Proceedings of the Tomsk Order of the October Revolution
and Order of the Red Banner of Labor
Polytechnic Institute named after S. M. Kirov

Vol. 284 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

INFLUENCE OF ELECTROMAGNETIC SHIELDS ON THE PARAMETERS OF THE END WINDINGS OF THE STATOR WINDING OF A PULSE GENERATOR

L. O. OGANYAN

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

The amplitude of the current pulse of a pulse generator is determined to a considerable extent by its transient inductive reactance, a large part of which is the leakage inductive reactance of the end windings. In generators with a normal tooth zone it amounts to about 50% of the total leakage inductive reactance of the stator winding, while in generators with the stator winding brought out into the air gap it is 70–80%.

A reduction in the leakage of the end windings of the stator winding of a pulse generator is achieved by enclosing the end portions of the jumpers in shields made of a material with high electrical conductivity (copper, bronze, etc.).

The effect of shields on the parameters of the end zone of the stator winding of a pulse generator can be taken into account by damping and shielding coefficients.

The first takes into account the reaction of the eddy currents induced in the body of the shield to the leakage field of the end windings, leading to a decrease in their inductive reactance; the second determines the degree of shielding, from the leakage fluxes of the end windings, of the structural elements of the end zone of the pulse generator (shields, bearings) made of ferromagnetic material. The latter substantially reduces the magnitude of losses in the specified elements.

In addition, it should be taken into account that shields make it possible to fasten the end winding reliably and protect it from destruction by electrodynamic forces.

Determination of the above-mentioned coefficients and investigation of their dependence on the factors characterizing the end zone of the machine (the geometry of the end windings, the thickness of the shield, the frequency of the exciting field, etc.) are of great interest at the design stage of pulse generators.

In the present work, the study of the damping and shielding coefficients of the leakage fluxes of the end windings is carried out using the example of a single-phase pulse generator having a single-layer concentric stator winding with subdivided end windings bent through an angle of 90° relative to the machine axis.

Calculation of the leakage inductance of end windings enclosed in shields leads to the investigation of the electromagnetic fields of the end zone—

of a pulsed generator, since the inductance is determined through the energy of the magnetic field according to the well-known law [1].

It should be noted that in modern pulsed generators the width of the copper in a slot reaches about 20 mm. At a voltage at the stator-winding terminals of 15.7 kV, the double-sided insulation thickness in the end-winding region is also about 20 mm. The thickness of the copper shields ranges from 10 to 30 mm. That is, the dimensions of the elements entering the region under consideration are comparable. Neglecting any one of them leads to large inaccuracies in the final results. For this reason, it becomes impossible to use the results of investigations by a number of authors in the field of shielding as applied to the end portions of the stator winding of a pulsed generator.

The assumptions adopted by us in solving the problem relate mainly to the configuration of the end space and the end connections.

The configuration of the end connections is taken to be somewhat simplified: the absence of rectilinear sections of the stator winding that serve as a continuation of its active part is assumed; the end space is regarded as unfolded in the tangential direction, i.e., in the form of a rectangular infinite parallelepiped.

Fig. 1. Calculation scheme for solving the problem:
I — shields, II — end jumpers, III — end shield,
IV — pressure plate, V — insulation, VI — stator iron

The adopted assumptions do not introduce an appreciable error into the field calculation for the given problem, since the end connections of the concentric winding used in pulsed generators, made within limiting dimensions, especially multipole ones, have only a small curvature, and the size of the rectilinear sections of the stator-winding end connections, which depends on the machine voltage and technological factors, is small for pulsed generators with end connections bent through an angle of 90° in comparison with the curvilinear part.

The adopted assumptions concerning the “superconductivity” ($\gamma = \infty$) of the shields along the $y$ axis make it possible to represent the field in the form of a trigonometric series. Representation of the field as a sum of a series of spatial harmonics forms the basis of the subsequent solution of the problem.

