Vol. 97

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues

The Capacitance of the Circular Parallel Plate Capacitor Obtained by Solving the Love Integral Equation Using an Analytic Expansion of the Kernel

By Martin Karl Norgren and Lars Jonsson
Progress In Electromagnetics Research, Vol. 97, 357-372, 2009


The capacitance of the circular parallel plate capacitor is calculated by expanding the solution to the Love integral equation into a Fourier cosine series. Previously, this kind of expansion has been carried out numerically, resulting in accuracy problems at small plate separations. We show that this bottleneck can be alleviated, by calculating all expansion integrals analytically in terms of the Sine and Cosine integrals. Hence, we can, in the approximation of the kernel, use considerably larger matrices, resulting in improved numerical accuracy for the capacitance. In order to improve the accuracy at the smallest separations, we develop a heuristic extrapolation scheme that takes into account the convergence properties of the algorithm. Our results are compared with other numerical results from the literature and with the Kirchhoff result. Error estimates are presented, from which we conclude that our results is a substantial improvement compared with earlier numerical results.


Martin Karl Norgren and Lars Jonsson, "The Capacitance of the Circular Parallel Plate Capacitor Obtained by Solving the Love Integral Equation Using an Analytic Expansion of the Kernel," Progress In Electromagnetics Research, Vol. 97, 357-372, 2009.


    1. Love, E. R., "The electrostatic field of two equal circular co-axial conducting disks," Quart. J. Mech. and Appl. Math., Vol. 2, 428-451, 1949.

    2. Wintle, H. J. and S. Kurylowicz, "IEEE Trans. Instr. and Meas.," Edge corrections for strip and disc capacitors, Vol. 34, No. 1, 41-47, March 1995.

    3. Carlson, G. T. and B. L. Illman, "The circular disk parallel plate capacitor," Am. J. Phys., Vol. 62, No. 12, 1099-1105, December 1994.

    4. Nishiyama, H. and M. Nakamura, "Capacitance of disk capacitors," IEEE Trans. Comp., Hybrids., and Manuf., Vol. 16, No. 3, 360-366, 1993.

    5. Donolato, C., "Approximate evaluation of capacitances by means of Green's reciprocal theorem," Am. J. Phys., Vol. 64, No. 8, 1049-1054, August 1996.

    6. Hwang, C. O. and J. A. Given, "Last-passage monte carlo algorithm for mutual capacitance," Phys. Rev. E, Vol. 74, 027701, 2006.

    7. El-Gendi, S. E., "Chebyshev solution of differential equations," Comput. J., Vol. 12, 282-287, 1969.

    8., http://en.wikipedia.org/wiki/Clenshaw-Curtis quadrature..

    9. Carlson, G. T. and B. L. Illman, "Series capacitors and the inverse sum rule," Am. J. Phys., Vol. 70, No. 11, 1122-1128, November 2002.

    10. Kirchhoff, G., "Zur theorie des kondensators," Monatsb. Acad. Wiss. Berlin, 731-734, 1877.

    11. Hutson, V., "The circular plate condenser at small separations," Proc. Camb. Phil. Soc., Vol. 59, 221-224, 1963.

    12. Sneddon, I. N., Mixed Boundary Value Problems in Potential Theory, Wiley, New York, 1966.

    13. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1964.

    14. Gradshteyn, I. S. and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980.

    15. Ignatowsky, W., "Kreisscheibenkondensator," Acad. Sci. URSS Trav. Inst. Steklov, Series 2, Vol. 3, 1-104, 1932.

    16. Polya, G. and G. Szego, Isoperimetric Inequalities in Mathematical Physics, Princeton, 1951.