Direct Instantaneous Torque Control of Switched Reluctance Motors

16 IE(I) Journal – EL

Direct Instantaneous Torque Control of SwitchedReluctance Motors

K V Reddy, Associate Member

M V Kumar, Fellow

The main disadvantage of switched reluctance motors (SRM) is the higher torque ripple compared toconventional machines, which results in acoustic noise and vibration. The torque pulsations in an SRM aredue to the highly non-linear and discrete nature of torque production mechanism. This paper proposes theconcept of direct instantaneous torque control (DITC) for SRM to overcome the drawbacks of the indirectinstantaneous torque control (IITC) like complexity in torque-to-current conversion and the requirement oflarge amount of memory space. In DITC, the instantaneous torque, estimated from the motor terminalquantities, is considered directly as a control variable. Hence, the torque-to-current conversion and currentcontrollers are no longer required. Therefore, DITC is expected to counteract the torque error instantaneouslywith fast dynamic response and effectively minimize the inherent torque ripple.

Keywords : Direct instantaneous torque control; Torque ripple minimization, Switched reluctance motor; Torque sharing functions;DTC

INTRODUCTION

The switched reluctance motor (SRM) is a promisingcandidate for various adjustable speed drives in industrialand consumer product applications. The primarydisadvantage of a SRM is the higher torque ripple comparedto conventional machines, which results in acoustic noiseand vibration. The torque pulsations in a SRM are due tothe highly non-linear and discrete nature of torqueproduction mechanism. Therefore, without proper control,the inherent torque ripples, vibrations and acoustic noisecan become major problems of the SRM drives. With theview to achieve high-performance servo drives for SRM,several instantaneous torque control techniques includingtorque ripple minimization features have been successivelyproposed in the last three decades1.

In the conventional scheme called indirect instantaneoustorque control (IITC), SRM torque is regulated by controllingthe instantaneous phase currents in a cascaded fashion.The reference torque is converted to equivalent referencephase currents so that they can be tracked in the innercurrent control loops. By using the phase current profilingtechnique, optimal phase torques corresponding to torquesharing functions can be generated and torque ripples canbe minimized. Various indirect instantaneous torque control(IITC) techniques for the minimizations of the torque ripplein SRM were surveyed by Husain1. However, torque-to-current conversion in SRM is complex and becomes non-

trivial due to their nonlinear relationship. Analyticalexpression of such conversion is complicated and leads tointensive on-line computation. On the other hand, thecurrent profiles are to be pre-calculated and pre-stored inthe controller memory. Hence, this method also requireslarge amount of on-line memory space.

This paper proposes the concept of direct instantaneoustorque control (DITC) for SRM to overcome the earliermentioned drawbacks of the indirect instantaneous torquecontrol (IITC). In DITC, the instantaneous torque isestimated from the motor terminal quantities and isconsidered directly as a control variable. Hence, the torque-to-current conversion and current controllers are no longerrequired. Therefore, DITC is expected to counteract thetorque error instantaneously with fast dynamic responseand effectively minimize the inherent torque ripple. A DITCscheme for SRM using the concept of a short flux patternthat links two separate poles of the stator was proposed byJinupun and Luk2. This method was expensive andinconvenient, as it involved motor winding alteration andbipolar current requirement. Cheok and Fukuda3 proposeda DITC strategy for a three-phase SRM, which followedthe conventional DTC technique for three-phase acmachines. In this scheme, torque is directly controlledthrough the control of the magnitude of the flux linkage andthe change in speed (acceleration or deceleration) of thestator flux vector. However, the technique requires additionalflux linkage controller.

The main aim of this paper is to present a simplified directinstantaneous torque control for a four phase SRM. Here,torque sharing functions are used to generate the referencevalues of phase torques from the required torque. Theswitching signals are directly generated from thecomparison between the reference phase torques and theestimated phase torques using hysteresis controller. The

K V Reddy is with the Department of Electrical and ElectronicsEngineering, University College of Engineering, JNTU Kakinada,Kakinada 533 003, Andhra Pradesh; and M V Kumar is with theDepartment of Electrical Engineering, Jawaharlal NehruTechnological University Anantapur, Anantapur 515 002, AndhraPradesh.

This paper (modified) was received on April 05, 2011. Written discussionon the paper will be entertained till September 30, 2011.

