A comparison between star and delta connected induction motors when supplied by current source inverters

Electric Power Systems Research, 8 (1984/85) 41 - 51

41

A Comparison between Star and Delta Connected Induction Motors when Supplied by Current Source Inverters P. PILLAY, R. G. HARLEY and E. J. ODENDAL

Department of Electrical Engineering, University of Natal, King George V Avenue, Durban 4001 (South Africa)

(Received April 9, 1984)

SUMMARY

cy. The DC link between rectifier and inverter can be Operated with the link voltage held constant, or with the link current held constant; the latter m e t h o d is illustrated in Fig. 1 and is usually referred to as a current source inverter (CSI). Sinusoidal currents cannot be applied to the m o t o r from a CSI inverter; instead quasisquare blocks [1] of current are applied which adversely affect the m o t o r operation [ 1 - 3]. Some investigations [2, 3] have considered star connected induction m o t o r stators while others [4, 5] have evaluated delta connections. A comparison of the behaviour due to either a star or delta connected stator has n o t been presented. The purpose of this paper is therefore to investigate whether any differences in behavior arise due to the m e t h o d of stator connection of a CSI-fed m o t o r with special reference to the torque harmonics. The paper firstly evaluates the behaviour of the m o t o r with phases A, B and C connected in delta, and uses the current waveforms as shown in Fig. 2. It then evaluates the performance of this

This p a p e r analyses w h e t h e r a n y differe n c e s in behaviour arise d u e to an i n d u c t i o n m o t o r being star or delta c o n n e c t e d w h e n s u p p l i e d by a c u r r e n t source inverter. T w o c o m p a r i s o n s are considered. T h e first is that o f a delta c o n n e c t e d m o t o r c o m p a r e d to a star c o n n e c t e d m o t o r o f the same p o w e r a n d voltage ratings; the star c o n n e c t e d m o t o r in this case is t h e n m a t h e m a t i c a l l y the delta c o n n e c t e d m o t o r ' s e q u i v a l e n t star c o n n e c t i o n . T h e s e c o n d case c o n s i d e r e d is that o f a delta c o n n e c t e d m o t o r rewired as a star c o n n e c t e d motor.

1. INTRODUCTION

The induction m o t o r is robust and cheap and with the advent of p o w e r semiconductors is being used widely in variable speed applications. Frequency conversion usually takes place by first rectifying the fixed AC mains and then inverting to a new variable frequen-

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© Elsevier Sequoia/Printed in The Netherlands

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Fig. 2. Line and phase current waveforms in a delta connected induction motor.

same delta c o n n e c t e d m o t o r , b u t m a t h e m a t ically r e p r e s e n t s t h e d e l t a b y its star equivalent as s h o w n in Fig. 3. T h e b e n e f i t of using an equivalent star f o r s i m u l a t i o n p u r p o s e s is t h e simpler 3-step c u r r e n t w a v e f o r m o f Fig. 3

c o m p a r e d to the 4-step c u r r e n t w a v e f o r m of Fig. 2. T h e p a p e r t h e n c o n t i n u e s to analyse the same m o t o r , b u t with t h e t h r e e s t a t o r phases physically r e c o n n e c t e d into star as s h o w n in

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Fig. 4 in o r d e r t o establish w h e t h e r t h e actual c o n n e c t i o n has any e f f e c t on t h e m o t o r performance. In these c o m p a r i s o n s p a r t i c u l a r a t t e n t i o n is p a i d to t o r q u e h a r m o n i c s .

2. T H E O R Y

2.1. Analysis o f the current waveforms T a b l e 1 s h o w s a list o f m o t o r p a r a m e t e r s in t e r m s o f t h e original d e l t a winding, t h e equiv-

44 TABLE 1 Motor parameters

Line voltage (V) Phase voltage (V) Line current (A) Phase current (A) Base voltage (phase) (V) Base current (phase) (A) Base power (3Vphlph) (kVA) Base impedance ( Vph/ Iph ) ( ~ ) Stator resistance (~) Rotor resistance (~) Stator leakage reactance (~) Rotor leakage reactance (~) Magnetizing reactance ( ~ ) Stator resistance (p.u.) Rotor resistance (p.u.) Stator leakage reactance (p.u.) Rotor leakage reactance (p.u.) Magnetizing reactance (p.u.)

