IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 4, APRIL 2014 7300104

Phenomenological Hysteresis Modeling Based on AsymmetricTransition Probability of Magnetization

Chahn Lee1, Kenji Miyata1, Chikara Kaido2, and Tetsuji Matsuo2

1Hitachi Research Laboratory, Hitachi, Ltd., Omika-cho, Hitachi 319-1292, Japan2Kitakyushu National College of Technology, Kitakyushu 802-0985, Japan

3Kyoto University, Kyoto 615-8510, Japan

An accurate and efficient static hysteresis model is proposed that phenomenologically describes hysteretic properties based onthe asymmetric transition probability of magnetization. The transition probability function is determined from first-order reversalcurves. The model provides magnetic hysteresis curves using a simple function calculation without any assembly of hysterons.A comparison with measured properties of silicon steel shows that this model represents hysteretic properties, including minor loopsmore accurately than the play-hysteron-based model.

Index Terms— First-order reversal curves, minor hysteresis loop, silicon steel, transition probability.

I. INTRODUCTION

MAGNETIC field analysis of open-type permanentmagnet-type magnetic resonance imaging scanner using

iron cores requires a highly accurate hysteresis model [1]because minor hysteresis loops can affect the image resolution.Several hysteresis models have been proposed to representcomplex hysteresis properties, including minor loops. Forexample, the Preisach model [2], [3] can provide detailedminor loop descriptions using an assembly of a large numberof magnetic particles, commonly called hysterons. However,their minor-loop description is not very accurate quantita-tively even though their computational cost is not small. Theplay model [4], [5] is an efficient hysteresis model that ismathematically equivalent to the static scalar Preisach model[6]. Originally, the play model was developed to describe thehysteretic function providing the output of the magnetic fluxdensity B from the input of magnetic field H . Similar to thePreisach model using the inverse distribution function method[7], the play model can provide output H from input B [5].However, the play model also requires a large number of playhysterons for a precise description of the minor loops. Inaddition, the play model with input B is not the exact inverseof the play model with input H .

This paper proposes an accurate and efficient hystere-sis model that phenomenologically describes the hystereticproperty based on the asymmetric transition probability ofmagnetization. This model aims to provide magnetic hysteresiscurves (or more commonly, BH curves) using a simple func-tion calculation without any assembly of hysterons. Severalfunction-based hysteresis models, such as [8]–[10], have beenproposed already based on the similarity or congruency ofthe BH curves. However, these curves are not always accuratefor arbitrary magnetic-field input. The proposed model derivesthe probability function of magnetization from the observedmagnetization, which depends on the concise storage of the

Manuscript received August 1, 2013; revised October 8, 2013; acceptedOctober 13, 2013. Date of current version April 4, 2014. Correspondingauthor: T. Matsuo (e-mail: tmatsuo@kuee.kyoto-u.ac.jp).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2013.2287030

Fig. 1. Asymmetric transition probability of magnetization. (a) p±(H ).(b) B±(H ).

magnetization history. The representation capability of themodel is compared with that of a hysteron-based model.

II. ASYMMETRIC PROBABILITY MODEL

A. Asymmetric Transition of Magnetization

Macroscopic magnetization results from the accumulationof microscopic magnetization, which can be statisticallydescribed with the transition probability of magnetization. Thistransition probability depends on the direction and historyof the magnetization process. Using the transition probabilityfunction, the ascending and descending BH curves B+(H )and B−(H ) are described as dB±(H )/dH = p±(H ), wherep±(H ) is the transition probability function depending onwhether the applied magnetic field is increasing or decreasing.Fig. 1 gives sketches of the probability function and B±(H ),where p+(H ) (or p−(H ) = p+(−H )) is asymmetric aboutline H = 0. This model is hereafter called the asymmetricprobability (ASP) model because the pair of asymmetricprobabilities p±(H ) create the conditions for hystersis.

B. Determination of Probability Function and BH Curves

The function p±(H ) depending on the magnetizationhistory is determined phenomenologically from measured BHcurves. The simplest construction of p±(H ) requires only themajor hysteresis loop. This paper presents a simple determi-nation method using the first-order reversal curves (FORCs)assuming a similarity in the BH curves, which gives higher-order reversal curves in a deterministic manner.

The probability function is switched at branching pointsdepending on the magnetization process. For scalar hysteresis,

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7300104 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 4, APRIL 2014

Fig. 2. Branching points and BH curves.

the branching points are given by the extrema of H . Threerules are assumed.

1) Every BH curve is defined using the last two branchingpoints called, respectively, the initial point Pi: (Hi, Bi)and the terminal point Pf : (Hf , Bf), which are takento be the last and second-to-last branching points. Theprobability function is defined as

p±(H : Pi, Pf) = dB±(H )

dH. (1)

2) The branching points are given by the extrema of H .When H begins to decrease (or increase) after increasing(decreasing), the local maximum (minimum) of H givesthe new branching point, which becomes the initialpoint; the former initial point becomes the terminalpoint.

3) When H passes through Hf , the branching points Pi andPf are deleted and the previous two branching pointsreturn as the initial and terminal points. This property issimilar to the wiping-out or deletion property governingthe Preisach model [2], [3].

Fig. 2 shows branching points Pk at (Hk, Bk) (k = 1, …) andP±S at ±(HS, BS), where PS and P−S are positive and negativesaturation points. The probability function for the descendingBH curve from P1 is defined by the branching points P1(initial point) and P−S (terminal point), which is denoted byp−(H : P1, P−S). When H begins to increase at P2 beforereaching P−S , P2 becomes a new branching point (initial point)and the ascending BH curve (second order reversal curve) fromP2 toward P1 (terminal point) is determined by the probabilityfunction p+(H : P2, P1). Similarly, P3 and P4 are branchingpoints generated after P2. When H exceeds H3, P3 and P4 aredeleted and P1 and P2 become the terminal and initial points,respectively, again.

The ascending/descending FORC from B = B0 is denotedby B = g±(H : B0). Using g±, the probability function for anarbitrary ascending/descending BH curve (first or higher orderreversal curve) from branching point Pi toward Pf is given as

p±(H : Pi, Pf) = (Bf − Bi)dφ±(H : Hi, Bi, Hf)

dH(2)

where

φ±(H : Hi, Bi, Hf) = g±(H : Bi) − g±(Hi : Bi)

g±(Hf : Bi) − g±(Hi : Bi). (3)

Consequently, the ascending/descending BH curve isexpressed as

B±(H : Pi, Pf) = Bi + (Bf − Bi)φ±(H : Hi, Bi, Hf) (4)

where φ±(Hi: Hi, Bi, Hf) = 0 and φ±(Hf : Hi, Bi, Hf) = 1.Accordingly, the BH curve is constructed by the FORCcompressed along the B-direction by ratio c± given by

c± = Bf − Bi

g±(Hf : Bi) − g±(Hi : Bi). (5)

Thus, the ASP model provides BH curves using a simplefunction calculation without any assembly of hysterons andintegration procedure, as required for the Preisach and playmodels. The magnetization history is concisely stored in thesequence of branching points. Accordingly, it is an efficientmodel able to be applied to the finite element magneticfield analysis. Practically, g±(H : Bi) is obtained from afinite number of measured FORCs g±(H : Bm) (m = 1, 2,…) using an interpolation (see Appendix) or a polynomialapproximation.

If only the major hysteresis loop is available, g±(H : B0)are replaced by the ascending and descending curves of themajor loop.

It is possible to interpret the Preisach distribution function asa differentiation of another kind of probability function that isbasically depends only on H . However, it is not very practicalto define the probability function of the ASP model as anintegration of the Preisach-type distribution function becauseof the dependence of p± on Bi and Bf as in (2). Owing to thisdependence, the ASP model does not possess the congruencyproperty that is observed in the classical Preisach model[2], [3].

C. Normal Magnetization Curve and Symmetric BH Loops

This scheme yields higher-order reversal curves but doesnot give the normal magnetization curve and symmetric BHloops explicitly. This section derives the normal magnetizationcurve through the demagnetizing process.

Decaying back-and-forth input of H is applied for thedemagnetization, where the extreme sequence of (H , B) =(H0, B0), (H1, B1), and so on, are assumed to satisfy

|Hn+1| = |Hn| − �H. (6)

If H decreases from a local maximum Hn, the descendingcurve is given as

B = Bn + (Bn−1 − Bn)g−(H : Bn) − g−(Hn : Bn)

g−(Hn−1 : Bn) − g−(Hn : Bn). (7)

Because the preceding and following local minima are Hn−1 =−Hn − �H and Hn+1 = −Hn + �H , the following holds:

(Bn+1 − Bn)[g−(−Hn − �H : Bn) − g−(Hn : Bn)]= (Bn−1 − Bn)[g−(−Hn + �H : Bn) − g−(Hn : Bn)] . (8)

It is assumed that (Hi, Bi) is on the normal magnetizationcurve that is denoted by B = b(H ) (= −b(−H )). Using

g−(−Hn ± �H : Bn)≈ g−(−Hn : Bn) ± g′−(−Hn : Bn)�H

(9)

Bn±1 = b(−Hn ± �H ) ≈ −Bn ± b′(Hn)�H (10)

g′−(H : B) = ∂g−(H : B)

∂ H, b′(H ) = db(H )

dH(11)

LEE et al.: PHENOMENOLOGICAL HYSTERESIS MODELING BASED ON ASYMMETRIC TRANSITION PROBABILITY 7300104

Fig. 3. Simulated and measured normal magnetization curves.

