基于电磁场数值计算的永磁电机性能分析方法研究
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摘要
永磁电机的设计与应用是电机领域的前沿课题。本文从电磁场基本理论出
发，采用有限元法围绕永磁电机参数计算和性能分析进行了系统地研究。建立了
比较完整的理论分析模型和计算方法，对永磁电机的设计与优化有着重要的意
义。全文共分六章。
第一章阐述了本课题研究的目的和意义。分析了永磁电机和永磁材料的现
状与发展趋势，归纳了永磁电机的各种性能分析方法，指出有限元计算方法在电
机性能分析方面所具有的优越性。总结了永磁电机电磁场数值计算中的研究进展
与热点问题，并介绍了本文的研究内容。
第二章研究了永磁电机磁场计算中永磁体的数学模型。在全面分析解析计
算和数值计算中所采用的数学模型基础上，提出一种改进的边界等效面电流离散
化处理法，该方法不仅适用于任意复杂形状永磁体，而且可以统一方便的处理平
行、径向和周向几种永磁磁化方向，对实现永磁电机有限元程序的通用性具有重
要的意义。
第三章对永磁直流电动机有限元分析方法进行了研究。建立了合理的永磁
直流电动机动态有限元分析模型，该模型中考虑了齿槽效应和绕组换向对电机性
能的影响。首次对小功率奇数槽永磁直流电动机的性能进行了精确计算，并提出
计算斜槽或斜极电机齿槽转矩的简捷高效的有限元方法。分析了转矩脉动的原
因，讨论了削弱转矩脉动的方法。
第四章对异步起动永磁同步电动机进行了研究。当采用通用有限元软件对永
磁同步电动机进行分析时，存在着电机定转子轴线相对位置未知的问题，而确定
这个相对位置是进行电机负载磁场计算的前提。本文从研究永磁同步电动机物理
量的电磁关系出发，设计了高效的搜索算法，通过求解电磁场逆问题确定定、转
子空间位置。采用负载法计算永磁同步电动机的参数，使参数中的材料特性真正
反映了实际负载条件下磁介质的饱和情况。改进了现有的电枢反应电抗计算方
法，使得计算时间更短、计算精度更高。
第五章采用场路耦合时步有限元法分析了永磁同步电动机的稳态运行性
能。建立了考虑转子电路方程的永磁同步电动机场路耦合新模型，弥补了以往模
型中没有将转子电路与磁场方程耦合的不足。运用牛顿一拉夫逊法求解场路耦合
非线性方程组，推导了单元分析时雅克比矩阵和剩余矢量的计算格式。本文提出
的模型不仅改进了永磁同步电机的稳态性能分析，而且对暂态性能分析也具有重
要意义。
第六章中采用场路耦合时步有限元法分析了无刷直流电动机的稳态和暂态
东南大学博卜学位论义
性能。提出适用于无刷直流电机的分时段电路方程与磁场耦合的时步有限元分析
模型。通过深入研究无刷直流电动机的运行特性，导出各个运行区间和运行模式
的电路方程。研究了分时段电路与磁场方程耦合时有限元单元分析的处理方法。
本文提出的分析模型对无刷电机准确性能分析具有重要的意义。
Design and application of Permanent Magnet (PM) motors have been a forward research topic in the study of electrical machines. From the basic electromagnetic field theory, the performance and parameters of PM motors are investigated using Finite Element Method (FEM). A series of computation methods and analysis models that are more complete and more accurate are presented. The research work is important step in the design and optimization of PM motors. This thesis contains six chapters.
The purpose and meaning of this topic are introduced in chapter 1. It mainly describes the present state and perspective of PM motors and PM material. Various methods used in analyzing PM motors are summarized. It is pointed that FEM posses the advantage in accurate performance analysis. Key problems and research status in PM motor area are summed up. The main research contents of this thesis are presented here.
Mathematical model of PM in electromagnetic field computation in PM motors is investigated in chapter 2. Various models adopted in analysis method and numerical method calculation are summed up. An improved discrete method of boarder equivalent surface current model is presented in this thesis; which is suitable to arbitrarily complex shapes, and can conveniently solve different magnetizing direction problem, such as parallel, radial and circumferential direction. It is important to utilize the PM motor FEM program.
