基于双曲余切变换的Halbach阵列表贴式永磁电机转子偏心气隙磁场解析模型

doi:10.19595/j.cnki.1000-6753.tces.221813

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摘要

精确计算气隙磁场是设计和分析永磁电机的关键。Halbach阵列永磁电机具有良好的转矩输出特性。转子偏心会对永磁电机产生不良影响,准确获得其偏心气隙磁场分布具有重要意义。采用双曲余切变换解析计算Halbach阵列表贴式永磁电机转子偏心气隙磁场。该文建立了Halbach阵列定子开槽、转子偏心永磁电机二维模型。将同心解析模型分为三类子区域,利用各区域边界条件求解拉普拉斯方程和泊松方程,通过矢量磁位求得同心气隙磁场,利用相对磁导函数对其进行修正,从而获得偏心空载气隙磁场,对其进行研究,并利用回归评价指标进行评估。不同偏心率下气隙磁通密度、不平衡磁拉力、齿槽转矩的解析结果同有限元结果吻合,验证了该文解析模型的有效性和正确性。

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	刘蓉晖
	刘锦坤
	章君达
	孙改平

关键词 ： Halbach阵列永磁电机, 双曲余切变换, 转子偏心, 矢量磁位, 空载气隙磁场

Abstract：

Accurate calculation of air-gap magnetic field is the key to design and analyse the permanent magnet (PM) motors. The Halbach arrays PM motor has outstanding torque output performance. Rotor eccentricity will cause the noise, rotor loss and torque ripple, which will have adverse effects on the PM motors. Therefore, it is of great significance to solve and analyse the air-gap magnetic field of the Halbach arrays surface-mounted (HASM) PM motor with rotor eccentricity.
This paper combines the hyperbolic cotangent transformation and the relative permeance function, the air-gap magnetic field of the HASM PM motor with rotor eccentricity is analytically calculated. A two-dimensional model of the HASM PM motor with rotor eccentricity is established. As shown in Fig.A1, two groups of orthogonal circles representing the equipotential and magnetic force lines in the w plane are obtained by using the above transformation for the rotor eccentricity region in the z plane.
The hyperbolic cotangent transformation can be expressed as follows:
$z\text{=}\lambda \frac{{{\text{e}}^{w}}\text{+}1}{{{\text{e}}^{w}}-1}=\lambda \coth \frac{w}{2}$ （1）
where λ is a constant related to the rotor radius, the stator radius and the relative deviation distance between the stator and the rotor center.
The radial magnetic flux density of the eccentric magnetic field when the magnetic potential difference is 1 is calculated in the w plane, and the radial air-gap relative permeance function can be obtained. The radial air-gap relative permeance function f_r can be expressed as
${{f}_{\text{r}}}=\frac{{{{{B}'}}_{\text{r }\!\!\_\!\!\text{ ecc}}}}{{{{{B}'}}_{\text{r }\!\!\_\!\!\text{ noecc}}}}$ （2）
where ${{{B}'}_{\text{r }\!\!\_\!\!\text{ ecc}}}$ is the radial magnetic flux density when the rotor is eccentric, and ${{{B}'}_{\text{r }\!\!\_\!\!\text{ noecc}}}$ is the radial magnetic flux density when the stator and the rotor are concentric.
The concentric analytical model is divided into three sub-regions: Halbach PMs, air-gap and slots. The Laplace equations and the Poisson equations are solved by using the boundary conditions of each region, and the air-gap magnetic field when the stator and the rotor are concentric is obtained by the vector magnetic potential.
By modifying the concentric air-gap magnetic field with the relative permeance function, the eccentric no-load air-gap magnetic field of the HASM PM motor is obtained.

Fig.A1 Curves coordinates u and v in the z plane
When the ratio of the eccentricity is 0.2, the comparison between the analytical solutions and the finite element (FE) solutions of no-load air-gap magnetic field is shown in Fig.A2. B_r and B_θ represent the radial and the tangential components of air-gap magnetic flux densities, respectively.

Fig.A2 Comparison of no-load air-gap magnetic flux density waveforms, e=0.2
It can be seen that the analytical solutions are consistent with the FE solutions, which proves the effectiveness and the correctness of the proposed analytical model.
In addition, the air-gap magnetic flux densities, unbalanced magnetic forces and cogging torque at different eccentricities are calculated and compared with the FE solutions, and the regression evaluation indicators are used for evaluation. The analytical method is compared with the boundary perturbation method, which is verified that the analytical model is suitable for the eccentric air-gap magnetic field with large eccentricity. The proposed method can be used for the design and the optimization of the HASM PM motor with rotor eccentricity.

Key words： Halbach arrays permanent magnet motor hyperbolic cotangent transformation rotor eccentricity magnetic vector potential no-load air-gap magnetic field

收稿日期: 2022-09-25

PACS:

TM351

通讯作者: 刘蓉晖女,1975年生,博士,副教授,研究方向为电机设计与电机电磁场解析计算。E-mail: liuronghuiyzy@126.com

作者简介: 刘锦坤男,1999年生,硕士研究生,研究方向为电机电磁场解析计算。E-mail: a1229805586@qq.com

引用本文:

刘蓉晖, 刘锦坤, 章君达, 孙改平. 基于双曲余切变换的Halbach阵列表贴式永磁电机转子偏心气隙磁场解析模型[J]. 电工技术学报, 2023, 38(6): 1433-1446. Liu Ronghui, Liu Jinkun, Zhang Junda, Sun Gaiping. Analytical Model for Air-Gap Magnetic Field in Halbach Arrays Surface-Mounted Permanent Magnet Motor with Rotor Eccentricity Based on Hyperbolic Cotangent Transformation. Transactions of China Electrotechnical Society, 2023, 38(6): 1433-1446.

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