SMSITIVITY OF Faz A R c m PITCH RATIO CN

SWITCHED RELuCIwa m P"a

R. Arumugam J.F. Lindsay R. Krishnan Dept. of Electrical & Computer Engr. Dept. of Electrical En-. Concordia University Montreal, Canada, H3G 1M8

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In this paper, the sensitivity of the pole arc/pole pitch ratio of the stator and rotor on the performance of a switched reluctance [SRI motor is investigated. An analytical method based on "magnetic flux path" and a two dimensional finite element analysis are used for the study. The method of sensitivity study is performed by comparing the average torque developed for different stator as well as rotor pole arc/pole pitch ratios and choosing the ratio combination that produces the greatest value of ,average torque.

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In recent years a number of papers on SR motors has been published in the literature [11-[141. Only a few of them address the design aspects [l, 6, 9, 101. The design philosophy published so far, is based on the knowledge of variable reluctance stepper motor designs. In these papers, the pole arc/pole pitch ratio has been either derived from the permeance values assuming parallel sided teeth and slots or taken to be slightly less than that used for variable reluctance stepper motors. Unlike these variable reluctance stepper motors, SR motors have smaller, unequal but even numbers of poles on the stator and rotor. Moreover, SR motors can be used for higher power applications requiring larger sizes. Hence, the change in motor performance due to variations of the stator and rotor pole arc/pole pitch ratios will be of interest to the SR motor designer. With the above considerations the present study has been undertaken.

lumped magnetic circuit model at different judiciously selected sections of the magnetic circuit has been proposed by Corda and Stephenson 1151. This method is used, as a first approximation, to determine the suitable pole arc/pole pitch ratio that enables the .SR motor to develop the greatest value of the average torque.

The advantage of using finite element analysis for electromagnetic field analysis and particularly for the SR motor, in which high levels of saturation are encountered, has already been reported.[16,171 A two dimensional finite element analysis is used in this study to calculate the energy stored in the motor when its rotor is in the aligned and unaligned positions. The coenergy change between the above two positions of the rotor is computed from which the average torque developed is determined. As in the analytical method, the best choice for the pole arc/pole pitch ratio is determined by considering the average torques for different pole arc/pole pitch ratio combinations.

An analytical method based on the

Basis of Sensitivity Stub!

In SR motors the air-gap geometry plays a vital role on their performance. A suitable choice of the air-gap length, pole width and pole height is necessary for a better design of the motor. Therefore, attention

VPI & su Blacksburg, VA 240612

has been directed to optimize the air-gap geometry so that an SR motor with a greater value of torque is desiqned. The radial lenqtth of the air-gap is made as small as mechanically possible so that the torque developed is maximum when ail other air gap parameters are held constant. In this study it is assumed that the height of the pole is fixed. The pole arcs on the stator and rotor are changed in steps of about 0.05 of a pole pitch, starting from a pole arc/pole pitch ratio of 0.25 up to 0.55. Values below 0.25 and above 0.55 are not required in this study as will become evident from the trend of the results which follow.

Due to the nonlinear nature of the magnetic fields in the SR motor under normal operating conditions, the virtual displacement principle provides the most convenient method of calculating the average torque. The flux linkages are calculated for different excitation currents when the rotor is in the aligned and unaligned positions. The coenergy change is obtained from which the average torque is calculated. The range of pole arc/pole pitch ratios which produces the greatest average torque is chosen as.the preferred values for a better design.

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In this study, two methods have been used for torque prediction. One is an analytical method which is used to calculate the aligned and unaligned flux linkages for various pole arc/pole pitch ratios. The other is a finite element analysis. The following sections give a brief description of these methods.

Analytical method

The analytical method describes the determination of the minimum and maximum inductances when the configuration of the motor is known. The minimum inductance is calculated using the assumption that the magnetic fields in the interpolar and air-gap regions consist of straight line segments and circular arcs. Unlike earlier methods, this one takes into account the actual distribution of the winding on the stator poles. The flux linkages when the rotor is in the unaligned position are obtained assuming that the minimum inductance remains constant for the range of excitation used in the analysis. When the rotor is in the aligned position there is considerable mmf drop in the stator and rotor cores compared with that in the air-gap. Hence, the conventional magnetic circuit analysis is used to determine the flux linkages.

