An Investigation Into the Torque Behavior of A

8/14/2019 An Investigation Into the Torque Behavior of A

1/8

AN INVESTIGATION INTO THE TORQUE BEHAVIOR OF ABRUSHLESS DC MOTOR DRIVE

P. illayDepartment of Electricaland Electronic EngineeringUniversity of Newcastle upon TyneNE1 7RU, England

ABSTRACTThis paper uses a previously developed model forthe brushless dc motor (BDCM) to investigate itstorque behavior. When the input currents and motorflux linkages are perfect, no torque pulsations areproduced in this motor. However imperfections in thecurrent arise due to finite commutation times whileimperfections in the flux linkage can arise due to thephase spread, finite slot numbers and manufacturingtolerances. Using an harmonic analysis, the effectsof these imperfections on the production of torque ina BDCM are investigated. It is shown that torquepulsations and a reduction in the average value oftorque is produced, both of which can affect theperformance of torque, speed and position servos.1.INTRODUCTION

AC servo drives are commanding a larger share ofthe servo market each year. The advantages of acservo drives over dc include increased robustness,reduced maintenance and higher torque and speedbandwidths. The ac motors that are used include theinduction, permanent magnet synchronous and permanentmagnet brushless dc machines [I]. Some of theadvantages of permanent magnet machines over induction[ 2 ] include higher torque to inertia ratios and powerdensities, lower rated rectifier and inverter ratingsduring constant torque operation and higherefficiencies. Hence permanent magnet machines may bepreferable for applications where weight or efficiencyis of importance, for example in the aerospaceindustry or electric vehicles.The permanent magnet synchronous motor (PMSM)and the brushless dc motor (BDCM) have manysimilarities. They both have permanent magnets on therotor and require alternating stator currents toproduce constant torque. The difference [3,4] etweenthem is that the PMSM has a sinusoidal back emf whilethe BDCM has a trapezoidal back emf. This leads todifferent operating and control requirement for thesetwo machines as explained in [4]. The dynamicbehavior of a BDCM has been studied [SI and theresults indicate that the BDCM can be subject tosevere torque pulsations [ 6 ] due to the statorcurrents commutating from one phase to another.Torque pulsations are also created by the magnet fluxlinkage deviating from the ideal. The aboveimperfections also create the possibility of areduction in the average value of torque. Thesephenomena can affect the performance of torque, speedor position servos.The object of this paper is to quantitativelydetermine the motor characteristics that affect theproduction of torque in a BDCM. Attention is paid tothe overall torque pulsations as well as individualtorque harmonics. The torque behavior during fluxweakening operation is also addressed. Thenonsinusoidal currents and flux linkages arerepresented by Fourier series and the impact of

R.KrishnanElectrical Engineering DepartmentVirginia Polytechnic Institute6 State University, Blacksburg,VA, 24061, USA

different flux and current harmonics on the motortorque are investigated.The paper is organized as follows: Section I1presents the mathematical model of a BDCM. Section111 discusses torque production in a BDCM and showshow the dynamic mathematical model in section I1 canbe used to study the steady state torque behavior of aBDCM. Section IV discusses the machine parametersthat affect the production of torque in a BDCM.Finally section V and VI have the results andconclusions of this investigation.11. MATHEMATICAL MODEL OF THE BRUSHLESS DC MOTOR

The BDCM has three stator windings and apermanent magnet on thr rotor. Since both the magnetand the stainless steel retaining sleeves have highresistivity, rotor induced currents can be neglectedand no damper windings are modelled. Hence thecircuit equations of the three windings in phasevariables are

where it has been assumed that the stator resistanceof all the windings are equal. The back emfs ea, eband ec have trapezoidal shapes as shown in figure 1.Assuming further that there is no change in the rotorreluctance with angle, thenLa - $ - Lc - LLab - = Lcb -Hence

B a c k em f o f t h e b r u s h l e s s DC m o t o r&I UC u r r e n t w a v e fo r m r e q u i r e d f o r c on s t a n t t o r q u eFigure 1. Back em f and currents of a BDCM

