Rev. Roum. Sci. Techn.– Électrotechn. et Énerg. Vol. 61, 3, pp. 217–220, Bucarest, 2016 University 8 Mai 1945, Laboratoire LGEG, BP. 401, Guelma, 24000, Algeria, E-mail :aziz_ladjimi@yahoo.fr 1 MODELING OF FREQUENCY EFFECTS IN A JILES-ATHERTON MAGNETIC HYSTERESIS MODEL ABDELAZIZ LADJIMI, ABDESSELAM BABOURI 11 Key words: Hysteresis model, Jiles-Atherton theory, Ferromagnetic materials, Frequency dependence. A frequency-dependent model is necessary, to understand the dynamic behavior of hysteresis phenomenon in ferromagnetic materials. In this study, the hysteresis model based on Jiles-Atherton theory was developed, to simulate the frequency effects on the magnetic hysteresis loop. The frequency effects have been integrated in the model, by introducing the frequency behavior of the parameter k from Jiles-Aterton theory. The proposed model was validated, by comparing the results with those provided by the dynamic Jiles model, and the results are in good agreement. 1. INTRODUCTION The modeling of the magnetic hysteresis is very important in the design of electromagnetic devices [1]. The hysteresis loop of a ferromagnetic material depends on several factors such as: temperature, shape of the sample, type of the excitation [2] and frequency. However, the magnetic behavior of ferromagnetic materials can be significantly modified by the variation of frequency and, therefore, the hysteresis loop can be changed substantially. Several models are proposed in literature, in order to analytically describe the hysteretic behavior of the materials such as: the Bertotti model [3], the Bernard model [4] and the dynamic Jiles model [5]. These three models are often quite difficult to implement and they are very expensive regarding the calculation time. In addition, they do not clearly consider the effect of frequency on the hysteresis loop. In this paper it is suggested a change in the Jiles-Atherton (JA) model, by introducing the frequency effect. 2. THE JILES-ATHERTON MODEL In the JA model, the magnetization M is decomposed into two parts: the reversible component M rev and the irreversible component M irr [6]. The reversible component represents the translation and the movement of the domain walls (DWs) and the reversible rotation of the magne- tization vector in ferromagnetic materials. The irreversible component represents the irreversible dis-placement of the magnetic domain walls. The relationship between these two components and the anhysteretic magnetization M an (the ideal case is considered without losses) is based on physical considerations of the magnetization processes. The equations of the JA model are: , M M M rev irr = + (1) ( ), M cM M rev an irr = − (2) , (coth ) H a e M M an s a H e = + (3) , d d δ M M M an irr irr H k e − = (4) where k is proportional to hysteresis losses, a is the shape parameter for M an , c is the reversibility coefficient and M S is the saturation magnetization. All the model parameters are determined from an experimental hysteresis loop and δ is a factor that takes the value +1, when dH ≥ 0, and –1 when dH < 0. The term H e is the effective field, which is the sum of the applied field H and the molecular field αM as described by eq (5): M H H e α + = , (5) where the parameter α must be determined experimentally. Using Eqs. (1–4), and after substitution and simplifi- cation, we obtain the following differential equation for JA model. . d d ) 1 ( α d d α 1 d d d d ) 1 ( d d e irr e an e an e irr H M c H M c H M c H M c H M − − − + − = (6) The JA model has already been extended to include magneto-elastic effects [7–8], anisotropy [9], temperature [10–16]. The previous works on extending the model to include frequency effects [17] were based on the coupling between the JA model and the mathematical model of Chua. On the other hand, a new extension of JA model [6] was developed by Jiles in [5]. This new energy approach, based on the separation of losses, helps one to develop the Jiles dynamic model. The losses in a magnetic material consist of hysteresis, eddy current (eq. 7) and anomalous (excess) losses (eq. 8): ( ) , d d ρβ 2 dμ d d 2 2 0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = t M t W ec (7) , d d ρ d d 2 3 2 1 0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = t B GdwH t W a (8) where ρ is the resistivity, d is the cross-section of the sample, ß is a geometrical factor, G = 0.1356, w is the width of the lamination and H 0 is a parameter representing the internal magnetic field, who acts on the domain walls. By using the equations of JA model and adding the two components of losses (eddy current and excess losses) we can deduce the differential equation of the Jiles dynamic model: