IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014 7009304

Loss Calculation Method Considering Hysteretic Property WithPlay Model in Finite Element Magnetic Field Analysis

Junji Kitao1, Yoshimi Takeda1, Yasuhito Takahashi1, Koji Fujiwara1, Akira Ahagon2, and Tetsuji Matsuo3

1Department of Electrical Engineering, Doshisha University, Kyoto 610-0321, Japan2Science Solutions International Laboratory, Inc., Tokyo 153-0065, Japan

3Graduate School of Engineering, Kyoto University, Kyoto 615-8510, JapanThis paper proposes a novel estimation method for iron loss considering hysteretic property. In the proposed method, iron loss

considering hysteretic property is estimated with play model as a postprocessing of usual finite element magnetic field analysis basedon ordinary magnetization curve. Numerical results are compared with measured results to demonstrate the effectiveness of theproposed estimation method for iron losses.

Index Terms— FEMs, inductor, magnetic hysteresis, magnetic loss.

I. INTRODUCTION

TO DESIGN the energy-efficient electric machinery, it isimportant to accurately evaluate iron loss and copper

loss [1]. A lot of papers reported the estimation method foriron loss in finite element magnetic field analysis [2]–[4].For the highly accurate calculation of iron loss and copperloss, it is essential to consider the hysteretic properties ofthe magnetic materials. However, it is not easy to accuratelyrepresent magnetic hysteresis due to the complexity of physicalphenomena and computational costs.

In this paper, we propose a novel estimation method ofiron loss considering hysteretic property. In the proposedmethod, the iron loss is estimated with play model [5], whichcan provide the hysteretic property, as a postprocessing ofusual finite element magnetic field analysis based on ordinarymagnetization curve. Although the computational cost of theproposed method is almost the same as that of conventionalones, the accuracy is expected to be improved drastically byconsidering hysteretic property. To demonstrate the effective-ness of the proposed estimation method of iron loss, numericaland experimental results are compared in a ring specimenof dust core. Finally, as a practical application, the iron lossanalysis of an inductor for a choke coil is performed.

II. IRON LOSS CALCULATION IN FINITE

ELEMENT ANALYSIS

A. Conventional Iron Loss Estimation Method

The iron loss w can be classified into an eddy current losswe and a hysteresis loss wh as follows:

w = we + wh = ke(Bm) f 2 + kh(Bm) f (1)

where ke, Bm , f , and kh are an eddy current loss coefficient,maximum flux density of a hysteresis loop, frequency, andhysteresis loss coefficient, respectively. By approximating w/ fas a linear function, these loss coefficients are determined [1].In the conventional iron loss estimation method, a hysteresis

Manuscript received June 28, 2013; revised September 17, 2013 andSeptember 25, 2013; accepted September 25, 2013. Date of current versionFebruary 21, 2014. Corresponding author: Y. Takahashi (e-mail:ytakahashi@mail.doshisha.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.2013.2284341

loss is evaluated by kh–Bm curve or constant hysteresis losscoefficient at Bm = 1.0 T.

B. 3-D Vector Hysteresis Represented by Vector Play Model

A discretized form of the vector play model [5] representsthe hysteretic properties as follows:

pζn(B) = B − ζn(B − p∗ζn)

max(|B − p∗

ζn |, ζn) (2)

H =M∑

n=1

fζn(|pζn(B)|) pζn(B)

|pζn(B)| (3)

where pζn is the play hysteron operator with a width of ζn ,ζn = (n − 1)Bs/M , Bs is the maximum measurable magneticflux density, p∗

ζn is the value of the play hysteron operatorat previous time step, M is the number of hysterons, andfζn is the shape function for the play hysteron operatorpζn . The identification method for the Preisach model canbe applied to the play model [6], because the play model ismathematically equivalent to the Preisach model [7].

In a dust core inductor, it is necessary to consider vec-tor hysteresis model because rotating magnetic fields occurgenerally in the region where the direction of magnetic fluxvaries such as the corner of magnetic circuits, although itsinfluence is very small and 1-D hysteresis model could beenough. In [8], 3-D vector hysteresis model was presentedbased on the superposition of stop and play hysterons. In thispaper, as a different approach, 3-D geometrical vectorization of(2) and (3) is proposed for the analysis of a dust core inductor.

The geometrical meaning of the 3-D vector play hysteronis shown in Fig. 1. A point B is located in a sphere whosecenter and radius are p and ζn , respectively. When B movesinside the sphere, p does not move [Fig. 1(a)]. When B reachesthe circumference of the sphere and pulls the sphere, p movestogether with B [Fig. 1(b)]. The proposed geometrically vec-torized 3-D play model is more efficient than 3-D vector modelbased on the superposition of scalar models because it doesnot require azimuthal integration.

