PARTIAL CAPACITANCES BETWEEN ESG ELEMENTS

A. P. Kononov, V. V. Patsevich, V. D. Eskov

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)

The development of an engineering calculation methodology is one of the most important problems in the further development of electrostatic generators (ESG). This problem can be solved only if sufficiently simple relationships are found between the principal characteristics of the generator (voltage, specific power, etc.) and the geometric dimensions of its structural elements.

As regards ESG with a cascade connection of conductor carriers, the currently existing methods for determining their characteristics are based on the use of electrostatic equations with lumped parameters, the application of which is possible because the electric field in an ESG satisfies quasistationary conditions [1].

Generators of this type, having a uniform distribution of potential over the stator, are more conveniently studied with the aid of equations with partial capacitances and the subsequent application of the method of electrostatic analogy. In this way one can obtain quite compact expressions in which the output parameters of the generator are related to the partial capacitances between the generator elements and to the quantitative characteristics of these elements (the number of carriers, the number of pole pairs, etc.). Analysis of these expressions shows that, among the partial capacitances, the greatest influence on the characteristics of an ESG is exerted by the partial capacitances between neighboring carriers \(C_2\) and between a carrier and the stator \(C_3\).

The usual methods of direct capacitance measurement (for example, with the aid of a ballistic galvanometer or various measuring bridges), when measuring the indicated partial capacitances, give an error comparable with the magnitude of the latter (units of picofarads). Therefore it appears more expedient to use, for this purpose, modeling methods based on mathematical similarity.

Among such methods is the electrolytic-tank method, which makes it possible to reduce the measurement error to tenths of a percent. This method is based on electrostatic analogy, i.e., on the formal similarity of the equations describing a static field in a dielectric and a stationary current field in a conducting medium.

Under identical boundary conditions, identical distribution of potential on the boundaries of the region, and geometric similarity of the original and the model, the following expression is valid:

\[ \frac{C}{\varepsilon}=\frac{g}{\gamma}m_l, \tag{1} \]

where \(C\) is the capacitance between the electrodes of the original,
\(\varepsilon\) is the dielectric permittivity of the medium surrounding them,
\(g\) is the conductance between the electrodes of the model,
\(\gamma\) is the specific electrical conductivity of the electrolyte,
\(m_l\) is the scale factor.

The measurement of partial conductances is carried out in accordance with the determination of partial capacitances. Thus, for example, in order to measure, in a certain electrode system, the partial conductance \(g_{12}\) between electrodes 1 and 2, electrode 1 should be connected to one terminal of the voltage source, and all the remaining electrodes should be connected to the second terminal of the latter. In this case the ratio of the current flowing to the second electrode to the applied voltage will be equal to the desired partial conductance. The corresponding partial capacitance is found with the aid of equation (1).

As a first step in measuring the partial capacitances between elements of the ESG structure, we carried out experiments in a system of parallel cylinders located at equal distances from two infinitely extended planes (with a uniform arrangement of the cylindrical electrodes). This model was chosen for the following reasons:

1) the conditions for obtaining maximum specific power in an ESG require a large number of carriers per pole pitch of the stator [2], which in turn entails a reduction in the transverse dimensions of the latter; from the point of view of manufacturing technology, the most expedient form will probably be a circular cross section of the carrier (with its length considerably exceeding its diameter);

2) in a cylindrical generator the axes of the carriers are parallel to one another, but are arranged along a circumference; in a disk generator, however, they lie in one plane but are not parallel; deviations from the chosen model will decrease as the number of carriers increases.

Without dwelling on the method of modeling in an electrolytic bath, which is described in sufficient detail in [3, 4], we note that the measurements were performed with alternating current at frequencies of 50 and 1500 cps; a weak solution of copper sulfate in distilled water was used as the electrolyte (\(\gamma\) of the order of \(10^{-2}\ 1/\Omega\cdot\text{m}\)), and the electrodes were made of brass. From the measurement results the following empirical formulas were derived (measurement error no more than 2%, approximation error within 5%):

\[ \frac{C_{\mathrm{e}}}{\varepsilon l}=\frac{2.15a}{d_{\mathrm{pr}}+r}, \tag{2} \]

\[ \frac{C_2}{C_{\mathrm{e}}}=\frac{d_{\mathrm{pr}}}{kr}\left(\mathrm{SC}\frac{\pi r}{a}-1\right)^{0.8}, \tag{3} \]

where \(r\) is the radius of the carrier,
\(l\) is its length,
\(a\) is the distance between the axes of the carriers,
\(\varepsilon\) is the dielectric permittivity of the solid dielectric surrounding them;

\[ d_{\mathrm{pr}} = d + \frac{\varepsilon}{\varepsilon_0} d_0, \]
where \(d\) is the thickness of the layer of this dielectric above the surface of the transporters, and \(d_0\) is the thickness of the gas gap; \(k\) is a coefficient equal to 3 for a single-row arrangement of the transporters (Fig. 1, \(a\)) and \(4/3\) for their two-row arrangement (Fig. 1, \(b\)).

The indicated quantities in our experiments varied within the following limits:
\[ 1.25 \leq \frac{a}{2r} \leq 3.75;\quad 1.25 \leq \frac{d_{\mathrm{pr}}}{2r} \leq 15.75;\quad 0.5 \leq \frac{d}{2r} \leq 5; \]
\[ 0 \leq \frac{d_0}{2r} \leq 2;\quad 1 \leq \frac{\varepsilon}{\varepsilon_0} \leq 6. \]

Fig. 1. Schematic representation of a single-row (\(a\)) and two-row (\(b\)) arrangement of transporters embedded in a dielectric:
1—transporters, 2—solid dielectric, 3—gas, 4—stator.

For the two-row arrangement of the transporters, formulas (2) and (3) are applicable only when the distance between the rows exceeds the diameter of the transporter. Since such an arrangement is expedient only when

small diameters of the transporters, then in actual generators this ratio is satisfied.

The application of expressions (2) and (3) to the calculation of the capacitance between the rotor and stator and of the short-circuit current of experimental generators made at TPI, and direct measurement of these quantities, showed good agreement of the results (within the experimental error). This gives grounds to hope that the formulas obtained can be used in engineering calculations of ESGs with a cascade connection of transporter-conductors.

REFERENCES

1. D. White and G. Woodson. Electromechanical Energy Conversion. “Energiya,” 1964.

2. Edited by V. I. Levitov and A. G. Lyapin. Application of Electric-Field Forces in Industry and Agriculture, VNIIEM. Moscow, p. 15, 1964.

3. I. I. Tetelbaum. Electrical Modeling. Fizmatgiz, 1959.

4. W. Karplus. Analog Simulation: Solution of Field Problems. IL, 1962.