ORIGINAL PAPER

Djordje Stojic Æ Slobodan Vukosavic

Induction motor drive based on the stator flux vector control

Received: 11 August 2003 / Accepted: 3 November 2003 / Published online: 23 December 2003� Springer-Verlag 2003

Abstract This paper presents a new torque controlalgorithm for induction motors, based on the stator fluxvector control. For each sampling period, the value of thestator voltage is calculated to keep the stator flux equal tothe reference vector, while the stator flux reference vectoris calculated to keep the rotor flux amplitude constant atall operating conditions. An improved stator and rotorflux estimation algorithm is proposed, enabling robustand stable operation of the drive, even at low speeds. Theinduction motor torque is manipulated by variations ofthe flux angular velocity, enabling drive operation withfixed switching frequency and ripple-free torque in thesteady state. The performance of the proposed algorithmis tested through various experimental runs, provinggood behavior of the drive in both transient and steady-state operating conditions.

Keywords Sensorless torque control Æ Flux vectorcontrol Æ Induction motor

1 Introduction

The efforts in developing novel induction motor (IM)control algorithms are focused on drives with the mini-mum number of sensors required for operation. In re-cent years, a wide variety of speed sensorless solutionshas been proposed, contributing performance increase athigher speeds, but failing to improve the drive behaviorbelow 1% of the rated speed.

The concept of direct torque control (DTC) [1, 2, 3]algorithms for induction motors was introduced, sig-nificantly improving the torque control performancewith algorithms that achieve fast and robust torquecontrol of IM without use of the current controllers andshaft position sensors. The DTC control strategies em-body algorithms in which the voltage source inverter(VSI) switching patterns are generated directly as afunction of the torque and flux errors (with the corre-lation between the torque, flux error, and switchingstates defined by means of the switching table combinedwith the sliding mode control strategy). The basic DTCconcept with a hysteresis regulator [1, 2] results in a veryfast torque response, unforeseen in conventional driveswith a current controller in the minor loop. Nonlinearhysteresis-based control results in a high level of torqueripple and in irregular inverter switching, with thecommutation frequency dependent upon the modulationindex, speed, flux, and current level. Various attempts tominimize the torque ripple and switching frequencyvariations where done by using the switching tables withadaptive sliding mode control [3].

Another attempt to minimize the torque chatteringand switching frequency variations was introduced bythe stator flux vector control (SFVC) schemes [4, 5, 6].The aim of SFVC is to drive the estimated stator fluxvector toward the reference value by indirect control ofthe inverter switching states, through pulse-width mod-ulation (PWM). The SFVC algorithms are based onvarious control strategies (‘‘dead-beat’’, feed-forward,PI, sliding mode controllers, etc.), with the stator fluxerror used as the input variable. The output of SFVCscheme is fed into the space vector modulator (SVM),achieving the DTC with constant switching frequencyand smooth torque and flux waveforms.

Papers [4, 5, 6] present three different SFVC schemes,proving that the direct linear control of torque andstator flux vectors enables fast torque dynamics, im-proved stator flux estimation at low speeds, and ripple-free drive operation. The common drawback of theseSFVC algorithms is that they rely on the calculation of

Electrical Engineering (2005) 87: 23–32DOI 10.1007/s00202-003-0214-2

D. Stojic (&)Electrical Institute INT,Golsvortijeva 13, 11000 Belgrade,Serbia and MontenegroE-mail: djs@ieent.org

S. VukosavicElectrical Engineering Faculty of Belgrade,Bulevar Kralja Aleksandra 73, 11000 Belgrade,Serbia and Montenegro

the field vector position from the estimated stator fluxvector and on the estimation of field velocity by differ-entiation of the calculated field vector angle. This leadsto an increase of drive sensitivity to the measurementnoise and to a decrease in drive performance, especiallyat low speeds (with poor field angle estimation). Also,SFVC drives [4] and [6] exhibit high sensitivity to the IMparameter variations, since these algorithms are de-signed in the dead-beat fashion.

