fast modelling techniques for microprocessor-based optimal pulse-width-modulated control of...

Fast modelling techniques formicroprocessor-based optimal

pulse-width-modulated control ofcurrent-fed inverter drives

S.R. Bowes, Ph.D.. C.Eng.. M.I.Mech.E., F.I.E.E.. and R.I. Bullough, Ph.D.

Indexing terms: Inverters, Microprocessors, Control systems, Computer-aided design

Abstract: Quasi-square-wave current-fed inverter drives suffer from low-speed torque pulsations, which cancause mechanical resonances and speed-ripple problems. Recent developments in microprocessor-based PWMcontrol techniques offer the possibility of improving the low-speed performance of CFI drives, thereby signifi-cantly extending their range of applications. This paper describes the development of fast computer modellingtechniques which will allow wide ranging computer-aided-design investigations to be performed on optimalPWM current-fed inverter drives. The validity and accuracy of these fast modelling techniques are confirmedusing practical results obtained from a microprocessor controlled current-fed PWM inverter-drive system.

List of principal symbols

/ = phasor currentL = inductanceM = mutual inductanceP = pole pairsSn = harmonic slipT = torquei = instantaneous currentk = integern = harmonic orderr = suffix denoting rotors = suffix denoting statort = instantaneous timea = switching anglesT = time variable¥ = fluxCDS = stator angular frequency (electrical)

1 Introduction

During the last decade quasi-square-wave current-fedinverters (CFIs) have been developed, and gained popu-larity as a result of their relatively simple and ruggedpower-circuit topology [1-6]. In addition, as a direct con-sequence of using a current source 'front-end', CFIs haveinherent short-circuit protection and regenerative capabil-ities, and also provide effective buffering of the inverteroutput from supply-voltage variations. Also, because of thedirect control of stator current, precise closed-loop controlof CFI drives can be implemented with relative ease.

Of course, CFIs also have disadvantages; for example,to provide the current source 'front-end' a large DC linkinductance is generally necessary and to suppress voltagespikes during commutation, large commutating capacitorsare required. Also, as a direct result of the quasi-square-wave current, low-speed torque pulsations, which canexcite mechanical resonances and cause noticeable speedripple, are present in the CFI drive.

Research is continuing to minimise these problems, and,in the past, the presence of low-speed torque pulsationshas generally been considered to the most serious dis-

Paper 3225B (PI, P6) received 8th August 1983

The authors are with the Department of Electrical and Electronic Engineering, Uni-versity Engineering Laboratories, University of Bristol, University Walk, BristolBS8 1TR

advantage. This problem has severely restricted the viablerange of operations of CFI drives, and prevented their usein many industrial applications requiring smooth rotormotion down to zero speed.

1.1 Optimal PWM techniquesMore recently PWM techniques have been proposed andused with the aim of reducing the adverse affects of torquepulsations [7-14]. While PWM is, of course, a well knowntechnique for achieving fundamental voltage control involtage-fed inverters (VFIs) [15-20], its application toCFIs is relatively recent.

The application of PWM techniques to CFI drivesdiffers in a number of important aspects from those used inPWM VFI drives. For example, the relatively long com-mutation times associated with CFIs impose practical limi-tations on the switching frequency that may be achieved inPWM operation. Also since current harmonics areimpressed upon the drive motor from the CFI, the filteringaction of the motor inductances cannot improve the har-monic structure of the PWM switching strategy; as is nor-mally the case in VFI drives. In addition, in the CFI drive,fundamental current control can be achieved by control-ling the DC link current, and the PWM inverter controlcan then be used solely for controlling the harmoniccontent of the PWM current waveform [14]. The separa-tion of the current and harmonic control functions allowsthe choice of PWM control to be directed solely towardsimproving the motor torque profile, while keeping theadditional losses due to the PWM motor current to aminimum.

As a direct consequence of the separation of thecurrent- and harmonic-control functions, the lack of filter-ing by the motor of the PWM switching strategy, and therelatively low switching rates, typically less than 25 Hz [5,7], there are distinct advantages to be gained in using theso-called 'optimal' PWM control strategies.

These optimal PWM strategies are arranged to produceswitching patterns in the PWM stator current waveformwhich optimise some specified performance criteria. Themajority of PWM techniques used in CFI drives to datehave been based on 'harmonic-elimination' techniques [8,10], which are designed to eliminate the low-order stator-current harmonics and thereby reduce the low-ordertorque harmonics.

While this approach can be highly effective in prevent-

IEE PROCEEDINGS, Vol. 131, Pt. B, No. 4, JULY 1984 149

ing low-frequency mechanical resonances of the motor-load system, it is not necessarily optimal in the context ofimproving the quality of the rotor motion of the drivemotor at low speeds. For example, in traction applicationsthe rotor speed ripple resulting from motor torque pulsa-tions can detract from passenger comfort, and excessivespeed fluctuation at low speeds is clearly unacceptable inshunting applications. The same is true for steel-milldrives, paper mills, extrusion plants, mine hoists etc., whereprecise speed regulation is important.

