Improving the open-loop torque step response ofinduction motors

N.C. Enslin, MSc, PhD, FIEEProf. F.S. van der Merwe, BEng, PhD, DIC, MSc(Eng)

Indexing terms: Induction motors, Torque

Abstract: Driving functions which requirechanges of torque and speed of induction motorsproduce DC offsets or transients in the torque-producing flux and current waves in the machine.There are two time constants controlling therespective rates at which they disappear and ifthese are short compared with the rate at whichtorque and speed are required to change, they canbe neglected. Robotic systems place critical con-straints on the time taken by the motor to executeits instruction. The flux wave transient has a muchlonger time constant than the current wave andseriously limits the application of the motor in acontrol system. It is shown how it can be avoidedby prior injection of a direct current and sub-sequent control of the driving functions to keepthe flux constant. The motor becomes comparablein its response to DC servomotors when operatedin this manner. The application of a single-phasehigh rate of rise current pulse to the stator at theappropriate space orientation, can improve theresponse so that it is better than would normallybe allowed by the leakage reactance of the motor.

List of symbols

Rt = distributed stator resistance, Q/mR2 = distributed rotor resistance, Q/mLy = distributed stator inductance, H/mL2 = distributed rotor inductance, H/mM = distributed stator to rotor mutual inductance,

H/ma = leakage coefficient = 1 — M2/(LlL2)Pi = Ri/faLi) = inverse stator time constantPi — ^2/(^2) = inverse rotor time constantv' = real part of frequency of stator transient currentp' = imaginary part of frequency of stator transient

currentv" = real part of frequency of rotor transient currentp" = imaginary part of frequency of rotor transient

current(f> = initial angular location of component6 = angular displacement of stator steady-state dis-

tributions during time between two switchings

Paper 5629B (PI), first received 17th February and in revised form 21stJuly 1987The authors are with the Department of Electrical and ElectronicEngineering, University of Stellenbosch, Stellenbosch, 7600 SouthAfricaDr. Enslin is also a drives and power systems consultant working fromPO Box 250, Newlands Cape, 7725 South Africa

co = angular velocity of rotorcos = angular velocity of stator steady-state distribu-

tionsG = steady-state torqueV = stator phase voltagek = torque constant, rad s "1 N ~ * m ~i

d/dt = time differential with respect to a circuitd/dt' = time differential with respect to a coupled circuit

with relative velocity

1 Introduction

Induction motors develop their normal torque by theinteraction of sinusoidal rotor current and stator fluxwaves. Under idealised steady-state excitation of thestator they are of constant amplitude, stationary relativeto one another and rotate synchronously relative to thestator according to the supply frequency and motor polenumber. The corresponding steady-state torque is adirect function of the product of their wave amplitudesmultiplied by the cosine of the space angle between theirpeaks.

The merit of such a machine as part of a roboticsystem is measured by its ability to execute input instruc-tions accurately for output torque without delay or over-shoot, or simply, its linearity and frequency response.Other considerations are the power/weight ratio, itsability to operate in harsh environments and the cost ofthe motor and associated electrical driving and controlequipment.

Distortions can be removed and response speeded upby operating the motor in a closed loop configurationwith feedback to iron out the imperfections of the motor.Flux orientation control of induction motors [1-4]applies this approach by driving the stator windings withindividual amplifiers which, on account of the feedbacksignal, enforce the transfer function set by the feedbacknetwork.

Electronic inverters have become more common as thesource of variable freqeuncy supply to change speedinduction motor drives. An improvement in the dynamicresponse of such a combination by minor modification tothe driving function of the inverter may make it moreacceptable in the field of robotics. It should result in asimplification compared with existing methods and allowlarger machines to be used.

2 Causes of the torque-step distortion ofinduction motors

Changes in motor torque can only be produced by alter-ing either or both the amplitudes or relative angular loca-tion of the steady-state stator borne flux and rotor

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987 317

current waves. The flux wave flows mainly in the corecircuit of the motor whereas the rotor current wave, withits opposing image on the other side of the airgap, sets upflux through the stator to rotor leakage path, which ispredominantly air. Both these magnetic fields representenergy stores which have to be changed in magnitudeand relative location as the torque requirement is altered.

Section 8 analyses the behaviour of the motor whensubjected to changes in its steady-state stored energies bya solution of the stator and rotor voltage differentialequations. Contrary to normal phase values, flux, currentand voltage are expressed as sinusoidal waves or dis-tributions with individual amplitudes and angular velo-cities. The time differential operators are based onrelative angular velocities between the windings and thecorresponding current waves. Motor parameters of statorand rotor resistance, inductance and leakage inductanceare stated as distributed values per metre of the airgapcircumference. They are independent of their angularlocation and not influenced by rotation of the rotor.

