STUDY OF THE LEAKAGE INDUCTANCE OF THE END-WINDING SECTIONS OF THE STATOR WINDING OF A PULSED GENERATOR

G. A. SIPAILOV, V. Z. KHOR'KOVA

(Presented by the joint scientific seminar of the departments of electrical machines and general electrical engineering)

An exact calculation of the inductive reactance of the end-winding section of a stator winding presents considerable difficulties, associated with the complex pattern of the magnetic field in the region of the end windings. In the practice of designing electrical machines, when determining the leakage inductance of the end-winding sections of windings, as a rule empirical formulas are used, which make it possible to calculate with sufficient accuracy only under the condition of geometric similarity of the end-winding overhangs. When the configuration of the end-winding section changes, these formulas require correction. In addition, these formulas do not reflect the physical picture of the leakage fields; therefore, with their aid it is impossible to assess the influence of individual factors on the magnitude of the leakage reactance of the end-winding sections—in particular, it is impossible to assess the influence of the ferromagnetic surfaces surrounding the end-winding section.

Assessment of the influence of individual factors on the leakage inductance of end-winding sections is especially necessary in designing the end-winding sections of the stator winding of a pulsed generator, when it is required to obtain a minimum value of leakage inductance.

In the present work the results are set forth of a study of the influence of a ferromagnetic medium on the leakage inductance of the end-winding sections of a single-layer concentric stator winding of a pulsed generator with the number of pole pairs \(2p = 2 \div 8\). The theoretical part of the study was carried out on digital computers; the experimental part, on physical models constructed with allowance for the laws of physical modeling. The calculation method for the leakage inductance of the end-winding sections is based on expressions for the inductances of complex contours, determined by the method of sections.

The self-inductance of a contour consisting of \(n\) sections and the mutual inductance with other contours consisting of \(m\) sections may be represented in the form

\[ L = \sum_{k=1}^{n} L_k + \sum_{i=1}^{m} M_{ki}, \tag{1} \]

where \(L_k\) is the self-inductance of the \(k\)-th section,
\(M_{ki}\) is the mutual inductance of the \(k\)-th and \(i\)-th sections.

The influence of the end iron on the leakage inductance of the end-winding sections is taken into account by the mutual inductance of the segments of the end-winding section with their mirror images in the ferromagnetic surface. In doing so, the generally accepted assumptions are made:

1. The lengths of the end-winding sections are much greater than the linear dimensions of their cross sections.

2. The distances from the elements of one contour to the elements of another (in particular, to the mirror images of the segments) are much greater than the linear dimensions of the cross sections of the segments.

3. The magnetic permeability of the end surface of the stator is infinite.

4. Eddy currents in the steel of the end surface are absent.

Compilation of the calculation program

The overhang of the coil end winding, with the adopted designations of the geometric quantities, is shown in Fig. 1. The distance from the axial line of the end winding to the end iron is equal to:

\[ a = l_1 + l_2 \cos \alpha . \tag{2} \]

Fig. 1. Overhang of the end winding of the stator winding of a pulsed generator

The diameter along the axial line of the curvilinear end-winding section

\[ D_l = D_i + h_p + l_2 \sin \alpha . \tag{3} \]

The central angle

\[ \Theta = \frac{\tau}{2}\left(1 - 0.5\,\frac{b_p}{z}\right). \tag{4} \]

When calculating the mutual inductance of an end-winding part with its mirror image, the end-winding part is approximated by three segments of length

\[ l=\frac{1}{3}\frac{\pi D_i}{2p}\left(1-0.5\frac{b_{\text{п}}}{\tau}\right). \tag{5} \]

In these expressions:

\(D_i\) — diameter of the stator bore;
\(h_{\text{п}}\) — height of the stator slot;
\(b_{\text{п}}\) — width of the stator slot;
\(p\) — number of pole pairs;
\(\alpha\) — angle of inclination of the end-winding part to the machine axis.

