Incorporating control trajectories with the direct torque control scheme of interior permanent magnet synchronous motor drive

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Published in IET Electric Power ApplicationsReceived on 29th December 2007Revised on 27th June 2008doi: 10.1049/iet-epa:20070518

ISSN 1751-8660

Incorporating control trajectories with thedirect torque control scheme of interiorpermanent magnet synchronous motor driveM.E. Haque1 M.F. Rahman2

1School of Engineering, University of Tasmania, GPO Box 252-65, Hobart, Tas 7001, Australia2School of Electrical Engineering and Telecommunication, The University of New South Wales, Sydney NSW 2052, AustraliaE-mail: [email protected]

Abstract: Interior permanent magnet (IPM) synchronous machines have gained increased attention forapplications in electric vehicle, variable speed wind turbine, industrial drives, etc., because of their hightorque density, wider speed range and compact construction. The authors present a detailed analysis andmodelling of control trajectories and incorporate those trajectories in the direct torque control (DTC) schemeof an IPM synchronous motor drive, for constant-torque and constant-power operating regions. The controltrajectories are implemented on a real-time digital signal processor. Because the inputs to the inner torquecontrol loop of DTC are the references for the torque and the amplitude of the stator flux linkage (ls), theyare transformed in the T–ls plane, than in the id – iq plane in the indirect control. The modelling andexperimental results are presented. Results show very good dynamic and steady-state performances of directtorque controller, incorporating these control trajectories.

1 IntroductionPermanent magnet synchronous motors (PMSMs) offerhigher torque/volume when compared with other motorsby exploiting their rotor saliency. The IPM synchronousmotor also offers the possibility of wider speed range,compared with their surface magnet version, because themagnets are securely embedded inside the rotor iron. Asubstantially increased speed range can be obtained by fluxweakening, thus allowing a constant power-like operationat speeds higher than the base speed level.

Since the last decade, development has occurred in twomain areas of permanent magnet (PM) motor drive: First,various PM motor configurations of suitable values of thestator d- and q-axes inductances (Ld, Lq and Lq/Ld ratio)and method of reducing cogging torque; second, variouscontrol strategies for the maximum torque-per-ampere(MTPA) characteristic and maximum flux-weakening rangehave been studied [1–7]. All these research works usedindirect torque and flux control strategies in which thesetwo quantities were regulated by current controllers in the

Electr. Power Appl., 2009, Vol. 3, Iss. 2, pp. 93–10110.1049/iet-epa:20070518

rotor dq-reference frame. A shaft encoder associatedcoordinate transformation and current control networks aremandatory in these schemes. The delays suffered throughthese networks by the control signals are known to beresponsible for similar delay in the torque responses.

With the appearance of high-speed digital signalprocessor, a control method called (DTC has becomepopular both in induction and PMSM drives [8–11].Direct torque controlled IPM synchronous motor drive is apotential candidate in high-performance and wide-speedapplications such as in electric vehicle. However, muchattention has not yet been paid to the trajectory controlwith DTC. A good number of papers on the trajectorycontrol of PMSM under PWM (pulse width modulation)current control in the constant torque (MTPA trajectory)and the field-weakening region has been reported in theliterature [1–7]. In [12, 13], DTC of induction motor inthe over-modulation and field-weakening region has beenpresented. DTC of PMSM with field weakening wasreported in [14, 15]. However, these works emphasised onhigh-speed region and the complete trajectory control from

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zero speed to field-weakening region has not beeninvestigated.

This paper presents a complete trajectory control (constanttorque operation and field-weakening operation) from zerospeed to field-weakening region under DTC usingKollmorgen industrial IPM motor. The analyses of controltrajectories are discussed in detail. The MTPA trajectory,current and voltage constraints are expressed in the T–ls

plane. These have been incorporated into the DTC scheme.To improve the low-speed performance of the DTCincorporating control trajectories, the compensationtechniques explained in [16] is used, which is not repeated here.

