IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 7, JULY 2013 3175

An Efficient Inverted Hysteresis Model with ModifiedSwitch Operator and Differentiable Weight FunctionShasha Bi , Alexander Sutor , Reinhard Lerch , Fellow, IEEE, and Yunshi Xiao

Chair of Sensor Technology, University of Erlangen-Nuremberg, Erlangen, Bayern 91052 GermanyDepartment of Control Science and Engineering, Tongji University, Shanghai 201804, China

This paper proposes a different inverted hysteresis model with modification of the classic Preisach switch operator. By using this newswitch operator, the inverted model remains the wiping out and congruency properties. It also guarantees the symmetry and total posi-tiveness of weight function in the Preisach plane. According to the change pattern of branches, a differentiable weight function isintroduced in the invertedmodel. The weight function performs with good continuity and symmetry. This makes it possible to implementthe inverted model in numerical analysis without iterative procedure. The identification work is done by means of the measured majorloops. Here the Newton method algorithm is applied to optimize the mean squared error (MSE) between the measured and simulateddata. By this way, the limited number of parameters can be determined. The inverted model was verified for both soft and hard magneticmaterials. Besides major hysteresis loops, minor loops and first-order reversal curves (FORCs) can also be simulated. By comparison,the simulation results produced by the inverted hysteresis model show good approximation to the measurement data.

Index Terms—Differentiable weight function, finite element method, inverted hysteresis model, switch operator.

I. INTRODUCTION

W ITH the development of computers, numerical tech-niques have been commonly utilized to solve elec-

tromagnetic fields problems. However, in finite element (FE)formulations, the magnetic vector potential is introduced asunknown. The magnetic flux can be directly obtained, butthe nonlinear hysteretic relation is usually approximated bya single-valued magnetization curve [1]. To realize accurateelectromagnetic field simulation, it is necessary to implementthe hysteresis model in FE codes, especially when anisotropicmagnetic materials are involved in rotational electromagneticfield [2], [3].In the past few years, much research effort has been put into

the implementation of hysteresis model in FE codes [1], [2],[4], [5]. However, most of hysteresis models, such as Preisachmodel, employ magnetic field as an input variable and takemagnetic flux (or magnetization ) as the output quantity[4], [6]. Therefore, to compute the magnetic field, iterativemethods are mostly utilized in the local inversion of forwardhysteresis models [5]. Much computation for converged solu-tions is ineluctable in the simulation process.In order to circumvent iterative procedure, Takahashi et al.

proposed to compute the inverse of the Preisach distributionfunction in [7]. With this method the magnetic field can bedirectly calculated from the flux density. However, no differ-entiable weight function was proposed in the inversion work.What’s more, curve [ see Fig. 1(a)] and curve[see Fig. 1(b)] have different ascending and descending paths.The application of the classic switch operator [see Fig. 1(c)]in the inverted model can not guarantee the total positiveness

Manuscript received November 05, 2012; revised January 12, 2013; acceptedJanuary 18, 2013. Date of current version July 15, 2013. Corresponding author:S. Bi (e-mail: shasha.bi@lse.e-technik.uni-erlangen.de).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2244583

Fig. 1. Forward and inverted hysteresis models. (a), (c), and (e) are the generalcurve, switch operator and Preisach plane of the forward hysteresis

model. (b), (d), and (f) are the general curve, switch operator andPreisach plane of the inverted hysteresis model.

Fig. 2. curve and the Preisach plane when is applied in the invertedmodel. The input firstly increases from to , then decreases fromto .

of the weight function in whole Preisach plane (see Fig. 2 inSection II).In this paper, we introduce a new switch operator [see

Fig. 1(d)] in Section II. With the inverted switch operator, thecorresponding differentiable weight function is explained inSection III. Section IV presents the obtained simulation results.Their good approximation to the measurement data can beobserved in the figures.