Volume 92, June 2011 17

hysteresis torque controller generates gating signals to thepower converter. The instantaneous phase torques can becalculated from the phase currents and rotor position.Simulations were carried out on four phase 8/6 SRM usingMatlab/Simulink. Simulation results really validate theapplicability of the proposed direct instantaneous torquecontrol scheme.

INDUCTANCE MODEL OF SRM

Since the phase inductance of SRM changes periodicallywith the rotor position, it can be expressed as a Fourierseries with respect to rotor position θ4,5

( ) ( )0

, cosm

k rk

L i L i kN=

θ = θ∑ (1)

where Nr is the number of rotor poles.

A three-dimensional (3-D) plot of inductance shown inFigure 1 depicts the profile of inductance variation with rotorposition and phase current. At 0o and 60o, phase A is at itsaligned positions and has the highest value of inductance.It decreases when the phase current increases. At 30o,phase A is at its unaligned position and has lowest value ofinductance. The inductance at unaligned position does notchange much with the phase current and can be treated asa constant. The inductance at midway and aligned positiondecrease when current increases due to saturation.

The torque equation of SRM is given by5

( )( )

( ) ( )( )

,const 0

1 0

,di

sin di

ico

e ji

im

r r kk

iWT

kN kN L i i

=

=

∂∂= =∂∂

= −

∫

∑ ∫

λ θ

θθ

θ (2)

The Figure 2 shows the torque variation with the rotorangular position at different values of phase current. Sincethe inductance gradient is negative from 0o to 30o the torqueis negative and when the inductance gradient is positivefrom 30o to 60o the torque is positive.

The torque-speed equation can be expressed as

d

dt e lJ T T= −ω

(3)

where J is the moment of inertia of rotor; Te, theelectromagnetic torque; and Tl, the load torque. Theelectromagnetic torque, Te can be computed as follows,

,1

n

e e jj

T T=

= ∑ (4)

where Te, j is the electromagnetic torque of the jth phaseand n, the number of phases.

TORQUE SHARING FUNCTIONS

At any time, the resultant output torque of SRM is thesummation of the torque in all phases. If the phase currentis fixed, the torque of an 8/6 SRM will have a profile asshown in Figure 3.

From Figure 3, it is clear that high torque is not availablenear aligned/unaligned position even when high phasecurrent is presented. To generate a ripple-free output torque,there must be overlapping between phases. During phaseoverlapping, the current in one phase is decreasing, andthat in the other phase is increasing. To obtain a constanttorque, the summation of the torque generated by thesecurrents must be equal to the torque generated in non-overlapping period. To determine the desired torqueproduced by each phase, torque sharing functions (TSFs)Figure 1 Profile of nonlinear phase inductance

3020

100Current, A 0

2040

60

θ, degree

Ind

uct

anc

e,

H

6

5

4

3

2

1

0

× 10–3

Figure 2 Torque at different currents and different rotor positions

3020

100

Current, A0

2040 60

θ, degree

Torq

ue,

N-m

2

1.5

1

0.5

0

– 0.5

– 1

– 1.5

– 2


are introduced, which are defined as

ref1 1

TSF ( )N N

j jj j

T T T= =

= = θ∑ ∑ (5)

where TSFj(θ) is the torque sharing function for phase j atrotor position θ, and Tref is the reference torque.

The motor used in this work is a four phase 8/6 switchedreluctance motor. To generate desired torque, the torquesharing function must meet the following requirements:

4

1TSF ( ) 1j

j=θ =∑ (6)

TSF ( ) TSF3j jπ θ = θ +

(7)

TSF ( ) TSF ( )12j k j kπ θ = θ − −

(8)

For 8/6 SRM, the inductance increasing/decreasing periodfor each phase is π/6. In this work, the conduction angle ischosen as π/8. This means

off on 8

πθ − θ = (9)

So the phase overlapping for each two adjacent phases isπ/8 – π/12 = π/24.

