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

A Comparison between Star and Delta Connected Induction Motors whenSupplied 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)

SUMMAR Y

This paper analyses whether any differ-ences in behaviour arise due to an inductionmotor being star or delta connected whensupplied by a current source inverter. Twocomparisons are considered. The first is thatof a delta connected motor compared to astar connected motor of the same power andvoltage ratings; the star connected motor inthis case is then mathematically the deltaconnected motor's equivalent star connection.The second case considered is that of a deltaconnected mo tor rewired as a star connectedmotor.

1. INTRODUCTION

The induction motor is robust and cheapand with the advent of power semiconductorsis being used widely in variable speed applica-tions. Frequency conversion usually takesplace by first rectifying the fixed AC mainsand then inverting to a new variable frequen-

R

w

B

CONTROL

CI RCU!TRY

CONTROL

CI RCU!TRY

Fig. 1. Current source inverter-fed induction motor drive.

0378-7796/84/$3.00

cy. The DC link between rectifier and invertercan be operated with the link voltage heldconstant, or with the link current held con-stant; the latter method is illustrated in Fig. 1and is usually referred to as a current sourceinverter (CSI).

Sinusoidal currents cannot be applied tothe motor from a CSI inverter; instead quasi-square blocks [1] of current are appliedwhich adversely affect the motor operation[1 - 3]. Some investigations [2,3] have con-sidered star connected induction motorstators while others [4,5] have evaluateddelta connections. A comparison of thebehaviour due to either a star or delta

connected stator has not been presented.The purpose of this paper is therefore to

investigate whether any differences in behav-ior arise due to the method of stator connec-

tion of a CSI-fed motor with special referenceto the torque harmonics. The paper firstlyevaluates the behaviour of. the motor with

phases A, Band c..c..Qllnected in delta, anduses the current waveforms as shown in Fig.2. It then evaluates th~ performance of this

"

.

@ Elsevier Sequoia/Printed in The Netherlands

'i~

; .'U- ~-_.

I"nd.. amenta!

-",

- ..-..

-. .-- .- - ..-

.- .-- .. -_u. -- -0 -0 .-,- fundamental

- ~'311 . _. --.

I

I,

iAL 0

fundamental

. --- - _n_- --_.

--u_-_u- .-- ---. - , . -

-I,r

IDC

1600

1L2o°

11800

12400

13000

13600

L. --. ---4200

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

same delta connected motor, but mathemat-ically represents the delta by its star equiv-alent as shown in Fig. 3. The benefit of usingan equivalent star for simulation purposes isthe simpler 3-step current waveform of Fig. 3

compared to the 4-step current waveform ofFig. 2.

The paper then continues to analyse thesame motor, but with the three stator phasesphysically reconnected into star as shown in

111:.

!-sII

ias 0

-1II

-3-:31,

I,

IIi 0cs

43

iAL

iSL.-ibs

iCLics

h

iAL

ias

0

--- - - - - --- - -- - --

-12,.. - h .- ----

j~

00

I

6~ ltoo

Fig. 3. Line and phase current waveforms in a star connected induction motor.

u_- -----

fundamental

--- -- - --

-- __bO- --- ----

t180°

.L.-3000

13600

-1420.

12400

Fig. 4 in order to establish whether the actualconnection has any effect on the motor per-formance.

In these comparisons particular attention ispaid to torque harmonics.

2. THEORY

2.1. Analysis of the current waveformsTable 1 shows a list of motor parameters in

terms of the original delta winding, the equiv-

iSL. I .

B

..lCL

iAL tias

44

TABLE 1

Motor parameters

Fig. 4. Delta winding reconnected in star.

alent star (or in other words a star connectedmotor of the same power and voltage ratings)and the motor reconnected in a star connec-tion; for convenience they are referred to asmotors A, Band C respectively. An importantresult from Table 1 is that the per unit im-pedances are equal for all three motors [6].

