Tellegen's theorem

Tellegen's theorem is one of the most powerful theorems in network theory. Most of the energy distribution theorems and extremum principles in network theory can be derived from it. It was published in 1952 by Bernard Tellegen. Fundamentally, Tellegen's theorem gives a simple relation between magnitudes that satisfy Kirchhoff's laws of electrical circuit theory.

The Tellegen theorem is applicable to a multitude of network systems. The basic assumptions for the systems are the conservation of flow of extensive quantities (Kirchhoff's current law, KCL) and the uniqueness of the potentials at the network nodes (Kirchhoff's voltage law, KVL). The Tellegen theorem provides a useful tool to analyze complex network systems including electrical circuits, biological and metabolic networks, pipeline transport networks, and chemical process networks.

Consider an arbitrary lumped network whose graph ${\displaystyle G}$ has ${\displaystyle b}$ branches and ${\displaystyle n_{t}}$ nodes. In an electrical network, the branches are two-terminal components and the nodes are points of interconnection. Suppose that to each branch of the graph we assign arbitrarily a branch potential difference ${\displaystyle W_{k}}$ and a branch current ${\displaystyle F_{k}}$ for ${\displaystyle k=1,2,\dots ,b}$, and suppose that they are measured with respect to arbitrarily picked associated reference directions. If the branch potential differences ${\displaystyle W_{1},W_{2},\dots ,W_{b}}$ satisfy all the constraints imposed by KVL and if the branch currents ${\displaystyle F_{1},F_{2},\dots ,F_{b}}$ satisfy all the constraints imposed by KCL, then

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