On the effective impedance of finite and infinite networks

TL;DR

This paper investigates the effective impedance of AC networks modeled as graphs, establishing estimates for finite networks and demonstrating convergence of impedance sequences to a holomorphic function for infinite networks.

Contribution

It introduces a method to define the effective impedance of infinite networks via convergence of finite network impedances, extending classical results to complex-valued weights.

Findings

Finite network impedances are effectively estimated.

Impedance sequences converge to a holomorphic function.

A framework for defining infinite network impedance is established.

Abstract

In this paper we deal with the notion of the effective impedance of AC networks consisting of resistances, coils and capacitors. Mathematically such a network is a locally finite graph whose edges are endowed with complex-valued weights depending on a complex parameter $\lambda $ (by the physical meaning, $\lambda =i\omega $, where $\omega $ is the frequency of the AC). For finite networks, we prove some estimates of the effective impedance. Using these estimates, we show that, for infinite networks, the sequence of impedances of finite graph approximations converges in certain domains in $\mathbb{C}$ to a holomorphic function of $\lambda$, which allows us to define the effective impedance of the infinite network.