(8) is rewritten as

b′(Hn) = 2Bng′−(−Hn : Bn)

g−(Hn : Bn) − g−(−Hn : Bn). (12)

Consequently, b(H ) satisfies the ordinary differential equation

db(H )

dH= 2b(H )g′−(−H : b(H ))

g−(H : b(H )) − g−(−H : b(H )). (13)

The integration of (13) gives b(H ). The symmetric BH curvesare obtained by setting the two branching points ± (H j , B j )on B = b(H ).

D. Expression With Input B

Magnetic field analysis often requires a relation betweeninput B and output H . Using the one-to-one correspondencebetween B and H given by (4), the ASP model is redefinedwith a hysteretic function having an input B and an output H .

By setting g±(Hi: Bi) = Bi0, (4) is rewritten with (5) as

g±(H : Bi) = Bi0 + B − Bi

c±. (14)

Using (14), the BH curve between Pi and Pf is expressed as

H±(B : Hi, Bi, Hf) = g−1±(

Bi0 + B − Bi

c±: Bi

). (15)

Thus, B-input ASP model is derived by expressing the FORCsas functions g−1± of B .

III. SIMULATION RESULTS

A nonoriented silicon steel sheet JIS: 35A300 is measuredusing the Epstein frame with a slowly varying current excita-tion. As is generally observed in silicon steel, the increase ordecrease in B is steep when B ≈ 0. Consequently, choosingH as the input variable generates rapid changes in output B .Accordingly, this paper compares the ASP model with ahysteresis model having input B that can be convenientlytreated in the magnetic field analysis using a magnetic vectorpotential.

For comparison, the play model [4], [5] is examined, whichis mathematically equivalent to the static Preisach model.The B-input play model using 60 play hysterons is identifiedfrom FORCs and from symmetric BH loops, called here theplay models (F) and (S), respectively. Note that (F) requiresthe symmetrization of shape functions after the identificationfrom FORCs, which corresponds to the symmetrization of thedistribution function in the Preisach model. The ASP modeldoes not need this symmetrization.

Fig. 4. Simulated FORCs. (a) ASP model. (b) Play model identified fromFORCs. (c) Play model identified from symmetric loops.

Fig. 5. Simulated symmetric BH loops. (a) ASP model. (b) Play modelidentified from FORCs. (c) Play model identified from symmetric loops.

Fig. 3 shows the normal magnetization curve given by (13).The agreement with the measured curve justifies the ASPmodel constructed from the FORCs.

7300104 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 4, APRIL 2014

Fig. 6. Simulated asymmetric minor loops having minimum points at 1 T.(a) ASP model. (b) Play model identified from FORCs. (c) Play modelidentified from symmetric loops.

Fig. 7. Simulated asymmetric minor loops having minimum points at 0.5 T.(a) ASP model. (b) Play model identified from FORCs. (c) Play modelidentified from symmetric loops.

Fig. 4 compares simulated FORCs given by the ASP modeland play models (F) and (S). The ASP model accuratelyreconstructs the FORCs, whereas those given by the playmodels disagree with measurement data. This discrepancy bythe play model (F) results from the symmetrization above,which may be improved by an advanced symmetrizationmethod (see [11]).

Fig. 5 compares simulated symmetric BH loops where theASP model gives more accurate symmetric loops than the play

model (F). The play model (S) also represents symmetric loopsaccurately.

Figs. 6 and 7 compare simulated asymmetric minor loops.The ASP model reconstructs minor loops more accurately thanthose given by play models (F) and (S). This is because theASP model can fully exploit the information of magnetizationbehavior included in the FORCs.

IV. CONCLUSION

This paper described a phenomenological hysteresis modelbased on the transition probability of magnetization. Thismodel provides an output B (or H ) from an input H (or B)using a simple function calculation without any assembly ofhysterons. The transition probability function is determinedfrom the FORCs, which derives the normal magnetizationcurve theoretically. It is an accurate, efficient, and convenientmodel able to be applied to the finite element magnetic fieldanalysis. Based on this scalar model, a vector hysteresis modelcan be constructed, which will be reported in near future.

APPENDIX

For example, a descending FORC g−(H : Bi) is obtainedfrom an interpolation between measured g−(H : Bm) andg−(H : Bm+1) (Bm ≤ Bi <Bm+1) as

g−(H : Bi) = Bi−Bm+1Bm−Bm+1

g−(H ∗m : Bm)

+ Bi−BmBm+1−Bm

g−(H ∗m+1 : Bm+1)

(16)

H ∗j =

{min(H, H j max) (min(H, H j max)<0)H j max H/Himax (min(H, H j max)≥0)

( j = m, m + 1)

(17)where H j max and Himax are the maximal H of g−(H : B j )and g−(H : Bi), respectively; Himax is known because (Himax,Bi) is on the ascending curve of major loop.

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