近年來，隨著人們對生活品質的要求提昇，創造舒適的生活空間一直是家電與馬達業者所努力的目標，而馬達之振動噪音也因此逐漸受到重視。本論文針對電磁力對於永磁無刷馬達振動之影響進行研究，包含四個主要的研究議題，分別為：一、切線電磁力；由於切線方向電磁力之擾動易造成輸出轉矩不穩定，進而形成轉矩漣波、造成馬達振動。本項研究針對永磁無刷馬達之反電動勢波形，進 行實測與分析，並提出雙段(Two-segment)轉子結構，可降低頓轉扭矩，並消除反電動勢高次諧波，以減少轉矩漣波之產生。二、徑向電磁力；此電磁力沿著半徑方向直接作用於馬達定子與轉子，易於馬達運作時，造成馬達形變而引起振動。由於徑向電磁力深受馬達結構影響，本研究則以定子細部尺寸做變化，以電磁力軌跡圖觀察馬達運轉時，定子之受力狀況，並找出可降低馬達徑向電磁力，仍能維持額定轉矩輸出之最佳尺寸。三、馬達偏心；此為馬達製造上常見的問題，造成馬達機械與電氣之不平衡，進而使馬達振動。本論文則針對馬達偏心程度的不同，分析轉矩漣波及徑向力之變化，並採用不同繞組連結方式以降低轉矩漣波，同時分析馬達各項電氣參數，以了解馬達偏心對性能所造成的影響。四、馬達對稱與不對稱結構；由於馬達切線與徑向電磁力皆影響馬達振動量，因此分別針對對稱與不對稱之馬達結構，進行轉矩漣波與徑向電磁力分析，並模擬與實測馬達之振動大小，以釐清真正主要影響馬達振動的電磁力。

Creating a comfortable environment has become a sought-after goal for both home appliances and motor manufacturers due to the increasing consumer demand for an improved quality of life in recent years. Thus, the issues on how to minimize the vibration and noise of motors have gradually received substantial attention. This dissertation explores the effects of electromagnetic forces on the vibration found in permanent magnet brushless motors. Four research topics are investigated in this dissertation and they are: 1. Tangential force: the fluctuation of tangential force causes unstable output torque, and thus torque ripple and motor vibration are created. This study analyzes the main factor that results in torque ripple of permanent magnet brushless motors, i.e., back-EMF waveforms, and proposes a two-segment rotor structure to reduce cogging torque, higher-order harmonics, and torque ripple. 2. Radial force: this electromagnetic force acts on the stator and rotor in the radial direction and causes motor vibration as the rotor rotates. In this study, the effect of motor structure on the fluctuation of radial force is obtained by recording force loci figures. Moreover, the structure parameters which can reduce the radial force with the same output torque are examined. 3. Rotor eccentricity: rotor eccentricity is a common problem which induces motor vibration and noise. This dissertation presents the variation of torque ripple and radial force with various levels of rotor eccentricity. Furthermore, the electrical parameters are also analyzed to show the influence of rotor eccentricity. 4. Symmetric and asymmetric motor structures: in this study, the examination of torque ripple and radial force are presented with the symmetric and asymmetric motors. In order to recognize the main factor which seriously causes the motor vibration, the simulation and measurement of motor vibration are reported. Overall, this dissertation provides a good reference for motor designers to reduce motor vibration in the design procedures.

[1] Andriamalala R. N., Razik H., Baghli L., and Sargos F. M., “Eccentricity fault diagnosis of a dual-stator winding induction machine drive considering the slotting effects,” IEEE Transactions on Industrial Electronics, vol. 55, pp. 4238-4251, Dec. 2008.
[2] Atallah K. and Howe D., “The application of Halbach cylinders to brushless AC servo motors,” IEEE Transactions on Magnetics, vol. 34, pp. 2060-2062, July 1998.
[3] Babu A. R. C. S., Rajagopal K. R., and Upadhyay P. R., “Performance prediction of multiphase doubly salient permanent magnet motor having nonuniform air gap,” IEEE Transactions on Magnetics, vol. 42, pp. 3503-3505, Oct. 2006.
[4] Beaty, H. W. and Kirtley, J. L., Electric Motor Handbook, McGraw-Hill, New York, 1998.
