Optimal PWM microprocessor-controlled current-source inverter drives

Optimal PWM microprocessor-controlledcurrent-source inverter drives

S.R. Bowes, PhD, DSc, CEng, FIEER.I. Bullough, PhD

Indexing terms: Inverters, Power electronics, Computer applications

Abstract: Optimal PWM switching strategies arepresented for the current-source inverter (CSI)induction motor drive. These optimal PWM stra-tegies are designed with the objective of improvingthe quality of the rotational motion of the drive atlow operating speeds, thereby extending the viablerange of operation and application of CSI drives.Optimal PWM switching strategies designed tominimise rotor speed ripple and rotor positionerror are presented, based on both 1/4-wave and1/2-wave symmetric PWM stator current wave-forms, and their superiority demonstrated usingboth experimental and computed CSI drive per-formance characteristics. It is shown that 1/2-wavesymmetric PWM strategies designed to minimiserotor position error are superior to all previouslydeveloped PWM strategies, in terms of producingsmooth rotor motion and reduced mechanical res-onances, during low speed operation of the CSIdrive. In contrast to previously developed CSIPWM strategies based on harmonic elimination/minimisation techniques, which are essentiallyopen-loop strategies, it is shown that all theoptimal PWM strategies are dependent uponmotor operating conditions and are thereforeclosed-loop strategies.

1 Introduction

List of symbols

F(a)/1*JTi, k, m, qnrstausAa>r

Ap£o

= performance function= current amplitude= D C link current= inertia= torque= integers= harmonic order= suffix denoting rotor= suffix denoting stator= instantaneous time= switching angles= stator angular frequency (electrical)= rotor speed ripple= rotor position error= harmonic current distortion factor

Paper 581 OB (P6), first received 16th April, and in revised form 3rdSeptember 1987Dr. Bowes is with the Department of Electrical and Electronic Engin-eering, University of Bristol, Bristol BS8 1TRDr. Bullough was with the Department of Electrical and ElectronicEngineering, University of Bristol, and is now with the Power Engineer-ing Division, ERA Technology Ltd., Cleeve Road, Leatherhead, SurreyKT22 7SA

Microprocessor control of PWM inverter drives offersthe possibility of using sophisticated optimal PWMswitching strategies to improve drive performance[1-13]. These optimal PWM switching strategies can bedesigned to optimise some specific performance criteria,for example, minimisation of torque harmonics, speedripple, positional error etc. to improve the low speeddrive performance.

A recent paper [5] presented a range of harmonicminimisation PWM techniques which, as demonstratedcan be used to reduce the rotor speed ripple and rotorpositional errors normally associated with quasi-squarewave operation of current source inverter (CSI) drives[14-16]. It was shown that these new PWM harmonicminimisation switching strategies produced superior CSIdrive performance, resulting in smoother rotor motiondown to zero speed, thereby significantly extending theviable range of operation and application of CSI drives.These harmonic minimisation and harmonic eliminationPWM techniques were designed to minimise or eliminateparticular current harmonics, and thereby indirectlyreduce the torque harmonics, resulting in reduced rotorspeed ripple and rotor positional errors.

Since the performance criteria involved in the develop-ment of these PWM switching strategies involves onlystator current harmonics, for example, based on the mini-misation of a weighted function of stator current harmo-nics [5] the resulting PWM strategies are independent ofthe motor operating conditions. This significantly sim-plifies the computation of the PWM switching angles,and, in addition, considerably simplifies the micro-processor implementation and generation of these PWMstrategies, since feedback from the rotor of the drivemotor is not required. Thus these harmonic current mini-misation and harmonic current elimination PWM stra-tegies are essentially generated 'open-loop', and thereforeare not optimised with respect to the CSI drive operatingconditions.

In contrast, PWM strategies designed to minimisemotor speed ripple and rotor position error directly,involving the direct minimisation of a function of torque,are highly dependent on the motor operating conditions,and therefore considerably more complex. For example,since the optimal performance criteria in this case is nowa function of the electrical torque profile, which in turn isa function of the motor operating conditions, principallyslip frequency or phase-angle between flux and current,the resulting pulse-widths defining the optimal PWMstrategies are also a function of slip frequency. Thereforeto perform wide ranging investigations and detailedassessment of the effects on drive performance of usingthese optimal PWM control strategies requires a fast

IEE PROCEEDINGS, Vol. 135, Pt. B, No. 2, MARCH 1988 59

accurate model of the CSI drive system, which allowsrapid and efficient computation of the optimal PWMswitching strategies. Fast computer modelling techniqueswhich allow these more complex optimal PWM stra-tegies to be developed and assessed have recently beendeveloped [4], and will be used as a basis for the optimalPWM strategy development presented in the followingsections.

This paper presents the results of a detailed researchstudy into the low-speed performance of optimal PWMcontrolled CSI drives, based on various optimal per-formance criteria designed to achieve smoother rotormotion at low speeds. Section 2 presents computed andexperimental results which demonstrate the performancecharacteristics of the CSI drive, when controlled usingoptimal PWM strategies designed to minimise rotorspeed ripple. Further, it is shown in Sections 3 and 4 howthe half-wave symmetric optimal PWM strategies, whichincorporate more degrees of freedom, are superior toquarter-wave symmetric optimal PWM strategies, pro-viding greatly reduced low order torque harmonicsresulting in smoother rotor motion.

It is also shown that whilst a minimised speed rippleperformance criteria would appear to be a logicalapproach to achieving smoother rotor motion at lowspeeds, experimental observation suggests that minimisedrotor positional error is a more appropriate performancecriteria for achieving smooth rotor motion. Noting thatthe use of a minimised speed ripple criteria does not nec-essarily give minimised positional error, and vice-versa,as demonstrated in the results presented.

Computed and experimental results confirming theseconclusions are presented in Section 5, where the CSIdrive performance characteristics for optimal PWMoperation based on minimised rotor positional error arepresented.

2 Rotor speed ripple minimisation

The electrical torque pulsations produced in the CSIdrive result in a periodic ripple component of rotor speedwhich prevents smooth rotor motion, particularly at lowoperating speeds. This rotor speed ripple is clearly unac-ceptable in many applications, for example, traction, milldrives, machine tools, robotics, etc, where precise speedcontrol down to zero speed is essential. In these, andsimilar applications, the minimisation of the extent ofrotor speed is of overriding importance, and thereforerepresents a relevant practical performance criteria forcomparing and assessing the performance of PWMcurrent strategies.

The reasons for adopting a performance criterionbased on rotor speed ripple would appear, at first sight,to be logical. Since during steady-state operation therotor should ideally rotate at constant speed, providingsmooth rotor motion, and speed ripple represents a devi-ation from this ideal and detracts from the smoothness ofrotor motion. Thus, speed ripple provides, to someextent, a quantitative measure of the smoothness of rota-tion. However, practical experience of operating theexperimental CSI drive suggests that rotor positionalerror provides a more appropriate measure of the rotormotion. This conclusion is further discussed and con-firmed in Section 3 where positional error minimisationis discussed in detail, and compared with the speed mini-misation results of this Section.

