PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 149 1966

ON THE SPECIFIC ENERGY CHARACTERISTICS OF ELECTROSTATIC GENERATORS AND THE ANALOGY BETWEEN ELECTROSTATIC AND ELECTROMAGNETIC MACHINES

V. A. LUKUTIN, V. V. PATSEVICH

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

An analysis of the present state of the theory and practice of electrostatic generators (ESG) shows that the conditions of maximum power (total and specific with respect to volume and weight) are most fully met by disk ESGs with conductor conveyors; multidisk ESGs with parallel connection of the poles; and multidisk ESGs with cascade (series) connection of the conveyors.

For generators with parallel connection of the conveyors in the maximum-power regime we have [1, 2]

\[ U_{\text{n}} = U_{\text{v}} = \frac{U_0}{2} = \frac{1}{2} E_{\text{pr}} d, \]

\[ I_{\text{n}} = 2pnU_0 \frac{\varepsilon_0 S}{d}, \]

that is,

\[ P = 2pndS \frac{\varepsilon_0 E_{\text{pr}}^{2}}{2}, \]

where \(p\) is the number of pairs of ESG poles,
\(n\) is the number of revolutions of the rotor per second,
\(d\) is the gap between the rotor and stator,
\(S\) is the one-sided working surface of the rotor disk,
\(\varepsilon_0\) and \(E_{\text{pr}}\) are the dielectric permittivity and the permissible electric-field strength in the rotor–stator gap.

Let us determine the specific power of such a generator. Proceeding from the condition of preserving the uniformity of the generator electric field, we shall assume [1, 2] that the thickness of the rotor conveyors and the stator inductors \(h\) is equal to or somewhat greater than twice the rotor–stator gap \(d\), i.e., \(h \gg 2d\), and, consequently, the volume corresponding to one rotor disk will be \(V = \pi r_2^2 \cdot 6d = 6dS_1\). Here \(r_2\) is the outer radius of the rotor disk.

The number of generator poles is limited by the influence of the parasitic capacitances of the generator, which depend on the size of the end surfaces and on their ratio to the working surface of the conveyors. In practice it is assumed [3] that the influence of parasitic capacitances will still be insignificant if the equality \(l_1 = (4 \div 5)h\) is satisfied, where

\[ l=\frac{\pi r_1}{p_1}-a \]
is the size of the carrier along the arc of the inner radius \(r_1\),

\(a\) is the gap between carriers. Taking \(a \simeq d\), we obtain

\[ \frac{\pi r_1}{p_0}=10d \quad \text{or} \quad p_0=\frac{\pi}{10}\frac{r_1}{d}. \]

The value \(r_1\) is determined by the conditions of reliable fastening of the carriers, reliable insulation between them, and the need to preserve rational shapes of their surface. In generators that have been practically implemented or are currently being developed, it is assumed [3] that \(r_1=\left(\frac{1}{2}\div\frac{1}{3}\right)r_2\); consequently, in this case

\[ p_0=\left(\frac{1}{2}\div\frac{1}{3}\right)\frac{\pi}{10}\frac{r_2}{d}=k_1\frac{r_2}{d}. \]

The coefficient \(K_1=\left(\frac{1}{2}\div\frac{1}{3}\right)\frac{\pi}{10}=0.1\div0.16\) is called the pole coefficient, determining the optimal number of pairs of poles of the generator \(p_0\).

