ON METHODS FOR CALCULATING THE LEAKAGE OF END WINDINGS OF SINGLE-PHASE WINDINGS

G. A. SIPAILOV, K. A. KHORKOV

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

The leakage of end windings, along with slot leakage, is one of the principal components of the total leakage of the windings of electrical machines. Therefore, in those cases where, because of the operating conditions of an electrical machine, it is necessary for the leakage reactance to be as small as possible, efforts are made to reduce both slot and end-winding leakage. For example, in generators for interrupting and surge power, in order to obtain the maximum value of the short-circuit current in a machine of specified dimensions, the reduction of leakage reactance is achieved by reducing the number of turns of the stator winding, reducing the depth of the stator slot and increasing its width, shielding the end windings, reducing the air gap, etc.

The leakage reactance of the end windings, like the slot-leakage reactance, is determined by the known formula

\[ X_\lambda = 4\pi f \mu_0 \frac{w_1^2}{p q_1}\lambda_\lambda, \tag{1} \]

where \(\lambda_\lambda\) is the leakage permeance of the end windings.

To determine the end-winding leakage permeance, there exist empirical formulas valid for a particular type of winding; for example, for a single-phase, single-layer, concentric winding this formula has the form

\[ \lambda_\lambda = (0.47 l_{\mathrm{av}} - 0.3\tau)\frac{q_1}{l}. \tag{2} \]

Calculation by formula (2) gives values of \(\lambda_\lambda\) that are sufficiently close to the actual ones. A disadvantage of formula (2) is that it does not reflect the physical essence of the complex picture of the leakage field of the end windings. Therefore, when calculating by formula (2), it is impossible to assess the influence of individual factors on the magnitude of the permeance; in particular, it does not show what influence on the leakage field is exerted by the ferromagnetic surfaces surrounding the winding, the angle of inclination of the end part of the winding, etc.

To determine the most effective methods for reducing the end-winding leakage permeance, it is necessary to find the values of each component of the permeance. For this purpose, in the present work, on the basis of

using the methods of segments and mirror images with the aid of the “Minsk-1” computing machine, calculations were carried out, the results of which make it possible to analyze the influence of individual components on the magnitude of the conductance of end leakage. The calculation was performed as applied to impulse-power generators, for which the leakage reactance is one of the principal parameters and in which a single-phase, single-layer winding has tangible advantages over other types of windings [6].

The segment method and the method of mirror images

Attempts are known to have been made to develop a procedure for calculating the leakage of the end portions of windings of electrical machines by the segment method, i.e., by calculating the inductances and mutual inductances of individual sections of the winding coils with their subsequent summation and by allowing for the influence of the stator steel by the method of mirror images [1]. The procedure of [1] is based on classical expressions for the inductance and mutual inductance of two segments [2].

Since, with a large number of coils, calculation by the segment method becomes very laborious, in order to simplify it the author of [1] replaces the mean geometric distances between the segments by their actual distances and combines the group of conductors constituting the end portion into one equivalent conductor. Such simplifications lead to large errors in the calculation of mutual inductances.

Modern computing technology, using electronic computers, makes it possible to calculate the leakage of the end portions by the segment method for any number of coils without the above simplifications. Such a calculation was carried out by us on the Minsk-1 computing machine for a number of generators having single-phase, single-layer concentric windings on the stator. The configuration of the winding end portions is shown in Fig. 1.

Fig. 1. End portion of the stator winding.

The leakage reactance of the end portions is regarded as the sum of the self-induction reactances \(L\) of all segments of each of the three zones outlined in Fig. 1 by dashed lines, and of the mutual inductance \(M\) of all segments both within each zone and between zones. In doing so, the increase in the total inductance of the segments due to the presence of the steel surfaces of the stator \(M_{\mathrm{cc}}\), the frame \(M_{\mathrm{ck}}\), the rotor \(M_{\mathrm{cp}}\), and the end shields \(M_{\mathrm{ct}}\) is taken into account.

The self-inductance of the end portion of the winding shown in Fig. 1 can be determined from its geometrical dimensions [2]:

\[ L = 0.8 \cdot 10^{-8} w_1^2 \sum_{n=1}^{n_{\max}} \left[ 2l_{1n}\left(\ln \frac{2l_{1n}}{g}+\frac{g}{l_{1n}}-1\right) + l_{2n}\left(\ln \frac{2l_{2n}}{g}+\frac{g}{l_{2n}}-1\right) \right], \tag{3} \]

where \(w_1\) is the number of series-connected turns of the winding,
\(n\) is the ordinal number of the coil, starting with the smallest,
\(g\) is the mean geometric distance of the coil cross section from itself; for a rectangular cross section,

\[ g = 0.224(h_{\mathrm{m}}+b_{\mathrm{m}}), \]

\(h_{\mathrm{m}}\) and \(b_{\mathrm{m}}\) are the height and width of the insulated copper in the slot, with

\[ b_{\mathrm{m}}=\pi k_{\mathrm{p}}\frac{D}{z_1'}, \]

\(k_{\mathrm{p}}\) is the ratio of the slot width to the tooth pitch,
\(l_{1n}\) and \(l_{2n}\) are the lengths of the segments of the \(n\)-th coil along the middle of its cross section, respectively along the machine axis and in the tangential direction (Fig. 1).

