m] 2 w 0 - ~ 2 ' (6.139)

which is the equation for the ascending branch. To find function v, we shall invoke formula (1.28) from Chapter 1. For

the case when property (6.138) is valid, function )v(c~, fl) also depends on


ot - fl, and formula (1.28) takes the form

d2F v(ot) = dot2. (6.140)

For the ascending branch (6.139), function 9 v and argument ot can be identified as follows:

~ = Wo, ot - ~. (6.141)

By substituting the last equalities into expression (6.139) and then by using formula (6.140), we derive

(7

v(ot - /3) = 2Bm = const. (6.142)

In the general case, function v(ot - / 3 ) can be found experimentally. The best way to see how it can be accomplished is to write formula (6.137) in terms of current i(t) and voltage v(t). To this end, we employ formulas (6.101), (6.106), and (6.105) and after simple transformations we arrive at the following expression:

//. E'/o' l -~ fir) dr - v(ot -/3)}9~ ~ v(r) dr + ~o dot d~. (6.143)

The last expression can be construed as a terminal voltage-current relation for a coil placed around a conducting magnetic cylinder. The difference of this relation from the one given by formula (6.107) is that terminal relation (6.143) is in a "voltage controlled" form. This suggests that by applying any positive voltage v(t) (for instance v(t) = s(t)) and by measuring i(t), we can find corresponding functions ~(t) and wo(t). By excluding time t from those functions, we can find a relation w0(~), which represents the ascending branch. This relation can be used for the determination of

~.(Ho)

-H c

/ v

H C

FIGURE 6.19

6.3 EDDY CURRENT LOSSES 411

v(c~ - /J ) , or it can be directly used to predict current i(t) for arbitrary variations of voltage v(t).

Finally, we remark that formula (6.122) can be generalized to the case when abrupt magnetic transitions are described by rectangular hysteresis loops. It can be easily shown that in this case formula (6.122) can be modified as follows:

II (/o ) �9 (t) - dd~>~ #(c~,/~)G~ X(H0(r)) dr dc~ dfl, (6.144)

where function X(H0) is defined as (see Fig. 6.19)

X(Ho) = (Ho - Hc)s(Ho - He) + (No + H c ) s ( - H o - He), (6.145)

and s(.) is the unit step function. As approximations, the last two formulas can be used in situations

when actual hysteresis loops are close to rectangular ones. In those situations, the same simple experiments as described before can be used for the identification of function/~.

6.3 EDDY CURRENT LOSSES IN M A G N E T I C C O N D U C T O R S WITH ABRUPT M A G N E T I C T R A N S I T I O N S . EXCESS HYSTERESIS LOSSES

It is well known that classical eddy-current losses in magnetic conducting laminations are proportional to the square of frequency (f2) of time- varying magnetic flux. This result is usually derived under the following two assumptions: (1) distribution of the magnetic flux density over lamination cross-section is uniform, and (2) "edge" effects are neglected (one- dimensional problems).

It will be shown below that the f2-1aw for eddy-current losses is universal in the case of abrupt magnetic transitions. The universality of the f2-1aw manifests itself in the fact that this law holds for cylindrical conductors of arbitrary cross-sections and for highly nonuniform distributions of magnetic flux density. It should be pointed out that even if eddy- current losses for abrupt magnetic transitions meet thef2-1aw, the amount of losses for a given frequency is larger than the corresponding classical eddy-current losses. This increase in eddy-current losses, usually called "excess losses", depends on the shape of the conductor.

The presence of excess losses has been traditionally attributed to the existence of domains within the magnetic conductors [10, 11]. It turns out that experimentally observed frequency dependence of eddy-current


losses may have quite different qualitative features depending on the mi- crostructural properties of the materials. In this respect, materials that display the f2-1aw for eddy-current losses generally exhibit a very fine domain structure [12], while for other materials, the deviation from the f2-1aw can be appreciable [12]. The analytical treatment presented below (see [13]) is based on the use of constitutive relation. This is tantamount to the assumption that a macroscopically small portion of the media can be described by a local relation between magnetic field H and magnetic flux density B. This assumption can be considered appropriate for materials with very fine domain structures, where, even for very small portion of the media, the total magnetic moment is the average of the magnetic moments of a large number of domains.

In the sequel, we assume sinusoidal variations of the voltage applied to the coil, which is the common case for transformers and electrical ma- chines. In this case, eddy-current losses per unit vertical length are given by

1 ft~ ~ Pe - - ~ H0(t) (tic E. d l )dt

1 fto+T d~ - -T ,,to Ho(t)-d- f dt

co [to+T de' 2re Jto Ho(t)-d- f dt, (6.146)

where to is an arbitrary instant of time and P is the contour around the cylinder cross-section. The general form of the flux waveform ~(t) will be

�9 (t)-- ~0 - ~ cos(cot), (6.147)

where �9 and ~0 are, respectively, the amplitude of oscillation and the constant component of the flux waveform. The latter can be determined by meeting the following condition:

~ 0 - - ~ m + ~ (6.148)

which guarantees that for t - 0 , the flux assumes the value -~m- Let us consider one period of the function ~(t) and assume to - 0. In

the first half of the period, the flux increases from its minimum - ~ m to

its maximum ~0 4- ~. The increase in the flux results in the formation of a positive rectangular front of magnetic flux density and its inward motion within the cylinder cross-section. This inward motion will produce eddy- current losses. In the second half of the period, the flux decreases from its maximum ~0 4- ~ to its minimum -~m. The decrease in the flux occurs


through the formation and the inward motion of a negative rectangular front of magnetic flux density. There is intrinsic symmetry between propagation of positive and negative rectangular fronts of magnetic flux density. For this reason, the change of the front polarity will only change the sign of the electric field. However, eddy-current losses are proportional to the square of the electric field and will not be affected by this change in sign. Thus, the energy losses in the second half of the period are exactly the same as the energy losses in the first half. As a result, eddy-current losses can be computed as follows:

Pe = ~yr L T/asO(t) d-ff-tt dt. (6.149)

Next, we express Ho(t) in terms of ~(t). This can be achieved by differentiating both sides of (6.137):

d d{i/w v(ot, fl)~(~(t))doldfl}. (6.150) Ho(t) = ~wo(t)- -~ (*m,--~m)

Substituting (6.150) into (6.149) and appropriately arranging the order of integrations and differentiation, one obtains

{/o ''2' P< - 7- at (6.151) Yr ( C~ m , _ CI) m )

The time derivative of } 9 ~ ( t ) y i l l be different from zero onlyfor those }9~ that satisfy the condition ~ + ~0 ~> a ~> fl ~> - ~ m = ~0 - ~- During the first half of the period, these }9~ will be switched "up" in the time instants (see Fig. 6.20)

t r

cI)0

A

FIGURE 6.20


1 ( . - ,0 ) f~ = -- arccos - A (6.152)

co

and c o n s e q u e n t l y the t ime d e r i v a t i v e of }9~fi (1)(f) is g i v e n b y

d d--t 1~# [O(t)] = 2 a ( t - t , ) , 0 <~ t <<. T /2 . (6.153)

Using this equation and taking into account the fact that

dO(t) = coo sin(cot) (6.154)

dt

we can rewrite (6.151) in the following form:

Pe 2co2~ f = v(c~ - /3) sin(cot~) dc~ d3, (6.155) Jr

where Tm is the triangle T(~', 3') represented in Fig. 6.21 with ~' = ~ + Oo and 3' = - ~ + Oo. Then, by using (6.152) in (6.155), we find

Pe 2coZOm f = v(c~ - 3)~/1 - [(c~ - O0)/$]2 dot dfl. (6.156) 7/" m

Equation (6.156) can be further simplified by introducing new integration variables

c~ - ~ - O0, fi - fl - O0 (6.157)

which yields

2 c o 2 ~ / ~ Pe : V(Ot -- fi)~l -- (~ / $)2 dc~ d/~, (6.158)

re ($,_$)

(X

T(~', ~')

S

FIGURE 6.21


where T ( ~ , - ~ ) is now the triangle in the plane (&, fi) defined by the inequality ~ ~> & ~> fi ~> - ~ . It is interesting to note that the integral in (6.158) does not depend on ~. Therefore, losses are proportional to the square of the frequency. We next prove how this integral can be evaluated directly in terms of simple experimental data. This is achieved by using step response characterization of eddy-current hysteresis.

For the sake of notational simplicity, we introduce the following function:

G(~, (I)) = ffl - (~/(I)) 2. (6.159)

By taking into account (6.140) and the fact that Y'(~ + (I)m) = 1}/~d(ot), where W(~) describes the ascending branch of eddy current hysteresis, one obtains from (6.158) the following integral expression for losses:

092~ ~m)G(ol, dfi Pe - rr - - - -

J i l l

(6.160)

o2~f__~ f__:~ By integrating with respect to fi and using (6.148), we can transform (6.160) as follows:

P e - O92~ { f_ ~ r r $ W'(& + ~o)G(6l,'~)d~

-- wt(--(I)m) f G(~, ~) d~ . (6.161) $

Then, integration by part in the first integral yields

Pe r { []/~(~ -t- ~o)G(~, ~)]~$ - f_~ - w(~ + | ~ c(~, ~)a~

+ W'(-*m) [ G(~, ~) a~ . (6.162) j_ $

A A

The term in square brackets is zero since G(+~, q~)= 0. The last term is also zero because W ' ( - ~ m ) = 0 regardless of the cylinder's cross-section shape. This fact can be easily verified in the case of lamination. For generic cross-sections, the following reasoning can be used. The value of W ' ( - q~m) depends on the initial stage of the diffusion of magnetic flux density within the conductor. In this initial stage, the penetration of magnetic flux density is confined to a very thin layer adjacent to the boundary of the cylinder, and nonlinear diffusion occurs almost in the same way as in the


case of plane boundaries. As a result, the initial slope of the ascending branch of the major loop of eddy-current hysteresis, i.e., W'(-~m), vanishes for any smooth shape of the cylinder cross-section.

Finally, by invoking (6.159), we arrive at the following formula:

O92 f cI) C~ Pe = - - W ( ~ Jr- ~o)

"~ V/ '~ 2 __ ~t 2 d&. (6.163)

This is the final expression of eddy-current losses in terms of the measured ascending branch W(*) of eddy-current hysteresis.

Next, we study the scaling laws of eddy-current losses with respect to the conductivity rr of the conductor, the saturation magnetic flux density Bm, and the shape and the dimension of the cross-section. These scaling laws are the direct consequence of the fact that the Preisach representation of eddy-current hysteresis is based on the solution of the following nonlinear diffusion equation:

32H 32H 3 3x---- i -~- ~ = o'Bm -~ sign(H), (6.164)

which obeys very general scaling laws with respect to quantities rr Bm and cylinder cross-sectional area A. Indeed, let us introduce the following dimensionless spatial variables:

_ x y (6.165) x = / X , 9 = , / 7

and dimensionless magnetic field

,-, H

rrBmA

H

rY~m (6.166)

In these new variables, (6.164) becomes

02~--1 02~--:1 O sign(H) (6.167) + . O~: 2 Oy 2

This is the nonlinear diffusion equation for a conducting cylinder with abrupt magnetic transition with unit saturation magnetic flux density, unit conductivity, and unit cross-sectional area. The solutions of (6.167) are in one-to-one correspondence with the solutions of (6.164). From this correspondence, one can easily derive that the ascending branch of eddy- current hysteresis for actual cylinder is related to the ascending branch W(.) of eddy-current hysteresis for the "normalized" cylinder through the following scaling law:

W ( ~ ) - o ' B m A W ( ~ / ~ m ) = ry~m]'~(~/~m). (6.168)


By using this result, formula (6.163) can be written in a more convenient form. In fact, by replacing the integration variable ~ with ~ = ~/Om, we find

Pe - r~~ ~ ) r r ~m ' (6.169)

where

L(X) = W(~ + X - 1) d~ (6.170) X VX 2 -- ~""""--~

and X = O / Om. The function L(X) is a dimensionless function, which takes into account the effect of the shape of the cylinder cross-section on eddy- current losses.

The discussion presented above can be further generalized to the case of the constitutive relation described by the rectangular hysteresis loop. In this case, the relationship between the magnetic flux and the magnetic field on the boundary H0 is given by formulas (6.144) and (6.145).

By inverting the Preisach operator (6.144), one obtains

f0 t / /T wo(t) = X (H0(r)) dr = v(~, 3)G~O(t) dc~ d3. (6.171) (--~m,~m)

We assume, as before, that the flux waveform O(t) is given by (6.147). Moreover, we assume that the hysteretic relation (6.171) between wo(t) and O(t) has only strictly increasing branches. In order to facilitate the analysis, let us introduce the fictitious magnetic field Ho(t ) defined by

d Ho(t ) = X(Ho(t)) - -~wo(t)

d = d~lffr(.m,_.m) V(~,3)f,~(oo(t))d~d3}. (6.172)

Now, let us focus our attention on (6.171). When the flux is increasing, wo(t) is also increasing, and this implies that X(Ho(t)) > 0. When this is the case, Ho(t ) = X(Ho(t)) = Ho(t) - Hc. On the other hand, when the flux is decreasing, wo(t) is decreasing too and X(H0(t)) < 0. In this case, Ho(t ) = X(Ho(t)) - Ho(t) + He. We can summarize this by

Ho(t) = Ho(t) + He sign(dO/dt), (6.173)

which expresses the field on the boundary Ho(t) in terms of the fictitious field Ho(t ).