For the adopted ratio of the radii \(r_1\) and \(r_2\), the ratio between the useful area of the carriers \(S\) and the total surface \(S_1\) is

\[ \frac{S}{S_1}=1-\frac{\pi r_1^2+2p_0a(r_2-r_1)}{\pi r_2^2}=(0.75\div0.9) \]

and for the specific power of the generator we obtain

\[ P_{\text{sp. max}}=\frac{k_1S}{12\pi S_1}\cdot\frac{v}{d}\,\varepsilon_0E_{\text{bd}}^2 =k_2\frac{v}{d}\,\varepsilon_0E_{\text{bd}}^2 =(2\div3.8)10^{-3}\frac{v}{d}\,\varepsilon_0E_{\text{bd}}^2. \]

Under conditions (for example, in vacuum) \(v=100\ \text{m/sec}\), \(d\simeq1\ \text{mm}\), \(E_{\text{bd}}=10^3\ \text{kV/cm}\), and \(\rho\simeq4\ \text{kg/dm}^3\) (\(\rho\) is the specific density of the generator volume), close to those adopted in [3], we obtain

\[ P_{\text{sp. max}}=4\div8\ \text{kW/kg}. \]

For generators with cascade connection of carriers, when the number of the latter is large, the maximum power per disk, without taking recharging into account, is equal to [4].

\[ P_{\text{k. max}}=nlS\varepsilon_0E_{\text{bd}}E_\tau, \]

where \(E_\tau\) is the permissible value of the electric-field intensity in the plane of motion of the carriers (the tangential component),

\(l\) is the distance between the grounded and high-voltage brushes along the path of motion of the carriers.

The magnitude of the single-sided working surface of the rotor is \(S=\pi(r_2^2-r_1^2)\), where \(r_1\) and \(r_2\) are, respectively, the inner and outer radii of the rotor disk with respect to the carriers. Taking into account that, in order to ensure the necess—

of the insulation over the surface of the disk, the equality \(l=\pi r_1\) must be satisfied; we obtain

\[ P_{\mathrm{k.max}}=2\pi^2 n \varepsilon_0 E_{\mathrm{pr}}E_\tau (r_2^2-r_1^2)r_1. \]

The value of the radius \(r_2\) is usually determined from the conditions of mechanical strength of the rotor and may be regarded as given. The value corresponding to the maximum power is determined from the equation

\[ \frac{dP_{\mathrm{k.max}}}{dr_1}=0 \]

and is equal to \(r_1=\dfrac{r_2}{\sqrt{3}}\).

Consequently, the value of the maximum power is

\[ P_{\mathrm{k.max}}=\frac{2\pi}{3\sqrt{3}}\,r_2^2 v \varepsilon_0 E_{\mathrm{pr}}E_\tau . \]

Assuming, as before, that the volume per disk is \(v=6d\pi r_2^2\), for the specific power we obtain

\[ P_{\mathrm{k.sp.max}}=\frac{1}{9\sqrt{3}}\cdot\frac{v}{d}\varepsilon_0 E_{\mathrm{pr}}E_\tau =6.4\cdot10^{-2}\frac{v}{d}\varepsilon_0 E_{\mathrm{pr}}E_\tau =k_3\frac{v}{d}\varepsilon_0 E_{\mathrm{pr}}E_\tau . \]

The ratio of the specific powers of the compared types of generators under analogous conditions, on the basis of the equations obtained, is determined by the equality

\[ \frac{P_{\mathrm{k.sp.max}}}{P_{\mathrm{sp.max}}} =\frac{k_3 E_\tau}{k_2 E_{\mathrm{pr}}}. \]

The coefficients \(k_2=(2\div3.8)10^{-3}\) and \(k_3=6.4\ 10^{-2}\) may be called the volume-utilization coefficients of the generators. From the equality obtained above it follows that the power of generators for which \(P=E_{\mathrm{pr}}^2\) will be equal to or greater than the power of generators for which \(P=E_{\mathrm{pr}}E_\tau\), when

\[ \frac{E_{\mathrm{pr}}}{E_\tau}>\frac{k_3}{k_2}=(20\div30). \]

Thus, for \(E_\tau=20\div25\ \mathrm{kV/cm}\), it is necessary that \(E_{\mathrm{pr}}>400\div800\ \mathrm{kV/cm}\).