2.1 Computation of optimal PWM pulse - widthsThe rotor speed ripple Acor(t) can be defined in terms ofthe pulsating component of torque, Tp(t), neglecting fric-tion and windage, as

Acor(t)=-T [Vp(T)rfrJ Jo

where Tp(t) is defined by eqn. 5 given in Appendix 9.The performance function adopted to minimise the

extent of rotor speed ripple can therefore be defined as

F(cck) = peak-peak value of/s

2i Jo TJx) dx (1)

Noting that the inclusion of the / s l term in F(cck) ensuresthat the fundamental component of the resulting optimalPWM stator current is maintained at the highest possiblelevel, so avoiding excessive harmonic losses. Thus theminimisation of F(ak) provides the optimal PWM stra-tegies, in terms of the optimal switching angles afc, whichminimises the extent of speed ripple as a fraction ofaverage torque, whilst maintaining high fundamentalstator current.

To determine F(ak) numerically, it is necessary toprovide a subroutine to evaluate the integral of thetorque pulsation at its maxima and minima. Since thissubroutine may be accesssed many times during thecourse of the numerical minimisation procedure, it isessential that this subroutine is computationally efficient.This is particularly important when the computation ofhigh pulse number PWM strategies, involving manydegrees of freedom, are being considered.

Fast modelling techniques [4], for the CSI inductionmotor drive, have previously been developed to cater forthis situation, the main results of which are included inAppendix 9. Using these fast modelling results, as shownin Appendix 9, the torque pulsation Tp(z) (eqn. 5 ofAppendix 9) can be integrated to provide a very simple'closed-form' analytic expression of the form

TJx)dx = - ^ 2 ! \ilk cos (x)

2n\ / 4?r+ i2k cos ( x + y I + i3k cos I x + -j

+ — Vu cos (xj) + i2k cos I x,- + —Ms I \ 3 /

+ i3k cos (xt + -j)\ - C(t - tk.x)

+for t < tk (2)

where

x = cost — ($!

xi = (oatk..1 - fit

C = f / , A i cos (y, - fix)

Noting that the instantaneous PWM current levels inphases 1, 2 and 3 are piece-wise constant between suc-cessive switching instants, and are denoted by ilk, i2k, i3k

within a time interval (£*_!, tk), defined by the switchinginstants <xk in each phase. Thus to evaluate the per-formance function F(ak), the integral, eqn. 2, need only beevaluated at the switching instants of stator current, andat the zeros of the torque pulsations that occur between

60 IEE PROCEEDINGS, Vol. 135, Pt. B, No. 2, MARCH 1988

switching instants. The zeros of Tp(x) can be easily deter-mined by equating eqn. 5 (given in Appendix 9) to zerofor each combination of stator current vector (ilk, i2k,i2k). Also since the PWM strategies are constrained topossess half-wave symmetry, then only one sixth of acycle of PWM stator current need be considered. Thus,since only a mimimum number of points on the torquewaveform need be considered, the performance functionF(ak) may be computed extremely quickly; which whenincluded into the numerical minimisation procedure pro-vides an extremely efficient optimal solution.

It is important to note, from inspection of eqn. 2, thatthe shape of the integral of torque pulsation is deter-mined solely by the phase angle (/?! — y j between thefundamentals of flux and stator current. Thus the per-formance function F(cck), and hence the optimal PWMstrategies, are dependent on motor operating conditions.Since the phase angle (fit — yx) is a function of slip fre-quency, the CSI drive performance characteristics havebeen presented as functions of slip frequency in the fol-lowing Sections. To implement these optimal PWM stra-tegies it is therefore necessary to have available either slipfrequency, phase angle, or some related variable as aninput to the microprocessor PWM waveform generator.Such variables will, in general, be available in the major-ity of closed-loop CSI drive control schemes.

In the experimental CSI drive, rotor frequency is mea-sured and compared with stator frequency to provide slipfrequency. This slip frequency is input to microprocessorPWM controller and used to select the pulse widths cor-responding to a particular optimal PWM strategy.

2.2 1/4- wave symmetric PWM strategiesUsing the minimisation procedure described above, thepulse widths defining 3, 5, 7, 9 and 11-pulses per 1/2cycle, 1/4-wave symmetric speed ripple minimisationPWM strategies can be computed as a function of slipfrequency, as illustrated in Fig. 1.

10

jf 6

12

KM0

0)

a, 8

Noting that pulse widths rather than switching angleshave been displayed in Fig. 1 since these provide a morecompact and accurate representation of the character-istics, providing minimum pulse width information andare also used in the practical microporcessor PWM con-troller. The equivalent switching angle characteristics canof course be straight forwardly derived from Fig. 1, usingthe modulation rules presented earlier [6], if required.

The characteristics of Fig. 1 cover the complete loadrange, no-load (zero slip frequency) to full-load (1 Hz slipfrequency) of the 10 kW induction motor used in theexperimental CSI drive system. The equivalent circuitmotor parameters for the 10 kW experimental motor,used to compute all the results of this paper, are given inAppendix 9. An interesting feature of the characteristicsshown in Fig. 1 is the step discontinuity in pulse widthswhich occur for the higher pulse number PWM stra-tegies, 7 to 11-pulse, at a particular slip frequency. Theseabrupt changes in pulse width result from the numericalspeed ripple minimisation, outlined in Section 2.1, search-ing for both the optimum widths and optimum positionsof the pulses as the slip frequency is increased. This mini-misation search procedure is constrained by the 1/4 wavesymmetry imposed on the PWM current waveform, andthereform only a limited range of optimum pulse width(positions) are feasible. Thus at some stage (slipfrequency) in the minimisation search procedure thepulse-widths (and positions) must be automatically re-adjusted, resulting in a step discontinuity, to maintain 1/4wave symmetry whilst ensuring minimised speed ripple.This feature is not unique to CSI optimal PWM stra-tegies and has been demonstrated and described in detailearlier [3] for VSI optimal PWM strategies.

The practical implementation of these optimal PWMstrategies on the experimental drive is demonstrated,using a 5-pulse strategies as an example, in Figs. 2 and 4for a rotor frequency of 4 Hz, at zero (no-load) and 1 Hz(full-load) slip frequencies, respectively. The correspond-

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IEE PROCEEDINGS, Vol. 135, Pt. B, No. 2, MARCH 1988

Fig. 1 Pulse widths defining 1/4-wavesymmetric speed ripple minimisation PWMstrategies

a 3-pulse PWMb 5-pulse PWMc 7-pulse PWMd 9-pulse PWMell-pulse PWM

61

ing computed results are shown in Figs. 3 and 5, respec-tively.

