Vector Preisach modelingKurt Wiesen and S. H. Charap Citation: Journal of Applied Physics 61, 4019 (1987); doi: 10.1063/1.338966 View online: http://dx.doi.org/10.1063/1.338966 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/61/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Vector Preisach modeling (invited) J. Appl. Phys. 73, 5818 (1993); 10.1063/1.353538 Experimental verification of a vector Preisach model J. Appl. Phys. 69, 4502 (1991); 10.1063/1.348338 Generalization of the vector Preisach hysteresis model J. Appl. Phys. 69, 4832 (1991); 10.1063/1.348247 Vector Preisach and the moving model J. Appl. Phys. 63, 3004 (1988); 10.1063/1.340928 Isotropic vector Preisach model of hysteresis J. Appl. Phys. 61, 4022 (1987); 10.1063/1.338565

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Vector Preisach modeling Kurt Wiesen and S. H. Charap Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

A vector hysteresis model for particulate media has been developed and applied to a two­dimensional, self-consistent simulation of the magnetic recording process. Vector modeling is accomplished using a scalar Preisach model for each of the two perpendicular directions, with each model responding to the corresponding component of the applied field. The Preisach density function is factored to permit description of hysteretic behavior based solely on the major loop. Each scalar Preisach model is partially demagnetized by the corresponding perpendicular field component. This feature couples the models and provides a substantial degree of reversibility with respect to rotating applied fields. Vector model performance was tested by the degradation of the longitudinal remanent moment as a function of vertical applied field. This model has been successfully used in a recording simulation.

INTRODUCTION

Vector hysteresis simulations are useful if they are accu­rate or fast. The former quality is associated chiefly with models which respond as a collection of Stoner-Wohlfarth­like particles, ; while the latter quality is the best attribute of models based on the superposition of scalar model re­sponses.2 Our vector model is an extension of the scalar mod­el approach. It consists of scalar Preisach models acting in orthogonal directions, but adds the feature of allowing the scalar models to respond to perpendicular field components as weH as those parallel to the model axis. By this device, our model strives to keep the computational quickness of the scalar model approach while increasing its accuracy,

We are not the first to propose vector modeling via the Preisach formalism. Barker et al. 3 proposed using three in­dependent scalar Preisach models for a 3D model. Also, Mayergoyz4 has proposed independent scalar Preisach mod­els uniformly distributed in two or three dimensions. Our proposal differs from these in that the scalar Preisach models respond to perpendicular field components.

SCALAR PREISACH MODELING

The formalism of Preisach has been discussed in many papers,s and will be repeated here only to the extent neces­sary to clarify our modifications to the conventional ap­proach. We adopt the notation of Del Vecchio.6

According to Preisach, complex hysteretic behavior re­suits from the superposition of elemental hysteresis loops. Each square elemental loop represents unit magnetization and is completely characterized by the upper and lower switching fields, a and b. The aggregate behavior ofthe ele­ments is determined by defining the density function, p(a,b), which when integrated gives the change in magneti­zation for corresponding changes in field. It is thus more convenient to work with the equivalent function, E(H"Hz), defined as

(1)

which win hereafter be referred to as Everett's7 function. The hatched region of Fig. 1(a) illustrates that portion of the Preisach plane over which the density function is inte­grated to give Everett's function.

Experimentally, Everett's function is defined by the family of curves which originate from points on the ascend­ing half of the major loop. Having determined Everett's function, the hysteresis behavior of any magnetic material can be predicted as a sum of Everett's functions if a list of appropriate turning point field values is maintained.

We depart from conventional Preisach theory by assum­ing that the density function factors, according to

p(a,b) = p(a)p( - b) (2)

(Biord and Pescetti 8 assumed a more general factorization), This permits us to define Everett's function solely from the major loop, which can be seen by substituting Eq. (2) into Eq. (1) and performing the integration. The result is

E{H!,H2 ) = [q(H2 ) - q(H!) J

x[q(-H1)-q(-H2 )], (3)

where

p(H) = !i~q(H). dH

Everett's function of two field variables is now expressed in terms of a function q(H) of a single field variable.

The limiting values of q(H) are q( - H..at) = 0 and

q( + H sat ) = J2M,a, . Knowing this, q(H) can be expressed as a quadratic equation in terms of M(H) and M( - H), where these are magnetization values on the ascending half of the major loop. Solving the quadratic yields the constraint

(-Msat)[Msat +M(H) +M( -H)]

+HM(H)-M(-H)]2;>O, (4)

which must be satisfied to keep q(H) real. Within the mar­gin of experimental error, Eq. (4) was always satisfied.

