大气隙混合磁悬浮轴承相关理论及设计方法的研究
|
摘要
磁悬浮支承技术是利用可控电磁力将物体悬浮起来,使物体与基础之间没有机械接触,从而实现无机械摩擦、无磨损、无需润滑、功耗低和速度高等优点,同时其支承特性(刚度和阻尼等)可通过控制参数实现在线调整,从而增强系统的稳定性,因此,磁悬浮支承技术被誉为支承技术的一场革命。混合磁悬浮轴承是将电磁轴承与永磁轴承组合起来形成的一种磁悬浮轴承系统,它兼顾了电磁轴承和永磁轴承的优点,目前己被广泛研究。但在风力发电机和人工心脏泵等应用场合中,混合磁悬浮轴承的工作气隙大,常规磁悬浮轴承设计理论中磁场均匀、无漏磁等基本假设不成立,需要建立新的设计理论与方法。
本文针对一种典型的混合磁悬浮轴承,利用常规的磁路法对不同气隙长度的混合磁悬浮轴承进行了结构设计,建立了其承载力解析模型,并采用有限元分析软件对其承载特性进行了分析；对比了不同气隙长度下的两种混合磁悬浮轴承的解析法结果和有限元结果,结果表明在混合磁悬浮轴承中,当气隙长度较大时采用常规磁路法无论是在设计还是在分析时都存在较大的误差。
本文对大气隙混合磁悬浮轴承中磁路法模型计算误差的特性及变化规律进行了研究,并研究了导致磁路法计算误差的主要影响因数和各因数的影响规律。
针对忽略漏磁导致的磁路法计算误差这一主要影响因数,采用漏磁系数对漏磁进行了描述,并建立了基于漏磁系数的混合磁悬浮轴承磁路模型；研究了混合磁悬浮轴承的气隙漏磁随结构参数的变化规律,提出了混合磁悬浮轴承漏磁系数的描述方法,并对漏磁系数描述方法的适用范围进行了研究；基于该漏磁系数的描述方法对大气隙和常规气隙混合磁悬浮轴承进行了界定。
利用场路结合的方法建立了一种基于漏磁导的气隙磁场漏磁模型,并建立了基于漏磁导修正的混合磁悬浮轴承磁路法承载力模型；对模型的适用范围进行了研究,结果表明该模型能满足大多数工程实际中混合磁悬浮轴承承载力分析的需求；设计了一种单自由度混合磁悬浮实验装置,对该模型进行了验证。
最后,对常规磁路法不能进行大气隙混合磁悬浮轴承结构设计的主要原因进行了分析,并基于漏磁系数修正模型和漏磁导修正模型提出了大气隙混合磁悬浮轴承结构设计方法,利用改进后的磁路法对大气隙混合磁悬浮轴承进行了结构设计。
Magnetic bearing technology uses controllable electromagnetic force to levitate the object, realizing no mechanical contact between the object and the basis. Thereby it has many advantages, such as no mechanical friction, no wear, no lubrication, low power consumption and high speed. Meanwhile its supporting characteristics (stiffness and damping) can be adjusted online by changing control parameters, which can enhance the system stability. Therefore, magnetic bearing technology is known as a revolution of supporting technology. Hybrid magnetic bearing is a kind of magnetic bearing system which formed by combining electromagnetic bearings and permanent magnetic bearings. It takes into account the advantages of both electromagnetic bearings and permanent magnetic bearing. So it has been widely studied at present. However, in the applications such as wind turbines and artificial heart pump, the working air-gap of hybrid magnetic bearing is large, which causes basic assumption in the conventional magnetic bearing design theory such as uniform magnetic field, no magnetic flux leakage are no longer correct. Therefore, a new design theory and method need to be developed.
In this paper, a typical kind of hybrid magnetic bearing has been discussed, and the structures of hybrid magnetic bearings with different air-gap have been designed by using conventional equivalent magnetic circuit method. The bearing capacity models of those hybrid magnetic bearings have been built, and the bearing forces have been analyzed by ANSYS. By comparing the calculation results of traditional equivalent magnetic circuit method and ANSYS, it shows that the conventional equivalent magnetic circuit method cannot meet the requirement of structural design and bearing capacity analysis of large air-gap hybrid magnetic bearings.
The characteristics and variation rules of equivalent magnetic circuit method model calculation error in large air-gap hybrid magnetic bearings are studied. What's more, impact factors which caused equivalent magnetic circuit method calculation error and impact rules of each factor on the error are studied.
For the main factor, magnetic flux leakage that causes errors of conventional magnetic circuit method, leakage coefficients are introduced to describe magnetic flux leakage. Furthermore, the magnetic circuit model of large air-gap hybrid magnetic bearings based on leakage coefficients is estabilished. The changing laws between air-gap magnetic flux leakage and structural parameters are studied. The description method of leakage coefficients is presented and application range of this description method is studied. Based on the description method, large air-gap hybrid magnetic bearings are demarcated from conventional air-gap ones.
An air-gap magnetic flux leakage model based on leakage permeance has been built by using a combination method of magnetic field and an equivalent magnetic circuit bearing capacity model of hybrid magnetic bearings based on magnetic flux leakage has been built. On this basis, the application range of this model is studied and the results show that this model can satisfy the majority of practical engineering hybrid magnetic bearing capacity analysis needs. Then, a single degree of freedom hybrid magnetic experimental device is designed and manufactured. And the model is validated by utilizing that device.
Finally, the main reason that conventional equivalent magnetic circuit method can not be used for structure design of large air-gap hybrid magnetic bearings is analyzed. Furthermore, the structure design method of large air-gap hybrid magnetic bearings based on leakage coefficient correction model and leakage permeance correction model is proposed. And the structural of large air-gap hybrid magnetic bearings is designed by using improved magnetic circuit method.