Motor A Delta

Motor B Equivalent star

Motor C Star

380 380 43.20 24.94 380 24.94 28.43 15.24 0.333 0.897 0.762 2.052 47.4 0.021 0.057 0.049 0.132 3.038

380 220 43.20 43.20 220 43.20 28.43 5.08 0.111 0.299 0.254 0.684 15.8 0.021 0.057 0.049 0.132 3.038

658 380 24.94 24.94 380 24.94 28.43 15.24 0.333 0.897 0.762 2.052 47.4 0.021 0.057 0.049 0.132 3.038

Fig. 4. Delta winding reconnected in star. alent star (or in o t h e r w o r d s a star c o n n e c t e d m o t o r o f t h e s a m e p o w e r a n d v o l t a g e ratings) a n d t h e m o t o r r e c o n n e c t e d in a star c o n n e c t i o n ; f o r c o n v e n i e n c e t h e y are r e f e r r e d t o as m o t o r s A, B a n d C r e s p e c t i v e l y . An i m p o r t a n t result f r o m T a b l e 1 is t h a t t h e p e r u n i t imp e d a n c e s are e q u a l f o r all t h r e e m o t o r s [ 6 ]. The characteristic current waveform of e a c h w i n d i n g (Figs. 2 - 4) is a p p l i e d t o e a c h m o t o r t o d e t e r m i n e such q u a n t i t i e s as r u n - u p time, terminal voltage, speed and torque. The c u r r e n t s in Figs. 2 - 4 can be d e s c r i b e d in t e r m s o f t h e i r F o u r i e r series, a n d in e a c h case t h e p e a k value o f t h e f u n d a m e n t a l is c h o s e n as 1 p.u. This e n s u r e s t h a t all t h r e e m o t o r s are able t o s u p p l y t h e s a m e o u t p u t p o w e r . Also it is k n o w n [5, 7] t h a t t h e m a g n i t u d e o f t h e fundamental component of current determ i n e s t h e average r e s p o n s e o f the i n d u c t i o n motor (for example run-up time), with the h a r m o n i c c u r r e n t s h a v i n g o n l y parasitic e f f e c t s . F o r e x a m p l e , t h e average t o r q u e t h a t

is r e s p o n s i b l e f o r r u n n i n g t h e m a c h i n e u p to s p e e d is d u e to i n t e r a c t i o n b e t w e e n t h e f u n d a m e n t a l air gap f l u x a n d f u n d a m e n t a l rotor current; the harmonic rotor currents do n o t a f f e c t this t o r q u e , t h e y m e r e l y p r o d u c e p u l s a t i n g t o r q u e s t h a t have average values of zero. T h e p h a s e c u r r e n t in Fig. 2 has t h e following F o u r i e r series:

ias = ( 3 / n ) I i [ s i n ( n e t )

+ sin(5et)/5

+ sin(7et)/7 +...] (1) w h e r e n is the h a r m o n i c n u m b e r . T h e c u r r e n t s in t h e o t h e r t w o phases o f t h e m o t o r , i b s ( e t ) a n d i c s ( e t ) are d e s c r i b e d b y similar expressions e x c e p t t h a t t h e s i n ( n e t ) in eqn. (1) is r e p l a c e d b y s i n ( n e t - - k) a n d s i n ( n e t + k) respectively. T h e p h a s e c u r r e n t in Fig. 3 on the o t h e r h a n d h a s t h e f o l l o w i n g F o u r i e r series: /as = ( 2 X / ~ /Tr )I2[ s i n ( n e t ) - - s i n ( 5 e t ) / 5 -- sin(7et)/7 + ...]

(2)

E q u a t i o n s (1) a n d (2) h a v e t h e s a m e h a r m o n i c c o m p o n e n t s e x c e p t f o r 1 8 0 ° p h a s e shifts in t h e 5th, 7th, 17th, 1 9 t h , etc. h a r m o n i c s . T h e f u n d a m e n t a l c o m p o n e n t s of t h e t w o p h a s e c u r r e n t s are d r a w n in Figs. 2 - 4; in t h e d e l t a case the m a g n i t u d e of t h e f u n d a m e n t a l c o m p o n e n t is (3/7r)I 1 = 0 . 9 5 5 I1, w h e r e a s in t h e star case t h e m a g n i t u d e of t h e f u n d a m e n t a l c o m p o n e n t is ( 2 x / ~ / T r ) I 2 = 1 . 1 0 3 12; t h a t is,

45

the fundamental c o m p o n e n t of current is smaller than the peak value of the delta waveform b u t larger than the peak value of the star waveform. Hence to ensure a 1 p.u. fundamental c o m p o n e n t in the delta case (motor A), I~ is chosen equal to 1.047 p.u., while to ensure a 1 p.u. fundamental c o m p o n e n t in the star cases (B and C),/2 is chosen equal to 0.91 p.u.

