Theory of impedance networks: The two-point impedance and LC resonances

TL;DR

This paper develops a mathematical framework to compute the impedance between any two nodes in an impedance network, revealing resonance conditions in LC circuits and potential practical applications.

Contribution

It introduces a novel formulation using the Laplacian matrix to determine two-point impedance and analyze LC resonances in complex networks.

Findings

Impedance between nodes can be expressed via Laplacian eigenvalues and eigenvectors.

Resonances occur at frequencies where eigenvalues vanish in LC networks.

The approach is demonstrated with explicit network examples.

Abstract

We present a formulation of the determination of the impedance between any two nodes in an impedance network. An impedance network is described by its Laplacian matrix L which has generally complex matrix elements. We show that by solving the equation L u_a = lambda_a u_a^* with orthonormal vectors u_a, the effective impedance between nodes p and q of the network is Z = Sum_a [u_{a,p} - u_{a,q}]^2/lambda_a where the summation is over all lambda_a not identically equal to zero and u_{a,p} is the p-th component of u_a. For networks consisting of inductances (L) and capacitances (C), the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of lambda_a. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples.