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 $\phi$.

Answer 1

The average power consumed in a series LCR circuit is VrIrcosФ

Explanation :

Let the voltage in the circuit,

v = Vsinwt

and the current in the circuit,

i = Isin(wt-Ф)

Hence total work done in a complete cycle,

W = $\int\limits^T_0 {vi} \, dt$

=> W = ∫ Vsinwt . Isin(wt-Ф) dt

solving this we get,

W = VIT cosФ/2

we know that power is defined as the work done per unit time,

=> P = W/T = VI cosФ/2

=> P = V/√2 x I/√2 x cosФ

=> P = VrIrcosФ

where Vr and Ir are the rms values respectively.