For these reasons optimal PWM switching strategieshave recently been proposed [12] which minimise theextent of the rotor speed ripple resulting from theseperiodic torque pulsations. The adoption of performancecriteria based on minimised speed ripple, for comparingand assessing the performance on PWM current strategies,would appear to be a logical step towards achievingsmoother rotor motion at low speeds. There are, however,other performance criteria which can equally be used; forexample, rotor positional error, the differential of whichgives speed ripple. More sophisticated performance criteriacan also be used which involve the minimisation of bothspeed ripple and harmonic losses together etc. All thesevarious optimal performance criteria can be tailored tomeet the system performance requirements, and theoptimal PWM current strategies designed to achieve them.

However, to perform wide ranging investigations andassessment of the effects of using optimal PWM controltechniques on the steady-state performance of CFI drivesrequires a fast accurate model of the system, which willallow rapid and efficient computation of the optimalPWM switching strategies. This is particularly true whennumerical minimisation techniques are used to minimisethe performance criteria objective function.

For example, one possible objective function for thespeed ripple case can be defined [12] as F(a) = peak topeak value of (J'o TP(T) dx), where the torque pulsationfunction Tp is defined in terms of a set of PWM currentswitching angles a, and an approximate model of the CFIdrive induction motor. Minimisation of the performancefunction F(a), with respect to a, provides the optimalPWM switching strategy which minimises the extent of therotor speed ripple. As can be appreciated, to minimise F(a),it is generally necessary to access the subroutine which cal-culates Tp many times during the course of the mini-misation procedure, thus the Tp subroutine must becomputationally efficient. This is particularly so whendetermining the optimal switching angles for high pulsenumber PWM strategies, where many degrees of freedomare involved.

The study described in this paper was concerned withthe development of an extremely efficient, and numericallystable, computer-based model of the CFI drive, whichwould allow optimisation studies to be performed on rela-tively small, 'short-wordlength' microcomputers in thefuture.

2 Development of PWM CFI drive models

A recent research study [12] into the development of suit-able models for determining optimal PWM switching stra-tegies has been reported. This study used thefrequency-domain harmonic model for the CFI drive, andpresented results solely for optimal PWM strategies basedon three pulses per half cycle. Our research experience hasshown the method to be inappropriate for computation ofoptimal PWM strategies for high pulse numbers, due tocertain computational difficulties. These difficulties were

centred around the computational load involved in theevaluation and summation of individual harmonics: thiswas found to be prohibitive when applied to high pulsenumber PWM strategies. In addition, during the mini-misation process, false local minima tended to be detectedas a result of ill conditioning arising from the truncation ofthe infinite harmonic series.

To overcome these difficulties an approximate torquecalculation method is proposed, based upon an harmonicmodel of the PWM CFI drive. It will be shown that thisapproach provides an extremely rapid, numerically stable,and efficient means for computing the optimal per-formance function and resulting optimal PWM switchingstrategy for any pulse number. It will also be demonstratedthat, although certain approximations have been made, theaccuracy of the method is extremely good over a widespeed range down to very nearly zero speed.

2.1 Fast modelling techniquesThe fast torque-calculation routine developed in the nextSection is based on the induction-motor harmonic modeldescribed in the Appendix.

Two basic assumptions are made to simplify the model;namely

(i) All the stator current harmonics flow in the rotor ofthe induction motor, thus implying that the stator androtor harmonic currents are considered equal.

(ii) Only the fundamental component of airgap fluxcontributes to the harmonic torques. This implies that theharmonic flux component contributions to harmonictorque are negligible in comparison to the fundamentalflux contributions.

The range of validity and implications of each of theseapproximations will be considered below.

As shown in the Appendix, eqn. 19, the amplitude of thenth harmonic of rotor current can be expressed as

Irn = Isr, ™>, Sn Mntfn + (riW, Sn(L,n + MJ)2] " ^ (1)The difference in the amplitude of the stator and rotorcurrent harmonics of order n is therefore small, as shownin eqn. 1, if

Lrn < Mn and rrn <̂ (ncos Sn Mn)

The first condition is generally satisfied in practice, and thevalidity of the second condition increases with increasingharmonic frequency. If the application and scope of themodel is restricted to stator current switching strategiesdisplaying halfwave symmetry, then only odd harmonicsexist and the lowest harmonic which needs to be con-sidered is the fifth.

