USING THE MEAN FIELD MODEL TO ANALYZE THE INFLUENCE OF TEXTURE ON THE HYSTERESIS BEHAVIOUR OF SILICON STEELS

A critical study of the Jiles and Atherton Mean Field Model was done. to determine the validity of the modal, a tool in describing and understanding the magnetization process in textured silicon steels.

proposed for paramagneti materials and lheefore does not describe hysteresis.
Before introducing the Langevin model, it is cenvenient to introduce the following shorthand notatien: Le(x) is the Lanevin functien and is defined as: L(x) com(x) (X/x) We shall now define successive derivatives of the Lansevin functions as follows: If we define M at saturation u M,, thin the I.angevin equation for magnetization is: where A is constant dcpcnding on the msnetic lOeies of the materiaL In fact A is kThn whcrc m is the magnctic mamcnt.T is the tanpcraturc in Kclvin and k s is thc Boltnnann's constant (1.38 x 10 "a J/K).
Let us now madify the Langcvin model in two ways by assuming that the magnetization is affected by the magnetization of ncighbouring grains as well as the applied external field and by assuming that the magnedzatim phenomenon also involves irreversible steps.
Let us discuss and justify these two assumlxions; the magnetic dnains are nat enly affected by the external field strength, but also, er y, by the magnetism of nearby magnetic damains.Furnnot, neighbon8 domains are physically uple through the domains walls.Thus, the resultant magnetization is not only a function of the applied field strength but also of the bulk magnetizatim of the specimen.In ether words the resultant magnetizatkm is a funcdm of a modified field instead.If we want the simplest relationship des the modified external field, then it will be of the form Cs*M + C*H where CI and Ca are two Domain movement and rotation involve an energy loss.Let us define a general resistance R that magnetizatien, always mntraty to lhe change in magnetizatim.If we want a very simple re2ttiamhip then we let this resistanee be pmportienal to the incremental magnetic susceptib'dity.The istance term then beeemes: Thus we get a modified Langevin model of the form: M M. Lm(C*M + C*H) -Cs dM/dH Such an equatim be very useful to us for studying magnetic materiab.Jules and Athertcm proposed such an equaticm assuming that the mcrgy, which is supplied to the specimen during a magnen process, is split into two parts.One part of the enersy is dissipated against pinning and the other pa is responsible for eversible changes in magnclizatim.This energy balan can be expressed by:.

Sf(a *M+H) dB= fMdl+fk
By successive integrating by parts, and seuing limits for a hysteresis loap, the equation can bc rcwriucn into nm-diHerendal equadm.

THE FrITING PROCEDURE
Semi-numerical technique has been developed to analyze the experimental hysteresis loops in various textured materials by Sztmnar and $atts.An approximate mlution of the hysteresis equation was obtained from which analytical formula relatin parameten of the model to the merive force, the remanenee and lhe initial susceptibility were derived.In order to obtain the best fit. the parameters k and at were derived analytically from the equatien for the susceptibility md the cercive force.IrmalIy enly me parameter has been fined.There is a need however to develep new tedmique which will be used reutinely to analyze many different samples.Still anether advantage of having fully numerical technique is being able to add more parameter terms in the future to aemmodate improved models.
In the program the following five incoming data peints were used: M,, the suscepuility at 3 cersteds, a, I and the eoereivity.The program developed is fully autemati, and the citeria for fitting is specified by the user.The secant method was used throughout to solve all equations.The result of the program is a data file which can be ploued if desired.The M was fixed since it main parameter of the mean field model as well as being directly measurable.Specifying M, greatly simplified the solution of the problem.Funlnnore, we wish to use the secant method wherever possible to speed up program execution, and the secam method F'th-e Hysteresis loops measm and calculated ( ) for directim at 0e, for various maximum field strength.
The two points for the secant method were the endpoint and the pseudo.eeenfiv.Three points were not used because the mean field model equation does not guarantee a tlnee peint solution.In this technique, a pair of secant loops were used, to solve for k and at simultanmusly, at was found by calculating the error in titling and sem:hing for the corresponding minimum error conditim.In lhe prepmed improved mean field model, where M, is allowed to vary from experimental data, M, and at were found by calculating the enor in fitting.Various cembinadons of the fining parameters were atteml lrtrst we amaned that M, must be taken from data.gh the computer program worked suussfully, the fittings were often poor.
To improve the fittings, we tried vmying M. Such an assumlon has resulted in a much better fitting.This was expected, as the model then had one extra parmneter.
One may argue that M should remain ccmstant for the same muterJd, because at inf'mite field strmsth. 811magnetic moments c/' atoms align alon$ the don of the field.However.we would like to argue that it takes a very large field to achieve dosc-to-saturatiaa levels, and in fact even at field strengths of I00 Oersteds.differences in M, are still in the order of 40.lmthermore, we aru tady discmtin results up to $ Oersteds.
Figure 2 Hysteresis loops measured and calo.dated for rolling direction ct 40.for various maximum field strengths EXPERIMENTAL RESULTS AND ANALYSIS The measurements of magnetic hysteresis were carried out on silicon ste using an Epstein apparatus at various angles with respect to the roiling direction for the maximum fieM strength tg I Oe.. 30e.and $ Oe.
The hysteresis cuxves were in 10 degree intervals.The texture of the steel was measured using x- my diffraction, to obtain three pole figures (110), ( 200), (211), frem which the cyml orientation distribution function ODF was detmnined."Ihe analysis of the ODF shows that there is a sux GOSS type orientat/ present having at most 30 randtaxt units.Figure 3 Hysteresis loops measured and calculated for rolling direction et 80, for various maximum field strengths Exemplary results of the 1,3, and $ Oe. fittings are presented in Figures 1,2,3.Such experimental results are allowing us to amdyze not only the texture influence on the pinning constant and other parameten of the model, but also to in the influence of field and magnetization on these constants.The results of the analysis are given in Figures 4 and $ in the form of relationships between the parameters of the model and the field strength as well as the direm angles from RD.We will now analyze the data and discuss the results in order to assess just how much the model can be used to provide useful information about magnetization processes in textured silicon steel The pinning constant, k, i]insuated in Figure 4 is most stnmgly aff by the existing texture.In order to explain the changes of k with directien we must stress that in a broad sense this pamneter is a. restraining force which inhibits changes in magne/izaticn and may not necessarily be related to demain wall movemenl.The resulls obtaiz show that the pinning constant for three different field strengths is smallest along the rolling directim and highest along 60 to 70 degrees frem the rolling direction.The resulls oblained ccnespond to the variation of the hysteresis loss in the Goss oriented silicon steels, where losses are lowest at 0 degrees from the rolling directim and hiest at 60 degrees from the toning direction.The parameters a and at fluctuate quite significantly.The hi sensilivity of these parameters to changes in angle from the rolling direction does not have a physical meaning but simply illustrates the fact that the shape of the hysteresis curve is a function of M + H)/a.Thus the ratio (a/a) could be responsible for a change in the shape of the hystere.uhcurve.
The maximum field strength affects the variation of k with the field strength cely at a Oe. and the differences between the values at I Oe. and those obtained for 30e.and 50e.. are related to the difficulty in estimating the saturation frem measurements of magnetization at such low fields.For field strengths at 3 and 50e., and we believe higher as well, lhe pinning constant is not affected by the field stnmgth, and therefore one can consider that it is truly a constant.Such constant values characterize the specimen investigated in a concise way, and there is no strong need to introduce the variation of k with the field strength and magnetizatien, as was previonsly suggested by Jiles and Atherum.