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Dynamic Analysis of Synchronous Reluctance Motor Drives Based onSimulink and Finite Element Model

Francesco Parasiliti, Marco Villani Alessandro Tassi Department of Electrical and Information Engineering Spin Applicazioni Magnetiche S.r.l.

University of LAquila 67100 Poggio di Roio, LAquila 29010 Pianello Val Tidone, Piacenza

ITALY ITALY villani@ing.univaq.it info@spinmag.it

Abstract A fine motor analysis that takes the driving controlinto account allows to evaluate with a good accuracy thedynamic performance. The use of Simulink

together with FEA

has allowed to investigate deeply the dynamic behaviour of theSynchronous Reluctance Motor in different operatingconditions, giving results closer to the actual motorperformance.

I. INTRODUCTION

Synchronous Reluctance Motors (SRM) could beconsidered as alternative to its counterparts, namelyPermanent Magnet, Switched Reluctance and InductionMotors. Moreover, it has been demonstrated that SRM hasthe same or higher power density than the Induction Motorsin the low and medium power level.The fine analyses of these motors are very difficult becauseof their highly saturated operation conditions and theirsalient structure. The use of linear models in evaluating the

performances of the considered motors can lead to seriouserrors since the SRM present notable nonlinearcharacteristics due to the effects of the saturation and cross-coupling phenomena occurring in the magnetic circuits.

These phenomena can be taken into account only by anaccurate nonlinear analysis which can be performed by FiniteElement programs that allow an accurate prediction ofmachine parameters and performances [1].Moreover, the growing demand of high dynamic

performance motors (e.g. with fast torque response) requiresa fine motor analysis taking into account the driving control.Usually, to simplify the controller, constant values of axisinductances ( Ld and Lq) are considered, but it is well knownthat the axis currents and the stator-rotor relative positionhave a considerable influence on the inductances, withsignificant effects on torque behaviour. For these reasons, inorder to predict with a good accuracy the dynamic

performance, it is suitable to develop a model that allows tolink the drive scheme with a fine motor model.With this aim, the link between Simulink

code (by

Mathworks) and a Finite Element software Flux2D byCedrat has been employed to simulate the SRM drive. Thisapproach allows to simulate, with good accuracy, the motor

behaviour in different operating conditions related toimposed control strategy [2], giving to the designer usefulindications (e.g. the torque ripple) in view of the motor andcontrol design refinements.

The proposed study concerns with a Synchronous ReluctanceMotor with two flux barriers 4 pole, 200 V, 20 Nm: a view ofthe rotor structure is shown in Fig.1The paper presents a preliminary analysis with constantvalues of axis inductances; then, more accurate models have

been carried out tacking into account the parametersvariation.

Fig. 1 - View of the two flux barriers rotor

II. THE CONTROL STRATEGIES OF THE SRM

The considered torque expressions of the SRM, withreference to the rotating d-q frame synchronized with therotor, are:

( ) 223 2 sin I L L pT sqd = , (1)

( ) qd qd I I L L pT = 23

, (2)

where p are the pole pairs, I s the space vector of the stator

current, I d and I q the direct and quadrature axis currents and the angle between d-axis current and the vector I s.A suitable control strategy is to maximize the Torque-Current ratio and this can be achieved if is equal 45: itcorresponds to impose the same values for the axis currents.Since for this type of motors Ld is usually 78 times higherthan Lq, the above mentioned strategy cannot be adoptedwhen I d component increases; then, the d-axis current should

be kept constant and only the q-axis component increases.

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Alternatively, the d-axis current can be always fixedconstant.Therefore, the control strategies could be (Fig.2):- d-q current angle control (A, B, C trajectory);- constant I d control (A, B, C trajectory).

In this paper the constant I d control only has beeninvestigated, and the control scheme is shown in Fig.3.

=45A DCB

id

iq

idmax

Fig. 2 - Locus of the stator current vector

III. THE FE ANALYSIS OF THE SRM

The proposed study concerns the SRM whose main data are presented in Tab.I. The investigated motor has a flux barriersrotor that presents, respect to the axially laminated one [2], asimplicity in mechanical construction, lower manufacturingcost, and the rotor skewing possibility; on the other hand ithas a quite large level of torque ripple due to the inductancesvariation with rotor position that can cause, particularly atlow-speed, inaccuracy in motion control, noise andvibrations.