To confirm the validity of assumption (i), the computedamplitude ratio Irn/Isn and phase difference based on eqn.1, for the six lowest-order harmonics are shown in Fig. 1,for a stator frequency range 0-1 Hz. As illustrated in thisFigure, the current amplitude ratio is within 5% for statorfrequencies greater than 0.2 Hz, thus confirming the valid-ity of the approximation implied by assumption (i). Inaddition, the amplitude ratio is greater than 0.98 for statorfrequencies greater than 0.5 Hz, noting that the harmonicslip is always close to unity and therefore the effects offundamental slip frequency are negligible.

It is evident from these results that the rotor-currentassumption is satisfied for stator frequencies down tobelow 0.5 Hz for the induction motor used in the experi-mental drive. It is important to note that when PWM stra-tegies are used the low-order harmonics are generallyeither small or eliminated, which further extends the lowfrequency range of the model.

150 IEE PROCEEDINGS, Vol. 131, Pt. B, No. 4, JULY 1984

The validity of assumption (ii) can be assessed usingeqn. 22, given in the Appendix, for the amplitude of the

A 6 8 10stator frequency, Hz x10~1

xiO1

6 8stator frequency, Hz

10x10'

Fig. 1 Variation of error between rotor and stator current harmonicswith frequencya Amplitude ratiob Phase difference

airgap flux harmonic of order n; thus

yVan = Kn{irrnlncosSn)2 + L2

rnV12 (2)

The computed amplitude of the six lowest-order harmo-nics, with respect to y¥l, for quasi-square-wave operationis shown in Fig. 2. As illustrated in Fig. 2a and 2b theflux harmonics are extremely small for stator frequenciesgreater than 0.2 Hz; noting that in PWM operations thefifth and seventh current harmonics are usually small,resulting in a flux waveform of very low harmonic content.Fig. 2c indicates that although the harmonic contentincreases with slip frequency, it still remains small over thecomplete slip-frequency range. The experimental confirma-tion of this approximately sinusoidal flux waveform will bedemonstrated in Section 4.

Having established that assumptions (i) and (ii) result inextremely small errors in the approximate motor models, itis now possible to considerably simplify the computationalprocedure for determining the electrical-torque waveforms.

3 Fast computational procedure forelectrical torque

The torque component produced in one phase of themotor, due to the interaction of the nth current harmonicand the mth airgap flux harmonic, can be expressed, as

shown in the Appendix by eqn. 23, as

Tnm{t) = lrn sin (ncost - an)Vam sin {rncoj -

, - 2 .

(3)

0a

x10~2

5 r

A 6 8stator frequency , Hz x10"

n = 13-n=17--n=19T

x10

25

20

1 5

10

5

1 2stator frequency , Hz

A 6slip frequency , Hz

ft 10x10'1

Fig. 2 Variation of flux harmonics with stator and slip frequenciesa Slip frequency = 1 Hz b Slip frequency = 0 Hz c Rotor frequency = 2 Hz

Applying the airgap-flux approximation, which impliesthat the effects of harmonics of airgap flux are neglected,and only the fundamental component of airgap flux is con-sidered.

The total torque component produced in phase 1,resulting from the rotor current harmonic of order n, is

Tnl(t) = Irn sin (ncost - a^¥al sin (cost - 0t) (4)

The total pulsating component of torque produced inphase 1 by the combined effects of the rotor current har-monics is, therefore,

Tpl(t) = V.i sin (co, t-fijf, Irn sin K t - an) (5)n= 5

When the rotor current approximation is applied to eqn. 5,all the rotor current harmonics are replaced by the correisponding stator current harmonics; thus

Tpl(t) = Vfll sin K t-fijf, Isn sin (ncos t - yn) (6)n=5


Eqn. 6 can be expressed more simply as

Tpl(t) = ¥ a l sin (cost - PJUsiit) - I si sin (cost - yj]

(7)

Where isl(t) is the total instantaneous stator current inphase 1.

Similarly the pulsating component of torque due tophases 2 and 3 are

Tp2(t) = Tpl(t + 2n/3)

Tp3{t) = Tpl(t + 4TT/3)

(8)

(9)

assuming identical stator current waveforms in each phase,with a phase sequence 3-2-1. The total pulsating com-ponent of torque is the sum of the contributions of eachphase; namely

Tp(t) = Tpl(t) + Tp2(t) + Tp3(t) (10)

Substitution of eqns. 7-9 into eqn. 10 yields, after manipu-lation,

TJt) = isl(t) sin (cost-

+ is2(t) sin (o)s t + — -

+ is3(t) sin [cos t + — - (11)

cos (7l -

The total electrical torque produced can be obtained bythe addition of the average torque due to the fundamentalflux and rotor current and Tp(t), and multiplication by thepole pairs P, such that

TJLt) = + 3/2«Fal/rl cos («t - (12)