88CH2565-0/88/01$01.oo o 1988 IEEE


2/8

O O RBut

ia + ib + ic = 0Therefore

Mib + Mic = -MiaHence

(3)

(4)

Hence in state space form the equations are arrangedas follows:

and the electromagnetic torque is,

The equation of motion is

The above equations can be used to examine thedetailed behavior of the BDCM.III.TORQUE PRODUCTION IN A BDCM

The back emf and the required currents in orderto produce constant torque are shown in figure 1in anideal machine. In (7) it was shown that the torque isgiven by the product of the back emf and statorcurrent waveform divided by the speed. The back emfdivided by the speed is a constant and represents theflux linkage which has the same waveform as the backemf in figure 1in an ideal machine. The flux linkageis horizontal (constant) for 120' and for constanttorque it is necessary to supply a rectangular shapedcurrent to the phase during this period. When theflux linkage is negative, a negative current isneeded in order to produce constant positive torque.In addition, at any given instant, only two phasesconduct current with the phase carrying the positivecurrent using the phase carrying the negative currentas a return path.In figure 1, consider an instant when

ea = Eia = Ieb = -Eib - -1;ic - 0Then the electric torque becomes

At any other instant, it will always be foundfrom figure 1 that only two phases conduct, with thethird being zero and then (10) holds. The torquepredicted from this idealized machine is thereforeconstant with no torque pulsations.Any periodic wave can be expressed as a Fourierseries. Hence both the flux linkage and currentwaveforms of the BDCM in figure 1can be expressed as

a Fourier series as well. The Fourier series of theflux linkage is given byX (x) = 4(sinFsinx + (sin3Fsin3~)/3~ +(:in5Fsin5~)/5~ +. . .)/nF (11)While that o f rectangular current isia(x) = 4(cosHsinx + (cos3Hsin3x)/3 +where F and H are defined in figure 1.

It is known [ 6 ] that current and flux linkageharmonics of the same order interact to produceconstant torque while if they are of different ordersthey produce pulsating torques. However it has beenshown in (10) that the output torque is constant forthe waveforms in figure 1, hence there are nopulsating torques. The steady torque is given by theinteraction of the fundamental of the flux linkagewith the fundamental component of current plus the 5thharmonic of the flux with the 5th harmonic of currentetc. That is all odd harmonics of flux which interactwith current harmonics of the same order (exepttriplen) produce a constant torque. The contributionof the fundamental component of flux linkage with thefundamental component of current gives (after addingall 3 phases)

161 X ( s in (F ) s in (w t )co s (H)s in (w t ) +Tels;n(F)!iR(wt - 2n/3)cos (H) sin(wt - 2n/3) +sin( F) sin(wt+2%/3) cos H) in(wt+2a/3) )/n2FNow H = F = n/6 in figure 1, thereforeTel= 96(sin2x + sin2(x-2n/3)

(cos5Hsin5x)/5+ . . . /a (12)

(13)

+ sin2(x+2r/3)I X )/n3- 1.341 (3/2)IpXp = 2.0111pXp 847Using the technique above it can be shown thatthe interaction of the 5th harmonic of flux linkagewith the 5th harmonic of current gives a steady torquewith a magnitude of -0.01607X while the interactionof the 7th flux linkage and current harmonics give apositive constant torque of 0.005859Xpfp. Hence thecontribution of the 1st and 5th harmonics contribute1.99493 + 0.005859 = 2.00079X I It is thereforeclear that the contribution 'of' the higher orderharmonics to the steady torque is negligible. Thecontribution of the fundamental components of currentand flux linkage is essentially responsible for thesteady torque of the machine.The interaction of flux linkages and currents ofdifferent orders produce pulsating torques (6,7].However it was shown in (10) that when the 120'trapezoidal flux density waveform interacts with therectangular current that only a steady torque isproduced with no torque pulsations. Therefore it canbe deduced that the pulsating torques produced by theinteraction of current and flux linkage harmonics ofdifferent orders must all cancel to produce zero netpulsating torque for the waveforms shown in figure 1.The BDCM can therefore be regarded as ageneralization of the PMSM or alternately, the PMSMcan be regarded as a special case of the BDCM whereonly the fundamental components of flux and currentare present. Hence if only the steady torque of theBDCM is under study, the possibility exists of usingjust the fundamental component of flux and current inthe analysis. A transformation can then be made tod,q variables as is done for the PMSM. Great care

should be taken however whenever this simplifiedapproach is used.Up to now a BDCM with the idealized waveforms infigure1 has been considered. In practice, deviationsfrom the idealized current and flux linkage waveformsshown in figure 1occur. Some of the deviations fromthe idealized machine are discussed in the nextsection.202


