Vector Preisach Models of Hysteresis

CHAPTER 4

Vector Preisach Models of Hysteresis Isaak D. Mayergoyz, Electrical and Computer Engineering Department and UMIACS, University of Maryland, College Park MD 20742, USA

Contents 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Classical Stoner-Wohlfarth Model Vector Hysteresis Definition of Vector Preisach Models of Hysteresis and their Numerical Implementation Some Basic Properties of Vector Preisach Hysteresis Models Identification Problem for Isotropic Vector Preisach Models Identification Problem for Anisotropic Vector Preisach Models Dynamic Vector Preisach Models of Hysteresis Generalized Vector Preisach Models of Hysteresis. Experimental Testing References

4.1 C L A S S I C A L S T O N E R - W O H L F A R T H V E C T O R HYSTERESIS

447 458 467 485 494 504 511 526

MODEL

In magnetics, research on the modeling of scalar and vector hysteresis has been pursued along two quite distinct lines. Modeling 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 modeling of vector hysteresis has long been centered around the classical Stoner-Wohlfarth model (S-W) model [2]. As a result, this model has further been developed and used in the area of magnetic recording [3-5]. The The Science of Hysteresis Volume I, II and III G. Bertotti and I. Mayergoyz Copyright 9 2005 Elsevier Inc. All rights reserved.

448

CHAPTER 4 Vector Preisach Models of Hysteresis

attractiveness of the S-W 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. Owing to its popularity in magnetics, the S-W model is a natural benchmark for comparison with other vector hysteresis models. This is the main reason w h y 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 (see Fig. 4.1). The orientation-dependent part of free energy g of the S-W particle is given by g = K sin 2 0 -/VI./7/,

(4.1)

where K is the anisotropy constant and 0 is the angle between the easy axis and M. The first term in the right-hand side of (4.1) represents the anisotropy energy, while the second term in (4.1) is the energy of interaction of magnetic m o m e n t u m M with the applied magnetic field. By using the Cartesian coordinates shown in Fig. 4.1, the expression (4.1) can be represented as g = K sin 2 0 - MHx cos 0 - MHy sin 0.

(4.2)

Equilibrium orientations of/VI correspond to minima of s and they can be found from the equations ~g --

~0 (~2g

=0,

(4.3)

~0 2 ~ 0.

(4.4)

2K sin 0 cos 0 + MHx sin 0 - MHy cos 0 - 0.

(4.5)

From (4.2) and (4.3) we derive

4.1 CLASSICAL STONER-WOHLFARTH MODEL VECTOR HYSTERESIS

449

M

H

,,

•

F I G U R E 4.1

By introducing the so-called switching field a: 2K - M'

(4.6)

the expression (4.5) can be rewritten as sin 0 cos 0 + Hx sin 0 - Hy cos 0 = O,

(4.7)

which is equivalent to sin 0

cos 0

= a.

(4.8)

Equation (4.7) (as well as (4.8)) is a quartic equation with respect to cos 0. For this reason, this equation m a y 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 m i n i m u m 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 the H-plane there are two different regions where M has one and two equilibrium orientations, respectively. On the b o u n d a r y between the above two regions one m i n i m u m and one m a x i m u m merge together. In the case of a minim u m , the inequality (4.4) holds, while for a m a x i m u m we Consequently, on the above b o u n d a r y we have 0g 0---0 = 0

have (~2~/~02~0.

02g and

~02

= 0

9

(4.9)

The first condition in (4.9) leads to Eqn. (4.8). Now, let us derive the second equation by using both conditions in (4.9). From (4.2), by using the notation (4.6), we derive

1

egg

sin 0 cos 0 00

= a -t

Hx

Hy

cos 0

sin 0"

(4.10)

CHAPTER 4 VectorPreisach Models of Hysteresis

450

Hy (Hxl ,

H

FIGURE 4.2

By differentiating (4.10), we obtain c9( 1 )egg 1 cgagHxsinOHycosO 0---0 sin 0 cos 0 ~ + sin 0 cos 0 002 = cos 2 0 -ff ~ ' s i n 2 0

(4.11)

By using both conditions (4.9), from (4.11) we find

HI

cos3---~ +

Hy sin 3 0

= 0.

(4.12)

Thus, on the boundary which separates regions with one and two minima, Eqns (4.8) and (4.12) are satisfied. These are two linear equations with respect to Hx and Hy. By solving these equations, we find

Hx =

- a c o s 3 (9,

Hy

-- 0r s i n 3 0.

(4.13)

From (4.13) we find the equation for the boundary separating the above two regions: Hx2/3+--y /_/2/3 _. 0~2/3 (4.14) This equation represents the astroid curve shown in Fig. 4.2. This astroid curve helps to visualize the solution of the quartic equation (4.7). This 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 Eqn. (4.7) are parallel to those tangent lines to the astroid that pass through the point H (see Fig. 4.2). The proof of the above statement proceeds as follows. Let Hx and Hy be Cartesian components of f/, and let Hxl and Hy1 be Cartesian coordinates of the point on the astroid at which the abovementioned tangent line touches the astroid. For the slope of this tangent

4.1 CLASSICAL STONER-WOHLFARTH MODEL VECTOR HYSTERESIS 451 line we have m=

I-ly1 - I-Iy Hx~ - H ~ '

(4.15)

which is tantamount to Hy 1 - Hy = m(Hxl - Hx).

(4.16)

Since the point (Sx1 , S y 1) belongs to the astroid, we have 2/3

/_/2/3 __ ~2/3

(4.17)

By using implicit differentiation of (4.17) with respect to Hxl, we derive 2 H~11/3 + 2-H,,--11/3 dHyl = 0 -3 3-dHxl "

(4.18)

From (4.18) we obtain m ~-~

ally 1 Hy I 1/3 " dHxl = - - ( ~ )

(4.19)

L e t / / b e the angle formed by the above tangent line with the easy axis of the particle. Then, m = tan//, (4.20) and from (4.19) and (4.20) we find HYl -- - tan 3 ft.

(4.21)

Hx I

Expressions (4.17) and (4.21) can be construed as two simultaneous equations with respect to two unknowns: Hxl and Hy 1. By solving these equations, we obtain Hxl -- - ~ c o s 3 fl,

Hy 1 -- cxsin 3 ft.

(4.22)

By substituting (4.20) and (4.22) into (4.16), we find cxsin 3 f l - Hy = tan/J(-cx cos 3 f l - Hx).

(4.23)

A trivial transformation leads to 0r

3 fl cos/J + cos 3 fl sin fl) q- Hx sin fl - Hy cos fl - 0,

(4.24)

CHAPTER 4 VectorPreisach Models of Hysteresis

452

Hy 6

a

Hx

FIGURE 4.3

which can be simplified as sin fl cos fl 4- Hx sin fl - Hy cos fl = 0.

(4.25)

It is clear that Eqns (4.7) and (4.25) are identical. Consequently, fl = 0.

(4.26)

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. 4.3) and that the magnetic field is first monotonically increased from its value H_ corresponding to point I to the value H+ corresponding to point 6 and then is monotonically decreased back to H_. The dependence of the magnetization projection along the line a-a t on the value of the magnetic field H exhibits hysteresis that is shown in Fig. 4.4. This is clear from Fig. 4.3. Indeed, as we move up along the line a-a t, the equilibrium (stable) orientations of M coincide with directions of tangent lines to the right-hand side of the astroid until we reach point 5.

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 453 Ma_a,

H 8-8'

/

FIGURE 4.4

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 point 6, we use the tangent lines to the lefthand side of the astroid to determine the stable direction of/~'i. 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 - d , which are inside the astroid, there are two stable orientations of M that result in two different branches of the hysteresis loop shown in Fig. 4.4. It is clear from the preceding discussion that if the applied field varies along the easy axis x then a S-W particle exhibits the rectangular hysteresis loop shown in Fig. 4.5. It is also clear that if the applied magnetic field is varied along the direction perpendicular to the easy axis, then due to symmetry there is no hysteresis effect and a S-W particle exhibits the single-valued magnetization curve shown in Fig. 4.6. Thus, the shapes of hysteresis loops depend on the direction along which the applied field is being varied. It has been shown in Chapter 3 (see formula (3.221)) that the loop shown in Fig. 4.4 can be represented in terms of the rectangular ~-loop. G. Friedman found a very interesting generalization of the formula (3.221) (see reference [8]). This generalization represents the magnetization of a S-W particle for arbitrary (not only collinear) variations of magnetic fields in terms of rectangular ~-loops. The basis for this representation is the notion that there are two distinct states (vector branches) for any S-W

454

CHAPTER 4 VectorPreisach Models of Hysteresis

MX

HX

F I G U R E 4.5

,'H

F I G U R E 4.6

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/vl + (r 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 the 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-(q~(t), H(t)) will be used for the magnetization in the second state. It is clear that ~r + (q~(t), H(t)) and/~I- (~(t), H(t)) can be found geometrically by using the previously described 'astroid' rule or by solving the quartic equation (4.7).

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 455 The rectangular loop operator ~a,_~ 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 q0(t) [Hx(t)2/3 + Hy(t)2/313/2 I cosq~(t)l

v(t)-

(4.27)

It is clear from (4.27) and (4.14) that v(t) reaches the value 0r as the tip of /7/(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 ~a,_~ 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

~r

= -~(t)~a,_~v(t) + ~t(t),

(4.28)

where 1

3(t) - ~[/~+ (~0(t), H(t)) - ~l-(qo(t), H(t))],

(4.29)

-. - ~[/~I 1 d(t) + (q0(t) , H(t)) + ~ i - (q0(t) , H ( t ) ) ] .

(4.30)

It is clear from (4.28), (4.29) and (4.30) that/~(t) =/VI + (q0(t), H(t)) when ~a _~v(t) - 1, and M(t) - M - (qo(t), H(t)) when ~a,_~v(t) - - 1 . Switchings of r from I to - 1 and vice versa occur at the times when the tip of H(t) crosses the astroid. This proves that formulas (4.27)-(4.30) 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 axes and different values of switching field c~. The notation So, a will be used for a S-W particle whose switching field is equal to c~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

A/i(t) - f f ~(0, a)'So,~H(t) dO da,

(4.31)

where ~(0, ~) is a distribution function that should be determined by fitting the model to some experimental data.

456

CHAPTER 4 VectorPreisach Models of Hysteresis

The expression (4.31) 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

A f" (t) - fl ~(0, a)So,~(t) dO da,

(4.32)

wheref(t) is the vector output, while fi(t) is the vector input. The output of the operator So,~ can be formally determined by using the astroid rule or the quartic equation (4.7) 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, singledomain 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 (4.28)-(4.30) 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, a and by introducing the operators So,~,~ with shifted astroids (Fig. 4.7). Then, the generalized S-W model can be represented as A

f(t) - f f / f r

~, ~)'So,~,~u(t) dO da d~.

(4.33)

However, this generalization will require much more computational work for the numerical implementation of the S-W model. Even without this

4.1 CLASSICALSTONER-WOHLFARTH MODEL VECTOR HYSTERESIS 457 Y -(X \\

,

FIGURE 4.7

generalization, the S-W model is computationally slow. This is in partbecause 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 integrals in the case of the classical two-dimensional S-W model (4.32). The last difficulty is magnified in the case of the generalized S-W model (4.33). Furthermore, the identification problem of finding the distribution function ~(0, a) 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 viable alternative to the S-W model.

458

CHAPTER 4 VectorPreisach Models of Hysteresis

4.2 D E F I N I T I O N O F V E C T O R P R E I S A C H M O D E L S OF HYSTERESIS AND THEIR NUMERICAL IMPLEMENTATION For the sake of generality, a vector hysteresis nonlinearity will be characterized below by a vector input ~(t) and a vector outputf(t). In magnetic 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 the 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 the 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 projections along the chosen direction sample 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 the 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

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

459

i1 iI i

s

s

s S

,'

I III

Lr

" J"

o ~. *" ~, ~

.~ ~ ' '

s

i I

. . o

i S

i

II

i i

s

II

s

i

t

%

I

% %

i

s 9

...."

i

s s s

s

s S

]-~r-~(-~.U-~(t)).. ,~, *

r

..

A

%

i

s

t

i

I

II

9

I

i i s i

I

e

I

FIGURE 4.8 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 outputs 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. 4.8). Thus scalar Preisach models are the main building blocks of the vector model, which is constructed as a superposition of scalar models. This can be expressed mathematically in two dimensions as )7(t) - ~~1=1 ~F~(~. ~(t)) ds

(4.34)

and the integration in (4.34) is performed over a unit circle. Similarly, a three-dimensional vector Preisach model can be written in the form f(t) - ~~1 =1 ~ FF(r. ~(t))dsr, where the integration is performed over a unit sphere.

(4.35)

460

CHAPTER 4 VectorPreisach Models of Hysteresis A

The scalar Preisach models F~ are defined by

F~(~ 9 fi(t)) = fl)]~ v(e, fl)~/~(~, fi(t))de dfl

(4.36)

for isotropic vector models, and

F~(~ 9 fi(t)) - ff~>/~ v(e, fl, ~)~//(? 9 fi(t)) de dfl

(4.37)

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 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 (4.35) 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 ~ + ~ C + there is a corresponding opposite point ~- ~ C- (see Fig. 4.9) ~+ = - ~ - . (4.38) From (4.35) we find

f(t) = ffc+ ?+F~+ (? +" fi(t)) ds+ + fJc- ?-F~- (?-" fi(t)) ds_,

(4.39)

where F~+ (~ +. fi(t)) - ff~

v(e, fl, ~ +)~a/~(~+. fi(t)) de dfl,

(4.40)

F ~ - ( ~ - . fi(t)) = ff~

v(e, fl, ~ - ) ~ ( ~ - .

