Calculation Of The Flux-linkage Characteristics Of A Switched Reluctance Motor By Flux Tube Method

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

4069

Calculation of the Flux-Linkage Characteristics of a Switched 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 fluxlinkage is a function of both the excitation and the rotor position. Due to the nonlinear nature of this motor, estimation of the flux-linkage characteristics 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) fully unaligned 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 the results 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

C

ALCULATION of accurate flux-linkage characteristics is necessary for predicting the torque developed by a switched reluctance motor (SRM). The flux-linkage being a rotor position and stator excitation dependent quantity, merely considering its values for the fully aligned and unaligned conditions of the stator and rotor poles for calculating the torque developed by the motor, will lead to erratic results. Methods for calculation of the fully aligned and unaligned inductances and flux linkages are available in [1] and [2]. The finite element (FE) method can be used to compute the inductances at various rotor positions of the SRM [3], but this necessitates a package and more time for modeling the motor. In this paper, equations for calculating the phase inductances and flux-linkages of a multiphase SRM for all the rotor positions and stator excitations using analytical method are presented. II. METHODOLOGY After conducting FE analyses of various configurations of multiphase SRM, it is observed that the angle between the unaligned and aligned positions which is equal to half the rotor pole pitch can be divided in to three regions: a) fully unaligned to starting of the pole overlap, b) starting of the pole overlap to full pole overlap, and c) full pole over lap to fully aligned condition of stator and rotor poles. These three regions are identified in such a way that each one of these can be accurately modeled using a predetermined number of flux tubes, which is decided after analyzing the flux plots of the motor obtained from the FE analysis for all rotor positions and various excitations. Using the flow chart shown in Fig. 1, inductance for each tube is calculated and summed up to get the phase inductance. Once the phase inis known, it can be multiplied with the phase ductance to get the phase flux linkage , where, is current the angle of displacement of the rotor from the fully unaligned condition.

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 the motor. The first region, i.e., from fully unaligned to the starting of pole overlap, can be shown as . Fig. 3(a) shows the flux line plot for the fully unaligned position obtained from the FE analysis. Nine flux tubes as shown in Fig. 3(b) will be sufficient to represent the actual flux paths of this region. The dotted line in Fig. 4 shows the flux tube 1. Considering and

Digital Object Identifier 10.1109/TMAG.2005.854865 0018-9464/$20.00 © 2005 IEEE

4070

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

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

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

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

and

(5)

These calculations made for the tube passing through the right side of the upper portion of the stator is valid for the left side of the lower portion also. The total reluctance of the tube is the summation of the reluctances of both these halves, and following the procedure shown in Fig. 1, the inductance contributed by tube 1 is calculated. Similarly inductances contributed by all the tubes are calculated and summed up to . get the phase inductance

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

IV. INDUCTANCE FOR REGION II

Fig. 4. Flux tube 1 for region I.

coordinates of points B and C and hence the airgap length of the tube 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 pole areas. The path lengths through the rotor pole and stator pole are and , respectively. In the stator

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 from the FE analysis. Five flux tubes will be sufficient to represent the actual flux paths of this region. Fig. 5(b) shows three of the assumed flux tubes for this region. The remaining two flux tubes are similar to the flux tube 8 and 9 of the region I, shown in Fig. 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 actual airgap length of the motor. Length of the stator pole, length and area of the stator yoke and rotor core are as same as given in the Section III. The inductance for tube 1 is calculated as given in the flow chart of Fig. 1, and also for other tubes. The total phase . inductance 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 will be sufficient to represent the actual flux paths of this region. Fig. 6(a) shows flux line plot for the fully aligned condition and Fig. 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. Fig. 7. Comparison of flux-linkage characteristics computed by the analytical and the FE methods for motor I.

TABLE I MOTOR DIMENSIONS

TABLE II FLUX-LINKAGE AT VARIOUS ROTOR POSITIONS FOR 1781 A EXCITATION

Mean path length through the airgap and the rotor core and Reluctance of the circuit

(7) (8)

Using (6)–(8) and appropriate areas, inductance is calculated for the tube-1 using the procedure given in Fig. 1, and also for other 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 SRM rotor position as shown above. The flux-linkage characteristics of two designs of 5 hp, four-phase, 8/6 SRM with the dimensions given in Table I, are computed using the developed program and is compared with the corresponding characteristics obtained by the FE analysis for the motors. Fig. 7 shows the flux-linkage characteristics computed by both the methods for motor I. Table II gives a comparison of the unaligned, halfaligned, and fully aligned flux-linkages of both the motors calculated using the two methods. It can be observed that the results obtained using the new method is fairly matching with those obtained from the FE analysis. However, it is interesting to note that the error at the unaligned region is more than the aligned and partially aligned regions. In the unaligned condition, the leakage flux passes through all the nearby area; thereby necessitating more and more flux tubes for computation of the actual

inductance. But as the absolute values of the phase inductance and the flux-linkage at the unaligned position are too low, the effect of this error in calculation of the developed torque will be negligible. VI. CONCLUSION A simple analytical method for accurately calculating the phase inductance and flux-linkage characteristics of a multiphase 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 new method is as good as those obtained from the FE analysis. REFERENCES [1] J. Corda and J. M. Stephenson, “An analytical estimation of the minimum 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 of minimum and maximum inductance in single and two phase switched reluctance motor,” IEEE Trans. Magn., vol. 33, no. 2, pp. 2037–2040, Mar. 1997. [3] A. Radun, “Analytical calculation of the switched reluctance motor’s unaligned inductance,” IEEE Tran. Magn., vol. 35, no. 6, pp. 4473–4481, Nov. 1997.

Manuscript received February 5, 2005.