In this paper, a novel SFVC algorithm is proposedfor controlling the flux and torque of induction motorsoperating without shaft sensors. The improvementsintroduced in this SFVC algorithm include decreasedsensitivity to IM parameter variations and improvedstability of drive operation, even at speeds close to zero.The increase in the drive performance is achieved by thefollowing changes in the SFVC algorithm: implementa-tion of a simplified feed-forward stator flux regulatorwith a reduced set of control parameters, the directmanipulation of the field velocity (as opposed to solu-tions that rely on the field position and field velocityestimation), and by using the closed-loop estimator ofthe stator and rotor flux vectors.

The proposed solution is verified through a set ofexperimental tests on a setup having a 10-HP 4-poleindustrial motor. The results obtained confirm theability of the proposed controller to ensure both torqueand speed control in any practical operation condition,including standstill.

2 Basic drive operating principle

In the proposed algorithm, the torque control relies onthe fundamental behavior of the squirrel cage inductionmotor. Namely, the torque generation is proportional tothe square of the rotor flux and slip frequency, and isinversely proportional to the rotor resistance. Hence, the

torque control loop derives the torque error and in-creases/decreases the speed of the field rotation. Varia-tions of the field rotation velocity have an impact on theslip frequency that will, in turn, result in a desired torquechange and steer the torque error toward zero.

The proposed flux estimator and associated control-ler are located in the stationary reference frame. Theexplicit flux control deals with the stator flux in order toachieve a faster response and boost the robustnessagainst changes of IM parameters. On the other hand,the stator flux reference is calculated so as to take intoaccount the feed-forward component, related to theleakage flux. In turn, the rotor flux amplitude isguaranteed to be constant in the steady-state andconstant-torque operating mode, contributing to theease of torque control by making the slip-torque relationlinear. By operating in the constant-torque mode, theproposed solution ensures a low torque ripple and aconstant switching frequency. The algorithm requiressetting a relatively small number of parameters and issimple to implement.

Figure 1 shows the block diagram of the proposedDTC algorithm. The algorithm inputs are the referencerotor flux vector w*DQ, reference torque T*e, and statorcurrent iabs. The main control loop consists of the torquecontroller and block for the stator flux reference calcu-lation. The amplitude of the stator flux reference iscalculated from the rotor flux and torque references,ensuring a constant rotor flux amplitude in the constant-torque operating mode. The rotational frequency xe ofreference vector w*abs is determined by using the torquePI controller, in order to drive the torque error towardzero. Consequently, the angle of the stator flux referencevector w*abs is calculated by integration of xe.

The local inner loop, within the dotted rectangle,represents the linear stator flux controller. The fluxcontroller is realized in the stationary reference frame,with zero error of the stator flux vector in the stationary

Fig. 1 Block diagram of SFVCalgorithm

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state. The zero flux error is guaranteed by implementingfeed-forward actions that compensate voltage dropsacross the stationary resistance and induced back-elec-tromotive force (back-EMF). The stator flux regulatorproduces reference stator voltage vector v* abs, fed to thePWM block. The proposed drive also includes com-pensation of ‘‘dead-time’’ effects in the three-phase ACinverter.

The DTC algorithm of Fig. 1 includes a flux vectorestimator. In this paper, the problems associated withflux estimation are resolved by using the closed-loopobserver, with the rotor flux used as a feedback signal.The closed-loop observer introduces the significantimprovement in the drive behavior, especially at lowspeeds. The closed-loop flux observer also resolves theproblems of estimator output DC-drift, which is presentin the flux estimators based on back-EMF integration.The stator flux estimation is described in more detail inthe section that follows.

3 Flux vector estimation

The flux vector estimation uses the stator current vectorand stator voltage vector as input variables. The statorvoltage vector may be directly measured at the motorterminals, whereas this solution requires the electricalinsulation between the power circuit and control hard-ware with considerably large bandwidth. On the otherhand, the voltage reference values, fed to the spacevector modulator, could be used instead of the voltagemeasurements, thus avoiding the use of expensive mea-surement hardware. When using the reference instead ofthe measured voltage value, care must be taken of thefact that the former does not represent the latter to thefull extent, due to distortions exhibited by the inverternonlinearities [7]. If not compensated, effects of inverternonlinearities are amplified, especially at low speeds andwith a low level of voltage fundamental.

In this paper, voltage reference vector v* abs is usedfor the flux vector estimation, as suggested in [7]. Toavoid irregular drive operation at low speeds because ofinaccurate stator voltage samples, the inverter nonlin-earities are compensated (‘‘dead-time’’ compensationblock in Fig. 1) by using the algorithm proposed in [8].

