Calculation of the Stator Field of an Electrostatic Generator
Abstract Generated abstract
The paper develops an analytical method for calculating the stator electric field of a cylindrical electrostatic generator with conducting carriers. Under assumptions of a two-dimensional field, negligible end effects, and a homogeneous rotor dielectric without local conductor disturbances, the problem is reduced to an internal Dirichlet problem for Laplace’s equation in a three-layer annular dielectric region. Using separation of variables and Fourier expansion of the prescribed stator surface potential, the authors derive series expressions for the potential in each dielectric layer, with coefficients determined by continuity of potential and normal electric displacement at interfaces. The resulting formulas are specialized for conduction and induction excitation schemes with approximately linear stator potential distributions, and the authors note that the series converge rapidly enough for practical engineering calculations.
Full Text
CALCULATION OF THE STATOR FIELD OF AN ELECTROSTATIC GENERATOR
I. P. Guk, V. A. Lukutin.
(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)
The power of an electrostatic generator (ESG) is determined by the energy of the electric field in the working volume of the machine; therefore, calculation of this field is highly desirable. In the general case this problem is rather complex, if all structural elements of the generator and the conditions of its operation are taken into account.
Figure 1 shows a schematic cross-section of a cylindrical generator with conducting conveyors. Here the stator is represented by two concentric surfaces of radii \(r_1\) and \(r_4\) and by two pairs of inductors. Between the stator surfaces there is a dielectric rotor with metal rods—conveyors—placed inside it.
We shall consider the electric field in the region between the stator surfaces; this field is produced by the charged conveyors and inductors. To simplify the problem, it is expedient, using the principle of superposition, to calculate separately the fields produced by the stator and by the rotor. In the present work a variant is proposed for determining the potential function of the stator field.
We shall neglect the fringe effect at the ends of the machine; then the field will be two-dimensional. For a number of reasons, which we shall not discuss here, the cross-section of the conveyors is chosen to be very small; therefore the metallic inclusions in the rotor dielectric distort the field only in their immediate vicinity. In any case, they do not affect the character of the field in the gas gap; therefore below we shall assume the rotor dielectric to be homogeneous and shall not take the conducting conveyors into account.
Taking into account the assumptions made, the problem reduces to finding the potential function in a plane annular region with a three-layer dielectric. The potential of the high-voltage inductor is known \((U_{\mathrm{n}})\), and we assume its distribution over the stator surface to be prescribed:
\[ \varphi(r_1;\theta)=\varphi(r_4;\theta)=U(\theta). \tag{1} \]
If there are no free charges in the dielectric, then in the annular region the potential function satisfies Laplace’s equation
\[ \Delta\varphi=0 \tag{2} \]
Fig. 1. Basic schematic of a cylindrical-type electrostatic generator with conducting carriers:
1 — excitation inductor,
2 — high-voltage inductor,
3 — semiconducting layer of the stator,
4 — conducting carriers,
5 — rotor body with \(\varepsilon=\varepsilon_{\mathrm{d}}\),
6 — working medium with \(\varepsilon=\varepsilon_{\mathrm{s}}\).
and the calculation of the stator field of an electrostatic generator is reduced to the solution of the internal Dirichlet problem.
It is known that the potential \(\varphi^{(i)}\) in a region filled with a medium of dielectric permittivity \(\varepsilon_i\) also satisfies the Laplace equation
\[ \Delta \varphi^{(i)}=0, \tag{3} \]
where \(i=1,2,3\) is the number of the annular region.
Applying the method of separation of variables to the solution of equation (3), for a cylindrical coordinate system one may write the formula for the potential in the form of a series
\[ \varphi^{(i)}(r;\theta)=\frac{a_0^{(i)}}{2}+\sum_{k=1}^{\infty}\left(a_k^{(i)}\cos k\theta+b_k^{(i)}\sin k\theta\right). \tag{4} \]
It remains only to determine the coefficients \(a_0^{(i)}\), \(a_k^{(i)}\), and \(b_k^{(i)}\), and the problem will be completely solved.
The coefficients of the sine component \(b_k^{(i)}\) are equal to zero, since the potential distribution is symmetric with respect to the diameter passing through the middles of the inductors \((\theta=0)\).
The cosine coefficients \(a_0^{(i)}\) and \(a_k^{(i)}\) will be determined using the boundary conditions at the interface between media (continuity of the potential and of the normal component of the dielectric displacement vector).
tion at the boundary of two media \(r=r_2,\ r=r_3\)) and taking into account the potential distribution on the outer circles.
Substituting the values of the coefficients \(a_0^{(i)}\), \(a_k^{(i)}\) and \(b_k^{(i)}\), determined from the above conditions, into series (4) and introducing the notation
\[ \varepsilon_{\mathrm{pr}}=(\varepsilon_{\mathrm{d}}+\varepsilon_{\mathrm{c}}) -(\varepsilon_{\mathrm{d}}-\varepsilon_{\mathrm{c}})\lambda^{2k}, \]
\[ \varepsilon_{\mathrm{pr}}^{*}=(\varepsilon_{\mathrm{d}}+\varepsilon_{\mathrm{c}}) -(\varepsilon_{\mathrm{d}}-\varepsilon_{\mathrm{c}})\lambda^{-2k}, \]
where
\[ \lambda=\frac{r_1}{r_2}\simeq \frac{r_3}{r_4}, \]
we obtain an expression for calculating the potential function describing the field of the ESG stator.
