Tellegen's Theorem

Tellegen's theorem is a tool for analyzing electrical circuits and networks.  Tellegen's theorem stated for any linear or nonlinear, passive or active network, the summation of the power at each element in the network is zero, provided that Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL) are satisfied.  In other words, the algebraic sum of power supplied or absorbed by all elements in the network is always zero.

• \( \sum P_{in} = \sum P_{out}\)
• \( P_{in} \)​ is the power supplied to the network by voltage sources and absorbed by current sources.
• \( P_{out} \) is the power absorbed by passive elements (resistors, capacitors, and inductors).

More formally, consider a network with \(n\) branches.  For each branch \(Ii\), let:

• $\backslash (Vi\backslash )$ be the voltage across the branch.
• $\backslash (Ii\backslash )$ be the current through the branch.

Tellegen's Theorem states that:

• \( \textstyle \sum_{i=1}^n \; V_i \; I_i  \;=\; 0  \)

Key Points about Tellegen's Theorem

• KCL and KVL Satisfaction  -  The theorem assumes that KCL and KVL hold for the network.  KCL states that the algebraic sum of currents at any node is zero, and KVL states that the algebraic sum of voltages around any closed loop is zero.
• General Applicability  -  The theorem is valid for any network configuration, regardless of whether the components are linear or nonlinear, passive or active.  This makes it a very general and widely applicable theorem in circuit analysis.
• Power Conservation  -  The essence of Tellegen's theorem is power conservation.  It implies that the total power supplied by the sources is equal to the total power absorbed by the passive components in the network.  This is consistent with the conservation of energy principle.

Applications

• Verification of Network Equations  -  Tellegen's theorem can be used to verify the correctness of network equations derived from KCL and KVL.
• Network Analysis  -  It simplifies the analysis of complex networks by providing a straightforward power balance check.
• Circuit Simulation  -  In circuit simulation, Tellegen's theorem can be used to check the consistency and accuracy of simulation results.
• Error Detection  -  It can help detect errors in the formulation of network equations or in the measurement of network parameters.