The characteristic current waveform ofeach winding (Figs. 2 -4) is applied to eachmotor to determine such quantities as run-uptime, terminal voltage, speed and torque. Thecurrents in Figs. 2 - 4 can be described interms of their Fourier series, and in each casethe peak value of the fundamental is chosenas 1 p.u. This ensures that all three motors areable to supply the same output power. Also itis known [5, 7] that the magnitude of thefundamental component of current deter-mines the average response of the inductionmotor (for example run-up time), with theharmonic currents having only parasiticeffects. For example, the average torque that

is responsible for running the machine up tospeed is due to interaction between thefundamental air gap flux and fundamentalrotor current; the harmonic rotor currents donot affect this torque, they merely producepulsating torques that have average values ofzero.

The phase current in Fig. 2 has the follow-ing Fourier series:

ias = (3j1T)II[sin(nwt) + sin(5wt)j5

+ sin(7wt)j7 + . . .] (1)where n is the harmonic number. The currentsin the other two phases of the motor, ibs(wt)and ics(wt) are described by similar expres-sions except that the sin(nwt) in eqn. (1) isreplaced by sin(hwt - A) and sin(nwt + A) res-pectively.

The phase current in Fig. 3 on the otherhand has the. following Fourier series:

ias = (2V3j1T)I2[sin(nwt) - sin(5wt)j5- sin(7wt)j7 + . . .] (2)

Equations (1) and (2) have the same harmoniccomponents except for 1800 phase shifts inthe 5th, 7th, 17th, 19th, etc. harmonics. Thefundamental components of the two phasecurrents are drawn in Figs. 2 - 4; in the deltacase the magnitude of the fundamental com-ponent is (3j1T)II= 0.955 II, whereas in thestar case the magnitude of the fundamentalcomponent is (2V3j1T)I2 = 1.103 12; that is,

Motor A Motor B Motor CDelta Equivalent star Star

Line voltage (V) 380 380 658 1Phase voltage (V) 380 220 380Line current (A) 43.20 43.20 24.94

Phase current (A) 24.94 43.20 24.94

Base voltage (phase) (V) 380 220 380

Base current (phase) (A) 24.94 43.20 24.94

Base power (3Vphlph) (kV A) 28.43 28.43 28.43

Base impedance (Vphllph) (it) 15.24 5.08 15.24

Stator resistance (it) 0.333 0.111 0.333

Rotor resistance (it) 0.897 0.299 0.897

Stator leakage reactance (it) 0.762 0.254 0.762

Rotor leakage reactance (it) 2.052 0.684 2.052

Magnetizing reactance (it) 47.4 15.8 47.4

Stator resistance (p.u.) 0.021 0.021 0.021

Rotor resistance (p.u.) 0.057 0.057 0.057

Stator leakage reactance (p.u.) 0.049 0.049 0.049

Rotor leakage reactance (p.u.) 0.132 0.132 0.132

Magnetizing reactance (p.u.) 3.038 3.038 3.038

the fundamental component of current issmaller than the peak value of the delta wave-form but larger than the peak value of the starwaveform. Hence to ensure a 1 p.u. funda-mental component in the delta case (motorA), II is chosen equal to 1.047 p.u., while toensure a 1 p.u. fundamental component in thestar cases (B and C), Iz is chosen equal to 0.91p.u.

Since all the harmonics have amplitudesthat are inversely proportional to their order,the amplitudes of the corresponding harmon-ics of the delta and star waveforms are equalif their fundamental components are equal.

2.2. Two-axis analysis of the CSI-fedinduction motor

The analysis of the CSI-fed inductionmotor, summarized below, is based on thewell-known [8] two-axis theory. An idealizedsymmetric motor is assumed with a balancedsinusoidal airgap mmf and a linear magneticcircuit. Iron and mechanical losses, stray loadlosses and mechanical damping are all neg-lected. All motor resistances and inductancesare independent of frequency, which limitsthe usefulness of these models to wound rotorand single cage rotors with shallow bars.

