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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

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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.

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`

  Is there an error in this question or solution?
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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. Concept: Ac Voltage Applied to a Series Lcr Circuit.
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