Construction of External Characteristics of Electrostatic Machines

V. A. Lukutin

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

For every electrostatic machine, as for any induction machine, it is possible to construct an external or load characteristic. This characteristic makes it possible to judge the main features of the machine and the possible areas of its application. For a projected prototype, it is desirable to calculate the characteristic theoretically, using as initial data the dimensions of the article and the physical properties of the materials used. It is especially important to be able to design an electrostatic generator (ESG) that will satisfy a previously specified external characteristic.

In the present article the basic principles are set forth for determining analytical expressions for the load characteristics of electrostatic (capacitive) machines. The formulas must take into account the influence of the geometrical dimensions of the structural elements of the article and of their physical properties on the electrical parameters of the generator being designed. The stated problem is quite complex, since the electrical data of the machine depend on very many factors, and taking them all fully into account leads to very cumbersome relations. However, with the aid of the assumptions and simplifications usually adopted, one can obtain formulas sufficiently convenient for engineering calculations. The method is equally suitable for calculating the load characteristics of machines with a dielectric rotor and with a rotor having conducting carriers.

An electrostatic generator consists of a stator and a rotating rotor that carries charges either on its own surface (a dielectric rotor) or on conducting carriers fastened in the body of the rotor. The voltage at the output of the machine is determined by the potentials of the carriers, and its current by the charges of the rods or of the rotor surface. Consequently, in order to determine the output parameters of the machine, it is necessary to be able to determine the charges and potentials of the electrodes of the rotor and stator.

Below an ESG with conducting carriers is considered; using its example it is more convenient to explain the derivation of the formulas, although all the reasoning can also be extended to generators with a dielectric rotor.

Figure 1 shows a section of the working volume of the machine, where the region with dielectric permittivity $\varepsilon_1$ is the body of the rotor with the carrier rods, which is separated from the stator by a gas or liquid medium with permittivity $\varepsilon_0$. At the upper and lower boundaries of the region, inductors are visible, creating an electric field.

of the stator. The brushes used for applying and removing charges are not shown in the figure.

If the dielectric of the rotor is assumed ideal, then the charge of a carrier during rotation of the machine remains unchanged and equal to \(q\), and after commutation (removal of the charge by a brush) its value changes abruptly to \(q'\). The potentials of the rods change as they move, but in the established operating regime the potential of any rod, located in one and the same position relative to the stator, will be identical. In this connection it becomes possible to consider a stationary system of electrodes on which charges are placed at specified potentials. In other words, it is necessary to calculate the electrostatic field in the working space of the machine. Since the space is bounded by parallel stator plates, the distance between which is less than the axial length of the conducting carriers, the electric field in this case may be regarded as two-dimensional. The boundary conditions usually can be formulated for each generator design.

Fig. 1

We shall use the method of superposition; for this purpose we first consider the field produced by the stator in a region with a three-layer dielectric, where on the boundary \(y=\pm d\) the potential is known and equal to \(U_{\mathrm{st}}(x,d)\), and then calculate the field in the same region due to the charges of the carriers under zero boundary conditions (the rotor field). In the first case it is necessary to solve the Dirichlet problem for the Laplace equation under the given boundary conditions; in the second, the analogous problem for the Poisson equation under zero conditions.

In the present article the actual calculation of the field is not considered, since this is an independent and complicated question. We shall assume that the corresponding analytical solutions already exist [2]; here it is entirely immaterial by what method they were obtained and what form they have. In the general case, the potentials of the stator and rotor fields will be functions of the coordinates, the geometrical dimensions \((a,d,n,r,b)\), the physical properties of the media \((\varepsilon_1,\varepsilon_0)\), the excitation voltage \(U_{\mathrm{v}}\), and the load \(U_{\mathrm{n}}\).

The electric field of the stator is produced by charged high-voltage inductors, at the load voltage \(U_{\mathrm{n}}\), and by semiconducting disks or cylinders forming the stator itself. The potential of points of the region \(y=0\), where the carriers are placed (Fig. 2, a), will be less than at the boundary of the region and is expressed by the function \(U_{\mathrm{st}}(x,0)\).