The performance of PMDC motors is analyzed in chapter 3. A precise moving FEM analysis model is presented, in which the cogging effect and commutation are included in performance calculation. For the first time the performance of odd number slots of small size motors is precisely calculated. A convenient method is presented to calculate the cogging torque in skew slot and skew pole structure motors. The reasons of torque fluctuation are researched and the methods to decrease fluctuation are discussed.
Asynchronous start permanent magnet synchronous motors (PMSM) are investigated in chapter 4. When using universal finite element software to analyze PMSM, the relative position of stator and rotor axis is unknown, which is the precondition for load field calculation. From the basic relationship of electromagnetic variables in PMSM, a simple and efficient searching strategy is presented. Through solving inverse electromagnetic field problem, an iteration method is used to acquire the relative position. Load method is used to calculate the parameters of PMSM, in which the material characteristics reflect the magnetic saturation under actual operation. An improved calculation method of reactance of armature reaction is put forward, which need less computation time and have higher accuracy degree.
in
Steady state performance of PMSM is analyzed using circuit-field coupled FEM time-stepping method in chapter 5. For the first time, a complete field-circuit FEM time stepping model including both rotor and stator circuits is presented, which is more accurate and cover the shortage of lacking rotor circuit in past model. The Newton-Laphson method is taken to solve circuit-field coupled FEM time-stepping nonlinear problem. Jacobi matrix and the residue vector are derived in element analysis. The model used in this thesis not only improves the steady state analysis in PMSM, but also is helpful in transient state performance analysis.
Steady and transient state performance of brushless DC (BLDC) motors are calculated in chapter 6. For the first time segment circuits and field coupled FEM model is put forward, which is very suitable to analyze the BLDC motors. Through analyzing the performance characteristics of BLDC motors, the circuits at each operating interval and operating mode are obtained. The element analysis of segment circuits and field coupled FEM is studied. It is significant to accurately analyzing BLDC motors.
发，采用有限元法围绕永磁电机参数计算和性能分析进行了系统地研究。建立了
比较完整的理论分析模型和计算方法，对永磁电机的设计与优化有着重要的意
义。全文共分六章。
第一章阐述了本课题研究的目的和意义。分析了永磁电机和永磁材料的现
状与发展趋势，归纳了永磁电机的各种性能分析方法，指出有限元计算方法在电
机性能分析方面所具有的优越性。总结了永磁电机电磁场数值计算中的研究进展
与热点问题，并介绍了本文的研究内容。
第二章研究了永磁电机磁场计算中永磁体的数学模型。在全面分析解析计
算和数值计算中所采用的数学模型基础上，提出一种改进的边界等效面电流离散
化处理法，该方法不仅适用于任意复杂形状永磁体，而且可以统一方便的处理平
行、径向和周向几种永磁磁化方向，对实现永磁电机有限元程序的通用性具有重
要的意义。
第三章对永磁直流电动机有限元分析方法进行了研究。建立了合理的永磁
直流电动机动态有限元分析模型，该模型中考虑了齿槽效应和绕组换向对电机性
能的影响。首次对小功率奇数槽永磁直流电动机的性能进行了精确计算，并提出
计算斜槽或斜极电机齿槽转矩的简捷高效的有限元方法。分析了转矩脉动的原
因，讨论了削弱转矩脉动的方法。
第四章对异步起动永磁同步电动机进行了研究。当采用通用有限元软件对永
磁同步电动机进行分析时，存在着电机定转子轴线相对位置未知的问题，而确定
这个相对位置是进行电机负载磁场计算的前提。本文从研究永磁同步电动机物理
量的电磁关系出发，设计了高效的搜索算法，通过求解电磁场逆问题确定定、转
子空间位置。采用负载法计算永磁同步电动机的参数，使参数中的材料特性真正
反映了实际负载条件下磁介质的饱和情况。改进了现有的电枢反应电抗计算方
法，使得计算时间更短、计算精度更高。
第五章采用场路耦合时步有限元法分析了永磁同步电动机的稳态运行性
能。建立了考虑转子电路方程的永磁同步电动机场路耦合新模型，弥补了以往模
型中没有将转子电路与磁场方程耦合的不足。运用牛顿一拉夫逊法求解场路耦合
非线性方程组，推导了单元分析时雅克比矩阵和剩余矢量的计算格式。本文提出
的模型不仅改进了永磁同步电机的稳态性能分析，而且对暂态性能分析也具有重
要意义。
第六章中采用场路耦合时步有限元法分析了无刷直流电动机的稳态和暂态
东南大学博卜学位论义
性能。提出适用于无刷直流电机的分时段电路方程与磁场耦合的时步有限元分析
模型。通过深入研究无刷直流电动机的运行特性，导出各个运行区间和运行模式
的电路方程。研究了分时段电路与磁场方程耦合时有限元单元分析的处理方法。
本文提出的分析模型对无刷电机准确性能分析具有重要的意义。
Design and application of Permanent Magnet (PM) motors have been a forward research topic in the study of electrical machines. From the basic electromagnetic field theory, the performance and parameters of PM motors are investigated using Finite Element Method (FEM). A series of computation methods and analysis models that are more complete and more accurate are presented. The research work is important step in the design and optimization of PM motors. This thesis contains six chapters.