The lengths of the segments \(l_{1n}\) and \(l_{2n}\) can be expressed in terms of parameters that are most characteristic of the overhang of the end portions (Fig. 1):

\[ l_{1n}=C_2+0.5b_{\mathrm{m}}+(n-1)(b_{\mathrm{m}}+C_2'), \tag{4} \]

\[ l_{2n}=\frac{\pi}{z_1'}(D+k_1k_2b_{\mathrm{m}}) \left(\frac{z_1'}{6}-1+2n\right), \tag{5} \]

where \(k_1\) is the ratio of the slot height to its width;
\(k_2\) is a coefficient accounting for the angle of inclination of the end portion to the abscissa axis, with

\[ k_2=1+\frac{2l_{1n}\operatorname{tg}\alpha}{k_1b_{\mathrm{m}}}. \]

For two-pole machines with a single-phase single-layer winding on the stator, when the number of slots filled by the winding is equal to \(2/3\) of the total number of slots, the maximum number of segments of one zone is

\[ n_{\max}=\frac{z_1'}{6}. \]

The average length of the segments of each zone, \(l_{1\mathrm{cp}}\) and \(l_{2\mathrm{cp}}\), can be determined from expressions (4) and (5) by substituting for \(n\) its value

\[ n_{\mathrm{cp}}=\frac{z_1'}{12}+\frac{1}{2}. \]

Then the average length of the end portion will be equal to

\[ l_{\mathrm{cp}}= 2C_2+\frac{\pi}{3}(D+k_1k_2b_{\mathrm{m}}) +(C_2'+b_{\mathrm{m}})\left(\frac{z_1'}{6}-1\right) + \]

\[ +0.5b_{\mathrm{m}}(\pi-2)+R(\pi-4). \tag{6} \]

The values of \(C_2\), \(C'_2\), and \(R\) are given in the form of tables as functions of the stator-bore diameter and the generator voltage [3, 4]. In the case considered, it was taken that \(C_2 = 0.14D\), \(C'_2 = 0.5 b_m\), \(R = 0.05D\). The results of calculating the mean length of the end part of the winding are presented in Table 1 in the form of the ratio of \(l_{\text{e,avg}}\) to the pole pitch. From Table 1 it follows that, for \(\alpha < 10^\circ\),

\[ l_{\text{e,avg}} = (1.07 \div 1.17)\tau . \]

Table 1

The mutual inductance of any two parallel segments (Fig. 2a) is, in the general case, equal to

\[ M = \frac{\mu_0}{4\pi}\int_0^a \int_0^b \frac{dx_1 dx_2}{\sqrt{(x_2 - x_1)^2 + h^2}} . \tag{7} \]

After integration of (7), very cumbersome expressions are obtained; however, they become simpler for the particular relative positions of the segments that occur in the case under consideration:

Fig. 2. For calculating the mutual inductance of segments.

1) for equal segments issuing from one boundary plane (Fig. 2b),

\[ M_1 = 16 \cdot 10^{-8} w_1^2 b \left[ \ln \frac{b + q_1}{h_1} + \frac{h_1}{b} - \frac{q_1}{b} \right]; \tag{8} \]

2) for unequal segments issuing from one boundary plane (Fig. 2c)

\[ M_2 = 0.1 \cdot 10^{-8} w_1^2 \left[ a \ln \frac{q_1 + a}{h_1} + b \ln \frac{r_1 + b}{h_1} - \delta \ln \frac{p_1 + \delta}{h_1} + p_1 + h_1 - q_1 - r_1 \right]; \tag{9} \]

c) for unequal segments arranged symmetrically (Fig. 2,d)

\[ M_3 = 0.2 \cdot 10^{-8} w_1^2 \left[ (b + \delta) \ln \frac{b + \delta + q_1}{h_1} - \delta \ln \frac{\delta + p_1}{h_1} + p_1 - q_1 \right], \tag{10} \]

where \(h_1\) is the mean geometric distance between the segments; \(p_1, q_1\), and \(r_1\) are segments analogous to the segments \(p, q\), and \(r\) (Fig. 2), calculated by replacing the actual distance \(h\) between the segments by the mean geometric distance \(h_1\).