The two-axis voltage equations of a voltage-fed induction motor can be summarized asfollows in terms of a reference frame rotatingin synchronism with the fundamental com-ponent of the stator current:

[v] = [R][i] + [L]p[i] + Wi[F][i] +swi[G][i]

(3)where

[v] = [Vd\> Vq\> VdZ, lIqZ]T

[i] = [idl> iq\> idz, iqzF

s = (Wi - Wr)/Wi

(4)

(5)

(6)

The other matrices in eqn. (3) appear inAppendix C. Moreover, in the case of a cur-rent source inverter, idl and iql are indepen-dent predefined variables (obtained fromPark's transform of ias, ibs and ics) and differ-ential equations are only required for therotor or secondary currents idz and iqz suchthat [4]

[vz] = [Rz] [iz] + [Lz]p[iz] + [Lm]p[id

+ Wi[Gd [id + swJ Gz] [iz]where

(7)

45

[Vz] = [VdZ,vqzF

[iz] = [idZ, iqzF

[id = [idl, iqlF

(8)

The other matrices in eqn. (7) appear inAppendix C. In a short-circuited rotor VdZandVqZare zero, in which case eqn. (8) can be re-arranged to yield the following differentialequations for the rotor currents in state spaceform:

p[iz] = -[B]{[Rz] + sWi[Gz][iz]

+ SWi[Gd [id + [Lm]P [id} (9)

where [B] = [Lzr1. Equation (9) is nonlinearand is integrated numerically step by step toyield values for idz and iqz, and together withidl and iql these are used to compute the elec-trical torque Te from

Te = LmWb(idZiql - iqzidd/3

The mechanical motion is described by

(10)

PWr = (Te - Td/J (11)

For a sinusoidal line current to the motor idland iql are constant quantities in a synchro-nously rotating reference frame. However, inthe case of an inverter-fed motor, where theline current consists of a series of harmonics

(eqns.(l) and (2», idl and iql are functions oftime and are defined by the Park transformoperating on each harmonic component andsumming the result. Expressions for pidl> piqlare required in eqn. (9) and are found by dif-ferentiating the summed series expressions foridl and iql , as shown in Appendix D.

A computer program was developed topredict the dynamic behaviour of the CSI-fedinduction motor drive. The independentvariables are idl> iq\>pidl and piql, and theiraccuracy depends on the number n ofharmonics used in the Fourier series. Thevalue of n has to be infinity in order to rep-resent the waves in Figs. 2(b) and 3(b) exact-ly. However, it was shown elsewhere [4] thatthirty-one (n = 31) current harmonics yieldsufficient accuracy with the advantage ofkeeping computation time down.

The program starts by finding the mag-nitude of each of the thirty-one harmoniccomponents. It then calculates idl, iql> pidland piql. From this it calculates idz, iqz, Teand wr as stated above at each step of theintegration process.

46

3. RESULTS

This section uses the above techniques toevaluate the response of the delta connectedmotor A (which has a 4-step waveform) andof the star connected motors Band C (whichhave 3-step waveforms). Note that a simula-tion done for motor B is also true for motor Cif the analysis is carried out in p.u. form, be-cause the current waveform is the same andthey have the sanle p.u. impedances; however,the physical magnitudes of the differentvariables will be different because they havedifferent voltage and current base values. Thesignificance of this will be explained later.

Figure 5 shows the no-load start-up resultswhen the motor is supplied from the 10 Hz,4-step current waveform of Fig. 2, while Fig.6 shows the corresponding waveforms whenthe motor is started up from the 3-stepcurrent waveform of Figs. 3 and 4. Theseresults show that the run-up time and torquepulsations in p.u. are exactly the same for allthree motors. Howeyer, since all three ma-chines have the same power and frequencybases, the physical magnitudes of the motortorque pulsations are also equal. The mag-nitude of the terminal voltages are equal, butthe 3-step current waveform with four currenttransitions per cycle produces four voltagespikes per cycle, while the 4-step currentwaveform produces six voltage spikes percycle. Since these voltage spikes stress themotor insulation, the star connected motorwhich produces the lower number of voltagespikes may be preferable.

