induction motor is a dynamic systems

7/30/2019 Induction Motor is a Dynamic Systems

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Pulsewidth Modulation in Motion ControlJOACHIMHOLTZ,enior Member, IEEE

University of Wuppertal, Germany

1. IntroductionMotion control is the most challenging field of applicationfor electrical drives. Highest performance is required as re-gards dynamic behaviour, stead y-state accuracy, and uniform-ity of movem ent at crawling speed. Optimal utilization of themachine and the power converter, and high power densitiy,are as important as good efficiency of the overall system.These stringent performance criteria have to be met coveringthe full operating range from no-load to overload and fromzero speed up to high speeds in the field weakening range.

Typical applications are in m achine tool control and robotics.Pulsewidth modulation (PW M) techniques play a importantrole in satisfying the above demands. They have been thesubject of intensive research during the last few decades. Alarge variety of methods, different in their concepts and per-formance, have been developed and analysed. The choice ofan adequate method to implement in a motion control systemdepends on performance and cost criteria, the machine type,and the semiconductor devices used in the powe r converter.The basic requirements can be easily met when d c machinesare used. Here, the m ean armature current i s controlled by theduty cycle, while its harmonic content depends on the switch-ing frequency. Similar conditions are encountered with brush-less dc motors, i. e. synchronous motors having trapezoidalback-e.m.f. w aveforms, and with switched reluctance motors.The problem is different and more involved in the case ofinduction motor drives, and synchronous motor drives usingmachines with permanent magnet excitation and sinusoidalback-e.m.f. The revolving field of these machines should bealmost free from other components than the fundamental. Th isnecessitates the reduction of the lower order harm onics of thestator currents. The w ays to achieve this objective are greatlydependent on the chosen pulsewidth modulation technique.

2. Performance CriteriaOperation in the switched mode is a very efficient means ofcontrolling the power flow in the machine. The switchingharmonics are suppressed to a large extent by the low-passcharacteristic of the machine inductances, a nd by the intertiaof the mechanical system. The remaining distortions of thecurrent waveforms and the electromagnetic torque can bevalued by performance criteria. - 1-412.1 Current harmonicsThe harmonic cur rents basically determ ine the copper loss-es of the machine, which account for a major portion of themachine losses. The r.m.s. harmonic curren t

does not only depend on the performance of the pulsewidthmodulator, but in addition on the internal impedance of themachine. This influence is eliminated when the distortionfactor

=Ih rms l Z h ms six-step (2)is used as a figure a merit. Here, the distortion current I h rms(Eqn. 1) of a given switching sequence is referred to th edistortion current I ) , rm s six.step of same machine operated inth e six-step mode, i. e. with unpulsed rectangular waveforms.2.2 Harmonic spectrumIn a more detailed manner than by the global distortionfactor, contributions of individual frequency components areexpressed in the harmonic current spectrum. We have discretecurrent spectra hi(k.fl)~n he cas e of synchronized P WM ,where the switching frequencyfs=N fl s an integral multipleof the fundamental. N is the pulse number, or gear ratio.Nonsynchronized pulse (sequences produce continuou s har-monic current density spectra hd(fl. ote that all spectra inthis paper are normalized, e. g.

hiO =~ t zn r s v ) /Itz rm s six-step. (3 )They desribe the properties of a pulse modulation schemeindependently from the parameters of the connected m achine.

2.3 Torque harmonicsThe torque ripple produced by a given switching sequencecan be expressed as(4)T =(Ti,, - Tav)/TR,

where T,, maximum air-gap torqueTav average aLr-gap torque, andT R rated torque.Although the torque harmonics are produced by the har-monic currents, there is no stringent relationship between bothof them. Lower torque ripple can go along with higher currentharmonics, and vice versa.2 .4 Switching frequericyAnother impo rtant parcameter s the switching frequency f,.The harmonic distortion of the ac currents reduce almostlinearly with this frequency. Yet, the switching frequencycannot be deliberately increased as the switching losses ofsemiconductor devices increase proportional to the switchingfrequency. The regulations regarding electromagnetic com-patibility (EMC) are stricter for power conversion equipmentoperating at higher sw itching frequency [ 5 ] .This is primarilya cost problem.2.6 Dynamic performanceThe dynamics of a motion control system depend greatly onthe pulsewidth modulation method. There is always a currentcontrol scheme incorporated in a motion control drive, theresponse time of which dietermines the dynamic perform anceof the system. It is influenced by the switching frequency and/or the PWM method used. Some schemes require feedbacksignals that are free from c urrent harmonics. Filtering of feed-back signals increases the response time of the loop.

