Effect of the ESG circuit on its external characteristic

INFLUENCE OF THE ESG CIRCUIT ON ITS EXTERNAL CHARACTERISTIC

A. P. Kononov, V. V. Patsevich, V. D. Eskov

(Submitted by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)

The operating principle and the fundamentals of the theory of electrostatic generators (ESG) with a cascade connection of carrier conductors are described in [1] and in a number of other works. It has been established that if, in the working volume of the generator, the potential gradients do not reach critical values (at which substantial leakage of charges from the carriers, intense ionization in the gap between the rotor and stator, etc. arise), then the voltage and current at the generator output are related by the well-known linear equation

\[ I_{\mathrm{n}} = I_{\mathrm{kz}}\left(1 - \frac{U_{\mathrm{n}}}{U_{\mathrm{xx}}}\right), \tag{1} \]

where \(I_{\mathrm{kz}}\) and \(U_{\mathrm{xx}}\) are the short-circuit current and the no-load voltage of the ESG.

In this case the ratios \(\dfrac{I_{\mathrm{kz}}}{nU_{\mathrm{v}}}\) and \(\dfrac{U_{\mathrm{xx}}}{U_{\mathrm{v}}}\) (\(n\) is the rotor speed, \(U_{\mathrm{v}}\) is the excitation voltage) prove to be constant quantities for each particular machine, depending on its geometry and circuit.

The present article is devoted to the study of one aspect of this dependence—the comparison of the external characteristics of an ESG when using various connection circuits for the excitation source and brushes of different width.

Fig. 1. Principal electrical circuits of an ESG.

Three variants of connecting the elements of an ESG with two pairs of poles are considered; they are shown schematically in Fig. 1. Here the rotor is depicted as a circle, the brushes as arrows, and the inductors as arcs. The individual carriers and the stator elements that equalize the field are not shown. The direction of rotation of the rotor coincides with the clockwise direction. Circuits “a” and “b” correspond to a unipolar generator with conduction and induction excitation circuits; circuit “c” corresponds to a bipolar generator. The brushes may cover one carrier each (“narrow”) or several carriers simultaneously (“wide”).

For known values of the partial capacitances between ESG elements (the most important are the partial capacitances of any of the carriers relative to: the grounded elements of the ESG structure—\(C_0\), the opposite stator element—\(C_1\), the neighboring carrier—\(C_2\), and the stator element opposite the neighboring carrier—\(C_3\)), the calculation of the external characteristic can be carried out rather simply by the method set forth in [2]. For this it is sufficient to calculate three quantities: the coefficient of natural overcharging \(k_i\), the voltage multiplication coefficient \(k_u\), and the short-circuit current of the generator without overcharging \(I_0\).

Introduce the notation: \(2m\) is the total number of carriers, \(p\) is the number of pairs of poles, and \(s\) is the number of carriers covered by each of the inductors. Then, for

\[ \frac{m}{p} > \frac{C_2}{C_3} \]

according to [2] it is not difficult to obtain

\[ I_0 = 2mnC_\mathrm{e}U_\mathrm{B}, \tag{2} \]

\[ k_i = \sqrt{1 + \frac{4C_2}{C_3}} \tag{3} \]

\[ k_u = \left[ \frac{C_0}{C_\mathrm{e}}+ \frac{F}{\dfrac{m}{p}-s+1} \right]^{-1}, \tag{4} \]

where \(C_\mathrm{e}=C_0+C_1+2C_3\), and \(F\) is a complicated function of \(m\), \(p\), \(s\), \(C_1\), \(C_2\), \(C_3\), \(C_0\) \((F<k_i)\).

Taking these parameters into account, the ratios \(I_{\mathrm{kz}}/I_0\) and \(U_{\mathrm{xx}}/U_\mathrm{B}\), depending on the circuit under consideration, acquire the values given in Table 1. Let us compare generators differing by only one factor.

It is easy to see that the use of wide brushes eliminates the conditions for the manifestation of natural overcharging: in this case the short-circuit current of unipolar ESGs and the no-load voltage of bipolar ESGs turn out to be \(k_i\) times smaller than the corresponding quantities for generators with narrow brushes. At the same time, \(I_{\mathrm{kz}}\) of bipolar generators also decreases, while the no-load voltage of unipolar generators changes practically not at all. Hence it is clear that the use of narrow brushes contributes to an increase in ESG power.