By using (6.173), the total losses per unit vertical length can be expressed as

1 [to+T dO = Ho(r)--d-ddr P -T Jto

~o [tO+THo(t)dO 2re j to --~ dt

H ftO+T (aeo) ao sign dt. +2zr c j t 0 --~ -~

(6.174)

The first term takes into account eddy-current losses. By using (6.172), this term can be evaluated by following the same line of reasoning as in the previous discussion. The second term is related to the hysteretic nature of the constitutive relation and takes into account hysteresis losses. In fact, the second integral in (6.174) can be computed as follows:

o)2rcH c [to+T sign( ) dO (6.175) -27 -27

where A = O/Bin is the cross-sectional area swept by the front of magnetic flux density. It is important to notice that A does not depend on the frequency but only on the flux waveform and on the cross-section shape. Therefore, the integral (6.175) is linearly dependent on the frequency. Equation (6.175) has the following transparent interpretation: the product 4HcBm corresponds to the area of hysteresis loop. The quantity 4HcBmA is the energy dissipated per unit vertical length. Therefore, (6.174) expresses the separation of total losses into eddy-current losses and hysteresis losses. Eddy-current losses are proportional to the square of the frequency, while hysteresis losses are proportional to the frequency.

It is customary to speak of excess eddy current losses, while excess hysteresis losses are rarely (or ever) mentioned. Nevertheless, these excess hysteresis losses do exist and they may be prevalent and dominant at very low frequencies.

It is demonstrated below that the origin of excess hysteresis losses can be traced back to some intrinsic nonlinear dynamics underlying bistable (multistable) hysteretic behavior. This intrinsic dynamics leads to in- evitable delay of switching between metastable states. As a result of this switching delay, a traced hysteresis loop is somewhat broader than the underlying static (rate independent) hysteresis loop. This broadening is the manifestation of excess hysteresis losses. This broadening may occur for nonconducting hysteretic materials, which clearly suggests that it cannot be attributed to eddy currents.

The purpose of the following discussion is to demonstrate the universal nature of excess hysteresis losses as well as their universal dependence


ec/f actual losses /

" ~ classical losses "abrupt" increase

A

B/-- , l

13

\


~ U

on frequency [14]. By using the mathematical formalism of the Preisach model of hysteresis and some known results [15, 16] on nonlinear bistable dynamics, it is demonstrated that for small frequencies excess hysteretic losses are increased with frequency as 0)2/3. This universal low frequency dependence of excess hysteresis losses may explain a puzzling "abrupt" increase of excess core losses observed for low frequencies (see Fig. 6.22). This fairly abrupt, low frequency increase cannot be easuly attributed to excess eddy current losses, while it can be easily understood by using the notion of excess hysteresis losses.

To start the discussion, consider systems with hysteresis that can be represented (in a static limit) by the following Preisach model:

f f ~ A f (t) - /x(~, ~)F'~u(t) d~ d~. (6.176)

Here/x(~,/~) is a weight function determined from the identification pro- A cedure, while P~ are the elementary hysteresis operators defined by the loops shown in Fig. 6.23. It is also tacitly assumed that, in the case of magnetics, input u(t) and output f(t) can be construed as the magnetic field and the magnetic flux density (or magnetization), respectively. It can be easily seen that I '~u(t) can be mathematically represented as follows:

v~(t) = PA~u(t)= f~(u) -f~(u) 9~u(t)+f~+~(u)+fs (u), (6.177) 2

where i3~ are rectangular loop operators with output levels -t-1 and with and/J being "up" and "down" thresholds, respectively, whi lef~ andf~


are continuous parts of descending and ascending branches of the loops that define V'~#u(t). It is apparent from (6.177) that hysteresis model (6.176) is a particular case of the "input dependent" Preisach model (see Chap- ter 2). It is also clear that in the case when f ~ and f ~ are flat, the model (6.176) coincides with the classical Preisach model.

With each rate independent operator P~#u(t) we shall associate the rate dependent (dynamic) hysteresis operator described by the first order nonlinear dynamical system

dv~# dt

-F V~#(v~#) - u(cot). (6.178)

Here, function V~#(v) is a continuously differentiable function and consists of three branches: f ~ (v~#),f~ (v~#) and & (v~#) (see Fig. 6.23).

It is self-evident that in the limit of co ~ 0 the rate dependent (dynamic) hysteresis operator (6.178) is reduced to the rate independent hysteretic operator (6.177). Thus, one can say that nonlinear differential equation describes the underlying dynamics of bistable switching implied by the static loop shown in Fig. 6.23.

Next, we shall assume that the function u(cot) is periodic with exactly one maximum and one minimum within the period T - 2__~. We shall also

CO

assume that the peak values of input u(cot) are appreciably larger than the switching thresholds ~ and # of all operators P~#u(t). This implies that the major hysteresis loop of hysteretic nonlinearity (6.176) will be traced. Finally, let us suppose that input u(cot) takes its minimum value at the beginning of period T. In this situation, the solution of differential equation (6.178) will first follow the ascending branchf~ (see Fig. 6.24). When input u(cot) reaches the switching threshold c~, the switching will not occur immediately. This is because of dynamic delay that causes the output value to be less than v~. Thus, some input increase (input increment) will be necessary to affect the switching. In other words, the switching will

o

fo,13"~,, I) ', 0(,

_ ALi c

= U

FIGURE 6.24


commence at some input value that is larger than the static switching threshold c~. This phenomenon clearly results in the broadening of hysteresis loop and leads to excess hysteresis losses.

The input increment necessary to affect the switching can be evaluated for low frequencies by using the following asymptotic analysis. We shall use the following expansion for function V ~ ( v ) and input u(cot) around the switching point A (see Fig. 6.24):

W~fl (v)-~ vc~- fi 4- ~ G / ? (v~-]~) (v - V~-~) 2, (6.179) 2

! u(co t ) - UA 4- UACOt. (6.180)

By substituting formulas (6.179) and (6.180) into differential equation (6.178), we end up with the nonlinear first order Ricatti differential equation. By using the appropriate change of variables, this nonlinear Ricatti equation can be reduced to linear second order Mathieu equation with variable in time coefficients. The solution of this Mathieu equation can be given in terms of Airy function. By using the asymptotic analysis of Airy function and its derivative, it can be shown that the input increment A C (see Fig. 6.24) necessary to affect the switching is given by

A C - ko~flco 2/3. (6.181)

The mathematical details of the described analysis can be found in [15] and [16]. Similar broadening will occur at the switching point B (see Fig. 6.23). There will be additional broadening of the dynamic hysteresis loop caused by the finite time of switching. However, since the switching is a fairly fast process and the input varies slowly in the limit of low frequencies, this broadening will be small in comparison with one caused by an output lag around the switching points. Thus, it can be accurately assumed that in the limit of low frequencies the broadening of hysteresis loop area ~ for each elementary operator P ~ u ( t ) (for each a and ~) is proportional to a92/3:

,A~ (a)) ~ ,A~x/? (0) 4-/Cc~ (-o 2/3. (6.182)

Formula (6.182) has been verified by using the following numerical experiment. Differential equation (6.178) has been solved for the case of V ~ ( v ~ ) - 2v3~ - 3/2(v~) + 1/2 (or its diffeomorphic deformations) and u(cot) = -urn cos cot. It is apparent that the chosen function V ~ ( v ~ ) is not symmetric with respect to the origin v~ - 0 and for this reason, it leads to the "shifted" static operator P~u(t). This, in turn, results in nonequal (nonsymmetric) broadening in the case of "up" and "down" switchings, which is demonstrated in Fig. 6.25. After the numerical solution of differ-


1.2 , , ,

1 ~ , - , - - . - -~

o ~ . . . . . . . . . . . . . . . i . . . . . . . i . . . . . . . . . . i . . . . . . . . .

o ~ . . . . . . I - - - o , - ~ . , o - ~ t . . ,~ . . . . . . . . . i

0 . 4 . . . . . . . I - - u=V(v ) ~ [ ! . . . . . . . . . . t

0 . 2 " ' . . . . . . . .

0 :

I I

�9

, .

. . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . .

-O.6 ! . .

~.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 input (u)

0.8

-0 .2

0.6

0.4

0.2

0

~ -0 .2

~ --0.4

-0 .6

-0 .8

- 1 . . . . . - 0 " : . . . . ~ . o

�9 . , .

i i ! ' . . . .

-0 .15 -0.1 -0 .05 0 0.05 0.1 0.15 0.2 input (u)

FIGURE 6.25

ential equation (6.178) was performed, the broadening of hysteresis loop has been evaluated for various frequencies. The results of these calculations for different values of Um are shown in Fig. 6.26 and compared with formula (6.182). It is apparent from this figure that formula (6.182) predicts very accurately the low frequency broadening of hysteresis loop.

Now, we can compute the broadening of the major loop of hysteresis nonlinearity described by the model (6.176). The area enclosed by a hysteresis loop is given by the formula

A(co) = f u(cot)dr(t). (6.183)

5.3 EDDY CURRENT LOSSES

~ 1 0 -~

1 0 - 2

10 -5

. . . . . . . . . . . . .

. . . . . . . . .

} ! ! i i " ! : , ; ~ . ,J . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . !...i..~...~.~: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :: . . . . . . . i.~.. i ~ . ' ~

i i i ! ! i i i i ! ! ~ ' " 0 " " : - ' : r :

. . . . . . . . . . . . . . .', . . . . . . . . i . . . . . . : . . . . i . . . . " . . . . . . - , - , - . . . . . . . . . . . . . . , ~ . - , , , O f f , : , . . ~ . . . . . : - - - ; �9 : ' " " -

........... : i .... :: : : : i : : : : i : i : ! : : i : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

. . . . . . . . . . . . . . . . . " . . . . . . . ) r 0 " . . . . . .

i i i - : : : : , X ~ , , ' ~ : : : ' : . . : . . i . . . . . . . . . . . . . . . i . . . . . . . . i . . . . . . ~ - - . . . . ! - . . ! ~ , ~ ! ~ . ~ . . . . . ~ , o . . . . i . . . . . . . i . . . . . . i . . . . i i ! : !

; X " , G r - , - . . . . . . . . . .

~ - ' ; r - ' ! . " . ~ . i i i i i i i ! ! ! . . . . . . . ~ " " ; . . . . . . . . ; ~ . . . . - . . . . : . . . . ; , . , ; , . ; , . . : , . ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . , , ; , . . ; . . ;

" o " "~." ! i i i i i ] i i ! ! i ! ! ," ~ " i ! i ! : ! : i ! : : : : : : : :

~ " i ! ! : i i i i ~ i i i i i i i " r . . . . . . . . . . . ' . . . . . . . . ' . . . . . . ." . . . . ; . . . . ; " ' " " " ; " " ' ; " " ; . . . . . . . . . . . . . . . ; . . . . . . . . . : . . . . . . " . . . . ' " " ! " : " ' : " ! -

. . . . . . . . . . . . . 2, . . . . . . . . : . . . . . . ; . . . . , . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . , . . . . . . . . . . . .

; ; ; ~ ~ . . . . . . : : : i : ; ; . . . . ; ; ; ;

l 0 - 4 1 0 - 3

Frequency

FIGURE 6.26

423

A

By taking into account that each rate independent operator P~ in (6.176) is associated with rate dependent operator defined by (6.178), we can write

dr(t) = / / ~ # ( ~ , ~)dv~(t) dc~ dfl. (6.184)

By substituting formulas (6.184) into formula (6.183) and by changing the order of integration, we obtain

A(co)-/~>~l~(~,~)(fu(cot)dv~(t))dc~d~. (6.185)

Since the internal integral in (6.185) is equal to the broadened area of the elementary hysteresis loop F~u(t), from formula (6.182) we find

f U(cot) ~" A(0) q- K~CO 2/3. (6.186) dv~(t) By substituting the last formula into the expression (6.176), we arrive at

A(co) ~" A ( 0 ) q - ]Coo 2/3. (6.187)

The last formula describes the frequency dependence of broadening of the area of major hysteresis loop. It is clear that the term K~CO 2/3 accounts for excess hysteresis power losses. The last formula is valid for the entire class of hysteretic nonlinearities described by the model (6.176). This clearly


reveals the universality of low frequency dependence of excess hysteretic losses.

6.4 EDDY CURRENT ANALYSIS IN THE CASE OF G R A D U A L M A G N E T I C T R A N S I T I O N S

In Section 6.1, nonlinear diffusion of electromagnetic fields in conducting media with abrupt magnetic transitions was discussed and simple analytical solutions were derived. However, these solutions do not allow one to understand how actual gradual magnetic transitions (or actual shapes of hysteresis loops) may affect the diffusion process. For this reason, the analytical study of nonlinear diffusion of electromagnetic fields in conducting media with gradual (and more or less realistic) magnetic transitions is an important problem. It has been extensively discussed in [9] and [17- 20]. Next, an attempt will be made to solve this problem for the case of hysteresis loops that are exemplified by Fig. 6.27. These hysteresis loops are characterized by the property that their ascending (and descending) branches can be subdivided into two distinct parts: part I of slow increase of magnetic flux density B from -Bin to -Bc and part II of steep increase

B

Bmq . . . . . .