(4.41)

fi(t))dedfl.

The following identity can be verified for any continuous piecewise monotonic function v(t)" ~ v ( t ) = -~_~,_~ (-v(t) ). (4.42)

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

461

+

FIGURE 4.9

Indeed, if v(t) < fi, t h e n - v ( t ) > -fl and

~aflv(t)- -1,

while

~_~,_a(-v(t)) - +1.

(4.43)

If v(t) increases and exceeds ~, then -v(t) decreases and becomes smaller than-c~. Consequently,

~v(t)-

1,

while

~_~,_~(-v(t)) - -1.

(4.44)

Similarly, it can be shown that the identity (4.42) holds for any monotonic decrease of v(t). In the last integral in (4.39) we change the variable ~- to -~ +. As a result, we find

l / c - -f -F~- (~- " ~(t)) as_ - - f/c+ ~+F_~+ ( - ~ + . ~(t))ds+,

(4.45)

where F_~+ (-~ + 9~(t)) -

)/~ v(e, fl, -~ +)~c~(-~ +. ~(t)) de aft.

(4.46)

By using the identity (4.42), the last integral in (4.46) can be transformed as follows" ff~ ~>~v(~,fl,-~ - - ~ ddc~

+. ~(t)) d~ dfl

v(c~, fl, -~+)~_1~_~(~+. ~(t)) da dfi.

(4.47)

CHAPTER 4 VectorPreisach Models of Hysteresis

462

In the last integral we will change a to -]7 and ]7 to -a, then according to (4.46) and (4.47) we obtain F_~+ ( - 7 + . fi(t)) = - ~

dda >~

v(-fl, -~, -~+)~3(7+. fi(t)) d~ aft. (4.48)

By substituting (4.48) into (4.45), we find as_

fc

=

fZ >.3

+

ds+"

(4.49)

By substituting (4.40) into the first integral in (4.39), we have

f f + ~+?~+ (F+. fi(t)) ds+ ,tIC =ffc~+(fZ +

>.3

v(e, fl,-f+)~aB(-f+.fi(t))&zdfl)ds +.

(4.50)

From (4.39), (4.49) and (4.50) we obtain

f"(t) =

+ ff + 7[fZ tvt=, >>.

fl, ~ +) + v(-~, -~, --f +))

x9~3(~ +. fi(t)) dad//] ds+.

(4.51)

By introducing a new function

v' (o~,fl,-f +) = v(o~, fl,-f +) + v(-fl, -a, -~ +),

(4.52)

from (4.51) we derive

f(t) -

ffc 7 +"F~+ (~+. fi(t)) ds+, +

(4.53)

where

~'~+(~+ 9~(t)) - f L ~ v'(~, fl, ~+)~3(~+. fi(t)) da dfl. This completes the proof.

(4.54) []

The proven proposition suggests that (4.53) and (4.54) can be regarded as an equivalent definition of the vector Preisach model. This definition will

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

463

be used in our subsequent discussions and the superscript ' will be omitted. It is also clear that a similar proposition is valid for the two-dimensional vector Preisach model (4.34) as well. Consequently, this model can be represented as )7(t) = f

~ +F~+ (~ +. fi(t)) dl+,

(4.55)

,/L +

where L+ is a semicircle and F'~+ is defined by (4.54). It is apparent from the above proof that particular choices of semispheres and semicircles in (4.53) and (4.55), 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 (4.52) we change a to - f l , / / t o - a , and ~ + to - ~ +, then we find =

(4.56)

+

From (4.56) and (4.52), we obtain (4.57) This expresses the property of mirror symmetry of the function vf with respect to the line a - -//. This also extends the definition of the function vI 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 (4.57) can be simplified as follows: v' (a, fl) = v'(-fl, - a ) , (4.58) which is similar to the symmetry property of the #-function for the classical scalar Preisach model. Up to this point, the vector Preisach models have been defined in coordinate invariant forms (4.34), (4.35), (4.53) and (4.55). 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,

~+

- e 0 ,-~ r

-~ r +

.fi(t)

-

uo,~o

(t) ,

(4.59)

where e0,~0 is a unit vector along the direction specified by angles q~ and 0, and uo,~o(t) is the projection of fi(t) along the direction of e0,~0. By using (4.59), the three-dimensional vector Preisach model (4.53)(4.54) can be written as

f (t) =

f0270

A euo, e(t) sin 0 dO dq~, -~o,eFo,

(4.60)

CHAPTER 4 VectorPreisach Models of Hysteresis

464

where A

Fo, q)UO,q)(t) =

v(e, fl, O, q))4&flUO,q)(t) dedfl. >>.~

(4.61)

The last two formulas can be combined into one expression:

( ff~>~ v(e,fl, O,q));&~uo,~o(t)dedfl)

f(t) =

-do,~o

x sin 0 dO dq~.

(4.62)

Similarly, the two-dimensional vector Preisach model (4.55) can be represented in polar coordinates as A

f(t) =

~FCu~(t) dq~,

(4.63)

where A

Fq~uq~(t) -

v(e, fl, q))~flUq~(t) de dfl.

(4.64)

In the previous formulas, ~r is a unit vector along the direction specified by a polar angle q~, Fr is the scalar Preisach model (operator) for this direction, and uq~(t) is the projection of fi(t) along the direction of ~q~. By combining (4.63) and (4.64), we have

f (t) =

-do

v(e, fl, qo)~flu~o(t) de dfl dq).

(4.65)

The expressions (4.62) and (4.65) are written for anisotropic models. In the isotropic case, the function v should be independent of 0 and q~ (or of q~in the two-dimensional case). This leads to the following three-dimensional and two-dimensional isotropic vector Preisach models:

-"eo,~o( ff~ >~ v(e, fl)4&~uo,~o(t)dedfl) sinOdOdqo, (4.66)

f(t) = f (t) -

-@

v(a, fl)~fluq~(t) doe dfl dq~.

(4.67)

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

4.2 DEFINITION OF VECTOR PREISACH MODELS OF HYSTERESIS

465

d

(~, ~)

T| (0, ,~)

FIGURE 4.10

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(~, fl, O, q~). For any fixed 0 and q~, consider a triangle T(~, fl) shown in Fig. 4.10. Then by definition, we have

P ( o~, fl, O, r ) -//T{~,fl) v(a', fl', O, q~)da' dfl'.

(4.68)

By using (4.68), it can easily be shown that P is related to v by the formula

v(c~, fl, O, qo) = -

a2P(~, fl, O, q~)

a~afl

.

(4.69)

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 a and fl in expressions (4.62), (4.65), (4.66) and (4.67) 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 a-fl diagram. A typical example of such a diagram is shown in Fig. 4.11, where Mo, q~,kand mo, q~,kform 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

CHAPTER 4 VectorPreisach Models of Hysteresis

466

0'~'k)

FIGURE 4.11

of the scalar Preisach model Fo, q~uo,q~(t) associated with the direction e0,q~ can be evaluated as follows A

Fo, q~uo,r

= - P ( a o , flo) no,r +2 ~ [P(Mo, q~,k, mo, q,,k-1, O, r -- P(Mo, q~,k, mo, q~,k, O, r k=l

(4.70) The proof of (4.70) literally repeats the proof of the similar expression for the classical Preisach model and, for this reason, it is omitted. In the case of the two-dimensional vector Preisach model (4.63)-(4.65), a similar expression is valid for Fq~ur A

Feud(t) = - P ( a 0 , rio) %(t) +2 y~ [P(M~,k, mq~,k-1, q~) -- P(Mq~,k, mq~,k, q~)].

(4.71)

k=l

By using formulas (4.70) and (4.71) the numerical implementation of the three-dimensional vector model (4.60)-(4.62) and the two-dimensional vector model (4.63)-(4.65) can be reduced to the evaluation of double and single integrals, respectively. We note here that the numerical implementation of the classical two-dimensional Stoner-Wohlfarth model (4.32) requires the evaluation of double integrals. In this respect, numerical implementation of the two-dimensional vector Preisach model can be accomplished more efficiently than the numerical implementation of the

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

467

two-dimensional Stoner-Wohlfarth model. Another advantage of using the formulas (4.70) and (4.71) is that the function P can be directly related to experimental data. This will be demonstrated when we study the identification problems for the vector Preisach models. Using the expressions (4.70) and (4.71), digital codes that implement the vector Preisach models (4.60)-(4.62) and (4.63)-(4.65) have been developed. In these codes finite meshes of directions e0,~ and ~ are used to evaluate double and single integrals in (4.60) and (4.63), respectively. For each mesh direction the integrands in (4.60) and (4.63) are computed by using (4.70) and (4.71). Some numerical examples computed by using the developed digital codes will be given in the next section.

4.3 S O M E B A S I C PROPERTIES OF V E C T O R P R E I S A C H HYSTERESIS MODELS 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. This property is experimentally observed when an input is restricted to vary along an 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 two-dimensional Preisach model (4.63)-(4.65), although it holds for the three-dimensional Preisach model (4.60)-(4.62) as well. REDUCTION PROPERTYOF THE VECTOR PREISACH MODEL TO THE SCALAR PREISACH

MODEL.Consider an input fi(t) restricted to vary along some direction -~ofor times t>~to. Suppose that during t>~to, Ur (t) = u(t) consecutively reaches values u+ and u_ (with u+ > u_) and remains thereafter within these bounds. Then,for the Preisach vector model (4.63)-(4.65), 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.

CHAPTER 4 Vector Preisach Models of Hysteresis

468

(z+cos~ ~mj 'ucos~o FIGURE 4.12

PROOF. Without impairing the generality of our discussion, we can assume that q~0 = 0. Then, for any r the input projection u~(t) varies between u+ cos ~ and u_ cos ~. This means that for any ~ input variations m a y affect a - ] / d i a g r a m s only within the triangle T(u+ cos ~, u_ cos r (see Fig. 4.12). It is also clear that all input projections ur reach m a x i m u m or minim u m values at the same time and that these extremum values are 'cos qamultiples' of the corresponding extremum values of uo(t) = u(t). Consequently, if {Mk} and {ink} constitute an alternating series of dominant extrema of u(t), then {M~,k} and {m~,k} defined as

Mr

= Mk cos q~,

m~,k = mk cos ~

(4.72)

constitute the corresponding alternating series of dominant extrema of

u~(t). From (4.63) we find A

fx(t) -

cos ~FCu~(t) de.

(4.73)

2

From (4.71), (4.72) and (4.73) we conclude that only the alternating series of dominant extrema Mk and mk of u(t) affect the value offx(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 m a y have different past histories I these inputs vary for to~t~t~o . However, starting from instant of time t 0,

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

469

(x

u~cos~o .

.

.

.

u cosq~

FIGURE 4.13

u+cosq~

/ ulcos "13 u_cosq~

FIGURE 4.14

back and forth between the same two consecutive extrema, u+I and u t_. 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 of outputs fx(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) = Au (2) - Au. As a result of these increases, the identical triangles T~1) and T~2) are added to positive sets S + qo,1 (t) and S + qo,2 (t) and subtracted from the negative sets S~, 1(t) and S~, 2 (t) (see Figs 4.13 and 4.14). Since

F~ou~o(t) -

v(o~, fl, qo)de d f l +(t)

;(t)

v(~, fl, ~o)de dfl,

(4.74)

CHAPTER 4

470

Vector Preisach Models of Hysteresis

from (4.73) we derive

Af(x1) _ 2 f _ ~ (cos / / T q~ ( l V ( ~ f l ),

,

qo)dadfl)dq ~,

(4.75)

2

Afx(2) = 2

f

_~COS q)

(2)

)

v(~, fl, q~) da dfl dq~.

2

(4.76)

Since T (1) = T (2) for any q~, we conclude that Af(1) -- Af(2).

(4.77)

The equality (4.77) 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 m i n i m u m value u'_. Thus, this equality shows the congruency for the ascending branches of the above minor loops. By literally repeating the previous reasoning, we can prove that the same equality (4.77) holds when the inputs u O) (t) and u (2) (t) are monotonically decreased by the same amount Au after achieving the same maximum value u~. This 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. [] In our discussion of the reduction property, we have proved the congruency of 'scalar' minor loops described by the vector Preisach models. The last result admits the following generalization. CONGRUENCYPROPERTYOFVECTORMINORLOOPS. Let the tips of two inputs Ft(1)(t) and ~(2)(t) trace the same closed curve for t>~to (see Fig. 4.15). Then the tips of the corresponding outputs ](1)(t) and/(2)(t) of the vector Preisach models trace congruent closed curves for t>~to (see Fig. 4.16). These curves may be noncollocated in space because of possibly different past input histories prior to to. PROOF. Consider the three-dimensional Preisach model (4.60)-(4.62). Since the tips of both inputs ~(1)(t) and ~(2)(t) trace the same closed curve, we conclude that for any direction eo,~0 the corresponding a-fl diagrams are modified with time in an identical way within the same triangles T(u+,o,~o, u-,o,~o). For instance, as the tips of the inputs ~ (1) (t) a n d v~(2) (t ) move from the point a to the point b (see Fig. 4.15), the same regions ftl,0,~0 and f22,0,~0m a y b e added to the positive sets S + 0,(p,1 and S + 0,q~,2 and subtracted

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

471

Uz

1

, Uy

Ux

FIGURE 4.15

from the negative

sets So,(pI and So, q~,2. Since

-Fo, q~UO,q~(t) - I/s ~,+~o~(t) v(c~,fl, O, qo)de d[3 -

[[ JJs ~,r

v(c~, fl, 0, ~o)d~ dfl,

(4.78)

we conclude that AFo, q~A u(ll0,q~(t) -- 2

ffolo

AFo, q ~ u(2)O,q~(t) = 2 [[_ J d.s

v(or fl, 0, q~) de dfl,

(4.79)

v(c~, fl, 0, qo) de dfl.

