﻿ 开关磁阻电机矩角特性模型非线性拟合方法<sup>*</sup>
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1. 空间物理重点实验室, 北京 100076;
2. 北京航空航天大学 自动化科学与电气工程学院, 北京 100083

Nonlinear fitting method for torque-angle characteristic model of switched reluctance motor
YE Wei1,2, MA Qishuang2, XU Ping2, ZHANG Poming2
1. Key Laboratory of Space Physics, Beijing 100076, China;
2. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China
Received: 2018-04-23; Accepted: 2018-07-27; Published online: 2018-09-17 16:59
Corresponding author. MA Qishuang, E-mail: qsma304@buaa.edu.cn
Abstract: The modeling method of the switched reluctance motor is different from the traditional motor. By analyzing the principle of electromagnetic torque generation, the relationship between electromagnetic torque and phase inductance derivatives under different motor saturation conditions is determined. The inductance derivative curve is modeled by piecewise function nonlinear fitting. And then, the analytical model for the reversible torque-angle characteristics of switched reluctance motor is obtained. The parameters of the model are determined and optimized by the structural parameters and constraint condition of the motor. The reversible torque-angle model makes the calculation of the magnetization curve and the instantaneous flux linkage possible. It brings great convenience for the design of the motor and the drive control of the motor. The accuracy of the analytical results is verified by the measured data and the finite element method with two prototypes.
Keywords: switched reluctance motor     modeling     nonlinear     torque-angle characteristics     curve fitting

1 可逆矩角特性的数学模型的建立 1.1 转矩模型的建立

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1) 在电流很小时，新的转矩公式需与式(2)等效，即转矩与电流平方成正比。

2) 当电流逐渐增大，电机变得饱和时，新转矩方程将从电流的二次函数逐渐变为线性函数。

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1.2 电感模型的建立

 图 1 电感L0(θ)及其导数L0p(θ)的理想和实际曲线示意图 Fig. 1 Schematic of ideal and real curves of inductance L0(θ) and its derivative L0p(θ)

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 图 2 开关磁阻电机几何尺寸 Fig. 2 Physical dimensioning of switched reluctance motor

 图 3 L0pN由分段函数grise(x)、gtop(x)和gfall(x)拟合 Fig. 3 L0pN is fitted by piecewise function grise(x), gtop(x) and gfall(x)

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1) 在x=x1位置时(x1θ1的归一化)，L0p取最大值，则可以确定gtop(x=x1)=1。

2) 相较于区间xbox≤1，在区间0≤xxbo上转矩值及其变化率都很小，所以精确确定函数grise(x)表达式显得并不是十分重要，一般可以将grise(x)简单定义为x的线性函数。

3) gfall(x)是为了确保L0p(x)在对齐位置θ=θal附近的快速衰减为0，所以在定、转子极互相偏离较远时，函数gfall(x)的值应等于1，而当定、转子极接近对齐位置即x→1时，函数gfall(x)的值应从1迅速衰减为0。

2 函数L0pN(θ)参数的确定

2.1 m值的选择

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 图 4 不同的m值所对应的过渡区间 Fig. 4 Transition intervals for different m values
2.2 grise(x)的选择

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 图 5 不同α与μrise下的grise(x)曲线 Fig. 5 Curves of grise(x) for different α and μrise
2.3 gtop(x)的选择

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 图 6 不同μtop下的gtop(x)曲线 Fig. 6 Curves of gtop(x) for different μtop

2.4 gfall(x)的选择

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 图 7 不同k下的gfall(1)(x)和gfall(2)(x)的曲线 Fig. 7 Curves of gfall(1)(x) and gfall(2)(x) for different k
3 模型参数n与函数f(θ)的确定

3.1 参数n的确定

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 图 8 Lal/Lu值与n的关系曲线 Fig. 8 Curve of relationship between value of Lal/Lu and n

