Calculation of the Field in a Three-Layer Dielectric of Annular Cross-Section with a Specified Potential Distribution on the Outer Boundary Surfaces
Abstract Generated abstract
The paper derives a general solution for the electrostatic potential in a long three-layer cylindrical dielectric of annular cross section with prescribed angular potential distributions on the inner and outer boundary surfaces. Treating the field as plane-parallel and charge-free, it solves Laplace’s equation in each layer by separation of variables, expands the boundary data in Fourier series, and imposes continuity of potential and normal electric displacement at the internal interfaces. The resulting formulas give the Fourier coefficients for the potential in all three layers, including harmonic and logarithmic terms, for arbitrary radii, permittivities, and boundary potential functions. A reduction to the two-layer case is indicated, and the authors note the usefulness of the expressions for machine computation and for assessing approximate field-calculation methods.
Full Text
CALCULATION OF THE FIELD IN A THREE-LAYER DIELECTRIC OF ANNULAR CROSS SECTION WITH A GIVEN POTENTIAL DISTRIBUTION ON THE OUTER BOUNDARY SURFACES
I. P. Guk, V. A. Lukutin, V. S. Sokolov.
(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)
Determining the potential in a single-layer dielectric of annular cross section (a long tube made of dielectric material), with a given potential distribution on the circumferences of the boundary surfaces, is directly reduced to solving the internal Dirichlet problem for an annulus. In the literature [1] this case has been considered in sufficient detail; the solution is written in the form of a trigonometric series, and formulas are given for determining its coefficients.
For a two-layer and three-layer dielectric, a similar solution of the problem is absent from the literature, and there are no ready formulas for determining the indicated coefficients; meanwhile, in connection with the design and study of cylindrical-type electrostatic generators and certain other electrophysical apparatus, the general solution of this problem is of definite practical interest.
In the present work the potential distribution in each of the layers of a three-layer dielectric of annular cross section is found in the general case, i.e., for an arbitrary ratio of the radii of the boundary surfaces, the permittivities of the media, and an arbitrary form of the potential-distribution functions on the outer boundary surfaces.
Let us consider a dielectric that is a long tube consisting of three concentrically arranged layers of different dielectric materials with permittivities \(\varepsilon_1, \varepsilon_2, \varepsilon_3\). The cross section of such a three-layer tube is shown in Fig. 1.
We assume that the axial length of the tube is sufficiently large in comparison with the outer diameter of the cross section, so that the field may be regarded as plane-parallel; that there are no free charges in the dielectrics or on the surfaces separating them; and that the distribution of potentials along the circumferences of the outer boundary surfaces is given by certain functions
\[ U_1 = \varphi(r_1;\theta) \quad \text{and} \quad U_4 = \varphi(r_4;\theta) \tag{1} \]
The potential in each of the dielectric layers is determined by Laplace’s equation
\[ \Delta \varphi^{(i)} = 0, \tag{2} \]
where \(i=1, 2, 3\) is the number of the layer.
Solving equation (2) by the method of separation of variables, we obtain an expression for the potential in the form of a trigonometric series [1]
Fig. 1.
\[
\varphi^{(i)}(r;\theta)=A_0^{(i)}\ln r+B_0^{(i)}+\sum_{k=1}^{\infty}\left[(A_k^{(i)}r^k+B_k^{(i)}r^{-k})\cos k\theta+\right.
\]
\[
\left.+(C_k^{(i)}r^k+D_k^{(i)}r^{-k})\sin k\theta\right].
\tag{3}
\]
This solution is valid for any layer.