[/n - H m Hc]

I i Hm I I I I I I I I I I

- - ~ - - i - /7"-- - ~ - B m

~ H

FIGURE 6.27

6.4 CASE OF GRADUAL MAGNETIC TRANSITIONS

B m qr"

- H m H c 9 "r

�89 - B m

~ H H m

425

F I G U R E 6.28

of B from -Bc to Bm. Such hysteresis loops are typical for most ferromagnetic materials in the case of sufficiently large values of Hm and they are encountered in many applications.

To attempt the analytical solution of nonlinear diffusion equation (6.1), we adopt a "flat-power" approximation of a hysteresis loop shown in Fig. 6.27. This approximation is illustrated in Fig. 6.28 and it is analytically described by the following equations:

B = - B m if - H m < H ~ H c , (6.188)

1

B + Bm = [ k ( H - Hc)] -~ if Hc ~ H ~ Hm (6.189)

in the case of the ascending branch, and

B = B m if - H c ~ H ~ H m , (6.190)

1

Bm - B - [k(H + Hc)] -~ if - H m ~ H ~ - H c (6.191)

in the case of the descending branch. In other words, part I of the ascending branch is approximated by a "flat" straight line parallel to the H-axis, while part II is approximated by the "power" expression (6.189).

In the above formulas, coefficient k coordinates the dimensions of both sides of expressions (6.189) and (6.191), while the exponent n is a measure of the sharpness of magnetic transition. It is important to note that


in applications the exponent n is usually larger than 7 (n ~ 7). This fact is essential and it will be used in our subsequent discussions in order to simplify relevant analytical expressions and to achieve some universality in the final form of the solution to the nonlinear diffusion equation. By introducing the "shifted" magnetic field h and magnetic flux density b

h - H - Hc, b = B 4- Bm, (6.192)

expression (6.189) can be rewritten as follows:

b n h - -~-. (6.193)

Next, we shall consider the following "model" problem. It will be assumed that at time t - 0 the magnetic flux density B is equal to -B in throughout the conducting half-space:

B(z, 0) --- -Bin . (6.194)

It will also be assumed that the magnetic flux density at the boundary of the conducting half-space is monotonically increased with time as follows:

B(0, t )= - B m + ct p (p > 0). (6.195)

By using the nonlinear diffusion equation (6.1) as well as expressions (6.192) and (6.193), the stated model problem can be reduced to the following initial boundary value problem: find the solution of the nonlinear diffusion equation

O2b n 3b 3z 2 -- kr~ O---t" (6.196)

subject to the following initial and boundary conditions:

b(z, 0) = 0, (6.197)

b(0, t) = ctP. (6.198)

It is worthwhile to mention that these boundary conditions are chosen for the following two reasons. First, it will be demonstrated that it is possible to find simple analytical solutions for these boundary conditions. Second, these boundary conditions describe a broad class of monotonically increasing functions as p varies from 0 to oo (see Fig. 6.29). It will be shown in the sequel that for all these monotonically increasing boundary conditions the distribution (profile) of the magnetic flux density as a function of z remains practically the same. This observation will suggest using the same profile of magnetic flux density for arbitrary monotonically increasing (between -Bin and Bin) boundary conditions. This, in turn, will


b(O,t)

~ p< 1 /

~ p>l

~ - t

F I G U R E 6.29

427

lead to very general and simple analytical solutions, which can then be extended to periodic in time boundary conditions.

It turns out that the "model" problem (6.196)-(6.198) can be reduced to the boundary value problem for a certain ordinary differential equation. To accomplish this, we introduce the following dimensionless variable:

z

= ~ / k _ l f f _ l c n _ l tin, (6.199)

where p ( n - 1) + 1

m - 2 " (6.200)

By using this variable ~, we shall look for the solution of the initial boundary value problem (6.196)-(6.198) in the following form:

b(z, t) - ctPf(~), (6.201)

where f(~) is some dimensionless function of variable ~. By substituting formula (6.201) in Eq. (6.196), we arrive at the following differential equation for f(~)

d2f n d f d~ 2 -}- m~ - ~ - p f -- O. (6.202)

By using expressions (6.199) and (6.201), we can easily conclude that b(z, t) given by (6.201) will satisfy the initial and boundary conditions (6.197) and (6.198), respectively, if the function f(~) satisfies the boundary conditions:

r i O ) - 1, (6.203)

f ( ~ ) = 0. (6.204)

Thus, the initial boundary value problem (6.196)-(6.198) is reduced to the boundary value problem (6.202)-(6.204) for nonlinear differential equation (6.202). This nonlinear equation has some interesting properties. For


instance, it can be proved that if f(~) is a solution to Eq. (6.202), then the function

2 F(~) = X-~-l f0~ ) (6.205)

is also a solution to the same equation for any constant ~. This fact can be utilized as follows. Suppose we can find some solution

f(~) to Eq. (6.202) that satisfies the boundary condition (6.204), but does not satisfy the boundary condition (6.203):

f(0) = q # 1. (6.206)

Then, by using n - 1

= q 2 , (6.207)

we observe that the function

F(~) - ~ f ( q ~ ) (6.208)

will be the solution to Eq. (6.202), that satisfies the boundary condition (6.204) and, in addition,

F(0)_ _1 f (0)= 1. (6.209) q-

This demonstrates that we can first find a solution to Eq. (6.202) satisfying the boundary condition (6.204), and then, by using the transformation (6.205), we can always map this solution into the solution that satisfies the boundary condition (6.203) as well. It can be shown (see [9]) that a solution to Eq. (6.202) satisfying the boundary condition (6.204) has the form

/ a(1 - ~)~[1 4- a1(1 - ~) 4- a2(1 - ~)2 4 - . . . ] f(~) / 0

if 0K~ ~1 , (6.210)

if ~ 1 .

By substituting the last formula into Eq. (6.202), after simple but lengthy transformations, we find:

1 c~ - (6.211)

n - l ' 1 [m nl lnl

a - , (6.212) n

p ( n - 1 ) - m al - 2 m n ( n - 1 ) ' (6.213)

1 4- 0.5a1[(2n - 1)(3n - 2) - 4n] a2 = - a l . (6.214)

3 ( 2 n - 1)

6.4 CASE OF GRADUAL MAGNETIC TRANSITIONS 429

It is clear that

riO) = a(1 + al + a2 + . . . ) # 1. (6.215)

This can be corrected by using transformation (6.205) with

n-1 X = [a(1 + al q- a2 -+-...)] 2 . (6.216)

This leads to the following solution of the boundary value problem (6.202)-(6.204):

[ (1 - ; ~ ) 1 l+ai(1_x~)+a2(1_~)2+.., if 0 ~< i~ ~< 1, f(~) -- 1+a1+a2+.-- (6.217) / 0 if X ~ > l .

It is clear from (6.213) and (6.214) that al and a2 depend on n and p. How- ever, it is possible to derive the following inequalities for these coefficients expressed only in terms of n:

1 lall ~< 2 n ( n - 1)' (6.218)

1 1 la21 ~ + �9 (6.219)

6(n - 1)(2n - 1)n 8n2(n - 1)

It has been previously stressed that the exponent n in the "power" approximation (6.193) is usually larger than 7. By using this fact, from inequalities (6.218) and (6.219)we derive

lall < 0.012, la21 < 0.00075. (6.220)

The above estimates suggest the following simplification of formula (6.216) and solution (6.217):

n-1 ~ l ~ z _ _ - X = a 2 - - , (6.221)

(1 - ~/m(n-1)~)~_l if 0 ~< ~ ~< ~/m(n 1), f(~) - (6.222)

0 if ~ ~ ~/m(nnl).

By substituting the last expression into formula (6.201) and taking into account definition (6.199) of ~, we end up with the following analytical solution for the model problem:

{ 1 _ d-/-~) ~ if 0 ~ z <~ dt m, (6.223) b(z,t) ctP(1- z

0 if z >~ dt m,


where 1 (ncnl)

d-- kcrm(n- 1) (6.224)

It is easy to observe that solutions of the model problem exhibit an interesting property. It is clear from formulas (6.199) and (6.201) that z-profiles of magnetic flux density at various instants of time can be obtained from one another by dilation (or contraction) along b- and z-axes. In other words, those z-profiles remain similar to one another. This explains why solutions of the type (6.201) and (6.223) are called self-similar solutions. The property of self-similarity is closely related to the choice of "power" approximation (6.193) and boundary conditions (6.198) that makes the problem amenable to the dimensional analysis. The intrinsic property of the self-similar solutions is that they are dimensionally deficient. This property allowed us to reduce the nonlinear partial differential equation (6.196) to the ordinary differential equation (6.191). It is also clear that the self-similar solutions are invariant under certain scaling transformations. For this reason, they are often called group-invariant solutions.

The self-similar solutions discussed in this section have been derived by using dimensional analysis. For this reason, they are regarded as self- similar solutions of the first kind. There are, however, self-similar solutions that cannot be obtained by using dimensional analysis alone. These solutions contain additional parameters, which are called anomalous dimensions. These are self-similar solutions of the second kind, and they are physically significant because they describe intermediate asymptotics [21]. The interesting treatment of these solutions by using the machinery of the renormalization group is presented in the book [22].

The self-similar solutions for nonlinear diffusion equation (6.196) were first studied by Ya. Zeldovich and A. Kompaneyets [23] for the ra- diative heat conduction problem and by G. Barenblatt [24] for problems of gas flow in porous media. The discussion presented in this section closely parallels in some respects the work of G. Barenblatt.

A brief examination of the obtained self-similar solutions (6.223) leads to the following observation. Profiles of magnetic flux density b(z,t) as functions of z remain approximately the same (see Fig. 6.30 as well as formula (6.223)) for wide-ranging variations of the boundary conditions (6.198) (see Fig. 6.29). For typical values of n (n ~ 7), those profiles are very close to rectangular ones. This insensitivity of self-similar solutions profiles to a particular boundary condition suggests that actual profiles of magnetic flux density will be close to rectangular ones for any monotonically increasing boundary conditions bo(t) = b(0, t). Thus, we arrive at the following generalization of self-similar solutions (6.233).


b ( tl< t2<t 3)

1 . Zo(t,) Zo(t )

b

-/--ta bo(t3)

bo(t2)

bo(tl)

~Z

/t2

/t3

A i �9 w w i , i ,

Zo(t3) 0 Zo(tl) Zo(t2) Zo(t3)


431

~-z

The actual profiles of magnetic flux density b(z, t) are approximated by rectangular ones with the height equal to the instantaneous boundary value bo(t)"

b0(t) if 0 ~< z ~ z0(t), b(z, t )= 0 if z >~ zo(t). (6.225)

This generalization is illustrated by Fig. 6.31. We recall that rectangular profiles of magnetic flux density were en-

countered in Section 6.1 when we discussed nonlinear diffusion in media with abrupt magnetic transitions. For those transitions, rectangular profiles of magnetic flux density can be attributed to abrupt magnetic saturation. The self-similar solutions (6.223) show that b-profiles are close to rectangular ones even if media are not saturated. Rectangular-like shapes of b-profiles can be explained as follows. In the process of diffusion, magnetic field h is attenuated as z is increased. The attenuation of h results in the increase in magnetic permeability (defined as # = b _ kh 1-1). This increase, at first, compensates for the decrease in h and leads to more or less "flat" values of b. When values of z are sufficiently close to z0, the very fast attenuation of h cannot be compensated for by the increase in # and this results in the precipitous drop in magnetic flux density b.

Next, we shall derive the following expression for the zero front zo(t) of b(z, t) in (6.225):

1

zo(t) = abo(t) " (6.226)

The last formula can be derived by using the first moment relation for nonlinear diffusion equation


32h 3b = a m (6.227)

3z 2 3t"

that is easily obtainable from (6.196) and (6.193). Indeed, let us multiply Eq. (6.227) by z and integrate from 0 to z0(t):

fo ZO(t) 02h ;zo(t) 3b

z-d~z2 dz = ~ z m dz. (6.228) JO 3t

By integrating twice by parts in the left-hand side of Eq. (6.228) and by taking into account that

Oh (zo(t), t) - O, h(zo(t), t) - O, (6.229)

3z we obtain

~o zo(t) 82h Z-~z 2 dz = h(O,t) = ho(t). (6.230)

We remark here that the first equality in (6.228) comes from the fact that the electric field is equal to zero at z - zo(t) and oh oH _ - a E . -O--~ -- 0 z -

By using the formula of differentiation of integral dependent on parameter, we obtain

d ~oZ~ ;z~ Ob d~ zb(z, t) dz - z m dz 4- zo(t)b(zo(t), t) dzo(t____~) (6.231)

JO 3t dt "

Since b(zo(t),t) - 0 , (6.232)

from formula (6.231), we derive

f zo(t) 3b d fzo(t) z m dz - zb(z, t) dz.

