Digital Computer Simulation of Three-Phase IM Dynamics-A Generalized Approach

8/14/2019 Digital Computer Simulation of Three-Phase IM Dynamics-A Generalized Approach

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106 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL 21 . NO I , J A N U A R Y / F E B R U A R Y 198R

Digital Computer Simulation of Three-phaseInduction Machine Dynamics-AGeneralized ApproachSAYEED NURUL GHANI

Abstract-A new unified and universally applicable method is pro-posed for simulating the dynamic behavior of a large, interconnected,nonlinear, time-varying physical system containing one or more three-phase induction machines as subsystems. The total system may alsocontain static power conv ersion, conditioning, or control equipment inwhich switching is achieved by devices of one kind or another. Theprocedure is simple yet accurate and independent of any specific system.A new time-domain static network model of a three-phase inductionmotor of wound rotor variety is presented. The model can account forunbalance on both sides of the airgap provided the structure of thewindings remains unaltered. It has inherent capability for accounting ironlosses as well. Similar network models can also be derived to represent amachine with asymmetrical internal faults that split one or more phases,on either or both sides of the airgap, into multiple sections with orwithout interconnections. A previously published model o f a triac hasalso been extended.

I . INTRODUCTIONOWER electronic equipment containing solid-stateP witching devices is nowadays almost universally used for

controlling the speed of induction m achines [I]. Developmentof a new product or a system often involves expensive trial-and-error experimentation with costly prototypes. The less isknown about a device or a system the greater is the number ofprototypes required. Hen ce implementation of a fresh idea intoa useful new product is usually an expensive, long, tedious,and involved process. A scientifically sound approach toaccurate simulation of expensive prototypes, using advancedapplied mathematical modeling techniques and solution proce-dures, allows realization of preferred designs with minimumtime, expenditure, and effort.

To determine the ratings of the components used in powerelectronic equipment it is essential to know, amongst otherthings, the worst-case instantaneous voltages and currents atvarious parts of the system. T hey shou ld be evaluated for both

Paper IPCSD 87-22, approved by the Electric Machines Committee of th eIEEE Industry Applications Society for publication in this TRANSACTIONS.This work supported by Lucas Research Centre, Solihull, West Midlands,B9 0 4J J U .K . Manuscript released for publication May 1, 1987.The author iswith the Department of Electrical and Electronic Engineering,Newcastle Upon Tyne Polytechnic, Newcastle Upon Tyne, NE1 8ST, U.K.IEEE Log Number 8716205.

normal and faulty operation, ideally by simulating the entiresystem. Also the effects of control loops on th e transient andsteady-state performance of the total system can be rapidlyevaluated from the results of simulation. If the performance isunsatisfactory then the feedback may be readjusted or totallydifferent structures used until satisfactory behavior isachieved. This is specially true for adaptive control systems,which are essentially no nlinea r time-varying sy stems. A largefamily of adaptation law s (for continuous-time systems) andalgorithms (for discrete-time systems) exist which guaranteeglobal asymptotic stability of the system. It is only throughefficient and accurate simulation techniques that the mostappropriate control structure according to a specific index ofperformance can be economically chosen. Furthermore, if theopen-loop system is unstable then experimental adjustment o fthe closed-loop system for stability may not be feasible forsafety reasons, and simulation would be the only alternativeleft. This is also true fo r those nonlinear feedback systems thatare stable with no reference input but that lose stability undersuch an input, and vice-versa.

11. T H EPROPOSEDETHODOR SIMULATIONTime-domain analyses have so far advanced in basicallyfour directions. They are: a) digital computer solution of statemodels, b) analogue computer simulation of dynam ic models,c j dynamic models fo r small perturbations and transferfunctions, and dj solutions for dynamic behavior in closedsymbolic form.

The new simulation procedure proposed in this paper is anaddition to this list. It is the most sophisticated and advancedtechnique for simulation of the nonlinear dynamics of anylarge interconnected physical system containing one or moreinduction machines. The system may contain, among otheritems, static power electronic equipment as well. Time-domain static network models of thyristors [ 2 ] , riacs [3] , ndbipolar transistors [4], [SI xist. Simple modifications to thesemodels allow switching characteristics or diodes, GTO s, andpower MOSFETs to be simulated. Time-domain static net-work models of dc m achines have also been published [6]-[8].Furthermore, the well-known Electronic Circuit Analysis

0093-9994/88/0100-0106$01 OO O 1988 I EEE

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1 0 7H ANI 5IMIJLAIION OF THREE-PHASE INDUCTION DYNAMICSProgram (EC AP ) has been modified du ring the last decade tospecifically perform numerical analysis of the transient behav-ior of power inverters. and renamed the Bristol TransientAnalysis Package (BTRAP) [9].

