APPROXIMATE ANALYTICAL CALCULATION OF THE ELECTRIC FIELD OF THE ROTOR OF A CAPACITIVE GENERATOR

V. A. LUKUTIN, E. D. MOGILEVSKAYA

(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering and the Laboratory of Rotor ECGs of the YaPh Scientific Research Institute)

The study of a capacitive—or, as it is also called, electrostatic—generator (ECG), the diagram of which is shown in Fig. 1, is connected with the investigation of electric fields in the working volume of this machine. In this connection, it becomes necessary to have information on the magnitude of the potential and its gradients at individual points of the rotor body, the stator, and the gas gap under various operating modes of the generator.

The search for methods of electrical calculation of the generator on the basis of the potential functions of the electric field appears to be a very attractive undertaking, but along this path enormous difficulties are encountered, arising when classical methods of field calculation are used for a region of complex configuration with no less complex boundary conditions. Below an approximate analytical method is proposed for calculating the field of an electrostatic generator whose rotor is equipped with conducting transporter electrodes.

Fig. 1. Annular working region of the generator

If the dielectric from which the machine parts are made is assumed ideal and with constant dielectric permittivity, then for such a medium the principle of superposition will be valid, and the complex problem of studying the field can be divided into a number of simpler ones. Thus, the potential of any point of the space between the stator surfaces can be represented as the sum of two functions, one of which is due to the field created by the charges of the stator electrodes \((U_{\mathrm{st}})\), and the other by the charges of the rotor transporters \((U_{\mathrm{r}})\):

\[ U(x,y)=U_{\mathrm{st}}(x,y)+U_{\mathrm{r}}(x,y). \tag{1} \]

Since the charges of the transporters and inductors do not have time to change appreciably during the interval of time required for electromagnetic disturbances to propagate between the two most distant points of the system under consideration, the elec-

tric field in the working space of the ESG may be regarded as quasi-stationary.

Let us construct a computational model of the machine; to this end we shall make a number of generally accepted assumptions. We shall neglect leakage of charges through the rotor dielectric and across the gas gap, and shall not take into account corona discharge of the electrodes. As a rule, the length of the carriers is considerably greater than the thickness of the rotor; therefore the electric field between the stator surfaces may be considered two-dimensional. Finally, we shall cut the annular working region of the generator along the plane \(oa\) (see Fig. 1) and unfold it into a rectangular region, as is customary in a number of works [3, 4, 5].

Let the coordinate axes be arranged so that the \(z\)-axis is parallel to the axis of the carriers, while the \(x\)-axis coincides with the plane in which they lie.

The advancing section of the rotor carries \(m\) carriers with identical charges \(q\), and on each rod of the receding part of the rotor there remains a certain charge \(q'\). We replace these discrete rotor charges by continuous uniformly charged strips with volume charge densities

\[ \rho=\frac{mq}{4lrz}\quad \text{and}\quad \rho'=\frac{mq'}{4lrz}, \tag{2} \]

where \(2m\) is the number of rotor carriers belonging to one pair of poles,

\[ \begin{aligned} 2l&\text{ is the length of the rotor arc between adjacent poles,}\\ 2r&\text{ is the diameter of the carriers,}\\ z&\text{ is their axial length.} \end{aligned} \]

Having performed the above transformations, we obtain the computational diagram shown in Fig. 2, which represents the cross section of the generator working space corresponding to one pair of poles. Here

Fig. 2. Computational diagram

\(2d\) is the distance between the inner surfaces of the stator, \(2n\) is the rotor thickness, \(a\) is the distance between the axes of adjacent carriers, \(\varepsilon_0\) is the dielectric permittivity of the gas gap, and \(\varepsilon_1\) is the permittivity of the rotor dielectric.

We shall seek the potential function of the electric field of the rotor, \(U_p(x,y)\), caused only by the charges of the carriers. It will correspond to the short-circuit regime of an ESG operating according to a conduction excitation circuit. In this case the potential of the stator plates will be zero, since they are made of a poorly conducting material.

The field in the region under consideration satisfies Poisson’s equation

\[ \Delta U_p(x,y)=-\frac{\rho(x,y)}{\varepsilon}. \tag{3} \]

The boundary conditions are determined more simply if the volume charge density is decomposed into two components:

\[ \rho=\rho_a+\rho_b,\qquad \rho'=-\rho_a+\rho_b, \tag{4} \]

which does not contradict the principle of superposition. The latter equations mean that the potential function of the rotor, in turn, can be represented as two terms:

\[ U_p(x,y)=U_{pa}(x,y)+U_{pb}(x,y). \tag{5} \]

The component \(U_{pb}\) is found in an elementary way, since the field between the stator plates is produced by a uniformly charged strip with charge density \(\rho_b\) and will be homogeneous. It is quite obvious that, under the assumptions made,

\[ U_{pb_0}(x,y)=U_{pb_0}(y)=\frac{\rho_b r}{\varepsilon_0}(d-|y|) \quad \text{for } n\le |y|\le d, \]

\[ U_{pb_1}(x,y)=U_{pb_1}(y)=\frac{\rho_b r}{\varepsilon_0}(d-n)+ \frac{\rho_b r}{\varepsilon_1}(n-|y|) \quad \text{for } r\le |y|\le n. \tag{6} \]

