General analytical solutions for DC/AC circuit network analysis

TL;DR

This paper introduces a general analytical method for calculating currents in complex DC/AC networks using eigenvalues of the Laplacian matrix, simplifying analysis of current redistribution and impedance.

Contribution

It provides a novel eigenvalue-based analytical solution for network currents, enabling easier computation of impedance and current redistribution compared to traditional Kirchhoff's methods.

Findings

Derived current solutions in terms of Laplacian eigenvalues and eigenvectors.

Enabled calculation of equivalent impedance between nodes.

Showed solutions match traditional circuit analysis results.

Abstract

In this work, we present novel general analytical solutions for the currents that are developed in the edges of network-like circuits when some nodes of the network act as sources/sinks of DC or AC current. We assume that Ohm's law is valid at every edge and that charge at every node is conserved (with the exception of the source/sink nodes). The resistive, capacitive, and/or inductive properties of the lines in the circuit define a complex network structure with given impedances for each edge. Our solution for the currents at each edge is derived in terms of the eigenvalues and eigenvectors of the Laplacian matrix of the network defined from the impedances. This derivation also allows us to compute the equivalent impedance between any two nodes of the circuit and relate it to currents in a closed circuit which has a single voltage generator instead of many input/output source/sink nodes. Contrary to solving Kirchhoff's equations, our derivation allows to easily calculate the redistribution of currents that occurs when the location of sources and sinks changes within the network. Finally, we show that our solutions are identical to the ones found from Circuit Theory node analysis.