JO at ~ J O (6.233)

By substituting expressions (6.230) and (6.233) into formula (6.228), we arrive at the following first moment equation:

fot ;zo(t) h0(r) dr - r~ zb(z, t) dz. do

(6.234)

By using the rectangular profile approximation (6.225) in formula (6.234), we obtain

o t z2(t) (6.235) h0(r) dr = cr b0(t)--~---,

which leads to formula (6.220). The "rectangular profile" approximation just discussed is very suit-

able for the derivation of time periodic (steady state) solutions of nonlinear diffusion problems. Consider periodic time variations of magnetic


field Ho(t) at the boundary of magnetically nonlinear conducting half- space. Suppose that at time to initial condition B(z, to) = -Bin is in effect. Furthermore, suppose that magnetic field Ho(t) is increased from He to Hm during the time interval to ~ t ~ tin, then it is decreased from Hm to Hc during the time interval tm ~ t ~ to, and finally it is decreased from Hc to -Hc during the time interval t0 ~ t K ~ + to, where T is a period of Ho(t). As the magnetic field Ho(t) is increased from Hc to Hm, the rectangular profile of magnetic flux density is formed and it moves inside the conducting media. The front zo(t) of this profile can be found by using formula (6.226), which can be rewritten in terms of Ho(t) and Bo(t) as follows (see expressions (6.192)):

[2 ftto[Ho(r) - Hc] dr ] 1 zo(t) = . (6.236)

r~(Bo(t) + em) When the magnetic field at the boundary reaches the value of Hm, the height of the rectangular profile becomes equal to Bin. As the magnetic field at the boundary is decreased from Hm to Hc, the inward progress of the rectangular profile is continued and its height remains the same and equal to Bin. The latter is in accordance with the "flat-power" approximation of hysteresis loops (see Fig. 6.28). The front, z0(t), of the rectangular profile can now be found by replacing Bo(t) in formula (6.236) by Bm, which leads to

z~ - I ~t~176 - Hc] dr ] " (6.237)

As the magnetic field at the boundary is further reduced from Hc to -Hc, nothing happens. This means that the rectangular profile of magnetic flux density remains still because induced eddy currents and electric fields are equal to zero.

As the magnetic field at the boundary is reduced from -Hc to -Hm and then increased from -Hm to Hc during the time interval to + ~ K t to + T, the rectangular profile of "negative" polarity is formed and it moves inside the conducting media. Its inward progress is fully analogous to the progress of the rectangular profile of "positive" polarity described above for the time interval to ~ t K to + ~. The front, z0(t), of the rectangular profile of "negative" polarity can be determined by using the formula

12~t ] zo(t) = t0+~- [H0(r) + Sc] dr �89 Bm) " (6.238)

During subsequent cycles, the situation repeats itself. In formulas (6.236)-(6.238) for the front zo(t), magnetic field Ho(t) and

magnetic flux density Bo(t) at the boundary are related by the "flat-power"


approximation of the hysteresis loop. Thus, if Ho(t) is known, then, by using formulas (6.188)-(6.191), we can find B0(t), which, in turn, can be used for the calculations of zo(t).

The rectangular profile approximation can be further extended to make it directly applicable to actual hysteresis loops of the type shown in Fig. 6.27. In this extension, it is assumed that as the magnetic field at the boundary is increased from Hc to Hm, the rectangular profile of magnetic flux density is formed and it moves inside the media. This assumption is supported by the derived self-similar solutions and their "rectangular profile" approximation. As the magnetic field at the boundary is decreased from Hm to -Hc, it is assumed that the profile of magnetic flux density retains its rectangular shape as well as its inward progress (see Fig. 6.32a). That assumption is justified by the fact that the magnetic flux density varies slightly as the magnetic field varies from Hm to -Hc. This prevents appreciable deformations of magnetic flux density profiles. Actually, this profile deformation may even improve the resemblance of actual magnetic flux density profiles to rectangular ones. Indeed, when the magnetic field at the boundary is increased from Hc to Hm, the boundary values of magnetic flux density are larger than those within the media and a "flat" part of magnetic flux density profile exhibits some small "downward" slope. As the magnetic field at the boundary is decreased from Hm to -Hc, the magnetic flux density at the boundary is reduced faster than within the conducting media, and this may result in the "flattening" of the above downward slope and in better resemblance of actual profiles to rectangular ones.

Diffusion of rectangular profiles of magnetic flux density of opposite polarity occurs in a similar way during the next half-period. This is shown in Fig. 6.32b.

The front, z0(t), of the rectangular profiles can be determined by using formulas (6.236) and (6.238). However, in these formulas Ho(r) and Bo(r) are now related through the actual shapes of hysteresis loops rather than by their "flat-power" approximations. This is justified on the grounds that the derivation of formulas (6.236) and (6.238) was based on general nonlinear diffusion equation (6.227) and the "rectangular" profile assumption. This derivation did not use the "flat-power" approximation of hysteresis loops. The latter approximation was instrumental in the derivation of self-similar solutions and, in this way, it paved the road for the notion of rectangular profiles of magnetic flux density. Now, the "flat-power" approximation of hysteresis loops can be passed into oblivion.

The described model of nonlinear diffusion implies that at every point of conducting media the magnetic field and magnetic flux density are related by the same hysteresis loop as at the boundary. This is a natural


Bm "t' 7

Bo ' i "

-Bo tl_l _' ,-

t6 = to + T

; Z

( t l < t2< t3< t4< t5 < t 6)

I---

,--Z

\ t ~ t, 5 s t)+T

( t l < t2< t3< t4< t5 < t6)

{a) (b)

FIGURE 6.32

435

consequence of rectangular profile approximation. In reality, at different points of conducting media the magnetic field and magnetic flux density are related by different hysteresis loops. However, because actual profiles of magnetic flux density are close to rectangular ones, these hysteresis loops are almost the same as the loop at the boundary. This is true everywhere within the conducting media except for a very narrow region where the precipitous drop in the magnetic flux density occurs.

Next, we shall derive the impedance-type relation between electric field Eo(t) and magnetic field Ho(t) at the boundary of conducting media. To this end, we consider the half-cycle to ~< t ~< to 4- ~ and recall that

1 OH 1 Oh Eo(t) = E(0, t) . . . . (0, t) . . . . (0, t). (6.239)

r~ 3z r~ 3z

By using the law of electromagnetic induction, we find

d E o ( t ) - ~[bo(t)zo(t)]. (6.240)

By taking into account that

bo(t) - Bo(t) 4- Be, ho(t) - Ho(t) - Hc, (6.241)


and by using formula (6.226), from the last expression we derive 1

d t E0(t) -- -~[2 (Bo(t)+Bc) fo (Ho(r) -Hc)dr] ~ . (6.242)

Now, we introduce the following functions"

Ho(t) - He fH(t) = , (6.243) Hm Bo(t) + Be

fB(t) - �9 (6.244) Bm It is assumed that magnetic field Ho(t) at the boundary is known. Then, by using the actual shape of the hysteresis loop, we can find Bo(t). Next, by employing the last two formulas, the functions fH(t) and fB(t) can be fig- ured out. Thus, it will be assumed in the subsequent discussion that func- tionsfH(t) andfB(t) are known. By using the definitions (6.243) and (6.244) of these functions as well as the following definition of magnetic permeability #m"

Bm ]l,m(Hm)-- Hm" (6.245)

the formula (6.242) can be transformed as follows: 1

~ f ~ d t 2 Eo(t)=Hmv-~---~[2fB(t) ftofH(r)dr ] . (6.246)

Next, we shall scale the time t by using the formula

2Jrt t' -- = cot, (6.247)

T

where 27r

co = (6.248) T

is the frequency of the fundamental (first) harmonic of H0(t), which is periodic (but may or may not be sinusoidal).

By using the scaling defined by (6.247), formula (6.246) can be further transformed as follows:

t' 1

~co~m d [a~B(t, ) fo fH(r')dr'12 (6.249) E 0 ( f ' ) - - Hm dt--- 7

where we used the following notations:

fB (t ' )=fB ( ~ ) , fn(t')=fg(~). (6.250)


By introducing the function 1

fE(t')= ~---~[afB(t') f i ' fS(r ')dr' l ~,

we present the formula (6.249) as follows:

(6.251)

Ho(t) = Hm sin o)t. (6.253)

In this case, we will be interested in the first harmonic of E0(t), which can be written as follows:

~Og~m t' , E~l)(t')-Hm (acos + bsint') (6.254)

2 ft~ +t; t' a=-- fE(t')cos dt', Jr

(6.255)

b - 2 ft ~+t~ -~ fE (t') sin t' dt'. (6.256)

To simplify the calculations, we assume that He << Hm, which is typical in many applications. Because time to is determined by the equation Ho(to) = He, from the last inequality and formula (6.253) we conclude that to -~ 0. We shall also adopt the following power approximations:

1

Bo(t) + Bc -(Bm -Jr- Bc) ( H~ -Hc ) ~ \ Hm -Hc (n > 1), (6.257)

/ E0 (t') - Hm ~/O)~m fE (t'). (6.252)

v O"

Formulas (6.252) and (6.251) constitute one of the most important results of this chapter. These formulas represent a nonlinear impedance-type relation between tangential components of electric and magnetic fields at the boundary of conducting media. This relation is nonlinear because # m is a function of Hm. Formulas (6.252) and (6.251) are very general in nature. They are valid for arbitrary periodic (not only sinusoidal) boundary condition Ho(t) with only one restriction: the total cycle T can be subdivided into half-cycles of monotonic variations of Ho(t). Another distinct feature of the above impedance-type relation is that it directly relates the time variations of E0(t') to the time variations of H0(t') as well as to actual shapes of hysteresis loops. The latter is accomplished through functiond~B(t').

To illustrate how formulas (6.252) and (6.251) can be used in calculations, consider a particular case when


for the "steep" part of hysteresis loop traced when Ho(t) is increased from Hc to Hm, and

em - eo(t) = (Bm - Bc) ( 1 - No(t)q-gc) nl Hm + Hc (nl > 1), (6.258)

for the "flat" part of hysteresis loop traced when H is reduced from Hm to -He.

By using approximations (6.257) and (6.258) and the assumption Hc << Hm, from formulas (6.243), (6.244), (6.250), and (6.253), we derive

fi' f0t' ( t')2 fH(r') d r ' = sinr' dr' = 1 - cos t' = 2 sin ~ , (6.259)

1 tt Jr )?B(t') = (1 + X)(sint') ~ if 0 ~ <~ ~ , (6.260)

re t' , )?B(t') = (1 + X ) - (1 -- X)(1 - sint ') nl if ~ ~< ~< 7r (6.261)

where Bc

X = (6.262) Bm

is the "squareness" factor. By substituting expression (6.259), (6.260), and (6.261) into formula

(6.251) and then plugging the result of substitution into formulas (6.255) and (6.256), after integration by parts we derive:

If0 t b= ---4 ~ ( l + x ) l ( s i n t , ) ~ s i n ~ c o s t , dt, 7g

/2 t, l + [(1 + X) - (1 - X)(1 - sint ')nl] �89 sin g cost'dt' , (6.263)

4 4 [ f 0 ~ t' a - - - - ~ / 2 - X + -- (l + x ) l (s int ' )~ s in-~sint 'dt ' 7r 7r

/2 t ] + [(1 -t- X) - (1 - X)(1 -- sint')nl] ~ sin ~ sint ' dt' .

(6.264) Formula (6.254) can be represented in the phasor form

= P0, (6.265)

where the surface impedance 7/is given by

r I = v/a 2 + b2,/c~ e j~~ (6.266) V O"


a tan e - ~. (6.267)

By using formulas (6.263) and (6.264), we can compute tan tp and v/a 2 + b 2 for various values of x ,n , and nl. In this way, we can evaluate to wha t extent the surface impedance depends on a part icular shape of hysteresis loop. Computat ions show that the surface impedance is not very sensitive to variations of n and nl, whereas variations of X may appreciably affect the surface impedance, especially the value of tanr The results of cal-

culations of tan ~a and v/a 2 + b 2 as functions of X are shown in Figs. 6.33a and 6.33b, respectively. These calculations have been performed for n -- 10 and nl = 4. It is apparent from Fig. 6.33a that tan tp varies from 0.5 to 0.71. There is an extensive body of experimental data publ ished in Russian lit-

0.63

tan q), f

0.75~ . . . . . . . . . . . . . . . . . . . . ' . . . . . . . . . . . . . . . . . .

L 0.69 ~

0.57

0.51

0.45

0.60 0.70 0.80 0.90 1.00

Va2.b b 2

1.35

1.33

1.31

1.29

1 . 2 7 .

0.60

. . . . . . . . . i . . . . . . . . . i . . . . . . . . . i . . . . . . . . . , ~

0.70 0.80 0.90 1.00

(a)

(b)

FIGURE 6.33


erature [25], which suggests that tan ~a varies between 0.5 and 0.69. Thus, our computational results are consistent with these experimental data.

The curves shown in Figs. 6.33a and 6.33b can be fairly accurately approximated by the following expressions:

tan ~ "" 1.01 - 0.53X,

v/a 2 q- b 2 ~" 1.16 + 0.19X.

This leads to the following simple formula for the surface impedance:

, - (1 .16 + 0.19X)~C~ m e jtan-l(l'01-0"53x). (6.268)

In the last formula, the dependence of the surface impedance ~ on the shape of the hysteresis loop is represented by the two parameters only: "squareness" of the loop X and magnetic permeability #m.

