Abstract Grain-oriented electrical steel sheets are widely used in high power transformers because of their good magnetic properties in the rolling direction. However, the core of transformer will be affected by mechanical stress during assembly and operation, resulting in significant changes in the hysteresis characteristics of electrical steel sheets. The Sablik-Jiles-Atherton (S-J-A) hysteresis model is widely used to simulate the hysteresis characteristics under mechanical stress because of its simple calculation and clear physical meaning. Due to the crystal orientation of the grain-oriented electrical steel sheets, its magnetic properties exhibit strong anisotropic characteristics, and the anisotropic energy has a strong dependence on the applied external stress. The original S-J-A model doesn't consider the influence of anisotropic energy, which makes the S-J-A model significantly inaccurate in modeling the hysteresis return of grain-oriented electrical steel sheets laminations. To solve these problems, this paper proposes an improved S-J-A model that takes into account the anisotropic energy and the stress dependence of the model parameters. In this paper, the anisotropic energy is considered in the anhysteretic magnetization., including the anisotropic energy of the material itself as well as the anisotropic energy due to mechanical stress. At the same time, the magnetostriction and pinning coefficient k are regarded as the result of the joint action of the magnetization M and the applied mechanical stress σ. In addition, in the relevant literature, only the parameter k is considered to be dependent on the mechanical stress, and other parameters are not affected by the mechanical stress. In the actual simulation of the model, it is found that the shape parameter of anhysteretic magnetization a is also dependent on the mechanical stress, so an improved s-j-a model is obtained. The simulation results show that the improved S-J-A hysteresis model fits better under tensile stress and smaller compressive stress, and the mean square error does not exceed 3A/m, which proves the accuracy and reasonableness of the model in this paper. When the compressive stress is greater than 7.14 MPa, the error becomes larger, reaching a maximum of 8.21 A/m, but the overall fit is acceptable. Comparing this method with the original S-J-A hysteresis model, it is found that the mean square error of the original model increases with the increase of compressive stress, reaching 162.91 A/m at σ = -10.63 MPa, which is too large to be acceptable. Under the condition of tensile stress, the mean square error of the original model is also larger than that of the proposed method under the same tensile stress. Through the above analysis, it can be seen that the fitting accuracy obtained by this method is higher than that of the original S-J-A hysteresis model, which proves the superiority of the proposed method. The following conclusions are obtained from the simulation of the hysteresis line under stress: (1) The simulation results of the improved model and the original S-J-A model for electrical steel sheets are compared and analyzed. The results show that the simulation accuracy of the improved model is higher, which proves its superiority. (2) The model parameters under different mechanical stresses are extracted, and the results show that the average anisotropic energy density Kan, the pinning coefficient k and the shape parameter of anhysteretic magnetization a in the S-J-A model have obvious dependence on the mechanical stresses, and these parameters have different dependence on the compressive and tensile stresses.
Zhu Yuying,Li Lin. An Improved Sablik-Jiles-Atherton Hysteresis Model Considering Anisotropy and Stress Dependence of Model Parameters[J]. Transactions of China Electrotechnical Society, 2023, 38(17): 4586-4596.