Let us now turn to the potential function of the rotor field. In essence, we shall consider the electric field of a generator operating in the short-circuit regime, when the stator potential is equal to zero. This is true for the case of a conduction excitation circuit and when the stator plates are made of a semiconducting material.

We shall assume that removal of charges from the conveyor takes place under each high-voltage inductor only once per revolution of the rotor, as is the case in generators with narrow brushes of any shape. The discharge of the conveyor occurs so rapidly that during this time the rotor does not have time to move appreciably. In such a rapid process the conveyor that has approached the brush successively has two charges: immediately before commutation, \(q\), and immediately after its completion, \(q'\).

In Fig. 2,b a graph is shown of the distribution of charges over the conveyors, with the origin of coordinates placed between rods carrying charges of different sign. For convenience of calculation, the initial charges may be represented in the form of sums

\[ q=q_a+q_b,\qquad q'=-q_a+q_b. \tag{1} \]

Such a decomposition does not contradict the principle of superposition and makes it possible somewhat to simplify the boundary conditions.

Fig. 2. Distribution of charges and potentials over the rotor conveyors:
\(a\)—potentials of the conveyors in the stator field; \(b\)—potentials and charges of the conveyors in the short-circuit mode

We first find the potential functions for the region under consideration from the charges \(q_a\), and then from \(q_b\). In general form they may be written as

\[ U_{pa}=q_a N(x,\ y,\ a,\ d,\ n,\ r,\ \varepsilon_0,\ \varepsilon_1), \]

\[ U_{pb}=q_b M(x,\ y,\ a,\ d,\ n,\ r,\ \varepsilon_0,\ \varepsilon_1). \tag{2} \]

The potential of any point of the working space of the machine can be found by summing the field functions of the stator and rotor. If, in the formula written in this way, we put \(y=0\), we obtain the potential of the points at which the conveyors are placed:

\[ U_{\mathrm{tr}}(x)=U_{\mathrm{st}}(x,0)+U_{pa}(x,0)+U_{pb}(x,0). \tag{3} \]

This last equality cannot be used directly, since it contains unknown charges, whose magnitude is determined by the excitation and load voltages.

In Fig. 2,a, c, d are shown approximate curves of the distribution of potentials over the conveyors. If they are summed, we obtain the graph of the distribution of potentials over the rotor conveyors (Fig. 3,a), corresponding to equation (3). The next conveyor approaching the high-voltage inductor has a potential higher than that of the brush connected to the inductor and therefore at the voltage \(U_{\mathrm{n}}\). At a certain instant, when the aforementioned rod is at a distance \(x=-\gamma\) from the brush, it is discharged by a spark. During the discharge the rotor practically does not have time to move, and the discharged conveyor comes to be at the brush potential \(U_{\mathrm{n}}\). The potentials of the conveyors immediately before commutation and immediately after its completion are represented respectively by curves 1 and 2 in Fig. 3,a. The equality follows directly from the figure:

\[ U_{pb}(-0.5a,0)=U_{\mathrm{st}}(-\gamma,\pm d)-U_{\mathrm{st}}(-\gamma,0)+U_{pa}(-0.5a,0). \tag{4} \]

Let us now turn to the zone of the excitation inductor. With the conduction scheme of excitation, the nature of the stator field under the low-voltage and high-voltage inductors will be the same. The potential functions of the rotor (short-circuit mode) will also be quite analogous. These statements will be valid if the designs of the brushes and inductors on the high and low sides are completely identical. Under these conditions, the potentials of the carriers located in the excitation zone can be found with the aid of the functions used above and by the same methods. The results of the calculations are shown in Fig. 3, b, from which the following relation is obtained:

Fig. 3. Potentials of the carriers in the zone of the high-voltage inductor (a) and in the zone of the low-voltage inductor (b):
1 — before commutation,
2 — after charge removal

\[ \begin{aligned} U_b &= U_{pb}(-0.5a,\,0) + {}\\ &\quad + U_{\mathrm{st}}(-\gamma,\,0) - U_{\mathrm{st}}(-\gamma,\,\pm a) + {}\\ &\quad + U_{pa}(-0.5a,\,0). \end{aligned} \tag{5} \]