To find the constants in equation (3), we expand the boundary functions in a Fourier series
\[ U_1(\theta)=\frac{\alpha_0^{(1)}}{2}+\sum_{k=1}^{\infty}\left(\alpha_k^{(1)}\cos k\theta+\beta_k^{(1)}\sin k\theta\right) \tag{4} \]
\[ U_4(\theta)=\frac{\alpha_0^{(4)}}{2}+\sum_{k=1}^{\infty}\left(\alpha_k^{(4)}\cos k\theta+\beta_k^{(4)}\sin k\theta\right). \tag{5} \]
Equating the coefficients of the cosines and sines in equation (3), written for the first layer \((i=1)\), to the corresponding coefficients of the trigonometric series (4), and the coefficients of the cosines and sines in equation (3), written for the third layer, to the corresponding coefficients of the series (5), we obtain six equations.
| For \(r=r_1\) | For \(r=r_4\) |
|---|---|
| \(\displaystyle A_0^{(1)}\ln r_1+B_0^{(1)}=\frac{\alpha_0^{(1)}}{2},\) \(\tag{6}\) | \(\displaystyle A_0^{(3)}\ln r_4+B_0^{(3)}=\frac{\alpha_0^{(4)}}{2},\) \(\tag{9}\) |
| \(\displaystyle A_k^{(1)}r_1^k+B_k^{(1)}r_1^{-k}=\alpha_k^{(1)},\) \(\tag{7}\) | \(\displaystyle A_k^{(3)}r_4^k+B_k^{(3)}r_4^{-k}=\alpha_k^{(4)},\) \(\tag{10}\) |
| \(\displaystyle C_k^{(1)}r_1^k+D_k^{(1)}r_1^{-k}=\beta_k^{(1)}.\) \(\tag{8}\) | \(\displaystyle C_k^{(3)}r_4^k+D_k^{(3)}r_4^{-k}=\beta_k^{(4)}.\) \(\tag{11}\) |
The remaining twelve equations are obtained from the conditions of continuity of the potential and equality of the normal components of the electric displacement vector at the interfaces between the dielectrics.
For \(r=r_2\)
\[
\varphi^{(1)}\big|_{r=r_2}=\varphi^{(2)}\big|_{r=r_2},
\tag{12}
\]
\[
\varepsilon_1\frac{\partial\varphi^{(1)}}{\partial r}\bigg|_{r=r_2}
=
\varepsilon_2\frac{\partial\varphi^{(2)}}{\partial r}\bigg|_{r=r_2}
\tag{13}
\]
For \(r=r_3\)
\[
\varphi^{(2)}\big|_{r=r_3}=\varphi^{(3)}\big|_{r=r_3},
\tag{14}
\]
\[
\varepsilon_2\frac{\partial\varphi^{(2)}}{\partial r}\bigg|_{r=r_3}
=
\varepsilon_3\frac{\partial\varphi^{(3)}}{\partial r}\bigg|_{r=r_3},
\tag{15}
\]
Equating term by term the coefficients of the sines and cosines, as well as the nonharmonic components in equations (12)—(15), we obtain the remaining twelve equations for determining the constants. These latter equations are simple, and we do not give them here. Solving the resulting system and introducing the reduced permittivities
\[ \varepsilon_{12}=(\varepsilon_2+\varepsilon_1)-(\varepsilon_2-\varepsilon_1)\left(\frac{r_1}{r_2}\right)^{2k}, \]
\[ \varepsilon_{21}=(\varepsilon_2+\varepsilon_1)-(\varepsilon_2-\varepsilon_1)\left(\frac{r_2}{r_1}\right)^{2k}. \]
\[ \varepsilon_{34}=(\varepsilon_2+\varepsilon_3)-(\varepsilon_2-\varepsilon_3)\left(\frac{r_3}{r_4}\right)^{2k}, \]
\[ \varepsilon_{43}=(\varepsilon_2+\varepsilon_3)-(\varepsilon_2-\varepsilon_3)\left(\frac{r_4}{r_3}\right)^{2k}, \]
we find the following values of the constants for the cosines:
\[ A_k^{(1)}= \frac{ d_k^{(4)}r_4^k\,4\varepsilon_2\varepsilon_3 - \alpha_k^{(1)}r_1^k \left[ (\varepsilon_2+\varepsilon_1)\varepsilon_{43} + (\varepsilon_2-\varepsilon_1)\varepsilon_{34} \left(\frac{r_1}{r_2}\right)^{2k} \right] }{ r_4^{2k}\varepsilon_{12}\varepsilon_{34} - r_1^{2k}\varepsilon_{21}\varepsilon_{43} }, \]
