I Current waveform required for constant torque

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1 AN INVESTIGATION INTO THE TORQUE BEHAVIOR OF A BRUSHLESS DC MOTOR DRIVE. illy Deprtment of Electricl nd Electronic Engineering University of Newcstle upon Tyne NE 7RU, Englnd R.Krishnn Electricl Engineering Deprtment Virgini olytechnic Institute 6 Stte University, Blcksburg, VA, 206, USA ABSTRACT This pper uses previously developed model for the brushless dc motor (BDCM) to investigte its torque behvior. When the input currents nd motor flux linkges re perfect, no torque pulstions re produced in this motor. However imperfections in the current rise due to finite commuttion times while imperfections in the flux linkge cn rise due to the phse spred, finite slot numbers nd mnufcturing tolernces. Using n hrmonic nlysis, the effects of these imperfections on the production of torque in BDCM re investigted. It is shown tht torque pulstions nd reduction in the verge vlue of torque is produced, both of which cn ffect the performnce of torque, speed nd position servos..introduction AC servo drives re commnding lrger shre of the servo mrket ech yer. The dvntges of c servo drives over dc include incresed robustness, reduced mintennce nd higher torque nd speed bndwidths. The c motors tht re used include the induction, permnent mgnet synchronous nd permnent mgnet brushless dc mchines [I]. Some of the dvntges of permnent mgnet mchines over induction [2] include higher torque to inerti rtios nd power densities, lower rted rectifier nd inverter rtings during constnt torque opertion nd higher efficiencies. Hence permnent mgnet mchines my be preferble for pplictions where weight or efficiency is of importnce, for exmple in the erospce industry or electric vehicles. The permnent mgnet synchronous motor (MSM) nd the brushless dc motor (BDCM) hve mny similrities. They both hve permnent mgnets on the rotor nd require lternting sttor currents to produce constnt torque. The difference [3,] between them is tht the MSM hs sinusoidl bck emf while the BDCM hs trpezoidl bck emf. This leds to different operting nd control requirement for these two mchines s explined in []. The dynmic behvior of BDCM hs been studied [SI nd the results indicte tht the BDCM cn be subject to severe torque pulstions [6] due to the sttor currents commutting from one phse to nother. Torque pulstions re lso creted by the mgnet flux linkge deviting from the idel. The bove imperfections lso crete the possibility of reduction in the verge vlue of torque. These phenomen cn ffect the performnce of torque, speed or position servos. The object of this pper is to quntittively determine the motor chrcteristics tht ffect the production of torque in BDCM. Attention is pid to the overll torque pulstions s well s individul torque hrmonics. The torque behvior during flux wekening opertion is lso ddressed. The nonsinusoidl currents nd flux linkges re represented by Fourier series nd the impct of different flux nd current hrmonics on the motor torque re investigted. The pper is orgnized s follows: Section I presents the mthemticl model of BDCM. Section discusses torque production in BDCM nd shows how the dynmic mthemticl model in section I cn be used to study the stedy stte torque behvior of BDCM. Section IV discusses the mchine prmeters tht ffect the production of torque in BDCM. Finlly section V nd VI hve the results nd conclusions of this investigtion.. MATHEMATICAL MODEL OF THE BRUSHLESS DC MOTOR The BDCM hs three sttor windings nd permnent mgnet on thr rotor. Since both the mgnet nd the stinless steel retining sleeves hve high resistivity, rotor induced currents cn be neglected nd no dmper windings re modelled. Hence the circuit equtions of the three windings in phse vribles re where it hs been ssumed tht the sttor resistnce of ll the windings re equl. The bck emfs e, eb nd ec hve trpezoidl shpes s shown in figure. Assuming further tht there is no chnge in the rotor reluctnce with ngle, then Hence L - $ - Lc - L Lb - = Lcb - &I Bck emf of the brushless DC motor U I Current wveform required for constnt torque Figure. 88CH2565-0/88/0$0.oo o 988 IEEE Bck emf nd currents of BDCM

