Approximate Calculation of the Capacitance of Some Centered Systems
Abstract Generated abstract
This paper proposes an approximate method for calculating the capacitance of a centered electrode system consisting of a long rectangular busbar inside a circular cylindrical outer electrode, where exact analytical solutions are difficult or unavailable. The approach replaces the rectangular inner conductor with an equivalent elliptic cylinder whose perimeter matches that of the rectangle and whose semiaxis ratio matches the rectangle side ratio, then applies the known capacitance formula for a confocal elliptic capacitor with a circular outer electrode approximation. The authors derive a compact capacitance expression, including a simplified form for a square inner conductor. Comparison with electrolytic bath measurements indicates agreement within about 7 percent for the stated geometric range, with decreasing error as the inner conductor size becomes smaller relative to the outer radius.
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APPROXIMATE CALCULATION OF THE CAPACITANCE OF SOME CENTERED SYSTEMS
V. A. Lukutin, V. S. Sokolov, P. P. Galinskii
(Presented by the scientific seminar of the Department of Theoretical Fundamentals of Electrical Engineering)
An exact analytical calculation of the capacitance and conductance of centered systems consisting of two electrodes is possible for a very limited number of cases, for example:
1) a spherical capacitor,
2) a capacitor formed by confocal ellipsoids of revolution,
3) a cylindrical capacitor,
4) a capacitor formed by confocal elliptic cylinders, etc.
However, in practice one encounters a number of other centered systems for which an exact analytical solution is absent or is associated with great mathematical difficulties.
In designing electrical-engineering devices and in preliminary calculations, the creation of a model for the direct measurement of capacitance (conductance), or modeling in an electric bath, is in a number of cases impractical, since it involves a large expenditure of labor and time. At the same time, high accuracy at this stage of design may not be required. For this purpose it is desirable to have approximate formulas for electrode configurations frequently encountered in practice.
Below we consider one of the centered systems, consisting of a busbar of rectangular cross section \((2a \times 2b)\), placed inside a circular cylinder of radius \(R\).
We take the length of the system \(l\) to be sufficiently large in comparison with the radius of the outer cylinder \(R\), so that we neglect the edge effect and regard the field as plane-parallel.
It is obvious that for each system of electrodes, and in particular for the centered cylinders in Fig. 1, one can select such cylindrical, elliptic, plane, and other capacitors whose capacitance is equal to the capacitance of the system under consideration.
When constructing sketch field diagrams it was found that the shape of the cross section of the inner electrode has only a slight effect on the capacitance of the system if the perimeter of its cross section remains unchanged. This feature is especially clearly manifested if \(R \ge 3R_1\), where \(R_1 = \sqrt{A^2 + B^2}\) is the radius of the circle into which the cross section of the inner electrode is inscribed.
As the equivalent capacitor to which the given system is reduced, a confocal elliptic capacitor was adopted.
Its capacitance is determined by formula [1]
\[ C=\frac{2\pi\varepsilon_a}{\ln\frac{a_2+b_2}{a_1+b_1}}, \tag{1} \]
where \(a_1, b_1\) and \(a_2, b_2\) are the semiaxes of the cross sections of the inner and outer elliptical electrodes.
For \(a_2 \geqslant 3a_1\), the shape of the cross section of the outer elliptical electrode is very close to a circle.
For \(b_1=0\), the cross section of the inner electrode degenerates into a line of length \(2a_1\).
In this most unfavorable case, the deviation of the shape of the cross section of the outer electrode from a circle, determined by the expression
\[ \delta=\frac{a_2-b_2}{a_2}\cdot 100\%, \]
is about \(6\%\).
As \(b_1\) is increased from \(0\) to \(a_1\), the deviation \(\delta\) decreases.
Fig. 1.
Fig. 2.
Thus, in all subsequent calculations we take the outer electrode to be a circular cylinder, that is, \(a_2=b_2=R\).
We replace the inner electrode of rectangular cross section \((2a\times 2b)\) by an elliptic cylinder with semiaxes \(a_1\) and \(b_1\), the perimeter of whose cross section is equal to the perimeter of the rectangle, and the ratio of whose semiaxes is equal to the ratio of the sides of the rectangle.
We obtain the equations
\[ 4(a+b)=L_{\mathrm{e}}; \tag{2} \]
\[ \frac{a}{b}=\frac{a_1}{b_1}, \tag{3} \]
where \(L_{\mathrm{e}}\) is the perimeter of the ellipse, determined by the approximate formula
\[ L_{\mathrm{e}}=\pi\cdot\left[1.5\cdot(a_1+b_1)-\sqrt{a_1\cdot b_1}\right]. \tag{4} \]
Substituting the expression for \(L_{\mathrm{e}}\) into equation (2) and solving the system with respect to the required semiaxes \(a_1\) and \(b_1\), we obtain,
\[ a_1=\frac{4(a+b)}{\pi[1.5(1+\lambda)-\sqrt{\lambda}]}; \]
\[ b_1=\frac{4(a+b)\cdot\lambda}{\pi[1.5(1+\lambda)-\sqrt{\lambda}]}, \]
where
\[ \lambda=\frac{b}{a}. \]
Substituting the values of \(a_1\) and \(b_1\) into formula (1), we obtain an approximate expression for the capacitance of the cylinders, Fig. 1, for the specified dimensional ratios \((R\geq 3a)\):
\[ C\approx \frac{2\pi\varepsilon_a} {\ln \frac{\pi R[1.5(1+\lambda)-\sqrt{\lambda}]}{2(a+b)(1+\lambda)}} = \frac{2\pi\varepsilon_a} {\ln\left[\frac{\pi R}{2(a+b)}\left(1.5-\frac{\sqrt{\lambda}}{1+\lambda}\right)\right]}. \tag{5} \]
If \(b=a\), \(\lambda=1\), that is, the inner electrode is a square with side \(2a\), then the capacitance formula is considerably simplified:
\[ C=\frac{2\pi\varepsilon_a}{\ln\frac{\pi\cdot R}{4a}}. \]
Experimental verification of the calculation results in an electrolytic bath showed sufficiently good agreement between the experimental and calculated data.
For \(\sqrt{a^2+b^2}\leq 0.9R\), the maximum relative error did not exceed \(7\%\); moreover, as the ratio \(\frac{\sqrt{a^2+b^2}}{R}\) decreases, the relative error decreases.
REFERENCES
- G. Buchgolts, Calculation of Electric and Magnetic Fields. Moscow, IL, 1961.
- I. N. Bronshtein and K. A. Semendyaev, Handbook of Mathematics. Moscow, State Publishing House of Technical-Theoretical Literature, 1953.