ACCOUNTING FOR CONVEYORS IN THE CALCULATION OF THE ELECTRIC FIELD OF A CAPACITIVE DC GENERATOR

V. A. LUKUTIN, E. D. MOGILEVSKAYA

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

An exact analytical calculation of the field of a capacitive generator is very laborious and complicated. However, by applying a number of simplifying assumptions, this problem can be solved approximately. Thus, in [2, 3], on the basis of the variational method and the principle of superposition, potential functions were obtained that make it possible to determine the potential of the electric field at any point of the gas gap and of the rotor surface. In this calculation, the presence of metallic rods—conveyors—in the rotor body was not taken into account. The solution was written in the form

\[ U(x,y)=U_c(x,y)+U_p(x,y). \tag{1} \]

Here \(U_c(x,y)\) is the component of the potential function obtained from the charges of the stator (the inductors and the semiconducting surface of the stator). This component characterizes the stator field. \(U_p(x,y)\) is the component induced by the charges of the rotor conveyors and describing the rotor field.

Function (1) contains the principal geometric dimensions of the machine, the excitation voltages, the loads, and takes into account the dielectric properties of the medium. There is no need to write out this formula in detail here.

Fig. 1

In the present article we shall dwell on questions related to the influence of metallic rods on the potential function of the stator. Figure 1 shows a cross section of the working space of a capacitive machine, more precisely only one quarter of this working space, since the stator field is symmetric with respect to the middle of the inductors.

Here \(2n\) is the rotor thickness, \(2r\) the diameter of the conveyors, \(2d\) the distance between the stator surfaces, \(2b\) the width of the inductors, and \(2l\) the distance between the centers of adjacent inductors.

Modeling the electrostatic field for the indicated region in an electrolytic bath showed that the approximate analytic function \(U_c(x,y)\) gives a sufficiently accurate representation, for engineering purposes, of the field potential in the rotor–stator gap. The calculated distribution of potentials along the conveyors themselves (or along the axis \(y=0\)) is the same as in the experiment only in the case of a small conveyor radius and for a zone far removed from the origin of coordinates. In the region between the inductors, however, the discrepancy between the calculated and experimental values of the potentials becomes more substantial the greater the number of conveyors and their radius.

It is easy to see that the presence of metallic conveyors leads to an increase in the potential difference between the inductor and a rod located at the point \(x=y=0\). If there are many conveyors and they are in contact with one another, this potential difference increases to \(0.5\,U_{\mathrm{n}}\).

The energy characteristics of a capacitive direct-current generator are determined to a significant extent by the electrophysical processes in the inductor zone, where charge commutation takes place. In this connection it becomes clear that the investigation of the electric field in this zone should be carried out especially carefully.

A dielectric with conducting rods embedded in it can be replaced by an equivalent layer with a certain dielectric permittivity \(\varepsilon_2>\varepsilon_1\) and thickness \(2r\). The permittivity \(\varepsilon_2\) will depend on the radius of the conveyors and their number. For its quantitative determination one can make use of simple experiments.

The entire space between the inductors is a plane capacitor with a multilayer dielectric. If the area of an inductor is \(S\), then the capacitance between the plates of the inductors without taking the conveyors into account will be

\[ c=\frac{S\varepsilon_0\varepsilon_1}{2(d-n)\varepsilon_1+2n\varepsilon_0}. \tag{2} \]

When the conveyors are replaced by a dielectric layer with permittivity \(\varepsilon_2\), the capacitance of the inductors will be

\[ c_{\mathrm{e}}=\frac{S\varepsilon_0\varepsilon_1\varepsilon_2}{2(d-n)\varepsilon_1\varepsilon_2+2(n-r)\varepsilon_0\varepsilon_2+2r\varepsilon_0\varepsilon_1}. \tag{3} \]

Now let us model the zone between the inductors and measure the conductivity between the plates, first for the case without conveyors and then with conveyors. The conductivities obtained on the basis of the field-analogy method will correspond to the capacitances in formulas (2) and (3).

