Question

(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) Without making any other change, find the value of the additional capacitor C₁, to be connected in parallel with the capacitor C, in order to make the power factor of the circuit unity.

Answer 1

(i) Given V = V₀sin(1000t + ϕ)
ω = 1000 s^-1

Given,
L = 100 mH
​C = 2 μF
R = 400 Ω

Phase difference ϕ= tan-1((XL−XC)/R)

X_L = ωL = 1000 × 100 × 10​^-3 = 100 Ω
`X_C=1/(omegaC)=1/(1000xx2xx10^-6)=500Omega`

`ϕ= tan^-1((100−500)/400)= tan^-1(-1)`

ϕ = -450 and the current is leading the voltage.

(ii)
For power factor to be unity, R = Z

or X_L = X_C
`ω^2 =1/(LC)` (C = resultant capacitance)
`10^6 = 1/(100xx10^-3xxC')`

⇒C' = 10^-5 F

For two capacitance in parallel, resultant capacitance C'= C + C1

10^-5 = 0.2 x 10^-5 + C₁
⇒C₁ = 8 μF