The mean geometric distance between the segments is determined by the magnitude of the segments and by their mutual arrangement (Fig. 2)

\[ \ln h_1 = \frac{h^2}{ab} \ln \frac{hp}{qr} - \frac{1}{2ab} \left[ p^2 \ln p + h^2 \ln h - r^2 \ln r - q^2 \ln q \right] - \frac{h}{ab} \left[ \delta \varphi_1 - b\varphi_2 - a\varphi_3 \right] - \frac{3}{2}, \tag{11} \]

where

\[ \varphi_1 = \operatorname{arctg}\frac{\delta}{h}, \qquad \varphi_2 = \operatorname{arctg}\frac{b}{h}, \]

\[ \varphi_3 = \operatorname{arc\ tg}\frac{a}{h}. \]

The investigation of the influence of ferromagnetic surfaces on the magnitude of the leakage fields of the end portions of windings of electrical machines is not a new problem. The increase in inductance due to the presence of steel surfaces surrounding the end portions is taken into account by the method of mirror images; in this case the direction of the current in the mirror image of a segment retains the direction of the current of the real segment, and the distance between the segments is taken equal to twice the distance from the real segment to the ferromagnetic surface. A. I. Voldek examined fairly fully the influence of the end surface of the stator steel; however, he neglected the influence of the frame surface, assuming that the distance between the end portion and the frame is large, without giving any quantitative data [5]. There is an opinion that the method of mirror images is not applicable for calculating the inductances of windings located near ferromagnetic cylindrical surfaces.

Allowance for the influence of the rotor and the frame on the leakage inductance of the end portions

The investigations we carried out showed that, by introducing a correction coefficient, the consideration of a curved conductor located near a cylindrical ferromagnetic surface can be reduced to a straight conductor and a plane ferromagnetic surface. This coefficient depends on the width of the coil and the distance from it to the ferromagnetic surface.

For the experimental determination of the correction factor, three groups of coils were made, with three coils in each group. One coil of a group was made bent to a radius \(R=\dfrac{5}{\pi}\) and had the form of the end portion of a concentric winding; moreover, in the first group the width of the coil was taken equal to the full pitch \(\tau\), in the second group to \(\dfrac{2}{3}\tau\), and in the third to \(\dfrac{1}{3}\tau\). In all the bent coils the straight bars had a shorter length than the bent bars. The other two coils of each group were made square, with sides respectively equal to the larger and smaller side of the bent coil.

Fig. 3. Diagram of the experimental setup, where
TC denotes the test coils installed in succession.

The experiment was carried out on the setup whose schematic circuit is shown in Fig. 3. As the alternating-current source, a PT-1100 generator was used, making it possible to obtain an adjustable voltage at a frequency of 1000 Hz. The use of an increased frequency was due to the desire to obtain noticeable changes in the reactance of coils having comparatively small geometrical dimensions. In the course of the experiments, for all the coils in turn the total impedance \(Z\) and the active resistance \(r\) were measured. The total impedance was measured by the voltmeter-and-ammeter method using instruments of class 0.2, while the active resistance was measured with a direct-current bridge of the UMB type. The reactance of the coils was determined by calculation:

\[ x=\sqrt{z^{2}-r^{2}}. \]

Initially, the reactance of the bent and flat coils of each group, placed in air, was determined; then the resistance of the same coils was determined when they were placed at a distance \(h\) from a ferromagnetic surface. In this case, for each coil the difference \(X_m\) between its resistance near the ferromagnetic surface and its resistance in air was recorded.

If by \(X_{m1}\) we denote the increase in inductive resistance due to the presence of steel near the flat square coil whose side is equal to the smaller (straight-line) side of the bent coil, by \(X_{m2}\) the same for the larger square coil, and by \(X_{m3}\) the same for the bent coil, then the increment of inductive-

of the reactance of one bent bar due to the presence of steel will be equal to

\[ \frac{X_{m3}-0.5X_{m1}}{2}, \]

and the increment of the reactance of a straight bar, equal in length to the bent bar, is equal to \(0.25X_{m2}\). Hence the required correction factor can be found:

\[ k=\frac{2X_{m3}-X_{m1}}{X_{m2}}. \tag{12} \]

The results of calculating the correction factor from experimental data for one gap between the coil and the ferromagnetic surface are presented in Table 2. In determining the coefficient \(k\).

Table 2

In the case where the coil embraces the steel surface, the data for flat coils were taken from the rows of Table 2 that refer to coils embraced by steel. Similar calculations were carried out for other gaps. The dependences of the coefficient \(k\) on the distance between the coil and the ferromagnetic surface of both the rotor and the frame are presented in Fig. 4.

Fig. 4. Dependence of the correction coefficient \(k\) on the distance between the coil and the ferromagnetic cylindrical surface: a) the coil embraces the surface; b) the coil is embraced by the surface.

