A Comparison Between Star And Delta Connected Induction .
Electric Power SystemsResearch,8 (1984/85)4141 - 51A Comparison between Star and Delta ConnectedSupplied by Current Source InvertersInductionMotors whenP. PILLAY, R. G. HARLEY and E. J. ODENDALDepartment of Electrical Engineering, University of Natal, King George V Avenue, Durban 4001 (South Africa)(Received April 9,1984)SUMMAR YThis paper analyses whether any differences 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. INTRODUCTIONThe induction motor is robust and cheapand with the advent of power semiconductorsis being used widely in variable speed applications. Frequency conversion usually takesplace by first rectifying the fixed AC mainsand then inverting to a new variable frequen-cy. The DC link between rectifier and invertercan be operated with the link voltage heldconstant, or with the link current held constant; 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 quasisquare blocks [1] of current are appliedwhich adversely affect the motor operation[1 - 3]. Some investigations [2,3] have consideredstar connectedinductionmotorstators while others [4,5] have evaluateddelta connections.A comparisonof thebehaviourdue to either a star or deltaconnected stator has not been presented.The purpose of this paper is therefore toinvestigate whether any differences in behavior arise due to the method of stator connection of a CSI-fed motor with special referenceto the torque harmonics. The paper firstlyevaluates the behaviour of. the motor withphases A, Band c.c.Qllnected in delta, anduses the current waveforms as shown in Fig.2. It then evaluates th performance of thisRw"B.Fig. 1. CurrentCONTROLCONTROLC I RCU!TRYC I RCU!TRYsource inverter-fed0378-7796/84/ 3.00inductionmotordrive.@ Elsevier Sequoia/Printedin The Netherlands
'i ; .'U- - .111:.I". nd amenta!!-sII0ias-",-1II--3-:31,.-.-. .-.- .-- . - u.I,--- -0 -0.- - .-,fundamentalIIi0cs. .- '311--.II,fundamental. --- - n - -- .iAL0--u - u-.-----.-, . --I,rIDC1600Fig. 2, Line and phase current11L2o 1800waveforms12400in a delta connectedsame delta connected motor, but mathematically represents the delta by its star equivalent as shown in Fig. 3. The benefit of usingan equivalent star for simulation purposes isthe simpler 3-step current waveform of Fig. 31300013600inductionL. --.4200---motor.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
43iALiSL.IB.lCLtiALias.-iSLibsiCLi cs-----u -hfundamentaliALias0--- -- - ---- ---- - -- - ---- . - h-12j 00,.---bO- --- --------I6 ltoot180 12400.L.3000Fig. 3. Line and phase current waveforms in a star connected inductionFig. 4 in order to establish whether the actualconnectionhas any effect on the motor performance.In these comparisons particularpaid to torque harmonics.attentionis13600-1420.motor.2. THEORY2.1. Analysis of the current waveformsTable 1 shows a list of motor parameters interms of the original delta winding, the equiv-
44TABLE 1Motor parametersLine voltage (V)Phase voltage (V)Line current (A)Phase current (A)Base voltage (phase) (V)Base current (phase) (A)Base power (3Vphlph) (kV A)Base impedance (Vphllph) (it)Stator resistance (it)Rotor resistance (it)Stator leakage reactance (it)Rotor leakage reactance (it)Magnetizing reactance (it)Stator resistance (p.u.)Rotor resistance (p.u.)Stator leakage reactance (p.u.)Rotor leakage reactance (p.u.)Magnetizing reactance (p.u.)Motor ADeltaMotor BEquivalent starMotor 038Fig. 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 connection; for convenience they are referred to asmotors A, Band C respectively. An importantresult from Table 1 is that the per unit impedances 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 determines the average response of the inductionmotor (for example run-up time), with theharmonic currents having only parasiticeffects. For example, the average torque that1is 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 following Fourier series:ias (3j1T)II[sin(nwt) sin(7wt)j7 sin(5wt)j5. . .](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 expressions except that the sin(nwt) in eqn. (1) isreplacedby 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(7wt)j7 . . .]sin(5wt)j5(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 component is (3j1T)II 0.955 II, whereas in thestar case the magnitude of the fundamentalcomponent is (2V3j1T)I2 1.103 12; that is,