The application of the technique proposed by Corda et al. may lead to large errors in the minimum inductance calculation unless care is exercised. The formulae derived for permeance components are based on the assumption that the ratio of pole arc/pole pitch is 0.5 for both the stator and rotor. Also, the windings are taken to be extended U? to the middle of the stator interpolar regions. Hence a suitable modification of the formulae is required when this technique is used for a particular motor configuration and winding arrangement.

88CH2565-0/88/050$01.00 0 1988 IEEE

In the maximum inductance position the cores are saturated even at lower excitations due to the shorter air-gap length. Hence the mmf drops in various sections of the core and the nonlinear magnetization characteristic of the core are taken into account when the flux linkages vs current characteristic of the motor is computed.

The data for the magnetization characteristic of the iron (M-19 steel) is taken from the manufacturer's data sheets. The characteristic is divided into 20 line segments and each segment is represented using cubic spline polynomials. For flux densities beyond the range of available values, a linear extrapolation is used.

In order to determine the flux linkages vs current characteristic in the aligned position, the flux linkages are assumed and the excitations required to establish the assumed flux linkages are calculated. From the known number of turns, the flux and flux densities in various sections of the magnetic circuit are computed. Using the magnetization characteristic and the flux densities, the mmf drops in the different sections of the core are determined. The excitation current is obtained as the ratio of the total mmf to the number of turns.

Figure 1: Typlcal SR motor configuration

The flux linkages vs current characteristic for the aligned and unaligned positions are stored in a data file. The area between these curves is obtained using numerical integration. Having obtained the coenergy change for the chosen ratios of pole arc,/pole pitch in the stator and rotor, the average torque developed is calculated.

The procedure used to determine the optimum pole arc/pole pitch ratio is as follows:

1 Set the pole arc/pole pitch ratio of the stator to 0.25.

2 Set the pole arc/pole pitch ratio of the rotor to 0.25.

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Determine the flux linkages vs current characteristics for the aligned and unaligned positions.

Calculate the change In coenergy and hence the average torque develope.'.

Tncrernent the pole arc/pole pitch ratio of the rotor by 0.05.

If the rotor pole arc/pole pitch ratio is greater than 0.55, then go to step 7. Otherwise, go to step 3 .

Increment the stator pole arc/pole pitch ratio by 0.05.

If the stator pole arc/pole pitch ratio is greater than 0.55, then go to step 9. Otherwise, go to step 2.

Tabulate the average torque developed as a function of the stator and rotor pole enclosures for different excitations.

10 Choose the pole arc/pole pitch ratios corresponding to the greatest average torque for a given current.

It can be realized that there are 49 combinations of pole arc/pole pitch ratio when both the stator and rotor pole arcs are wried in this manner. For each combination, one flux linkages vs current characteristic is obtained for both aligned and unaligned positions. When five excitation currents are considered for each combination, there are 245 average torque values computed. It is presumed that this will give sufficient informtion on the nature of torque development for the changes in stator and rotor pole widths. Any number of average torque values can be calculated if a suitable number of excitations is considered. On the other hand, when finite element analysis is used, separate problems are to be set up and solved for each excitation and pole arc/pole pitch ratio combination in the aligned and unaligned positions. Thus, for the chosen five excitation currents 490 problems are solved and post-processed to obtain the coenergy for both aligned and unaligned positions. The coenergy is directly computed in the finite element analysis and hence the flux linkages vs current characteristic need not be plotted to determine the average torque.

lement Analvsis

The finite element formulation used for field analysis has already been described in [161. The configuration of the SR motor considered for investigation is shown in Fig. 1.

For each stator pole arc/pole pitch ratio, the rotor pole arc/pole pitch ratio is changed from 0.25 to 0.55 in steps of 0.05. Thus, seven finite element analysis mdels are developed for the aligned position, each model having provision to change the rotor pole arc/pole pitch ratio. Since, the field solutions are obtained for five chosen excitations, 35 problems are set up for each stator pole arc/pole pitch ratio. Similarly, for the unaligned position, seven different models have been used, each time solving and postprocessing 35 problems.

