Calculation of the Field of an Electrostatic Generator

I. P. Guk.

(Presented by the scientific seminar of the TOE department and the RESG division of the YaF Research Institute).

Electrostatic machines, which have recently been finding ever broader application in various fields of science and technology, still do not have an established calculation methodology. Usually an electrostatic generator (ESG) is regarded as a quasistationary device, and its calculation is carried out using lumped parameters obtained from static field equations (partial capacitances [3], potential coefficients [5], various generalized capacitances, etc.). All these methods make it possible to determine only the integral characteristics of the generator, from which it is impossible to quantitatively assess the nature of the field; and, as is known, the design of an ESG requires knowledge of its properties. Nevertheless, the possibilities of classical methods for solving problems of this type—which make it possible to determine not only integral but also differential characteristics, and thereby to calculate the ESG field completely—have not yet been exhausted.

The purpose of the present article is to show the possibility of applying one of these methods to the calculation of the field of a cylindrical electrostatic generator with a cascade connection of transport conductors of circular cross section.

Description of the Electrostatic Machine

Considering an ESG with a cascade connection of transport conductors, whose schematic diagram is shown in Fig. 1, we shall confine ourselves to a description only of its principal elements, since the operating principle and the fundamentals of ESG construction are described in detail in [4].

The dielectric stator is coated with a semiconducting layer 9 in order to create a more uniform field in the generator gap. The excitation system consists of low-voltage inductors 8, brushes 4, and an excitation source. The load resistance 5 is connected to the high-voltage inductors 7, to which brush 6 is also connected, removing charge from the approaching transporters.

The rotor consists of \(2m\) transport conductors insulated from one another. The dielectric permittivity of the medium surrounding the transporters, \(\varepsilon\), is assumed constant throughout the working space of the ESG.

At the moment when transporters 1 and \(m+1\) break away from the corresponding brushes, the transporters 1, 2, 3, … \(m\) have charge \(q\), while \(m+1\), \(m+2\) … \(2m\)—

Fig. 1. Schematic diagram of an electrostatic generator

charge \(q'\). The difference of these charges is delivered to the load, whose resistance is \(R_{\mathrm{n}}\), by means of brush 6. The current \(I_{\mathrm{n}}\) is produced by this difference of charges.

For electrostatic generators of this type, two excitation circuits are possible: conduction excitation—the low-voltage inductors are grounded, and an excitation source with voltage \(U_{\mathrm{v}}\) is connected to brush 4—and induction excitation—the brush is grounded, and the inductors are connected to the excitation source.

The subject of the calculation presented in this article is the determination of the scalar potential of the field of an electrostatic generator as a function of the geometric parameters and the operating conditions of its mode of operation. In this calculation, the edge effect due to the finite length of the conveyors and cylinders is not taken into account, and it is assumed that:

a) there are no insulation leakages and no corona losses,

b) the radius of the conveyor is much smaller than its length,

c) there is no displacement of the electrical axes of the charged conveyors.

Calculation of the field of an electrostatic generator with a conduction excitation circuit

To solve the problem we shall use the method of superposition, assuming that the field produced by a linear distribution of potential in the semiconducting layer on the stator is superposed with the field of the linear charges of the conveyors, i.e.

\[ \varphi_{(r;\theta)}=\varphi_{(r;\theta)}^{(c)}+\varphi_{(r;\theta)}^{(p)}. \tag{1} \]

a) Determination of the scalar potential

In the present case it is necessary to solve the Dirichlet problem for the Laplace equation

\[ \Delta \varphi = 0 \]

in the case of the region enclosed between two concentric circles of radii \(R_1\) and \(R_2\), i.e., it is required to find a function \(\varphi^{(c)}\), harmonic in the working space of the generator and taking on the closed contours that bound this space the following values:

\[ f(\theta)=\varphi^{(c)}(R_1;\theta)=\varphi^{(c)}(R_2;\theta) \]