(4.80)

2,0,tO

By using (4.60), (4.79) and (4.80), we find that the corresponding output increments which connect the points A (1), B (1) and A (2), B (2) (see Fig. 4.16) are given by

af(l) -2 f2~f~o,~( ff n JO

Af (2) = 2

JO

fOarCfo rce0,~o(ff

v(a, fl, O,q))do~dfl)sinOdOdq),

(4.81)

1,0,rp

)

v(c~, fl, 0, qo) d~ dfl sin 0 dO dqo. (4.82) 2,0,(p

Since f21,o,~o - f22,o,~o for any 0 and q~, from (4.81) and (4.82) we conclude that A f ( l ) -- A f (2) . (4.83) The equality (4.83) holds for arbitrary chosen points a and b, and this proves that the vector minor loops shown in Fig. 4.16 are congruent. []

CHAPTER 4

472

Vector Preisach Models of Hysteresis

B2

2~

A2 " fy

fx FIGURE 4.16

We next proceed to the discussion of one remarkable property which is valid for the two-dimensional isotropic vector Preisach model (4.67). ROTATIONALSYMMETRYPROPERTY. Consider a uniformly rotating input (that is, one of constant magnitude and angular velocity)

Ft(t) - {ux(t) - UmCOscot, Uy - - Um Sin cot}.

(4.84)

Then the output of the two-dimensional isotropic Preisach model (4.67) can be represented as N

f (t) =f0 + f (t),

(4.85)

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. 4.17 that

u~o(t) -

U m

cos(cot - (p).

(4.86)

According to the proposition proved in Section 4.2, the integration over any semicircle can be used in the definition of the two-dimensional vector Preisach model. This fact allows one to modify the definition (4.67) of the two-dimensional isotropic Preisach model as follows: -

1

2re

f (t) - -~ f o

-G( ff~>~v(c~,fl)P~u~o(t)d~dfl) d~~

(4.87)

It is clear from (4.86) that for all directions e~0 the corresponding a-fl diagrams are modified with time within the same triangles T - T(um, -urn). Outside these triangles, the a-fl diagrams remain unchanged. These unchanged parts of c~-fl diagrams contribute to the term f0 in (4.85), while the

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

~

473

~(t)

FIGURE 4.17

time-varying parts of ~-fl diagrams result in the time-varying term By using the above comment as well as (4.86) and (4.87), we find: 1

f(t).

2~

f (t) - -i f ~ -dq,( f f v(a, fl)~t3Um COS(O~t- q)) da dfl) dq).

(4.88)

The expression (4.88) can be represented in terms of Cartesian components as

1 fO2re cos q)(ff v(~, fl)~Um COS(Ogt- qo)dadfl) dq~, fx(t)-~

(4.89)

1 fO2re sin q)(/f v(~, fl)~UmCOS(O~t- q~)d~dfl) dq~. fy(t) -- -i

(4.90)

Consider some instant of time t. For this instant of time, all directions ~ can be divided into two sets such that O~
(4.91)

~ < o 0 t - q~<2~.

(4.92)

and For the first set, all input projections u~o(t) are monotonically decreasing, and this results in the 0r diagram shown in Fig. 4.18. For the second set, all input projections uq,(t) are monotonically increasing, and this results in the ~-fl diagram shown in Fig. 4.19. Next we introduce new variable 0 - ogt- q~.

(4.93)

It is apparent that the double integral over T in (4.89) and (4.90) is the function of 0. This justifies the following notation

G(O) = ff~ v(e, fl)F~flUm A cos(o~t -

q~) d~ dfl.

(4.94)

CHAPTER 4 Vector Preisach Models of Hysteresis

474

(X

FIGURE 4.18

If 0~0~,

(4.95)

then the diagram shown in Fig. 4.18 is valid, and from this diagram and formulas (4.68) and (4.94) we find G(O) = P(um,

- - U m ) --

2P(um,

Um

COS 0).

(4.96)

If ~KOK2z~,

(4.97)

then the diagram shown in Fig. 4.19 is valid, and as before we find G(O) = 2P(um cos O, -urn) - P(um, -urn).

(4.98)

By using the change of variables (4.93), the notation (4.94), and by taking into account that dO - -d~o, (4.99) and c o t - 2z~O~cot,

(4.100)

we transform (4.89) as follows:

lj02

fx(t) - -~

where

cos(cot - O)G(O) dO - A cos cot + B sin cot,

lf02

(4.101)

cos OG(O) dO,

(4.102)

B -- ~1 f02~ sin OG(O) dO.

(4.103)

A- ~

475

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

/

FIGURE 4.19

Similarly, the expression (4.90) can be reduced to the form (4.104)

fy(t) - A sin cot - B cos cot. The formulas (4.101) and (4.104) can be modified as follows:

VIA2 q- B 2 cos(cot- ~),

(4.105)

y~y( t ) - v/A 2 + B2 s i n ( c o t - ~),

(4.106)

B tan ~ = A"

(4.107)

fx(t)

-

where

From (4.105) and (4.106), we conclude t h a t f is a uniformly rotating vector with the magnitude equal to v/A 2 + B2. We next express A and B in terms of the function P and prove that the angle ~ in (4.105) and (4.106) is acute. From (4.95)-(4.98) and (4.102) we find

1{/0 ~2~

cos O[P(um, -urn) - 2P(um, Um cos 0)] dO

A- ~

q-

}

cos O[2P(um cos 0, -urn) - P(um, -urn)] dO . (4.108)

In the second integral in (4.108), we will use the change of variables 0'= 0- ~

(4.109)

CHAPTER 4 Vector Preisach Models of Hysteresis

476

and take into account that

P(-um cos 0, -urn) ~- P(um, Um cos 0).

(4.110)

The last formula easily follows from the symmetry property (4.58) and the definition (4.68) of the function P. By using (4.109) and (4.110), we can transform (4.108) as follows: A = -2

~0 ~

cos OP(um, Um COS 0) d0.

(4.111)

From (4.95)-(4.98) and (4.103), we obtain

11/0

B= ~

+

sin O[P(um, -urn) - 2P(um, Um cos 0)] dO sin O[2P(um cos 0, -urn) - P(um, -urn)] d0 . (4.112)

To transform the second integral in (4.112), we will use the change of variables (4.109) and the identity (4.110). This eventually leads to the following expression for B:

/z

B - 2P(um, --Urn) -- 2

f0

sin OP(um, Um COS 0) d0.

(4.113)

Thus, we have found explicit expressions (4.111) and (4.113) 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 vectorf(t) in terms of these data. Now, we will use formulas (4.111) and (4.113) to prove that the phase angle ~ is acute under some general conditions. The first condition can be expressed mathematically as

P(a, fl)~P(a, fl')

if fl>~fl'.

(4.114)

The condition (4.114) means that P(a, fl) is a monotonically decreasing function of fl for any fixed ~. It is clear from the definition (4.68) of P that the condition (4.114) is satisfied if v(a, fl) is positive. From (4.111), we find

A 2[/0

cos OP(um, Um cos 0) dO q-

cos OP(um, Um cos 0) dO . (4.115)

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

477

By using the change of variables 0t = ~ - 0 in the second integral in (4.115), we derive /T

A

-

-2

f0

cos O[P(um, Um COS 0)

-- P ( u m , - u r n c o s 0)] dO.

(4.116)

From (4.114) and (4.116), we conclude A > 0.

(4.117)

O[P(um, -Um) -- 2P(um, Umcos 0)] d0 > 0.

(4.118)

The second condition is f0 ~rsin

From (4.114) and (4.118), we find B > 0.

(4.119)

From (4.107), (4.117) and (4.119), we conclude that the phase angle ~ is acute. This completes the proof of the rotational symmetry property. [] It is worthwhile noting that in spite of the nonlinear structure of the vector Preisach model (4.67) the time harmonic input (4.84) produces (up to a history-dependent constant term f0) 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 (4.67) has a mathematical form which is invariant with respect to any rotation of Cartesian coordinates. The mathematical form of uniformly rotating input (4.84) is also invariant with respect to any rotation of Cartesian coordinates. Thus, on 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 (4.67) 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. 4.20. This input gradually approaches the regime of uniform rotation (4.84). The results of computations shown in Fig. 4.21 demonstrate that the output also gradually attains the regime of uniform rotation. The property of rotational symmetry has also been observed in the experiments. The --4

CHAPTER 4 VectorPreisach Models of Hysteresis

478

5.0

2.5

_

o.o_

t,,._j

,

,

,)/, i

-5.0 -5.0

-2.5

0.0

2.5

5.0

FIGURE 4.20

results of these experiments are shown in Fig. 4.22. In these experiments, the applied magnetic fields were stationary, while the magnetizable 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. 4.22 clearly suggest that for large applied fields the whole magnetization of the sample moves in synchronism with the uniformly rotating coordinate frame. This means that we are dealing 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 COMPONENTS OF

OUTPUTANDINPUT. Suppose that the input ~(t) is first restricted to vary along the y-axis. It is increased from an infinitely negative value to some positive value u+, and then it is decreased to zero. Some remnant value of output fr = -eyfr results from the above input variations. After reaching zero, the input is restricted to vary

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH 20

I

I

0

10

479

10

-10

-20

-10

20

F I G U R E 4.21

along the orthogonal x-direction (see Fig. 4.23). It is asserted that, by increasing the input in x-direction, it is possible to reduce the orthogonal remnant component of the output to any, however small, value. PROOF. All directions ~r can be subdivided into sets: 0Kq~K~

and

~q0K0.

(4.120)

The input variations along the y-axis result in the a-fl diagrams shown in Figs 4.24 and 4.25 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 dashed lines. They result in continuous expansions of positive sets S~ (t) at the expense of negative sets S~ (t). From (4.67), we find

fy(t) = f_~ sinq~( / L 2

v(~, fl)~[~Ux(t)cosq~d~dfl) dq~.

(4.121)

~fl

By introducing the notation

G(qo, t) - ~ v(c~,fl)~ux(t) cos (p d~ dfl, dJa>~

(4.122)

480

'

I'

'

'

'

I

'

'

'

'

I

I

'

'

. . . .

'

'

8 o

II

....

I''

i .... ~

~sa~Asu~.q

'

''

o I

o

8o II :=

t~

Vector Preisach Models of Hysteresis

,I ~

I'''

uop, uzp, a u g g t u

, , , , l , , , I

''''I''

CHAPTER 4

o -~

-

'~

,

.

,

~

,,

,~

I

-

,

I ' ' ' '

~

_

,

l~ll~.~d

. . . .

,

,

,

UOI~Z~U~gUZ

I , ,

o

o uo!lwzT1~u]luuz

,

,

u~ /

~

I , ,

aS.ZOASU~.Z I

,,

~s1~Asug.zl

~,

I

o

t~

-~ o

o

r,.,

~J

o ~

~J

~J

A

H U~

r

o

Cl

,

-

"4

,,

~oo

-

,

o uop,~zrlauli~u~

I

....

-

-

_

-~ o

c~ fl

_

I,

~

. . . .

~

i

~

_

_

'I

o

i ....

uop, gZ!laUli~cu lallV.z~d

....

o laII~.md

II

_

. . . .

''I

'!''i

I,

_

o~

o

I-

!

i

--I -

o

uop,~znauil~ux

e,l e,l

o 9-~

1

!

1

,

' ' '

,

,

,

I ' ' ' '

o

IOlI~aud

! ' ' ' '

,

I

tD

. . . .

,,,

I ' ' ' '

i,

?

I

lali~.tecI

,

1 , , , , I , , , ,

i,,

uo!~lez!laugum

o

uo!~ez!leugum

I ' ' ' '

~

I

i

....

IeIW.tud

v

I

N

~-,9

ii

0

#

-~ o

I

o

o

84

I

~

_

-

o o

_ ~

-- o

~

o

o

!

~ , , , I ,

I'

'

uo!luzTl~uSum

''I

~,,I

I ' ' ' '

esaeAsuual

I,,,,It,I, ~

....

I

I'

I

'

' ' '

....

....

~ , ~ 1 ~ , ~

asa~a$

I

~

. . . .

o

as.taAsuu.x I

'I

UOD~zt.1~U~tatLt

' ' ' ' I ' '

u0!$ez[~ugu~

' ' ' ' ! ' '

i l l l l

' ' ' '

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

' ' ~ '

!

,

i i i i i

,

,I

uo!~ez!lau~tu

-

~,

o

-

o

7

0 0

o

o

o

II

-~

o o II

o ~0

~0

II

-'-"

~

s 8 .~

_

o v t~

..4

o

.~

481

-o

O

.4=a

v eq e,4

L~

o

482

,,(,I

. . . .

ill

o

. . . .

J i l l l

I

~

. . . .

' ' 1

uotl~z~u~m

I

o

uo1'~'~z~eu~'~m

I'

uo!:~wz~u~um

. . . .

I

1

i i i

. . . .

. . . .

' I ' ' ' '

leITe.n~d

l ~ d

. . . .

l l i l i l

l~l~.r~d

fl

o

0

"J"

'

....

,,

'"

,

'

'

'

'

o

I

'

'

I

,

'

i i i

'

,,

I''''

I

?

/

~

l l l l l l l

''

,

u~,

~ex~AsuvJ)

'

'I''

O

'

_

_

I

at

8•

o

II v

II

o o

fi

"o

o

a

,---,

8"" N

o

8~

o ~

o

T

o

Vector Preisach Models of Hysteresis

tO

I

I ....