Since all the harmonics have amplitudes that are inversely proportional to their order, the amplitudes of the corresponding harmonics of the delta and star waveforms are equal if their fundamental components are equal.

2.2. Two~xis analysis o f the CSI-fed induction motor The analysis of the CSI-fed induction motor, summarized below, is based on the well-known [ 8] two-axis theory. An idealized symmetric m o t o r is assumed with a balanced sinusoidal airgap m m f and a linear magnetic circuit. Iron and mechanical losses, stray load losses and mechanical damping are all neglected. All m o t o r resistances and inductances are independent of frequency, which limits the usefulness of these models to wound rotor and single cage rotors with shallow bars. The two-axis voltage equations of a voltagefed induction m o t o r can be summarized as follows in terms of a reference frame rotating in synchronism with the fundamental component of the stator current: Iv] = [R ] [i] + [Lip [i] + ogi[F] [i] + swi [G] [i] (3) where [U] ---- [Udl , Vql , Vd2 , Uq2] T

(4)

[ i ] = [idl , iql , id2 , iq2] T

(5)

s = (¢oi -

(6)

¢Or)/O~i

The other matrices in eqn. (3) appear in Appendix C. Moreover, in the case of a current source inverter, idl and iql are independent predefined variables (obtained from Park's transform of i,~, ibs and ic~) and differential equations are only required for the rotor or secondary currents id2 and iq2 such that [4] [v2] = [R2][i2] + [L2]p[i2] + [Lm]p[il] + C O l [ V i i [ i l ] + s o g i [ G 2 ] [i2]

where

(7)

[v~] = [Vd2, vq2] T

[i21 = lid2, iq2] T

(8)

[ i l ] = [idl , iql] T

The other matrices in eqn. (7) appear in Appendix C. In a short~ircuited rotor Vd2 and Vq2 are zero, in which case eqn. (8) can be rearranged to yield the following differential equations for the rotor currents in state space form:

p[i2] = - - [ S ] ([R2] + s¢oi[G2] [i2] + scoi[G1][il] + [Lm]p[il]) (9) where [B] = [L2] -1. Equation (9) is nonlinear and is integrated numerically step by step to yield values for id2 and iq2, and together with idl and iql these are used to c o m p u t e the electrical torque Te from T e = LmOgb(id2iql

-- iq2idl)/3

(I0)

The mechanical motion is described by POOr= (Te

--

TL)/J

(11)

For a sinusoidal line current to the m o t o r idl and iql are constant quantities in a synchronously rotating reference frame. However, in the case of an inverter-fed motor, where the line current consists of a series of harmonics (eqns. (1) and (2)), idl and iql are functions of time and are defined by the Park transform operating on each harmonic c o m p o n e n t and summing the result. Expressions for Pidl, piql are required in eqn. (9) and are found by differentiating the summed series expressions for ial and iql , as shown in Appendix D. A c o m p u t e r program was developed to predict the dynamic behaviour of the CSI-fed induction m o t o r drive. The independent variables a r e idl , iql , P i d l a n d p i q l , and their accuracy depends on the number n of harmonics used in the Fourier series. The value of n has to be infinity in order to represent the waves in Figs. 2(b) and 3(b) exactly. However, it was shown elsewhere [4] that thirty-one (n = 31) current harmonics yield sufficient accuracy with the advantage of keeping computation time down. The program starts by finding the magnitude of each of the thirty-one harmonic components. It then calculates idl, iql, Pidl and piql. From this it calculates id2, iq2, Te and cot as stated above at each step of the integration process.

