基于傅里叶级数方法的超导电动磁浮半解析电磁模型及刚度特性分析

doi:10.3901/JME.2025.14.261

摘要/Abstract

摘要： 超导电动磁浮具有悬浮间隙大、自稳定等优势，是目前唯一实现时速600 km载人运行的地面运载工具，更是高速轨道交通的重要发展方向之一。模型采用超导电动磁浮MLX01悬浮架的基本结构，基于傅里叶级数法建立车载跑道形线圈的空间磁场分布半解析模型，在该模型基础上，计算悬浮架的电磁性能及刚度特性。该模型建立在三维坐标系下，在跑道形磁体外区域采用标量磁位拉普拉斯方程，并利用区域之间的边界条件得到谐波系数；采用数值计算得到车载磁体空间磁场分布；随后，结合动态电路模型及能量法，计算车辆在不同运行条件下的时域电磁力，通过与有限元和试验结果相比较，验证了该模型的准确性。最后，分析超导电动磁浮的单个悬浮架的电磁力及其刚度特性；探究速度和位置对横、垂向及旋转刚度的影响规律，分析变化趋势的原因并给出拟合表达式。该半解析模型充分考虑车载超导磁体的包含厚度和宽度的跑道形结构及悬浮架之间的极距，表达悬浮架多个车载磁体的空间磁场分布；该模型和分析方法可用于含跑道形线圈的直线电机等系统的电磁分析，刚度拟合表达式可用于超导电动磁浮车辆动力学性能分析，为车辆紧固件和支撑的机械结构设计提供参考。

关键词: 超导电动磁浮, 傅里叶级数, 跑道形磁体, 时域电磁力, 刚度特性

Abstract: The superconducting electrodynamic suspension(EDS) has a large levitation gap and self-stability, and is currently the only rail transit to achieve a speed of 600 km/h for manned operation. It is one of the important development directions for high-speed rail transport. A semi-analytical model based on the Fourier series method is established to calculate the spatial magnetic field of onboard racetrack magnets of the EDS train. Firstly, based on this semi-analytical model, the electromagnetic properties and stiffness performance of the bogie of the EDS system are simulated. The model is established in a 3-D coordinate. The Laplace equation is established in the outer region of the racetrack-shaped magnets, with the scalar magnetic potential as the solution variable. The boundary conditions between the regions are used to obtain the harmonic coefficients. The partial differential equation is solved to obtain the spatial magnetic field using numerical methods. Secondly, the electromagnetic forces at different operation conditions are solved by combining the dynamic circuit model and the energy method. This model is verified through the comparison of the published experiment and finite element analysis results. Finally, the electromagnetic force and stiffness characteristics of the suspension bogie are analyzed in detail at different operation velocities, lateral offset, vertical offset, and rotation angles. The fitting expressions obtained included the aforementioned factors. The semi-analytical model in this article takes account of the racetrack-shaped structure of the onboard superconducting magnets containing thickness and width. It also takes into account the pole distance between the suspension bogie. The magnetic field of the four onboard magnets on one side of the bogie can be expressed exactly. The model and analysis method in this article can be used for the electromagnetic analysis of systems such as linear motors containing racetrack-shaped coils. Most important, the fitting equation of the stiffness can be used for the analysis of the dynamics of the EDS train. It provided the reference for the mechanical structural design of the vehicle fasteners and supports.

Key words: superconducting electrodynamic suspension, fourier series method, racetrack magnet, transient electromagnetic force, stiffness characteristic

中图分类号:

U237
TM153

黄欢, 李海涛, 祝翰林, 程言行, 邓自刚, 郑珺. 基于傅里叶级数方法的超导电动磁浮半解析电磁模型及刚度特性分析[J]. 机械工程学报, 2025, 61(14): 261-272.

HUANG Huan, LI Haitao, ZHU Hanlin, CHENG Yanxing, DENG Zigang, ZHENG Jun. Semi-analytical Electromagnetic Model and Stiffness Characteristics of Superconducting Electrodynamic Suspension Based on the Fourier Series Method[J]. Journal of Mechanical Engineering, 2025, 61(14): 261-272.

