IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 10, OCTOBER 2005 4069

Calculation of the Flux-Linkage Characteristics of aSwitched Reluctance Motor by Flux Tube Method

N. K. Sheth, Student Member, IEEE, and K. R. Rajagopal, Senior Member, IEEE

Electrical Engineering Department, Indian Institute of Technology Delhi, New Delhi 110016, India

The torque developed by a switched reluctance motor (SRM) is dependent on the change of flux-linkage and rotor position. The flux-linkage is a function of both the excitation and the rotor position. Due to the nonlinear nature of this motor, estimation of the flux-linkagecharacteristics is cumbersome. In this paper, a simple analytical method to estimate the flux-linkage characteristics of SRM is presented.Here, equations for the calculation of inductance and the flux-linkage for three identified regions based on the rotor position; a) fullyunaligned to starting of pole overlap, b) starting of pole overlap to full pole overlap, and c) full pole overlap to fully aligned conditions,are derived in terms of motor dimensions, the magnetic properties of the materials used, and the stator excitation. The validation of theresults obtained from this new analytical method is carried out using the finite element analysis of the motors.

Index Terms—CAD, finite-element (FE) analysis, flux-linkage, inductance, MATLAB, motor, switched reluctance motor (SRM).

I. INTRODUCTION

CALCULATION of accurate flux-linkage characteristicsis necessary for predicting the torque developed by a

switched reluctance motor (SRM). The flux-linkage being arotor position and stator excitation dependent quantity, merelyconsidering its values for the fully aligned and unaligned con-ditions of the stator and rotor poles for calculating the torquedeveloped by the motor, will lead to erratic results. Methodsfor calculation of the fully aligned and unaligned inductancesand flux linkages are available in [1] and [2]. The finite element(FE) method can be used to compute the inductances at variousrotor positions of the SRM [3], but this necessitates a packageand more time for modeling the motor. In this paper, equationsfor calculating the phase inductances and flux-linkages of amultiphase SRM for all the rotor positions and stator excitationsusing analytical method are presented.

II. METHODOLOGY

After conducting FE analyses of various configurations ofmultiphase SRM, it is observed that the angle between the un-aligned and aligned positions which is equal to half the rotorpole pitch can be divided in to three regions: a) fully unalignedto starting of the pole overlap, b) starting of the pole overlap tofull pole overlap, and c) full pole over lap to fully aligned condi-tion of stator and rotor poles. These three regions are identifiedin such a way that each one of these can be accurately modeledusing a predetermined number of flux tubes, which is decidedafter analyzing the flux plots of the motor obtained from the FEanalysis for all rotor positions and various excitations. Using theflow chart shown in Fig. 1, inductance for each tube is calculatedand summed up to get the phase inductance. Once the phase in-ductance is known, it can be multiplied with the phasecurrent to get the phase flux linkage , where, isthe angle of displacement of the rotor from the fully unalignedcondition.

Digital Object Identifier 10.1109/TMAG.2005.854865

Fig. 1. Flow-chart to calculate the inductances and flux-linkage of SRM.

III. INDUCTANCE FOR REGION I

Fig. 2 shows the nomenclature used for dimensions of themotor. The first region, i.e., from fully unaligned to the startingof pole overlap, can be shown as

.Fig. 3(a) shows the flux line plot for the fully unaligned po-

sition obtained from the FE analysis. Nine flux tubes as shownin Fig. 3(b) will be sufficient to represent the actual flux pathsof this region.

The dotted line in Fig. 4 shows the flux tube 1. Considering

and

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4070 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 10, OCTOBER 2005

Fig. 2. Nomenclature used for physical dimensions of SRM.

Fig. 3. Flux line plot and assumed flux tubes for region I.

Fig. 4. Flux tube 1 for region I.

coordinates of points B and C and hence the airgap length of thetube is calculated as

(1)

(2)

Airgap length of the tube (3)

Rotor and stator pole areas

and (4)

The tube area is half of the sum of the stator and rotor poleareas. The path lengths through the rotor pole and stator pole are

and , respectively. In the stator

Fig. 5. Flux line plot and assumed three flux tubes for region II.

yoke, the length of the flux tube is andits corresponding area is . In the rotor core, the lengthand the area of the flux tube can be worked out as

and (5)

These calculations made for the tube passing through theright side of the upper portion of the stator is valid for theleft side of the lower portion also. The total reluctance of thetube is the summation of the reluctances of both these halves,and following the procedure shown in Fig. 1, the inductancecontributed by tube 1 is calculated. Similarly inductancescontributed by all the tubes are calculated and summed up toget the phase inductance .