The coenergy for each pole arc/pole pitch ratio combination and current, when the rotor is in a given position, is directly computed as described earlier. The change in coenergy and the average torque are determined.

Results

The average torque values calculated by the analytical method are given in Table I. To illustrate the nature of the results that lead to these values, a typical flux linkages vs current characteristic for a pole arc/pole pitch ratio of 0.4 on the stator and 0.35 on the rotor is shown in Fig. 2. The variation of the average torque with changes in stator pole arc/pole pitch ratio is plotted for different rotor ?le arc/pole pitch ratios and is shown in Fig. 3. Keeping the pole arc/pole pitch ratios for the stator and rotor the same, average torque values are calculated for various excitation currents. The resulting average torque variation for different pole arc/pole pitch ratios is shown in Fig. 4 .

0 - L 0 *, 6 - 0 m -

: 4 - a -

2 -

U .7>

2 -

c . 4 l - I

L// c LL 0

0 2 4 6 E 1 0 1 2 1 4 E x c i t a t i o n i n amperes

Figure 2: Flux linkages vs excitation current (Analytical Method)

The average torque is proportional to the change in coenergy =lues when the rotor moves from the unaligned to the aligned position. Hence, the results obtained by the finite element method are presented in terms of coenergy itself. Excitation currents of 1, 2, 4, 8 and 12 amperes are used for the investigation. The results for 1 A excitation are. not presented as they are very similar to the 2 A results. Fig. 5 shows the variation of coenergy change with varyingstator and rotor pole arc/pole pitch ratios when the excitation is 2 A. The results for excitations of 4, 8 and 12 amperes are shown in Figs. 6, 7 and 8, respectively.

Conclusions The changes in stator pole arc greatly influence

the average torque compared with the changes in rotor pole arc. At lower excitation currents, when the saturation in the core is not appreciable, the developed torque increases invariably with increase in stator pole arc for a given rotor pole enclosure. Instead, at higher excitations, the increase in average torque with stator pole enclosure is less and at higher stator pole arcs the average torque decreases. It is particularly evident at a steady excitation of 12 A where the average torque is less for a stator pole arc/pole pitch ratio of 0.55 compared with that for 0.5. When the stator pole arc/pole pitch ratio is less than 0.35, there is distinctly less developed torque at all excitations. Hence, it may be recommended that the stator pole arc/pole pitch ratio be chosen in the range of 0.35 to 0.5. Higher pole arc/pole pitch ratios for

the stator m y be limited by considerations of winding space and the necessary clearance &tween the windings.

At a l l excitation currents, the average torque value increases with increase in rotor pole arc, reaches a peak value and then decreases when the stator pole arc is held constant. It is observed that there is no increase in torque developed when the rotor pole arc/pole pitch is increased beyond 0.45. Higher excitation currents produce maximum values of average torques when the ratio is slightly less than 0.4 , whereas, at lower currents the average torque values reach peaks when the ratio is slightly above 0.4. Considering the nature of currents, normally encountered in SR motors, choosing the rotor pole arc/pole pitch ratio around 0.4 may produce the highest torques. The average torque is less for a rotor pole ardpole pitch ratio of 0.25 compared with that for higher ratios when the stator pole arc/pole pitch ratio is in the range of 0.35 to 0.5.

Rotor p o l e enclosure

LEGEND - . 25

0 . 3

- _ . 0.40 - - 0.45 0 .50 --- 0 . 5 5

-1 0.35

6:s .> .A5 .4 .:5 .5 .:5 S t a t o r P o l e arc/Pole p t t c h R a t t o

Figure 3: Average torque vs stator pole arc/pole pitch ratio (Excitation at 12 A)

E z C -

Figure 4: Average torque vs pole arc/pole pitch ratio (Stator and Rotor ratios are equal)

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1. s q 13

' vQ .2s .38 .3s . 4 8 . 4 s .sa . s s

STATOR

Figure 5: Average torque vs pole arc/pole pitch ratios of stator and rotor (Excitation at 2 A )