\[ \left. \begin{aligned} f(\theta)&=0 &&\text{for } 0 \leq \theta \leq \theta_0 \text{ and } 2\pi-\theta_0 \leq \theta \leq 2\pi,\\[4pt] f(\theta)&=\frac{\theta-\theta_0}{\pi-2\theta_0}U_{\mathrm n} &&\text{for } \theta_0 \leq \theta \leq \pi-\theta_0,\\[4pt] f(\theta)&=U_{\mathrm n} &&\text{for } \pi-\theta_0 \leq \theta \leq \pi+\theta_0,\\[4pt] f(\theta)&=\frac{2\pi-\theta_0-\theta}{\pi-2\theta_0}U_{\mathrm n} &&\text{for } \pi+\theta_0 \leq \theta \leq 2\pi-\theta_0. \end{aligned} \right\} \tag{2} \]

Figure 2 shows the linear distribution of the potential in the semiconducting layer in accordance with equations (2). The boundary conditions are the prescribed values of the function \(f(\theta)\) on the circles of radii \(R_1\) and \(R_2\) (Fig. 2) and the periodicity of the sought solution as a function of \(\theta\), i.e.,

\[ \varphi^{(c)}(r;\theta+2\pi)=\varphi^{(c)}(r;\theta). \]

To solve this problem we apply the method of separation of variables (the Fourier method), for which we substitute \(\varphi=T(\theta)\cdot P(r)\) into Laplace’s equation in the cylindrical coordinate system,

\[ r^2\frac{\partial^2\varphi}{\partial r^2} +r\frac{\partial\varphi}{\partial r} +\frac{\partial^2\varphi}{\partial\theta^2}=0. \]

After the corresponding transformations (the general solution of this problem is given in [2]) we obtain

\[ \varphi^{(c)}(r;\theta)= \frac{a_0}{2} + \sum_{k=1}^{\infty} \left[ \frac{a_k\left(r^{2k}+R_1^k\cdot R_2^k\right)} {r^k\left(R_1^k+R_2^k\right)} \cos k\theta + \frac{\beta_k\left(r^{2k}+R_1^k\cdot R_2^k\right)} {r^k\left(R_1^k+R_2^k\right)} \sin k\theta \right], \tag{3} \]

where the constants \(a_0\), \(a_k\), and \(\beta_k\) are the coefficients of the corresponding sines and cosines in the trigonometric series

\[ f(\theta)=\frac{a_0}{2}+\sum_{k=1}^{\infty} a_k\cos k\theta+\beta_k\sin k\theta. \tag{4} \]

Fig. 2. Law of variation of the stator potential

Calculation of a series of elementary integrals gives

\[ \alpha_0=\frac{1}{\pi}\int_{0}^{2\pi} f(\theta)d\theta=U_{\mathrm{n}}, \]

\[ \alpha_k=\frac{1}{\pi}\int_{0}^{2\pi} f(\theta)\cos k\theta\,d\theta = -\,2\,\frac{1-(-1)^k}{\pi(\pi-2\theta_0)}\,\frac{\cos k\theta_0}{k^2}, \]

\[ \beta_k=\frac{1}{\pi}\int_{0}^{2\pi} f(\theta)\sin k\theta\,d\theta=0. \]

After substituting the values of \(\alpha_0\), \(\alpha_k\), and \(\beta_k\) into equation (3) and carrying out a series of transformations, we obtain that the desired potential is equal to

\[ \varphi_{(r;\theta)}^{(c)} = \left[ 1-\frac{8}{\pi(\pi-2\theta_0)} \sum_{k}^{1,3,5,\ldots} \frac{\sinh k\beta+\sinh k\gamma}{k^2\sinh k\alpha} \cos k\theta_0\cos k\theta \right]\frac{U_{\mathrm{n}}}{2}, \tag{5} \]

where

\[ \alpha=\ln\frac{R_2}{R_1},\qquad \beta=\ln\frac{r}{R_1},\qquad \gamma=\ln\frac{R_2}{r}. \]

b) Determination of the potential \(\varphi^{(p)}(r;\theta)\).

To determine the potential \(\varphi^{(p)}\) of the field formed by the line charges \(q\) and \(q'\) of the corresponding conveyors, let us first consider the field of a unit line charge in a circular ring. Fig. 3 illustrates this case. Since the outer and inner cylinders have potential equal to zero, in this case the same conditions are obtained as in the calculation of the Green’s function. As is known from [1], the Green’s function for a circular ring has the following value:

\[ G'(\zeta;\zeta') = \frac{1}{2\pi}\ln \left\{ \frac{ \vartheta_1\left[ \frac{1}{2\alpha}\ln\left(\frac{\zeta}{\zeta'}\right);\pi \right] }{ \vartheta_1\left[ \frac{1}{2\alpha}\ln\left(\frac{\zeta\bar{\zeta}'}{R_1^2}\right);\pi \right] } \right\}, \tag{6} \]

where

\[ \zeta=re^{j\theta}, \]

\[ \zeta'=R_0e^{j\theta'},\qquad \bar{\zeta}'=R_0e^{-j\theta'}, \]

\[ |\theta-\theta'|\leq \pi. \]

\(\vartheta_1\) is a theta function with arguments

\[ \frac{1}{2\alpha}\ln\frac{\zeta}{\zeta'},\qquad \frac{1}{2\alpha}\ln\frac{\zeta\bar{\zeta}'}{R_1^2} \]

and modulus

\[ j\frac{\pi}{\alpha}. \]

The unit line source is located at the point \(\zeta'=R_0e^{j\theta'}\). If the function \(G'(\zeta;\zeta')\) is expanded in a series and its real part is taken, we obtain an expression more convenient for computational purposes:

\[ \operatorname{Re}G'(\zeta;\zeta') = \frac{1}{2\pi} \left\{ \frac{1}{2}\ln \frac{ \cosh\left[\frac{\pi}{\alpha}(\theta-\theta')\right] -\cos\left(\frac{\pi}{\alpha}\ln\frac{r}{R_0}\right) }{ \cosh\left[\frac{\pi}{\alpha}(\theta-\theta')\right] -\cos\left(\frac{\pi}{\alpha}\ln\frac{rR_0}{R_1^2}\right) } \right. \]

\[ \left. +4\sum_{\lambda=1}^{\infty} \frac{ \cosh\left[\frac{\pi\lambda}{\alpha}(\theta-\theta')\right] }{ \lambda\left[\exp\left(\frac{2\pi^2\lambda}{\alpha}\right)-1\right] } \sin\left(\frac{\pi\lambda}{\alpha}\ln\frac{R_0}{R_1}\right) \sin\left(\frac{\pi\lambda}{\alpha}\ln\frac{r}{R_1}\right) \right\}. \tag{7} \]

Fig. 3

It is not difficult to verify that the expression given has all the properties that Green’s function must possess. Inside the ring it has a single singularity at \(\zeta=\zeta'\) and is equal to zero if \(\zeta\) is located on the inner or outer circumference.

The scalar potential in a circular ring in which there is one carrier with charge \(q\) and coordinates \(R_0 e^{j\theta'}\) is equal to

\[ \varphi(r;\theta)=-\frac{q}{2\pi\varepsilon}\operatorname{Re}G(re^{j\theta};R_0e^{j\theta'}), \tag{8} \]

where

\[ R_0=\frac{R_1+R_2}{2},\qquad G(re^{j\theta};R_0e^{j\theta'})=2\pi G'(re^{j\theta};R_0e^{j\theta'}). \]

To determine the potential of a carrier, in equation (8) one must set \(\zeta=(R_0+\rho_0)e^{j\theta}\); then

\[ \varphi_{\mathrm t}\simeq \frac{q}{2\pi\varepsilon}\ln \frac{ 2aR_0\sin\left(-\frac{\pi}{z}\ln\frac{R_0}{R_1}\right) }{ \pi\rho_0 }, \tag{8*} \]

where \(\rho_0\) is the carrier radius, \(\rho_0\ll R_0\).

