A novel model for magnetic hysteresis of silicon-iron sheets S. Boukhtache1, B. Azoui1, M. Féliachi2 1) Laboratoire LEB, Département Electrotechnique, Faculté des Sciences de l'Ingénieur, Université de Batna, Rue Chahid Med Elhadi Boukhlouf, 05000 Batna, Algérie, sebti_boukhtache @yahoo.fr , [email protected] 2) GE44/LRTI, CRTT Bd de l'Université, BP 406, 44602 Saint-Nazaire, France [email protected]

A new approach to calculate the magnetic hysteresis, based on the Brillouin theory associated with the Jiles-Atherton approach, is presented. This study represents a general model compared with the classical Jiles-Atherton one. A r Brillouin function, using the kinetic atomic moment J , allows to determine accurately the value of the anhysteretic magnetization. The obtained results are compared with experimental ones of the silicon- iron sheets. PACS numbers: 81.40.Rs, 85.70.Ay, 74.25.Ha

2. Anhysteretic magnetization in classical Jiles- Atherton Model

1. Introduction To determine the magnetic hysteresis loop and magnetization curve, several mathematical models are proposed in literature [1]. We can quote analytical models, Preisach model and the Jiles-Atherton model. The more reliable model is the one taking into account the physical phenomenon at atomic scale. At first sight, the model which takes into consideration this criteria, among the models quoted above, is the Jiles–Atherton one [2, 3, 4, 5]. However, the use of this model is not always adequate to reproduce the hysteresis loop [6]. Then, to reproduce the initial magnetization curves, hysteresis loops and the symmetrical minor loops, the Jiles-Atherton approach is refined by modifing the expression of the anhysteretic magnetization. The Brillouin function instead of the Langevin one is used, bearing in mind that the theory of these functions is discussed in [7, 8]. The obtained results are then compared with the curves applied to the silicon-iron sheets given by the research group (GDR) [9].

The anhysteretic magnetization M an used in Jiles model is as follows [2]:

M an = M s L(y) L(y) = coth(y) − y =

a=

(1)

1 y

µ 0 m(H + αM)

k BT

k BT M s = 3 µ0 m

(2) =

H + αM a

⎞ ⎛ 1 ⎜⎜ + α ⎟⎟ ⎠ ⎝ χ an

(3) (4)

where M s is the saturation magnetization, L(y) the µ0 Langevin function, the vacuum permeability, m the magnetic moment, H the applied field, α the averaging parameter of the magnetic field, M the magnetization of the material, k B the Boltzmann constant, T the considered temperature, χ an the anhysteretic

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susceptibility at the origin and α the parameter of the anhysteretic curve shape. Equation (4) is obtained from Langevin function [2,7]. This function takes into account the atomic structure of the magnetic material which has been deduced from the classical physics, while the atomic system is studied by the quantum physics. However, the Langevin function is not adequate for the most transition metals [7].

dM irr M an − M irr = dH kδ − α (M an − M irr )

Knowing that the anhysteretic magnetization is the main part in Jiles-Atherton model, using the Langevin function which has shown its limitations, a new expression is needed to fulfil this requirement. Hence, the anhysteretic magnetization is determined by using Brillouin function [10]. It is reminded that this function is deduced from the quantum physics which describes well the atomic magnetic moment [7,8]. In this case, the anhysteretic magnetization is given by:

B J (y) = y =

k BT k BT a= µ 0 Jgm B

(H + αM ) = H + αM a

(10)

M = M rev + M irr

(11)

⎧+ 1 ⎩− 1

δ =⎨

for dH dt > 0 for dH dt < 0

(12)

a , α , c and k are the modified Jiles parameters. Their Expressions are determined by following the same procedure presented in [2]. The parameter of the anhysteretic curve shape a is given by:

a=

(5)

2J + 1 ⎛ 2J + 1 ⎞ 1 ⎛1 ⎞ coth ⎜ y ⎟ − coth ⎜ y ⎟ 2J ⎝ 2J ⎠ 2J ⎝ 2J ⎠

µ 0 Jgm B

M rev = c(M an − M irr )

where M irr is the irreversible magnetization, M rev the reversible magnetization and M is the total magnetization. δ is the directional parameter given by:

3. Proposed Model

M an = M s B J (y)

(9)

⎞ J +1 M s ⎛ 1 ⎜⎜ + α ⎟⎟ J 3 ⎝ χ an ⎠

(13)

The parameter of the reversible movement of the domain wall c is given by:

(6) (7)

c=

χ in χ an

(14)

(8) where χ in the initial normal susceptibility. The pinning parameter of the domain wall k ( A / m ) is:

where J is the quantum number, g the Landé splitting factor and mB the Bohr magneton. In equation (8), a is a parameter with dimensions of magnetic field (A/m) which characterises the shape of the anhysteretic magnetization. To complete the proposed model, equation (5) is integrated in the Jiles-Atherton model as follows:

⎧ ⎫ ⎪ ⎪⎪ M (H ) ⎪ 1 k = an c ⎨α + ⎬ 1 c M an (H c ) ⎪ 1− c ⎪ χ max − ⎪⎩ 1− c 1 − c dH ⎪⎭ (15)

2

where

χ max is the differential susceptibility at the

Start

coercive point and H c the coercive field. For very soft magnetic materials it is found that k = Hc . The averaging parameter of the magnetic field α representing interdomain coupling is:

Data Input : M s , χ an , χ in , χ r , χ max , M r , H c Calculate c using equation (14)

⎡ ⎤ ⎢ ⎥ k 1 α = (1 − c )⎢ − ⎥ ⎢ M r − M an (M r ) χ − c dM an (M r ) ⎥ r dH ⎣⎢ ⎦⎥ (16)

α = 0.001

Calculate a using equation (13)

where M r and χ r are the magnetization and differential susceptibility at the remanence point respectively. It is noticed that the above equations gouverning the modified Jiles parameters are coupled. Therefore, a numerical method is necessary for calculating the different values of these parameters by using successive iteration as it is shown in the flowschart (Fig.1). Comparison of values obtained in successive iterations is made, and the procedure is completed if there is no significant change, and the solution obtained is considered as a final values of the parameters.

Calculate k using equation (15) Calculate α using equation (16)

No Converging test

Yes Solution End Fig. 1. Flowchart of the parameters calculation

4. Determination of the quantum number J The experimental results show that the origin of magnetic phenomena in 3d transition metals series are the spin kinetic moments [7, 8]. Hence, the quantum number is calculated as follows: the material used is the (3wt%) siliconiron and the silicon atoms have a zero magnetic moment. The material is considered as an isotropic one. So, the substitution of iron atoms

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by silicon atoms reduces the magnetic moments density in the material. Therefore, the saturation magnetization of the material at 0°K decreases linearly with the concentration of alloy elements according to the relationship [8]:

M s = 2.2µ0 (1 − 1.8C )

the modified Jiles parameters. Figs. 2, 3, 4 and 5 show respectively the anhysteretic magnetization curves, the initial magnetization curves, the hysteresis major loops and the hysteresis symmetrical minor loops without saturation. The results show a good agreement between both the proposed model (PM) and the measured data. Whereas, there is no agreement between the classical Jiles-Atherton model and the measured data. This disagreement is due to the presence of the Langevin function in the model. In fact this appears obvious, since the Langevin function is a particular case of the Brillouin function for which J becomes infinite [7, 8]. According to [8], both functions coincide for J = 20 . However, this value appears high for the proposed (3wt%) siliconiron material where, the J value is equal to 1. This is a proof of the choice of the Brillouin function instead of the Langevin one.

(17)

where C indicates the weighted concentration of silicon in alloy. At 0°K, the saturation magnetization of the (3wt%) silicon-iron alloy is close to 1.656×106A/m instead of 1.7×106A/m in pure iron material and the number of moments par unit volume of iron is 9.455×1028/m3. For (3wt%) silicon-iron alloy, the number of moments per unit volume is 9.171×1028/m3 and the average magnetic atomic moment (m) is 18.056×10-24 Am2 and it is equal to 1.947 mB in term of Bohr magneton. Following the classical law [7, 8], the magnetic moment is given by: m = JgmB

Table 1. Modified Jiles Parameters Model A(A/m) c k(A/m) α JM 105.86 2.53x10-4 8.54x10-3 54.28 BM 159.03 1.61x10-4 8.54x10-3 54.11

(18)

The magnetic moment in 3d transition metals series is caused by the spin motion. Under this condition, the Landé splitting factor is equal to 2 which gives J equals to 0.974. As J values are either integer or half integer, thus, the value of J can be approximated by J = 1.