The purpose and meaning of this topic are introduced in chapter 1. It mainly describes the present state and perspective of PM motors and PM material. Various methods used in analyzing PM motors are summarized. It is pointed that FEM posses the advantage in accurate performance analysis. Key problems and research status in PM motor area are summed up. The main research contents of this thesis are presented here.
Mathematical model of PM in electromagnetic field computation in PM motors is investigated in chapter 2. Various models adopted in analysis method and numerical method calculation are summed up. An improved discrete method of boarder equivalent surface current model is presented in this thesis; which is suitable to arbitrarily complex shapes, and can conveniently solve different magnetizing direction problem, such as parallel, radial and circumferential direction. It is important to utilize the PM motor FEM program.
The performance of PMDC motors is analyzed in chapter 3. A precise moving FEM analysis model is presented, in which the cogging effect and commutation are included in performance calculation. For the first time the performance of odd number slots of small size motors is precisely calculated. A convenient method is presented to calculate the cogging torque in skew slot and skew pole structure motors. The reasons of torque fluctuation are researched and the methods to decrease fluctuation are discussed.
Asynchronous start permanent magnet synchronous motors (PMSM) are investigated in chapter 4. When using universal finite element software to analyze PMSM, the relative position of stator and rotor axis is unknown, which is the precondition for load field calculation. From the basic relationship of electromagnetic variables in PMSM, a simple and efficient searching strategy is presented. Through solving inverse electromagnetic field problem, an iteration method is used to acquire the relative position. Load method is used to calculate the parameters of PMSM, in which the material characteristics reflect the magnetic saturation under actual operation. An improved calculation method of reactance of armature reaction is put forward, which need less computation time and have higher accuracy degree.
in
Steady state performance of PMSM is analyzed using circuit-field coupled FEM time-stepping method in chapter 5. For the first time, a complete field-circuit FEM time stepping model including both rotor and stator circuits is presented, which is more accurate and cover the shortage of lacking rotor circuit in past model. The Newton-Laphson method is taken to solve circuit-field coupled FEM time-stepping nonlinear problem. Jacobi matrix and the residue vector are derived in element analysis. The model used in this thesis not only improves the steady state analysis in PMSM, but also is helpful in transient state performance analysis.
Steady and transient state performance of brushless DC (BLDC) motors are calculated in chapter 6. For the first time segment circuits and field coupled FEM model is put forward, which is very suitable to analyze the BLDC motors. Through analyzing the performance characteristics of BLDC motors, the circuits at each operating interval and operating mode are obtained. The element analysis of segment circuits and field coupled FEM is studied. It is significant to accurately analyzing BLDC motors.
