Analysis of Permanent Magnet Synchronous Motors with Integer-slot and Fractional-slot Windings

Transcription

1 Analyi of Permanent Magnet Synchronou Motor with Integer-lot an Fractional-lot Wining J. A. Güeme #, A. M. Iraolagoitia #2, M. P. Donión *3, P. Fernánez +4 # Department of Electrical Engineering, E. U. e Ingenieria Técnica Inutrial, Univerity of the Baque Country Plaza e la Cailla 3, 482 Bilbao, Spain joeantonio.gueme@ehu.e 2 ana.iraolagoitia@ehu.e * Department of Electrical Engineering, Univerity of Vigo Campu of Lagoa Marcoene, 363 Vigo (Spain) 3 onion@uvigo.e + Department of Electronic an Telecommunication, E.U. e Ingenieria Técnica Inutrial, Univerity of the Baque Country Plaza e la Cailla 3, 482 Bilbao, Spain 4 pablo.fernanezr@ehu.e Abtract Thi paper examine torque ripple, cogging torque, an -axi an q-axi reactance variation in permanent magnet ynchronou motor (PMSM). Effect of number of pole on electromagnetic motor torque (incluing cogging torque), an - an q-axi reactance, ha been analyze for two ifferent PMSM configuration having the ame envelop imenion an output requirement. Finite element technique i ue for computation of the machine characteritic. Maxwell tre tenor i ue to fin the torque. I. INTRODUCTION PMSM are wiely ue in many inutrial application for their high efficiency, compact tructure, high torque-to-current an torque-to-volume ratio, fat ynamic repone, imple mechanical contruction, abence of moving contact an eay maintenance. A their cot continue to ecreae they have the opportunity to become a ominant force in the inutrial application market [] - [3]. One of the mot important problem in permanent magnet motor i the torque ripple which i inherent in their eign. Thi ripple i paraitic, an can lea to mechanical vibration, acoutic noie, an problem in rive ytem. Minimizing thi ripple i of great importance in the eign of PMSM. Pulating torque houl be pecially analyze for the application of contant pee or high-preciion poition control, epecially at low pee []-[2], [4]-[5]. There are three ource of torque ripple coming from the machine: a) cogging torque, b) ifference between permeance of the air gap in the - an q-axi (reluctance torque), an c) itortion of the magnetic flux enity waveform in the air gap [6]. Cogging torque i the conequence of interaction (magnetic attraction) between rotor-mounte permanent magnet fiel an the tator teeth, which prouce reluctance variation epening on rotor poition; it i inepenent of tator current. It manifet itelf by the tenency of the rotor to align in a number of table poition even when machine i unexcite, reulting in a pulating torque which oe not contribute to the net effective torque of the motor. In the literature, numerou metho for reucing the cogging torque, uch a employing a fractional number of lot per pole, king of magnet or tator lamination tack (lot), iplacing an haping the magnet, optimizing the magnet pole-arc-to-pole-pitch ratio, introucing auxiliary lot or teeth, etc, have been propoe [3]-[4], [7]-[2], However, optimizing the cogging torque to a low value i not ufficient to obtain a low torque ripple [3], [], [3]. A a conequence, in motor eign proce, the problem of the torque ripple reuction houl be coniere a a whole. Minimizing torque ripple i of utmot importance in many inutrial application an thi i the reaon it ha receive much attention in recent year. Different technique for torque ripple minimization have been propoe in literature. Broaly peaking, thee technique can be ivie into two main group. The firt cla concentrate on the optimum motor eign ubject to the contraint of minimum pulating torque an maximum average torque. Several torque ripple minimization technique for urface-mounte magnet motor have been propoe [3], [], [4]-[5]. Ue of the ame approach for interior-mounte magnet motor i troubleome an imprecie, an i almot abent in literature. The econ cla i bae on the rive an control concept to generate an appropriate control effort to minimize the torque ripple [6]-[2]. In PMSM, apart from etimating torque ripple, i alo important to calculate equivalent circuit parameter of the motor. Knowlege of the ifferent parameter of the equivalent circuit allow u to know the behaviour of the machine, by applying the law of electrical circuit. It i well known that the - an q-axi reactance are the mot ignificant parameter when ealing with teay tate an/or ynamic performance analyi of PMSM. Thee //$26. 2 IEEE 499