I n the field of asynchronous machines it has been shownhow general-purpose circuit analysis software can be used

2 ) Electromechanical Description: Since the referenceframe is holonomic in nature the instantaneous electromag-netic motoring torque can be obtained from

1T~~ = ? i T ( & L ) i N . m. (3)together with appropriate time-domain static network modelsfor simulation of machine [lo]-[12] and total system [I31dynamics. Another such time-domain static network modelhas recently been reported [14]. The method in generalrequires static network models of the various constituentdevices and subsystems. The macromodel of the overallsystem is numerically generated and solved by a digitalcomputer under the control of a general-purpose circuitanalysis program. Models of different complexity for variousdevices and subsystems are stored in the program library. As aresult the simulation technique has the flexibility to call mod elsof appropriate complexity for a particular problem at hand.

A new and original time-domain static network model of athree-phase induction machine with wound rotor is developedhere. The model caters to unbalance of all windings so long asthe topology remains unaltered. The theory presented shouldaid easy modeling and simulation in the time domainof the most difficult internal asymmetrical faults [15], [16] thatsplit one or more phases, o n either or both sides of the airgap,into multiple sectio ns, with or without interconnections.Modern circuit analysis software like SCEPTRE [I71 can beused to perform transient analysis of the network and therebysimulate the dynamic behavior of the machine.

111, T I M F - D O M A I NA TH F M. A TICA L MODELBased on nswmptions presented elsewhere [IS], the dy-namics of the six coupled windings can be quantified along a

holonomic reference frame containing axes A BC for the statorand abc for the rotor as follows.

I ) Electrical D e . w i p t i o n :u=z i v (1)o r

Extracting L from 2 in ( 2 ) and substituting in (3), th eexpression for the instantaneous electromagnetic motoringtorque becomes

Te.m -![ i d [ia(Lda+LuA)in or+ ib(LAb+LbA) in+ic(LA,+L,) si n

+iB [i'(LBa+LaB) si n o r - -+ ib(LBb LbB) in O r+ i LBc+LcB)si n

( :

+ic [ia(Lcu+Luc) in

+ib(LCb+ hC) in O r - -( 311ic(Lcc Lcc) si n Or . (4)

3 ) Mechanical Description: Instantaneous rotor speed inthe anticlockwise direction can be obtained from

T,, - T \ = J O r + K f 0 , N m. ( 5 )Equation (5) quantifies the dynamics of an inertia load withlinear viscous friction. A time-domain static network modelthat has this mathematical description is simply a series R-Lcircuit. The friction coefficient Kf is represented by theresistance, and the inertia J by the inductance. The circuitwhen fed from a voltage source of magnitude (T, - T,)results in a current that represents the rotor velocity 9,.

The electrical and the electromechanical descriptions con-tained in ( 2 ) an d (4) are completely general. They can becons iderab ly simplified if the followin g practical cons traintsare introduced: 1) reciprocity of mutual inductances; 2 )identical winding structures for the stator group and the rotorgroup; 3) balanced machine doubly fed through unequalexternal impedances with no magnetic coupling betw een them;an d 4) both stator and rotor neutrals disconnected from theneutrals of their respective supplies.

The simplified mathematical model then is the following.

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108 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL 24, NO I . J A N U A R Y I FE B R U A R Y 19RXElectrical description:

R + pLb b1 1

c cR + pL

The various new self-inductances in (6) ar e

L ' = L ' - L 'an rm L ' = L 'bb -L ' m Lc'=Lc'c-Lrm. (7 )The electromechanical description is as follows:

( i ' a i A + i ' b i B + i ' c i C )in Or

+ ( i ' a i c + i ' b i A + i ' c i B ) in d r + - . (8)( 31IV. TIME-DOMAINTATIC ETWORKODELThe availability of various second-, third-, and fourth-

generation software for circuit analysis, e.g. SCEPTRE (andits variations: SUPER *SCEPTRE, EXTENDED SCEPTRE),ECAP, PCAP, SPICE, CANDY [3], has reduced the problemof numerical solution of the time-domain mathematical modelto a relatively simple matter. The software accepts thetopological description of any static network, ev en withnonlinear parameters. F acilities exist in ECA P, PC AP, andSCEPTRE to define these parameters directly as nonlinearfunctions due to saturation and hysteresis in the magneticcircuits, and skin effects in a deep bar rotor. The othersoftware needs multipliers and function generators of one formor another to be programmed by the user. The nonlinearvariation of the machine parameters can be easily obtaineddirectly from the load test data by solving an optimizationproblem [12] for parameter estimation.

