APPLICATION OF A CHAIN EQUIVALENT CIRCUIT TO THE CALCULATION OF ESG

V. A. Lukutin, V. D. Eskov

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

The electric field in electrostatic generators (ESG) satisfies the conditions of quasistationarity [1]; consequently, its investigation can be carried out with the aid of the equations of electrostatics.

Bobkovskii developed a rigorous theory of ESG [2], using equations with potential coefficients. Unfortunately, calculation of the external characteristic of a generator by his method becomes very cumbersome if one takes into account the specified uniform distribution of potential over the stator, which is required by the conditions for obtaining maximum power from the generator.

Investigations of ESG by equations with partial capacitances are presented in [3], but here as well, with a large number of transporters, it is necessary to solve a system consisting of many equations.

In the present article a method is set forth for calculating the external characteristic of an ESG with a cascade connection of transporters, based on the use of the method of electrostatic analogy.

Considering the ESG as a system of conducting bodies, one can construct the equivalent circuit for a generator with one pair of poles [3], shown in Fig. 1. Here \(c_2\) is the partial capacitance between neighboring transporters, and \(c_1\) is the partial capacitance of a transporter with respect to the stator. In this case the remaining partial capacitances are not taken into account, and the presence of edge—

Fig. 1.

of the effect at the boundary between the inductor and the semiconducting layer, and it is assumed that the dielectric permittivity of the medium surrounding the transporter does not depend on the electric-field strength and that there are no insulation leakages or corona losses.

The generator has a number of transporters equal to \(2m\), the brushes spanning one transporter each. The load is connected at one end to the high-voltage inductors and the brush, and at the other end is grounded. In the excitation system the brush may be grounded; then the inductors are connected to the excitation source with voltage \(U_{\mathrm{в}}\), the other terminal of which is also grounded (the induction circuit); conversely, the brush may be connected to the excitation source, while the inductors are grounded (the conduction circuit). The semiconducting layer on the stator, which provides a uniform distribution of potential, is represented as a voltage divider with a voltage drop \(U_0\) across each of its elements.

If we consider the instant at which transporters 1 and \(m+1\) separate from the corresponding brushes, then transporters \(1, 2,\ldots, m\) have charge \(q\), and transporters \(m+1, m+2,\ldots, 2m\) have charge \(q'\). The difference of these charges is delivered to the load with conductance \(G_{\mathrm{н}}\). The current \(I_{\mathrm{н}}\) flowing in the load produces across it a voltage drop \(U_{\mathrm{н}}\).

We shall now use the method of electrostatic analogy. The partial capacitance \(c_1\) is replaced by a conductance \(2g\), and the partial capacitance \(c_2\) by a conductance

\[ \frac{1}{R}. \]

The sum of the currents flowing into a node through these conductances must be equal either to \(I\) or to \(I'\), where the current \(I\) corresponds to the charge \(q\), and \(I'\) to the charge \(q'\). Consequently, in order to satisfy Kirchhoff’s first law, a current source of the same magnitude must be connected to each node. The voltage of the excitation source and the voltage drops on the stator elements will correspond to voltage sources \(U_{\mathrm{в}}\) and \(U_0\).

Fig. 2.

As a result, the chain circuit shown in Fig. 2 is obtained. If the inductors span \(s\) transporters (\(s\) is an odd number), then \(s-1\) quadripoles on both the excitation side and the load side will be passive, while the remaining ones will be active. The chain circuit has two identical branches, the beginnings of which are connected to the voltage source \(U_{\mathrm{в}}\), and the ends are short-circuited.

The circuit of the active four-terminal network and the result of its transformation according to [4] are shown in Fig. 3. The values of the currents \(I_{1\mathrm{k}}\) and \(I_{2\mathrm{k}}\) are determined when the four-terminal network is short-circuited on both sides.

\[ I_{1\mathrm{k}}=I_{2\mathrm{k}}=\frac{U_0}{R}. \tag{1} \]

In the chain circuit, the current sources \(I_{1\mathrm{k}}\) and \(I_{2\mathrm{k}}\) of adjacent four-terminal networks cancel one another. Therefore, in the calculation it is necessary to take into account only the outermost of them, i.e., those connected to nodes \(1+s'\), \(m+1+s'\), \(m+1-s'\), \(2m+1-s'\), where \(s'=\dfrac{s-1}{2}\).

Let us find the currents in the branch of the voltage source \(U_{\mathrm{v}} — I_{\mathrm{v}}\) and in the short-circuited branch — \(I_{\mathrm{k}}\) (Fig. 2). The calculation will be carried out by the superposition method, using the known formulas of circuit theory [4].

Fig. 3.