The unidirectional characteristic of 3-D vector play hysteronin (2) and H in (3) are equivalent to scalar play model,and therefore the same identification method for a scalarplay model is applied. As for an identification of rotating

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7009304 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 1. 3-D vector play hysteron. (a) |B − pζn∗| ≤ ζn . (b) |B − pζn

∗| > ζn .

losses, the same method in [10] can be adopted, in which theperpendicular component in (3) is corrected by the ratio ofthe measured rotational hysteresis loss to the simulated loss.Although a vector hysteresis model is essential even in theanalysis of inductors as we mentioned before, its influence oniron loss is expected to be negligibly small. Therefore, rotatinglosses are not identified in this paper. Because the main focusof this paper is the verification of an estimation method ofiron loss considering hysteretic property, we will investigatethe effectiveness of the 3-D vector play model in the analysisof a motor, in which the rotational loss is relatively large, in thefuture. The play model can represent not only the symmetrichysteresis loops, but also dc-biased and pulsewidth modulationexcited minor loops appropriately [9].

C. Novel Iron Loss Estimation Method

The procedure of magnetic field analyses is shown in Fig. 2.The magnetization curve is considered in usual finite elementmagnetic field analysis and the iron loss is estimated by kh–Bmcurve as a postprocessing based on the amplitude of flux den-sity waveforms. In this paper, this method is called Method I.By treating hysteretic property directly, further accurate mag-netic field analysis can be achieved. The method is calledMethod III. As is well known, however, the computationalcost increases drastically by considering hysteretic property.

To improve the accuracy of calculated iron losses withinacceptable computational costs, we propose the iron lossestimation method considering hysteretic property only in apostprocessing. In the proposed method, first, magnetic fieldanalysis is performed by magnetization curve. Then, hysteresisloops in each element are simulated based on the numericalresults obtained from usual magnetic field analysis as apostprocessing, as shown in Fig. 2. The proposed methodis called Method II. Because hysteretic property is notconsidered in magnetic field analysis, it is expected that thecomputational cost of the proposed method is almost thesame as that of Method I.

III. COMPARISON OF ESTIMATION METHODS

FOR IRON LOSS

A. Specification of Ring Specimen

The effectiveness of the proposed estimation method foran iron loss is investigated. Table I shows the comparedanalysis methods. Methods I and II evaluate the iron lossby kh–Bm curve and the proposed method, respectively, asa postprocessing of the finite element magnetic field analysisbased on magnetization curve. Method III evaluates the ironloss directly by considering hysteretic property in the finiteelement magnetic field analysis.

Fig. 2. Analysis Method.TABLE I

ANALYSIS METHOD

TABLE II

SPECIFICATIONS OF RING SPECIMEN

Table II shows the specifications of a ring specimen of dustcore. We do not consider an eddy current loss because theeffect of eddy current is negligibly small in dust core. The playmodel is identified from 40 measured symmetric loops of thedust core at intervals of 0.05 T from 0.05 to 2.0 T at 50 Hz.

B. Numerical Results for Symmetric Hysteresis Loops

Fig. 3 shows the numerical results for symmetric hysteresisloops. The same voltage waveform as the measurement, whichis controlled under the condition that induced voltage of B-coilis sinusoidal, is applied as an input in magnetic field analysis.

KITAO et al.: LOSS CALCULATION METHOD CONSIDERING HYSTERETIC PROPERTY 7009304

Fig. 3. Numerical results obtained from Methods I–III (symmetric hysteresisloops). (a) Hysteresis loss. (b) Copper loss. (c) Flux density waveforms (Bm =1.4 T). (d) Current waveforms (Bm = 1.4 T).

The frequency used in the experimental test is 50 Hz. Fig. 5(a)shows the mesh of an analyzed ring core.

As shown in Fig. 3(c), the flux density waveforms obtainedfrom Methods I–III agree well with the measured resultsregardless of whether the hysteretic property is considered ornot. Therefore, as shown in Fig. 3(a), numerical results of ironloss wh obtained from Methods I–III are very close to themeasured results. On the other hand, the hysteretic magneticproperty has also an influence on current waveforms, as shownin Fig. 3(d). The copper loss wcu obtained from Methods Iand II shown in Fig. 3(b) differs slightly from the measuredresults due to the difference of current waveforms.