I

uop,~z!~u~ux

,o

lllilli~i'i'i'i'i'i'i'i~li~ll,

i'll

CHAPTER 4

,

I

o

oc

!

'

0

uo!'twzr~;~u~tu ~r~;~Asue.rt

o G;

e,i r

4.3 SOME BASIC PROPERTIES OF VECTOR PREISACH

f~

483

/~(t) ~X

FIGURE

4.23

u cos~

(~o)

u.~in~o

FIGURE 4.24

•u

Usn'q l

cos~

(~o)

FIGURE 4.25

formula (4.121) can be represented as follows:

fy(t) =/o ~ sin ~G(~,

t)do +/0

sin qaG(~, t)do. 2

(4.123)

CHAPTER 4 Vector Preisach Models of Hysteresis

484

Changing q~ to -q~ in the second integral in (4.123), we obtain 2

fy(t) -

Jo

sin q~[G((p, t) - G(-q~, t)] dq~.

(4.124)

If Ux cos q~a0, then according to the diagrams shown in Figs 4.24 and 4.25 we have G(q~, t) - G(-q~, t). (4.125) From (4.124) and (4.125), we find

fy (t) =

rccos

=0

sin q~[G(q~, t)

-

G(-q~, t)] dq~.

(4.126)

ux(t)

Since arccos Ux(t) ~0 ~ ~ as ux(t) ~ cx~,we conclude that

fy(t) ~ O.

(4.127)

This completes the proof of the above property. This property has been confirmed by numerical computations performed by using a digital code that implements the model (4.67). The results of the computations are shown in Fig. 4.26 for different remnant values of the

5

0

--5

-lO

1

~,~1~1,~,~1,,,,i,,,,1,,,,

0

1000

2000

3000

F I G U R E 4.26

4000

-

5000

6000

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

485

o

-5

-lo

j L i i 1 , l , l I i l , , 1 l o

.05

.1

.15

FIGURE 4.27

output. The property of correlation between orthogonal components of input and o u t p u t has also been observed in the experiments. The results of these experiments are presented in Fig. 4.27. There is an apparent qualitative similarity between the computational and experimental results. It is worthwhile noting here that the above property of orthogonal correlation has long been regarded as an important 'testing' property for vector hysteresis models in magnetics.

4.4

IDENTIFICATION PROBLEM FOR VECTOR PREISACH MODELS

ISOTROPIC

The essence of the identification problem is in determining the function v(a, fl) or P(a, fl) from some experimental data. It turns out that this problem can be reduced to the solution of a special integral equation that relates the function P(~, fl) to some 'unidirectional' (scalar) hysteresis data. We first present the derivation of this equation for the two-dimensional model (4.67). The derivation proceeds as follows. Consider the projectionfx(t) off(t) along the direction q~ = 0. According to Eqn. (4.67), we have

fx(t)

=

I' (o(iL'>~' v(a', fl')~,~,u~o(t) d~' aft' )de. cos

(4.128)

CHAPTER 4 Vector Preisach Models of Hysteresis

486

d

s:,

FIGURE 4.28

Now we restrict ~(t) to vary along the direction r = 0, which means that Ft(t) = -~xu(t). First, we assume that u(t) is made 'infinitely negative'. Then for any a,I fl! and (p we have ~,/~,u,(t) = -1. Next, we assume that the input is monotonically increased until it reaches some value a. Let fa denote the corresponding value offx(t). As a result of the above input increase, we find that for any q~ we have ~,/~,u,(t) = +1 if a' < a cos q~ and ~a,/~,u,(t) = - 1 if a t > a cos q~. This is shown geometrically in Fig. 4.28. Consequently, for any r we have

ff~,>>.~,v(a', fl')~,ffUq~(t) da' dfl' = ffs+ * v(a', i f ) d a ' d f f - ffs;.,

v(a', fl') da' dff.

(4.129)

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 offx(t). The above input decrease modifies the previous geometric diagram that now for any q~ assumes the form shown in Fig. 4.29. According to this figure, we have ff~ ,>>.~'v(a',

= ffsL

fl')~,~,ur

da' dfl'

v(a', fl')da' d f l ' - f f

dis ~.~

v(a', fl')da' dfl'.

(4.130)

Consider the function 1

F(a, t) = ~(f~ -f~/~).

(4.131)

By using formulas (4.68), (4.128)-(4.131), it is straightforward to show that

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

487

Td (~) / ,

13cos~ s~*~,,I-

FIGURE 4.29

P(a, fl) is related to F(a, fl) by the expression

f

~ cos q~P(a cos q~, fl cos q~) dq~ - F(a, fl).

(4.132)

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 negative as was assumed in the derivation. However, by making the input 'sufficiently negative', a reasonable accuracy can be secured. The formula (4.132) can be construed as the integral equation which relates P(~, 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

AP = / ~ cos qoP(a cos q~, fl cos q~) d~0.

(4.133)

2

Monomials ~k fls are eigenfunctions of the operator A. PROOF. From (4.133) we find -

~ cos Oa k cos k Oti s cos s r d o 2

~xkfls f ~ cos k+s+l q~dq~ - 2k+lak fls, d-

(4.134)

CHAPTER 4 VectorPreisach Models of Hysteresis

488

where

(2n-l)!!

ifk + s + 1 = 2n, (4.135)

2k+s --

2 (2n)!!

ifk + s + 1 - 2n + 1.

By using the above theorem, the solution of Eqn. (4.132) can be easily found when the right-hand side F(a, fl) is a polynomial M F(a, f l ) = ~ ~-(m)akfls aks m=0 k+s=m

(4.136)

Indeed, looking for the solution of the integral equation in the form M P(a, fl) -- y ~ ~-~__"(m)akfls,laks m=0 k+s=m

(4.137)

from the above theorem we find ~(m) p~m) = Uks 2m"

(4.138)

By using this fact, the following algorithm for the solution of Eqn. (4.132) can be suggested. We first extend the function F(a, fl) from the triangle -a0~
(4.139)

where the Chebyshev polynomials are defined as To(a) = 1,

1

Tt(a) - ~--;--ccos(s

(4.140)

By assuming for the sake of notational simplicity that a0 = 1, for the expansion coefficients agq in (4.139) we have the formula (see [12])

agq=

2e+qf; f/ F(a,fl)Tt(a)Tq(fl)dadfl. /z2

1

1

v/l_a2v/l_fl2

(4.141)

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

489

Let teq(O~,fl) be the solution of the integral equation

/

7[

~ cos (pts

cos (p, fl cos q~) dq~ - Te(~) Tq(fl),

(4.142)

then the solution of the integral equation (4.132) can be represented in the form P(a' fl) - E as163 fl). (4.143)

s Thus, there are two major steps in finding the solution (4.143) of the integral equation (4.132). The first step is to solve Eqn. (4.142). This can be accomplished by using the technique (4.136)-(4.138). The second step is to evaluate the expansion coefficients as This can be handled by using the formula (4.141). However, for the sake of computation it is desirable to transform this formula as follows. By introducing new variables 0 and 0I = cos 0,

fl = cos 0',

(4.144)

cos qO' 2q_1 ,

(4.145)

F(cos 0, cosO') cosf.OcosqO' d0d0'.

(4.146)

we find cos t~0 , 2s

Ts

4/070

as - -~

Tq(fl)

-

Thus the expansion coefficients as can be evaluated by using fast cosinetransform schemes. There is another approach to the solution of the integral equation (4.132) that leads to a closed-form expression for P(a, fl). This approach is based on the following change of variables: x = a cos q~,

2 = ilia,

(4.147)

which after simple transformation yields the following integral equation of the Abel type:

f0 40r x-- X2 P(x, 2x) dx = ~F(~, 0~ 2a).

(4.148)

Some simplification of the above equation is achieved by introducing new auxiliary functions:

N(x) - xP(x, 2x),

R(a) - 2F(a, 2a).

(4.149)

CHAPTER 4 Vector Preisach Models of Hysteresis

490

oc

x

FIGURE 4.30

Then, from (4.148) we obtain

fo ~ N(x) dx

= R(~).

(4.150)

/0r _ x 2

The solution of the above equation can be found by using the following trick. We multiply both sides of (4.150) by ~/s2_~2 and integrate with respect to e from 0 to s:

f0s

o~ x/S 2 __ CX2

(f0

N(x) dx

) d ~ = f0

da.

(4.151)

~zdcz ~ dx. V/(S2_~2)(-~2_x 2) ]

(4.152)

~/0r __ X 2

&2

__ 0r

By using Fig. 4.30, we find

fos

cx (fo~ N(x)dX)dcz V/S2 -- 0~2

X/C(2 -- X 2

[~ ~N(x)dx d~ JJ. v/(S2 -- 0r162 2 -- X2) s =f0

s N(x)(L

Next, by using simple transformations, we obtain fx s

0rdcz

v/(S2_ CX2)(CX2_X2)

_ 1 fx s 2 ~

d(cz2)

v/(s22x2_)2 _(cx 2 -

~

(4.153)

s2+x22)2

By introducing a new variable of integration S2 + X2 09 -- O~2

(4.154)

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

491

from (4.153) we derive s2_x 2

/x s

o~da v/(S2 -- X2)(0r2 -- X 2)

_ 1 ~x22s2 2

do~

~ ~/ (S2-X2) 2 __ 092

1 209 = - arcsin 2 s 2 -- 09 2

s2 _x 2 09----

--. x2 -s 2 09----

2

(4.155)

From (4.151), (4.152) and (4.155) we obtain f0 s N(x) dx - -2 f0 s

c~R(c0 da.

(4.156)

v/S2 _ ~2

By differentiating (4.156) with respect to s and then by replacing s by and a by s, we find N(~) -

2 d [~ d~ ~0

sR(s)

ds.

(4.157)

V/or2 -- S2

This is the closed-form solution to the integral Eqn. (4.150). Now, by substituting (4.149) into (4.157), we find the closed-form expression for P:

P(c~, ;ccO- 1 d fo ~ s2F(s, ,~s) ds. ~:o~d~x

(4.158)

V/0r _ s 2

The last expression is valid for any value of 2. By varying 2 from 0 to 1 and using (4.158), we can compute all required values of P. However, for actual computations, it is convenient to transform (4.158) by integration by parts. The final expression is then given by

P(a, 2a)

1~

F(s, 2s) + s dF(s, ,ls)

7C Y0

V/0r 2 -- S 2

ds.

(4.159)

Now, we turn to the three-dimensional isotropic model (4.66). Consider the projection fz(t) of f(t) along the direction 0 - 0. According to (4.66), we have

fz(t) -

cos 0 x sin 0 dO dcp.

'~>/~'v(a', fl )?a,~,Uo, q~(t) da' aft' (4.160)

We restrict ~(t) to vary along the direction 0 - 0, which means that ~(t) -dzu(t). As before, we first assume that u(t) is made infinitely negative. Then

CHAPTER 4 Vector Preisach Models of Hysteresis

492

d ..

+

l~coso

/ FIGURE 4.31

I d (o , , ) / , +

FIGURE 4.32

for any a, fl, 0 and q~ we have r q~(t) - - 1 . Next, we assume that the input is monotonically increased until it reaches some value ~. Letf~ denote the corresponding value of fz(t). As a result of the above input increase, we find that for any 0 and r we have ~a,~,UO,q~(t) = +1 if a' < c~cos 0 and r q~(t) - - 1 if a' > a cos 0. This is illustrated geometrically in Fig. 4.31. Consequently, for any 0 and (p we have

fL ,>>.~'v(a', ff

dis

do~'dfl'

v(a', fl')da' a f t ' - f f

JJs;,o

v(a', fl')da' aft'.

(4.161)

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-mentioned input decrease results in geometric diagrams shown in Fig. 4.32. According to this

4.4 IDENTIFICATION PROBLEM FOR ISOTROPIC VECTOR

493

figure, we have /f~ '>~]~'

- dffi s

fl')~,/~, , u 0,q~(t) da' dfl'

v(~x',fl')&x' aft'- [[ v(~x',fl')d~x' dfl'. JJs;, ,o

(4.162)

By substituting (4.161) and (4.162) into (4.160), then by subtracting one expression from another and by using (4.68) and (4.131), we derive

f0

2~0 ~ COS OP(a cos 0, fl cos 0) sin 0 dO d o = F(a, fl).

(4.163)

Since the integrand does not depend on q~, we obtain 7r

/0

cos 0 sin OP(a cos O, fl cos O) dO -

1

F(a, fl).

(4.164)

This is the integral equation that relates P(a, fl) 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~cos O,

2 = fl/~,

(4.165)

Eqn. (4.164) can be represented in the form

fo

C~XP(x, 2x) dx - ~nnF(a, 200.

(4.166)

By differentiating the last equation with respect to c~, we find 1 d [~2F(~ ' ,~)] c~P(~, 2~) - 2~z dc~

(4.167)

which leads to the following final expression: P(c~, 2cr -

1 d [0r162 ' ~0r 2rcc~ dc~

(4.168)

It is remarkable that the solution of the identification problem for threedimensional isotropic Preisach models of vector hysteresis turns out to be m u c h simpler than the solution of the same problem for two-dimensional isotropic models.