2 reactance are alo very important for eigning control ytem, in orer to maximize the efficiency, power factor, etc. In the literature, variou technique have been ecribe to preict the - an q-axi inuctance [2]-[26]. The main objective of thi paper i to invetigate the influence of pole number on the torque ripple (incluing cogging torque) an - an q-reactance of two 36-lot threephae PMSM with the ame envelop imenion an output requirement. Magnetic vector potential waveform in the air gap are alo obtaine an compare. Finite element technique i ue for the computation of the machine characteritic. Core aturation i irectly coniere in the magnetic fiel calculation. Maxwell tre metho i ue to fin the torque. The - an q-reactance etermination i carrie out through input energy calculation of the moel (effect uch a tator en-wining leakage an magnet k can be eaily inclue uing claical formula). Implementation can be conveniently realize by uing commercially available an technically mature finite element package. II. ANALYZED MOTORS Cro ection of two 36-lot PMSM, icue in thi paper, i hown in Fig.. Each rotor pole contain a permanent magnet of the neoymium-iron-boron type (NFeB) that i magnetize acro their horter imenion along the - axi. III. MODEL AND METHOD Finite element technique i ue for machine characteritic computation. Finite element moel are two-imenional plane. A non-lineal fiel analyi i carrie out for calculating the magnetic vector potential an magnetic flux enity in each noe of the finite element moel. Stator wining i excite with three-phae balance inuoial current to prouce the characteritic ynchronouly rotating magnetomotive force waveform. The moel i contitute by a tranvere ection through the mile of the motor (36º geometry). The pace of air urrouning the motor ha to be taken into account too. Figure 2 how the finite element meh ue. Fig. 2. Two-imenional mehing moel (zoom) Torque i calculate uing the Maxwell tre metho [27]- [29]. Thi metho require the local flux enity itribution along a pecific line or contour. Total force can be calculate by mean of the following expreion: ( B ) n 2 2 B t n BnBt t F = + 2μo μo () Fig.. Cro ection of the motor Table I how the main eign parameter of the tuie motor. TABLE I DETAILS OF THE ANALYZED MOTORS Connection Star Power rating 6 kw Number of pole 2 Frequency of the motor input voltage Hz Number of tator lot 36 Rotor outie iameter 58 mm Stator inner iameter 6 mm Stator outie iameter 24 mm Magnet thickne 6 mm With of magnet mm Magnet pole-arc-to-pole-pitch ratio, Coercive magnetizing intenity 92 ka/m Remanent flux enity.6 T where B n an B t, are the normal an tangential component of the magnetic flux enity, n an t are the normal an tangential unit vector to the urface of integration. Taking into account that in rotating electric machine only the tangential component of the Maxwell tree generate torque, we have: F t = B nbt (2) μo For two-imenional plane moel the integration urface i tranforme into a cloe contour of raiu r (air gap region centre). Calculating the tangential force in each point of the roun path, by mean of the following expreion: Bn Bt Ft = l (3) μ where i the length of the path between two conecutive noe an l i the axial length of the magnetic heet core. o 5

