Determination of Partial Capacitances in a System of Cylindrical Electrodes Taking into Account the Edge Effect
Abstract Generated abstract
The paper develops approximate engineering relationships for partial capacitances in rod type electrostatic generators with cylindrical transporter electrodes, including finite length edge effects. It uses electrolytic bath modeling under geometrically similar boundary conditions to measure partial conductances, which are then converted to capacitances, focusing on capacitance between a transporter and stator elements and between neighboring transporters. Empirical formulas are derived over specified ranges of electrode spacing, stator separation, and transporter length, with reported approximation errors of about 8 percent for large numbers of carriers. Additional experiments for radially arranged rods yield correction formulas for disk electrostatic machines, with a maximum reported error of 10 percent.
Full Text
DETERMINATION OF PARTIAL CAPACITANCES
IN A SYSTEM OF CYLINDRICAL ELECTRODES
WITH ALLOWANCE FOR THE EDGE EFFECT
V. A. LUKUTIN, V. D. ESKOV, Yu. A. ROMANOV, A. F. KOCHEVA
(Presented by the scientific seminar on ESG)
For the development of an engineering method for calculating electrostatic generators (ESG), it is necessary to establish sufficiently simple relationships between the principal energy characteristics of the generator and the geometric dimensions of the electrode structures. Such relationships, as applied to rod-type ESGs, can be obtained by using equations with partial capacitances [1], if the dependence of the latter on the geometry of the ESG is known.
The purpose of the present article is to indicate such a dependence for an approximate model of a rod generator with transporters of circular cross section and finite dimensions, the schematic diagram of which is shown in Fig. 1, where
Fig. 1. Schematic diagram of a model of a rod generator:
1, 2, 3, ... are transporters; 0, 0 are stator plates.
\(r\) — radius of the transporter,
\(a\) — distance between the axes of neighboring transporters,
\(d\) — distance between the stator planes.
The determination of partial capacitances in this scheme is carried out quite simply and with sufficient accuracy by modeling in an electrolytic bath [2], with partial conductances being measured instead of partial capacitances. Under identical boundary conditions and geometric similarity of the original and the model, the transition to electrostatic quantities is performed by the formula:
\[ \frac{C_{nk}}{\varepsilon b_0}=\frac{g_{nk}}{\gamma_v b_M}, \tag{1} \]
where \(C_{nk}\) is the partial capacitance between electrodes \(n\) and \(k\),
\(g_{nk}\) is the partial conductance between the corresponding electrodes of the model,
\(\gamma_v\) is the specific volume conductivity of the electrolyte,
\(b_0/b_M\) is the ratio of the geometrical dimensions of the original and the model,
\(\varepsilon\) is the dielectric permittivity of the medium.
The boundary conditions are then established in accordance with equations of the form:
\[ g_n=\sum_{\substack{k=1\\(k\ne n)}}^{k=m} C_{nk}(U_n-U_k), \tag{2} \]
\[ I_n=\sum_{\substack{k=1\\(k\ne n)}}^{k=m} C_{nk}(U'_n-U'_k). \tag{3} \]
To determine, for example, the quantity \(C_{nm}\), all electrodes except the \(m\)-th must have the same potential \(U_n\), different from the potential of the \(m\)-th electrode \(U_m\).
Then the ratio of the charge of the \(m\)-th electrode to the potential difference \(U_n-U_m\) gives the value \(C_{nm}\). Similarly, electrodes of the model 1, 2, 3, …, \(m-1\) must have the same potential \(U'_n\), and the \(m\)-th electrode the potential \(U'_m\); then
\[ g_{mn}=\frac{U'_n-U'_m}{I_n}, \tag{4} \]
where \(I_n\) is the current flowing to the \(n\)-th electrode.
Without dwelling on the modeling procedure in an electrolytic bath, which is described in sufficient detail in [3], we note that the measurements were carried out with alternating current at a frequency of 50 Hz; distilled water was used as the electrolyte,
\[ \gamma=[0.8\cdot 10^{-5}\div 1.2\cdot 10^{-5}]\,1/\mathrm{ohm\,cm}, \]
and the electrodes were made of brass.
The measurement error was determined on standard systems (a plane capacitor, a cylindrical capacitor, etc.) and amounted to 2–3 percent.
In the course of the experiments, the main attention was devoted to determining the sum of the partial capacitance of the transporter with respect to all stator elements \(C_{10}\) and the partial capacitance between neighboring transporters \(C_{12}\), which, as noted in [1], have the greatest influence on the characteristics of an electrostatic transducer. The measurements were carried out according to the electrical circuit shown in Fig. 2. When measuring the partial capacitance \(C_{10}\), the switch is in position \(2'—2''\).
