Torque ripple minimisation control method for a four-phase brushless DC motor with non-ideal back-electromotive force

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Published in IET Electric Power ApplicationsReceived on 29th August 2012Revised on 31st January 2013Accepted on 10th March 2013doi: 10.1049/iet-epa.2012.0269

60The Institution of Engineering and Technology 2013

ISSN 1751-8660

Torque ripple minimisation control method for a four-phase brushless DC motor with non-ideal back-electromotive forceSeyed Mohammad Shakouhi1, Mustafa Mohamadian1, Ebrahim Afjei2

1Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran2Department of Electrical and Computer Engineering, University of Shahid Beheshti, Tehran, Iran

E-mail: [email protected]

Abstract: In conventional control methods of brushless DC (BLDC) motor drives, back-electromotive force (EMF) is assumed tobe in ideal form and the controller injects rectangular phase current commands to produce the desired constant torque. However,real back-EMF waveform might not be exactly trapezoidal because of non-ideality of magnetic material, design considerationsand manufacturing limitations. This makes the generated electromagnetic torque contain ripples in its waveform which is notdesirable in motor operation performance especially, in sensitive industries. Moreover, commutation states affect the qualityof generated torque by producing torque pulsations because of changes of conducting phases. In this study a control strategyfor a four-phase BLDC motor with non-ideal back-EMF to reduce electromagnetic torque ripples is presented. Basis of theproposed method is to inject phase currents considering back-EMF instantaneous magnitude. For this purpose, an on-lineback-EMF estimation technique is used to inject appropriate phase currents to compensate non-ideality of back-EMFwaveform. Moreover, the estimated back-EMF is also used for commutation torque ripple reduction. The experimental resultsindicate performance of the proposed control strategy in torque pulsations reduction compared with conventional control method.

1 Introduction

Brushless DC (BLDC) motors are widely used in industriessuch as appliances, automotive, aerospace, consumer,medical, industrial automation equipment and instrumentation[1–4]. BLDC motors have many advantages. A few of theseare: better speed against torque characteristics, high dynamicresponse, high efficiency, long operating life, noiselessoperation and higher speed ranges [5–8].Owing to wide applications of BLDC motors in different

industries, performance of these motors seems to be quiteimportant. It is quite possible that back-electromotive force(EMF) waveform of a BLDC motor is not exactlytrapezoidal because of non-ideality of magnetic material,design considerations and manufacturing limitations. Thismakes the generated electromagnetic torque contain ripplesin its waveform.In sensitive industries where electromagnetic torque with

minimum ripple is required, application of conventionalcontrol method that does not take non-ideality of back-EMFwaveform into account, is not efficient. This is because ofinjecting similar rectangular current commands to statorphases without considering non-uniformity of back-EMFwaveform which may lead to considerable electromagnetictorque ripples. Hence, application of a control strategywhich minimises torque pulsations is essential.In recent years, several control methods for reducing torque

ripples of BLDC motors with non-ideal back-EMF, have been

presented. In [9, 10], harmonic injection methods to minimisetorque ripples, because of back EMF harmonics, werepresented. These methods ignore higher-order Fourier seriesterms because of complexity and time-consumingcalculation. Moreover, harmonics calculations, complicatesthe real-time implementation.In [11], torque control of multiphase brushless motors

based on inequality constraints via Kuhn–Tucker theorem ispresented which leads to copper loss and torque ripplesreduction. However, in this method feedback sensors suchas high resolution encoder and torque transducer are required.In [12, 13], direct torque and indirect flux control of BLDC

motor with non-sinusoidal back EMF has been investigated.However, in this method both Clarke and Parktransformations are used and then flux and torque areestimated. These transformations and estimations affect theaccuracy of the method and would be time consuming.In [14], instantaneous torque control of small inductance

BLDC motor was investigated, where, instantaneous torqueand back-EMF coefficient were estimated using neuralnetwork fitting. However, using neural network analysis ispractically time consuming and back-EMF constant doesnot fully express the real back-EMF waveform characteristics.In [15–19], torque ripple reduction in BLDC motor with

non-ideal back-EMF was investigated. Off-line measuringof back-EMF waveform in different speeds for one or allphases, back-EMF pre-stored shape functions andback-EMF waveform approximation with limited harmonic

IET Electr. Power Appl., 2013, Vol. 7, Iss. 5, pp. 360–368doi: 10.1049/iet-epa.2012.0269

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orders (less than fifth harmonic) have been used in thesepapers. In [20], the duty cycle of pulses used to operate theinverter gates, is calculated in the torque controller.However, back-EMF calculations are not discussed in thepaper. Moreover, it is assumed that the motor armatureresistance is relatively small and its affect is neglected.In this paper an algorithm for reduction of electromagnetic
torque pulsation of a BLDC motor with non-ideal back-EMFwaveform is proposed. Basis of this concept is phase-to-phaseback-EMF estimation. The control is performed in stationaryreference frame and no d–q or other transformation is applied.Torque ripple minimisation is achieved by injecting statorcurrents, considering the instantaneous estimated back-EMFmagnitudes. As an example when the back-EMF magnitudeincreases compared with previous estimated value, theinjected current magnitude reduces and vice versa.Moreover, the estimated back-EMF is also used forcommutation torque ripple reduction which is based onenergising all phases at a specific time before the end ofeach conduction interval with proper commands. Thisstrategy reduces instantaneous increase or decrease ofelectromagnetic torque values in commutation states.Thereupon, torque ripples will be reduced and its waveformbecomes smoother. Experimental results are presented toillustrate the validity and performance of the proposedcontrol strategy.

2 Phase-to-phase back-EMF estimation

In most BLDC motors, back-EMF is not ideal trapezoidal andhas non-uniformity in its waveform shape. This can affect theperformance of operation such as generating considerableripples in its electromagnetic torque waveform.Hence, obtaining actual waveform of back-EMF provides

the controller the necessary information to inject appropriatecurrents to reduce torque ripples. For this purpose, in thissection a phase-to-phase back-EMF estimation method ispresented.In a four-phase, four-pole BLDC motor, in each 45° of

rotation, two phases are conducting (eight conductionsections in one motor rotation). This means that phases ‘a’and ‘c’ which have 180 electrical degrees phase difference,conduct in four mechanical 45° sections in one motorrotation. In two of these sections, phase ‘a’ carries thepositive current (connected to positive voltage) and phase‘c’ carries the negative current (connected to negative

Fig. 1 Equivalent circuit of a four-phase BLDC motor and the inverter


voltage). In the other two sections, phase ‘c’ carries thepositive current and phase ‘a’ carries the negative current.This is also valid for the other phases ‘b’ and ‘d’, whichconduct in the other four mechanical 45° sections ofrotation. In Fig. 1, a four-phase BLDC motor equivalentcircuit and the current path related to a 45° of rotationduring which phase ‘a’ (connected to positive voltage) andphase ‘c’ (connected to negative voltage) are conducting isshown. Phase voltages are given as

Van = Ria + Ldia/dt + ea (1)

Vcn = Ric + Ldic/dt + ec (2)

In above equations, stator winding resistances and phaseinductances are assumed to be equal. On the other hand,DC bus voltage and phase currents are as

Vdc = Van − Vcn (3)

ia = −ic (4)

From (1–4) Vdc is determined as

Vdc = 2Ria + 2Ldia/dt + ea − ec( )

(5)

and

ea − ec = Vdc − 2Ria − 2Ldia/dt (6)

Using (6), phase-to-phase induced voltage ea–ec can beestimated. Using the same algorithm in the other 45°intervals, eb− ed can be estimated.It should be noted that for this control strategy, only

positive and negative peak portions of phase-to-phaseback-EMFs (top and bottom sections of back-EMFwaveform, which participate in producing electromagnetictorque) are required and the rest of the waveform is not ofinterest.The derivative term in (6) may affect the accuracy of

back-EMF estimation because of presence of measurementnoise in stator currents. To tackle this problem, (6) can beintegrated to make the waveforms smoother because oflow-pass nature of integration. This yields the following

361& The Institution of Engineering and Technology 2013

Fig. 2 Phase currents in commutation intervals

a Conventional commutationb Proposed method commutation

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expression
∫ea − ec( )

dt =∫Vdcdt −

∫2Riadt −

∫2L dia/dt

( )dt (7)