.wroQ .8s .38 . 3 5 . 4 8 . 4 5 . 5 8 . 5 5 STATOR

F i ~ u r e 6

8 . 5 8 W U

7 . 8 4

I U 7 . 1 8 t 0

Z

0

6 . 5 2

5 . 8 6

5 . 2 h . 5 5

Aver3Se torque vs pole of stator and rotor (ExcitJtion at 4A)

arc/pole pitch ratios

. 3 5 y , Qo7 OQ . 2 5 .38 .3S . 4 8 . 4 5 . S E - 5 5

STATOR

Figure 7: Average torque vs pole arc/pole pitch ratios of statcr and rotor (Excitation at 8A)

It can be concluded that the pole arc/pole pitch ratio on the stator and rotor of an SR motor need not be the same. The range of values that m y be used f o r the pole arc/pole pitch ratio of the rotor can be C.3 to 0.45 and that of the stator 0.35 to 0.5.

53

-

W U Z

I

t U

Z W

U

[r 12

I 1

5 i a 0 9

8 . 5

Figure 8: Average torque vs pole arc/pole pitch ratios of stator and rotor (Excitation at 12A)

Average Torque Calculated Using Analytical Method . . _ - _ _ .. - __-- . _ _

Rotor pole enclosure - - - . - . - . - - - .-

0.30 0.35 0.40 0.45 0.50

. -. .- . .

! stator I pole j y 'ps u-r e I 1 0.25

0.30 i 0.35 1 0.40 ' 0.45

0.50 i 0.55 ' 0.25 : 0.30 ' 0.35 : 0.40 ' 0.45

0.50 0.55

0.25 0.30 0.35 0.40 0.45 0.50 0.55

0.25 0.30 0.35 0.40 0.45 0.50 0.55

0.25 0.30 0.35 0.40 0.45 0.50 0.55

0.354 0.415 0.472 0.524 0.562 0.561 0.560

1.254 1.434 1.607 I. 719 1.782 1.779 1.774

3.158 3.872 4.123 4.342 4.424 4.406 4.380

6.010 7.255 9.353 9.722 9.810 9.732 9.619

8.136 12.816 14.514 15.030 15.102 14.922 14.666

Excitation = 1 A 0.355 0.357 0.357 0.416 0.418 0.413 0.475 0.476 0.476 0.528 0.530 0.530 0.629 0.686 0.732 0.629 0.685 0.731 0.628 0.683 0.728

Excitation = 2 A 1.256 1.257 1.257 1.436 1.437 1.437 1.610 1.511 1.509 1.72: 1 . 7 2 3 1.729 1.976 1.945 1.997 1.872 1.940 1.990 1.865 1.931 1.979

Excitation = 4 A 3.149 3.141 3.133 3.868 3.862 3.953 4.118 4.109 4.096 4.338 4.326 4.308 4.522 4.588 4.629 4.501 4.561 4.593 4.469 4.521 4.542

Excitation = 8 A 5.964 5.919 5.875 8.312 8.273 8.226 9.313 9.266 9.206 9.675 9.614 9.533 9.873 9.888 9.855 9.779 9.771 9.704 9.645 9.606 9.493

Excitation = 12 A 8.019 7.915 7.810 12.727 12.635 12.528 14.414 14.301 14.162 14.913 14.770 14.585 15.090 15.009 14.846 14.874 14.740 14.500 14.570 14.367 14.023

0.357 0.418 0.476 0.530 0.769 0.768 0.764

1.257 1.435 1 . 5 0 6 I. 715 2.034 2.024 2.007

I

3.123 i 3.840 ! 4.077 ' 4.282 4.645 j 4.596 I 4.526 '

5.824 8.166 9.124 9.422 9.764 9.559 9.273

7.690 12.391 13.975 14.331 14.568 14.102 13.454

Wknovledaements

This research forms part of a project receiving financial support from the Natural Science and mgineering Research Council of Canada (NSERC). The authors are also grateful for the computational support made available by Infolytica Corporation of Montreal.

Ppfprpncps

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Element Model" , . , Vol

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