[5] Bi C., Liu Z. J., and Low T. S., “Effects of unbalanced magnetic pull in spindle motors,” IEEE Transactions on Magnetics, vol. 33, pp. 4080-4082, Sep. 1997.
[6] Bianchi N. and Bolognani S., “Design techniques for reducing the cogging torque in surface-mounted PM motors,” IEEE Transactions on Industry Applications, vol. 38, pp. 1259-1265, Sep.-Oct. 2002.
[7] Bogh D., Crowell J., and Amstutz R., “IEEE 841 motor vibration,” IEEE Industry Applications Magazine, vol. 11, pp. 32-37, Nov.-Dec. 2005.
[8] Burakov A. and Arkkio A., “Comparison of the unbalanced magnetic pull mitigation by the parallel paths in the stator and rotor windings,” IEEE Transactions on Magnetics, vol. 43, pp. 4083-4088, Dec. 2007.
[9] Chen Y. S., Zhu Z. Q., and Howe D., “Vibration of PM brushless machines having a fractional number of slots per pole,” IEEE Transactions on Magnetics, vol. 42, pp. 3395-3397, Oct. 2006.
[10] Dai M., Keyhani A., and Sebastian T., “Torque ripple analysis of a PM brushless DC motor using finite element method,” IEEE Transactions on Energy Conversion, vol. 19, pp. 40-45, Mar. 2004.
[11] Debortoli M. J., Salon S. J., Burow D. W., and Slavik C. J., “Effects of rotor eccentricity and parallel windings on induction machine behavior - a study using finite-element analysis,” IEEE Transactions on Magnetics, vol. 29, pp. 1676-1682, Mar. 1993.
[12] Dorrell D. G. and Smith A. C., “Calculation of UMP in induction-motors with series or parallel winding connections,” IEEE Transactions on Energy Conversion, vol. 9, pp. 304-310, June 1994.
[13] Dorrell D. G., “Experimental behaviour of unbalanced magnetic pull in 3-phase induction motors with eccentric rotors and the relationship with tooth saturation,” IEEE Transactions on Energy Conversion, vol. 14, pp. 304-309, Sep. 1999.
[14] Dorrell D.G., Popescu M. Cossar C., Ionel, D., “Unbalanced magnetic pull in fractional-slot brushless PM motors,” Industry Application Society Annual Meeting, pp. 1-8, Oct. 2008.
[15] Dorrell D. G. and Cossar C., “A vibration-based condition monitoring system for switched reluctance machine rotor eccentricity detection,” IEEE Transactions on Magnetics, vol. 44, pp. 2204-2214, Sep. 2008.
[16] Ebrahimi B. M., Faiz J., Javan-Roshtkhari M., and Nejhad A. Z., “Static Eccentricity Fault Diagnosis in Permanent Magnet Synchronous Motor Using Time Stepping Finite Element Method,” IEEE Transactions on Magnetics, vol. 44, pp. 4297-4300, Nov. 2008.
[17] F. Liang, Rotor skew methods for permanent magnet motors, U.S. Patent 245880, Dec. 9, 2004.
[18] Faiz J., Ardekanei I. T., and Toliyat H. A., “An evaluation of inductances of a squirrel-cage induction motor under mixed eccentric conditions,” IEEE Transactions on Energy Conversion, vol. 18, pp. 252-258, June 2003.
[19] Faiz J., Ebrahimi B. M., Akin B., and Toliyat H. A., “Comprehensive eccentricity fault diagnosis in induction motors using finite element method,” IEEE Transactions on Magnetics, vol. 45, pp. 1764-1767, Mar. 2009.
[20] Faiz J., Ebrahimi B. M., Akin B., and Toliyat H. A., “Finite-element transient analysis of induction motors under mixed eccentricity fault,” IEEE Transactions on Magnetics, vol. 44, pp. 66-74, Jane 2008.
[21] Frauman P., Burakov A., and Arkkio A., “Effects of the slot harmonics on the unbalanced magnetic pull in an induction motor with an eccentric rotor,” IEEE Transactions on Magnetics, vol. 43, pp. 3441-3444, Aug. 2007.
[22] Gieras J.F., Wang C., and Lai J.C., Noise of Polyphase Electric Motors, Taylor & Francis, Boca Raton, 2006.
