The average power in LCR series circuit is

Explanation

Consider a general ac circuit made up of inductor coils, capacitors and resistors to which a sinusoidally alternating emf is applied. Suppose there is a phase difference `phi` between the current `I` and emf E.
Then, let
`I=I_(0) sin omega t and E=E_(0) sin (omega t +phi) " " ` ...(1)
where `I_(0) and E_(0)` are the peak values of the current and emf, respectively, and `f=omega//2pi` is the frequency of the ac supply.
The instantaneous power supplied is,
`P=EI=E_(0)I_(0) sin omega t * sin(omega t +phi) " " ` ....(2)
To find the average power supplied `P_(av)`, we use the trigonometrical identity,
`sin A sinB =(1)/(2)[cos(B-A)-cos(B+A)]`
`therefore P=(1)/(2)E_(0)I_(0)[cos phi -cos(2 omega t +phi)] " " ` ...(3)
The first term in the bracket is constant, while the second term is a sinusoidally oscillating quantity whose mean value taken over one cycle is zero.
`therefore P_(av)=(1)/(2) E_(0)I_(0)cos phi =E_("rms")I_("rms") cos phi " " ` ...(4)
as `E_("rms") =(E_(0))/(sqrt(2)) and I_("rms")=(I_(0))/(sqrt(2)).`
The product `E_("rms")*I_("rms") ` is sometimes called the apparent power. It is equal to the true power `P_(av)` in the special case of `phi=0`, i.e., when E and `I` are in phase as with a purely resistive circuit. The cosine of the phase angle, `cos phi`, is known as the power factor of a circuit.
The power factor of a circuit is defined as the ratio of the true power averaged over one cycle to the apparent power.
Power factor `=("true power averaged over one cycle")/("apparent power")`
`=("true power averaged over one cycle")/(E_("rms")I_("rms")) " " ` ...(5)
If a sinusoidal alternating emf is applied to an LCR series circuit containing a resistance R, the power factor is
`cos phi =R//Z " " ` ...(6)
where Z is the impedance of the circuit.