引文
第一章
[1] 唐任远．现代永磁电机理论与设计．北京：机械工业出版社，1997．
[2] 王宗培．永磁直流微电机．南京：东南大学出版社，1992．
[3] 陈峻峰．永磁电机．北京：机械工业出版社 1982．
[4] 王海黎．永磁微特电机发展动向．微电机．1994，Vol．27，No．33：24-25．
[5] 高勋章，罗飞路．稀土永磁电机研究现状与发展．中小型电机．1998，Vol．25，No．1：16-19．
[6] 耿联发，吴延忠．现代永磁电机发展趋势．沈阳工业大学学报．1995，Vol．17，No．1：25-28．
[7] 杨大伟．钕铁硼稀土永磁材料及其在微特电机中的应用．电子工程师．1996，No．2．
[8] 都有为．磁性材料进展．物理．2000，Vol．29，No．6：324-332．
[9] 林河成．我国稀土永磁材料的新进展．粉末冶金工业．2001，Vol．11，No．2：35-37．
[10] 杨大伟．稀土永磁材料发展动向及其在微特电机中应用概况．电子工艺技术．1997， Vol．7，No．3：186-190．
[11] 周洁，谢卫，汪国梁．用等效磁网络法与有限元法计算永磁电机参数的比较．西安交通 大学学报．1998，Vol．32，No．4．
[12] 张伟雄，郑婵君，张敬华．基于磁网络模型范本永磁电机气隙磁场的研究．第四界中国 小电机技术研讨会．上海．1999：71-75．
[13] Gu Qishan, Gao Hongzhan. Air gap field for PM electric machines. Electric Machines and Power Systems. 1985, No.10: 459-470．
[14] Gu Qishan, Gao Hongzhan. The fringing effect in PM electric machines. Electric Machines and Power Systems. 1986,No.11: 159-169．
[15] Gu Qishan, Gao Hongzhan. Effect of slotting in PM electric machines. Electric Machines and Power Systems. 1985, No.10 : 273-284．
[16] Z. Q. Zhu and D. Howe. Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅰ open-circuit field. IEEE Transactions on Magnetics. 1993, Vol. 29, No.1: 124-135．
[17] Z. Q. Zhu and D. Howe. Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅱ Armature-reaction field. IEEE Transactions on Magnetics. 1993, Vol. 29, No.1: 136-142．
[18] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅲ Effect of stator slotting. IEEE Transactions on Magnetics. 1993, Vol. 29,No1: 143-151．
[19] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅳ Magnetic field on load. IEEE Transactions on Magnetics. 1993, Vol. 29,No.1: 152-158．
[20] Atallah K, Zhu Z. Q. Curvature effect in radial permanent magnet machines. Electric Machines and Power systems. 1994, No.22: 511-520．
[21] 盛剑霓．工程电磁场数值分析．西安：西安交通大学出版社．1991．
[22] 张榴晨，徐松．有限元法在电磁计算中的应用．北京：中国铁道出版社．1996．
[23] Craiu, O., Dan, N., Badea, E.A. Numerical analysis of permanent magnet DC motor performances. IEEE Transactions on Magnetics. 1995, Vol.31, No.6, Part.2: 3500-3502．
[24] Z. Q. Zhu and D. Howe Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors. IEEE Transactions on Magnetics. 1992, Vol.28, No.2, 1371-74．
[25] Z. Q. Zhu and D. Howe. Influence of design parameters on congging torque in permanent magnet machines. IEEE Transactions on Energy Conversion. 2000, Vol.15, No.4: 407-412．
[26] R. P. Deodhar, David A. Station, T.M.Jahns. Prediction of Cogging Torque Using the Flux MMF Diagram Technique. IEEE Transactions on Industry Applications. 1996, Vol.32, No.3: 569-576．
[27] Takeo Ishikawa, Gordon R. Slemon A method of Reducing Ripple Torque in Permanent Magnet Motors without Skewing. IEEE Transactions on Magnetics. 1993, Vol.29, No.2: 2028-2031．
[28] Touzhu Li, Gordon Slemon. Reduction of Cogging Torque in Permanent Magnet Motors. IEEE Transactions on Magnetics. 1988, Vol.24, No.6: 2901-2903．
[29] Zhou, P., Rahman, M.A., Jabbar, M.A. Field circuit analysis of permanent magnet synchronous motors. IEEE Transactions on Magnetics. 1994, Vol.30, No.4, Part.2: 1350-1359
[30] Rahman, M.A., Ping Zhou. Field-based analysis for permanent magnet motors. IEEE Transactions on Magnetics. 1994, Vol.30, No.5, Part.2: 3664-3667．
[31] Kurihara, K.., Wakui, G; Kubota, T. Steady-state performance analysis of permanent magnet synchronous motors including space harmonics. IEEE Transactions on Magnetics. 1994, Vol.30, No.3: 1306-1315．
