Abstract：The rotor magnetic field distribution in copper-coated solid rotor induction motors is complex, making it difficult to visualize the internal field distribution using traditional lumped-parameter magnetic circuit analysis or to account for the influence of slip on equivalent circuit parameters. This paper focuses on solving the electromagnetic field within the copper-coated solid rotor. A subdomain analytical method in cylindrical coordinates is employed for the effects of stator slotting and slot-opening shape on the electromagnetic field. The analytical solutions for the air gap, rotor copper layer, and rotor steel are derived using 2D eddy-current finite-element analysis. Compared with the finite element method, the proposed analytical approach significantly reduces computation time by selecting appropriate harmonic orders. For the copper-coated solid-rotor induction motor, the subdomain analytical method and the traditional wave impedance method are compared with respect to equivalent circuit models, input impedance parameters, and output characteristics. The results demonstrate that the subdomain analytical method achieves greater consistency with finite element calculations across different slip conditions than the traditional wave impedance method. This paper lists the Laplace equations (for the air-gap and slot-opening subdomains), the Poisson equation (for the slot subdomain), and the Helmholtz equations (for the rotor copper layer and iron core subdomains). The general solutions for the vector magnetic potential in each subdomain are obtained. The harmonic coefficients in the Fourier series of the vector magnetic potential are determined through eight boundary conditions applied at four interfaces. Taking a 2-pole 18-slot copper-coated solid rotor induction motor as the model, the radial and tangential air-gap flux densities, rotor magnetic flux line distributions, and rotor eddy current density distributions under no-load, load, and locked-rotor conditions are calculated. The Maxwell stress torque under current source excitation is derived. The analytical results are validated using a two-dimensional eddy-current field finite-element analysis. To investigate the influence of harmonic orders on computation time and numerical errors, the harmonic orders for the air gap, slot, and slot opening are incrementally increased, and the resulting discrepancies between analytical results and finite element simulations are observed. A comparison is made with the traditional wave impedance method regarding the form of the equivalent circuit and the approach to calculating impedance parameters. The reasons for errors in the equivalent circuit parameters computed by the wave impedance method and the subdomain analytical method are compared. The following conclusions are drawn. (1) The subdomain analytical method can clearly and accurately represent the electromagnetic field distribution in copper-clad solid rotor induction motors. Selecting appropriate harmonic orders maintains computational efficiency while achieving accuracy. (2) Compared to the wave impedance method, the proposed subdomain analytical method provides more accurate calculations of equivalent circuit parameters and remains unaffected by variations in slip.
朱利伟, 邓智泉, 王澳华, 高志国. 覆铜实心转子感应电机二维磁场解析及等效电路参数计算[J]. 电工技术学报, 2026, 41(6): 1907-1921.
Zhu Liwei, Deng Zhiquan, Wang Aohua, Gao Zhiguo. Two-Dimensional Analytical Calculation of Magnetic Field and Equivalent Circuit Parameters for Copper-Coated Solid Rotor Induction Motors. Transactions of China Electrotechnical Society, 2026, 41(6): 1907-1921.
[1] 陈鑫杰, 王慧贞. 航空电源系统用高速实心式永磁同步电机设计[J]. 电机与控制应用, 2024, 51(2): 61-70. Chen Xinjie, Wang Huizhen.Design of high-speed solid permanent magnet synchronous motor for aviation power system[J]. Electric Machines & Control Application, 2024, 51(2): 61-70. [2] 唐孝镐, 宁玉泉, 傅丰礼. 实心转子异步电机及其应用[M]. 北京: 机械工业出版社, 1991. [3] Gerada D, Mebarki A, Brown N L, et al.High-speed electrical machines: technologies, trends, and developments[J]. IEEE Transactions on Industrial Electronics, 2014, 61(6): 2946-2959. [4] Gieras J F, Saari J.Performance calculation for a high-speed solid-rotor induction motor[J]. IEEE Transactions on Industrial Electronics, 2012, 59(6): 2689-2700. [5] 黄子果, 王善铭, 倪守辉. 光滑实心转子异步电机等效电路参数的二维计算方法[J]. 中国电机工程学报, 2016, 36(9): 2505-2512. Huang Ziguo, Wang Shanming, Ni Shouhui.2D calculation methods of equivalent-circuit parameters in smooth solid rotor induction machines[J]. Proceedings of the CSEE, 2016, 36(9): 2505-2512. [6] 洪岑岑, 张驰, 滕福林, 等. 自轴承式双定子实心转子盘式异步飞轮电机设计[J]. 微电机, 2021, 54(8): 33-37. Hong Cencen, Zhang Chi, Teng Fulin, et al.The design of a novel self-bearing dual stator solid rotor axial flux induction motor[J]. Micromotors, 2021, 54(8): 33-37. [7] 朱利伟, 邓智泉. 