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Verification of Tellegen's Theorem
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Objective : To Verify Tellegen's Theorem.

 

             Fig.1. Tellegen's Theorem 

For any given time, the sum of power delivered to each branch of any electric network is zero.

Thus for K^{th} branch , this theorem states that,

 \tiny \small \sum_{K=1}^{n}{V_k}{i_k}=0'

n being the number of branches, {V_K} the drop in the branch and {i_K} the through current.

Explanation:

let {i_K}=Kth branch through current.

   {V_K}={V_p-V_q} is the Voltage drop in branch K, while {V_p} and {V_q} are the respective node voltages at p and q nodes.

We have ,

V_Ki_p_q=(V_p-V_q)i_p_q=V_Ki_K .....(1)

Also, V_Ki_p_q=(V_q-V_p)i_q_p.....(2)

Obviously,i_p_q=-i_q_p.....(3)

Summing equations (1) and (2),

2V_Ki_K=(V_p-V_q)i_p_q+(V_q-V_p)i_q_p

V_Ki_K= \frac{(V_p-V_q)i_p_q+(V_q-V_p)i_q_p}{\2}-----(4)

Equation (4) can be written for every branch of the network. Assuming n branches, generalisation yields

\sum_{K=1}^{n}{V_K}{i_K}=\frac{1}{\2}\sum_{p=1}^{n}\sum_{q=1}^{n}{(V_p-V_q)}{i_p_q}

= \frac{1}{\2}\sum_{p=1}^{n}{V_p}\sum_{q=1}^{n}{{i_p_q}}- \frac{1}{\2}\sum_{q=1}^{n}{V_q}\sum_{p=1}^{n}{{i_p_q}} .....(5)    

However, following Kirchhoff's current laws, the algebric sum of currents at each node is equal to zero.

\sum_{p=1}^{n}{{i_p_q}}=0.....(6)

\sum_{q=1}^{n}{{i_q_p}}=0.....(7)

Substituting equ. (6) and (7) in equ. (5), we obtain

 \sum_{K=1}^{n}{V_K}{i_K}=0.....(8)

Equations (8) shows that the sum of power delivered to a closed network is zero. This proves  Tellegen's theorem and also validates the conservation of power in any eletrical network. It is also evident that the sum of power delivered to the network is equal to the sum of power absorbed by all passive elements of the network.

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