When PWM stator-current waveforms are used, furthersimplification of eqn. 11 for Tp(t) is possible. For example,if the instantaneous PWM current levels in phases 1, 2 and3 are denoted by ilk, i2k, i3fc within a time interval (tfc_l5 tk)defined by the switching instants in each phase, then eqn.11 becomes

TJt) =

+ i2k sin

+ hk sinAn

cos (7l - (13)

Since ilk, i2k and i3k are constant between successiveswitching instants, eqn. 13 is extremely simple, consistingonly of the addition of segments of three sine waves andthe subtraction of a constant. Thus, only fundamentalquantities need be computed. The computational efficiencyof this torque calculation routine is evident from inspec-tion of Fig. 3, which illustrates the steps involved in thecomputation of the pulsating component of torque for ageneral PWM piecewise constant stator current waveform.

3.1 Computation of optimal performance criteriaUsing the fast computational models developed in pre-vious Sections, it is now possible to formulate a variety ofperformance functions which can be used to determine

optimal PWM switching strategies to achieve any desiredsystem performance specification.

1-r

kl

e

f

^ UP

Fig. 3 Computation of electrical torque

a Phase 1b Phase 1 torque componentc Phase 2d Phase 2 torque componente Phase 3/ Phase 3 torque component

stator current— fundamental component of stator current

fundamental component of airgap flux

Possibly the simplest performance criterion involves theelimination of individual low-frequency harmonicsresulting in the so called 'harmonic-elimination' PWMcurrent strategies [8-10]. The procedure in this case doesnot involve a performance function, as such, but onlyrequires the solution of a set of nonlinear simultaneousequations, defined by the PWM harmonics in terms of theset of unknown PWM switching angles. Computed andexperimental results which illustrate the approach aregiven in Section 4.

The simplest optimal PWM strategy involves the mini-misation of the current harmonics using a performancefunction based on the current distortion defined, in thegeneral case, as

r2

If(14)

as a basis for comparing switching strategies. This per-formance criterion gives a good indication of the quality ofa PWM strategy in terms of the harmonic I2R losses; par-ticularly in the case of the low-skin-effect motor, which isdesirable in CFI. drives [21], since the variation of motorresistances with respect to harmonic order is relativelysmall. It is worthy of note that similar performance criteria


to those mentioned have also been used for developingPWM switching strategies for VFI drives [17-20].

More recently, in the context of CFI drives, the mini-misation of a performance function based on rotor speedripple has been used to develop optimal PWM currentstrategies [12].

Note that the method previously proposed [12]involves a prohibitive amount of computation, and istherefore considered inappropriate for the investigation ofoptimal PWM strategies with pulse numbers greater thanthree-pulses per halfcycle. Using the fast models describedin this paper has eliminated these computational diffi-culties, thereby allowing high-pulse-number optimal PWMstrategies to be developed.

If friction, windage and transmission losses areneglected in the CFI drive, the rotor speed ripple Acor(t),resulting from the pulsating component of torque Tp(x),can be expressed as [12]

(15)

where Tp(x) is determined from eqn. 13.It is important to note that the approximate torque cal-

culation method described previously is based on thesteady-state induction motor harmonic model, whichassumes constant rotor speed. However, if the calculatedtorque waveform is sufficiently accurate (this will be con-firmed in Section 4), then it is permissible to use Tp(x\ eqn.13, to predict the rotor-speed ripple resulting from thePWM stator currents.

Another optimal performance criterion, which has beenof interest in our research, is the minimisation of rotorpositional error. This performance criterion has particularrelevance at very low speeds, where it has been found fromexperimental observation to be more indicative of smoothrotor motion. Noting that during steady-state sinusoidalcurrent operation the rotor speed should ideally be con-stant, and therefore the rotor angular position increaseslinearly with time. However, when the motor is suppliedwith PWM piecewise-constant stator currents the rotorposition is subjected to periodic deviations from the ideallinear response, due to the rotor speed ripple discussedpreviously. Thus the rotor positional error may beexpressed as the integral of rotor speed ripple; thus

ApJLt) = I Acor(r) dx = -J J o Jo

Tp(z) dt' dx (16)

Both eqn. 15 for Acor(t) and eqn. 16 for Ape(t) can be com-puted extremely rapidly in the minimisation routine, bynoting that the minimisation objective function F(a) isdefined as the peak to peak value of Acor(t) or Ape(t) forminimised speed ripple or positional error, respectively.Thus the integrals associated with eqns. 15 and 16 needonly be evaluated at the switching instants of the PWMstator current.

This allows the equations for Tp(t) (eqn. 13) to be inte-grated to provide a very simple closed-form analyticexpression for Aa>r(t) and Ape(t), which can be includedinto the minimisation procedure to provide an extremelyefficient optimal solution. It is hoped to report more fullyon the results of this research in a companion paper in thenear future. However, the next Section gives some typicalexamples of both computed and experimental results basedon this research.

