ON THE INFLUENCE OF SHIELDING THE END WINDINGS OF A SHOCK GENERATOR

G. A. Sipaylov, V. F. Kulakov, V. I. Popov

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

The leakage inductive reactance of the end windings of the stator winding of a shock generator constitutes a significant part of the subtransient reactance, especially in designs with a large ratio of pole pitch to length. In order to obtain maximum shock power, the leakage of the end windings should be reduced; this is achieved, on the one hand, by choosing such a winding configuration in which the leakage flux will be as small as possible, and, on the other hand, by shielding the end windings with electrically conducting caps or screens.

The eddy currents of the shield weaken the field of the end winding, which leads to a reduction in its inductance and to a reduction in electrodynamic forces.

The question of the necessity and effectiveness of shielding the leakage flux of the end windings of shock-generator windings has not yet been clarified. Some authors consider the use of current-conducting shields inexpedient and recommend fastening the end windings of the stator windings of generators of interrupting and shock power with getinax plates [1]. Others suppose that the leakage of the end windings can be reduced by 50–60% [2, 3]. Experimental investigations of the shielding and damping action of shields made of materials with different conductivities, carried out at the Tomsk Polytechnic Institute [4], show their considerably greater effectiveness.

Attempts to solve the shielding problem by representing the shield as a circuit with definite inductance, mutual inductance, etc., lead to very approximate results [5]. The general method of solution consists in treating the shielded space and the shield as a region in which the magnetic field is investigated and calculated. Such a method of investigation leads to integration of Maxwell’s equations in a region of which the body of the shield is also a part.

In the present case we solve the problem with the aid of the vector-potential function \(A\) in a rectangular coordinate system, by a method in principle similar to that set forth in [6]. The action of the shield is most effective if the latter is located in the immediate vicinity of the end windings. With a small curvature of the end region and with a sufficiently small distance from the shields to the winding, the field distortion introduced by the nonrectilinearity proves insignificant. Therefore, in a first approximation, the field distortion due to the nonrectilinearity of the end structure when shields are used may be neglected.

Let us consider a group of bars of one phase of the end part of a single-layer...

of the winding, surrounded by a nonmagnetic shield with electrical conductivity \(\gamma\) and magnetic permeability \(\mu=\mu_0\) (Fig. 1). The bars have the same length, equal cross section, and are uniformly distributed over the width \(2a\) of the shield.

Let us divide the space under consideration into three regions (Fig. 1), in which the potential functions \(A_{\mathrm{I}}, A_{\mathrm{II}}, A_{\mathrm{III}}\) must satisfy the equations

\[ \frac{\partial^2 A_{\mathrm{I}}}{\partial x^2} + \frac{\partial^2 A_{\mathrm{I}}}{\partial y^2} = -\mu_0 \Delta \tag{1} \]

\[ \frac{\partial^2 A_{\mathrm{II}}}{\partial x^2} + \frac{\partial^2 A_{\mathrm{II}}}{\partial y^2} = 0 \tag{2} \]

\[ \frac{\partial^2 A_{\mathrm{III}}}{\partial x^2} + \frac{\partial^2 A_{\mathrm{III}}}{\partial y^2} = j k^2 A_{\mathrm{III}} \tag{3} \]

Fig. 1

Since the origin of coordinates is located at the center, it is convenient to specify the current density \(\Delta\) by expansion in a cosine series:

\[ \Delta = \frac{a}{c}\Delta_0 + \sum \frac{2}{\pi}\Delta_0 \frac{1}{n}\cos ma \cdot \cos mx, \tag{4} \]

where

\[ \Delta_0 = I e^{j\omega t}\frac{N}{4ab}; \]

\[ m=\frac{\pi}{c}\cdot n;\qquad k=\sqrt{\omega\gamma\mu_0}; \]

\(i\) — current amplitude,
\(N\) — number of shielded rods,
\(\omega\) — angular frequency,
\(n\) — any integer,
\(c=a+h_{\mathrm{i}}+d\) — half-period of the expansion,
\(h_{\mathrm{i}}\) — insulation thickness.