In this it was assumed that both types of generators have equal rotor–stator gaps. If the rotor–stator gap is approximately \(1\ \mathrm{mm}\), then the maximum load voltage of a generator with parallel connection of the poles cannot exceed the value

\[ \frac{E_{\mathrm{pr}}}{2}=20\div40\ \mathrm{kV/cm}, \]

whereas in generators with an increased number of carriers the voltage is almost independent of the magnitude of the rotor–stator gap, which in this case may be made arbitrarily small. In this connection, the question of comparing the specific characteristics of the generators under investigation can be considered only in theoretical terms. In a practical consideration of the question, the decisive role is played by the magnitude of the load voltage that must be obtained from the generator.

The physical interpretation of the results obtained is that the electric field, not requiring a closed flux-conducting circuit, makes it possible to utilize more effectively the volume occupied by an electrostatic generator (ESG), as compared with the magnetic field in electromagnetic machines (EMM).

The multidisk ESGs considered, with cascade connection of the carriers, differ from EMMs not only in the character of the field. From the energy point of view, their operating principle is directly opposite to the operating principle of EMMs. Indeed, in an ESG with cascade connection of the carriers, the main part of the electrical

the energy of the individual conveyors, and consequently of the entire generator, is obtained at the expense of mechanical work when

\[ i=-\frac{dq}{dt}=-\frac{dN}{dt}=0. \]

In an EMM, electrical energy from the individual turns, and consequently from the entire generator, can be obtained only when

\[ e=\frac{d\psi}{dt}\ne 0. \]

The analogue of these ESGs from the energy point of view could evidently be an EMG with turns in the superconducting state. When such a turn is moved from a strong magnetic field into a weak one, or conversely, currents are induced in it that maintain constant the magnetic flux linked with the given turns. Thus, in this case, as in an ESG,

\[ \frac{d\psi}{dt}=0. \]

The mechanical work expended on moving the turn is converted into the energy of the magnetic field, which can be used if, at the maximum possible current of the turn, determined by the magnitude of the possible magnetic flux, a load resistance is connected into its leakage. Then, only at this moment of energy transfer to the load,

\[ \frac{d\psi}{dt}\ne 0. \]

Just as in an ESG, for the conveyor at this moment

\[ \frac{dN}{dt}\ne 0. \]

The absence of superconducting turns under ordinary conditions does not permit this method of converting mechanical energy into electrical energy in an EMM to be used with any effectiveness.

At the same time, the presence of good dielectrics makes it possible to retain charges on a surface for a sufficiently long time at high voltages. It is precisely for this reason that in ESMs the most widespread generators are those in which the conversion of mechanical energy into electrical energy occurs mainly when \(\frac{dN}{dt}=0\), and generators in which the energy conversion occurs when

\[ \frac{dN}{dt}=0. \]

are not used.

Thus, ESMs and EMMs that correspond to each other according to the duality principle do not correspond to each other in efficiency from the energy point of view.

The areas of possible application of ESGs are already quite varied at the present time [5, 6]. Such qualities as the possibility of obtaining high stability and an exceptionally weak ripple of the output voltage make it possible to obtain intense monochromatic radiation; the limited short-circuit current makes them almost safe for use in the textile industry, in electrostatic painting, and in other branches of industry. Finally,

With the expansion of research in the field of developing electrostatic generators with high specific characteristics, a multitude of new ideas and possibilities will emerge, opening up new areas of application that are unsuspected, or that at present seem only like science fiction.

REFERENCES

1. Edited by A. A. Vorob’ev. High-voltage testing equipment and measurements. GEI, 1960.

2. Morel R., Contribution à l’étude rationelle des machines électrostatique (dissertation), Grenoble, 1948.

3. Gignoux D., Electrostatic generators for space application, 102-e’m colloque du SNRC, Grenoble, 1960.

4. A. F. Kalganova, V. V. Patsevich. Izvestiya vuzov, “Elektromekhanika,” No. 8, 1963.

5. Edited by Academician A. K. Val’ter. Electrostatic generators. Atomizdat, 1959.

6. N. Felisi. Elektrichestvo, No. 1, 1962.