3/8

IV.DEVIATION F THE PRACTICAL MACHINE FROM THEIDEALIZEDIn a practical machine it is impossible to forcea rectangular current to flow into the machinewindings. This is because the motor inductance limitsthe rate of change of current [5]. In the steadystate, the rise time of the current depends on thevoltage differential between the dc bus and the backemf, and the time constant of the stator winding whichis given by the ratio of the stator leakage inductance

to resistance. The higher this ratio, the longer isthe rise time of the current and the greater thedeviation from the idealized value. For an accuratecalculation, (6), (7) and (8) must be solved indetail. Although the rise is given by (1-exp-(L-M)t/R)in pu at a constant speed, it can be approximated [7]by a straight line so that the actual currentresembles a trapezoid as shown in figure 2 . Thisavoids the detailed calculation of (6), (7) and (8).Note that high frequency switching with eitherhysteresis or PWM logic is used to track therectangular references. This is not shown in figure 2since the effect of this on the torque has alreadybeen examined and shown to be of secondary importance[5] to that produced by the commutation of current.The torque behavior as a result of using trapezoidalcurrents instead of rectangular can be studied withthe aid of the Fourier series of the trapezoidalcurrent given below:i (x) = 4((sin H - sin h)sin x + (sin3H - sin3h)s?n3~/3~+..)/n(H-h) (15)

The second deviation from the idealized is theangle for which the flux density remains constant. Infigure 1 it is assumed to be of 120'. This isdesirable for a three phase machine. In practice thisangle may range from l o o o to 150' depending on thephase spread, the effects of the number of slots perphase and manufacturing tolerances.Torque pulsations result from the motor currentsor flux linkage deviating from the ideal. Themagnitude of individual torque harmonics can becalculated from equations (ll), (12) and (15) whichrepresent the flux linkage, idealized rectangularcurrent and nonideal trapezoidal current respectively.In (ll), F = 30' densityfor a three phase machine. Nonideal flux densitywaveforms produced when the flux density is less orgreater than 120' can be represented in (11) byincreasing or decreasing F respectively. Similarlynonideal currents can be represented in (15) byvarying h relative to H.In the BDCM, the 6th harmonic of torque isdominant [6]. This can be produced by a variety ofcurrent-flux interactions. For example the

for the idealized 120' flux

Figure 2. Trapezoidal shaped current

fundamental of the flux can interact with the 5th and7th current harmonics to produce a 6th harmonic torquepulsation and vice-versa. (11) gives the relevantequation for any flux harmonic while (15) gives therelevant equation for any nonideal current harmonic.(13) shows how multiplication of the fundamental fluxwith the fundamental current gives a steady torqueafter adding the effects of the three phases. In asimilar manner, the multiplication of a flux harmonicof one order (except triplen) with a current harmonicof a different order (except triplen) so that thedifference in the order is 6, esults inTe6=24sin(N2.F)(sin(N1.H)-sin(Nl.h))I P P~os(6wt)/(N2~Nl*Fx*(H-h) (16)where N2 is the order of the flux linkage harmonic, N1is the order of the current harmonic, I is the peakof the current, X is the peak of the'flux linkagewaveform, F is dgfined in figure 1, H and h aredefined in figure 2. If N2 and N1 are chosen suchthat they add to 6 then the negative of (16) should beused in the calculation of the 6th torque harmonic.Since (11) and (15) have been used in thecalculation of (16), it turns out that (16) is alsovalid for the 12th torque harmonic by using theappropriate N2, N1 and sign of (16) and replacing 6wtby 12wt etc.In this investigation, the effects of thecommutation of the stator current as well as theeffects of different flux density distributions on thetorque of a BDCM are investigated. Attention is paidto the overall torque pulsations in addition toindividual torque harmonics. The effects of magnitudeof the current in addition to phase advancing are alsoexamined.