CHAPTER 4 VectorPreisach Models of Hysteresis

494

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR PREISACH MODELS We shall first discuss the identification problem for the two-dimensional anisotropic model (4.65). As before, we shall use the function P(~, fl, q0) defined by (4.68). We shall relate this function to the experimental data represented by the sets of first-order transition curves measured along all directions e~0. These experimental data can be characterized by the function 1

F(or fl, r = ~(f~r -f~/~r

(4.169)

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 e~0. However, it is possible to measure these curves for some finite meshes of directions e~0and to use subsequent interpolation for computing F(~, ]~, (p) for all 9). Consider local polar coordinates (p, ~) with the polar axis directed along the vector e~o'. The model (4.65) in local polar coordinates can be written as

f (t) =

-de/

v(~, fl, q~' + d/)~ue/(t) d~ dfl dO,

(4.170)

where the relationship between the angles q0, opt and ~ is illustrated by Fig. 4.33. By using the expression (4.170) and by repeating almost literally the same reasoning as in the derivation of the integral Eqn. (4.132), we find that the function P(a, fl, q0) is related to the function F(a, fl, q0) by the expression

f

~ _~cos OP(a cos ~, fl cos ~, q0' + O)dO - F(a, fl, (p'). 2

FIGURE 4.33

(4.171)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

495

Next we shall use the following Fourier series expansions: (x)

F ( ~ , fl, (p' ) -

E

Fn ( o~, fl) e in~~,

(4.172)

Pn((X, fl)e inq~

(4.173)

Vn ( Ot, fl) e inq).

(4.174)

tl-----r

(x)

P(r fl, q ) ) -

E tl'----(X)

oo

V( Ot, fl, q)) -

E tl-'--O0

It is clear from (4.68) and (4.69) that Pn and

Pn(ot, fl) - / f T Vn(O~, f l ) - -

(~,~)

Vn are related by the expressions:

Vn(O~',fl') da' dfl',

~2pn(~, fl)

.

(4.175)

(4.176)

By substituting (4.172) and (4.173) into (4.171), we find

E

einqr / ~~ einr

n----oo

I[IPn(c~cos ~, fl cos I]/) dO

2 oo

-- E

Fn(c~, fl) e inq~

(4.177)

tl----(X)

From (4.177) we obtain

f

2 einr/ COSI~Pn(or cos ~, fl cos ~) dO = Fn(or fl) (n -- O, +1, 4-2 . . . . ).

(4.178)

Since sin n~ cos 6Pn(a cos 6, fl cos ~) is an odd function of ~, from (4.178) we derive

/2

_n

cos $ cos n$Pn((Z cos $, fl cos $) d $ - Fn(O~, fl)

(n = 0, 4-1, +2 . . . . ).

(4.179)

Thus we have obtained the infinite set of decoupled (separate) integral equations for Pn. The right-hand sides Fn(~X, fl) of these equations can be computed by using fast Fourier transform (FFT) algorithms.

CHAPTER 4 Vector Preisach Models of Hysteresis

496

For the case n = 0, from (4.179) we find _n 2 COS ~/P0( 0~c o s ~/, ]~ c o s ~/)

d6

= F0(a,

fl),

(4.180)

which naturally coincides with the integral Eqn. (4.132) for the isotropic vector Preisach model. If n = 4-1, from (4.179) we obtain

f_

~ c o s 2 ~P+l(a cos ~, fl cos ~) d~ = F+l ((x, fl). _n 2

(4.181)

Closed-form solutions to integral Eqns (4.180) and (4.181) can be found by using the change of variables x - a cos ~,

2=

fl/~

(4.182)

and by reducing the above equations to the Abel-type integral equations. In the case of Eqn. (4.180), the result is readily available and can be expressed by the formula P0(~, 2~)

1 d fa~ s2Fo(s, 2s) ~a d~ vu ~--~2~i2s as.

(4.183)

In the case of Eqn. (4.181), the change of variables (4.182) leads to the integral equation f0 a

X 2 dx 0r2 P+I (x, 2x) ~/0r2 _ x 2 -- -~-F+I (0r 200.

(4.184)

These equations can be solved in exactly the same way as Eqn. (4.148). The final result is given by the following formula P+I (a, 2a)

1

d [ ~ s3F+l(S, 2s) ds.

~0~2 d~ J0

(4.185)

V/o~2 -- S 2

Thus, if w e are interested only in the firstthree 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 (4.183) and (4.185). In the case of higher-order terms, the change of variables (4.182) leads to the following Abel-type integral equations

fo ~ Pn(x, 2x) ~/O~ xTn(X) ~ 2 _ X2 d x - -i-dFn((X, 2a),

(4.186)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

497

where Tn are Chebyshev polynomials defined by (4.190). 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 Eqn. (4.179) can also be found by exploiting the following property of this equation. THEOREM. Consider operators lr

~4nP - / 2~ cos ~ cos n~P(~ cos Ifi, fl cos ~) d~.

(4.187)

2

Monomials ~kfls are eigenfunctions of these operators. PROOF. It is straightforward to check the validity of the above statement and to get the following expressions for the eigenvalues "~k+s" j(n) . 2k+s (n) - 2

fo

cos nO cos k+s+l 0 dO

(4.188)

The explicit formula for the integral in (4.188) can be found in the literature (see [13]). [] By using the above theorem, polynomial solutions to the integral Eqn. (4.179) can be easily found. Indeed, if the right-hand sides of these equations are polynomials in the form (4.136), then the solutions to these equations will be polynomials in the form (4.137). Polynomial coefficients of the right-hand sides and the solutions will be related by the expression (4.138) where instead of eigenvalues ~m the eigenvalues ~(n) should "~k+s be employed. By taking advantage of this fact, Chebyshev polynomial expansions of type (4.139) for Fn(~, fl) can be used and the solutions can be found in the form (4.143), where the expansion coefficients can be computed by using expressions similar to (4.146). Now, we proceed to the identification problem for the three-dimensional anisotropic vector Preisach model (4.62). This problem is technically more complicated than the corresponding problem for two-dimensional models and, for this reason, it requires 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(~, fl, 0, q~) defined by (4.68) and we shall relate this function to the first-order transition curves experimentally measured along all possible directions e0,~0. These curves can be used

CHAPTER 4

498

Vector Preisach Models of Hysteresis

to define the function 1

F(o~, fl, O, (p) = -~(f~or -f~oq,).

(4.189)

It is convenient to use spherical harmonic expansions for the functions v, P andF: oo

V(O~,fl, O, q))= E k=0

k

E Vkm(O~'fl)Ykm(O' q))' m=-k

(4.190)

k

P(~, fl, O, r

E

E Pkm(~ fl)Ykm(O, r k=O m=-k o0 k

F(~, ,8, O,q~)= E

E Fkm(a,fl)Ykm(O,q~),

k=O

(4.191)

(4.192)

m=-k

where Ykm(O, q~) are spherical harmonics, while Vkm, Pkm a n d Fkm are corresponding expansion coefficients. It is clear from (4.68) and (4.69) that Pkm and Vkm are related by the following expressions:

Pkm(O~,fl) = f f Vkm(Off, if) ,liT (~,~) Vkm(e, f l ) = -

a2pkm(O~,fl)

d e ' dfl',

9

(4.193)

(4.194)

It turns out that the identification problem can be reduced to the solution of special integral equations which relate Pkm(~, fl) to Fkm(Ct, fl). The derivation of these equations proceeds as follows. Consider an arbitrary direction specified by the angles ff and q~'. We shall use a local coordinate system xyz, which is obtained from the system XYZ by Euler rotations Rz(0), Rx(0') and Rz(g7~ + q~') (see Fig. 4.34). The Euler angles for the inverse rotation from xyz to XYZ are equal to grc - ~0', 0 f and 7r, respectively. It is clear that the direction of the z-axis coincides with the direction of-do,e,. By using local spherical coordinates ~ and ~, we can represent the model (4.62) as

(/I., x sin r d~ d~.

o,,

d.d,) (4.195)

By using the last expression and by repeating almost literally the reasoning that was used in the derivation of the integral Eqn. (4.163), we can show

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

499

z z

y

o! i

x

,u

X

FIGURE 4.34

that the model (4.195) matches first-order transition curves measured along the direction e0'~0' if the function P satisfies the integral equation

J0270: cos ~P(a cos ~, fl cos ~, ~, ~, 0t, q~') sin ~ de d6 = F(~, fl, 0', ~0').

(4.196)

Next, we shall relate the function P in local coordinates to the same function in spherical coordinates 0 and q0. To achieve this, we shall use the spherical harmonic expansion (4.191) and the following facts from the theory of irreducible representations of the group of rotations (see [14-16]). Linear combinations of spherical harmonics Wkmof 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, q)) in coordinates 0 and ~0 can be represented as a linear combination of the spherical harmonics Ykm(~, ~) of the same order k: k

Ykm(O,,q),)

-

-

~ k atom, ( l'r. -- q)') Ykm'(~, 0), ~, 0', -~ ml'--k

(4.197)

where atom, k 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 XYZ system. These functions are sometimes called generalized spherical harmonics because they are reduced to spherical harmonics when m or m t is equal to zero.

CHAPTER 4 Vector Preisach Models of Hysteresis

500

By using (4.197) and (4.191), we find the expression for P in local coordinates

P(~, ~, ~, r 0', ~') k

k

--

amm, ~z, , -~ - q;' Pkm(O~,fl)Ykm'(~, 6). (4.198) m=-k m'=-k By substituting (4.198) into (4.196), we find k=0

F(~, ~, 0', ~') c~ -

k

k amm,

-

k=0

~,

, -~ -- qJ

m=-k m'=-k

x ~O2~z~O~ cos~Pkm(O~cos~, ~cos~)Ykm,(~, 0 ) s i n ~ d ~ d 0 .

(4.199)

In (4.199) the integral with respect to 6 is equal to zero if m' r 0, and it is equal to 2Ir if m' = 0. Thus we obtain

F(~, ~, 0', ~') oo

k

k=0

m=-k

= 27r

x

)

amo re, 0', ~ - q~'

~0 arc

COS~Pkm(~Xcos ~, fl cos ~)Yko(~) sin ~ d~.

(4.200)

It is known from group theory (see [16]) that

k ( 7r ) m / 4re amo ~z, 0',~--q0' - - ( - - 1 ) ~ 2 k + 1

Ykm(O', ~o').

(4.201)

It is also known (see [12]) that ~/2k + 1 Lk(cos ~), 4~r

Yko(~) =

(4.202)

where Lk are Legendre polynomials. From (4.200), (4.201) and (4.202), we derive oo

F(~, fl, 0', ~0') - 2Ir ~ k=0

k

~ (-1)mykm(O ', q)') m=-k

/[

x

Pkm(O~cos ~, ~ cos ~)Lk(cOS ~) cos ~ sin ~ d~.

(4.203)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

501

By comparing (4.203) with (4.192), we find f0 ~ Pkm(~Xcos ~, fl COS ~)Lk(cos ~) cos ~ sin ~ d~ = (-1)m 2----~Fkm(O~,fl). (4.204) 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 - ~ cos ~,

2 = -fl.

(4.205)

Indeed, after simple transformations, we derive

f0 ~ Pkm(X, 2x)XLk (X) dx --

~2 (--1)m-~-~Fkm(C~, ~,~x).

(4.206)

Explicit solutions to Eqn. (4.206) can be found for the cases k = 0 and k - 1. If k = 0, from (4.206) we obtain

~0~xPoo(x, 2x) dx = ~-~F00(c~, ~ 2~).

(4.207)

By differentiating (4.207) with respect to ~, we find P00(a 2a) -

1

d [a2F00(~ ,2~)].

2 ~ d~

'

(4.208)

As expected, this result coincides with the one obtained for the isotropic model (see (4.168)). If k = 1, according to (4.206) we have

f0 c~X2plm(X,

0r3 Flm(~X, ,~00 )~x) dx - ( - 1 ) m -~--~

(m = 4-1, 0).

(4.209)

By differentiating (4.209), we derive Plm( 0~, ,~,~) _

(-1) m d 27r~ 2

d~

[~Z3Flm(0~' ~)]

(m - +1 0).

(4.210)

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 (4.208) and (4.210). For k > 1, Eqn. (4.206) can be solved numerically. Discretization procedures can be applied directly to Eqn. (4.206), or the equation can first be

502

CHAPTER 4 VectorPreisach Models of Hysteresis

reduced by differentiation to the Volterra equation of the second kind

Pkm(~Z,,~0~)-- fo ~Pkm(X, ,~x)-~x Lk,(X)d x = (-1) m d [o~2Fkm(~,~)]. (4.211) 2z~a da 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 of integral equation may be desirable as far as computational stability is concerned. Finally, the integral Eqn. (4.204) can be solved analytically if polynomial approximations for the right-hand side are employed. These analytical solutions can be found in exactly the same manner as for twodimensional identification problems discussed before. Up to this point, we have used scalar hysteresis data (4.169) and (4.189) 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 two- and three-dimensional problems can be represented in the following forms, respectively 1

F(~, fl, ~#)= ~ ( f ~ -)?~/~),

(4.212)

F(a, ]/, 0, q~)= ~ 07~0~ -)?~//0r

(4.213)

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:

f(t) - ~lTl:l ( Sf~>>.~7(~'fl")~a~(-f "u(t))d~

dsr"

(4.214)

By repeating the same line of reasoning as in Section 4.2, we can show that by means of redefining ~ the integration over a unit sphere in (4.214) can be reduced to the integration over a unit hemisphere

f(t) - //c+ ( ff~>>.~v(~z'fl'-f )~;a~(-f"u(t))d~

dsr"

(4.215)

It can also be shown that the redefined function ~ has the following symmetry property ~(~, fl, ~) - -~(-fl, -~, - ~ ) . (4.216)

4.5 IDENTIFICATION PROBLEM FOR ANISOTROPIC VECTOR

503

By using spherical coordinates, the generalized model (4.215) can be represented as

-* - ~02rc~0~( f ~ f(t)

~(a, fl, O, q~)~a~uoq~(t) da dfl ) sin 0 dO dq~.

(4.217)

Similarly, the two-dimensional model can be expressed in the form /r

f (t) =

~(a, fl, q~)~13uo(t) dadfl

dq~.