It is shown that attempts to alter the energy storingfluxes and their currents, by changing the steady-statestator applied voltages or currents for a new torque, areinitially ignored by the motor. Its subsequent behaviourcan be explained in terms of transients arising instantane-ously to bridge the discontinuity between the precedingand new energy states which the driving functionsattempt to implement. Although hidden at the instant ofvoltage change, they become evident as relative angularvelocity separates them from their initial spatial super-position on the new steady-state values. Their magni-tudes are determined by the height of the implied energysteps and their attenuations by exponential functionswith time constants which normally restrict the torqueresponse of the motor.

There are two complete sets of transients excited inde-pendently by the respective demands for energy changes.They are replicas of the steady-state set of stator androtor currents and fluxes with the usual ampere-conductor balance and a distinction of magnetisingcurrent. Their initial magnitudes, and locations are deter-mined by the step between the original and the newlydemanded steady-state conditions, the instant of itsimplementation, the rotor angular velocity and theparameters of the motor. Using the rotor angular veloc-ity and motor parameters, the common angular velocityand exponential rate of fall of the amplitudes for each setare determined.

Fortunately for typical motors the transient set causedby the leakage reactance stored energy change is virtuallystationary relative to the stator and is based on a statorborne transient magnetising current. The main magne-tising flux step yields the other set of transients, station-ary relative to the rotor with their own common rotorborne transient magnetising current and flux.

The transient sets, within themselves, produce expo-nentially decaying braking torques if the rotor is not atstandstill. The interaction between these transient setsand their interference with the new steady-state set ismore undesirable, whether the motor is stationary or not.Comparatively high pulsating torques are generated dueto their relative velocity and they have durations whichmay be long in terms of the response required from themotor. There are ten components of torque. They areexpressed in terms of the currents and fluxes followingfrom the solutions of Section 8 and the initial and newsteady-state conditions which are applied to the motor.

As the variables and parameters defined are not the

normal phase values, they cannot be measured directly.The Appendix shows, however, that they can be scaled interms of the X-phase magnitudes and angles providedthat the reference axis is chosen suitably. Calculation ofthe transient complex frequencies involves only the ratiosof the distributed parameters allowing phase values to beused, as the common scaling constant cancels. Thenumerical results given refer to the distributions defined,but, for convenience, are written in terms of the X-phasecurrents and voltages. The B- and C-phase values aregiven by the projections of the distributions on theirrespective axes at 120° and 240° to the chosen reference.Negative sequence components have these angles trans-posed. The approach adopted in Section 8 is only validfor balanced polyphase excitation so that in the case ofsingle phase, a resolution into positive and negativesequence sets is necessary.

Instantaneous torque and the three stator transientcurrents for a standard 3 kW, 4-pole, 400 V, 50 Hzinduction motor are shown in Fig. 1. A strategic fre-quency of 15 Hz was chosen to enable the motor to

200 250 300 350 400 450 500time, ms

Fig. 1 Pulsating torque and exponential stator transient currents fordirect-on connection to 15 Hz supplyNo transient suppression and motor initially unexcited

develop a torque of 55 Nm at standstill for direct-onstarting by simultaneous connection of the three lines toa 130 V supply. Calculation shows that the instantaneoustorque achieves a peak of twice the steady-state valueafter 40 ms. The pulsation persists beyond 500 ms corre-sponding to the longer time constant of 360 ms for therotor borne transient flux. As the motor was initiallyunexcited, the transient current amplitudes are almostequal to the steady-state values which follow the connec-tion of the motor to the supply. Two exponential cur-rents flow, one with a magnitude equal to theshort-circuit current and a short time constant of 4.6 msand the other equal to the magnetising current with thelonger time constant. This behaviour of induction motorsis well known [5-8] but is restated with a distinctionbetween the two transient sets described by polar vectorsto facilitate the explanation of the methods used in theirelimination.

3 Avoiding the rotor transient

Inputs to the motor which imply step changes to itsmutual core path flux will induce a rotor borne transientwith a time constant which is so long that it will precludeits application from most position control systems. In astop-start configuration, flux would have to be inducedin the motor for every start. It becomes obvious that theflux should be held constant and torque needs imple-mented by current changes thus involving only the statortransient which has a much shorter time constant.

318 IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987

Synchronous, stepping and DC machines allow thismode of operation quite naturally for, whether deliveringtorque or not, their working flux can be maintained con-tinuously by means of permenant magnets or separateDC field excitation coils. They are an obvious choice asservomotors with response times involving leakage ratherthan magnetising inductance.

