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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Operating points in infinite nonlinear networks approximated by finite networks

Authors: Bruce D. Calvert and Armen H. Zemanian
Journal: Trans. Amer. Math. Soc. 352 (2000), 753-780
MSC (1991): Primary 31C20, 94C05
Published electronically: October 6, 1999
MathSciNet review: 1487608
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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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Additional Information

Bruce D. Calvert
Affiliation: Department of Mathematics, University of Aukland, Aukland, New Zealand

Armen H. Zemanian
Affiliation: Electrical Engineering Department, SUNY at Stony Brook, Stony Brook, New York 11794–2350

Keywords: Infinite resistive networks, ends, metrizing infinite networks, monotone networks, 1-nodes, Kirchhoff’s voltage law, Kirchhoff’s current law
Received by editor(s): August 5, 1996
Received by editor(s) in revised form: October 17, 1997
Published electronically: October 6, 1999
Additional Notes: This work was partially supported by the National Science Foundation under Grants DMS-9200738 and MIP-9423732.
Article copyright: © Copyright 1999 American Mathematical Society