(4.218)

To solve the identification problems for the models (4.217) and (4.218), we as before introduce the following auxiliary functions P(~, fl, O, q~) = P(a, fl, q~) =

(~,/~) (~,/~)

?(a', fl', O, q~) dor aft,

(4.219)

~(a', fl', q~) de' dfl'.

(4.220)

These functions can be related to the experimental data (4.213) and (4.212) by the following equations, respectively

fo2O

P(0r cos ~, fl cos ~, ~, ~, 0', q~') sin ~ d~ dO = F'(a, fl, 0', q/), (4.221) /r

f ~ P(acos$, flcos$, q~' + ~ ) d $ - F ( ~ , fl, q~').

2

(4.222)

By employing spherical harmonic expansions in the three-dimensional case and Fourier expansions in the two-dimensional case, we derive as before the following integral equations cos O, fl cos O)Lk(cos ~) sin ~ d~ = (-1) mF-km(~, [3) (4.223) f0~_Pkm(O~ . 2~r /r

2

COSnl~Pn(o~ cos ~, fl cos ~) d o = Fn(o~, fl).

(4.224)

By using the change of variables (4.182) and (4.205), the above integral equations can be reduced to the following forms, respectively

f0

()

(4.225)

~0 ~ -" Zn(X) d x - 1 F-n(O~,,~,o0 Pn(x, ~,x) V/0~2 _ x 2 n2 .

(4.226)

Pkm(X, 2x)Lk -dx d x - (--1) m a ~_km(a ' 2~x),

504

CHAPTER 4 Vector Preisach Models of Hysteresis

When k and n are equal to one, the following explicit solutions of Eqns (4.225) and (4.226) can be derived: Plm(~, ,~or = (--1)m d [o~2~71m(Or ~,or 2rc~ dc~

(m = O, +1),

P1 (a, 2a) = 1 d f0 ~ SaFl(S, 2s) ~ dc~ v/---2-2 82 ds.

(4.227)

(4.228)

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 (4.212) and (4.213) are reduced to (4.131). It can be shown that in this case ~(~, fl) = ~q~V(~,fl) and thus the generalized model (4.218) is reduced to the model (4.67). The proof of the above statement is left to the reader as a useful exercise.

4.6 D Y N A M I C

VECTOR PREISACH MODELS

OF

HYSTERESIS The vector Preisach models 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 two-dimensional 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 models for all directions q0 on the speed of output variations, dd~, along these directions. This can mathematically be expressed as

f (t)

-

f ~ -de ff~ >>-~v a, fl, c3t ,I ~ u ~ o ( t )

d~ dfl) dqg.

(4.229)

The direct utilization of the above model is associated with some untractable difficulties. These difficulties can be circumvented by using the power series expansion of the v-function with respect to df~. dt

v(~, B, --~ df~) = vo(~,~) + --~-Vl(~, df~ fl) q - . - . .

(4.230)

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

505

By retaining only the first two terms of the above expansion, we arrive at the following dynamic model: 7[

f(t) =fo(t)+ L~-dodf~ ( f L 2

"-~

Vl(a, fl)~;afluo(t)dlzdfl)dq~,

(4.231)

~fl

where

fo(t) -

-do

vo(cc,fl)~3uo(t) d~ dfl dq).

(4.232)

It is clear that in the case of very slow input variations the second term in the right-hand side of (4.231) becomes negligible. Thus fo(t) can be construed as a rate-independent component. This means that the function v0(~, fl) should coincide with the v-function of the rate-independent model (4.67). In other words, the function v0(a, fl) can be determined by matching rate-independent first-order transition curves measured for unidirectional variations of the input ~(t). We next represent the model (4.231) in Cartesian coordinates. To this end, we shall use the following expressions: eo - ex cos q~ + ~y sin q~, (4.233)

df 0 =G df dt

dfx

9 ~-~ = cos q~~

dfy

(4.234)

+ sin q~ dt "

By substituting (4.233) and (4.234) into (4.231), after simple transformations we arrive at the following form of the model (4.231): .

-

--~ -- f (t) - fo(t),

(4.235)

where the matrix,4 is given by

fii_(?Zxx(FZ(t)) ?Zxy(~(t))) ?Tyx(~(t)) ~yy(~(t)) '

(4.236)

and the matrix entries are specified by the expressions: --

f

COS 2 q9

(If

Vl(a,

)

fl)9~3uo(t) da dfl dq~,

(4.237)

/z

--f

sin2 r ( / L

Vl(~Z,fl)7afluo(t)d~zdfl)dcP,

(4.238)

?Zxy(F~(t))- ?Zyx(~(t)) 7[

cos O sin q~

/z

2

Vl

>/3

(0~,fl)~flUo(t) de dfl)

do.

(4.239)

CHAPTER 4 Vector Preisach Models of Hysteresis

506

Thus the dynamic model (4.231) can be interpreted as a set of two coupled ordinary differential equations (4.235) with hysteretic coefficients (4.237)-(4.239). The expression (4.235) 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 Vl(a, t) by fitting the model (4.235) to some experimental data. The following experiments are used to solve this problem. We restrict ~(t) to vary along the direction r = 0, which means that ~(t) = -~xu(t). First, we assume that u(t) is made 'infinitely negative' and then it is monotonically increased until it reaches some value a 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 value f0~. Owing to symmetry, we have fo(t) - -~xfo(t),

f (t) - -~xf(t).

(4.240)

Thus, according to the model (4.234)-(4.239), the above relaxation process is described by the differential equation ~T~ ~d f

= f i t ) - foa

(4.241)

where "~a =

COS2 (p

(ff

V1(a~, fl') da t dfl'

- ffs+ ~ vl (a', fl')de ~dfl') do,

(4.242)

and geometrical configurations of S+~,r and S;, r are the same as in Fig. 4.28. The solution to Eqn. (4.241) is given by t

fit) -- (f~ - f o ~ ) e - ~

+ fo~.

(4.243)

Thus z~ 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 a. Afterwards, the input is ! monotonically decreased until it reaches some value fl at time t = t o and it is kept constant for t > t~. As the input is being kept constant, the output relaxes from its value f~/~ at t - t~ to its 'static' value f0~3. Owing to

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

507

symmetry, expressions (4.240) hold and the model (4.235)-(4.239) yields the following differential equation for the above relaxation process:

df =fit) - fo~fl,

(4.244)

-r~-d-i where 7C

~ COS 2 q)

~/~ --

2

(ffs

Vl(~Z', fl') &z' dfl'

_

~,fl,~o

~,/~,~

and the geometry of regions S + 0~,(p and S~,~ is the same as in Fig. 4.29. By solving (4.244), we find: t

f(t) = (f~/~-f0~/~)e ~ + f 0 ~ .

(4.246)

Thus, %/~ has the meaning of relaxation time and can be experimentally measured. We next show that by knowing relaxation times % and %/~ for all possible a and fl we can determine the function Vl (~, fl). To this end, we introduce the functions 1

q(~, fl)= ~(% - %/~),

(4.247)

P1 (~, fl) - I f

Vl (~, if) d~ ~dff, JJT (~,fl) where T(~, fl) is the triangle shown in Fig. 4.10. It is clear as before that P1 and Vl are related by the formula Vl(~, fl) - - ~2P1 (~' fl).

(4.248)

(4.249)

8~ 8fl Thus, if the function P1 (~, fl) is found, then the function Vl (~, fl) can be retrieved. However, from the computational point of view, it is more convenient to deal with the function PI(~, fl) rather than with Vl(~, fl). This is because the double integral with respect to ~ and fl in expressions (4.237)(4.239) can be explicitly expressed in terms of PI(~, fl) by using formulas similar to (4.71). Another advantage of using PI(~, fl) is that this function can be directly related to the experimental data (4.247). Indeed, by using the expressions (4.242), (4.245), (4.247), (4.248) and Figs 4.28 and 4.29, we derive

f2

n COS2 (PP1 (a c o s r

fl c o s ( p ) d o

= q(a, fl).

(4.250)

CHAPTER 4 VectorPreisach Models of Hysteresis

508

The expression (4.250) is the integral equation that relates the function P1 (a, fl) to the experimental data q(a, ]/). This equation is similar to the integral Eqn. (4.132) and, consequently, the same techniques can be used for the solution of Eqn. (4.250) as for Eqn. (4.132). Namely, by using the change of variables x = ~ cos q), 2 = ]//~, (4.251) Eqn. (4.250) can be reduced to the following Abel-type integral equation: fo ~

x 2 dx 0r2 P1 (x, ,,Ix)V/(x2_ x2 -- --~-q(~x,,~).

(4.252)

By using the technique discussed in Section 4.4, the following closed-form solution of the above equation can be found:

PI (O~,,~o0 -

1 d fo ~ s3q(s, ,~s) ds. ~0r2 d ~

(4.253)

~/0r _ s2

We next turn to the discussion of three-dimensional dynamic vector Preisach models of hysteresis. Similar to (4.231), these models can be represented in the following mathematical form

df~176 ( fl >~ vl(a,fl)%~uo~o(t)dadfl)

f_.(t) = fo(t) _. + foaZ~fO~ -dOq)-~-[x sin 0 dO do,

(4.254)

where f0 (t) represents a'static' component of hysteresis nonlinearities defined by the expression ?~

- f02~f0~eoq~-"( /f~)~ Vl(~, fl)~Uoq~(t) d~ d]/ ) sin 0 dO dq~. (4.255)

By using Cartesian coordinates, we have e0~o- ex cos ~osin 0 + ~y sin q~sin 0 + ez cos 0,

dfo, dt

= cos ~0sin 0~-~f; + sin q0sin 0~-~f~4- cos 0 dfz dt "

(4.256) (4.257)

By substituting (4.256) and (4.257) into (4.254), after simple transformations we find df - f ( t ) -f0(t),

(4.258)

4.6 DYNAMIC VECTOR PREISACH MODELS OF HYSTERESIS

509

where the matrix,4 and its entries are given by

~l-

(

?lxx(~(t)) ?Zxy(~(t)) ?Zxz(~(t))) ?Zyx(~(t)) ~yy(~(t)) ?lyz(~(t)), ?Zzx(~(t)) ?~z~(F~(tl) ?~zz(~(t))

(4.259)

?~xx(F~(t))= foar~fO~ cos 2 qosin 3 0 X (/~

~yy(~(t)) =

Vl(~ ,

~>~

f020 x ( /L

(4.260)

sin 2 qosin 3 0

x (/f~

~lzz(U(t))--

fl)~&~uo~o(t)dc~d~)dOdq0,

Vl(~x,fl)~fluo~o(t)doraft)dO dqo,

(4.261)

COS2 0 sin 0

vl (a, fl)~a~uom(t)do~dfl) dOdq~,

(4.262)

?~xy(~(t))= ?Zyx(~(t))- ~O2'~fO-~cos q~sin q~sin 3 0

axz (u(t)) -- azx (u(t)) --

x (ff~

Vl(~X,fl)~&BUOq~(t)&xdfl)dOdq~, (4.263) ~>B

f02f0

cos q~cos 0 sin 2 0 Vl(C~,fl)~BUOq~(t)&xd~)dOdq~, (4.264)

x (/f~ ~yz (~(t)) - ~zy (~(t)) =

f0210

sin q~cos 0 sin 2 0

x (/f~

Vl(C~,~)}~,Uoq)(t)&xd[3)dOd~0. (4.265) ~>~

Thus the three-dimensional dynamic vector Preisach model of hysteresis (4.254) can be interpreted as a set of three coupled ordinary differential equations (4.258) with hysteretic coefficients (4.260)-(4.265). We next proceed to the discussion of the identification problem for the model (4.258)-(4.265). The solution to this problem is very similar to

510

CHAPTER 4 Vector Preisach Models of Hysteresis

that for the model (4.235)-(4.239) 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 (4.258)-(4.265) are measured when the input is restricted to vary along the z-axis, that is, when ~(t) - -~zU(t). As before, two types of relaxation processes are considered. The relaxation processes of the first type occur when the input is made 'infinitely negative' and then monotonically increased to some value a and kept constant thereafter. The relaxation processes of the second type occur when, starting from the state of negative saturation along the z-axis, the input is first monotonically increased to some value a, 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, fl) defined by (4.247). On the grounds of symmetry, for both types of relaxation processes we have f0 - ~zf0(t),

f(t) - -~zflt).

(4.266)

This results in the reduction of the model (4.258)-(4.265) to the following equation

df

~zz (~(t)) ~ = fit) - fo (t).

(4.267)

Now, by using the same line of reasoning as in the derivation of Eqn. (4.250), we can show that the function P1 (cx, fl), defined by (4.248), is related to the experimental data q(a, fl) by the integral equation f0 ~ Pl(a cos 0, fl cos 0) cos 2 0 sin 0 dO =

q(a, 2~

"

(4.268)

By using the change of variables x = a cos 0,

;~ = -fl,

(4.269)

the integral Eqn. (4.268) is reduced to the form ~0 a x2p1 (x, ,~x) dx - -~-~ ~3 q(a, ,~),

(4.270)

from which we derive the final expression PI(X, ,~x) --

1 d [cx3q(or ,~or 27ca2 da

(4.271)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

511

4.7 GENERALIZED VECTOR PREISACH M O D E L S 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 Uq~(t). We begin with the first approach. To be specific, we shall use the nonlinear (input-dependent) scalar Preisach models. 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 two-dimensional isotropic models that can be represented in the following mathematical form

f (t) =

-

f (ff-~q~

v(~, fl, u~o(t))~Uq~(t) d~ dfl dq)

)

u,p(t)

4- ~(u(t))fi(t).