V. ESULTSRectangular Current

In order to test the validity of using theFourier Series approach to study the steady statebehavior of a BDCM, a program was written to determinethe output torque given that the motor current andflux waveforms are idealized as shown in figure 1.The phase A current and flux linkage waveforms areshown in figure 3, the waveforms of the other phasesbeing similar and phase shifted from that of phase Aby 120'. The output torque, which is essentiallyconstant, is also given in figure 3. 21 current andflux linkage harmonics are used in the simulation.The slight ripple is due to the truncation in theFourier Series that is necessary in any practicalimplementation. However, the ripple is small enoughas to have negligible engineering significance. Theseresults indicate the suitability of using a Fourierseries to examine the torque behavior in a BDCM.Trapezoidal Current

Keeping the flux density waveform the same, theslope of the current waveform was varied from O o to15' to simulate different stator time constants andoperating speeds. Figure 4 shows the flux linkage,current and torque for a 5 slope in current and 120'flux linkage waveform. 21 flux and current harmonicsare used. Because of the nonrectangular current,torque pulsations are produced during the commutationof the current, the fundamental frequency of which is6 times that of the current. The slope of the currentwaveform was varied and the corresponding torquepulsations determined. From these results, a graph oftorque pulsations vs commutation angle was drawn asshown in figure 5 for different current magnitudes.The magnitude of the torque pulsations increases

203


4/8

N

N

TORQUE

Figure 3. Rectangular sh@ Eurrml& for 21 h a " i a Figure 4. Trapezoidalshaped current results for 21 current harmonics

rapidly initially up to a commutation angle of 5 withthe rate of increase being lower after 5' . Althoughthe magnitude of the torque pulsations level off withan increase in the commutation angle, the width ofthese pulsations increases continually thus reducingthe average value. The increased torque pulsationsand reduction in the average torque can affect theperformance of torque, speed and position servos. Theincrease in the magnitude of the torque pulsation withcurrent for a given commutation angle is linear. Thatis doubling the magnitude of the current beingcommutated doubles the magnitude of the resultingtorque pulsation as well.

A graph of the average torque vs the commutationangle is given in figure 6 . The average torque can beas low as 1 . 7 5 X I for a commutation angle of 15'which is a fair PrEduction from the expected 2.0XpI p.This reduction can have consequences for torque servoperformance since the commanded torque will not bemet. This torque reduction is independent of machineparameter changes due to temperature or saturationwhich can cause further reductions. The average valuewas obtained by averaging the instantaneous torqueover a complete cycle. If a complete cycle were notused it is possible for the average torque to beeither less than or greater than the complete cyclevalue depending on the section of the torque profileused to calculate the average value. This can have

4

Ha 3a4Xc

'r l

: 2311

aJ1G-

12

5 O omutation oongle W-h) 15'Figure 5. Torque pulsations vs commutation angle

204


5/8

consequences in position servo performance where it i'spossible that the rotor is commanded to stop beforecompleting a full revolution.The results presented thus far show the effectsof changing commutation times on the overall magnitudeof the torque pulsations and the consequent reductionin the average torque. The flux waveform was keptconstant for 120 as shown in figure 1. In the nextsection, the effects of the different commutationtimes on individual torque harmonics are examined.

Torque HarmonicsThe electric torque Te is given by the productof the flux in (11) and the rectangular current in(12) or the trapezoidal current in (15). It must alsobe remembered to include the current-flux interactionof the other two phases. The above equations can alsobe used to calculate the magnitude of individualtorque harmonics as well by considering theappropriate current and flux harmonics. From (ll),(12) and (15) it is clear that there is a largereduction in both the current and flux harmonics asthe order increases so that the higher order harmonicshave less of an effect on the torque profile.From the waveforms presented earlier it is clearthat the torque pulsations have a fundamental of 6times the fundamental frequency of the current. Thismeans that the 6th torque harmonic is predominant.The 6th torque harmonic is given by the interaction ofthe 1st harmonic of flux with the 5th and 7th current