[1] Hubert O, Daniel L.Multiscale modeling of the magneto-mechanical behavior of grain-oriented silicon steels[J]. Journal of Magnetism and Magnetic Materials, 2008, 320(7): 1412-1422. [2] Anderson P I, Moses A J, Stanbury H J.Assessment of the stress sensitivity of magnetostriction in grain-oriented silicon steel[J]. IEEE Transactions on Magnetics, 2007, 43(8): 3467-3476. [3] 赵志刚, 徐曼, 胡鑫剑. 基于改进损耗分离模型的铁磁材料损耗特性研究[J]. 电工技术学报, 2021, 36(13): 2782-2790. Zhao Zhigang, Xu Man, Hu Xinjian.Research on magnetic losses characteristics of ferromagnetic materials based on improvement loss separation model[J]. Transactions of China Electrotechnical Society, 2021, 36(13): 2782-2790. [4] 陈彬, 秦小彬, 万妮娜, 等. 基于R-L型分数阶导数与损耗统计理论的铁磁材料高频损耗计算方法[J]. 电工技术学报, 2022, 37(2): 299-310. Chen Bin, Qin Xiaobin, Wan Nina, et al.Calculation method of high-frequency loss of ferromagnetic materials based on R-L type fractional derivative and loss statistical theory[J]. Transactions of China Electrotechnical Society, 2022, 37(2): 299-310. [5] Dias M B S, Landgraf F J G. Compressive stress effects on magnetic properties of uncoated grain oriented electrical steel[J]. Journal of Magnetism and Magnetic Materials, 2020, 504: 166566. [6] Zhao Wei, Wang Shuting, Xie Xianda, et al.A simplified multiscale magneto-mechanical model for magnetic materials[J]. Journal of Magnetism and Magnetic Materials, 2021, 526: 167695. [7] Daniel L, Rekik M, Hubert O.A multiscale model for magneto-elastic behaviour including hysteresis effects[J]. Archive of Applied Mechanics, 2014, 84(9): 1307-1323. [8] Daniel L, Hubert O, Buiron N, et al.Reversible magneto-elastic behavior: a multiscale approach[J]. Journal of the Mechanics and Physics of Solids, 2008, 56(3): 1018-1042. [9] Bergqvist A, Engdahl G.A stress-dependent magnetic Preisach hysteresis model[J]. IEEE Transactions on Magnetics, 1991, 27(6): 4796-4798. [10] 李伊玲, 李琳, 刘任. 机械应力作用下电工钢片静态磁滞特性模拟方法研究[J]. 中国电力, 2020, 53(10): 10-18. Li Yiling, Li Lin, Liu Ren.Modeling methods of static hysteresis characteristics of electrical steel sheets under stress[J]. Electric Power, 2020, 53(10): 10-18. [11] Ktena A.Vector Preisach modeling of magnetic materials under stress[J]. Journal of Physics: Conference Series, 2015, 585: 012001. [12] 赵小军, 刘小娜, 肖帆, 等. 基于Preisach模型的取向硅钢片直流偏磁磁滞及损耗特性模拟[J]. 电工技术学报, 2020, 35(9): 1849-1857. Zhao Xiaojun, Liu Xiaona, Xiao Fan, et al.Hysteretic and loss modeling of silicon steel sheet under the DC biased magnetization based on the preisach model[J]. Transactions of China Electrotechnical Society, 2020, 35(9): 1849-1857. [13] Sablik M J, Jiles D C.A model for hysteresis in magnetostriction[J]. Journal of Applied Physics, 1988, 64(10): 5402-5404. [14] Sablik M J, Kwun H, Burkhardt G L, et al.Model for the effect of tensile and compressive stress on ferromagnetic hysteresis[J]. Journal of Applied Physics, 1987, 61(8): 3799-3801. [15] Jiles D C.Theory of the magnetomechanical effect[J]. Journal of Physics D: Applied Physics, 1995, 28(8): 1537-1546. [16] Singh D, Martin F, Rasilo P, et al.Magnetomechanical model for hysteresis in electrical steel sheet[J]. IEEE Transactions on Magnetics, 2016, 52(11): 1-9. [17] 贲彤, 陈芳媛, 陈龙, 等. 考虑力-磁耦合效应的无取向电工钢片磁致伸缩模型的改进[J]. 中国电机工程学报, 2021, 41(15): 5361-5371. Ben Tong, Chen Fangyuan, Chen Long, et al.An improved magnetostrictive model of non-oriented electrical steel sheet considering force-magnetic coupling effect[J]. Proceedings of the CSEE, 2021, 41(15): 5361-5371. [18] Liu Qingyou, Luo Xu, Zhu Haiyan, et al.Modeling plastic deformation effect on the hysteresis loops of ferromagnetic materials based on modified Jiles-Atherton model[J]. Acta Physica Sinica, 2017, 66(10): 107501. [19] 祝丽花, 李晶晶, 朱建国. 服役条件下取向硅钢磁致伸缩模型的研究[J]. 电工技术学报, 2020, 35(19): 4131-4138. Zhu Lihua, Li Jingjing, Zhu Jianguo.Research on magnetostrictive model for oriented silicon steel under service conditions[J]. Transactions of China Electrotechnical Society, 2020, 35(19): 4131-4138. [20] Kumar A, Arockiarajan A.Evolution of nonlinear magneto-elastic constitutive laws in ferromagnetic materials: a comprehensive review[J]. Journal of Magnetism and Magnetic Materials, 2022, 546: 168821. [21] Jiles D C, Thoelke J B, Devine M K.Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis[J]. IEEE Transactions on Magnetics, 1992, 28(1): 27-35. [22] Jiles D C, Atherton D L.Theory of ferromagnetic hysteresis (invited)[J]. Journal of Applied Physics, 1984, 55(6): 2115-2120. [23] Sablik M J, Rubin S W, Riley L A, et al.A model for hysteretic magnetic properties under the application of noncoaxial stress and field[J]. Journal of Applied Physics, 1993, 74(1): 480-488. [24] Sablik M J.A model for asymmetry in magnetic property behavior under tensile and compressive stress in steel[J]. IEEE Transactions on Magnetics, 1997, 33(5): 3958-3960. [25] Ikhlef M, Bendjerad A, Boukhtache S, et al.Refined approach in Jiles-Atherton model for ferromagnetic sheet under the tensile stress[J]. Journal of Superconductivity and Novel Magnetism, 2021, 34(1): 227-234. [26] Ramesh A, Jiles D C, Bi Y.Generalization of hysteresis modeling to anisotropic materials[J]. Journal of Applied Physics, 1997, 81(8): 5585-5587. [27] Jiles D C, Ramesh A, Shi Y, et al.Application of the anisotropic extension of the theory of hysteresis to the magnetization curves of crystalline and textured magnetic materials[J]. IEEE Transactions on Magnetics, 1997, 33(5): 3961-3963. [28] Zirka S E, Moroz Y I, Moses A J, et al.Static and dynamic hysteresis models for studying transformer transients[J]. IEEE Transactions on Power Delivery, 2011, 26(4): 2352-2362. [29] Shin S, Schaefer R, DeCooman B C. Anisotropic magnetic properties and domain structure in Fe-3%Si (110) steel sheet[J]. Journal of Applied Physics, 2011, 109(7): 07A307. [30] Ramesh A, Jiles D C, Roderick J M.A model of anisotropic anhysteretic magnetization[J]. IEEE Transactions on Magnetics, 1996, 32(5): 4234-4236. [31] Jiles D C, Ramesh A, Shi Y, et al. Application of the anisotropic extension of the theory of hysteresis to the magnetization curves of crystalline and textured magnetic materials[C]//1997 IEEE International Magnetics Conference (INTERMAG), Orleans, LA, USA, 1997: FD-08. [32] Celasco M, Mazzetti P.Saturation approach law for grain-oriented polycrystalline magnetic materials[J]. IEEE Transactions on Magnetics, 1969, 5(3): 372-378. [33] Kuruzar M E, Cullity B D.The magnetostriction of iron under tensile and compressive stress[J]. International Journal of Magnetism, 1971, 1(4): 323-325. [34] Li Jianwei, Xu Minqiang.Modified Jiles-Atherton- Sablik model for asymmetry in magnetomechanical effect under tensile and compressive stress[J]. Journal of Applied Physics, 2011, 110(6): 063918. [35] Wlodarski Z.The Jiles-Atherton model with variable pinning parameter[J]. IEEE Transactions on Magnetics, 2003, 39(4): 1990-1992. [36] Lo C C H, Lee S J, Li L, et al. Modeling of stress effects on magnetic hysteresis and Barkhausen emission using an integrated hysteretic-stochastic model[C]//2002 IEEE International Magnetics Conference (INTERMAG), Amsterdam, Netherlands, 2002: FS07. [37] Sai Ram B, Baghel A P S, Kulkarni S V, et al. A frequency-dependent scalar magneto-elastic hysteresis model derived using multi-scale and Jiles-Atherton approaches[J]. IEEE Transactions on Magnetics, 2020, 56(3): 1-5. [38] Jiles D C, Atherton D L.Theory of ferromagnetic hysteresis[J]. Journal of Magnetism and Magnetic Materials, 1986, 61(1-2): 48-60. [39] Perevertov O, Schäfer R.Magnetic properties and magnetic domain structure of grain-oriented Fe-3%Si steel under compression[J]. Materials Research Express, 2016, 3(9): 096103. [40] Alatawneh N, Rahman T, Hussain S, et al.Accuracy of time domain extension formulae of core losses in non-oriented electrical steel laminations under non-sinusoidal excitation[J]. IET Electric Power Applications, 2017, 11(6): 1131-1139. [41] 曹祎, 王路, 雷民, 等. 基于改进粒子群算法的电流互感器J-A模型参数辨识[J]. 电测与仪表, 2021, 58(5): 70-77. Cao Yi, Wang Lu, Lei Min, et al.Parameter identification for J-a hysteresis model of current transformer based on improved particle swarm optimization algorithm[J]. Electrical Measurement & Instrumentation, 2021, 58(5): 70-77.
2023, Vol. 38 