2 But Therefore Hence O O R i + ib + ic = 0 Mib + Mic = -Mi (3) () Hence in stte spce form the equtions re rrnged s follows: nd the electromgnetic torque is, The eqution of motion is Fourier series s well. flux linkge is given by The Fourier series of the X (x) = (sinfsinx + (sin3fsin3~)/3~ + (:in5fsin5~)/5~ +...)/nf () While tht of rectngulr current is i(x) = (coshsinx + (cos3hsin3x)/3 + (cos5hsin5x)/5+...)/ (2) where F nd H re defined in figure. It is known [6] tht current nd flux linkge hrmonics of the sme order interct to produce constnt torque while if they re of different orders they produce pulsting torques. However it hs been shown in (0) tht the output torque is constnt for the wveforms in figure, hence there re no pulsting torques. The stedy torque is given by the interction of the fundmentl of the flux linkge with the fundmentl component of current plus the 5th hrmonic of the flux with the 5th hrmonic of current etc. Tht is ll odd hrmonics of flux which interct with current hrmonics of the sme order (exept triplen) produce constnt torque. The contribution of the fundmentl component of flux linkge with the fundmentl component of current gives (fter dding ll 3 phses) 6 X (sin(f)sin(wt)cos(h)sin(wt) + Tels;n(F)!iR(wt - 2n/3)cos (H) sin(wt - 2n/3) + sin( F) sin(wt+2%/3) cos (H) sin(wt+2/3) )/n2f (3) The bove equtions cn be used to exmine the detiled behvior of the BDCM. III.TORQUE RODUCTION IN A BDCM The bck emf nd the required currents in order to produce constnt torque re shown in figure in n idel mchine. In (7) it ws shown tht the torque is given by the product of the bck emf nd sttor current wveform divided by the speed. The bck emf divided by the speed is constnt nd represents the flux linkge which hs the sme wveform s the bck emf in figure in n idel mchine. The flux linkge is horizontl (constnt) for 20' nd for constnt torque it is necessry to supply rectngulr shped current to the phse during this period. When the flux linkge is negtive, negtive current is needed in order to produce constnt positive torque. In ddition, t ny given instnt, only two phses conduct current with the phse crrying the positive current using the phse crrying the negtive current s return pth. In figure, consider n instnt when e = E i = I' eb = -E ib - -; ic - 0 Then the electric torque becomes At ny other instnt, it will lwys be found from figure tht only two phses conduct, with the third being zero nd then (0) holds. The torque predicted from this idelized mchine is therefore constnt with no torque pulstions. Any periodic wve cn be expressed s Fourier series. Hence both the flux linkge nd current wveforms of the BDCM in figure cn be expressed s Now H = F = n/6 in figure, therefore Tel= 96(sin2x + sin2(x-2n/3) + sin2(x+2r/3)i X )/n3 -.3 (3/2)IpXp = 2.0pXp 87 Using the technique bove it cn be shown tht the interction of the 5th hrmonic of flux linkge with the 5th hrmonic of current gives stedy torque with mgnitude of X I while the interction of the 7th flux linkge nd current hrmonics give positive constnt torque of Xpfp. Hence the contribution of the st nd 5th hrmonics contribute = X I It is therefore cler tht the contribution 'of' the higher order hrmonics to the stedy torque is negligible. The contribution of the fundmentl components of current nd flux linkge is essentilly responsible for the stedy torque of the mchine. The interction of flux linkges nd currents of different orders produce pulsting torques (6,7]. However it ws shown in (0) tht when the 20' trpezoidl flux density wveform intercts with the rectngulr current tht only stedy torque is produced with no torque pulstions. Therefore it cn be deduced tht the pulsting torques produced by the interction of current nd flux linkge hrmonics of different orders must ll cncel to produce zero net pulsting torque for the wveforms shown in figure. The BDCM cn therefore be regrded s generliztion of the MSM or lterntely, the MSM cn be regrded s specil cse of the BDCM where only the fundmentl components of flux nd current re present. Hence if only the stedy torque of the BDCM is under study, the possibility exists of using just the fundmentl component of flux nd current in the nlysis. A trnsformtion cn then be mde to d,q vribles s is done for the MSM. Gret cre should be tken however whenever this simplified pproch is used. Up to now BDCM with the idelized wveforms in figure hs been considered. In prctice, devitions from the idelized current nd flux linkge wveforms shown in figure occur. Some of the devitions from the idelized mchine re discussed in the next section. 202