Integration of back-EMFs results in corresponding fluxlinkages, so the above equation can be written as

la − lc =∫Vdc dt − 2R

∫ia dt − 2Lia + lo (8)

where λ0 is flux linkage initial value in each mechanical 45°interval. It should be noted that in every sampling time, DCsupply voltage and conducting phases current are sampled.The discrete integral form of (8) is calculated as

la[n]− lc[n] =∑Nn=1

1

2Vdc[n− 1]+ Vdc[n]( )

tsamp

− 2R∑Nn=1

1

2ia[n− 1]+ ia[n]( )

tsamp

− 2Lia[n]+ l0 (9)

where tsamp is the A/D converter sampling time period and Nis the number of samples obtained in the related mechanical45° interval of conduction.Instantaneous phase-to-phase back-EMF is the slope of

instantaneous flux linkage. Since the integration is onlyevaluated in the related 45° of phase ‘a’ and ‘b’ conductioninterval, the value of λ0 is not known. However, this valueis not needed, because we are only interested in slope offlux linkage, which leads to elimination of term λ0. Theslope of flux linkage is determined as

ea(t)− ec(t) =[(la(t)− lc(t)+ lo) − (la(t − tsamp)

− lc(t − tsamp) + lo)]/tsamp (10)

This equation can be rewritten to obtain instantaneousphase-to-phase back-EMF value

ea(t)− ec(t)

= la(t)− lc(t) la t − tsamp

( )− lc t − tsamp

( )( )[ ]/tsamp

(11)

3 Proposed control strategy

To reduce torque pulsations, reference currents should begenerated considering back-EMFs. Electromagnetic torquecan be expressed as a function of phase currents andback-EMFs

Te = eaia + ebib + ecic + edid( )

/vm (12)

Assuming an interval that phases ‘a’ and ‘c’ are conducting.Electromagnetic torque and phase currents are as

Te = eaia + ecic( )

/vm (13)

ia = −ic (14)


So

Tevm = ia ea − ec( )

(15)

Considering electromagnetic torque command as Te*, from(15) current reference commands are obtained

i∗a = −i∗c = vmT∗e / ea − ec( )

(16)

In (11), phase-to-phase back-EMF is estimated as describedin Section 2, and rotor mechanical speed is measured usingincremental encoder. Hence, in every 45° of rotation,reference currents of conducting phases can be calculated.In this method, it is not necessary to know each phaseback-EMF individually. This is an advantage, since theneutral point of stator windings is not always accessible.In this paper, phase-to-phase back-EMF estimation is also

used for torque ripple reduction in commutation intervals. Atthe beginning of each conduction interval, commutationtorque ripples are generated because of change ofconducting phases and the difference between current falltime of out-going phases and current rise time of in-comingphases. If phase currents change linearly with the sameslope during commutation period, by simultaneousswitching of both pairs of phases ‘a/c’ and ‘b/d’,commutation torque ripples would be reduced considerably.However, the rate of phase current changes is exponentialduring commutation intervals and rise and fall timedurations are not equal as shown in Fig. 2a. This can beexplained by means of equivalent circuits of conductingphases during commutation period.


Fig. 3 Equivalent circuit during commutation period

a Phases ‘b’ and ‘d’ switching onb Phases ‘a’ and ‘c’ switching off

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Assume a conduction interval in which, phases ‘a’ and ‘c’are conducting and the in-coming and out-going phases riseand fall time durations are measured. Fig. 3a shows phase‘b’ and ‘d’ (in-coming phases) equivalent circuit and risetime current path during commutation interval. It can beseen that phase-to-phase back-EMF voltage polarity isagainst the DC voltage supply polarity. However, for theout-going phases ‘a’ and ‘c’, as shown in Fig. 3b, duringcommutation interval, current closes its path throughfreewheeling diodes in the inverter circuit. This makes theDC supply appear in reverse polarity in equivalent circuit ofFig. 3b. Hence, the rise time current increases with a slowerrate compared with fall time current rate. In [21],commutation torque ripple of a BLDC motor was reducedby applying direct model adaptive control. In this methodreference currents are forced to have equal rising and fallingdurations, which are adjusted adaptively based on operatingpoint with the minimum torque ripple.In [22, 23], methods for commutation torque ripple

reduction are presented. However, in these papers,back-EMF values are assumed to be constant or obtainedby simple approximations in the commutation intervals.Hence, actual back-EMF values in commutation interval arenot considered which limits accuracy of these methods.In this paper, for reduction of commutation torque ripples,

the in-coming phases are energised before the out-goingphases are switched off. In this time interval, all phases areconducting. During this short time interval the in-comingphase current increases from zero. Hence, the in-comingphase current initial value at the beginning of the maincommutation interval is not zero anymore. This providesthe current controller enough time to inject the rightamounts of current considering instantaneous currentmagnitudes of out-going phases in their fall time.This time interval (t0) is selected considering out-going

phase current fall time and in-coming phase current risetime durations. By measuring each phase current rise andfall time durations after a change in speed or motor load,the difference can be determined as

t0 = tr − tf (17)