The calculation scheme of the end region, compiled on the basis of the adopted assumptions, is shown in Fig. 1. The field in the space between the end portions and the shields in the resulting system may be regarded as plane-parallel. The solution of the problem is carried out with the aid of the vector-potential function of the magnetic field $A$. With the adopted configuration of the system under consideration, it is convenient to study the electromagnetic field in a Cartesian coordinate system; and since the system is arranged symmetrically with respect to the $x$ axis, it is sufficient to calculate and investigate the field in only one of the parts of the space under consideration. The distribution of induction in the end region can be found by first determining the vector potential of the magnetic field as a function of the coordinates. The determination of the vector potential is carried out in four regions, on

which divide the space under study [2]. In region I, for \(0<y<\dfrac{b}{2}\), the equation

\[ \frac{\partial^2 \dot A_1}{\partial x^2}+\frac{\partial^2 \dot A_1}{\partial y^2}=-\mu_0 \dot\delta . \tag{1} \]

is valid. In regions II and IV, for \(\dfrac{b}{2}<y<\dfrac{H}{2}\) and \(\dfrac{H}{2}+d<y<\infty\), the equation

\[ \frac{\partial^2 \dot A_{2,4}}{\partial x^2}+\frac{\partial^2 \dot A_{2,4}}{\partial y^2}=0 . \tag{2} \]

is valid. In region III, for \(\dfrac{H}{2}<y<\dfrac{H}{2}+d\), the equation

\[ \frac{\partial^2 \dot A_3}{\partial x^2}+\frac{\partial^2 \dot A_3}{\partial y^2}=-j\omega\gamma\mu \dot A_3 , \tag{3} \]

is valid, where

\[ \begin{aligned} \dot\delta&\text{ is the current density,}\\ \gamma,\ \mu&\text{ are the electrical and magnetic conductivity of the shield material,}\\ \omega&\text{ is the angular frequency.} \end{aligned} \]

Represent the current density in the form of a trigonometric series

\[ \dot\delta=\sum_{n=1}^{\infty}\dot\delta_n(y)\sin kx, \tag{4} \]

where \(\dot\delta_n(y)\), in the general case, depends on \(y\) and is determined by the known Fourier formulas.

The vector potential of the magnetic field is a periodic function of \(x\) with the same period as \(\dot\delta\); then the general expression for the vector potential will be

\[ \dot A=\sum_{n=1}^{\infty}\dot A_n(y)\sin kx. \tag{5} \]

Solving equations (1)–(3), taking into account (4) and (5), presents no particular difficulties. Using the equalities of the normal and tangential components of the induction vectors at the boundaries of adjacent zones, and also taking into account that, as \(y\to\infty\), \(\dot A_4=0\), and at \(y=0\), \(\dfrac{\partial \dot A_1}{\partial y}=0\), the integration constants for the vector potentials of the corresponding regions are determined.

The components of the induction at any point of the space under consideration are determined as

\[ \dot B_x=\frac{\partial \dot A}{\partial y},\qquad \dot B_y=-\frac{\partial \dot A}{\partial x}. \tag{6} \]

The inductance of the shielded system of conductors is determined by integrating the square of the induction over the space occupied by the field, that is, in regions 1, 2, 3, and 4.

\[ \dot L_{\mathrm{e}}=\frac{1}{\mu_0 I^2}\sum_{i=1}^{4}\int_v \dot B_i^2\,dv, \tag{7} \]

where

\[ \dot B_i^2=\dot B_{xi}^2+\dot B_{yi}^2 \]

is the induction in the region under consideration;
\(i\) is the number of the region under consideration.

After transformations we obtain

\[ \dot L_{\mathrm{e}}=8\mu_0 l\,\frac{N}{I^2}\,\sum \dot\lambda_{\mathrm{e}i}, \tag{8} \]

where

\[ \lambda \]

is the permeance of the leakage flux of the end windings.

\[ \sum \dot\lambda_{\mathrm{e}i}=\dot\lambda_{\mathrm{e}1}+\dot\lambda_{\mathrm{e}2}+\dot\lambda_{\mathrm{e}3}+\dot\lambda_{\mathrm{e}4}; \]

\[ \dot\lambda_{\mathrm{e}1} =\sum_{n=1,3}^{\infty}\frac{1}{n^5} \left[ M\left(1+\frac{M}{2}\operatorname{ch}k\frac{b}{2}\right) \operatorname{sh}k\frac{b}{2} +k\frac{b}{2} \right]\cos^2 k\Delta; \tag{9} \]

\[ \dot\lambda_{\mathrm{e}2} =\sum_{n=1,3,\ldots}^{\infty}\frac{1}{n^5} \left\{ \left[2m^2\operatorname{sh}kh+(1+m^2)\operatorname{ch}kh\right] \operatorname{sh}^2\beta d\,\operatorname{sh}kh +\right. \]

\[ \left. +me^{2kh}\left[m\operatorname{ch}\beta d+(1+m^2)\operatorname{sh}\beta d\right] \operatorname{ch}\beta d \right\} \frac{\operatorname{ch}^2 k\frac{b}{2}}{M^2}\cos^2 k\Delta; \tag{10} \]

