ON CALCULATING INSTANTANEOUS CURRENT VALUES IN THE ESGP UNDER SHORT-CIRCUIT CONDITIONS

V. M. FEDYAKIN, V. D. ESKOV

(Presented by the scientific seminar of the Department of TOE)

When calculating the energy characteristics of an electrostatic generator with conveyor conductors (ESGP), the latter is usually considered as a quasistatic system described at each instant of time by electrostatic equations with lumped parameters, relating the charges and potentials of the individual elements of the machine [1, 2]. Knowing the law of variation of these parameters with time, one can obtain an idea of the dynamics of the processes occurring in the ESGP. This approach is also used in the present work to determine the instantaneous values of currents in the electrical circuits of the generator, using as an example the short-circuit mode of an ESGP with a uniform distribution of potential over the stator, one pair of poles, and an induction excitation scheme, the schematic diagram of which is shown in Fig. 1.

Fig. 1. Schematic diagram of the ESGP under short-circuit conditions

Here \(1, 2, 3, \ldots, 2m\) are conducting conveyors symmetrically arranged in a dielectric rotor, which makes \(n\) revolutions per second in the direction indicated by the arrow. The stator consists of \(2(m-s)\) conducting plates connected to a voltage divider (the use of a continuous semiconducting stator does not change the schematic diagram [2]). The inductors cover \(s\) conveyors each and may be regarded as \(s\) stator plates short-circuited—

[[unclear: ending of preceding word]]. Commutation is carried out either by means of a mechanical contact or by means of a spark discharge. Each of the brushes is arranged in such a way that it commutates only with the transporter located opposite the middle of the corresponding inductor. The excitation brush has potential \(U_{\mathrm{v}}\), while the load brush in the short-circuit mode has a potential equal to zero, as do all the elements of the stator.

Owing to the symmetry of the system, all processes occurring in it repeat with the commutation period \(T=\dfrac{1}{2mn}\); therefore it is sufficient to consider only one such period, taking \(t=0\) as the instant at which the next commutation begins.

The commutation time is small (it differs by several orders of magnitude from \(T\) [3]), that is, it occurs with the rotor practically stationary. Therefore, the commutation processes may be considered on a static model, and, in analyzing the dynamic model, the commutation time need not be taken into account.

Under these conditions let us determine the currents in the generator circuits (denoted in the diagram of Fig. 1):

\[ i_{\mathrm{shv}},\ i_{\mathrm{shn}} \]

—the currents in the excitation and load brushes,

\[ i_{\mathrm{iv}},\ i_{\mathrm{in}} \]

—the currents in the circuits of the corresponding inductors,

\[ i_{\mathrm{kz}} \]

—the ESGP short-circuit current,

summing, according to the superposition principle, their components in the dynamic and static models.

Currents in the dynamic model

Here the currents in the brushes are equal to zero, and the currents in the inductor circuits can be found as the rate of change of the charges of these inductors:

\[ i_{\mathrm{iv}}=\frac{dQ_{\mathrm{v}}}{dt};\qquad i_{\mathrm{in}}=-\frac{dQ_{\mathrm{n}}}{dt}. \tag{1} \]

To determine the charges, as shown in [2], it is advisable to use equations with partial capacitances; moreover, without introducing substantial errors, it is sufficient in the calculation to take into account only the following:

\[ C_0 \]

—the partial capacitance of a transporter relative to the grounded structural elements of the ESGP,

\[ C_2 \]

—the partial capacitance between adjacent transporters,

\[ C'_3,\ C_1,\ C_3,\ C_5 \]

—the partial capacitances between the stator element opposite, at the instant \(t=0\), the \(n\)-th transporter and the transporters \(n+1\), \(n\), \(n-1\), \(n-2\), respectively.

In the first approximation one may regard the partial capacitances \(C_0\), \(C_2\), and the sum of all the others as independent of time (although each of the latter changes individually); then the quantity

\[ C_{\mathrm{e}}=C_0+C_1+C_3+C'_3+C_5=\mathrm{const} \]

is determined as in [2].

With the chosen notation,

\[ \left. \begin{aligned} -\,Q_{\mathrm{n}}={}& C_1 \sum_{n=m+1-\frac{s-1}{2}}^{m+1+\frac{s-1}{2}} U_n + C_3 \sum_{n=m-\frac{s-1}{2}}^{m+\frac{s-1}{2}} U_n + C'_3 \sum_{n=m+2-\frac{s-1}{2}}^{m+2+\frac{s-1}{2}} U_n + C_5 \sum_{n=m-1-\frac{s-1}{2}}^{m-1+\frac{s-1}{2}} U_n, \\[1.2em] Q_{\mathrm{v}}={}& C_1 \sum_{n=2m+1-\frac{s-1}{2}}^{2m+1+\frac{s-1}{2}} U_n + C_3 \sum_{n=2m-\frac{s-1}{2}}^{2m+\frac{s-1}{2}} U_n + C'_3 \sum_{n=2m+2-\frac{s-1}{2}}^{2m+2+\frac{s-1}{2}} U_n + C_5 \sum_{n=2m-1-\frac{s-1}{2}}^{2m-1+\frac{s-1}{2}} U_n. \end{aligned} \right\} \tag{2} \]

