Electrical Engineering ⇒ Topic : Compensation Theorem with sinusoidal excitation

Samual
 
Compensation Theorem Compensation Theorem The compensation theorem states that any impedance having voltage across its terminal in the linear, bilateral network, may be replaced by a voltage source of zero internal impedance equal to the current passing through the impedance multiplied by the value of the impedance, provided the currents and voltages in other part of the network remain unaltered. Let a branch of a network contain impedance Z_{1 }and Z_{2}. If the current in this branch is I, the voltage drop across Z_{1} is IZ_{1} with polarity as shown in Fig. 1 (a). Fig. 8.80(b) shows the compensation source Vc = IZ_{1} which replace Z_{1}. However V_{c} must have polarity as shown in Fig. 8.80(b). If any chance which should effect /occurs in the network then the compensation source must be changed accordingly. The compensation is often referred as substitution theorem. This theorem is of use, when it is required to evaluate the changes in magnitudes of currents and voltages in the different branches of a network, due to a small change in the impedance of one of the branches
(a) (b) FIGURE (1) Consider a network shown in Fig. 1 (a). Let the impedance of branch AB change from Z_{1} to (Z_{1} + δZ_{1}). Let I_{1} be the new current. The impedance Z_{1} of the network shown in Fig. 1 (a) may be replaced by a voltage source, V_{C} By substitution theorem V_{c} = IZ_{1} with polarity as shown in Fig. 1 (c). Similarly, the network shown in Fig. 1 (b) can be replaced by the network shown in Fig. 1(d). Let δI_{1} denote the small change in current, due to the small change in the impedance value by δZ_{1}. figure (d) figure (e) The network for which the above relationship holds good is as shown in Fig. 1 (e). By compensation theorem the small change in the magnitude of current due to a small change in a branch impedance is given by Therefore, the original voltage source should be set equal to zero and a new voltage source IδZ_{1 }must be introduced with correct Fig.(e) polarity  
 
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