ON THE CALCULATION OF THE SLOT LEAKAGE PERMEANCE OF THE STATOR WINDING OF A SHOCK GENERATOR

V. Z. KHOR'KOVA

(Presented by the scientific seminar of the Department of Electrical Machines and the Electrical Machines Division of the Nuclear Research Institute of Electromechanics)

In calculating the slot leakage permeance of the stator winding of a shock generator, it is customary to use known expressions [1]. For example, for a single-layer winding lying in an open rectangular slot,

\[ \lambda_{\mathrm{p}}=\frac{1}{3}\frac{h_1}{b_{\mathrm{p}}}+\frac{h_2}{b_{\mathrm{p}}}, \tag{1} \]

where \(h_1\) is the height of the winding copper,
\(h_2\) is the height of the wedge and of the adjacent insulation above the copper,
\(b_{\mathrm{p}}\) is the slot width.

Expression (1) was obtained from the condition that the leakage flux passes only across the slot, perpendicular to its walls. In [2] we showed that, with a considerable decrease in height, the leakage flux through the bottom of the slot begins to have an effect; this flux increases substantially, and for \(h_{\mathrm{p}}<0.5b_{\mathrm{p}}\) it becomes impossible to neglect this flux in calculations.

It is known that the electromagnetic energy of the field of a current-carrying conductor in some volume with \(\mu=\mathrm{const}\) can be determined as

\[ W=\frac{1}{2}\int_v \bar{A}\bar{\delta}\,dv, \tag{2} \]

where \(\bar{A}\) is the vector potential of the magnetic field,
\(\bar{\delta}\) is the current-density vector,
\(v\) is the volume occupied by the magnetic field.

At the same time,

\[ W=\frac{1}{2}LI^2, \]

where \(L\) is the inductance of the conductor,
\(I\) is the current flowing through the conductor.

Hence

\[ L=\frac{2W}{I^2}. \tag{3} \]

We shall use the expressions for the current density and the vector potential given by us in [2]:

\[ \delta=a_0+\sum_n a_n\cos nky, \]

where

\[ a_0=\frac{I\,2b}{ab\,b_{\mathrm{п}}}; \qquad \kappa=\frac{2\pi}{b_{\mathrm{п}}}; \]

\[ a_n=\frac{I}{ab}\frac{2}{n\pi}\left[\sin nk\Delta+(-1)^n\sin nk\Delta\right], \]

\[ \Delta \text{ is the thickness of the slot insulation,} \]

\[ a,b \text{ are the dimensions of the stator-winding bar without insulation,} \]

\[ n \text{ is any integer} \]

\[ A_{\mathrm{I}}=\sum_n B_n\left(e^{nkx}+e^{-nkx}\right)\cos nky. \]

\[ A_{\mathrm{II}}=-2\pi a_0(x-s_1)^2+ \sum_n\left(D_ne^{nkx}+E_ne^{-nkx}+\frac{4\pi a_n}{n^2k^2}\right)\cos nky \]

\[ A_{\mathrm{III}}=-2\pi a_0(s_2-s_1)\left[2x-(s_1+s_2)\right] +\sum_n\left(G_ne^{nkx}+\right. \]

\[ \left.+F_ne^{-nkx}\right)\cos nky \]

\[ B_n=\frac{2\pi a_n}{n^2k^2}\left(e^{-nks_1}-e^{-nks_2}\right) \]

\[ E_n=B_n-\frac{2\pi a_n}{n^2k^2}e^{nks_1} \]

\[ G_n=B_n-\frac{2\pi a_n}{n^2k^2}\left(e^{-nks_1}-e^{-nks_2}\right) \]

\[ F_n=B_n-\frac{2\pi a_n}{n^2k^2}\left(e^{nks_1}-e^{nks_2}\right). \]

\[ s_1;\ s_2;\ s_3 \text{ are sections along the slot height, Fig. 1.} \]