From the data of Fig. 4, expressions have been obtained that make it possible to determine the correction coefficient analytically in the course of machine design:

a) for the case where the coil embraces the ferromagnetic surface,

\[ k = 1 + 1.1 \frac{y}{z} - 0.5 \frac{h}{D}. \tag{13} \]

b) for the case where the coil is embraced by the ferromagnetic surface,

\[ k = 1 - 0.65 \frac{y}{z} - 0.2 \frac{h}{D}. \tag{13a} \]

The use of the correction coefficient makes it possible to employ the method of mirror images in the case where the coil is located near a cylindrical ferromagnetic surface.

The effect of the end shields on the leakage of the end portions of the stator winding is taken into account in the same way as the end surface of the stator.

Analysis of the Results of Calculating the Leakage Reactance of End Portions

Using the method set forth above, with expressions (3–13), calculations were carried out on the “Minsk-1” electronic digital computer for the individual components and the resultant leakage reactance of the end portions. The block diagram of the calculation is presented in Fig. 5. In the calculations the following ranges of variation of the variables were adopted:

stator bore diameter

\[ D = 0.5 + k \cdot 0.2, \qquad \text{where } k = 0;\ 1;\ 2;\ 3;\ 4, \]

total number of stator slots

\[ z'_1 = 24 + k \cdot 6, \qquad \text{where } k = 0;\ 1;\ 2 \ldots 7. \]

Text in the block diagram: Input of \(k_1\) (0); comparison with \(k_{1\max}\); if \(k_1 > k_{1\max}\), stop; input of \(D\) (0); increment of \(k_1\); if \(D > D_{\max}\), comparison with \(D_{\max}\); input of \(z'_1\) (0); comparison with \(z'_{1\max}\); if \(z'_1 > z'_{1\max}\), increment of \(D\); increment of \(z'_1\); calculation of constants: \(b_n, n_{\max}, \ell_{\min}, \ell_{\max}, \tau\); input of \(n\) (0); comparison with \(n_{\max}\); if \(n > n_{\max}\), printout of calculation results; calculation of \(\ell_1\) and \(\ell_2\); calculation of \(a\); comparison with \(\ell_{\max}\); if \(a > \ell_{\max}\), increment of \(n\); calculation of \(h\); calculation of \(N_x\); increment of \(a\); calculation of \(x_{\mathrm{sa}}\).

Fig. 5. Block diagram for calculating the end leakage reactance on the “Minsk-1” electronic digital computer.

In this case, the distance from the outermost coil to the end shield was taken as equal to \(0.6D\), the distance from the coils to the frame as equal to \(D\), and the distance from the coils to the rotor, depending on the number of stator and rotor teeth and on the value of the diameter, as equal to \((0.1 \div 0.3)D\).

The results of the calculation for one of the diameters \((D = 0.9\ \text{m})\) are presented in Table 3; for comparison, the values of the end leakage reactance calculated by formulas (1) and (2) are also given there. For the values of the self-inductance and mutual inductance in henries, the values \(L\) and \(M\) in Table 3 must be multiplied by \(10^{-8}\).

For all the diameters considered, the pattern of distribution of the components of the leakage reactance of the end portions is the same as for \(D = 0.9\ \text{m}\); therefore, the calculation data for the other diameters are not given.

Analyzing the obtained results, partially presented in Table 3, one may conclude that the end leakage reactance con-

Table 3

consists of two approximately equal parts: the reactance that any end portion would have in the absence of steel, and the reactance added due to the presence of ferromagnetic surfaces surrounding the winding. In this case, the increase in reactance due to the influence of the end surface of the stator steel is about 20%, that of the end surface of the shields about 10%, that of the cylindrical surface of the rotor about 20%, and that of the housing surface about 4% of the total leakage reactance of the end portions.

Comparing the results of the computer calculation with the data obtained by calculation using formulas (1) and (2), one may note their good agreement. Consequently, in those cases where a quantitative evaluation of the individual components of the leakage reactance of the end portions is not required, it is advisable to carry out the calculation using empirical formulas.

REFERENCES

1. I. V. Shekhman. Elektrichestvo, No. 3, 1950.
2. P. L. Kalantarov, L. A. Tseitlin. Calculation of Inductances (reference book), 1955.
3. I. M. Postnikov. Design of Electrical Machines. GITL UkrSSR, Kiev, 1960.
4. A. E. Alekseev. Construction of Electrical Machines. GEI, 1958.
5. A. I. Voldek. Elektrichestvo, No. 1, 1963.
6. G. A. Sigaĭlo, K. A. Khorkov. On the choice of the stator winding of a shock generator. Izvestiya TPI, vol. 132, 1965.