An Investigation Into The Torque Behavior Of A

AN INVESTIGATION INTO THE TORQUE BEHAVIOR OF A

BRUSHLESS DC MOTOR DRIVE

P.Pillay Department of Electrical and Electronic Engineering University of Newcastle upon Tyne NE1 7RU, England

ABSTRACT This paper uses a previously developed model for the brushless dc motor (BDCM) to investigate its torque behavior. When the input currents and motor flux linkages are perfect, no torque pulsations are produced in this motor. However imperfections in the current arise due to finite commutation times while imperfections in the flux linkage can arise due to the phase spread, finite slot numbers and manufacturing tolerances. Using an harmonic analysis, the effects of these imperfections on the production of torque in a BDCM are investigated. It is shown that torque pulsations and a reduction in the average value of torque is produced, both of which can affect the performance of torque, speed and position servos. 1.INTRODUCTION

AC servo drives are commanding a larger share of the servo market each year. The advantages of ac servo drives over dc include increased robustness, reduced maintenance and higher torque and speed bandwidths. The ac motors that are used include the induction, permanent magnet synchronous and permanent magnet brushless dc machines [I]. Some of the advantages of permanent magnet machines over induction [ 2 ] include higher torque to inertia ratios and power densities, lower rated rectifier and inverter ratings during constant torque operation and higher efficiencies. Hence permanent magnet machines may be preferable for applications where weight or efficiency is of importance, for example in the aerospace industry or electric vehicles. The permanent magnet synchronous motor (PMSM) and the brushless dc motor (BDCM) have many similarities. They both have permanent magnets on the rotor and require alternating stator currents to produce constant torque. The difference [3,4]between them is that the PMSM has a sinusoidal back emf while the BDCM has a trapezoidal back emf. This leads to different operating and control requirement for these two machines as explained in [4]. The dynamic behavior of a BDCM has been studied [SI and the results indicate that the BDCM can be subject to severe torque pulsations [ 6 ] due to the stator currents commutating from one phase to another. Torque pulsations are also created by the magnet flux linkage deviating from the ideal. The above imperfections also create the possibility of a reduction in the average value of torque. These phenomena can affect the performance of torque, speed or position servos. The object of this paper is to quantitatively determine the motor characteristics that affect the production of torque in a BDCM. Attention is paid to the overall torque pulsations as well as individual torque harmonics. The torque behavior during flux weakening operation is also addressed. The nonsinusoidal currents and flux linkages are represented by Fourier series and the impact of

R.Krishnan Electrical Engineering Department Virginia Polytechnic Institute 6 State University, Blacksburg, VA, 24061, USA

different flux and current harmonics on the motor torque are investigated. The paper is organized as follows: Section I1 presents the mathematical model of a BDCM. Section 111 discusses torque production in a BDCM and shows how the dynamic mathematical model in section I1 can be used to study the steady state torque behavior of a BDCM. Section IV discusses the machine parameters that affect the production of torque in a BDCM. Finally section V and VI have the results and conclusions of this investigation. 11. MATHEMATICAL MODEL OF THE BRUSHLESS DC MOTOR

The BDCM has three stator windings and a permanent magnet on thr rotor. Since both the magnet and the stainless steel retaining sleeves have high resistivity, rotor induced currents can be neglected and no damper windings are modelled. Hence the circuit equations of the three windings in phase variables are

where it has been assumed that the stator resistance of all the windings are equal. The back emfs ea, eb and ec have trapezoidal shapes as shown in figure 1. Assuming further that there is no change in the rotor reluctance with angle, then

- $ - Lc - L Lcb -

La Lab

=

Hence

Back emf o f t h e b r u s h l e s s DC motor &I

I

U

C u r r e n t waveform r e q u i r e d f o r c o n s t a n t t o r q u e

Figure 1.

88CH2565-0/88/01$01 .oo

Back emf and currents of a BDCM

o 1988 IEEE

O

O

R

+ ib + ic

=

0

+ Mic

(x) = 4(sinFsinx (:in5Fsin5~)/5~ +. . .)/nF

+

=

-Mia

While that of rectangular current is

(4)

ia(x) = 4(cosHsinx + (cos3Hsin3x)/3 (cos5Hsin5x)/5+ . . . )/a (12) where F and H are defined in figure 1.