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"0 .2 .4 .6 .8 1 0 2 4 6kHzrn - f-

b) waveforms, phase ad) harmonic spectrumFig. 1: Suboscillation methoda) signal flow diagram,c) distortion factor

PWM methods for the most commonly used voltage-sourceinverters are either impressing the voltages, or the currentsinto the stator windings of the machine. The respective ap-proach also influences upon the structure of the drive controlsystem: The methods of the first category operate in an open-loop feed-forward fashion. C losed-loop PWM schemes injectthe stator currents into the mach ine and require different drivecontrol structures.3. FeedforwardSchemes

Feedforward schemes generate switched three-phase volt-ages, the normalized fundamental space vector gl(r)of whichequals a given reference vector &*(t) . The ratio m =u*/ul s i x -is called the modulation index, where uI sin-step is thefundamental voltage of a six-step waveform. We have m c 1for pulsewidth control and m =1 in the six-step m ode.3.1 Carrier based pulsewidth modulatorsThese are the classical and most widely used methods ofpulsewidth modulation. Their common characteristic are sub-cycles of constant time duration, a subcycle being defined asthe time interval T , during which any active inverter legassumes two consecu tive switching states of opposite voltagepolarity. Operation at constant duration of subcycles is re-flected in the harmonic spectrum by two salient sidebands,centered aroun d the carrie r frequency, and additional frequen-cy bands arou nd integral multiples of the carrier.3.1 I Suboscillation methodThis method employs individual modulators in each of thethree phases, Fig. la . Exemplified waveforms for phase a areshown in Fig. lb, consisting of the sinusoidal reference volt-ag e uu* and the triang ular carrier signal U , of frequencyf,. Theswitched output waveform is U;. The distortion factor d atconstant carrier frequency as a function of the modulationindex is displayed in Fig. IC ,and a harmonic spec trum in Fig.Id. Note that the maxim um modulation index mm a I =n/4 =85 occurs at a point where the amplitudes of the reference

and the carrier signal becomeequal. A distorted referencewaveform , containing only tri-plen harmonics to an extentthat its maximum assumes aflat-top shape (Fig. 2b), in-creases the maximum modu-lation index to a second limitmmm27 x/6..\/3 =0.907. Theadded triplen harm onics mod-ify the switched voltage w ave-forms; they do not produce harmonic currents in a three-phasesystem. - 6,713.1.2 Space vecto r modulationThe space vector modulation averages three consecutiveswitching state vectors ya, & an d & (Fig. 3a) over the timeinterval of a subcycle T,=1/2f,. The resultant average voltagevector must equ al the sampled reference vector &* * ( t ) , hence

"BFig. 2: Reference waveformsa) sinusoidalb) triplen harmonics added

2fs ' &a 4- tb &) =&*(ts) (5)determines the on -durations t , of ya, an d tb of &,, respectively.&*(t,) is the reference vector at sampling instant r (Fig. 3a),the sampling frequency being 2fs. The remaining portion of asubcycle (6 )is the on-duration of the ze ro vector&.For minimum harmon-ic distortion, ya an d &, are chosen as the two active switchingstate vectors adjacent to the space vector U*.The minimumnumber of inverter commutations is obtained in the typicalsequence ya -& -&,- &, - &a -& At higher modulation index,the sequence & - gP - ur- gP yields lower current harmonics,where 4 is the nonzero sw itching state vector in proximity toU*,& th e vector in the more remote location.The modulator Fig. 3b ouputs & as the k-th switching statevector. T he maximum m odulation index is mm m =0.907. -18, 10,221

