Electrical Engineering ⇒ Topic : Compensation Theorem with sinusoidal excitation
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 Z1 and Z2. If the current in this branch is I, the voltage drop across Z1 is IZ1 with polarity as shown in Fig. 1 (a). Fig. 8.80(b) shows the compensation source Vc = IZ1 which replace Z1. However Vc 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
Consider a network shown in Fig. 1 (a).
Let the impedance of branch AB change from Z1 to (Z1 + δZ1). Let I1 be the new current.
The impedance Z1 of the network shown in Fig. 1 (a) may be replaced by a voltage source, VC By substitution theorem Vc = IZ1 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 δI1 denote the small change in current, due to the small change in the impedance value by δZ1.
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 IδZ1 must be introduced with correct Fig.(e) polarity
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