Induction motors, particularly when operated fromstandstill, must therefore have the necessary flux insertedinto their mutual core circuit prior to demanding torquefrom them. This flux should have a magnitude com-mensurate with the voltage and frequency of the supplyrequired to meet the torque demand. If the motor is notat rest and connected to the supply, it will already haveflux in its working circuit and the transient, on demandof a torque change, is avoided by frequency and voltagechanges not requiring a step change in flux. Thisamounts to slip control with a constant voltage/frequency ratio.

Fig. 2 shows the instantaneous stator currents, withthe corresponding transient components superimposed,

30

20

10

0

-10

-20

,'A'

0 12 18 24 30 36 42 48 54 60time, ms

Fig. 2 Instantaneous and exponential transient 3 phase currents for

direct-on connection to 15 Hz supply

Rotor transient suppression with initial direct current

for the 3 kW motor when connected to a 130 V 15 Hzsupply by simultaneous closing to the stator. A priordirect current of 2.18 A is arranged to flow from line C toB by means of the programmed function generator of thesupply electronic inverter. Care has to be taken that theinstant of connection of the 3-phase 15 Hz supplyinvokes a steady-state flux which lines up in position andmagnitude with the initial DC excitation for which themotor has been prepared. Initial and steady-state scaledvalues of polar distribution amplitudes and angles withthe X-phase as the reference axis are as follows:

voltage = 75.06/-7.1°stator steady-state current = 18.34/ —30.64°rotor steady-state current = 17.46/154.69°magnetising steady-state current = 1.885/ —90°initial effective DC stator current = 1.885/-90°

The rotor transient has been eliminated leaving the statortransient current wave. The latter is reflected by the3-phase currents shown by the dashed curves in Fig. 2.The x-axis has been expanded because of the shorter timeconstant involved. Curves (iii) and (iv) in Fig. 3 must becompared to appreciate the remarkable improvementachieved by the DC injection. The remaining stator tran-sient current wave, with a time constant of 4.6 ms,accounts for the delay in the torque to rise to its steady-state value of 1 p.u. This performance can be comparedwith a typical electrical time constant of 9.3 ms of abrushless DC motor used as a robot drive [9]. In thiscase the torque-producing flux wave is installed by per-

manent magnets and the longer time constant does notarise.

4 Dealing with the stator transient

Setting up the machine energy stores prior to its torquecommand is feasible for either the core or leakage flux

1.25r

^ - " 12.5 15.0 175 20.0 22.5 25.0time, ms

Fig. 3 Torque time curves for motor at standstill

(i) both transients removed by using DC and 100 Hz pulse(ii) both transients removed using DC and 50 Hz

(iii) rotor transient removed using DC(iv) both transients present as normally when starting

but obviously not both. It would seem sensible to set theiron flux initially and maintain it constant for all sub-sequent speeds and torques. The only reasons for thischoice are the longer time constant of the rotor transientrequiring more time for its installation and the ease withwhich the function can be implemented by the inverter.

When the flux has been set, the stator and rotorcurrent waves and the associated leakage energy storehave to be induced following the torque instruction tothe motor. The time this takes determines the response ofthe motor. It has to be effected as quickly as possiblewithout leaving a transient which will cause subsequentinterference with the desired output torque.

A balanced polyphase system has a constant non-pulsating stored energy for a given steady-state flux wave.This cannot possibly be changed in a time less than thecircuit time constant without causing a transient. Non-rotating pulsating flux waves on the other hand, havesteady-state energies which vary sinusoidally at twice thefrequency of their excitation. It is therefore possible tochoose an instant when the continuously varying pulsa-ting steady-state stored energy commanded by the single-phase connection of the inverter lines up in magnitudeand space orientation with the initial value which existsin the motor. For instance, if the motor is initially unex-cited, the single-phase connection is made as the 'wouldbe' pulsating steady-state energy for this connectionsweeps through zero. It is then allowed to change undersingle-phase excitation without inducing a transient at arate limited only by the voltage and frequency capabil-ities of the inverter. On reaching the magnitudes of thecurrent wave distributions demanded by the new torque,the inverter switches the motor back into a 3-phase con-figuration. Its applied excitation must then be set so thatit requires new steady-state waves which correspond inmagnitude and space orientation with those for which themotor has been prepared. An alternative explanation isoffered by considering the single-phase excitation to bemade up of equal positive and negative sequence com-ponents. For instance, a single-phase voltage applied toan initially unexcited motor between its 4-phase terminaland the B- and C-phase terminals, with the B- and

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987 319

C-phase terminals connected, would result in two tran-sients because of the applied energy changes of the posi-tive and negative sequence leakage flux distributions. Asthe flux waves rotate in opposite directions the transientscan be placed opposite to one another to cancel bychoosing the correct time for switching. Time is thenallowed to elapse while the remaining steady-state wavesrotate until they coincide, add and produce the desiredcurrent distributions ready for the reconnection of themotor to the three phases.