4 Comparison of computed andexperimental results

It is important to note that although the CAD packagecan be used to produce results at any desired operatingfrequency, the following results have been included tohighlight typical practical effects at low-frequency oper-ation. Particular emphasis is given to low-frequency oper-ation below 10 Hz, where the instantaneous torquepulsations can produce significant speed variations andsystem resonances, particularly for quasi-square-waveoperation. At higher operating frequencies the rotationalinertia of the load tends to damp out effects of torque pul-sations.• The computed results using the approximate induction-

motor model are shown in Figs. 4, 6 and 8 for quasi-square-wave harmonic-elimination five-pulse PWM

xiO1

2

1

0

-1

-2

x1010

5

0

-5

10 15 20

x10

25-2

-1 time , s

- 1 0

x1054321

A 6t i me , s x10"

Fig. 4 Computed quasi-square-wave operationfrequency = 5 Hz; slip frequency = 1 Hz; DI = 19.24

operation, and harmonic minimisation nine-pulse PWMoperation, respectively. The corresponding experimentalresults are shown in Fig. 5, 7 and 9, respectively. A com-parison of the experimental and computer waveforms illus-trates that very little discernable difference exists betweenthem, and confirms the validity and accuracy of theapproximate modelling techniques used in the CADpackage. Note that the harmonic ripple present in theexperimental current and torque traces is principally at300 Hz, and results from the phase-controlled bridge usedto produce the CFI DC link current source. This ripplecomponent is of a sufficiently high frequency to have negli-gible effect on the rotor motion or system resonances.Additionally, it should be noted that although the airgapflux is assumed to be sinusoidal, the flux waveforms pre-sented in the Figures are all stator flux. Therefore, thesmall deviations from the fundamental sinusoidal wave-form shown in the Figures are due to an additional Lis(t)term, which accounts for the stator leakage flux.

The stator flux results were obtained using search coilsattached to the stator of the CFI drive motor. The experi-mental torque results were obtained using an analoguetorque calculator [6] based on the motor transient torqueequations. This torque calculator uses the measured stator


Fig. 5 Experimental results corresponding to Fig. 4

a upper trace: stator current, 20 A/div. b torque, 10 Nm/div.lower trace: flux, 0.5 Wb/div. horizontal scale: 10 ms/div.horizontal scale: 20 ms/div.



SJ - 5

3 -10

n ni int ime , s

5 x,a<6

1510

< 5c 0£ -5D -10

time, s

x10.-1 5

Fig. 6 Computed PWM operation for five-pulse harmonic eliminationstrategy

frequency = 2 Hz; slip frequency = 0.04 Hz; DI = 6.87

time , s

101

2 1

10time, s

15x10

20

Fig. 8 Computed PWM operation for nine-pulse harmonic minimisationstrategy

frequency = 2.4 Hz; slip frequency = 0.5 Hz; DI = 12.1

current and flux as inputs, and therefore does not dependupon, or make any assumptions regarding, the motor par-ameter values. Commercially available torque transducersof the strain-gauge torsional-tube design have been used tomeasure the instantaneous torque. Unfortunately, the res-onance frequencies of these transducers occur in the power

range of interest, and therefore produce results inferior tothose obtained using the analogue torque calculator.

As further confirmation of the accuracy of the approx-imate modelling techniques, rotor current waveforms cor-responding to the previously used operating conditions areshown in Figs. 10 and 11. Fig. 10 illustrates the computed




X1032

o - 'o -2w -3

. r — ^

:

5

10

15

— •

time, s

J 2° -22

< 10

f 55 °2 " 5

2 -10x10"

time , s

Fig. 11 Computed rotor current waveforms using approximate harmonicmotor model

Waveforms correspond to operating conditions of Figs. 5-9a Quasi-square-wave operation (as Fig. 5)b Five-pulse PWM operation (as Fig. 7)c Nine-pulse PWM operation (as Fig. 9)

xiO1

10

-1-2-3

D 5

10

t i me , s

ti me, s

t ime , s

Fig. 10 Computed rotor current waveforms using exact harmonic model

Waveforms ccorrespond to operating conditions of Figs. 5-9a Quasi-square-wave operation (as Fig. 5)

b Five-pulse PWM operation (as Fig. 7)c Nine-pulse PWM operation (as Fig. 9)

Fig. 12 Computed results for rotor speed ripple and position error

Stator frequency = 1 Hz; zero load; 1/3-cycle showna Quasi-square-wave operationb Three-pulse PWM operation, minimised position error


