Compensation Theorem

In many circuits, after the circuit is analysed, it is realized that only a small change need to be made to a component to get a desired result. In such a case we would normally have to recalculate.The compensation theorem allows us to compensate properly for such changes without sacrificing accuracy. 

In any linear bilateral active network, if any branch carrying a current I has its impedance Z changed by an amount ∆Z, the resulting changes that occur in the other branches are the same as those which would have been caused by the injection of a voltage source of (-) I . ∆Z in the modified branch. 

Consider the voltage drop across the modified branch. 

V +∆V = (Z +∆Z)( I +∆I) = Z . I + ∆Z . I + (Z +∆Z) . ∆I 

from the original network, V = Z . I 

∴  ∆V = ∆Z . I + (Z +∆Z) . ∆I

Since the value I is already known from the earlier analysis, and the change required in the 

impedance, ∆Z , is also known, I .∆Z is a known fixed value of voltage and may thus be represented by a source of emf I.∆Z . 

Using superposition theorem, we can easily see that the original sources in the active network give rise to the original current I, while the change corresponding to the emf I.∆Z must produce the remaining changes in the network.