Graphical-Analytical Method for Approximate Calculation of the Magnetic Permeance Curve Between the Stator Tooth and Rotor of a Reluctance Machine

A. N. Korogodsky, S. L. Medovar, A. M. Menshikova

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

The analysis of a reluctance machine with pronounced saliency on the stator is in a number of cases carried out by reducing the magnetic field in the air gap to a system of lumped parameters (an equivalent magnetic-circuit diagram) [1, 2, 3].

The principal parameter of the equivalent circuit is the magnetic permeance, varying in time or as a function of the electrical angle of rotation of the rotor, between the stator region enclosed by the elementary contour of the armature winding (the stator tooth pitch) and the equipotential body of the rotor.

The problem is to construct the curve of the indicated permeance as a function of the electrical angle of rotation of the rotor. The field in a reluctance machine is plane in character; therefore it is expedient to confine oneself to determining the specific permeance (per unit axial length of the stator).

In most cases, because of the relatively complicated configuration of the boundaries, the problem of determining the specific permeance cannot be solved analytically. Numerical methods of field calculation require the use of a digital computer. In engineering practice, the most rational approach appears to be the use of relatively simple approximate graphical-analytical methods of calculation, which permit, within a selected range of the main parameters of the tooth zone, correction according to experimental data.

We take as a basis the graphical-analytical method of Pohl [4], which, as shown in [5] and [6], is especially convenient in determining the integral magnetic permeances per tooth pitch of the stator and rotor.

According to Pohl, the fictitious angles of inclination of the walls of actually rectangular stator and rotor teeth are taken equal to one another, \(\beta_1=\beta_2=\beta\), and amount to \(1—1.1\) radians to the horizontal. In this case the flux tubes of the excitation field in the slot region are replaced by parts of rings bounded by arcs of concentric circles drawn from the vertex of the angle formed by the active part and the wall of the nearest stator tooth. In the air gap, the lines of force are drawn perpendicular to the circumference of the armature and are segments of length \(\delta\). The real field pattern differs from the adopted one, in particular, in that the lines of force in the slot region are not parallel to one another and are not directed along concentric circles. The real flux tube becomes thicker as it approaches the slot boundaries, since the flux tubes fill all sections of the slot. Consequently, the specific permeance (per unit length of the line of force) of a real elementary flux tube conduc-

bridges in the slot region is greater than the specific permeance of the calculated elementary tube. The introduction of fictitious angles of inclination of the tooth walls, i.e., shortening of the calculated flux tubes, is intended to compensate for the indicated error when calculating the permeance of the tube.

Thus, the fictitious angles of inclination of the tooth walls are, in essence, empirical correction coefficients. For a rectangular slot shape, the choice of fictitious inclination angles of 1.1 radians is successful. As has been noted by a number of authors, the error in calculating integral permeances by the Field method for \(\tau/\delta \geqslant 25\) is very small. A. A. Terzyan [7] showed the possibility of applying the Field method to the calculation of the permeance of a comb-shaped tooth zone in the range \(3 \leqslant \tau/\delta \leqslant 10\), with a corresponding correction of the empirical coefficient \(\beta\) depending on the geometry of the zone and the relative position of the stator and rotor. When the slot shape deviates from rectangular, the relationships between the specific permeances of the real and calculated elementary field tubes change; consequently, Field’s empirical coefficients require refinement. In [8] and [9] it was shown that direct application of the Field method to the calculation of tooth zones with a nonrectangular slot shape gives a significant error.

Meanwhile, in micromachines with a small stator bore diameter, considerable deviations of the shape of the stator and rotor slots from rectangular or segmental often occur. In particular, in tractor induction generators the walls of the stator and rotor teeth are usually parallel (Fig. 1, \(a\)), and under conformal mapping of the air-gap region into a strip with projections (Fig. 1, \(b\)) they are transformed into converging curves for the stator and diverging curves for the rotor.