Fig. 13 Computed results for rotor speed ripple and position error

Stator frequency = 1 Hz; zero load; 113-cycle showna Five-pulse PWM operation, minimised speed rippleb Five-pulse PWM operation, minimised position error

Fig. 14 Experimental results corresponding to Fig. 12Unloaded motor; 1 Hz supply; 6 A DCa Quasi-square operation

upper trace: utr,2.\ rads~'/div.lower trace: Ape, 0.105 rad/div.horizontal scale: 0.1 s/div.

b PWM operation with a three-pulse Ape minimisation strategyupper trace: cor, 2.1 rad s~ l/div.lower trace: Ape, 0.021 rad/div.horizontal scale: 0.1 s/div.

r i r

« ' A ' A ' 1 '

Fig. 15 Experimental results corresponding to Fig. 13Unloaded motor; 1 Hz supply; 6 A DCa PWM operation with a five-pulse Acor minimisation strategy

upper trace: cor, 2.1 rad s~7div.lower trace: Ape, 0.021 rad/div.horizontal scale: 0.1 s/div.

fc PWM operation with a five-pulse Ape minimisation strategyupper trace: a>r, 2.1 rad s~ '/dw-lower trace: Ape, 0.011 rad/div.horizontal scale: 0.1 s/div.

rotor current waveforms using the 'exact' harmonic motordeveloped in the Appendix, and Fig. 11 computer resultsare based on the approximate motor model of Section 2.As shown in these Figures the discrepancies between thetwo sets of results are slight, thereby justifying the assump-tions of Section 2 concerning the nature of the rotorcurrent waveforms.

Finally, computed and experimental results for rotorspeed ripple and rotor positional error are illustrated inFigs. 12 and 13 (computed) and Figs. 14 and 15(experimental). These results correspond to no-load oper-ation at a stator frequency of 1 Hz using quasi-square-wave five-pulse rotor speed minimisation PWM strategy,and three-pulse and five-pulse rotor positional error stra-tegies.

Comparison of these Figures demonstrates that evenunder quite difficult experimental conditions is possible toachieve reasonably good agreement between computedand experimental results, provided that the computersimulation and experimental conditions are precisely con-trolled. In particular, when the motor is supplied with con-stant stator current the slope of the torque/slipcharacteristic in the low slip region is extremely steep. It istherefore important that accurate measurement of the slipfrequency be achieved, since small errors in slip canproduce large errors in the torque waveforms. This isachieved in the experimental CFI drive by measuring therotor speed digitally, and feeding this measurement backto the microprocessor controller. The microprocessor is


then used to calculate the appropriate stator frequencycorresponding to the average slip frequency required.Using this arrangement it is possible to precisely controlthe operation of the drive on any part of the torque/slipcharacteristic.

There are also practical difficulties in measuring rotorspeed ripple and rotor position error, since the sensitivityof analogue tachometers to small deviations of speedabout an average value is limited. Noting that thepositional-error experimental waveforms were obtained byintegrating the AC component of the analogue tachometeroutput. It was found in practice that to obtain a reason-ably meaningful measurement, using the analogue tachom-eter, it was necessary to operate the unloaded inductionmotor at a low stator frequency of 1 Hz, with no addi-tional connected inertia and using low PWM pulsenumbers. Under these operating conditions the rotor speedripple is relatively large and can therefore be measuredreasonably accurately.

Note that, when extra load inertia is added and thestator frequency and/or PWM pulse number is increased,the rotor speed ripple is correspondingly reduced which,because of the tachometer sensitivity, makes experimentalmeasurement more difficult. Research is continuing toimprove the accuracy of this measuring system; it is hopedthat a report on the results of these investigations willfollow in due course.

The important conclusions to be deduced from Figs.12-15 is that both rotor speed and positional error can besignificantly reduced using optimal PWM current stra-tegies, and also both speed ripple and positional errorreduce with increasing PWM pulse number. This suggeststhat the optimal PWM pulse number should be increasedas the stator frequency is reduced to achieve maximumeffect; this is equivalent to the so-called 'gear-changing'strategy used in VFI drives.

5 Conclusions

Quasi-square-wave CFI drives suffer from the presence oflow-speed periodic torque pulsations, which can excitelow-frequency mechanical resonances in the motortransmission-load system, and cause noticeable speedripple. This problem has, in the past, severely restricted theviable range of operation of CFI drives, and preventedtheir use in many industrial applications requiring smoothrotor motion down to zero speed.

Recent developments in microprocessor-based PWMcontrol techniques now offer the possibility of improvingthe low-speed performance of CFI drives, which shouldsignificantly extend their range of application.

Microprocessor-based optimal PWM switching stra-tegies based upon sophisticated performance criteria, forexample, minimisation of torque ripple, speed ripple andpositional error, can now be designed using the previouslydeveloped CAD facilities described in this paper.