2 Direct torque control schemeof interior permanent magnetsynchronous motor driveThe main objective of this paper is to incorporate controltrajectories with the DTC scheme of IPM synchronousmotor. So, a basic DTC scheme is used in this paper insteadof the recent modified version of this control scheme. InDTC scheme, the stator flux linkage and the electromagnetictorque are controlled directly by applying voltage-switchingvectors of the inverter. It is a principal goal to select thosevoltage-switching vectors that yield the fastest electromagnetictorque response. The DTC scheme of IPM motor suppliedby a voltage source inverter is shown in Fig. 1. The three-phase variables are transformed into stationary dq-axesvariables. In Fig. 1, the electromagnetic torque error and thestator flux linkage errors are inputs to the respective fluxlinkage and torque hysteresis comparators. The discretisedoutputs of the hysteresis comparators (t, l) are inputs to thevoltage-switching selection look-up table. However, theinformation on the position of the stator flux-linkage space

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vector (region number) is also an input to the look-up table.The scheme also contains an estimator, which providesestimation of stator flux linkage and electromagnetic torque.Also note that all calculations are in the stator dq-referenceframe and no coordinate transformation is involved.

3 Control trajectories for thedirect torque control schemeof an interior permanent magnetsynchronous motor driveTrajectory control techniques are employed to produce thetorque or current references to achieve different goals inconstant torque and field-weakening ranges of operation.The control of the stator flux linkage should be based on thetorque, and according to the control trajectories (MTPA andfield-weakening trajectories). These control trajectories inid– iq plane can also be adopted in the DTC of the IPMsynchronous motor drives as shown in later sections. Becausethe inputs to the inner torque control loop are the referencesfor the torque and the amplitude of the stator flux linkage,the control trajectories in id– iq plane are transformedinto the torque and ls plane, rather than in id and iq. Basedon these trajectories in the new plane, ls is determined fromthe torque, which is the output of the speed controller, forconstant torque and field-weakening operations.

3.1 Maximum torque-per-amperetrajectory in torque–ls plane

The torque for IPM synchronous motor in terms of statorflux linkage is given by

T ¼3

2P ld id � lqiq

� �¼

3

2P[lf iq þ (Ld � Lq)id iq] (1)

Figure 1 DTC scheme of an IPM synchronous motor drive

IET Electr. Power Appl., 2009, Vol. 3, Iss. 2, pp. 93–101doi: 10.1049/iet-epa:20070518

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Figure 2 Torque and d as a function of the amplitude of stator flux linkage for the MTPA trajectory

The motor developed torque, in terms of the stator and rotorflux linkage amplitudes, is also given by

T (k) ¼3pl̂s(k)

4Ld Lq

[2lf Lq sin {d(k)}� l̂s(k)(Lq � Ld ) sin 2{d(k)}]

(2)The stator flux linkage and d are given by

ls

�� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

d þ l2q

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(Ld id þ lf )2

þ (Lqiq)2q

d ¼ tan�1Lqiq

Ld id þ lf

� �8>><>>: (3)

As the IPM motor has a saliency (Lq . Ld ) and the reluctancetorque Tr is available, the armature current vector is controlled


Table 1 Parameters of the IPM synchronous motor used

number of pole pairs (P) 2

stator resistance (R) 5.8 V

magnet flux linkage (lf) 0.377 Wb

d- and q-axes inductance (Ld, Lq) 0.0448 H and 0.1024 H

phase voltage and current(V and I)

132 V and 3 A

base speed (vb) 1260 rpm

crossover speed (vc) 1460 rpm

rated torque (Tb) 3.7 Nm

Figure 3 Control trajectories in torque–ls plane

a Torque against ls

b d against ls

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to produce the MTPA. The relationship between id and iq forthe MTPA can be derived as [2]

id ¼lf

2(Lq � Ld )�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

f

4(Lq � Ld )2þ i2

q

vuut (4)

Figure 4 Flowchart for the control mode selection

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If id and iq are controlled according to the MTPA trajectoryas shown in (4), then (3) is rewritten as

ls ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

f �L2

d

Lq � Ld

þ Lq � Ld

!lf id þ (L2

d þ L2q)i2

d

vuut

d ¼ tan�1Lq

lf þ Ld id

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2d �

lf

Lq � Ld

id

s !8>>>>><>>>>>:

(5)

It is possible to solve id from (4) and substitute it together with(3) into (1) to obtain the expressions for ls and d in terms oftorque. These expressions will be very complicated anddifficult to solve in real time. The relationships among ls, dand positive torque for the IPM motor can be found from theoff-line calculation as shown in Fig. 2 for the IPM motor inTable 1. It is seen from Fig. 2 that both the amplitude ls andangle d increase with the increase in torque. When the torqueis zero, the angle is zero and the stator flux is equal to themagnet flux. It is also seen from this figure that the load angled is always below dm with the MTPA control when thetorque is limited below the maximum torque that the motorcan produce. According to the torque equation (2), if two ofthe three variables, namely T, ls and d, are known, the thirdone is uniquely determined. Given that the torque is known,MTPA control is achieved if the amplitude or the angle ofthe stator flux is determined from the above figure, which canbe stored in a look-up table. For the DTC, it is obvious thatthe amplitude of the stator flux, rather than its angle, should

Figure 5 Direct torque control of IPM motor drive, incorporating control trajectories in T–ls plane


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Figure 6 Dynamic responses of the DTC drive, incorporating control trajectories under constant torque operation; simulationresults

a Speed responseb Torque responsec Locus of the stator fluxd Torque against ls

be controlled. When the torque and ls are controlled in thisway, the angle d will be automatically controlled and dm willnot be exceeded. This is the requirement for the applicationof DTC in IPM synchronous motor drives.

3.2 Current and voltage constraintsin torque–ls plane

The current and voltage constraints can be written as

id

�� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI 2

sm � i2q

q(6)

vd

�� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2

sm � v2q

q(7)


The current constraint is plotted in the torque–ls plane andd–ls plane as shown in Fig. 3 for the IPM motor of Table 1.It is seen from Fig. 3b that the current limit trajectory isalways below dm. If the stator resistance is neglected, thestator voltage is given by

Vs ¼ vsls (8)

where vs is the rotational speed of the stator flux linkage. Inthe steady state, the rotational speeds of the stator flux linkageand the rotor magnet flux linkage are the same and (8) can berewritten as

Vs ¼ vr ls ¼ vb lsr ¼ vc lf (9)

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Figure 7 Dynamic responses of the DTC drive, incorporating control trajectories under field-weakening operation; simulationresults


where vr, vb and vc are the rotor speed, base speed andcrossover speed, respectively. lsr is the rated stator fluxlinkage. Maximum voltage trajectories for a motor can bedetermined by each (id, iq) pair and a given speed, usingKirchhoff’s voltage equation. For simplicity, the maximumvoltage limit for each speed is indicated as a vertical line, asdefined in (9). The current limit and the MTPAtrajectories for the IPM motor of Table 1 are superimposedin Fig. 3. The current limit is satisfied if the torque and ls

are controlled below the current limit trajectory. Theintersection of the current limit and MTPA trajectories isPoint A, which corresponds to the operating point with themaximum torque and current. If the torque is limitedbelow the value at the operating point A for the MTPAcontrol, the current limit is always satisfied. For constanttorque operation, the amplitude of the stator flux isindependent of the speed and is only dependent on thetorque, whereas for field-weakening operation, it isindependent of the torque and is determined only by therotor speed.


3.3 Control mode selection for constanttorque and field-weakening operation

For operation below the base speed, constant torque controlshould be selected. For the operation above the crossoverspeed, field-weakening control is undoubtedly selectedbecause the voltage limit will no longer be satisfied if thetorque and ls are controlled along the MTPA trajectory.However, for the operation between the base speed andcrossover speed, the torque determines the control mode.With the MTPA control, for instance, if the verticaldashed line in Fig. 3 represents the voltage limitcorresponding to the operation with the rotor speedbetween vb and vc, there is an intersection of this line andthe MTPA trajectory, and at this point, the torque is TB. Ifthe motor torque is greater than TB, field-weakeningcontrol is selected. Otherwise, if the motor torque issmaller than TB, constant torque control is selected eventhough the rotor speed is above the base speed. Fig. 4shows the flowchart for the control mode selection.


