(i) Find the value of the phase difference between the current and the voltage in the series LCR circuit shown below. Which one leads in phase : current or voltage ?

(ii) What making any other change, find the value of the additional capacitor C, to be connected in parallel with the capacitor C, in order to make the power factor of the circuit unity.

Explanation

Gives `V = V_(0) sin (1000t + phi)`
`omega = 1000s^(-1) ,L = 100 mH C = 2muF , R = 400Omega`
Phase difference `theta = tan^(-1) ((X_(L) - X_(C))/(R))`
`X_(L) = omega L = 1000 xx 100 xx 10^(-3) Omega = 100 Omega`
`X_(C )= 1/(omegaC) =1/(1000 x 2 xx 10^(-6)) = 500 Omega`
`phi = tan^(-1) ((100-500)/(400)) = tan^(-1)(-1)`
`rArr phi = -45^(@)` Current leads voltage .
(ii) For `P.F. = 1 , R = Z` or `X_(L) = X_(C)`
`rArr Omega^(2) = 1/(LC) rArr 10^(6) = 1/(100 xx 10^(-3) xx C')`
`rArr C' = 10^(-5)F`
Resultant capacitance in parallel.
`C' = C_(2) + C_(1)`
`C_(2)= 10^(-5) - .2 xx10^(-5) = 8 muF`.