The closed-loop flux estimation is developed from therotational dq frame model of IM expressed by the fol-lowing equations

vdqs ¼ idqsRs þ jxwdqsþdw

dqs

dt; ð1Þ

0 ¼ iDQRr þ j x� xrð ÞwDQþdw

DQ

dt; ð2Þ

wdqs¼ Lsidqs þMiDQ; ð3Þ

wDQ¼ LriDQ þMidqs; ð4Þ

T ¼ 3

2p wdsiqs � wqsids

� �; ð5Þ

where p represents the number of pole pairs, x is thereference frame angular velocity, and x r is the rotorangular speed. For the stationary ab reference frame, xis equal to zero, while for the dq reference frame syn-chronous with the rotor flux vector, x is equal to the fluxvector angular velocity xe.

The stator flux could be calculated by solving differ-ential Eq. (1). This type of flux estimation introducespositive current feedback into the DTC algorithm,causing unstable drive operation at low speeds. Theinstabilities originate from the inaccurate value of statorresistance in Eq. (1) and nonlinearities of the inverteramplified by the positive stator current feedback.

The flux estimation can be improved by introducingnegative current feedback into the calculation of fluxvector. In this paper, the negative current feedback isindirectly introduced by using the closed-loop flux ob-server. The operating principle of the closed-loop fluxestimation is based on feeding back the difference be-tween the reference and estimated rotor flux vector. Thedifference signal is then used to correct the voltagemodel of Eq. (1), consequently minimizing the error offlux estimation. The difference signal also includes thenegative feedback path for stator current, canceling outthe bad influence of positive current feedback, present inEq. (1), on the dynamic behavior of the drive. Thebehavior of the flux estimator is described by followingequations:

w_

abs¼Z

vabs � Rsiabs þ G w�abr� wabr

� �h idt; ð6Þ

w_

abr¼ Lr

Mw

abs� r Ls Lr

Miabs; ð7Þ

where w* abr represents the rotor flux reference vector,and G is the observer gain. The rotor flux reference w*abr is calculated by using the rotor flux reference vectorw*DQ in the rotational reference frame synchronous withthe stator flux. Reference w* DQ is defined by compo-nents w* Q =0 and w* D (component w* D can be freelyadjusted). Reference vector w* abr is calculated fromvector w* DQ by using the following inverse rotationaltransformation:

w�arw�br

� �¼¼ cos heð Þ � sin heð Þ

sin heð Þ cos heð Þ

� �w�D0

� �: ð8Þ

The d -axis of the rotational reference frame is in thedirection of rotor flux reference vector w*DQ. In Eq. (8),h e represents the angle of the rotational frame position.Since the angular velocity of the rotational referenceframe is chosen to be equal to xe, which is set by thetorque regulation, the reference frame angle is calculatedas

he kð Þ ¼ he k � 1ð Þ þ Kxx kð Þ: ð9Þ

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The observer gain G is chosen according to the cri-terion that stable drive operation is to be maintained forvariations of Rs within the range of 25% around itsnominal value, where this region may be extended ifnecessary. Since the analytical evaluation of G thatmatches this criterion is rather complicated, a moreconvenient procedure for setting G is to tune its value onthe experimental setup in the following manner. First,the value of Rs is intentionally detuned for 25% of itsnominal value in Eqs. (6) and (10). Then, for the de-tuned value of Rs, the value of G is increased until stabledrive operation is obtained. Finally, the value of Rs inEqs. (6) and (10) is returned to its nominal value, afterthe proper value of parameter G is chosen in the secondstep. By using the above three-step procedure, theappropriate value of observer gain G is achieved thatguarantees stable drive operation with respect to anundesirable influence of stator resistance variations.

The flux observer, Eqs. (6) and (7), enables stabledrive operation, even at very low speeds. It makes thedrive less sensitive to variations of the drive parametersand nonlinearities of the voltage inverter.

4 The stator flux regulator

The stator flux regulator determines the stator voltagereference value, which is fed into the PWM block. Thevoltage value is calculated in order to achieve the statorflux equal to reference vector w*abs within a finite set-tling time. In doing so, the control is designed to achievezero flux error in less than three sampling periods (set-tling time is tst=2.5 T s, where Ts is the digital controllersampling period, Ts=200 ls).

