On the Analysis of the Short-Circuit Characteristic of the ESGPD

V. V. PATSEVICH, V. D. ESKOV, I. P. GUK

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering and the Rotor ESG Laboratory of the Nuclear Physics Research Institute)

The short-circuit characteristic of electrostatic generators with carrier-conductors placed in a dielectric (ESGPD), under contact commutation, has the form shown in Fig. 1 (curve 1). On the initial linear portion of the characteristic, the current of an ESGPD with one pair of poles is determined from the known equation

\[ I_{\mathrm{kz}} = 2mn(q - q'), \tag{1} \]

where \(2m\) is the number of rotor carriers,

\(q\) and \(q'\) are the charges of a carrier during its forward travel (from the excitation brush to the load brush) and reverse travel (from the load brush to the excitation brush), respectively.

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

The magnitudes of the charges \(q\) and \(q'\) can be determined by using the equations of electrostatics with potential coefficients, electrostatic induction coefficients, partial capacitances, suitably introduced generalized capacitances, or by direct calculation of the electric field of the generator.

If the latter is known, then the charges of the carriers can be found from Gauss’s theorem in integral form. Choosing the surface of integration so that it partly coincides with the dielectric surface of the rotor \(S\), and is partly perpendicular to the latter, so as to enclose the carriers of only the forward or only the reverse travel, we obtain, with sufficient accuracy,

\[ mq = \frac{1}{2}\varepsilon_0 E'_{n\,\mathrm{av}} S, \]

\[ mq' = \frac{1}{2}\varepsilon_0 E''_{n\,\mathrm{av}} S, \]

then

\[ I_{\mathrm{kz}} = n\varepsilon_0 S\left(E'_{n\,\mathrm{av}} - E''_{n\,\mathrm{av}}\right), \tag{2} \]

where

\(E'_{n\,\mathrm{av}}\) and \(E''_{n\,\mathrm{av}}\) are the mean values of the normal component of the electric-field intensity at the rotor surface, respectively in the regions of forward and reverse carrier travel.

Equation (2) establishes a linear dependence of the generator current on the electric-field intensity in the rotor–stator gap. In those cases where the maximum value of the short-circuit current

of the generator is limited by discharges in the rotor–stator gap [1]; this equation also makes it possible to determine the limiting value of the current.

Indeed, the rotor–stator gap may be regarded as a slot bounded by one or two dielectric barriers, for which the limiting value of the field intensity \(E_k\) is equal to the electric strength of the corresponding gaps between metallic electrodes in a uniform electric field [2, 3].

Assuming \(E'_{n\,\mathrm{av}}=-E''_{n\,\mathrm{av}}=E_k\), we obtain the limiting value of the generator current as

\[ I_{\mathrm{lim}}=2nS\varepsilon_0 E_k . \tag{3} \]

The short-circuit characteristic according to equations (2) and (3) is represented by two rectilinear segments (broken line 2, Fig. 1).

Fig. 1

The actual field in the gap, because of the discrete distribution of the charges carried by the carriers, and also under the influence of a number of other factors, will be nonuniform. In this case the average values of the normal component of the electric-field intensity will be lower than the corresponding maximum values \(E'_{\max}\) and \(E''_{\max}\), which in turn cannot exceed the value \(E_k\). For practical use of equation (3) in calculating and analyzing short-circuit characteristics, even when the values of \(E_k\) and \(E_{\max}\) are known, it is necessary to know the dependence of \(E_{\max}\) on the magnitude of the generator excitation voltage and to determine the conditions under which the magnitude of the maximum current is attained at practically acceptable values of the excitation voltage. As analysis of the results of investigations presented in [4, 5, 6] shows, the electric-field intensities and the carrier potentials in the short-circuit mode attain their greatest values in the middle part both of the region of the direct stroke of the carriers and in the region of the return stroke. Transforming the expressions obtained in [6], taking into account the simplification for the coefficient of natural recharging \(k_i\) made in [7], for the potentials of the corresponding carriers we have

\[ U'_{\mathrm{tr}}=0.5(1+k_i)U_v, \tag{4} \]

\[ U''_{\mathrm{tr}}=0.5(1-k_i)U_v. \tag{5} \]

Taking the potential on the surface of the rotor dielectric in the region of the maximum intensity values to be

\[ U_{\max}=k_p U_{\mathrm{tr}}, \tag{6} \]

the maximum value of the intensity can be determined, with a sufficiently high degree of accuracy, as

\[ E_{\max}=\frac{U_{\max}}{\delta}. \tag{7} \]

Here \(\kappa_p\) is a coefficient depending on the rotor geometry and on the electrical characteristics of the rotor dielectric;

\(\delta\) is the magnitude of the rotor–stator gap.