Figure 7 shows the FFTs of the phase cur-rents for the star (3-step) and delta (4-step)waveforms. The FFT of the star waveform inFig. 7(a) shows that the fundamental com-ponent of the current is 2 p.u. peak-to-peak(the magnitude of the fundamental compo-nent of current was chosen as 1 p.u. peak);the magnitude of the fundamental componentof the delta waveform in Fig. 7(b) is also 2p.u. peak-to-peak, and the magnitudes of allthe relative current harmonics in Fig. 7(b) areequal to those in Fig. 7(a). The FFTs oftorque in Figs. 7(c) and (d) show that theindividual torque harmonics which add up toproduce the torque pulsations in Figs. 5(c)and 6(c) are also equal.

These results show that the run-up timeand the magnitude of the torque pulsations

are the same for the delta and star connectedmotors when expressed in p.u. The line cur-rent needed (this specifies the link current in-directly) is 43.2 A rms for both the deltaconnected motor A and the star connectedmotor B of the same power and voltage rat-ings. Hence, a given current source inverterable to supply current to motor A would alsobe able to supply it to motor B. The only dif-ference in operation between these twomotors is the presence of a different numberof voltage spikes per cycle. However, motor Conly requires 24.94 A in its line in order tosupply full power. This means that thecurrent ratings of its current source invertercan be less. Nevertheless, its terminal voltage(when it draws 24.94 A) is 658 rms, and inany practical system this voltage gets reflectedback to the input voltages of the rectifierwhich must now be rated at 658 V rms.Hence the induction motor winding cannot bechanged from delta to star when driven by acurrent source inverter without changing theratings of the current source inverter itself(Le. a reduction of its current rating by .J3and an increase of its voltage rating by .J31.These results also show that, as far as thedynamics of the motor are concerned, a deltaconnected motor can be analysed in terms ofits equivalent star connection.

4. CONCLUSIONS

This paper has compared the differences inbehaviour between a delta connected motor,its star equivalent of the same power andvoltage ratings and the original delta recon-nected into star. Simulations have beencarried out with the .motors represented bytheir two-axis equations and their currentwaveforms by Fourier: series. The magnitudesof the fundamental components of the phasecurrents for all three eases were chosen to beequal to 1 p.u. The following conclusions canbe drawn:

(a) The harmonic' current componentspresent in the phase ~urrent of a delta con-nected inductionmobor fed from a CSI arethe same as those present in a star connectedmotor.

(b) Provided the magnitudes of the funda-mental components of these currents arespecified as equal in p.u., the delta connected

1.PHASE CURRENT

:i

n:

".

I-ZIII0::0:: -1."::>u

-1.5iii."" .5" .15

(a)

1."" 1.25

SEC

2.

.25

TIME

PHASE VOLTAGE

:i

n: 1.

".

IIIU -1.<I-J0>

-2."iii."" .25

TIME.5" .15

(c)1.25

SEC

:i

n:

III::>CJ0::0I-

1.5"

2.'"TORQUE

1.""

-1. "Jii."" .25 .50

TIME

250.-- SPEED (1)>/1>>.)

(b)

iii 50. ",IIIa.U)

1.5"

Fig. 5.Predicted start-upof a delta connected CSI.fed induction motor.

".",iii."" . 25

TIME (d)1.25

SEC1.50.15 1."".5"

1.1!JPHASE CURRENT

::in:

I-ZW0:0:::JU

-1.1!J9.1!J1!J

---+---

1.25

SEC

1.5

.25TIME

PHASE VOLTAGE. .