Next, consider the penetration depth, 3, in the case of boundary condition (6.253). This depth can be defined as follows:

8 - z0 ~ + to .

By using formula (6.226) and expressions (6.241) and taking into account that Hc << Hm, to ~ 0, and B0(T + to) = Be, we derive

T 1

8=[f~H~ ~ r ~ B c " (6.269)

By using boundary condition (6.253) and the definition (6.262) of the "squareness" factor X, we obtain

T 1

[Hmf~ sino~rdr]~ 8 = . (6.270) XaBm

By performing integration in formula (6.270) and recalling that /~m = Bm/Hm, we finally arrive at

~ 2 8 = ~ . (6.271)

XcoCr#m

As in the case of surface impedance, we can observe that the dependence of penetration depth 8 on the shape of the hysteresis loop is represented by two parameters only: "squareness" of the loop X and magnetic permeability #m. The latter is a nonlinear function of Hm, which is determined by a main magnetization curve that passes through vertices of symmetric hysteresis loops. This makes the penetration depth field dependent.


B

I _ _I_A

B

Bo . . . . . . . . I

z - - " z z

2

FIGURE 6.34

441

The previous analysis can be easily extended to the important case of magnetically nonlinear conducting laminations. Indeed, during initial stages of positive half-cycles, nonlinear diffusion of magnetic fields at both sides of laminations occurs in the same way as in the case of conducting half-space (see Fig. 6.34a). The motion of front zo(t) can be calculated by using formula (6.226). At the instant of time ta such that

A z0( ta ) - ~- (6.272)

two rectangular fronts are merged together and the distribution of magnetic flux density over a lamination cross-section is uniform. It remains this way until the commencement of negative half-cycle (see the same Fig. 6.34a). During negative half-cycles, the situation reverses itself (see Fig. 6.34b). It is apparent from this discussion that in the case of gradual magnetic transitions eddy currents are being induced all the time, whereas in the case of abrupt magnetic transitions eddy currents are limited in time and only induced during initial stages of half-cycles, that is, before the rectangular fronts merge.

The rectangular front model of nonlinear diffusion described above has been used for the calculation of eddy current losses in ferromagnetic conducting laminations [26]. The losses predicted by this model have been compared with experimental data as well as with losses computed by using finite elements. The experiments were performed on a ferromagnetic conducting laminations (1 mm thick Fe-Si 2 wt%, conductivity of 8.85.106 S/m) by testing (30 mm wide, 300 mm long) lamination pieces, inserted in an Epstein frame. Power losses were measured under controlled magnetic flux density waveform by a digital feedback wattmeter. The current was adjusted in order to obtain sinusoidal flux trough the


�9 Experimental data 0 v Finite element computations ~

s0o o rectangular front model . . . . . . . . Classical losses

�9

V ..- ..... 4 0 O

. . .. , , . j . . . . .

, i .... ..

. . i . ' .... �9 ,,,,,,,

..... �9 . . ..... .

~ 3 0 o . . . . . . . ' ... - j . . . . . i . . . .

�9 , .....

2 0 O

- .

. .,~.. , . z " i

�9 . ,

100 . . . . . . ..- ..

P " . Bp = i l .ST

0 �9 . "

o i i , i 0 50 100 150 200 250 300

f [Hz ]

F I G U R E 6.35

lamination. The comparison of the experimental data with both numerical computations and rectangular front model predictions, in the case of 1.5 T average flux density, is shown in Fig. 6.35. For this value of the average magnetic flux density, the hysteresis loop in any location inside the lamination is almost the same and approximately coincides with the loop shown in Fig. 6.36. In this situation, the rectangular front model gives remarkably accurate predictions.

Thus, one can reach the conclusion that for sufficiently strong field, the rectangular front model is an accurate and simple one to describe the diffusion of the electromagnetic field inside conducting lamination. This model provides simple formulas for the computation of eddy current losses. Moreover, by using this simple model, one can overcome the difficulties of numerical solution of nonlinear diffusion in the presence of sharp front propagation which requires very fine meshes in order to accurately resolve these front profiles.

It is well known that the actual eddy current losses in ferromagnetic laminations are appreciably higher than the classical eddy current losses computed under the assumption of uniform distribution of magnetic flux density. This increase in eddy current losses, usually called "excess losses," has been traditionally attributed to the existence of domains within the magnetic conductors. The purpose of the preceding discussion has been to experimentally confirm the validity of a different explanation of the excess losses, which has been usually overlook in the

6.5 CASE OF CIRCULARLY POLARIZED FIELDS

2 i ! ! i i i i

o experirnentaJ da, a I ! ' !

i ! ~ o~ , Z

~ O.5[ ! I

-~ ! :: i o i i

-1.5[ I I I I - 2 I

- 8 0 0 -600 -400 - 2 0 0 0 2 0 0 4 0 0 600 8 0 0 m a g n e t i c f i e l d [ A / m ]

FIGURE 6.36

443

previous studies of this topic. This explanation has been first suggested in [19] and it is based on the peculiar nature of nonlinear diffusion of electromagnetic fields in magnetically nonlinear conducting laminations. The essence of this peculiar nature is that nonlinear diffusion occurs as an inward progress of almost rectangular profiles of magnetic flux density of variable height. As a consequence of the front motion, the magnetic flux density is not uniform even for relatively low frequency and this results in the increase of eddy current losses [27].

6.5 E D D Y C U R R E N T A N A L Y S I S I N T H E C A S E O F C I R C U L A R L Y P O L A R I Z E D E L E C T R O M A G N E T I C F I E L D S . R O T A T I O N A L E D D Y C U R R E N T L O S S E S

In this section, we shall continue our discussion of nonlinear diffusion of plane electromagnetic waves in the conducting half-space. In the previous section, the treatment of this problem was carried out for the case of linearly polarized magnetic fields. In that case, the analysis was reduced to the solution of a scalar nonlinear diffusion equation. In many applications, the magnetic field is not linearly polarized. For this reason, it is of importance to consider nonlinear diffusion of arbitrary polarized electromagnetic fields. It can be shown that the above problem is reduced to the


solution of the vector nonlinear diffusion equation

32H 3B(H) = a ~ (6.273)

3z 2 3t "

or two coupled scalar nonlinear diffusion equations:

02Hx OBx(Hx, Hy) = a (6.274)

3Z 2 3t "

32Hy 3By(Hx, Hy) =r~ . (6.275)

3Z 2 3t

This obviously raises the level of mathematical difficulties. However, these difficulties can be completely circumvented in the case of isotropic media and circular polarization of electromagnetic fields. This is due to the high degree of symmetry of the problem in the above case. The analysis of nonlinear diffusion for the circular polarization of an electromagnetic field is of interest for the following two reasons. First, linear and circular polarizations can be viewed as two limiting cases for other types of polarization. Therefore, the solution of the problem for these two limiting cases may provide some insights in how the surface impedance of magnetically nonlinear conducting media depends on the type of polarization. Second, elliptical polarizations of electromagnetic fields can be treated as perturbations of the circular polarization. For this reason, the perturbation technique can be extensively used for the solution of the vector nonlinear diffusion equation (6.273) in the case of elliptical polarizations of electromagnetic fields.

We shall proceed with the discussion of nonlinear diffusion in the case of circular polarization and isotropic media. For isotropic media, the Cartesian components of the magnetic flux density are related to the Cartesian components of the magnetic field by the formulas

- , ( ( . 2 +.;).x. (6.276)

,(j 2 + .; (6.277)

where #([H]) = #(y/H 2 4-H~) is the magnetic permeability of isotropic

conducting media. It is clear that we deal with the case of unhysteretic media. The case

of isotropic hysteretic media will be treated later in this section. By substituting formulas (6.276) and (6.277) into Eqs. (6.274) and

(6.275), respectively, we end up with the following coupled nonlinear dif-

6.5 CASE OF CIRCULARLY POLARIZED FIELDS 445

fusion equations:

32Hx - 3 II~(v/H2 4- H ~ ) H x ] (6.278) O z 2 - rr -~

32Hy - 3 []z(v/H2 4- H ~ ) H y ] (6.279) Oz 2 - ry -~

We shall be interested in time-periodic solutions to the above equations subject to the following boundary conditions:

Hx(0, t) = Hm cos(~ot 4- 00), (6.280)

Hy(O, t) - Hm sin(~ot 4- 00), (6.281)

Hx(oe, t) - O, (6.282)

Hy(~ , t ) = 0. (6.283)

The boundary conditions (6.280) and (6.281) correspond to the circular polarization of the magnetic field, whereas the boundary conditions (6.282) and (6.283) reflect the fact that the magnetic field decays to zero.

Now we shall make the following very important observation. The mathematical structure of nonlinear partial differential equations (6.278) and (6.279) as well as of boundary conditions (6.280)-(6.283) is invariant with respect to rotations of x- and y-axes around the x-axis. In other words, the mathematical form of the above equations and boundary conditions will remain the same for any choice of x- and y-axes in the plane z - 0. This suggests that the solution of the boundary value problem (6.278)- (6.283) should also be invariant with respect to the rotations of the x- and y-axes. 1 This, in turn, implies that the magnetic field is circularly polarized everywhere within the conducting media:

Hx(z, t) - H(z) cos(~ot 4- 0(z)), (6.284)

Hy(z, t) = H(z) sin(~t 4- O(z)). (6.285)

Next, we shall formally show that the circularly polarized solution (6.284)-(6.285) is consistent with the mathematical form of the boundary

1Strictly speaking, this statement is valid when the solution to the boundary value problem (6.278)-(6.283) is unique, which is assumed here on physical grounds. In the case when there are many (or infinite number) of solutions, the symmetry of equations may not be reflected in the symmetry of each individual solution, but rather in the symmetry of the overall pattern of all solutions. This is the so-called "spontaneous symmetry breaking" phenomenon.


value problem (6.278)-(6.283). First, it is clear from formulas (6.284) and (6.285) that

[H(z)[- v/H2(z) + H~(z) - H(z). (6.286)

This means that the magnitude of the magnetic field and, consequently, the magnetic permeability/z(HI) does not change with time at every point within the conducting media.

Next, we represent formulas (6.284) and (6.285) in the phasor form:

Hx(z) - H(z)e j~ (6.287)

Hy (z) - - j H (z)e jO (z), (6.288)

where, as before, the symbol ..... is used for the notation of phasors, while

It is apparent from expressions (6.286), (6.287), and (6.288) that

and

HKz) = I xKz) l = I Kz) l,

(IH(z)l) = -

(6.289)

By using phasor (6.287) and (6.288) as well as the formula (6.290), it is easy to transform the boundary value problem (6.278)-(6.283) into the following boundary value problems:

a2 x(Z) dz 2 = jcocr i~ (]Hx (z)[) Hx(z), (6.291)

and

Hx(O) - Hme j~176 (6.292) A

Hx(cX~) - 0, (6.293)

d2Hy(z)

dz 2

Hy(O) - - j H m e j~176 (6.295) A

Hy(oo)-0. (6.296)

This exact transformation of the boundary value problem (6.278)- (6.283) into boundary value problems (6.691)-(6.293) and (6.294)-(6.296) can be construed as a mathematical proof that the circular polarization of the incident wave is preserved everywhere within the magnetically nonlinear conducting media. This also proves the remarkable fact that there

(6.294)

(6.290)


are no higher-order time-harmonics of the magnetic field anywhere within the media despite its nonlinear magnetic properties.

From the purely mathematical point of view, the achieved simplification of the boundary value problems (6.278)-(6.283) is quite extensive. First, partial differential equations (6.278) and (6.279) are exactly reduced to ordinary differential equations (6.291) and (6.294), respectively. Sec- ond, the boundary value problem (6.278)-(6.283) for coupled equations is reduced to two completely decoupled boundary value problems (6.291)- (6.293) and (6.294)-(6.296). Finally, the decoupled boundary value problems have identical mathematical structures. As a result, the same solution technique can be applied to both of them.

It turns out that simple analytical solutions of Eqs. (6.291) and (6.294) can be found in the case of a power law approximation of a magnetization curve. This approximation is given by the expression:

(B) n H = ; (n > 1), (6.297)

which can also be rewritten as follows:

S = k H 1. (6.298)

This approximation implies the following formula for the magnetic permeability:

I~(H) = kH ~-1. (6.299)

Sketches of the B vs H and # vs H relations corresponding to the power law approximation are shown in Figs. 6.37 and 6.38, respectively. It is clear that, for the above approximation, the magnetic permeability is decreased as the magnetic field is increased. Thus, this approximation takes magnetic saturation of media into account. However, this approximation idealizes the actual magnetic properties of media for very small values of magnetic field. Namely, the permeability approaches infinity as the field approaches zero. The physical implications of this idealization will be discussed later.

