# Compensation Theorem

Before understanding Compensation Theorem, one question that comes in your mind.

Why Compensation Theorem? i.e. why do we need this theorem?

Compensation Theorem is a useful theorem. We can apply it to directly find the change in the magnitude of current when the resistance of a branch is changed from RL to RL+△R

NOTE: Compensation Theorem is applicable to linear, time-invariant networks only. (We have provided the definitions of linear networks and time-invariant networks at the end of the article).

### Definition of Compensation Theorem According to this theorem, when the resistance (RL) of a branch, in a linear time-invariant circuit, changes to RL + △RL, there is a change in the current (△I ) in the same branch. And we can obtain this change in current (△I ),

(i) by adding a compensating voltage source Vc (= I.△RL) in series with RL + △RL and

(ii) by replacing all other sources in the source network by their internal resistance.

To directly find △I using Compensation Theorem for the circuit (shown in figure(ii)), we have to replace that circuit by the circuit given below (see figure(iii)). Then we apply the following steps from the definition.

1. Adding a compensating voltage Vc (=I.△RL) in series with RL + △RL in the circuit shown in figure(ii),
2. Finding the Thevenin Resistance RTH of the source network of that circuit,
3. Replacing the entire source network by RTH only. Here we replace the voltage sources of the source network with the short circuit. Then we replace the current sources with open circuits. Here we also replace the Thevenin’s voltage of source network with a short circuit. The final transformed circuit is shown in figure(iii).

Where, RTH = Thevenin’s Resistance of the source network.
Applying KVL to the above circuit gives, We can see that △I can be found directly using the equation(i).

### Proof/Derivation of Compensation Theorem After converting the network shown in figure(i) into its Thevenin’s equivalent shown in figure(iv), we can write the current I as, Where

VTH = Thevenin’s voltage of the Source network and

RTH = Thevenin resistance of the source network.

Similarly, converting the network shown in figure(ii) into its Thevenin Equivalent circuit shown in figure(v) and calculating for I'(= I + △I) gives, Substituting the value of I from equation(ii) and simplifying gives Which is the same as given by equation(i) hence it proves the Compensation Theorem.

### Linear networks and Time-Invariant Networks

• Linear Network: In short, a network which contains only linear elements (those elements which have linear V-I characteristics like resistors) is a linear network. For example, a network containing only resistors is a linear network.
• Time-invariant Network: A network whose parameters do not vary with time is known as Time-invariant Network.

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