A numerical example follows which illustrates thefunctions which avoid both transients. The magnitudesand angles of the inverter excitations are critical. Theyconsist of an initial direct current of 1.877 A flowing fromphase C to B, followed by a single-phase voltage pulseapplied between phases A and B and C (B and C areconnected) for a quarter of a cycle of a 100 Hz supplyfrom the inverter. These values are tabulated as positiveand negative sequence sets and refer to the instant ofapplying the single-phase voltage.

(a) Positive sequence:voltagestator steady-state currentrotor steady-state current

= 131.4/- 15.43C

= 9.174/-90°= 8.797/90.33°

initial effective stator DC current = 0.813/ -90°(b) Negative sequence:same as for positive sequence except

initial effective stator current = 0.813/90°

The rate of rise of the currents and hence the torqueduring the single-phase connection will depend on theinverter voltage and frequency used. In this case with455 V and 100 Hz, a risetime of 2.5 ms is achieved, whichis less than the time constant of 4.6 ms corresponding tothe leakage reactance of the motor.

The voltage and new steady-state currents required bythe connection of the 15 Hz supply and the initial valuesprovided by the application of the 100 Hz single-phasevoltage are as follows:

voltage = 75.06/23.45°stator steady-state current = 18.348/0°rotor steady-state current = 17.463/185.33°stator initial current = 18.431/ —4.41°rotor initial current = 17.523/-179.36°

The alignments in magnitude and position which havebeen achieved are evident from the tabulations of thesteady-state values demanded by the 15 Hz connectionand the initial values left by the single-phase excitation.

Fig. 4 shows the corresponding instantaneous andtransient currents during the implementation of the sup-

0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.5 22.5 25.0time, ms

Fig. 4 Instantaneous and exponential transient 3-phase currents forconnection to a 15 Hz supplyBoth transients depressed with initial DC and 100 Hz single phase pulse

320

pression of both transients when starting the motor inthe manner described. The transient currents are verymuch smaller than in Fig. 2 and the 100 Hz single-phasecurrent wave injection has improved the torque responsefrom that shown by curve 3 to curve 1 in Fig. 3. Thecurrents in phases A and phases B and C during the first2.5 ms in Fig. 4 are consistent with single-phase excita-tion. The small constant difference in the B- and C-phasecurrents is due to the DC injection. Using 50 rather than100 Hz for the single-phase injection with the same initialdirect current resulted in curve (ii) which is not a worth-while improvement on curve (iii). This is to be expectedas the lower frequency has a quarter cycle of 5 mswhereas the stator transient has a time constant of4.6 ms.

5 Implementation of transient suppression by theinverter control functions

Control functions can be developed to speed up theresponse of the motor by taking into account the originand effects of the transients. These must avoid the excita-tion of transients emphasised by the higher rates ofchange of stored energy that would be required.

As the core flux stored energy has the longer time con-stant, particularly at standstill, it is wise to induce thestored energy initially and keep it constant. If the motoris at standstill, this amounts to circulating a directcurrent between two of the phases with a polarity gov-erned by the direction of rotation and hence the phasesequence of the stator voltages. The motor must be con-trolled in a manner in which, irrespective of speed ortorque requirements, the mutual flux wave remainsunchanged [10].

If stator impedances are neglected, the following sim-plified expressions apply for constant mutual flux:

inverter angularfrequency cos = co + kG

inverter voltage = cos/cosmax x voltage at cosmax

The inverter voltage is thus set by the value of cos derivedfrom the desired shaft speed and torque.

Torque demands must be limited by the inverter notapplying stator frequencies which will take the rotor fre-quency outside preset values. These retain a satisfactorylinear relation between slip and torque and limit changesof airgap flux which cause rotor borne transients. Thisrestricts the amplitude of the torque excursions but notits rate of change.

The time response of torque under these circumstanceswill be limited by the stator transient left by the currentwave changes. It has been shown how these may beavoided at standstill and the motor response improved sothat it is faster than allowed by the time constant of thestator transient. The single-phase injection which sets upthe new steady-state current waves is readily achieved asa virtual step function on demand of a torque step butcannot be applied as a continuous function. It has to beset in spatial relation to the instantaneous position of theairgap flux at the instant the torque is demanded, aproblem which does not arise at standstill. This determi-nation probably yields a control system as involved asthat used for flux orientation with similar limitations setby the processing time of the control signal.

A control system which eliminates both transientsbecomes feasible and may be justifiable in fast positioncontrol applications. Fixed instructions can be set in thecontrol functions of the motor for starting and stopping.

