Toward the Calculation of the External Characteristic of a Self-Excited ESG

Yu. G. Leleko, V. D. Es'kov

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering and by the Department of Rotor ESGs of the Research Institute of Nuclear Physics of the Siberian Branch of the USSR Academy of Sciences at TPI)

In connection with the rapid development of electron-ion technology in recent years, interest has increased in electrostatic generators as high-voltage direct-current sources. The greatest attention is being given to generators with independent excitation. Nevertheless, in autonomous systems and in small-size devices it is expedient to use self-excited electrostatic generators with self-starting (SESG), which do not require an external source of excitation for their operation and, in this connection, possess higher specific energy characteristics.

The principle of operation of generators with self-starting has been described in sufficient detail in the literature (for example, [1], [2]); however, there is practically no information there on relations connecting the output parameters of the machine with its geometry, without which scientifically substantiated design of generators is impossible.

At the same time, several methods already exist for obtaining such relations for generators with independent excitation [3, 4]. The advisability of applying these sufficiently well-developed methods also to the calculation of an SESG is obvious. It is precisely this approach that is used in the present article to determine a method for constructing and deriving the equation of the external characteristic of a small-size SESG, the schematic diagram of which is shown in Fig. 1, where 1 is the stator made of insulating material; 2 is a layer of weakly conducting material, ensuring a uniform distribution of potential over the stator outside the in-

Fig. 1. Schematic diagram of an SESG

inductors; 3—the dielectric base of the rotor; 4—the gap between the rotor and the stator; 5—the inductor of the load system; 6—the inductor of the excitation system; 7—the main transporters; 8—the auxiliary transporters.

The brushes by means of which contact commutation is effected are shown in the figure as triangles. In addition, the figure indicates: \(U_{\mathrm{n}}\) and \(I_{\mathrm{n}}\)—the voltage and current of the generator load; \(U_{\mathrm{v}}\)—the excitation voltage; \(I_{\mathrm{v}}\)—leakage of charges in the excitation system (mainly through a special corona gap, which serves to regulate the excitation voltage). Leakages in the load system and along the stator are not taken into account in the first approximation.

The system described can be regarded as a combination of two generators (main and auxiliary), the output voltage of each serving as the excitation voltage for the other (the excitation circuit is inductive). Accordingly, their currents and voltages can be related by equations obtained in [4] for an ESG with external excitation and having, in the present case, the following form:

\[ I_{\mathrm{n}} = 2mnK_i C_{\mathrm{e}}\left(U_{\mathrm{v}} - \frac{U_{\mathrm{n}} + U_{\mathrm{v}}}{K_u}\right). \tag{1} \]

\[ I_{\mathrm{v}} = 2mnK_i' C_{\mathrm{e}}'\left(U_{\mathrm{n}} - \frac{U_{\mathrm{v}} + U_{\mathrm{n}}}{K_u'}\right), \tag{2} \]

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

\(2m\)—the number of transporters of each of the generators;

\(C_{\mathrm{e}}\)—the sum of the partial capacitances of the main transporter with respect to the stator elements and the grounded elements of the SESG;

\(K_u, K_i\)—dimensionless coefficients that may either be expressed through the partial capacitances between the elements of the main generator or determined experimentally;

\(C_{\mathrm{e}}', K_u', K_i'\)—the corresponding quantities for the auxiliary generator.

To simplify the notation, introduce the designations:

\[ R_0 = \frac{K_u}{2mnK_i C_{\mathrm{e}}}, \]

\[ R_0' = \frac{K_u'}{2mnK_i' C_{\mathrm{e}}'}, \]

(these quantities have the dimensions of resistances and characterize the slope of the external characteristic of the corresponding generator when operating with a constant external excitation voltage). Then equations (1) and (2) can be rewritten as follows:

\[ I_{\mathrm{n}}\cdot R_0 = U_{\mathrm{v}}\cdot (K_u - 1) - U_{\mathrm{n}}, \tag{3} \]

\[ I_{\mathrm{v}}\cdot R_0' = U_{\mathrm{n}}\cdot (K_u' - 1) - U_{\mathrm{v}}. \tag{4} \]