45the fundamentalcomponentof current issmaller than the peak value of the delta waveform but larger than the peak value of the starwaveform. Hence to ensure a 1 p.u. fundamental componentin 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 harmonics of the delta and star waveforms are equalif their fundamental components are equal.2.2. Two-axis analysis of the CSI-fedinduction motorThe 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 neglected. All motor resistances and inductancesare independentof frequency, which limitsthe usefulness of these models to wound rotorand single cage rotors with shallow bars.The two-axis voltage equations of a voltagefed induction motor can be summarized asfollows in terms of a reference frame rotatingin synchronism with the fundamentalcomponent of the stator current:[v] [R][i] [L]p[i] Wi[F][i] swi[G][i](3)where[v] [Vd\ Vq\ VdZ, lIqZ]T(4)[i] [idl iq\ idz, iqzF(5)s (Wi - Wr)/Wi(6)The other matrices in eqn. (3) appear inAppendix C. Moreover, in the case of a current source inverter, idl and iql are independent predefinedvariables (obtainedfromPark's transform of ias, ibs and ics) and differential 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)[Vz] [VdZ,vqzF[iz] [idZ, iqzF(8)[id [idl, iqlFThe other matrices in eqn. (7) appear inAppendix C. In a short-circuited rotor VdZandVqZare zero, in which case eqn. (8) can be rearranged 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 electrical torque Te fromTe LmWb(idZiql-iqzidd/3(10)The mechanical motion is described byPWr (Te - Td/J(11)For a sinusoidal line current to the motor idland iql are constant quantities in a synchronously 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 differentiating the summed series expressions foridl and iql, as shown in Appendix D.A computerprogram was developed topredict the dynamic behaviour of the CSI-fedinductionmotordrive. The independentvariables are idl iq\ pidl and piql, and theiraccuracydependson the number n ofharmonics used in the Fourier series. Thevalue 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] thatthirty-one(n 31) current harmonics yieldsufficientaccuracy with the advantage ofkeeping computation time down.The program starts by finding the magnitude 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.
463. RESULTSThis 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 simulation done for motor B is also true for motor Cif the analysis is carried out in p.u. form, because the current waveform is the same andthey have the sanle p.u. impedances; however,the physical magnitudesof 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 machines have the same power and frequencybases, the physical magnitudes of the motortorque pulsations are also equal. The magnitude 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 currents for the star (3-step) and delta (4-step)waveforms. The FFT of the star waveform inFig. 7(a) shows that the fundamental component 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 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 pulsationsare the same for the delta and star connectedmotors when expressed in p.u. The line current needed (this specifies the link current indirectly) is 43.2 A rms for both the deltaconnected motor A and the star connectedmotor B of the same power and voltage ratings. Hence, a given current source inverterable to supply current to motor A would alsobe able to supply it to motor B. The only differencein operationbetween 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 byand 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.J34. CONCLUSIONSThis paper has compared the differences inbehaviour between a delta connected motor,its star equivalent of the same power andvoltage ratings and the original delta reconnectedinto star. Simulationshave 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'currentcomponentspresent in the phase urrent of a delta connected inductionmoborfed from a CSI arethe same as those present in a star connectedmotor.(b) Provided the magnitudes of the fundamental componentsof these currents arespecified as equal in p.u., the delta connected
PHASE1.CURRENT2.'":i:in:n:TORQUE".IZIII0::0:: -1.":: III:: CJ0::0I-u-1.5-1. ".25.50TIMEVOLTAGE250.--SPEED(b)(1) /1 .):in:1.".IIIU -1. IJ0 iii50. ",IIIa.U)-2."iii."".25.5"TIMEFig. 5. Predicted start-up ofa.15(c)1.""delta connected1.25SEC".",1.5"CSI.fed induction motor.iii."". 25TIME.5"1."".15(d)1.25SEC1.50
1.1!JPHASE CURRENTTORQUL-- ------ ----f--H 00- 51!JTIME1.5PHASE.751. I!JI!J(a). VOLTAGE -. 1.25I.251.51/1SECI.51!JTIME251!J.1!J:- --SPEEDI.75I1.1!J1!J(b)1. 25SEC1. 51!J(w/ws)::in:-1. 59.1!J1!J.25TIME.51!J.75(c)1.01!J1.251.51!JSECFig. 6. Predicted start-up of a star connected CSI-fed induction SEC1. 50
FFT OF CURR-STAR IfFFT OF 5.JILIII.JIL 2931 35 37110.00.00100.00FREO.30"- 000.0aILl"- 00400.00I1LJLJ/\/\1\ 1\,200.00(b)FREO.29 31/\,/\300.0035 371\ 1\400.00HZ.FFT OF TORO-DELTA----- n:1\1\ ;A-A100.00HZ.FFT OF TORO-STAR::J5::Jrh,6th1000. 1111-112th12th18th24,thl1J0''' h" I.I18thE 400.0i1II24th::JI-.JIL.JIL 0.1/!1/!0.01/!HZ.Fig. 7. FFTs of current and torque for the star and delta connected induction Z.
