Determine the current in the given circuit, if the source voltage is v_s = 12 cos (1000t + 15°).

Concept:

Consider a series RLC circuit as shown below,

The impedance of the circuit can be given as,

Z = R + j(X_L - X_C) Ω

XL = ωL Ω       -------- (1)

XC = 1/ωC Ω       -------- (2)

Where, R = resistance, XL = inductive reactance, XC = capacitive reactance

The magnitude of impedance   Ω ------- (3)

Impedance angle        -------- (4)

Then the impedance of the circuit can be represented in polar form as,

Z = |Z| ∠θ Ω

Let the voltage be represented as

V(t) = V_m cos (ωt + ϕ) V

Where V_m is peak value,

ω is the angular velocity in rad/sec, 

ϕ is the voltage phase angle in degree.

Then the current through the circuit can be given as,

 A         ---------- (5)

Calculation:

Given:

vs = 12 cos (1000t + 15°) V

ω = 1000 rad/sec , R = 30 Ω, L = 65 mH, C = 40 µF

Form equations(1) & (2) we can write

X_L = ωL = 1000 × 65 × 10^-3 = 65 Ω

X_C = 25 ​Ω

From equations(3) & (4) we can write

Impedance magnitude 

|Z| = 50 Ω

Impedance angle  degree

Then the current through the circuit can be given from equation(5) as,

 A

i(t) = 0.24 cos (1000t + 15° - tan-1 4/3) A