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Modified JilesAtherton model and parameters identification using falseposition method

M. Hamimid a,b,, M. Feliachi a, S.M. Mimoune b

a IREENA-IUT, CRTT, 37 Boulevard de lUniversite, BP 406, 44602 Saint Nazaire Cedex, Franceb Laboratoire de modelisation des systemes energetiques LMSE, Universite de Biskra, 07000 Biskra, Algeria

a r t i c l e i n f o

Article history:

Received 10 December 2009Accepted 12 January 2010

Keywords:

Hysteresis

Classical and modified JilesAtherton

model

Parameters identification

False position method

a b s t r a c t

In this paper, a modified JilesAtherton model is proposed. This model uses a physical meaning by

introducing the magnetizationMinstead of the irreversible magnetization Mirrin the effective magneticfieldHe-magnetic fieldHrelationship. The false position method is coupled to the iterative algorithm to

identify the JilesAtherton parameters for both classical and modified JilesAtherton model. These

parameters are evaluated by the resolution of three nonlinear equations obtained from three

conditions. The validity of the modified model is done by comparing the obtained hysteresis loops to

the experimental ones.

& 2010 Elsevier B.V. All rights reserved.

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1947

2. Modified JilesAtherton hysteresis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1948

3. Determination of the modified JilesAtherton parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1948

4. Identification technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19495. Results and discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1949

6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1950

Work context. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1950

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1950

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1950

1. Introduction

Different models have been proposed to describe the hyster-

esis phenomenon[13]. Some of them are physical in nature[1,2]

and others ignore the physical behavior of materials [3]. Amongthe physical models, the JilesAtherton is the most and widely

used[1].

To evaluate the effective magnetic field, Jiles and Atherton use

the irreversible magnetization instead of the total magnetization

to derive a simplified formula of the total differential suscept-

ibility [4]. Our purpose in this work is to present a modified

JilesAtherton (MJA) model to take into account the total

magnetization to evaluate the equivalent magnetic field. The

expression of the total differential susceptibility obtained from

this modification is completely different to that obtained by Jiles

and Atherton (JA). The integration of this model in a calculationcode requires the knowledge of the exact parameters. The

parameters of both models JA and MJA require an identification

technique. Several techniques are used to identify the JA model

parameters [59].

In this work, we have used a technique which combines an

iterative algorithm and a false position method FPM for both

models. The parameters that we want to identify in the MJA are

grouped into three nonlinear equations and solved by the FPM.

This technique needs some measurements data such as the

initial, the coercive and the remanence susceptibilities. It requires

also the introduction of the coercivity field intensity, the remanence

magnetization and the coordinates of the saturation point.

ARTICLE IN PRESS

Contents lists available atScienceDirect

journal homepage:www .elsevier.com/locate/physb

Physica B

0921-4526/$- see front matter& 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.physb.2010.01.078

Corresponding author at: IREENA-IUT, CRTT, 37 Boulevard de lUniversite, BP

406, 44602 Saint Nazaire Cedex, France.

E-mail addresses: [email protected] (M. Hamimid), mouloud.

[email protected] (M. Feliachi),[email protected] (S.M. Mimoune).

Physica B 405 (2010) 19471950
http://-/?-http://www.elsevier.com/locate/physbhttp://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.physb.2010.01.078mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.physb.2010.01.078http://www.elsevier.com/locate/physbhttp://-/?-


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Experimental results of the hysteresis loops are conducted to

validate the MJA model with comparison to the JA model.

2. Modified JilesAtherton hysteresis model

In the JilesAtherton model, the magnetization M is decom-

posed into reversible component Mrev, representing the transla-

tion and the reversible rotation of the walls within ferromagneticmaterials, and the irreversible componentMirrwhich corresponds

to domain wall displacement against the pinning effect, M MrevMirr. In their model, Jiles and Atherton suppose that the

magnetization M is equivalent to the irreversible magnetizationMirrin the relation which binds the magnetic effective fieldHeand

the magnetic field H:

He HaMirr 1

In this relation, they suppose that the reversible magnetization is

negligible compared to the irreversible magnetization Mirr and

they obtained the differential irreversible susceptibility in the

simplified form:

dMirr

dH

ManMirr

kdaManMirr 2

And finally they determined the total differential susceptibilitydM=dH in the usually form:

dM

dH

1cManMirr

kdaManMirrc

dManHe

dH 3

The hypothesis of neglecting Mrev in the relation (1) has not a

physical meaning and conduct to imprecisely result in general

case. Our purpose is to use the physical relation relating the

magnetic effective field He and the magnetic field H:

He HaM 4

By using Eq. (4) the differential irreversible susceptibility is

obtained as

dMirrdH

ManMirr 1acdMandH

kda1cManMirr

5

We can see clearly that ifc=0 in Eq. (5) this relation becomes as

the same form as Eq. (2) because the parameter cis related to the

reversible magnetization as

Mrev cManMirr 6

After several manipulations, given in Appendix A, we arrived

at the following relation of magnetic total differential suscept-

ibility:

dM

dH

1cManMirr kcddMandHe

kda1cManMirrkcaddMan

dHe

7

As we can see, this relation (7) of MJA model is completely

different from the relation (3) of JA model.

