Electrical network

Builds on Graph Discrete harmonic function

Definition

An electrical network is a pair $(G, c)$ of a connected graph $G = (V, E)$ , in which every vertex has finitely many neighbours, and a function $c \colon E \to (0, \infty)$ . The number $c(e)$ is the conductance of the edge $e$ , and $r(e) = 1/c(e)$ is its resistance. For $x, y \in V$ , write $c(x, y) = c(\set{x, y})$ if $\set{x, y} \in E$ and $c(x, y) = 0$ otherwise. A voltage is a function $v \colon V \to \R$ . The current of $v$ is the function on ordered pairs of neighbours given by Ohm's law: $i(x, y) = c(x, y)\big(v(x) - v(y)\big)$ .

Remarks

The current is antisymmetric: $i(y, x) = -i(x, y)$ . The net current out of $x$ is $\sum_y i(x, y) = \sum_y c(x, y)\big(v(x) - v(y)\big)$ . Kirchhoff's node law at $x$ says that this net current is $0$ , which holds exactly when $v$ is harmonic at $x$ . Example: the path $a - m - b$ with two edges of conductance $1$ is a network. The voltage with $v(a) = 1$ , $v(m) = \tfrac12$ , $v(b) = 0$ is harmonic at $m$ , and it sends the current $\tfrac12$ along each edge from $a$ to $b$ . Non-example: giving an edge conductance $0$ is not allowed; such an edge carries no current and should be removed from the graph instead. Kirchhoff's cycle law, that voltage drops add to $0$ around each cycle, is automatic here because the current comes from a voltage.

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Tag	`PHTRQKMAC8`
URI	`m2://PHTRQKMAC8`
Kind	definition
Writing	W0 stub
MSC	31C20, 94C05, 05C81
Defines	Electrical network
Uses	Graph, Discrete harmonic function

Electrical network

Definition

Remarks

Reading path

Further reading

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