In the proposed work, a sinusoidal torque sharing functionsare used. The sinusoidal torque sharing function meansthat the torque produced by the phases, during phase

commutation, changes with the rotor position in terms ofthe sinusoidal function. On the basis of the sinusoidal torquesharing function presented by Husain and Ehsani6, in thiswork, the improved sinusoidal torque sharing functions havebeen developed and used. The sinusoidal torque sharingfunction for phase j in a rotor period can be expressed as

on on on

on

off off of

1 1cos24( ) for

2 2 24

1 for24 24Fact ( )

1 1cos24 for

2 2 24 24

0 otherwise

j j j

j off jj

j j f j

π − θ−θ θ ≤θ< θ +

π π θ + ≤θ< θ − θ =

π π + θ−θ − θ − ≤θ<θ

(10)

The Figure 4 shows the sinusoidal torque sharing functionsof all four phases in forward motoring operation. In thisfigure, the dotted line represents the summation of all thefour torque sharing functions which is equal to 1 at anyrotor position.

DIRECT INSTANTANEOUS TORQUE CONTROL OFSRM

The block diagram of the direct instantaneous torque controlis shown in Figure 5. First, the phase torques are calculatedfrom the measured phase currents and rotor position usingthe torque expression [equation (2)]. The magnitudes ofthe reference phase torques are calculated using the torquesharing functions. The input reference phase torques arecompared with the feedback estimated phase torques usinghysteresis controller. The hysteresis controller outputs threediscrete voltage levels +Vdc, 0, –Vdc to be applied to themotor.

Sinusoidal torque sharing functions

Figure 4 Sinusoidal TSFs for forward motoring operation

– 30 – 20 – 10 0 10 20 30

θ, degree

1.2

1

0.8

0.6

0.4

0.2

0

– 0.2

A B C D

Figure 3 Phase torque profile under fixed current

– 30 – 20 – 10 0 10 20 30

θ

Torq

ue

, N

0.8

0.6

0.4

0.2

0

– 0.2

– 0.4

– 0.6

– 0.8


The justification of using applied phase voltage to directlycontrol the instantaneous torque of switched reluctancemotor is explained here.

The nonlinear instantaneous torque of SRM can be foundfrom co-energy principle, as expressed in equation (2)earlier. For simplification, the torque equation becomes

( ) ( ),

0

, ,di

i

e ji i

T i i∂λ θ ∂λ θ

≈ ≈∂θ ∂θ∫ (11)

Thus, the instantaneous torque of the saturated SRM canbe found from the product of the flux linkage derivative (withrespect to rotor position) and the phase current, as shownin equation (11).

The phase voltage equation of SRM is given by

d ( , )

dtd d

dt dtd

( , )dt

iV R i

iR i

ii

R i l i

λ θ= +

∂λ ∂λ θ= + +

∂ ∂θ∂λ

= + θ + ω∂θ

(12)

This incremental inductance l(i,θ) has significantly largevalue such that phase current can be assumed unchangedin one sampling period. Once the resistive drop is negligibleand the current is considered constant, the phase voltageequation can be approximated as equation (13)

V∂λ

≈ ω∂θ

(13)

Consequently, the instantaneous phase torque equation ofSRM can be simplified to:

iT i V

∂λ≈ ≈

∂θ ω(14)

The rotor speed and the phase current are assumedconstant during the control cycle. As a result, the torqueexpression has been linearized and the phase voltage Vbecomes an effective control variable for the DITC.

SIMULATION RESULTS

The proposed direct instantaneous torque control schemeis implemented and simulated in the Matlab / Simulinkenvironment for a four phase switched reluctance motorwith eight stator poles and six rotor poles. Figure 6 describesthe profile of the total torque produced by SRM withouttorque control for turn-on angle θon = 30o, turn-off angleθoff = 52.5o, constant speed N = 500 rpm and a currentreference of Iref = 12 A with hysteresis current control. FromFigure 6, it is clear that the total torque produced by theSRM has high amount of torque ripple. The magnitude ofthe torque is varying between 0.6 Nm and 0.9 Nm.

Here a torque ripple factor (TRF) is defined and used tomeasure the ripple of the torque produced by the motor.

Figure 6 Total torque produced by SRM without DITC

Total torque

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time, s

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Torq

ue,

N-m

Figure 5 Block diagram of the direct instantaneous torque control

Torque sharingfunctions

Direct torquecontroller

Torque calculation from T(i,θ)equation

Power converter SRM

Positionencoder

Gatingsignals

Tref

TAref

TBref

TCref

TDref

iaibicid

TA TB TC TD

θ

θ

θon θoff


the maximum extent. With the implementation of theproposed direct instantaneous torque control, the TRF ofSRM is less than 2% which is acceptable.