本文针对一种典型的混合磁悬浮轴承,利用常规的磁路法对不同气隙长度的混合磁悬浮轴承进行了结构设计,建立了其承载力解析模型,并采用有限元分析软件对其承载特性进行了分析；对比了不同气隙长度下的两种混合磁悬浮轴承的解析法结果和有限元结果,结果表明在混合磁悬浮轴承中,当气隙长度较大时采用常规磁路法无论是在设计还是在分析时都存在较大的误差。
本文对大气隙混合磁悬浮轴承中磁路法模型计算误差的特性及变化规律进行了研究,并研究了导致磁路法计算误差的主要影响因数和各因数的影响规律。
针对忽略漏磁导致的磁路法计算误差这一主要影响因数,采用漏磁系数对漏磁进行了描述,并建立了基于漏磁系数的混合磁悬浮轴承磁路模型；研究了混合磁悬浮轴承的气隙漏磁随结构参数的变化规律,提出了混合磁悬浮轴承漏磁系数的描述方法,并对漏磁系数描述方法的适用范围进行了研究；基于该漏磁系数的描述方法对大气隙和常规气隙混合磁悬浮轴承进行了界定。
利用场路结合的方法建立了一种基于漏磁导的气隙磁场漏磁模型,并建立了基于漏磁导修正的混合磁悬浮轴承磁路法承载力模型；对模型的适用范围进行了研究,结果表明该模型能满足大多数工程实际中混合磁悬浮轴承承载力分析的需求；设计了一种单自由度混合磁悬浮实验装置,对该模型进行了验证。
最后,对常规磁路法不能进行大气隙混合磁悬浮轴承结构设计的主要原因进行了分析,并基于漏磁系数修正模型和漏磁导修正模型提出了大气隙混合磁悬浮轴承结构设计方法,利用改进后的磁路法对大气隙混合磁悬浮轴承进行了结构设计。
Magnetic bearing technology uses controllable electromagnetic force to levitate the object, realizing no mechanical contact between the object and the basis. Thereby it has many advantages, such as no mechanical friction, no wear, no lubrication, low power consumption and high speed. Meanwhile its supporting characteristics (stiffness and damping) can be adjusted online by changing control parameters, which can enhance the system stability. Therefore, magnetic bearing technology is known as a revolution of supporting technology. Hybrid magnetic bearing is a kind of magnetic bearing system which formed by combining electromagnetic bearings and permanent magnetic bearings. It takes into account the advantages of both electromagnetic bearings and permanent magnetic bearing. So it has been widely studied at present. However, in the applications such as wind turbines and artificial heart pump, the working air-gap of hybrid magnetic bearing is large, which causes basic assumption in the conventional magnetic bearing design theory such as uniform magnetic field, no magnetic flux leakage are no longer correct. Therefore, a new design theory and method need to be developed.
In this paper, a typical kind of hybrid magnetic bearing has been discussed, and the structures of hybrid magnetic bearings with different air-gap have been designed by using conventional equivalent magnetic circuit method. The bearing capacity models of those hybrid magnetic bearings have been built, and the bearing forces have been analyzed by ANSYS. By comparing the calculation results of traditional equivalent magnetic circuit method and ANSYS, it shows that the conventional equivalent magnetic circuit method cannot meet the requirement of structural design and bearing capacity analysis of large air-gap hybrid magnetic bearings.
The characteristics and variation rules of equivalent magnetic circuit method model calculation error in large air-gap hybrid magnetic bearings are studied. What's more, impact factors which caused equivalent magnetic circuit method calculation error and impact rules of each factor on the error are studied.
For the main factor, magnetic flux leakage that causes errors of conventional magnetic circuit method, leakage coefficients are introduced to describe magnetic flux leakage. Furthermore, the magnetic circuit model of large air-gap hybrid magnetic bearings based on leakage coefficients is estabilished. The changing laws between air-gap magnetic flux leakage and structural parameters are studied. The description method of leakage coefficients is presented and application range of this description method is studied. Based on the description method, large air-gap hybrid magnetic bearings are demarcated from conventional air-gap ones.
An air-gap magnetic flux leakage model based on leakage permeance has been built by using a combination method of magnetic field and an equivalent magnetic circuit bearing capacity model of hybrid magnetic bearings based on magnetic flux leakage has been built. On this basis, the application range of this model is studied and the results show that this model can satisfy the majority of practical engineering hybrid magnetic bearing capacity analysis needs. Then, a single degree of freedom hybrid magnetic experimental device is designed and manufactured. And the model is validated by utilizing that device.
Finally, the main reason that conventional equivalent magnetic circuit method can not be used for structure design of large air-gap hybrid magnetic bearings is analyzed. Furthermore, the structure design method of large air-gap hybrid magnetic bearings based on leakage coefficient correction model and leakage permeance correction model is proposed. And the structural of large air-gap hybrid magnetic bearings is designed by using improved magnetic circuit method.
引文
[1]胡业发,周祖德,江征风.磁力轴承的基础理论与应用[M].机械工业出版社，2006.
[2]Gerhard Schweitzer, Eric H. Maslen. Magnetic bearings:theory, design, and application to rotating machinery [M]. Springer,2009
[3]Hannes Bleuler. A survey of magnetic levitation and magnetic bearing types[J], JSME Intemat. Journal Series Ⅲ,1992.35(3):335-342
[4]武汉理工大学机电工程学院.磁悬浮轴承应用前景广泛[J].国际学术动态，2011,1：32-34
[5]徐龙样,周波.磁浮多电航空发动机的研究现状及关键技术[J].航空动力学报.2003,18(1)：51-59.
[6]张刚,汪希平.实验用航空发动机磁悬浮轴承样机的稳定性分析田.航空发动机.2002,2：1-5.
[7]R.弗雷泽,P.J.基林森，G.A.奥伯贝克.磁悬浮和电悬浮[M].北京：国防工业出版社,1982.
[8]M. A. Pichot, J. P. Kajs, B. R. Murphy, et al. Active Magnetic Bearings for Energy Storage Systems for Combat Vehicles[J]. IEEE Transactions on magnetics,2001,37 (1):318-323.
[9]吴刚.混合磁轴承飞轮系统设计与控制方法研究[D].长沙：国防科学技术大学,2006.10
[10]Tau Meng Lim, Dongsheng Zhang,Juanjuan Yang, Etc. Design and parameter estimation of hybrid magnetic bearings for blood pump applications[J]. Mechanical Systems and Signal Processing,23(2009): 2352-2382
[11]G. Shrestha, D. Bang, Jan Abraham Ferreira. Structural Flexibility:A Solution for Weight Reduction of Large Direct-Drive Wind-Turbine Generators [J]. IEEE TRANSACTIONS ON ENERGY CONVERSION,2010.25(3):732-740
[12]G. Shrestha, H, Polinder, D. J. Bang and J. A. Ferreira. Direct Drive Wind Turbine Generator with Magnetic Bearing [C]. Proceedings Offshore wind 2007,2007
[13]G. Shrestha, H. Polinder, D.J. Bang, J.A. Ferreira. A.S. McDonald. A New Concept for Weight Reduction of Large Direct Drive Machines [C]. Proceedings of the 2008 International Conference on Electrical Machines,2008
[14]Nianxian Wang, Jinguang Zhang, Guoping Ding. Influence of Magnetic Bearing Stiffness on Rotor in Wind Turbine Generator. Applied Mechanics and Materials,150(2012):57-62
[15]褚海英.飞轮电池及其混合磁悬浮控制系统的研究[D].北京：北京交通大学,2008.12
[16]Weiwei Zhang, Yefa Hu. A Prototype of Flywheel Energy Storage System Suspended by Active Magnetic Bearings with PID controller[C]. Proceedings of the 2009 IEEE International Conference on Asia-Pacific Power and Energy Engineering(2009 APPEEC), March 28-31,2009, Wuhan, China,2328-2331.