II. THE MODIFIED SWITCH OPERATOR

The classical Preisach model (CPM) has been widely appliedwithin FE applications, since the model approaches the require-

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3176 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 7, JULY 2013

ments of accuracy, simplicity, and efficiency [1]. In CPM, themagnetic flux (or magnetization ) is calculated from mag-netic field at time as

(1)

where the switch operator [see Fig. 1(c)] works as

(2)

The pseudo-inverted model shown in (3) is a simplified def-inition of the inverse of CPM [5], [6], [8]. It employs the mag-netic flux (or magnetization ) as input and takes the mag-netic field as output

(3)

Suppose the classic switch operator and the corre-sponding weight function are applied in the invertedmodel

As it is shown in Fig. 2, the input increases fromto , then decreases back to . The generated reversalcurve is shown in Fig. 2(a).Based on the work principle of CPM, the change of the outputcan be calculated by the integration of weight function in

the Preisach plane. The integration in rectangular S2 shown inFig. 2(b) can be calculated as

The negative distribution in some parts of Preisach plane willcause inaccuracy and nonsmoothness of the modeling. There-fore, we propose a new switch operator for the invertedmodel in (4)

(4)

operates based on the ascending and descending paths ofcurve [shown in Fig. 1(b) and (d)]. It can be mathemati-

cally expressed as

(5)

where , .

Fig. 3. Work principle of in Preisach plane. The curve (a), stateof switch operator and (b and c) when the input increases from

to . The curve (d), state of switch operator and (e andf) when decreases from to . Dark areas with symbol representsthe 1 state, light area with symbol present the 1 state.

The work principle of is shown in Fig. 3. In the range ofand , works similar with . However,

in the range of , where the forward model keepsthe old value in memory, has an opposite behavior to .As it is shown in Fig. 3(a), when increases from to, keeps 1 in the memory part [area with skew lines in

Fig. 3(c)]. In contrast, switches to 1 in the correspondingpart [area with skew lines in Fig. 3(b)]. From to , it formsa first-order reversal curve shown in Fig. 3(d). In the memoryarea with skew lines of Fig. 3(f), keeps the old state value.Whereas in the area with skew lines in Fig. 3(e), negatesthe state value of (5).With the new switch operator , the inverted model still

keeps the congruency and wiping out properties [4], [6]. As itis shown in Fig. 3, the Preisach planes of forward and invertedmodel are symmetric with line . In calculation, the switchmatrix can be formed as

......

......

Here the state value of is saved in lower triangular partof matrix , while value is stored in upper triangular part.In the area . The state value of in areaswith skew lines in Fig. 3 can be obtained according to thevalue in symmetric area. Since the state values can only be 1or 1, the calculation of switch matrix will not increase thecomputation time.

III. THE DIFFERENTIABLE WEIGHT FUNCTION

In order to find a reasonable weight function for the invertedhysteresis model, we analyze the changing pattern of ascendingand descending branches [blue solid line in Fig. 4(a)].The curve is obtained by interpolation ofmeasureddata [red dashed line in Fig. 4(a)]. Both of the inputs [ for

BI et al.: AN EFFICIENT INVERTED HYSTERESIS MODEL WITH MODIFIED SWITCH OPERATOR AND DIFFERENTIABLE WEIGHT FUNCTION 3177

Fig. 4. The ascending and descending branches of and curves (a)and their derivatives (b). Where specifies the sample number. The magneticflux (red dashed) was measured on a FeCo thin film material by means ofvibrating sample magnetometer (VSM).

Fig. 5. Normalized weight functions composed from main loop of a FeCo thinfilm material. The forward hysteresis model (a), (b) and inverted hysteresismodel (c) (d). Both of the weight functions are symmetric with line .

curve, for curve] change as the black dashedline shown in Fig. 4(a).The change rate can be observed by means of derivatives

and . As it is shown in Fig. 4(b), (reddashed) achieves the maximal value near coercive field strength.

