CALCULATION OF THE ELECTRIC FIELD OF THE STATOR OF A CAPACITIVE GENERATOR WITH A DIELECTRIC ROTOR

V. A. LUKUTIN, Yu. I. ALEKSEEV

(Presented by the scientific seminar of the Department of TOE)

A capacitive, or, as it is often called [2], electrostatic generator with a dielectric rotor (ESGD), is a source of high voltage, i.e., it is a high-voltage apparatus that converts mechanical energy into electrical energy by transferring charges from a charging system, called the excitation system, to a discharge system, called the high-voltage system. Consequently, like any high-voltage device, a capacitive generator must be designed so that the potential gradient at any point of its working volume does not reach the critical value at which the ionization process begins. In this connection there arises the need to calculate electric fields of rather complex configuration. An exact solution of the problem posed presents certain difficulties; therefore, an approximate analytical solution is proposed below.

The basic diagram of a generator with a dielectric rotor is shown in [2]. The principle of operation of such a device is described there as well. It should be noted that if charge is deposited on the surface of the rotor from one side, which corresponds to a one-sided arrangement of the ionizer, then in the working process only one half of the lateral surface of the rotor actually participates. Such a generator may be regarded as a machine with an asymmetrical arrangement of the ionizer and inductor relative to the rotor (AESGD). If, however, the inductors are arranged on both sides of the rotor, and the ionizer is placed in its middle, then it is convenient to regard it as a machine with a symmetrical arrangement of the ionizer and inductors relative to the rotor (SESGD).

In both cases the potential at any point of the generator’s working volume will be determined by the charges of the excitation ionizer (the ionizer field), the charges of the rotor (the rotor field), and the charges of the semiconducting stator (the stator field), as has been partially adopted in [3, 4].

In this work the following assumptions are adopted:

1. There is no leakage of charge from the rotor surface when an element of its surface is moved from the excitation zone to the high-voltage zone.

2. The generator field satisfies the conditions of quasistationarity.

3. The distribution of potential along the stator surface between the inductors is uniform.

Fig. 1

According to the method of superposition and the assumptions made, the potential at any point of the working volume of the machine can be written as:

\[ U(x,y)=U_c(x,y)+U_p(x,y)+U_u(x,y), \tag{1} \]

where

\(U(x,y)\) is the total value of the potential at the given point;
\(U_c(x,y)\) is the component of the electric field created by the stator charges;
\(U_p(x,y)\) is the component of the potential obtained from the rotor charges;
\(U_u(x,y)\) is the component of the field due to the ionizer charges.

In the present work we shall restrict ourselves to consideration of the stator field. The calculation region may be obtained by cutting a cylindrical or disk generator by a radial plane and then unfolding it into a strip, taking as the basis \(2l\)—the distance between the centers of the excitation inductor and the high-voltage inductor, as shown in Fig. 1. The \(y\)-axis is directed through the middle of the high-voltage inductor, which has the potential \(U_n\). The \(x\)-axis is along the surface of the stator plate of the generator. In Fig. 1 the following are denoted:

\(2b\) — the width of the inductor;
\(2l\) — the distance between adjacent high-voltage and low-voltage inductors along the stator;
\(E_0\) — the dielectric permittivity in the gap between the rotor and the generator casing;
\(E_1\) — the dielectric permittivity of the rotor body;
\(E_2\) — the dielectric permittivity of the rotor–stator gap;
\(2n\) — the thickness of the rotor;
\(d\) — half the rotor thickness plus the rotor–stator gap;
\(d'\) — the rotor–generator casing gap.

According to [2, 3, 4], in any load regime the change in potential along the inner surface of the stator (facing the rotor) obeys a definite law. To equalize the potential over the rotor, as indicated in [2, 3], it is necessary to ensure its uniform distribution over the stator. In this case the tangential component of the electric-field strength along the stator surface has one and the same magnitude.

Taking the above into account, the law of potential distribution along the stator surface at \(y=0\) and \(-2l \le x \le 2l\) can be written as follows:

\[ f(x)=C_0+C_2\cdot P_2+C_4\cdot P_4+\ldots+C_k\cdot P_k, \tag{2} \]

where

\(P_k\) are Legendre polynomials of the first kind,
\(C_k\) are unknown coefficients determined in the usual way.

On the other boundaries of the region the potential is equal to zero. It is appropriate to note that at \(x=\pm 2l\) on the line inductor excitation–generator casing the potential is different from zero. The practice of modeling in electrolytic-

in the electrolytic bath showed that the magnitude of the potential will be so small that it does not affect the distribution of the potential in the zone adjacent to the high-voltage inductor.

Consequently, determination of the potential inside the computational region in this case reduces to solving the Dirichlet problem for the Laplace equation with the prescribed boundary conditions:

\[ \Delta U = 0,\quad \text{for } U = \begin{cases} f(x), & y=0,\\ 0, & x=\pm 2l,\\ 0, & y=d+d' + \text{п}. \end{cases} \tag{3} \]

From [1, 3] it is known that integration of a differential equation can be replaced by the problem of finding a function that gives a minimum value to a certain functional (the variational method). In finding the minimizing sequence we shall use the Ritz method. For this it is necessary to choose a combination of coordinate functions satisfying the following conditions:

1. The functions must be twice differentiable.
2. They must vanish on the boundary of the region.
3. They must be complete in energy.

Fig. 2

Because the law of variation of the potential is sought in three media with different dielectric permittivities, the solution for each of the media will be valid within the limits of a homogeneous dielectric. A qualitative graph of the potential distribution for \(x=0\) and \(U=f(y)\) is shown in Fig. 2, and the coordinate functions for each of the dielectrics, satisfying the conditions listed above, can be written in the following form:

\[ U_2(x,y)=U_{\mathrm{H}}\cdot f(x)\frac{(a_2^2-y^2)}{a_2^2} + A_1(a_2^2-y^2)\cdot y^2\cdot\bigl[(2l)^4-x^4\bigr] + \tag{4} \]

\[ + A_2\cdot(a_2^2-y^2)\cdot y^2\cdot\bigl[(2l)^8-x^8\bigr], \quad \text{for } 0\le y\le \kappa_2; \]

\[ U_1(x,y)=M\cdot U_{\mathrm{H}}\cdot f(x)\cdot(a_1^2-y^2) + B_1(a_1^2-y^2)y^2\cdot\bigl[(2l)^4-x^4\bigr] + \tag{5} \]

\[ + B_2(a_1^2-y^2)\cdot y^2\cdot\bigl[(2l)^8-x^8\bigr], \quad \text{for } \kappa_2\le y\le \kappa_1; \]

\[ U_0(x,y)=N\cdot U_{\mathrm{H}}\cdot f(x)\cdot(\kappa^2-y^2) + C_1(\kappa^2-y^2)\cdot y^2\cdot\bigl[(2l)^4-x^4\bigr] + \tag{6} \]

\[ + C_2\cdot(\kappa^2-y^2)\cdot y^2\cdot\bigl[(2l)^8-x^8\bigr], \quad \text{for } \kappa_1\le y\le \kappa. \]

Equations (4), (5), (6) contain the unknown coefficients of the coordinate functions \(A_1, A_2, B_1, B_2, C_1, C_2\), which are to be determined, as well as \(M\) and \(N\)—coefficients relating the magnitude of the potential at the boundary of adjacent regions with different dielectric permittivities. Each of the sequences (4), (5), (6) will be minimizing for the functional

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

To determine the coefficients, let us first compose four equations. Two equations can be written on the basis of equality of the functions at the boundaries \(k_2=(d-n)\) and \(k_1=(n+d)\):

\[ U_2(x,y)=U_1(x,y)\quad \text{for } y=k_2, \]

\[ U_1(x,y)=U_0(x,y)\quad \text{for } y=k_1. \]

Two more equations can be written from the condition of equality of the normal components of the electric-field strength at the boundaries of adjacent regions:

\[ \varepsilon_2\left.\frac{\partial U_2}{\partial y}\right|_{y=k_2} = \varepsilon_1\left.\frac{\partial U_1}{\partial y}\right|_{y=k_2} \]

\[ \varepsilon_1\left.\frac{\partial U_1}{\partial y}\right|_{y=k_1} = \varepsilon_0\left.\frac{\partial U_0}{\partial y}\right|_{y=k_1} \]

If the indicated operations are carried out, it is not difficult to obtain the following equations:

\[ M=\frac{\varepsilon_2}{\varepsilon_1}\cdot \frac{1}{a_2^2};\qquad N=\frac{\varepsilon_2}{\varepsilon_0}\cdot \frac{1}{a_2^2}; \tag{7} \]

\[ a_1^2=(n+d)^2\cdot\left(1-\frac{\varepsilon_1}{\varepsilon_0}\right) +\kappa^2\cdot\frac{\varepsilon_1}{\varepsilon_0}; \tag{8} \]

\[ a_2^2=(d-n)^2\cdot\left(1-\frac{\varepsilon_2}{\varepsilon_1}\right) +(d+n)^2\cdot\left(\frac{\varepsilon_2}{\varepsilon_1}-\frac{\varepsilon_2}{\varepsilon_0}\right) +\frac{\varepsilon_2}{\varepsilon_0}\kappa^2; \tag{9} \]

\[ B_1=A_1\cdot\frac{\varepsilon_2}{\varepsilon_1}\cdot \frac{a_2^2-2(d-n)^2}{a_1^2-2(d-n)^2}; \tag{10} \]

\[ C_1=B_1\cdot\frac{\varepsilon_1}{\varepsilon_0}\cdot \frac{a_1^2-2(d+n)^2}{\kappa^2-2(d+n)^2}; \tag{11} \]

\[ \frac{A_2}{A_1}=\frac{B_2}{B_1}=\frac{C_2}{C_1}. \tag{12} \]