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987

Should the response be adequate when set by the statortransient time constant, transient suppression may beconfined to the rotor component, which is much simplerand continuously applicable over the whole speed range.It achieves a profound improvement in the motorresponse particularly at standstill as in the case of the3 kW motor in which the time constants of 360 and4.6 ms should be compared. The shorter stator adherenttransients merely cause an exponential delay in torquechanges, whereas the rotor transient induces pulsationwhich may even result in reversals of torque. Most servosystems would tolerate the former but should avoid therotor transient which, fortunately, is easier to remove.

6 Experimental verification of transientsuppression

A 5.5 kW, 50 Hz, 3-phase, delta connected motor wascoupled to a locked transmission dynamometer. Tran-sient torque measurement proved to be difficult becauseof the compromise which had to be sought between thesignal/noise ratio and the mechanical resonant frequencyof the system. A figure of 109 Hz was achieved. Themotor phase voltages were reduced by reconnecting themachine in the star configuration thus avoiding satura-tion and allowing better correlation with the calculatedperformance.

The initial investigation was based on electricaldriving functions which could be achieved without theconstruction of an electronic inverter. Three anti-parallelthyristor groups were arranged so that the A and B con-nections to the supply could be switched independentlyat any desired instant relative to the positive maximum ofthe A-phase supply voltage. To inject the proposed directcurrent flowing from C to B prior to the torque demandfrom the motor, a battery in series with a thyristor and a5 H inductance, was connected between the C and Bphases of the motor. The application of the alternatingsupply voltage conveniently opened this circuit as thethyristor gate voltage was not maintained.

The calculated rotor and stator time constants usingmeasured motor parameters were 294 and 4.0 ms, respec-tively. Fig. 5 compares the torque step of an initially

a/ 2.0 -

time, msFig. 5 Suppression of rotor transient by prior injection of a directcurrent of 0.7 A

with direct-current and simultaneous closing of three phasesno DC simultaneous closing giving maximum transient

unexcited motor shown by the dashed curve, with thetorque it delivers when a prior direct current of 0.7 A isinjected as described. The simultaneous connection to the50 Hz supply was found to give the best result whenmade at an instant for which the A-phase voltage was at4.5 rather than 0° as required theoretically. A significant

improvement is achieved although the effect is spoilt bythe behaviour of the torquemeter. The result should becompared with curve (iii) in Fig. 3, calculated for a 3 kWmotor with similar parameters.

Verification of the method of reducing the stator tran-sient by a single-phase injection of current was not pos-sible with the available torque recording equipmentbecause of the much shorter time constant of 4 ms. Toremain within the range of the torquemeter's response,the single-phase current injection used to suppress thelonger rotor transient is shown in Fig. 6. The principles

Fig. 6 Suppression of rotor transient by single phase alternatingcurrent injection

single-phase connection of A and C at 26.1° followed by B at 109°simultaneous closing giving maximum transient

involved are exactly the same as set out earlier, but tosimplify the switching operation, the A and C terminalsare connected at the instant at which the correspondingsingle-phase circuit has zero mutual flux stored energy. Itremains connected in this manner for quarter of a cycleto allow the stored energy to build up without leaving atransient, whereupon the B-phase is connected. The besttiming sequence relative to the maximum voltage in theA-phase derived by trial and error, gave the closing of Aand C as 26.1° and that of B as 109°. The calculatedvalues are 22.4° and 112°, respectively. The torque pulsa-tion which persists is 100 Hz generated by the motorunder steady-state conditions possibly due to slightimbalance and enhanced by the mechanical systemresponse. There is initial interference between this com-ponent and a 109 Hz transient excited by the torque step.The improvement is striking and has been referred to inearlier work [6, 7, 12].

7 References

1 BLASCHKE, F.: The principle of field orientation as applied to thenew Transvektor closed-loop control system for rotating-fieldmachines', Siemens Rev., May 1972, pp. 217-220

2 GABRIEL, R., LEONHARD, W, and NORDBY, C.J.: 'Field-oriented control of a standard AC motor using microprocessors',IEEE Trans., 1980, IA-16, pp. 186-192

3 NABAE, A., OTSUKA, K., UCHINO, H., and KUROSAWA, R.:'An approach to flux control of induction motors operated withvariable-frequency power supply', ibid., 1980, IA-16, pp. 342-349

4 DE FORNEL, B., PIETRZAK-DAVID, M., and GAUVRIT, M.:'State-space compensation applied to the control of a current-fedinduction motor', IEE Proc. B, Electr. Power AppL, 1982, 129, (4),pp. 221-226

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987 321

5 SLATER, R.D., and WOOD, W.S.: 'Constant-speed solutionsapplied to the evaluation of induction-motor transient torquepeaks', Proc. IEE, 1967,114, (10), pp. 1429-1435