[17]Steyn, S.J.M., Van Vuuren, P.A., Van Schoor, G., Multivariable H∞ control for an active magnetic bearing flywheel system[C]. UKACC International Conference on CONTROL 2010,2010,1014-1019
[18]http://www.turbocor.com/
[19]Se Young Yoon, Zongli Lin, Chris Goyne, Paul E. Allaire. Control of Compressor Surge with Active Magnetic Bearings[C].49th IEEE Conference on Decision and Control,2010,4323-4328
[20]吴刚，刘昆，张育林.磁悬浮飞轮技术及其应用研究[J].宇航学报，2005,26(3)： 385-390
[21]吴刚,刘昆,张育林等.磁悬浮动量轮设计与实验研究[J].轴承,2005,8：4-7
[22]Martin Gronek, Torsten Rottenbach, Frank Worlitz. A contribution on the investigation of the dynamic behavior of rotating shafts with a Hybrid Magnetic Bearing Concept (HMBC) for blower application [J]. Nuclear Engineering and Design,2010, (240):2436-2442
[23]褚海英,范瑜.飞轮电池磁悬浮控制系统的仿真和实验研究[J].系统仿真学报,2007,19(23)：5473-5486
[24]褚海英,范瑜.用于地铁机车的飞轮电池磁悬浮控制系统的设计[J].北京交通大学学报,2007,31(2)：80-84
[25]马宏忠，王斌,鞠平.混合磁悬浮水轮发电机组转子承重系统设计与建模[J].河海大学学报(自然科学版),2010,38(3)：342-346
[26]Nobuyuki Kurita, Keisuke Ohshio, Takeo Ishikawa. Basic Analysis and Design of the Permanent Magnet Hybrid Type Magnetic Bearing for Small-sized Hydraulic Generator [C]. The 2010 International Power Electronics Conference,2010:1506-1511
[27]Xu Yanliang, Xin Feng. Model Establishment and Parameter Calculation of Permanent Magnet-Biased Hybrid Magnetic Bearing with Magnet in Rotor [C].2007 Second IEEE Conference on Industrial Electronics and Applications,2007:41-45
[28]Xu Yanliang, Dun Yueqin, Wang Xiuhe, and Kong Yu. Analysis of Hybrid Magnetic Bearing With a Permanent Magnet in the Rotor by FEM [J]. IEEE TRANSACTIONS ON MAGNETICS, 2010,42(4):1363-2006
[29]Qinghai Wu, Wei Ni, Tao Zhang. Parameter Design and Finite Element Analysis for AC 2-degree Freedom Hybrid Magnetic Bearing [C]. Proceedings of the 2010 Chinese Control and Decision Conference, CCDC 2010,280-284
[30]Qinghai Wu, Wei Ni, Tao Zhang, Etc. Performance analysis and digital control system design for AC 2-degree freedom hybrid magnetic bearing [C]. Proceedings of the 2011 Chinese Control and Decision Conference, CCDC 2011,2783-2787
[31]Yohji OKADA, Kouji SAGAWA, Eisaku SUZUKI, Ryou KONDO. Development and Application of Parallel PM Type Hybrid Magnetic Bearings [J]. Journal of System Design and Dynamics,2009, 3(4):530-539
[32]HAN Bangcheng. Modeling and Analysis of Novel Integrated Radial Hybrid Magnetic Bearing for Magnetic Bearing Reaction Wheel [J]. CHINESE JOURNAL OF MECHANICAL ENGINEERING,2010, 23(5):655-662
[33]邬清海,朱熀秋.三自由度双磁极面混合磁轴承参数设计与性能分析[J].中国机械工程,2009,20(12)：1477-1483
[34]孙玉坤,朱烷秋,蔡兰.三自由度混合磁悬浮轴承耦合特性[J].江苏大学学报(自然科学版),2006,27(4)：342-346
[35]王冠,徐龙样.永磁偏置五自由度磁轴承结构及磁路分析[J].机电产品开发与创新,2003,6：16-23
[36]高殿荣,杜世渊,阎淑丽.具有较大气隙的锥形螺旋血泵磁悬浮结构的设计与分析[J].机械设计,2007,24(12)：33-35
[37]刘强.姿态可控磁悬浮轴承系统的设计与研究[D].北京：北京邮电大学,2008.3
[38]汶涛.电磁永磁悬浮系统的设计及特性研究[D].西安：西安电子科技大学,2011.12
[39]张亦静.大气隙磁悬浮装置的力学特性研究[D].武汉：武汉理工大学,2012.5
[40]Dellinger S J, Smith A O, Strnat K J. Field and force calculations for use in passive magnetic bearing systems employing rare earth magnets [C]. In:The 8th International Workshop on Rare Earth-Cobalt Permanent Magnets and Their Applications,1985,6:153-164
[41]R. Ravaud, G Lemarquand. V. Lemarquand. Force and Stiffness of Passive Magnetic Bearings Using PermanentMagnets. Part I: Axial Magnetization[J]. IEEE TRANSACTIONS ON MAGNETICS, 2009.45 (7):2996-3002
[42]R. Ravaud, G. Lemarquand. V. Lemarquand. Force and Stiffness of Passive Magnetic Bearings Using PermanentMagnets. Part 2:Radial Magnetization[J]. IEEE TRANSACTIONS ON MAGNETICS, 2009.45 (9):3334-3342
[43]孙立军,张涛，赵兵.永磁磁轴承数学模型的研究[J].机械工程学报,2005,41(4)：69-74
[44]谭庆昌,刘明浩,孟慧琴，修世超.永磁向心轴承承载能力与刚度的计算[J].摩擦学学报,1994，14(4)：337-344
[45]修世超,谭庆昌,孟慧琴.同轴环形磁铁磁作用力计算的等效磁荷法[J].沈阳黄金学院学报，1995,14(3)：359-363
[46]田录林.永磁轴承和导轨磁力解析模型的研究[D].西安：西安理工大学,2008,2
[47]姚海.永磁轴承力学特性的研究[D].杭州：浙江工业大学,2004，5
[48]寇尊权.不均匀磁化对永磁推力轴承及其转子系统性能影响的研究[D].长春：吉林大学,2006,5
[49]冯慈璋.电磁场[M].高等教育出版社，1983
[50]陈重,崔正勤.电磁场理论基础[M].北京理工大学出版社,2003
[51]H. Okuda, T. Abukawa, K. Arai, M. Ito. CHARACTERISTICS OF RING PERMANENT MAGNET BEARING [J]. IEEE TRANSACTIONS ON MAGNETICS.1984,20(5):1693-1695
[52]王洪昌，蒋书运,梁玉飞.基于分子电流法轴向永磁轴承轴向刚度的分析[J].机械工程学报,2009，45(5)： 102-107
