comparisonofpreisachandjiles–athertonmodelstotake ...boulph.free.fr/krzysztof/2003 comparison of...

Journal of Magnetism and Magnetic Materials 261 (2003) 139–160

Comparison of Preisach and Jiles–Atherton models to takeinto account hysteresis phenomenon for finite element analysis

A. Benabou, S. Cl!enet, F. Piriou*

Laboratoire d’Electrotechnique et d’Electronique de Puissance de Lille, B #atiment P2, Universit!e des Sciences et Technologies de Lille,

Equipe Mecosyel, 59655 Villeneuve d’Ascq, France

Received 21 March 2002; received in revised form 6 November 2002

Abstract

In electrical engineering, study and design of electromagnetic systems require more and more accurate models. To

improve the accuracy of field calculation code, hysteresis phenomenon has to be taken into account to model

ferromagnetic material. This material model has to be accurate and fast. In that context, two macroscopic models are

often used: the Preisach and the Jiles–Atherton (J–A) models.

In this paper, both models are presented. Field calculation requires a model giving the magnetization M versus either

the magnetic field H or the magnetic flux density B: Consequently, from the classical Preisach and J–A, two sub-models

MðHÞ and MðBÞ are deduced. Then, we aim at comparing these models in terms of identification procedure facilities,

accuracy, numerical implementation and computational effort. This study is carried out for three kinds of materials,

which have different magnetic features: ferrites, FeSi sheets and a soft magnetic composite material. Then, the

implementation of these models in a finite element code is presented. As example of application, a high-frequency

transformer supplied by a rectangular voltage is studied.

r 2002 Elsevier Science B.V. All rights reserved.

PACS: 75.50.Bb; 75.50.Cc; 75.60.�d; 75.60.Ej

Keywords: Hysteresis curve; Ferromagnetic material; Preisach model; Jiles–Atherton model; FEM

1. Introduction

To design and to study electromagnetic systems,field calculation codes are more and more used.We have at our disposal a virtual prototype inwhich geometrical dimensions and material char-acteristics can easily be modified. But, to achievean efficient design procedure, the used models have

to be accurate and fast. Satisfying these twocriteria simultaneously is not easy, therefore acompromise has to be made.Lots of electrical devices are made up of

ferromagnetic materials. To represent the behaviorof such materials in field computation codes, non-linear univoc function are generally used. But, inthis case, the hysteresis phenomenon is neglected.In that context, the use of a constitutive relation-ship which takes into account the hysteresisphenomenon would be more useful to improvethe accuracy.

*Corresponding author: Tel.: +33-0-320337114; fax: +33-0-

320436967.

E-mail address: [email protected] (F. Piriou).

0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0304-8853(02)01463-4

The aim of this work is the integration of asimple and few time-consuming hysteresis modelin a field computation code based on the finiteelement method.Different models have been proposed to repre-

sent the hysteresis phenomenon and many of themare mathematical models which ignore the under-lying physics of the material behavior [1,2]. In ourcase, we are interested in the use of a model basedon physical assumptions. Two macroscopic hys-teresis models have been chosen for this study: thePreisach [3,4] and Jiles–Atherton (J–A) [5] models.The Preisach model, based on a phenomenologicaldescription of the ferromagnetic materials, is themost used in electrical engineering. The J–A modeldescribes the hysteresis phenomenon as the con-sequence of a frictional force which opposes toBloch domain-wall motion. Previous works haveshown that both models can be implemented infinite element analysis. Nevertheless, in a 2D fieldcomputation code, two formulations are used: thescalar potential formulation (O-formulation) andthe vector potential formulation (A-formulation),the most used. To take the material behavior intoaccount, the O-formulation requires a model withthe magnetization versus the magnetic field(MðHÞ) whereas the A-formulation requires amodel with the magnetization versus the magneticflux density (MðBÞ). As both original models arepresented with H as entry, to implement them in aA-formulation calculation code it is necessary toinvert them numerically. For this purpose, di-chotomic or Newton numerical schemes are usedbut it leads to an increase of the computation time.A solution is to use a direct model with B as entry.It is possible to use the J–A model and the Preisachmodel with B as entry but this is not much studiedin the literature.In this paper, we compare the Preisach and J–A

models with both H and B as entry in terms ofaccuracy and time computation. They are appliedto model the behavior of three materials. Thesehave been chosen for their use in different areas ofelectrical engineering: ferrites which are used inpower electronics, FeSi sheets for electrical ma-chines at industrial frequencies and iron powder, asoft magnetic composite (SMC), which offers newpossibilities for electrical machine design [7]. First

of all, both models are presented with theirrespective procedures of identification. To evaluatetheir accuracy, they are tested for different kindsof excitation and using different criterions (hyster-esis losses, coercive and remanence fields evolu-tion). They are also compared in terms ofcomputational time. Finally, implementation ofthe models in a finite element code is presented. Asexample of application, a high-frequency transfor-mer, made of ferrites, is also studied.

2. Presentation of the models

The constitutive relationship of a magneticmaterial can be described by

B ¼ m0ðH þ MÞ; ð1Þ

where B is the magnetic flux density, H themagnetic field, M the magnetization and m0 thevacuum permeability. In this section, the Preisachand J–A models are presented. In their originalform, they give a relationship between themagnetic field H and the magnetization M : Themagnetic flux density B is then obtained fromEq. (1). Nevertheless, models of the magnetizationwith B as entry, giving the HðBÞ constitutiverelationship, can also be deduced from the originalform. Both models MðHÞ and MðBÞ are presentedin the following. They are restricted to the study ofisotropic magnetic materials and in the case of aquasi-static behavior. Dynamic effects (eddy cur-rent, dynamic of domain-wall motion, etc.) are nottaken into account in the original form of thesemodels.