6 FLYNN, F.P., SLATER, R.D., and WOOD, W.S.: 'Transient nega-tive torques in induction motors due to rapid reconnection of thesupply', ibid., 1969,116, (12), pp. 2009-2014

7 SMITH, I.R., and SRIHARAN, S.: 'Induction-motor reswitchingtransients', ibid., 1967, 114, (4), pp. 503-509; discussion: ibid., 1968,115, (1), p. 133

8 ENSLIN, N.C., KAPLAN, W.M., and DA VIES, J.L.: 'Influence oftransient switching currents and fluxes on the torque developed by asquirrel-cage induction motors' ibid., 1966, 113, (6), pp. 1035-1043;discussion: ibid., 1968,115, (1), p. 133

9 WILLIAMS, S.: 'Direct drive system for an industrial robot using abrushless DC motor', IEE Proc. B, Electr. Power Appl., 1985, 132,(1), pp. 53-56

10 AL-NIMMA, D.A., and WILLIAMS, S.: 'Study of rapid speed-changing methods in AC motor drives', ibid., 1980,127, (6), pp. 382-385

11 RUDENBERG, R.: 'Transient performance of electric power cir-cuits' (MIT Press, 1969)

12 BLOM, J.F.: 'The influence of the switch on the transients of thethree-phase induction machine', Elektrotechniek, 1963, 16, pp. 181—183

8 Appendix: Solution for transient currents

The currents are found by solving the voltage differentialequations written in terms of the sinusoidally distributedwaves of current per metre of the stator and rotor for agiven rotor angular velocity. The parameters of induc-tance, mutual inductance and resistance per metre, usedin these equations, are for a uniform airgap machine inwhich space and time harmonics have been neglected.Flux, current and voltage waves are written as symmetri-cal component vectors in the polar form. As explained inSection 2, steady-state changes result in two completeadditional transient sets of current waves defined asfollows in terms of the initial values and parametersderived from the solutions of the differential equations:

hi

= total stator current distribution!= total rotor current distribution j

_ J gJW>Sl +a>st)

(1)

s l

hz =

= IoieJ<t>01

^ 0 2 eJ<t>02

rie~p'teji<i>i'(2)

= new stator steady-statecurrent distribution

IS2eJl4>S2+icos-{O)t] = new rotor steady-state

current distribution— initial stator current

distribution= initial rotor current

distribution= stator transient

current distribution/'2g-"V(*2'+V2'° = rotor image current

of statortransient currentdistribution

J '^-PV^ 1 "" 1 "" 1 " 0 = stator image currentof rotortransient currentdistribution

/^ e -p 'V w + V 2 ' r ) = rotor transientcurrent distributionThe characteristic differential equations are written in

terms of the distributed parameters of the motor whichare independent of the spatial angle between rotor andstator as they are not defined relative to the phase wind-ings. They are, however, influenced by relative velocity.

hz

*'oi

'02 =

i\ =

i'2 =

i[ =

Lx dijdt + Ryiy + M dijdt' = 0

L2 dijdt + R2 i2 + M dijdt' = 0

(3)

(4)

Earlier work [8] develops the roots of these equations asthe complex frequencies of the transients which occur inthe stator and rotor when a stepped steady state isapplied to the motor. They are given by

+ P2)

(5)+These expressions are confined to the angular velocity ofthe rotor and ratios of distributed resistance and mutualand self inductance. It allows the calculation of the tran-sient frequencies using parameters derived from phasecurrents and voltages provided that they are balancedpositive or negative sequence values and are the puresinusoids corresponding to the distributions and polenumber involved. Solutions to eqn. 5 are written as

stator transientfrequencies = (v\ + jp') and (v',[ + jp") (6)

rotor transient

frequencies = (v'2 + jp') = (v\ — co + jp')

and (v'2 + jp") = (v'i — a> + jp") (7)

The complete instantaneous current distributions become

x pj(<f>2" + V2"t) I J gj(tf>S2 + COS' ~ <Ot) /m

The A-phase instantaneous stator current is obtained bythe projection of the current vector, expressed by eqn. 8,on its axis taken as the common reference for all threephases. Defined in this manner, the wave magnitudes can,for convenience, be expressed by A-phase values as theyonly differ by a constant.

Solutions for the initial values of the transient distribu-tions

l\ei+1' J?e*r l'2^%' and I^e**2"

are found by substituting t = 0 in eqns. 8 and 9 andequating them to the known initial distributions imme-diately prior to switching, to satisfy the conditions forcontinuity.

i0l = rxeM + l\e»f + /sle»« (10)

i02 = IS2

The differential equations allow the transient current dis-tribution ratios [8] to be expressed as

-SUP'= a

-KJP" + v'i')M'± = b

(12)

(13)

Solutions for the magnitudes and angular locations of thefour transient current distributions are found from eqns.

