ARTICLE IN PRESS
Physica B 343 (2004) 59–64
Parameter identification of Jiles–Atherton model with nonlinear least-square method Pe! ter Kis, Ama! lia Iva! nyi* Department of Electromagnetic Theory, Budapest University of Technology and Economics, Budapest 1521, Hungary
Abstract A new method to the parameter identification of the widely used scalar Jiles–Atherton (J–A) model of hysteresis is detailed in this paper. The extended J–A model is also investigated including the eddy-current and the anomalous loss terms, which are taken into account by modeling the frequency dependence of the hysteresis. The five parameters of the classical J–A model can be determined from low-frequency hysteresis measurement. At higher frequency the effect of the eddy currents is not negligible, the J–A model must be extended. The loss of the hysteresis characteristics and the coercitive field are increasing with the frequency. Nonlinear least-squares method is used for parameter fitting of classical and extended J–A model, as well. The curve fitting is executed automatically based on the initial parameters and the measured data. r 2003 Elsevier B.V. All rights reserved. Keywords: Hysteresis modeling; Jiles–Atherton model; Eddy current effect
1. Introduction Two kinds of methods are used for determining the parameters of J–A model. Some step-by-step methodologies are available for parameter identification of J–A model from experimental hysteresis curves. These methods are based on the physical meaning of the determinable parameters [1]. Another approach to the problem consists in mathematical description of the parameter identification disregarding the physical background. This mathematical approach works in the statespace of the J–A model parameters and tries to
*Corresponding author. Tel.: +36-1-463-2915; fax: +36-1463-3189. E-mail addresses:
[email protected] (P. Kis),
[email protected] (A. Iv!anyi).
find the optimal combination of the parameters, which leads to the global minimum. An extension of the J–A model of hysteresis that takes into account the effects on the hysteresis curves of eddy current in electrically conducting media has been applied. In the derivation presented it is assumed that the frequency of the applied field is low enough that the skin effect can be ignored so that the magnetic field penetrates uniformly throughout the material. In this case, the DC hysteresis equation is extended by the addition of classical eddy-current-loss term depending on the rate of change of magnetization with time, the resistivity of the material, and the shape of the specimen, and on an anomalous eddycurrent-loss term, which depends on ðdB=dtÞ3=2 : In the limit as the frequency of the magnetic field tends to zero, the frequency-dependent hysteresis curve approaches the DC curve.
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.041
ARTICLE IN PRESS ! P. Kis, A. Ivanyi / Physica B 343 (2004) 59–64
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The classical quasi-static and extended scalar J–A model is detailed, and a possible procedure of parameter identification is presented in this paper.
2. Jiles–Atherton model equations 2.1. Quasi-static model The average magnetic moment per unit volume M is comprised of an irreversible component Mirr and a reversible component Mrev ðM ¼ Mrev þ Mirr Þ [1–4]. The following differential equation must be solved for obtaining Mirr: Mirr Man þ kd
dMirr ¼ 0; dHe
ð1Þ
where He ¼ H þ aM is the effective field taking into account the domain interactions, a is the molecular field parameter, k parameter is linked to the coercitive field (Table 1), d parameter is d ¼ signðHe Þ and finally Man is the anhysteretic curve, which follows the Langevin function H þ aM a Man ¼ MS coth : ð2Þ a H þ aM The differential susceptibility according to the J–A model can be expressed as follows: dM Man M dMan þc ¼ dM ; dHe kd dHe
ð3Þ
where dM parameter precludes the possibility of the non-physical solutions of (1): 8 ’ > < 0 : He o0 and Man M > 0; dM ¼ 0 : H’ e > 0 and Man Mo0; ð4Þ > : 1 : otherwise: J–A model with eddy current and anomalous loss. Table 1 Physical properties of J–A model parameters Parameter
Physical property
Ms a a c k
Saturation magnetization Domaininteraction Shape parameter of Man Reversibility coefficient Linked to hysteresis loss
The eddy current and the anomalous losses can be taken into account as dWbat ¼ dWmag þ dLmag þ dLEC þ dLA ;
ð5Þ