TABLE IMAIN DATA OF THE MOTOR

Number of poles 4Stack length (mm) 130Outside stator diameter (mm) 152Inner stator diameter (mm) 90

Stator slots 36 Number of flux barriers per pole 2 Number of turns per phase 198

Phase voltage (V) 200Phase current (rms) (A) 7.7Torque (Nm) 20Frequency (Hz) 50

An accurate bi-dimensional FE model has been carried out by the software Flux2D ver.9 by Cedrat, [3], introducing a parametric model of motor in order to modify the geometricdimensions of stator and rotor shape, the rotor position ( )and the currents.The magnetostatic analyses allow to evaluate the axisinductances, whose values could depend on the axis currentsonly (if one rotor position only is analyzed):

- L d = f (Id, Iq); (3)- L q= f (Id, Iq);

or on currents and rotor position (see Fig.4):

- L d = f ( , Id, Iq); (4)- L q= f ( , Id, Iq).

If a transient analysis is carried out, the calculatedinductances depend also on the variation of currents withtime:

- L d = f ( , Id(t), I q(t)); (5)- L q= f ( , Id(t), I q(t)).

A further FE model has been developed, taking into accountthe rotor skewing. This design solution allows to drasticallyreduce the high torque pulsation in flux barriers type SRM[1], [4].Referring to Finite Element analysis, skewing presentsdifficulties in computing the magnetic field distribution, aswell as machine parameters and performances evaluation.These difficulties are overcome considering a simplifiedseveral-blocks equivalent rotor and by using several crosssectional 2D FEA [5]. As alternative the Skew module ofFlux2D can also be used.

Fig.4 shows the axis inductances profiles for a skewed rotor(one stator slot pitch): in this case a skewing of one pole

pitch has been considered and the rotor has been divided into10 blocks. The inductances profiles have been compared withthe no-skewed rotor ones (4).

A

Fig. 3 - Scheme of the constant I d control

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Comparison Ld

0,00

0,05

0,10

0,15

0,20

0,25

0 5 10 15 20 25 30 35 4 0 4 5 50 55 60 65 7 0 7 5 8 0 85 90

Position [degrees]

I n d u c

t a n c e

[ H ]

LdLd Skewed

Comparison Lq

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,035

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Position [degrees]

I n d u c

t a n c e

[ H ]

LqLq Skewed

Fig. 4 - Axis inductances profiles (I d=3.7 A; I q=10.2 A)

IV. DINAMIC ANALYSIS OF SRM BY SIMULINK

In this step the SRM drive (Fig.3) is simulated by Simulinkcode, solving the dynamic equations of the SRM in d-q reference frame, where inductances values are thoseevaluated by preliminary (out of line) FEA; the SMR is thensimulated by lumped parameters like inductances andresistances. The analyses concern the no-skewed rotor.Two different tests have been carried out on the grounds of

two inductance evaluation hypotheses:a) variable inductances with currents but independent of

stator-rotor relative position (3); b) variable inductances with currents and stator-rotor

relative position (4).Simulink solves the SRM model using the preliminaryevaluated inductances values corresponding to the referencecurrents imposed by the controller.Speed responses between 0 and 1500 rpm with constant 22

Nm load torque have been simulated. Steady-state axiscurrents were: I d=3.7 A and I q=10.2 A.The time torque response for both cases are shown in Fig.5.In the case (a), since the axis inductances are constant, the

torque pulsation is absent. In the case (b) a significant rippleappears, that is about 11% (defined like as the ratio betweenthe difference of the maximum and minimum values oftorque and the average one).

V. DINAMIC ANALYSIS BY SIMULINK AND FEM

With the aim to link the drive scheme with a fine motormodel, the Simulink control scheme has been directlyinterfaced with the FE model of the SRM.

Fig. 5 - Torque waveforms (I d=3.7 A; I q=10.2 A).