Where tr and tf are phase current rise and fall time durations,


respectively. During commutation interval, we have

Te = ((ea − ec)ifa + (eb − ed)irb)/vm (18)

Where ifa and irb are instantaneous phase ‘a’ current in its falltime and phase ‘b’ current in its rise time, respectively.Assuming equal values of phase-to-phase back-EMFs forboth pairs of phases in this short-time interval, because ofsymmetry of back-EMF waveforms, we obtain

ea − ec = eb − ed (19)

and

Tevm = ea − ec( )

ifa + irb( )

(20)

For reducing commutation torque ripples in this interval(beginning from t0 to the end of commutation interval),phases ‘b’ and ‘d’ current reference commands are obtainedfrom (20). Hence, irb is controlled considering instantaneousmagnitude of ifa

i∗rb = −i∗rd = vmT∗e / ea − ec( )[ ]− ifa (21)

During fall time interval, the controller should generatecurrent commands considering measured instantaneouscurrent value of out-going phases as expressed in (21). Thismakes commutation torque ripples reduce notably becauseof balance between in-coming and out-going phasescurrents as shown in Fig. 2b. The precision of this methodcan be improved by increasing fall and rise time durationmeasurement sampling rate.Another alternative is to adjust the out-going phase current

based on the in-coming phase current. Unlike theconventional method where the out-going phase IGBTs areturned off instantly, in this alternative, the out-going phasecurrent is reduced with the switching of the IGBTs withfollowing current command

i∗fa = −i∗fc = vmT∗e / ea − ec( )[ ]− irb (22)

Since the rate of current, decreasing in out-going phases isfaster than the rate of current increasing, in the in-comingphases, there is no need to energise any of the phases earlier.The proposed control method can be applied to the

three-phase BLDC motor. The main difference is the periodof continuous conduction for each phase, which is increasedfrom 90 to 120 electrical degrees.Moreover, the proposed control method can be applied to

BLDC motors which use hall sensors to generatecommutation signals. The flux linkage and back-EMFestimations are the same for both cases. Motor speed canalso be estimated by hall sensor signals. However, thecommutation torque ripple minimisation can beimplemented by early energising the incoming phase, withapproximation. Assuming almost constant motor speed, thetime between the two hall sensor edges contains N tsamp,where tsamp is the microprocessor sampling time. Since rise/fall time durations are measured with tsamp unit, thedifference (t0) is obtained in tsamp unit. In order to energisethe incoming phase t0 = k tsamp seconds earlier, theincoming phase has to be energised (N–k) tsamp after eachhall sensor edge.Motor field weakening operation to achieve higher motor

speeds can be implemented by increasing the phase angle


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by which the current leads the back-EMF voltage, calledphase-advanced drive method. This allows fast current risebefore the occurrence of the back-EMF peak portions. Inthe proposed control strategy, regardless of phase angleposition at which current is injected, phase-to-phaseback-EMF can be estimated. This enables the controller toinject the required current level considering phase-to-phaseback-EMF and hence achieve reduced torque ripples athigher speed in field weakening region.Block diagram of the proposed control method is shown in
Fig. 4. Control system consists of speed controller and currentcontroller blocks. Speed controller compares the referencespeed with actual speed feedback. The error signal goesthrough a proportional integral (PI) block to generate torquereference signal. The torque reference is the input to thecurrent reference generator block. On the other hand, usingphase currents (by two current sensors) and DC voltage busfeedbacks, phase-to-phase flux linkages and thenphase-to-phase back-EMFs are obtained and passed tocurrent reference generator block as inputs. Moreover, forreducing commutation torque pulsations, phase current fall/rise time durations are measured and passed to currentreference generator. In reference generator block, using (16)in normal states and (21) in commutation states, referencecurrents are generated and passed to current controllerblock. In this block, each phase reference current iscompared with the phase actual current and using hysteresisband control, inverter gate signals are generated.The main difference between the proposed control scheme

and the conventional control method is their current referencegeneration blocks. In conventional method torque reference isdivided by a coefficient (torque coefficient) to producecurrent commands. This is effective when the back-EMFsare trapezoidal. When back-EMF waveforms are not idealor trapezoidal, using conventional method, this non-idealityleads to considerable torque ripples which affectsperformance of the system. In the proposed method, currentcommands are generated proportional to instantaneousestimated values of back-EMFs. So, non-ideality in

