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# 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. - Physics

#### 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.

#### 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), wherei_m =v_m/Zand

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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