Axial Unbalanced Magnetic Force in a Permanent Magnet Motor Due to a Skewed Magnet and Rotor Eccentricities

Transcription

1 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER Axial Unbalanced Magnetic Force in a Permanent Magnet Motor Due to a Skewed Magnet and Rotor Eccentricitie Chi Ho Kang, Kyung Jin Kang, Jeong Yong Song, Young Jin Cho, and Gun Hee Jang Preciion Rotating Electromechanical Machine Laboratory, Department of Mechanical Convergence Engineering, Hanyang Univerity, Seoul 4763, South Korea We invetigate the effect of a kewed permanent magnet (PM) with overhang on the axial unbalanced magnetic force (UMF) of a PM motor, epecially in the preence of tatic and dynamic eccentricitie of a rotor. We derive the mathematical equation of the axial UMF acting on PM motor due to the kewed PM with overhang and rotor eccentricitie. We develop a 3-D finite-element model of the PM motor, not only to verify the frequency component of the propoed equation but alo to determine the amplitude of the axial UMF. Further, we develop an experimental etup to meaure the axial UMF caued by a kewed PM with overhang and rotor eccentricitie. We confirm that the kewed PM with overhang generate an axial UMF with the harmonic of the leat common multiple of pole and lot number. In addition, the tatic and dynamic eccentricitie with the interaction of the kewed PM and overhang generate the axial UMF with the pole harmonic and lot harmonic, repectively. Index Term Axial unbalanced magnetic force (UMF), eccentricity, permanent magnet (PM) motor, kew. I. P INTRODUCTION ERMANENT magnet (PM) motor have been widely ued in many application for power generation due to their high-power denity [1]. PM motor upply power uing a PM and applied current, and everal deign of PM have been propoed to increae their power denity. The overhang of a PM ha been ignificantly applied to increae the power denity of PM motor becaue a PM that ha a longer height than that of the tator core concentrate a magnetic flux in the air gap and the tator core [2]. However, it alo generate magnetically excited vibration and noie in PM motor, epecially when the PM ha axial aymmetry with repect to the tator core, i.e., the height of the upper PM overhang i different from that of the lower PM overhang [3]. A kewed PM ha alo been applied to reduce the cogging torque and torque ripple. Zhu and Howe [4] tudied the effect of the pole arc to pole-pitch ratio, pole and lot number combination, and optimum kew angle on the cogging torque of PM motor. Bianchi and Bolognani [5] preented variou deign technique for reducing the cogging torque of PM motor uch a kewed PM, dummy lot, and hifted PM. Ilam et al. [6] propoed a method to reduce the cogging torque and torque ripple of PM motor by uing kewed PM and variou pole and lot number combination. Jung et al. [7] analyzed the optimal length of the kew and overhang of a PM in which the normal and tangential electromagnetic force of the linear PM motor are minimized. Even though the ue of kewed PM i a well-known method for reducing the cogging torque and torque ripple of PM motor [8], [9] in uch a way that decreae the radial and tangential magnetic excitation force of PM motor, the kewed PM with overhang of PM motor break the Manucript received March 1, 217; revied May 15, 217; accepted May 19, 217. Date of publication May 25, 217; date of current verion October 24, 217. Correponding author: G. H. Jang ( ghjang@ hanyang.ac.kr). Color verion of one or more of the figure in thi paper are available online at Digital Object Identifier 1.119/TMAG magnetic ymmetry in the axial direction and thu generate a reultant axial unbalanced magnetic force (UMF) acting on the PM motor. We mathematically, numerically, and experimentally invetigated the characteritic of the axial UMF originating from the kewed PM with overhang, a well a it reulting interaction with the tatic and dynamic eccentricitie of a rotor in PM motor ince the rotational aymmetry of a motor i unavoidable in real production due to manufacturing tolerance or in variou operating condition [1] [12]. Thi paper i organized a follow. In Section II, we mathematically derived the frequency equation of the axial UMF of PM motor due to the kewed PM with overhang and it interaction with the tatic and dynamic eccentricitie of a rotor uing the Fourier erie. In Section III, we performed finite-element analye to validate the frequency component of the axial UMF in the propoed mathematical equation. Then, we invetigated the effect of kewed PM with overhang and rotor eccentricitie on the axial UMF of PM motor. In Section IV, we conducted an experiment to meaure and verify the characteritic of the axial UMF of PM motor in the preence of rotor eccentricitie. Finally, Section V give the concluding remark. II. MATHEMATICAL DERIVATION OF AXIAL UMF DUE TO SKEWED PM WITH OVERHANG AND ROTOR ECCENTRICITIES The axial magnetic flux denity of the air gap correponding to the nth tooth can be repreented by uing the Fourier erie a follow [13]: ( B zn = B k co k=1 kpθ +kpωt + 2πkp (n 1) + θ ) kew z φ k h PM where B k, p, θ, ω, t,, n, θ kew, h PM, z, andφ k are the amplitude of magnetic flux denity, number of pole pair, angular poition along the air gap, angular peed, time, number IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information. (1)

