APPROXIMATE METHOD FOR CALCULATING THE FIELD OF AN ELECTROSTATIC GENERATOR

A. P. Kononov, V. A. Lukutin, V. M. Osipov

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

In the working volumes of an electrostatic generator (ESG), the electric-field strength reaches a considerable magnitude, since these machines are intended to operate at voltages of tens and hundreds of kilovolts. The designs of such machines will be successful if the individual parts are made so that the potential gradients on them do not reach critical values. In this connection, there arises the need to calculate electric fields of a very complex configuration. Exact analytical solutions of such a problem are very difficult; therefore an approximate solution is proposed below.

The basic diagram of a generator with conducting transporters is shown in the figures in the preceding articles. A transporter located under the excitation brush, whose potential is \(U_{\mathrm{v}}\), receives a charge \(q\) and moves toward the high-voltage inductor, while its potential increases to the value \(U_{\mathrm{n}}\).

At the moment when the transporter is connected to the high-voltage brush, the charge, in accordance with the known laws of electrical engineering, moves onto the inductors (plates). Since the latter do not completely enclose the transporter, and the neighboring conductors are charged, a certain charge \(q'\) remains on the transporter leaving the high-voltage brush; the magnitude and sign of this charge depend on the operating mode of the machine and on its geometrical dimensions.

In solving the problem we shall make the assumptions that the field in the gap between transporter and stator is plane-parallel, that this gap has a homogeneous dielectric, and that commutation is only by contact. We cut, along a radius passing through the middle of the high-voltage inductor, the cylindrical region between the outer and inner surfaces of the stator and unroll it into a strip. In so doing, the field pattern in the region will not change appreciably, since the width of the strip is many times smaller than its length.

After the transformations made, we shall seek the potential function of the electric field in the rectangle shown in Fig. 1.

In this problem we shall consider the law of distribution of the potential along the boundary of the region to be known. Along the \(x\)-axis there are uniformly charged rods; the linear charges on them will be, up to the high-voltage brush, \(q\), and after the brush \((x<0)\), \(q'\).

Since the problem posed is linear, we shall use the method of superposition, first calculating the electric field of the stator-

Fig. 1

ra, and then the field produced by the charged transporters (the rotor field). It is more convenient to carry out the calculations in relative units; for this purpose we divide all geometrical dimensions by \(l\)—the length of the rotor semicircumference.

Calculation of the electric field of the stator

In load operation the potential varies along the inner surface of the stator according to a definite law. From the standpoint of field equalization, it would be desirable that the tangential component of the electric-field strength on the inner surface of the stator be the same everywhere. This corresponds to a uniform distribution of the potential on this surface. Designers take a number of measures to achieve such a distribution, including the creation of special voltage dividers.

Without going into detail, in the calculations we shall take the potential of the excitation inductor to be zero, and that of the high-voltage electrode to be \(U_{\mathrm{н}}\). Between them the potential increases according to a linear law (Fig. 2).

The adopted law (another may also be specified) of potential distribution within the limits \(-1 \le x \le 1\) can be approximately represented in the form of a series

\[ f(x)=b_0+b_2T_2+\cdots+b_{2k}T_{2k}+\cdots,\quad k=1,2,\ldots, \tag{1} \]

where \(T_{2k}\) are Chebyshev polynomials of the first kind, and \(b_{2k}\) are coefficients to be determined in the usual way.

On the other two sides of the region \((x=\pm 1)\) the potential is equal to \(U_{\mathrm{н}}\). Thus, determination of the potential inside the closed region reduces to solving the Dirichlet problem for Laplace’s equation under the above boundary conditions:

\[ \Delta U=0,\quad U= \begin{cases} f(x), & y=\pm d,\\ U_{\mathrm{н}}, & x=\pm 1. \end{cases} \tag{2} \]

It is known that integration of a differential equation can be replaced by the problem of finding a function that gives a certain functional its minimum value (a variational problem). We shall seek the minimizing sequence by the Ritz method.