Let us extend formula (8) to a sequence of \(2m\) linear charges with the distance between them remaining unchanged, \(a=\dfrac{\pi}{m}R_0\) (Fig. 1). And since in the ESG carriers \(1,2,3,\ldots,m\) have charge \(q\), while carriers \(m+1,m+2,\ldots,2m\) have charge \(q'\), then, applying the superposition method, for the potential \(\varphi^{(p)}\) one may write the following equation:

\[ \varphi(r;\theta)^{(p)} = -\frac{q}{2\pi\varepsilon} \sum_{n=1}^{m} \operatorname{Re}G\left(\zeta;\,R_0 e^{j\frac{n-1}{m}\pi}\right) -\frac{q'}{2\pi\varepsilon} \sum_{n=m+1}^{2m} \operatorname{Re}G\left(\zeta;\,R_0 e^{j\frac{n-1}{m}\pi}\right), \tag{9} \]

where \(n\) is the ordinal number of the conveyor.

c) Determination of the potential \(\varphi(r;\theta)\).

Thus, summing (5) and (9), we obtain that the scalar potential in the annular gap of the generator is equal to

\[ \begin{aligned} \varphi(r;\theta) ={}& \left[ 1-\frac{8}{\pi(\pi-2\theta_0)} \sum_{k=1}^{\infty} \frac{\operatorname{sh}k\beta+\operatorname{sh}k\gamma} {k^2\operatorname{sh}k\alpha} \cos k\theta_0\cos k\theta \right]\frac{U_{\mathrm{H}}}{2} \\ &-\frac{q}{2\pi\varepsilon} \sum_{n=1}^{m} \operatorname{Re}G\left(\zeta;\,R_0 e^{j\frac{n-1}{m}\pi}\right) -\frac{q'}{2\pi\varepsilon} \sum_{n=m+1}^{2m} \operatorname{Re}G\left(\zeta;\,R_0 e^{j\frac{n-1}{m}\pi}\right), \end{aligned} \tag{10} \]

where \(k=1,3,5,\ldots\).

To shorten the notation, let us set

\[ \operatorname{Re}G\left(\zeta;\,R_0 e^{j\frac{n-1}{m}\pi}\right)=F_n(\zeta). \]

To determine the charges \(q\) and \(q'\), we shall use the boundary conditions in the different operating modes of the electrostatic generator:

a) no-load operation: \(q=q'\)

\[ \begin{aligned} U_{\mathrm{B}}=\varphi(R_0;0) ={}& \left[ 1-\frac{8}{\pi(\pi-2\theta_0)} \sum_{k}^{1,3,5,\ldots} \frac{\operatorname{sh}k\beta_0+\operatorname{sh}k\gamma_0} {k^2\operatorname{sh}k\alpha} \cos k\theta_0 \right]\frac{U_{\mathrm{xx}}}{2} \\ &-\frac{q}{2\pi\varepsilon} \sum_{n=1}^{2m}F_n(R_0;0) \end{aligned} \tag{11} \]

\[ \begin{aligned} U_{\mathrm{xx}}=-\varphi(R_0;\pi) ={}& \left[ 1+\frac{8}{\pi(\pi-2\theta_0)} \sum_{k}^{1,3,5,\ldots} \frac{\operatorname{sh}k\beta_0+\operatorname{sh}k\gamma_0} {k^2\operatorname{sh}k\alpha} \cos k\theta_0 \right]\frac{U_{\mathrm{xx}}}{2} \\ &-\frac{q}{2\pi\varepsilon} \sum_{n=1}^{2m}F_n(R_0;\pi), \end{aligned} \tag{12} \]

where

\[ \beta_0=\ln\frac{R_0}{R_1};\qquad \gamma_0=\ln\frac{R_2}{R_0}. \]

Summing (11) and (12), we obtain an expression for the charge in the no-load regime

\[ q_{\mathrm{xx}}=-\pi\varepsilon\, \frac{U_{\mathrm{B}}} {\displaystyle \sum_{n=1}^{2m}F_n(R_0;0)}. \tag{13} \]

b) short circuit:

\[ U_{\mathrm{B}}=\varphi(R_0;0) = -\frac{q}{2\pi\varepsilon} \sum_{n=1}^{m}F_n(R_0;0) -\frac{q'}{2\pi\varepsilon} \sum_{n=m+1}^{2m}F_n(R_0;0), \tag{14} \]

\[ 0=\varphi(R_0;\pi)=-\frac{q}{2\pi\varepsilon}\sum_{n=1}^{m}F_n(R_0;\pi) -\frac{q'}{2\pi\varepsilon}\sum_{n=m+1}^{2m}F_n(R_0;\pi). \tag{15} \]