As is seen from equations (4)—(7), the field strength in the forward-motion zone of the carriers is greater, in absolute value, than in the reverse-motion zone of the carriers, and the signs of the strengths are opposite. Consequently, the limiting value of the field strength is reached first in the forward-motion zone of the carriers.

Denoting the excitation voltage at which \(E'_{\max}=E_k\) by \(U_{\mathrm{vk}}\), from (4), (6), and (7) we obtain

\[ E'_{\max}=\kappa_p \frac{K_i+1}{2\delta}\,U_{\mathrm{vk}}=E_k. \tag{8} \]

For \(U_v>U_{\mathrm{vk}}\), an electric discharge arises in the gap in the region of the forward motion of the carriers, similar to a discharge in a dielectric slot. During the discharge, a charge of opposite sign relative to the carrier charge will be deposited on the dielectric surface. Therefore, in the zone of maximum field strength the magnitude of the latter will remain constant and approximately equal to \(E_k\), if the discharge ignition voltage \(E_{\mathrm{kz}}\) and the discharge extinction voltage \(E_{\mathrm{pk}}\) differ only slightly. At present there are no sufficiently substantiated data on the ratio of the quantities \(E_{\mathrm{zk}}\) and \(E_{\mathrm{pk}}\) even for slots with a uniform field. Since a change in the absolute value of \(E_k\) does not affect the qualitative aspect of the phenomena under consideration, in what follows we shall take the value \(E_k\) to be equal to the maximum field strength at which extinction of the discharge occurs in a dielectric slot analogous to the rotor–stator gap.

In the presence of charges on the dielectric surface of the rotor, the linearity of the short-circuit characteristic will be preserved until the field strength in the reverse-motion zone of the carriers reaches the value \(-E_k\) [1]. Assuming the conductivity of the rotor dielectric to be zero, one may assert that, after charges have been deposited on the dielectric surface of the rotor at \(U_v=\mathrm{const}>U_{\mathrm{vk}}\), the ionization processes in the gap will cease, and the superposition principle will be applicable to the system under consideration when calculating the field. The resulting maximum field strength will be determined as the algebraic sum of two components. The first is due only to the charges of the carriers, \(E_m\), which can be found from formulas (6), (7); the second is due only to the charge on the dielectric surface, \(E_{\sigma m}\). In calculating \(E_m\), it should be taken into account that the resulting potentials of the carriers, also in the presence of surface charge, are described by equations (4), (5). Therefore, denoting the component of these potentials due to the charge on the dielectric surface by \(U_\sigma\), we obtain for the regions of forward and reverse motion

\[ E'_{\max}=\frac{\kappa_p}{\delta}\left(\frac{1+K_i}{2}U_v-U_\sigma\right)+E_{\sigma m}=E_k, \tag{9} \]

\[ E''_{\max}=\frac{\kappa_p}{\delta}\left(\frac{1-K_i}{2}U_v-U_\sigma\right)+E_{\sigma m}. \tag{10} \]

With a further increase in the excitation voltage, the critical gradients will also be reached in the reverse-motion zone of the carriers, i.e. \(E''_{\max}=-E_k\). Denoting the corresponding excitation voltage by \(U_{vm}\) and subtracting equation (10) from (9), we obtain

\[ E'_{\max}-E''_{\max}=E_k-E''_m=\frac{\kappa_p K_i}{\delta}U_{vm}=2E_k, \]

whence

\[ E_{\mathrm{k}}=\frac{\kappa_{\mathrm{p}}\kappa_i U_{vm}}{2\delta}. \tag{11} \]

Comparison of expressions (8) and (11) gives

\[ U_{vm}=\left(1+\frac{1}{\kappa_i}\right)U_{vk}, \tag{12} \]

i.e., the excitation voltage at which \(E_{\mathrm{k}}\) is reached on the return path can exceed by no more than a factor of two the voltage at which ionization arises on the forward path of the transporters (the case \(\kappa_i=1\)).