2.1. The Preisach model and the Everett function

2.1.1. The Preisach model

The Preisach model associates to a ferromag-netic material a set of bistable units defined by thefunction ga;b ¼ 71 (Fig. 1(a)). The switching fieldcouple ða; bÞ characterizing a bistable unit mustrespect some conditions. If Hsat represents thesaturation magnetic field and Msat the correspond-ing magnetization of the ferromagnetic material,when H > Hsat; all bistable units are positive andthe magnetization is M ¼ Msat: On the opposite

A. Benabou et al. / Journal of Magnetism and Magnetic Materials 261 (2003) 139–160140

side, if Ho� Hsat; all bistable units are negativeand M ¼ �Msat: Both previous assumptions leadto the following conditions for the couple ða; bÞ [4]:

apHsat

bX� Hsat

As the hysteresis phenomenon is energeticallydissipative, we must also have aXb: These threeconditions allow us to define a triangle D (Fig. 1),called the Preisach plane. Each couple ða; bÞcharacterizing a bistable unit must belong to thisplane. A ferromagnetic material is then determinedby a statistic distribution pða; bÞ of the switchingfield couples ða; bÞ belonging to the triangle D:The total magnetization is then given by

M ¼ Msat

Z ZD

pða; bÞga;b da db ð2Þ

The demagnetized state is represented by theequation b ¼ �a and the Preisach plane is splittedinto two equal surfaces Sþ and S � : Sþ is thesurface of couples (a; b) which are such that ga;b ¼þ1 and S� the surface of couples (a; b) such thatga;b ¼ �1: For any other state of the system, thetriangle D is splitted into two surfaces Sþ and S�separated by a broken line as shown in Fig. 2.Expression (2) can be rearranged as

M ¼Msat

Z ZSþ

pða; bÞ da db

�

�Z Z

S�pða; bÞ da db

�ð3Þ

The magnetic state of the system is totallycharacterized by the broken line. This latter isdefined by a memory vector h which includes some

extrema of the excitation magnetic field Hi; i.e.some return points of the magnetic field. Thememory vector coordinates must verify the follow-ing conditions [10]:

H0 ¼ 0; h ¼ fH0;H1;H2;H3;y;Hng

for i ¼ 1; y; n � 1 and di ¼ Hi � Hi�1

di � diþ1o0;

jdiþ1jojdij

ð4Þ

where Hn; the last component, is the current valueof the magnetic field. Using these relations, thememory vector can be easily determined. Forexample, the magnetic state of Fig. 2 is given bythe memory vector h ¼ f0;þHsat;H1;H2;H3g:The knowing of the Preisach density function is

sufficient to represent a ferromagnetic material.Several methods for the determination of thisfunction from experimental results are proposed inthe literature [8–10]. All these methods requiregenerally numerical derivation and integration,which adds extra numerical errors to experimentalones. So, in contrast of attempts at analyticalmodels, the aim of this work is a fully numericalrepresentation of the Everett function [11] definedby the following equation:

Eðx; yÞ ¼ Msat

Z ZTðx;yÞ

pða; bÞ da db: ð5Þ

The surface Tðx; yÞ is defined by the right-angledtriangle in the Preisach plane (Fig. 1) with ðx; yÞthe vertex coordinates corresponding to the rightangle and the hypotenuse is supported by thestraight line a ¼ b: The two other legs of thetriangle are parallel to a- and b-axis, respectively.Then, if the Everett function is known we can

Fig. 1. (a) Bistable unit, (b) triangle D; and (c) triangle Tðx; yÞ:

A. Benabou et al. / Journal of Magnetism and Magnetic Materials 261 (2003) 139–160 141

calculate the magnetization M : In fact, we canshow that

if H > Hm; MðHÞ ¼ MðHmÞ þ 2EðH;HmÞ

if HoHm; MðHÞ ¼ MðHmÞ � 2EðHm;HÞ ð6Þ

where Hm is the last return point of the magneticfield (the next to last value of vector h).Then, the Everett function and the magnetiza-

tion M are linked by a relation which requires nonumerical derivation or integration. Experimentaldetermination of the Everett function is presentedin the next section.

2.1.2. Identification of the Everett function

In the following, we propose a method for theEverett function identification that needs a set ofexperimental centered minor hysteresis loops.Using these data of MðHÞ; the functionEðHm;HÞ is determined for values of H belongingto ½�Hm;Hm�: Fig. 3 gives the descending part of acentered hysteresis loop and the corresponding

function EðHm;HÞ obtained from Eq. (6) forHm ¼ 530 Am�1:From n measured centered loops ði ¼ 1; nÞ; a

curve set EðHmi;HÞ supporting the Everett func-tion is obtained (Appendix A). Now, we have todetermine this function for an arbitrary pointðH 0

m;H0Þ of the triangle D: This is done by an

interpolation method using the previous curve set.This interpolation method must respect the Ever-ett function continuity (this function is a primitive)on the whole studied domain and then forhysteresis curves. The proposed method, alreadypresented in Ref. [12], satisfies this condition. It isbased on shape functions used to interpolate thefield in 2D finite element method [13]. Theexpressions of these shape functions are detailedin Appendix A.The Preisach model can also be adapted to

obtain a model with B as entry [14]. In this case,the method used for the Everett function identifi-cation is detailed in Appendix B. We can note that

Fig. 2. Example of a magnetization process and the corresponding surfaces Sþ and S � :

Fig. 3. Descending part of a centered hysteresis curve and the corresponding Everett function (MðHÞ model).


for each model MðHÞ or MðBÞ an Everett functionEðHm;HÞ or EðBm;BÞ must be determined.

2.2. The J–A model

2.2.1. The model equation

The original J–A model presented in Ref. [5]gives the magnetization M versus the externalmagnetic field H: This model is based on themagnetic material response without hysteresislosses. This is the anhysteretic behavior whichManðHÞ curve can be described with a modifiedLangevin equation:

ManðHÞ ¼ Msat cothHe

a

� ��

a

He

� �� ; ð7Þ

where He ¼ H þ aM is the effective field experi-enced by the domains: H is the external appliedfield and a the mean field parameter representinginter-domain coupling. The constant a is anincreasing function of the temperature. Theanhysteretic magnetization represents the effectsof moment rotation within domains but does nottake into account losses induced by domain wallmotions. Then, by considering rigid and planardomain walls, the energy dissipated throughpinning sites during a domain wall displacementis calculated [5]. The expression of the magnetiza-tion energy is obtained under the assumption of auniform distribution of pinning sites. The magne-tization energy is assumed to be the differencebetween the energy which would be obtained in theanhysteretic case minus the energy due to thelosses induced by domain wall motions. Conse-quently, after some algebraic operations, thedifferential susceptibility of the irreversible mag-netization Mirr can be written as

dMirr

dHe¼

ðMan � MirrÞkd

ð8Þ

where the constant k is linked to the averagepinning site energy. The parameter d takes thevalue þ1 when dH=dt > 0 and �1 when dH=dto0with respect to the force which opposes variationsof magnetization. However, during the magnetiza-tion process, domain walls do not only jump fromone pinning site to another: they are flexible andbend when being held on pinning sites. Domain

wall bending is associated to reversible changes inthe magnetization process. Then, by some physicalenergy assumptions on the domain wall bending,the obtained reversible magnetization is linearlydependent on Man � Mirr [5]:

Mrev ¼ cðMan � MirrÞ; ð9Þ

where the reversibility coefficient c belongs to theinterval ½0; 1�: Assuming that the total magnetiza-tion is the sum of the reversible and irreversiblecomponents, we have the following expression:

M ¼ Mrev þ Mirr ð10Þ

with Mirr and Mrev defined by Eqs. (8) and (9).Using Eqs. (10) and (9) we can write

M ¼ Mirr þ cðMan � MirrÞ: ð11Þ

Then, by differentiating this equation withrespect to H; the total differential susceptibilityof the system is given by the following expressionwhich has already been presented in Ref. [14]:

dM

dH¼

ð1� cÞ dMirr=dHe þ c dMan=dHe

1� ac dMan=dHe � að1� cÞ dMirr=dHe:

ð12Þ

This is the model differential equation that givesthe magnetization as a function of the magneticfield H ; where Man is given by Eq. (7). The modelcan also be adapted with B as entry [15]. As for theprevious model and using the fact that Be ¼ m0He;Eq. (11) is differentiated with respect to B:

dM

dB¼

ð1� cÞ dMirr=dBe þ c dMan=dBe

1þ m0ð1� cÞð1� aÞ dMirr dBe þ m0cð1� aÞ dMan=dBe:

ð13Þ

In both cases, five parameters a; a; k; c and Ms

have to be determined from experimental results.It is important to notice that the J–A parametersare theoretically the same whether B or H is theentry of the model. At the opposite, the Preisachmodel requires the determination of two indepen-dent functions EðHm;HÞ and EðBm;BÞ: Thephysical properties of the five parameters arepresented in Table 1.


2.2.2. The parameter identification

An identification procedure from experimentaldata has been described in Ref. [6]. Aftermathematical developments using Eqs. (7), (8)and (11) for some points of the hysteresis loop,the implicit expressions of the five parameters areobtained in function of experimental data whichare:

* Hc and Mr the coercive magnetic field and theremanent magnetization,

* Msat the saturation magnetization,* w0ini and w0an the normal and the anhysteretic

differential susceptibilities,* w0c and w0r the coercive and remanent differential

susceptibilities.

Determination of parameters c; a; k and arequires an iterative procedure presented in Ref.[6]. This method is numerically sensitive and doesnot systematically converge, so we use a slightlydifferent procedure. First of all, we define anobjective function Fobj which evaluates the gapbetween experimental and theoretical results.

Fobj ¼Z T

0

½MexpðtÞ � MJ2AðtÞ�2 dt ð14Þ

where MexpðtÞ is the experimental magnetizationand MJ2AðtÞ the calculated one, obtained with thesame periodic excitation field HðtÞ with a period T :The data ðHðtÞ;MexpðtÞÞ corresponds to the majorloop, i.e. the centered loop with the highestmagnitude. In the first step, we use the followingprocedure to calculate a first set of the fiveparameters:

(1) systematic choice of ai in the interval [amin;amax] by steps Da;

(2) calculation of ai; ci; ki from experimental data(Hc; Mr; Msat; w0ini; etc.),

(3) calculation of the objective function Fiobj;

(4) back to (1) until ai ¼ amax;(5) determination of (aj ; aj ; cj ; kjÞ using F

jobj ¼

minðFiobjÞ:

The choice of the interval [amin; amax] depends onthe materials. But, it can be chosen sufficientlylarge because this procedure is relatively fast. Thisfirst calculation step gives a good estimation of theparameter values but these can be still improved.In this second step, we use an optimizationprocedure which minimizes the objective functionFobj by modifying independently the five para-meters without any constraint. The minimizationis carried out considering a set of centeredhysteresis loops [16] and not only the majorhysteresis loop as for the first step of theidentification. Then, the obtained parameters fitwell hysteresis loops for small and large magni-tudes of excitation. We can note that numericaltests have shown that the second step does not leadto good results without the first step. In fact, thislatter enables us to have a set of parameters closeto the best solution (in the sense of the chosenobjective function) which makes easier the con-vergence of the optimization procedure.

3. Experimental bench

The parameter identification and the compar-ison of both models can be achieved withexperimental data. An experimental bench formagnetic materials characterization is used. Thisone is described on the synoptic scheme (Fig. 4).The sample is a torus with primary and

secondary windings. Excitation is applied usingan arbitrary function generator which allows us toimpose current or voltage on the primary winding.A computer remotely controls these devices. Usingthe Amp"ere and Faraday laws, the field H and themagnetic flux density B are calculated from themeasurement of the current in the primary wind-ing and the secondary winding voltage. The aim ofthis study is to compare both models withmaterials used in different areas of electricalengineering.

Table 1

Physical properties of model parameters a; a; k; c; and Ms

Parameter Physical property

a Linked to domain interaction

a Shape parameter for Man

k Linked to hysteresis losses

c Reversibility coefficient

Msat Saturation magnetization


These materials are :

* N30 ferrites, used in power electronics;* FeSi sheets, used in electrical machines;* SMC materials (iron powder) which focuses

attention for the new possibilities of electricalmachines design that they offer [7].

As the presented models are aimed to be used inthe quasi-static case, the measurements were doneat the frequency f ¼ 0:5 Hz: At this frequency, wecan consider that the three materials are in thequasi-static conditions. In fact, dynamical effectsbecome non-negligible above 100 kHz for the N30ferrites, 5 Hz for the FeSi sheets (laminationthickness is 0.5mm) and 1 kHz for the consideredSMC material.