The time-domain static network equivalent (Fig. 1) of th eelectrical description given by (6) is deduced as follows. The

i 4 R L A 2@RA

Fig. 1. Time-domain static network model of three-phase induction machinewith unbalance in self-impedance of all electrical circuits on both sides ofthe airgap.

stator A-phase voltage isU A = R A A+pLA i A+pLI,

= RA i A+pLA ( A+ J A ). (9 )To avoid "computational delay" [17] (computation at the nthtime step begins with inde penden t-variable values valid at the(n - 1)th step) the line current i A is referred to as

(10)A= L A - A


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G H A N I . S I MU L A T I O N O F T H R E E - PH A SE I N D U C T I O N D Y N A MI C S 109where

I , current through the inductor L A and, being a statevariable. has been updated at the start of each timestep.time-varying current source across the inductor, and

i co s 8r+ib os+ i c c o s O r - - . ( 1 1 )( 231

The circuit between nodes 12-2-3-1 shown in Fig. 1, withoutthe dummy resistance RL A , has the mathematical descriptionof (9).The six dummy resistances RLA , RLB, RLC, RLD,RLE, and RLF have been provided to eliminate the two all-inductor time-varying current sou rce cutsets [171 prese nt in th eneighborhood of nodes 1 and 10. They also prevent inadver-tent generation of such cutsets when an induction motor is asubsystem of a large total system.

The topological sum mary of the time-domain static networkmodel excluding the mechanical load is 14 nodes, 24 branches,I2 rebistors. 6 inductors, 6 current-controlled current sources,and 7 function generators, which include multipliers. Alldummy shunt resistances have been assigned a value of 1 0 kQ.This time-domain static network model of an inductionmachine can be used for simulation of a variety of normal andabnormal operating conditions.

A doubly fed induction machine supplied by two three-phase three-wire systems requires four state variables for itselectrical description. This is also true for the time-domainstatic network model, provided the six dummy shunt resistorsare removed. Their presence is required, however; otherwisea general-purpose circuit analysis program would ceaseexecuting and return an error flag. Although the presence ofthe dummy resistances gives rise to six state variables with noredundancy [ lo] , the possibility exists to account for the ironlosses in the machine core. In this investigation the dummyresistances are of equal but high value, the upper limit beingdictated by the numerical accuracy required of the solution.

Similar network models can also be derived to represent amachine with any asymmetrical internal fault that splits one ormore phases on either or both sides of the airgap into multiplesections with or without interconnections [151. This wouldrequire the use of suitable connection matrices [16].

V . D Y N A M I CTUDIESThe time-domain static network model shown in Fig. 1 has

been used to study the dynamics of two completely differentbalanced three-phase induction machines. I The first machinehas a wound rotor that is short-circuited. It is rotating at aconstant speed but suddenly switches on to balanced three-phase three-wire supplies of the following descriptions: a)sinusoidal voltage waveform, and b) six-step line-to-neutralvoltage waveform as produced by some static thyristor

Two internal reports on this topic, comprehensive in nature and containingdata tiles. ar e obtainable on application from either the author or theLibmrian. Newcastle Upon Tyne Polytechnic, Newcastle Upon Tyne, NE1XST. U . K .

inverters. Study c) involved the second machine with cagerotor supplied as in a), but thro ugh triac-controlled soft-startequipme nt. The circuit is as shown in Fig. 1 of [19] except thatthe inverse parallel in-line thyristors were considered to betriacs. Such a controller provides a very effective form ofvoltage control and has wide commercial acceptance. Theprocedure for the calculation of steady-state performance isknown [ 19 ] . The present paper contains the results ofsimulation, verified by measurement, of the transients thatexist during the first cycle. Machine dynamics with a similarcontroller have been simulated (Fig. 4 in [20]), but underdifferent operating conditions, and the results of simulationhave not been experimentally checked. Computer simulationrequires validation bec ause accumulated roundoff and appro x-imation errors may cause the reality to be completelydifferent. Dynam ic study c) and the operating conditions wereselected for thorough verification of the procedure, device,and the machine models. To focus purely on the electricalaspects, measurements were taken with the rotor locked.