Observation of these figures demonstrates the closecorrelation which exists between experimental and com-puted current, and torque waveforms, and confirms thevalidity and accuracy of the modelling techniques used.The slight discrepancies observed between the amplitudesof the higher order torque harmonics results from thedigitisation of the pulse-widths in the microprocessor-based PWM generator, and the slight modulation of theDC-link current in the experimental results. Experimen-tal traces for rotor speed have not been included, owingto the limited definition of analogue tachometers.However, the close correlation shown in the figures,between experimental and computed torque waveformsand spectra, provides confidence in the accuracy of thecomputed speed ripple results of Figs. 3 and 5, notingthat this has been confirmed earlier [4] for the case of anunloaded motor.

A number of interesting features emerge from closeinspection of these figures. For example, at no-load

5A/div

1Nm/dlv ,

6 121824 c

Fig. 2 Experimental results for 5-pulse 1/4-wave symmetric speedripple minimisation strategy, no-loadStator frequency 4 HzSlip frequency 0.05 HzStator current 5 A RMSa Stator current and torqueb Current spectrumc Torque spectrum

62

(Figures 2 and 3), the amplitude of the speed ripple isshown to vary between an upper and lower bound, witheach speed 'swing' coinciding with a current (or torque)pulse. Thus, each stator current pulse produces an equaleffect on speed ripple, giving a result which is virtuallyidentical to 'hysteresis' control of speed ripple. This effectis generally only obtained at no-load for any PWM pulsenumber, as further confirmed, using an example, the 19-pulse strategy shown in Figs. 6 (experimental) and 7(computed). This 19-pulse strategy is important since itdemonstrates the practical feasibility of implementinghigh pulse number optimal strategies, which allow thelow frequency operation of the CSI drive to be consider-ably extended.

At no-load, with no connected inertia and very lowstator frequencies, the speed ripple is a maximum, andtherefore it is possible to measure the speed ripple rea-sonably accurately. This is shown in the experimentalspeed ripple results for 3-pulse to 13-pulse PWM stra-tegies of Fig. 8, which as illustrated, confirms the char-acteristics of hysteresis control in each case for no-loadoperation (zero slip). Fig. 8 also confirms the practicalimplementation of these PWM strategies and demon-strates the reduction in speed ripple with increasingPWM pulse number.

This hysteresis characteristic is not however fullyexhibited at higher slip frequencies. This is demonstratedin the results shown in Figs. 4 (experimental) and 5(computed) for a slip frequency of 1 Hz. As shown in Fig.5, the speed ripple waveform displays four zero-crossingsin each sixth of a cycle, as compared with six in the

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Fig. 7 Computed results corresponding to Fig. 6

does not have an equal effect on the speed ripple. This isclearly demonstrated in Fig. 5, where, as shown, the firstcurrent pulse makes a negligible contribution to thespeed ripple waveform. In general, it has been confirmedthat, for any PWM pulse numbers, the speed ripplewaveform approximately exhibits the characteristic ofhysteresis control, but this characteristic diminishes asthe slip frequency increases.

From these observations it is clear that 1/4-wave sym-metric speed ripple minimisation strategies are moreeffective at the lower slip frequencies than at the higherslip frequencies. This is further confirmed in Fig. 9, whichshows the speed ripple against slip frequency character-istics for rated flux operation, with 5-pulse and 9-pulse

0.9rad/s/div

a 50ms/div

0.9rad/s/div

b 50ms/div

Fig. 8 Experimental rotor speed ripple results for speed ripple mini-misation PWM strategies, no-loadStator frequency 1 HzStator current 3.8 A RMSno-load connecteda 3-pulse stator currentb 13-pulse stator current

strategies. As shown in this figure, the effects of opti-mising each strategy at slip frequencies of 0, 0.2, 0.4, 0.6,0.8 and 1 Hz are presented. As illustrated, similar trendsare evident for both 5-pulse and 9-pulse strategies, withmost significant reduction in speed ripple, compared toquasi-square wave operation, taking place at low slip fre-quency. For example, at no-load the 5-pulse and 9-pulsestrategies reduce speed ripple by 60% and 74%, respec-tively, compared with 20% and 35%, respectively at 1 Hzslip frequency. Furthermore, as clearly illustrated in Fig.9, the optimal strategies in both cases maintain the abso-lute magnitude of the speed ripple virtually constant overthe complete load range, for a given rotor frequency.These results therefore suggest that minimal speed ripplecan be achieved by implementing relatively few optimalstrategies. Indeed, strategies optimised at the six slip fre-quencies shown in Fig. 9 clearly represent sufficientsamples for a practical microprocessor PWM controllerimplementation.

2.3 Effect on torque harmonicsThe torque spectra of Figs. 2 to 5 all exhibit noticeable6th harmonic torque components, resulting in a corre-sponding 6th harmonic in the speed ripple waveforms.This is also apparent for the no-load case of Fig. 3,which, as described earlier, is a well formulated mini-misation problem, exhibiting all the characteristics ofspeed ripple hysteresis control. It can therefore be con-

lOr

Hz

10slip frequency, Hzx10

10

optimal slip frequency, Hz

4 6slip frequency, Hzx10"

b

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Fig. 9 Computed speed ripple for 1/4-wave speed ripple minimisationPWM strategiesRated flux operation, 1 Hz rotor frequency, no additional load inertia. Each curverepresents the performance of a strategy that is optimised at the indicated slipfrequencya S-pulseb 9-pulse


eluded that as a direct result of the nature of the speedripple waveforms produced by the 1/4-wave symmetricPWM strategies, the 6th harmonic will be present, invarying degrees, if peak-peak speed ripple is minimised.Noting that this is further confirmed, for high PWMpulse numbers in the 19-pulse results of Figs. 6 and 7.

To demonstrate the above conclusions, Fig. 10 showsthe amplitudes of the low order torque harmonics for a9-pulse strategy, for rated flux operation. As illustrated,the effects of optimising the strategy at 0, 0.2, 0.4, 0.6, 0.8and 1 Hz slip frequency is shown, with the strategy oper-ated over a slip frequency range of +0.1 Hz about itsoptimal slip frequency. For the 9-pulse strategy, theamplitudes of the 6th and 12th torque harmonics are sig-nificantly lower than in quasi-square wave operation,although the 6th harmonic is still quite large at slip fre-quencies greater than approximately 0.5 Hz. It is possibleto show, using similar results (not shown), for the 5-pulsestrategy, that in the region of rated slip frequency the 6thharmonic shows little improvement over quasi-squarewave operation. This, together with the increased higherorder harmonics, results in a torque spectrum for the5-pulse strategy which is arguably inferior to that pro-duced in quasi-square wave operation, particularly at thehigher slip frequencies.

It is therefore possible to conclude that 1/4-wave sym-metric PWM speed ripple minimisation strategies arehighly effective at light loads, producing good low ordertorque harmonic content and providing a significantreduction in speed ripple. However, at high slip fre-quencies these strategies are not as effective, although it isnoted that the absolute value of speed ripple is relativelyconstant over the slip frequency range. Unfortunately, thelow order torque harmonic content increases as slip fre-quency increases, resulting in little improvement over

quasi-square wave operation in terms of avoiding pos-sible mechanical resonances.

