Unbalanced magnetic force and cogging torque of PM motors due to the interaction between PM magnetization and stator eccentricity

Transcription

1 Microsyst Technol (2016) 22: DOI /s x TECHNICAL PAPER Unbalanced magnetic force and cogging torque of PM motors due to the interaction between PM magnetization and stator eccentricity Bosung Seo 1 Sangin Sung 2 Kyungin Kang 2 Jeongyong Song 2 Gunhee Jang 1,2 Received: 31 August 2015 / Accepted: 19 January 2016 / Published online: 5 February 2016 Springer-Verlag Berlin Heidelberg 2016 Abstract We investigate the characteristics of cogging torque and unbalanced magnetic forces (UMF) due to the interaction between magnetization of permanent magnets (PM) and stator eccentricity in PM motors. We mathematically and numerically show that the third harmonic of PM magnetization with stator eccentricity decreas the amplitude of the pole harmonic of the cogging torque and UMF while at the same time increasing the amplitude of the least common multiple harmonic of the pole and slot numbers of cogging torque and UMF in PM motors. The effects of the third harmonic of PM magnetization in the PM motors with stator eccentricity were experimentally verified by measuring UMF and acoustic noi. 1 Introduction Maor sources of magnetically-induced acoustic noi in permanent magnet (PM) motors are cogging torque and unbalanced magnetic forces (UMF) (Sung et al. 201). Cogging torque is generated by the interaction between the poles and slots of PM motors. The fundamental harmonic of cogging torque is the least common multiple (LCM) harmonic of the pole and slot numbers (Jang and Lieu 199). UMF is not generated in rotational symmetric motors with respect to pole, slot, and winding configurations (Jang et al. 1996). However, stator eccentricity generates the pole harmonics of the cogging torque and UMF of PM motors (Koh and Seol 2003). Stator eccentricity is defined as a constant offt of the shaft with respect to the geometric center of the motor and is not related to the rotation of the PM motor, as shown in Fig. 1. Many rearchers have addresd veral design methods to reduce cogging torque and UMF (Koh and Seol 2003; Bianchi and Bolognani 2002; Zhu 2000). Some rearchers have reported that PM with pure sine magnetization reduces the magnitude of LCM harmonics of cogging torque in PM motors (Hong et al. 1999). Several rearchers have also reported acoustic noi and vibration reduction methods for PM motors that u a combination of harmonic frequencies of PM magnetization (Hur et al. 2011). Previous rearchers, however, have not investigated the effects of PM magnetization with stator eccentricity on the UMF and cogging torque of PM motors. We mathematically derive the harmonic equations for cogging torque and UMF generated by the interaction between the PM magnetization and stator eccentricity in a PM motor. Frequency components of cogging torque and UMF are investigated using the derived harmonic equations. We develop a finite element (FE) model of a fan motor with 12 poles and 9 slots in order to investigate the characteristics of UMF and cogging torque. The cogging torque and UMF are calculated by the virtual work method. Their characteristics are investigated through spectral analysis. The mathematical and simulated results are experimentally verified by measuring the UMF and acoustic noi of the PM motor. * Gunhee Jang ghang@hanyang.ac.kr 1 Department of Advanced Power Conversion System Engineering, Hanyang University, 17 Haengdang dong, Seongdong gu, Seoul , Republic of Korea 2 Department of Mechanical Convergence Engineering, Hanyang University, 17 Haengdang dong, Seongdong gu, Seoul , Republic of Korea 2 Equations of cogging torque and UMF The equations for cogging torque and UMF are derived by neglecting the magnetic saturation of the stator core. For the PM motors with stator eccentricity, the slot permeance function, λ (θ), and magneto-motive force (MMF) function, f(θ,α), are expresd as follows:

2 1250 Microsyst Technol (2016) 22: where k, Λ and Λ are the tooth number, Fourier coefficient of slot permeance without stator eccentricity and Fourier coefficient of slot permeance with stator eccentricity, respectively. When PM motors have stator eccentricity, tooth reluctance, R, which is reprented as the air gap length divided by the product of permeability and area, can be reprented as shown in Eq. (5). K c g R = ( ) µ 0 hr (5) Fig. 1 Stator eccentricity of a PM motor λ (θ) = f (θ, α) = λ i cos (i θ)(1 + ε cosθ) f sin (p(θ α)) =1 where λ i,, ε, f, p and α are the Fourier coefficient of slot permeance, number of slots, relative stator eccentricity, Fourier coefficient of MMF, number of pole pairs and rotating angle, respectively (Jacek et al. 2005). The air gap flux density, B(θ,α), can be determined by multiplying the slot permeance function and MMF function. B(θ, α) = λ (θ)f (θ, α) λ i f = 2 =1 [ sin {(p ± i )θ pα} + 1 ] 2 ε sin {(p ± i 1)θ pα} The magnetic flux acting on a tooth, Φ k, is derived by integrating the air gap flux density along a tooth. π Φk {2(k 1)+1} = B(θ, α)dθ π {2(k 1) 1} [ ] = f Λ + ελ (k 1) p(k 1) cos sin pα =1 p = i : 2λ i ( 1) i ( ) (p±i ) sin πp Λ = p = i : πλ i p ± i = ±1 and p = i : Λ = p ± i = ±1 and p = i : 0 p ± i =±1 : (1) (2) (3) () λ i ( 1) i (p±i 1) sin π(p 1) πλ i )} (k 1) R k = R 1 + ε cos where K c, g, μ 0, h, r, and R k are Carter s coefficient, the length of the air gap, permeability of air, height of the stator, radius of the stator and the reluctance on the kth tooth with stator eccentricity, respectively. When the electric energy input is equal to zero, the radial magnetic force of PM motors can be determined as the derivative of the magnetic energy stored in the air gap with respect to the radial direction using the virtual work method as follows: W f = 1 2 R k Fr,k = W f r = 1 2 ( ) Φ 2 k ( Φ k ) 2 R k r where W f and F r,k reprent the magnetic energy stored in the air gap and the radial magnetic force acting on kth tooth of PM motors with stator eccentricity, respectively (Hanlman 2003). The radial magnetic force acting on the kth tooth of PM motors can be expresd as shown in Eq. (9). Fr, k = ± 1 r f if i=1 =1 Rε 2{ Λ i Λ + 1 ( 2 Λ i Λ + Λ i Λ + Λ i Λ )} + Rε Λ i Λ + Λ i Λ + Λ i Λ + 3Λ i Λ )} ε 2 cos (k 1) { + Rε 2 ( 2 Λ i Λ + Λ i Λ + Λ i Λ ) } cos π(k 1) ( + ε 3 ) RΛ i Λ cos 6π(k 1) p(i ± )(k 1) cos (i ± )pα (9) The cogging torque and tangential magnetic force of PM motors can be expresd, using virtual work method, as the derivative of the magnetic energy stored in the air gap, with respect to the rotating angle. (6) (7) (8)

3 Microsyst Technol (2016) 22: T cog = W f α = Φk R dφk k dα k=1 Ft,k = 1 W f r α (10) (11) where T cog and F t,k are the cogging torque and tangential magnetic force, respectively. The cogging torque and tangential magnetic force of PM motors can then be reprented as shown in Eq. (12) and (13) torque and the X and Y-directional UMFs can be derived as follows: 3 T cog = 8 Rε3 f 1 f 1 Λ 1 Λ sin 12α Rε3 f 1 f 3 Λ 1 Λ Rε3 f 1 f 3 Λ 1 Λ 3 sin 2α 9 2 R { Λ 3 Λ ε2( Λ 3 Λ 3 + 2Λ 3 Λ 3) } sin 36α (16) T cog = p 2 Rε 2 )} Λ i Λ Λ i Λ + Λ i Λ + Λ i Λ + Rε Λ f i f i Λ + Λ i Λ + Λ i Λ + 3Λ i Λ ε 2)} cos )} i=1 =1 k=1 + Rε 2 2 Λ i Λ + Λ i Λ + Λ i Λ cos π(k 1) ( ) + ε 3 RΛ i Λ cos 6π(k 1) p(i ± )(k 1) sin (i ± )pα Rε 2 )} Λ i Λ Λ i Λ + Λ i Λ + Λ i Λ Ft,k = p + Rε Λ f