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4 Implementation of the controltrajectories with the direct torquecontrol scheme4.1 System configuration

Fig. 5 shows the block diagram of the direct torquecontrolled IPM synchronous motor drive incorporatingcontrol trajectories. The lookup table is used todetermine the amplitude of the stator flux linkageaccording to the MTPA trajectory for constant torquecontrol. The amplitude of the stator flux linkage isdetermined by the inverse of the speed for field-weakening operation. The parameters of the IPMsynchronous motor are given in Table 1. The Ld and Lq

parameters indicated in Table 1 are for the unsaturatedmachine. These were measured using the technique asdescribed in [17, 18].

4.2 Simulation results and discussion

The trajectory control under DTC was modelled using dynamicsystem simulation software SIMNON for both constant torqueand field-weakening operations for the IPM motor shown inTable 1. The sampling time is 100 ms for the inner torqueand flux control loops and 500 ms for speed control loop. The


rated speed and crossover speeds are 1260 and 1460 rpm,respectively. A look-up table was used to represent theMTPA trajectory, and the torque–speed characteristic infield-weakening region was represented by mathematicalequation based on the torque–speed characteristic. The drivesystem of Fig. 5 with speed loop and trajectory controller ismodelled in SIMNON environment.

Fig. 6 shows the dynamic responses of the system with respectto a step change in speed reference from 0 to 1260 rpm. Themaximum torque of the IPM motor under MTPA control is3.7 N m. Figs. 6a–6c show the speed response, torqueresponse and flux locus, respectively. Fig. 6d shows the actualtorque and the amplitude of stator flux in torque–ls plane. Itis seen that the locus of MTPA in the T–ls plane in Fig. 6dagrees well with the analysis in Section 3.2.

The dynamic responses of the drive system in the field-weakening region are shown in Fig. 7, with respect to a stepchange in speed reference from 0 to 1700 rpm. It is seen fromFig. 7a that the transitions between the constant torque andfield-weakening operations are very smooth. It is also seenfrom Fig. 7c that the amplitude of the stator flux is decreasingwith the increase in speed above the base speed. Fig. 7dshows the torque and stator flux in torque–ls plane. Field-weakening operation occurs at the intersections of MTPA

Figure 8 Dynamic responses of the DTC drive, incorporating control trajectory under constant torque operation; experimentalresults


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Figure 9 Dynamic responses of the DTC drive, incorporating control trajectory under field-weakening operation; experimentalresults


T

and current limit trajectories. All these results agree with theanalysis in the previous section.

4.3 Experimental results and discussion

The DTC drive under trajectory control was implementedwith the IPM synchronous motor of Table 1. A digitalsignal processor TMS320C31 was used to carry out theDTC and trajectory control algorithms. The sampling timeis 100 ms for the inner torque and flux control loops and500 ms for the speed control loop.

Fig. 8 shows experimental results under constant torqueoperation. Figs. 8a and 8b show the speed and torqueresponses, respectively, for a step change in speed referencefrom 0 to 1260 rpm. Figs. 8c and 8d show the stator fluxlocus and T–ls trajectory, respectively.

Fig. 9 shows the dynamic responses of the drive system withrespect to a step change in speed reference from 0 to 1700 rpm.Figs. 9b and 9c show the actual torque and ls waveforms.Fig. 9d shows the locus of the stator flux linkage vector,which is a circle in both constant torque and field-weakening operations. Fig. 9d shows the torque and statorflux in the torque–ls plane for field-weakening operation.

The simulation and experimental results demonstrate thatthe drive is capable of working from zero speed to field-


weakening region and shows very good dynamic andsteady-state performance. It is seen that the transitionbetween the constant torque and field-weakeningoperations are very smooth.