Substituting (14) into (15) and taking into account that the Green’s function is symmetric for constant \(r\), we obtain

\[ q_{\mathrm{k.z.}}=-\pi\varepsilon \left[ \frac{1}{\displaystyle\sum_{n=1}^{2m}F_n(R_0;0)} +\frac{1}{F_1(R_0;0)} \right]U_{\mathrm{v}}, \tag{16} \]

\[ q'_{\mathrm{k.z.}}=-\pi\varepsilon \left[ \frac{1}{\displaystyle\sum_{n=1}^{2m}F_n(R_0;0)} -\frac{1}{F_1(R_0;0)} \right]U_{\mathrm{v}}. \tag{17} \]

c) load mode

\[ U_{\mathrm{v}}= \left[ 1-\frac{8}{\pi(\pi-2\theta_0)} \sum_k^{1,3,5,\ldots} \frac{\operatorname{sh}k\beta_0+\operatorname{sh}k\gamma_0}{k^2\operatorname{sh}ka} \cos k\theta_0 \right]\frac{U_{\mathrm{n}}}{2} - \]

\[ -\frac{q}{2\pi\varepsilon}\sum_{n=1}^{m}F_n(R_0;0) -\frac{q'}{2\pi\varepsilon}\sum_{n=m+1}^{2m}F_n(R_0;0), \tag{18} \]

\[ U_{\mathrm{n}}= \left[ 1+\frac{8}{\pi(\pi-2\theta_0)} \sum_k^{1,3,5,\ldots} \frac{\operatorname{sh}k\beta_0+\operatorname{sh}k\gamma_0}{k^2\operatorname{sh}ka} \cos k\theta_0 \right]\frac{U_{\mathrm{n}}}{2} - \]

\[ -\frac{q}{2\pi\varepsilon}\sum_{n=1}^{m}F_n(R_0;\pi) -\frac{q'}{2\pi\varepsilon}\sum_{n=m+1}^{2m}F_n(R_0;\pi). \tag{19} \]

Solving (18) and (19) simultaneously, we find that in the load mode the charges are equal to

\[ q=-\pi\varepsilon \left[ \frac{1}{\displaystyle\sum_{n=1}^{2m}F_n(R_0;0)} + \frac{1-A\cdot U_{\mathrm{n}}/U_{\mathrm{v}}}{F_1(R_0;0)} \right]U_{\mathrm{v}}, \tag{20} \]

\[ q'=-\pi\varepsilon \left[ \frac{1}{\displaystyle\sum_{n=1}^{2m}F_n(R_0;0)} - \frac{1-U_{\mathrm{n}}/U_{\mathrm{v}}}{F_1(R_0;0)} \right]U_{\mathrm{v}}, \tag{21} \]

where

\[ A=1-\frac{8}{\pi(\pi-2\theta_0)} \sum_k^{1,3,5,\ldots} \frac{\operatorname{sh}k\beta_0+\operatorname{sh}k\gamma_0}{k^2\operatorname{sh}ka} \cos k\theta_0. \]

It is obvious that \(U_{\mathrm{v}}/A=U_{\mathrm{xx}}\) is the theoretical value of the no-load voltage; then for \(U_{\mathrm{n}}=0\) (short-circuit mode), formulas (16) and (17) are obtained from (20) and (21), while for \(U_{\mathrm{n}}=U_{\mathrm{xx}}\) (no-load mode) formula (13) is obtained.