2 82155 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 217 of lot, tooth number, kew angle, height of PM, axial poition, and phae of the magnetic flux denity, repectively. We aume that the axial magnetic flux i generated only in the overhang area due to the height difference between the PM and tator, and the equation are derived by neglecting the magnetic aturation effect of the tator core. The tatic and dynamic rotor eccentricity function can be repreented a follow [14]: E ta = 1 + ε co (θ + 2π ) g (n 1) ϕ (2) E dyn = 1 + ε co (θ + ωt + 2π ) g (n 1) ϕ (3) Fig D finite-element model of the three-phae PM motor with 12 pole and 9 lot. where ε, g,andϕ are the magnitude of eccentricity, initial airgap length, and phae of eccentric direction, repectively. For the cae with tatic and dynamic eccentricitie, the magnetic flux denitie through a tooth can be calculated by multiplying the magnetic flux denity in (1) by the eccentricity function in (2) and (3), repectively. The magnetic force denity of the nth tooth in the axial direction can be expreed by uing the Maxwell tre tenor method a follow: Fig. 2. Magnetized pattern of (a) rectangular PM and (b) kewed PM. df zn = B2 zn (4) 2μ where μ i the permeability of free pace. The axial magnetic force induced in the nth tooth can be calculated by integrating the magnetic force denity a follow: h PM h tooth 2 2π F zn = df zn dθ dz 2π df zn dθ dz (5) h PM h tooth 2 where h tooth i the height of the tator core. The axial UMF can be given by umming the magnetic force induced in each tooth a follow: F z = F zn. (6) n=1 In the cae of the concentric and eccentric motion of the PM motor with a rectangular PM (non-kewed PM) and overhang, the axial UMF are not generated becaue the axial magnetic force acting on the upper and lower PM overhang have the ame magnitude in oppoite direction a follow: F z = F z_ta = F z_dyn =. (7) In the cae of the concentric motion of the PM motor with a kewed PM and overhang, the axial UMF can be repreented a follow: F z = w { ( ) } 4μ θ kew h PM B k B l in(lcm(2 p, )mωt (φ k +φ l )) where w tooth i the width of the tooth. The kewed PM with overhang generate the axial UMF with the harmonic of the leat common multiple (LCM) of the pole and lot number. (8) In the cae of the tatic eccentric motion of the PM motor with a kewed PM and overhang, the axial UMF i generated with pole harmonic a follow: F z_ta = w { ( ) } 4μ θ kew h PM B k B l in(mpωt (φ k +φ l ± mϕ)). In the cae of the dynamic eccentric motion of the PM motor with a kewed PM and overhang, the axial UMF i generated with lot harmonic a follow: F z_dyn = w { ( ) } 4μ θ kew h PM B k B l in(mωt (φ k +φ l ± mϕ)). (9) (1) III. FINITE-ELEMENT ANALYSES A. Finite-Element Model The propoed equation in Section II can explain the frequency component of the axial UMF generated by the kewed PM with overhang and it interaction with rotor eccentricitie. However, it i difficult to predict the magnitude of the axial UMF due to the complex geometry of PM motor and the nonlinearity of magnetic material. Therefore, we ued finiteelement analyi to validate the frequency component of the axial UMF of the propoed equation, a well a to determine the amplitude of component of the axial UMF acting on PM motor. Fig. 1 how a 3-D finite-element model of a three-phae PM motor with 12 pole and 9 lot, ued a a cooling fan in a refrigerator. It conit of hexahedron element with eight node. Fig. 2 how the imulated reidual flux denity

3 KANG et al.: AXIAL UNBALANCED MAGNETIC FORCE IN A PM MOTOR Fig. 3. (a) Simulated axial UMF and (b) it frequency pectrum for the concentric motion of the PM motor with a kewed PM and overhang. Fig. 5. (a) Simulated axial UMF and (b) it frequency pectrum for the dynamic eccentric motion of the PM motor with a kewed PM and overhang. Fig. 4. (a) Simulated axial UMF and (b) it frequency pectrum for the tatic eccentric motion of the PM motor with a kewed PM and overhang. for the rectangular and kewed PM. The Nd-Fe-B bonded PM i radially magnetized with a maximum reidual flux denity of.7 T. The kewed PM i axially inclined by 1, a hown in Fig. 2(b). The finite-element analye utilizing FLUX3D are performed by conidering both tatic and dynamic eccentric motion of the rotor, which are aumed to be 2% in the air gap of 5 μm between the tator and rotor. Alo, nonlinear B H characteritic curve of the tator core i conidered. The numerical accuracy of the developed finite-element model i verified by comparing the imulated back electro-motive force (EMF) with the meaured back EMF at an operating peed of 2 rpm. The imulated and meaured back EMF are 4.78 and 4.85 V, repectively, with a 1.4% error. B. Simulated Axial UMF The axial UMF of the PM motor are numerically imulated by uing the virtual work method, and the frequency characteritic are invetigated via pectral analyi. For