The self-inductance of the rectilinear sections of the end-winding part \(l_1\) and \(l_2\) is determined from the expression:

\[ L_{1(2)}=\frac{\mu_0}{2\pi} w_1 l_{1(2)} \left[ \ln \frac{2l_{1(2)}}{g_{1(2)}}+\frac{a_{1(2)}}{l_{1(2)}}-\frac{q_{1(2)}^{2}}{4l_{1(2)}^{2}}-1 \right], \tag{6} \]

where \(g_{1(2)}\), \(a_{1(2)}\), \(q_{1(2)}\) are, respectively, the geometric-mean, arithmetic-mean, and root-mean-square distance of the segments from one another.

The mutual inductance of the end-winding part and its mirror image is considered as the sum of the mutual inductances of a number of rectilinear sections, taking into account their mutual spatial arrangement (Fig. 2):

Fig. 2. For calculation of the mutual inductance of the end-winding part and its mirror image

\[ M_{\text{fe}}=\frac{\mu_0}{2\pi}w_1^2\frac{\cos\varphi}{2} \sum_{p=1}^{n}\sum_{q=1}^{n}(-1)^{p+q}F_{pq}, \tag{7} \]

\[ F_{pq}=x_p \ln \left(y_q-x_p\cos\varphi+D_{pq}\right)+y_q \ln \left(x_p-y_q\cos\varphi+D_{pq}\right)+ \]
\[ +\frac{2a}{\sin\varphi}\operatorname{arctg}\left(\frac{x_p+y_q+D_{pq}}{a}\operatorname{tg}\frac{\varphi}{2}\right), \tag{8} \]

\(x_p; x_q; y_p; y_q\) are the coordinates corresponding to the beginnings and ends of the segments,

\(D_{pq}\) is the geometric mean distance between the segments. Thus, the inductance of the end-winding part of the coil will be equal to

\[ L_k=2L_1+2L_2+2M_{11}+2M_{22}+M_{\mathrm{Fe}}+L_{\mathrm{kr}} . \tag{9} \]

In compiling the calculation programs, the following parameters were taken as variables:

\[ \begin{aligned} \tau &\text{— pole pitch of the machine;}\\ p &\text{— number of pole pairs;}\\ l_1 &\text{— length of the rectilinear section;}\\ \alpha &\text{— angle of inclination of the end-winding part to the machine axis.} \end{aligned} \]

The block diagram of the calculation program is presented in Fig. 3.

Fig. 3. Block diagram of the calculation program:
a) mutual inductance of the end-winding part with its mirror image, b) self-inductance of the end-winding part

Calculation Results

The results of calculating the inductance of the end-winding part and its mutual inductance with the mirror image for a number of values of the pole pitch ...

for different angles of inclination of the end-winding part for 2-, 4-, 6-, and 8-pole pulsed generators made with identical diameters of the stator bore are presented in Figs. 4 and 5. The number of parallel branches in all the cases considered is taken equal to 1.

Fig. 4. Dependence of the inductance of the end-winding part on the angle of inclination to the machine axis

Calculations show that, at an angle of inclination of the end-winding part \(\alpha=0\), the leakage inductance of the stator-winding end parts is practically inversely proportional to the number of pole pairs. However, as \(\alpha\) increases, the growth of the leakage inductance of the end parts is different. The greatest increase in inductance is observed in two-pole machines: when \(\alpha\) changes from 0 to \(90^\circ\), the inductance increases by a factor of 2.5. With increasing polarity, the growth of the inductance of the end-winding part decreases, and for \(2p=8\) the inductance increases by only 30%. This is explained by the fact that, in the case of two-pole machines, the absolute increase in the length of the arc of the end-winding part as the angle of inclination increases proves to be more significant. Naturally, the self-inductance of the end-winding part (the inductance in air) and the mutual inductance with its mirror image prove to be considerably larger. In relative units, the dependence of \(L_k\) on the angle \(\alpha\) for a series of \(P\) is shown in Fig. 4.

Calculations show that the leakage inductance due to the influence of the end iron in the total end leakage inductance differs for machines of different polarity. The inductance is greatest for two-pole machines. At \(\alpha=0\), \(L_{\mathrm{Fe}}\) amounts to about 12% of the total leakage inductance, and at \(\alpha=90^\circ\) it increases to 37%. As the number of poles increases, the share of \(L_{\mathrm{Fe}}\) decreases. For eight-pole machines, at \(\alpha=0\), \(L_{\mathrm{Fe}}\) is about 6%, and at \(\alpha=90^\circ\) reaches 20%. However, the relative increase in the mutual inductance of the end-winding part with its mirror image as \(\alpha\) increases is the same for machines of all polarities (Fig. 5).