Hence in state space form the equations are arranged as follows:

and the electromagnetic torque is,

The equation of motion is

Now H to

examine the

+

=

n/6 in figure 1, therefore

+

sin2(x-2n/3) = 2.0111pXp

+

sin2(x+2r/3)I

X )/n3

847

Using the technique above it can be shown that the interaction of the 5th harmonic of flux linkage with the 5th harmonic of current gives a steady torque with a magnitude of -0.01607X I while the interaction of the 7th flux linkage and harmonics give a positive constant torque of 0.005859Xpfp. Hence the contribution of the 1st and 5th harmonics contribute 1.99493 + 0.005859 = 2.00079X I It is therefore clear that the contribution 'of' the higher order harmonics to the steady torque is negligible. The contribution of the fundamental components of current and flux linkage is essentially responsible for the steady torque of the machine. The interaction of flux linkages and currents of different orders produce pulsating torques (6,7]. However it was shown in (10) that when the 120' trapezoidal flux density waveform interacts with the rectangular current that only a steady torque is produced with no torque pulsations. Therefore it can be deduced that the pulsating torques produced by the interaction of current and flux linkage harmonics of different orders must all cancel to produce zero net pulsating torque for the waveforms shown in figure 1. The BDCM can therefore be regarded as a generalization of the PMSM or alternately, the PMSM can be regarded as a special case of the BDCM where only the fundamental components of flux and current are present. Hence if only the steady torque of the BDCM is under study, the possibility exists of using just the fundamental component of flux and current in the analysis. A transformation can then be made to d,q variables as is done for the PMSM. Great care should be taken however whenever this simplified approach is used. Up to now a BDCM with the idealized waveforms in figure 1 has been considered. In practice, deviations from the idealized current and flux linkage waveforms shown in figure 1 occur. Some of the deviations from the idealized machine are discussed in the next section.

current

In figure 1, consider an instant when =

F

Tel= 96(sin2x

The back emf and the required currents in order to produce constant torque are shown in figure 1 in an ideal machine. In (7) it was shown that the torque is given by the product of the back emf and stator current waveform divided by the speed. The back emf divided by the speed is a constant and represents the flux linkage which has the same waveform as the back emf in figure 1 in an ideal machine. The flux linkage is horizontal (constant) for 120' and for constant torque it is necessary to supply a rectangular shaped current to the phase during this period. When the flux linkage is negative, a negative current is needed in order to produce constant positive torque. In addition, at any given instant, only two phases conduct current with the phase carrying the positive current using the phase carrying the negative current as a return path.

=

=

- 1.341 (3/2)IpXp

III.TORQUE PRODUCTION IN A BDCM

ea ia eb ib ic

+

It is known [ 6 ] that current and flux linkage harmonics of the same order interact to produce constant torque while if they are of different orders they produce pulsating torques. However it has been shown in (10) that the output torque is constant for the waveforms in figure 1, hence there are no pulsating torques. The steady torque is given by the interaction of the fundamental of the flux linkage with the fundamental component of current plus the 5th harmonic of the flux with the 5th harmonic of current etc. That is all odd harmonics of flux which interact with current harmonics of the same order (exept triplen) produce a constant torque. The contribution of the fundamental component of flux linkage with the fundamental component of current gives (after adding all 3 phases) 161 X ( s i n ( F ) s i n ( w t ) c o s ( H ) s i n ( w t ) + Tels;n(F)!iR(wt - 2n/3)cos (H) sin(wt - 2n/3) + sin( F) sin(wt+2%/3) cos (H)sin(wt+2a/3) )/n2F (13)

Hence

The above equations can be used detailed behavior of the BDCM.

(sin3Fsin3~)/3~ (11)

(3)

Therefore Mib

The Fourier series of the

X

But ia

a Fourier series as well. flux linkage is given by

E 'I

-- 0-E-1; =

Then the electric torque becomes

At any other instant, it will always be found from figure 1 that only two phases conduct, with the third being zero and then (10) holds. The torque predicted from this idealized machine is therefore constant with no torque pulsations. Any periodic wave can be expressed as a Fourier series. Hence both the flux linkage and current waveforms of the BDCM in figure 1 can be expressed as

202

fundamental of the flux can interact with the 5th and 7th current harmonics to produce a 6th harmonic torque pulsation and vice-versa. (11) gives the relevant equation for any flux harmonic while (15) gives the relevant equation for any nonideal current harmonic. (13) shows how multiplication of the fundamental flux with the fundamental current gives a steady torque after adding the effects of the three phases. In a similar manner, the multiplication of a flux harmonic of one order (except triplen) with a current harmonic of a different order (except triplen) so that the difference in the order is 6, results in