4. Comparison of the models

4.1. Identification procedure

4.1.1. Everett function

As it was described in Section 2.1.2, theidentification of the Everett function needs themeasurement of several centered hysteresis loops.The experimental measurement procedure has tobe very accurate, especially for high excitationfields where the hysteresis loops have to berigorously centered otherwise the interpolationmethod will fail. Experimentally, this condition

cannot be fulfilled and especially for loops withhigh magnetic field magnitude. Consequently, it isnecessary to adjust numerically the measured datain order to center the loops. The more hysteresisloops we use, the more accurate the simulationresults. In our case, about 20 centered hysteresisloops have been measured for the three materials.For low excitation fields, it is recommended tohave the most important hysteresis loops concen-tration. In fact, in this area the shape of thehysteresis loop changes a lot (it corresponds to aconcavity modification of the first magnetizationcurve). The Everett functions of the three materi-als, corresponding to the MðHÞ model, arepresented on Figs. 5–7. Their shapes are quitedifferent: the smoother surface is obtained with theSMC material. Indeed, it is characteristic of thehysteresis loop smoothness around the coercivepoint (Fig. 10). Consequently, it would be difficultto find an analytical expression which enables us tomodel accurately all these Everett functions. Thisis one of the advantage of the interpolationmethod proposed in Section 2.1.2 that does notdepend on the studied material.In order to verify the interpolation method

robustness, we simulate the same hysteresis loopsused for the identification of the Everett function.Simulation results and measured loops are thesame. For other magnitudes of excitation, experi-mental and theoretical results are in good agree-ments. The gaps between simulation andmeasurements for the MðHÞ model are shown onFigs. 8–10. These results are similar to thoseobtained with the MðBÞ model.

Fig. 4. Synoptic scheme of the experimental bench.

Fig. 5. Everett function for N30 ferrites.


4.1.2. Parameters of the J–A model

The optimization procedure presented in Sec-tion 2.2.2 is applied for the three materials. Theparameters are presented in Table 2. The gapsbetween the calculated major hysteresis loops and

the measurements are shown in Figs. 8–10. Onecan note that experimental results are well fittedusing the optimized parameters.Comparison of the parameters between two

different materials is not easy (Table 2). In fact,despite the physical signification of the modelparameters, an optimization procedure can lead togood results for the hysteresis loops restitutionwith parameter values disconnected from anyphysical signification. Nevertheless, consideringthe case of N30 ferrites and FeSi sheets, we cannote that the parameters lie within the same orderof magnitude. Comparison of their parameters cangive a good first approach for their physicalproperties. The parameter a is greater for the FeSisheets, it is linked to the fact that hysteresis loopsfor this material have a higher slope at the coercivefield. And parameter k shows that hysteresis lossesare more important for the FeSi sheets.In the following, we aim at comparing both

models for different kinds of excitation waveshapes.

4.2. Sinusoidal excitation

4.2.1. Hysteresis losses

Usually, an hysteresis model is often used toquantify the hysteresis losses, so this criterion waschosen for the comparison. Figs. 11–13 show theevolution of hysteresis losses versus Hmax; themagnitude of the sinusoidal excitation field. Firstof all, we can note that the Preisach model is moreaccurate than the J–A model. The J–A model gives

Fig. 6. Everett function for FeSi sheets.

Fig. 7. Everett function for the SMC material.

Fig. 8. N30 ferrites; measured hysteresis loop and gap between both models and measurements for the decreasing part of the loop.


good results for the three materials even ifhysteresis losses are over-estimated for the ferritesand under-estimated for the FeSi sheets. For theSMC material, there is no noticeable differencebetween both models. In this case, the J–A modelgives hysteresis loops in good agreements withexperience although the coercive and remanencepoints evolutions are under-estimated as it will beshown further in Section 4.2.2. This can be

explained by the observation of the hysteresis loop(Fig. 14). Indeed, the over-estimated losses at thehysteresis loop bends (P1 in Fig. 14) are compen-sated by the under-estimated losses at low field (P2

in Fig. 14).A summary of these results, the maximum

relative errors for both models, is presented inTable 3. For the J–A model, the MðHÞ and theMðBÞ models are equivalent, so only one result isgiven. It can be noted that this latter is welladapted for the SMC material study. At theopposite, both Preisach models (MðHÞ and theMðBÞ) give different results which are neverthelessclose. Globally, the Preisach model is the mostaccurate.

4.2.2. Remanent and coercive point evolutions

Another criterion for testing the model robust-ness is the evolution of two characteristic points of

Fig. 9. FeSi sheets; measured hysteresis loop and gap between both models and measurements for the decreasing part of the loop.

Fig. 10. SMC material; measured hysteresis loop and gap between both models and measurements for the decreasing part of the loop.

Table 2

Parameters values for J–A model

Parameter N30 ferrites FeSi sheets SMC material

a 9:8 10�5 1:3 10�4 1:8 10�3

a 20 59 1642

k 56 99 1865

c 0:9 0:55 0:8MsðA=mÞ 282 100 1 145 500 1 122 600


a hysteresis loop: the remanence and the coercivepoints. Remanent and coercive points evolutionversus Hmax are presented in Figs. 15–20. Simula-

tions for ferrites and FeSi sheets give results closeto the measurements. An important aspect of theJ–A model is the behavior of the coercive point

Fig. 11. N30 ferrites; comparison of hysteresis losses for both models with measurement at 0.5Hz.

Fig. 12. FeSi sheets; comparison of hysteresis losses for both models with measurement at 0.5Hz.

Fig. 13. SMC material; comparison of hysteresis losses for both models with measurement at 0.5Hz.


which becomes constant for low excitation fieldswhereas this asymptotical evolution appears ex-perimentally at higher excitation fields. As it willbe explained in Section 4.4, this behavior comesfrom the numerical solution of the differentialEq. (12).