The full-load power output of the first machine is 5 bhp at1440 r/min when supplied with the rated line voltage of 110 Vat 50 Hz. The machine parameters are

RA=Rs= Rc= Rs= 0.0815 QLA = Lg= Lc= Ls = L6 = LL

=L i = Lr = (19.55 + 0.8754) mHR=R=R=R=0.13b c r Q Llr=13.03 mH.

A voltage of 81.65-V peak line-to-neutral is used for a). Thecorresponding value for b) is 85.5 V so that the peak value ofthe fundamental component remains unaltered at 81.65-V line-to-neutral. Figs. 2(a) and 2(b) show the predicted instantaneouscurrent in the stator A-phase and torque with sinusoidalsupply of 50Hz suddenly c onnected. Th e operating conditionsare zero sl ip and 90 switching angle. The zero-degree angleis defined to be the instant when the A-phase line-to-neutralvoltage becomes zero with positive rate of change. Figs. 3(a)and 3(b) are those obtained with the six-step line-to-neutralsupply voltage at 50 Hz with zero slip and zero-degreeswitching angle. Fig. 4(a) shows the oscillograms of the linevoltage and current at the input terminals of the inverter-fedmachine. Fig . 4(b) gives the correspond ing simulated current.The error in calculation for the electromagnetic motoringtorque is entirely numerical in origin. The torque is evaluatedfrom the difference between large quantities, and smallround-off errors in these have considerable effect on the torquefigure. For the six-step supply voltage waveform, maximumerrors occur during the instants when the current peaks occur,and the rate of change of current becom es discontinuous. Thecalculation for torque can be conducted at a greater accuracythan that presented, at the expense of central processing unit(CPU) time, by de mandin g more accurate calculations for thevarious currents. The largest CPU time encountered on anIBM 370/168 computer was 337 s for seven cycles of statorcurrent waveform . Th e calculation was cond ucted for six-stepline-to-neutral voltage input, 10-percent slip, and 30 switch-ing delay.


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110 IEEE TRANSAC TIONS ON INDUSTRY APPLICATIONS. VO L 24 NO I J A h U A K Y / F F B R U A R Y I Y X X

120Predicted steady state current = 8.80 A rms.Measured steady state current = 8.60 rms.

Time-s (~0.01)Computer simulation.-40

-80 (a)Computer simubtion.ff+j---ime-s (~0.011

Predicted steady state torque = -0.06N.m.Measured steady state torque = 0.0 N.m.

-801 "(b )

Fig. 2. Predicted transients in induction motor suddenly energized fromsource delivering sinusoidal voltage wav eform. (a) Stator A-phase current.(b) Electromagnetic motoring torque.

t 8011 I Computer simulation.I5 0

u- 40I*' 401 Time-s (x0.01i(a )

Computer simulation

Predicted averagesteady state torque = 0.05 N.m.Measured averagesteady state torque = 0.0 N.m.

(b)Fig. 3 . Predicted transients in induction motor suddenly energized frominverter delivering six-step line-to-neutral voltage waveform. (a) Stator A-phase current. (b) Electromagnetic motoring torque.

The second induction motor fed through a soft-startequipme nt was at standstill, and the cyclical triggering of thein-line triacs was delayed by 90". The parameters of th emachine used for this simulation areR~=Re=Rc=Rs=O.8970

L~=L~=Lc=Ls=(100.9$3.79) HR ' = R ' = R ' = R ' = 0 . 9 7 7 Qa b c r

L ; = L ; = L ; = L ; =(100.9+3.79) m HL;,= 67.27 m H .

The rating of this machine is 3 kW at 1420 r/min whensupplied with the rated line voltage of 240 V at 50 Hz.Oscillogram s of instantaneous voltages and cu rrents during the

(d)

b

Compu7er stmu btion.Measured value =

t i20 11 1

( b )Fig. 4. Induction motor rotating with 1.07-percent slip and fed from a ninverter d elivering six-step line-to-neutral voltage waveforin at SO Hz . l a )To p trace: line-to-line voltage: scale 54 V/div . Bottom trace: line current:scale 14.2 A/div. (b) Prcdicted line current.