harmonics, the 1st harmonic of current with the 5thand 7th flux harmonics etc. These results aresummarized in figure 7 for different commutationangles. The results were obtained by keeping Fconstant in (16) and varying (H-h) and choosing N2 andN1 appropriately.Firstly it should be noted that the 6th torqueharmonic is produced by other flux currentinteractions than those listed in figure 7. Howeversince the magnitude of the flux and current harmonicsreduce either as a function of their order or thesquare of the order, the contribution of the higherorder harmonics to the 6th torque harmonic areinsignificant.The results for the slope of 0 corresponds tothe results for the rectangular current, for which itwas previously shown that should be no torqueharmonics except for the constant torque. From Figure7 it is clear that the 1st harmonic of flux interactswith the 5th harmonic of current to produce thelargest contribution to the 6th torque harmonic thanany other flux-current interactions. The other flux-current interactions shown go towards neutralizing themagnitude of the 6th torque harmonic produced by the1st flux and 5th current harmonics such that the netresultant after adding the contributions of the fourlowest current and flux harmonics is almost zero.The 1st flux and 5th current harmonics are alsothe largest contributors to the 6th torque harmonicwhen the current is trapezoidal rather thanrectangular as shown in figure 7for 5O, 10' and 15'current commutation slopes. However in these casesthe other flux current interactions shown are unableto completely neutralize the torque harmonic producedby the 1st flux and 5th current harmonics. Instead,there is a residual which finally shows up as thetorque pulsations presented in figure 4 . In addition,the larger the slope of the current waveform, thelarger is the residual 6th torque harmonic.

Similar results are shown for the 12th torqueharmonic in figure 8 . Here also the 12th torqueharmonic was calculated from (16) by fixing F at 30'and varying (H-h). N2 and N1 are chosen so as theirsum or difference gives 12. From the results for therectangular current ( 0 ' ) it is again evident that the

Flux1 51 75 17 1

x =1 puI =1 puPP

current 0" 5" IO" 15"+ 0.4022 + 0.4394 +0.4482 +0.4283-0.2873 -0.2207 -0.1317 -0.4283-0.0804 -0.0782 -0.0760 -0.0734-0.0410 -0.0400 -0.0388 -0.0375

I I 1So 1 o 1 oCommutation angle (H-h )

Total

Figure 6. Average torqw vs commutation angle

-0.0065 +O.IOOO +0.2017 +0.2798

FluxI1I 113

Figure 7. 6th Harmonic torque pulsations

current 0" 5" IO" 15'1 1 -0.1828 -0.2029 -0.1633 -0.088513 +0.1 547 +0.0781 -0.0124 -0.06341 +0.0166 +0.0162 +0.0157 +0.01511 +0.0119 +0.0116 +0.0112 +O.OIOS

Total +O.o004 -0,097 -0.1488 -0.1258

Figure 8. 12th Harmonic torque pulsations

20 5


6/8

largest contributor to the 12th torque harmonic is thefirst flux and llth current harmonics. Just as forthe 6th harmonic, the torque harmonics produced by theother flux-current interactions go towards nullifyingthat produced by the first flux and llth currentharmonics. From figure 8it is also clear that as thecurrent commutation angle increases, the residualtorque harmonic increases up to a commutation angle of1 For larger commutation angles the 12th torqueharmonic decreases. This probably explains why themagnitude of the torque pulsation which is aninstantaneous sum of all the torque harmonics tends tolevel off as the commutation angle is increased beyondl o o as shown in figure 5 . The timing shown in Figure 1is used up to therated speed of the machine. High speed operation isobtainable by phase advancing of the current relativeto the back emf. In this study it is assumed thatthere is sufficient bus voltage available to force thecurrents to be as close to the desired rectangularshape as when operated below the maximum speed. Thisis done s o that the effects of phase advancing aloneon the torque profile can be determined. Figure 9shows the results when a trapezoidal current with acommutation slope of 5 is phase advanced by 20, 40,60' and 80'. These curves should be compared with thecurve for torque presented in figure 4 . As the phaseis advanced, the torque pulsations increase at theexpense of the duration for which the torque remainsconstant. This is an extremely undesirable feature o fphase advancing. In addition the average value oftorque is reduced greatly as shown in figure 10. Itshould be remembered that the average value isapproximately 1 . 9 1 7X I for a commutation slope of 5Oand when there is n$ 'phase advance. The magnitudeand shape of the current are kept the same during thephase advancing and the dramatic reduction in theaverage torque is due only to the phase advance.