4.3. Non-sinusoidal excitation

In electrical engineering, systems are not neces-sarily studied under sinusoidal excitations (pulsewidth modulation supply or during transientstate). To test the capabilities of models torepresent the magnetic material behavior undersuch kinds of excitation, we now consider twotypes of excitations:

Excitation A : HðtÞ ¼ H0 sinðotÞexpð�t=tÞ; ð15Þ

Excitation B : HðtÞ ¼H0 sinðotÞ

þ H 00 sinð5ot þ fÞ; ð16Þ

with t being a decreasing time constant, H0 themagnetic field magnitude, H 0

0 the fifth harmonicmagnitude and f its phase lag.The demagnetization-like excitation, denoted

excitation A, (Eq. (15)) is often met in the transientstate of a system supplied by sinusoidal voltage.However, it allows us to sweep through a greatnumber of magnetic states of the material. In theother hand, as electrical devices are more andmore supplied by power electronic converterswhich induce periodic but non-sinusoidal voltageand current, the second kind of excitation, denotedexcitation B, (Eq. (16)) enables us to test theaccuracy of both models in such conditions.Results for both excitations are shown inFigs. 21–26.In the case of excitation A, both models are in

good agreements with the experimental measure-ments. These results could be predicted because

Fig. 14. Over- and under-estimated losses for a major hysteresis

loop in the SMC material case.

Fig. 15. N30 ferrites; comparison with experience of remanent point evolution for both models.

Table 3

Hysteresis losses maximum relative errors (in %) for both

models

Material J–A model Preisach

M(H) model

Preisach M(B)

model

N30

ferrites

+12 �4.5 +6.6

FeSi sheets �9 +3.7 +3.7

SMC

material

+2.5 �2 +8


this excitation can be interpreted as a sequence ofcentered hysteresis loops with a decreasing magni-tude. In fact, we have seen in the previous sectionthat both models are accurate in the case ofcentered loops. Only, a shift between measure-

ments and simulations appears at low excitationfields for FeSi sheets.When applying excitation B, important differ-

ence between the two models appears. We can seethat the Preisach model is relatively accurate

Fig. 16. N30 ferrites; comparison with experience of coercive point evolution for both models.

Fig. 17. FeSi sheets; comparison with experience of remanent point evolution for both models.

Fig. 18. FeSi sheets; comparison with experience of coercive point evolution for both models.


whereas the J–A model gives an incoherentbehavior. This behavior is more noticeable in thecase of FeSi sheets and SMC material. Figs. 25

and 26 include a zoom of the field return-pointsregion where minor loops are not closed. Indeed,this model does not systematically ensure the

Fig. 19. SMC material; comparison with experience of remanent point evolution for both models.

Fig. 20. SMC material; comparison with experience of coercive point evolution for both models.

Fig. 21. N30 ferrites; comparison of theoretical and experi-

mental results using excitation A.Fig. 22. FeSi sheets; comparison of theoretical and experimen-

tal results using excitation A.


enclosure of minor loops. Moreover, the minorloops are very flat in comparison with experi-mental ones. The phenomenon has already been

underlined in numerous papers [17]. Some solu-tions have been proposed to avoid this problem,but it makes the model more complicated andrequires to know a priori the magnitude of theloop. When simulating an electrical device withtime-stepping finite elements analysis and using anhysteresis model, the magnitude of each loop oneach element of the mesh is not initially known. Inthis case, the modified J–A model is not adapted.The Preisach model is more adapted in this case

thanks to its intrinsic memory properties. In fact,there is a return-point memory aspect of thismodel which is clearly emphasized by the brokenline in the Preisach plane (cf. Section 2.1.1). Eachangle of this broken line is a return-point, i.e. anextremum, of the applied field. For this model, theresults are quite correct in the case of the N30ferrites and SMC material. For FeSi sheets, thesimulated minor loops are flatter than the experi-mental ones. Several methods have been proposedto improve the account for minor loops. Forexample, E. Della Torre has proposed a methodcalled ‘‘moving Preisach model’’ which insures abetter restitution for minor loops [18]. An interac-tion field, that depends on the magnetization, isintroduced as a feed-back to the initial model. But,this method increases the complexity of the modeland the calculation time.

4.4. Computational effort

To use the J–A model, we have to solve a first-order differential equation which can be done by

Fig. 23. SMC material; comparison of theoretical and experi-

mental results using excitation A.

Fig. 24. N30 ferrites; comparison of theoretical and experi-

mental results using excitation B.

Fig. 25. FeSi sheets; comparison of theoretical and experimen-

tal results using excitation B.

Fig. 26. SMC material; comparison of theoretical and experi-

mental results using excitation B.


the Euler implicit scheme. It has been observedthat this model is very sensitive to the chosenmagnetic field step dH to solve the equation. Theuse of a large dH can lead to an importantdeformation of hysteresis loops. For example, inthe case of a sinusoidal excitation, we can see inFig. 27 a comparison between two dH giving 200points and 2000 points for a major hysteresis loop.The surface of the loop increases with the magneticfield step dH: Then, it exists a minimal dH beyondwhich hysteresis loops become stable. Our experi-ence shows that a time step allowing the calcula-tion of about 2000 points for an hysteresis loop isrequired to have a good compromise betweenaccuracy and computational time for the threematerials. As it was presented previously for theminor loops, the numerical sensitivity of thismodel does not allow calculation of hysteresisloops in the Rayleigh zone. These are not closedand often not centered. It is necessary to calculateseveral hysteresis curves in order to obtain asatisfying result. On the other hand, the Preisachmodel is not sensitive to the time step, in fieldcalculation about only 100 points or less arecalculated for this model. In this case, consideringthe same return points for the excitation field, thetime step has no effect on the hysteresis loop. Infact, at the opposite of the J–A model, nodifferential equation is solved (Eq. (6)).Computational times of both models are re-

ported in Table 4. Simulations are made for theFeSi sheets on a major hysteresis loop with time

steps giving 2000 points in both cases and 100points in the Preisach model case. First of all, wecan see that computation time for the Preisachmodel is not proportional to the number of points.In the beginning of the code execution, the curvesset supporting the Everett function must beloaded. The time duration of this process isincluded in the time computation.The Preisach model is more time consuming

which is explained by two important points. First,the memory vector management is a heavynumerical method. Moreover, the interpolationmethod requires a great number of numericaloperations. At the opposite, the J–A modelrequires a reduced number of operations at eachcalculation step. Despite the number of 2000points required for this model, it is slightly lesstime consuming than the Preisach model with 100points. This slight difference can give advantage tothe J–A model when using the finite elementsanalysis providing no minor-loops are calculated(to avoid the minor-loops non-enclosure). In fact,the magnetization calculation is done on eachelement and for each time step as it will bepresented in the next section.