This study was concerned with developing fast com-puter models for the CFI drive, which would allow efficientwide ranging CAD investigations of optimal PWM currentstrategies to be performed. It has been shown that fastcomputational models for the CFI drive can be construc-tion based on certain approximating assumptions, and thatthese approximate models are sufficiently accurate foroptimal PWM performance studies over a wide speed(frequency) range down to nearly zero speed. It has alsobeen shown that investigations of optimal PWM currentstrategies of any pulse number can now be performed withrelative ease, thus eliminating the computational diffi-

culties experienced previously, and allowing wide rangingcomparative studies of optimal PWM strategies to be per-formed.

Demonstration CAD examples have been presentedusing quasi-square-wave control and PWM control stra-tegies involving harmonic elimination, and performancecriteria based on the minimisation of current distortion,torque ripple, speed ripple and positional error. Thesecomputer results have been compared with practicalresults from an experimental microprocessor-based PWMCFI drive system. This comparison has further confirmedthe validity and accuracy of the modelling techniques usedin the CAD package and provides sufficient evidence tosupport the view that CAD modelling can be applied withconfidence for designing new optimal PWM CFI drives.

The results have demonstrated that significant improve-ments in low-speed CFI drive performance can beachieved by judicious positioning of the PWM switchingedges. It is only as a result of recent developments inmicroprocessors that the feasibility of implementing theseoptimal PWM control strategies has now become a realpossibility. It is hoped to report more fully on the effects ofusing these microprocessor-based optimal PWM switchingstrategies on the performance of CFI drive systems in thefuture.

6 Acknowledgments

The authors gratefully acknowledge the financial supportof the UK Science & Engineering Research Council andthe University of Bristol for providing excellent computingand experimental facilities.

7 References

1 FARRER, W., and MISKIN, J.D.: 'Quasi-sine-wave fully regenerativeinverter', Proc. IEE, 1973,120, pp. 969-976

2 MANN, S.: 'A current source converter for multi-motor application'.IEEE IAS Conf. Rec, 1975, pp. 980-984

3 LIENAU, W.: 'Commutation modes of a current source inverter'. 2ndIFAC symposium on control in power electronics and electric drives,1977, pp. 219-229

4 ESPELAGE, P.M., and WALKER, L.H.: 'A high performance con-trolled current inverter drive', IEEE Trans., 1980, IA-16, pp. 193-202

5 AKAMATSU, M., IKEDA, H., TOMEI, and YANO, S.: 'High per-formance IM drive by co-ordinates control using a controlled currentinverter', ibid., 1982, IA-18, No. 4, July/Aug 1982

6 LIPO, T.A.: 'Analysis and control of torque pulsations in current fedinduction motor drives'. IEEE Power Elect. Spec. Conf. Rec, 1978,pp. 89-95

7 BLUMENTHAL, M.K.: 'Current source inverter drive system withlow speed pulse operation'. IEE Conf. Publ. 154, 1977, pp. 88-91

8 LIENAU, W.: 'Torque oscillations in traction drives with current-fedasynchronous machines'. IEE Conf. Publ. 179, 1979, pp. 102-107

9 LIENAU, W., MULLER-HELLMANN, A., and SKUDELBY, H.C.:'Power converters for feeding asynchronous traction motors of single-phase a.c. vehicles'. IEEE IAS Conf. Rec, 1977, pp. 295-304

10 LIENAU, W., and MULLER-HELLMANN, A.: 'Moglichkeiten zu,Betnib von stromeinpregenden Wechselrichtern ohne niederfrequerijteOberschwingungen'. ETZ-Arch., 1976, 97, pp. 663-667

11 ZUBEK, J.: 'Evaluation of techniques for reducing shaft cogging incurrent-fed a.c. drives'. IEEE IAS 78 Conf. Rec: 16B, pp. 517-524

12 CHIN, T.H., and TOMITA, H.: 'The principles of eliminating pulsa-ting torque in current source inverter-induction motor systems'. IEEEIAS 78 Conf. Rec, 30F, pp. 910-917

13 TUNG-HAI CHIN and HIDEO, T.: 'Elimination of torque pulsationby the current instantaneous value control in squirrel cage inductionmotors fed with controlled-current inverters', Electr. Eng. Jpn., 1978,98, pp. 105-112

14 BOWES, S.R., and BULLOUGH, R.: 'Steady-state performance ofcurrent-fed pulse-width-modulated inverter drives', IEE Proc. B,Electr. Power Appl., 1984, 131, (3), pp. 113-132

15 BOWES, S.R., and BIRD, B.M.: 'Novel approach to the analysis andsynthesis of modulation processes in power converters', Proc. IEE,1975, 122,(5), pp. 507-513