\[ \dot\lambda_{\mathrm{e}3} =\sum_{n=1,3,\ldots}^{\infty}\frac{1}{n^5} \left[ \frac{1}{4}(1+m^4)(2\beta d-\operatorname{sh}2\beta d) -\frac{m}{2}(2\beta d+\operatorname{sh}2\beta d) +\right. \]

\[ \left. +(1+m^2)(\operatorname{ch}2\beta d-1) \right] \frac{\operatorname{sh}^2 k\frac{b}{2}}{M^2}\cos^2 k\Delta; \tag{11} \]

\[ \dot\lambda_{\mathrm{e}4} =\sum_{n=1,3,\ldots}^{\infty}\frac{1}{n^5} m^2\frac{\operatorname{sh}^2 k\frac{b}{2}}{M^2}\cos^2 k\Delta. \tag{12} \]

In the expressions obtained above (8)—(12), the following notation has been adopted:

\[ M=me^{k\frac{H}{2}}\operatorname{ch}\beta d+ \left( \operatorname{sh}k\frac{H}{2} +m^2\operatorname{ch}k\frac{H}{2} \right)\operatorname{sh}\beta d; \]

\[ M_1= \frac{\operatorname{sh}k\frac{b}{2}}{Me^{k\frac{H}{4}}} (1-m^2)\operatorname{sh}\beta d -e^{-k\frac{b}{2}}; \]

\[ N=\frac{T^4}{\pi^5}\frac{I^2}{(ab)^2}; \qquad k=\frac{n\pi}{T}; \qquad m=\frac{\beta}{k}; \]

\[ \beta=\sqrt{k^2+p^2} \qquad p^2=j\omega\gamma\mu. \]

The inductance is a complex quantity. Its imaginary part is due to losses from eddy currents in the shield. It amounts to as much as 10% of the total inductance of the end windings.

In the absence of shields 1 (Fig. 1), the expressions for (9) and (10) take the form

\[ \dot{\lambda}_1=\sum_{n=1,3,\ldots}^{\infty}\frac{1}{n^5} \left[ 2e^{-k\frac{b}{2}} \left( \frac{e^{-k\frac{b}{2}}}{2}\operatorname{ch} k\frac{b}{2}-1 \right) \operatorname{sh} k\frac{b}{2} +k\frac{b}{2} \right]\cos^2 k\Delta; \tag{13} \]

\[ \dot{\lambda}_2=\sum_{n=1,3,\ldots}^{\infty}\frac{1}{n^5}e^{-kb}\operatorname{sh}^2 k\frac{b}{2}\cos^2 k\Delta . \tag{14} \]

The inductance in this case, according to (8), is equal to

\[ \dot{L}_1=8\mu_0 l\frac{N}{I^2}\left(\dot{\lambda}_1+\dot{\lambda}_2\right). \tag{15} \]

From equations (8) and (15) one can find a general expression for the coefficient that accounts for the reduction of any leakage inductance,

\[ k_g=Re\,\frac{\dot{L}_э}{\dot{L}_1} =Re\,\frac{\displaystyle\sum_{i=1}^{4}\dot{\lambda}_{эi}} {\dot{\lambda}_1+\dot{\lambda}_2}, \tag{16} \]

and the shielding coefficient

\[ k_э=Re\,\frac{\dot{\lambda}_{э4}}{\dot{\lambda}_{э2}} . \tag{17} \]

It should be noted that expression (17), under the assumptions adopted in solving the problem, is valid only in the case when the ferromagnetic surfaces of the shields are removed to a comparatively large distance from the surface of the end jumpers. In this connection it is expedient to characterize, by the shielding coefficient, the degree of reduction of the leakage fluxes of the end portions in the region of the end shield and other ferromagnetic surfaces in the end zone of the pulse generator,

\[ k_э=\frac{\left|B_{э4}\right|}{\left|B_{э2}\right|}, \tag{18} \]

where

\[ \left|B_{э4}\right|,\ \left|B_{э2}\right| \]

are, respectively, the moduli of the induction in regions 4 and 2 when the end portions are enclosed in shields.

