Question

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?

Answer 1

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)`