On the Calculation of the Potentials of ESGS Transporters before Switching

V. M. Fedyakin, V. D. Eskov

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

To calculate switching processes in rod-type electrostatic generators (ESGS), it is necessary to know the distribution of potentials over the transporters immediately before switching.

Below, one variant of the calculation is set forth using as an example an ESGS with one pair of poles and a uniform distribution of potential over the stator, in which switching is carried out when the corresponding transporters pass the midpoint of the inductors (simultaneously in the excitation and load systems).

In the steady operating mode of the generator, all processes occurring in it repeat with the switching period \(T\); therefore it is sufficient to consider one such period, taking \(t=0\) as the instant at which the next switching is completed, and \(t=T\) as the instant at which the following one begins. In a first approximation, the duration of switching may be neglected because of its short time [1].

If, in the working volume of the machine, the gradients of the potential have not reached critical values, then the ESGS may be regarded as a linear system, and the potential of the \(n\)-th transporter may be represented as the sum of two components:

\[ U_n = U_n^{p} + U_n^{c}, \tag{1} \]

the first of which is determined only by the charges of the transporters with the stator grounded (the rotor field), and the second only by the field of the stator in the absence of charges on the transporters.

Since the variable component of the stator field is considerably smaller than its constant component [2], in the interval \(0 \le t < T\), \(U_n^{c}\) depends only on the position of the transporter. The component \(U_n^{p}\), on the contrary, in this interval does not depend on the displacement of the transporter, since the charges of all transporters remain unchanged.

Therefore, assigning the number \(n=1\) to the transporter that has just switched in the excitation system, and continuing the numbering in the direction of rotation of the rotor, we may write

\[ U_n(T) = U_n^{p}(0) + U_{n+1}^{c}(0), \tag{2} \]

since the \(n\)-th transporter at the moment \(t=T\) occupies the position which, at \(t=0\), was occupied by the \((n+1)\)-st transporter.

The method for determining the potentials of the transporters at the moment of termination of commutation \((t=0)\) is set forth in sufficient detail in [3]. There, expressions are also given for \(U_n(0)\), which in general form take into account the values of the principal partial capacitances of the transporter (of the stator element \(C_1\) opposite to it; of the neighboring transporter \(C_2\); of the stator element opposite to the neighboring transporter \(C_3\); of the grounded structural elements \(C_0\)), the total number of transporters \(2m\), and the number of transporters overlapped by each inductor \(S\), the excitation circuit, the value of the excitation voltage \(U_{\mathrm{v}}\), and the load \(U_{\mathrm{n}}\).

Separating in these expressions the components of the rotor field and the stator field, for a conduction excitation circuit we have

\[ U_n^p(0)=U_{\mathrm{v}}K_1\left(1-\frac{U_{\mathrm{n}}}{K_uU_{\mathrm{v}}}\right) +\frac{U_{\mathrm{n}}}{2}\left(\frac{C_0}{C_3}+\frac{1}{K_u}\right), \tag{3} \]

where

\[ K_1= \begin{cases} \dfrac{\operatorname{sh}\dfrac{m-n+1}{2}\gamma\cdot \operatorname{sh}\dfrac{n\gamma}{2}} {\operatorname{sh}\dfrac{m\gamma}{2}\cdot \operatorname{sh}\dfrac{\gamma}{2}}, & 1\leq n\leq m,\\[2.2ex] \dfrac{\operatorname{sh}\dfrac{m-n+1}{2}\gamma\cdot \operatorname{sh}\dfrac{2m-n}{2}\gamma} {\operatorname{sh}\dfrac{m\gamma}{2}\cdot \operatorname{sh}\dfrac{\gamma}{2}}, & m+1\leq n\leq 2m, \end{cases} \]

\[ \frac{1}{K_u}= \frac{C_0}{C_3}+ \frac{2(C_1+2C_3\cdot \operatorname{ch}\gamma)\cdot \operatorname{sh}\dfrac{m-s+1}{2}\gamma} {C_3(m-s+1)\cdot \operatorname{sh}\gamma\cdot \operatorname{ch}\dfrac{m\gamma}{2}}; \]

Table

\[ C_{\mathrm{e}} = C_1 + 2C_3 + C_0; \]

\[ \gamma = \ln \frac{ \sqrt{1+\dfrac{4C_2}{C_{\mathrm{e}}}}+1 }{ \sqrt{1+\dfrac{4C_2}{C_{\mathrm{e}}}}-1 }; \]

\[ U^{c}_{n+1}(0) = \frac{U_{\mathrm{n}}}{2} \left[ K_2\left(1-\frac{C_0}{C_{\mathrm{e}}}\right) + K_3\left(\frac{1}{K_{\mathrm{i}}}-\frac{C_0}{C_{\mathrm{e}}}\right) \right], \tag{4} \]

and the values of the coefficients \(K_2\) and \(K_3\), depending on the position of the transporter, are given in the table.

Substitution of (3,4) into (2) gives an expression for the potential of any transporter immediately before switching in a generator with a conduction excitation circuit. To obtain an analogous expression for an ESGS with an induction excitation circuit, it is sufficient to subtract the quantity \(U_{\mathrm{v}}\cdot C_0/C_{\mathrm{e}}\) from (3), and \(U_{\mathrm{v}}(1-C_0/C_{\mathrm{e}})\) from (4), and in both formulas to replace \(U_{\mathrm{n}}\) by \(U_{\mathrm{n}}+U_{\mathrm{v}}\).

The expressions obtained may be used in calculating commutation processes as independent initial conditions.

REFERENCES

1. A. M. Kuptsov, V. V. Patsevich. Analysis of commutation in electrostatic generators. Elektrichestvo, 1968, No. 7.

2. A. M. Kuptsov, V. A. Lukutin, V. M. Fedyakin. Dependence of ESG current pulsations on the width of the inductors. Izvestiya TPI, vol. 227, 1972.

3. V. A. Lukutin, V. D. Eskov. On the calculation of electrostatic generators with a cascade connection of transporter-conductors. Izv. AN SSSR, “Energetika i transport”, 1967, No. 2.