3 IV.DEVIATION OF THE RACTICAL MACHINE FROM THE IDEALIZED In prcticl mchine it is impossible to force rectngulr current to flow into the mchine windings. This is becuse the motor inductnce limits the rte of chnge of current [5]. In the stedy stte, the rise time of the current depends on the voltge differentil between the dc bus nd the bck emf, nd the time constnt of the sttor winding which is given by the rtio of the sttor lekge inductnce to resistnce. The higher this rtio, the longer is the rise time of the current nd the greter the devition from the idelized vlue. For n ccurte clcultion, (6), (7) nd (8) must be solved in detil. Although the rise is given by (-exp-(l-m)t/r) in pu t constnt speed, it cn be pproximted [7] by stright line so tht the ctul current resembles trpezoid s shown in figure 2. This voids the detiled clcultion of (6), (7) nd (8). Note tht high frequency switching with either hysteresis or WM logic is used to trck the rectngulr references. This is not shown in figure 2 since the effect of this on the torque hs lredy been exmined nd shown to be of secondry importnce [5] to tht produced by the commuttion of current. The torque behvior s result of using trpezoidl currents insted of rectngulr cn be studied with the id of the Fourier series of the trpezoidl current given below: i (x) = ((sin H - sin h)sin x + (sin3h - sin3h) s?n3~/3~+...)/n(h-h) (5) The second devition from the idelized is the ngle for which the flux density remins constnt. In figure it is ssumed to be of 20'. This is desirble for three phse mchine. In prctice this ngle my rnge from looo to 50' depending on the phse spred, the effects of the number of slots per phse nd mnufcturing tolernces. Torque pulstions result from the motor currents or flux linkge deviting from the idel. The mgnitude of individul torque hrmonics cn be clculted from equtions (ll), (2) nd (5) which represent the flux linkge, idelized rectngulr current nd nonidel trpezoidl current respectively. In (ll), F = 30' for the idelized 20' flux density for three phse mchine. Nonidel flux density wveforms produced when the flux density is less or greter thn 20' cn be represented in () by incresing or decresing F respectively. Similrly nonidel currents cn be represented in (5) by vrying h reltive to H. In the BDCM, the 6th hrmonic of torque is dominnt [6]. This cn be produced by vriety of current-flux interctions. For exmple the fundmentl of the flux cn interct with the 5th nd 7th current hrmonics to produce 6th hrmonic torque pulstion nd vice-vers. () gives the relevnt eqution for ny flux hrmonic while (5) gives the relevnt eqution for ny nonidel current hrmonic. (3) shows how multipliction of the fundmentl flux with the fundmentl current gives stedy torque fter dding the effects of the three phses. In similr mnner, the multipliction of flux hrmonic of one order (except triplen) with current hrmonic of different order (except triplen) so tht the difference in the order is 6, results in Te6=2sin(N2.F)(sin(N.H)- sin(nl.h))i X ~os(6wt)/(n2~nl*fx*(h-h) (6) where N2 is the order of the flux linkge hrmonic, N is the order of the current hrmonic, I is the pek of the current, X is the pek of the'flux linkge wveform, F is dgfined in figure, H nd h re defined in figure 2. If N2 nd N re chosen such tht they dd to 6 then the negtive of (6) should be used in the clcultion of the 6th torque hrmonic. Since () nd (5) hve been used in the clcultion of (6), it turns out tht (6) is lso vlid for the 2th torque hrmonic by using the