(blue solid) changes extraordinary sharply at the begin-ning and the end of branches ( near 0, 100, and 200).As the general shape of major curves are often tangent

like, the weight function for inverted model is supposed to besimilar to the derivative of the tangent function. We propose thedifferentiable weight function as

(6)

The exponent controls the sharpness of at the begin-ning and the end of branches. It makes this function suitable forboth hard and soft magnetic materials. The shape of inPreisach plane is shown in Fig. 5(c).Due to the new switch operator, keeps positive in

whole Preisach plane. It can be compared with an analytic

Fig. 6. Comparison of the measured and simulated hysteresis curves. Hys-teresis curve in (a) is measured with Epstein frame on soft magnetic materialM33035 [12] (Steel with thickness of 0.35 mm, coercive field strength49.396 A/m, remanent magnetization 1.387 T). Hysteresis curve in (b) ismeasured with VSM on material FeCo (Thin film with thickness of 2 ,coercivity field strength 8400 A/m, remanent magnetization 2.282 T).

weight function for forward hysteresis model [seeFig. 5(a)] [3], [9], [10]. The weight function of classic Preisachmodel is supposed to be mirror symmetric with respect tothe line [11]. It can be observed in contour maps inFig. 5(b) and (d), the new weight function also keepsthe symmetry like .As it happens in forward Preisach model, the forced magne-

tization effect [9] also needs to be modeled with an analyticalfunction. It is reasonable to choose the tangent function as abasis of the final formulation

(7)

where the parameter in (7) is limited to [0, 1), as the range oftangent function is limited to ( 1, 1).

IV. SIMULATION RESULTS

The identification of the inverted model is done with mea-sured major loops. The seven parameters in (6) are determinedby Newton method algorithm by optimization of the meansquared error (MSE) between the measured and simulated data.Fig. 6 shows the simulation results for soft [see Fig. 6(a)] andhard [see Fig. 6(b)] materials. Through comparisons the sim-ulation results exhibit good agreement with the measurementdata.Parameters for simulation of soft and hard materials are

listed in Table I. Parameter for soft material is muchgreater than hard material . It shows more obviousdifference in the comparison of identified weight functions(shown in Fig. 7). The weight function for soft materials shownin Fig. 7(a) has a distinct increase in areas where andapproach 0 and 1. The integration area is located where thebranches change sharply.Besides the major loops, the inverted model proposed in

this paper can also simulate the first-order reversal curves(FORCs) [see Fig. 8(a)] and minor loops [see Fig. 8(b)]. Noextra identification work is required here. As it has been men-tioned above, the inverted model has only to be identified with

3178 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 7, JULY 2013

TABLE ICOMPARISON OF PARAMETERS FOR WEIGHT FUNCTION

Fig. 7. Comparison of the simulated weight functions. (a) Weight function ofinverted hysteresis model for M33035 steel. (b) Weight function of invertedhysteresis model for 2 FeCo thin film.

Fig. 8. Comparison of the measured and simulated FORCs and minor loops.Measurement was carried out with VSMon 2 thin film FeCo. FORCs shownin (a): Magnetic field H starts from 60 kA/m. Reversal points are 0 kA/m,10 kA/m, 20 kA/m, 30 kA/m and 40 kA/m. (b) shows the simulated minor loops.

measured major loops. It can be seen in the comparison, thesimulated FORCs and minor loops behave very closed to themeasurement data.

V. CONCLUSION AND DISCUSSION

In this paper, a different inverted hysteresis model has beenproposed for implementing hysteresis model into FE codes. Ac-cording to different behavior of curve from curve,

a modified switch operator is introduced in the integration calcu-lation. By means of the new switch operator, the inverted modelkeeps the wiping out and congruency properties. It also guaran-tees the total positivity of the weight function in Preisach plane,which is necessary for maintaining the monotony of ascendingand descending branches. The corresponding weight function isformed with differentiable analytic function, which is suitablefor both soft and hard materials. The simulation results showgood approximation to the measurement data. The algorithmcan be implemented into FE codes without iteration procedure.By introduction of an additional rotation operator, the invertedhysteresis model can be applied to solve numerical problems inrotational electromagnetic field.

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