参考文献

[1] POWELL J，DANBY G. A 300 mph magnetically suspended train[J]. Mechanical Engineering，1967，89:30-35.
[2] POWELL J，DANBY G. High-speed transport by magnetically suspended trains[C/CD]// American Soc. of Mech. Eng. Winter Annual Meeting，America，New York，1966.
[3] HAN H，KIM D. Magnetic levitation[M]. Dordrecht:Springer，2016.
[4] LIM J，LEE C，LEE J，et al. Design model of null-flux coil electrodynamic suspension for the hyperloop[J]. Energies，2020，13(19):5075.
[5] 马光同，杨文姣，王志涛，等. 超导磁浮交通研究进展[J]. 华南理工大学学报(自然科学版)，2019，47(7):68-74. MA Guangtong，YANG Wenjiao，WANG Zhitao，et al. Research development of superconducting Maglev transportation[J]. Journal of South China University of Technology(Natural Science Edition)，2019，47(7):68-74.
[6] 邓自刚，刘宗鑫，李海涛，等. 磁悬浮列车发展现状与展望[J]. 西南交通大学学报，2022，57(3):455-474，530. DENG Zigang，LIU Zongxin，LI Haitao，et al. Development status and prospect of maglev train[J]. Journal of Southwest Jiaotong University，2022，57(3):455-474，530.
[7] 李家志，索红莉，王毅，等. 超导材料在磁悬浮列车上的应用进展(上)[J]. 铁道技术监督，2020，48(3):38-44. LI Jiazhi，SUO Hongli，WANG Yi，et al. Progress in the application of superconducting materials on maglev trains (Part 1 of 2)[J]. Railway Quality Control，2020，48(3):38-44.
[8] 李家志，索红莉，王毅，等. 超导材料在磁悬浮列车上的应用进展(下)[J]. 铁道技术监督，2020，48(4):51-57. LI Jiazhi，SUO Hongli，WANG Yi，et al. Progress in the application of superconducting materials on maglev trains (Part 2 of 2)[J]. Railway Quality Control，2020，48(4):51-57.
[9] HE J L，ROTE D M，COFFEY H T. Applications of the dynamic circuit theory to maglev suspension systems[J]. IEEE Transactions on Magnetics，1993，29(6):4153-4163.
[10] 龚夕霞，卢琴芬. 超导电动悬浮系统基于有限元-解析耦合算法的力特性分析[J]. 微电机，2020，53(9):1-7. GONG Xixia，LU Qinfen. Force performance analysis of superconducting EDS system by finite element-analytical coupling algorithm[J]. Micromotors，2020，53(9):1-7.
[11] GONG T，MA G，WANG R，et al. 3-D FEM modeling of the superconducting EDS train with cross-connected figure-eight-shaped suspension coils[J]. IEEE Transactions on Applied Superconductivity，2021，31(3):3600213.
[12] LÜ G，ZHANG Z，LIU Y，et al. Analysis of forces in linear synchronous motor with propulsion，levitation and guidance for high-speed maglev[J]. IEEE Journal of Emerging and Selected Topics in Power Electronics，2022，10(3):2903-2911.
[13] AZUKIZAWA T. Persistent current analysis of superconducting coils in an electrodynamic suspension system[J]. Electrical Engineering in Japan，1995，115(2):432-437.
[14] SU Z，LUO J，MA G，et al. Fast and precise calculation of mutual inductance for electrodynamic suspension:Methodology and validation[J]. IEEE Transactions on Industrial Electronics，2022，69(6):6046-6057.
[15] NONAKA S，HIROSAKI T，KAWAKAMI E. Analysis of characteristics of repulsive magnetic levitated train using a space harmonic technique[J]. Electrical Engineering in Japan，1980，100(10):601-608.
[16] LÜ G，ZHANG Z，LIU Y，et al. Characteristics analysis of linear synchronous motor integrated with propulsion，levitation and guidance in high-speed maglev system[J]. IEEE Transactions on Transportation Electrification，2021，7(4):3185-3193.
[17] FUJIWARA S. Characteristics of EDS magnetic levitation with ground coils for levitation arranged on the side wall[J]. Electrical engineering in Japan，1988，108(3):101-110.
[18] ANDRIOLLO M，MARTINELLI G，MORINI A，et al. 3D analysis of forces in the shields of SC coils in EDS-MAGLEV transport systems[J]. Elsevier Studies in Applied Electromagnetics in Materials，1995(6):371-374.
