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Tensor-geometric methods for problems of the circuit theory

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Abstract

It is proved that to any electric circuit, there correspond two pairs of conjugate linear vector spaces. One of these pairs is generated by a homology group, while the other is generated by a cohomology group. A new method of analysis of mechanical and electric circuits is proposed, which consists of representing the main variables and matrices of oscillatory circuits in terms of multidimensional tensor objects. A solution for the problem of defining eigenvalues of pure-loop and pure-node circuits is obtained. A new method is developed for defining a full range of eigenvalues of linear oscillatory systems with a great number of degrees of freedom.

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  1. Georgian Technical University, 77, Kostava St., Tbilisi, 0175, Georgia

    A. Mylnikov

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.

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Mylnikov, A. Tensor-geometric methods for problems of the circuit theory. J Math Sci 148, 192–258 (2008). https://doi.org/10.1007/s10958-008-0004-5

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