B ~t

~ H ~ H



By using formula (6.299), we find the following expression for the magnetic permeability # m at the boundary of media:

1 1

[d.m = kHrn . (6.300)

By combining formulas (6.299) and (6.300), we can exclude the coefficient k from expression (6.300):

1_ 1

# ( H ) = tim Hmm " (6 .301)

From formulas (6.290) and (6.301), we obtain A 1 1 Hx(z) ~ (6.302)

# , , . . .Z . , , ( [ n r { ' ) [ } = # m Hm "

~ ( ] H y ( z ) I ) = # m

A

Hy(z) Hm

1

�9 (6.303)

respectively, we arrive at the following boundary value problems:

d 2 H x ( z )

dz 2 = jo~a tim

A

Hx(z) Hm

1_ 1 ..~ Hx(z), (6.304)

Hx(O) = Hme j~176 (6.305)

Hx(oO) = 0, (6.306)

and

where

(6.306). This solution will be sought in the form

H x ( z ) - Hm(1-~)~ if O<~z<~zo,

0 if z >~ zo,

Hm - H m e jO~ , (6.311)

I jo/ll = c~ + . (6.312)

d 2 ~ ( z )

dz 2 = j~oer tim

A

Hy(z) Hm

1_ 1 A Hy(z), (6.307/

~Iy( O ) = - jHme J~ (6.308) A

Hy(oo)-0. (6.309)

First, we find the solution to the boundary value problem (6.304)-

(6.310)

By substituting expressions (6.302) and (6.303) into Eqs. (6.291) and (6.294),


It is clear that the function (6.310) satisfies the boundary conditions (6.305) and (6.306). Next, we shall choose parameters z0,c~' and c~" in the way that Eq. (6.304) will also be satisfied. To this end, we shall rewrite formula (6.310) as follows:

A

H x ( z ) ( 1 - ~ O ) ~ ----- -- . (6.313) Hm

The last expression is writ ten in an abbreviated form with the tacit under- s tanding that it is valid for 0 ~< z K z0, while for z ~> z0 the r ight-hand side is equal to zero. Similar abbreviations will be tacitly used in subsequent formulas when they are appropriate.

From formulas (6.311)-(6.313), we find

which leads to

A Hx(z)

Hm Z t ~ = 1 - ~ , (6.314)

A ] 1 Hx(z) ~-

Hm

d(n-1) ( Z ~ ) n = 1 - . (6.315)

By substituting formulas (6.313) and (6.315) into Eq. (6.304), we arrive at

-~- ( Z ) Oe-2 ( Z~0) Oe ~ ot(oe- 1)Hm 1 - ~o - jo)a#mHmz 2 1 - n . (6.316)

It is clear that the last equality will hold, if the following two conditions are satisfied:

oe '(n- 1) 2 ~ - - I H

and

(6.317)

c~(c~- 1)=jr, or (6.318)

From formula (6.317), we find

, 2n oe = . (6.319)

n - 1

Next, we shall use the characteristic equation (6.318) to determine ~" and z0. To this end, we shall represent this equation in the form

(c~' + jc~")(c~' - 1 + jr - jo)cr #mZ 2, (6.320)

which is equivalent to the following two equations:

~'(c~' - 1 ) - (E') 2 = 0 , (6.321)


c~" (2c~' - 1) = co~ ~mZ~. (6.322)

By using formulas (6.319) and (6.321), we find

,, v/2n(n + 1) c~ = . (6.323)

n - 1

Next, by substituting this expression for ~" into Eq. (6.322), we arrive at

(2c~' - 1)~/~'(c~' - 1 )= corY~m Z2. (6.324)

By taking formula (6.319) into account in the last equation, we finally obtain:

[2n(n 4- 1)(3n + 1)2] z0 - . (6.325)

~/coCr #m(n - 1)

Formulas (6.310), (6.319), (6.323), and (6.325) completely define the solution to the boundary value problem (6.304)-(6.306). The boundary value problem (6.307)-(6.309)is identical (up to notations) to the boundary value problem (6.304)-(6.306). For this reason, the solution to the boundary value problem (6.307)-(6.309) can be written in the form

H y ( z ) - - j H m 1 - ~ , (6.326)

where c~ and z0 are given by the expressions (6.312), (6.319), (6.323), and (6.325).

Solutions (6.310) and (6.326) are written in terms of phasors. We shall next transform them into time-domain forms (6.284) and (6.285). To this end, we shall first use the following transformations:

Hx(Z) = Hme j~176 1 - G

~' (6.327)

--Hm (1-~oZ) eJ[Oo+o~"ln(X-~o)]

Now, by using the standard expression

Hx(z, t) - Re[Hx(z)e j~~ (6.328)

and formulas (6.327), (6.319), and (6.323), we derive

2n

( Z ) ~-=-~- 1 [ v/an(n4-1) ( z ) ] Hx(z, t) - Hm 1 - ~0 cos ~ot 4- 00 4- n - 1 In 1 - ~ . (6.329)


By repeating the same line of reasoning, we arrive at the following expression for Hy(z, t):

Hy(z,t) Hm(1 ff-~O) n2-~n-1 [ v /an(n+l ) ( z~) l = - sin ~ot + 00 + In 1 - . (6.330) n - 1

In the last two formulas, parameter z0 is given by expression (6.325). Formulas (6.329) and (6.330) give the exact analytical solution to the

boundary value problem (6.278)-(6.283) in the case of power law relation (6.299). Next, we shall analyze this solution. If we fix time t in the last two formulas and consider Hx and Hy as functions of z, then we can easily observe that on the interval 0 ~< z ~< z0 these functions have infinite numbers of zeros (infinite numbers of oscillations). It is also clear that the sequences of Hx-zeros and Hy-zeros converge to z0. This is the result of logarithmic dependence of phase O(z) and z. Indeed, by comparing the last two formulas with expressions (6.284) and (6.283), we find

v/Sn(n+X) ( ~ ) O(z) - Oo + In 1 - (6.331)

n - 1

2//

mtl n-1 6332

By using the last equation and formula (6.301), we find the following expression for the magnitude of magnetic flux density as a function of z:

1 2 ~H(z)) ~-1 ( z ) n-1

e(z) --- #m~, Hm H(z)- Bm 1 - G " (6.333)

where the following notation is introduced

Bm -- #mHm. (6.334)

A typical plot of B(z) as a function of z is shown in Fig. 6.39. It is apparent from this figure (as well as from formula (6.333) and other previous formulas) that there is a finite depth z0 of penetration of electromagnetic fields into magnetically nonlinear conducting media. This can be explained by the fact that power law approximation (6.298) introduces idealization of magnetic properties of conducting media by allowing for the infinite growth of the magnetic permeability when the magnetic field tends to zero. This infinite growth in # causes the complete attenuation of the magnetic field at the finite distance z0. Actual z-variation of magnetic flux density B(z), schematically shown in Fig. 6.40, exhibits a tail "1" at small values of B(z). This tail is usually of no practical significance and

452

B

z0


Zo


~ Z

can be neglected. As a result, the depth z0 attains the physical meaning of the penetration depth of the "bulk" part of magnetic flux density.

Now, we proceed to the discussion of surface impedance in the case of circular polarization. To find this impedance, we shall use the following formulas for the electric field phasors:

E'x(Z)- 1 dHy(z) A 1 dHx(z) (6.335) a----7-' Ey(z) = a----7-

By using the last formulas as well as expressions (6.310) and (6.326), we find

Ex(0)= ~ Hy(0), E y ( 0 ) = - Hx(O). (6.336) ryz 0 ryz 0

From the last equations we obtain the following expression for the surface impedance:

EK(O) Ey(O) - ,- = - ~ = . (6.337)

Hy(0) Hx(O)

Now, by invoking formulas (6.312), (6.319), (6.323), and (6.325), we represent the surface impedance in the form

- ( a + jb)~/~ , (6.338) v o-

where a n

a - (6.339) [2n(n 4- 1)(3n 4-1)2] 1 '

b = v/2n(n 4- 1) (6.340)

[2n(n 4- 1)(3n 4- 1)2]�88 .


It is also convenient to represent the surface impedance in the polar form

~I = I rl le j~ , (6.341)

where

t )1~(.O~ b ~ n 4 - 1 2n ~ m tango - - (6.342) Irll = n + 1 ' a 2n "

By performing simple calculation in accordance with formulas (6.339), (6.340), and (6.342), we find that as n varies from 7 to cx~, coefficients a

and b as well as v/a 2 4- b 2 and tan ~0 vary within the following narrow limits:

.92 K a ~ .971, (6.343)

.687 K b ~ .694, (6.344)

1.15 ~ v/a 2 4- b 2 ~ 1.19, .707 ~ tan ~ K .76. (6.345)

This suggests that with fair accuracy the surface impedance rl can be represented by the formulas

945§ or

(6.346)

rl-1.17~/~ ej36.3o" (6.347)

It is instructive to compare these results with the expression for the surface impedance obtained in the previous section for the case of linear polarization of electromagnetic fields. This expression can be written in the same

form as (6.341)-(6.342), however, the limits of variations for v/a 2 4- b 2 and tan ~ are appreciably different and specified here:

1.28 ~ v/a 2 + b 2 ~ 1.35, (6.348)

.49 ~ tan ~ ~ .71. (6.349)

It was shown that in the case of linear polarization the value of surface impedance is most sensitive to the "squareness" X of hysteresis loops. There is no parameter like that in formulas (6.339) and (6.340). This makes the above comparison between the cases of circular and linear polarization somewhat ambiguous. This ambiguity can be completely removed in the case of abrupt magnetic transitions described by rectangular magnetization curves. These magnetization curves can be obtained from the power


l aw approximation (6.298) in the limit of n approaching infinity. In this limit, by using formulas (6.339), (6.340), and (6.342), we find

v/a 2 + b 2 - 1.19, (6.350)

tan ~o = 0.707. (6.351)

In the case of linear polarization, for the same quantities we have (see formula (6.39))

& 2 q_ b 2 = 1.34, (6.352)

tan ~o = 0.5. (6.353)

The last two expressions have been obtained for sinusoidal variations of magnetic field at the boundary of conducting half-space. In the case of sinusoidal variations of electric field at the boundary, the last two expressions are modified as follows (see formulas (6.55) and (6.56)):

v/a 2 Jr- b 2 - 1.47, (6.354)

tan ~ - 0.424, (6.355)

which makes the discrepancy with the case of circular polarization even more pronounced. This discrepancy suggests the importance of polarization effects on the surface impedance.

Next, we shall extend the results of this section to the case of isotropic hysteretic media. In the case of isotropic hysteresis, a uniformly rotating magnetic field results in a uniformly rotating component of magnetic flux density. However, due to hysteresis, this uniformly rotating component of magnetic flux density lags behind the magnetic field (see Fig. 6.41). Apart from the above-mentioned uniformly rotating component of magnetic flux density, there can be a component B0 of the magnetic flux density that does not change with time. This constant component is usually

B(t)

03.4--

--)

H(t)

FIGURE 6.41


dependent on past history and it is not essential as far as eddy currents are concerned. The existence of the uniformly rotating component B(t) of magnetic flux density has been observed in numerous experiments and it can be justified on the symmetry grounds. Indeed, the isotropicity of hysteretic media means that the mathematical description of the properties of this media should be invariant with respect to rotations of Cartesian coordinates. In particular, the description of the media properties must be invariant with respect to rotations of x- and y-axes. Since the mathematical form of a uniformly rotating (in x - y plane) magnetic field is also invariant with respect to rotations of the same axes, we conclude that the time-varying component of magnetic flux density should have the mathematical form that is invariant with respect to these rotations as well. This implies that the time-varying component of magnetic flux density is a uniformly rotating vector.

By using Fig. 6.41, the relation between the uniformly rotating vectors B(t) and H(t) can be expressed in the mathematical form as follows:

= #(]H])e-J~H. (6.356)

Here: B and H are vector phasors corresponding to B(t) and H(t), respectively, while ]z(IHI)e -j~ can be construed as a complex magnetic permeability with 8 being a "loss" (lag) angle.

In the sequel, we shall use the approximation (model) (6.299) for #, which can also be written in the forms (6.302) and (6.303):

/- / ' (]HI)--#m Hx(z)

H m

1 --1 1 --1 -fi -fi

= # m

A

Hy(z) H m

(6.357)

In addition, it will be assumed that

8 = const. (6.358)

The last assumption will be justified (to some extent) after the derivation of the expression for the magnetic flux density.