Fig. 1

In this case, there are different deviations of the slot shape from rectangular for the stator slot and the rotor slot. The integral permeance in the slot region of the stator decreases in comparison with a rectangular slot, while the integral permeance of the rotor slot region increases. Consequently, for this case Field’s empirical coefficients must be corrected as follows:

for the stator, \(\beta_1 \geqslant \beta\),

for the rotor, \(\beta_2 \leqslant \beta\),

where \(\beta_1\) is the fictitious angle of inclination of the stator-tooth walls,

\(\beta_2\) is the fictitious angle of inclination of the rotor-tooth walls,

\(\beta\) is the Polya coefficient for a rectangular slot, \(\beta = 1.1\) rad.

Figure 2 gives the results of calculating the tooth zone according to Fig. 1 by the graphical-analytical method on the basis of Polya’s method; dependences are plotted of the specific magnetic permeance between a stator tooth and the rotor on the electrical angle of rotor rotation. The empirical coefficients were taken as follows:

Fig. 2

curve II: \(\beta_1 = 1.1\) radians, \(\beta_2 = 0.5\) radians.

curve I: \(\beta_1 = \beta_2 = 1\) radian.

The points plotted on the graph were obtained experimentally. As is seen from comparison of the calculated and experimental data, in this case the most successful choice is \(\beta_1 = 1.1\), \(\beta_2 = 0.5\).

The authors experimentally investigated the tooth zones of machines with parallel walls of the stator and rotor teeth in the following range of the main parameters: \(25 < \dfrac{\tau}{\delta} < 80\) and \(0.1 < \dfrac{b}{D} < 0.35\). As a result, the following empirical formulas are proposed for the calculation coefficients \(\beta_1\) and \(\beta_2\):

\[ \beta_1 \simeq 1 + \frac{b_1}{D}. \tag{1} \]

\[ \beta_2 \simeq 1 - 2\,\frac{b_2}{D}, \tag{2} \]

where \(b_1\) is the width of the stator tooth,

\(b_2\) is the width of the rotor tooth,

\(D\) is the diameter of the stator bore.

According to Polya, the length of a section of a field line in the slot region is equal to \(\beta X\) (\(X\) is the distance from the edge of the nearest tooth). Since only the length and width of the elementary flux tube enter into the calculation, while its curvature does not appear in the calculation formulas for magnetic permeance, the following simplification suggests itself in constructing the approximate field pattern: the shape of the field line over its entire extent from the stator to

we take as a straight-line segment perpendicular to the stator-bore line; the fictitious angle of inclination of the stator tooth is taken as \(\beta'_1\), and that of the rotor, correspondingly, as \(\beta'_2\).

\[ \beta'_1=\operatorname{arc\,tg}\,\beta_1, \tag{3} \]

\[ \beta'_2=\operatorname{arc\,tg}\,\beta_2. \tag{4} \]

It is easy to verify that the length and width of the elementary flux tubes do not change in this case. The division of the field into figures is carried out very simply (Fig. 3). From the vertices of all angles, vertical straight lines are drawn until they intersect the opposite boundary of the region.

The figures—trapezoids—are formed by two segments of vertical straight lines and the corresponding portions of the stator and rotor contours.

The permeance of each trapezoid is calculated from the formulas:

\[ \text{for}\qquad l_1=l_2=l \qquad \lambda=\frac{c}{l}, \tag{5} \]

\[ \text{for}\qquad l_1\ne l_2 \qquad \lambda=\frac{c}{l_2-l_1}\ln\frac{l_2}{l_1}. \tag{6} \]

These formulas are convenient when the dimensions are taken from a field picture plotted on graph paper, similar to that shown in Fig. 3. The integral permeance per stator or rotor tooth pitch is equal to the sum of the permeances of the figures falling within the given tooth pitch.

Fig. 3

To construct the main permeance curve between the stator tooth and the rotor, it is usually sufficient to calculate the field for seven positions of the stator-tooth axis relative to the rotor-tooth axis \((0, 30, 60, 90, 120, 150\) and \(180\) electrical degrees). The magnetic-permeance curve possesses symmetry of the first kind.

REFERENCES

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2. M. M. Krasnoshapka. High-Frequency Inductor Alternators. Publishing House of the VVIA named after N. E. Zhukovsky, 1948.

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4. R. Pohl. Theory of pulsating-field machines. The Journal of the IEE, vol. 93, II, 1946, No. 31.

5. L. E. Dombur. Magnetic field in the air gap of an axial induc-

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