322 IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987

10-13:

(b

(b

(a

(a

b—

b

ab—

(b

a

-

(a

a)

1-

a)

a

-

b)

1

b

[

a)

a)

1

b)

(14)

account of the initial values and the new steady-state cur-rents.

Substituting P and S as the current discontinuities forthese expressions, the initial current transients are rewrit-ten as

hie**2} (15)

(16)

75

(a-b)" vz ^ }

In each of the solutions for the transient currents (eqns.14-17), the braces group the expressions for the currentsteps in the stator and rotor, respectively. They have tobe calculated for each switching operation and must take

I'^^2" = K^P - K3S

The values of K are calculated from the ratios defined ineqns. 12 and 13:

K, = l/(a - b)

K2 = a/(a - b)

K3 = b/(a - b)

X4 = ab/(a - b)

The stator and rotor currents for each transient set areadded to obtain the magnetising currents which bear thetransient fluxes. Taking the two transient and the steady-state fluxes, the torques they produce with the currentwaves yield ten expressions. These are in terms of com-ponent initial amplitudes, space position, angular velocityand exponential attentuation [8].

IEE Conference Publication 279Electric railway systems for a new century(22nd-25th September 1987, Institution of ElectricalEngineers, Savoy Place, London)

The following contributions appear in IEE ConferencePublication 279:

General review of traction schemesPrototype trains for the Central Line. R. Minter,

C. Llewellyn and R. KingChopper control equipment for Singapore MRT.

T.C. ChewHeavy haul 25 kV electric locomotives. B.M. Stephens

and I.C. Rossow'Hi-tech' electric locomotive developments of the South

African Transport Services. A.R. MercadoSNCF experience with power electronics — experiments

of AC drives and their consequences on the choicesmade for the design of new electric units. D. Brun

Central Line modernisation economics. R.W. Aylwardand Ms. H.J. McCormick

Electrification: why not? J. Bouley

Three-phase traction controlApplications of modern power semiconductors. B. MiillerControl of a three-phase traction drive. J.K. NiiranenThe new locomotives of French railways with synchro-

nous motors. F. JonardThree-phase technology for electric and diesel electric

vehicles and locomotives. C. UrbankeThree-phase drives for multiple units. D.M. Brooks, W.D.

Hodgson and M.P. Ford

Drives and control AC and DC railwaysAnti-slip control of voltage source inverter-controlled

railway vehicle. K. Aburaya

79 papers, 399 pp., 297 x 210 mm, photolitho, soft covers, 1987. Price£45.00, $90.00 in the USA. Orders, with remittances, should be sent to:IEE, PO Box 26, Hitchin, Herts. SG5 ISA, United Kingdom

On-line implementation of optimised PWM schemes fortraction inverter drives. J.A. Taufiq, R.J. Chanceand C.J. Goodman

System design of voltage fed inverter drives with mini-misation of signalling interference. G. Crawshaw

Torque control of AC induction motor traction drives.B.J. Cardwell

25 kV AC locomotives. A.S. KingApplication of GTO thyristors in electrical multiple

units. B.Y.M. Marogy

Linear motorsNational project of developing a linear-motor driven

metro system. A. Matsumoto and S. SoneThe research and development programme 'magnetically

suspended high speed transport systems' in theFederal Republic of Germany. P.G. Hartmann

An overview of linear motor propulsion: present applica-tions and future prospects. E.E. Riches

The development of a 350 kVA traction inverter system.P.D. Cross

Specialised AC railway topicsA frequency domain model of an autotransformer trac-

tion system for harmonic penetration and resonantovervoltage studies. P.T. Griffiths

A comparison between autotransformers and boostertransformers for AC traction supply. H. Bozkaya,Z.Y. Shao, J. Allan, B. Mellitt and C.J. Goodman

The application of shunt compensation on AC railways.R.E. Morrison, K. Warburton, A. Singh and D.J.Young

xnduced noise calculations in AC railways using CADtechniques. B. Mellitt, C.J. Goodman, Z.Y. Shaoand W.B. Johnston

Power factor and harmonic aspects of AC supplied DCdrives for traction. P.J. Donnison

IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987 323

The immunisation of signalling systems for main lineelectrification on Queensland Railways.D.G. Whisson

Electrification schemesThe evaluation of the feasibility of a 12 kV DC railway

system. L. Mayer and O. VenturaFuture power supplies for the London Underground.

J.H. TomkinsNew concepts for DC traction substations.