Fig. 4 Proposed control strategy block diagram


back-EMF waveform is compensated by applying propercontroller current commands. Moreover, by energising newconducting phases at an appropriate time with suitablecommands during commutation states, commutation torquepulsations are reduced considerably.

4 Experimental results

In this section a four-pole, four-phase, 40 W, BLDC motor isused for experimental tests with phase resistance of 4 Ω andphase inductance of 7.5 mH with non-ideal back-EMF.Since back-EMF is not trapezoidal and has a sinusoidalshape, this makes the motor suitable to be studied fornon-ideal back-EMF effects on motor operation.Motor starts up with 1800 rpm speed command, 0.11 N.m

load torque (low load) using conventional control method.Fig. 5a shows the measured motor current and generatedelectromagnetic torque of the motor.As seen in Fig. 5a, the motor current has a rectangular shape

in each conducting interval because of constant currentcommand. However, non-ideality of back-EMF waveformwhich is shown in Fig. 6a (measured phase-to-phaseback-EMF), is not considered. Since back-EMF is nottrapezoidal and has a sine waveform, injection of rectangularcurrent generates electromagnetic torque cups. Observingelectromagnetic torque waveform carefully, one can notethat the waveform consists of high-frequency ripples becauseof inverter switching and low-frequency sigmoid portionsbecause of non-uniformity of back-EMF.Now the proposed control strategy is applied to the motor.

For this purpose, BLDC motor starts up with the same load asin conventional control test. The basis of this method isinjection of phase currents, considering instantaneousphase-to-phase back-EMF values. For instantaneousphase-to-phase back-EMF estimation, phase currents andDC bus voltage information are sampled via two currentsensors and a voltage sensor by digital signal processor(DSP) board interface.


Fig. 5 Current and electromagnetic torque waveforms (referencetorque 0.11 N.m)

a Conventional control methodb Proposed control method

Fig. 6 Non-ideality of back-EMF waveform and estimated phase‘a’ to phase ‘c’a Measured phase-to-phase back-EMF waveform (In order to demonstratemotor back-EMF waveform, inverter gate signals are turned off for onecycle.)b Estimated phase-to-phase flux linkage and back-EMF

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Fig. 6b illustrates estimated phase ‘a’ to phase ‘c’ fluxlinkage and back-EMF in peak portions where phasecurrents are injected. As seen in Fig. 6b, estimatedback-EMF is similar to the measured motor phase-to-phaseback-EMF in Fig. 6a (indicated by circles). Flux linkage isestimated only in peak portions of back-EMF waveformwhere phase currents are injected. One can observe thatinitial value of flux linkage in each interval of estimation isconsidered as zero. This does not affect the estimation,because initial value only shifts the flux curve in vertical upor down direction, which will be cancelled as explainedearlier in (11).In conventional control algorithm, current command is

generated by dividing torque reference by a coefficient(torque coefficient). However, the proposed control methoduses estimated back-EMF to produce current commands.This is implemented by dividing electromagnetic torquereference by instantaneous estimated back-EMF considering


the mechanical rotor speed. Fig. 5b shows measured motorcurrent and generated electromagnetic torque resulted fromthe proposed method.As can be seen in Fig. 5b, the current waveform is not

rectangular as in Fig. 5a. This is because of the shape ofback-EMF shown in Fig. 6. The controller generates phasecurrent commands to compensate back-EMF curvenon-ideality. So, convex form of phase-to-phase back-EMFvoltage makes the concave shape of motor current.One can observe the electromagnetic torque waveform

resulted from proposed method (Fig. 5b) and compare itwith the electromagnetic torque resulted from conventionalcontrol (Fig. 5a). Although, average torque still remains thesame, quality of generated electromagnetic torque becomesmuch better compared with electromagnetic torquewaveform of conventional method. Unlike conventionalmethod, sigmoid portions cannot be seen in electromagnetictorque waveform generated by the proposed control methodanymore. Moreover, commutation torque ripples arereduced considerably because of proposed commutationtime interval control strategy.Now, performance of the proposed method in full load