(4.272)

The numerical implementation of the above model is substantially facilitated by the use of the following function P(~, fl, u) - f f

v(~', fl', u) d~' dfl',

(4.273)

, I l K aflu

where R~/~u is the rectangle shown in Fig. 4.12. By employing the above function we can find explicit expressions for the double integral with respect to a and fl in (4.272). In addition, the function P(a, fl, u) can be directly used for the identification of the model (4.272). First- and second-order transition curves measured along any fixed direction (for instance, along the direction (0 -- 0) will be utilized for the solution of the identification problem. By using these curves, the following function can be constructed

1

F(a, fl, u) - -~(f~u - f~u),

(4.274)

where f~u and f~u have the meaning of output values along the first- and second-order transition curves, respectively. It can be shown that the function P(a, fl, u) is related to the experimental data (4.274) by the following integral equation 7~

f

~ cos q~P(a cos r 2

cos q~, u cos q~) dq~

F(a, fl, u).

(4.275)

CHAPTER 4 Vector Preisach Models of Hysteresis

512

This equation is very similar to the integral Eqn. (4.132). Thus, the same techniques can be employed for the solution of Eqn. (4.275) as for the solution of Eqn. (4.132). 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 akfl sum are eigenfunctions of the operator ti: 7~

AiP = f 2 cos q~P(~ cos ~0, fl cos ~o, u cos q~) d~o.

(4.276)

7~

2

In other words, we have A o~kfl s U m -- ,~k+s+mO~k fl s u m,

(4.277)

,~k+s+m -- /_~ (COScp)k+s+m+l dcp.

(4.278)

where: 2

Now, by expanding the right-hand side of the integral Eqn. (4.275) into the series of Chebyshev polynomials

F(a, fl, u) = y ~ a~qmT~(~)Tq(fl)Tm(u),

(4.279)

f.,q,m

we can represent the solution of the above equation in the form P(a, fl, u) = y ~ aeqmt~qm(a, fl, u),

(4.280)

~,q,m

where the polynomials t~qm(a, fl, u) are m a p p e d by operator A into polynomials T~(~)Tq(fl)Tm(u). Polynomials t~qm(a, fl, u) can be determined by using expressions (4.277) and (4.278). As far as expansion coefficients a~qm are concerned, they can be computed by using a formula similar to (4.146):

8 f0/0/0 F (cos 0, cos 0', cos

ae.qm = ~

x cos s cos qO~cos toO" dO dO~dO".

(4.281)

It is also possible to find a closed-form solution to Eqn. (4.275). To this end, the following change of variables is used: x = ~ cos q~,

2 = fl/~,

;~ = u/~

(4.282)

and the above equation is reduced to the Abel-type integral equation fo x v/cz2x_ x 2 P(x, 2x, ;~x) dx = ~F(~, ~ 2a, ;~a).

(4.283)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

513

Using the same reasoning as in Section 4.4, we obtain the following solution of Eqn. (4.283):

1 d fo ~ s2F(s, 2s, Zs) ds. P(~, 2~, Z~) - ~ d~ V~2 _ S2

(4.284)

As is seen from the above discussion, function P(~, fl, u) (and consequently function v(a, fl, u)) can be determined by matching the increments (4.274) between the first-order and second-order transition curves. The function ~(u) in (4.272) can be found by matching an ascending branch of a major loop. This leads to the expression ~(u) - f + + f u . 2u

(4.285)

We next proceed to the discussion of three-dimensional generalized vector Preisach models of hysteresis. These models can be represented as follows

f-"(t) -

/0270rt-dOq)(/[

v(~, fl, UOq)(t))~UOq~(t) d~ aft ) sin 0 dO dq0 uoq~(t)

+ ~(u(t))F~(t).

(4.286)

By introducing function P defined by (4.273) and by using experimental data (4.274) the identification problem for the model (4.286) can be reduced to the following integral equation 2re f0 ~ cos 0 sin OP(oecos O, fl cos O, u cos O) dO - ~(~,/~, u) "

(4.287)

The explicit solution of the above equation can be obtained by using the change of variables

u/~

(4.288)

f0~xP(x, 2x, Zx) dx - ~-~F(a, 0~2 2a, Za).

(4.289)

x = ~ cos O,

2 =///~,

Z=

and by reducing (4.287) to the integral equation

Differentiation of (4.288) yields 1 d [a2F(~ ' 2a, Za)]. P(~, 2~, Z~) - 2 ~ dc~

(4.290)

We next turn to generalized dynamic vector Preisach models of hysteresis. For the sake of notational simplicity, we consider only two-dimensional

CHAPTER 4 VectorPreisach Models of Hysteresis

514

models; extensions to three-dimensional models are straightforward. The two-dimensional generalized dynamic Preisach models can be defined as follows \

f (t) - fo(t) +

Vl (e, fl, Uq)(t))~BUq~(t) de dfl) do,

-b~o --~

u~o(t)

(4.291) where f0 is a 'static' component of hysteresis nonlinearity which coincides with (4.272). By using Cartesian coordinates, the model (4.291) can be reduced to the following ordinary differential equations df -. -. A~-~ =f(t) -fo(t),

(4.292)

where the matrix,4 has the following entries:

xxlO t t/ =

Vl (e, fl, U~o(t))~aBUq)(t)de dfl) dq~, (4.293)

COS 2 (p

u~o(t)

V1(e, fl, Uq~(t))~;~BUq~(t)de dfl) do, (4.294) 2

U~o(t)

?Zxy(~(t)) = ~yx(u(t)) = / _ ~ cos ~osin qo 2

X (ffR

uq~(t)

vl(e, fl, Uq~(t))~;aflUq)(t)dedfl)dq~.

(4.295)

To solve the identification problem for the model (4.292)-(4.295), we restrict input to vary along the x-axis. Then, on the grounds of symmetry, it is easy to conclude that f (t) = -Grit), fo(t) - -exfo(t). (4.296) This results in the reduction of the model (4.292)-(4.295) to the following equation

^ df[ =fit) -fo(t) axx(U(t))--d-

(4.297)

We next introduce the function Pl(e, fl, u): P1 (e, fl, u) - / f R

Vl(e',/J', u) de' dfl', aflu

(4.298)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

515

and experimental data: 1

Q(:r fl, u) = ~(%u - %flu),

(4.299)

where %u and r~flu are relaxation times for the processes, which are similar to those described in detail in the previous section. By using the expressions (4.298)-(4.299) and the same line of reasoning as in the previous section, it can be shown that the function P1 (cx, fl, u) is related to the experimental data (4.299) by the following integral equation

f

2 COS2 (PP1(~zcos (.p, fl cos q~, u cos qo) dq~ - Q(a, fl, u).

(4.300)

/c

By using the change of variables (4.282), the above equation can be reduced to the Abel-type integral equation f0 a

V/~x2X2_ x2P1 (x, ~x, )~x) d x

-

~X2 -~--Q(a, 2a, za),

(4.301)

whose solution is given by 1 d fo a s3Q(s, ,~s, zs) ds. P1 (0r 20r Z~x) - ~0r d~ ?~-2 _ s2

(4.302)

Finally, we shall discuss generalized anisotropic vector Preisach models. Again, for the sake of notational simplicity, we consider only twodimensional models. These models can be defined as /-r ..+

f(t)-L2(ffR

u,p(t)

~(oe, fl, cp, ur

d~ dfl d~p + ~(~(t)). (4.303)

The experimental data defined by the function -*

1 -*

fl, u,

-*

- i(f u ,

(4.304)

will be used for the identification of the model (4.303). As before, we introduce the auxiliary function

P(o~, fl, cp, u) = ffR

:r

~(~'' fl'' cp, u) da' dfl'.

(4.305)

CHAPTER 4 Vector Preisach Models of Hysteresis

516

We employ the following Fourier expansions: OO

P(a, fl, u, q)) -- ~

rJn(a, fl, u)e imp,

(4.306)

F-(a, fl, u, q))= y~ F-n(a, fl, u)e inc.

(4.307)

n~--OO OO

By using (4.304)-(4.307), it can be shown that Pn is related to Fn by the following integral equation cos

n~Pn(a cos ~, fl cos ~, u cos ~) d~ - Fn(a, fl, u).

(4.308)

This equation can be reduced to the Abel-type integral equation /0 ~ Pn(x, -" ,~x, Zx) ~rTn(X) 2 _ x 2 dx = ~1 Fn(a, ,~a, Za)

(4.309)

Various numerical techniques can be used for the solution of this equation. In particular cases when n --- 4-1, the following closed-form solutions can be found: -~ 1 d f ~ s2F+l(S, ,~s, Zs) P+I (a, 2a, Za) = ~a da ]0 2---7~/a - s2 ds. (4.310) As far as the function ~(~) in (4.303) is concerned, the following expression (similar to (4.285)) can be derived: ~(~pU) = f+~p +f~cp 2 '

(4.311)

where the notations in (4.311) 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)

= ,i-~12

-J4~

~fl

v(a, fl)~l~(t)lg[O(t) - ~b] dad]/ d~b,

(4.312)

where O(t) is the angle between ~(t) and the polar axis. In the case when g(O-q5) = cos(0-q)), the above model is reduced to vector Preisach models extensively studied in this chapter. This justifies the following' cosine-type'

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

517

constraints on the function g(~b): g'(~b)~<0 for 0~r g(0) - 1, g(~) = - 1 ,

g(2)

(4.313) = 0,

(4.314)

g ( ~ - ~b) = - g ( r

(4.315)

and the product Ifi(t)Ig(O- q5) can be construed as a generalized projection of vector input fi(t) on the direction specified by the angle r Function

1

g(r = I cos r (cos ~) is an example of a 'cosine-type' function that satisfies the constraints (4.313)-(4.315). In the above model, functions v(a, fl) and g(r 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. (a) First-order transition curves that are measured when the input fi(t) is restricted to vary along one, arbitrary fixed direction. By using these curves, we can introduce the function:

1

(4.316)

7(~,/~) = ~(f~ - f ~ ) .

(b) 'Rotational' experimental data measured for the case when the input is a uniformly rotating vector: fi(t) = {urn cos cot, um sin cot}. It can be shown that for isotropic hysteretic media, the output has the form f(t) =f0 +fl(t), where f0 does not change with time, whilef~ (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) =

fi(t) .fl(t)

Um

,

(4.317)

that has the meaning of the projection of fl (t) on the direction of input. Functions v(a, fl) and g(r will be recovered from experimental data (4.316) and (4.317). For the identification as well as computational purposes, it is convenient to introduce the function P(~, fl):

P(~' fl) --/fT (~,~) v(a', fl')d~' dfl',

v(a, fl) = -

O2p(~, fl)

.

(4.318)

518

CHAPTER 4 Vector Preisach Models of Hysteresis

By using the same line of reasoning as before, it can be shown that model (4.312) will match the experimental data (4.316) and (4.317) if functions P(a,//) and g(~b) satisfy the equations: 2

cos ~bP(eg(4~), ~g(~b)) d4~ = ~'(c~, ~),

(4.319)

-2

cos d~P(um, umg(d~)) dd~ = R(um).

(4.320)

Since v(e,/~) is nonzero only within the triangle r = {-e0~
.Jz'(O~,fl) "- y ~ n--0

y~ Uks-(2n+l)o~kfls. k+s=2n+l

(4.321)

Approximation (4.321) contains only odd terms because of the mentioned symmetry properties of ~'(a, ]/). Owing to these symmetry properties, it is also clear that Uks _(2n+1) -- --Usk ..(2n+1)9 An actual polynomial approximation (4.321) can be found by using a Chebyshev polynomial series expansion of ~(~, ]/) as described before. We shall also use the odd extension of R(um) from [0, a0] to [-a0, a0] and the following polynomial approximation: N

a(um)

(4.322)

= y ~ R2n+lU 2n+1 m 9 n--0

We shall next look for P(cr ]/) in the form N

P(a, fl) - y~ n=0

y~.

C:ks"(an+l)o~k~s.

(4.323)

k+s--2n+l

By substituting (4.321), (4.322) and (4.323) into (4.319) and (4.320) and by equating similar terms, we derive ..(2n+1) f ~ ..(2n+1) 2~'ks JO COS ~bg2n+l (~b) d~b = Uks

-2 ~ k+s=2n+l

_(2n+1)~07rCOS~gS (~b) d~b = Cks

(4.324) a2n+l.

(4.325)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

519

From (4.315) we find that the integrals in (4.325) are equal to zero if s is even and

f0

cos qbg s ( q5) d q5 = 2

/0

cos ~gS (~b) d~b

if s is odd.

(4.326)

Next, from (4.324) we obtain (2n+1) ao ' 2n+1 ~0 ~ COS ~g2n+l (~b) d~b. (2n+1) -- ~(2n+1) = 2 Cks ~0,2n+1

(2n+1) aks

(4.327)

From (4.325), (4.326) and (4.327) we derive . (2n+1) ~0 ~ cos ~g2n+l (~) d~b R2n+l -- -~c0,2n+l

-

~0 ~ cos c~gs (~b) d~b, 4 y ~ Ic(2n+1) ks

(4.328)

where Y-~' stands for the s u m m a t i o n over all s that are odd and less than 2n + 1 and such that k + s = 2n + 1. By using (4.327) in (4.328), we find .. ( 2 n + 1 ) _ 2 ~ , . , ( 2 n + l ) ~ a ~ 2 s n + l ) ) ( a ~ ) s ' ~ R2n+l -- -za0,2n+l ~'0,2n+1/ (2n+1) ~(s) ] ' \a0,2n+l COs / which leads to ~(2n+1) c0,2n+l - - _

., (2n+1) R2n+l q- za0,2n+l _(2n+1) (s) " ) ( a6s~ 2 ~ ' ( Uks _(2n+1) (s) ,] UO,2n+l COs-

(4.329)

(4.330)

By assuming that C0,1(1) 1 = 2,

(4.331)

where 2 is some u n k n o w n constant, from (4.330) we derive c(2n+1) _ ~h(2n+l) (4.332) 0,2n+1 . . . . 0,2n+1" ~(2n+1) According to (4.330), constants u0,2n+l can be recursively determined by using the formula ~,(2n+1) u0,2n+l - - _

_(1) and the fact that ,o0,1

_

1.

,. (2n+1) R2n+l -}- za0,2n+l

2E'(a(2n+l))(aoS)s ' k s "~ ..(2n+1) (s) u0,2n+l bos ,]

(4.333)

CHAPTER 4 Vector Preisach Models of Hysteresis

520

After h(2n+l) ~'0,2n+1 are computed, from (4.326), (4.327) and (4.332) we find f0 z~COS ~bg2n+1 (~) d~b - $2n+1 2 '

(4.334)

(2n+1) a0'2n+l $2n+1 -- ~(2n+1) " u0,2n+l

(4.335)

where

According to (4.313)-(4.315), there exists a function z(g) which is inverse to g(~b). By using this function, we shall use the following change of variables in integrals (4.334): ~b = ;~(g),

dq~ = Z ( g ) d g ,

z(g(~b)) = ~b.

(4.336)

By substituting (4.336) into (4.334) and integrating by parts, we transform (4.334) as follows: f_ +1 g2n sin )~(g) dg = 2(2n $2n+1 + 1)" 1

(4.337)

We shall next make another change of variables: g=cos~,

dg=-sin~d~,

(4.338)

which leads to f0 z~COSan ~ sin

6T(6) d~

=

$2n+1 ;~(2n + 1)

(4.339)

where T(6) - sin Z(cos ~).

(4.340)

We shall extend T(~) from [0, ~] as a periodic odd function with half-wave symmetry. As a result, this function can be expressed as N T(~t) - E t2n+l sin(2n + 1)~, n=0 where

2f0 T(6)

t2n+l -- -7~

sin(2n + 1)~ d~.

(4.341)

(4.342)

4.7 GENERALIZED VECTOR PREISACH MODELS OF HYSTERESIS.

521

It is known (see [13]), that

sin(an+1)~_sin~[a2ncos2n~_(n-1)a2n-2 1 COS2n-2 if/ (n--3)22n-4cos2n-4d/

+ 2

--(n--4)aan-6cosan-6~4-...]

3

where (Pk) --

(4.343)

P! . From (4.339), (4.342) and (4.343), we derive k,(p-k)

t2n+l

1[

tn-)aan_2.S2n-1 2

~ 22n

2n 4- 1

1

2n- 1

4-(n-3) $ 2 n22n-4" - 3 2 an-3 --

(n-4) $ 2 n22n-6 - 5 3"2-dn~-511 {- . . . . ~s

(4.344)

From (4.314), (4.336) and (4.340) we find T(2)-sinz(0)-sin(2)=1.

(4.345)

From (4.341), (4.344) and (4.345) we obtain N , ~ - ~--~(--1)ns n--0

(4.346)

Now, the identification procedure can be summarized as follows. First, we find polynomial approximations (4.321) and (4.322). Then, by using coefficients , a2n+l and (4.333), we recursively compute b(2n+1)0,2n+1and $2n+1. Next, by using (4.344) and (4.346), we determine s ,~ and t2n+l. By using (4.327) and (4.332), we compute C ~(2n+1) ks 9 Finally, by using (4.341), (4.340) and (4.336), we compute T(q), ;~(g) and then retrieve the inverse function g(~b). In the particular case when

Uks"(2n+l)

g(~b) = [cos ~p[1/nsign(cos ~)

CHAPTER 4 Vector Preisach Models of Hysteresis

522

the solution of the identification problem can be simplified. Indeed, in this case, the integral Eqn. (4.319) can be written as follows: F(a, fl) = 2

f0~cos (~)P(a COS1/n t~, fl COS1/n r

d~b.

(4.347)

By introducing the following change of variables, ,~ =

X = a COS1/n t~,

fl/a,

(4.348)

and by using the same transformations as before, the following integral equation is obtained from (4.347):

a(a) -

fo ~

(4.349)

N(x) dx, v/a2n _ x 2n

where the following notations are adopted:

N(x) = x2n-lp(x, J,x),

(a2/2n)F(a, ,a.a).

R(a) =

(4.350)

Multiplying both sides of Eqn. (4.349) by a 2n-1/v/S 2n -- a 2n and integrating with respect to a from 0 to s, and then interchanging the order of the double integration for the right-hand side, the following result can be derived: ~0 s

a2n-1

/S2n _ a2n

R(a) da

/ o sN(x) (/x lf0S

an

2n-ld

x/(san _ a2n)( a2n _ xan)

N(x)(~x s ~/( san-xana2 )

)dx

d(a 2n) -

-

(a 2n

(s2n+x2n)2 ) dx. (4.351) 2

Then, by introducing the following variable of integration,

W - a 2n -- (S2n if- x2n ) / 2 ,

(4.352)

we obtain fO s

a 2n-1 R(a) d a /S2n _ a2n

1loS N(x) t f ~n

(s2n-x2n)/2

dw

"J(xRn-s2n)/2 V/( s2n-xan ) _ W2 ) dx

~f0 s N(x) dx. 2n

(4.353)

4.7 GENERALIZEDVECTOR PREISACH MODELS OF HYSTERESIS.

523

By using the notation of (4.350), we obtain

P(o~, 200

1 ['~ nsn-lF(s, 2s) + s n (d/ds)F(s, ,~s) I"C

Jo

ds.

(4.354)

V/~2n _ s2n

From Eqn. (4.354) 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 result (4.354) coincides with the final result for the case of the identification of vector models discussed in Section 4.4. 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-squares' approximation. Next, we shall illustrate the above discussion by the following experimental identification and testing of the model (4.312). The identification of the model has been performed for a typical Ampex-641 (7Fe203) 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 was measured (see Fig. 4.35). Then, the unknown function P(a, fl) was found as a function of n. Next, the set of 'rotational' experimental data were measured. This was 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 reached a certain positive value where it was kept constant while sample (in-plane) rotation was introduced. This was repeated for different positive field values and the o u t p u t / i n p u t phase-lag relationship as a function of the rotating input amplitude was obtained. It was 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 o u t p u t / i n p u t phase-lag values is shown in Fig. 4.36 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 was 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 was carried out as follows. First, the input field was restricted to the y-axis and was increased from some negative

CHAPTER 4

524

Vector Preisach M o d e l s of Hysteresis

0.06

0.04

3' E .~.

0.02-

r

.9

0.m

(1) E o)

-0.02

--

r

- i

-0.04

-0.06

I , , , I , , , I , , , I , , , I , , , I , -400 -200 0 200 400 600

-600

Magnetic

field

[Oe]

F I G U R E 4.35

60

I

I

t

I

I

I

I

I

I

I

I ,I

I

I

I

I

I

I

I

I

Experiment

(/)

o o

_

,

....

Model

when

n=l

Model

when

n=3

-

I I \X

o

40

__

/

~ \

E~r) _.1

--

6ffl

--

e.12.

--

II

\\\

II

\ \

II / ~

\ x x

"5 20 e~ E Q

//

d--,

0

--

d/

0 I

0

I

I

I

t

t

0.2

t

I

i

i

t

0.4

Rotating

I

0.6

field amplitude

F I G U R E 4.36

i

i

t

I

0.8 [KOe]

t

I

I

1

4.7 G E N E R A L I Z E D VECTOR PREISACH MODELS OF HYSTERESIS.

0.06 ~, E .o

I:' '~'r"'i'r'''

I ''

''

I~

r l ' '-i'~: !

0.04 0.02

0n -0.02

.

.

0

.

.

.

0.5

.

.

.

.

.

.

.

.

.

.

.

.

.

1 1.5 2 Orthogonal Field H x [KOe]

2.5

0.04

O E 5;

"11

0.02 0~

~~'e=~--'~-

'~'~"~'"-"~

- ~

_oo~;,,,, i i~-i i ~i i,,,,,, ii ii, ,l 0

0.5

1

1.5

2

2.5

Orthogonal Field H x [KOe] 0.01 0 E -0.01 :~ -0.02 -0.03

,...,. ,_.I. ,_, _ L , I , _ _ , , -0.04 ----__,., J~_!. ,., j_j_l, 0 0.5 1 1.5 2 Orthogonal Field H x [KOe] 0.02

'' E 0

J'l'

~'~I

,-'2.5

" " '-"'-~[') i--i-'l ~-~ I ~ .-4

~, E 9~. -0.02

~; -0.04 -0.06 0

0.5

1 1.5 2 Orthogonal Field H x [KOe]

[] Experiment

Z~ n = 1

n=3 F I G U R E 4.37

2.5

525

526

CHAPTER 4 VectorPreisach Models of Hysteresis

value (sufficient to drive the sample into the negative saturation state) to some positive value Hy+, and then decreased to zero. This resulted in some remnant magnetization Myr. Then, the magnetic field was restricted to vary along the x-axis. By increasing the field value (now Hx) gradually from zero to very high values, the orthogonal remnant component Myrw a s reduced and the Myr versus Hx relationship was recorded. This experiment was repeated for different values of Hy+ (and consequently different values of Myr). Then by using the model (4.312) the computational results were 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. 4.37. It is clear from Fig. 4.37 that the model (4.312) 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 an additional unknown function g(~0), which can be determined from the identification process. As a result, the opportunity appears to incorporate more experimental data in the identification process, leading to a more accurate model. The presentation of the material in this chapter is largely based on publications [17-27].

References 1. F. Preisach, Uber die magnetische nachwirkung, Zeitschrift fiir Physik 94, 227-302 (1935). 2. E.C. Stoner and E. P. Wolhfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Philosophical Transactions of the Royal Society of London Series A-Mathematical and Physical Sciences 240 (826), 599-642 (1948). 3. I.A. Beardsley, Modeling the record process, IEEE Trans. Magn. 22 (5), 454-459 (1986). 4. T. R. Koehler, A computationally fast, two-dimensional vector hysteresis model, J. App1. Phys. 61 (4), 1568-1578 (1987). 5. G. Bertotti, Hysteresis in Magnetism, Academic Press, New York (1998). 6. J. C. Slonczewski, IBM Research Memorandum, No. RM 003.111.224 (1956). 7. M. Prutton, Thin Ferromagnetic Films, Butterworth, Washington, DC (1964). 8. G. Friedman, New formulation of the Stoner-Wohlfarth hysteresis model and the identification problem, J. Appl. Phys. 67 (9), 5361-5363 (1990).

REFERENCES

527

9. M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis, Nauka, Moscow (1983). 10. J. A. Barker, D. E. Schreiber, B. G. Huth, and D. H. Everett, Magnetic hysteresis and Minor loops-Models and Experiments, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 386 (1791), 251-261 (1983). 11. A. Damlamian and A. Visintin, A multidimensional generalization of the Preisach model for hysteresis, Comptes Rendus De L Academie Des Sciences Serie I-Mathematique 297 (7), 437-440 (1983). 12. C. Lubich, Fractional linear multistep methods for Abel-Volterra integral-equations of the 1st kind, IMA J. Numer. Anal. 7 (1), 97-106 (1987). 13. I. S. Gradstein and I. M. Ryzkik, Tables of Integrals, Series and Products, Academic Press, New York (1980). 14. A.W. Joshi, Elements of Group Theoryfor Physicists, Wiley Eastern, New Delhi (1982). 15. N.J. Vilenkin, Special Functions and the Theory of Group Representation, American Mathematical Society, Providence, RI (1968). 16. G.Y. Lyubarski, Applications of Group Theory in Physics, Pergamon Press, Oxford (1960). 17. I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, Berlin (1991). 18. I.D. Mayergoyz, Mathematical-Models of hysteresis, IEEE Trans. Mag. 22 (5), 603-608 (1986). 19. I. D. Mayergoyz and G. Friedman, Isotropic vector Preisach model of hysteresis, J. Appl. Phys. 61 (8), 4022-4024 (1987). 20. I. D. Mayergoyz and G. Friedman, On the Integral-Equation of the vector Preisach hysteresis model, IEEE Trans. Mag. 23 (5), 2638-2640 (1987). 21. I. D. Mayergoyz, Vector Preisach hysteresis models, J. Appl. Phys. 63 (8), 2995-3000 (1988). 22. I. D. Mayergoyz, Identification problem for 3-D anisotropic Preisach model of vector hysteresis, IEEE Trans. Magn. 24 (6), 2928-2930 (1988). 23. A.A. Adly and I. D. Mayergoyz, A new vector Preisach-type model of hysteresis, J. App1. Phys. 73 (10), 5824-5826 (1993). 24. I. D. Mayergoyz and A. A. Adly, A new isotropic vector Preisach-type model of hysteresis and its identification, IEEE Trans. Magn. 29 (6), 2377-2379 (1993).

528

CHAPTER 4 VectorPreisach Models of Hysteresis

25. I.D. Mayergoyz, 2D vector Preisach models and rotational hysteretic losses, J. Appl. Phys. 75 (10), 5686-5688 (1993). 26. A.A. Adly and I. D. Mayergoyz, Accurate modeling of vector hysteresis using a superposition of Preisach-type models, IEEE Trans. Magn. 33 (5), 4155-4157 (1997). 27. A. A. Adly, I. D. Mayergoyz, and A. Bergqvist, Utilizing anisotropic Preisach-type models in the accurate simulation of magnetostriction, IEEE Trans. Magn. 33 (5), 3931-3933 (1997).