To further reduce speed ripple at the high slip fre-quencies, and also maintain an acceptable low ordertorque harmonic content, 1/2-wave symmetric PWMstrategies may be employed. These are discussed in detailin Section 4, however, it is of interest at this stage tobriefly consider the reasons underlying the greater effec-tiveness of 1/2-wave symmetric PWM strategies.

3 Comparison of 1 /4-wave and 1 /2-wavesymmetric modulation

The reduction in the effectiveness of 1/4-wave symmetricstrategies at higher slip frequencies is explained with ref-erence to Fig. 11. At zero slip frequency, a quasi-squarewave stator current produces a saw-tooth torque pulsa-tion that is anti-symmetric about the zero axis, as illus-trated in Fig. lib (dashed line). This results from thetorque-producing component of flux leading the funda-mental of stator current by 90°. Introduction of 5-pulse1/4-wave symmetric modulation of stator current, shownin Fig. lla, involves the addition of pulse a and theextraction of pulse b of equal width, in accordance withthe modulation rules given earlier [6]. Both pulses con-tribute equally to the shape of the torque pulsation wave-form, as illustrated in Fig. lib, due to the symmetrydisplayed by the torque pulsation, hence both pulsesmake equal contribution to the shape of speed ripple. Athigher slip frequencies, the phase angle between the fluxand current changes, such that the torque pulsation is nolonger anti-symmetric about zero, as shown in Fig. lie.The contribution of pulse a to the torque pulsation wave-form, and more importantly to its integral (speed ripple),is now considerably less than the contribution of pulse b,

2 0 r

15

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2 0 r

15

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slip frequency, Hzx10-110

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2 A 6 8 10slip frequency, Hz x 10"

2 0 r

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10

20 r

15

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0 10slip frequency, Hz x10"

Fig. 10 Effect of rotor speed ripple mimimisation strategies on low order torque harmonic content, for 9-pulse strategyRated flux operation1 Hz rotor frequencyEach strategy is operated over a slip frequency range of 0.1 Hz either side of the slip frequency at which it is optimised


thus the potential for reduction of speed ripple has sig-nificantly decreased. This is consistent with the observa-tions made earlier regarding Figs. 3 and 5 in Section 2.2.

These considerations suggest that 1/2-wave symmetricstrategies may be more effectively employed at higher slipfrequencies. This is illustrated in Figs. 1 Id and e for thetype of 3-level, 1/2-wave symmetric, modulation whichmay be implemented in the CSI (in accordance with themodulation rules given earlier [6]). The stator currentstrategy may now be arranged such that pulses a and bcontribute equally to the shape of the torque pulsation,and hence its integral (speed ripple). There are now essen-tially double the number of degrees of freedom associatedwith the stator current waveform, and consequently thepotential for reduction of speed ripple at higher slip fre-quencies is greatly increased, in comparison with 1/4-wave symmetric modulation. This statement is supportedby the results shown in Figs. 14 and 15, which are dis-cussed in more detail in Section 4.

The greater effectiveness of 1/2-wave symmetric modu-lation is also reflected in the low order harmonic torquecontent. This can be simply explained by considering theharmonic effects in the frequency domain. For example,with a 1/4-wave symmetry constraint, PWM providesonly a means of manipulating the amplitudes of individ-ual stator current harmonics. The phase of the harmonicswith respect to the fundamental is fixed at either zero or180°, thus torque harmonics can be reduced only byminimisation of the current harmonic amplitudes.

In contrast, with a 1/2-wave symmetry constraint,PWM provides a means of manipulating both the ampli-tude and the phase of individual current harmonics.

JLTD [TUL

a b

Fig. 11 Relative effectiveness of 1/4-wave and 1/2-wave symmetricstrategies on the shape of torque pulsations

effect of quasi-square operationa 1/4-wave symmetrical PWM stator currentb No load torque pulsation waveformc Torque pulsation waveform at high loadd 1/2-wave symmetrical PWM stator currente Torque pulsation waveform at high load

Thus, torque harmonics can still be reduced by areduction of current harmonics. In addition, they can befurther reduced by phase adjustment of the individualcurrent harmonics. This provides increased cancellationbetween the Tn + 1 x and Tn_l<x components principallyresponsible for the torque harmonic of order n. This har-monic cancellation mechanism will be further demon-strated in the results of the following Section.

4 1 /2-wave symmetric speed ripple minimisationstrategies

The pulse widths defining 3-, 5-, 7-, 9- and 11-pulse, 1/2-wave symmetric, speed ripple minimisation strategies areshown as functions of slip frequency, in Fig. 12. As can beobserved from this figure, the discontinuities evident inthe equivalent 1/4-wave symmetric characteristics, shownearlier in Fig. 1, are not present in Fig. 12. This resultsfrom the improved conditioning of the minimisationproblem, as described in the preceding Sections, in partic-ular, the additional degrees of freedom in pulse posi-tioning resulting from the less severe 1/2-wave symmetryconstraints. It is also noted that the optimal pulse widthsat zero slip frequency are identical to those for 1/4-wave symmetry. This is due to the symmetrical nature ofthe torque pulsation at zero load, as illustrated in pre-vious figures. It is also important to note, from compari-son of Figs. 1 and 12, that there is no significantdifference in the practical minimum pulse width con-straints for each PWM strategy shown, with the excep-tion of the 3-pulse strategies. Thus, both the 1/4-waveand 1/2-wave symmetric 5- to 11-pulse PWM strategiescan be used over the same frequency range with approx-imately the same minimum pulse width constraints.Noting that it is usual to use the higher pulse numberstrategies at the lower inverter frequencies [6].

Figs. 13 (experimental) and 14 (computed) demon-strate the practical implementation of 1/2-wave sym-metric strategies, using the operating conditionscorresponding to those of Figs. 4 and 5 for 1/4-wave sym-metric operation.

4.1 Effects on speed rippleFor a slip frequency of 1 Hz, Fig. 14 demonstrates thatspeed ripple now exhibits fully the characteristics of hys-teresis control, as noted earlier for 1/4-wave symmetricoperation at no-load. As shown, each pulse in the statorcurrent results in a swing of speed ripple between upperand lower bounds, and as a result the speed ripple isapproximately 23% lower than in the equivalent 1/4-wave symmetric case of Fig. 5.

Fig. 15 demonstrates that the above comments applyover the complete load range, noting that these resultscorrespond to Fig. 9 for 1/4-wave symmetric modulation.It can be seen from these figures that 1/2-wave symmetryis increasingly more effective as slip frequency increases,both for the 5-pulse and 9-pulse strategies, and in generalfor higher pulse number PWM strategies.

4.2 Effect on torque harmonicsThe low order torque harmonic content shown in Figs.13 and 14 is also superior to that obtained from the 1/4-wave symmetric strategies, shown earlier in Figs. 4 and 5.The improvement is largely in the amplitude of the 6thharmonic, which is substantially reduced as a result ofusing 1/2-wave symmetry. It has been confirmed (resultsnot included) that the 5-pulse strategies provide a consis-tent reduction of 75% in the amplitude of the 6th harmo-nic throughout the load range. The 12th and 18th

66 1EE PROCEEDINGS, Vol. 135, Pt. B, No. 2, MARCH 1988

harmonics are also lower than in quasi-square operation.This represents a significant improvement over the per-formance of the 1/4-wave symmetric PWM strategies.

