Tellegen Theorem

This theorem has been introduced in the year of 1952 by Dutch Electrical Engineer Bernard D.H. Tellegen. This is very useful theorem in network analysis. According to Tellegen theorem the summation of instantaneous powers for the n number of branches in an electrical network is zero. Are you confused ? Let’s explain. Suppose n number of branches in an electrical network have i₁, i₂, i₃, ………….i_n respective instantaneous currents through them. These currents satisfy Kirchhoff current law. Again, suppose these branches have instantaneous voltages across them are v₁, v₂, v₃, ……….. v_n respectively. If these voltages across these elements satisfy Kirchhoff Voltage law then,

Where v_k is the instantaneous voltage across the k^th branch and i_k is the instantaneous current flowing through this branch. Tellegen theorem is applicable mainly general class of lumped networks consists of linear, non-linear, active, passive, time variant and time variant elements. This theorem can easily be explained by the following example.

In the network shown, arbitrary reference directions have been selected for all of the branch currents, and the corresponding branch voltages have been indicated, with positive reference direction at the tail of the current arrow. For this network, we will assume a set of branch voltages satisfy the Kirchhoff voltage law and a set of branch current satisfy Kirchhoff current law at each node. We will then show that these arbitrary assumed voltage and currents satisfy the equation

and it is the condition of Tellegen theorem,

In the network shown in the figure, let v₁, v₂ and v₃ be 7, 2 and 3 volts respectively. Applying Kirchhoff voltage law around loop ABCDEA. We see that v₄ = 2 volt is required. Around loop CDFC, v₅ is required to be 3 volt and around loop DFED, v₆ is required to be 2. We next apply Kirchhoff current law successively to nodes B, C and D.

At node B let i_i = 5 A, then it is required that i₂ = − 5 A. At node C let i₃ = 3 A and then i₅ is required to be − 8. At node D assume i₄ to be 4 then i₆ is required to be − 9. Carrying out the operation of equation,

we get, 7 X 5 + 2 X ( − 5) + 3 X 3 + 2 X 4 + 3 X ( − 8) + 2 X ( − 9) = 0
Hence Tellegen theorem is verified.