A.A. Ershov. Contact resistance of a square contact ... P. 105-113.

We consider a conductive body in the form of a parallelepiped with small square contacts attached to its ends. The potential of the electric current is modelled by a boundary value problem for the Laplace equation in a parallelepiped. The zero normal derivative is assigned on the boundary except for the areas under the contacts, where the derivative is a nonzero constant. Physically, this condition corresponds to the presence of a low-conductivity film on the surface of the contacts. The problem is solved by separation of variables, and then the electrical resistance is found as a functional of the solution in the form of the sum of a double series. Our main aim is to study the dependence of the resistance on a small parameter characterizing the size of the contacts. The leading term of the asymptotics that expresses this dependence is the contact resistance. The mathematical problem is to treat the singular dependence of the sum of the series corresponding to the resistance on the small parameter: the series diverges as the small parameter vanishes. We solve this problem by replacing the series with a two-dimensional integral. We find the leading term of the asymptotics and estimate the remainder. It turns out that the main contribution to the remainder is made by the difference between the two-dimensional integral and the double sum.

Keywords: contact resistance, boundary value problem, electric potential, Laplace equation, small parameter.

The paper was received by the Editorial Office on February, 31, 2017}

Aleksandr Anatol'evich Ershov, Cand. Phys.-Math. Sci., Krasovskii Institute of Mathe\-matics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Chelyabinsk State University, Chelya\-binsk,  454001 Russia, e-mail: ale10919@yandex.ru 


1.   Holm R. ber Kontaktwiderstnde, besonders bei Kohlekontakten. Zeitschrift fr technische Physik, 1922, vol. 3, no. 9, pp. 290–294; no. 10, pp. 320–327; no. 11, pp. 349–357.

2.   Holm R. Strmer R. Eine Kontrolle des metallischen Charakters von gereinigten Platinkontakten. Wissenschaftliche Verffentlichungen aus dem Siemens-Konzern, 1930, Band 9, heft 2, pp. 323–330.

3.   Pavleino O.M., Pavlov V.A., Pavleino M.A. Verification of the boundaries of the applicability of the holm approximation for the calculation of the resistance of electric contacts. Surf. Engin. Appl.Electrochem., 2010, vol. 46, no. 5, pp. 440–446. doi: 10.3103/S1068375510050078 .

4.   Zatovsky V.G., Minakov N.V. Experimental modeling of the resistance of the retraction. Elektricheskie kontaktyi i elektrodyi, 2010, no. 10, pp. 132–139 (in Russian).

5.   Holm R. Electric contacts handbook. Berlin: Springer-Verlag, 1958, 527 p. Translated under the title Elektricheskie kontakty, Moscow, Inostrannaya literatura Publ., 1961, 314 p.

6.   Silvester P.P., Ferrari R.L. Finite elements for electrical engineers. New York: Cambridge University Press, 1983, 512 p. Translated under the title Metod konechnykh elementov dlya radioinzhenerov iinzhenerov-elektrikov, Moscow, Mir Publ., 1986, 229 p.

7.   Gadyl’shin R.R., Ershov A.A., Repyevsky S.V. On asymptotic formula for electric resistance of condactor with small contacts. Ufa Math. J., 2015, vol. 7, no. 3, pp. 15–27. doi: 10.13108/2015-7-3-15 .

8.   PДolya G., SzegЈo G. Isoperimetric inequalities in mathematical physics, Princeton: Princeton Univ. Press, 1951, 279 p. Translated under the title Izoperimetricheskie neravenstva v matematicheskoi fizike, Moscow, Gos. Izd-vo Fiz.-Mat. Lit. Publ., 1962, 336 p.

9.   Landkof N.S. Foundations of Modern Potential Theory, Berlin, Springer, 1973, 424 p. Original Russian text published in Osnovy sovremennoi teorii potentsiala, Moscow, Nauka Publ., 1966, 515 p.

10.   Landau L.D., Lifshitz E.M. Course of Theoretical Physics. Vol. 8: Electrodynamics of Continuous Media, 1rst ed., Oxford: Butterworth-Heinemann, 1984, 460 p. ISBN 978-0-7506-2634-7.

11.   Ershov A.A. Asymptotics of the solution to the Neumann problem with a delta-function-like boundary function. Comp. Math. Math. Phys., 2010, vol. 50, no. 3, pp. 457–463. doi:10.1134/S0965542510030073 .

12.   Ershov A.A. Asymptotics of the solution of Laplace’s equation with mixed boundary conditions. Comp. Math. Math. Phys., 2011, vol. 51, no. 7, pp. 994–1010. doi: 10.1134/S0965542511060066 .

13.   Ershov A.A. On measurement of electrical conductivity. Comp. Math. Math. Phys., 2013, vol. 53, no. 6, pp. 823–826. doi: 10.1134/S0965542513060079 .

14.   Bateman H., Erdelyi A. Higher transcendental functions: vol. 2. New York: McGraw-Hill Book Company Inc., 1953, 414 p.