UDC 621.319.3.001.024

MATHEMATICAL DESCRIPTION OF THE EXTERNAL FIELD OF ESGS

I. P. Guk, V. A. Lukutin, V. S. Sokolov

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

With the emergence of new forms of electron-ion technology based on the use of the forces of an electric field, electrostatic generators are finding ever wider application; a variety of these are generators with conductor carriers placed in a dielectric (ESGS). In the latter, energy conversion is carried out by transporting charged metallic carriers against the forces of an external electric field. This external field—the stator field—arises when the generator operates under load, owing to leakage currents flowing through a weakly conducting layer from the load electrodes to the excitation electrodes. The stator field leads to the appearance of the so-called “armature reaction” [1], which in ESGS manifests itself, first, in a decrease in the load current as its voltage increases (in theory this is characterized by the voltage increase coefficient $k_U$), and, second, in a limitation of the maximum specific power of the generator owing to the normal component of this field.

The purpose of the present article is to determine the components of the stator-field strength vector and the voltage increase coefficient as functions of the geometric and electrical characteristics of the rotor and of the gas gap, for an arbitrary distribution of potential along the stator surface. In the work, the assumptions customary in such cases, for example in [2], [3], are adopted:

1. The electric field is considered quasistatic.
2. The system is assumed linear; the dielectric of the rotor and the working medium are homogeneous and isotropic.
3. The field is considered plane-parallel.
4. The influence of metallic inclusions (carriers) on the field in the gap is not taken into account.

In accordance with the assumptions listed and similarly to how this is done in [2], the ESGS model shown in Fig. 1 is adopted for calculating the stator field. Here the rotor corresponds to a layer of thickness $2d$ with dielectric permittivity $\varepsilon_2=\varepsilon_0\varepsilon_r$, and the gas gaps correspond to layers of thickness $\delta$ with $\varepsilon_1=\varepsilon_3=\varepsilon_0\varepsilon_g$. The influence of the stator on the electric field is taken into account in the computational model by the boundary functions

$$ \varphi(d+\delta, x)=\varphi(-d-\delta, x)=U(x), \tag{1} $$

$$ \varphi(y,0)=\varphi(y,l)=U(y). \tag{2} $$

Structurally, the stator is, as a rule, made symmetric, and then the boundary function \(U(x)\) has period \(2l\), where \(l\) is the distance between adjacent poles, and its Fourier expansion contains no even harmonics. If the ordinate axis is placed midway between the poles, that is, along the line of field symmetry, then

\[ U(y)=\frac{U_1-U_2}{2}=\text{const}, \tag{3} \]

where \((U_1-U_2)\) is the potential difference between points of the stator surface \((y=d+\delta)\) lying under adjacent poles.

Fig. 1

Thus, the calculation of the stator field reduces to solving the Dirichlet problem for a rectangle with a three-layer dielectric, with the given functions \(U(x)\) and \(U(y)\) on the boundaries.

Taking into account the properties of the boundary functions, the solution of the problem posed can be obtained as the sum \(\varphi_1+\varphi_2\) of the solutions of two simpler problems for the given region under new boundary conditions:

\[ \begin{cases} \varphi_1[(d+\delta),x]=\varphi_1[-(d+\delta),x]=U(x)-U(y)=U^*(x),\\ \varphi_1(y,0)=\varphi_1(y,l)=0, \end{cases} \tag{4} \]

\[ \begin{cases} \varphi_2[(d+\delta),x]=\varphi_2[-(d+\delta),x]=U(y)=\dfrac{1}{2}(U_1-U_2),\\ \varphi_2(y,0)=\varphi_2(y,l)=U(y)=\dfrac{1}{2}(U_1-U_2). \end{cases} \tag{5} \]

Since \(\varphi_2=\text{const}\) on the entire boundary of the region, all points inside the region will have the same potential \(\dfrac{U_1-U_2}{2}\), and the problem reduces to determining \(\varphi_1(y,x)\).

The potential in the \(i\)-th layer satisfies Laplace’s equation

\[ \nabla^2 \varphi_1^{(i)}=0,\quad (i=1,2,3). \tag{6} \]

By the method of separation of variables, we shall seek the solution of equation (6), according to [4], in the form of a series:

\[ \varphi_1^{(i)}(y,x)=\sum_{k=1}^{\infty} \left[ A_k^{(i)}\operatorname{ch}\frac{k\pi}{l}y+ B_k^{(i)}\operatorname{sh}\frac{k\pi}{l}y \right]\sin\frac{k\pi}{l}x, \tag{7} \]

the coefficients \(A_k^{(i)}\) and \(B_k^{(i)}\) of which are determined from the boundary conditions (4) and the matching conditions at the dielectric interfaces:

\[ \varphi_1^{(1)}(d,x)=\varphi_1^{(2)}(d,x),\quad \varepsilon_1\left.\frac{\partial\varphi_1^{(1)}}{\partial y}\right|_{y=d} = \varepsilon_2\left.\frac{\partial\varphi_1^{(2)}}{\partial y}\right|_{y=d}, \tag{8} \]

\[ \varphi_1^{(2)}(-d,x)=\varphi_1^{(3)}(-d,x),\quad \varepsilon_2\left.\frac{\partial \varphi_1^{(2)}}{\partial y}\right|_{y=-d} = \varepsilon_3\left.\frac{\partial \varphi_1^{(3)}}{\partial y}\right|_{y=-d}. \tag{9} \]