Now, substituting (20) and (21) into equation (10), we obtain a formula for the scalar potential at any point of the field of an electrostatic generator with a conduction excitation circuit:

\[ \varphi(r;\theta)= \left[ 1-\frac{8}{\pi(\pi-2\theta_0)} \sum_k^{1,3,5,\ldots} \frac{\operatorname{sh}k\beta+\operatorname{sh}k\gamma}{k^2\operatorname{sh}k\alpha} \cos k\theta_0 \cos k\theta \right]\frac{U_{\mathrm{n}}}{2} + \left[ \frac{\displaystyle \sum_{n=1}^{2m} F_n(r;\theta)} {\displaystyle \sum_{n=1}^{2m} F_n(r;\theta)} + \frac{\displaystyle \sum_{n=1}^{m} F_n(r;\theta)-\sum_{n=m+1}^{2m} F_n(r;\theta)} {F_1(R_0;0)} \left(1-\frac{U_{\mathrm{n}}}{U_{\mathrm{xx}}}\right) \right]\frac{U_{\mathrm{v}}}{2}. \tag{22} \]

Calculation of the field of an ESG with an induction excitation circuit

An ESG with an induction excitation circuit differs from a generator with a conduction excitation circuit in that its excitation source is connected to the low-voltage inductors, while brush 4 is grounded (Fig. 1).

Carrying out a calculation analogous to that presented above, we obtain the expression for the scalar potential:

\[ \varphi(r;\theta)= \left[ \frac{1}{2} - \frac{4}{\pi(\pi-2\theta_0)} \sum_k^{1,3,5,\ldots} \frac{\operatorname{sh}k\beta+\operatorname{sh}k\gamma}{k^2\operatorname{sh}k\alpha} \cos k\theta_0 \cos k\theta \right] \times \]

\[ (U_{\mathrm{n}}+U_{\mathrm{v}})-U_{\mathrm{v}} + \left[ \frac{\displaystyle \sum_{n=1}^{2m} F_n(r;\theta)} {\displaystyle \sum_{n=1}^{2m} F_n(R_0;0)} + \frac{\displaystyle \sum_{n=1}^{m} F_n(r;\theta)-\sum_{n=m+1}^{2m} F_n(r;\theta)} {F_1(R_0;0)} \left( 1- \frac{U_{\mathrm{n}}+U_{\mathrm{v}}}{U_{\mathrm{xx}}+U_{\mathrm{v}}} \right) \right]\frac{U_{\mathrm{v}}}{2}. \tag{23} \]

Formulas (22) and (23) make it possible to determine, with sufficient accuracy, the potential and hence the field strength at any point of the ESG. At special points, i.e., at those points where the axes of the transporters are located, the potential must be determined as follows: in equations (22) and (23) one must substitute not \(\zeta=R_0 e^{j\frac{n-1}{m}\pi}\), but \(\zeta=(R_0+\rho_0)e^{j\frac{n-1}{m}\pi}\), similarly to how this was done in determining the transporter potential in formula (8*).

Numerical calculations have shown that for narrow rings, i.e., when \(\alpha<1\), the second term in (7) may be neglected. And since in real generators \(\alpha<0.5\), the quantity

\[ \exp\left(\frac{2n^2\lambda}{\alpha}\right)>\exp(20\lambda), \]

which stands in the denominator in formula (7), leads to the second term having a very small value. Consequently, it is sufficient to restrict consideration to only the logarithmic terms, i.e.,

\[ F_n(r;\theta)=R_e G\left(\zeta;\,R_0e^{j\frac{n-1}{m}\pi}\right)\simeq \]

\[ \simeq \frac{1}{2}\ln \frac{ \operatorname{ch}\left[\frac{\pi}{\alpha}\left(\theta-\frac{n-1}{m}\pi\right)\right] - \cos\left[\frac{\pi}{\alpha}\ln\frac{r}{R_0}\right] }{ \operatorname{ch}\left[\frac{\pi}{\alpha}\left(\theta-\frac{n-1}{m}\pi\right)\right] - \cos\left[\frac{\pi}{\alpha}\ln\frac{rR_0}{R_1^2}\right] }. \tag{24} \]