Dividing the right- and left-hand sides of equations (2) by the corresponding parts of equation (3), and replacing capacitances by conductivities, we obtain a formula for determining the equivalent dielectric permittivity

\[ \varepsilon_2=\frac{G_{\mathrm{e}}r\varepsilon_0\varepsilon_1}{(G-G_{\mathrm{e}})\left[\varepsilon_1(d-n)+\varepsilon_0 n\right]+G_{\mathrm{e}}r\varepsilon_0}, \tag{4} \]

where

\(G_{\mathrm{e}}\) is the conductivity between the plates in the presence of conveyors,

\(G\) is the conductivity without conveyors.

On the basis of numerous experiments it has been established that equation (4) may be replaced by the empirical formula

\[ \varepsilon_2=\varepsilon_1\left(\frac{a}{a-2r}\right)^{\frac{3}{2}}. \tag{5} \]

After the equivalent replacement of the conducting rods by a dielectric layer, one can proceed to the calculation of the electric field in the region with a multilayer dielectric (Fig. 1). The calculation procedure is set out in detail in [1, 2, 3], and here we shall confine ourselves only to the final formulas.

We solve the internal Dirichlet problem by the variational method, the potential functions for each dielectric layer being obtained separately:

\[ U_{c0}(x,y)=f_c(x)+\left[A(l^2-x^2)^2+B(l^2-x^2)^8\right](d^2-y^2)\frac{\varepsilon_2}{\varepsilon_0}, \tag{6} \]

\[ U_{c1}(x,y)=f_c(x)+\left[A(l^2-x^2)^2+B(l^2-x^2)^8\right]\left[\frac{\varepsilon_2}{\varepsilon_0}(d^2-n^2)+ \frac{\varepsilon_2}{\varepsilon_1}(n^2-y^2)\right], \tag{7} \]

\[ U_{c2}(x,y)=f_c(x)+\left[A(l^2-x^2)^2+B(l^2-x^2)^8\right][C+r^2-y^2]. \tag{8} \]

In the last equations the following notation is used:

\[ A=\frac{U_n\cdot 5}{128^2\cdot 4^{16}K} \left[34.4(88r^2+25+110C)(7D_2+6D_4)- 2.96(112r^2+75+140C)(19D_2+6D_4)\right], \]

\[ B=\frac{U_n5}{128^4/18K} \left[100(24r^2+20+5C)(19D_2+6D_4)- 14.3(112r^2+75+140C)(7D_2+6D_4)\right], \]

\[ C=\frac{\varepsilon_2}{\varepsilon_0}(d^2-n^2)+ \frac{\varepsilon_2}{\varepsilon_1}(n^2-r^2), \]

\[ K=16.45r^4+18r^2+3.62+4.9C(4.62r^2+3.54)+2.6C^2. \]

In Fig. 2, the solid curves show the equipotential lines obtained from formulas (6), (7), (8); for clarity, not the entire computational region is shown here, but only the part located under the inductors. The dashed lines were constructed by modeling this same region with conveyors present in it. In the immediate vicinity of the conducting rods, the experimental equipotentials differ from the calculated ones, and this is quite natural; however, the potentials of the conveyors themselves agree quite satisfactorily with the calculated potentials of the points at which these conveyors are placed. In the part of the region not shown in Fig. 2, the calculated and experimental equipotentials differ even less.

The foregoing indicates that the approximate analytical calculation of the electric field of the stator on the basis of the variational method gives good results even when metallic rods are present in the rotor body. These rods are replaced by an equivalent dielectric layer with a permittivity calculated from equation (5).

REFERENCES

1. S. G. Mikhlin. Variational Methods in Mathematical Physics. Nauka, 1970.
2. A. P. Kononov, V. A. Lukutin, V. M. Osipov. Approximate method for calculating the field of an electrostatic generator. Izv. TPI, vol. 191, 1969.
3. V. A. Lukutin, Yu. I. Alekseev, A. F. Kochneva. Potential function for the stator field of an electrostatic generator. Izv. TPI, vol. 210, 1971.