[53]高刚,张钢,赵志峰,梁世颇.永磁轴承的刚度研究[J].轴承,2008,3(3)：5-7
[54]Fang Jiancheng, Sun Jinji, Xu Yanliang, Wang Xi. A New Structure for Permanent-Magnet-Biased Axial Hybrid Magnetic Bearings[J]. IEEE TRANSACTIONS ON MAGNETICS,2009,45(12):5319-5325
[55]唐任远.现代永磁电机理论与设计[M].机械工业出版社,1997
[56]Marlene Marinescu, Nicolae Marinescu. A New Improved Method for Computation of Radial Stiffness of Permanent Magnet Bearings [J]. IEEE TRANSACTIONS ON MAGNETICS.1994.30(5):3491-3494
[57]Lstrok;agodzinski, Jakub. Modelling of magnetic fields with the finite element method in machine diagnostic systems [J]. Diffusion and Defect Data Pt.B:Solid State Phenomena,2009,v 147-149, p 155-160
[58]P. Bolognesi, F. Papini, L. Papini, O. Bruno. Effects of Saturation on Eccentricity Force in Permanent Magnet Synchronous Machines [C]. SPEEDAM 2010-International Symposium on Power Electronics, Electrical Drives, Automation and Motion,2010, p 324-330
[59]高颖.江东,杨嘉祥.混合磁悬浮球系统磁场数值计算[J].哈尔滨理工大学学报,2008,13(5)：99-102
[60]Shanbao Cheng, Mark W. Olles, Aaron F. Burger, Steven W. Day. Optimization of a hybrid magnetic bearing for a magnetically levitated blood pump via 3-D FEA[J]. Mechatronics,21 (2011):1163-1169
[61]Yong Li, Wei Li, and Yongping Lu. Computer-Aided Simulation Analysis of a Novel Structure Hybrid Magnetic Bearing[J]. IEEE TRANSACTIONS ON MAGNETICS,2008,44(10):2283-2287
[62]Erich Schmidt, Matthias Hofer. Inductances and Forces of a Three Phase Permanent Magnet Biased Radial Active Magnetic Bearing in Dependence on the Rotor Eccentricity[C]. Electric Machines and Drives Conference,2009, p 1788-1794
[63]Sun Jinjin, Fang Jiancheng. A novel structure of permanent-magnet-biased radial hybrid magnetic bearing[J]. Journal of Magnetism and Magnetic Materials,2011,323:202-208
[64]Carlos R. Morrison, Mark W. Siebert, Eric J. Ho. Electromagnetic Forces in a Hybrid Magnetic-Bearing Switched-Reluctance Motor [J]. IEEE TRANSACTIONS ON MAGNETICS,2008,44(12):4626-4638
[65]S. PARK, C. LEE. Design and control of hybrid-type three-pole active magnetic bearings using redundant coordinates [J]. Journal of Vibration and Control,2010,16(4):601-614
[66]S. PARK, C. LEE. Decoupled Control of a Disk-Type Rotor Equipped With a Three-Pole Hybrid Magnetic Bearing [J]. IEEE/ASME TRANSACTIONS ON MECHATRONICS,2010,15(5):793-804
[67]Yong Li, Wei Li, and Yongping Lu. Computer-Aided Simulation Analysis of a Novel Structure Hybrid Magnetic Bearing [J]. IEEE TRANSACTIONS ON MAGNETICS,2008,44(10):2283-2287
[68]Han Bangcheng, LID Bin. The Influences of Parameters on Performance of Hybrid Axial Magnetic Bearing [C].2008 2nd International Symposium on Systems and Control in Aerospace and Astronautics, ISSCAA 2008,2008
[69]田录林,李言,杨国清等.径向磁化的双筒永磁轴承轴向磁力研究[J].机械科学与技术,2007,26(9)：1216-1219
[70]田录林,李言,田琦 等.轴向磁化的双环永磁轴承轴向磁力研究[J].中国电机工程学报,2007,27(36)：41-45
[71]Chunting Mi, Mariano Filippa, Weiguo Liu, Ruiqing Ma. Analytical Method for Predicting the Air-Gap Flux of Interior-Type Permanent-Magnet Machines [J]. IEEE TRANSACTIONS ON MAGNETICS,2004, 40(l):50-58
[72]吴刚,张育林,刘昆等.永磁电磁轴承的磁路设计方法[J].轴承,2004,3：4-7
[73]Chang-Chou Hwang, Y. H. Cho. Effects of Leakage Flux on Magnetic Fields of Interior Permanent Magnet Synchronous Motors [J]. IEEE TRANSACTIONS ON MAGNETICS,2001,37(4):3021-3024
[74]梅磊,邓智泉,赵旭升,王晓刚,王晓琳.基于磁通量计算的混合型轴向-径向磁悬浮轴承参数设计[J].中国电机工程学报,2009,29(24)：115-121
[75]Lei Huang, Guangzhou Zhao, Heng Nian, Yikang He. Modeling and Design of Permanent Magnet Biased Radial-Axial Magnetic Bearing by Extended Circuit Theory[J]. Proceeding of International Conference on Electrical Machines and Systems 2007,2007, p 1502-1507
[76]Hou Eryong, Liu Kun. Investigation of Axial Carrying Capacity of Radial Hybrid Magnetic Bearing[J]. IEEE TRANSACTIONS ON MAGNETICS,2012,48(1):38-46
[77]Jagu S. Rao, R. Tiwari. Design optimization of double-acting hybrid magnetic thrust bearings with control integration using multi-objective evolutionary algorithms[J]. Mechatronics,19(2009):945-964
[78]Hou Eryong, Liu Kun. A Novel Structure for Low-Loss Radial Hybrid Magnetic Bearing[J]. IEEE TRANSACTIONS ON MAGNETICS,2011,47(12):4725-4733
