Research on the Equivalent Surface Current Model for the Spatial Magnetic Field of Permanent Magnets
Zhou Rui1,2, Xu Shuai1,3, Wang Qunjing2,3,4, Li Guoli2,3,4
1. School of Electrical Engineering and Automation Anhui University Hefei 230601 China; 2. Anhui Collaborative Innovation Center of Industrial Energy-Saving and Power Quality Control Anhui University Hefei 230601 China; 3. National Engineering Laboratory of Energy-Saving Motor & Control Technology Anhui University Hefei 230601 China; 4. Anhui Key Laboratory of Industrial Energy-Saving and Safety Anhui University Hefei 230601 China
Abstract：Permanent magnets find extensive applications in modern electric motors, sensors, transmission systems, and other fields. Their three-dimensional magnetic field values in non-ferromagnetic material environments form the foundation for establishing relevant electromagnetic models. The finite element method (FEM) applies to various permanent magnet models, particularly complex geometries, but suffers from lengthy computation times and unstable mesh convergence. Other computational approaches exhibit low accuracy and significant limitations due to the influence of the permanent magnet's physical and geometric characteristics. Based on Ampère's molecular current hypothesis, this paper proposes an analytical model for the three-dimensional spatial magnetic field of permanent magnets using an equivalent current surface integration method. The fundamental principle of the equivalent surface current model for permanent magnets was first established. The magnetic field of a permanent magnet equals that generated by a surface current on its magnetized-direction side surfaces, with the current density related to the remanent magnetization. This principle serves as the foundational tool for calculating the magnetic field via surface currents. Based on the Biot-Savart law, magnetic field calculation formulas for spatial straight-line and circular-arc currents were derived. Subsequently, using these formulas in conjunction with a limiting segmentation and summation approach, spatial magnetic field models were developed for cylindrical, cubic, truncated cone-shaped, and axially/radially magnetized circular arc-shaped permanent magnets. Comparisons with finite element simulation results verified the accuracy of the proposed models and the feasibility of the equivalent surface current method for calculating permanent magnet fields. Furthermore, the magnetic field model of a cubic permanent magnet confirmed the hypothesis that the equivalent surface currents on each side surface of a permanent magnet can be calculated independently. The proposed method, with minor modifications, can be extended to calculate the magnetic fields of permanent magnets with other shapes, such as sector or spherical surfaces. Since permanent magnets with complex shapes cannot be directly modeled using the magnetic field formulas for spatial straight-line and circular-arc currents, their magnetic fields were obtained by adding or subtracting the fields from simple-shaped permanent magnets (e.g., cylinders, cubes) based on the equivalent magnetization direction and shape. Its flexible application enables the spatial magnetic field calculation of permanent magnets with arbitrary shapes. Actual measurements of the spatial magnetic fields of three types of permanent magnets were conducted. The magnetic flux density was measured along a diameter of a ring and cylindrical permanent magnets and along the central line in the length direction of a rectangular permanent magnet. The small errors between the measured and calculated values confirm the proposed spatial magnetic flux density model for permanent magnets.
周睿, 徐帅, 王群京, 李国丽. 永磁体空间磁场的等效面电流模型研究[J]. 电工技术学报, 2026, 41(20): 6791-6806.
Zhou Rui, Xu Shuai, Wang Qunjing, Li Guoli. Research on the Equivalent Surface Current Model for the Spatial Magnetic Field of Permanent Magnets. Transactions of China Electrotechnical Society, 2026, 41(20): 6791-6806.