the PM motor with a rectangular PM and overhang, we confirmed that the axial UMF of the PM motor are not generated, irrepective of the preence of tatic and dynamic eccentricitie. However, for the PM motor with a kewed PM and overhang, the axial UMF with an LCM harmonic of pole and lot number i generated, even in the cae of concentric motion, a hown in Fig. 3. In the cae of the tatic eccentric motion of the PM motor with a kewed PM and overhang, the axial UMF with pole harmonic i generated, a hown in Fig. 4. In the cae of the dynamic eccentric motion of the Fig. 6. Experimental etup to meaure the axial UMF of the PM motor due to rotor eccentricitie. PM motor with a kewed PM and overhang, the axial UMF with lot harmonic i generated, a hown in Fig. 5. IV. EXPERIMENTAL VERIFICATION A. Experimental Setup Fig. 6 how an experimental etup to meaure the axial UMF of the three-phae PM motor with 12 pole and 9 lot, ued a a cooling fan in a refrigerator. The tator and rotor are decoupled from an aembled PM motor ince the UMF of the PM motor cannot be meaured in the aembled tate of the PM motor [3]. The diaembled tator from the PM motor i mounted to a haft on an xy-tage in which the poition of the diaembled tator i adjutable, and they are connected to the ervo motor to rotate an experimental rotor aembly. The aembly conit of the diaembled tator, haft, and xy-tage. The diaembled rotor from the PM motor i clamped to a load cell and they are fixed to an xyz-tage. The center and axial poition of the experimental tator aembly, which conit of the diaembled rotor, load cell, and xyz-tage, are eparately adjuted by controlling the micrometer of the xyz-tage. The rotating poition of the haft i meaured by uing an optical enor. The axial UMF of the PM motor with the tatic and dynamic eccentricitie (2% in

4 82155 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 11, NOVEMBER 217 Fig. 7. (a) Meaured axial UMF and (b) it frequency pectrum for the concentric motion of the PM motor with a rectangular PM and overhang. Fig. 9. Frequency pectra of the meaured axial UMF of the PM motor with (a) rectangular PM and overhang and (b) kewed PM and overhang in the preence of 2% tatic eccentricity. Fig. 8. (a) Meaured axial UMF and (b) it frequency pectrum for the concentric motion of the PM motor with a kewed PM and overhang. the air gap of 5 μm) are meaured by uing the load cell on the diaembled rotor, with a meauring range between 25 and 25 g. B. Meaured Axial UMF Fig. 7 how the axial UMF and it frequency pectrum for the concentric motion of the PM motor with a rectangular PM and overhang. Even though it i uppoed to have no frequency component in the axial UMF, the 1t, 9th, 12th, and 18th harmonic are preent becaue the PM ha uneven magnetization and mall tatic and dynamic eccentricitie, even after attempt to minimize the eccentricitie. Kim et al. [3] reported that the imultaneou preence of the uneven magnetization of the PM and tatic eccentricity generate the firt and the pole harmonic in the axial UMF (1t and 12th in thi paper), and that the imultaneou preence of uneven magnetization of the PM and dynamic eccentricity generate the lot harmonic of the axial UMF (9th and 18th in thi paper). Fig. 8 how the axial UMF and it frequency pectrum for the concentric motion of the PM motor with a kewed PM and overhang. The meaured axial UMF ha the 36th harmonic (the harmonic of the LCM of pole and lot number), in addition to the 1t, 9th, 12th, and 18th harmonic. Fig. 9(a) how the frequency pectrum of the axial UMF of the PM motor with a rectangular PM and overhang in the preence of 2% tatic eccentricity. Even in the preence Fig. 1. Frequency pectra of the meaured axial UMF of the PM motor with (a) rectangular PM and overhang and (b) kewed PM and overhang in the preence of 2% dynamic eccentricity. of 2% tatic eccentricity, the meaured axial UMF of the PM motor with a rectangular PM and overhang i very cloe to one without rotor eccentricity, a hown in Fig. 7(b), becaue tatic eccentricity doe not affect the axial UMF of the PM motor with a rectangular PM and overhang. Fig. 9(b) how the frequency pectrum of the axial UMF of the PM motor with a kewed PM and overhang in the preence of 2% tatic eccentricity. The meaured axial UMF ha the dominant 12th and 36th harmonic (the harmonic of the pole number), in addition to the 1t, 9th, and 18th harmonic. Fig. 1(a) how the frequency pectrum of the axial UMF of the PM motor with a rectangular PM and overhang in the preence of 2% dynamic eccentricity. Even in the preence of 2% dynamic eccentricity, the meaured axial UMF of the PM motor with a rectangular PM and overhang i very cloe to one without rotor eccentricity, a hown in Fig. 7(b), becaue dynamic eccentricity doe not affect the axial UMF of the PM motor with a rectangular PM and overhang. Fig. 1(b) how the frequency pectrum of the axial UMF of the PM motor with a kewed PM and overhang in the preence of 2% dynamic eccentricity. The meaured axial UMF ha dominant 9th and 36th harmonic (the harmonic of the lot number), in addition to the 1t, 12th, and 18th harmonic.