pproprite N2, N nd sign of (6) nd replcing 6wt by 2wt etc. In this investigtion, the effects of the commuttion of the sttor current s well s the effects of different flux density distributions on the torque of BDCM re investigted. Attention is pid to the overll torque pulstions in ddition to individul torque hrmonics. The effects of mgnitude of the current in ddition to phse dvncing re lso exmined. Rectngulr Current V. RESULTS In order to test the vlidity of using the Fourier Series pproch to study the stedy stte behvior of BDCM, progrm ws written to determine the output torque given tht the motor current nd flux wveforms re idelized s shown in figure. The phse A current nd flux linkge wveforms re shown in figure 3, the wveforms of the other phses being similr nd phse shifted from tht of phse A by 20'. The output torque, which is essentilly constnt, is lso given in figure 3. 2 current nd flux linkge hrmonics re used in the simultion. The slight ripple is due to the trunction in the Fourier Series tht is necessry in ny prcticl implementtion. However, the ripple is smll enough s to hve negligible engineering significnce. These results indicte the suitbility of using Fourier series to exmine the torque behvior in BDCM. Figure 2. Trpezoidl shped current Trpezoidl Current Keeping the flux density wveform the sme, the slope of the current wveform ws vried from Oo to 5' to simulte different sttor time constnts nd operting speeds. Figure shows the flux linkge, current nd torque for 5' slope in current nd 20' flux linkge wveform. 2 flux nd current hrmonics re used. Becuse of the nonrectngulr current, torque pulstions re produced during the commuttion of the current, the fundmentl frequency of which is 6 times tht of the current. The slope of the current wveform ws vried nd the corresponding torque pulstions determined. From these results, grph of torque pulstions vs commuttion ngle ws drwn s shown in figure 5 for different current mgnitudes. The mgnitude of the torque pulstions increses 203

4 N N TORQUE Figure 3. Rectngulr sh@ Eurrml& for 2 h "i Figure. Trpezoidl shped current results for 2 current hrmonics rpidly initilly up to commuttion ngle of 5' with the rte of increse being lower fter 5'. Although the mgnitude of the torque pulstions level off with n increse in the commuttion ngle, the width of these pulstions increses continully thus reducing the verge vlue. The incresed torque pulstions nd reduction in the verge torque cn ffect the performnce of torque, speed nd position servos. The increse in the mgnitude of the torque pulstion with current for given commuttion ngle is liner. Tht is doubling the mgnitude of the current being commutted doubles the mgnitude of the resulting torque pulstion s well. A grph of the verge torque vs the commuttion ngle is given in figure 6. The verge torque cn be s low s.75x I for commuttion ngle of 5' which is fir reduction from the expected 2.0XpIp. This reduction cn hve consequences for torque servo performnce since the commnded torque will not be met. This torque reduction is independent of mchine prmeter chnges due to temperture or sturtion which cn cuse further reductions. The verge vlue ws obtined by verging the instntneous torque over complete cycle. If complete cycle were not used it is possible for the verge torque to be either less thn or greter thn the complete cycle vlue depending on the section of the torque profile used to clculte the verge vlue. This cn hve H 3 X c 'rl : 2 3 c J b.l G- r2 5O Comuttion loo ngle W-h) 5' Figure 5. Torque pulstions vs commuttion ngle 20

