IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 5, SEPTEMBER 1990 2837

COMPARISON BETWEEN VARIOUS HYSTERESIS HODELS AND EXPERIAENTAL DATA.

F.OSSART * - G.MEUNIER * * * CENG/ LETI/SMEM 85X 38041 GRENOBLE CEDEX ** LEG (CNRS ,URA 3 5 5 ) ENSIEG BP46 38042 SAINT MARTIN D'HERES FRANCE

Abstract : We propose a comparison between results predicted by 4 mathematical models of hysteresis and experimental data measured on a CoNiCr alloy for longitudinal recording. Two of those models are simple analytical formula and can be used in a first approach of the phenomena, but are deficient to model the behavior of minor loops. Hore sophisticated models are analysed: the first assumes a differential equation between B and A, the other is derived from the Preisach model. We give some results showing that, for the material we studied, the latest model gives the best fitting to experimental data.

INTRODUCTION

Many models of magnetic hysteresis exist in the literature. Some of them exhibit very exciting properties, but they have not been applied yet to various materials and compared. In this paper, we analyze two mathematical scalar models which give good results. After a summary of the principles and the properties of the models, we will compare the experimental behavior of a hard magnetic material to those predicted by the models.

PRINCIPLES OF TEE MODELS

Analytical models We will describe here two very simple

models. The first was used by Potter to model the writing in magnetic recording[l]. The magnetization curves are represented by the formula :

M(H;a) =

M, (sgn a - a [ l + tanh (Hc - H s m q x tanh-l s 1 ) Hc

where sgn is the signe function, Hc the coercive field, Ms the saturation magnetization, S-Mr/Ms the squareness parameter, and Mr the remanent magnetization. After each extremum Hm of the field, the change of magnetization curve is made by calculating a new value of a given by :

a' =

2 sgn a - a ( 1 + tanh [ ( 1 - H, sgn a/Hc)tanh-l S ) I

1 + tanh [ ( I + H, sgn a/HC) tanh-l S I

we propose a simpler expression for the magnetization curves. The experimental data used are the field and the magnetization at the closure point of the major loop, the remanent magnetization and the coercive field. This model assumes a hard material with a zero slope after the closure point.

---&-r I

Fig.1 : the "polynomial" model.

The major loop is approached by the polynomial functions P1, P2 and P3 respectively of degree 2,2 and 3 . The minor loops are deduced from the major loop by homothetic transformations (Fig.1).

It is obvious that such analytical models cannot be accurate, nevertheless it may be useful to know how far they are from the experimental behavior.

Bodgdon's model This model [21 assumes a constitutive

relation between H and B given by the differential equation

dH/dB = a.sgn(dH/dt).[f(B)-HI + g(B) (3)

In this relation, a is a real constant, sgn is the sign function, f and g are material functions and must obey some conditions of regularity. With a proper choice of U and of those functions, it is possible to describe hysteresis loops of various ferromagnetic materials. Hodgdon proposed a formula using exponential and tangent functions for f and g. The experimental data used are measured on the major loop, as shown on Fig.2.

Fig.2 : experimental data for Hodgdon's model.

The most interesting properties of this model concern the behavior of the minor loops [3]. If H oscillates between two values Hmin and Hmax, the resulting loop moves towards a stable loop independent of the initial state. This behavior is experimentally observed on some materials and is known as the accomodation phenomenon. Furthermore, infinitely small stable cycles are localized on the graph of the function f, thus representing the anhysteretic curve.

0018-9464/90/0900-2837$01 .OO O 1990 IEEE

2838

Mayergoyz's model : This model is much more complex than the

former ones. It is an extension of the Preisach model developped by Mayergoyz I 4 1 and Friedman. It is a weighted sum of elementary hysteresis fully reversible step Fig.3) according., to the

The description of the functions p ( H,a, 6 ) and determined experimental

'4 r t r

cycles y a p and of operators A, ( see formula

lr e l m n t q cycle operetor totally revenlble operator

Fig.3 : definition of the operators y a e and A,.

We will recall here only the most important properties of this model, from the point of view of the user : 1. The model stores the extreme values of the applied field, and reproduces the erasure of past extremum values of the field by fields of larger amplitude. 2. Minor loops are stable. 3 . The identification requires the first and second order reversal curves defined on Fig.4. Those data are directly used in a formula equivalent to ( 4 ) but it requires only arithmetic operators and without any integration [41.

