On the Calculation of the Electric Field Strengths of a Capacitive Generator

V. A. Lukutin, E. D. Mogilevskaya

(Presented by the scientific seminar of the Department of TOE)

A direct-current capacitive generator is a high-voltage apparatus, and for its reliable operation it is necessary to ensure high electrical strength of all components of this machine. The electrical strength of particular structural elements is determined not only by the quality of the dielectrics from which they are made, but also by the electric-field strengths in all portions of these elements. When designing machines, a number of measures should be taken to create the most uniform distribution of potential gradients and to reduce them in those zones where the electrical strength of the medium is lowest.

To study the influence of the design features of generators on the magnitude of the field strength in zones with the lowest electrical strength, and also to study the dependence of these strengths on the operating regime of the machine, an appropriate mathematical apparatus is required that makes it possible to solve the problem posed.

In this article approximate analytical functions are proposed, with the aid of which the electric-field strengths in the working space of the generator are readily determined.

In a number of works [1, 2], approximate analytical calculations of the electric field of the machine were carried out on the basis of the superposition principle and using the Ritz variational method.

Here we shall restrict ourselves to considering the field only in the rotor–stator gap, where the electrical strength of the medium is the lowest. The potential function for this region was found [1, 2] in the form of a sum of components due to the charges of the stator and rotor. Below we shall determine separately the projection of the electric-field-strength vector onto the rotor surface (the tangential projection \(E_x\)) and the projection perpendicular to this surface (the normal component \(E_y\)). To compute them, we differentiate the potential function for the gap with respect to \(x\) and \(y\), respectively.

\[ E_y(x,y) = -\frac{\partial U_0(x,y)}{\partial y} = -\frac{\partial U_{\mathrm{co}}}{\partial y} -\frac{\partial U_{\mathrm{pao}}}{\partial y} -\frac{\partial U_{\mathrm{pbo}}}{\partial y} = \tag{1} \]

\[ = E_{yc}+E_{ypa}+E_{ypb}. \]

The subscripts denote the components due to the charges of the stator and rotor, and \(U_0(x,y)\) is the potential function for the rotor–stator gap region. The following functions enter into (1):

\[ E_{yc}(x,y)= U_{\mathrm{H}} \frac{0.25\,\varepsilon_1 y}{\varepsilon_0\,1638\,l\Delta} \left[ F(1-x^2)^2+L(1-x^2)^8 \right], \tag{2} \]

\[ E_{y\mathrm{ra}}(x,y)= y\,\frac{(U_{\mathrm{v}}+U_{\mathrm{n}}K_1(m+n^2)(F+L))\{M[1-(x\mp1)^4]+N[1-(x\mp1)^{12}]\}} {(m+n^2)\{M[1-(1-a)^4]+L[1-(1-a)^{12}]\}l}, \tag{3} \]

\[ E_{y\mathrm{rb}}= \frac{U_{\mathrm{v}}\varepsilon_1}{2[\varepsilon_1(d-n)+\varepsilon_0(n-r)]l}. \tag{4} \]

The last equations are valid for the region \(n\le |y|\le d\), \(-1\le x\le 1\).

Taking into account the symmetry of the field, the coordinate axis \(x\) is directed along the axes of the carriers in the direction of rotation of the rotor, while the axis \(y\) is perpendicular to the rotor surface and passes through the middle of the high-voltage inductor.

In formulas (2, 3, 4) the following notation has been adopted:

\[ F=34.3(88n^2+25+110m)(7D_2+6D_4)- \]

\[ -2.96(112n^2+75+140m)(19D_2+6D_4); \]

\[ L=100(24n^2+20+5m)(19D_2+6D_4)- \]

\[ -14.32(112n^2+75+140m)(7D_2+6D_4); \]

\[ \Delta=16.45n^4+18.1n^2+3.62+4.9m(4.62n^2+3.54)+2.6m^2; \]

\[ M=(260n^2+92+325m)[5-(1-a)^4]- \]

\[ -5.77(8.84n^2+5.4+11.05m)[13-(1-a)^{12}]; \]

\[ N=(36.5n^2+28.4+45.6m)[13-(1-a)^{12}]- \]

\[ -15(8.84n^2+5.4+11.05m)[5-(1-a)^4]; \]

\[ K_1=\frac{0.25\varepsilon_1}{\varepsilon_0 1638\Delta}, \qquad m=\frac{\varepsilon_1}{\varepsilon_0}(d^2-n^2). \]

The listed coefficients depend only on the dimensions of the machine and on the properties of the dielectrics from which it is made. These dimensions, as well as the coordinates \(x,y\), are taken relative to the dimension \(l\), equal to the distance between the centers of adjacent low-voltage and high-voltage inductors. In the formulas the notation is:

\(2d\) — distance between the stator surfaces;
\(2n\) — rotor thickness;
\(2b\) — width of an inductor;
\(2r\) — diameter of the carriers;
\(2a\) — distance between the axes of adjacent carriers;
\(\varepsilon_0\) — absolute dielectric permittivity of the medium in the rotor–stator gap;
\(\varepsilon_1\) — absolute permittivity of the rotor dielectric.