[79]Bangcheng Han, Shiqiang Zheng, Xiaofei Hu. Dynamic Factor Models of a Thrust Magnetic Bearing With Permanent Magnet Bias and Subsidiary Air Gap[J]. IEEE TRANSACTIONS ON MAGNETICS, 2013,49(3):1221-1230
[80]J.P. YONNET. PASSIVE MAGNETIC BEARINGS WITH PERMANENT MAGNETS [J]. IEEE TRANSACTIONS ON MAGNETICS,1978.14(5):803-805
[81]J.P. YONNET. Permanent Magnet Bearings and Couplings [J]. IEEE TRANSACTIONS ON MAGNETICS,1981,17(1):1169-1173
[82]G. AKOUN, J. P. YONNET.3D ANALYTICAL CALCULATION OF THE FORCES EXERTED BETWEEN TWO CUBOIDAL MAGNETS [J]. IEEE TRANSACTIONS ON MAGNETICS,1984, MAG-20(5):1962-1964
[83]J.P. Yonnet, G. Lemarquand, S. Hemmerlin, E. Rulliere. Stacked structures of passive magnetic bearings [J]. J. Appl. Phys.,1991.70(10):6633-6635
[84]J.P. Yonnet, S. Hemmmerlin, E. Rulliere, G Lemarquand. Analytical calculation of permanent magnet couplings [J]. IEEE Transactions on Magnetics,1993,29(6):2932-2934
[85]V. Fernandez, J. Fandino, C. Sauvey, J. P. Yonnet, G. Reyne, O. Cugat. A Design Methodology for Permanent Magnet Microbearings [J]. IEEE TRANSACTIONS ON MAGNETICS,2000, 36(4):1919-1922
[86]田录林.李言,安源等.轴向放置轴向磁化的双环永磁轴承径向磁力研究[J].中国机械工程，2007,18(23)：2926-2929
[87]F. T. Backers. A Magnetic Journal Bearing [J]. Philips Tech. Rev.,1960-1961,2:232-238
[88]Brad Paden, Nelson Groom, James F. Antaki. Design Formulas for Permanent-Magnet Bearings[J]. Transactions of the ASME,2003,125:734-738
[89]张二攀，石卡.利用麦克斯韦张量法分析永磁电动机齿槽转矩[J].微特电机,2010,12：27-30
[90]王宝国.无轴承电机磁悬浮机理及其控制方法研究[D].沈阳：沈阳工业大学,2002,5
[91]A. Hamler, V. Gorican, B. Stumberger, Etc. Passive magnetic bearing [J]. Journal of Magnetism and Magnetic Materials,2004,272-276:2379-2380
[92]T.J. Hoffmann, M. Chudzicka-Adamczak. The Maxwell stress tensor for magnetoelastic materials [J]. International Journal of Engineering Science,2009,47:735-739
[93]J. T. Marshall, M. E. F. Kasarda, J. Imlach. A Multipoint Measurement Technique for the Enhancement of Force Measurement With Active Magnetic Bearings [J]. Transactions of the ASME,2003,125:90-94
[94]崔鹏,张锟,李杰.基于许-克变换的悬浮电磁铁力与转矩解析计算[J].中国电机工程学报.2010,3(24)：129-134
[95]Miroslav Markovic, Marcel Jufer, Yves Perriard. Analyzing an Electromechanical Actuator by Schwarz-Christoffel Mapping[J]. IEEE TRANSACTIONS ON MAGNETICS,2004,40(4):1858-1863
[96]Damir Zarko, Drago Ban, Thomas A. Lipo. Analytical Calculation of Magnetic Field Distribution in the Slotted Air Gap of a Surface Permanent-Magnet Motor Using Complex Relative Air-Gap Permeance[J]. IEEE TRANSACTIONS ON MAGNETICS,2006,42(7):1828-1837
[97]K. Boughrara, D. Zarko, R. Ibtiouen, O. Touhaml, A. Rezzoug. Magnetic Field Analysis of Inset and Surface-Mounted Permanent-Magnet Synchronous Motors Using Schwarz-Christoffel Transformation[J]. IEEE TRANSACTIONS ON MAGNETICS,2009,45(8):3166-3178
[98]华中工学院步进电机科研组.步进电动机的气隙磁导及其导数计算[J].微电机,1977,Z1：23-30
[99]Li Zhu, S. Z. Jiang, Z. Q. Zhu, C. C. Chan. Analysis and Modeling of Open-circuit Airgap Field Distributions in Multi-segment and Multilayer Interior Permanent Magnet Machines[C]. IEEE Vehicle Power and Propulsion Conference (VPPC),2008, p 1-6
[100]Ronghai Qu, Thomas A. Lipo. Analysis and Modeling of Air-Gap and Zigzag Leakage Fluxes in a Surface-Mounted Permanent-Magnet Machine [J]. IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS,2004,40(1):121-127
[101]WenBin Tsai, TingYu Chang. Analysis of Flux Leakage in a Brushless Permanent-Magnet Motor with Embedded Magnets [J]. IEEE TRANSACTIONS ON MAGNETICS,1999,35(1):543-547
[2]Gerhard Schweitzer, Eric H. Maslen. Magnetic bearings:theory, design, and application to rotating machinery [M]. Springer,2009
[3]Hannes Bleuler. A survey of magnetic levitation and magnetic bearing types[J], JSME Intemat. Journal Series Ⅲ,1992.35(3):335-342
[4]武汉理工大学机电工程学院.磁悬浮轴承应用前景广泛[J].国际学术动态，2011,1：32-34
[5]徐龙样,周波.磁浮多电航空发动机的研究现状及关键技术[J].航空动力学报.2003,18(1)：51-59.
[6]张刚,汪希平.实验用航空发动机磁悬浮轴承样机的稳定性分析田.航空发动机.2002,2：1-5.
[7]R.弗雷泽,P.J.基林森，G.A.奥伯贝克.磁悬浮和电悬浮[M].北京：国防工业出版社,1982.
[8]M. A. Pichot, J. P. Kajs, B. R. Murphy, et al. Active Magnetic Bearings for Energy Storage Systems for Combat Vehicles[J]. IEEE Transactions on magnetics,2001,37 (1):318-323.