C. Numerical Results for DC-Biased Minor Loops

Fig. 4 and Table III show the numerical results obtainedfrom Methods I–III for dc-biased minor loops. The values inparentheses indicate the relative differences of each loss withrespect to measurement values. Bmax indicates the maximumflux density value of hysteresis loop including superimposeddc component. The same input voltage waveform as theexperiment at Bm = 0.3 T is used, where Bm is the amplitudeof the ac component of flux density. The frequency used inexperimental test is 50 Hz.

As shown in Fig. 4(a), the waveform of flux density differsdepending on whether the hysteretic property is considered ornot. In the case of Bmax = 0.5 T, the iron losses obtainedfrom each method are almost the same in Table III. However,as Bmax increases, the difference of hysteresis loss betweenthe numerical result obtained from Method I and experimentalvalue also increases. Method I, in which the hysteresis loss isevaluated by kh–Bm curve, is based on the assumption thatthe shape of minor loop is similar to that of major loop.However, the shape of minor loop differs substantially asthe dc component of the flux density increases, as shown inFig. 4(c). The most accurate hysteresis loss can be obtainedfrom Method III regardless of Bmax as is expected. Method IIcan obtain better analysis accuracy than Method I, as shownin Table III although the computational accuracy is almost

Fig. 4. Numerical results obtained from Methods I–III (minor loops). (a) Fluxdensity waveforms (Bmax = 1.5 T). (b) Current waveforms. (c) Minor loops.

TABLE III

IRON LOSS AND COPPER LOSS

the same. From the above results, the advantages of theproposed method over the conventional method based onhysteresis loss coefficient can be confirmed.

In the case of copper losses, same analysis results areobtained from Methods I and II because the current wave-form is calculated from the magnetic field analysis basedon magnetization curve. There is no air gap in this ringspecimen, and therefore the hysteretic property has a greatinfluence on current waveform prominently. However, practi-cal electric machines and devices such as motors and inductorsfrequently have air gaps. In such a case, because the influenceof hysteretic property gets relatively small, as mentioned inSection IV-B, it is expected that the copper loss obtained fromthe proposed method does not differ significantly from thatobtained from Method III.

IV. IRON LOSS ESTIMATION OF INDUCTOR

A. Specification of Inductor

The three methods shown in Table I are applied to theinductor shown in Fig. 5(b) to confirm the effectiveness ofthe proposed estimation method for an iron loss. As shown inSection III, Method III achieves the most accurate evaluationof iron and copper losses. Here, we compare the numericalresults obtained from Methods I and II with Method III. In the

7009304 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 5. Analyzed mesh. (a) Ring specimen (1/720 model). (b) Inductor(1/8 model).

TABLE IV

ANALYSIS CONDITION (REACTOR)

TABLE V

CALCULATION TIME (Vmax = 6 V)

Fig. 6. Numerical results of inductor (Vmax = 6 V). (a) Flux densitywaveforms. (b) Current waveforms.

analyzed inductor, the dust core is used for yoke and leg. Wedo not consider an eddy current loss. Table IV shows the spec-ifications of analyzed model and analysis condition. Identifica-tion of the play model is the same, as mentioned in Section III.

B. Numerical Result for Reactor Fed by Under Bias Supply

The applied voltage is sinusoidal wave with dc component.Table V shows the calculation time. Figs. 6 and 7 show thenumerical results obtained from Methods I–III. Vmax indicatesthe maximum value of voltage as input voltage superimposeddc component. The amplitude of the ac component is 2 V, andthe drive frequency is 50 Hz.

As shown in Fig. 6(a), flux density and current waveformsobtained from Method I or II are in good agreement withthose obtained from Method III. However, the iron lossesobtained from Method I differ significantly compared withthose obtained from Method III. Method II obtains almost thesame hysteresis loss as Method III. Furthermore, because thismodel has air gap, nearly the same copper losses are obtained

Fig. 7. Copper loss and iron loss (reactor). (a) Vmax = 5 V. (b) Vmax = 6 V.(c) Vmax = 7 V.

from each method. As shown in Table V, the calculation timeof the proposed method is shorter than that of Method IIIand almost the same as that of Method I, which indicate theeffectiveness of the proposed method.

V. CONCLUSION

We propose a novel estimation method of iron loss usingthe play model as a postprocessing of usual finite elementanalysis based on the magnetization curve. The effectivenessof the proposed method is investigated in the analysis ofring specimen and a practical inductor for a choke coil.The proposed method is fairly effective especially in the casethat the magnetic circuit includes an air gap such as ordinaryelectric machines and devices.

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