Tellegen’s Theorem

Tellegen’s Theorem states that the summation of power delivered is zero for each branch of any electrical network at any instant of time.

Contents:

Explanation of Tellegen’s Theorem

Tellegen’s Theorem can also be stated in another word as, in any linear, nonlinear, passive, active, time variant or time invariant network the summation of power (instantaneous or complex power of sources) is zero.

Thus, for the Kth branch, this theorem states that

tellegen's theorem eq1

Where,
n is the number of branches
vK is the voltage in the branch

iK is the current flowing through the branch

Let
tellegen's theorem eq2

Equation (1) shows the Kth branch through current

vK is the voltage drop in branch K and is given as
tellegen's theorem eq3

Where, vp and vq are the respective node voltage at p and q nodes.
TELLEGEN'S-THEOREMWe have,

tellegen's theorem eq4

Also
tellegen's theorem eq5

Obviously
tellegen's theorem eq6

Summing the above two equations (2) and (3) we get
tellegen's theorem eq7

Such equations can be written for every branch of the network.

Assuming n branches the equation will be
tellegen's theorem eq8

However, according to the Kirchhoff’s current law (KCL), the algebraic sum of currents at each node is equal to zero.

Therefore,
tellegen's theorem eq9

Thus, from the above equation (4) finally we obtain
tellegen's theorem eq10

Thus, it has been observed that the sum of power delivered to a closed network is zero. This proves that the Tellegen’s Theorem and also proves the conservation of power in any electrical network. It is also evident that the sum of power delivered to the network by an independent sources is equal to the sum of power absorbed by all passive elements of the network.

Steps for Solving Networks Using Tellegen’s Theorem

Step 1 – The following steps are given below to solve any electrical network by Tellegen’s Theorem

Step 2 – In order to justify this theorem in an electrical network, the first step is to find the branch voltage drops.

Step 3 – Find the corresponding branch currents using conventional analysis methods.

Step 4 – Tellegen’s Theorem can then be justified by summing the products of all branch voltages and currents.

For example, if a network having a number of branches “b” then

tellegen's theorem eq11

Now if the set of voltages and currents is taken, corresponding the two different instants of time, t1 and t2, the Tellegen’s Theorem is also applicable where we get the equation as shown below

tellegen's theorem eq12

Application of Tellegen’s Theorem

The various applications of the Tellegen’s theorem are as follows

  • It is used in the digital signal processing system for designing of filters.
  • In the area of biological and chemical process.
  • In topology and structure of reaction network analysis.
  • Theorem is used in chemical plants and oil industries to determine the stability of any complex systems.

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