[9]吴刚.混合磁轴承飞轮系统设计与控制方法研究[D].长沙：国防科学技术大学,2006.10
[10]Tau Meng Lim, Dongsheng Zhang,Juanjuan Yang, Etc. Design and parameter estimation of hybrid magnetic bearings for blood pump applications[J]. Mechanical Systems and Signal Processing,23(2009): 2352-2382
[11]G. Shrestha, D. Bang, Jan Abraham Ferreira. Structural Flexibility:A Solution for Weight Reduction of Large Direct-Drive Wind-Turbine Generators [J]. IEEE TRANSACTIONS ON ENERGY CONVERSION,2010.25(3):732-740
[12]G. Shrestha, H, Polinder, D. J. Bang and J. A. Ferreira. Direct Drive Wind Turbine Generator with Magnetic Bearing [C]. Proceedings Offshore wind 2007,2007
[13]G. Shrestha, H. Polinder, D.J. Bang, J.A. Ferreira. A.S. McDonald. A New Concept for Weight Reduction of Large Direct Drive Machines [C]. Proceedings of the 2008 International Conference on Electrical Machines,2008
[14]Nianxian Wang, Jinguang Zhang, Guoping Ding. Influence of Magnetic Bearing Stiffness on Rotor in Wind Turbine Generator. Applied Mechanics and Materials,150(2012):57-62
[15]褚海英.飞轮电池及其混合磁悬浮控制系统的研究[D].北京：北京交通大学,2008.12
[16]Weiwei Zhang, Yefa Hu. A Prototype of Flywheel Energy Storage System Suspended by Active Magnetic Bearings with PID controller[C]. Proceedings of the 2009 IEEE International Conference on Asia-Pacific Power and Energy Engineering(2009 APPEEC), March 28-31,2009, Wuhan, China,2328-2331.
[17]Steyn, S.J.M., Van Vuuren, P.A., Van Schoor, G., Multivariable H∞ control for an active magnetic bearing flywheel system[C]. UKACC International Conference on CONTROL 2010,2010,1014-1019
[18]http://www.turbocor.com/
[19]Se Young Yoon, Zongli Lin, Chris Goyne, Paul E. Allaire. Control of Compressor Surge with Active Magnetic Bearings[C].49th IEEE Conference on Decision and Control,2010,4323-4328
[20]吴刚，刘昆，张育林.磁悬浮飞轮技术及其应用研究[J].宇航学报，2005,26(3)： 385-390
[21]吴刚,刘昆,张育林等.磁悬浮动量轮设计与实验研究[J].轴承,2005,8：4-7
[22]Martin Gronek, Torsten Rottenbach, Frank Worlitz. A contribution on the investigation of the dynamic behavior of rotating shafts with a Hybrid Magnetic Bearing Concept (HMBC) for blower application [J]. Nuclear Engineering and Design,2010, (240):2436-2442
[23]褚海英,范瑜.飞轮电池磁悬浮控制系统的仿真和实验研究[J].系统仿真学报,2007,19(23)：5473-5486
[24]褚海英,范瑜.用于地铁机车的飞轮电池磁悬浮控制系统的设计[J].北京交通大学学报,2007,31(2)：80-84
[25]马宏忠，王斌,鞠平.混合磁悬浮水轮发电机组转子承重系统设计与建模[J].河海大学学报(自然科学版),2010,38(3)：342-346
[26]Nobuyuki Kurita, Keisuke Ohshio, Takeo Ishikawa. Basic Analysis and Design of the Permanent Magnet Hybrid Type Magnetic Bearing for Small-sized Hydraulic Generator [C]. The 2010 International Power Electronics Conference,2010:1506-1511
[27]Xu Yanliang, Xin Feng. Model Establishment and Parameter Calculation of Permanent Magnet-Biased Hybrid Magnetic Bearing with Magnet in Rotor [C].2007 Second IEEE Conference on Industrial Electronics and Applications,2007:41-45
[28]Xu Yanliang, Dun Yueqin, Wang Xiuhe, and Kong Yu. Analysis of Hybrid Magnetic Bearing With a Permanent Magnet in the Rotor by FEM [J]. IEEE TRANSACTIONS ON MAGNETICS, 2010,42(4):1363-2006
[29]Qinghai Wu, Wei Ni, Tao Zhang. Parameter Design and Finite Element Analysis for AC 2-degree Freedom Hybrid Magnetic Bearing [C]. Proceedings of the 2010 Chinese Control and Decision Conference, CCDC 2010,280-284
[30]Qinghai Wu, Wei Ni, Tao Zhang, Etc. Performance analysis and digital control system design for AC 2-degree freedom hybrid magnetic bearing [C]. Proceedings of the 2011 Chinese Control and Decision Conference, CCDC 2011,2783-2787
[31]Yohji OKADA, Kouji SAGAWA, Eisaku SUZUKI, Ryou KONDO. Development and Application of Parallel PM Type Hybrid Magnetic Bearings [J]. Journal of System Design and Dynamics,2009, 3(4):530-539
[32]HAN Bangcheng. Modeling and Analysis of Novel Integrated Radial Hybrid Magnetic Bearing for Magnetic Bearing Reaction Wheel [J]. CHINESE JOURNAL OF MECHANICAL ENGINEERING,2010, 23(5):655-662
[33]邬清海,朱熀秋.三自由度双磁极面混合磁轴承参数设计与性能分析[J].中国机械工程,2009,20(12)：1477-1483
[34]孙玉坤,朱烷秋,蔡兰.三自由度混合磁悬浮轴承耦合特性[J].江苏大学学报(自然科学版),2006,27(4)：342-346
[35]王冠,徐龙样.永磁偏置五自由度磁轴承结构及磁路分析[J].机电产品开发与创新,2003,6：16-23
[36]高殿荣,杜世渊,阎淑丽.具有较大气隙的锥形螺旋血泵磁悬浮结构的设计与分析[J].机械设计,2007,24(12)：33-35
[37]刘强.姿态可控磁悬浮轴承系统的设计与研究[D].北京：北京邮电大学,2008.3
[38]汶涛.电磁永磁悬浮系统的设计及特性研究[D].西安：西安电子科技大学,2011.12
[39]张亦静.大气隙磁悬浮装置的力学特性研究[D].武汉：武汉理工大学,2012.5