where \(U_n\) is the potential of the \(n\)-th transporter, which, according to [2], is determined by the expression:

\[ U_n = \begin{cases} \displaystyle U_{\mathrm{в}}\, \frac{ \operatorname{sh}\dfrac{m-n+1}{2}\gamma\, \operatorname{sh}\dfrac{n}{2}\gamma }{ \operatorname{sh}\dfrac{m}{2}\gamma\, \operatorname{sh}\dfrac{\gamma}{2} }, & 1 \leq n \leq m, \\[2.2ex] \displaystyle U_{\mathrm{в}}\, \frac{ \operatorname{sh}\dfrac{m-n+1}{2}\gamma\, \operatorname{sh}\dfrac{2m-n}{2}\gamma }{ \operatorname{sh}\dfrac{m}{2}\gamma\, \operatorname{sh}\dfrac{\gamma}{2} }, & m+1 \leq n \leq 2m . \end{cases} \tag{3} \]

Here

\[ \gamma = \ln \frac{ \sqrt{1+\dfrac{4C_2}{C_3}}+1 }{ \sqrt{1+\dfrac{4C_2}{C_3}}-1 } \simeq \ln \frac{\kappa_i+1}{\kappa_i-1}, \]

\[ \kappa_i = \frac{1}{ \operatorname{th}\dfrac{m}{2}\gamma\, \operatorname{th}\dfrac{\gamma}{2} } \]

are dimensionless coefficients independent of time, and therefore the potentials of the transporters do not change in the interval between commutations.

Fig. 2

The law of variation of the partial capacitance of a transporter relative to a stator surface element was determined by modeling in an electrolytic bath while changing the geometry of the model within limits corresponding to real ESGP designs. The generalized curve characterizing this law is shown in Fig. 2. Here \(x\) is the displacement of the transporter axis relative to the midpoint of the stator element, and \(a\) is the distance between the axes of adjacent transporters or the midpoints of adjacent stator elements (\(x\) and \(a\) are measured along the same circumference). Obviously, at \(t=0\)

\[ C_1 = C_m(x=0);\quad C_3 = C'_3 = C_m-\Delta C(x=a);\quad C_5 \simeq 0(x=2a); \]

and at \(t=T\)

\[ C_1 = C_5 = C_m-\Delta C(x=a);\quad C_3 = C_m(x=0);\quad C'_3 \simeq 0(x=2a). \]

To simplify further calculations, the curve in Fig. 2 can, with a sufficient degree of accuracy, be approximated by two straight-line segments ...

segments whose common point has abscissa \(x=2a\). In this case one may write:

\[ \left. \begin{aligned} C_1(t) &= C_{\mathrm m}-\Delta C\,\frac{t}{T},\\ C_3(t) &= C_{\mathrm m}-\Delta C\left(1-\frac{t}{T}\right),\\ C'_3(t) &= (C_{\mathrm m}-\Delta C)\left(1-\frac{t}{T}\right),\\ C_5(t) &= (C_{\mathrm m}-\Delta C)\,\frac{t}{T} \end{aligned} \right\}, \tag{4} \]

where

\[ C_{\mathrm e}-C_0=C_1+C_3+C'_3+C_5=3C_{\mathrm m}-2\Delta C. \]

Then from equations (1—4) it follows that

\[ i_{\mathrm{in}}=-i_{\mathrm{out}} = \left[AC_{\mathrm m}-(A-B)\Delta C\right]\frac{\varkappa_i U_{\mathrm v}}{T}, \tag{5} \]

where

\[ A=3-\frac{\operatorname{ch}\frac{m-s}{2}\gamma\,(2\operatorname{ch}\gamma+1)} {\operatorname{ch}\frac{m}{2}\gamma\,\operatorname{ch}\frac{\gamma}{2}}; \qquad B=1-\frac{\operatorname{ch}\frac{m-s}{2}\gamma} {\operatorname{ch}\frac{m}{2}\gamma\,\operatorname{ch}\frac{\gamma}{2}}. \]

Analysis of the last expressions shows that in real ESGPs the condition \(A\approx 3\) and \(B\approx 1\) is satisfied; then equation (5) takes the form

\[ i_{\mathrm{in}}=-i_{\mathrm{out}} = (3C_{\mathrm m}-2\Delta C)\frac{\varkappa_i U_{\mathrm v}}{T} = 2mn\varkappa_i(C_{\mathrm e}-C_0)U_{\mathrm v} = I. \tag{6} \]

Thus, under a piecewise-linear approximation of the curve \(C(x)\), the inductor currents do not depend on time. However, the use of other approximating functions also gives a variable component, incommensurable with the constant one.

Currents in the Static Model

When calculating commutation processes, in the first approximation we neglect the inductance of the circuit (a small parameter) and the nonlinearity of the current-voltage characteristic of the spark discharge. We shall assume the resistance of the latter, \(R=\mathrm{const}\), to be the same in the charging and discharging systems.

