Tellegen's theorem is one of the most powerful theorems in
network theory
Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. In computer science and network science, network theory is a part of graph theory: a network can be defi ...
. 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
Bernard D.H. Tellegen (24 June 1900 – 30 August 1990) was a Dutch electrical engineer and inventor of the pentode and the gyrator. He is also known for a theorem in circuit theory, Tellegen's theorem.
He obtained a master's degree in electri ...
.
Fundamentally, Tellegen's theorem gives a simple relation between magnitudes that satisfy
Kirchhoff's laws of electrical
circuit theory
Circuit may refer to:
Science and technology
Electrical engineering
* Electrical circuit, a complete electrical network with a closed-loop giving a return path for current
** Analog circuit, uses continuous signal levels
** Balanced circui ...
.
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
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhof ...
, KCL) and the uniqueness of the potentials at the network nodes (
Kirchhoff's voltage law
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchho ...
, KVL). The Tellegen theorem provides a useful tool to analyze complex network systems including electrical circuits,
biological
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary in ...
and
metabolic network
A metabolic network is the complete set of metabolic and physical processes that determine the physiological and biochemical properties of a cell. As such, these networks comprise the chemical reactions of metabolism, the metabolic pathways, as w ...
s,
pipeline transport
Pipeline transport is the long-distance transportation of a liquid or gas through a system of pipes—a pipeline—typically to a market area for consumption. The latest data from 2014 gives a total of slightly less than of pipeline in 120 countr ...
networks, and
chemical process
In a scientific sense, a chemical process is a method or means of somehow changing one or more chemicals or chemical compounds. Such a chemical process can occur by itself or be caused by an outside force, and involves a chemical reaction of some ...
networks.
The theorem
Consider an arbitrary
lumped network that 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 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
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
'', ''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
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
. 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
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
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element ...
) Tellegen's theorem remains true. This extension of Tellegen's Theorem leads to many theorems relating to two-port networks.
[''Tellegen's Theorem and Electrical Networks'' by Paul Penfield, Jr., Robert Spence, and Simon Duinker, The MIT Press, Cambridge, MA, 1970]
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.
References
; In-line references
; General 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
{{refend
External links
Circuit example for Tellegen's theorem*G.F. Oster and C.A. Desoer
Tellegen's Theorem and Thermodynamic InequalitiesNetwork thermodynamics
Circuit theorems