Examination of the results of Fig. 16 shows that thesame conclusions are true for the 9-pulse strategies. Inthis case, the 6th harmonic is further reduced, the 12th isreduced by 55%, and both the 18th and 24th harmonics

15

HO

10sl ip f requency, H z x 1 0 "

o

2 A 6

s l i p f r e q u e n c y , H z x 1 0

d

10-1

12

10

are smaller than in quasi-square operation. The reasonsunderlying this considerable improvement in low ordertorque harmonic content can be simply explained asfollows. The 1/4-wave symmetric PWM strategies relytotally on a reduction in low order torque harmonics.However, it is well-known [5] that a reduction in the loworder harmonics of a PWM current waveform is inevit-

10

I/)d) gQ) O

o>

^ 6

. C

T>

dJJ/l

Q. z

010s l i p f r e q u e n c y , Hz x 1 0 "

b

2 A 6 8 10

s l i p f r e q u e n c y , H z x 1 0

c

slip frequency, Hz x 10-110

Fig. 12 Pu/se widths defining 1/2-wavesymmetric speed ripple minimisation strategies,as a function of slip frequencya 3-pulse PWMb 5-pulse PWMc 7-pulse PWMd 9-pulse PWMell-pulse PWM

(a) Sfator current and torque

20A/di

20Nm/div

a 20ms/div

(c) Torque spectrum

0.5Nm/div

(b) Current spectrum

2A/dlv

1 5 7 11 13

if

6 12 18 24

Fig. 13 Experimental results for 5-pulse, 1/2-wave symmetric speedripple minimisation strategy, 1 Hz slip frequency

Stator frequency 5 HzStator current 13 A RMSa Stator current and torqueb Current spectrumc Torque spectrum


10r

f l 15

I I I .1 1 II Vs

time.II

s x10'2

20 25

4 6time, s x10"2

o y- o - time, s x10

\7time, sx10-2

to

cr

<

c

'oX

crE

15

10

50

20

10

current spectrum

1 6 11 16 21 26 31 36A1 46 5156 61 6671 76 81 86 91 96N harmonic order

[] n torque spectrum

D n n H fl n n n n n

F i g . 1 4

2 0 r

15

". 10

6 18 30 42 54 66 78 90 102 114 126 138 150 162 174N harmonic order

Computed results corresponding to Fig. 13

20 r

15

Q5W 10

10

O

0


0 0.2 0.4 0.6 0.81.0

7 / / \/ \

0

10r

10slip frequency, Hzx10"

j" 8

Q.woo 2

0


0 0.2 0.4 0.6 0.8 1.0-' \

\

0 2 4 6 8 10s l ip frequency, Hzx10

Fig. 15 Effect of 1/2-wave symmetric speed ripple minimisation stra-tegies on speed ripple

Rated flux operation, 1 Hz rotor frequency, no additional load inertia. Each curerepresents the performance of a strategy that is optimised at the indicated slipfrequencya 5-pulseb 9-pulse

QSW

10slip frequency, Hzx10" slip frequency, Hz x10"

20

15

- 10

0

20

15

10

QSW

0

QSW

2 4 6 8 10slip frequency, Hz x10"

0 10slip frequency, Hzx10"

Fig. 16 Effect of rotor speed ripple minimisation strategies on low order torque harmonic content for 9-pulse, 1/2-wave symmetric strategy1 Hz rotor frequencyEach strategy is operated over a slip frequency range of 0.1 Hz either side of the slip frequency at which it is optimised


ably accompanied by an increase in the amplitudes of thehigher order current harmonics. Therefore, higher orderharmonics of current, and hence of torque, are increased.In contrast, the 1/2-wave symmetric PWM strategies relylargely upon cancellation between the Tnm components ofthe same frequency, to reduce low order torque harmo-nics. This can be deduced from the stator current spectraof Figs. 13 and 14, from which it is seen that the loworder current harmonics are not particularly small,however, the pairs of harmonics of order (6n + 1) tend todisplay similar amplitudes. For example, in Fig. 13, 75and / 7 , J n and /1 3 , etc are approximately equal, thusthe low order torque harmonics are reduced largely bycancellation. Since the reduction of low order currentharmonics is less pronounced, compared with 1/4-wavesymmetric modulation, the higher order current harmo-nics are not significantly increased. Hence the amplitudesof higher order torque harmonics are generally lowerthan those obtained with 1/4-wave symmetric modula-tion.

5 Philosophy of position error minimisationstrategies

It has been demonstrated in the preceding Sections thatspeed ripple minimisation PWM strategies can provide asubstantial reduction in rotor speed ripple. In theory, thisshould improve the quality of rotor motion, since speedripple is ideally zero for steady state operation. Duringpractical operation of the experimental drive, however,the rotor motion was observed to be less smooth thanhad been expected. In particular, at light loads, the speedripple minimisation strategies did not provide theexpected improvement over other harmonic minimisationPWM strategies [5].

To explain these observations, the concept of constantspeed needs to be considered. In steady state operation,constant rotor speed is desired, although the primaryobjective is to achieve linear variation of the rotorangular position with time, which is characterised by aconstant rotor speed. However, an appreciable deviationof rotor speed from its average value will have little effecton rotor position if it only persists for a comparativelyshort period of time, and therefore will have little notice-able effect on the smoothness of rotor motion. Alterna-tively, should the deviation of rotor speed be such as toappreciably deflect the rotor position from its linearvariation with time, the effect on rotor motion will bequite noticeable.

Fig. 17 provides an appropriate demonstration of thisimportant point. The 19-pulse speed ripple minimisationstrategy, Fig. 17a, provides, by definition, minimal speedripple. As shown in the figure, between points a and b thearea enclosed by each segment of the speed ripple wave-form is approximately equal, such that each will producea small, equal change in position error. However,between points b and c the area enclosed by the speedripple waveform is much greater, therefore a large changein position occurs. This results in a 6th harmonic com-ponent of position error, which was observed to be veryprominent during operation of the experimental drive atlow frequency.

It is apparent from these observations that mini-misation of speed ripple is an inappropriate criterion forassessing the performance of PWM stator current stra-tegies. In contrast, Fig. lib shows the effect of a 19-pulsestrategy designed to minimise rotor position error. Asshown in this figure, a dramatic decrease in the extent of


position error, by a factor of approximately 6, is possiblewith the speed ripple minimisation stratgey, Fig. 17a.

106

1 & 1 -4time, sx10'

c <=>o s ^in If x>o >~ aa d ) " time, sx10"

Fig. 17 Comparative effects of speed ripple minimisation and positionerror minimisation strategies, no-load

Stator frequency 1 HzSlip frequency 0 Hz5 ARMSa 19-pulse speed ripple b 19-pulse position error

This represents a substantial improvement in the smooth-ness of rotor motion, which has been confirmed fromexperimental observation of the low speed operation ofthe practical drive system. It is also important to notethat the position error minimisation strategy produces aslight increase in speed ripple, as shown in Fig. \lb.

