General analytical solutions for DC/AC circuit-network analysis

Abstract

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. This simplifies the treatment that could be done via Thévenin's theorem. 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 nodal analysis.

Calculating Nodal Voltages Using the Admittance Matrix Spectrum of an Electrical Network

“…We will refer to G as the Laplacian matrix. Note that the rows of G sum to zero, i.e., the matrix has the zero eigenvalue (see [3,14]). The algebraic multiplicity of the zero eigenvalue in the Laplacian is the number of connected components in the network.…”

“…The purpose of the present paper is to clarify the derivations provided in [3]. The scope of the present work is narrowly theoretical: Linear algebra is used to articulate the correct relationship between the variables treated in [3].…”

“…It is well established that the eigenvalues and eigenvectors (deemed the spectrum) of a Laplacian matrix encode meaningful information about a network's structure [1]. Recent work in [2,3] indicates that, in electrical networks, this spectrum can be directly related to nodal voltages and branch current flows. The purpose of the present paper is to clarify the derivations provided in [3].…”

“…Recent work in [2,3] indicates that, in electrical networks, this spectrum can be directly related to nodal voltages and branch current flows. The purpose of the present paper is to clarify the derivations provided in [3]. The scope of the present work is narrowly theoretical: Linear algebra is used to articulate the correct relationship between the variables treated in [3].…”

Festschrift on the occasion of Ulrike Feudel’s 60th birthday

“…In this way they find coexistence of in-phase and anti-phase limit cycles, noise-induced transitions between these states, and the influence of high period limit cycles on the complex dynamics. In [13] Rubido et al present general solutions for current conservative DC/AC circuit networks with resistive, capacitive, and/or inductive edge characteristics, proposing an alternative to Kirchhoff's equations that is advantageous for constantly changing locations of inputs and outputs. Their novel approach provides a rigorous link between network topology and the steady-state currents of a conservative circuit network.…”