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 between a star connected and a delta connectedmotor supplied from a current source inverteris the reduction in voltage spikes in the caseof the star connections.ACKNOWLEDGEMENTSThe authors acknowledge the assistance ofD.C. Levy, R.C.S. Peplow and H.L. Nattrassin the Digital Processes Laboratory of theDepartmentof ElectronicEngineering,University of Natal. They are also grateful forfinancial support received from the CSIR andthe University of Natal.NOMENCLATURECSIcurrent source inverterphase A stator currentline current corresponding to phase Apeak value of ias for the delta connected motor12peak value of ias for the star connected motoridl 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 voltageJinertia of motorL11stator self inductanceL22rotor self inductanceLmmutual inductancepderivative operator d/dtR 1, R 2 stator and rotor phase resistancessslipTeelectrical torqueTLload torqueA2rr/3()arbitrary angle wtwarbitrary frequencyiasiallIInominalfrequencyat which p.u.torque p.u. powerfundamental frequency of inverterrotor speedWbWiWrREFERENCES1 J. M. D. Murphy, Thyristor Control of ACMotors,PergamonPress,1973,ISBN0080169430.2 T. A. Lipo, Analysis and control of torque pulsations in current fed induction motor drives, IEEEIAS Annual Meeting, 1978, Paper No. CH 13375/78.3 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, Analysing 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. Raamaswami, 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, ISBN 0412155605.9 L. P. Heulsman, Basic Circuit Theory with DigitalComputations, Prentice Hall, Englewood Cliffs,NJ, 1972, ISBN 0131574309.APPENDIX ADerivation of the Fourier series of line currentThe stator phase current waveform iasapplied to the star connected motor is shownin Fig. 3. This waveform is symmetrical aboutthe wt axis and f(wt) -f(-wt).Henceiascan be represented as a Fourier series [9]such thatias B1 sin(wt) B2 sin(2wt) . Bn sin(nwt)whereBn [cos(nrr/6) - cos(n5rr/6)]2Idnrr(A-I)(A-2)Henceias Bn sin(nwt)n 1(A-3)
51Libs -Bn sin(nwtn lAPPENDIX CA)(A-3 )iesL Bn sin(nwt Definitionof Park's orthogonaltion matrixtransforma- A)n l[FOdqdThis analysis is similar for the phase currentwaveform in Fig. 2. [PO][Fabed(C-l)whereyTf2 yT72APPENDIXB[Po] v'213Elements of matrices000 I0Rl000000R20LlI0Rl0[R]yT72cos ()cos«() - A)cos«() A)[sin ()sin«()- A)sin«() A)l(C-2)[F abed [Por1[Fodqd(C-3)In the synchronouslyrotating referenceframe where the frame rotates at speed w, theangle () wt.R2 -[L] 10Lm00[F] ;L110LmLlI0I-Lll000Lm0L22Lm00L220APPENDIX DDerivatives of the stator currentsFromPark'sorthogonal(Appendix C),Lm0-Lm000000idl v'213[iALCOS () iBL cos«()transform- A)(D-l) iCL cos«() A)]Hence[G] IL0000-Lm00000 piCL cos«() A) - iALW sin ()Lm00L22- iBLW sin«() - A) - iCLW sin«() A)]-L220 -pidl v'213[PiAL COS() piBL cos«() - A)(D-2)'[R2] [R'0,][G,]-L22[0iql v'213[iAL sin () iBLsin«()- A)(D-3) iCL sin«() A)]Hence[G,]0-Lml[Lm] [mJpiql v'213[piALsin () piBL sin«() - A) piCL sin«() A) iALW cos () iBLW cos«()[L,]["2J-A) iCLW cos«() A)](D-4 )
motor being star or delta connected when supplied by a current source inverter. Two comparisons are considered. The first is that of a delta connected motor compared to a star connected motor of the same power and voltage ratings; the star connected motor in this case is then mathematically the delta conne
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