5 consequences in position servo performnce where it i's possible tht the rotor is commnded to stop before completing full revolution. The results presented thus fr show the effects of chnging commuttion times on the overll mgnitude of the torque pulstions nd the consequent reduction in the verge torque. The flux wveform ws kept constnt for 20 s shown in figure. In the next section, the effects of the different commuttion times on individul torque hrmonics re exmined. x = pu I = pu Torque Hrmonics The electric torque Te is given by the product of the flux in () nd the rectngulr current in (2) or the trpezoidl current in (5). It must lso be remembered to include the current-flux interction of the other two phses. The bove equtions cn lso be used to clculte the mgnitude of individul torque hrmonics s well by considering the pproprite current nd flux hrmonics. From (ll), (2) nd (5) it is cler tht there is lrge reduction in both the current nd flux hrmonics s the order increses so tht the higher order hrmonics hve less of n effect on the torque profile. From the wveforms presented erlier it is cler tht the torque pulstions hve fundmentl of 6 times the fundmentl frequency of the current. This mens tht the 6th torque hrmonic is predominnt. The 6th torque hrmonic is given by the interction of the st hrmonic of flux with the 5th nd 7th current hrmonics, the st hrmonic of current with the 5th nd 7th flux hrmonics etc. These results re summrized in figure 7 for different commuttion ngles. The results were obtined by keeping F constnt in (6) nd vrying (H-h) nd choosing N2 nd N ppropritely. Firstly it should be noted tht the 6th torque hrmonic is produced by other flux current interctions thn those listed in figure 7. However since the mgnitude of the flux nd current hrmonics reduce either s function of their order or the squre of the order, the contribution of the higher order hrmonics to the 6th torque hrmonic re insignificnt. The results for the slope of 0' corresponds to the results for the rectngulr current, for which it ws previously shown tht should be no torque hrmonics except for the constnt torque. From Figure 7 it is cler tht the st hrmonic of flux intercts with the 5th hrmonic of current to produce the lrgest contribution to the 6th torque hrmonic thn ny other flux-current interctions. The other fluxcurrent interctions shown go towrds neutrlizing the mgnitude of the 6th torque hrmonic produced by the st flux nd 5th current hrmonics such tht the net resultnt fter dding the contributions of the four lowest current nd flux hrmonics is lmost zero. The st flux nd 5th current hrmonics re lso the lrgest contributors to the 6th torque hrmonic when the current is trpezoidl rther thn rectngulr s shown in figure 7 for 5O, 0' nd 5' current commuttion slopes. However in these cses the other flux current interctions shown re unble to completely neutrlize the torque hrmonic produced by the st flux nd 5th current hrmonics. Insted, there is residul which finlly shows up s the torque pulstions presented in figure. In ddition, the lrger the slope of the current wveform, the lrger is the residul 6th torque hrmonic. Flux current 0" 5" IO" 5" Totl Figure 7. I I So oo so Commuttion ngle (H-h) Figure 6. Averge torqw vs commuttion ngle O.IOOO th Hrmonic torque pulstions Flux I I 3 current 3 0" " IO" ' O.OIOS Totl +O.o00-0, Similr results re shown for the 2th torque hrmonic in figure 8. Here lso the 2th torque hrmonic ws clculted from (6) by fixing F t 30' nd vrying (H-h). N2 nd N re chosen so s their sum or difference gives 2. From the results for the rectngulr current (0') it is gin evident tht the 205 Figure 8. 2th Hrmonic torque pulstions

6 lrgest contributor to the 2th torque hrmonic is the first flux nd llth current hrmonics. Just s for the 6th hrmonic, the torque hrmonics produced by the other flux-current interctions go towrds nullifying tht produced by the first flux nd llth current hrmonics. From figure 8 it is lso cler tht s the current commuttion ngle increses, the residul torque hrmonic increses up to commuttion ngle of 0'. For lrger commuttion ngles the 2th torque hrmonic decreses. This probbly explins why the mgnitude of the torque pulstion which is n instntneous sum of ll the torque hrmonics tends to level off s the commuttion ngle is incresed beyond loo s shown in figure 5. The timing shown in Figure is used up to the rted speed of the mchine. High speed opertion is obtinble by phse dvncing of the current reltive to the bck emf. In this study it is ssumed tht there is sufficient bus voltge vilble to force the currents to be s close to the