3. Determination of the modified JilesAtherton parameters

The total differential susceptibility can be written by using

1cManMirr ManMas

w dM

dH

ManM kcddMandHe

kdaManMkcaddMan

dHe

8

or

w Z

kdaZ 9

with the following complementary relationships:

Z ManM kcddMandHe

10a

Man Ms coth He

a

a

He

10b

dMandHe

Ms

a 1coth2

Hea

a

He

2 ! 10c

d 1;

dH

dtZ0

1;dH

dto0

8>>>: 10d

The parameters of the MJA model have different expressions

compared to those of JA model and are determined from

experimental data using the differential susceptibilitywinat initialpoint (H=0,M=0), the differential susceptibility wrat remanence

point (H=0, M=Mr), the differential susceptibility wc at thecoercive point (H=Hc, M=0) and the differential susceptibility

wmat the loop tip (H=Hm, M=Mm).From these measured magnetic properties, the parameters

governing the hysteresis equations are obtained from the new

expression of the total differential susceptibility (9).

The initial susceptibility is calculated at the initial point (H=0

andM=0) of the initial magnetization curve with d=+1:

win dM

dH

H 0; M 0

ZinkaZin

11

The parameter c is associated to the reversible magnetization, at

the beginning of the initial magnetization curve and is deter-mined using the initial susceptibility

With

dMandHe

H 0;M 0

Ms3a

In this conditionZin kcMs=3a, whena and a are initially known,the parameter c is determined by using the Eq. (11) and is

expressed as

c 3awin

1awinMs12

At the coercive point (H=Hc, M=0) with d= +1, the differential

susceptibilitywcis given by

wc dM

dH

H Hc;M 0

ZckaZc

13

with

Zc ManHc ckdManHc

dHe

The relationship between the parameters k and wc is obtainedusing Eq. (13), whena,aandcare previously determined, Eq. (13)can be written as follows:

fck Zc1awckwc 0 14

This is the first nonlinear equation, the term Zcis function of the

parametersk, cand a .

M. Hamimid et al. / Physica B 405 (2010) 194719501948


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At remanence point (H=0, M=Mr), the differential suscept-

ibilitywris given with d= 1:

wr dM

dH

H 0;M Mr

Zr

kdaZr15

with

Zr ManMrMr ckddManMr

dHe The relationship between the parameter a and the susceptibilityat the remanence point wris given by

fra Zr1awr kwr 0 16

frrepresents the second nonlinear equation and is extracted from

Eq. (15), where the term Zr is also function ofk ,a , cand a.Finally, the use of the coordinates of loop tip (H=Hm,M=Mm),

withd= +1, the differential susceptibilitywmcan be written as

wm dM

dH

H Hm ;M Mm

ZmkaZm

17

with

Zm ManHm; MmMm ckdManHm; Mm

dHe

The parameterais determined using the differential susceptibilitywm at the loop tip; on the other hand, parameters c, a and k areknown and the parametera is obtained by solving:

fma Zm1awmkwm 0 18

The last nonlinear equation is given by Eq. (18) where the termZmis function ofk ,c,a and a .

The parameter Ms is the magnetization value at strong

saturation. It is probably the easiest parameter to obtain; this is

often known for a particular material; in the most cases it is given

by the constructor of the metallic sheet.

The parameters c, k, a and a are nonlinearly coupled. Todetermine them, we propose to solve successively the Eqs. (12),

(14), (16), and (18) using an iterative algorithm coupled with theFPM.

4. Identification technique

This technique is based on the coupling of an iterative

algorithm with the FPM to identify the parameters of both MJA

and JA models. The expressions giving the JA model parameters

are the same as in Ref.[4]. In the MJA the parametercis obtained

by using Eq. (12) with a given initial values of the parameters aanda, and we introduce all of them into Eq. (14) which is resolved

by FPM to find a first estimation ofk.

The parametersa and a are calculated from Eqs. (16) and (18)using the same FPM. The current values ofa and a are used to

determine the new value of the parameter c. The calculationprocess is repeated until the satisfaction of the criterion

jcnewcold=cnewjre, wheree is a given small number.The FPM starts with two pointsaandb such thatc(a) andc(b)

are of opposite sign, implying that the function c has at least one

zero in the interval [a, b] [10]. The method has to produce a

decreasing sequence of intervals [ak,bk], which all contain a zero

ofc . In step k, the numberbk is calculated by

bk ak akbk

cakcbkcak 19

Ifc(ak) andc(bk) have the same sign, thenak+1=bkand bk+1=bk,

elsebk+1=bkandak+1=ak. This process is repeated until c(bk)E0.

For example we calculate the parameter a with FPM using

Eq. (19): we have from Eq. (16), fra Zr1awr kwr and we

seek a such as f(a)=0; let start with two points a1=a anda2=bsuch asf(a1) andf(a2) are of opposite sign. In step k, the numberbk, where bk is the solution in step k, is calculated by

1. bk akakbk=fakfbkfak,

2. iff(ak) f(bk)40 thenak+1=bkand bk+1=bk; else bk+1=bk andak+1=ak,

3. if |f(bk)|rethen end; else go to 1 (eis a given small number).