考虑材料非线性覆铜实心转子感应电机转子阻抗参数的二维解析计算[J]. 电工技术学报, 2024, 39(24): 7688-7699. Zhu Liwei, Deng Zhiquan.2D analytic calculation of rotor-impedance parameters in copper-coated solid rotor induction motors considering the nonlinear material[J]. Transactions of China Electrotechnical Society, 2024, 39(24): 7688-7699. [8] 黄子果, 王善铭, 孙宇光, 等. 开槽实心转子电机的等效电路参数计算与性能分析[J]. 中国电机工程学报, 2017, 37(4): 1208-1216. Huang Ziguo, Wang Shanming, Sun Yuguang, et al.Equivalent-circuit parameter calculations and performance analysis of slit solid rotor asynchronous machines[J]. Proceedings of the CSEE, 2017, 37(4): 1208-1216. [9] Feng Haichao, Cui Xu, Si Jikai, et al.Equivalent circuit model of novel solid rotor induction motor with toroidal winding applying composite multilayer theory[J]. Applied Sciences, 2019, 9(16): 3288. [10] Greig J, Freeman E M.Travelling-wave problem in electrical machines[J]. Proceedings of the Institution of Electrical Engineers, 1967, 114(11): 1681. [11] Pyrhonen J.Calculating the effects of solid-rotor material on high-speed induction motor characteristics[J]. European Transactions on Electrical Power, 1991, 1(6): 301-310. [12] 凌在汛, 周理兵, 张毅, 等. 笼型实心转子屏蔽感应电机电磁场及参数研究(二): 二维多层电磁场模型及其解析计算[J]. 电工技术学报, 2018, 33(17): 4016-4028. Ling Zaixun, Zhou Libing, Zhang Yi, et al.Parameters determination and electromagnetic field analysis of canned solid-rotor induction motor(2): 2D-multilayer electromagnetic model and its analytical calculation[J]. Transactions of China Electrotechnical Society, 2018, 33(17): 4016-4028. [13] Räisänen V, Suuriniemi S, Kurz S, et al.Rapid computation of harmonic eddy-current losses in high-speed solid-rotor induction machines[J]. IEEE Transactions on Energy Conversion, 2013, 28(3): 782-790. [14] 李世奇, 佟文明, 贾建国, 等. 考虑磁桥非线性的内置式永磁同步电机空载电磁性能通用解析模型[J]. 电工技术学报, 2023, 38(6): 1421-1432. Li Shiqi, Tong Wenming, Jia Jianguo, et al.General analytical model of No-load electromagnetic performance of interior permanent magnet synchronous motors considering nonlinearity of magnetic bridges[J]. Transactions of China Electrotechnical Society, 2023, 38(6): 1421-1432. [15] 佟文明, 杜绍雨, 贾建国, 等. 基于改进复相对磁导函数的开槽轴向磁通永磁电机气隙磁场解析模型[J]. 电工技术学报, 2024, 39(24): 7700-7711. Tong Wenming, Du Shaoyu, Jia Jianguo, et al.Analytical model of air-gap magnetic field of slotted axial flux permanent magnet motor based on improved complex relative permeance function[J]. Transactions of China Electrotechnical Society, 2024, 39(24): 7700-7711. [16] 佟文明, 杨先凯, 鹿吉文, 等. 双层永磁体结构高速永磁电机转子涡流损耗解析模型[J]. 电工技术学报, 2024, 39(20): 6293-6304. Tong Wenming, Yang Xiankai, Lu Jiwen, et al.Rotor eddy current loss analytical model for high-speed permanent magnet motor based on double layer permanent magnet structure[J]. Transactions of China Electrotechnical Society, 2024, 39(20): 6293-6304. [17] 佟文明, 田野, 李晓健, 等. 双层复合护套高速永磁电机转子涡流损耗解析模型[J]. 电工技术学报, 2024, 39(14): 4328-4340. Tong Wenming, Tian Ye, Li Xiaojian, et al.Analytical modeling for rotor eddy current loss of high-speed surface-mounted permanent magnet motor with double-layer compound retaining sleeve[J]. Transactions of China Electrotechnical Society, 2024, 39(14): 4328-4340. [18] 于博学, 刘光伟, 宋志环, 等. 内置式双V型断桥结构永磁同步电机空载电磁性能解析计算[J]. 电机与控制学报, 2025, 29(5): 31-40. Yu Boxue, Liu Guangwei, Song Zhihuan, et al.Analytical calculation of no-load electromagnetic performance for interior permanent magnet synchronous motor with double-V magnetic bridge break shape[J]. Electric Machines and Control, 2025, 29(5): 31-40. [19] Mirzaei M.Computational modeling of an axial airgap induction motor with a solid rotor[J]. IEEE Transactions on Magnetics, 2022, 58(6): 8105109. [20] 张守首, 郭思源. 基于子域分析模型的实心转子感应电机磁场解析[J]. 电工技术学报, 2021, 36(20): 4285-4296. Zhang Shoushou, Guo Siyuan.Analytical solution of magnetic field in solid rotor induction machine based on subdomain model[J]. Transactions of China Electrotechnical Society, 2021, 36(20): 4285-4296. [21] Boughrara K, Dubas F, Ibtiouen R.2-D analytical prediction of eddy currents, circuit model parameters, and steady-state performances in solid rotor induction motors[J]. IEEE Transactions on Magnetics, 2014, 50(12): 7028214. [22] 许浩, 赵镜红, 熊义勇, 等. 五相实心转子感应电机磁场及参数计算[J]. 华中科技大学学报(自然科学版), 2025, 53(1): 28-34. Xu Hao, Zhao Jinghong, Xiong Yiyong, et al.Calculation of magnetic field and parameters of five phase solid rotor induction motor[J]. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2025, 53(1): 28-34. [23] Lubin T, Mezani S, Rezzoug A.Analytical computation of the magnetic field distribution in a magnetic gear[J]. IEEE Transactions on Magnetics, 2010, 46(7): 2611-2621. [24] 陈世坤. 电机设计[M]. 2版. 北京: 机械工业出版社, 1990.
2026, Vol. 41 
