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* = *i*_{m} sin (*ω**t* +*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 (2*ω**t* + *Φ*) is equal to zero.

Thus,

Average power,

`p = (v_mi_m)/2 cosphi = VI cos phi`

Concept: Ac Voltage Applied to a Series Lcr Circuit

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