[32] Ho, S.L., Fu, W.N., Li, H.L. Performance analysis of brushless DC motors including features of the control loop in the finite element modeling. IEEE Transactions on Magnetics. 2001, Vol.37, No.5, Part. 1: 3370-3374．
[33] Seung-Chan Park, Byung-Ⅱ Kwon, Hee-Soo Yoon. Analysis of exterior-rotor BLDC motor considering the eddy current effect in the rotor steel shell. IEEE Transactions on Magnetics. 1999, Vol.35, No.3, Part.1: 1302-1305．
[34] 邱捷，励庆乎．分数槽永磁同步电动机稳态参数的计算．西安交通大学学报．1996， V01． 30，No．11：33-38．
[35] 邱捷，励庆孚．实心转子永磁同步电动机的动态转子参数及起动特性的计算．中国电机 工程学报．1996，Vol．19． No．6：6-10．
[36] 李声晋，芦刚，马瑞卿．稀土永磁电机磁场有限元分析自动化技术．微电机．1994， Vol．27，No．3：27-28．
[37] 吴延忠，孙建中，白凤仙．面向对象技术的永磁电机二维电磁场计算软件．沈阳工业大 学学报．1998，Vol．20 No．2：29-33．
[38] 林鹤云，朱震莲．稀土永磁电机磁场有限元分析系统．微电机．1998，Vol．31，No．4：7-10． 第二章
[1] Gu Qishan, Gao Hongzhan. Air gap field for PM electric machines. Electric Machines and Power Systems. 1985, No.10 : 459-470．
[2] Gu Qishan, Gao Hongzhan. The fringing effect in PM electric machines. Electric Machines and Power Systems. 1986,No.11: 159-169．
[3] Gu Qishan, Gao Hongzhan. Effect of slotting in PM electric machines. Electric Machines and
Power Systems. 1985, No.10 : 273-284．
[4] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅰ open-circuit field. IEEE Transactions on Magnetics. 1993, Vol.29, No.1: 124-135．
[5] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅱ Armature-reaction field. IEEE Transactions on Magnetics. 1993, Vol.29, No.l: 136-142．
[6] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅲ Effect of stator slotting. IEEE Transactions on Magnetics. 1993, Vol.29, No.1: 143-151．
[7] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅳ Magnetic field on load. IEEE Transactions on Magnetics. 1993, Vol.29, No.l: 152-158．
[8] Atallah K, Zhu Z. Q, et al. Curvature effect in radial permanent magnet machines. Electric Machines and Power Systems. 1994, No.22: 511-520．
[9] Z. Q. Zhu and D. Howe Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors. IEEE Transactions on Magnetics. 1992, Vol.28, No.2: 1371-74．
[10] Z. Q. Zhu and D. Howe Influence of design parameters on congging torque in permanent magnet machines. IEEE Transactions on Energy Conversion. 2000, Vol.15, No.4: 407-412．
[11] Nady Boules Prediction of no-load flux density distribution in permanent magnet machines. IEEE Transactions on Industry Applications. 1985, Vol.21, No.4: 633-643．
[12] Jamime De Ree and Nady Boules Torque production in permanent magnet synchronous motors. IEEE Transactions on Industry Applications. 1989, Vol. 25, No.1: 107-112．
[13] 熊光煜．用磁荷模型计算永磁电机的气隙磁场分布．太原工业大学学报．1990，Vol．21 NO．2：69-75．
[14] 胡之光．永磁同步电动机的研究．微特电机．1983，No．3：14-16．
[15] 李世作，肖仁山．钕铁硼永磁无刷直流电动机电磁场有限元分析．微特电机．1996，No．3： 10-13．
[16] 林鹤云，朱震莲．稀土永磁电机磁场有限元分析系统．微电机．1998，No．4，7-10．
[17] Zhu, Z.Q., Howe, D. Halbach permanent magnet machines and applications: a review. IEE Proceedings-Electric Power Applications. 2001, Vol. 148 No. 4 : 299-308．
[18] 唐任远．现代永磁电机理论与设计．北京：机械工业出版社，1997．
[19] 尹华杰，金振荣．有限元方法中永磁励磁的自动处理．微特电机．1993，Vol．21，No．4： 9-10． 第三章
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[2] 唐任远．现代永磁电机理论与设计．北京：机械工业出版社，1997．
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第四章
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[19] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅳ Magnetic field on load. IEEE Transactions on Magnetics. 1993, Vol. 29,No.1: 152-158．