[40]Dellinger S J, Smith A O, Strnat K J. Field and force calculations for use in passive magnetic bearing systems employing rare earth magnets [C]. In:The 8th International Workshop on Rare Earth-Cobalt Permanent Magnets and Their Applications,1985,6:153-164
[41]R. Ravaud, G Lemarquand. V. Lemarquand. Force and Stiffness of Passive Magnetic Bearings Using PermanentMagnets. Part I: Axial Magnetization[J]. IEEE TRANSACTIONS ON MAGNETICS, 2009.45 (7):2996-3002
[42]R. Ravaud, G. Lemarquand. V. Lemarquand. Force and Stiffness of Passive Magnetic Bearings Using PermanentMagnets. Part 2:Radial Magnetization[J]. IEEE TRANSACTIONS ON MAGNETICS, 2009.45 (9):3334-3342
[43]孙立军,张涛，赵兵.永磁磁轴承数学模型的研究[J].机械工程学报,2005,41(4)：69-74
[44]谭庆昌,刘明浩,孟慧琴，修世超.永磁向心轴承承载能力与刚度的计算[J].摩擦学学报,1994，14(4)：337-344
[45]修世超,谭庆昌,孟慧琴.同轴环形磁铁磁作用力计算的等效磁荷法[J].沈阳黄金学院学报，1995,14(3)：359-363
[46]田录林.永磁轴承和导轨磁力解析模型的研究[D].西安：西安理工大学,2008,2
[47]姚海.永磁轴承力学特性的研究[D].杭州：浙江工业大学,2004，5
[48]寇尊权.不均匀磁化对永磁推力轴承及其转子系统性能影响的研究[D].长春：吉林大学,2006,5
[49]冯慈璋.电磁场[M].高等教育出版社，1983
[50]陈重,崔正勤.电磁场理论基础[M].北京理工大学出版社,2003
[51]H. Okuda, T. Abukawa, K. Arai, M. Ito. CHARACTERISTICS OF RING PERMANENT MAGNET BEARING [J]. IEEE TRANSACTIONS ON MAGNETICS.1984,20(5):1693-1695
[52]王洪昌，蒋书运,梁玉飞.基于分子电流法轴向永磁轴承轴向刚度的分析[J].机械工程学报,2009，45(5)： 102-107
[53]高刚,张钢,赵志峰,梁世颇.永磁轴承的刚度研究[J].轴承,2008,3(3)：5-7
[54]Fang Jiancheng, Sun Jinji, Xu Yanliang, Wang Xi. A New Structure for Permanent-Magnet-Biased Axial Hybrid Magnetic Bearings[J]. IEEE TRANSACTIONS ON MAGNETICS,2009,45(12):5319-5325
[55]唐任远.现代永磁电机理论与设计[M].机械工业出版社,1997
[56]Marlene Marinescu, Nicolae Marinescu. A New Improved Method for Computation of Radial Stiffness of Permanent Magnet Bearings [J]. IEEE TRANSACTIONS ON MAGNETICS.1994.30(5):3491-3494
[57]Lstrok;agodzinski, Jakub. Modelling of magnetic fields with the finite element method in machine diagnostic systems [J]. Diffusion and Defect Data Pt.B:Solid State Phenomena,2009,v 147-149, p 155-160
[58]P. Bolognesi, F. Papini, L. Papini, O. Bruno. Effects of Saturation on Eccentricity Force in Permanent Magnet Synchronous Machines [C]. SPEEDAM 2010-International Symposium on Power Electronics, Electrical Drives, Automation and Motion,2010, p 324-330
[59]高颖.江东,杨嘉祥.混合磁悬浮球系统磁场数值计算[J].哈尔滨理工大学学报,2008,13(5)：99-102
[60]Shanbao Cheng, Mark W. Olles, Aaron F. Burger, Steven W. Day. Optimization of a hybrid magnetic bearing for a magnetically levitated blood pump via 3-D FEA[J]. Mechatronics,21 (2011):1163-1169
[61]Yong Li, Wei Li, and Yongping Lu. Computer-Aided Simulation Analysis of a Novel Structure Hybrid Magnetic Bearing[J]. IEEE TRANSACTIONS ON MAGNETICS,2008,44(10):2283-2287
[62]Erich Schmidt, Matthias Hofer. Inductances and Forces of a Three Phase Permanent Magnet Biased Radial Active Magnetic Bearing in Dependence on the Rotor Eccentricity[C]. Electric Machines and Drives Conference,2009, p 1788-1794
[63]Sun Jinjin, Fang Jiancheng. A novel structure of permanent-magnet-biased radial hybrid magnetic bearing[J]. Journal of Magnetism and Magnetic Materials,2011,323:202-208
[64]Carlos R. Morrison, Mark W. Siebert, Eric J. Ho. Electromagnetic Forces in a Hybrid Magnetic-Bearing Switched-Reluctance Motor [J]. IEEE TRANSACTIONS ON MAGNETICS,2008,44(12):4626-4638
[65]S. PARK, C. LEE. Design and control of hybrid-type three-pole active magnetic bearings using redundant coordinates [J]. Journal of Vibration and Control,2010,16(4):601-614
[66]S. PARK, C. LEE. Decoupled Control of a Disk-Type Rotor Equipped With a Three-Pole Hybrid Magnetic Bearing [J]. IEEE/ASME TRANSACTIONS ON MECHATRONICS,2010,15(5):793-804
[67]Yong Li, Wei Li, and Yongping Lu. Computer-Aided Simulation Analysis of a Novel Structure Hybrid Magnetic Bearing [J]. IEEE TRANSACTIONS ON MAGNETICS,2008,44(10):2283-2287
[68]Han Bangcheng, LID Bin. The Influences of Parameters on Performance of Hybrid Axial Magnetic Bearing [C].2008 2nd International Symposium on Systems and Control in Aerospace and Astronautics, ISSCAA 2008,2008
[69]田录林,李言,杨国清等.径向磁化的双筒永磁轴承轴向磁力研究[J].机械科学与技术,2007,26(9)：1216-1219
[70]田录林,李言,田琦 等.轴向磁化的双环永磁轴承轴向磁力研究[J].中国电机工程学报,2007,27(36)：41-45
[71]Chunting Mi, Mariano Filippa, Weiguo Liu, Ruiqing Ma. Analytical Method for Predicting the Air-Gap Flux of Interior-Type Permanent-Magnet Machines [J]. IEEE TRANSACTIONS ON MAGNETICS,2004, 40(l):50-58
[72]吴刚,张育林,刘昆等.永磁电磁轴承的磁路设计方法[J].轴承,2004,3：4-7
[73]Chang-Chou Hwang, Y. H. Cho. Effects of Leakage Flux on Magnetic Fields of Interior Permanent Magnet Synchronous Motors [J]. IEEE TRANSACTIONS ON MAGNETICS,2001,37(4):3021-3024
