ON THE QUESTION OF THE DISTRIBUTION OF POTENTIAL OVER THE ESG ROTOR*

V. D. ESKOV

Presented by the joint scientific seminar of the Department of TOE of TPI and the Rotor ESG Section of the NII YaF at TPI

Electrostatic generators (ESG) are at present finding ever wider application in the national economy.

The principle of operation and the fundamentals of the design of ESGs with a cascade connection of transporter-conductors are set forth in sufficient detail in [1].

In order to obtain in these generators the greatest voltage (and, consequently, the greatest power for a given load), it is necessary to have a uniform distribution of potential over the rotor. To a certain extent this problem is solved by the presence on the stator of a semiconducting layer, which ensures uniformity of the distribution of potential over the stator. The present article is devoted to determining the potentials of the transporters in generators of this type, which is of interest from the standpoint of solving problems of commutation and of choosing the optimum generator design.

Figure 1 shows the schematic diagram of an ESG with a cascade connection of transporter-conductors, having one pair of poles and a semiconducting layer on the stator. The number of generator transporters is \(2m\); in this case the inductors each cover \(s\) transporters (\(s\) is an odd number), and the brushes cover one transporter each. Two excitation schemes are possible: conduction excitation (the low-voltage inductors are grounded, and the brush is connected to an excitation source with voltage \(U_{\mathrm{B}}\), the other terminal of which is also grounded) and induction excitation (the brush is grounded, and the inductors are connected to the excitation source). The load is connected by one lead to the high-voltage inductors and the brush, and the other lead is grounded. The semiconducting layer on the stator is represented in the form of a voltage divider, the number of stages of which is \(m - s + 1\), and the voltage drop across each stage is \(U_0\). The error from such a replacement does not exceed 5% for \(m - s \geqslant 5\) [2].

At the instant when transporters 1 and \(m+1\) break away from the corresponding brushes, transporters \(1, 2, \ldots, m\) have charge \(q\), while transporters \(m+1, m+2, \ldots, 2m\) have charge \(q'\). The difference of these charges is delivered to the load, while the current \(I_{\mathrm{n}}\) flowing in the load produces across it a voltage drop \(U_{\mathrm{n}}\).

* The work was carried out under the supervision of Candidate of Technical Sciences V. A. Lukutin.

Fig. 1. Schematic diagram of an ESG with cascade connection of carrier conductors.

The electric field in an ESG satisfies the conditions of quasi-stationarity; therefore, it can be studied by means of the equations of electrostatics.

Considering the ESG as a system of conducting bodies, on the basis of equations with partial capacitances one can construct an equivalent circuit of the generator, an element of which is shown in Fig. 2. Here \(C_1\) is the partial capacitance of the carrier relative to the opposite stator element, \(C_2\) relative to the neighboring carrier, \(C_3\) relative to the stator element opposite the neighboring carrier, and \(C_4\) relative to the grounded elements of the generator structure. The remaining partial capacitances are not taken into account, and it is assumed that the dielectric permittivity of the medium surrounding the carriers does not depend on the electric-field strength and that there are no insulation leakages or corona losses.

The problem of determining the potentials of the carriers by the shortest route can be solved by means of the method of electrostatic analogy. Let us replace the partial capacitances by conductances: \(C_1\) by \(2g_1\), \(C_2\) by \(g_2\), \(C_3\) by \(g_3\), \(C_4\) by \(2g_4\); the voltage of the excitation source and the voltage drops on

the voltages at the stages of the divider will correspond to voltage sources \(U_{\mathrm{в}}\) and \(U_0\), and to the charges \(q\) and \(q'\)—current sources \(I\) and \(I'\). As a result, instead of the equivalent circuit with partial capacitances, for the conduction excitation circuit one obtains the ladder circuit of sextipoles shown in Fig. 3. In this case \(s-1\) sextipoles, both on the excitation side and on the load side, will be passive, while the remaining ones will be active. The circuit of an active sextipole is shown in Fig. 4, and a passive one differs from it by the absence of the voltage source \(U_0\).

Let us determine in this circuit the potential of the \(n\)-th carrier and the currents \(I_{\mathrm{в}}\) and \(I_{\mathrm{к}}\). The calculation will be carried out by the method of superposition.

Fig. 2. Element of the generator equivalent circuit.

Fig. 3. Computational ladder circuit.

Fig. 4. Circuits of an active six-terminal network and a passive four-terminal network.

When determining the components of the quantities named above due to \(U_{\mathrm{v}}\), \(I\), and \(I'\), the circuit of Fig. 3 is quite simply reduced to a chain circuit made up of passive four-terminal networks (the circuit of such a four-terminal network is shown in Fig. 4), and the calculation is carried out by the known formulas of circuit theory [3]. The same circuit can also be obtained when calculating the components due to \(U_0\), by successively applying the principle of reciprocity and the condition of equivalence of energy sources [3].

As a result of the calculation, the following expressions are obtained:

a) for the potential of the \(n\)-th transporter at \(1 < n \le m+1\)

\[ U_{\mathrm{p}} = U_{\mathrm{v}} \frac{\operatorname{sh}(m-n+1)\beta}{\operatorname{sh} m\beta} + \frac{I\beta}{2g_0} \left[ 1- \frac{ \operatorname{ch}\left(\frac{m}{2}-n+1\right)\beta }{ \operatorname{ch}\frac{m\beta}{2} } \right] + U_{\mathrm{p}0}, \tag{1} \]

where

\[ U_{\mathrm{p}0}= \begin{cases} \displaystyle \frac{U_{\mathrm{n}}\operatorname{sh}(n-1)\beta}{g_0\operatorname{sh}m\beta}(g_4+g_5), & \displaystyle \text{for } 1<n<\frac{s+1}{2}, \\[2.0ex] \displaystyle \frac{U_{\mathrm{n}}}{g_0} \left[ (g_0-g_4)\frac{\,n-\frac{s+1}{2}\,}{m-s+1} + g_4\frac{\operatorname{sh}(n-1)\beta}{\operatorname{sh}m\beta} + g_5 \frac{ \operatorname{sh}\frac{s-1}{2}\beta\, \operatorname{sh}\left(\frac{m}{2}-n+1\right)\beta }{ \operatorname{sh}\frac{m-s+1}{2}\beta\,\operatorname{sh}m\beta } \right], & \displaystyle \text{for } \frac{s+1}{2}<n<m-\frac{s-3}{2}, \\[2.0ex] \displaystyle \frac{U_{\mathrm{n}}}{g_0} \left[ g_0-g_4 + g_4 \frac{ \operatorname{ch}\left(\frac{m}{2}-n+1\right)\beta }{ \operatorname{ch}\frac{m\beta}{2} } - (g_4+g_5) \frac{\operatorname{sh}(m-n+1)\beta}{\operatorname{sh}m\beta} \right], & \displaystyle \text{for } m-\frac{s-3}{2}<n<m+1. \end{cases} \]

\[ U_{\mathrm{n}}=U_0(m-s+1),\qquad g_5=\frac{2(g_1+g_3\operatorname{ch}\beta)}{(m-s+1)\operatorname{sh}\xi}\operatorname{ch}\frac{m\beta}{2}\operatorname{sh}\frac{m-s+1}{2}\beta, \]

\[ \beta=\ln \frac{\sqrt{1+\dfrac{2g_2}{g_0}}+1} {\sqrt{1+\dfrac{2g_2}{g_0}}-1}, \qquad g_0=g_1+g_3+g_4; \]

for \(m+1\le n\le 2m+1\), formula (1) is valid if in it \(I\) is replaced by \(I'\) and \(n\) by \(n'=2m+2-n\);