1,8 1,5

Man(T)

1,2 5. Modelling results As far as the magnetization versus the temperature M(T) is concerned the results obtained by Brillouin function are best suited for Fe Ni and Co materials [7, 8]. According to this idea, a novel expression of anhysteretic magnetization is proposed to plot the bulk magnetisation versus the magnetic field M(H) for the silicon-iron sheets. The equations system of the proposed model have been solved according to [2]. The obtained results are compared with those measured. Table 1 shows

0,9 0,6 PM J AM Meas.

0,3 0,0 0

600 800 1000 1200 H( A/ m) Fig. 2. Anhysteretic magnetization curves: Measured, modelled with classical JilesAtherton model (JAM) and Proposed model (PM, J=1)

4

200

400

1,5

1,5

1,0

1,2

0,5

B(T)

B(T)

1,8

0,9

0,0

0,6

-0,5

PM J AM Me a s.

0,3 0,0 0

PM J AM Me a s.

200

400

600 800 H( A/ m)

1000

-1,0 -1,5 -120

1200

Fig. 3. Initial magnetization curves: Measured, modelled with classical Jiles-Atherton model (JAM) and Proposed model (PM, J=1)

-80

-40

0

40 H ( A / m)

80

120

Fig. 5. Hysteresis symmetric loops without saturation: Measured, modelled with classical Jiles-Atherton model (JAM) and Proposed model (PM, J=1)

1,8

B(T)

0,9

PM J AM Me a s.

6. Conclusion A novel model of magnetic hysteresis is developped by using the Brillouin function associated with the Jiles-Atherton Model. By this approach, we have improved the Jiles model. A good agreement is obtained between measured data and modelling results applied to (3wt%) silicon-iron sheets. The proposed model can be applied to different magnetic hysteresis curves generated in order to perform its validity. However, an optimisition algoritm, such as the simulated annealing algorithm, is suitable to minimise the knee part in the curves.

0,0

-0,9

-1,8 -1200

-800

-400

0

400

800

1200

H( A/ m)

Fig. 4. Hysteresis major loops: Measured, modelled with classical Jiles-Atherton model (JAM) and Proposed model (PM, J=1)

Acknowledgment The authors gratefully thank the Research Group (GDR) especially the CEGELY (Centre de Génie Electrique de Lyon, France) for their cooperation by supplying measured data. They also thank Dr. Fouad Azzouz and Dr. Yassine Ouled Amor (Ge44 Lab., France) for their help.

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References 1. A. Ivany, Hysteresis Models in Electromagnetic Computation, (Akademiai kiadio, Budapest, Hungary, 1997). 2. D. C. Jiles, J. B. Theolke, and M. K. Devine, IEEE Transaction on Magnetics, 28, 27 (1992). 3. D. C. Jiles and J. B. Theolke, IEEE Transaction on Magnetics, 25, 3928 (1989). 4. D. C. Jiles and D. L. Atherton, Journal of Magnetism and Magnetic Materials, 61, 48 (1986). 5. D. C. Jiles and D. L. Atherton, IEEE Transaction on Magnetics, 19, 2183 (1983). 6. D. Lederer, H. Igarashi, A. Kost and T. Honma, IEEE Transaction on magnetics, 35, 1211 (1999). 7. D. C. Jiles, Introduction to Magnetism and Magnetic Materials, 2nd edn., (Chapman & Hall, London, 1998). 8. P. Brissonneau, Magnétisme et Matériaux Magnétiques pour l’Electrotechnique, (Hermès, Paris, 1997). 9. F. Alves, et al. , Journées SDSE, 23, (Lyon, 20/21/Janvier 2000). 10. S. Boukhtache and M. Feliachi, in Proceedings II of 13th COMPUMAG Conference on Computation of Electromagnetic Fields, 2001, Evian, France, p. 14.

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