5. Implementation in field computation

In the following, the 2D implementation of anhysteresis model in a field calculation code, withthe coupling with external circuit equations, ispresented. We consider a domain D bounded by asurface S in the case of magnetostatics. Theequations to be solved are:

div B ¼ 0;

curl H ¼ J;Fig. 27. Two loops calculated with 200 points and 2000 points

per period with the J–A model (N30 ferrites).

Table 4

Computational times for FeSi sheets (400 MHz Digital Work-

Station)

Model Time (ms) for

2000 points

Time (ms) for 100

points

Preisach (Everett) 250 100

J–A 80 n.a.


with n � B ¼ 0 on Sb;

and n 4 H ¼ 0 on Sh; ð17Þ

with Sb and Sh being two complementary parts ofS; J the current density and n the outward normalvector of S: To take into account the materialbehavior, the constitutive relationship H ¼ fðBÞ; isadded. To model ferromagnetic material, thisrelationship can be one of the two modelspreviously presented. Nevertheless, these modelsare scalar models whereas we need a vectorialmodel in Eqs. (17). Then, to evolve this latter, weassume that the material is isotropic and that Band H are collinear. The magnitude of H iscalculated from the one of B and the direction ofH is the same as the one of B: All these equationsare generally solved using a potential formulation.In the 2D case, the vector potential formulation isgenerally preferred to the scalar potential one,then we have

curl fðcurl AÞ½ � ¼ J;

with n 4 A ¼ 0 onSb;

and n 4 H ¼ 0 on Sh; ð18Þ

where A represents the magnetic vector potential(i.e. B ¼ curl A). One can note that for thisformulation a constitutive relationship with B asentry is required, but if a scalar potentialformulation (H ¼ �grad j) is chosen, a modelwith H as entry is necessary. The numericalsolution of Eq. (18) including hysteresis cannotbe done with the same method as the one used withunivoc functions (Newton–Raphson scheme forexample). We have chosen the fixed-point methodalready presented in Ref. [19]. The hystereticconstitutive relationship is then rewritten underthe form

H ¼ fðBÞ ¼ nFPBþMFPðBÞ: ð19Þ

The reluctivity nFP is a constant and mustrespect some conditions to achieve convergence[20]. The studied hysteretic models assume B andH collinear, consequently the magnetization MFP

has the same direction as nFPB: Its magnitude isobtained by calculating MPF ¼ f ðBÞ � nFPB:Finally, the partial differential equation (18)

becomes

curl nFP curl A ¼ J� curlMFP ð20Þ

The discretization with nodal shape functionsfor the potential vector of Eq. (20) using theGalerkin method leads to the matrix system

½SFP�½A� ¼ ½J� � ½MFP�; ð21Þ

where the vector ½A� represents the nodal values ofvector potential, ½SFP� a square matrix calledstiffness matrix, ½MFP� and ½J� the vectors whichtake into account the magnetization MFP and thecurrent density J. One can note that the matrix½SFP� is constant because the permeability nFP isconstant as well. The non-linearities introduced byferromagnetic materials are reported in thesource term ½MFP� which depends on B (i.e. A).To take into account the coupling with theexternal circuit of a coil made up of strandedconductors flowed by a current i; a vector ½D� isintroduced such that ½J� ¼ ½D�i [21]. Then, weobtain the system

SFP �D

0 R

" #A

i

" #þ

0 0

Dt 0

" #d

dt

A

i

" #

¼0

u

" #þ

MFP

0

" #:

This system can be time discretized using an Eulerimplicit scheme.

6. Example of application

To compare the performances of both hysteresismodels implemented in finite element analysis, apower electronic transformer, which core iscomposed of N30 ferrites, has been studied.Conditions of simulation are such that the primarywinding is supplied by a square wave voltage V ¼400 V at a frequency of 80 kHz: In this case, as thecore is made of ferrites, dynamic effects can beneglected. The secondary winding is in series witha resistance R ¼ 10O: The 2D geometry of thesystem is given in Fig. 28. Due to the symmetryproperties, only a quarter of the structure isstudied. The structure mesh is chosen relativelyfine with 1801 elements (Fig. 29). We aim at


comparing local and global quantities obtainedfrom both models.

6.1. Calculation of local and global quantities

First, as local quantity, we can compare thehysteresis loops obtained from both models for agiven element of the mesh. Both models give closeresults for different elements of the mesh. Asexample, Fig. 30 gives the loops obtained on theelement located at the point P (see Fig. 28).The global quantities to be compared are

the primary and the secondary winding currents.As both models give very close results, onlywave shapes of currents and, for more con-venience, the difference DI between the currents,given by both models, are reported in Figs. 31and 32.As we can see, difference between both models is

negligible: the maximum gap is about 0.9% of theamplitude for the primary current and about 0.3%for the secondary current. The second global

quantity to be compared is the evaluation ofhysteresis losses. These are very important inelectrical engineering for machines design. Hyster-esis losses for both models have been determinedfrom the evolution of the density of losses dhversus Bmax:

Ph ¼ lX

elements

SedhðBemaxÞ ð22Þ

where the sum is over all the elements of the mesh.Se is the surface of the element e; l the depth of thesystem (in the third dimension) and Be

max themaximum magnitude of the magnetic flux densityon the element e. Results of the calculation arereported in Table 5.From these results, current and hysteresis losses

evaluation, we can conclude that, in this case, bothmodels give close results for global quantitiescalculation. Now, it is important to choose themodel with the best performances in terms ofcalculation time.

6.2. Calculation times

In Table 6, the ratio of computational times arereported for both models. The J–A model compu-tation time is taken as reference. Calculationhave been made for 1 period and 3 periods. Infact, the Preisach model first numerical step isthe loading of the experimental Everett functionswhich are used for the interpolation method. Itappears that, after some periods the ratio

Fig. 30. Hysteresis loops obtained from both models for an

element of the mesh.

Fig. 28. Geometry of the transformer.