The effects of phase advancing on individualtorque harmonics was also examined. Since the fluxlinkage and current waveforms were maintained the sameas during zero phase advance, it has been calculatedthat the individual torque harmonics are exactly thesame magnitude as with zero phase advance for zero,5 O , loo and 15' commutation angles. In other wordseven when a rectangular current is phase advanced, theindividual torque harmonics are exactly the samemagnitude as when there is no phase advance. In thezero phase advance case all the 6th torque harmonicsfor example are either in phaseand sum to zero as shown in figure 7. However whenthe current is phase advanced relative to the backemf, the individual torque harmonics of a given orderare not all in phase samemagnitude) such that the cancellation shown in Table 1for example does not occur. This results in anincrease in the overall torque pulsations as shown infigure 9 for the 5 trapezoidal current. Similarresults occur for the other commutation angles.Nonideal F l u x Linkage Effects

phase or 180 out of

(even though they have the

Up to now the effects of the current commutationtimes on the profile has been examined. Here theeffects of the flux density waveform on the torque areexamined. Torque pulsations are produced by the fluxdensity waveform being less than the desired 120'.Note that if the constant portion is greater than120, but 120' currents are still used, then theresults will be the same as if the flux were constantfor 120' only provided the timing in figure 1is used.However when the flux density waveforms are constantfor less than 120, torque pulsations are produced.The magnitude of individual torque harmonics can becalculated from (16) by fixing (H-h) and varying F.The entire torque profile can be obtained by choosing

a4x s

E?0

Z0

70.00 18.00 36.0 5q.00 72.(DEGREES) a 10

Figure 9 . Phase advancing results

2 .

aa4XB25 1.01MmLl2

no

I = lpux = lpuPP

I I I I20' 40 60' EO0

Phase advance angleFigure 10. Phase advancing results

206


7/8

a certain number of harmonics and evaluating (11) fora given F. Similarly, (15) is evaluated for a given(H-h) and multiplication of these instantaneous fluxand current waveforms gives the instantaneous torqueprofile.The torque pulsation produced when the fluxdensity is constant for l o oo and llOo instead of 120'are shown in figure 11. In this study, rectangularcurrents were used so as to determine the contributionof the flux density alone on the torque pulsations.From these results, a graph of torque pulsation as afunction of F can be drawn as shown in figure 1 2 . Thegraph is linear indicating a linear increase in torquepulsation with decrease in the duration of theconstant flux density. In practice, the combinationof the effects of the flux density and trapezoidalshaped currents would increase the torque pulsationbeyond the contribution of each.

A s the torque pulsations increase, so theaverage values decrease. However the reduction in theaverage torque is minimal when compared to thatproduced by the commutation of current as presentedearlier and a graph is therefore not drawn.VI. CONCLUSIONS

A detailed investigation into the torquebehavior of a BDCM drive has been done in this paper.A previously published model to study the dynamicbehavior was used to analyze the steady state behavioras well. It was shown that a constant output torqueis produced only if the flux density and currentwaveforms of the BDCM are idealized.In the practical and hence nonideal case, torquepulsations arise as a result of the actual currentbeing trapezoidal instead of rectangular, from the

I = lpu i =1 puP P .E F=35""cI

4 F=40a

-t

Figure I I . Torque pulsations for nonideal flux linkages

aa4

Xg 0. 6

0. 4

VI

4

,-I3

a, 0.2$ 0.0J0-

I = lpuh = lpuPP

30' 3 5 O 40F

~j~~~~ 12. Torque pulsations as a function ofFflux linka e being constant for less than 120 instead

