Abstract
In this paper, we study the properties of effective impedances of finite electrical networks, considering them as weighted graphs over an ordered field. We prove that a star-mesh transform of finite network does not change its effective impedance. Moreover, we consider two particular examples of infinite ladder networks (Feynman’s network or LC-network and CL-network, both with zero at infinity) as networks over the ordered Levi–Cività field R. We show that the sequence of effective impedances of finite LC-networks converges to the limit in the order topology of R, but the sequence of effective impedances of finite CL-networks does not converge in the same topology. We calculate an effective impedance of a finite ladder network as an auxiliary result.
Highlights
It is known that electrical networks with resistances are related to weighted graphs, where weights of edges are positive real numbers
An effective resistance is tightly related to the random walk and Dirichlet problem on a graph, which are described in many papers and books (e.g., Refs. 2, 13, 22, and 26)
We present a new application of the Levi–Cività fields whose other applications to mathematical physics are described in Refs. 3 and 4
Summary
It is known that electrical networks with resistances are related to weighted graphs, where weights of edges are positive real numbers (see, e.g., Refs. 9, 19, and 22). It shows that a sequence of effective admittances of finite networks, exhausted a given infinite network, decreases It does not give a convergence over a non-Archimedean field. We show that in the case of the LC-network, the sequence of effective admittances of finite networks converges in ordered topology of the Levi–Cività field (Theorem 20). The latest example shows that, in general, it is not possible to generalize the notion of effective resistance for infinite networks for the case of non-Archimedean weights, or, more precisely, in the case of R -weights, there is some other possibility for a network, apart from being transient or recurrent This again shows that computations over the Levi–Cività field are more precise than over the real numbers. We give a sufficient condition on an infinite ladder network for convergence of the sequence of admittances (Remark 22)
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