By using formula (6.356), (6.357), and the above assumption, we can modify the boundary value problems (6.304)-(6.306)and (6.307)-(6.309) as follows:

1_ 1 d2Hx(z) = join#me -j~ Hx(z) -~ Hx(z), (6.359)

dz 2 Hm

Hx(O) - Hme j~176 (6.360)

Hx(o~) =0, (6.361)


and

d2Hy(z) -- j~ocr l.tm e-j'~ I~-Iy(z) 1 -1A dz 2 Hm Hy(z), (6.362)

Hy ( O ) = - j H m e j~176 . (6.363)

To solve these boundary value problems, we shall use the same approach as before. Namely, we shall look for the solution of Eq. (6.359) in the form

Hx(z) Hm(1 z-~) ~ - - , (6.364)

where Hm = Hme jO~ and ~ - ~' + j~". It is clear that function (6.364) satisfies the boundary conditions (6.360)

and (6.361). From the last formula, we also find

1 --1 d(n-1) IHx(z) l -ff ( ~ 0 ) n Hm -- 1 - . (6.365)

Now, by substituting formulas (6.364) and (6.365) into Eq. (6.359), we find that this equation will be satisfied if c~ is the root of the following characteristic equation:

ot(ol - 1) = joxr l,tm e-j`~ z 2, (6.366)

and

The last relation yields:

a ' ( n - 1) 2 - ~ . (6.367)

an a' = ~ . (6.368)

n - 1

We shall next proceed to the determination of imaginary part a" and z0 from Eq. (6.366). To this end, we shall rewrite the complex equation (6.366) as the following two coupled equations:

a ' (a ' - 1 ) - (Ol') 2 - (_oo'].tmZ~Sinr~, (6.369)

a" (2a' - 1) - ~oa ],tm z2 COS 8. (6.370)

By using formula (6.368) in the last two equations, we find

(a,,)2 _ 2n(n + 1) _ - ~ 2 s ing (6.371) ( n - 1) 2

,,_ n - 1 a, - 3n + 1 (.orr [d, mZ~ COS 3. (6.372)


The last expression yields

3n + 1 d ' . (6.373) COCr ~m z2 ~-~ (n - 1) cos 8

By substituting the last formula into Eq. (6.371), we arrive at

(o~,,)2 + d , 3 n + 1 2n(n + 1) n - 1 t a n 8 - ( n - 1) 2 = 0. (6.374)

By solving the last quadratic equation, we find

d ' = 3n + 1 . / (3n + 1) 2 2n(n + 1) (6.375) - 2 ( n - 1~-----) tan8 + V4(n _ 1) 2 tan 2 8 + (n - 1) 2 "

Finally, by substituting formula (6.375) into Eq. (6.373), we derive

j 3n+l J gn+l 2 2n n+l, gn+l (n-1)cos3 4(n_1) 2 tan 2 8 + (n_1)2 2(n-1) t a n 8

z0 = . (6.376) ~/coCr~m

It is apparent that in the particular case of 8 = 0, formulas (6.375), and (6.376) are reduced to formulas (6.323) and (6.325).

Similar to formula (6.365), we find

t z; Hy(z) =-jHm 1 - ~0 ' (6.377)

where oe and zo are specified by the expressions (6.368), (6.375), and (6.376), respectively.

By using formulas (6.356), (6.357), (6.364), and (6.377), we derive

2

B ( z ) = B m ( l - ~ O ) n - I �9 (6.378)

The last expression suggests that for sufficiently large (and typical) values of n, the magni tude of magnetic flux density is fairly close to its boundary value Bm almost everywhere within the conducting media (see Fig. 6.39). Therefore, the loss angle 8, which is a function of B(z), is also close to its boundary value almost everywhere within the media. This justifies the assumption that 8 is constant. Furthermore, the values of 8 are usually small (and tend to zero) for sufficiently large values of Bin.

By using formulas (6.364), (6.375), and (6.376), this impedance can be written in the form (6.338) with a and b given by the following "messy"

458

expression:


an a - , (6.379)

~ 3n+1 (~(3n+1) 2 2n(n+l) 3n+1 ) ( f / - 1) (n-1)cos3 4(n_1) 2 tan 2 3 + (n_1)2 2(n-1) tan3

/

3n+1 ,/(3n+1) 2 2n(n+l) 2(n-1) tan 3 4- v4(n-1) 2 tan 2 3 4- (n_1)2

b = . (6.380)

~ 3n+1 (/(3n+1)2 tan2 3 . tan3) 2n(n+l) 3n+1 (n-l) cos3 \V4(n_l) 2 (n-l) 2 2(n-1)

As was pointed out before, coefficients a and b are not very sensitive to particular values of n for n ~> 7. Thus, assuming that n is sufficiently large, we can simplify the last two equations:

2 a = , (6.381)

~co--~ (~//9 tan2 3 4- 2 - 3 tan3)

-3- t a n 3 + ~ 9 tan2 3 + 2 b = 2 . (6.382)

~co-~ (V/9 tan2 3 + 2 - 3 tan3)

The results of calculations of v/a 2 4-b 2 and tan ~ as functions of the loss angle 3 are shown in Figs. 6.42 and 6.43.

%/a2+ b 2

1.250] ' ' ' I

1.220

1.210

1.200

1" 1.190

1.180 0.0

i i i

2.5 5.0 7.5 = 6 (degrees)

10.0

FIGURE 6.42

6.5 CASE OF CIRCULARLY POLARIZED FIELDS

tan rf

0.75

0.70

0.65

0.60

0 . 5 5 , , , , , , ,

0 2

I , | i I , , , I , �9 ,

4 6 8 o

FIGURE 6.43

= 6 (degrees)

459

In concluding this discussion, we shall make the following histori- cal remark. Many formulas presented in this section are similar to those published by the Russian scientist L. R. Neumann [25] in 1949. However, in [25] these formulas had entirely different meaning. They were obtained for the problem of nonlinear diffusion of linearly polarized electromagnetic fields by using the method of "equivalent sinusoids," that is, by neglecting higher-order harmonics. For this reason, these formulas were approximate in nature. Many years passed before it was understood and proved (see [9, 28, 29]) that these formulas give the exact solution to a different problem, namely, that of nonlinear diffusion of circularly polarized electromagnetic fields.

The above analytical solutions have been obtained for the case of isotropic media and circular polarization of electromagnetic fields. Ellip- tical polarizations and anisotropic media can be treated as perturbations of circular polarizations and isotropic media, respectively. On the basis of this treatment, the perturbation technique can be developed in the same way as discussed in the last section of Chapter 5 and simple analytical solutions of perturbed problems can be found. The detailed exposition of these results is presented in [9]. Next, we proceed to the discussion of eddy current losses in ferromagnetic laminations subject to rotating magnetic fields.

It is well known that rotating magnetic fields occur in various types of electric machinery, actuators and other devices. It was realized that eddy current losses in steel laminations caused by these fields are appreciably higher than losses associated with unidirectional alterating magnetic fields of comparable magnitude. For this reason, "rotational" eddy current


losses have been a focus of active research for many years. These losses were first investigated experimentally for some specific field values and flux patterns (see [30-32]). Then some efforts were made to study these losses theoretically [33] by using numerical techniques for the solution of nonlinear diffusion equations. Below, we shall revisit the issue of "rotational" eddy current losses. By using the results obtained in the previous sections, we shall derive analytical expressions for these losses and clarify certain aspects related to this matter [34].

We begin with the discussion of eddy current losses under the assumption that a distribution of magnetic flux density over a lamination cross-section is uniform. We start with the case of unidirectional alternating magnetic fields. This is a classical problem that has been extensively treated in the literature. Then, the discussion of this classical problem will be generalized to the case of rotating magnetic fields.

Consider a magnetic conducting lamination with height h, width w, and thickness A (see Fig. 6.44). It is assumed that the magnetic flux density is uniform over the lamination cross-section and has only the x-component

B(t) = exBm cos cot. (6.383)

i / / / / -

i / /

h .... ~2 . . . . . . . . . ~ z

I I

A

FIGURE 6.44


This time-varying magnetic flux density induces eddy currents whose closed lines lie in planes normal to the x-axis. Let Lx be one of these eddy current lines. By applying Faraday's Law of electromagnetic induction to path Lx, we find

~L E. dl d~x(Z , t) (6.384) m ~

x dt '

where ~x(Z, t) is the flux that links Lx. By taking into account that the lamination is thin (A ( (h) , the left-

hand and right-hand sides of formula (6.384) can be approximated as follows:

~L E. dl ~ t). 2h, (6.385) Ey(z, x

�9 x(Z, t) "" 2hzBm cos cot. (6.386)

By substituting (6.385) and (6.386) into (6.384), we arrive at

Ey(z, t) = cozBm sin cot, (6.387)

which suggests that Emy(Z) = cozBm. (6.388)

By using expression (6.387), we can compute local power loss density p(z , t ) :

2_2,-,2 p( z , t ) -- r~E~(z,t) -- r~co z Dmsin 2 cot. (6.389)

This power loss density varies with time. For this reason, it is customary to characterize eddy current losses by the average power loss density ~(z), which can be computed as follows:

1 2 p(z ) = -~crEmy. (6.390)

By taking into account formula (6.388), the last equation can be written in the form

co2cr 2 2 p( z ) - - - - ~ B m z . (6.391)

The eddy current power losses per unit surface area of lamination can be obtained by integration of ~(z) with respect to z from --~ to -~"

~0 a A3 -~ (6.392) lin _ 2 ~(z) dz - (.02 cr B 2 24'

where superscript "lin" indicates that the losses are computed for the case of linear polarization of the magnetic field.


By using the relation ~m = Bm A, (6.393)

formula (6.391) can be reduced to the form

~O2ry A (6.394) pl in - - ( I )2 2----4-"

Now suppose that the magnetic flux density is uniform over the lamination cross-section and circularly polarized:

B - exBm cos ~ot + eyBm sin ~ot. (6.395)

This magnetic flux density induces the electric field, which has x- and y- components. The y-component of this field can be computed in the same way as before. In other words, formula (6.387) is valid for this component. To compute the x-component of the electric field, we consider a path Ly in a plane normal to y-axis (see Fig. 6.44) and apply Faraday's Law to this path:

~ E . dl dq~y(z, t) (6.396) y dt "

where ~y(z, t) stands for the magnetic flux that links Ly. By taking into account that the lamination is thin (A ((w) we find

~ E. Ex(z, t). (6.397) dl 2w, Y

�9 y(z, t) ~_ 2a~zBm sin ~ot. (6.398)

By substituting the last two equations into formula (6.396), we obtain

Ex(z, t) = -ogzBm cos ~ot. (6.399)

By comparing formulas (6.387) and (6.399) we conclude that the induced electric field is circularly polarized. This is expected because the magnetic flux density is circularly polarized. By using formulas (6.387) and (6.399), we can compute the instantaneous power loss density

p(z,t) - rr[E2x(Z,t) 4- E~(z,t)] - rr~o2zRB 2. (6.400)

It is apparent from the last equation that this power loss density is constant in time. In other words, in the case of circular polarization of the magnetic flux density the "eddy current" energy dissipation occurs at a constant rate in time. This clearly explains why the rotational eddy current losses are higher than those for unidirectional magnetic fields.

From formula (6.400) we conclude that

p(z) p ( z , t ) - 22 2 - rra~ z Bm. (6.401)


By integrating the last expression with respect to z, we obtain losses per unit surface area of lamination:

~0 A A3 2- 2 (6.402) ~cir = 2 ~(z) dz -- o92 cr Bm ~-~,

where superscript "cir" indicates that the losses are computed for the case of circular polarization of magnetic field.

By using formula (6.393), the last equation can be transformed as follows:

pCir - - - ~ m 2 c~ 1- - - -~ " A (6.403)

By comparing expressions (6.394) and (6.403), we conclude that

~cir __ 2~lin. (6.404)

The above discussion can be easily generalized to the case of elliptical polarization of magnetic flux density

B(t) - exBmx cos ~t + eyBmy sin cot. (6.405)

For this case, formulas (6.399) and (6.387) can be, respectively, written as follows:

Ex(z, t) = -cozBmy cos cot, (6.406)

Ey(z, t) = cozBmx sin cot. (6.407)

This leads to the following expressions for the instantaneous power loss density

p(z,t) - cr(E2x § E~) = crcoaza(B2xsin2 cot + B2y cos 2 cot), (6.408)

and average power loss density

1 crco2Z 2 p ( z ) - -~r~(E2mx § E2my) = 2 (B2K § B2my)" (6.409)

Now, by literally repeating the same line of reasoning as before, we obtain the following expression for the power losses per unit surface area of lamination:

~el_ ( ~ 2 x § ~2y)coarY_____A A (6.410) 24 "

By comparing formulas (6.394) and (6.410) we conclude that the last equation can be written in the form

~el --px=lin § . .-: lin (6.411)


The last expression clearly suggests that eddy current losses in the case of elliptical polarizations of the magnetic flux density are equal to the sum of eddy current losses associated with two unidirectional and orthogonal components of magnetic flux density acting separately. This fact was first observed experimentally (see [30, 31]) and later was confirmed by numerical computations [33]. It is important to point out that we have analytically derived this fact without invoking any assumptions concerning magnetic properties of laminations. For this reason, this fact as well as formulas (6.403) and (6.410) hold for magnetically isotropic and anisotropic laminations with (and without) hysteresis. The main limitation of our derivation is the assumption that the magnetic flux density is uniform over a lamination cross-section. If this assumption does not hold, the above fact and formulas (6.403) and (6.410) are not valid. In other words, in the case of nonuniform distributions of magnetic flux density, eddy current power losses are affected by the magnetic properties of laminations.

To treat the case of nonuniform magnetic flux density, we shall use the results obtained in this section. These results have been derived for nonlinear diffusion of electromagnetic fields in magnetically nonlinear conducting half-space, and the existence of finite penetration depth z0 has been established. Therefore, it is obvious that if the thickness of lamination exceeds 2z0, then nonlinear diffusion of electromagnetic fields at each side of the lamination will occur in the same way as in the case of the semi-infinite conducting half-space. Consequently, we can use the previously derived results for the case of conducting lamination. Namely, when the magnetic field is circularly polarized and conducting media are magnetically isotropic, we can write the following formulas for the phasors of the magnetic field:

-~+z~ Hm(1- --G-0, if --~ ~ z ~ - - ~ +z0,

Hx(z) - 0 if --~ + z0 ; z ; -~ - z0, (6.412)

Ha(1 ~--z~ ~ a -Go " if - ~ - z o K z ~ 7 ,

-jHm(1 - -~+z)~ if --~ K z K --~ + z0, -Go G ( z ) - 0 if --~ + z0 K z K -~ - z0, (6.413)

-jHm (1- )~ if -~-z0~<z~<g,

where z0 and o~ are given by expressions (6.325) and (6.312), (6.319), (6.323), respectively.

