Compensation Theorem

Compensation Theoremexpresses that in a direct time-invariant system when the opposition (R) of an uncoupled branch, conveying a current (I), is changed by (ΔR), at that point the flows in all the branches would change and can be acquired by accepting that a perfect voltage wellspring of (VC) has been associated to such an extent that VC = I (ΔR) in arrangement with (R + ΔR) when every other source in the system are supplanted by their inside protections. 

In Compensation Theorem, the source voltage (VC) contradicts the first current. In straightforward words, remuneration hypothesis can be expressed as – the obstruction of any system can be supplanted by a voltage source, having a similar voltage as the voltage drop over the opposition which is supplanted.

 

Clarification 

Let us accept a heap RL be associated with a DC source arrange whose Thevenin's proportionate gives V0 as the Thevenin's voltage and RTH as the Thevenin's obstruction as appeared in the figure beneath: 

Here,

Let the heap opposition RL be changed to (RL + ΔRL). Since the remainder of the circuit stays unaltered, the Thevenin's proportional system continues as before as appeared in the circuit chart underneath: 

Here,

The change of current being termed as ΔI

Therefore,

Putting the estimation of I' and I from the condition (1) and (2) in the condition (3) we will get the accompanying condition: 

Presently, putting the estimation of I from the condition (1) in condition (4), we will get the accompanying condition: 

As we probably am aware, VC = I ΔRL and is known as remunerating voltage. 

Along these lines, condition (5) becomes,

Consequently, Compensation hypothesis tells that with the difference in branch opposition, branch flows changes and the change is proportional to a perfect remunerating voltage source in arrangement with the branch contradicting the first current, where every other source in the system being supplanted by their inward protections.