Fig. 3. Equivalent circuit of the ESGP when calculating the commutation current.

Considering the ESGP as a certain active four-terminal network with respect to the commutation intervals (brush–transporter), we obtain the equivalent circuit shown in Fig. 3a, which, after being reduced to zero initial conditions, takes the form shown in Fig. 3b. The emfs \(U_{\mathrm{v}}\) appearing in the latter circuit are equal to the potential differences between the brushes and the transporters \(m\) and \(2m\) approaching them (this follows from equations (3)). The parameters of the Π-shaped equivalent circuit of the passive four-terminal network (Fig. 3c) can be obtained from the equivalent circuit of the ESGP proposed in [2]:

\[ C'=\kappa_1 C_{\mathrm{e}}\operatorname{th}^2\frac{m}{2}\gamma \simeq \kappa_1 C_{\mathrm{e}}, \]

\[ C''=\kappa_1 C_{\mathrm{e}}\frac{\operatorname{th}\frac{m}{2}\gamma}{\operatorname{sh}m\gamma}\simeq 0, \]

since for real ESGPs \(\operatorname{th}\frac{m}{2}\gamma \simeq 1\), while \(\operatorname{sh}m\gamma \gg 1\).

Then

\[ i_{\mathrm{shn}}=i_{\mathrm{shv}}=\frac{U_{\mathrm{v}}}{R}\exp\left(-\frac{t}{\tau}\right), \tag{7} \]

where \(\tau=\kappa_1 C_{\mathrm{e}}R\).

Each of these currents can be represented as the sum of two components, one of which is closed by the displacement currents in the system transporters–inductor (\(i'\), proportional to the quantity \(C_{\mathrm{e}}\equiv C_0\)), and the other in the system transporters–grounded structural elements of the ESGP (\(i''\), proportional to \(C_0\)):

\[ i'=\frac{U_{\mathrm{v}}(C_{\mathrm{e}}-C_0)}{C_{\mathrm{e}}R}\exp\left(-\frac{t}{\tau}\right), \tag{8} \]

\[ i''=\frac{U_{\mathrm{v}}C_0}{C_{\mathrm{e}}R}\exp\left(-\frac{t}{\tau}\right). \tag{9} \]

Instantaneous and Average Current Values

For the selected positive directions of currents in the generator circuits (Fig. 1), their instantaneous values in the interval \(0\le t\le T\) are determined by the superposition principle, taking formulas (6–9) into account, in the form:

\[ i_{\mathrm{shn}}=i_{\mathrm{shv}}=i'+i''=\frac{U_{\mathrm{v}}}{R}\exp\left(-\frac{t}{\tau}\right), \]

\[ i_{\mathrm{in}}=I-i'=2mn\kappa_1(C_{\mathrm{e}}-C_0)-\frac{U_{\mathrm{v}}(C_{\mathrm{e}}-C_0)}{RC_{\mathrm{e}}}\exp\left(-\frac{t}{\tau}\right), \]

\[ i_{\mathrm{iv}}=-I+i'=-2mn\kappa_1(C_{\mathrm{e}}-C_0)+\frac{U_{\mathrm{v}}(C_{\mathrm{e}}-C_0)}{RC_{\mathrm{e}}}\exp\left(-\frac{t}{\tau}\right), \]

\[ i_{\mathrm{kz}}=I+i''=2mn\kappa_1(C_{\mathrm{e}}-C_0)+\frac{U_{\mathrm{v}}C_0}{RC_{\mathrm{e}}}\exp\left(-\frac{t}{\tau}\right). \]

It follows from this that the currents in the brush circuits have a pulsed character (pulse duration \(\Delta t\ll T\)). The remaining currents, in addition to the pulsed component, also contain a constant (more precisely, slowly varying) component. Moreover, the polarities of these components are the same in the short-circuit current and opposite in the inductor currents.

The values of the currents averaged over a period \(\left(I_{\mathrm{avg}}=\dfrac{1}{T}\int\limits_0^T i\,dt\right)\) are equal to:

\[ I_{\mathrm{sc}}=I_{\mathrm{shn}}=I_{\mathrm{shv}}=2mnkiC_{\mathrm{g}}U_{\mathrm{v}}, \]

\[ I_{\mathrm{iv}}=I_{\mathrm{in}}=0. \]

The expression for the average value of the short-circuit current coincides with the analogous expression obtained in [2]. The equality of the average values of the brush currents and the short-circuit current, as well as the equality to zero of the average values of the inductor currents, is confirmed by the experiments described in [3]. This makes it possible to conclude that the assumptions adopted are valid and that the above-described method can be used for analyzing the ESGP as a dynamic system.

REFERENCES

1. V. I. Levintov, A. G. Larin. Electrostatic Generators with a Rigid Rotor, part 1, TsINTIElektroprom, 1963.
2. V. A. Lukutin, V. D. Eskov. Izv. AN SSSR, “Energetika i transport,” No. 2, 1967.
3. A. M. Kuptsov, V. V. Patsevich. “Elektrichestvo,” No. 7, 1968.