( b )Induction motor fed through soft-start equipment from 50-Hr powersupply. The rotor is at standstill and triacs are cyclicully triggered to r 90gphase delay. oscillogram^ of first cycle of ( a) tator v:dtage betwcen line5 Aand C (vertical scale: 60 Vidiv.: horizontal scale: 5 ins). and i b ~corresponding stator input current in line C ( v e r t i c a l ~ a k0 A / ~ I Lhorizontal scale: 5 msidiv. 1.

Fig. 5 .

first cycle are shown in Figs. S(a) and 5 ( b ) and thecorresponding simulated waveforms in Figs. 6(a) -h(c) . I n -spection of Figs. S an d 6 reveals that the shapes of' actualwaveforms of the stator voltages are similar to those obtainedby computer simulation. The magnitudes of the instantaneous


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G H A N I SIMUI.ATION OF T H R E E - PH A SE I N D U C T I O N D Y N A MI C S 1 11

(c )Fig. 6. Predicted waveforms in induction motor fed through soft-startequipment. ( a) First cycle of stator voltage between lines A andC. (b) Firstcycle cf stator input current in line C. (c) First two cycles of electromag-netic motoring torque.

voltages at certain instants of time, howev er, differ considera-bly, This is due to finite inductances of the supply lines thathave not been accounted for in the computer simulation. Atthose instants of time when the triacs are triggered and the rateof change of current in the corresponding line is high,significant amounts of voltage dro p occur in the supply lines ofthe actual system. Investigation in this direction will beconducted in the near future. This departure diminishes afterthe first cycle when all three lines are fully energized. Thesmall discrepancies in the curren t waveform s are due solely tothe fact that. because of the analogue nature of the firingcircuits. the triacs were not gated at precise instants of time.This resulted in slightly asy mme trical triggering. Electromag-netic motoring torque (in (8)) is a simple nonlinear algebraicfunction of the machine currents. Validity of the currentstherefore implies the same for the torque, provided the spaceharmonics of the airgap M MF a re negligible. Parasitic torquesamount to only a few percent, and the core loss in the machineat standstill with reduced supply voltage has virtually no effecto n the torque dev eloped. The simulated torque (Fig. 6 (c)) wasnot verified experimentally because the associated problemswould have diverted the research from its present objectives.Fo r a solution of acceptable degree of accuracy for two cyclesof both current and the torque, a CPU time of 17 3 s wa srcquircd. To ensure that turn-offs occ ur at the neighborhood ofthe holding current, higher accuracy of solution is required.This needs an increased CPU time of 630 s due to drasticreduction of the size of the time step taken by the integrationroutine. which is completely unnecessary for the machinedynamics. The worst-case solution for the voltage waveformduring a turn off can be easily obtained using the procedurediscussed in the Appendix. The CPU times quoted are thoseobtained using an AMDAHL 5860 digital computer, which isapproximately three an d one-half times faster than an IB M3701168 computer. All overvoltage data during turn-off was

deleted from the data file used in generating the plot file. T heplot file was then used to draw the waveforms of Fig. 6 . Thevoltage waveforms of Fig. 6 therefore do not show anyovervoltage transients.

VI. CONCLUSI ONA unified and universally applicable method for simulation

of dynam ic behavior of a large interconnected nonlinear time-varying physical system containing one or more three-phaseinduction machines as subsystems has been proposed. Thislarge system may also contain static power conversion,conditioning, or control equipment in which power switchingis achieved by devices like diodes, transistors, thyristors,GTOs, tr iacs, and power MOSFETs. The new procedure issimple yet accurate and flexible, and results in considerablesavings in highly s killed, hence expen sive, manho urs requiredto achieve a successful simulation. The power of the methodlies in its complete generality, in that it is not specific to anyparticular system. Furthermore, the method allows flexibleusage of device and subsystem models of various degrees ofsophistication and complexity with the greatest ease. The onlydisadvantage of the method lies in its inherent computingoverheads. Specially designed programs for specific systemsscore on this point and are in general faster. However, codingfor use in a design office is not an easy task and may not becost effective since maintenance and documentation is expen-sive. Also, the life span of specialized p rograms can be rath erlimited because of rapid innovation. Research is at presentbeing directed towards increasing the execution speed ofgeneral-purpose circuit analysis programs. Two approachesare being used: a) development of better numerical methods,and b) replacement of those portions of the program that arerepeatedly used in a loop written in high-level language bytheir equivalent in the assembler. Approach b) is costly andbecomes economically feasible only for large expensive


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1 1 2 lEEE TRANSACTIG NS ON I N D U ST R Y A PPL I C A T I O N S, VOL. 24, NO . 1. J A N U A R Y I FE B R U A R Y 1988i Computer simulation.