3 A the magnetic flux enity i calculate at icrete point in the air gap region, the torque can be obtaine by mean of the following expreion: T = r Bn Bt l (4) μ o where r i the raiu of the circular path taken. Torque calculation by mean of (4) epen trongly on the path an meh tructure an a conierable care i require with meh icretization in the air gap in orer to achieve high accuracy. To minimize error integration contour wa etablihe in the mile of air gap with a four-layer meh (ee Fig. 2). IV. TORQUE Variation in cogging torque, average torque an torque ripple for two motor configuration inicate above are analyze in thi ection. A. Cogging torque The combination of pole an lot number of a motor have influence on frequency an peak value of cogging torque waveform. Cogging torque i the ocillatory torque conequence of interaction between rotor-mounte permanent magnet fiel an the tator teeth, which prouce reluctance variation epening on rotor poition. Cogging torque exit even when there i no tator current. The mallet common multiple (LCM) of tator lot number (K) an pole number (2p) repreent number of cogging torque cycle per mechanical revolution (n r ), thu cycle number per lot pitch (n p ) i the ratio between LCM an K: ( K 2 p) LCM, n p = (5) K If iviion between tator lot number an pole number i an integer, only one cogging torque cycle per lot appear an o, cycle number per mechanical revolution i the ame than lot number. n p houl be a high a poible ince a higher perio number of cogging torque reult in a maller peak value. Generally, a high n p value alo ha a mitigating effect on the current-epenent ripple torque. Another inex for checking the level of cogging torque i the greatet common ivior (GCD) of K an 2p. Structural perioicity exit aroun the air gap every 36/(GCD) mechanical egree. The maller GCD(K,2p) i, maller will be cogging torque. Table II how number of lot (K), number of pole (2p), number of lot per pole an phae (q), wining factor (k w ), lot pitch (p ), pole pitch (p p ), leat common multiple between the pole number an the tator lot number LCM(K,2p), number of perio of the cogging torque waveform per lot pitch (n p ) an greatet common ivior between lot number an pole number GCD(K,2p) for two ifferent PMSM configuration analyze in thi paper. TABLE II PARAMETERS OF ANALYZED MOTORS K 2p p q k w p p LCM n p GCD 36 2 º 3º º º To calculate cogging torque, behaviour of the motor for ifferent rotor poition (when there i no current in the wining) i imulate; at each rotor poition the mehing i rene. Figure 3 how the cogging torque a a function of mechanical angle for the two ifferent motor configuration. Cogging torque (Nm) Cogging torque (Nm) ,3,2, -,2 -,3 -,4 a) 2 pole motor 2 pole -, pole b) pole motor Fig. 3. Cogging torque in a lot pitch Table III how peak-to-peak value of the cogging torque (T cpp ), in two analyze motor. TABLE III PEAK-TO-PEAK VALUE OF COGGING TORQUE Motor T cpp (Nm) 2 pole 46 pole.53 From the reult, it can be een that: a) the two motor configuration have the ame lot pitch ( mechanical egree), but iffer in cogging-torque waveform perio number (five in pole motor an one in 2 pole motor), an b) the peak-to-peak value of the cogging torque i reuce by 98.8 % for pole motor compare to the peak-to-peak value obtaine with the 2 pole motor. B. Torque ripple Electromagnetic torque evelope by the motor i calculate imulating loa behaviour of the motor, for ifferent poition of the rotor, when armature flux linkage an excitation are perpenicular (I =, I q = I). 5

4 Figure 4 how the torque ripple a a function of mechanical angle for the two analyze motor. It i clear that the pulating torque perio i 2 mechanical egree in pole motor, an egree in 2 pole motor (both cae 3 cycle per pole pitch), an the torque ripple i maller in fractional wining motor than in integer wining motor. Torque ripple (Nm) pole 2 pole Fig. 4. Torque ripple Torque ripple factor (t r ) can been efine a following [6]: t r T = max T T av min where T max,, T min an T av are repectively, maximum, minimum an average value of the torque. Cogging torque factor (t c ) can be efine a follow: t r (6) Tcpp = (7) T av Table IV how maximum, minimum, average value of the torque (T max, T min an T av, repectively), torque ripple factor (t r ), peak-to-peak value of the cogging torque (T cpp ) an cogging torque factor (t c ) for the two analyze motor configuration. TABLE IV TORQUE RIPPLE FACTOR AND COGGING TORQUE FACTOR T max T min Motor (Nm) (Nm) (Nm) t r (%) (Nm) t c (%) 2 pole pole T av We can oberve that pole motor average evelope torque (293 Nm) i 4.9% lower than 2 pole motor average torque (38 Nm), an magnitue of the torque ripple factor i reuce by 72 % for pole motor compare to 2 pole motor. Bae on the reult hown above, the pole motor with a fractional-lot wining an a high number of cogging torque cycle per mechanical revolution, have a lower torque ripple that the 2 pole motor. However pole motor ha a low GDC which point out raial magnetic-force unbalance, vibration an magnetic noie [3]. Depening on the pecific application an operating requirement of the motor, thi effect houl be tuie. T cpp V. DIRECT AND QUADRATURE REACTANCES Electrical machine are uually repreente, in electrical engineering, by mean of their equivalent circuit. The knowlege of the ifferent parameter of the equivalent circuit allow u to know the behaviour of the machine, by applying the law of electrical circuit. The circuit equivalent parameter of a PMSM are the reitance an the - an q-axi reactance. Etimation of the machine phae reitance at room temperature i relatively eay an there i no nee to etimate it by FEM. For PMSM with urface mounte magnet, the - an q- axi reactance are approximately equal (the moern permanent magnet have relative permeability cloe to unity, therefore the effective air gap een from the tator i nearly inepenent of the poition). For alient pole permanent magnet motor X q i larger than X ince the flux path pae through a magnet (which have a relative permeability cloe to unity) in the -axi, but through magnetic iron in the q-axi. The -an q-axi inuctance, L an L q, repectively, can be calculate a follow [6]-[8]: L Ψ Ψ m = ; I Ψ q L q = (8) Iq where Ψ an Ψ q are the total fluxe in the -axi an q-axi, I an I q are the -axi an q-axi armature current, an Ψ m i the flux-linkage ue to the permanent magnet. The whole proceure to calculate the -axi an q-axi reactance may be ecribe in term of the following equence of tep: (a) to calculate the -axi an q-axi inuctance, L an L q, repectively, by imulating the magnetic performance of the motor without coniering the 3D effect; (b) to calculate the en-wining leakage inuctance; (c) to etermine the -axi an q-axi inuctance, L an L q, repectively, by aing the inuctance calculate in the tep (a) an (b). By uing the finite element metho an by mean of two behaviour analyi of the motor, the -axi inuctance can be calculate. In a firt analyi the no-loa behaviour of the motor i imulate. In a econ analyi the loa behaviour of the motor when the armature flux an the excitation flux are in the ame irection (I = I; I q = ) i imulate. The q-axi inuctance can be calculate, by imulating the loa behaviour of the motor when the armature flux i perpenicular to excitation flux (I = ; I q = I) The en-wining inuctance can be calculate from [3], a follow: L n (. 67l 43y τ ) = 8π p q Z. (9) where Z n i the number of conuctor per lot, l i the average length of en-wining, y c i the coil pitch in lot pitche an τ i the lot pith. c 52