.51!J .75

(a)

1. I!JI!J

-+--

::in:

-1. 59.1!J1!J

~ -.

1.25

SEC

.25

TIME.51!J .75

(c)

1.01!J

-+-f--TORQUL--+---

::in:

W::J00:QI-

1.51/1I I I I.25 .51!J .75 1.1!J1!J

TIME (b)

SPEED (w/ws)251!J.1!J:

1.51!JI!J.I!J,

9.1!J1!J .75

(d)

.25TIME

.51!J

Fig. 6. Predicted start-up of a star connected CSI-fed induction motor.

H>-00

1. 25SEC

1. 51!J

1.1/10 1.25

SEC

1. 50

::Jn:

FFT OF CURR-STAR+ f f--

IFFT OF CURR-DELTA

t--1

::Jn:

Fig. 7. FFTs of current and torque for the star and delta connected induction motors.

1.0

l1J l1J0 0::J ::JI- .5 I-

5 III 5...J ...JIL IL

< 29 31 35 37 <

0.0aILl 1\1\ ;A-A11 1 LJ LJ 29 31 35 37

0.0 1\ 1\, /\/\ /\,/\ 1\ 1\0.00 100.00 30"- 00 400.00 "- 00 100.00 200.00 300.00 400.00

FREO. HZ. FREO. (b) HZ.

FFT OF TORO-STAR

1000. 1111

FFT OF TORO-DELTA-----+

-1rh

,6th::J

I

::Jn:

12th

I E400.0i II

12th

18th 18th24,th '''_h "_I. 1 24th

l1J0::JI-

]' IL-J,L-JL!

...J ...JIL IL< <

-200.0.00 100.00 200.00 300.00 4Q10.1/!1/! 0.01/! 11/!0.1/!0 201/!.1/!0 300.1/!0 400.00

FREO. (c) HZ. FREO. (d) HZ.

motor, its star equivalent and the reconnectedstar have the same run-up time. This alsomeans that any delta connected motor can beanalysed in terms of its star equivalent fordynamic studies.

(c) The torque pulsations produced in allthree motors are the same because they allhave the same power and frequency basevalues. Individual torque harmonics whichadd up to make the torque pulsations are alsoequal in magnitude for all three motors.

(d) The only difference in response be-tween a star connected and a delta connectedmotor supplied from a current source inverteris the reduction in voltage spikes in the caseof the star connections.

ACKNOWLEDGEMENTS

The authors acknowledge the assistance ofD.C. Levy, R.C.S. Peplow and H.L. Nattrassin the Digital Processes Laboratory of theDepartment of Electronic Engineering,University of Natal. They are also grateful forfinancial support received from the CSIR andthe University of Natal.

NOMENCLATURE

CSI current source inverterphase A stator currentline current corresponding to phase Apeak value of ias for the delta con-nected motorpeak value of ias for the star con-nected motor

idl> Vdl stator d-axis current and voltageid2, Vd2 rotor d-axis current and voltageiql> Vql stator q-axis current and voltageiq2, Vq2rotor q-axis current and voltageJ inertia of motorL11 stator self inductanceL22 rotor self inductanceLm mutual inductancep derivative operator d/dtR 1, R 2 stator and rotor phase resistancess slipTe electrical torqueTL load torqueA 2rr/3() arbitrary angle = wtw arbitrary frequency

iasiallII

12

Wb nominal frequency at which p.u.torque =p.u. powerfundamental frequency of inverterrotor speed

Wi

Wr

REFERENCES

1 J. M. D. Murphy, Thyristor Control of ACMotors, Pergamon Press, 1973, ISBN0080169430.

2 T. A. Lipo, Analysis and control of torque pulsa-tions in current fed induction motor drives, IEEEIAS Annual Meeting, 1978, Paper No. CH 1337-5/78.