Determination of the function \(U_{pa}(x,y)\) reduces to solving the interior Dirichlet problem for the Poisson equation

\[ \Delta U_{pa}=-\frac{\rho_a(x,y)}{\varepsilon} \tag{7} \]

with zero conditions on the boundary of the region

\[ x=\pm l,\qquad y=\pm d. \]

We solve the problem by a variational method, replacing the integration of the partial differential equation (7) by finding a function that realizes the minimum of the functional \([1,2]\).

\[ I=\iint\left[\left(\frac{\partial\varphi}{\partial x}\right)^2+ \left(\frac{\partial\varphi}{\partial y}\right)^2-2P\varphi\right]\,dx\,dy. \tag{8} \]

We shall seek the minimizing sequence of functions by the Ritz method.

In our case the region has an inhomogeneous dielectric; therefore, for the medium with dielectric permittivity \(\varepsilon_0\), we choose the coordinate functions as follows: for \(-l\le x\le l,\ n\le |y|\le d\),

\[ U_{pa_0}(x,y)\simeq A_0(l^4-x^4)(d^2-y^2)+ B_0(l^{12}-x^{12})(d^2-y^2), \tag{9} \]

and for the region occupied by the rotor dielectric \(\varepsilon_1\),

\[ \begin{aligned} U_{pa_1}(x,y)\simeq{}& A_0(l^4-x^4)(d^2-n^2)+B_0(l^{12}-x^{12})(d^2-n^2)\\ &+A_1(l^4-x^4)(n^2-y^2)+B_1(l^{12}-x^{12})(n^2-y^2) \end{aligned} \tag{10} \]

for \(-l\le x\le l,\ r\le |y|\le n\).

In the first case, on the boundary of the region the potential is equal to zero, whereas in the second the boundary conditions are nonhomogeneous. By introducing a new variable and substituting into (7),

\[ U_{pa_1}(x,y)=\varphi(x,y)+\varphi_0(x,y) \]

we again obtain the Poisson equation, but with a different right-hand side and with zero boundary conditions,

\[ \Delta \varphi(x,y)=-P(x,y), \tag{11} \]

where

\[ \varphi=A_1\varphi_1+B_1\varphi_1 =A_1(l^4-x^4)(n^2-y^2)+B_1(l^{12}-x^{12})(n^2-y^2), \]

\[ P=\frac{\rho_a(x,y)}{\varepsilon_1}+\Delta\varphi_0, \]

\[ \varphi_0=A_0(l^4-x^4)(d^2-n^2)+B_0(l^{12}-x^{12})(d^2-n^2). \]

To determine the coefficients \(A_0, A_1, B_0, B_1\), we compose a system of equations. The first two equations are Ritz equations:

\[ \begin{aligned} A_1[\varphi_1,\varphi_1]+B_1[\varphi_1,\varphi_2]-(P,\varphi_1)&=0;\\ A_1[\varphi_1,\varphi_2]+B_1[\varphi_2,\varphi_2]-(P,\varphi_2)&=0. \end{aligned} \tag{12} \]

We shall use the conditions, known from electrostatics, at the boundary of separation of two dielectrics:

\[ \begin{aligned} \left.\frac{\partial U_{\mathrm{pa}_0}}{\partial x}\right|_{y=\pm n} &= \left.\frac{\partial U_{\mathrm{pa}_1}}{\partial x}\right|_{y=\pm n},\\ \varepsilon_0\left.\frac{\partial U_{\mathrm{pa}_0}}{\partial y}\right|_{y=\pm n} &= \varepsilon_1\left.\frac{\partial U_{\mathrm{pa}_1}}{\partial y}\right|_{y=\pm n}. \end{aligned} \tag{13} \]

In equations (12) the following are denoted:

\[ [\varphi_1,\varphi_2]\text{ — the energy product of the functions;} \]

\[ [\varphi_1,\varphi_2] = \iint\left[ \left(\frac{\partial\varphi_1}{\partial x}\cdot\frac{\partial\varphi_2}{\partial x}\right) + \left(\frac{\partial\varphi_1}{\partial y}\cdot\frac{\partial\varphi_2}{\partial y}\right) \right]\,dx\,dy; \]

\[ (P,\varphi)=\iint P\cdot\varphi\,dx\,dy \text{ — the scalar product of the functions.} \]