Based on the results of Fig. 17 (and other results notshown), and observation of the experimental drive, it isconsidered that the minimisation of rotor position erroris a more appropriate optimal performance criteria forassessing the quality of rotor motion. The developmentand effects on drive performance of PWM strategiesdesigned to minimise position error are described in thefollowing Sections.

5.1 Computation of optimal PWM pulse widthsThe rotor position error may be expressed [4] as theintegral of rotor speed ripple, thus,

Ape(t) = f \cor(x) dx = - I' I Tp{x) dt' dxJO J JO JO

The performance function adopted to minimise the extentof rotor speed ripple is therefore defined as

F(ak) = peak-peak value of 4 r r^sl JO Jo

Tlx)dt'dx\ (3)

Eqn. 3 can be evaluated to provide a relatively simple

69

analytic closed-form solution, as shown in Appendix 9,eqn. 6; which, when incorporated into the numericalminimisation routine, provides an extremely efficientoptimal solution.

5.2 1/4- wave symmetric position error minimisationstrategies

Examples of the effects of implementing these optimalPWM strategies on the position error are provided inFigs. 18 (experimental), 19 (computed) for 5-pulse, andFigs. 20 (experimental), 21 (computed) for 19-pulse oper-ation. These results correspond to Figs. 2, 3, and 6, 7respectively, for speed ripple minimisation. Comparisionof the computed and experimental results of Figs 18 to 21shows good correlation, with only slight discrepanciesbetween the harmonic spectra caused by digitisation ofthe pulse widths in the microprocessor implementation.

Under no-load operation, as shown in Fig. 19 for the5-pulse strategy, the position error displays the character-istics of hysteresis control, as previously demonstrated forthe speed ripple minimisation in Section 2.2. Experimen-tal confirmation of this no-load hysteresis characteristic

10A/div

a 20ms/div

1A/dlv

1Nm/dlv

6 121824

Fig. 18 Experimental results for 5-pulse 1/4-wave symmetric positionerror minimisation strategy, no-loadStator frequency 4 HzSlip frequency 0.02 Hz (no load)Stator current 5 A RMSa Stator current and torqueb Current spectrumc Torque spectrum

70

is provided in Fig. 22 for quasi-square wave, 3-pulse, and5-pulse stator currents. Noting that the position errortraces, shown in Fig. 22, were obtained by integrating theripple component of a tachometer output, and whilst notparticularly accurate, provides a prediction of the generaltrends. As clearly shown in this figure, PWM operationprovides a substantial reduction in position error.

At the higher slip frequencies, results have shown (notincluded) that the position error approximately exhibitscharacteristics of hysteresis control, but with only fourzero-crossings as compared with six shown in theno-load case of Figs. 19 and 22. These characteristics areequivalent to those presented earlier in Section 2.2 for1/4-wave symmetric speed ripple minimisation.

Comparing the results for speed ripple minimisation,Figs. 3 and 7, with position error minimisation, Figs. 19and 21, confirms that the latter provides much smootherrotor motion. For example, from these figures (and othersnot included) it is confirmed that the position error isreduced by factors of 2.5 and 2 for operation at no-loadand 1 Hz slip frequency, respectively. This is furtherdemonstrated in Fig. 23, which show position error, as afunction of slip frequency for quasi-square, 5-pulse, and9-pulse operation at rated flux. As shown, a significantimprovement in position error results in PWM oper-ation, particularly at no-load, when the position error isreduced by 87% and 95% for 5-pulse and 9-pulse oper-ation, respectively, compared to quasi-square wave oper-ation. The quality of rotation indicated by Fig. 23represents a significant improvement over quasi-squarewave operation, which was confirmed in the experimentaldrive performance for rotor frequencies down to below0.5 Hz at rated flux.

spee

dro

tor

UO

Il

posi

<

rent

cur

'oX

E

torq

ue, b

'o. X

ripp

iro

d/:

o_̂- X

10

0

-10

2

0

-2

4

0

- 4

10

0

-10

10 20 25 30

time, s x10"

10

time, sx10

V Vtime. sx10'2

r\ \7 2A

10

time sx10 -2

, * 2cc: 0

current spectrum

1 6 11 16 21 26 31 3641 46 51 5661 6671 7681 86 91 96N harmonic order

L 0

torque spectrum

m t i i ,6 18 30 42 54 66 78 90 102 114 126 138150 162174

N harmonic order


1EE PROCEEDINGS, Vol. 135, Pt. B, No. 2, MARCH 1988

5.3 Effect on torque harmonicsIn contrast to the 1/4-wave symmetric speed rippleresults of Figs. 2 to 7, the harmonic torque spectra forposition error minimisation, Figs. 18 to 21, display verylow 6th harmonic torque content. This is to be expectedsince the position error is proportional to to the double

10A/d!0.1s/div

Oms/dlv

1Nm/div

6 12 18 24

10

0

-10

mini 2 1114 111time.

Illsx 10"

* I1IINII 12

in

cr< «:£ 2c

~. 0

in

h* - 2= 0

time, sx 10

current spectrum

1 6 11 16 21 26 31 3641 4651 5661 66 71 76 81 86 91 96N harmonic order

„ n n ft n n

torque spectrum

n - (1 fl n fl I n 1 0 n n n n n- 6 18 30 42 54 66 78 90 102 114 126138150 162174

N harmonic order



integral of torque pulsation, whereas the speed ripple isonly proportional to the integral of torque pulsation.Thus, the low order torque harmonics are more havilyweighted in the performance function adopted for mini-misation of position error.

This feature is further confirmed using a 9-pulse strat-egy as an example, as shown in Fig. 24, which is theequivalent of Fig. 10 for the speed ripple minimisation

Fig. 20 Experimental results for 19-pulse 1/4-wave symmetric positionerror minimisation strategy, no-load

Stator frequency 1 Hz a Stator current and torqueSlip frequency 0.02 Hz b Current spectrumStator current 5 A RMS c Torque spectrum

strategy. Comparing these figures, it is evident that amuch lower 6th harmonic torque is present with positionerror strategies at all slip frequencies. In addition, the12th harmonic is also reduced over most of the slip fre-quency range, and the 18th harmonic is comparable tothat produced in the speed ripple strategies.

