Modified Static Hysteresis Simulation Method for Electrical Equipment with Silicon Steel Core

doi:10.11930/j.issn.1004-9649.202503012

Abstract

Abstract:

Compared to traditional hysteresis models like Preisach and Jiles-Atherton (J-A), the micromagnetic Landau-Lifshitz-Gilbert (LLG) equation offers advantages such as clearer physical interpretation and higher simulation accuracy. However, its application to macroscopic static hysteresis simulation of electrical steel sheets has been limited due to computational intensity and memory constraints. This paper first proposes a geometric model discretization method for the LLG equation based on the Representative Volume Element (RVE) approach, tailored to the structural characteristics of electrical steel sheets. Subsequently, simplified expressions for the energy terms within the total Gibbs free energy are developed for the millimeter scale. A rapid parameter identification method for the relevant parameters is also introduced. This leads to the derivation of a simplified LLG equation specifically for macroscopic static hysteresis simulation in electrical steel sheets. Finally, the proposed simplified LLG equation is used to simulate static hysteresis loops of industry-grade grain-oriented (GO) and non-oriented (NO) silicon steel sheet samples under various operating conditions. The simulations exhibit strong agreement with measurements. Comparisons in simulation accuracy with the widely applied Preisach and J-A hysteresis models further validate the superiority of the proposed model and methodology.

Key words: micromagnetics, LLG equation, electrical steel sheet, representative volume element, static hysteresis

LIU Ren, ZENG Yu, HU Anlong, TANG Bo. Modified Static Hysteresis Simulation Method for Electrical Equipment with Silicon Steel Core[J]. Electric Power, 2025, 58(7): 137-146.

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URL: https://www.electricpower.com.cn/EN/10.11930/j.issn.1004-9649.202503012

https://www.electricpower.com.cn/EN/Y2025/V58/I7/137

Figures/Tables 16

Fig.1 Schematic of magnetisation vector Mi of magnetic moment around an active field Heff_i

Fig.2 Analysis subject and analysis space of the LLG equation

Fig.3 Relationship between the parameters of the LLG equation and the shape of the hysteresis return line

Fig.4 Flow of hysteresis loop simulation based on the proposed improved simplified LLG model

Fig.5 Measurement platform for the magnetic properties

Table 1 Parameters of oriented and non-oriented silicon steel samples

类型	牌号	长度/ mm	宽度/ mm	厚度/ mm	密度/ (kg·m^–3)
取向硅钢样品	23QG100	300	30	0.23	7650
无取向硅钢样品	35W233	300	30	0.35	7650

Table 2 Parameters of the simplified LLG equation for the non-oriented silicon steel samples in high magnetic flux density regions (Bm=1.3~1.5 T)

M_s/T		h_Zee/T		μ_ani/(A·m^–1)		b₂/J		b₄/J		b₆/J		b₈/J		b₁₀/J		b₁₂/J
1.52		2.58		93.53		26.6		116		–625.9		1610.1		–1870.2		866.7

Table 3 Parameters of the simplified LLG equation for the grain oriented silicon steel samples in high magnetic flux density regions (Bm=1.2~1.8 T)

M_s/T		h_Zee/T		μ_ani/(A·m^–1)		b₂/J		b₄/J		b₆/J		b₈/J		b₁₀/J		b₁₂/J
1.82		3.17		28.33		31.8		32.9		–125.7		281.3		–337.4		185.6

Table 4 Fixed parameters of J-A models of the oriented and non-oriented silicon steel samples

参数	取向硅钢片	无取向硅钢片
饱和磁化强度M_s/(A·m^–1)	1.21×10⁶	1.45×10⁶
形状参数a/(A·m^–1)	26.924	15.284
可逆磁化参数c	0.3081	0.5825

Fig.6 Comparison of static hysteresis loops calculated based on simplified LLG equation for non-oriented silicon steel wafers with measured values

Fig.7 Comparison of static hysteresis loops calculated based on simplified LLG equation for oriented silicon steel wafers with measured values

Fig.8 Simulated static hysteresis loops based on Preisach model, J-A model and simplified LLG equation respectively and the measured ones

Table 5 Simulated and measured values of static hysteresis loss of unoriented silicon steel wafer samples simulated based on simplified LLG equations

磁密幅值B_m/T	计算值W_i,calc/(J·m^–3)	实测值W_i,meas/(J·m^–3)	相对误差σ/%
1.5	271.95	278.14	2.96
1.4	219.64	229.47	4.28
1.2	153.68	153.68	1.31
1.0	116.04	107.23	4.41
0.8	74.00	73.99	0.01
0.6	46.62	46.67	0.08
0.4	23.34	24.20	3.52
0.2	6.38	6.54	2.30

Table 6 Simulated and measured values of static hysteresis loss in samples of oriented silicon steel wafers simulated based on simplified LLG equations

磁密幅值B_m/T	计算值W_i,calc/(J·m^–3)	实测值W_i,meas/(J·m^–3)	相对误差σ/%
1.8	138.46	150.25	7.84
1.7	86.75	98.18	11.88
1.5	47.87	55.42	13.62
1.3	37.51	34.91	10.78
1.1	26.63	26.38	0.95
0.9	16.46	17.32	4.96
0.7	10.58	10.31	2.66
0.5	5.35	4.94	8.41
0.3	2.29	1.87	18.33

Table 7 Mean relative errors of non-oriented silicon steel sample based on Preisach model, J-A model and simplified LLG equation

模型		平均相对误差/%
LLG方程		2.36
Preisach模型		8.44
J-A模型		9.57

Table 8 Mean relative errors of oriented silicon steel sample based on Preisach model, J-A model and simplified LLG equation

模型		平均相对误差/%
LLG方程		8.82
Preisach模型		12.51
J-A模型		14.65

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