On the Energy Relationship Between Rod ESGs and ESGs with a Dielectric Conveyor

A. F. Kalganov

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering and the sector of rotor ESGs of the YaFÉA Research Institute)

As shown in [1], the power of electrostatic generators with conductor conveyors in the form of rods located inside the dielectric body of the rotor (ESGS) can theoretically be brought close to the power of an electrostatic generator with a dielectric conveyor (ESGD), if the ESGS rotor carries a sufficiently large number of small-diameter rods, all other conditions being equal. However, whereas in an ESGD only one of the two surfaces of the disk or the wall of a cylindrical rotor is the working surface, in an ESGS both surfaces may be used as working surfaces if a second stator is employed. The use of a second stator practically does not increase the dimensions of the ESGS, while in the ideal case the generator power doubles. This served as an incentive for further intensive study of generators of this type. A method was proposed for calculating ESGS on the basis of equations with partial capacitances [2, 3]. The method of [3] proved to be more complete, making it possible to express analytically the linear portions of the short-circuit characteristic and the load characteristic as functions of the geometry of the system (the magnitude of the rotor–stator gas gap \(d_1\), the depth of embedding of the rods in the dielectric body of the rotor \(d_2\), the distance between the rod axes \(a\), the rod radius \(r\), the number of conveyors per pole pair \(2m\), the number of conveyors “covered” by inductors), the electrophysical properties of the rotor (the relative dielectric permittivity of the rotor material \(\varepsilon_r\)), the commutation conditions (narrow or wide brushes), and the electrical operating circuit of the generator (induction or conduction excitation circuits, the number of pole pairs \(p\), etc.).

As calculations subsequently confirmed experimentally have shown, a rod ESG is characterized by a phenomenon analogous to the additional natural self-excitation in an ESGD, called natural recharging in an ESGS [4]. This indicated the existence of a deep analogy between the processes occurring in an ESGS and in an ESGD.

A study of the physical processes accompanying the operation of an ESGS showed [5] that in an ESGS, as in an ESGD, a double natural transfer of charges takes place. This eliminated the need to apply artificial recharging in an ESGS in order to double the current.

In the works carried out, however, the question of the level of the maximum current in an ESGS and the conditions for its approach to the level of the maximum current of an ESGD was not addressed. Further studies showed

It has been shown [6] that, as in the ESGD, the maximum short-circuit current of an ESGS is determined by the value attainable, under the given conditions, of the normal component of the electric-field strength in the rotor—stator gas gap and does not depend on the degree of natural recharging.

In calculating an ESGD it is assumed that the charges on the rotor surface are distributed uniformly, and this corresponds to the idea of uniformity of the electric field in the rotor—stator gas gap. In an ESGS, however, owing to the discrete distribution of charges concentrated on rod conveyors, the field is nonuniform. However, the location of the conveyors inside the dielectric body of the rotor makes the field nonuniformity weakly expressed if the embedding depth of the conveyors is comparatively large (relative to the size of the gas gap) [7]. In connection with this, it becomes possible to represent the charge distribution in an ESGS not as discrete, but as if over the entire surface of the rotor, and to define the average maximum “surface” charge density as

\[ \sigma'_m=\gamma\sigma_m=\varepsilon_0 E'_{\text{lim}}=\gamma\varepsilon_0 E_{\text{lim}}=\gamma\varepsilon_0 E_{\text{br}}, \]

where \(\sigma_m\) is the maximum surface charge density in an ESGD (an averaged value);

\(E_{\text{lim}}\) is the average maximum permissible value of the normal component of the electric-field strength \(E_n\) in an ESGD, which in calculations is usually taken equal to the electric strength of the working medium \(E_{\text{br}}\);

\(E'_{\text{lim}}\) is the average maximum permissible value of \(E_n\) in an ESGS;

\(\gamma<1\) is a coefficient characterizing the degree of uniformity of the charge distribution over the rotor surface, or the degree of uniformity of the electric field in the gas gap of the ESGS.

Then the expression for the maximum short-circuit current of an ESGS having a double stator, by analogy with the expression for the short-circuit current of an ESGD, is written as

\[ I'_{\text{s.c.}0}=2\sigma'_m\cdot 2S_{\text{geom}}\cdot n\cdot p =2\gamma\sigma_m\,2S_{\text{geom}}\cdot n\cdot p, \]

or

\[ I'_{\text{s.c.}0}=2I_{\text{s.c.}0}\cdot \gamma, \tag{1} \]

where \(I_{\text{s.c.}0}=2\sigma_m S_{\text{geom}}\,n\cdot p\) is the maximum short-circuit current of the ESGD,

\(S_{\text{geom}}\) is the working geometrical surface of one side of the ESGS and ESGD rotor,

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

On the other hand, analysis of the expression for the maximum short-circuit current of an ESGS on the basis of the concepts set forth in [6] shows that it can be written in the form

\[ I'_{\text{s.c.}0}=2I_{\text{s.c.}0}\cdot \frac{C_{\text{rs}}}{C_{\text{rs}0}} =2I_{\text{s.c.}0}\cdot k_0, \tag{2} \]

where \(k_0=\dfrac{C_{\text{rs}}}{C_{\text{rs}0}}\) is a coefficient less than unity, taking into account the degree to which the value of the actual capacitance of the system of ESGS rotor conveyors relative to the stator approaches the value of the maximum possible capacitance \(C_{\text{rs}0}\) (for the given conditions). The latter is equal to twice the capacitance of the ESGD rotor—stator system under equivalent conditions and is determined as

\[ C_{\text{rs}0}=\frac{\varepsilon_0\varepsilon_r\,2S_{\text{geom}}}{d_{\text{av}}}, \]

where \(d_{\mathrm{pr}} = d_2 + \varepsilon_{r2} d_1\) is the reduced rotor–stator gap. The coefficient \(k_0\) to a certain extent characterizes the “filling” of the rotor by conveyors and, in this sense, may be called the filling coefficient.

