Operating points in infinite nonlinear networks approximated by finite networks

Calvert, Bruce; Zemanian, Armen

doi:10.1090/S0002-9947-99-02228-X

Operating points in infinite nonlinear networks approximated by finite networks
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by Bruce D. Calvert and Armen H. Zemanian

Trans. Amer. Math. Soc. 352 (2000), 753-780

DOI: https://doi.org/10.1090/S0002-9947-99-02228-X

Published electronically: October 6, 1999

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Abstract:

Given a nonlinear infinite resistive network, an operating point can be determined by approximating the network by finite networks obtained by shorting together various infinite sets of nodes, and then taking a limit of the nodal potential functions of the finite networks. Initially, by taking a completion of the node set of the infinite network under a metric given by the resistances, limit points are obtained that represent generalized ends, which we call “terminals,” of the infinite network. These terminals can be shorted together to obtain a generalized kind of node, a special case of a 1-node. An operating point will involve Kirchhoff’s current law holding at 1-nodes, and so the flow of current into these terminals is studied. We give existence and bounds for an operating point that also has a nodal potential function, which is continuous at the 1-nodes. The existence is derived from the said approximations.

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