16 BOWES, S.R.: 'New sinusoidal pulsewidth—modulated inverter' ibid.,1975, 122, (11), pp. 1276-1285

17 BOWES, S.R., and CLEMENTS, R.R.: 'Computer-aided design ofPWM inverter system', IEE Proc. B, Electr. Power Appl., 1982, 129,(1), pp. 1-17

18 BOWES, S.R., and CLARE, J.C.: 'Steady-state performance of PWMinverter drives', ibid., 1983,130, (4), pp. 229-244

19 BOWES, S.R., and MOUNT, M.J.: 'Microprocessor control of PWMinverters', ibid., 1981,128, (6), pp. 293-305

20 BOWES, S.R., and CLEMENTS, R.R.: 'Digital computer simulationof variable speed PWM inverter-machine drives', ibid., 1983, 130, (3),pp. 149-160

21 LARGIADER, H.: 'Design aspects of induction motor for tractionapplications with supply through static frequency changers', BrownBoveri Rev., 1970, 57, pp. 152-167

22 CHALMERS, B.J., and SARKER, B.R.: 'Induction motor losses dueto nonsinusoidal supply waveforms', Proc. IEE, 1968, 115, (12), pp.1777-1782

23 JAIN, G.C.: 'The effect of voltage waveshape on the performance of a3-phase induction motor', IEEE Trans., 1974, PAS-83, pp. 561-566

24 ROBERTSON, S.D.T, and HEBBAR, K.M.: 'Torque pulsations ininduction motors with inverter drives', ibid., 1971, IGA-7, pp. 318-323

8 Appendix

The assumptions implicit in the derivation of the steady-state single-phase equivalent circuit of the three-phaseinduction motor are well known [21-24] and therefore arenot repeated.

The nth harmonic equivalent circuit is illustrated in Fig.16, where, as shown, the frequency dependence of the

I rn

Fig. 16 Harmonic single-phase equivalent circuit of induction motor

equivalent circuitimpedances to nthharmonic of stator current

= harmonic of stator current= harmonic of rotor current= harmonic of airgap EMF= harmonic slip

various circuit parameters has been included in the model[21-24].

The PWM stator current of fundamental frequency cas

can be expressed in terms of its harmonic spectrum as

U0= l U O (17)

where

hn = Isn sin (ncost-yn)

If the PWM stator current waveform posses halfwave sym-metry then only odd harmonics exist and n is an oddinteger. The PWM current waveform is characterised byconstant current levels ik, over the angular intervals

ak _ x < (Ds t < ak, and therefore Isn can be defined in termsof ik and an arbitrary set of switching angles a. Noting thatthese switching angles a are determined in the mini-misation algorithm described in Section 3.

Applying the PWM stator current is(t), eqn. 17, to theharmonic equivalent circuit of Fig. 16 produces a rotorcurrent of the form

(18)

where

(ncos t - <xn)

and

Kn = hn ncos Sn Mn\r2rn + (ncos Sn(Lrn + M J ) 2 ]

cn = yn- t a n " 1 [rj{noisSn(Lrn + MB))]

2] " (19)

The harmonic slip Sn can be defined in terms of the funda-mental slip Sx as [21-24]

S =n±(l-Sl) (20)

where the negative and positive signs correspond to posi-tive and negative sequence harmonics, respectively. Thetotal instantaneous rotor current is obtained by summa-tion of the individual instantaneous harmonic amplitudes.

The individual airgap flux harmonics may be expressedin terms of /„, as [21-24]:

= * « . s i n (ncos t - P H - n/2)

where

(21)

(22)

and

= « „ - tan"1 (ncosSnLJrrn)

The instantaneous torque component produced in phase 1by the interaction of the nth harmonic of rotor current andthe nth harmonic of airgap flux can be expressed as

TnJt) = Irn sin (ncost - aJ^F^ sin (ma>st - j8J (23)

Similar expressions can be deduced for phases 2 and 3,which when combined gives the total three-phase instanta-neous torque component for a two-pole induction motoras

Tnm(t) =* ± V 2 I r n V a m c o s ( ( n - m)cos t - ( a H - (24)

for Irn and Tam of opposite phase sequence. Note that thenegative sign in eqns. 24 and 25 appears when *¥am is ofnegative phase sequence.

The total torque harmonic for any required harmonicorder can be calculated using eqns. 24 and 25, by the trigo-nometric addition of 'the Tnm components of that order.The time-domain waveforms for rotor current, airgap fluxand electromagnetic torque can be computed by summa-tion of the individual harmonic components at any instantin time.

Eqns. 17-25 form the basis of the frequency-domaincomputer simulation used to provide the 'exact' harmonicmodel results of Fig. 10, for comparison with the approx-imate model results of Fig. 11, presented in Section 4.


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