The basic concept of the Tellegen’s theorem being identical in d.c. or a.c. systems, for the application in a.c. systems, it can be stated as follows:

*In any linear, non-linear, passive, active time variant network, exited by alternating sources, the summation of instantaneous or complex power of the source is zero.*

For the network exited by sinusoidal sources, if the number of branch be “b”,

Where v_{b} and i_b represent the instantaneous voltage and current of source of each branch.

When considering the complex power, if V_{b} and I_{b} be the voltage and current of each branch, as per this theorem,

Where I_{b}^{*} is a complex conjugate of I_{b}.

## Proof of Tellegen’s Theorem

With reference to the network shown in figure 1, let the node voltage be V_{1}, V_{2} and V_{3} at nodes 1, 2 and 3 respectively. The current directions are shown arbitrarily.

The summation of instantaneous power in the network is given by

[where, V_{C1}, V_{C2}, V_{L}, V_{R1} etc. indicate the respective voltage across the elements C_{1}, C_{2}, L, R_{1} etc.]

Or,

Application of KCL at node (1) reveals that i_{1} + i_{2} + i_{6} = 0, at node (2), i_{3} + i_{4} – i_{2} = 0 and at node (3), i_{5} – i_{4} – i_{6} = 0.

Thus finally,

Hence the theorem is proved.