HYSTERESIS MODELLING

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HYSTERESIS MODELLING

D. Pescetti

To cite this version:

(2)

JOURNAL DE PHYSIQUE

Colloque C8, Supplbment au no 12, Tome 49, dkcembre 1988

HYSTERESIS MODELLING

D. Pescetti

Istituto di Fisica dell'univ., P. Ie Kennedy, 1-16129 Genoa, Italy

Abstract. - A generalization of the Preisach representation including both irreversible and reversible components of magnetization is presented. The Mayergoyz's theorem is extended to the generalized Preisach representation, and the effect of the shape-demagnetizing-field-transformation on the relative weight of reversible and irreversible components of magnetization is considered.

A scalar hysteretic system can be considered as a hysteresis transducer [I] whose output y (t) not only is a function of the present value of the input x (t)

,

a t any given time t, but it is also a function of the input's past values. Let xo be the value of x corresponding to a state independent of the past history of the system (for example, the saturation state, f x., in magnetic hysteresis). Let the input x (t) be subsequently and monotonically varied between xo, xi,

...,

3,. For static hysteresis transducers HT, the speed v d a t i o n of the input x (t)', has no influence on y. Output y is a function only of the sequence of input extrema (so,

...,

x,,)

.

The Preisach model P M [I-51 of a HT is defined by an assembly of hysterons +ab, with a weight function cp (a, b)

.

The hysterons are the elementary hysteresis operators characterized by a rectangular loop in input- output plane. The critical values a and b correspond to upward and downward jumps. The output T a b % (t) of these operators assumes only two values (+1 and -1 in magnetic hysteresis); more precisely qabx (t) is equalto -1, if x

5

b;f 1, if b

<

x

<

a ; + 1 , if x

2

a. The output y (t) is then given by

Y(t)=

11

cp (a, b) %bx ( t ) da db (1) -z.<b<a<o.

We shall call Preisach hysteresis transducer (PHT) a HT whose input-output relations are exactly described by the PM.

The generalized Preisach model GPM of a HT can be defined as an assembly of hysterons

kb

with a weight function cp (a, b)

,

and an assembly of degenerate hysterous (a = b) with a weight function f (a) [5, 61. The output of the GPM is then given by Y ( Q =

/

J

cp (a, b) Taax ( t ) da, db

- x e < b < a < z s

r x .

+

Lh

f

(a) Taax (t) da. (2) We shall call generalized Preisach hysteresis transducer (GPHT) a HT whose input-output relations are exactly described by the GPM. The class of GPHT includes as a subclass the PHT.

Let us consider the following properties of y us. x

Property B (congruency property): all hysteresis loops corresponding t o the same extremum values of input are congruent in the geometrical sense.

Property C (generalized congruency property): all hysteresis loops having the same input-amplitude are

congruent in the geometrical sense.

Thwwm 1 (existence of the generalized Preisach representation GPR): properties A and B constitute necessary and sufficient conditions for a HT t o be r e p resented by the GPM.

Theorem 2 (effect of the shape-demagnetizing-field- transformation): let (HT)

'

be the hysteresis transducer obtained by submitting a GPHT to transformation

where k is a constant. The hysteresis loops on plane (x', y') do not satisfy in general property B. (HT)

'

is not, in general, a GPHT.

Theorem 3 (shape-demagnetizing-field-

transformation and invariance of existence of GPR): properties A and C constitute necessary and sufficient conditions for a HT t o be represented by a GPHT also when it is submitted t o the linear transformation [3]. Theorem 4: a GPHT satisfies property C if and only if its weight functions have the form: cp (a, b) = cp (a

-

b, 0) ; f (a) = constant.

Theorems 1-4 are an extension t o GPHT of theo- rems previously established for the PHT. For PHT, theorem 1 has been proved by Mayergoyz [I], and theo- rems 2-4 by Arca, Grandis and Pescetti [7]. The esten- sion t o GPHT derives from the following arguments: i) the output y of a GPHT is given by y = yrev+ Yirrev, l/irre., and yrev being the outputs of the assem- blies of hysterons and degenerate hysterons (a = b ) fe- spectively; ii) yirrev is described by the PM; iii) y,,, satisfies properties A and B; iv) yrev satisfies property C if and only if f (a) =constant.

We shall define ACHT a HT whose y us. x relation- ship satisfy properties A and C. Clearly, the ACHT are a sublass of the GPHT.

curves. _{Let us consider the function ya} ₌_{y (so}₌_-x,,

Property A (wiping out property): let zi be: z, = XI = a ) representing the ascending limit cycle, and x, - xi-r. A z cancels the effects (on the final value the family of paths gab = y (xo = -xs, XI = a, 2 2 = of y) of the preceding z which have smaller or equal b), -2.

5

b

5

a

5

XS, representing the first order absolute value. descending reversal curves. Let us define the Everett

(3)

C8

-

1924 JOURNAL DE PHYSIQUE function by

F (a1 b) = (ya

-

yab) /2. (4) The Everett function F (a, b)

,

defined by equation (4),

lumps together the effects of both irreversible and reversible components of the output.