With some approximation, the change in the leakage inductance

Fig. 5. Relative mutual inductance of an end-winding section with its mirror image

of the end-winding sections with an increase in the angle of inclination to the machine axis can be expressed by the following dependence:

\[ L_{\ell}=L_{\ell 0}\left[1+\frac{2.4}{\pi}\alpha(2.17-K_p)\right], \tag{10} \]

where \(L_{\ell 0}\) is the leakage inductance of the end-winding sections at zero angle of inclination.

\(K_p\) is a coefficient accounting for the polarity of the machine. For two-pole machines \(K_p=1\); for multipole machines

\[ K_p=2-\frac{2}{2^{p-1}}. \tag{11} \]

Experimental studies

For an experimental verification of the obtained dependences of the leakage inductance of the end-winding sections on the angle of inclination of the end-winding section to the machine axis, an experimental setup was created. The setup consists of a current source with a frequency of 1000 Hz and three models, made in accordance with the laws of physical modeling and representing stator stacks having different active-part lengths in the ratio 1:2:3 and a concentric winding placed in them with identical end-winding sections. The method of separating the inductance

the leakage of the end-winding part of the section from its total inductance was adopted according to [2, 3]. The experimental results are presented in Fig. 6.

Fig. 6. Experimental dependences \(x_l = f(\alpha)\)

Comparison of the experimental and calculated data confirms the validity of the theoretical analysis carried out. The influence of end shields on the leakage inductance of the end-winding parts can theoretically be calculated analogously to the influence of the end face of the packet. Indeed, experiments show that when the distances from the end-winding part to the end face of the stator and to the end shield are equal to each other, their influence on the leakage inductance is the same. The dependence of the leakage inductance of the end-winding part, in fractions of the inductance of the end-winding part in the absence of shields, as a function of the ratio of the distance from the axial line of the end-winding part to the surface of the shield to the distance from the axial line of the end-winding part to the end face of the frame, is presented in Fig. 7.

In practice, the ratio \(\frac{l_2}{l_1}\) in the design of an end-winding part bent through an angle of \(90^\circ\) lies within the range 3–4; consequently, the inductance of the end-winding part, owing to the influence of the bearing shield, increases by no more than 10%, which is in good agreement with the results of investigations by other authors [4, 5].

A study of the influence of the machine frame on the leakage inductance of the end-winding parts showed that even at an inclination angle of the end-winding part of \(90^\circ\), i.e., at the minimum distance between the end-winding part and the cylindrical surface of the frame, the increase in leakage inductance does not exceed 4%.

The studies carried out of the leakage inductance of end-winding parts, with analysis of the various factors affecting \(x_l\), can be used in the design of pulsed generators for a preliminary assessment of various structural embodiments of the end-winding parts of the stator winding.

![Diagram with labels: stator, shield, \(l_1\), \(l_2\).]

![Graph: vertical axis \(x_l\); horizontal axis \(\frac{l_2}{l_1}\).]

Fig. 7. Dependence of \(x_l\) on the ratio \(\frac{l_2}{l_1}\)

REFERENCES

1. P. A. Kalantarov, L. A. Tseytlin. Calculation of Inductances. Reference book. “Energiya,” 1970.

2. V. P. Shuyskii. Calculation of Electrical Machines. “Energiya,” 1968.

3. I. Mays. Calculation of the end-winding leakage inductance of electrical machines using physical modeling to determine magnetic-conductivity coefficients. Abstract of Candidate’s dissertation, Moscow Power Engineering Institute, 1966.

4. E. C. Barnes. An Experimental Study of Induction Machine End–Turn Leakage Reactance, TATEE, 1951.

5. P. I. Lawrenson. Calculation of machine end-winding inductances with special reference to turbogenerators. Proc. Inst. Electr. Eng. 117, No. 6, 1970.