i f i Λ + Λ i Λ + Λ i Λ + 3Λ i Λ ε 2)} cos )} 2r i=1 =1 + Rε 2 2 Λ i Λ + Λ i Λ + Λ i Λ cos π(k 1) ( ) + ε 3 RΛ i Λ cos 6π(k 1) p(i ± )(k 1) sin (i ± )pα (k 1) (k 1) (12) (13) The X and Y-directional UMFs induced in a PM motor, F x and F y, can be determined through a coordinate transformation as follows: F x = k=1 Fy = k=1 ) ( )} F r,k cos (k 1) + F N t,k sin (k 1) s ) ( )} F r,k sin (k 1) + F t,k cos (k 1) (1) (15) Using Eqs. (12), (1) and (15), we can calculate the harmonics for the cogging torque and the X and Y-directional UMFs in PM motors developed as a result of the interaction between PM magnetization and stator eccentricity. When a PM motor with 12 poles, 9 slots and stator eccentricity has a third harmonic of PM magnetization, the cogging Fx = 11 ( 32r Rε2 f 1 f 1 Λ 1 Λ 1 + 2Λ 1 Λ ) r Rε2 f 1 f 3 ( Λ 1 Λ 3 + 2Λ 1 Λ 3 cos 12α ) 23 ( 16r Rε2 f 1 f 3 Λ 1 Λ 3 + 2Λ 1 Λ 3) cos 2α + 1 ( 8r Rεf 3f 3 Λ 3 Λ 3 + 2Λ 3 Λ 3 + 3Λ 3 Λ ) 3 ε 2 cos 36α (17) 13 ( Fy = 32r Rε2 f 1 f 1 Λ 1 Λ 1 + 2Λ 1 Λ ) 1 sin 12α 13 ( 16r Rε2 f 1 f 3 Λ 1 Λ 3 + 2Λ 1 Λ ) 3 23 ( 16r Rε2 f 1 f 3 Λ 1 Λ 3 + 2Λ 1 Λ 3) sin 2α 9 ( 2r Rεf 3f 3 Λ 3 Λ 3 + 2Λ 3 Λ 3 + 3Λ 3 Λ ) 3 ε 2 sin 36α (18)

4 1252 Microsyst Technol (2016) 22: The 12th and 2th harmonics for the cogging torque and UMF are generated by the interaction between the fundamental harmonic and the third harmonic of the PM magnetization. Also, the 36th harmonic of the cogging torque and UMF are generated by the third harmonics of PM magnetization. The 12th harmonics of the cogging torque and UMF decrea due to the third harmonic of the PM magnetization. 3 Finite element analysis A two-dimensional FE model of a fan motor with 12 poles and 9 slots was developed to investigate cogging torque and UMF, as shown in Fig. 2. The air gap between the stator and the rotor is 500 μm and the stator eccentricity of 200 μm is applied to the FE model. Three magnetization forms of the PMs are modeled in the FE analysis, as shown in Fig. 3: a pure sinusoidal magnetization, which has the fundamental 6th harmonic only; and two sinusoidal magnetizations with 5 % and 10 % addition to the third harmonic (18th). All root mean square (RMS) values of the residual flux densities are 0.9 T. The FE model has approximately two million tetragonal elements with four nodes. The magnetic field is calculated using the FLUX2D software with nonlinear B-H characteristics for the stator core. The FE analysis is performed every 0.5 rotation of the rotor from 0 to 360. The cogging torque and UMF are determined by the virtual work method, and their frequency characteristics are investigated by spectral analysis. Figure shows the frequency spectra of the simulated cogging torque according to the magnetization forms applied to the PM motor with stator eccentricity of 200 μm. The PM magnetization using only a pure sine wave reduces the magnitude of the 36th harmonic of the cogging torque, which is the LCM harmonic of the pole and slot numbers. However, it increas the magnitude of the 12th harmonic of the cogging torque, which is the pole number harmonic Fig. 2 Two-dimensional FE model of a fan motor with 12 poles and 9 slots Fig. 3 Modeled magnetization forms of PMs with 12 poles Fig. Frequency spectra of the simulated cogging torque according to the magnetization forms of the PM motor with stator eccentricity of 200 μm generated by the stator eccentricity. The magnitude of 2th harmonic of the cogging torque is very small in comparison with tho of the 12th and 36th harmonics of the cogging torque. Therefore, we can neglect the 2th harmonic of the cogging torque. A UMF is not generated in ideal PM motors due to the rotational symmetry with respect to the poles, slots and winding configurations, however, additional harmonics of the UMF are generated by the stator eccentricity. Figure 5 shows the frequency spectra of the simulated UMF according to the PM magnetization forms when the stator eccentricity was 200 μm. The PM magnetization using pure sine waves reduces the magnitude of the 36th harmonic of the UMF, but it increas the magnitude of the 12th harmonic of the UMF. In this motor, the magnitude of the 12th harmonic is much greater than tho of the 2th and 36th harmonics of the UMF.