[1] 杨庆新, 李永建. 先进电工磁性材料特性与应用发展研究综述[J]. 电工技术学报, 2016, 31(20): 1-12. Yang Qingxin, Li Yongjian.Characteristics and developments of advanced magnetic materials in electrical engineering: a review[J]. Transactions of China Electrotechnical Society, 2016, 31(20): 1-12. [2] 张缘为, 尤晶晶, 张显著. 并联式六维加速度感知机构的灵敏度及静刚度研究[J]. 传感器与微系统, 2025, 44(11): 47-51, 56. Zhang Yuanwei, You Jingjing, Zhang Xianzhu.Study on sensitivity and static stiffness of parallel six-axis acceleration sensing mechanisms[J]. Transducer and Microsystem Technologies, 2025, 44(11): 47-51, 56. [3] 王洪明, 陈亮亮, 蒋科坚. 圆盘转子永磁悬浮轴承支承刚度特性研究[J]. 振动工程学报, 2025, 38(7): 1496-1502. Wang Hongming, Chen Liangliang, Jiang Kejian.Study on the support stiffness characteristics of a disc rotor permanent magnetic levitation bearing[J]. Journal of Vibration Engineering, 2025, 38(7): 1496-1502. [4] 周扬忠, 陈东远, 梁彤伟, 等. 基于悬浮电流重构的无轴承磁通切换电机转子磁悬浮内模控制[J]. 中国电机工程学报, 2025, 45(21): 8594-8604. Zhou Yangzhong, Chen Dongyuan, Liang Tongwei, et al.Internal model control of bearingless flux switching permanent magnet synchronous motor with rotor magnetic suspension based on suspension current reconstruction[J]. Proceedings of the CSEE, 2025, 45(21): 8594-8604. [5] 周嗣理, 李国丽, 王群京, 等. 基于改进粒子群优化算法的永磁球形电机驱动策略研究[J]. 电工技术学报, 2023, 38(1): 166-176, 189. Zhou Sili, Li Guoli, Wang Qunjing, et al.Improved particle swarm optimization algorithm based driving strategy research for permanent magnet spherical motor[J]. Transactions of China Electrotechnical Society, 2023, 38(1): 166-176, 189. [6] 许扬帆, 李国丽, 孙泽辉, 等. 降低永磁球形电机铜耗的通电策略研究[J]. 电气工程学报, 2024, 19(3): 274-285. Xu Yangfan, Li Guoli, Sun Zehui, et al.Research on energization strategy to reduce the copper loss of the permanent-magnet spherical motor[J]. Journal of Electrical Engineering, 2024, 19(3): 274-285. [7] Song Weiye, Shuai Changgeng, Zhai Qi.A magnetic field model of rectangular permanent magnets in a magnetic pendulum array[J]. AIP Advances, 2023, 13(10): 105009. [8] Nguyen V T, Lu T F.Modelling of magnetic field distributions of elliptical cylinder permanent magnets with diametrical magnetization[J]. Journal of Mag-netism and Magnetic Materials, 2019, 491: 165569. [9] Nguyen V T.Magnetic field distribution of a conical permanent magnet with an application in magnetic resonance imaging[J]. Journal of Magnetism and Magnetic Materials, 2020, 498: 166136. [10] Guo Zhiming, Li Tiefeng, Fang Qihong, et al.Modeling and measurement of non-uniform magnetic fields of a diametrically magnetized annular per-manent magnet[J]. Measurement, 2025, 253: 117495. [11] 李博, 李辉杰, 赵晓龙, 等. 基于改进等效面电流法的梯形Halbach永磁直线同步电机气隙磁场特性研究[J]. 电机与控制应用, 2025, 52(6): 658-668. Li Bo, Li Huijie, Zhao Xiaolong, et al.Research on air gap magnetic field characteristics of trapezoidal Halbach permanent magnet linear synchronous motor based on improved equivalent surface current method[J]. Electric Machines & Control Application, 2025, 52(6): 658-668. [12] 高锋阳, 齐晓东, 李晓峰, 等. 不等宽不等厚Halbach部分分段永磁同步电机电磁性能解析计算与优化分析[J]. 电工技术学报, 2022, 37(6): 1398-1414. Gao Fengyang, Qi Xiaodong, Li Xiaofeng, et al.Analytical calculation and optimization analysis of electromagnetic performance of halbach partially-segmented permanent magnet synchronous motors with unequal width and thickness[J]. Transactions of China Electrotechnical Society, 2022, 37(6): 1398-1414. [13] 周恩权, 郑仲桥, 张燕红, 等. 圆柱形永磁体磁场建模及仿真研究[J]. 河南科技, 2017, 36(21): 139-143. Zhou Enquan, Zheng Zhongqiao, Zhang Yanhong, et al.Modeling and simulation for cylinder permanent magnetic field[J]. Henan Science and Technology, 2017, 36(21): 139-143. [14] 宋承林, 吴志鹏, 黎明, 等. 永磁体偏移对永磁同步电机电磁噪声的影响研究[J]. 