Fig. 29. Mesh of the studied system: 1 quarter of the structure.


between the models calculation times stabilizes at2.25 in favor for the J–A model. Moreover, theconvergence of non-linear FEM procedure is quitethe same.These results show how important is the choice

of the user for the ratio (accuracy/calculation time)for the use of one of these models. Of course, thischoice has to be made in the case of a sinusoidalexcitation. In other cases, the Preisach model isrecommended.

7. Conclusion

Main results of the comparison are summarizedin Table 7. The Preisach model has been presentedwith an identification method based on the Everettfunction determination. This method needs themeasurement of rigorously centered hysteresis

Table 5

Hysteresis losses calculated from both models

J–A model Preisach model

Hysteresis losses (W) 5.69 5.94

Table 6

Ratio of computational times for 1 and 3 periods field

calculation (400MHz Digital WorkStation)

J–A model Preisach model

1 period 1 2.5

3 periods 1 2.25

(J–A model computational time is taken as reference.)

Fig. 31. Primary winding current wave shape and difference between the models.

Fig. 32. Secondary winding current wave shape and difference between the models.


loops otherwise gaps between experience andsimulation can be observed for the remanent andcoercive points, especially in the case of the MðBÞmodel. Nevertheless, results for the Preisach modelare more accurate whatever the kind of excitationused in this paper. At the opposite, although the J–A model has more physical basis, to obtain goodresults on a wide range of magnitudes, theparameters are calculated using an optimizationprocedure based on several centered loops. Interms of identification effort, both models areequivalent.Magnetic materials used for this study were

chosen for their different areas of application andfor their different hysteretic behavior. Then,results give us a good idea of which model to usein relation to the studied electromagnetic system(Table 7). For example, in the case of a sinusoidalexcitation, the use of the J–A model gives resultsclose to those of the Preisach model. Nevertheless,it has been observed that this model is less accuratethan the Preisach model for the representation ofthe N30 ferrites and FeSi sheets. In fact, thePreisach model is well suited for the study of thethree materials. But, results show that, in the caseof a sinusoidal excitation with a SMC material, theuse of the J–A model is recommended. The J–Amodel main drawback is the non-closure of minorloops when applying excitations with superposi-

tions of higher harmonics. In this case, thePreisach model should be preferred.Concerning the computational aspect, the J–A

model is simple to be implemented in a fieldcalculation code and it requires less computationaltime and memory. In fact, the Preisach modelrequires the memory vector to be stored inaddition to the current magnetic field. Whenstudying a complex electromagnetic system, espe-cially in finite element analysis, the magneticconstitutive relationship calculation must be fewtime consuming. The example of application withthe transformer show how it is important as thePreisach model requires more than 2 timescomputational time for similar results to those ofthe J–A model. Nevertheless, this point has to becompensated by the fact that the Preisach model ismore accurate than the J–A model in case of non-sinusoidal excitation.

Appendix A. Identification of the Everett function

for the M(H) model of Preisach

In the following, we present the method todetermine the Everett function from n centeredhysteresis loops whose magnitudes are between 0and Hmax: It is then possible to calculate EðHm;HÞfor ðHm;HÞA½0;Hmax� ½0;Hmax� with HoHm:

Table 7

Summary of the results

Excitation J–A model Preisach model (Everett)

Identification Two steps: iterative and Interpolation method for

effort optimization procedures MðHÞ and MðBÞ models.for five parameters. Experimental loops have

Set of experimental to be rigorously centered

centered hysteresis loops. for the MðBÞ model.N30 ferrites Sinusoidal ++ ++

Non-sinusoidal + ++

FeSi sheets Sinusoidal + ++

Non-sinusoidal + +

SMC material Sinusoidal ++ ++

Non-sinusoidal + ++

Computational effort � Require 2000 points per loop but fast � 100 points are sufficient but more time consuming

� Easy to be numerically implemented � Difficult to implement

++ good results

+ correct results


Only the model with H as entry is detailed.Nevertheless, it is possible to have a similarapproach to obtain the Everett function with themagnetic flux density B as entry. In this case, weobtain the magnetization versus the magnetic fluxdensity.If we consider a centered hysteresis loop of

magnitude Hm; from relation (6), the magnetiza-tion M along the descending part of the hysteresisloop is:

MðHÞ ¼ MðHmÞ � 2EðHm;HÞ: ðA:1Þ

From n measured hysteresis loops ði ¼ 1; nÞ with amagnitude Hmi; the function EðHmi;HÞ is calcu-lated for values of H belonging to ½�Hmi;Hmi� andfor each loop i: We obtain the set of functionsEðHmi;HÞ supporting the 3D Everett function(Fig. 33) with H between �Hmi and Hmi:We can notice that for the particular case where

H ¼ Hmi; the Everett function is equal to zero. Inthis case, the triangle Tðx; yÞ is a single point.Now, we have to calculate the Everett function forany point ðH 0

m;H0Þ belonging to the triangle D:

The proposed method is based on functions usedfor the magnetic field interpolation in the bidimen-sional finite element analysis method [13]. Usingthe symmetry property of the Everett functionEðHm;HÞ ¼ Eð�H;�HmÞ; it is possible to treatonly the case where the value of H is between �H 0

m

and þH 0m: We consider now the values Hmi�1 and

Hmi which correspond to the magnitudes of theknown curves set (Fig. 33) such asHmi�1oH 0

moHmi (Fig. 34).In the following, for more convenience, we

suppose that i ¼ 2 and the searched value will beinterpolated from the two curves with the lowestmagnitudes values Hm1 and Hm2: We have then todistinguish two cases:

* First case: jH 0jojHm1j: First, we search for thefields H11 and H12 whose Everett functionsvalues EðHm1;H11Þ and EðHm1;H12Þ are known(i.e. determined from experimental measure-ments). These fields verify H11oH 0oH12: Inthe same way, we determine H21 and H22 on thestraight line Hm ¼ Hm2 such as H21oH 0oH22:Then, four points a11; a12; a21; a22 are obtained(Fig. 35) which define a quadrilateral with the

point ðH 0m;H

0Þ inside. The value of the Everettfunction EðH 0

m;H0Þ corresponding to this point

is obtained by a quadratic interpolation methodfrom the value of the Everett function at points

Fig. 33. Curves set supporting the Everett function.