IV.DEVIATION OF THE PRACTICAL MACHINE FROM THE IDEALIZED In a practical machine it is impossible to force a rectangular current to flow into the machine windings. This is because the motor inductance limits the rate of change of current [5]. In the steady state, the rise time of the current depends on the voltage differential between the dc bus and the back emf, and the time constant of the stator winding which is given by the ratio of the stator leakage inductance to resistance. The higher this ratio, the longer is the rise time of the current and the greater the deviation from the idealized value. For an accurate calculation, (6), (7) and (8) must be solved in detail. 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 s o that the actual current resembles a trapezoid as shown in figure 2 . This avoids the detailed calculation of (6), (7) and (8). Note that high frequency switching with either is used to track the hysteresis or PWM logic rectangular references. This is not shown in figure 2 since the effect of this on the torque has already been 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 trapezoidal currents instead of rectangular can be studied with the aid of the Fourier series of the trapezoidal current given below: i (x) = 4((sin H - sin h)sin x . .)/n(H-h) s?n3~/3~+.

+ (sin3H -

Te6=24sin(N2.F)(sin(N1.H)sin(Nl.h))I PX P~os(6wt)/(N2~Nl*Fx*(H-h)

sin3h) (15)

The second deviation from the idealized is the angle for which the flux density remains constant. In This is figure 1 it is assumed to be of 120'. desirable for a three phase machine. In practice this depending on the angle may range from looo to 150' phase spread, the effects of the number of slots per phase and manufacturing tolerances. Torque pulsations result from the motor currents or flux linkage deviating from the ideal. The magnitude of individual torque harmonics can be calculated from equations (ll), (12) and (15) which represent the flux linkage, idealized rectangular current and nonideal trapezoidal current respectively. for the idealized 120' flux density In (ll), F = 30' for a three phase machine. Nonideal flux density waveforms produced when the flux density is less or greater than 120' can be represented in (11) by increasing or decreasing F respectively. Similarly nonideal currents can be represented in (15) by varying h relative to H. In the BDCM, the 6th harmonic of torque is dominant [6]. This can be produced by a variety of current-flux interactions. For example the

(16)

where N2 is the order of the flux linkage harmonic, N1 is the order of the current harmonic, I is the peak linkage of the current, X is the peak of the'flux waveform, F is dgfined in figure 1, H and h are defined in figure 2. If N2 and N1 are chosen such that they add to 6 then the negative of (16) should be used in the calculation of the 6th torque harmonic. Since (11) and (15) have been used in the calculation of (16), it turns out that (16) is also valid for the 12th torque harmonic by using the appropriate N2, N1 and sign of (16) and replacing 6wt by 12wt etc. In this investigation, the effects of the commutation of the stator current as well as the effects of different flux density distributions on the torque of a BDCM are investigated. Attention is paid to the overall torque pulsations in addition to individual torque harmonics. The effects of magnitude of the current in addition to phase advancing are also examined. V.RESULTS Rectangular Current In order to test the validity of using the Fourier Series approach to study the steady state behavior of a BDCM, a program was written to determine the output torque given that the motor current and flux waveforms are idealized as shown in figure 1. The phase A current and flux linkage waveforms are shown in figure 3, the waveforms of the other phases being similar and phase shifted from that of phase A by 120'. The output torque, which is essentially constant, is also given in figure 3. 21 current and flux linkage harmonics are used in the simulation. The slight ripple is due to the truncation in the Fourier Series that is necessary in any practical implementation. However, the ripple is small enough as to have negligible engineering significance. These results indicate the suitability of using a Fourier series to examine the torque behavior in a BDCM. Trapezoidal Current

Figure 2.

Keeping the flux density waveform the same, the slope of the current waveform was varied from Oo to 15' to simulate different stator time constants and operating 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 harmonics are used. Because of the nonrectangular current, torque pulsations are produced during the commutation of the current, the fundamental frequency of which is 6 times that of the current. The slope of the current waveform was varied and the corresponding torque pulsations determined. From these results, a graph of torque pulsations vs commutation angle was drawn as shown in figure 5 for different current magnitudes. The magnitude of the torque pulsations increases

Trapezoidal shaped current

203

N

N

TORQUE

Figure 3. Rectangular sh@

Eurrml&

for 21 h a " i a

Trapezoidal shaped current results for 21 current harmonics

Figure 4.