46 3. RESULTS This section uses the above techniques to evaluate the response of the delta connected m o t o r A (which has a 4-step waveform) and of the star connected motors B and C (which have 3-step waveforms). Note that a simulation done for m o t o r B is also true for m o t o r C if the analysis is carried out in p.u. form, because the current waveform is the same and they have the same p.u. impedances; however, the physical magnitudes o f the different variables will be different because they have different voltage and current base values. The significance of this will be explained later. Figure 5 shows the no-load start-up results when the m o t o r is supplied from the 10 Hz, 4-step current waveform of Fig. 2, while Fig. 6 shows the corresponding waveforms when the m o t o r is started up from the 3-step current waveform of Figs. 3 and 4. These results show that the run-up time and torque pulsations in p.u. are exactly the same for all three motors. However, since all three machines have the same power and frequency bases, the physical magnitudes of the m o t o r torque pulsations are also equal. The magnitude of the terminal voltages are equal, but the 3-step current waveform with four current transitions per cycle produces four voltage spikes per cycle, while the 4-step current waveform produces six voltage spikes per cycle. Since these voltage spikes stress the m o t o r insulation, the star connected m o t o r which produces the lower number of voltage spikes may be preferable. Figure 7 shows the FFTs of the phase currents for the star (3~step) and delta (4-step) waveforms. The FFT of the star waveform in Fig. 7(a) shows that the fundamental comp o n e n t of the current is 2 p.u. peak-to-peak (the magnitude of the fundamental component of current was chosen as 1 p.u. peak); the magnitude of the fundamental c o m p o n e n t of the delta waveform in Fig. 7(b) is also 2 p.u. peak-to-peak, and the magnitudes of all the relative current harmonics in Fig. 7(b) are equal to those in Fig. 7(a). The FFTs of torque in Figs. 7(c) and (d) show that the individual torque harmonics which add up to produce the torque pulsations in Figs. 5(c) and 6(c) are also equal. These results show that the run-up time and the magnitude of the torque pulsations

are the same for the delta and star connected motors when expressed in p.u. The line current needed (this specifies the link current indirectly) is 43.2 A rms for both the delta connected m o t o r A and the star connected m o t o r B of the same power and voltage ratings. Hence, a given current source inverter able to supply current to m o t o r A would also be able to supply it to m o t o r B. The only difference in operation between these two motors is the presence of a different number of voltage spikes per cycle. However, m o t o r C only requires 24.94 A in its line in order to supply full power. This means that the current ratings of its current source inverter can be less. Nevertheless, its terminal voltage {when it draws 24.94 A) is 658 rms, and in any practical system this voltage gets reflected back to the input voltages of the rectifier which must now be rated at 658 V rms. Hence the induction m o t o r winding cannot be changed from delta to star when driven by a current source inverter without changing the ratings of the current source inverter itself (i.e. a reduction of its current rating by x/3 and an increase of its voltage rating by xf3). These results also show that, as far as the dynamics of the m o t o r are concerned, a delta connected m o t o r can be analysed in terms of its equivalent star connection.

4. CONCLUSIONS This paper has compared the differences in behaviour between a delta connected motor, its star equivalent of the same power and voltage ratings and the original delta reconnected into star. Simulations have been carried out with the motors represented by their two-axis equations and their current waveforms by Fourier series. The magnitudes of the fundamental components of the phase currents for all three cases were chosen to be equal to 1 p.u. The following conclusions can be drawn: (a) The harmonic current components present in the phase current of a delta connected induction m o t o r fed from a CSI are the same as those present in a star connected motor. (b) Provided the magnitudes of the fundamental components of these currents are specified as equal in p.u., the delta connected

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motor, its star equivalent and the reconnected star have the same run-up time. This also means that any delta connected m o t o r can be analysed in terms of its star equivalent for dynamic studies. (c) The torque pulsations produced in all three motors are the same because they all have the same power and frequency base values. Individual torque harmonics which add up to make the torque pulsations are also equal in magnitude for all three motors. (d) The only difference in response between a star connected and a delta connected m o t o r supplied from a current source inverter is the reduction in voltage spikes in the case of the star connections.

ACKNOWLEDGEMENTS

The authors acknowledge the assistance of D.C. Levy, R.C.S. Peplow and H.L. Nattrass in the Digital Processes Laboratory of the Department of Electronic Engineering, University of Natal. They are also grateful for financial support received from the CSIR and the University of Natal.

NOMENCLATURE

CSI i~s iaQ I1

current source inverter phase A stator current line current corresponding to phase A peak value of /as for the delta connected m o t o r I2 peak value of i~ for the star connected m o t o r idl, Vdl stator d-axis current and voltage id2, Vd2 rotor d-axis current and voltage iql, Vql stator q-axis current and voltage iq~, vq2 rotor q-axis current and voltage J inertia of m o t o r L 11 stator self inductance L22 rotor self inductance Lm mutual inductance p derivative operator d/dt R 1, R 2 stator and rotor phase resistances s slip Te electrical torque TL load torque k 27r/3 0 arbitrary angle = cot co arbitrary frequency

nominal frequency at which p.u. torque = p.u. power fundamental frequency of inverter rotor speed

cob col cot

REFERENCES 1 J. M. D. Murphy, Thyristor Control o f AC Motors, Pergamon Press, 1973, ISBN 0080169430. 2 T. A. Lipo, Analysis and control of torque pulsations in current fed induction m o t o r drives, IEEE IAS Annual Meeting, 1978, Paper No. CH 1337-