5 ConclusionsThe analysis and modelling of control trajectories for the directcontrol of IPM synchronous motor drives have been presentedin this paper. The MTPA trajectory, current and voltageconstraints are expressed in the T–ls plane. Then, thecontrol trajectories are incorporated with the DTC schemeof IPM synchronous motor drive. The modelling andexperimental results are discussed for both constant torqueand field-weakening operations. The modelling andexperimental results show very good performance of directtorque controller, incorporating control trajectories fromzero speed to field-weakening range. It is seen that thetransition between the constant torque and field-weakeningoperations is very smooth and the drive is capable ofworking from zero speed to field-weakening region, andshows very good dynamic and steady-state performance.

6 References

[1] JAHNS T.M.: ‘Flux-weakening regime operation of aninterior permanent magnet synchronous motor drive’,IEEE Trans. Industry Appl., 1987, 23, pp. 398–407


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[2] MORIMOTO S., SANADA M., TAKEDA Y.: ‘Wide-speed operationof interior permanent magnet synchronous motors withhigh-performance current regulator’, IEEE Trans. IndustryAppl., 1994, 30, pp. 920–926

[3] PAN C., SUE S.: ‘A linear maximum torque per amperecontrol for IPMSM drives over full-speed range’, IEEETrans. Energy Conversion, 2005, 20, (2), pp. 359–366

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[5] KIM J.M., SUL S.K.: ‘Speed control of interior permanentmagnet synchronous motor drive for the flux-weakeningoperation’, IEEE Trans. Industry Appl., 1997, 33, pp. 43–48

[6] UDDIN M.N., RADWAN T.S., RAHMAN M.A.: ‘Performance ofinterior permanent magnet motor drive over wide speedrange’, IEEE Trans. Energy Conversion, 2002, 17, (1),pp. 79–84

[7] LENKE R.U., DONCKER R.W., MU-SHIN K., KWON T.S., SUL S.K.: ‘Fieldweakening control of interior permanent magnet machineusing improved current interpolation technique’. IEEEPower Electron. Specialist Conf., 2006, vol. 1, pp. 1–5

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[9] BAADER U., DEPENBROCK M., GIERSE G.: ‘Direct self control ofinverter-fed induction machine: a basis for speed controlwithout speed measurement’, IEEE Trans. Industry Appl.,1992, 28, pp. 581–588

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[11] RAHMAN M.F., ZHONG L., HU W.Y., LIM K.W., RAHMAN M.A.: ‘A directtorque controller for PM synchronous motor drives’, IEEETrans. Energy Convers., 1997, pp. 637–642

[12] TRIPATHI A., KHAMBADKONE A.M., PANDA S.K.: ‘Dynamic controlof torque in overmodulation and in the field weakeningregion’, IEEE Trans. Power Electron., 2006, 21, (4),pp. 1091–1098

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[14] RAHMAN M.F., ZHONG L., HU W.Y., LIM K.W., RAHMAN M.A.: ‘Adirect torque controlled interior permanent magnetsynchronous motor drive incorporating field weakening’,IEEE Trans. Industrial Appl., 1998, 34, (6), pp. 1246–1253

[15] ZORDAN M., VAS P., RASHED M., BOLOGNANI S., ZIGLIOTTO M.:‘Field-weakening in high-performance PMSM drives:a comparative analysis’, IEEE Industry Appl. Conf., 2000, 3,pp. 1718–1724

[16] RAHMAN M.F., HAQUE M.E., LIXIN T., ZHONG L.: ‘Problemsassociated with direct torque controller of an interiorpermanent magnet synchronous motor drive and theirremedies’, IEEE Trans. Industrial Electron., 2004, 51, (4),pp. 799–809

[17] HAQUE M.E.: ‘Improvements of a direct torque controllerof an interior permanent magnet synchronous motor drive’.PhD thesis, The University of New South Wales, Australia,2003

[18] DUTTA R., RAHMAN M.F.: ‘A comparative analysis of twotest methods of measuring d- and q-axes inductances ofinterior permanent-magnet machine’, IEEE Trans. Magn.,2006, 42, (11), pp. 3712–3718

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Incorporating control trajectories with the direct torque control scheme of interior permanent magnet synchronous motor drive