5 Microsyst Technol (2016) 22: Fig. 5 Frequency spectra of the simulated UMF according to the magnetization forms of the PM motor with stator eccentricity of 200 μm Fig. 7 Measured surface flux densities along the inner surface of the PMs by Gauss meter Table 1 Harmonic frequencies of measured surface flux densities for PM I and PM II Harmonic frequencies Magnitude (mt) PM I PM II 6th (fundamental harmonic) th (third harmonic) Fig. 6 Experimental tup to measure the surface flux densities of the PMs Experiments Using an experimental tup, the surface flux densities of the PMs were investigated to determine the effect of magnetization forms of PM I and PM II on the UMFs, as shown in Fig. 6. Figure 7 shows the measured surface flux densities along the inner surface of the PMs by Gauss meter. Table 1 shows the frequency spectra of the measured surface flux densities of the PMs. The RMS values of the measured surface flux densities for PM I and PM II are at the same level, and the magnitudes of third harmonics (18th) for PM I and PM II are 1.53 and.92 % of the magnitudes of the fundamental harmonics (6th), respectively. An experimental tup was developed to measure the UMF of the PM motor and to verify the simulated results Fig. 8 Experimental tup to measure UMF of PM motors of the UMF, as shown in Fig. 8. The rotor is parated from the stator in order to measure the UMF. The rotor is also attached to a load cell (Honeywell 31) capable of measuring between 250 g and 250 g of force. The load cell is fixed to an XYZ-table, and the disasmbled stator is fixed to an XY-table on a rotating shaft. The position of the stator can be controlled with respect to the center of the rotor so that we are able to change its eccentricity. The rotating shaft is connected to another driving motor through a belt. When the rotating shaft turns at constant speed, the UMF of the stationary disasmbled

6 125 Microsyst Technol (2016) 22: Fig. 11 Experimental tup to measure the acoustic noi of PM motors with PM I and PM II Fig. 9 Measured UMF of the PM motor with PM I when the stator eccentricity is 200 μm Fig. 12 Measured acoustic noi of the PM motor with PM I when the stator eccentricity is 129 μm Fig. 10 Measured UMF of the PM motor with PM II when the stator eccentricity is 200 μm rotor is measured by the load cell. Figures 9 and 10 show the measured UMF of the PM motor for PM I and PM II when the stator eccentricity is 200 μm. The 1st, 8th, and 10th harmonics of the measured UMF are generated by the uneven magnetization of the PM (Kang et al. 2013). The 12th harmonic of the measured UMF, which is the pole harmonic, is generated by the stator eccentricity. The magnitude of the 12th harmonic of PM I is greater than that of PM II becau the magnitude of the third harmonic in PM II is larger than that of PM I. The magnitude of the 36th harmonic is small in comparison with that of the 12th harmonic. The measured results match well with simulated ones. Fig. 13 Measured acoustic noi of the PM motor with PM II when the stator eccentricity is 129 μm