IV. INDUCTANCE FOR REGION II

The second region, i.e., start of pole overlap to full overlap,can be shown as

. Fig. 5(a) shows the flux line plot obtained fromthe FE analysis. Five flux tubes will be sufficient to representthe actual flux paths of this region. Fig. 5(b) shows three of theassumed flux tubes for this region. The remaining two flux tubesare similar to the flux tube 8 and 9 of the region I, shown inFig. 3(b). If represents the stator and rotor pole overlap angle,then the area of the tube for the overlap can be worked out as

. Airgap length of the tube is twice the actualairgap length of the motor. Length of the stator pole, length andarea of the stator yoke and rotor core are as same as given in theSection III. The inductance for tube 1 is calculated as given inthe flow chart of Fig. 1, and also for other tubes. The total phaseinductance is calculated as .

V. INDUCTANCE FOR REGION III

The third region, i.e., from full overlap to fully aligned condi-tion can be shown as .From the FE analysis it is observed that two flux tubes willbe sufficient to represent the actual flux paths of this region.Fig. 6(a) shows flux line plot for the fully aligned condition andFig. 6(b) shows the assumed flux paths for this region.

Mean path length through the stator and rotor poles

(6)

SHETH AND RAJAGOPAL: FLUX-LINKAGE CHARACTERISTICS OF A SWITCHED RELUCTANCE MOTOR 4071

Fig. 6. Flux tubes for the calculation of aligned inductance.

TABLE IMOTOR DIMENSIONS

Mean path length through the airgap and the rotor core

and (7)

Reluctance of the circuit (8)

Using (6)–(8) and appropriate areas, inductance is calculatedfor the tube-1 using the procedure given in Fig. 1, and also forother tubes. The total phase inductance is calculated as

.A program is developed in MATLAB to calculate the phase

inductance and flux-linkages for various regions of the SRMrotor position as shown above. The flux-linkage characteris-tics of two designs of 5 hp, four-phase, 8/6 SRM with the di-mensions given in Table I, are computed using the developedprogram and is compared with the corresponding characteris-tics obtained by the FE analysis for the motors. Fig. 7 showsthe flux-linkage characteristics computed by both the methodsfor motor I. Table II gives a comparison of the unaligned, half-aligned, and fully aligned flux-linkages of both the motors cal-culated using the two methods. It can be observed that the resultsobtained using the new method is fairly matching with those ob-tained from the FE analysis. However, it is interesting to notethat the error at the unaligned region is more than the alignedand partially aligned regions. In the unaligned condition, theleakage flux passes through all the nearby area; thereby neces-sitating more and more flux tubes for computation of the actual

Fig. 7. Comparison of flux-linkage characteristics computed by the analyticaland the FE methods for motor I.

TABLE IIFLUX-LINKAGE AT VARIOUS ROTOR POSITIONS FOR 1781 A EXCITATION

inductance. But as the absolute values of the phase inductanceand the flux-linkage at the unaligned position are too low, theeffect of this error in calculation of the developed torque will benegligible.

VI. CONCLUSION

A simple analytical method for accurately calculating thephase inductance and flux-linkage characteristics of a mul-tiphase SRM for all rotor positions and stator excitations,which can be used to calculate the actual developed torque,is presented in this paper. The results obtained using the newmethod is as good as those obtained from the FE analysis.

REFERENCES

[1] J. Corda and J. M. Stephenson, “An analytical estimation of the min-imum and the maximum inductances of doubly-salient motor,” in Proc.Int. Conf. Stepping Motors Systems, Leeds, U.K., Sep. 1979, pp. 50–59.

[2] A. Michaelides, C. Pollock, and C. Jolliffe, “Analytical computation ofminimum and maximum inductance in single and two phase switchedreluctance motor,” IEEE Trans. Magn., vol. 33, no. 2, pp. 2037–2040,Mar. 1997.

[3] A. Radun, “Analytical calculation of the switched reluctance motor’sunaligned inductance,” IEEE Tran. Magn., vol. 35, no. 6, pp. 4473–4481,Nov. 1997.

Manuscript received February 5, 2005.