[74]梅磊,邓智泉,赵旭升,王晓刚,王晓琳.基于磁通量计算的混合型轴向-径向磁悬浮轴承参数设计[J].中国电机工程学报,2009,29(24)：115-121
[75]Lei Huang, Guangzhou Zhao, Heng Nian, Yikang He. Modeling and Design of Permanent Magnet Biased Radial-Axial Magnetic Bearing by Extended Circuit Theory[J]. Proceeding of International Conference on Electrical Machines and Systems 2007,2007, p 1502-1507
[76]Hou Eryong, Liu Kun. Investigation of Axial Carrying Capacity of Radial Hybrid Magnetic Bearing[J]. IEEE TRANSACTIONS ON MAGNETICS,2012,48(1):38-46
[77]Jagu S. Rao, R. Tiwari. Design optimization of double-acting hybrid magnetic thrust bearings with control integration using multi-objective evolutionary algorithms[J]. Mechatronics,19(2009):945-964
[78]Hou Eryong, Liu Kun. A Novel Structure for Low-Loss Radial Hybrid Magnetic Bearing[J]. IEEE TRANSACTIONS ON MAGNETICS,2011,47(12):4725-4733
[79]Bangcheng Han, Shiqiang Zheng, Xiaofei Hu. Dynamic Factor Models of a Thrust Magnetic Bearing With Permanent Magnet Bias and Subsidiary Air Gap[J]. IEEE TRANSACTIONS ON MAGNETICS, 2013,49(3):1221-1230
[80]J.P. YONNET. PASSIVE MAGNETIC BEARINGS WITH PERMANENT MAGNETS [J]. IEEE TRANSACTIONS ON MAGNETICS,1978.14(5):803-805
[81]J.P. YONNET. Permanent Magnet Bearings and Couplings [J]. IEEE TRANSACTIONS ON MAGNETICS,1981,17(1):1169-1173
[82]G. AKOUN, J. P. YONNET.3D ANALYTICAL CALCULATION OF THE FORCES EXERTED BETWEEN TWO CUBOIDAL MAGNETS [J]. IEEE TRANSACTIONS ON MAGNETICS,1984, MAG-20(5):1962-1964
[83]J.P. Yonnet, G. Lemarquand, S. Hemmerlin, E. Rulliere. Stacked structures of passive magnetic bearings [J]. J. Appl. Phys.,1991.70(10):6633-6635
[84]J.P. Yonnet, S. Hemmmerlin, E. Rulliere, G Lemarquand. Analytical calculation of permanent magnet couplings [J]. IEEE Transactions on Magnetics,1993,29(6):2932-2934
[85]V. Fernandez, J. Fandino, C. Sauvey, J. P. Yonnet, G. Reyne, O. Cugat. A Design Methodology for Permanent Magnet Microbearings [J]. IEEE TRANSACTIONS ON MAGNETICS,2000, 36(4):1919-1922
[86]田录林.李言,安源等.轴向放置轴向磁化的双环永磁轴承径向磁力研究[J].中国机械工程，2007,18(23)：2926-2929
[87]F. T. Backers. A Magnetic Journal Bearing [J]. Philips Tech. Rev.,1960-1961,2:232-238
[88]Brad Paden, Nelson Groom, James F. Antaki. Design Formulas for Permanent-Magnet Bearings[J]. Transactions of the ASME,2003,125:734-738
[89]张二攀，石卡.利用麦克斯韦张量法分析永磁电动机齿槽转矩[J].微特电机,2010,12：27-30
[90]王宝国.无轴承电机磁悬浮机理及其控制方法研究[D].沈阳：沈阳工业大学,2002,5
[91]A. Hamler, V. Gorican, B. Stumberger, Etc. Passive magnetic bearing [J]. Journal of Magnetism and Magnetic Materials,2004,272-276:2379-2380
[92]T.J. Hoffmann, M. Chudzicka-Adamczak. The Maxwell stress tensor for magnetoelastic materials [J]. International Journal of Engineering Science,2009,47:735-739
[93]J. T. Marshall, M. E. F. Kasarda, J. Imlach. A Multipoint Measurement Technique for the Enhancement of Force Measurement With Active Magnetic Bearings [J]. Transactions of the ASME,2003,125:90-94
[94]崔鹏,张锟,李杰.基于许-克变换的悬浮电磁铁力与转矩解析计算[J].中国电机工程学报.2010,3(24)：129-134
[95]Miroslav Markovic, Marcel Jufer, Yves Perriard. Analyzing an Electromechanical Actuator by Schwarz-Christoffel Mapping[J]. IEEE TRANSACTIONS ON MAGNETICS,2004,40(4):1858-1863
[96]Damir Zarko, Drago Ban, Thomas A. Lipo. Analytical Calculation of Magnetic Field Distribution in the Slotted Air Gap of a Surface Permanent-Magnet Motor Using Complex Relative Air-Gap Permeance[J]. IEEE TRANSACTIONS ON MAGNETICS,2006,42(7):1828-1837
[97]K. Boughrara, D. Zarko, R. Ibtiouen, O. Touhaml, A. Rezzoug. Magnetic Field Analysis of Inset and Surface-Mounted Permanent-Magnet Synchronous Motors Using Schwarz-Christoffel Transformation[J]. IEEE TRANSACTIONS ON MAGNETICS,2009,45(8):3166-3178
[98]华中工学院步进电机科研组.步进电动机的气隙磁导及其导数计算[J].微电机,1977,Z1：23-30
[99]Li Zhu, S. Z. Jiang, Z. Q. Zhu, C. C. Chan. Analysis and Modeling of Open-circuit Airgap Field Distributions in Multi-segment and Multilayer Interior Permanent Magnet Machines[C]. IEEE Vehicle Power and Propulsion Conference (VPPC),2008, p 1-6
[100]Ronghai Qu, Thomas A. Lipo. Analysis and Modeling of Air-Gap and Zigzag Leakage Fluxes in a Surface-Mounted Permanent-Magnet Machine [J]. IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS,2004,40(1):121-127
[101]WenBin Tsai, TingYu Chang. Analysis of Flux Leakage in a Brushless Permanent-Magnet Motor with Embedded Magnets [J]. IEEE TRANSACTIONS ON MAGNETICS,1999,35(1):543-547