We define the reversible component of magnetization by numerically integrating the measured reversible susceptibil­ity. Reversible effects are accounted for by subtracting this

4019 J. Appl. Phys. 61 (8),15 April 1987 0021·8979/871064019-03$02.110 @ 1987 American Institute of Physics 4019 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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(a) (b)

FIG. I. Regions of the Preisach plane corresponding to (a) Everett's func­tion, and (b) demagnetization due to a perpendicular field.

component at the input stage of the Preisach model, then adding it at the output stage.

2D VECTOR PREISACH MODELING

It is the fundamental assumption of our vector model that field components perpendicular to the axis associated with a scalar Preisach model have the effect of a partial ac demagnetization on that model. The degree of demagnetiza­tion depends on the magnitude of the field, and is complete for large fields. Our two-dimensional vector simulation ap­plies parallel and perpendicular field components in se­quence to each of the orthogonally oriented Preisach mod­els. As a perpendicular field has a demagnetizing effect, this component must be applied first After the appropriate amount of demagnetization has been accounted for, the field component parallel to the corresponding Preisach model is applied and the resultant magnetization component deter­mined. This is repeated for the remaining Preisach model, and adding the two magnetization components provides the vector result.

The degree to which a perpendicular field demagnetizes a Preisach model depends on the weight given those elemen­tal loops with coercivities and interaction fields less than the magnitude of the perpendicular field. The coercivity He and interaction field Hi of an elemental loop are defined as

Hc=(a-b)/2, H;=(a+b)/2,

The shaded region of Fig. 1 (b) represents all elemental loops with coercivities less than H!. One expects that the elemental loops within this region, in particular those with small inter­action fields (i.e., near the line a = - b), will be most affect­ed by a perpendicular field. Considering that Everett's func­tion is determined over triangular regions, we assume that the hatched portion of Fig. 1 (b) becomes demagnetized. A perpendicular field of magnitude H! thus causes half of the hatched region to be magnetized in one direction while the symmetric other half is magnetized in the opposing direc­tion. This region does not contribute to the net magnetiza­tion which is now reduced from its value prior to applying the perpendicular field.

RESULTS

Modeling results and experimental data (obtained with a vibrating sample magnetometer) for two types of record-

4020 J. Appl. Phys., Vol. 61, No. e, 15 April 1987

M [emu/eel 9D.G r ---------. --~_ .. (~-= ................... ~ -----,

I solid ~ ac demag . f dots -Y de pf'rp demit@;

.1 .( .;

i / lJ H fOe] lO,COO

FIG. 2. Longitudinal magnetization curves following ac and vertical de de­magnetization, BASF.

ing media are presented: an ISOMAX flexible disk, and a BASF rigid disk. The ISOMAX medium responded isotro­pically, as expected, with the bulk parameters M. = 130 emu/em3

, S = 0.8, and He = 870 Oe. The parameters of the BASF longitudinal recording medium were measured to be: Ms = 96 emu/cm3

, S = 0.8, and He = 700 Oe (longitudi­nal); Ms = 96 emu/cm3

, S = 0.35, and He = 450 Oe (shear-corrected vertical).

The chief assumption of our vector model-that a per­pendicular field is equivalent to an ac demagnetization--can be checked experimentally. If a large enough vertical field has been applied, we assume that it is equivalent to a com­plete ac demagnetization in the longitudinal direction. Fig­ure 2 compares the shapes of the longitudinal magnetization curves following ac demagnetization and demagnetization by a large (9500 G) vertical field for the BASF medium. The two curves shown in the figure are similar, the difference being that the initial susceptibility following perpendicular demagnetization is somewhat larger.

Smaller perpendicular field components do not cause complete demagnetization. We determined the amount of demagnetization by measuring the amount of remanent mo­ment remaining in the longitudinal direction after the appli­cation of various sized fields in the vertical direction. Figures 3 and 4 illustrate the effects-both those found experimen-

M [emu/eel 76.li ------.------- ---.----- -----·-1

experiment

FIG. 3. Reduction of remanent moment by perpendicular fields, BASP,

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r.l [emu/eel

lOG

~

"m"~ __ ",""K", °r-~rr!lc----------~~~============~~1~0~,O(~m

__ Perpendicular Fif'ld [~~J j

FIG. 4. Reduction of remanent moment by perpendicular fields, ISO MAX.

tally and those predicted by the simulation-of perpendicu­lar fields on remanent moments. Figure 4 indicates a reversal of the longitudinal magnetization which we attribute to ex­perimental imprecision.