3 W. Farrer and T. D. Miskin, Quasi-sinewave fullyregenerative inverter, Proc. Inst. Electr. Eng., 120(1973) 969 - 976.

4 R. G. Harley, P. Pillay and E. J. Odendal, Analys-ing the dynamic behaviour of an induction motorfed from a current source inverter, Trans. S. Afr.Inst. Electr. Eng., 75 (3) (Nov./Dec.) (1984).

5 V. Subrahmanyam, S. Yuvarajan and B. Raamas-wami, Analysis of commutation of a currentinverter feeding an induction motor load, IEEETrans., IA-16 (1980) 332 - 341.

6 B. M. Weedy, Electric Power Systems, Wiley,Chichester, U.K_, 1979, ISBN 0471 27584 O.

7 M. L. Macdonald and P. C. Sen, Control loopstudy of induction motor drives using DQ model,Trans. IEEE, IECI-26 (Nov.) (1979) 237 - 243.

8 B. Adkins and R. G. Harley, The General Theoryof Alternating Current Machines: Application toPractical Problems, Chapman and Hall, London,1975, ISBN0412155605.

9 L. P. Heulsman, Basic Circuit Theory with DigitalComputations, Prentice Hall, Englewood Cliffs,NJ, 1972, ISBN 0131574309.

APPENDIX A

Derivation of the Fourier series of line currentThe stator phase current waveform ias

applied to the star connected motor is shownin Fig. 3. This waveform is symmetrical aboutthe wt axis and f(wt) = -f(-wt). Hence ias

can be represented as a Fourier series [9]such that

ias =B1 sin(wt) + B2 sin(2wt) + . . .+ Bn sin(nwt)

where(A-I)

Bn = [cos(nrr/6) - cos(n5rr/6)]2IdnrrHence

(A-2)

ias = ~ Bn sin(nwt)n=1

(A-3)

ibs= L Bn sin(nwt - A)n=l

(A-3 )

ies = L Bn sin(nwt + A)n=l

This analysis is similar for the phase currentwaveform in Fig. 2.

51

APPENDIX C

Definition of Park's orthogonal transforma-tion matrix

[FOdqd = [PO][Fabed (C-l)

where

[

yTf2 yT72 yT72

l[Po] =v'213 cos () cos«() - A) cos«() + A)

sin () sin«()- A) sin«()+ A)

[F abed = [Por1[Fodqd

(C-2)

(C-3)

In the synchronously rotating referenceframe where the frame rotates at speed w, theangle () = wt.

APPENDIX D

Derivatives of the stator currentsFrom Park's orthogonal

(Appendix C),

idl = v'213[iAL COS () + iBL cos«() - A)

+ iCL cos«() + A)]

transform

(D-l)

Hence

pidl = v'213[PiAL COS() + piBL cos«() - A)

+ piCL cos«() + A) - iALW sin ()

- iBLW sin«() - A) - iCLW sin«() + A)]

(D-2)

iql = v'213[iAL sin () + iBLsin«() - A)

+ iCL sin«() + A)] (D-3)

Hence

piql = v'213[piAL sin () + piBL sin«() - A)

+ piCL sin«() + A) + iALW cos ()

+ iBLW cos«() - A) + iCLW cos«() + A)]

(D-4 )

APPENDIXB

Elements of matrices

Rl 0 0 0

0 Rl 0 0[R] = I

0 0 R2 0

0 0 0 R2 -

-LlI 0 Lm 0

;

10L11 0 Lm

[L] =Lm 0 L22 0

0 Lm 0 L22

0 LlI 0 Lm

I-Lll0 -Lm 0

[F] =0 0 0 0

0 0 0 0

0 0 0 0

[G] = I0 0 0 0

0 Lm 0 L22

-Lm 0 -L22 0L -

[R' ,] [G,] [0

'[R2]= 0 -L22

[G,] 0l [Lm] [m J-Lm

[L,] [" 2J