Since the flux regulator is designed in the stationaryreference ab frame, with sinusoidal variable frequency

references, it is necessary to design a control structurethat guarantees zero error signals in the stationary state.To achieve zero steady-state error, the feed-forwardcontrol structure is included in the stator flux regulation.Figure 2 shows the stator flux regulator, together withthe space vector pulse-width modulator and dead-timecompensation.

The flux regulator equation is expressed as

v�abs kð Þ ¼ Rsis kð Þ þ jxe kð Þw�abs

kð Þ

þ Kp w�abs

kð Þ � wabs

kð Þh i

: ð10Þ

The first two terms on the right-hand side in Eq. (10)represent the feed-forward control actions, aimed atcompensating the voltage drops across the stator resis-tance and stator back-EMF. Notice, the feed-forwardaction for the back-EMF is proportional to referencestator flux vector w*abs, opposite to the existing SFVCalgorithm [5], which includes feed-forward action for theback-EMF proportional to the estimated stator flux vec-tor. Consequently, the back-EMF compensation pro-posed in [5] introduces a flux feedback signal with variablegain (flux velocity xe), which causes undesirable varia-tions in the controller dynamics. In Eq. (10), the variableflux feedback signal is avoided by introducing back-EMFcompensation proportional to vector w*abs.

The first two feed-forward terms in Eq. (10) deter-mine the steady-state value of the voltage command,which guarantees zero error for a given stator flux ref-erence vector w*abs. The third term in Eq. (10) representsthe feedback control action, which is proportional to thestator flux error. In Eq. (10), K p is the gain of the fluxregulator and xe ( k) represents the instantaneous statorflux velocity ( xe ( k) is determined by the outer torquecontrol loop). Notice that in Eq. (10) only one param-eter, Kp, is to be set, which makes the proposed designprocedure simple to implement.

The feed-forward term in Eq. (10), which compensatesthe voltage drop across the stator resistance, can be

Fig. 2 Block diagram of stator flux regulator

26

found in various SFVC algorithms [4, 5]. This feed-forward action represents the positive feedback path forthe stator current, thus causing an increased sensitivity ofthe drive to the nonlinearities of the VSI and to variationsof R s. As discussed in the previous section, the problemsrelated to the positive current feedback, inherent for basicDTC algorithms, are resolved by introducing a negativecurrent feedback indirectly through the closed-loop statorflux estimation algorithm.

Since one of the objectives of the proposed DTCalgorithm is to enable drive operation with constantrotor flux amplitude, it is necessary to calculate theadequate stator flux reference for each operating con-dition. By using the rotational reference frame model inEqs. (1)–(5) of IM, under the condition that the rotorflux vector is constant, the stator currents in the sta-tionary state can be derived from the electromagnetictorque. Hence, for the rotor flux and torque referencesw* Q =0, T* e, and w* D, the reference drive statorcurrents are calculated by

i�qs ¼2Lr

3pLmw�DT �e ; ð11Þ

i�ds ¼w�DLm

: ð12Þ

From Eqs. (7), (11), and (12), the stator flux referencevector is expressed as

w�abs¼ M

Lrw�

abrþ rLsi�abs; ð13Þ

where vector w*abr is given by Eq. (8), and vector i*abs iscalculated from i* dqs by using the following inverserotational transformation:

i�asi�bs

� �¼¼ cos heð Þ � sin heð Þ

sin heð Þ cos heð Þ

� �i�dsi�qs

� �: ð14Þ

In Eq. (14), the value of angle h e is given by Eq. (9).

The ‘‘dead-time’’ compensation is achieved by usingthe algorithm proposed in [8]. The compensation isbased on the calculation of the volt-seconds lost in theblanking period, averaged over the switching cycle.Since the voltage distortion introduced by the blankingtime is of opposite sign to the phase current, the com-pensation voltage added to the commanded voltage is ofthe same sign as the appropriate phase current.

The proposed linear controller enables fast and robuststator flux regulation, which, together with the torqueregulator, yields zero steady-state errors of the rotor fluxand torque for given rotor flux and torque references.