desired rectngulr shpe s when operted below the mximum speed. This is done so tht the effects of phse dvncing lone on the torque profile cn be determined. Figure 9 shows the results when trpezoidl current with commuttion slope of 5' is phse dvnced by 20, 0, 60' nd 80'. These curves should be compred with the curve for torque presented in figure. As the phse is dvnced, the torque pulstions increse t the expense of the durtion for which the torque remins constnt. This is n extremely undesirble feture of phse dvncing. In ddition the verge vlue of torque is reduced gretly s shown in figure 0. It should be remembered tht the verge vlue is pproximtely.97x I for commuttion slope of 5O nd when there is n$ 'phse dvnce. The mgnitude nd shpe of the current re kept the sme during the phse dvncing nd the drmtic reduction in the verge torque is due only to the phse dvnce. The effects of phse dvncing on individul torque hrmonics ws lso exmined. Since the flux linkge nd current wveforms were mintined the sme s during zero phse dvnce, it hs been clculted tht the individul torque hrmonics re exctly the sme mgnitude s with zero phse dvnce for zero, 5O, loo nd 5' commuttion ngles. In other words even when rectngulr current is phse dvnced, the individul torque hrmonics re exctly the sme mgnitude s when there is no phse dvnce. In the zero phse dvnce cse ll the 6th torque hrmonics for exmple re either in phse or 80 out of phse nd sum to zero s shown in figure 7. However when the current is phse dvnced reltive to the bck emf, the individul torque hrmonics of given order re not ll in phse (even though they hve the sme mgnitude) such tht the cncelltion shown in Tble for exmple does not occur. This results in n increse in the overll torque pulstions s shown in figure 9 for the 5' trpezoidl current. Similr results occur for the other commuttion ngles. Nonidel Flux Linkge Effects Up to now the effects of the current commuttion times on the profile hs been exmined. Here the effects of the flux density wveform on the torque re exmined. Torque pulstions re produced by the flux density wveform being less thn the desired 20'. Note tht if the constnt portion is greter thn 20, but 20' currents re still used, then the results will be the sme s if the flux were constnt for 20' only provided the timing in figure is used. However when the flux density wveforms re constnt for less thn 20, torque pulstions re produced. The mgnitude of individul torque hrmonics cn be clculted from (6) by fixing (H-h) nd vrying F. The entire torque profile cn be obtined by choosing xs 0 Z 0 E? 7; q (DEGREES) 0 Figure 9. hse dvncing results H X 2. B 2 W 5. 0 M m LlB 2 Figure 0. I = lpu x = lpu no I I I I 20' 0' 60' EO0 hse dvnce ngle hse dvncing results 206

7 certin number of hrmonics nd evluting () for given F. Similrly, (5) is evluted for given (H-h) nd multipliction of these instntneous flux nd current wveforms gives the instntneous torque profile. The torque pulstion produced when the flux density is constnt for looo nd lloo insted of 20' re shown in figure. In this study, rectngulr currents were used so s to determine the contribution of the flux density lone on the torque pulstions. From these results, grph of torque pulstion s function of F cn be drwn s shown in figure 2. The grph is liner indicting liner increse in torque pulstion with decrese in the durtion of the constnt flux density. In prctice, the combintion of the effects of the flux density nd trpezoidl shped currents would increse the torque pulstion beyond the contribution of ech. As the torque pulstions increse, so the verge vlues decrese. However the reduction in the verge torque is miniml when compred to tht produced by the commuttion of current s presented erlier nd grph is therefore not drwn. H X VI g 0.6 U,-I n 3 0., 0.2 L 0- J $ 0.0 I = lpu h = lpu 30' 35O 0 F VI. CONCLUSIONS A detiled investigtion into the torque behvior of BDCM drive hs been done in this pper. A previously published model to study the dynmic behvior ws used to nlyze the stedy stte behvior s well. It ws shown tht constnt output torque is produced