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.
Statement Consider an arbitrary lumped network that has b {\displaystyle b} branches and n {\displaystyle n} nodes. In an electrical network, the branches are two-terminal components and the nodes are points of interconnection. Suppose that to each branch we assign arbitrarily a branch potential difference W k {\displaystyle W_{k}} and a branch current F k {\displaystyle F_{k}} for k = 1 , 2 , … , b {\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 W 1 , W 2 , … , W b {\displaystyle W_{1},W_{2},\dots ,W_{b}} satisfy all the constraints imposed by KVL and if the branch currents F 1 , F 2 , … , F b {\displaystyle F_{1},F_{2},\dots ,F_{b}} satisfy all the constraints imposed by KCL, then
∑ k = 1 b W k F k = 0. {\displaystyle \sum _{k=1}^{b}W_{k}F_{k}=0.}
Tellegen's theorem is extremely general; it is valid for any lumped network that contains any elements, linear or nonlinear, passive or active, time-varying or time-invariant. The generality is extended when W k {\displaystyle W_{k}} and F k {\displaystyle F_{k}} are linear operations on the set of potential differences and on the set of branch currents (respectively) since linear operations don't affect KVL and KCL. For instance, the linear operation may be the average or the Laplace transform. More generally, operators that preserve KVL are called Kirchhoff voltage operators, operators that preserve KCL are called Kirchhoff current operators, and operators that preserve both are simply called Kirchhoff operators. These operators need not necessarily be linear for Tellegen's theorem to hold. The set of currents can also be sampled at a different time from the set of potential differences since KVL and KCL are true at all instants of time. Another extension is when the set of potential differences W k {\displaystyle W_{k}} is from one network and the set of currents F k {\displaystyle F_{k}} is from an entirely different network, so long as the two networks have the same topology (same incidence matrix) Tellegen's theorem remains true. This extension of Tellegen's Theorem leads to many theorems relating to two-port networks.
Definitions We need to introduce a few necessary network definitions to provide a compact proof. Incidence matrix: The n × b {\displaystyle n\times b} matrix A a {\displaystyle \mathbf {A_{a}} } is called node-to-branch incidence matrix for the matrix elements a i j {\displaystyle a_{ij}} being
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