CONCLUSIONS

This paper proposes a simplified direct instantaneous torquecontrol (DITC) for a four phase 8/6 SRM which overcomethe drawbacks of the indirect instantaneous torque control(IITC) schemes like complexity in torque-to-currentconversion and the requirement of massive memory spacefor storing the current values corresponding to torque andall position. In DITC, the phase torques are calculated fromthe measured phase currents and rotor position. Themagnitudes of the reference phase torques are calculatedusing the torque sharing functions. The input referencephase torques are compared with the feedback estimatedphase torques using hysteresis torque controller. Thehysteresis torque controller have three discrete voltagelevels outputs which are to be applied to the motor. As theinstantaneous torque is considered directly as a controlvariable, DITC counteracts the torque error instantaneously

Figure 10 Total torque produced by SRM in DITC

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time, s

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Torq

ue,

N-m

Actual Reference

Figure 8 Phase torques in DITC

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time, s

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Torq

ue,

N-m

Phase AA Phase B Phase CPhase D

Figure 9 Phase currents in DITC

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time, s

14

12

10

8

6

4

2

0

Torq

ue,

N-m

Phase AA Phase B Phase CPhase D

Figure 7 Reference phase torques in DITC

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time, s

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Torq

ue,

N-m

Phase AAPhase BPhase CPhase D

The torque ripple factor (TRF) is defined as follows.

avg

avg

RMS value of [ ( ) ]TRF,% 100

T t T

T

−= × (15)

Without the torque control the TRF of SRM is observed as10% to 15% which is not desirable for many applications.

The simulation results, shown in Figures 7 to 10, have beenachieved using the proposed direct instantaneous torquecontrol scheme for the same turn-on and turn-off anglesand at the same speed and with a step change in thereference torque. The reference torque is 0.4 N-m up to0.025 s and 0.6 N-m from 0.025 s to 0.05 s. Figure 7 showsthe reference phase torques of the SRM which are obtainedusing the torque sharing functions. Figure 8 depicts thetorques produced by each phase of the SRM. The phasecurrents of the motor are described in Figure 9. Figure 10illustrates the profile of the total torque produced by theSRM. In Figure 10, it can be observed that ripple in thetotal torque produced by the SRM has been minimized to


with fast dynamic response and effectively minimize theinherent torque ripple. The DITC scheme also eliminatesthe use of current controllers. With the implementation ofproposed direct instantaneous torque control (DITC)scheme, the torque ripple factor (TRF) of SRM is reducedto less than 2% which is tolerable. Simulation results reallyvalidate the applicability of the proposed directinstantaneous torque control scheme.

REFERENCES

1. I Husain. ‘Minimization of Torque Ripple in SRM Drives’. IEEETransactions on Industrial Electronics, vol 49, no 1, 2002, pp 28-39.

2. P Jinupun and P Chi-Kwong Luk. ‘Direct Torque Control for SensorlessSwitched Reluctance Motor Drives’. Proceedings of the SeventhInternational Conference on Power Electronics and Variable Speed

Drives, 1998, pp 329-334.

3. A D Cheok and Y Fukuda. ‘A New Torque and Flux Control Method forSwitched Reluctance Motor Drives’. IEEE Transactions on PowerElectronics, vol 17, no 4, 2002, pp 543-557.

4. B Fahimi, G Suresh, J Mahdavi and M Ehsani. ‘A New Approach toModel Switched Reluctance Motor Drive Application to DynamicPerformance Prediction, Control and Design’. PESC 98 Record 29thAnnual IEEE, vol 2, nos 17-22, May 1998, pp 2097- 2102.

5. K V Reddy and M V Kumar. ‘A Non-linear Model for SwitchedReluctance Motors’. Journal of The Institution of Engineers (India),Electrical Engineering Division, vol 89, no 4, March 2009, pp 1-8.

6. I Husain and M Ehsani. ‘Torque Ripple Minimization in SwitchedReluctance Drives by PWM Current Control’. IEEE Transactions onPower Electronics, vol 11, no 1, January 1996, pp 83-88.

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Direct Instantaneous Torque Control of Switched Reluctance Motors