where dWbat is the work done by an external source (battery), dWmag is the change in the internal energy of the material, dLmag is responsible for the losses in the magnetization process, dLEC is the eddy current energy loss and dLA is the energy loss for the domain wall motion (anomalous loss). The classical eddy current loss is obtained by solving the Maxwell equation r E ¼ dB=dt for a given geometry, assuming that the magnetic field penetrates uniformly throughout the material. The classical eddy current instantaneous power loss per unit volume is proportional to the square of the rate of change of magnetization [2,5,6]: dLEC d 2 dB 2 d 2 m20 dM 2 ¼ ¼ ; ð6Þ dt 2rb dt 2rb dt where r is the resistivity in O m, d is the crosssectional dimension in meters and b is a geometrical factor which varies from b ¼ 6 in laminations to b ¼ 16 in cylinders and b ¼ 20 in spheres. The anomalous loss results from domain-wall motion during changes in the domain configuration. In many cases, this component of the loss can be expressed as dLA GdwH0 1=2 dB 3=2 ¼ ; ð7Þ dt dt r where G is a dimensionless constant, w is the width of laminations, d is the thickness, r the resistivity and H0 is equivalent to a magnetic field. The following differential equation results Mirr Man þ kd
dMirr dM dM þ k1 dt dHe dHe
dM 1=2 dM þ k2 ¼ 0; dt dHe
ð8Þ
where the final two terms on the right-hand side are the classical eddy current loss and the anomalous loss k1 ¼ d 2 m0 =2rb and k2 ¼ ðGdwH0 m0 =rÞ1=2 :
ARTICLE IN PRESS ! P. Kis, A. Ivanyi / Physica B 343 (2004) 59–64
Using the above equations the differential susceptibility can be obtained after some mathematical manipulations dM dHe ¼
dM ðMan MÞ þ ckd dMan =dHe pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: kd þ dM ð1 cÞ k1 dM=dt þ k2 dM=dt ð9Þ
The frequency dependent hysteresis curve can be calculated by Eq. (9).
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3.1. Initial test First, let us check the parameter identification program. A hysteresis curve has been generated by the J–A model itself to check the agreement between the original and the fitted parameters. Known parameters are in this test shown in Table 2 as the first column. Then it is considered as measured data to be fitted by the identification program. The initial values can be seen in the second column of Table 2. The third column contains the result of the identification. Figs. 2 and 3 show the results of the initial test checking the efficiency of the parameter identification program.
3. Results of parameter fitting The parameters of J–A model are determined by nonlinear least-squares method. The block scheme of the parameter identification can be seen in Fig. 1. The measured experimental data and x0 = [Ms0, a0, a0, c0, k0, k10, k20] initial parameters are given. The J–A model generates hysteresis curve based on initial vector. The experimental and modeled hysteresis is compared by a criterion function, which provides an error. Depending on the error the nonlinear least-squares method modifies the input parameters of the J–A model. The cycle runs until the error gets small enough. The speed of convergence of the algorithm is depending on the initial parameters. In general, the method requires approximately 50 steps.
INPUT Measured H -B Data Initial parameters (x0)
Measured H -B
Criterion Function
Error
Jiles - Atherton Simulated H -B Model of Hysteresis
3.2. Fitting to measured curves at 0.2 Hz The program was used to extract fitted parameters of a measured hysteresis curve obtained from measurements of C19 structural steel toroid core (outer/inner diameters are 60/40 mm, and the height is 16 mm). The used measurements system consists of a PC with data acquisition A/D and D/A cards [7]. The measurements can be declared as quasi-static measurements. The effect of the eddy currents and any other additional losses can be neglected. Figs. 4–7 show the results of the parameter fitting, ‘o’ denoting some measured points and the simulated hysteresis curves are plotted by solid lines. The following parameter values resulted: Ms = 1.33 106, a = 4.7 104, a = 326.01, c = 0.00565, k = 468.985. The relative error can also be observed in Figs. 5 and 7. The relative error is less than 10% everywhere. It means the simulated and measured curves are in good agreement. Table 2 Initial test of the J–A model identification
x Nonlinear Least Squares Method OUTPUT J - A Model Parameters
Fig. 1. The block scheme of the J–A model identification.
Ms a a c k
Original parameters
Initial parameters
Fitted parameters
1.5 106 106 1500 0.1 1000
2.1 106 1.5 106 2000 0.5 1200
1.51 106 1.01 106 1531.44 0.09688 989.26
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1.5 simulated measured
1.5 1
1
B [T]
B [T]
0.5
simulated original
0.5
0
0
-0.5 -0.5
-1 -1
-1.5
-3
-2
-1
0
1
2
H [A/m]
-1.5
3 x 10
-3000
4
Fig. 2. Fitting to the self-generated hysteresis loop.