5 The reultant - an q-axi inuctance can be calculate by mean of the following expreion: L L L = L + L ; q = q + L () Finally, the - an q-axi reactance can be calculate a follow: X = 2π fl ; X = 2π fl () where f i the upply frequency. Table V how the reactance value obtaine for two analyze motor. q TABLE V REACTANCES Motor X (Ω) X q (Ω) 2 pole pole VI. MAGNETIC VECTOR POTENTIAL AND MAGNETIC FLUX DENSITY Figure 5 to 7 how the air-gap magnetic vector potential waveform obtaine, once the no-loa an loa analyi of the motor have been carrie out. In thi figure, the horizontal axi repreent the poition of an oberver inie the air gap of the machine. The origin of coorinate correpon, in both motor, to the beginning of a polar pitch. The peak value of the magnetic vector potential on no-loa by the pole motor i 25.9 % higher compare to magnetic vector potential on noloa by 2 pole motor (ee Fig. 5). The ue of fractional number of lot per pole increae the funamental orer (eformation of the magnetic-vector-potential waveform in the air gap of the machine), ince the tator lot are locate at ifferent relative circumferential poition with repect to the ege of the magnet (ee Fig. 6 an 7). Potential (Wb/m) Potential (Wb/m) Fig. 5. Magnetic vector potential in no-loa text q No loa, pole No loa, 2 pole No loa, 2 pole Loa, 2 pole Fig. 6. Magnetic vector potential in 2 pole motor Potential (Wb/m) No loa, pole Loa, pole Fig. 7. Magnetic vector potential in pole motor Figure 8 to how the normal component of the magnetic flux enity in the centre of air gap veru rotor poition for the two motor configuration invetigate. We can ee that magnetic flux enity in no-loa conition by pole motor i 4.5 % higher compare to 2 pole motor (ee Fig. 8). Magnetic flux enity (T) Magnetic flux enity (T) No loa, pole No loa, 2 pole Fig. 8. Component normal of flux enity (no-loa tet) No loa, 2 pole Loa, 2 pole Fig. 9. Component normal of flux enity in 2 pole motor Magnetic flux enity (T) No loa, pole Loa, pole Fig.. Component normal of flux enity in pole motor 53

6 VII. CONCLUSIONS The effect of the pole number on torque ripple (incluing the cogging torque) ha been invetigate for two ifferent PMSM configuration having the ame envelop imenion an output requirement. The motor with fractional-lot wining have a lower cogging torque an torque ripple but how a reuction of the average torque. The perio number by pole pitch of the torque ripple waveform i the ame in both motor. The pole motor ha a magnetic flux enity in no-loa conition higher that 2 pole motor however, the average torque i lower. The magnetic-flux-enity normal component an magnetic vector potential itribution along the air gap by two PMSM configuration have been invetigate. Auming a pure inuoial phae current itribution, the ue of fractional number of lot per pole, increae harmonic content of the magnetic vector potential an normal component of the magnetic flux enity waveform. REFERENCES [] L. Doiek, an P. Pillay, Cogging torque reuction in permanent magnet machine, IEEE Tran. 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