Expand the function \(U^*(x)\) in a Fourier sine series on the interval \(l\):

\[ U^*(x)=\sum_{k=1}^{\infty}\beta_k\sin\frac{k\pi}{l}x, \tag{10} \]

where

\[ \beta_k=\frac{2}{l}\int_0^l U^*(x)\sin\frac{k\pi}{l}x\,dx. \]

Now, in order to obtain the necessary system of equations for determining \(A_k^{(i)}\) and \(B_k^{(i)}\), we replace the functions \(\varphi_1^{(i)}\) and \(U^*\) in equations (4), (8), and (9) by their series and equate the corresponding amplitudes of the harmonics. The solution of this system gives:

\[ A_k^{(1)}=A_k^{(3)} = \frac{\beta_k}{N_k} \left[ 1-(\varepsilon_r-1)\operatorname{sh}^2\frac{k\pi}{l}d \right], \tag{11} \]

\[ A_k^{(2)}=\frac{\beta_k}{N_k},\quad B_k^{(2)}=0, \tag{12} \]

\[ B_k^{(1)}=-B_k^{(3)} = \frac{\beta_k}{N_k}\frac{(\varepsilon_r-1)}{2}\operatorname{sh}\frac{2k\pi}{l}d, \tag{13} \]

where

\[ N_k=\operatorname{ch}\frac{k\pi}{l}(d+\delta) +(\varepsilon_r-1)\operatorname{sh}\frac{k\pi}{l}d\,\operatorname{sh}\frac{k\pi}{l}\delta, \]

\[ \varepsilon_r=\varepsilon_2/\varepsilon_1=\varepsilon_2/\varepsilon_3=\varepsilon_p/\varepsilon_0. \]

And the expressions for determining the components of the electric-field strength vector in the gas gap will have the form

\[ E_n^{(1)} = -\frac{\partial(\varphi_1+\varphi_2)}{\partial y} = -\frac{\pi}{l}\sum_{k=1}^{\infty}k\beta_k \frac{ \operatorname{sh}\frac{k\pi}{l}y + (\varepsilon_r-1)\operatorname{sh}\frac{k\pi}{l}d\,\operatorname{ch}\frac{k\pi}{l}(y-d) }{ \operatorname{ch}\frac{k\pi}{l}(d+\delta) + (\varepsilon_r-1)\operatorname{sh}\frac{k\pi}{l}d\,\operatorname{sh}\frac{k\pi}{l}\delta } \sin\frac{k\pi}{l}x, \tag{14} \]

\[ E_\tau^{(1)} = -\frac{\partial(\varphi_1+\varphi_2)}{\partial x} = -\frac{\pi}{l}\sum_{k=1}^{\infty}k\beta_k \frac{ \operatorname{ch}\frac{k\pi}{l}y + (\varepsilon_r-1)\operatorname{sh}\frac{k\pi}{l}d\,\operatorname{sh}\frac{k\pi}{l}(y-d) }{ \operatorname{ch}\frac{k\pi}{l}(d+\delta) + (\varepsilon_r-1)\operatorname{sh}\frac{k\pi}{l}d\,\operatorname{sh}\frac{k\pi}{l}\delta } \cos\frac{k\pi}{l}x. \tag{15} \]

The voltage-increase coefficient \(k_U\) is determined in [3] from the calculation of the stator field, and in our case one may write

\[ k_U^{-1} = \frac{2\varphi^{(2)}(0,x_0)}{U_1-U_2} = 1+\frac{2}{U_1-U_2}\sum_{k=1}^{\infty}\frac{\beta_k}{N_k}\sin\frac{k\pi}{l}x_0, \tag{16} \]

where \(y=0,\ x=x_0\) are the coordinates of the transporter commutating in the excitation system. As is evident from the expression given, \(k_U\) will be the larger, the smaller the ratios \(d/l\), \(\delta/l\), and \(\varepsilon_p/\delta\).

It follows from (14) and (15) that the field-strength components attain their maximum values on the stator surfaces, \(y=\pm(d+\delta)\), and where \(E_\tau\) has a maximum, \(E_n\) has a minimum, and conversely, i.e., they are, as it were, in antiphase; moreover, \(E_n\) will be the smaller the smaller \(d/l\) and \(\varepsilon_p/\varepsilon_0\) are, and it depends substantially on the distribution of the potential in the semiconducting layer of the stator.

REFERENCES

1. V. I. Levintov, A. G. Lyapin. Electrostatic generators with a rigid rotor. Moscow, 1965.

2. V. A. Lukutin. On the calculation of the electric field of the rotor of a capacitive generator. Izv. AN SSSR, “Energetika i transport”, 1971, No. 4.

3. I. D. Guk. Calculation of the field of an electrostatic generator. Izv. TPI, vol. 191, 1969.

4. L. V. Kantorovich, V. I. Krylov. Approximate Methods of Higher Analysis. Moscow—Leningrad, Gostekh­teorizdat, 1952.