b) for the currents \(I_{\mathrm{v}}\) and \(I_{\mathrm{k}}\)

\[ I_{\mathrm{v}}= \frac{2g_0U_{\mathrm{v}}\operatorname{ch}m\beta-2U_{\mathrm{n}}(g_4+g_5)} {\operatorname{th}\dfrac{\beta}{2}\operatorname{sh}m\beta} - \frac{I+I'}{\operatorname{sh}m\beta} \sum_{k=1}^{m-1}\operatorname{sh}k\beta -I, \tag{2} \]

\[ I_{\mathrm{k}}= \frac{2g_0U_{\mathrm{v}}-2U_{\mathrm{n}}(g_4\operatorname{ch}m\beta+g_5)} {\operatorname{th}\dfrac{\beta}{2}\operatorname{sh}m\beta} + \frac{I+I'}{\operatorname{sh}m\beta} \sum_{k=1}^{m-1}\operatorname{sh}k\beta +I'. \tag{3} \]

Since the charges are connected with the corresponding potential differences only through partial capacitances, the currents to nodes \(I\) and \(m+1\) can flow only through the conductances \(g_1, g_2, g_3,\) and \(g_4\), whence it follows that the currents \(I_{\mathrm{v}}\) and \(I_{\mathrm{k}}\) are equal to zero. Then, from equations (2) and (3), the currents \(I\) and \(I'\) are determined in the following form:

\[ I=g_0U_{\mathrm{v}}+g_4U_{\mathrm{n}}+\Delta I; \tag{4} \]

\[ I'=g_0U_{\mathrm{v}}+g_1U_{\mathrm{n}}-\Delta I. \tag{5} \]

where

\[ \Delta I= \frac{1}{\operatorname{th}\dfrac{\beta}{2}\operatorname{th}\dfrac{m\beta}{2}} \left[ g_0U_{\mathrm{v}} - U_{\mathrm{n}} \left( g_1+ \frac{g_5}{\operatorname{ch}^2\dfrac{m\beta}{2}} \right) \right]. \]

If we now substitute the values of \(I\) and \(I'\) into equation (1) and replace the conductances by the corresponding capacitances, we obtain the final formula for the potential of the \(n\)-th transporter of the generator under the conduction excitation scheme:

\[ \frac{U_n}{U_{\mathrm{v}}} = f_1\left(1-\frac{U_{\mathrm{n}}}{k_{\mathrm{u}}U_{\mathrm{v}}}\right) + \frac{U_{\mathrm{n}}}{U_{\mathrm{v}}} \left[ f_2+f_3\frac{C_1}{C_0} + \frac{1+f_4}{2} \left( \frac{1}{k_{\mathrm{u}}}-\frac{C_4}{C_0} \right) \right] \tag{6} \]

The values of the coefficients \(f_1, f_2, f_3,\) and \(f_4\) are given in Table 1, and

\[ \frac{1}{k_{\mathrm{u}}} = \frac{C_4}{C_0} + \frac{ 2(C_1+2C_3\operatorname{ch}\gamma)\operatorname{sh}\dfrac{m-s+1}{2} }{ C_0(m-s+1)\operatorname{sh}\gamma\operatorname{ch}\dfrac{m\gamma}{2} }, \]

where

\[ C_0=C_1+2C_3+C_4, \qquad \gamma=\ln \frac{\sqrt{1+\dfrac{4C_2}{C_0}}+1} {\sqrt{1+\dfrac{4C_2}{C_0}}-1}. \]

The meaning of the coefficient \(k_u\) can be explained as follows. Replacing in equations (4) and (5) \(I\) by \(q\), \(I'\) by \(q'\), and the conductances \(g_1\), \(g_2\), \(g_3\), and \(g_4\), respectively, by the partial capacitances \(\dfrac{C_1}{2}\), \(C_2\), \(C_3\), \(\dfrac{C_4}{2}\), we find the charge difference:

\[ q-q' = k_i C_0 U_B \left( 1 - \frac{U_H}{k_u U_B} \right), \tag{7} \]

where

\[ \frac{1}{k_i}=\operatorname{th}\frac{\gamma}{2}\operatorname{th}\frac{m\gamma}{2}. \]