to =T, - fa - b

tjIm

Fig. 3: Space vector modulationa) switching vectors ya,b) signal flow diagram and k, nd reference U*3.1.3 Synchronized carrier modulationThe above methods operate at constant carrier frequency,while the fundamental frequency is varying. Hence the switch-ing sequence is nonperiodic in principle, and the Fourier spec -tra are continuous, containing also frequencies lower than thelowest carrier sideband. These subharmonics are undesired asthey produce additional losses and torque harmonics of low erfrequency. A synchronization between the carrier frequencyand the controlling funda mental avo ids these drawba cks whichare especially prominent if the frequency ratiof,fl[ is low.The performance of synchronized carrier PWM i s illustrat-ed in Fig. 4 for the suboscillation technique and the space

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vector modulation. The latter appears superior at lower pulsenumbers, the difference becoming insignificant as N increas-es. The curves exhibit no differences at lower modulationindex. This range is of l i t t le practical use for constant vlfiloads where d decreases if m is reduced, Fig. IC .- l13

"0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1m- m-Fig. 4: Distortion fa ctor versus m odulation indexa) synchronized suboscillation methodb) synchronized space vector m odulation3.1.4 Sampling techniquesThe suboscillation method is simple to implement usinganalogue integrators and comparators for the generation of thetriangular carrier and the switching instants. It is problematic,however, to detect the crossing instant of two time-variablesignals, Fig. lb, when a microprocessor is used for PWMcontrol. This difficulty can bebypassed with the help of sam-pling techniques. T he sinusoi-dal reference wave is sampledonly at certain time instants.To use synchronized modula-tion is advantageous in that thesampling instants tk =k/( f l .N) ,

k =1 ...N are known a priori.Hence the sampl ing valuess i n ( o l . t k )can be stored in theprocessor memory with theter. Based on these, the switch-t 4 ing instants are computed on-Fig. 5: PWM sampling line as the points where the tri-technique, waveforms angular slope reaches the re-spective sampled value, Fig. 5.

The performance of a pulsewidth m odulator based o n sam-pling techniques is slightly inferior than that of the suboscilla-tion method, but only at low pulse num bers. - 1213.2 Carrierless modulat ionThe typical harmonic spectrum of c arrier controlled pulse-width modulation exhibits prominent harmonic amplitudesaround the carrier frequency and its harmonics. Acoustic noisecan be generated by the machine at these frequencies throughthe effects of magnetostriction and mechanical resonances ofthe active iron and its sup porting mechanical structures. It canbe considered advantageous to have the harmonic energy dis-tributed over a larger frequency range instead of being con-centrated in few discrete frequencies.A way to achieve this is a modified spac e vector modulationin which the complex volt-second contribution of the actualswitching state vector ba c ts continuously monitored, Fig. 6.The on-time of this vector terminates w hen

(7)

t k tk+l0

- - - i _ modulation index as a parame-

&act laact +U1 tl +U 2 . V - tact - l ) =U*(t) .T

holds, where f ac t is the resultant on-time of kc, ,l is anothersolution of (7) which is disregarded, and T is a constant, theinverse of which adjusts the average switching frequency.Once the on-time tactof &,ct has elapsed, ul is chosen as bctfor the next switching interval, u2 becomes ul, and the cyclicprocess starts again.Since the reference vector enters into (7 ) not as a sampledconstant but as a time-varying quantity, the duration of theindividual subcycles becomes time-varying, too. This resultsin a frequency modulation of the carrier signal. The way howthe durations of the subcycles vary is exemplified in Fig. 6c.A comparison of the harmonic spectra Fig. 6d and Fig. Iddemonstrates the desired effect. - 13, 141

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00 2 4 6 kH z 1f1 -Fig. 6: Carrierless PWMa) signa l flow diagramc) subcycle duration T, measured b) voltage vectorsd) harmonic spectrumover a fundamental cycle