\[ A_k^{(3)}= \frac{ \alpha_k^{(4)}r_4^k \left[ (\varepsilon_2+\varepsilon_3)\varepsilon_{12} + (\varepsilon_2-\varepsilon_3)\varepsilon_{21} \left(\frac{r_1}{r_3}\right)^{2k} \right] - \alpha_k^{(1)}r_1^k\,4\varepsilon_1\varepsilon_2 }{ r_4^{2k}\varepsilon_{12}\varepsilon_{34} - r_1^{2k}\varepsilon_{21}\varepsilon_{43} }. \]
The remaining constants for the cosines are expressed in terms of \(A_k^{(1)}\) and \(A_k^{(3)}\),
\[ B_k^{(1)}=\alpha_k^{(1)}r_1^k-A_k^{(1)}r_1^{2k}; \qquad B_k^{(3)}=\alpha_k^{(4)}r_4^k-A_k^{(3)}r_4^{2k}; \]
\[ A_k^{(2)}= \frac{r_2^{-2k}}{2\varepsilon_2} \left[ A_k^{(1)}r_2^{2k}(\varepsilon_2+\varepsilon_1) + B_k^{(1)}(\varepsilon_2-\varepsilon_1) \right]; \]
\[ B_k^{(2)}= \frac{1}{2\varepsilon_2} \left[ A_k^{(1)}r_2^{2k}(\varepsilon_2-\varepsilon_1) + B_k^{(1)}(\varepsilon_2+\varepsilon_1) \right]. \]
The constants for the nonharmonic components are written in the following form:
\[ A_0^{(1)}= \frac{\alpha_0^{(4)}-\alpha_0^{(1)}}{ 2\left[ \ln\frac{r_2}{r_1} + \frac{\varepsilon_1}{\varepsilon_3}\ln\frac{r_4}{r_3} + \frac{\varepsilon_1}{\varepsilon_2}\ln\frac{r_3}{r_2} \right] }; \]
\[ A_0^{(2)}=\frac{\varepsilon_1}{\varepsilon_2}A_0^{(1)}; \qquad A_0^{(3)}=\frac{\varepsilon_1}{\varepsilon_3}A_0^{(1)}. \qquad B_0^{(1)}=\frac{\alpha_0^{(1)}}{2}-A_0^{(1)}\ln r_1; \]
\[ B_0^{(2)} = A_0^{(1)} \left( \ln\frac{r_2}{r_1} - \frac{\varepsilon_1}{\varepsilon_2}\ln r_2 \right) + \frac{\alpha_0^{(1)}}{2}; \qquad B_0^{(3)} = \frac{\alpha_0^{(4)}}{2} - \frac{\varepsilon_1}{\varepsilon_2}A_0^{(1)}\ln r_4. \]
It is not advisable to write separately the constants for the sines \(C_k^{(i)}\) and \(D_k^{(i)}\), since they are obtained very simply from the corresponding expressions for the constants for the cosines \(A_k^{(i)}\) and \(B_k^{(i)}\). In order to write the formula for \(C_k^{(i)}\), it is enough, in the expression for the constant \(A_k^{(i)}\) (relating to the same \(i\)-th layer), to replace \(\alpha_k^{(1)}\) by \(\beta_k^{(1)}\), and \(\alpha_k^{(4)}\) by \(\beta_k^{(4)}\).
In a completely analogous way, the constants \(D_k^{(i)}\) are obtained from the constants \(B_k^{(i)}\).
For a two-layer dielectric, the potential in each layer is likewise determined by expression (3) \((i=1,2)\).
To determine the constants in this case, one may use the formulas written above, replacing \(r_4\) by \(r_3\) in all expressions and equating \(\varepsilon_3\) and \(\varepsilon_2\).
It should be noted that the rate of convergence of series (3), which determines the potential, depends mainly on the form of the curves specifying the potential distribution on the outer boundaries. The formulas obtained are sufficiently simple for calculating fields with the use of mathematical machines. They may also be used to estimate the accuracy of approximate calculation methods.
REFERENCES
-
A. V. Kontorovich, V. I. Krylov. Approximate Methods of Higher Analysis. Gostekhizdat, 1949.
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V. I. Smirnov. A Course of Higher Mathematics. Moscow, vol. 11, 1953.