Tellegen's theorem
Encyclopedia
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 among them electrical circuits, biological and metabolic networks, pipeline flow networks, and chemical process networks.
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. 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). This extension of Tellegen's Theorem leads to many theorems relating to two-port networks.
Incidence matrix:
The
matrix 
is called node-to-branch incidence matrix for the matrix elements 
being
Network theory
Network theory is an area of computer science and network science and part of graph theory. It has application in many disciplines including statistical physics, particle physics, computer science, biology, economics, operations research, and sociology...
. 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
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff...
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 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. 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). 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
matrix is called node-to-branch incidence matrix for the matrix elements being-
A reference or datum node
is introduced to represent the environment and connected to all dynamic nodes and terminals. The 
matrix 
, where the row that contains the elements 
of the reference node 
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
are the absolute potentials at the nodes to the reference node 
.
Proof
Using KVL:
-
because
by KCL. So:
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:
where
are the production terms, 
are the terminal connections, and 
are the dynamic storage terms for the extensive variables.
External links
- Circuit example for Tellegen's theorem
- G.F. Oster and C.A. Desoer, Tellegen's Theorem and Thermodynamic Inequalities
- Network thermodynamics
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