For the first layer \((r_1\leq r\leq r_2,\ \varepsilon=\varepsilon_{\mathrm{c}})\)
\[ \varphi(r;\theta)=\frac{\alpha_0}{2}+ \sum_{k=1}^{\infty} \left[A_k^{(1)}(r^{2k}-r_1^{2k})+r_1^k\right] \frac{\alpha_k}{r^k}\cos k\theta, \tag{5} \]
where
\[ A_k^{(1)}= \frac{ r_4^k\,4\varepsilon_{\mathrm{d}}\varepsilon_{\mathrm{c}} -r_1^k\left[ (\varepsilon_{\mathrm{d}}+\varepsilon_{\mathrm{c}})\varepsilon_{\mathrm{pr}}^{*} +(\varepsilon_{\mathrm{d}}-\varepsilon_{\mathrm{c}})\varepsilon_{\mathrm{pr}} \left(\frac{r_4}{r_2}\right)^{2k} \right] }{ r_4^{2k}\varepsilon_{\mathrm{pr}}^{2} -r_1^{2k}\varepsilon_{\mathrm{pr}}^{*2} }. \]
For the second layer \((r_2\leq r\leq r_3,\ \varepsilon=\varepsilon_{\mathrm{d}})\)
\[ \varphi(r;\theta)=\frac{\alpha_0}{2}+ \sum_{k=1}^{\infty} \lambda_k^{(2)}(r^{2k}-r_1^k r_4^k) \frac{\alpha_k}{r^k}\cos k\theta \tag{6} \]
where
\[ \lambda_k^{(2)}= \frac{2\varepsilon_{\mathrm{c}}} {r_4^k\varepsilon_{\mathrm{pr}}+r_1^k\varepsilon_{\mathrm{pr}}^{*}}. \]
For the third layer \((r_3\leq r\leq r_4,\ \varepsilon=\varepsilon_{\mathrm{c}})\)
\[ \varphi(r;\theta)=\frac{\alpha_0}{2}+ \sum_{k=1}^{\infty} \left[A_k^{(3)}(r^{2k}-r_4^{2k})+r_4^k\right] \frac{\alpha_k}{r^k}\cos k\theta, \tag{7} \]
where
\[ A_k^{(3)}= \frac{ r_4^k\left[ (\varepsilon_{\mathrm{d}}+\varepsilon_{\mathrm{c}})\varepsilon_{\mathrm{pr}} +(\varepsilon_{\mathrm{d}}-\varepsilon_{\mathrm{c}})\varepsilon_{\mathrm{pr}}^{*} \left(\frac{r_1}{r_3}\right)^{2k} \right] -r_1^k\,4\varepsilon_{\mathrm{d}}\varepsilon_{\mathrm{c}} }{ r_4^{2k}\varepsilon_{\mathrm{pr}}^{2} -r_1^{2k}\varepsilon_{\mathrm{pr}}^{*2} }. \]
The constants \(\alpha_0\) and \(\alpha_k\) entering expressions (5), (6), (7) are obtained from the expansion of the boundary function \(U(\theta)\) in a Fourier series:
\[ U(\theta)=\frac{\alpha_0}{2}+ \sum_{k=1}^{\infty}(\alpha_k\cos k\theta+\beta_k\sin k\theta). \tag{8} \]
The boundary function \(U(\theta)\) in the general case may have the most arbitrary form, and it depends on many factors.
In electrostatic generators with conductor carriers, an attempt is made to prescribe forcibly a linear distribution of the potential along the stator in order to equalize the electrostatic field. In this case the function \(U(\theta)\) will have the form shown in Fig. 2.
Fig. 2.
Here curve \(a\) corresponds to the conduction excitation circuit, and curve \(b\) to the induction circuit.
The Fourier expansion of such functions gives the following values for the coefficients for the conduction excitation circuit (Fig. 2, curve \(a\)):
\[ a_0 = U_{\mathrm{н}}, \]
\[ \alpha_k = - \frac{4\cos k\theta_0}{\pi(\pi - 2\theta_0)k^2} U_{\mathrm{н}}, \]
where \(k = 1, 3, 5 \ldots\),
\[ \beta_k = 0; \]
for the induction excitation circuit (Fig. 2, curve \(b\)),
\[ a_0 = U_{\mathrm{н}} - U_{\mathrm{в}}, \]
\[ \alpha_k = - \frac{4\cos k\theta_0}{\pi(\pi - 2\theta_0)k^2}(U_{\mathrm{н}} + U_{\mathrm{в}}), \]
\[ \beta_k = 0. \]
Substituting now the values of these coefficients into (5), (6), (7), we obtain the final formulas for the potential function describing the stator field of a cylindrical electrostatic generator.
In conclusion, it should be noted that the series entering into the expressions for the potentials converge very rapidly, and in engineering calculations one may restrict oneself to the first and the doubled second terms of the series.
REFERENCES
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V. I. Levintov, A. G. Lyapin. Electrostatic Generators with a Rigid Rotor. Part I, Moscow, 1963.
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G. A. Grinberg. Selected Problems of the Mathematical Theory of Electric and Magnetic Phenomena. Moscow, 1948.