3.4 Inverter switching delayIt was assumed until now that the inverter switches behaveideally. This is not true for nearly all types of semiconductorswitches. Minority carrier devices in particular have theirturn-off delayed owing to the storage effect. Th e storage timevaries with the current and the device temperature. To avoidshort-circuits of the bridge legs, a delay time Td must beintroduced by the inverter control. T he delay time counts fromthe time instant at which one switch is turned off and termi-nates when the opposite switch is turned on. The delay time Tdis set as the maximum value of storage time T,, lus a safetytime interval.We have now two different cases. If the load current flowsthrough the active semiconductor switch, the phase voltagepolarity reverses at the beginning of the safety time interval. Itreverses at its end if the current has the opposite polarity,flowing through the antiparallel feedback diode of the oppo-site semiconductor switch. Consequently, the actual duty cy-cles of the bridge legs diffcr from the ones of their respectivecontrol signals. The difference can be described by an errorvoltage vector(8)= * - hV=(Td - T , t ) / T , * &(U.

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where iskaVhe average voltage vector of a subcycle. It is seenthat the magnitude of & s proportional to the safety timeinterval T d - Tst; its direction changes in discrete steps, de-pending on the polarity of the phase currents. This is ex-pressed by a polarity vector of constant magnitude&(U =2 / 3 [(sign(i,) +a sign(ib)+u 2 sign(i,)], (9)where a =ex p ( i2 d 3 ) . The notation &(L) indicates that thiscomplex function exhibits properties of the sign function.

hi4-3-2 -1 -0-

. i o 3

,A\', I

Fig. 7: The polarity vector a(l>4 1

Fig. 8: Trajectory of voltage haV

The grapc&(U is shownin Fig. 7 for all possiblevalues of 1. The nonline-ar distortion of the aver-age voltage vector bayyvariable switching timedelay is exemplified inF ig . 8. The dis tor t iondoes not depend on thefundamental voltage uIand hence its influenceis very strong in the lowspeed range where thefundamental voltage isalso low. The effect in-fluences the waveform ofthe currents and maycause electromechanicalstability problems in cer-tain operating ranges ofthe drive.A switching delaycompensat ion schemefor one phase is shownin Fig. 9. There is alwaysa constant time delay es-tablished by closed loopcontrol between the log-ic output signal U' of thepulse modulator and theactual switching instantof the bridge leg. Toachieve this, the actualinstant sign(uph)s meas-ured and used as a feedback signal. U " is the signal which isfed to the tim e delay logic of the bricf[e leg.The changes of the error voltage vector & ct as suddendisturbances on the current control loop. They are comp ensat-ed only at the next switching of the phase leg. T he remainingtransient error is mostly tol-erable in induction motordrive systems; synchronous ..-..machines having sinusoidal

mup I{ 'd l.-..1 b)t0

Fig. 9: Switching time delay compensation for on e phasea) circuit diagram b) waveforms

back-e.m.f. behave more sensitive to these effec ts as they tendto operate partly in the discontinuous current mode at lightloads. Th e reason for this adverse effect is the absence of amagnetizing current component in the stator currents. Suchmachines require more elaborate switching delay compensa-tion schemes when applied to high-performance motion con-trol systems.As an alternative solution, a d-axis current component canbe injected into the machine which shortens the discontinuouscurrent time intervals at light loads. - 15-181

4. Feedback P WM ControlThese schemes generate the switch ing sequences inherentlyin a closed control loop, which is established either for thestator currents or for the stator flux vector. Stator flux schemesare not customary in motion control systems.

4.1 Nonoptimal methods4 .1 .I Hysteresis current controlThe block diagram Fig. 10shows three hysteresis control-lers, one for each phase. Each c ontroller determines the sw itch-ing state of one in-verter leg such thatthe error of the cor-ud responding phasecurrent is main-tained within thehys teres is band.The control meth-od is simple to im-plement, and its dy-namic performanceis excellent. Hys-a)

ti0

- -0 i o 20 30 kHz 50c) f-Fig. 10: Hysteresis current controla) signal flow diagramb) waveforms c) amplitude density spectrum

teresis current control requires high switching frequency tocompensate for some inherent drawbacks:There 1s no intercommunication between the individualhysteresis controllers of the three phases and hence nostrategy to generate zero voltage vectors. This increases theswitching frequency at lower modulation index.There is a tendency at lower speed to lock into limit-cyclesof high-frequency switching which comprise only nonzerovoltage vectors.The current error is not strictly limited. It can assumedouble the value permitted by one hysteresis controller incertain operating conditions.Hysteresis controllers are preferred for ultrasonic sw itching