\[ \left|B_{э4}\right| = N_1\sum_{n=1,3,\ldots}^{\infty} \frac{im}{n^2} e^{-k\left[y-\left(\frac{H}{2}+d\right)\right]} \frac{\operatorname{sh} k\frac{b}{2}}{M} \cos k\Delta; \tag{19} \]

\[ \begin{aligned} \left|B_{э2}\right| &= N_1\sum_{n=1,3,\ldots}^{\infty} \frac{1}{n^2} \Biggl\{ \frac{1}{2} \left[ (1+m^4)\operatorname{ch}2k\left(\frac{H}{2}-y\right) +(1-m^4)\cos 2kx \right.\\ &\qquad\left. +2m^2\operatorname{sh}2k\left(\frac{H}{2}-y\right) \right]\operatorname{sh}^2\beta d +me^{k\left(\frac{H}{2}-y\right)} \Bigl[ (\sin^2 kx+m^2\cos kx)\, \operatorname{sh}k\left(\frac{H}{2}-y\right) \\ &\qquad +(\cos^2 kx+m^2\sin^2 kx)\operatorname{ch}k\left(\frac{H}{2}-y\right) \Bigr]\operatorname{sh}2\beta d + \end{aligned} \]

\[ + j m^2 e^{2k\left(\frac{H}{2}-y\right)} \operatorname{ch}^2 \beta d \, \frac{1}{2}\frac{\operatorname{sh} k \frac{b}{2}}{M}\cos k\Delta; \tag{20} \]

\[ N_1=-\frac{4\mu_0\delta_0}{\pi^2}T. \tag{21} \]

In particular, when determining the magnitude of the induction \(B_{e4}\) and \(B_{e2}\) at the boundaries (respectively \(y=\frac{H}{2}\), \(y=\frac{H}{2}+d\)) and substituting their values into expression (18), the coefficient \(k_e\) will characterize the penetration of the field into the body of the shield.

The analytical expressions obtained in determining the coefficients \(k_d\) and \(k\) from formulas (16) and (18) proved to be rather cumbersome; however, their implementation on an electronic digital computer did not present great difficulty.

Calculations show the exceptional influence of the distance between the end parts of the stator winding and the shields, as well as of the geometrical dimensions of the bars of the end jumpers, on the value of the leakage inductance.

Table 1

Table 1 gives calculated data for the leakage inductance as \(h\) is increased from \(0{,}1b\) (\(h=0{,}1b\) corresponds to the minimum permissible insulation thickness of the end part of the winding) to \(1b\) (where \(b\) is the width of the bar of the end jumper, taken as unity), for \(\Delta=0{,}5\) and \(0{,}6\) at ratios of the transverse dimensions of the bar

\[ \frac{a}{b}=3 \quad \text{and} \quad 6. \]

The calculations were carried out at an exciting-field frequency of \(1000\) Hz and a copper-shield thickness \(d=12\) mm.

As can be seen from the table, as the value of \(h\) increases within the indicated limits, the inductance of the system increases almost by a factor of \(4\)—\(4{,}5\).

Fig. 2. Dependence of the coefficient \(k_d\) on frequency

Fig. 3. Dependence of the coefficient \(k_e\) on shield thickness

Figure 2 shows the dependence of the damping coefficient on frequency for the ratio \(\dfrac{a}{b}=4\). As can be seen, as the frequency decreases, the effectiveness of using shields decreases. Thus, for the same shield thickness and end-winding geometry, the inductance decreases at \(50\ \text{Hz}\) by a factor of \(1.3\)–\(1.4\) compared with \(25\ \text{Hz}\), and by a factor of \(1.9\)–\(2\) compared with \(12.5\ \text{Hz}\). The dependence of the shielding coefficient \(k_{\mathrm{e}}\) on the shield thickness \(d\) is shown in Fig. 3. It follows from the figure that the use of shields with a thickness equal to 3–4 values of the penetration depth of the electromagnetic wave practically completely shields the surrounding space from the leakage fluxes of the end windings.

In order to verify the theoretical propositions and computer calculations, experimental investigations were carried out on a physical model of the end region of the stator winding of a pulse generator. For the ratios \(\dfrac{h}{b}=0.5\) and \(\dfrac{a}{b}=4\) (the model data), the results of the experimental investigations showed satisfactory agreement with the calculation results (in Fig. 2 the experimental data are indicated by an asterisk*).

Thus, the damping and shielding coefficients are complex functions of many quantities characterizing the end region of a pulsed-power generator, the principal ones being the shield material and its thickness, the frequency, and the distance between the shield walls and the end jumpers.

*

REFERENCES

1. V. A. Govorkov. Electric and Magnetic Fields. Moscow, Energiya, 1968.

2. K. Binns, P. Lawrenson. Analysis and Calculation of Electric and Magnetic Fields. Moscow, Energiya, 1970.