5 KANG et al.: AXIAL UNBALANCED MAGNETIC FORCE IN A PM MOTOR V. CONCLUSION The ue of a kewed PM in PM motor i one of the mot common method to reduce cogging torque and torque ripple. However, we mathematically, numerically, and experimentally confirmed that the kewed PM with overhang generate the axial UMF of a PM motor with the harmonic of the LCM of pole and lot number due to magnetic aymmetry in the axial direction. In the preence of the tatic and dynamic eccentricitie with the interaction of the kewed PM and overhang, we found that the axial UMF of a PM motor i generated with the pole harmonic and lot harmonic, repectively. Thi reearch can contribute to reducing magnetically induced vibration and noie originating from the kewed PM with overhang and rotor eccentricitie of PM motor. REFERENCES [1] J. R. Henderhot and T. J. E. Miller, Deign of Bruhle Permanent- Magnet Motor. Oxford, U.K.: Magna Phyic Publihing/Clarendon, [2] I.-S. Jung, J. Hur, and D.-S. Hyun, 3-D analyi of permanent magnet linear ynchronou motor with magnet arrangement uing equivalent magnetic circuit network method, IEEE Tran. Magn., vol. 35, no. 5, pp , Sep [3] J. Y. Kim, S. J. Sung, and G. H. Jang, Characterization and experimental verification of the axial unbalanced magnet force in bruhle DC motor, IEEE Tran. Magn., vol. 48, no. 11, pp , Nov [4] Z. Q. Zhu and D. Howe, Influence of deign parameter on cogging torque in permanent magnet machine, IEEE Tran. Energy Conver., vol. 15, no. 4, pp , Dec. 2. [5] N. Bianchi and S. Bolognani, Deign technique for reducing the cogging torque in urface-mounted PM motor, IEEE Tran. Ind. Appl., vol. 38, no. 5, pp , Sep. 22. [6] R. Ilam, I. Huain, A. Fardoun, and K. McLaughlin, Permanentmagnet ynchronou motor magnet deign with kewing for torque ripple and cogging torque reduction, IEEE Tran. Ind. Appl., vol. 45, no. 1, pp , Jan. 29. [7] I.-S. Jung, J. Hur, and D.-S. Hyun, Performance analyi of kewed PM linear ynchronou motor according to variou deign parameter, IEEE Tran. Magn., vol. 37, no. 5, pp , Sep. 21. [8] T. Li and G. Slemon, Reduction of cogging torque in permanent magnet motor, IEEE Tran. Magn., vol. 24, no. 6, pp , Nov [9] R. Akmee and J. F. Eatham, Deign of permanent magnet flat linear motor for tandtill application, IEEE Tran. Magn., vol. 28, no. 5, pp , Sep [1] S. J. Sung, G. H. Jang, and K. J. Kang, Noie and vibration due to rotor eccentricity in a HDD pindle ytem, Microyt. Technol., vol. 2, no. 8, pp , Aug [11] J. Y. Song, K. J. Kang, C. H. Kang, and G. H. Jang, Cogging torque and unbalanced magnetic pull due to imultaneou exitence of dynamic and tatic eccentricitie and uneven magnetization in permanent magnet motor, IEEE Tran. Magn., vol. 53, no. 3, Mar. 217, Art. no [12] A. P. Ortega and L. Xu, Invetigation of effect of aymmetrie on the performance of permanent magnet ynchronou machine, IEEE Tran. Energy Conver., to be publihed, doi: 1.119/TEC [13] D. C. Hanelman, Effect of kew, pole count and lot count on bruhle motor radial force, cogging torque and back EMF, IEE Proc.-Electr. Power Appl., vol. 144, no. 5, pp , Sep [14] D. Guo, F. Chu, and D. Chen, The unbalanced magnetic pull and it effect on vibration in a three-phae generator with eccentric rotor, J. Sound Vibrat., vol. 254, no. 2, pp , Jul. 22.