Currents from the voltage source \(U_{\mathrm{v}}\)

\[ I'_{\mathrm{v}}=\frac{2U_{\mathrm{v}}}{z_c\,\operatorname{th} m\gamma}, \tag{2} \]

\[ I'_{\mathrm{k}}=\frac{2U_{\mathrm{v}}}{z_c\,\operatorname{sh} m\gamma}, \tag{3} \]

where

\[ z_c=\frac{1}{g\sqrt{1+\dfrac{2}{Rg}}} \]

is the characteristic resistance of the four-terminal network,

\[ \gamma=\ln\left[1+Rg\left(1+\sqrt{1+\frac{2}{Rg}}\right)\right] \]

is its propagation constant.

Currents from the current sources \(I\), connected to nodes \(2, 3, \ldots, m\), and \(I'\), connected to nodes \(m+2, m+3, \ldots, 2m\):

\[ I''_{\mathrm{k}}=-I''_{\mathrm{v}}=\frac{I+I'}{\operatorname{sh} m\gamma}\sum_{n=1}^{m-1}\operatorname{sh} n\gamma. \tag{4} \]

The currents from the current sources \(I_{1\mathrm{k}}\) and \(I_{2\mathrm{k}}\), taking into account their direction and point of connection, are

\[ I'''_{\mathrm{v}}=I'''_{\mathrm{k}}=-\frac{2U_0}{R}\cdot \frac{\operatorname{sh}\dfrac{m-s+1}{2}\gamma} {\operatorname{sh}\dfrac{m\gamma}{2}}. \tag{5} \]

In addition, in the branch \(U_{\mathrm{v}}\) there flows the current \(I\) from the first current source, and in the short-circuited branch, the current \(I'\) from the \((m+1)\)-th current source.

The total currents in these branches are equal to the sum of the components

\[ I_{\mathrm{в}}=I'_{\mathrm{в}}+I''_{\mathrm{в}}+I'''_{\mathrm{в}}-I, \tag{6} \]

\[ I_{\mathrm{к}}=I'_{\mathrm{к}}+I''_{\mathrm{к}}+I'''_{\mathrm{к}}+I'. \tag{7} \]

Since the charges of the carriers (Fig. 1) are associated with the corresponding potential differences only by the partial capacitances \(c_1\) and \(c_2\), the currents in the circuit of Fig. 2 can flow only through the conductances \(g\) and \(\dfrac{1}{R}\). Hence it follows that the currents \(I_{\mathrm{в}}\) and \(I_{\mathrm{к}}\) are equal to zero. Then, taking equations (1—7) into account, we obtain

\[ \frac{2U_{\mathrm{в}}}{z_c\operatorname{th} m\gamma} -\frac{I+I'}{\operatorname{sh} m\gamma}\sum_{n=1}^{m-1}\operatorname{sh} n\gamma -\frac{2U_0}{R}\cdot \frac{\operatorname{sh}\dfrac{m-s+1}{2}\gamma} {\operatorname{sh}\dfrac{m\gamma}{2}} -I=0, \tag{8} \]

\[ \frac{2U_{\mathrm{в}}}{z_c\operatorname{th} m\gamma} +\frac{I+I'}{\operatorname{sh} m\gamma}\sum_{n=1}^{m-1}\operatorname{sh} n\gamma -\frac{2U_0}{R}\cdot \frac{\operatorname{sh}\dfrac{m-s+1}{2}\gamma} {\operatorname{sh}\dfrac{m\gamma}{2}} +I'=0. \tag{9} \]

Adding (8) and (9) gives

\[ I-I'= \frac{2U_{\mathrm{в}}}{z_c\operatorname{th}\dfrac{m\gamma}{2}} -\frac{4U_0}{R}\cdot \frac{\operatorname{sh}\dfrac{m-s+1}{2}\gamma} {\operatorname{sh}\dfrac{m\gamma}{2}}. \tag{10} \]

Let us now return to the electrostatic quantities, replacing in expression (10) \(g\) by \(\dfrac{c_1}{2}\), \(\dfrac{1}{R}\) by \(c_2\), \(I\) by \(q\), and \(I'\) by \(q'\). As a result we have

\[ q-q'=c_1U_{\mathrm{в}} \left[ A-B\frac{U_0}{U_{\mathrm{в}}}(m-s+1) \right]. \tag{11} \]

Here, to simplify the notation, the following designations have been introduced:

\[ A=\frac{\sqrt{1+y}}{\operatorname{th}\dfrac{m\beta}{2}}, \qquad B=\frac{y\,\operatorname{sh}\dfrac{m-s+1}{2}\beta} {(m-s+1)\operatorname{sh}\dfrac{m\beta}{2}}, \]

where

\[ y=\frac{4c_2}{c_1}, \qquad \beta=\ln\left[ 1+\frac{2}{y}\left(1+\sqrt{1+y}\right) \right]. \]

The current in the load circuit is determined by the expression

\[ I_{\mathrm{н}}=2mn(q-q'), \tag{12} \]

where \(n\) is the number of rotor revolutions per second.