5 The torque regulator

In the previous section it was shown that the control ofstator flux may be performed for any operating fre-quency. The stator flux control is extended to direct

torque control by introduction of a linear controller thatmanipulates the torque by using the field velocity as acommand variable.

The proposed torque regulation relies on the oper-ating principle of IM that the torque is directly pro-portional to the slip frequency and to the squaredamplitude of the rotor flux. This statement is proven bythe following set of equations. Namely, flux velocity xe

represents the rate of change of rotor flux angle he.Angle h e is defined by

he ¼ arctanwbr

war; ð15Þ

and the flux angular velocity can be calculated as

xe ¼dhe

dt¼

war_wbr � wbr

_war

wabr

������2

: ð16Þ

Using Eq. (2) the following relation is obtained:

Te ¼3p2

wabr

������2

Rrxe � xrð Þ ¼ 3p

2

wabr

������2

Rrxs; ð17Þ

where x s is slip frequency.The estimated torque signal, used as the input to the

torque regulator, is calculated by

Te ¼3

2p w

_

asibs � wbsias

� �: ð18Þ

The torque controller is designed to generate both thetransient and steady-state components of the slipvelocity. The transient component determines the set-tling time of torque, while the steady-state componentdetermines the torque steady-state value. The PI torquecontroller applied is defined by the following expression:

xe kð Þ ¼ xe k � 1ð Þ þ KT1 DTe kð Þ � KT2DTe k � 1ð Þ�

;

ð19Þ

where KT 1 and KT2 represent the control parameters,and DTe kð Þ represents the torque error, DTe kð Þ ¼T �e � Te kð Þ. The torque controller works with a samplingperiod of 200 ls. The conventional PI regulator ofEq. (19) enables a fast transient response of torque anddetermines the appropriate steady-state value of the fluxangular velocity, which varies with the torque referenceand operating speed of IM.

6 Experimental tests

The proposed control scheme was realized as an exper-imental drive, consisting of an IGBT inverter, digitallycontrolled by hardware based on a PC-platform runningreal-time control software. The SFVC drive was realizedusing the floating-point algorithm (including the statorflux controller, flux reference calculation, stator flux

27

estimation, torque estimation and control, anti-windupintegrators, limiters, and other diagnostic facilities), withsampling period Ts=200 ls.

The induction machine under test was coupled to aseparately controlled DC machine, used as a dynamicbreak. The 4-pole squirrel cage induction motor wascharacterized by the following data: P=7.5 kW,n =1500 rpm, V=220 V, f=50 Hz. The steady-stateand transient behavior of the drive was investigated byvarious sets of tests.

6.1 The steady-state operating conditions

The steady-state behaviour was investigated in differentoperating conditions, i.e. with locked rotor, and for mid-and high-speed regions. The drive behaviour in operat-ing conditions involving low flux angular velocities wastested by using the locked rotor tests. Figure 3 shows thecurrent and torque behaviour for the locked rotor, withthe rotor flux and torque set to their rated values. Thestator current is sinusoidal, while the estimated torquematches the reference value. The experimental results inFig. 3 prove that the proposed PI torque control algo-rithm enables zero torque error signal in the stationarystate. This shows that precise torque regulation can beachieved with simple PI control strategy, contrary to the‘‘dead-beat’’ structure based on solutions that requireknowledge of exact IM parameter values to calculate theadequate regulation commands (cf. [4] and [6]).

Figure 4 shows the test results for the torque refer-ence set to 0.4 pu. The current waveform exhibits some

distortions, while the torque matches the set value.Since, in this experiment, the drive operates at very a lowfrequency, the presented results show that the proposedcontrol algorithm keeps the machine magnetized, evenat zero speed.

Figure 5 presents the results for the mid-speed region,with torque set to 0.4 pu and rotor speed set to 0.5 pu bya DC motor coupled with the IM. Figure 6 shows tor-que and stator current behaviour for torque set to 0.2 puand speed set to 0.8 pu. The experimental results inFigs. 5 and 6 show that in the mid- and high-speed re-gions stator current is sinusoidal with small distortions,while torque matches the reference value.