The larger the commutation angle, the larger isthe magnitude of the torque pulsations and the loweris the average value of the torque over a cycle. Thisis an extreme disadvantage of large commutation times.The commutation time is determined by the ratio of thestator leakage inductance to resistance, the operatingspeed and the dc bus voltage. Hence for a .givenresistance, the leakage inductance should be minimizedor for a given leakage inductance, the statorresistance should be maximized. Increasing the statorresistance should be done with due regard to theefficiency and cooling of the machine. The increase inthe torque pulsations with increase in commutationangle is nonlinear with the increase being much largerup to 5 and then reducing after 5 The increase intorque pulsations is linear with increase in themagnitude of the current being commutated. Thesetorque pulsations may affect the accuracy andrepeatability of position servos. A s the torquepulsations increase, so the average value of thetorque decreases. A 12.5% decrease is possible over afull cycle and this can have consequences in theperformance of torque, speed or position servos whenusing this machine.Phase advancing of the current waveform relativeto the flux density can produce large torquepulsations with a resultant reduction in the averagevalue of torque as well.Deviation of the flux density waveforms from theideal also produce torque pulsations although themagnitude is not as large. In addition, the reductionin the average torque is not as severe as that due tothe commutation in current. If the flux density waveis constant for a duration longer than 120, then theoutput torque is not affected.REFERENCES

of the 120 , or phase advancing of the current.

(11 R. Krishnan, "Selection criteria for servo motordrives", E E E Trans., vol. IA-2 3, o. 2, March/April[2] D. Pauly, G . Pfaff and A.Weschta, "Brushless servo1987, pp. 270-275.I I II 1 I%.00 18.00 36.00 53-00 72 . drives with permanent magnet motors or squirrel cageQNGLE (DEGREES) m 1 0 induction motors - a comparison," I E E E IAS AnnualMeeting, 1984, pp. 503-509.(31 G, Pfaff, A. Weschta and A . Wick, "Design andexperimental results of a brushless ac servo-drive",I E E E IAS Annual Meeting, 1982, pp. 692-697.

207


8/8

[4] P. Pillay and R. Krishnan, "Applicationcharacteristics of permanent magnet sychronous andbrushless dc motors for servo drives", IEEE IAS AnnualMeeting, 1987, pp. 380-390.[5] P. Pillay and R. Krishnan, "Modeling, simulationand analysis of a permanent magnet brushless dc motordrive", IEEE IAS Annual Meeting, 1987, pp. 7-14.[6] T.M. Jahns, "Torque production in permanent magnetmotor drives with rectangular current excitation,"IEEE Trans., vol. IA- 20, No.4, July/August 1984, pp.[7] J.M.D. Murphy, "Thyristor control of ac motors",(book), Pergamon Press, 1973.[8] V. Subrahmanyam anl D. Subbarayudu, "Steady stateanalysis of an induction motor fed from a currentsource inverter using complex-state (Park's) Vector",Proc. IEE, vol. 126, No. 5, May 1979, pp. 421-425.[9] H.R. Bolton, Y.D. Liu and N.M. Mallison,"Investigation into a class of brushless dc motor withquasisquare voltages and currents", Proc. IEE, vol.133, Pt B, No. 2 , March 1986, pp. 103-111.[ l o ] E.K.Persson, "Brushless dc motors - a review ofthe state of the art" Proceedings of the MotorconConference, 1981, pp. 1-16.(111 T . Sebastian and G.R. Slemon, "Operating limitsof inverter driven permanent magnet motor drives,"IEEE IAS Annual Meeting, 1986, pp.800-805.

803-813.

List of SymbolsB damping constant, N/rad/sea,eb,ec a,b and c phase back emfs, VE peak value of back emf, Vi:,ib,ic a,b and c phase currentJ moment of inertia, kg-mtorque constant - 2ea/wrLa,$,Lc self inductance of a,b 6 c phases, Hmutual inductance betwen phases a & b ,LabP derivative operatorP number of pole pairsR stator resistance, ohms' electric torque, N-mTe1,Te5,Te7 lst,Sth and 7th torque harmonics,N-mTL load torque, N-mva,vb,vc a,b and c phase voltages, Vdc bus voltage, Vrotor speed, rad/sec'W S synchronous speed, rad/sec

h p AKt

208

Documents

An Investigation Into the Torque Behavior of A