Most of the energy distribution theorems and extremum principles in network theory can be derived from it.
[1] 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).
In an electrical network, the branches are two-terminal components and the nodes are points of interconnection.
, and suppose that they are measured with respect to arbitrarily picked associated reference directions.
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.
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.
These operators need not necessarily be linear for Tellegen's theorem to hold.
[2] 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.
[3] We need to introduce a few necessary network definitions to provide a compact proof.
is introduced to represent the environment and connected to all dynamic nodes and terminals.
is eliminated, is called reduced incidence matrix.
The conservation laws (KCL) in vector-matrix form: The uniqueness condition for the potentials (KVL) in vector-matrix form: where
So: Network analogs have been constructed for a wide variety of physical systems, and have proven extremely useful in analyzing their dynamic behavior.
It is mainly in use to design filters in signal processing applications.
A more recent application of Tellegen's theorem is in the area of chemical and biological processes.
The assumptions for electrical circuits (Kirchhoff laws) are generalized for dynamic systems obeying the laws of irreversible thermodynamics.
Another application of Tellegen's theorem is to determine stability and optimality of complex process systems such as chemical plants or oil production systems.
The Tellegen theorem can be formulated for process systems using process nodes, terminals, flow connections and allowing sinks and sources for production or destruction of extensive quantities.
A formulation for Tellegen's theorem of process systems: where
are the dynamic storage terms for the extensive variables.