By using these formulas, we can compute surface impedances on each side of the lamination. It is obvious that these impedances will be the same


and given by formulas (6.338)-(6.340). Now, by invoking the notion of the Poynting vector S, we can compute eddy current losses in the lamination as follows:

#ci r -aRe(S. e z ) = R e [ ( E ( 2 ) x H * ( 2 ) ) - e z ] (6.414)

where the symbol "*" is used for the notation of complex conjugate quantity.

By employing the relations

E ' x ( 2 ) - - r / H y ( 2 ) , E y ( 2 ) = - r / H x ( 2 ) , (6.415)

and by substituting them into formula (6.414), we derive

~cir Re(r/)[ H x ( 2 ) aft_ H y ( 2 ) 2 ] = = 2H 2 Re(~), (6.416)

where we used the fact that in the case of circular polarizations IHx(-~)l =

[Hy(~)[--Hm. Now, by recalling expressions (6.338) and (6.339), we arrive at

7a)~m 4n (6.417) ~cir • H 2 cr [2n(n + 1)(3n + 1)2] 1"

In this formula, "rotational" eddy current losses are expressed in terms of the magnetic field magnitude at the lamination boundary. In many applications, the total flux through the lamination is given. For this reason, it is desirable to express rotational eddy current losses in terms of this flux. To this end, we recall the equation

3Ex --, 3z - -jcoBy. (6.418)

By integrating this equation with respect to z, we obtain A A

~ i~ " "" 2 S ~ ~ OEK 2 E x ( A ) ~ # �9 y = 2 By(z) dz - j - 3z (z) dz = j -~ (6.419) _zo (,o _zo

where we used the fact that E'x(-~ - z0) = 0, and �9 is a flux per unit width (or height).

By utilizing the impedance relation (6.415) in the last formula, we derive

~ Y - J - ~ H y ( 2 ) " (6.420)


From the last equation, we find

21r/I ~ m m H m ,

(o

or by invoking expression (6.337), we have

2I~1 ~ m -- H m . (6.421)

r 0

Since z0 depends on ]Zm (see (6.325)), which in turn, is a function of Hm, the last formula can be construed as a nonlinear equation for Hm. By solving this equation, we can find Hm and/Zm for the given ~m. By plugging this value of/*m in the expression for Hm in terms of ~m"

~o0-z______p_0. (6.422) Hm = ~m 2lc~[

From formulas (6.312), (6.319), and (6.323), we easily derive

1~12 = 2n(3n + 1) (6.423) (n - 1) 2 "

By substituting formula (6.422) into expression (6.417) and by using Eqs. (2.53) and (2.490), after simple transformations we arrive at the following result:

fcir __ 2 . / ~ [2n(n + 1)(3n + 1)21�88

~m~~ 2(3n 4-1) (6.424)

So far, we have discussed the case when 2z0 < A. Consider the limiting case when

A z0 = ~-, (6.425)

and let us e x p r e s s ~cir explicitly in terms of z0. According to formulas (6.416), (6.337), and (6.312), we have

pcir 2H2m c~_~' - . (6.426)

o-z0

Now, by substituting relation (6.422) into the last equation, we obtain

cir = ~ 2 O92 0-z0cz' 21~12 . (6.427)

By taking into account conditions (6.425) in the last formula, we derive

~cir _ ~2 0920- Ac~I m 41c~ i--------~. (6.428)


From expression (6.319) and (6.423), we find

~I n - 1 = (6.429)

41al 2 4(3n + 1)'

and formula (6.428) can now be rewritten as follows:

~cir 2 2 n - 1 (6.430) - - ~mo9 0- A4(3n 1)" +

For sufficiently large n, we have

n - 1 1 4(3n 4- 1) 12' (6.431)

a~nd formula (6.430) assumes the form

o920 - A pcir 2 ~ (6.432) - = ~ m 12 "

The last equation coincides with formula (6.403) derived by using a different line of reasoning under the assumption that the magnetic flux density within the lamination is uniform. Al though encouraging, this coincidence is not very surprising. This is because under condition (6.425) the distribution of magnetic flux density is almost uniform. Thus, we can conclude that for z0 K -~ we can use formula (6.424) for the calculation of rotational

eddy current losses, while for z0 > -~ formula (6.403) is appropriate.

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Subject Index

2D vector Preisach model 164 3D vector Preisach model 164

r diagram 165 Abel type integral equations 193, 212 abrupt magnetic transitions 411 abrupt (sharp) magnetic transitions

378 accommodation 65, 117 accommodation process 117 action functional 276, 277 adsorption hysteresis 1 after effect 225 Airy function 421 alternating series of dominant input

extrema 17 alternating series of subsequent global

extrema 74 anhysteretic output value 229 anisotropic media 365 anisotropy constant 150 anisotropy energy xxiv astroid curve 151 autocovariance matrix 268 "average" model 134, 146

backward Kolmogorov equation Bean model 300 branching xvi, xvii, 133

247

Chebyshev polynomials 186 circular polarization 349, 384, 444 "clock-wise" hysteresis 72 coil 412 comparable minor loops 100 congruency of minor loops 65 congruency property 19, 402, 406

- for the superconducting hysteresis 312

- o f comparable minor loops 100 - of minor hysteresis loops 105 - of vector minor loops 170 core losses 419 "counter clock-wise" hysteresis 72 creep 225 - coefficient S(H, T) 326 - phenomena 299 critical state (Bean) model for

superconducting hysteresis 304

critical state model 300 curve 21

300,

data collapse 241, 329 data-dependent thermal relaxations

244 device realization of the Preisach model

3 diagram technique 4, 8, 12 diffusion process 245 discrete-time i.i.d, random process 245 distinct time scales xvi, 273 domain walls xxiv domains xxiv, 6 dynamic effects 406 dynamic Preisach models of hysteresis

108 dynamic vector Preisach models 201

eddy current 411 - hysteresis 377 - hysteresis and the Preisach model

397 - losses 418 "effective" distribution function 270

4 6 9

4 7 0

elementary hysteresis operators 2, 281 elliptical polarization 364, 444 energy of interaction with an applied

field xxiv entropy 59 - production 60 - production for hysteresis processes

61 equilibrium orientations 150 excess eddy current losses 418 excess hysteresis losses 411,418 excess losses 442 exchange energy xxiii exit problem 245, 256 - for stochastic processes 247 exit rate 278 expected value 226 experimental testing of Preisach-type

models 132 experimental testing of the Preisach

modeling of superconducting hysteresis 313

f2-1aw for eddy-current losses 411 "fading" memory 20 finite difference equation 227 first level-crossing time 287 first-order decreasing transition curves

25 first-order increasing transition curves

25 first-order reversal curves 321 first-order transition 21 -curves 23, 44, 194 flat-power approximation of a hysteresis

loop 425, 434 flux filaments 299 forward Kolmogorov equation 258, 269 Fourier series expansions 191 Fourier transform 270 Fourier transform (FFT) algorithms 192 functional (path) integration models

of hysteresis 281 fundamental models of hysteresis xxiii

generalization of the critical state model 335

generalized projection of vector input ~(t) 213

Subject index

generalized scalar Preisach models of hysteresis 65

generalized vector Preisach models 207 geometric interpretation of the Preisach

model 8 giant magnetostriction 124 gradual magnetic transitions 424

hierarchy of cycles 277 higher-order harmonics 364 higher-order reversal curves 133 homogeneous diffusion process 292 hysteresis 65 - loops xvi - losses 418 - nonlinearities with local memories

xvii, xviii - nonlinearities with nonlocal memories

xvii, xix hysteretic energy dissipation 5 hysteretic energy losses 49

/-graphs 277 identification 322 identification problem 4, 20, 165, 183,

190, 206 impedance-type relation 437 independent identically distributed

(i.i.d.) random process 226 integral equation of the Abel type 187 interaction between the particles 7 intermediate asymptotics 326 intermediate "ln t" asymptotics 236 intrinsic thermal relaxations 244 trreducible representation of the rotation

group 194, 196 irreversible component of the classical

Preisach model 68 irreversible thermodynamics 58 Ito stochastic differential equation 245,

285

joint probability density function Josephson-junction coupling 331

Laplace transforms 249 last level-crossing time 287 Legendre polynomials 197 limiting ascending branch 21 limiting descending branch 25 linear polarization 331

268

Subject index

In t-asymptotics 231 lower envelope 282

"magnetic" definition of the Preisach model 7

magnetically nonlinear conducting laminations 387

magnetization 149 magnetostatic self-energy xxiv magnetostrictive hysteresis 124 magnetostrictive materials 125 magnetostrictive phenomenon 124 main state 278 major hysteresis loop 69 major loop 21 Markovian processes on graphs 268 mathematical models of hysteresis

. . .

X X l l l

Mathieu equation 421 mechanism of memory formation 13,

20 memory effects 406 metastable states xxiv micromagnetic approach xxiv mirror symmetry 26 moving boundary 404 "moving" Preisach model 65, 68 multibranch nonlinearity xiv multiplicity of metastable 225

negative saturation 9, 14 neural network 19 noise induced hysteresis in dynamical

systems 280 noninteracting particle 225 - model 230 nonlinear diffusion 299, 391,434, 443 - equation 377, 426 - in isotropic superconductors with

gradual resistive transitions 348 - in superconductors with gradual

resistive transitions 331 - in the case of elliptical polarizations

and anisotropic media 359 "nonlinear" Preisach model 73 nonlinearities with local memories xix nonlocal memory 4 numerical implementation of the

Preisach model 37

4 7 1

Ornstein-Uhlenbeck process 250, 258, 260, 268

parabolic cylinder functions 250 particles 6 path independence property 127 penetration depth 385, 387, 440, 452 perturbation technique 359, 459 physical universality of the Preisach

model 30 pinning 299 positive saturation 14 power law 331, 348 Preisach formalism 267 Preisach hysteresis operator 3 Preisach model 3, 19 - and wavelet transforms 4 - of hysteresis with accommodation

117 - with input-dependent measure 73 - with stochastic input 225 Preisach models with two inputs 124 Preisach type models with two inputs

65 property of correlation between

mutually orthogonal components of output and input 180

property of equal vertical chords 75

random free energy landscape 295 rate independence property 405 rate-independent 65 rate-independent hysteresis xiv rectangular front 379, 383, 389, 397 rectangular hysteresis loop 389, 411,

417 rectangular loop 2 rectangular profile approximation 432,

434 reduction property of the vector Preisach

model 167 reduction theorem 83 relaxation time 110, 111, 203 representation theorem 27, 81, 131 reptation 65, 117 restricted nonlinear Preisach model

106 restricted Preisach model 93, 102 Ricatti equation 421 rotational eddy current losses 443, 460

4 7 2

"rotational" experimental data 220, 321 rotational symmetry property 171

sausaging 331 scalar hysteresis xiv scalar nonlinear diffusion equation 443 scaling and data collapse of viscosity

coefficient 237 scaling law 416 second-order reversal curves 73 self-similar solutions 430 series of iterated convolutions 249, 256 single-domain, uniaxial magnetic

particle 149 spectral decomposition 4 spectral density 270 - of noise 253 spectral noise density 267 spherical harmonics 195, 196 spin-stand imaging 245 spontaneous symmetry breaking 445 squareness factor 438 staircase interface 11, 13 standing mode 341, 342 - of nonlinear diffusion 340 stationary distribution function 268 stationary Gaussian thermal noise 251 statistical instability 31 step response 407 stochastic process 226 stochastic processes on graphs 256, 265 Stoner-Wohlfarth magnetic particles

36, 37, 91, 149, 260 Stoner-Wohlfarth (S-W) model 149,

156-158 strain 124 structural disorders 295 superconducting hysteresis 299 superconductors with resistive

transitions 299 surface impedance 355, 356, 384, 386,

391, 438, 440, 452 switching probabilities 227, 246 symmetry property 405

Subject index

temperature dependent hysteresis 272 Terfenol type materials 124, 125 theory of large deviations 273 thermal activation 299, 324 - theory 331 thermal activation type models 225,

229 thermal relaxations 225 thermodynamic aspects of hysteresis

62 third-order reversal curves 104 three-dimensional anisotropic vector

Preisach model 194 type-II superconductor 299

universality of intermediate asymptotics 251

universality of low frequency dependence of excess hysteretic losses 424

upper envelope 282

vector hysteresis xxi vector nonlinear diffusion equation

444 vector Preisach models of hysteresis

158 vector rate-independent hysteresis xxii "vertex" boundary conditions 269 vibrating sample magnetometer (VSM)

253 viscosity 225 - coefficient 236 viscosity (creep) experiments 326 voltage 412

white-noise 275 Wiener process 245, 275 wiping-out property 15, 65, 402, 406 - of superconducting hysteresis 311

YBaCuO superconducting materials 253

mathematical models of hysteresis and their applications

Documents

eddycurrent hysteresis

vector preisach models

preisach formalism

exposition of preisach

x preface hysteresis

landmark paper of preisach

mathematical description

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