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4-u

0 5 10 kHz 20 0 0.5 rns 1b) f- c) t -Fig. 11: Suboscillation current controla) signal flow diagramb) harmonic spectrum c) carrier and reference signalfrequency at lower power level using MOSFETs as semicon-ductor switches. - 19,2034 .1 .2 Suboscillation current controlA carrier based modulation scheme can be established toform a current control loop as shown in Fig. 11.A proportion-al-intergral (P I-type) controller is used to derive the referencevoltage U* for the pulsewidth modulator from the currenterror. The back-e.m.f. of the machine acts as a disturbance inthis control loop. This voltage co ntains only the fundamentalfrequency and varies continuously with time. It is thereforepossible to compensate its influence through the I-channel ofthe PI-controller. However, a steady-state current error willpersist (tracking error).This error should be kept low by choosing a high gain forthe PI-controller. The gain is limited, on the other hand, as italso amplifies the harmonic currents which must not impairthe proper operation of the pulsewidth modulator. This isensured if the maximum slope of the current error signal is

@Fig. 12: Space vector current control

less than the slope of the triangular carrier signal.This scheme cannot be simply looked at as a pulsewidthmodulator having a superim posed current control loop, as alsothe harmonic currents influence upon the switching instants,Fig. llc. This ensures a fast response of the current loop,provided that the modulator reacts on instantaneous changesof its reference signal U*.An analogue circuit implem entaionof the subo scillation method is an adequate solution. - 21 ]4.1.3 Space vector current controlThe nonzero current ei-ror, inherent to the above scheme,may be undesired in a high-performance vector controlleddrive. The error can be e liminated by deriving the back-e.m.f.voltage from a mach ine model and using it as a comp ensatingfeedforward sign al, Fig. '12. The controller corrects only m i-nor errors which may originate from a mismatch of the mode lstructur or the model paraimeters. Th e dynamic performance isimproved by feedforward control based on the derivative ofthe current reference. The influence of current harmonics onthe modulated waveform is eliminated if a space vector mod-ulator is implemented in c onjunction with digital current con-trollers, the algorithm s of both subsystems being executed instrict synchronism.The generated reference voltage U* is basically free fromharmonics. Hence a space vector m odulator with its discontin-uous sampling input can be used for modulation. - 22]4.1.4It is a common approach for high-dynamic performancedrives to control the components of the current vector after atransformation to field coordinates, Fig. 13.The current refer-ence then represents the torque and machine excitation com-mands. A low-pass filter is generally used for the feedbacksignals to reduce their harinonic content and to avoid irregularoperation of the pulse modulator. Th e reference voltage vec-tor U* generated by the cui-rent controllers i s transformed backinto stator coordinates arid acts on any convenient feedfor-ward PWM scheme. - 231

Current control in1field coordinates

statorcoordinates I coordinatesfieldFig. 13: Current control in field coordinates; 6: field angle4.2 Feedback PWM with real-time optimization4.2 .1 Predictive current controlIn a closed-loop current co ntrol schem e, the error signal is aspace vector, &(t) = *(t) I- i(t). Limiting the magnitude 141ofthis error vector by a boundary value is a means to terminatean actual switching state. This principle is illustrated in Fig.14b. The boundary has a circular shape, and its location in thecomplex plane is determined by the current reference vector

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i*. When t h e t r a j e c t o r y o f t h e c u r r e n t v e c t o r i t o u c h e s t h eb o u n d a r y line, a decisionon t h e n e x t s w i t c h in g s t a t e v e c t o r ismade b y p r e d i c t i o n and opt imiza t ion .To t h i s aim, t he t ra j ec tor i e s of the cur ren t vec tor are c o m -p u t e d , and t h e t i m e required t o reach t h e n e x t b o u n d a r y isp r e d ic t e d. N o t e t h a t t h e error b o u n d a r y moves i n t h e complexp l a n e since the c u r r e n t r e f e r e n c e is varying w i t h t i m e . Predic-t i ons of t h e on-time are made for al l error reducing s w i t c h i n gs ta t e vec tors . Finally, t h e s w i t c h in g s t a t e v e c t o r w h i c h pro-duces t h e maximum o n - t i m e is se l ec t ed . Th i s procedure mini-mizes t h e switching f r e q u e n c y . The opt imiza t ion can be alsoe x t e n d e d over the n e x t two swi t ch ing s t a t e in t e rva l s .