振动与冲击, 2025, 44(23): 95-102. Song Chenglin, Wu Zhipeng, Li Ming, et al.Effects of permanent magnet offset on electromagnetic noise of permanent magnet synchronous motor[J]. Journal of Vibration and Shock, 2025, 44(23): 95-102. [15] Wu Shasha, Wang Baojian, Zhang Tao, et al.Design optimization and electromagnetic performance analysis of an axial-flux permanent magnet brushless DC motor with unequal-thickness magnets[J]. Applied Sciences, 2022, 12(15): 7863. [16] 罗辞勇, 李竹田, 沈启平. 异形永磁体圆柱型直线电机的优化设计[J]. 电工技术学报, 2017, 32(17): 127-134. Luo Ciyong, Li Zhutian, Shen Qiping.Optimal design of a cylindrical linear motor with trapezoidal per-manent magnet[J]. Transactions of China Electro-technical Society, 2017, 32(17): 127-134. [17] 范璐, 李唯东, 杨胜强, 等. 基于液体磁性磨具的相对式磁极头设计及实验研究[J]. 表面技术, 2019, 48(6): 353-360, 369. Fan Lu, Li Weidong, Yang Shengqiang, et al.Design and experimental study of opposite magnet pole based on fluid magnetic abrasives[J]. Surface Technology, 2019, 48(6): 353-360, 369. [18] 康瑞, 郭彦青, 段志强, 等. 弧形永磁体外部磁场的均匀性分析[J]. 电子测量技术, 2024, 47(16): 1-7. Kang Rui, Guo Yanqing, Duan Zhiqiang, et al.Uniformity analysis of external magnetic field of curved permanent magnet[J]. Electronic Measurement Technology, 2024, 47(16): 1-7. [19] 雷银照. 关于电磁场解析方法的一些认识[J]. 电工技术学报, 2016, 31(19): 11-25. Lei Yinzhao.Reviews of analytical methods for electromagnetic fields[J]. Transactions of China Electrotechnical Society, 2016, 31(19): 11-25. [20] 刘有为, 武新军, 宋韵. 基于等效面磁荷的开放式永磁磁路解析计算与有限元仿真验证[J]. 磁性材料及器件, 2021, 52(2): 17-23. Liu Youwei, Wu Xinjun, Song Yun.Analytical solutions and finite element simulating verification to the open magnetic circuit of permanent magnet based on the equivalent surface magnetic charge[J]. Journal of Magnetic Materials and Devices, 2021, 52(2): 17-23. [21] Nguyen V T, Lu T F.Analytical expression of the magnetic field created by a permanent magnet with diametrical magnetization[J]. Progress in Electro-magnetics Research C, 2018, 87: 163-174. [22] Hart S, Hart K, Selvaggi J P.Analytical expressions for the magnetic field from axially magnetized and conically shaped permanent magnets[J]. IEEE Transa-ctions on Magnetics, 2020, 56(7): 2100809. [23] Mengelkamp J, Ziolkowski M, Weise K, et al.Permanent magnet modeling for Lorentz force evaluation[J]. IEEE Transactions on Magnetics, 2015, 51(7): 6201211. [24] 彭飞, 杨文英, 翟国富. 基于优化的传输线法的并行静磁场有限元方法[J]. 电工技术学报, 2019, 34(13): 2716-2725. Peng Fei, Yang Wenying, Zhai Guofu.An optimized parallel transmission line iteration for parallel finite element analysis in magnetostatic field[J]. Transa-ctions of China Electrotechnical Society, 2019, 34(13): 2716-2725. [25] Wu Lijian, Yin Hao, Wang Dong, et al.A nonlinear subdomain and magnetic circuit hybrid model for open-circuit field prediction in surface-mounted PM machines[J]. IEEE Transactions on Energy Con-version, 2019, 34(3): 1485-1495. [26] He Jingxiong, Li Guoli, Zhou Rui, et al.Torque modelling and validation for a spherical motor with stepped permanent magnets[J]. Journal of Electrical Engineering & Technology, 2020, 15(6): 2661-2673. [27] 周睿. 永磁球形电机解析建模和基于粒子群算法的优化控制研究[D]. 合肥: 安徽大学, 2020. Zhou Rui.Analytical modeling and PSO algorithm based optimal control research of permanent magnet spherical motor[D]. Hefei: Anhui University, 2020. [28] 赵轩哲, 张殿海, 史凯萌, 等. 考虑颗粒属性的软磁复合材料磁特性预测方法[J]. 电工技术学报, 2024, 39(23): 7309-7318. Zhao Xuanzhe, Zhang Dianhai, Shi Kaimeng, et al.Prediction of magnetic properties of soft magnetic composites with considering of particle properties[J]. Transactions of China Electrotechnical Society, 2024, 39(23): 7309-7318. [29] 迟明珺, 马西奎, 马亮, 等. 适用于有源区域电磁波时程精细积分仿真的大时间步长矩阵数值积分方法[J]. 电工技术学报, 2024, 39(21): 6604-6613, 6895. Chi Mingjun, Ma Xikui, Ma Liang, et al.A large time step size matrix numerical integral method for precise integration time domain simulation of electro-magnetic waves in active regions[J]. Transactions of China Electrotechnical Society, 2024, 39(21): 6604-6613, 6895.
2026, Vol. 41 