-1m/ II I ;

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time-s (~0.01)

1.0

-1.0 I u-1.5I- ' 10.0 0.5 1 0 1 5 2. 0 2.5 3.0 3.5 4.0Time-s (~0.01)(b)

Fig. 7. Predicted waveforms in single-phase inductive load controlled bytriac. (a) Voltage across load. @) Current through load.general-purpose programs. The possibility of using parallelprocessors to provide superior computational speed and poweris also under investigation.The proposed procedure uses a time-domain static networkmodel of an induction machine. The m odel has been developedusing the physically existing holonom ic three-phase referenceframe, thus allowing for unbalance of all electrical circuits onboth sides of the airgap. H owe ver, the magnetic circuits of themachine are considered to be completely balanced. The thirdand higher harmonic variations of these stator to rotor mutualinductances have been n eglected for simplicity. It is possible toaccount for such variations [2 I]-[23] by simply includingthem in the space functions defining the stator-to-rotor mu tualinductances.

The proposed model allows m achine dynamics to be studiedwith unbalance in the self-impedances of all electrical circuits.It is therefore suited for prediction of transient behavior underany conceivable asymmetrical external and internal fault oneither or both sides of the airgap as long as the structure of themachine windings remains unaltered. Similar models forsimulation of asymmetrical internal faults, on either or bothsides of the airgap, that split one or m ore phases into multiplesections with o r without interconnections [15], [16] can also bedeveloped based on principles presented in this paper. Thetime-domain static network models of all other subsystemsconnected to the machine must be developed along axescompatible with the reference frame used. The model pre-sented w ould allow static-power electronic equipment contain-ing transistors, thy ristors, triacs, etc. to be connected simulta-neously to both the stator and the ro tor electrical circuits. Sucha situation, though far-fetched, shows the versatility of themodel. The model has been used to simulate machinedynamics during a run-up. Successful simulation of a totalsystem comprising an inverter feeding an induction motor withvarious types of faults on the inverter has also been achieved[131.

0I

-1501

I A2001.w 1.05 1.10 1.15 1. MTime-r (xO.011

(a)---5.5! \ tt

I t ' I

z 45 ; ',\ IComputer simulation I351

2 5 ,I+5

\

0.5 1 ,-5 L - _ _ _ -100 105 110 115 120Time-s (XOOI )(b )

Fig. 8. Predicted worst-case turn-off transients in single-phase inductiveload controlled by triac. (a) Voltage aross load. (b) Current through load.

Although in this investigation machine parameters havebeen assumed to be constant quantities, the accounting for thisnonlinearity [12] is intended at a future date. For machineswith P poles, all expressions for the electromagnetic m otoringtorque should be multiplied by a factor P / 2 . Experimentalverification of the model and its applications are alsopresented.

The SCEPTRE program automatically selects the optimumtime steps for integration in order to meet the user-specifiedaccuracy requirement, hence obtaining least CPU time.Numerical accuracy of this program is extremely high.Referring to Fig. 8the comp uter prediction of the peak voltageacross the triac was 196.7 4367 V. Using an electroniccalculator with ten digits this voltage was calculated, purelyfrom physical considerations, to be 196.74313 V . The CPUtime required was 55.061 s. A demand for greater accuracythan this would have required higher CPU time. SCEPTRE israther a slow program . T his is due solely to the inefficiency ofthe integration routines used [24]. Nev ertheless , the validity ofthe simulation technique has been established, and other moreefficient circuit analysis programs [3], [25] can be used toovercome the shortcoming. The CPU time becomes a problemonly if a general purpose large mainframe computer is used.With a minicomputer dedicated solely for the purpose ofsimulation, the CPU time ceases to become a limiting factor.Most circuit analysis programs developed for use on largemainframe com puters also have versions suitable for minicom-puters.

NOMENCLATUREi, Y Column vectors of instantaneous current, volt-

age (A, V>.L , 2 Inducta nce, impedance matrices (H, 0).i, u , L Elements of i, u , an d L .p 4 d / d t Differentiation operator.