4

rapidly initially up to a commutation angle of 5' with the rate of increase being lower after 5'. Although the magnitude of the torque pulsations level off with an increase in the commutation angle, the width of these pulsations increases continually thus reducing the average value. The increased torque pulsations and reduction in the average torque can affect the performance of torque, speed and position servos. The increase in the magnitude of the torque pulsation with current for a given commutation angle is linear. That is doubling the magnitude of the current being commutated doubles the magnitude of the resulting torque pulsation as well. A graph of the average torque vs the commutation angle is given in figure 6 . The average torque can be as low as 1 . 7 5 X I for a commutation angle of 15' which is a fair PrEduction from the expected 2.0XpIp. This reduction can have consequences for torque servo performance since the commanded torque will not be met. This torque reduction is independent of machine parameter changes due to temperature or saturation which can cause further reductions. The average value was obtained by averaging the instantaneous torque over a complete cycle. If a complete cycle were not used it is possible for the average torque to be either less than or greater than the complete cycle value depending on the section of the torque profile used to calculate the average value. This can have 204

Ha 3

a

4

X

c 'rl

: 2 3

c 11 aJ

1

b.l G-

r2 1

5O Comutationloo angle

W-h) 15'

Figure 5. Torque pulsations vs commutation angle

consequences in position servo performance where it i's possible that the rotor is commanded to stop before completing a full revolution. The results presented thus far show the effects of changing commutation times on the overall magnitude of the torque pulsations and the consequent reduction in the average torque. The flux waveform was kept constant for 120° as shown in figure 1. In the next section, the effects of the different commutation times on individual torque harmonics are examined.

x

=1 pu

P I =1 pu P

Torque Harmonics The electric torque Te is given by the product of the flux in (11) and the rectangular current in (12) or the trapezoidal current in (15). It must also be remembered to include the current-flux interaction of the other two phases. The above equations can also be used to calculate the magnitude of individual torque harmonics as well by considering the appropriate current and flux harmonics. From (ll), (12) and (15) it is clear that there is a large reduction in both the current and flux harmonics as the order increases so that the higher order harmonics have less of an effect on the torque profile. From the waveforms presented earlier it is clear that the torque pulsations have a fundamental of 6 times the fundamental frequency of the current. This means that the 6th torque harmonic is predominant. The 6th torque harmonic is given by the interaction of the 1st harmonic of flux with the 5th and 7th current harmonics, the 1st harmonic of current with the 5th and 7th flux harmonics etc. These results are summarized in figure 7 for different commutation angles. The results were obtained by keeping F constant in (16) and varying (H-h) and choosing N2 and N1 appropriately. Firstly it should be noted that the 6th torque harmonic is produced by other flux current interactions than those listed in figure 7. However since the magnitude of the flux and current harmonics reduce either as a function of their order or the square of the order, the contribution of the higher order harmonics to the 6th torque harmonic are insignificant. The results for the slope of 0' corresponds to the results for the rectangular current, for which it was previously shown that should be no torque harmonics except for the constant torque. From Figure 7 it is clear that the 1st harmonic of flux interacts with the 5th harmonic of current to produce the largest contribution to the 6th torque harmonic than any other flux-current interactions. The other fluxcurrent interactions shown go towards neutralizing the magnitude of the 6th torque harmonic produced by the 1st flux and 5th current harmonics such that the net resultant after adding the contributions of the four lowest current and flux harmonics is almost zero. The 1st flux and 5th current harmonics are also the largest contributors to the 6th torque harmonic when the current is trapezoidal rather than rectangular as shown in figure 7 for 5O, 10' and 15' current commutation slopes. However in these cases the other flux current interactions shown are unable to completely neutralize the torque harmonic produced by the 1st flux and 5th current harmonics. Instead, there is a residual which finally shows up as the torque pulsations presented in figure 4 . In addition, the larger the slope of the current waveform, the larger is the residual 6th torque harmonic. Similar results are shown for the 12th torque harmonic in figure 8 . Here also the 12th torque harmonic was calculated from (16) by fixing F at 30' and varying (H-h). N2 and N1 are chosen so as their sum or difference gives 12. From the results for the rectangular current (0') it is again evident that the 205

I

I

1

So

1oo

1so

Commutation angle (H-h)

Figure 6.

Average torqw vs commutation angle

Flux current 0"

5"

IO"

15"

1

5

+ 0.4022

+ 0.4394

+ 0.4482

+ 0.4283

1

7

-0.2873

-0.2207

-0.1317

-0.4283

5

1

-0.0804

-0.0782

-0.0760

-0.0734

7

1

-0.0410

-0.0400

-0.0388

-0.0375

-0.0065

+O.IOOO

+0.2017

+0.2798

Total

Figure 7.

