A voltage `V = V_(0)sinomegat` is applied to a series LCR circuit. Derive expression for the average power dissipated over a cycle.
Under What condition (i) no power is dissipated even though the current flows through the circuit?
(ii) Maximum power is dissipated in the circuit?

Explanation

Voltage `V=V_(0)sinomegat` is applied to a series LCR circuit.
Current is
`I=Isin(omegat+phi)`
`I_(0)=V_(0)Zphi=tan^(-1)x_(c)-x_(LR)`
Instantaneous power supplied by the source is
`P=V I`
`=(V_(0)sinomegat)xx(I_(0)sin(omegat+phi)`
`=V_(0)I_(0)2cosphi*cos(2omegat+phi)`
The average power over a cycle is average of the two terms on the R.H.S of the above equation. The second term is time dependent, so, its average is zero.
`P=V_(0)I_(0)2cosphi=V_(0)I_(0).^(2)2cosphi`
`=VI2cosphi`
`P=I_(2)zcosphi`
`cosphi` is called the power factor.
Case I : For pure inductive circuit or pure capacitive circuit, the phase difference between current and voltage is `(pi)/(2)`
`phi=(pi)/(2),cosphi=0`
Therefore no power is dissipated. This current is sometimes referred to as wattless current.
Case II : For power dissipated at resonance in an LCR circuit,
`X_(c)-X_(1)=0,phi=0` " "therefore Cosphi=1`
So, maximum power is dissipated.