Fig. 34. Projection in the D triangle of the Everett functions.

Fig. 35. Interpolation method M1 used to calculate the Everett

function in the first case.


a11; a12; a21 and a22:

EðH 0m;H

0Þ ¼ðH 0

m � Hm1ÞðHm2 � Hm1Þ

½EðHm2;H0Þ

� EðHm1;H0Þ� þ EðHm1;H

0Þ

ðA:2Þ

with

EðHmi;H0Þ ¼

ðH 0 � Hi1ÞðHi2 � Hi1Þ

½EðHmi;Hi2Þ

� EðHmi;Hi1Þ� þ EðHmi;Hi1Þ

ðA:3Þ

for i ¼ 1; 2:* Second case : jH 0j > jHm1j: The previous proce-

dure has to be slightly modified in the casewhere H 0 is greater than Hm1: In this case, weconsider the straight line crossing the pointsðHm1;Hm1Þ and ðH 0

m;H0Þ: The intersection point

with the vertical line Hm ¼ Hm2 gives us a pointðHm2;HcÞ (Fig. 36).

Finally, we search for the points H21 and H22

such as H21oHcoH22 and the Everett functionvalue is then obtained by

EðH 0m;H

0Þ ¼ðH 0

m � Hm1ÞðHm2 � Hm1Þ

½EðHm2;HcÞ

� EðHm1; aHm1Þ� þ EðHm1; aHm1Þ

ðA:4Þ

with a ¼ 1 if H > 0; else a ¼ �1;

EðHm2;HcÞ ¼ðHc � H21ÞðH22 � H21Þ

½EðHm2;H22Þ

� EðHm2;H21Þ� þ EðHm2;H21Þ

ðA:5Þ

The presented interpolation method, denomi-nated as M1, allow us to insure the continuity ofthe Everett function on the domain. This char-acteristic is necessary to obtain continuous MðHÞloops. Moreover, we observe an exact restitutionof the experimental loops used for the interpola-tion method.

Appendix B. Identification of the Everett function

for the M(B) model of Preisach

The Preisach model can also be adapted with B

as entry [14]. In this case, the shape of the Everettfunction which has to be determined is totallydifferent from the one previously presented in theMðHÞ case. To calculate the Everett function fromthe MðBÞ model, we can still use the same methodas presented for the MðHÞ model. Nevertheless,the interpolation method has to be modified in thiscase. In fact, the use of the interpolation methodM1 gives deformed hysteresis loops at lowmagnitudes of excitation. Then, we have chosenanother interpolation method which is similar tothe previous one, and denominated as M2(Fig. 37). As for the M1 method, the Everett

Fig. 36. Interpolation method M1 used to calculate the Everett

function in the second case. Fig. 37. Presentation of the interpolation method M2.


function is determined for the point ðH 0m;H

0Þ andwe have the same definition for the fields Hm1 andHm2: Using the straight line d crossing the originand the point ðH 0

m;H0Þ; we determine the intersec-

tions b1ðH1;Hm1Þ and b2ðH2;Hm2Þ with the verticallines H ¼ Hm1 and H ¼ Hm2: Then, points a11;a12; a21 and a22 are determined such as presented inFig. 37. As for the method presented in AppendixA, these four points are used to calculate theEverett function EðH 0

m;H0Þ: The used interpola-

tion function is then given by the Eq. (A.2).It must be noted that, at the opposite to the

method M1, method M2 needs the treatment ofonly one case.

References

[1] L.O. Chua, K.A. Stromsmoe, IEEE Trans. Circuit Theory

CT-17 (4) (1970) 564.

[2] M.L. Hodgdon, IEEE Trans. Magn. 24 (6) (1988) 3120.

[3] F. Preisach, Z. Phy. 94 (1935) 277.

[4] I.D. Mayergoyz, Mathematical Models of Hysteresis,

Springer, New York, 1991.

[5] D.C. Jiles, D.L. Atherton, Theory of ferromagnetic

hysteresis, J. Magn. Magn. Mater. 61 (1986) 48.

[6] D.C. Jiles, J.B. Thoelke, M.K. Devine, IEEE Trans. Magn.

28 (1992) 27.

[7] J. Cros, P. Viarouge, ICEM’02 Bruges, Belgium 2002,

CDROM, paper No. 514.

[8] H. Debruyne, S. Cl!enet, F. Piriou, Math. Comput.

Simulation 46 (1998) 301.

[9] G. Bertotti, V. Basso, J. Appl. Phys. 73 (10) (1993) 5827.

[10] G. Biorci, D. Pescetti, Il Nuovo Cimento 7 (6) (1958) 829.

[11] D. Everett, Trans. Faraday Soc. 51 (1955) 1551.

[12] S. Cl!enet, F. Piriou, MGE 2000, France, Lille, December

13–14, 2000, pp. 71–74.

[13] O.C. Zienkiewicz, R.L. Taylor, The Finite Element

Method, McGraw-Hill, London, New York, 1989.

[14] G.S. Park, S.Y. Hahn, K.S. Lee, H.K. Jung, IEEE Trans.

Magn. 29 (2) (1993) 1542.

[15] N. Sadowski, N.J. Batistela, J.P.A. Bastos, M. Lajoie-

Mazenc, Compumag, France, Evian, Vol. 4, July 2–5,

2001, pp. 246–247.

[16] S. Cl!enet, J. Cros, F. Piriou, P. Viarouge, L.P. Lefebvre,

Compel 20 (1) (2001) 187.

[17] P.I. Koltermann, L.A. Righi, J.P.A. Bastos, R. Carlson, N.

Sadowski, N.J. Batistela, Physica B 275 (2000) 233.

[18] E. Della Torre, IEEE Trans. Magn. 27 (4) (1991)

3697.

[19] O. Bottauscio, D. Chiarabaglio, M. Chiampi, M. Repetto,

IEEE Trans. Magn. 31 (6) (1995) 3548.

[20] V. Ionita, B. Cranganu-Cretu, B. Iona, IEEE Trans.

Magn. 32 (3) (1996) 1128.

[21] F. Piriou, A. Razek, IEEE Trans. Magn. 28 (1992) 1295.


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