"0 -2 .4 .6 .8 1 0 .I .2 .3 .4 .5rn - m--,

Fig. 14: Predic t ive cur ren t cont ro la ) signal f l o w diagramb ) reference c u r r e n t and a c t u a l c u r r e n t v e c t o rc ) s w i t c h i n g f r e q u e n cy and d ) t o r q u e ripple

T h e p r e d ic t i o n s t a k e a b o u t 20 ps on a DSP. T h e c o m p u ta -t i o n t i m e can b e n e u t r a l i z e d b y double predic t ion . Even so, t h eappl i ca t ion is r e s t ri c t e d t o s w i t c h i n g f r e q u e n c i e s around 1k H z . S i n c e t h e d i s t o r ti o n f a c t o r d is a l m o s t f i x e d b y t h eb o u n d a r y c o n d i t i o n , t h e p e r f o r m a n c e a t v a r y i n g m o d u l a t i o ni n d e x is p r i m a r i ly r e f l e ct e d i n t h e s w i t c h i n g f re q u e n c y . T h i s isd e m o n s t r a t e d in Fig. 14c a n d Fig. 14d f o r t h e case o f a predic-t i ve cur ren t controller w h i c h a l w a y s maximizes t h e on-dura-t i on s o f t h e n e x t tw o swi t ch ing s t a t e vec tors . - 24-261

5. FutureTrendsA n a l o g m e t h o d s , l i k e h y s t e r e s is c o n t r o l an d subosc i l l a t ionc u r r e n t controlare s t i l l in use, e s p e c i a ll y a t lowerdrivepower.T h e s e m e t h o d s will be gradually subs t i tu t ed by microproces-sor or microcontroller b a s e d c o n tr o l s y st e m s . M a n y P W Ms c h e m e s lend t h e m s e l v e s t o an i m p l e m e n t a t io n w h e r e memo-ries a n d c o u n t e r s are u s e d as e x t e r n a l d i g i t a l c i r cu i t s t o h a n d l et h e h i g h s w i t c h i n g f re q u e n c y o f modern s e m i c o n d u c t o r devic-es w i t h a d e q u a t e t i m i n g a c c u ra c y . T h e u s e o f ASICs f o r P W Mappl i ca t ions will increase.