To measure \(C_{12}\), the switch must be moved to position \(1'—1''\).
The variable resistance \(R\) in the circuit serves to compensate for the voltage drop across the ammeter (all electrodes except the first must have the same potential). This is recorded by voltmeter \(V_2\) (voltmeter \(V_2\), like \(V_1\), had a high internal resistance).
It was not possible to compensate completely for the voltage drop across the ammeter because of the presence of inductance in the ammeter, but the readings of \(V_2\) did not exceed 30–40 millivolts when the readings of \(V_1\) were not less than 10 volts.
Since the field pattern is symmetric with respect to the horizontal
Fig. 2. Circuit for measuring partial capacitances
\(C_{10}, C_{12}, C_{13}, \ldots\): 1, 2, 3, 4, 5—carriers;
0, 0—stator plates
and two vertical planes, then one eighth of the space can be modeled.
When these conditions were satisfied, \(C_{10}\) and \(C_{12}\) were determined, with the ratio of the geometric dimensions of the model varied within the limits
\[ 1.5 < \frac{a}{2r} < 6, \]
\[ 2 < \frac{b}{2r} < 18, \qquad 2 < \frac{d}{2r} < 18. \]
From the measurement results, empirical formulas were derived:
\[ C_{10}=C_{\infty}\left[1+\frac{d}{2(b+r)}\right], \tag{5} \]
\[ C_{12}=C_{\infty}\cdot \frac{\pi}{0.05}\operatorname{Arth}\frac{2r}{a} \left[\left(\frac{d}{2r}-1\right)\cdot 0.01\right]^{\sqrt{0.7-\frac{a}{2}}}, \tag{6} \]
where
\[ \frac{C_{\infty}}{b} = \frac{2\pi\varepsilon}{\ln \frac{2d}{\pi r}} \left[ 1-e^{-\frac{\sqrt{3a}}{d}} \right]. \tag{7} \]
is the partial capacitance of the carrier relative to the stator per unit length as \(b \to \infty\).
The approximation error does not exceed 8 percent.
It should be noted that formulas (5) and (6), with accuracy sufficient for engineering calculations, can give values of the partial capacitances only for generators with a very large number of carriers. In this case the smaller error will be obtained for a cylindrical rotor, and the larger one for a disk rotor. In view of this, additional experiments were carried out using analogous circuits for radially arranged rods (Fig. 3), at a fixed ratio
\[ \frac{R_2}{R_1}=\sqrt{3}, \]
as the most optimal [4].
The ratios of the remaining geometric dimensions were varied within the limits
Fig. 3. Schematic diagram of the rotor model of a disk generator: \(R_2\) is the outer radius of the disk, \(R_1=R_2-b\)
\[ 2 < \frac{a}{2r} < 6, \qquad 2 < \frac{d}{2r} < 20, \]
\[ 2 < \frac{b}{2r} < 18, \]
where \(a\) was measured along the circumference of radius \(R=\sqrt{R_1R_2}\). From the experimental results, empirical formulas were derived:
\[ \frac{C_{10}'}{\varepsilon b} = \frac{C_{10}}{\varepsilon b} - k e^{-r/d}\cdot \frac{0.8b}{36r}, \tag{8} \]
where
\[ k = 10 + 1.5\frac{a}{r}, \]
and \(\dfrac{C_{10}}{\varepsilon b}\) is determined from formula (5).
\[ \frac{C_{12}'}{\varepsilon b} = \frac{C_{12}}{\varepsilon b} + k' e^{-r/d}\cdot \frac{36r}{b}, \tag{9} \]
where
\[ k' = 6 + \frac{a}{3r}, \]
and \(\dfrac{C_{12}}{\varepsilon b}\) is determined from formula (6).
The formulas obtained for determining \(C_{10}'\) and \(C_{12}'\) give a maximum error of 10 percent and may be used for the calculation of disk electrostatic machines.
References
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V. A. Lukutin, V. D. Eskov. On the calculation of an ESG with a cascade connection of conductor transporters. Izv. AN SSSR “Energetika i transport”, No. 2, 1967.
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A. P. Kononov, V. V. Patsevich, V. D. Eskov. Partial capacitances between elements of an ESG. Izv. TPI, vol. 162, 1967.
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I. M. Tetelbaum. Electrical Modeling. Fizmatgiz, 1959.
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L. S. Polotovskii. High-Voltage Direct-Current Capacitive Machines. GEI, 1960.