R.W. Sturland, Z.S. Mouneimne, D.W. Keay andL.R. Denning

A new approach to high voltage distribution systems forDC electric railways employing solid polymericinsulated cables and screened premoulded connec-tors. R.S. Care and E.J. Cox

Co-extruded aluminium steel contact rail for electric rail-ways. G. Mier

Specialised S and T topicsFail-safe microcomputer for railway interlocking.

A. LewinskiCommon hardware/software techniques for railway infor-

mation and control systems. A. St. JohnstonThe effect of high traction current on track monitoring

developments. A.C. KnightDevelopment in intermittent ATC with respect to trans-

mission and processing techniques. R.K. FoxTrack, track circuits and traction. A.J. Fisher

Overhead design and AC electrificationThe electrification of New Zealand Railways North

Island main trunk route. M.R. Denley,A.W. Gregor and W.B.S. Johnston

Railway system oscillatory overvoltages. T. Peterssonand M. Eitzmann

25 kV ac electrification design constraints. M. StuartFurther developments in the dynamic simulation of the

contact wire/pantograph interface. T.A. Willetts,A.E. Seddon and J.P. Topliss

Development of a pantograph for high speed running oneconomic overhead equipment. A.I. Betts,R.J. Gostling and A.E.W. Hobbs

Defining and measuring the quality of current collectionon overhead electrified railways. A.I. Betts,R. Holmes and J.H. Hall

Computer aided design topicsTraction equipment modelling and the power network

solution for DC supplied rapid transit powersystem studies. N.B. Rambukwella, B. Mellitt,C.J. Goodman and Z.S. Mouneimne

Advantages of a simulation model for railway powersupply. L. Kiper, M. Harmelin and R. Malfait

Computer simulation of AC/DC power converter fedinverter drives for traction. A.F. Durie, J.A. Taufiqand P.J. Donnison

Computer-based design of block layouts for rapid transitrailways. D.C. Gill and C.J. Goodman

Computer aided design of audio frequency track circuits.R.J. Hill and D.N. Weedon

DC power system studiesReassessment of power feeding systems at introducing

regenerative trains. S. Sone and S. IshizuContribution to energy saving in DC railways.

R. FilliatreAssessment of voltage controlled substations for DC trac-

tion applications. P.O. Barnwell, B. Mellitt,

C.J. Goodman and W.B. JohnstonDetermination of protection requirements for DC trac-

tion systems and the development of a solid stateprotection relay to optimise current availability.H. Puntis and P. Striibin

Electromechanical and electronic falling voltage trackimpedance devices for fault detection on DC tracksystems. J.E. Buttery, D.N. Ebenezer andB.P. McCormick

Reduction of supply unbalance when planning AC trac-tion supplies. D.C. Howroyd

Maintenance, reliability and design topicsIntegrated logistic support systems for 'High-tech' electric

locomotives for South African Transport Services.A. Veldsman

Some aspects of safety on an electrified railway.R. Holmes

Power supply — electrical safety on AC railway supplysystems. H.M. Harrison

High voltage polymeric systems taking railways into the21st century. A.D. Atkins and D.W.M. Thornley

Solid state interlocking and reviewsSolid-state interlocking in railway signalling, SMILE.

K. Akita, T. Watanabe and H. NakamuraThe PCI (microprocessor-controlled signal box). J. PoreMicrocomputer controllers introduce modern technology

in fail-safe signalling. K. CelinskiSoftware validation for railway signalling and train

control systems. R.C. ShortComputers in railway signalling on London Under-

ground. G.E. Clark

Signalling systems reviewContinuous automatic train control and safety systems

using microprocessors. C.R. Brown, R.D. Hollandsand D. Barton

Digitally coded track circuit. D. Poole and D. BarkerTrack monitoring and automatic block for lines carrying

low traffic volumes. H. UebelEZS 800: MC-interlocking and track circuits form a

closed train control system. K.H. WobigSingapore's train control system in the overall systems

context. A.N. McKillop

Passenger environmentTraction auxiliary converters using GTOs. J.R. ChildsAuxiliary power supply for railway vehicles. F. MeijerAir conditioning and auxiliary power supplies for railway

passenger vehicles. G.W. Graham, J.P. Whatleyand D. Wilks

Passenger information systems and control for futurepassenger stock. B.J. Watkins

Multiplex control and monitoring of traction and aux-iliary systems. N.G. Green

Automatic train controlAutomatic train operation — the Hong Kong experience.

F.W. HarrisDesign principles of the equipment for the Docklands

light railway signalling and control system.R.E.B. Barnard

SACEM: a new train control system with various appli-cations. P. Chapront

Train control systems for high-speed trains using micro-processors and optical fibre data transmission.H.P. Beck

Economics of automatic train control in Sweden.K.L. Forsgran

324 IEE PROCEEDINGS, Vol. 134, Pt. B, No. 6, NOVEMBER 1987