Now, differentiating the same potential function with respect to \(x\), we write the equation for the tangential component of the electric-field strength in the gap:

\[ E_x(x,y)=-\frac{\partial U_0(x,y)}{\partial x} =-\frac{\partial U_{\mathrm{co}}}{\partial x} -\frac{\partial U_{\mathrm{rao}}}{\partial x} =E_{x\mathrm{c}}+E_{x\mathrm{ra}}; \tag{5} \]

\[ E_{x\mathrm{c}}(x,y)=U_{\mathrm{n}}(D_2x+2D_4x^3)+ \tag{6} \]

\[ U_{\mathrm{n}}K_1(d^2-y^2)[F2x(1-x^2)+L8x(1-x^2)^7]; \]

\[ E_{x\mathrm{ra}}(x,y)= \frac{\varepsilon_1}{\varepsilon_0}(d^2-y^2)\cdot \left[ \frac{U_{\mathrm{v}}} {2(m+n^2)\{M[1-(1-a)^4]+N[1-(1-a)^{12}]\}} +\right. \tag{7} \]

\[ \left. +\frac{U_{\mathrm{n}}K_1(m+n^2)(F+L)} {2(m+n^2)\{M[1-(1-a)^4]+N[1-(1-a)^{12}]\}} \right]\,[4M(x-1)^3+12N(x-1)^{11}]. \]

Here,

\[ f_c=\frac{U_n}{2}\left(D_0+D_2x^2+D_4x^4\right) \]

is an approximate function approximating the distribution of the potential along the stator surface.

An analysis of the formulas for the field strengths shows that the components entering into these formulas are determined both by the geometrical parameters of the machine and by its operating mode. Thus, the normal component has a term \(E_{yrb}\), which is proportional to the excitation voltage, while \(E_{yc}\) is determined by the load voltage. The third term has a more complex dependence on both voltages and, when \(U_n=U_{xx}\), becomes zero.

Figure 1 presents graphs of the distribution of the normal components of the electric-field strength over the rotor surface \(y=n\) in the zone of the high-voltage inductor. They were calculated from formulas (2), (3), (4) for the specific mode \(U_n=100\ \mathrm{kV}\), \(U_b=10\ \mathrm{kV}\) of a generator with relative dimensions \(d=0.067\), \(n=0.06\), \(r=0.0033\), \(\varepsilon_1=4\varepsilon_0\), \(l=15\ \mathrm{cm}\). For points with negative ordinates \(x\), the lower signs should be taken in formula (3).

Fig. 1

For another operating mode of the machine, the relationships between the individual components of the field strength will change accordingly, but the character of the curves will be preserved.

If the generator has other dimensions, then, in addition to changing their magnitude, the components may also change their steepness.

The tangential components of the electric-field strength on the rotor surface in the same zone and for the same generator are determined by formulas (6), (7) and are plotted in Fig. 2 as solid lines. As in the preceding case, the field strength depends on the operating mode and on the geometrical dimensions of the machine. If \(E_{xr}\) and \(E_{xc}\) are summed, the resulting curve will have a very complex form, and several extremal points can be observed on it. The sections of the rotor surface with maximum field strengths are determined by the operating mode of the generator and will shift when it changes. Thus, judging from Fig. 2, the tangential component of the field strength will be greatest and approximately the same directly under the inductor and in the middle part of the rotor surface between the inductors.

In the same figure, the corresponding field-strength plots obtained on a model of the generator under consideration in an electrolytic bath are shown by dashed lines. The character of the curves and their absolute values proved to be very similar, which indicates the acceptable reliability of the proposed formulas.

Fig. 2

In addition, we note that \(E_{\mathrm{yc}}\) and \(E_{\mathrm{xc}}\) are determined to a considerable extent by the shape of the potential-distribution curve on the stator surface, or, more specifically, by the polynomial \(f_c\) approximating this distribution.

In conclusion, it may be inferred that, on the basis of the variational method for calculating the electric field, it is possible to obtain analytical formulas that are quite simple and convenient for practical use in engineering for investigating the electric-field strength in the rotor–stator gap of a direct-current capacitive generator.

References

1. A. P. Kononov, V. A. Lukutin, V. M. Osipov. An approximate method for calculating the field of an electrostatic generator. Izv. TPI, vol. 191, 1969.

2. V. A. Lukutin, E. D. Mogilevskaya. An approximate analytical calculation of the electric field of the rotor of a capacitive generator. Izv. TPI, vol. 210, 1971.