On the effective impedance of finite and infinite networks

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.