Derive an experssion for the average power in LCR circuit connected to a.c. supply. Hence define power factor. Show that average power cousumed per cycle in an a.c. circuit conataining an ideal capacitor is zero.

Explanation

Let an alternating emf `E = E_0 sin omegat` be appliedto an LCR circuit and the current lags behind by a phase `phi` i.e. `I = I_0 sin (omegat = phi)`
`:." Instantaneous power", P = (dW)/(dt) = E.I. = E_0 sin omegat. I_0 sin(omegat - phi)`
`or (dW)/(dt) = E_0I_0 sin omegat [ sin omegat sinphi]`
`or dW`
`=[E_0I_0 sin^2 omegat cosphi - (E_0I_0)/(2) 2 sinomega tsin phi]dt or W`
`= (E_0I_0)/2 cos phi underset(0)overset(T)int (1-cos2 omegat)dt-(E_0I_0)/2 sin phi underset(0)overset(T)int sin 2 omega t dt`
`or W = (E_0I_0)/2 cos phi [underset(0)overset(T)int" dt" - underset(0)overset(T)int " " sin 2 omega t dt]`
`-(E_0I_0)/2 sin phiunderset(0)overset(T)int sin 2 omegat dt`
`=(E_0I_0)/2 cos phi[T_2-0]-(E_0I_0)/2 sin phi0`
`W = (E_0I_0cosphi)/2T`
`[ :.underset(0)overset(T)int cos 2 omega t dt = underset(0)overset(T)int sin 2 omegadt = 0]`
`or (W)/T = (E_0)/sqrt2 I_0/sqrt2 cosphi T = E_(v)I_(v) cos phi`
`or p_(av) = E_vI_v cos phi`