For \(U_v>U_{vm}\), periodic or steady ionization processes take place in the gap, as a result of which, in the region of the forward travel of the transporters, charges are deposited on the dielectric surface of the rotor, while in the region of the return travel they are compensated. Qualitatively, these processes will be analogous to processes in a dielectric gap under a periodically varying voltage (for example, a sinusoidal voltage with an imposed constant component). The generator current in this case does not increase [1].

To determine the maximum short-circuit current, it is necessary to establish the relation between \(E'_{\mathrm{av}}\) and \(E''_{\mathrm{av}}\), which determine the magnitudes of the charges \(q\) and \(q'\), and the corresponding maximum field strengths.

If the rotor is divided into identical cells, the number of which is equal to the number of transporters, then, as analysis of the field patterns shows, the charges of the transporters are determined by Gauss’s theorem most simply when choosing integration surfaces coinciding with the boundaries of cells equidistant from the commutation zones: on the one hand, it is precisely in these cells, as noted above, that the maximum field-strength values are attained, and on the other hand, the average field strength over the entire surface of a cell is equal to the average field strength on the dielectric surface (without taking into account the flux of the vector \(E\) in the direction of the transporter axis).

In this case the average and maximum field strengths are related by the proportionality coefficient

\[ \kappa_0=\frac{E_{\mathrm{av}}}{E_{\max}}, \tag{13} \]

which characterizes the nonuniformity of the field in the cell and depends on the geometry of the ESGPD.

Then from equations (2), (9), (10), (13) we obtain, for \(U_v=U_{vm}\),

\[ I_m=2n\kappa_0\varepsilon_0 S E_{\mathrm{k}}, \tag{14} \]

whence, taking (11) into account, it follows that

\[ I_m=\frac{\varepsilon_0 S}{\delta}\,n\kappa_0\kappa_{\mathrm{p}}\kappa_i U_{vm}. \tag{15} \]

In accordance with the foregoing, the theoretical short-circuit characteristic has the form of broken line 3 (Fig. 1), the point of conjugation of whose segments is determined by equation (15).

The ratio of the maximum current to the limiting attainable current from equations (3) and (14) is obtained as

\[ \frac{I_m}{I_{\mathrm{lim}}}=\kappa_0. \tag{16} \]

It is therefore clear that, in order to obtain the greatest possible short-circuit current, one should first of all improve the uniformity of the field in the gap. At the same time, however, it is necessary to have a sufficiently large coefficient at \(U_{\text{vt}}\) in equation (11) in order to reach the desired current maximum at practically acceptable values of the excitation voltage.

Thus, it is necessary to establish the dependence of the coefficients entering equation (15) on the rotor geometry. This can be done either on the basis of a calculation of the electric field of the ESGPD, or by modeling this field. Given these dependences, an engineering procedure can be developed for the electrical design of the ESGPD for maximum parameters with optimal geometry.

LITERATURE

1. A. M. Kuptsov, V. V. Patsevich. Study of double natural transfer in an ESG. Izv. TPI, vol. 191, 1969.

2. B. N. Kanonykhin. Discharge in air interlayers enclosed within the bulk of a solid dielectric. ZhTF, vol. 33, issue 10, 1939.

3. A. V. Dmitriev. Study of discharge in a narrow cavity bounded on one or both sides by a dielectric. ZhTF, vol. 33, issue 9, 1963.

4. I. P. Guk, V. A. Lukutin. Determination of the field potential of the rotor of an electrostatic machine. Izv. TPI, vol. 191, 1969.

5. A. M. Kuptsov, V. V. Patsevich. Oscillographic study of the distribution of potentials over the rotor in an ESG. Izv. TPI, vol. 191, 1969.

6. V. D. Eskov. On the question of the distribution of potential over the rotor of an ESG. Izv. TPI, vol. 152, 1966.

7. A. P. Kononov, V. V. Patsevich, V. D. Eskov. Natural recharging in an ESG with cascade connection of carrier-conductors. Izv. AN SSSR, “Energetics and Transport,” No. 4, 1967.