6. References1. E. A. Klingshirn and H. E. Jordan: A Polyphase Induction MotorPerformance and Losses on Nonsinusoidal Voltage Sources. IEEETrans. Power Ap p. Sys t . (1968), pp. 624-631.2. J. M. D. Murphy and M. G. Egan: A Comparison of PWM Strate-gies for Inverter-Fed Induction Motors. IEEE Trans. Ind. Appl.3. T. A. Lipo, P. C. Krause and H. E. Jordan: Harmonic Torqu e andSpeed Pulsations in a Rectifier-Inverter Induction Motor Drive.IEEE Trans. Power A pp. Syst . (1969). pp. 579-587.4. J. T. Boys and P . G. Handley: Harmonic Analysis of Space VectorModulated PWM Waveforms. IEE Proc. B (1990), pp. 197-204.5. CISPRE 11and C ISPRE 14 International Standards6. A. SchBnung and H. Stemm ler: Static Frequency Ch angers withSubharmonic Control in Conjunction with Reversible Variable SpeedAC Drives. Brown Boveri Rev. (1964). pp. 555-577.7. D. Grant and J. Houldsworth: PWM AC Motor Drive EmployingUltrasonic Carrier. IEE Con f. PEVSD, Lond. (1984), pp. 234-240.8. A. Busse and J. Holtz: Multiloop Con trol of a Unity Power Fac torFast-Switching AC to DC Converter. IEEEIPESC, Cambridge9. J. Holtz, P. Lammert and W. Lotzkat: High-speed Drive Systemwith Ultrasonic MOSFET PWM Inverter and Single-Chip Micro-processor Control. IEEE Trans. Ind. Appl. (1987), pp. 1010-1015.10.0. Ogasawara, H. Akagi, A. Nabae: A Novel PWM Scheme ofVoltage Source Inverters Based on Space Vector Theory. EPEEurop. Conf . Pow . Elec. andApp l . ,Aachen (1989), pp. 1197-1202.11 . G. B. Kliman and A. B. Plunkett: Development of a ModulationStrategy for a PWM Inv erter Drive. IEEE T rans. Ind. App l. (1979),12 . S . R. Bowes and M. J. Mount: Microprocessor Control of PWMInverters. IEE Proc. B (1981), pp. 293-305.13 . I. Holtz and L. Springob: Reduced Harmonics PWM ControlledLine-Side Converter for Electric Drives. IEEEIIAS Ann. Meet.,Seattle (1990), pp. 959-964.14. T. G. Habetler, D. Divan: Aco ustic Noise Reduction in S inusoidalPWM Drives Using a Randomly Modulated Carrier. IEEE Trans.Power Electr. (1991), pp. 356-363.15. W. Lotzkat: Aufwandarme und netzausfallsichere Frequenzumrich-ter zur parameterunempfindlichenRegelung von Asynchronmaschi-nen fur industrielle Standardantriebe. Ph.D.-Thesis,Wuppertal Univ.(1991).16 . Y .Murai, T. Watanabe and H. Iwasaki: Waveform Distortion andCorrection Circuit for PWM Inverters with Switching Lag-Times.IEEE Trans. Ind. App l. (1987), pp. 881-886.17. R. D. Klug: Effects and Correction of Switching Dead-Times i n 3-Phase PWM Inverter Drives. EPE Europ. Con$ Pow er Electr . anaApp l., Aachen (1989), pp. 1261-1266.18. Y. Wang, H. Grotstollen: Control Strategies for the DiscontiuouaCurrent Mode of AC Drives with PWM Inverters. EPE Europ.Conf . Power Elec. andAp pl . , Florencell taly (1991), pp. 31217-22219. D.M. Brod and D. W. Novotny: Current Control of VSI-PWMInverters. IEEE Tra ns, Ind. Appl. (19 89 , pp. 562-570.20. S . Salama, S . Lemon: Overshoot and Limit Cycle Free CurreniControl Method for PW M Inverter. EPE Europ. Conf .P ower E l ec .tronics and Appl., Florencell taly (1991), pp. 31247-251.21. V. R. Stefanovic: Present Trends in Variable Speed AC Drives22. G. Pfaff, A. Weschta and A. Wick: Design and Experimental Results of a Brushless AC Servo Drive. IEEEIIAS Ann. Meet. SurFrancisco (1982), pp. 692-697.23. W. Leonhard: Control of AC Machines with the Help Microelectronics. IFAC Sym p., Lausanne (1983), pp. 35-58 Survey.24. J. Holtz and S . Stadtfeld: A Predictive Controller for the StatoCurrent Vector of AC Machines Fed from a Switched VoltagcSource. IPEC Tokyo (1983), pp. 1665-1675.25 . J. Holtz and S.Stadtfeld: A PWM Inverter Drive System with O nLine Optimized Pulse Patterns. EPE Europ. Conf. Pow er Elecrrand A ppl., Brussels (1985), pp. 3.21-3.25.26. U. Boelkens: Vergleichende Untersuchung von trajektorienorientierten Steuerverfahren fur dreiphasige Pulswechselrichter zur Sp ei

sung von Asynchronmaschinen. Ph.D.-Thesis, Wuppertal University (1989).

(1983), pp. 363-369.

(1982), pp. 171-179.

pp. 702-709.

IPEC Tokyo (1983), pp. 438-449.

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