7 Microsyst Technol (2016) 22: We measured the acoustic noi of the PM motor for PM I and PM II, as shown in Fig. 11. The acoustic noi and overall sound level were measured in a mi-anechoic chamber with background noi of 17 db-a using a microphone located 30 cm above the PM motors. Figures 12 and 13 show the measured acoustic noi of the PM motor for PM I and PM II at 1000 rpm when the stator eccentricity is 129 μm. The 9th, 12th and 36th harmonics are obrved in the measured acoustic noi. The 9th harmonic is generated by the rotor eccentricity and the uneven magnetization of the PMs (Kim et al. 2012). The magnitude of the 36th harmonic of acoustic noi in PM I is smaller than that in PM II. However, the magnitude of the 12th harmonic in PM I is greater than that of PM II. This shows that the magnetization of the PM affects the acoustic noi of the PM motors, and the magnetization form using pure sine waves decreas the magnitude of the 36th harmonic but increas the magnitude of the 12th harmonic of the acoustic noi for the PM motor. This result matches well with the mathematical equations and simulated cogging torque and UMF. 5 Conclusions We mathematically and numerically investigated the characteristics of cogging torque and UMF caud by the interaction between PM magnetization and stator eccentricity for PM motors. The PM with pure sine magnetization, which is generally thought to reduce cogging torque, increas the pole number harmonics of the UMF and the cogging torque in PM motors that have stator eccentricity. We mathematically and numerically show that the third harmonic of the PM magnetization with stator eccentricity decreas the amplitude of the pole harmonic of the cogging torque and the UMF while at the same time increasing the amplitude of the least common multiple harmonic of the pole and slot numbers for the cogging torque and UMF 1255 in PM motors. The effects of the third harmonic of PM magnetization in PM motors with stator eccentricity were experimentally verified by measuring UMF and acoustic noi. This rearch can contribute to reducing magnetically-induced vibration and acoustic noi in PM motors. References Bianchi N, Bolognani S (2002) Design techniques for reducing the cogging torque in surface-mounted PM motors. IEEE Ind Appl 38: Hanlman D (2003) Brushless permanent magnet motor design. The Writers Collective Hong SP, Cho HS, Lee HS, Cho HR, Lee HY (1999) Effect of magnetization direction in permanent magnet on motor characteristics. IEEE Trans Magn 35: Hur J, Reu JW, Kim BW, Kang HG (2011) Vibration reduction of IPM-type BLDC motor using negative third harmonic elimination method of air-gap flux density. IEEE Trans Ind Appl 7: Jacek FG, Chong W, Joph CL (2005) Noi of polypha electric motors. Taylor and Francis, Boca Raton Jang GH, Lieu DK (199) Modeling of the influence of coil winding pattern on tooth forces in brushless DC motors. IEEE Trans Magn 30: Jang GH, Yoon JW, Park NY, Jang SM (1996) Torque and unbalanced magnetic force in a rotational asymmetric brushless DC motors. IEEE Trans Magn 32: Kang KJ, Jang GH, Sang SJ (2013) Frequency characteristics of BEMF, cogging torque and UMF in a HDD spindle motor due to unevenly magnetized PM. IEEE Trans Magn 9: Kim JY, Sung SJ, Jang GH (2012) Characterization and experimental verification of the axial unbalanced magnetic force in brushless DC motors. IEEE Trans Magn 8: Koh CS, Seol JS (2003) New cogging-torque reduction method for brushless permanent-magnet motors. IEEE Trans Magn 39: Sung SJ, Jang GH, Kang KJ (201) Noi and vibration due to rotor eccentricity in a HDD spindle system. Microsyst Technol 20: Zhu ZQ (2000) Influence of design parameters on cogging torque in permanent magnet machines. IEEE Trans Energy Convers 15:07 12