The results presented in this section show that theproposed torque control technique enables the zerotorque error operation in the stationary state in a widerange of operating speeds. This proves that zero torqueerror operation can be achieved by using the proposedPI control strategy, which does not require preciseknowledge of motor parameters. This is an improvementcompared to the existing SFVC solutions [4, 6], whichrequire accurate knowledge of motor parameters in or-der to generate adeqate command values for zero torqueerror operation.

6.2 Transient operating conditions

The transient behavior of the proposed DTC algorithmwas investigated for locked rotor and the low- and mid-speed regions. Figure 7 shows torque and stator currentresponses for locked rotor and for torque reference step

Fig. 3 Estimated torque andstator current for locked rotorand nominal torque

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change from 0.1 to 0.2 pu. The results show that theresponse time of torque for the step excitation equals4–5 sampling periods ( T s=200 ls), with zero steady-state error signal. Moreover, the presented experimentalresults show that the proposed DTC algorithm has re-sponse times comparable with the ones achieved withbasic DTC algorithm [1, 2]. Also, the presented results

prove that achieved torque dynamic is faster whencompared to the existing SFVC strategies [4, 5, 6] withtorque response times above 3 ms. Figure 8 representsthe torque response for a step excitation for rotor speedset to 0.2 pu, while the results in Fig. 9 correspond tothe speed set to 0.4 pu. The measurements presented inFigs. 7, 8, and 9 show that the torque regulator retains

Fig. 4 Estimated torque andstator current for locked rotorand torque set to 0.4 pu

Fig. 5 Estimated torque andstator current for speed set to0.5 pu and torque set to 0.4 pu

29

the same response times in a wide range of operatingspeeds. These test results prove that the proposed DTCalgorithm is robust in relation to rotor speed variations.

The results in Fig. 10 represent the rotor speedbehavior for torque reversal, showing that rotor speedhas the typical ‘‘saw-tooth’’ waveform, without signifi-

cant distortions in the speed waveform. This experimentwas performed in order to enable further comparison ofthe proposed algorithm with existing SFVC drives [4, 5, 6].Namely, the torque and speed measurements in Fig. 10prove that the stability of drive operation in the low-speed region is improved, since the drives in [4, 5, 6]

Fig. 6 Estimated torque andstator current for speed set to0.8 pu and torque set to 0.2 pu

Fig. 7 Torque and statorcurrent responses for lockedrotor and for torque referencestep excitation from 0.1 to0.2 pu

30

exhibit higher speed and torque distortions at near-zerospeed operation. This improvement is accomplished byintroducing a torque control algorithm that directlycalculates the value of flux velocity xe, contrary to thestrategies in [4, 5, 6], which use the estimated value of xe.This improvement enables estimated torque values tomatch the real torque values in a wide range of operatingspeeds.

7 Conclusion

A sensorless IM drive based on stator flux vector controlis presented. The SFVC algorithm is derived from thebasic DTC strategies but has the advantage of operationwith fixed switching frequency and low ripple of torqueand flux. The improved stator flux estimation, based on

Fig. 8 Torque and statorcurrent responses for speed setto 0.2 pu and for torquereference step excitation from0.2 to 0.4 pu

Fig. 9 Torque and statorcurrent responses for speed setto 0.4 pu and for torquereference step excitation from0.2 to 0.4 pu

31

a closed-loop flux observer, enabled stable operation ofthe drive under a whole range of operating speeds andloads. The linear stator flux controller with a reduced setof control parameters enabled fast stator flux responseunder a wide range of operating conditions. The IMtorque is controlled by variations of flux angularvelocity, using linear PI control strategy, enabling fasttorque control with zero steady-state error and lowsteady-state ripple. When compared to the existingSFVC algorithms, the proposed control strategy intro-duces several improvements: the drive operation relieson the direct calculation of the flux angular velocity,which contributes to the increase of the overall stability;the control algorithm does not require knowledge of theexact IM parameter values to calculate precise commandsignals; the drive operates with decreased levels of tor-que distortions at low speeds; and the control strategyenables faster torque dynamics when compared toexisting SFVC drives.

Several experimental tests were carried out to verifythe drive performance in the steady-state and transientoperating conditions. The presented experimental resultsshowed that the fast torque response of the basic DTCtechnique is preserved. They also prove that the imple-mentation of SFVC reduces the torque and flux ripple,typical for the basic DTC strategies.

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Fig. 10 Estimated torque andactual rotor speed for torquereversal

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