Denoting further \(R_{\mathrm{n}} = \dfrac{U_{\mathrm{n}}}{I_{\mathrm{n}}}\) and \(R_{\mathrm{v}} = \dfrac{U_{\mathrm{v}}}{I_{\mathrm{v}}}\), from (3) and (4) it is easy to derive that

\[ \frac{R_{\mathrm{v}}}{R_0'} = \frac{R_{\mathrm{n}} + R_0}{KR_{\mathrm{n}} - R_0}, \tag{5} \]

where

\[ K = \left[(K_u - 1)\cdot (K_u' - 1) - 1\right]. \]

Equation (5) shows that each specific value of the load resistance \(R_{\mathrm{н}}\) must correspond to a completely definite value of the resistance in the excitation system \(R_{\mathrm{в}}\). The required change in \(R_{\mathrm{в}}\) is provided by leakage currents in the excitation system, which appear when the values of the potential gradient on the elements of the auxiliary rotor reach critical values. However, when the SESG operates in such a mode, as a result of intense ionization processes, contamination of the working volume occurs and its service life is substantially reduced. To avoid this, the above-mentioned corona gap is provided in the excitation system; it plays the role of the load of the auxiliary generator, limiting the voltage on the excitation inductor.

As experiments show, the volt-ampere characteristic of the corona gap has the form of the curve shown in Fig. 2. In this case the external characteristic of the SESG can be constructed on the basis of equations (3)—(5) and will have the form of the curve shown in Fig. 3. The corresponding special points of both characteristics are denoted by the same letters, which are used below as indices of voltages, currents, and resistances.

Point \(k\) characterizes the instant at which current appears in the corona gap; segment \(0k\) is the no-load condition of the auxiliary generator.

Fig. 2. Volt-ampere characteristic of the corona gap

Fig. 3. External characteristic of the SESG

of the generator \((R_{\mathrm{v}}\to\infty)\). Then from equation (5) the minimum value of the load resistance is found:

\[ R_{\mathrm{nk}}=\frac{R_0}{K}, \tag{6} \]

at which self-excitation of the ESG is still theoretically possible. Under these conditions, from equations (4) and (3) one obtains

\[ U_{\mathrm{nk}}=\frac{U_{\mathrm{vk}}}{K_u'-1} \quad \text{and} \quad I_{\mathrm{nk}}=\frac{U_{\mathrm{nk}}}{R_{\mathrm{nk}}} =\frac{U_{\mathrm{vk}}}{(K_u'-1)R_{\mathrm{nk}}}. \]

These equations show that the segment of the characteristic \(ok\) corresponds to the generator operating as a linear system, since the magnitudes of the potential gradients of the generator elements have not yet reached their critical values.

Point \(m'\) corresponds to no-load operation of the main generator \((I_{\mathrm{n}}=0,\ R_{\mathrm{n}}\to\infty)\). In this case, from equation (5) one can express the necessary minimum value of the static resistance of the corona gap:

\[ R_{\mathrm{vm}}=\frac{R_0'}{K}, \tag{7} \]

and then, from the curve \(U_{\mathrm{v}}=f(I_{\mathrm{v}})\), find the greatest value of the excitation voltage \(U_{\mathrm{vm}}\). Then the no-load voltage of the ESG, according to (3), is equal to

\[ U_{\mathrm{nm}}=U_{\mathrm{vm}}\cdot (K_u-1). \]

