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Verification of Compensation Theorem
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Objective: To verify the Compensation Theorem.

In a linear time invariant network when the resistance (R) of an uncoupled branch, carrying current (I), is changed by \Delta{R}, the currents in all the branches would change and can be obtained by assuming that an ideal voltage source of V_c has been connected such that V_c=I{\Delta{R}} in series with (R+{\Delta{R}}) when all other sources in the network are replaced by their internal resistances. 

Explanation :

Let us assume a load RL be connected to a dc source network whose thevenin's equivalent gives Vo as the Thevenin's voltage and RTh as Thevenin resistance. 

I = \frac{ V_0}{R_T_h+R_L} .....(1)

Fig. 1. Thevenin Equivalent Circuit.

 

   Fig.2 (a) . Explanation of Compensation theorem

      Fig.2 (b) . Explanation of Compensation theorem

Let the load resistance RL be changed to  (R_L+{\Delta{R_L}}). Since the rest of the circuit remains unchanged, the Thevenin equivalent network remains the same.

I {^'}= \frac{ V_0}{R_T_h+R_L+{\Delta{R_L}}}.....(2)

This Change of current being termed as \Delta{I}, we find 

\Delta{I}=I{^'}-I\\

= \frac{ V_0}{R_T_h+R_L+{\Delta{R_L}}} - \frac{ V_0}{R_T_h+R_L}.

=\frac{ V_0}{R_T_h+R_L} \frac{-\Delta R_L}{R_T_h+R_L+{\Delta{R_L}}}

= \frac{- V_c}{R_T_h+R_L+{\Delta{R_L}}}.....(3)

V_c=I{\Delta R_L}. and is termed as compensating voltage.

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