Or, taking equation (11) into account,

\[ I_{\mathrm{н}}=2mnc_1U_{\mathrm{в}} \left[ A-B\frac{U_0}{U_{\mathrm{в}}}(m-s+1) \right]. \tag{13} \]

The voltage drop across the load is related to this current by the conductance \(G_{\mathrm{l}}\)

\[ U_{\mathrm{l}}=\frac{I_{\mathrm{l}}}{G_{\mathrm{l}}}, \tag{14} \]

and, according to Kirchhoff’s second law, it is equal to

\[ U_{\mathrm{l}}=U_{0}(m-s+1)-aU_{\mathrm{e}}, \tag{15} \]

where \(a\) is a coefficient taking into account the excitation circuit:

\[ a= \begin{cases} 0\text{ — for the conduction circuit},\\ 1\text{ — for the induction circuit}. \end{cases} \]

Solving equations (13–15) simultaneously with respect to \(U_{\mathrm{l}}\) and \(I_{\mathrm{l}}\), we obtain:

\[ U_{\mathrm{l}}=I_{0}(A-aB)\left(G_{\mathrm{l}}+\frac{BI_{0}}{U_{\mathrm{e}}}\right)^{-1}, \tag{16} \]

\[ I_{\mathrm{l}}=I_{0}(A-aB)\left(1+\frac{BI_{0}}{G_{\mathrm{l}}U_{\mathrm{e}}}\right)^{-1}, \tag{17} \]

where \(I_{0}=2mnc_{1}U_{\mathrm{e}}\).

For \(G_{\mathrm{l}}=0\) we have the no-load operation of the generator. From (17) we find \(I_{\mathrm{nl}}=0\), and from (16)

\[ U_{\mathrm{nl}}=U_{\mathrm{e}}\left(\frac{A}{B}-a\right). \tag{18} \]

For \(G_{\mathrm{l}}\to\infty\) we obtain the short-circuit operation. From (17) it follows that \(U_{\mathrm{sc}}=0\), and from (17)

\[ I_{\mathrm{sc}}=I_{0}(A-aB). \tag{19} \]

Equation (17), taking into account (16), (18), and (19), can be reduced to the generally accepted form

\[ I_{\mathrm{l}}=I_{\mathrm{sc}}\left(1-\frac{U_{\mathrm{l}}}{U_{\mathrm{nl}}}\right). \tag{20} \]

From the standpoint of comparing different generator designs, two quantities are of interest: the ratio of the no-load voltage to the excitation voltage,

\[ k_{u}=\frac{U_{\mathrm{nl}}}{U_{\mathrm{e}}}=\frac{A}{B}-a, \tag{21} \]

which may be called the voltage-increase coefficient, and the ratio of the short-circuit current to the current \(I_{0}\),

\[ k_{i}=\frac{I_{\mathrm{sc}}}{I_{0}}=A-aB, \tag{22} \]

which may be called the coefficient of natural overcharging (in the absence of the influence of neighboring transporters on the charging and discharging transporters, i.e., when \(c_{2}\to0\), we would have \(I_{\mathrm{sc}}=I_{0}\)).

It is obvious that the calculation of a generator with several pairs of poles will be completely similar to that set out above, and as a result the same formulas will be obtained if \(m\) is replaced by \(m'\), where \(2m'\) is the number of transporters per pair of poles, and \(I_{0}\) is replaced by \(I_{0}=2m'p^{2}c_{1}U_{\mathrm{e}}\,n\), where \(p\) is the number of pole pairs.

The study of equations (16–22) gives qualitative conclusions analogous to those made in [3], which have been confirmed experimentally on generators constructed at the Tomsk Polytechnic Institute.

Thus, on the basis of the method proposed in the present article for calculating the external characteristic of an ESG using an electrostatic analogy, simple expressions have been obtained which simultaneously take into account the relationship between the partial capacitances, the total number of transporters, the number of transporters covered by one inductor, the excitation circuit, and the number of pole pairs. It should be noted that this method also makes it possible to take into account other features of the design and circuit of generators, for example, the overlapping by the brush of several transporters, the displacement of the brush relative to the middle of the inductor, etc. After sufficiently simple methods for determining the partial capacitances have been developed, the equations obtained can be used in the engineering calculation of an ESG with a cascade connection of transporter-conductors.

REFERENCES

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

2. S. A. Bobkovskii. General theory of electrostatic generators, Zh. T. F., 10, No. 17, 1404, 1940.

3. V. V. Patsevich. Some questions in the theory and calculation of ESGs with transporter-conductors, dissertation, Tomsk, 1964.

4. G. V. Zeveke, P. A. Ionkin, A. V. Netushil, S. V. Strakhov. Fundamentals of Circuit Theory, Gosenergoizdat, 1963.