5/78. 3 W. Farrer and T. D. Miskin, Quasi-sinewave fully regenerative inverter, Proc. Inst. Electr. Eng., 120 (1973) 9 6 9 - 976. 4 R. G. Harley, P. Pillay and E. J. Odendal, Analysing the dynamic behaviour of an induction motor fed from a current source inverter, Trans. S. Afr. Inst. Electr. Eng., 75 (3) (Nov./Dec.) (1984). 5 V. Subrahmanyam, S. Yuvarajan and B. Raamaswami, Analysis of commutation of a current inverter feeding an induction m o t o r load, IEEE Trans., IA-16 (1980) 332 - 341. 6 B. M. Weedy, Electric Power Systems, Wiley, Chichester, U.K., 1979, ISBN 0 471 27584 0. 7 M. L. Macdonald and P. C. Sen, Control loop study of induction m o t o r drives using DQ model, Trans. IEEE, IECI-26 (Nov.) (1979) 237 - 243. 8 B. Adkins and R. G. Harley, The General Theory o f Alternating Current Machines: Application to Practical Problems, Chapman and Hall, London, 1975, ISBN 0 412 155605. 9 L. P. Heulsman, Basic Circuit Theory with Digital Computations, Prentice Hall, Englewood Cliffs, NJ, 1972, ISBN 0 131 574309.

APPENDIX A

Derivation o f the Fourier series o f line current The stator phase current waveform i~s applied to the star connected m o t o r is shown in Fig. 3. This waveform is symmetrical about the cot axis and f(cot)=--f(--cot). Hence i~s can be represented as a Fourier series [9] such that i~s = B1 sin(cot) + B2 sin(2cot) + . . .

+ B, sin(ncot)

(A-l)

where

B, = [cos(n~'/6) -- cos(n5~'/6)] 211/nrr

(A-2)

Hence ias =

~ Bn sin(nogt) n=l

(A-3)

51 oo

ibs =

Sn sin(ncot- ),)

E

APPENDIX

C

n=l

(A-3) its = ~ B. sin(ncot + X) n=l

Definition o f Park's orthogonal transformation matrix

(C-l)

[F0dql] = [Po][Fabcl]

This analysis is similar for the phase current waveform in Fig. 2.

where

APPENDIX

t P o ] - - ~ Icon0 co~<0-x) co~(0 ÷ x) /

B

[_sin 0

Elements o f ma trices

sin(O-- X)

sin(O+ X) _]

(c-2) Ii I JR] =

[L] =

0 R,

00

0

R2

0 0 ] 0

0

0

R2

[Fabcl ] =

In the synchronously rotating reference frame where the frame rotates at speed co, the angle 0 = cot.

iol 11° i l ~ Lm

0 Lm

L22

APPENDIX D

L22J

0

(c-3)

[po]-1[Fodql]

Derivatives o f the stator currents

[F] =

I --Lll 0

Lll 0

From Park's (Appendix C),

Lm 0 --L m i l

o

o

0

0

orthogonal

transform

idl = ~ : - ~ [ i A L COS 0 + iBL COS(0 -- ~,) + iCL COS(0 + ~)]

(D-l)

Hence

[G] =

Ii °

0 Lm

--Lm

0

Pidl = V / ~ [ P i A L COS 0 +PiBL COS(0 -- ~,)

0

o

+ PicL COS(0 + X) -- iALCOsin 0

L~

-- iBL(-Osin(0 -- X) -- iCL6) sin(0 + )k)]

--L22 (D-2)

I0" .°1 [G1] =

Lm

[Gel =

iql = ~V/f-~[iAL sin 0 + iBL Sin(0 -- ~,)

L~2

+

[Lml =

E mJ Lm 0

0 L

iCL sin(0 + X)]

(D-3)

Hence piql = ~¢/~[PiAL Sin 0 +PisL sin(0 -- X)

+PicL sin(0 + X) + iAL09COS0 -6 iBLCX)COS(0 - - ~ ) "b iCLL0 COS(0 "6 ~k)]

(D-4)