# General analytical solutions for DC/AC circuit-network analysis

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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.
Original language English 1829-1844 16 The European Physical Journal. Special Topics 226 9 21 Jun 2017 https://doi.org/10.1140/epjst/e2017-70074-2 Published - Jun 2017

### Fingerprint

network analysis
Electric network analysis
alternating current
sinks
direct current
Networks (circuits)
impedance
derivation
Circuit theory
voltage generators
Complex networks
Ohms law
Eigenvalues and eigenfunctions
eigenvectors
eigenvalues
theorems
Electric potential
output
matrices

### Cite this

In: The European Physical Journal. Special Topics, Vol. 226, No. 9, 06.2017, p. 1829-1844.

Research output: Contribution to journalArticle

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title = "General analytical solutions for DC/AC circuit-network analysis",
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 Th{\'e}venin’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.",
author = "Nicolas Rubido and Celso Grebogi and Baptista, {Murilo S.}",
note = "All authors thank the Scottish University Physics Alliance (SUPA) support. NR also acknowledges de support of PEDECIBA, Uruguay. MSB acknowledges the support of EPSRC grant Ref. EP/I032606/1. Open access via Springer Compact Agreement.",
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AB - 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.

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