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 the 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 among them electrical circuits, biological and metabolic networks, pipeline flow networks, and chemical process networks.

The theorem

Consider an arbitrary lumped network whose graph $G$ has $b$ branches and $n\_\{t\}$ nodes. Suppose that to each branch of the graph we assign arbitrarily a branch potential difference $W\_\{k\}$ and a branch current $F\_\{k\}$ for $k=1,2,dots,b$, and suppose that they are measured with respect to arbitrarily picked "associated" reference directions. If the branch potential differences $W\_\{1\},W\_\{2\},dots,W\_\{b\}$ satisfy all the constraints imposed by KVL and if the branch currents $F\_\{1\},F\_\{2\},dots,F\_\{b\}$ satisfy all the contraints imposed by KCL, then

: $sum\_\{k=1\}^\{b\}\; W\_\{k\}\; F\_\{k\}\; =\; 0.$

The Tellegen 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 follows from the fact that Tellegen's theorem depends only on the two Kirchhoff laws.

Definitions

We need to introduce a few necessary network definitions to provide a compact proof.

Incident matrix:The $n\_\{t\}\; imes\; n\_\{f\}$ matrix $mathbf\{A\_a\}$ is called node-to-branch incidence matrix for the matrix elements $a\_\{ij\}$ being

:

A reference or datum node $P\_0$ is introduced to represent the environment and connected to all dynamic nodes and terminals. The $(n\_\{t\}-1)\; imes\; n\_\{f\}$ matrix $mathbf\{A\}$, where the row that contains the elements $a\_\{0j\}$ of the reference node $P\_\{0\}$ is eliminated, is called reduced incidence matrix.

The conservation laws (KCL) in vector-matrix form:

: $mathbf\{A\}\; mathbf\{F\}=\; mathbf\{0\}$

The uniqueness condition for the potentials (KVL) in vector-matrix form:

: $mathbf\{W\}\; =\; mathbf\{A^\{T\; mathbf\{w\}$

where $w\_\{k\}$ are the absolute potentials at the nodes to the reference node $P\_\{0\}$.

Proof

Starting with Tellegen's Theorem

: $sum\_\{k=1\}^\{b\}\; W\_\{k\}\; F\_\{k\}\; =\; mathbf\{W^T\}\; mathbf\{F\}\; =\; 0$

: $mathbf\{W^T\}\; mathbf\{F\}\; =$

using KVL

: $=\; mathbf\{(A^\{T\}\; w)^T\}\; mathbf\{F\}$

: $=\; mathbf\{(w^\{T\}\; A)\}\; mathbf\{F\}$

: $=\; mathbf\{w^\{T\}\; A\; F\}\; =\; mathbf\{0\}$

since ($mathbf\{A\; F\}\; =\; mathbf\{0\}$) using KCL.

Applications

Network analogs have been constructed for a wide variety of physical systems, and have proven extremely useful in analyzing their dynamic behavior. The classical application area for network theory and Tellegen's theorem is electrical circuit theory. 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. Topology and structure of reaction networks (reaction mechanisms, metabolic networks) can be analyzed using the Tellegen theorem.

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:

: $sum\_\{j=1\}^\{n\_\{P\; W\_\{j\}frac\{dZ\_\{j\{dt\}\; =\; sum\_\{k=1\}^\{n\_\{f\; W\_\{k\}\; f\_\{k\}\; +\; sum\_\{j=1\}^\{n\_\{P\; w\_\{j\}\; p\_\{j\}\; +\; sum\_\{j=1\}^\{n\_\{t\; w\_\{j\}\; t\_\{j\},quad\; j=1,dots,n\_\{p\}+n\_\{t\}$

where $p\_\{j\}$ are the production terms, $t\_\{j\}$ are the terminal connections, and $frac\{dZ\_\{j\{dt\}$ are the dynamic storage terms for the extensive variables.

References

* "Basic Circuit Theory" by C.A. Desoer and E.S. Kuh, McGraw-Hill, New York, 1969
*"Tellegen's Theorem and Thermodynamic Inequalities", G.F. Oster and C.A. Desoer, "J. theor. Biol" 32 (1971), 219–241
*"Network Methods in Models of Production", Donald Watson, "Networks", 10 (1980), 1–15