Since in general the low order torque harmonics havethe most significant effect on the mechanical drivesystem, since they are most likely to excite mechanicalresonances, these effects will tend to be reduced.However, in common with other 1/4-wave symmetricPWM strategies, the magnitudes of the low order torqueharmonics tend to increase at higher slip frequencies.This situation may be improved by implementing 1/2-

0.105 rad/div

0.021rad/div

t> 0.1s/div

0.011 rad/div

c 01s/div

Fig. 22 Experimental rotor position error results for position errorminimisation PWM strategies, no-loadStator frequency 1 HzDC link current 6 Aa Quasi-square operationb PWM operation, 3-pulse Apt

c PWM operation, 5-pulse Ap£

71

2 0 r

1 5

10

optimal slip frequency, Hz0 0. 2 0.4 06 0.8 1.0

s l ip frequency, H z x 1 0 -110

20

c 10o

optimal slip frequency, Hz0 0.2 0.4 0.6 0.8 1.0

I \I \

/ ! \

8 10s l ip f requency, H z x 1 0

Fig. 23 Effect of 1/4-wave symmetric position error minimisation stra-tegies on position error, for 5-pulse and 9-pulse strategiesRated flux operation, 1 Hz rotor frequency, no additional load inertia. Each curverepresents the performance of a strategy that is optimised at the indicated slipfrequencya 5-pulseb 9-pulse

2 0 r

15

- 10 QSW

2 0 r

15

10

sl ip frequency, Hz x10 -110

wave symmetric PWM strategies, as shown earlier forspeed ripple minimisation in Section 4, and furtherdemonstrated in the following Section.

5.4 112 - wave symmetric position error minimisationPWM strategies

At slip frequencies greater than zero, 1/2-wave symmetricPWM strategies can provide a more effective reduction inposition error, due to the increased number of degrees offreedom associated with these strategies, as explainedearlier in Section 4. Also, as described earlier, a furtherreduction in the low order torque harmonics is possibleusing the mechanism of cancellation between the Tnmcomponents of the same harmonic order.

To demonstrate this, Figs. 25 (experimental) and 26(computed) show the results for a 5-pulse strategy at aslip frequency of 1 Hz. Inspection of these figures againdemonstrate extremely good correlation between experi-mental and computed results. The results of Fig. 26, whencompared with the equivalent results for a 1/4-wave sym-metric 5-pulse strategy (not shown), confirm that a 40%decrease in position error can be achieved using 1/2-wavesymmetry. This is further confirmed by Fig. 27, whichshows the effects of 5-pulse and 9-pulse strategies onposition error, as a function of slip frequency. Com-parision with the equivalent results for 1/4-wave sym-metry, Fig. 23, demonstrates that a consistent reductionin position error is maintained over the complete rangeof slip frequencies, particularly for the 5-pulse strategies.Hence, although the 1/4-wave symmetric strategiesprovide very satisfactory performance in practice, the useof 1/2-wave symmetry further improves the quality ofrotation of the drive under all load conditions.

5.5 Effect on torque harmonicsThe stator current waveform of Fig. 25 contains notablelow order harmonics. In particular, the 5th and 7th har-monics are much more prominent than for the equivalent

QSW


20

15

o

20 r

15

i* 10

QSW

0

QSW

0 6 8 10 0 2 4 6 8s l ip f requency , Hzx10


Fig. 24 Effect of rotor position error minimisation strategies on low order torque harmonic content, for 9-pulse 1/4-wave symmetric strategy

Each strategy is operated over a slip frequency range of 0.1 Hz either side of the slip frequency at which it is optimised


operation, with 1/4-wave symmetric stator currents. Asshown in Fig. 25, the stator current harmonic pairs oforder q = (6n ± 1) are of similar amplitude. This featurewas also noted for 1/2-wave symmetric speed ripple stra-tegies in Section 4. As a result of cancellation between theTq t components, the low order torque harmonics in Fig.25c are reduced, particularly the 12th and 18th harmo-nics.

The computed torque spectra of Fig. 28 shows that the9-pulse 1/2-wave symmetric strategies maintain very low6th harmonic content under all operating conditions. The12th, 18th and 24th harmonics are also significantlylower than in quasi-square wave operation. As with the9-pulse 1/2-wave symmetric speed ripple minimisationstrategies of Fig. 16, this represents a significant improve-ment over the performance of any of the 9-pulse 1/4-wavesymmetric strategies considered previously [5]. The sameconclusions apply to the torque harmonics for 5-pulse 1/2-wave symmetric operation, and indeed, it is possible toshow, all PWM pulse numbers. It is worthy of note thatthe amplitudes of the four harmonics shown in Fig. 28

20A/div

20Nm/div

0

20ms/div

1Nm/div

12 18 24

Fig. 25 Experimental results for 5-pulse 1/2-wave symmetric positionerror minimisation strategy, 1 Hz slip frequencyStator frequency 5 HzSlip frequency 1 HzStator current 13 A RMSa Stator line current and torqueb Current spectrumc Torque spectrum

decrease with decreasing harmonic order. This featurewould generally be considered desirable, since low orderharmonics are usually more likely to excite mechanicalresonances.

In summary, position error minimisation strategiesproduce a very substantial improvement in the smooth-ness of rotor motion. This applies to both 1/4-wave and1/2-wave symmetric PWM strategies. The low order har-monic content of torque is also arguably superior to thatproduced by other types of PWM strategy discussed pre-viously [5]. This applies more to the 1/2-wave symmetri-cal strategies, whose principal advantage lies in thequality of torque spectra produced over the whole of theload range.

6 Conclusions

The CSI PWM strategies which have been developed andpresented in this paper were all designed with the aim ofimproving the quality of the rotational motion of thedrive motor, at low operating frequencies.

It has been shown that those PWM strategiesdesigned to minimise rotor speed ripple do not necessar-ily produce smooth rotor motion, as confirmed experi-mentally, and indeed, it has been demonstarted that theyare in general inappropriate for this purpose. In contrast,PWM strategies designed to minimise rotor positionerror have been shown to produce excellent drive per-formance, which has been practically confirmed by obser-vation of the operation of the experimental drive system.It has also been shown that particular advantages can begained from the implementation of 1/2-wave symmetric

2 2

-2

[no10 15 20 25

time, sx10-2

time, sx10-2

o P

time. sx10'

current spectrum

1 6 11 16 21 26 31 3641 46 51 5661 66 71 7681 8691 96N harmonic order

torque spectrum

18 30 42 54 66 78 90 102 114 126 138 150 162 174N harmonic order



20

S 15o

o

* 10co

in

optimal slip frequency, Hz0 0.2 0.4 0.6 0.8 1.0

10slip frequency, Hz x10"

a20

15

10

Hz

U & 8slip frequency, HzxiO"1

b

10

Fig. 27 Effect of 1/2-wave symmetric position error minimisation stra-tegies on position error, for 5-pulse and 9-pulse strategies

Rated flux operation, 1 Hz rotor frequency, no additional load inertia. Each curverepresents the performance of a strategy that is optimised at the indicated slipfrequencya 5-pulse b 9-pulse

2 0 r

15

- 10

20 r

15

Q5W - 10

PWM strategies which provide significantly reduced loworder torque harmonics and reduce the possibility ofmechanical resonance effects, in addition to producingsmoother rotor motion. In both these respects, the per-formance of position error minimisation PWM strategieshave been shown to exceed that of the harmonicelimination/minimisation PWM strategies consideredearlier [5].