Solving equations (4) and (5) simultaneously, one can obtain

\[ \begin{aligned} U_{pa}(-0.5a,\,0) &= q_a N(-0.5a,\,0,\,a,\,d,\,n,\,r,\,\varepsilon_0,\,\varepsilon_1) \\ &= 0.5U_b - [U_{\mathrm{st}}(-\gamma,\,\pm d) - U_{\mathrm{st}}(-\gamma,\,0)] . \end{aligned} \tag{6} \]

In the theory of electrostatic generators, a formula is widely used by which the load current is determined [1, 3],

\[ I_{\mathrm{n}} = (q - q')\kappa m n_0 = 2q_a \kappa m n_0, \tag{7} \]

where \(m\) is the number of carriers in the rotor,
\(\kappa\) is the number of pole pairs,
\(n_0\) is the number of revolutions of the rotor per sec.

We find the charge \(q_a\) from formula (6), substitute it into the last equation, and obtain, in general form, the equation of the linear part of the external characteristic of an electrostatic generator

\[ I_{\mathrm{n}} = \frac{ 2\kappa m n_0 \{0.5U_b - [\,U_{\mathrm{st}}(-\gamma,\,\pm d) - U_{\mathrm{st}}(-\gamma,\,0)\,]\} }{ N(-0.5a,\,0,\,a,\,d,\,n,\,r,\,\varepsilon_0,\,\varepsilon_1) }. \tag{8} \]

We carried out approximate analytical calculations of the potential functions of the stator and rotor. With their aid, on the basis of the method described, one of the variants of the characteristic equation for an ESG was obtained, given below.

\[ \begin{aligned} I_{\mathrm{n}} &= \frac{ \varepsilon_1 l\,4n^3 z n_0 \kappa^2 \left[92 + 260n^2 + 325c(d^2 - n^2)\right] }{ (3n^2 - r^2)(1 - 0.5a)\,[1 - (1 - 0.5a)^2]\,[13 - (1 - 0.5a)^2] } \times {}\\ &\quad \times \left\{ \frac{U_b}{n^2 + c(d^2 - n^2)} + \frac{ 8U_{\mathrm{n}}\sqrt{1-b^2}\,(8b^4 - b^2 - 7) }{ 5\pi(1-b)\,[4 + 4n^2 + 5c(d^2 - n^2)] } \right\}, \end{aligned} \tag{9} \]

where

\[ c = \frac{\varepsilon_1}{\varepsilon_0}, \]

\(z\) is the axial length of the carrier,

$l$ is one-half of the distance along the stator between the high-voltage and low-voltage inductors.

All geometric dimensions here are referred to $l$.

It is easy to see that equation (9) becomes an analytical expression for the short-circuit characteristic if one sets $U_n = 0$.

By setting the right-hand side of the equality equal to zero, one can obtain the voltage amplification coefficient of the electrostatic generator, understood as the ratio of the output voltage in no-load operation to the excitation voltage.

The formula is also suitable for calculating the external characteristic of a generator operating according to an induction excitation scheme. For such a machine, in (9) one should replace $U_n$ by $(U_n + U_v)$.

The characteristics calculated from the above equation were compared with the experimental characteristics of three prototype electrostatic generators of different designs. The errors do not exceed 10%, which may be considered acceptable in engineering calculations. More accurate analytical calculations of electric fields may make it possible to find more successful formulas for external characteristics; however, the one presented by us may also be useful to those engaged in the design of electrostatic machines.

REFERENCES

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

2. A. P. Kononov, V. A. Lukutin, V. M. Osipov. An approximate method for calculating the field of an electrostatic generator. Izv. TPI, vol. 191, 1969.

3. V. A. Lukutin, V. D. Eskov. On the calculation of electrostatic generators with cascade connection of conveyor-conductors. Energetika i transport, No. 2, 1967.