condition is investigated. First, for comparison, theconventional control method is applied. The applied loadtorque is 0.22 Nm. Fig. 7a shows motor current andgenerated electromagnetic torque waveforms.It can be obviously seen that electromagnetic torque,

contains significant ripples because of non-ideality of motorback-EMF waveform and commutation effects.Now, the proposed control method is applied to the motor

with the same load and results are depicted in Fig. 7b. Onecan see that the current waveform follows the changes inback-EMF magnitude. At the beginning and end of current


Fig. 7 Current and electromagnetic torque waveforms (referencetorque 0.22 N.m)

a Conventional control methodb Proposed control method

Fig. 8 Current and electromagnetic torque waveforms of theproposed method (reference torque 0.11 N.m, 300 rpm)

Fig. 9 Electromagnetic torque dynamic response

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pulses, current magnitudes are larger because ofcorresponding lower values of back-EMF. Moreover,commutation torque ripples are reduced significantly andthe resultant electromagnetic torque waveform is quite moreuniform compared with electromagnetic torque waveformshown in Fig. 7a.

Fig. 8 shows motor current and electromagnetic torqueobtained by applying the proposed control method at lowerspeed (300 rpm). It can be seen that non-ideal back-EMFeffects and commutation torque ripples are reducedconsiderably which shows the proposed control strategyperformance in low-speed operation.To evaluate the dynamic response of the proposed control

strategy, load torque is changed from 0.09 Nm to 0.12 Nm(35% increase in load torque) without change in motorspeed (1600 rpm). As can be seen in Fig. 9, theelectromagnetic torque follows the demanded torque. Onecan see that it only lasts 17 ms for the controller to followthe reference torque which is less than the time required forone revolution of the rotor in this motor speed.


Fig. 10 Actual and estimated phase-to-phase back-EMF waveform

a 30% phase resistance increaseb 50% phase inductance decrease

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5 Sensitivity to motor parameter variations

In this section the proposed method sensitivity to motorresistance and inductance variations is investigated. For thispurpose, effects of phase resistance and phase inductancechanges on the controller back-EMF estimationperformance (main block of the proposed method) arediscussed.Motor internal temperature may vary during continuous

operation. This may affect the value of phase resistance. Toshow the influence of this phenomenon, simulation resultsof a BLDC motor which its phase resistances haveincreased by 30%, is shown in Fig. 10a. Simulation resultsindicate that the difference in the waveforms of actualphase-to-phase back-EMF and its estimation is < 2.5% inthis case. It should be considered that in the proposedcontrol strategy, back-EMF is not estimated continuouslyover the whole operation time. It is only carried out whencurrent is injected into the phase. This prevents estimationerror from propagating over operation time.Now, the effect of phase inductance because of saturation is

investigated. The effect of this error is shown in Fig. 10b,where phase inductances have decreased by 50% in motorhighly saturated condition. The main difference between thewaveforms of actual phase-to-phase back-EMF and itsestimation occurs at the beginning and end of eachestimation interval (∼10%). So, the proposed controlstrategy is more sensitive to motor phase inductancevariations compared with phase resistance changes. In the


worst case (50% phase inductance change) the controllerestimation error does not exceed 10%.

6 Conclusions

This paper presents a new control method for electromagnetictorque ripple reduction for a BLDC motor with non-idealback-EMF waveform. The concept is based onphase-to-phase back-EMF estimation. In conventionalcontrol method, phase current commands are generated bydividing reference torque by a coefficient. However, in theproposed method, the desired torque is generated by currentcommands which are proportional to instantaneousback-EMF magnitudes of conducting phases. This meansthat when the back-EMF value increases, the injectedcurrent magnitude decreases and vice versa. Moreover, theestimated back-EMF is also used for commutation torqueripple reduction which is based on energising all phases ata specific time before the end of each conduction intervaland with suitable commands. This strategy reducesinstantaneous increase or decrease of electromagnetic torquevalues in commutation states. Thereupon, torque rippleswill be reduced and its waveform becomes smoother.Experimental results were presented to illustrate validityand performance of the proposed control strategy.

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Torque ripple minimisation control method for a four-phase brushless DC motor with non-ideal back-electromotive force