Finally, it has been shown that these optimal PWMstrategies are dependent upon the motor operating con-ditions, principally slip frequency or phase angle betweenflux and current, and therefore can only be implementedusing a closed-loop control scheme. This is not con-sidered to be a signficant disadvantage, since in generalsuch variables are available in the majority of CSI drivecontrol schemes.

7 Acknowledgments

The authors gratefully acknowledge the financial supportof the UK Science and Engineering Research Counciland the University of Bristol for providing excellent com-puting and experimental facilities.

8 References

1 BOWES, S.R., and DA VIES, T.: 'Microprocessor-based develop-ment system for PWM variable-speed drives', IEE Proc. B, 1985,132,(1), pp. 18-45

2 BOWES, S.R., and MIDOUN, A.: 'Suboptimal switching strategiesfor microprocessor-controlled PWM inverter drives', Ibid., 1985,132, (3), pp. 133-148

3 BOWES, S.R., and MIDOUN, A.: 'New PWM switching strategyfor microprocessor controlled inverter drives', Ibid, 1986, 133, (4),pp. 237-254

4 BOWES, S.R., and BULLOUGH, R.I.: 'Fast modelling techniquesfor microprocessor-based optimal pulse-width-modulated control ofcurrent-fed inverter drives', Ibid, 1984,131, (4), pp. 149-158

5 BOWES, S.R., and BULLOUGH, R.I.: 'Harmonic minimisation inmicroprocessor controlled current-fed PWM inverter drives', Ibid,1987,134,(1), pp. 25-41

Q5W

slip frequency, Hz x102 0 r

15

- 10

2 A 6 8 10sl ip f requency, Hz x10"

0

20 r

15

QSW

-110

0OSW

0 2 U 6 8 10sl ip f requency , H z x 1 0 " ' s l ip f requency, H z x 1 0 ' 1

Fig. 28 Effect of rotor position error minimisation strategies on low order torque harmonic content for 9-pulse 1/2-wave symmetric strategyEach strategy is operated over a slip frequency range of 0.1 Hz either side of the slip frequency at which it is optimised


6 BOWES, S.R., and BULLOUGH, R.I.: 'PWM switching strategiesfor current-fed inverter drives', Ibid, 1984,131, (5), pp. 195-202

7 BOWES, S.R., and BULLOUGH, R.I.: 'Steady-state performance ofcurrent-fed PWM inverter drives', Ibid, 1984,131, (4), pp. 113-132

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

9 LIENAU, W.: Torque oscillations in traction drives with current-fed asymchronous machines', IEE Conf. Publ. 179, pp. 102-107

10 LIENAU, W.: 'Power converters for feeding asynchronous tractionmotors of single-phase AC vehicles', IEEE Trans., 1980, IA-16, pp.103-110

11 ZUBEK, J.: 'Evaluation of techniques for reducing shaft cogging incurrent-fed AC drives'. IEEE IAS Annual Meeting, Toronto,Canada, 1978, pp. 517-524

12 CHIN, T.H., and TOMITA, H.: The principles of eleminating pul-sating torque in current source inverter-induction motor systems',IEEE, IAS, 78, Conf. Rec, 30F, pp. 910-917

13 KOMOTO, M., HASHII, M., YANASE, T., and NAKANO, T.:'Performance improvements of current source inverter-fed inductionmotor drives', IEEE Trans., 1982, IA-18, pp. 703-711

14 FARRAR, W., and MISKIN, J.D.: 'Quasi-sine-wave fully regener-ative inverter', Proc. IEE, 1973,120, (9), pp. 969-976

15 MANN, S.: 'A current source converter for multi-motor applica-tions', IEEE IAS Annual Meeting, Atlanta, GA, USA, Oct 1975, pp.980-983

16 PHILIPS, K.P.: 'Current source converter for AC motor drives',IEEE Trans., 1972, IA-8, pp. 679-683

9 Appendix

9.1 Fast torque calculationIt has been shown earlier [4] how the expression for thepulsating component of torque Tp(t) can be considerablysimplified when PWM stator current waveforms are used.For example, if the instantaneous PWM current levels inphases 1, 2, and 3 are denoted by i l k , i2k, i3k within atime interval (tk_l5 tk) defined by the switching instants ineach phase, then the pulsating component of torque canbe expressed as

Tp(t) = \j/al\ ilk sin (cost - fix) + i2k sin (cost

Insin {cost "->•)]

speed ripple Acor(t) expression, eqn. 2, given in Section2.1.

9.3 Computation of rotor position errorIn steady-state sinusoidal current operation the rotorspeed is ideally constant, hence the rotor angular positionincreases linearly with time. When the motor is suppliedwith piecewise constant PWM stator currents, the rotorposition is then subjected to periodic deviations from theideal linear response, due to the rotor speed ripple Acor(t)discussed in the previous section.

The resulting position error can therefore be definedby

=Jo

dx- A(DavA,

where A(oav is the average value of Acor, which is zero ifthe correct initial condition is used in equation 2 ofSection 2.1. Using this expression for Ape(t), in conjunc-tion with eqn. 2 of Section 2.1, results in the followingexpression for position error

= ~ ~r~i lik sm (x) + i2ki x + — IJOis |_ \ J /

+ Uk sin ( x + —-

Inh + y ) + *3* sin [Xi + —

sin

(cos I x, + — j + i3k cos I x, + —

2n

for tk-.t < t < tk (6)

where

for tk_i<t<tk (5)

Since i l k , i2k, and i3k are constant between successiveswitching instants, this expression is extremely simple,consisting only of the addition of segments of three sinewaves and the subtraction of a constant. Thus, only thefundamental quantities need be computed, as shownearlier [4].

9.2 Computation of rotor speed rippleThe pulsating component of torque, Tp(t), resulting fromstator current harmonics, produces a speed ripple com-ponent superimposed on the average rotor speed. Theextent of this rotor speed ripple is therefore dependentprimarily upon the inertia of motor and load.

If friction, windage and transmission losses areneglected, the rotor speed ripple Acor(t) resulting from thepulsating component of torque Tp(t) can be expressed as[4]

Acor(t) = -. Tp{x) dxJ Jo

Substituting eqn. 5 into this expression yields the rotor


Aco,_ 6 " C"

iv— T L,1 «=1 Jti-i

A0)r(T)dT,

and m = number of switches in one sixth of a cycle.This expression may be computed rapidly, since most

of the terms need only be computed at the switchinginstants in the PWM stator currents. It is used in Section5 to compute the PWM stator current strategies designedto minimise the extent of rotor position error.

9.4 Experimental motor parametersThe experimental 10 kW, 4 pole, 380 volt (line) inductionmotor used in the study had the following equivalentcircuit parameters

Stator resistance: 0.32 QRotor resistance: 0.234 QStator leakage inductance: 2.86 raHRotor leakage inductance: 3.18 mHMutual inductance: 0.13 HMoment of inertia of rotor: 0.071 Kgm2

Rated slip frequency: 1 Hz

75

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Optimal PWM microprocessor-controlled current-source inverter drives