The simultaneous solution of equations (12), (13), (14) makes it possible to determine the coefficients of interest to us; substituting them into formulas (9) and (10), we obtain approximate components of the analytical functions of the potential in the gas gap and in the dielectric of the rotor:

\[ U_{\mathrm{pa}_0}(x,y)\simeq \frac{\rho_a}{\varepsilon_0}\cdot \kappa(d^2-y^2)\,[M(1-x^4)+N(1-x^{12})], \tag{15} \]

\[ U_{\mathrm{pa}_1}(x,y)\simeq \frac{\rho_a}{\varepsilon_1}\,\kappa \left[ \frac{\varepsilon_1}{\varepsilon_0}(d^2-n^2)+(n^2-y^2) \right] [M(1-x^4)+N(1-x^{12})], \tag{16} \]

where

\[ \kappa= \frac{rm(3n^2-r^2)l^2} {10^4\left[4(141n^2+4.5)+705\,\frac{\varepsilon_1}{\varepsilon_0}(d^2-n^2)\right]}, \]

\[ M= \left[ 200+706\,\frac{\varepsilon_1}{\varepsilon_0}(d^2-n^2) \right][5-m^4] - \]

\[ -62.5 \left[ 1.08+2.21\,\frac{\varepsilon_1}{\varepsilon_0}(d^2-n^2) \right][13-m^{12}], \]

\[ N= \left[ 3+5\,\frac{\varepsilon_1}{\varepsilon_0}(d^2-n^2) \right]\cdot 10.13\,[13-m^{12}] - \]

\[ -81.4 \left[ 1.08+2.21\,\frac{\varepsilon_1}{\varepsilon_0}(d^2-n^2) \right][5-m^4], \]

\[ m=1-\frac{a}{2}. \]

Here all geometric dimensions are referred to \(l\), including the variables \(x, y\).

Now let us sum (6), (15), (16) and obtain approximate functions for calculating the electric-field potential at any point of the gas gap between the rotor and stator and in the body of the rotor:

\[ U_{p_0}(x,y)=U_{pb_0}(x,y)+U_{pa_0}(x,y), \tag{17} \]

\[ U_{p_1}(x,y)=U_{pb_1}(x,y)+U_{pa_1}(x,y). \tag{18} \]

In order to check the accuracy of the calculation performed on the basis of the formulas obtained, a large number of experiments were carried out on modeling electric fields in an electrolytic bath.

The experiments were conducted for a region whose dimensional ratios correspond to real generator samples with rod-type transporters of various cross sections. In Fig. 3, as an example, values of the potential calculated by formula (16) are presented for points

Fig. 3. Experimental and calculated curves of the distribution of potentials along the transporters and at the rotor--gap boundary

lying on the \(x\)-axis (\(1, 2\)—dashed) and on the rotor surface \((y=\pm n)\) (\(1a, 2a\)—dashed). The values of the potentials of the transporters and of the rotor surface obtained during modeling are also given here (solid curves; the experimental data are marked with crosses). Series of curves 1 was taken and calculated for the following geometric dimensions, referred to \(l\):

\[ d=8\cdot 10^{-2},\quad n=6\cdot 10^{-2},\quad a=8\cdot 10^{-2},\quad r=2\cdot 10^{-2} \quad \text{for}\quad \frac{\varepsilon_1}{\varepsilon_0}=4. \]

The second series is for

\[ d=8\cdot 10^{-2},\quad n=6.7\cdot 10^{-2},\quad a=10^{-1},\quad r=4\cdot 10^{-2} \quad \text{for}\quad \varepsilon_1=4\varepsilon_0. \]

The experimental and calculated curves are located sufficiently close to one another, which indicates the fairly high accuracy of the calculations provided by the proposed formulas.

Conclusions

1. Approximate formulas have been obtained that make it possible, very simply and with sufficient accuracy, to determine the electric-field potential of the rotor in the rotor--stator gap.

1. On the basis of these formulas it is easy to find the normal and tangential components of the electric-field strength at any point of this same space.

2. Formula (18) gives sufficiently accurate values of the potential on the surface of the rotor and in its body in the zone adjacent to this surface. The values of the potentials of the transporters themselves are obtained very accurately. In the zone between any neighboring transporters, the potential cannot be calculated from equation (18), since it does not take into account the character of the field in this region.

3. On the basis of the variational method, one can also obtain greater accuracy of the calculations if a larger number of terms of the minimizing sequence (10), (9) is chosen; however, in this case the formulas become more cumbersome. In addition, the accuracy is determined not only by the number of terms, but also by their form.

4. Using the potential functions (17), (18) and the materials from [5], one can easily obtain the equations of the linear sections of the external characteristic and the short-circuit characteristic of an electrostatic generator, and determine its principal energy indices.

5. The described method for calculating the electric potential can be successfully extended also to an ESG with a dielectric rotor.

REFERENCES

1. S. G. Mikhlin. Variational Methods in Mathematical Physics. State Publishing House of Technical-Theoretical Literature. Moscow, 1957.

2. L. V. Kantorovich, V. I. Krylov. Approximate Methods of Higher Analysis. State Publishing House of Technical-Theoretical Literature. Moscow, 1949.

3. F. Ollendorff. “Field theory of self-excited influence machines.” Topics Appl. Mech. Amsterdam—London—New York, 1965, 135—158.

4. 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.

5. V. A. Lukutin, Yu. I. Alekseev, A. F. Kochneva. A potential function for the field of the stator of an electrostatic generator. Izv. TPI, vol. 210 (in press).