[19] 徐杰. 超导磁悬浮列车用直线发电机的设计与特性解析分析[D]. 北京:北京交通大学，2020. XU Jie. Design and characteristic analysis of linear generator for superconducting Maglev train[D]. Beijing:Beijing Jiaotong University，2020.
[20] YAMADA T，IWAMOTO M，ITO T. Magnetic damping force in inductive magnetic levitation system for high-speed trains[J]. Electrical Engineering in Japan，1974，94(1):80-84.
[21] LIM J，LEE C，OH Y J，et al. Equivalent inductance model for the design analysis of electrodynamic suspension coils for the Hyperloop[J]. Scientific reports，2021，11:23499.
[22] YONEZU T，HOSHINO H，SUZUKI E，et al. Development of models for co-simulation of mechanics and electromagnetics for dynamic analysis of superconducting Maglev vehicles moving with large displacements[J]. RTRI Report，2012，26(5):5-10.
[23] 张伟海. 超导电动悬浮车辆横向悬挂减振策略研究[D]. 成都:西南交通大学，2021. ZHANG Weihai. Study on the vibration reduction startegies of lateral suspension in superconducting electrodynamic suspension vehicle[D]. Chengdu:Southwest Jiaotong University，2021.
[24] WANG X，HUANG J. Research on electromagnetic relationship and passive electromagnetic damping characteristics of superconducting electrodynamic Maglev train[J]. IEEE Transactions on Applied Superconductivity，2022，32(7):3602818.
[25] 张娟，赵春发，冯洋，等. 超导磁浮列车电动悬浮导向力学特性研究[J]. 机械，2020，47(9):25-32. ZHANG Juan，ZHAO Chunfa，FENG Yang，et al. Study on mechanical characteristics of the electrodynamic levitation and guidance system for the superconducting maglev train[J]. Machinery，2020，47(9):25-32.
[26] ZHANG Z，LIU Y，ZHOU T，et al. Analysis of electromechanical characteristics in air-core integrated linear synchronous motor for EDS maglev train with pitching operation condition[J]. IEEE Transactions on Vehicular Technology，2022，71(7):6938-6947.
[27] LV G，LIU Y，ZHANG Z，et al. Numerical analysis of the rotational magnetic springs for EDS maglev train[J]. CES Transactions on Electrical Machines and Systems，2022，6(1):60-66.
[28] LÜ G，ZHANG Z，LI X. Three-dimensional electromagnetic characteristics analysis of novel linear synchronous motor under lateral and yaw conditions of maglev[J]. CES Transactions on Electrical Machines and Systems，2022，6(1):29-36.
[29] 马光同，龚天勇，王瑞晨. 一种阶梯形超导磁体及具有其的电动悬浮系统:中国，CN201910025899.5[P]. 2020-07-17. MA Guangtong，GONG Tianyong，WANG Ruichen. A stepped superconducting magnet and the electric levitation system:China，CN201910025899.5[P]. 2020-07-17.
[30] SAITOH T，MAKI N，KOBAYASHI T，et al. Electromagnetic force and eddy current loss in dynamic behavior of a superconducting magnetically levitated vehicle[J]. IEEE Transactions on Applied Superconductivity，1993，3(1):417-420.
[31] MIZUNO K，SUGINO M，TANAKA M，et al. Experimental production of a real-scale REBCO magnet aimed at its application to Maglev[J]. IEEE Transactions on Applied Superconductivity，2017，27(4):3600205.
[32] HUANG H，ZHU H，COOMBS T，et al. FE Modeling of superconducting EDS system employing mixed formulations and field-circuit coupling method[J]. IEEE Transactions on Transportation Electrification，2023，9(1):1618-1628.
[33] FUJIMOTO T，AIBA M，SUZUKI H，et al. Characteristics of electromagnetic force of ground coil for levitation and guidance at the Yamanashi Maglev test line[J]. Quarterly reports of the Railway Technical Research Institute，2000，41(2):63-67.
[34] OHASHI S，OHSAKI H，MASADA E. Equivalent model of the side wall electrodynamic suspension system[J]. IEEJ Transactions on Industry Applications，1993，117(6):758-767.
[35] YONEZU T，WATANABE K，SUZUKI E. Characteristics of magnetic springs for guidance of superconducting maglev vehicles characteristics of magnetic springs for guidance[J]. QR of RTRI，2018，59(4):293-298.