Fig. 1. Stator slot

Having determined the energy of the electromagnetic field in each of the three regions along the slot height, we find the energy of the electromagnetic field of the slot. After a number of transformations,

\[ W=-0.1\mu_0 I^2 l \left\{ \left[ \frac{s_3-s_2}{b_{\mathrm{п}}} +\frac{s_3-s_1}{3b_{\mathrm{п}}} +\frac{(s_3-s_2)^2}{(s_2-s_1)b_{\mathrm{п}}} \right]\times \]

\[ {}\times \frac{1}{\pi^4}\cdot \frac{\delta_{\mathrm{p}}^3}{ab^2}\sum_{n=2,4\ldots}\frac{\sin^2 nk\Delta}{n^4} +\frac{1}{4\pi^5}\frac{\delta_{\mathrm{p}}^4}{a^2b^2}\sum_n\frac{\sin^2 nk\Delta}{n^5}\times \]
\[ {}\times \left[(e^{nks_3}-e^{-nks_2})-(e^{nks_1}-e^{-nks_1})\right]e^{-nks_3}\Bigg\}. \tag{4} \]

The minus sign in expression (4) indicates that the density of the intrinsic magnetic energy of the given contour, bounded by its dimensions, will decrease when the volume occupied by the magnetic field of this current expands. The energy expression should be taken in absolute value [3]. Denoting the expression in braces in (4) by \(\lambda_{nw}\), we obtain, according to expression (3),

\[ L_{\mathrm{p}}=0.2\,\mu_0\,l\lambda_{nw}, \tag{5} \]

where \(\lambda_{nw}\) is the slot-leakage permeance taking into account the actual field pattern in the slot,

\[ \lambda_{nw}= \frac{1}{3}\frac{h_1}{\delta_{\mathrm{p}}} +\frac{h_2}{\delta_{\mathrm{p}}} +\frac{h_2^2}{a\delta_{\mathrm{p}}}\frac{1}{\pi^4}\frac{\delta_{\mathrm{p}}^3}{ab^2} \sum_{n=2,4\ldots}\frac{\sin nk\Delta}{n^4} +\frac{1}{4\pi^5}\frac{\delta_{\mathrm{p}}^4}{ab^2}\times \]
\[ {}\times \sum_{n=2,4\ldots}\frac{\sin^2 nk\Delta}{n^5} \left[(e^{nks_3}-e^{-nks_2})-(e^{nks_1}-e^{-nks_1})\right]e^{-nks_3}. \tag{6} \]

It is of interest to compare the obtained value \(\lambda_{nw}\) with expression \(\lambda_{\mathrm{p}}\) (1). The calculation results are presented in Fig. 2. It is evident that for \(h_{\mathrm{p}}\ge 0.5\delta_{\mathrm{p}}\), expression (1) gives values practically coinciding with \(\lambda_{nw}\) (6).

For \(h_{\mathrm{p}}<0.5\delta_{\mathrm{p}}\), expression (1) gives underestimated values of the slot permeance, and the error increases sharply as the slot height decreases.

Fig. 2. Dependence of \(\dfrac{\lambda_{nw}}{\lambda_{\mathrm{p}}}\) on the ratio \(\dfrac{h_{\mathrm{p}}}{b_{\mathrm{p}}}\) and \(\Delta_{\mathrm{iz}}\)

of the slot height. It is also evident that at a small slot height the thickness of the bar insulation plays a significant role. Thus, for values

For \(h_{\text{п}} \ll 0.5\delta_{\text{п}}\), the permeance of slot leakage should be calculated with allowance for the leakage fluxes through the bottom of the slot according to expression (6).

REFERENCES

1. I. M. Postnikov. Design of Electrical Machines. GITL, Ukrainian SSR, 1960.
2. V. Z. Khorkova. On the calculation of the electromagnetic field of slot leakage in shock generators. Thematic collection of papers, Izvestiya TPI, 1972.
3. V. A. Govorkov. Electric and Magnetic Fields. “Energiya,” Moscow, 1968.