This measurement shows that until the perpendicular field reaches a value on the order of the coercive force it has little effect. Above the coercivity, the remanent moment faUs with increasing field so that at about five times the coercivity very little remanent moment remains. A comparison of sim­ulation predictions with experiment suggests that the model clearly demagnetizes too easily for RASF, but only slightly so for ISO MAX.

Simulating vector hysteresis with orthogonal scalar models which are independent has the flaw that a longitudi­nally established remanence is unaffected by vertical fields. This problem is eliminated by our approach. However, a different problem is endemic to both methods. Large field components in both the longitudinal and vertical directions cause correspondingly large magnetization components which result in a net magnetization greater than saturation.

An additional advantage to our approach is a high de­gree of reversibility with respect to rotating applied fields. This reversibility occurs because the Preisach models asso­ciated with smaner field components are continually erased by the larger perpendicular component. This negates the hysteretic effects in the direction of the smaner component. The rotating magnetization is coaxial with a large rotating applied field when the field is entirely longitudinal or verti.­cal; the magnetization lags or leads slightly for the off-axis angles.

The vector model was demonstrated in a 2D self-consis­tent simulation of the record process. These simulations have been discussed at great length in the literature, and since ours claims no significant innovation we will not revisit them here. The reader is referred to Suzuki9 who describes a simulation similar to ours.

Transiti.on zones and readback pulses (calculated via reciprocity) were simulated using both the vector model and

4021 J. Appl. Phys., Vol. 61, No.8, 15 April 1987

independent scalar Preisach models. Qualitatively, the tran­sition zones obtained using the two models differed very lit­tle. The only significant quantitative difference was that the readback pulse amplitude using the vector model was about 20% less than for the independent models. This results from the demagnetizing effects of the vertical head field compo­nents on the primarily longitudinal magnetizations. The self-consistent simulations using the vector model, which in­dudes the complicated algorithm needed to account for the demagnetizing effects of perpendicular field components, re­quired a 15% increase in computation time as compared to those using independent scalar Preisach models.

A deficiency of our model is that the transition zones for thick particulate media occur where the coercivity equals the head field's longitudinal component rather than its mag­nitude as expected. 10

CONCLUSIONS

We have developed a two-dimensional vector hysteresis model which needs as input only the major loops (longitudi­nal and vertical) of the recording medium and a measure of its reversible susceptibility, The model is constructed of or­thogonally oriented scalar Preisach models which respond to both parallel and perpendicular field components. It thus offers an improvement to conventional scalar-based vector models, while paying only a small price (relative to more computationally intensive vector models) in extra computa­tion time. Although the model has the advantages of having longitudinal remanent moments reduced by vertical fields and of having a high degree of reversibility with respect to rotating applied fields, it still has some of the problems asso­ciated with conventional scalar-based models.

ACKNOWLEDGMENTS

We thank Seagate Corp. for supplying the BASF rigid disks, and Alcoa for the use of their vibrating sample magne­tometer. Kurt Wiesen has submitted a report of this research to Carnegie Mellon University in partial fulfillment of the requirements for the degree of Master of Science in Electri­cal Engineering.

II. A. Beardsley, IEEE Trans. Magn. MAG-22, 454 (1986). 2Thomas A. Roscamp, Gary E. Roberts, and Paul D. Frank, IEEE Trans. Magn. MAG-16, 973 (1980).

'J. A. Barker, D. E. Schreiber, B. G, Huth, and D. H. Everett, Proc. R. Soc. London Ser. A 386,251 (l983).

4Isaak D. Mayergoyz, iEEE Trans. Magn. MAG-22, 603 (1986).

'I. D. Mayergoyz, J. Appl. Phys. 57, 3803 (1985). 6R, M. Del Vecchio, IEEE Trans. Magn. MAG·Ui, 809 (1980). 7D. H. Everett. Trans. Faraday Soc. 51,1551 (1955). "G. Biard and D. Pescetti, Nuovo Cimento 7,829 (J 958). 9Kenji Suzuki, IEEE Trans. Magn. MAG-n, 224 (1976). wlrene B. Ortenburger and Robert I. Potter, J. Appl. Phys. 50, 2393

(1979).

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