Comparing (1) with (2), we find:

\[ \gamma = k_0. \]

The result obtained makes it possible to draw the following conclusions:

1. In accordance with the initial conditions, the degree to which an ESGS approaches an ESGD (in terms of the maximum short-circuit current) is determined numerically by the coefficient of filling of the rotor dielectric body with conveyors, \(k_0\), defined by the ratio of the capacitances of the rotor–stator system of the ESGS and the ESGD under equivalent conditions.

2. Under these same conditions, the filling coefficient \(k_0\) simultaneously characterizes the degree of uniformity of the electric field in the gap of the ESGS.

For the special case of a two-row arrangement of rods and a limited range of variation of the geometric parameters of the ESGS rotor, an empirical formula was obtained in [3] for determining the value \(C_{\mathrm{rs}}\). Using this formula, the expression for \(k_0\) takes the form:

\[ k_0 = \frac{2\pi d_{\mathrm{pr}}} {a \ln \left[ 2\left(\frac{d_{\mathrm{pr}}}{r}+1\right)\right]} \left[ 1 - e^{-\frac{a}{2r}\left(\frac{0.9r}{d_{\mathrm{pr}}+r}+0.02\right)} \right]. \tag{3} \]

The quantities \(a\), \(r\), and \(d_{\mathrm{pr}}\), already mentioned above, enter into (3).

It follows from (3) that, in order to bring the value of \(k_0\) closer to unity (and this is precisely what is of primary interest to us), it is necessary to increase the values of \(d_{\mathrm{pr}}\) and \(r\) and to decrease the value of \(a\). This agrees well with existing ideas about the nature of the influence of the quantities entering into (3) on the degree of uniformity of the electric field in a system of electrodes analogous to the rotor–stator system of an ESGS.

A detailed analysis of (3) shows that, for given values of the gas gap \(d_1\) and of the distance between the axes of the conveyors \(a\), the quantities \(d_{\mathrm{pr}}\) and \(r\) have an “optimum.” By the “optimal” values of \(d_{\mathrm{pr}}\) and \(r\) are meant such values above which the dependence of \(k_0\) on \(d_{\mathrm{pr}}\) and \(r\) becomes weakly expressed. Thus, for \(d_1\) on the order of tenths of a millimeter, the optimal value of \(d_{\mathrm{pr}}\) is no more than several millimeters.

The quantity \(a\), in accordance with (3), should be as small as possible. The limit of its minimum value is \(a = 2r\), when the rods are placed closely adjacent to one another (\(a - 2r = 0\)). Obviously, this is an unrealistic case.

As follows from the data of [3], the maximum potential difference between the rods occurs when the ESGS operates in the short-circuit mode and, in magnitude, is equal to the excitation voltage \(U_b\). This voltage is applied to the interval \(a - 2r\). It follows from this that the choice of the value of \(a\) must be made from the condition of preventing breakdown between the rods both along the surface of the collector part and through the thickness of the dielectric body of the rotor. Obviously, the value of \(a\) must be no less than a certain value (for the given conditions). It will be the smaller, the smaller the maximum value \(U_b\), determined by the value \(d_{\mathrm{pr}}\), will be. From these considerations it is necessary to have a minimum value of \(d_{\mathrm{pr}}\). This once again confirms the need to provide, in an ESGS, minimum rotor–stator gas gaps \(d_1\) (while preserving the optimal ratio between \(d_1\) and \(d_{\mathrm{pr}}\)), since in this case the current level \(2I_{\mathrm{s.c.}}\) is increased and simultaneously increas-

the degree of approximation to it of the current \(I'_{\mathrm{k.z}}\), at the expense of reducing the optimal value of \(a\). However, this route is very limited, since in any case the excitation voltage amounts to tens of kilovolts.

It should also be noted that the tendency to reduce \(a\) may have an adverse effect on the load characteristics, since this leads to an increase in the influence of parasitic capacitances in the ESGS. Retaining large values of \(a\), however, is the reason why the value \(k_0\) proves to be noticeably less than unity.

The results presented have been obtained on the basis of an analysis of a particular embodiment of an ESGS and without taking many other factors into account. Their verification and refinement, as well as further generalization, must be carried out on the basis of a comprehensive investigation of the electric field of rotors of various designs, using experimental data from the study of experimental ESGS specimens.

REFERENCES

1. A. F. Kalganov, V. V. Patsevich. Izv. vuzov, Elektromekhanika, No. 8, 917, 1963.

2. V. V. Patsevich. Certain problems in the theory and calculation of ESGs with conveyor conductors. Dissertation, Tomsk, 1964.

3. E. D. Eskov. Investigation of electrostatic generators with cascade connection of conveyor conductors. Dissertation. Tomsk, 1966.

4. A. P. Kononov, V. V. Patsevich, V. D. Eskov. Izv. AN SSSR. Energetika i transport, No. 4, 52, 1967.

5. A. M. Kuptsov. Physical processes in electrostatic generators with conducting conveyors placed in a dielectric. Dissertation, Tomsk, 1967.

6. A. F. Kalganov, Yu. N. Sivkov. Izv. TPI, vol. 193, 1968.

7. N. J. Felici. Ann. telecomm., No. 9, 44, 1954.