[23] Hanselman D.C., Brushless Permanent-Magnet Motor Design, McGraw-Hill, 2nd ed., New York, 2003.
[24] Hanselman D. C., “Effect of skew, pole count and slot count on brushless motor radial force, cogging torque and back EMF,” IEE Proceedings-Electric Power Applications, vol. 144, pp. 325-330, Sep. 1997.
[25] Hendershot J.R. and Miller T.J.E., Design of Brushless Permanent-Magnet Motors, Oxford University Press, New York, 1994.
[26] Hsieh M. F. and Hsu Y. S., “An investigation on influence of magnet arc shaping upon back electromotive force waveforms for design of permanent-magnet brushless motors,” IEEE Transactions on Magnetics, vol. 41, pp. 3949-3951, Oct. 2005.
[27] Hwang S. M., Lee H. J., Kim T. S., Jung Y. H., and Hong J. P., “The influence of electromagnetic force upon the noise of an IPM motor used in a compressor,” IEEE Transactions on Magnetics, vol. 42, pp. 3494-3496, Oct. 2006.
[28] Hwang S. M., Kim K. T., Jeong W. B., Jung Y. H., and Kang B. S., “Comparison of vibration sources between symmetric and asymmetric HDD spindle motors with rotor eccentricity,” IEEE Transactions on Industry Applications, vol. 37, pp. 1727-1731, Nov.-Dec. 2001.
[29] Hwang S. M., Eom J. B., Hwang G. B., Jeong W. B., and Jung Y. H., “Cogging torque and acoustic noise reduction in permanent magnet motors by teeth pairing,” IEEE Transactions on Magnetics, vol. 36, pp. 3144-3146, Sep. 2000.
[30] Ishak D., Zhu Z. Q., and Howe D., “High torque density permanent magnet brushless machines with similar slot and pole numbers,” Journal of Magnetism and Magnetic Materials, vol. 272, pp. E1767-E1769, May 2004.
[31] Ishak D., Zhu Z. Q., and Howe D., “Permanent-magnet brushless machines with unequal tooth widths and similar slot and pole numbers,” IEEE Transactions on Industry Applications, vol. 41, pp. 584-590, Mar.-Apr. 2005.
[32] Islam M. S., Mir S., Sebastian T., and Underwood S., “Design considerations of sinusoidally excited permanent-magnet machines for low-torque-ripple applications,” IEEE Transactions on Industry Applications, vol. 41, pp. 955-962, Jul.-Aug. 2005.
[33] Jahns T. M. and Soong W. L., “Pulsating torque minimization techniques for permanent magnet AC motor drives - A review,” IEEE Transactions on Industrial Electronics, vol. 43, pp. 321-330, Apr. 1996.
[34] Jang G. H., Yoon J. W., Park N. Y., and Jang S. M., “Torque and unbalanced magnetic force in a rotational unsymmetric brushless DC motors,” IEEE Transactions on Magnetics, vol. 32, pp. 5157-5159, Sep. 1996.
[35] Kawase Y., Mimura N., and Ida K., “3-D electromagnetic force analysis of effects of off-center of rotor in interior permanent magnet synchronous motor,” IEEE Transactions on Magnetics, vol. 36, pp. 1858-1862, July 2000.
[36] Kikuchi S. and Tsurumoto K., “Design and Characteristics of a New Magnetic Worm Gear Using Permanent-Magnet,” IEEE Transactions on Magnetics, vol. 29, pp. 2923-2925, Nov. 1993.
[37] Kikuchi S. and Tsurumoto,K. “Trial Construction of a New Magnetic Skew Gear Using Permanent-Magnet,” IEEE Transactions on Magnetics, vol. 30, pp. 4767-4769, Nov. 1994.
[38] Kim D. H., Park I. H., Lee J. H., and Kim C. E., “Optimal shape design of iron core to reduce cogging torque of IPM motor,” IEEE Transactions on Magnetics, vol. 39, pp. 1456-1459, May 2003.
[39] Kim K. T., Kim K. S., Hwang S. M., Kim T. J., and Jung Y. H., “Comparison of magnetic forces for IPM and SPM motor with rotor eccentricity,” IEEE Transactions on Magnetics, vol. 37, pp. 3448-3451, Sep. 2001.