Let us compose equations by the Ritz method for the region enclosed between the stator and the rotor:

\[ \begin{cases} A_1[\varphi_1,\varphi_1]+A_2[\varphi_1,\varphi_2]=(P,\varphi_1),\\ A_1[\varphi_2,\varphi_1]+A_2[\varphi_2,\varphi_2]=(P,\varphi_2), \end{cases} \tag{13} \]

where

\[ \varphi_1=\left[(2l)^4-x^4\right](a_2^2-y^2)y^2, \]

\[ \varphi_2=\left[(2l)^8-x^8\right](a_2^2-y^2)y^2, \]

\[ P= \frac{\partial^2}{\partial x^2} \left[ \frac{U_{\mathrm{n}}}{a_2^2}\cdot f(x)\cdot(a_2^2-y^2) \right] + \frac{\partial^2}{\partial y^2} \left[ \frac{U_{\mathrm{n}}}{a^2}\cdot f(x)(a_2^2-y^2) \right]. \]

Solving system (13) makes it possible to obtain the coefficients \(A_1\) and \(A_2\), expressed through the geometrical parameters of the machine:

\[ A_1= \frac{ U_{\mathrm{n}}\cdot \left\{[-1.3\cdot a_2^2(2l)^2+0.3(2l)^4]\cdot D_0+ [0.78\cdot a_2^4-0.31\cdot a_2^2(2l)^2+0.68(2l)^4]D_2+\right. }{ a_2^4\cdot(2l)^4\cdot[0.2\cdot a_2^4+1.24\cdot a_2^2(2l)^2+0.355(2l)^4] } \]

\[ \left. \frac{ +[0.6\cdot a_2^4-2.376\cdot a_2^2(2l)^2+0.51\cdot(2l)^4]\cdot D_4 \right\} }{ } \tag{14} \]

\[ A_2= \frac{ U_{\mathrm{n}}\cdot \left\{[0.465\cdot a_2^2(2l)^2-0.52(2l)^4]\cdot D_0 - [0.27\cdot a_2^4-0.28\cdot a_2^2(2l)^2+(0.69(2l)^4]D_2+\right. }{ a_2^4\cdot(2l)^4\cdot[0.2\cdot a_2^4+1.24\cdot a_2^2\cdot(2l)^2+0.355(2l)^4] } \]

\[ \left. \frac{ +[2.336\cdot a_2^2\cdot(2l)^2-0.5(2l)^4]\cdot D_4 \right\} }{ } \tag{15} \]

The other coefficients are determined from the found \(A_1\) and \(A_2\) according to (10), (11), (12).

On the basis of the formulas obtained, the electric field was calculated in the region with the following dimensions and dielectric constants.

in relative units: \(2l=1;\quad n+d+d'=0.278,\quad b=0.114,\)
\(d-n=0.0278,\quad 2n=0.0556,\quad d'=0.1946,\quad E_0=E_1=E=1.\)

Fig. 3

The calculated pattern of equipotential lines is shown in Fig. 3 (solid lines). The results of modeling the field in an electrolytic bath are also plotted here (dashed lines).

In conclusion, it should be emphasized that, on the basis of the variational method, one can obtain very simple analytical formulas (4, 5, 6) for determining the potential at any point of the working volume of a capacitive generator. The formulas provide accuracy sufficient for engineering calculations in the zone of the high-voltage inductor. The accuracy can be increased by increasing the number of terms of the minimizing sequence, or by a more rational choice of the form of the coordinate functions.

REFERENCES

1. S. G. Mikhlin. Variational Methods in Mathematical Physics, GTI, 1957.
2. V. I. Levitov, A. G. Lyapin. Electrostatic Generators with a Rigid Rotor, Part II, Moscow, 1965.
3. A. P. Kononov, V. A. Lukutin, V. M. Osipov. Izv. TPI, vol. 191, pp. 17–22, 1969.
4. V. A. Lukutin, Yu. I. Alekseev, A. F. Kochneva. Izv. TPI, vol. 210, 1971.