5. Results and discussion

To identify MJA and JA parameters, the algorithm proposed

needs some measured parameters that are presented in Table 1.

These ones are extracted from the experimental loop. The

identified parameters are presented inTable 2.

The JA model and the MJA model hysteresis loops, using the

identified parameters, respectively, ofTable 1are compared to the

experimental ones.Figs. 1 and 2show the hysteresis loops of both

models. A good agreement between measured and simulated

hysteresis loops is obtained for both MJA and JA models; this

Table 1

Parameters extracted from experiment.

Parameters Measured

win 184.12wm 0.0443wr 1.9725 10

3

wc 9.974 105

Hm (A/m) 1.039 103

Hc(A/m) 69.37

Mm (A/m) 1.134 106

Mr(A/m) 8.905 105

Table 2

Identified JA model parameters.

Identified parameters JA model MJA model

Ms(A/m) 1.18 106 1.18 106

a (A/m) 46.858 46.9605

k(A/m) 81.10 79.001

a 1.4843 104 1.507 104

c 0.0219 0.0214

-1500 -1000 -500 0 500 1000 1500

-1.5

-1

-0.5

0

0.5

1

1.5

magnetic field H (A/m)

magneticfluxdensityB

(T)

measured

JA

MJA

Fig. 1. Measured and simulated magnetic flux density.

M. Hamimid et al. / Physica B 405 (2010) 19471950 1949


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ARTICLE IN PRESS

shows the effectiveness of the technique used to identify modelparameters on the one hand, on the other hand we can see that

the MJA model hysteresis loop fits better the experimental one.

The parameters obtained by this technique have a physical

interpretation, such as the parameter k, in the case of soft

magnetic materials, it is like similar to the coercive field.

6. Conclusion

The employment of the exact expression of the effective

magnetic field relation by using the total magnetization instead of

the irreversible one improves the hysteresis evaluation of the

JilesAtherton model. A technique based on iterative algorithm

coupled with the false position method is introduced to identify

the parameters of the MJA and the JA models to derive thehysteresis loops. These parameters are compared to the experi-

mental data. The MJA model hysteresis loop fits better the

experimental loop compared to the JA model hysteresis loop.

This model can be improved by taking into account the

dynamic behavior of soft and hard magnetic materials.

Work context

The present work is carried out within an AlgerianFrench

cooperation PHI-Tassili program.

Appendix A

The total magnetization given by JilesAtherton is

M MrevMirr A:1

By using Eq. (6) the total magnetization becomes

M 1cMirr cMan A:2

The total differential susceptibility is given by

dM

dH 1c

dMirrdH

cdMandH

A:3

We can also write this susceptibility as the following form:

dM

dH 1c

dMirrdHe

cdMandHe

dHedH

A:4

wheredMan=dHe is given by Eq. (10c), andHe is given by Eq. (4)

dMirrdHe

ManMirr

kd A:5

The expression (A.5) is given in Ref. [4], by insert Eq. (A.5) in

Eq. (A.4) we obtained

dM

dH 1c

ManMirrdHe

cdMandHe

dHedH

A:6

By replacing 1cManMirr ManM and dHe=dH by

1adM=dHthe expression (A.6) becomes

dM

dH

Zkd

1a

dM

dH

A:7

Z is given by Eq. (10a).Finally the total differential susceptibility is obtained as given

by Eq. (7).

References

[1] D.C. Jiles, D.L. Atherton, J. Appl. Phys. 55 (1984) 2115.[2] E. Dlala, J. Saitz, A. Arkkio, IEEE Trans. Magn. 42 (8) (2006) 1963.[3] M.L. Hodgdon, IEEE Trans. Magn. 24 (6) (1988) 3120.[4] D.C. Jiles, J.B. Thoelke, M.K. Devine, IEEE Trans. Magn. 28 (1992) 27.[5] J. Izydorczyk, J. Magn. Magn. Mater. 302 (2006) 517.[6] K. Chwastek, J. Szczyglowski, J. Magn. Magn. Mater. 314 (2007) 47.[7] E.D.M. Hernandez, C.S. Muranaka, J.R. Cardoso, Physica B 275 (2000) 212.[8] P. Andrei, L. Oniciuc, A. Stancu, L. Stoleriu, J. Magn. Magn. Mater. 316 (2007)

330.[9] A. Salvini, F. Riganti Fulginei, IEEE Trans. Magn. 38 (2002) 873.

[10] A. Quarteroni, A. Sacco, F. Saleri, Me thode Numeriques, Algorithmes, Analyseet Applications, Springer, 2007.

0 200 400 600 800 1000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

magnetic field H (A/m)

magneticflu

xdensityB(T)

measured

JA

MJA

Fig. 2. Measured and simulated magnetic flux density (zoom up).

M. Hamimid et al. / Physica B 405 (2010) 194719501950

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