3.2 函数f(θ)的确定

1) 在定转子极不重叠区间(即θuθθbo)，电机磁场未饱和，特别是在非对齐位置角θ=θu附近，电磁转矩的非线性方程式(4)将退化为线性方程式(3)，从而可知当θθu时，f(θ)→0。

2) 在位置x=xbox=xeo处气隙的饱和效应最为显著[19]，这意味着f(x)在这2个位置取极大值。

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1) 常数c2

x=xeo时，气隙磁链和电流偏离线性关系，磁路临界饱和，∂T/∂i将达到最大值，即

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2) 常数c1

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3) 常数c

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 (n=3, λ=1.2, m=13) (n=3, λ=1.2, m=13) 图 9 归一化的f(x)与f1(x)的曲线 Fig. 9 Normalized curves of f(x) and f1(x)
4 可逆矩角特性模型的有限元计算与实验验证

 参数 SRM1 SRM2 定转子极数 6-4 6-4 定子外径/mm 150 100 定子内径/mm 95 50 定子轭高/mm 13.75 12.2 定子极弧系数 0.433 3 0.433 3 转子外径/mm 94.4 49.4 转子内径/mm 40 10 转子极弧系数 0.3 0.3 转子轭高/mm 13.6 10 轴向长度/mm 40 40 相绕组匝数 19 50 硅钢片材料 D25_50 D25_50 电机容量 7.5 kW, 270 V, 3 000 r/min 200 W, 60 V, 3 000 r/min

 图 10 两个样机的有限元模型及SRM1的磁路仿真结果 Fig. 10 Finite element models of two prototypes and magnetic simulation result of SRM1

 图 11 SRM1有限元模型结果 Fig. 11 Finite element model results of SRM1
 图 12 样机SRM1有限元模型与不同模型参数的解析模型结果对比 Fig. 12 Result comparison between finite element model and analytical model with different model parameters for prototype SRM1

 模型 m n xbo μrise α x1 μtop β xeo μfall k a 13 3 0.39 0.23 2 0.4 0.5 2 0.98 0.95 35 b 25 3 0.39 0.23 2 0.4 0.5 2 0.98 0.95 20 c 25 3 0.39 0.23 2 0.4 0.5 2 0.98 0.95 35

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 模型 误差 Is=2 A Is=4 A Is=6 A Is=8 A a Terror/(N·m) 0.24 0.97 2.07 4.26 TR/% 2.92 2.92 2.90 2.95 b Terror/(N·m) 0.38 1.56 3.16 2.81 TR/% 2.90 2.90 2.86 2.66 c Terror/(N·m) 0.10 0.42 0.73 2.97 TR/% 2.55 2.55 2.54 2.60

 图 13 样机SRM1解析模型、有限元模型和实验测量结果对比 Fig. 13 Result comparison among analytical model, finite element model and experimental measurement

 参数 m n xbo μrise α x1 μtop β xeo μfall k 数值 40 3 0.4 0.23 2 0.41 0.5 2 0.99 0.95 20

 图 14 样机SRM2有限元模型与解析模型结果对比 Fig. 14 Result comparison between finite element model and analytical model for prototype SRM2

5 结论

1) 用分段函数拟合，避免了单一函数在拟合电机相电感时精度不问题，在模型精确度和复杂度之间达到平衡。

2) 在电机线性区和非线性区均有较高精度，与有限元仿真结果相比误差小于2.6%，与样机实测数据相比误差在3%以内。

3) 模型参数中n的值，受到电机中相绕组等效自感最大值与最小值之比的约束，一般取3比较合适。

4) 模型参数中的xbox1xeo由电机的几何形状决定，μriseμtopμfallαβ由电磁特性确定。不确定的参数只有mk模型参数的优化较为容易实现。

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文章信息

YE Wei, MA Qishuang, XU Ping, ZHANG Poming

Nonlinear fitting method for torque-angle characteristic model of switched reluctance motor

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(1): 83-92
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0223