A series RLC circuit with L = 160 mH, C = 100 μF and R = 40.0 Ω is connected to a sinusoidal voltage V(t) = (40.0 V) sin ωt, where ω = 200 rad/s. Find I₀ assumed that the current at any instant in the circuit is given by:

I(t) = I₀ sin (ωt - ϕ)   

Series RLC Circuit:

For a series RLC circuit, the net impedance is given by: 

Z = R + j (XL - XC)

XL = Inductive Reactance given by:

XL = ωL

XC = Capacitive Reactance given by:

XL = 1/ωC

The magnitude of the impedance is given by:

\(Z = \sqrt{R^2 + \left( {{X_L} - {X_C}} \right)^2}\)

Phase angle \(ϕ = \tan^{-1}\frac{X_L-X_c}{R}\)

 \(V = \sqrt{V_R^2 + \left( {{V_L} - {V_C}} \right)^2}\)

And I(t) = V(t) / |Z|

Calculation:

Given that,

Angular frequency ω = 200 rad/sec

L = 160 mH, C = 100 μF and R = 40.0 Ω

V(t) = 40 sin ωt = 40 sin 200t

X_L = ω L = 200 × 160 × 10^-3 = 32 Ω 

X_C = 1/ω C = 1/ (200 × 100 × 10^-6)

⇒ X_C = 50 Ω 

⇒ \(Z = \sqrt{40^2 + \left( {{32} - {50}} \right)^2}\)  = 43. 86 Ω 

And I(t) = (40 / 43.86) sin (200t - ϕ) = .911 sin (200t - ϕ) A