Let us choose a combination of coordinate functions satisfying the following conditions:

1) the functions must belong to the domain of definition of the Laplace operator,

2) they must satisfy the boundary conditions,

3) they must be complete with respect to energy.

The following sequence corresponds to the listed conditions:

Fig. 2

\[ U_c(x,y)=f(x)\left[1+A_1(x^2-1)(y^2-d^2)+A_2(x^2-1)^2(y^2-d^2)^2\right]. \tag{3} \]

It must minimize the functional

\[ F(u)=\iint (\operatorname{grad} U)^2\,dx\,dy. \]

Depending on the desired accuracy, one or another number of terms is chosen in the sequence (3). However, as the number of terms increases, the amount of computation grows. Taking this circumstance into account, we shall restrict ourselves to three terms, which contain the unknown coefficients \(A_1\) and \(A_2\). The latter are found from a system of two equations obtained as follows:

\[ \frac{\partial F}{\partial A_1}=0,\qquad \frac{\partial F}{\partial A_2}=0. \]

Solving this system, we find \(A_1\) and \(A_2\), substitute the coefficients found into formula (3), which will then be the approximate solution of the problem. By specifying the coordinates \(x,y\) of the points of interest in the region under consideration, it is not difficult, using (3), to determine the potentials of these points.

Calculation of the Rotor Field (Short-Circuit Mode)

Figure 1 shows the schematic diagram of the working elements of an electrostatic generator. In generators of cylindrical design, the transporters are usually made in the form of rods. Studies show that the energy characteristics of the machine will be better the greater the number of rods contained in the rotor. It is known that as the number of rods increases, the distance \(a\) between them and the radius of their cross section will decrease.

We shall regard the rotor dielectric as ideal; there will be no charge leakage, and therefore a rod which has received a charge \(q\) in the charging system retains it until it is connected to the high-voltage grid. Likewise, after discharging, it again reaches the charging grid with a residual charge \(q'\).

Consequently, the problem will reduce to calculating the plane-parallel field produced by a system of parallel uniformly charged cylinders. In order that the stator potential not change, it is necessary to satisfy the condition

\[ U=0,\quad \text{for } y=\pm d. \]

The potential of the transporter located under the high-voltage grid must be equal to the potential of the corresponding inductor. In the regi-

At the instant of short circuit the high-voltage brushes and inductors are grounded; therefore one may assume

\[ U=0,\quad \text{for } x=\pm 1. \]

To simplify the problem, let us replace the uniformly charged cylinders by a uniformly charged plane with charge density

\[ \sigma=nq_0,\quad k\sigma=nq'_0,\quad k=\frac{q'_0}{q_0}, \]

where \(n\) is the number of carriers per unit length of the rotor perimeter, and \(q_0, q'_0\) are the charges of the carriers per unit of their length.

As a result we obtain the computational model shown in Fig. 3.

Fig. 3

If the field in the electrostatic generator is assumed quasistationary, as is done by the overwhelming majority of authors, then the problem of calculating it reduces to solving Poisson’s equation

\[ -\Delta U=\psi(x,y) \tag{5} \]

with the boundary conditions \(U=0\) for \(x=\pm 1,\ y=\pm d\). In equation (5)

\[ \psi(x,y)=\frac{k\sigma}{\varepsilon}\,[1(x+1)-1(x+a)]\,\sigma(y) +\frac{\delta}{\varepsilon}\,[1(x)-1(x-1+a)]\,\delta(y), \]

where \(\delta(y)\) is the delta function, and \(1(x)\) is the unit function.

We shall seek the dependence \(U_p(x,y)\) of interest to us by the variational method, the essence of which reduces to finding the minimum of the functional

\[ \Phi(U)=\iint \{(\operatorname{grad} U)^2-2U\psi\}\,dx\,dy. \tag{6} \]

We shall solve the latter problem by the Ritz method. As the solution, let us choose a combination of three coordinate functions. (More may be used, but this will complicate the illustration of the solution.)

\[ U_p(x,y)=(x^2-1)(y^2-d^2)[a_1+xa_2+x^2a_3]. \tag{7} \]

These functions are linearly independent and complete in energy.

To find the minimum of the functional, we shall compute the derivatives with respect to the unknown variables

\[ \frac{\partial \phi}{\partial a_i}=0,\quad i=1,2,\ldots \]

which will lead to the following system of equations (Ritz equations):

\[ (\Delta\varphi_1,\varphi_1)a_1 =(\Delta\varphi_1,\varphi_2)a_2 +(\Delta\varphi_1,\varphi_3)a_3 =(\psi,\varphi_1), \]

\[ (\Delta\varphi_2,\varphi_1)a_1 +(\Delta\varphi_2,\varphi_2)a_2 +(\Delta\varphi_2,\varphi_3)a_3 =(\psi,\varphi_2). \]

\[ (\Delta \varphi_3,\varphi_1)a_1+(\Delta \varphi_3,\varphi_2)a_2+(\Delta \varphi_3,\varphi_3)a_3=(\psi,\varphi_3). \]

Here \(\varphi_1,\varphi_2,\varphi_3\) are the multipliers of \(a_j\) in equation (7).

Carrying out the corresponding calculations, we obtain a system of three equations

\[ \frac{128d^3}{45}\left(1+d^2\right)a_1+ \frac{128d^3}{45}\left(\frac{1}{7}+\frac{d^2}{5}\right)a_3 = \frac{\sigma}{\varepsilon}\,m, \]

\[ \frac{128d^3}{15}\left(\frac{1}{21}+\frac{d^2}{5}\right)a_2 = \frac{\sigma}{\varepsilon}\,n, \]

\[ \frac{128d^3}{15}\left(\frac{1}{7}+\frac{d^2}{5}\right)a_1+ \frac{128d^3}{315}\left(\frac{1}{3}+\frac{11d^3}{5}\right)a_3 = \frac{\sigma}{\varepsilon}\,p, \]

where the following notation has been introduced:

\[ m=\frac{d^2}{3}\left[k(2-3a+a^3)+(2-3a^2+a^3)\right], \]

\[ n=\frac{d^2}{4}\left[k(-1+2a^2-a^4)+(1+4a-4a^2-a^4)\right], \]

\[ p=\frac{d^2}{15}\left[k(2-5a^3+3a^5)+(2-15a^2+25a^3-15a^4+3a^5)\right]. \]

It is now not difficult to find the coefficients \(a_i\) of interest to us; substituting them into formula (7), we obtain the desired solution of the problem. However, in this formula it remains to determine the quantities \(\sigma\) and \(k\). For this purpose we shall use additional conditions; in particular, for the induction excitation circuit one may write that

\[ U_p=U_v \quad \text{for } x=y=0, \]

whence we obtain \(U_v=d^2a_1\).

Substituting the coefficient \(a_1\) found above, we obtain an equation by means of which \(k\) can be calculated.

Load condition

If we sum the potential functions of the stator and rotor fields (formulas 3 and 7), we obtain an equation that makes it possible to determine the resultant potential at the points of the machine working volume that are of interest to us:

\[ U(x,y)=U_c(x,y)+U_p(x,y). \tag{8} \]

Equation (8), written in general form, contains the principal geometrical dimensions of the machine, the excitation voltage, and the load voltage; it is therefore convenient for analyzing the operation of the machine. It becomes possible to determine the influence of various machine parameters on its electrical characteristics, which is of definite interest to design engineers.

It also becomes possible to find approximate analytical formulas for the tangential and normal components of the electric-field strength for various sections of the machine.

In conclusion, we note that the formulas presented in the article should be regarded as an illustration of the method for calculating the field in an electrostatic generator. Other, probably more successful, coordinate functions may be chosen for the stator and rotor fields; a larger number of terms in them may be taken, and it is quite possible to take into account a number of factors specific to generators, such as the presence of surface charges on the ro-

the rotor, commutation phenomena, and so on. It becomes possible to apply the approximate method also to the calculation of the field of electrostatic machines with a dielectric rotor. However, all the questions listed are in themselves of considerable interest and have not yet been fully studied; therefore, taking them into account in field calculations requires additional investigations.