Since the current in the load is determined as

\[ I_H = 2mn(q-q'), \tag{8} \]

where \(n\) is the number of rotor revolutions per second, from equations (7) and (8) we obtain an expression for the external characteristic of the generator:

\[ I_H = k_i I_0 \left( 1 - \frac{U_H}{k_u U_B} \right), \tag{9} \]

where \(I_0 = 2mnC_0U_B\).

It follows from this that for \(I_H=0\), \(k_u=\dfrac{U_{XX}}{U_B}\) and can be called the voltage-increase coefficient, while for \(U_H=0\), \(k_i=\dfrac{I_{SC}}{I_0}\) and can be called the natural recharge coefficient (for \(C_2 \to 0\), \(I_{SC}=I_0\)) [4].

The calculation of a generator with an induction excitation circuit is entirely similar to that set out above (only in the circuit of Fig. 3 the other pole of the voltage source \(U_B\) should be grounded).

As a result we have:

\[ \frac{U'_n}{U_B} = f_1 \left[ 1-\left(1+\frac{U_H}{U_B}\right)\frac{1}{k_u} \right] + \left(1+\frac{U_H}{U_B}\right) \left[ f_2+f_3\frac{C_4}{C_0} + \frac{1+f_2}{2} \left( \frac{1}{k_u}-\frac{C_4}{C_0} \right) \right] -1. \tag{10} \]

In this case the voltage-increase coefficient changes \((k'_u=k_u-1)\) and the natural recharge coefficient

\[ \left( k'_i=k_i-\frac{k'_u}{k_u} \right). \]

Investigation of equations (6) and (10) makes it possible to draw several conclusions. In operating modes close to generator no-load, the greatest potential difference between neighboring conveyors is usually less than the magnitude \(U_0\). In modes close to short-circuit operation, however, the greatest potential difference exists between conveyors \(m\) and \(m+1\), and also between 1 and \(2m\). This is also confirmed experimentally: observations of the commutation zone on generators built at the Tomsk Polytechnic Institute show that in the space between these conveyors and

Table V

the corona at the brush arises the more intensely, the lower the voltage at the generator output.

Thus, the equations obtained make it possible to estimate the nonuniformity of the potential distribution over the ESG rotor and to identify the sections with the greatest electric-field intensity for given partial capacitances and number of conveyors. On the other hand, one can choose optimal ratios of the latter quantities for a given external characteristic of the generator and operating point on it. In addition, the proposed calculation method makes it possible to take into account a number of design features of the ESG (the presence of several pairs of poles, overlap by the brush of several conveyors, its displacement relative to the middle of the inductor, etc.). Modeling the field of the entire generator is a very complicated problem; knowledge of the potentials of the conveyors makes it possible to model individual sections of it (for example, the commutation zone). After the development of a sufficiently simple method for determining the partial capacitances, this calculation can be used in the engineering design of the generator. It should also be noted that this calculation method is applicable to the calculation of the potential distribution along a string of insulators and in certain types of arresters.

REFERENCES

1. A. F. Ioffe, Electrostatic Generator, ZhTF, 9, No. 23, 1939.

2. G. M. Bernstein, I. N. Saliy, On the discrete approximation of a continuous function on the boundary of a region in electrical modeling, Electricity, No. 11, 1963.

3. G. V. Zeveke, P. A. Ionkin, A. V. Netushil, S. V. Strakhov, Fundamentals of Circuit Theory, Gosenergoizdat, pp. 15, 35, 249, 1963.

4. V. V. Patsevich, Some questions of the theory and calculation of ESGs with conveyor conductors. Dissertation, Tomsk, 1964.