This analysis has been carried out thanks to the link betweenthe FE software Flux2D and Simulink code [3]. By such co-simulation method, the transitory effects and the cross-

coupling effects are accurately reproduced.Respect to the analysis in the Section IV, the SMR is notsimulated by lumped parameters but it is accurately modelled

by on-line transient FE analysis where the axis inductancesdepend on the rotor position and the currents both variablewith time (5).The drive scheme of the SRM with constant I d control isshown in Fig.6, where it is evident the block for the FEA ofthe Synchronous Reluctance Motor.This co-simulation between Simulink (for the drive) andFlux2D (for the motor) needs an external circuit that allowsto link the two codes. Particularly, in order to fed the FEmodel of the SRM, three voltage generators have been

introduced, whose values (variable with time) depend on thecontroller. Then, starting from the motor phase resistance andthe inductances values (evaluated on-line by FEA), the phasecurrents are calculated by Flux2D. From these values, thecurrents in each stator slot are imposed according to thewinding distribution.

The time currents, torque and speed responses are presentedin Fig.7 and Fig.8, where a load torque of 22 Nm has beenimposed. After the transient operation, the speed reaches1500 rpm and the axis currents settle to the imposed values.

case (b)

case (a)

d-axis inductances

q-axis inductances

skewed

no-skewed

skewedno-skewed

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In this case, the torque ripple is higher than the previousanalysis one (Fig.5b), giving result closer to the actual motortorque.It is important to underline that knowledge of the torqueripple it is very important for the SRM since it can cause,

Current Id=costant

-25

-20

-15

-10

-5

0

5

10

15

20

0,00 0,05 0,10 0,15 0,20 0,25

Time [s]

C u r r e n

t [ A ]

Ia Ib Ic

Current dq Id=costant

0

5

10

15

20

25

0,00 0,05 0,10 0,15 0,20 0,25Time [s]

C u r r e n

t [ A ]

IdIqIdrif

Fig. 7 Phase currents and axis currents vs. time

particularly at low-speed, inaccuracy in motion control, noiseand vibrations.This example demonstrates how an accurate analysis of themotor by a suitable tool, allows to predict the real motor

performance and verify the dynamic behaviour for differentoperating conditions.

Torque Id=costant

0

5

10

15

20

25

30

35

40

45

0,00 0,05 0,10 0,15 0,20 0,25Time [s]

T o r q u e

[ N m

]

TorqueDrag Torque

Angular Velocity Id=costant

-200

0

200

400

600

800

1000

1200

1400

1600

1800

0,00 0,05 0,10 0,15 0,20 0,25

Time [s]

A n g u

l a r

V e

l o c

i t y [ r p m

]

Angular Velocity

Reference

Fig. 8 Torque and speed vs. time

Fig. 6 - SRM drive with constant I d control

FEA

Phase currents

Speed

speed

Axis currents

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VI. CONCLUSIONS

The growing demand of motors with high dynamic performance requires not only an adequate (and moreefficient) designs procedure but also a fine motor analysistaking into account the driving control.The use of Simulink directly interfaced with Flux FEsoftware allows to investigate deeply the dynamic behaviourof the SRM in different operating conditions and thisapproach can be defined like as a virtual experimental test.The goodness of the results depends not only on the proposedmethod but also on the accuracy in the construction of the FEand control scheme models.This approach could represent an effective tool for theanalysis of motor performance and the refinement of SRMdesign in order to improve and optimise [6] its dynamic

performance.

VII. REFERENCES

[1] G.Conti, F.Parasiliti and M.Villani, "Torque Ripple

Analysis in Synchronous Reluctance Motors", Internat.

Journal ELECTROMOTION , vol. n.3 1996, pp. 188-193.

[2] I. Boldea, Reluctance Synchronous Machines and Drives , Clarendon Press: Oxford, 1996.

[3] Cedrat, Flux Users Guide and Flux To SimulinkTechnology Manual, Vers. 9, 2005.

[4] E.Chiricozzi, F.Parasiliti and M.Villani, "DesignSolutions to Optimize Torque Ripple in SynchronousReluctance Motor", Intern. Conference on Electrical

Machine (ICEM96) , Vigo (Spain), Sept. 1996, pp.148-153.

[5] G.Conti, F.Parasiliti and M.Villani, Analysis ofSynchronous Reluctance Motor with Skewed Rotor byFinite Element Method, 8th Biennal IEEE

Conferenceon Electromagnetic Field Computation(CEFC'98) , Tucson (USA), June 1998.

[6] L.Cirio, S.Lucidi, F. Parasiliti and M. Villani, "A globaloptimization approach for the synchronous motorsdesign by finite element analysis", International

Journal of Applied Electromagnetics and Mechanics ,

vol. n. 16, 2002, pp. 13-27.

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