-2000
-1000
0
1000
2000
3000
H [A/m]
Fig. 4. Result of parameter identification of quasi-static J–A model at 0.2 Hz. Fitted curve is plotted solid line, and some points of the measured curve with ‘o’.
5 4 15
3 2 10
0
Error [%]
Error [%]
1
-1
5
-2 0
-3 -4 -5
-5
-3000
-3
-2
-1
0
1
H [A/m]
2
x 10
-2000
-1000
0
1000
2000
3000
H [A/m]
3 4
Fig. 3. Error of the fitting.
3.3. Fitting to measured curves at 5 Hz Cross-section of the measured toroid is relatively large therefore the eddy current effect is appearing at this frequency. In this case the quasistatic J–A model cannot model the experimental
Fig. 5. Error between the simulated and measured hysteresis loops.
curve as can be observed in Fig. 8. The extended J–A model must be introduced taking into consideration the effect of eddy current and domain wall motion. The extended model has been identified in the same way as the classical one choosing k1 = 1.8 106 and k2 = 0.01 and the
ARTICLE IN PRESS ! P. Kis, A. Ivanyi / Physica B 343 (2004) 59–64 1.5
1.5 simulated measured
simulated measured
0.5
0.5
B [T]
1
B [T]
1
0
0
-0.5
-0.5
-1
-1
-1.5
63
-3000
-2000
-1000
0
1000
2000
-1.5 -3000
3000
-2000
-1000
0
1000
2000
3000
H [A/m]
H [A/m]
Fig. 6. Result of parameter identification of quasi-static J–A model at 0.2 Hz on a symmetrical minor loop. Fitted curve is plotted solid line, and some points of the measured curve with ‘o’.
Fig. 8. Simulation of the measured hysteresis at 5 Hz with quasi-static J–A model.
simulated measured
15
1
0.5
B [T]
Error [%]
10
5
0
-0.5 0
-1 -5
-3000
-2000
-1000
0
1000
2000
3000
H [A/m]
-3000
Fig. 7. Error between the simulated and measured hysteresis loops.
-2000
-1000
0
1000
2000
3000
H [A/m]
Fig. 9. Simulation of the measured hysteresis at 5 Hz with extended J–A model.
result is given in Fig. 9. The top and the bottom of the curve is not fitted exactly because of the dM parameter but simulation without this parameter leads to a non-physical solution. It means the slope of the hysteresis curve at the peak becomes negative and therefore this is not recommended
for use. For instance, using the model with negative slope can lead to numerical instability in field computation depending on the applied numerical iteration technique. The approximation with the extended model (Fig. 9) is much better
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than the quasi-static approach (Fig. 8) but the accuracy of the fitting is worse than in the case of lower frequency simulations (Figs. 4–7).
Acknowledgements The research work is sponsored by the Hungarian Scientific Research Found, OTKA, Pr. No. T 034 164/2003.
4. Conclusions Both classical and excess eddy current losses have been included into the J–A model, and these appear as different terms in energy-balance equation. The results show some interesting features including the increase of coercitivity with frequency, the increase in energy loss per cycle and the approximate invariance of remanence with frequency. The parameter identification of the J–A model is performed successfully in both cases. The necessity of the extended model can be proofed in Figs. 8 and 9.
References [1] D.C. Jiles, J.B. Thoelke, M.K. Devine, IEEE Trans. Magn. 28 (1) (1992) 27. [2] D.C. Jiles, J. Appl. Phys. 76 (1994) 5849. [3] D.C. Jiles, IEEE Trans. Mag. 30 (1994) 4326. [4] D. Atherton, D.C. Jiles, IEEE Trans. Mag. 19 (1983) 2183. [5] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1996. [6] Iv!anyi, Hysteresis Models in Electromagnetic Computation, ! Budapest, 1997. Akad!emia Kiado, [7] P. Kis, A. Iv!anyi, J. Fuzi, . A. Iv!anyi, Proceedings of the XVIth Electromagnetic Fields and Materials Bratislava, Slovakia, 2002.
