Contact resistance of a square contact

A. A. Ershov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 105-113

Résumé

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.

Détail

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

Bibliographie

[1] Holm R., “Über Kontaktwiderstände, besonders bei Kohlekontakten”, Zeitschrift für technische Physik, 3:9 (1922), 290–294; 10, 320–327; 11, 349–357

[2] Holm R., Störmer R., “Eine Kontrolle des metallischen Charakters von gereinigten Platinkontakten”, Wissenschaftliche Verö ffentlichungen aus dem Siemens-Konzern, 9:2 (1930), 323–330

[3] Pavleino O.M., Pavlov V.A., Pavleino M.A., “Utochnenie granits primenimosti kholmovskogo priblizheniya dlya rascheta soprotivleniya elektricheskikh kontaktov”, Elektronnaya obrabotka materialov, 46:5 (2010), 56–62

[4] Zatovskii V.G., Minakov N.V., “Eksperimentalnoe modelirovanie soprotivleniya styagivaniya”, Elektricheskie kontakty i elektrody, 2010, no. 10, 132–139

[5] Kholm R., Elektricheskie kontakty, Inostrannaya literatura, M., 1961, 314 pp.

[6] Silvester P., Ferrari R., Metod konechnykh elementov dlya radioinzhenerov i inzhenerov-elektrikov, Mir, M., 1986, 229 pp.

[7] Gadylshin R.R., Ershov A.A., Repevskii S.V., “Ob asimptoticheskoi formule dlya elektricheskogo soprotivleniya v provodnike s malymi kontaktami”, Ufim. mat. zhurn., 7:3 (2015), 16–28

[8] Polia G., Segë G., Izoperimetricheskie neravenstva v matematicheskoi fizike, Gos. izd-vo fiz.-mat. lit., M., 1962, 336 pp. | MR

[9] Landkof N.S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966, 515 pp. | MR

[10] Landau L.D., Lifshits E.M., Teoreticheskaya fizika, (v 10 t), v. 8, Elektrodinamika sploshnykh sred, Fizmatlit, M., 2005, 656 pp.

[11] Ershov A.A., “Asimptotika resheniya zadachi Neimana s deltoobraznoi granichnoi funktsiei”, Zhurn. vychisl. matematiki i mat. fiziki, 50:3 (2010), 479–485 | MR | Zbl

[12] Ershov A.A., “Asimptotika resheniya uravneniya Laplasa so smeshannymi usloviyami na granitse”, Zhurn. vychisl. matematiki i mat. fiziki, 51:7 (2011), 1064–1080 | Zbl

[13] Ershov A.A., “K zadache ob izmerenii elektroprovodnosti”, Zhurn. vychisl. matematiki i mat. fiziki, 53:6 (2013), 1004–1007 | DOI | MR | Zbl

[14] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 2, Nauka, M., 1974, 296 pp. | MR