6th Harmonic torque pulsations

Flux current 0"

5"

IO"

15'

I

11

-0.1828

-0.2029

-0.1633

-0.0885

1

13

+0.1547

+0.0781

-0.0124

-0.0634

I1

1

+0.0166

+0.0162

+0.0157

+0.0151

13

1

+0.0119

+0.0116

+0.0112

+O.OIOS

+O.o004

-0,097

-0.1488

-0.1258

Total

Figure 8.

12th Harmonic torque pulsations

largest contributor to the 12th torque harmonic is the first flux and llth current harmonics. Just as for the 6th harmonic, the torque harmonics produced by the other flux-current interactions go towards nullifying that produced by the first flux and llth current harmonics. From figure 8 it is also clear that as the current commutation angle increases, the residual torque harmonic increases up to a commutation angle of 1 0 ' . For larger commutation angles the 12th torque harmonic decreases. This probably explains why the magnitude of the torque pulsation which is an instantaneous sum of all the torque harmonics tends to level off as the commutation angle is increased beyond loo as shown in figure 5 . The timing shown in Figure 1 is used up to the rated speed of the machine. High speed operation is obtainable by phase advancing of the current relative to the back emf. In this study it is assumed that there is sufficient bus voltage available to force the currents to be as close to the desired rectangular shape as when operated below the maximum speed. This is done so that the effects of phase advancing alone on the torque profile can be determined. Figure 9 shows the results when a trapezoidal current with a commutation slope of 5' is phase advanced by 20°, 40°, 60' and 80'. These curves should be compared with the curve for torque presented in figure 4 . As the phase is advanced, the torque pulsations increase at the expense of the duration for which the torque remains constant. This is an extremely undesirable feature o f phase advancing. In addition the average value of torque is reduced greatly as shown in figure 10. It should be remembered that the average value is approximately 1 . 9 1 7 X I for a commutation slope of 5 O and when there is n$ 'phase advance. The magnitude and shape of the current are kept the same during the phase advancing and the dramatic reduction in the average torque is due only to the phase advance.

a

4

xs

0

The effects of phase advancing on individual torque harmonics was also examined. Since the flux linkage and current waveforms were maintained the same as during zero phase advance, it has been calculated that the individual torque harmonics are exactly the same magnitude as with zero phase advance for zero, 5 O , loo and 15' commutation angles. In other words even when a rectangular current is phase advanced, the individual torque harmonics are exactly the same magnitude as when there is no phase advance. In the zero phase advance case all the 6th torque harmonics for example are either in phase or 180° out of phase and sum to zero as shown in figure 7. However when the current is phase advanced relative to the back emf, the individual torque harmonics of a given order are not all in phase (even though they have the same magnitude) such that the cancellation shown in Table 1 for example does not occur. This results in an increase in the overall torque pulsations as shown in trapezoidal current. Similar figure 9 for the 5' results occur for the other commutation angles. Nonideal F l u x Linkage Effects

Z

0

E? ;70.00

5q.00

72.

a 10

no

2.

a a 4

H

X B 2 W

5

1.

m

LB l

I = lpu

P

M

206

36.0

Figure 9. Phase advancing results

01

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

18.00

(DEGREES)

x

P

=

lpu

2

I

I

20'

40'

I

60'

Phase advance angle Figure 10. Phase advancing results

I

EO0

a certain number of harmonics and evaluating (11) for a given F. Similarly, (15) is evaluated for a given (H-h) and multiplication of these instantaneous flux and current waveforms gives the instantaneous torque profile. The torque pulsation produced when the flux density is constant for looo and llOo instead of 120' are shown in figure 11. In this study, rectangular currents were used s o as to determine the contribution of the flux density alone on the torque pulsations. From these results, a graph of torque pulsation as a function of F can be drawn as shown in figure 1 2 . The graph is linear indicating a linear increase in torque pulsation with decrease in the duration of the constant flux density. In practice, the combination of the effects of the flux density and trapezoidal shaped currents would increase the torque pulsation beyond the contribution of each. A s the torque pulsations increase, so the average values decrease. However the reduction in the average torque is minimal when compared to that produced by the commutation of current as presented earlier and a graph is therefore not drawn.

a

H

a

4

X

I

=

h

= lpu

P

VI

g

0.6

4 U

P

lpu

0.4 ,-I

n 3 a,

0.2

J

L 0-

$

0.0

35O

30'

