Operating Points in Infinite Nonlinear Networks Approximated by Finite Networks |
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Authors: | Bruce D. Calvert Armen H. Zemanian |
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Affiliation: | Department of Mathematics, University of Aukland, Aukland, New Zealand ; Electrical Engineering Department, SUNY at Stony Brook, Stony Brook, New York 11794--2350 |
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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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Keywords: | Infinite resistive networks ends metrizing infinite networks monotone networks 1-nodes Kirchhoff's voltage law Kirchhoff's current law |
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