Substituting (24) into equations (22) and (23), we obtain the calculation formula for determining the electric potential in the working gap of a cylindrical electrostatic generator with conductor conveyors:

\[ \begin{aligned} \varphi(r;\theta)={}& \left[ 1-\frac{8}{\pi(\pi-2\theta_0)} \sum_{k}^{1,3,5,\ldots} \frac{\operatorname{sh} k\beta+\operatorname{sh} k\gamma} {k^2 \operatorname{sh} k\alpha} \cos k\theta_0 \cos k\theta \right] \frac{U_{\mathrm{n}}+\nu U_{\mathrm{v}}}{2} \\ &-\nu U_{\mathrm{v}}+ \left[ \frac{\ln \displaystyle\prod_{n=1}^{2m}\Gamma_n(r;\theta)} {\ln \displaystyle\prod_{n=1}^{2m}\Gamma_n(R_0;\theta)} + \frac{1-\dfrac{U_{\mathrm{n}}+\nu U_{\mathrm{v}}}{U_{\mathrm{xx}}+\nu U_{\mathrm{v}}}} {\ln \Gamma_1(R_0;\theta)} \ln \frac{\displaystyle\prod_{n=1}^{2m}\Gamma_n(r;\theta)} {\displaystyle\prod_{n=m+1}^{2m}\Gamma_n(r;\theta)} \right] \frac{U_{\mathrm{v}}}{2}, \end{aligned} \tag{25} \]

where

\[ \nu= \begin{cases} 0 & \text{for a conduction excitation circuit,}\\ 1 & \text{for an induction excitation circuit;} \end{cases} \]

\(2\theta_0\) is the width of the inductor in radians;

\[ \alpha=\ln R_2/R_1;\quad \beta=\ln r/R_1;\quad \gamma=\ln R_2/r; \]

\(U_{\mathrm{n}}\) is the load voltage;
\(U_{\mathrm{v}}\) is the excitation voltage;
\(U_{\mathrm{xx}}\) is the theoretical no-load voltage;

\[ \Gamma_n(r;\theta)= \frac{ \operatorname{ch}\left[\dfrac{\pi}{\alpha}\left(\theta-\dfrac{n-1}{m}\pi\right)\right] -\cos\left[\dfrac{\pi}{\alpha}\ln\dfrac{r}{R_0}\right] }{ \operatorname{ch}\left[\dfrac{\pi}{\alpha}\left(\theta-\dfrac{n-1}{m}\pi\right)\right] -\cos\left[\dfrac{\pi}{\alpha}\ln\dfrac{rR_0}{R_2^2}\right] }; \]

\(2m\) is the number of conveyors;
\(n\) is the serial number of the conveyor;

\[ \left|\theta-\frac{n-1}{m}\pi\right|<\pi \]

\[ R_0=\frac{R_1+R_2}{2}. \]

Equation (25) has \(2m\) singular points with coordinates

\[ r=R_0,\qquad \theta'=\frac{n-1}{m}\pi. \]

Moreover, at these points one must take

\[ \ln \Gamma_n\left(R_0 e^{j\frac{n-1}{m}\pi}\right) = 2\ln \frac{\pi\rho_0} {2\alpha R_0 \sin\left[\dfrac{\pi}{\alpha}\ln\dfrac{R_0}{R_1}\right]}, \]

where \(\rho_0\) is the conveyor radius, \(\rho_0\ll R_0\).

In conclusion it should be noted that equation (25), intended for calculating the field in the working gap of an electrostatic generator, is comparatively cumbersome; however, since the series converges rapidly and the logarithmic terms are symmetric and rapidly decay with distance from the calculation point, its practical use will not be too laborious.

Literature

1. G. Buchholz. Calculation of Electric and Magnetic Fields, IL, 1961.
2. L. V. Kantorovich, V. I. Krylov. Approximate Methods of Higher Analysis, GTTI, 1952.

1. V. A. Lukutin, V. D. Eskov. Application of a chain equivalent circuit to the calculation of an ESG. Izv. TPI, vol. 149, 1966.

2. V. V. Patsevich. Certain questions in the theory and calculation of ESGs with conductor conveyors. Dissertation, TPI, 1964.

3. S. A. Bobkovsky. General theory of the ESG, ZhTF, 10, No. 17, 1404, 1940.