Point \(n\) determines the maximum-load operating mode of the ESG \((I_{\mathrm{n}}=I_{\max})\). Using equations (3) and (4), it is not difficult to show that the tangent to the curve \(U_{\mathrm{v}}=f(I_{\mathrm{v}})\) (Fig. 2) at this point is parallel to the segment \(om\) (since in Fig. 3 the tangent at point \(n\) is parallel to the corresponding segment), i.e., the differential resistance of the corona gap in the maximum-load mode is equal to its static resistance in the maximum-voltage mode. This circumstance makes it possible to determine the quantities \(U_{\mathrm{vn}}\), \(I_{\mathrm{vn}}\), and

\[ R_{\mathrm{vn}}=\frac{U_{\mathrm{vn}}}{I_{\mathrm{vn}}}, \]

and then, from equation (5),

\[ R_{\mathrm{nn}}=R_0\cdot \frac{R_{\mathrm{vn}}+R_0'}{K(R_{\mathrm{n}}-R_0')} =R_{\mathrm{nk}}\frac{R_{\mathrm{vn}}+R_0'}{R_{\mathrm{vn}}-R_{\mathrm{vm}}}, \tag{8} \]

Stable operation of the ESG is possible only when \(R_{\mathrm{n}}>R_{\mathrm{nn}}\), and the segment \(okn\) of the external characteristic can be obtained experimentally only when operating with a special nonlinear (for example, corona) load. In this case the load voltage and current are determined by the relations:

\[ I_{\mathrm{nn}}= \frac{U_{\mathrm{vn}}K-I_{\mathrm{vn}}\cdot R_0'}{R_0(K_u'-1)} \quad \text{and} \quad U_{\mathrm{nn}}=I_{\mathrm{nn}}\cdot R_{\mathrm{nn}}. \]

The working segment of the external characteristic is the curve \(nm\). It is not difficult to verify that point \(p\) also lies on it, corresponding to the maximum-power mode—the point at which the differential and static resistances of the generator are equal in absolute value and have opposite signs. The position of this point is determined both by the features of the generator itself and by the properties of the corona-producing

interval. Experiments show that its coordinates can, with sufficient accuracy for practical purposes, be taken as equal to

\[ U_{\mathrm{nr}}=\frac{U_{\mathrm{vt}}}{2}, \quad I_{\mathrm{nr}}=I_{\mathrm{nk}}, \]

and the segment \(pm\) may be replaced by a straight line. In this case, the load resistance corresponding to the maximum-power regime proves to be equal to

\[ R_{\mathrm{nr}}=\frac{U_{\mathrm{nr}}}{I_{\mathrm{nr}}} =R_{\mathrm{nk}}\cdot\frac{U_{\mathrm{vt}}\,(K_i'-1)}{2U_{\mathrm{vk}}}. \tag{9} \]

As for point \(p\) on the curve in Fig. 2, it can be found by drawing through point \(k\) a straight line parallel to the segment \(om\), up to its intersection with the curve, and determining \(U_{\mathrm{vr}}\) and \(I_{\mathrm{vr}}\).

In conclusion, it should be noted that if an active resistance \(R\) is introduced into the grounded-brush circuit of the auxiliary generator (to reduce the potential difference between the commutating conveyor and the load inductor), then formulas (3--9) retain their form when the quantity \(R'_0\) in them is replaced by \(R''_0=R'_0+R\cdot(K_u'-1)\).

Thus, by the method set forth above, one can construct the external characteristic of a self-excited electrostatic generator, investigate it, and identify special points on it, provided that the coefficients characterizing the geometry of the generator \((K_u, K_i, C_e)\) are determined, the rotor speed is specified, and the volt-ampere characteristic of the corona gap in the excitation system is known.

References

1. High-Voltage Testing Equipment and Measurements. Edited by A. A. Vorob'ev. GEI, 1960.

2. A. F. Kalganov. Development and Investigation of Certain Types of Rotor Electrostatic Generators. Dissertation. Tomsk, 1962.

3. V. I. Levitov, A. G. Lyapin. Electrostatic Generators with a Rigid Rotor. Part I. TsINTIElektroprom, Moscow, 1963.

4. V. A. Lukutin, V. D. Es'kov. On the Calculation of an Electrostatic Generator with Cascade Connection of Conveyor Conductors. Proceedings of the Academy of Sciences of the USSR, “Power Engineering and Transport,” No. 2, 1967.