[40] Kim U. and Lieu D. K., “Magnetic field calculation in permanent magnet motors with rotor eccentricity: Without slotting effect,” IEEE Transactions on Magnetics, vol. 34, pp. 2243-2252, July 1998.
[41] Ko H. S. and Kim K. J., “Characterization of noise and vibration sources in interior permanent-magnet brushless DC motors,” IEEE Transactions on Magnetics, vol. 40, pp. 3482-3489, Nov. 2004.
[42] Lee C. I. and Jang G. H., “Experimental measurement and simulated verification of the unbalanced magnetic force in brushless DC motors,” IEEE Transactions on Magnetics, vol. 44, pp. 4377-4380, Nov. 2008.
[43] Lee J. H., Kim D. H., and Park I. H., “Minimization of higher back-EMF harmonics in permanent magnet motor using shape design sensitivity with B-spline parameterization,” IEEE Transactions on Magnetics, vol. 39, pp. 1269-1272, May 2003.
[44] Li J. T., Liu Z. J., and Nay L. H. A., “Effect of radial magnetic forces in permanent magnet motors with rotor eccentricity,” IEEE Transactions on Magnetics, vol. 43, pp. 2525-2527, June 2007.
[45] Li X. D., Wu Q., and Nandi S., “Performance analysis of a three-phase induction machine with inclined static eccentricity,” IEEE Transactions on Industry Applications, vol. 43, pp. 531-541, Mar.-Apr. 2007.
[46] Markovic M., Jufer M., and Perriard Y., “Determination of tooth cogging force in a hard-disk brushless DC motor,” IEEE Transactions on Magnetics, vol. 41, pp. 4421-4426, Dec. 2005.
[47] Nandi S., Bharadwaj R. M., and Toliyat H. A., “Performance analysis of a three-phase induction motor under mixed eccentricity condition,” IEEE Transactions on Energy Conversion, vol. 17, pp. 392-399, Sep. 2002.
[48] NEMA, NEMA Standards Publication No. MG 1-2006－Motors and Generators, 2006
[49] Peters R., Lundin U., and Leijon M., “Saturation effects on unbalanced magnetic pull in a hydroelectric generator with an eccentric rotor,” IEEE Transactions on Magnetics, vol. 43, pp. 3884-3890, Oct. 2007.
[50] Sheth N. K., Sekharbabu A. R. C., and Rajagopal K. R., “Effect of skewing the rotor teeth on the performance of doubly salient permanent magnet motors,” Journal of Applied Physics, vol. 99, pp. -, Apr. 15 2006.
[51] Tenhunen A., Holopainen T. P., and Arkkio A., "Effects of saturation on the forces in induction motors with whirling cage rotor," IEEE Transactions on Magnetics, vol. 40, pp. 766-769, Mar. 2004.
[52] Tsai Jia-Gang, Machine Examination Handbook, Chinese Electronic Press, 1998.
[53] Wu L., Huang X., Habetler T. G., and Harley R. G., "Eliminating load oscillation effects for rotor eccentricity detection in closed-loop drive-connected induction motors," IEEE Transactions on Power Electronics, vol. 22, pp. 1543-1551, July 2007.
[54] Yoon T., "Magnetically induced vibration in a permanent-magnet brushless DC motor with symmetric pole-slot configuration," IEEE Transactions on Magnetics, vol. 41, pp. 2173-2179, June 2005.
[55] Zhu Z. Q., Ruangsinchaiwanich S., Ishak D., and Howe D., "Analysis of cogging torque in brushless machines having nonuniformly distributed stator slots and stepped rotor magnets," IEEE Transactions on Magnetics, vol. 41, pp. 3910-3912, Oct. 2005.
[56] Zhu Z. Q., Ishak D., Howe D., and Chen J., "Unbalanced magnetic forces in permanent-magnet brushless machines with diametrically asymmetric phase windings," IEEE Transactions on Industry Applications, vol. 43, pp. 1544-1553, Nov.-Dec. 2007.
[57] Zhu Z. Q., Xia Z. P., and Howe D., "Comparison of Halbach magnetized brushless machines based on discrete magnet segments or a single ring magnet," IEEE Transactions on Magnetics, vol. 38, pp. 2997-2999, Sep. 2002.