40°

F

VI. CONCLUSIONS ~j~~~~

detailed investigation into the torque behavior of a BDCM drive has been done in this paper. A previously published model to study the dynamic behavior was used to analyze the steady state behavior as well. It was shown that a constant output torque is produced only if the flux density and current waveforms of the BDCM are idealized. In the practical and hence nonideal case, torque pulsations arise as a result of the actual current being trapezoidal instead of rectangular, from the

12. Torque pulsations as a function of F

A

"c

I = lpu P

E

i

flux linka e being constant for less than 120° instead of the 1208 , or phase advancing of the current. The larger the commutation angle, the larger is the magnitude of the torque pulsations and the lower is the average value of the torque over a cycle. This is an extreme disadvantage of large commutation times. The commutation time is determined by the ratio of the stator leakage inductance to resistance, the operating speed and the dc bus voltage. Hence for a .given resistance, the leakage inductance should be minimized or for a given leakage inductance, the stator resistance should be maximized. Increasing the stator resistance should be done with due regard to the efficiency and cooling of the machine. The increase in the torque pulsations with increase in commutation angle is nonlinear with the increase being much larger up to '5 and then reducing after .'5 The increase in torque pulsations is linear with increase in the magnitude of the current being commutated. These torque pulsations may affect the accuracy and repeatability of position servos. A s the torque pulsations increase, so the average value of the torque decreases. A 12.5% decrease is possible over a full cycle and this can have consequences in the performance of torque, speed or position servos when using this machine. Phase advancing of the current waveform relative to the flux density can produce large torque pulsations with a resultant reduction in the average value of torque as well. Deviation of the flux density waveforms from the ideal also produce torque pulsations although the magnitude is not as large. In addition, the reduction in the average torque is not as severe as that due to the commutation in current. If the flux density wave is constant for a duration longer than 120°, then the output torque is not affected.

=1 pu

P.

F=35

"

H

I

F=40

4

a

-t

REFERENCES

I

%.00

I

18.00

QNGLE

I 1

36.00 (DEGREES)

I I

53-00

m10

72.

Figure I I . Torque pulsations for nonideal flux linkages

207

(11 R. Krishnan, "Selection criteria for servo motor drives", I E E E Trans., vol. IA-23, No. 2, March/April 1987, pp. 270-275. [2] D. Pauly, G . Pfaff and A.Weschta, "Brushless servo drives with permanent magnet motors or squirrel cage induction motors - a comparison," I E E E IAS Annual Meeting, 1984, pp. 503-509. (31 G, Pfaff, A. Weschta and A . Wick, "Design and experimental results of a brushless ac servo-drive", I E E E IAS Annual Meeting, 1982, pp. 692-697.

List of Symbols

[4] P. Pillay and R. Krishnan, "Application characteristics of permanent magnet sychronous and brushless dc motors for servo drives", IEEE IAS Annual Meeting, 1987, pp. 380-390. [5] P. Pillay and R. Krishnan, "Modeling, simulation and analysis of a permanent magnet brushless dc motor drive", IEEE IAS Annual Meeting, 1987, pp. 7-14. [6] T.M. Jahns, "Torque production in permanent magnet motor drives with rectangular current excitation," IEEE Trans., vol. IA-20, No.4, July/August 1984, pp. 803-813. [7] J.M.D. Murphy, "Thyristor control of ac motors", (book), Pergamon Press, 1973. [8] V. Subrahmanyam anl D. Subbarayudu, "Steady state analysis of an induction motor fed from a current source 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 with quasisquare 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 of the state of the art" Proceedings of the Motorcon Conference, 1981, pp. 1-16. (111 T. Sebastian and G.R. Slemon, "Operating limits of inverter driven permanent magnet motor drives," IEEE IAS Annual Meeting, 1986, pp.800-805.

B ea,eb,ec E i:,ib,ic

damping constant, N/rad/s a,b and c phase back emfs, V peak value of back emf, V a,b and c phase current J moment of inertia, kg-mh p A torque constant 2ea/wr Kt self inductance of a,b 6 c phases, H La,$,Lc mutual inductance betwen phases a & b , Lab P derivative operator P number of pole pairs R stator resistance, ohms e' electric torque, N-m Te1,Te5,Te7 lst,Sth and 7th torque harmonics,N-m TL load torque, N-m va,vb,vc a,b and c phase voltages, V dc bus voltage, V 'dc rotor speed, rad/sec WS synchronous speed, rad/sec

-

208