The solution of equations (1), (2), (3):

\[ A_{\mathrm{I}}=-\mu_0\Delta_0\left[ \frac{a}{2c}x^2+\sum\left(I_1 B\operatorname{ch}my-\frac{2\cos ma}{\pi\cdot n\cdot m^2}\right)\cos mx \right] \tag{5} \]

\[ A_{\mathrm{II}}=\mu_0\Delta_0\sum I_1(D\operatorname{sh}my+E\operatorname{ch}my)\cos mx \tag{6} \]

\[ A_{\mathrm{III}}=\mu_0\Delta_0\sum I_1 F e^{-\sqrt{m^2+jk^2}\,y}\cos mx, \tag{7} \]

where

\[ I_1=\frac{2}{cm^3}\,\kappa_1, \]

\[ \kappa_1=\frac{a}{c}(\sin ma-ma\cos ma)+\cos ma. \]

The unknown coefficients \(B, D, E, F\) can be found from the boundary conditions and, in the case when the penetration depth \(d=\dfrac{\sqrt{2}}{k}\ll c\), characteristic for the problem under consideration, take the form:

\[ B=\frac{1}{\operatorname{sh}mb+\operatorname{ch}mb}+\operatorname{sh}mb(1-\operatorname{th}mH) \tag{8} \]

\[ D=\operatorname{sh}mb; \tag{9} \]

\[ E=\operatorname{sh}mb\,\operatorname{th}mH \tag{10} \]

\[ F=\frac{m}{k}\operatorname{sh}mb\,e^{-\frac{H}{d}(1+j)}e^{-j\frac{\pi}{4}}, \tag{11} \]

The field pattern (Fig. 2), constructed in equipotential lines \(A=\mathrm{const}\) according to equations (5), (6), (7) for one quarter of the shielded space, gives an idea of the nature of the distribution of fluxes in the end part of the winding and the shield.

Fig. 2. Field pattern of shielded conductors

The inductance of the shielded system of conductors is determined by integrating the square of the induction over the space occupied by the field, i.e., in regions I, II, III.

\[ L=\frac{1\displaystyle\int^x\!\!\int^y B^2\,dy\,dx}{\mu_0 i^2}, \tag{12} \]

where \(B^2=B_x^2+B_y^2\);

\[ B_x=\frac{\partial A}{\partial y}; \qquad B_y=-\frac{\partial A}{\partial x}, \]

\(l\) is the length of the shielded bars.

After transformations we obtain:

\[ \dot L=\mu_0 lN^2\frac{c^4}{8a^2b^2}\sum \lambda, \tag{13} \]

where

\[ \sum\lambda=\lambda_{\mathrm I}+\lambda_{\mathrm{II}}+\lambda_{\mathrm{III}}, \]

\[ \lambda_{\mathrm I}=\frac{a^5b}{3c^6}+\sum \kappa_n\left[ B^2\operatorname{sh}2mb+2\kappa_a^2mb-4\kappa_a B\operatorname{sh}mb+ \frac{8a}{c}\frac{\cos \pi n}{l_1m^2}(B\operatorname{sh}mb-\kappa_a mb) \right] \tag{14} \]

\[ \lambda_{\mathrm{II}}=\sum \kappa_n\operatorname{sh}^2mb\left[(\operatorname{sh}2mH-\operatorname{sh}2mb)(1+\operatorname{th}^2mH)- 2\operatorname{th}mH(\operatorname{ch}2mH-\operatorname{ch}2mb)\right]; \tag{15} \]

\[ \lambda_{\mathrm{III}}=2\sqrt{2}\,d\,e^{-j\frac{\pi}{4}}\sum \kappa_n m\operatorname{sh}^2mb; \tag{16} \]

where

\[ \kappa_n=\frac{\kappa_1^2}{\pi^5n^5}; \qquad \kappa_a=\frac{\cos ma}{\kappa_1}. \]

The inductance \(\dot L\) is a complex quantity. Its imaginary part \(R_{\mathrm{vn}}=J_mL\) is due to eddy-current losses in the shield and, as calculations show, for practical designs amounts to from 1 to 10% of the total inductance of the end portions.

The total permeance \(\sum\lambda\) of the leakage flux of the shielded part of the winding is determined by the ratio of the dimensions of its cross section, the distance to the shield, the ratio of these dimensions, and the penetration depth. \(\lambda_{\mathrm I}\) determines the flux leakage directly in the zone of the winding conductors, \(\lambda_{\mathrm{II}}\) in the region between the conductors and the shield, occupied by insulation, and \(\operatorname{Re}\lambda_{\mathrm{III}}\) the leakage in the body of the shield. Practical calculation of the permeance coefficients for a specific geometry of the end portion by formulas (14, 15, 16) presents no particular difficulties, since the series converge well and it is usually sufficient to calculate 3–4 terms.

As an example illustrating the degree of shielding, Fig. 3 presents curves of the dependence of \(\sum\lambda\) on the ratios \(\frac{b}{c}\) and \(\frac{b}{H}\) for winding bars whose height \(2b=5\) cm is most typical for large shock generators. The penetration depth into the copper shield at a frequency of 50 Hz is \(d=0.94\) cm. For comparison, the leakage permeances for a frequency of 25 Hz were calculated (dashed curves).

