Tellegen's Theorem - The Theorem

The Theorem

Consider an arbitrary lumped network whose graph has branches and 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 and a branch current for, and suppose that they are measured with respect to arbitrarily picked associated reference directions. If the branch potential differences satisfy all the constraints imposed by KVL and if the branch currents satisfy all the constraints imposed by KCL, then

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 and 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. 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 is from one network and the set of currents 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.