Everett function, F (a, b)

,

and Preisach functions,

cp (a, b) and f (a), are related by

p (a, 6) =

-a2F

(a, b) l a a ab, (5)

~ ( a , b ) = [ l ~ c p ( a , ~ ) d ~ d a + [ f b ( o ) d a . (7)

Let us call Rayleigh hysteresis transducers RHT a GPHT such as: cp (a, b) = constant =s; f (a) = constant =r. The RHT are a subclass of the ACHT. For RHT, the Everett function has the analytical ex- pression

F(a,b) = s ( a

-

b ) 2 / 2 + r ( a - b). ₍₈₎ The internal field Hi, inside a magnetized sample, is not equal to the external field He applied to it. For special geometries (ellipsoids) with magnetization M uniform through the (homogeneous) sample, the internal field may be expressed as Hi=He+Hd, where Hd is the shape demagnetizing field, i.e. Hd= -NM, N being a constant called demagnetizing factor. Let M, be the value of M corresponding to the saturation, and Hc the coercive force. Let us put: x = Hi/Hc, x' = He/Hc, y = M/M., y' = M1/MS. The shape- demagnetizing-field transformation is the linear transformation (3), with k = NMs/Hc.

Let us submit a RHT to transformation (3). By theorem 3, (HT)

'

is a ACHT. The Everett function F' (a', b') of (HT)

'

can be found by equations (3, 4). Lengthy but straightforward calculations give

F' (a', b') = (2ks (a'

-

b')

+

2kr

+

1-

-

[aka (a'

-

6')

+

(1

+

2kr12] I")

/

(4k2s)

.

(9) By use of equations (5), (6) and (9) one finds the Preisach functions

cp'

(a', b') and f '

($)

of (HT)',

2 -3/2 (a', b') = s [4ks (a'

-

b')

+

( I

+

2kr)

]

_,

(10)

f' (a') = r/ (1

+

2kr)

.

(11) The ratio G =

/

(AY)~,,, is a function of the particular monotonic y vs. x curve along which (Ay),, and ( A Y ) ~ ~ are measured or calculated. For RHT, G oc (Ax)-

.

Let O ~ O be the magnetization's change from = -1 corresponding to the nega-

I I

tive saturation, t o yl(zO = -2,-k, x l = xs

+

k) = 1 corresponding t o the positive saturation, along the ascending limit cycle. Let

the reversible and

netization's change, Ay; =

irrev For RHT, one has

(a&)

=4r (x.+k)

/

(1

+

~ P T ) ,

rev (12)

=4sx:/ (1

+

2kr)

irrev (13)

and in terms of M and H, the ratio G ( N ) =

(AM:)

/

(AM:) is gjven by (introducing

xr

= rev irrev

rM,/Hc and c = SM~/H?)

G

(N)

= (Xr/cHs) (1

+

NMs/Hs) =

= G (0) (1 f N M s / H s )

.

(14) Therefore equation (14) indicates that the reversible component increases with N.

The behaviour of any real HT can be simulated, to a first approximation, by an ideal GPHT. In general to simulate a real HT we can impose on the LLequivalent" ideal GPHT only some conditions because not all the experimental y vs. 3; curves of the real HT are consis-

tent with the GPM. Theorems 2-4 hold true for positive and negative values of k. The interest of transformation (3) is not restricted to the description of shape demagnetizing effects. Our theoretical analysis suggests that by a linear feedback transformation (3)

( h

>

0, negative feedback; k

<

0, positive feedback), with a proper choice of k, and a proper choice of the Everett function, it is possible to improve the match- ing between the y us. x paths of the real HT and of the "equivalent" ideal feedback GPHT. Equation (14) is valid for a feedback RHT. It is well known that real magnetic materials are described by a RHT only at low magnetization. However, let us remark that the concepts of RHT and feedback RHT are certainly use- ful for getting rapid estimates about the behaviour of real HT in the whole domain of hysteretic phenom- ena, from -M, to +Ms. For istance, in some cases one can impose to the "equivalent" feedback RHT the same values of H,, of H,, of M, and of remanent mag-

netization M,, as the real HT.

One final remark. The transformation (3) has a relevant role in the mathematical modelling of three- dimensional vector hysteresis because, in this case, the demagnetizing effects can never be avoided.

[I] Mayergoyz, I. D., Phys. Rev. Lett. 56 (1986) 1518.

[2] Preisach, F., 2. Phys. 94 (1935) 277. [3] Nkel, L., Cah. Phys. 12 (1942) 1.

[4] Everett, D. H., Trans. Faraday Soc. 48 (1955) 1551.

[5] Biorci, G. and Pescetti, D., Nuovo Cimento 7

(1958) 829.

[6] Janssens, N., IEEE Bans. Magn. MAG-13 (1977) 1379.