As can be seen from the graph, there is a strong dependence of the inductance of the end portions on the distance between the walls of the shield and the winding copper, and when the ratio \(\frac{b}{H}\) approaches unity, the inductance is almost completely determined by the field inside the conductor system and by leakage in the skin layer.

Fig. 3. Dependence of the leakage conductance on the relative dimensions of the shielded system

Experimental verification of the inductance of the shielded conductor system was carried out on a simple model consisting of a flat annular coil with a diameter of about 35 cm. The ratio of the winding height to the width of its cross section was

\[ \frac{b}{a}=0.15. \]

The relatively small radial dimensions make it possible to neglect the curvature of the coil placed in the shield. The measurement was performed at a frequency of 1000 Hz, at which the penetration depth is \(d=2\) mm. The modeling condition

\[ \frac{b}{d}=\mathrm{idem} \]

makes it possible to compare the experimental results with the calculated data presented in Fig. 3. It is seen that the points corresponding to the experiment agree fairly well with the results of the calculation by the method described.

The known law of field distribution also makes it possible to find the electrodynamic forces acting on the conductors of the shielded part of the winding.

\[ F_x=\frac{lI}{S_{\mathrm{cm}}}\int_{-b}^{+b}\int_{x_1}^{x_2} B_{y_1}\,dx\,dy \tag{17} \]

\[ F_y=\frac{lI}{S_{\mathrm{cm}}}\int_{-b}^{+b}\int_{x_1}^{x_2} B_{x_1}\,dx\,dy . \tag{18} \]

The component of the forces \(F_y\) on any bar is equal to zero if the conductors are arranged symmetrically with respect to the walls of the shield.

After integration of expression (17), the tangential forces on the winding bars are determined as

\[ F_x=\mu_0 \frac{2I^2N}{\pi^4}\frac{c^3}{ab}\frac{l}{a_c b} \sum \frac{1}{n^4}\left(\frac{\pi nb}{c}-\operatorname{Bch} mb\right) k_1 \cdot \sin \frac{\pi n(x_2+x_1)}{2c}\sin \frac{\pi n(x_2-x_1)}{2c}, \tag{19} \]

where

\[ 2a_c\cdot b=S_{\mathrm{st}} \]

is the cross section of the bar,

\(x\) is the coordinate of the bar,

\(x_2-x_1=a_c\) is the width of the winding bar,

\[ x_2+x_1=\frac{2a}{N}(2n_c-1). \]

\(n_c\) is the number of the bar for which the force is determined, counting from the middle.

To estimate the influence of shielding on the magnitude of the electrodynamic forces, a model was made; its general view with the upper shield removed is shown in Fig. 4. The forces acting on conductors placed in massive copper shields were measured by means of strain gauges and a four-channel strain-gauge amplifier. The measurements showed fairly good agreement with the calculation by formula (19). The nature of the distribution of forces over the bars of the model winding

Fig. 4. View of the model for strain-gauge measurements of forces in shielded conductors

is shown in Fig. 1,b. The relative distance to the shield, \(в/Н\), was about 0.4; however, even with such a considerable removal of the shields, the forces decreased by approximately 40% (curve 1) in comparison with the forces on conductors placed in air (curve 2). The points corresponding to the calculation by formula (19) agree fairly well with experimental curve 2.

Theoretical and experimental investigations of the complete shielding of conductors in conducting shields make it possible to conclude that the use of nonmagnetic shields of high electrical conductivity, for example copper ones, is expedient and effective. The degree of shielding of end-winding parts depends on the proximity of the shields to the winding. The shield substantially relieves the end-winding part from electrodynamic forces, facilitating the problem of fastening the conductors. Increasing the shield thickness beyond \(3d\) does not enhance the shielding effect.

REFERENCES

1. I. Tittel. Der neue 4300 Mva Stoßleistunggenerator für das Schaltwerk der Siemensschuckertwerke, Siemens—Zeitschrift, 1962, H. 8, S. 562—575.

2. I. M. Postnikov. Trudy LPI, No. 2, 1937.

3. Mitsubishi Denki Giho, 1964, v. 38, No. 9, p. 9—10.

4. K. A. Khorkov. Problems in the design of impulse-power generators. Dissertation, Tomsk, 1964.

5. I. Hak. Abschirmung von eisenlosen Spulen durch Platten und geschlossene Gefäße, Hochfr. u. Electroak., 45, 1935.

6. V. F. Kulakov, G. A. Sipailov. Supertransient inductive reactance of an impulse generator with a nonmagnetic active layer, Izvestiya TPI, vol. 212.