only if the flux density nd current wveforms of the BDCM re idelized. In the prcticl nd hence nonidel cse, torque pulstions rise s result of the ctul current being trpezoidl insted of rectngulr, from the H "c I = lpu i = pu. E F=35 " I F=0 -t Figure I I. Torque pulstions for nonidel flux linkges ~j~~~~ 2. Torque pulstions s function of F flux link e being constnt for less thn 20 insted of the 208, or phse dvncing of the current. The lrger the commuttion ngle, the lrger is the mgnitude of the torque pulstions nd the lower is the verge vlue of the torque over cycle. This is n extreme disdvntge of lrge commuttion times. The commuttion time is determined by the rtio of the sttor lekge inductnce to resistnce, the operting speed nd the dc bus voltge. Hence for.given resistnce, the lekge inductnce should be minimized or for given lekge inductnce, the sttor resistnce should be mximized. Incresing the sttor resistnce should be done with due regrd to the efficiency nd cooling of the mchine. The increse in the torque pulstions with increse in commuttion ngle is nonliner with the increse being much lrger up to 5' nd then reducing fter 5'. The increse in torque pulstions is liner with increse in the mgnitude of the current being commutted. These torque pulstions my ffect the ccurcy nd repetbility of position servos. As the torque pulstions increse, so the verge vlue of the torque decreses. A 2.5% decrese is possible over full cycle nd this cn hve consequences in the performnce of torque, speed or position servos when using this mchine. hse dvncing of the current wveform reltive to the flux density cn produce lrge torque pulstions with resultnt reduction in the verge vlue of torque s well. Devition of the flux density wveforms from the idel lso produce torque pulstions lthough the mgnitude is not s lrge. In ddition, the reduction in the verge torque is not s severe s tht due to the commuttion in current. If the flux density wve is constnt for durtion longer thn 20, then the output torque is not ffected. REFERENCES ( R. Krishnn, "Selection criteri for servo motor drives", IEEE Trns., vol. IA-23, No. 2, Mrch/April 987, pp I I I I I [2] D. uly, G. fff nd A.Wescht, "Brushless servo % drives with permnent mgnet motors or squirrel cge QNGLE (DEGREES) m0 induction motors - comprison," IEEE IAS Annul Meeting, 98, pp (3 G, fff, A. Wescht nd A. Wick, "Design nd 207 experimentl results of brushless c servo-drive", IEEE IAS Annul Meeting, 982, pp

8 []. illy nd R. Krishnn, "Appliction chrcteristics of permnent mgnet sychronous nd brushless dc motors for servo drives", IEEE IAS Annul Meeting, 987, pp [5]. illy nd R. Krishnn, "Modeling, simultion nd nlysis of permnent mgnet brushless dc motor drive", IEEE IAS Annul Meeting, 987, pp. 7-. [6] T.M. Jhns, "Torque production in permnent mgnet motor drives with rectngulr current excittion," IEEE Trns., vol. IA-20, No., July/August 98, pp [7] J.M.D. Murphy, "Thyristor control of c motors", (book), ergmon ress, 973. [8] V. Subrhmnym nl D. Subbryudu, "Stedy stte nlysis of n induction motor fed from current source inverter using complex-stte (rk's) Vector", roc. IEE, vol. 26, No. 5, My 979, pp [9] H.R. Bolton, Y.D. Liu nd N.M. Mllison, "Investigtion into clss of brushless dc motor with qusisqure voltges nd currents", roc. IEE, vol. 33, t B, No. 2, Mrch 986, pp [lo] E.K.ersson, "Brushless dc motors - review of the stte of the rt" roceedings of the Motorcon Conference, 98, pp. -6. ( T. Sebstin nd G.R. Slemon, "Operting limits of inverter driven permnent mgnet motor drives," IEEE IAS Annul Meeting, 986, pp List of Symbols B dmping constnt, N/rd/s e,eb,ec,b nd c phse bck emfs, V E pek vlue of bck emf, V i:,ib,ic,b nd c phse current J moment of inerti, kg-m hp A Kt torque constnt - 2e/wr L,$,Lc self inductnce of,b 6 c phses, H Lb mutul inductnce betwen phses & b, derivtive opertor number of pole pirs R sttor resistnce, ohms 'e electric torque, N-m Te,Te5,Te7 lst,sth nd 7th torque hrmonics,n-m TL lod torque, N-m v,vb,vc,b nd c phse voltges, V 'dc dc bus voltge, V rotor speed, rd/sec WS synchronous speed, rd/sec 208