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

Current in this circuit will be `I=I_(0)sin (wt+phi)`
where
`phi=tan^(-1) ((x_(L)-x_(C))/x_(R))`
`x_(L)=wL,x=1/(WC),x_(R)=R`
`P_(inst)=V.I`
`=(v_(0)sin wt)[l_(0)sin(wt+phi)]`
`=v_(0)I_(0)(sin wt)[sin(wt+phi)]`
`=(v_(0)I_(0))/2 2 sin wt [sin(wt+phi)]`
`(v_(0)I_(0))/2 [cos -(wt-(wt+phi)-cos(2wt+phi)]`
`P_(avg)=I/Tunderset(0)overset(T)int P_(inst) dt`
`=I/Tunderset(0)overset(T)int (v_(0)I_(0))/2 [(cos phi)-cos (2wt+phi)]dt`
`T=(2pi)/omega , wt=theta`
`wdt=dtheta`
`1/(2pi)underset(0)overset(2pi)int (v_(0)I_(0))/2[(cos phi)-cos(2wt+phi)]d theta`
`rArr (v_(0)I_(0))/2 cos phi-O`
`(v_(0)I_(0))/2 cos phirArr v_(rms)-I_(rms)cos phi`
(i)no power is dissipated if `phi=pi/2` i.e. `R=0`.
(ii)Maximum power is dissipated if `phi=0` i.e. `w^(2)=1/(LC)`