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[26] R. P. Deodhar, David A. Station, T.M.Jahns. Prediction of Cogging Torque Using the Flux MMF Diagram Technique. IEEE Transactions on Industry Applications. 1996, Vol.32, No.3: 569-576．
[27] Takeo Ishikawa, Gordon R. Slemon A method of Reducing Ripple Torque in Permanent Magnet Motors without Skewing. IEEE Transactions on Magnetics. 1993, Vol.29, No.2: 2028-2031．
[28] Touzhu Li, Gordon Slemon. Reduction of Cogging Torque in Permanent Magnet Motors. IEEE Transactions on Magnetics. 1988, Vol.24, No.6: 2901-2903．
[29] Zhou, P., Rahman, M.A., Jabbar, M.A. Field circuit analysis of permanent magnet synchronous motors. IEEE Transactions on Magnetics. 1994, Vol.30, No.4, Part.2: 1350-1359
[30] Rahman, M.A., Ping Zhou. Field-based analysis for permanent magnet motors. IEEE Transactions on Magnetics. 1994, Vol.30, No.5, Part.2: 3664-3667．
[31] Kurihara, K.., Wakui, G; Kubota, T. Steady-state performance analysis of permanent magnet synchronous motors including space harmonics. IEEE Transactions on Magnetics. 1994, Vol.30, No.3: 1306-1315．
[32] Ho, S.L., Fu, W.N., Li, H.L. Performance analysis of brushless DC motors including features of the control loop in the finite element modeling. IEEE Transactions on Magnetics. 2001, Vol.37, No.5, Part. 1: 3370-3374．
[33] Seung-Chan Park, Byung-Ⅱ Kwon, Hee-Soo Yoon. Analysis of exterior-rotor BLDC motor considering the eddy current effect in the rotor steel shell. IEEE Transactions on Magnetics. 1999, Vol.35, No.3, Part.1: 1302-1305．
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[1] Gu Qishan, Gao Hongzhan. Air gap field for PM electric machines. Electric Machines and Power Systems. 1985, No.10 : 459-470．
[2] Gu Qishan, Gao Hongzhan. The fringing effect in PM electric machines. Electric Machines and Power Systems. 1986,No.11: 159-169．
[3] Gu Qishan, Gao Hongzhan. Effect of slotting in PM electric machines. Electric Machines and
Power Systems. 1985, No.10 : 273-284．
[4] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅰ open-circuit field. IEEE Transactions on Magnetics. 1993, Vol.29, No.1: 124-135．
[5] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅱ Armature-reaction field. IEEE Transactions on Magnetics. 1993, Vol.29, No.l: 136-142．
[6] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅲ Effect of stator slotting. IEEE Transactions on Magnetics. 1993, Vol.29, No.1: 143-151．
[7] Z. Q. Zhu and D. Howe Instantaneous magnetic field distribution in brushless permanent magnet dc motors, Part Ⅳ Magnetic field on load. IEEE Transactions on Magnetics. 1993, Vol.29, No.l: 152-158．
[8] Atallah K, Zhu Z. Q, et al. Curvature effect in radial permanent magnet machines. Electric Machines and Power Systems. 1994, No.22: 511-520．
[9] Z. Q. Zhu and D. Howe Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors. IEEE Transactions on Magnetics. 1992, Vol.28, No.2: 1371-74．
[10] Z. Q. Zhu and D. Howe Influence of design parameters on congging torque in permanent magnet machines. IEEE Transactions on Energy Conversion. 2000, Vol.15, No.4: 407-412．
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[17] Zhu, Z.Q., Howe, D. Halbach permanent magnet machines and applications: a review. IEE Proceedings-Electric Power Applications. 2001, Vol. 148 No. 4 : 299-308．
[18] 唐任远．现代永磁电机理论与设计．北京：机械工业出版社，1997．
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[2] 唐任远．现代永磁电机理论与设计．北京：机械工业出版社，1997．
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第四章
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