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Question
Derive an expression for the average power consumed in a series LCR circuit connected to a.c. source in which the phase difference between the voltage and the current in the circuit is Φ.
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Solution
Power in ac circuit
Voltage v in an ac circuit is:
`v =v_m sinomegat`
which drives through the circuit a current i
i = im sin (ωt +Φ), where`i_m =v_m/Z`and
`phi =tan^-1 ((X_c -X_L)/R)`
Power `p = v_i = (v_msin omegat)[i_m sin(omegat +phi)]`
`=(v_mi_m)/2 [cosphi - cos[2omegat +phi]]`
Calculating the average power, it is observed that the average of the term
cos (2ωt + Φ) is equal to zero.
Thus,
Average power,
`p = (v_mi_m)/2 cosphi = VI cos phi`
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