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GHANI: S IMU LA TIO N OF THREE-PHASE INDUCTION DYNAMICS 113J Moment of inertia, current source (kg m* m* ,A).K Coefficient.R Resistance (a).r Torque (N -m ); when used as superscript indi-

Cates transpose.8, Instantaneous angular position of rotor a-axis with

respect to stator A-axis (rad). (For a P polemachine, distinction should be made betweenelectrical and mechanical radians.)

Superscripts an d Subscripts2:a, b, cse .m.IrnrA , B , CSi

Rotor phases.Electromagnetic.Friction.Load.Mutual between two relatively stationary coils.Rotor.Stator phases.Stator.Referred to stator.

APPENDIXThe network model of a triac is shown in Fig. S of [3]. Inthis investigation a subprogram FUNCTION FR was used to

select the high or the low value for the resistances RS and R 6whenever the current through the device model fell below orexceeded the holding current. The original model containeddata in tabular form for this selection purpose. Appropriatevalues for the resistances RS and R6 were obtained fromTables 1 and 2 of [3], respectively. It was subsequently fou ndthat under certain operating conditions the model failed to turnoff. The problem lies in SCEPTREs inability to locate theprecise instant when the current has attained the holding valueand then to operate with sufficiently small time-steps to ensurea proper turn-off. In fact, the next time step was so large that acurrent in the reverse direction of magnitude greater than th eholding value was com puted. As a result the model was unableto pull out from the latched mod e. S ince the Princeton CircuitAnalysis Program (PCAP) [ 3 ] calculates the precise instantwhen the current attains the holding value, and thereafter thenext time step is exceedingly small, such a p roblem do es notarise. This subprogram FR is not for exclusive use withSCEPTRE. Modern circuit analysis programs usually have asimilar subprogram capability and FR can be used with allsuch software. With the thyristor model [2], use of FUNC-TION FR is not required since reverse conduction can neveroccur.

The transients can be calculated with considerable numeri-cal accuracy if the computation is allowed to proceed only inthe neighborhood of a transient by stating a STA RT TIME an da STOP TIME, along with a l imit to the MAXIMUM STEPSIZE. Such a computation fo r transients would require initialvalues of the state variables involved. They can be obtainedeasily from the data printed in tabular f orm from the previousnormal run. The simulated transients in a triac-controlled

Superscripts and subscripts with upper-case letter refer to the stator, withlower-case letter to the rotor.

single-phase inductive load are shown in Figs. 7 an d 8 . Th evoltage transient so simulated by the mod el is the worst-casetransient. It is obtained by presenting the blocking resistanceof the device to the flow of current when the magnitude hasjust decreased below the holding value.

ACKNOWLEDGMENTThis paper is a result of advice and suppo rt given by Dr. B.J. Cory, Rea der, Imperial College of Science and Technology ,

London, SW 7 2AZ; Messrs. L. Barnes, Principal Lecturer,and A. R. Shirley, Senior Lecturer, and members of theComputer Unit, Newcastle upon Tyne Polytechnic; and theanonymous referees.

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Sayeed Nurul Ghani is a native of Bangladesh andhas been in the United Kingdom since 1963 Hereceived the B Sc. degree in electrical engineeringfrom the University of Peshawar, Pakistan. in 1961with a Gold Medal and the President of PakistansAward for meritorious results From 1963 to 1966he was a postgraduate student at the ImperialCollege of Science and Technology. University ofLondon, where he received the D C and Ph Ddegrees for researching the design of impulse-commutated thyristor inverters for ac variable-speed drivesFrom 1966 to 1970 he was a Lecturer at K ingston Polytechnic, U.K , nd isat present a Senior Lecturer at Newcastle upon Tyne Polytechnic, U K Heworked with Industrial Instruments Ltd , Bromley, Kent, during 1966, onvariable-speed ac machine drives, with Thermal Electronics Ltd , Wimble-don, Surrey, during 1969, on thyristor inverters for induction heating; andwith Lucas Research C entre, Solihull, West Midlands. during 1978- 1983. onthe simulation of power electronic equipment In 1972 he participated in anAnglo-Am erican teacher s excha nge scheme and lectured at the StateUniversity of New York for a year His fields of interest are power systems,power electronics, control engineering, electrical m achines. systems simula-tion, and optimization He is an author of twelve technical publications.Dr Ghani is a Chartered Electrical Engineer and a Corporate Member ofthe Institution of Electrical Engineers, U K

Digital Computer Simulation of Three-Phase IM Dynamics-A Generalized Approach

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