This model is able to represent a nonzero slope after the closure point, so we will apply it to the flux density B instead of the magnetization M.

Fig.4 : definition of the first' and second order reversal curves.

COMPARISON BETWEEN EXPERIMENTAL AND COMPUTED LOOPS

Identification Simulations of the behavior of a CoNiCr

thin film for longitudinal recording were performed with those models. When one fits the models to experimental data, different problems arise. Hodgdon's model is very sensitive to measurement errors, especially for the slopes at the closure point and

beyond it and to the value chosen for a. The main difficulty of Mayergoyz's model, lies in the large number of experimental measurements, but the measurements to do are well defined and it is an important advantage. We made various tests to compare the behaviors predicted by the models and the experimental one. In all the plotted curves, the flux density is expressed in Gauss and the field is in Oersted.

major loops The major loops given by each model

were plotted on Fig.5. As the Mayergoyz model uses the experimental major loop as data and perfectly reproduces it, only the results of the models is plotted. The figure shows that there is no need to use complex funtion to reproduce the major loop. The polynomial functions give better results than hyperbolic tangent functions or than the Hodgdon's model.

Demagnetization of the sample By demagnetization of the sample in an

AC decreasing field, we measured the flux density at each turning point. The comparison with the models is plotted on Fig.6.

Behavior of minor loops We observed the stability of minor loops

(see Fig.7). The alloy studied doesn't exhibit the phenomenon of accomodation and Mayergoyz's model reproduces its experimental behavior. The minor loop predicted by Hodgdon's model moves toward a stable loop not far from the experimental one, but those predicted by the analytical models do not stabilize before the major loop.

DISCUSSION

Those tests and others show that a simple analytical model can give good results if used with an applied field alternatively positive and negative. It may be useful in a first approach of a problem with hysteresis, but more sophisticated models are to be considered to reproduce properly the behavior of minor loops.

The flux density predicted by Hodgdon is lower than the experimental one, except in the region of the demagnetized state. Mayergoyz's model is no longer so accurate near the demagnetized state : one is far from the experimental configuration.

2839

15000

5000

-5000

-15000

B

-2000 -1000 0 1000 2000

B

L

-2ooo -1000 0 1000 2000 -2ooo -lo00 0 1000 2Ooo

Fig.6 : demagnetization of the sample,

Due to the fact that it uses much more data, Mayergoyz's model follows better the experimental behavior of the sample than Hodgdon's model. But other arguments are to be considered before choosing one O K the other model.

Preisach model and its generalisation are not simple to understand, nor to implement : the experimental data needed are very numerous and delicate to interpolate and the control of the memorized turning points must also be done properly. But the final formula is simple to compute. It requires only "plus" and "minus" operators, and the results fit well experimental behavior. On the other hand, Hodgdon's model is much simpler to understand and to implement, but it requires a numerical integration and the identification is partially empirical in the choice of the parameter a and of the slopes ,uCl and ,us.

curves of the turning points.

CONCLUSION

We presented various models of magnetic hysteresis and applied them to a CoNiCr sample of magnetic media. Very simple analytical models are found to be "not SO bad". Hodgdon's model is interesting to model the accomodation of minor loops, and is worth being tried with other materials.The model developed by Mayergoyz can be seen as a method to interpolate hysteresis loops from the major loop and reversal curves. It is complex but is the only one to really fit the experimental behavior of our material.

REFERENCES

[I1 R.I.Pottar and R.J.Schmulian, "Self-consistently computed magnetization pattern8 in thin magnetic recording media", IEEE Trans. Magn. MAG-7, No.4, pp.873-879, december 1971.

[21 M.L.Hodgdon, "Application of a theory of ferromagnetic hysteresis",IEEE Trans. Magn. MAG-24, No.1, pp.218-221, January 1988.

[3] B.D.Coleman and M.L.Hodgdon, "On a class of constitutive relations for ferromagnetic hysteresis",Arch. Rational Mech. Anal., 99, No.4, pp.375-396, 1987.

[4] 1.D.Mayergoyz and G.Friedman, "Generalized Preisach model of hysteresis", IEEE Trans. Magn. MAG-24, No.1, pp.212-217, January 1988.

Fig.7 : behavior of minor loops.