Next, the current of generators with an induction excitation circuit is always less than the current of an ESG with a conduction circuit at the very same load voltage; the difference becomes insignificant when \(k_u \gg 1\). In the latter case preference should be given to the generator with an induction circuit, since its excitation source consumes practically no energy.

Finally, let us compare unipolar and bipolar generators with narrow brushes and a sufficiently high value of \(k_u\). The short-circuit current of the former is \(2\left(k_i+\frac{1}{k_i}\right)\) times greater, and the no-load voltage

Table 1.

of the first only by a factor of $k_i$ smaller than the corresponding parameters of the second. It is not difficult to see that, for

\[ U_{\mathrm{n}} < k_u U_{\mathrm{v}}\left(1-\frac{k_i-1}{2k_i^2+1}\right) \tag{5} \]

a unipolar ESG gives a larger current than a bipolar one. In this range there usually also lies the operating point corresponding to the nominal mode. Even with a twofold increase in the excitation voltage of the bipolar generator (which is probably possible with the same dielectric strength of the working medium in both generators), the unipolar ESG, as the analysis shows, remains more effective.

These theoretical conclusions were checked on an experimental single-disk generator with a rotor made of epoxy resin with metal conveyors of circular cross section baked into it, arranged in one row at an equal distance from both surfaces of the rotor. The rotor was located between two stator disks made of weakly conducting glass, with which the metal inductors made contact. The principal design characteristics were: rotor diameter 140 mm, its thickness 7 mm, number of pole pairs 2, number of conveyors 52, conveyor diameter 4 mm, its length 25 mm, dielectric constant of the rotor material 3.75, gap between rotor and stator 0.8 mm on each side.

For such a system the partial capacitances can be calculated from the formulas given in [3]. This gives the following numerical values: $C_{\mathrm{e}}=1.9\ \mu\mathrm{F}$, $\dfrac{C_2}{C_{\mathrm{e}}}=0.5$, and from (3) $k_i=1.73$. As for the quantity $\dfrac{C_0}{C_{\mathrm{e}}}$, it is so small that the errors in its experimental determination are comparable with it. To avoid error, it was decided to increase the value of this capacitance artificially; for this purpose one half of the stator was grounded in all operating modes. In this case, obviously, $C_0=0.5C_{\mathrm{e}}$. Then in the right-hand side of equation (4) the second term may be neglected and one may take $k_u=2$.

The calculated values of $I_{\mathrm{sc}}$ and $U_{\mathrm{oc}}$ for each of the circuits considered, at $n = 47$ rev/sec and $U_{\mathrm{e}} = 1$ kV, are also given in Table 1, as are the values of these quantities obtained experimentally. It should be noted that the generator tests were carried out at comparatively small values of the excitation voltage ($U_{\mathrm{e}} \le 2$ kV); at larger values of $U_{\mathrm{e}}$, the dependences $I_{\mathrm{sc}} = f(U_{\mathrm{e}})$, $U_{\mathrm{oc}} = f(U_{\mathrm{e}})$, and consequently also $I_{\mathrm{n}} = f(U_{\mathrm{n}})$, became nonlinear.

A comparison of the results of calculation and experiment shows, first, that the accuracy of the calculation is entirely satisfactory and far exceeds the accuracy of calculation by all other methods currently known. Second, it confirms the correctness of the quantitative assessment of the phenomena of natural recharging and armature reaction, which are inseparably connected with one another and, in the final analysis, prove to be different forms of manifestation of electrostatic induction.

Thus, the method set forth in [2] correctly takes into account most of the factors characterizing the design and circuit of an ESG and affecting its external characteristic, and can be used in the engineering calculation of generators with a cascade connection of carrier-conductors.

REFERENCES

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

2. V. A. Lukutin, V. D. Eskov. On the calculation of an ESG with a cascade connection of carrier-conductors. Izv. AN SSSR. Energetika i transport, No. 2, 1967.

3. A. P. Kononov, V. V. Patsevich, V. D. Eskov. Partial capacitances between the elements of an ESG. Izv. TPI, vol. 162, 1967.