5. The following a series RLC circuit with R = 3 k ohm, L = 10H , and C = 200uF has a constant voltage source V = 5theta*VAC signal with a frequency of 60 Hz
Calculate the capacitive reactance and the inductive reactance in the circuit?
b. Determine the impedance?
c. Calculate the rms consumed in the circuit?
d. Calculate the voltage across the resistor, the inductor and the capacitor?
e. How much power is consumed in the circuit?
f. What is the resonant frequency of the circuit
To calculate the capacitive reactance (Xc) and inductive reactance (Xl) in the circuit, we can use the formulas:
Xc = 1 / (2 * π * f * C)
Xl = 2 * π * f * L
Where:
f = frequency = 60 Hz
C = 200uF = 200 * 10^-6 F
L = 10H
Xc = 1 / (2 * π * 60 * 200 * 10^-6)
Xc = 1 / (2 * 3.1416 * 60 * 0.0002)
Xc = 1 / 0.037699
Xc ≈ 26.50 ohms
Xl = 2 * 3.1416 * 60 * 10
Xl = 3770.80 ohms
b. Impedance (Z) in the circuit can be calculated using the formula:
Z = sqrt(R^2 + (Xl - Xc)^2)
Z = sqrt((3000^2) + (3770.80 - 26.50)^2)
Z = sqrt(9000000 + 14196754.25)
Z = sqrt(23196754.25)
Z ≈ 4813.18 ohms
c. The rms consumed in the circuit can be calculated using the formula:
Irms = Vrms / Z
Vrms = 5 θ VAC
Irms = 5 / 4813.18
Irms ≈ 0.00104 A
d. To find the voltage across the resistor (VR), inductor (VL) and capacitor (VC), we can use the formulas:
VR = Irms * R
VR = 0.00104 * 3000
VR ≈ 3.12 V
VL = Irms * Xl
VL = 0.00104 * 3770.80
VL ≈ 3.92 V
VC = Irms * Xc
VC = 0.00104 * 26.50
VC ≈ 0.03 V
e. Power consumed in the circuit can be calculated as:
P = Irms^2 * R
P = (0.00104)^2 * 3000
P ≈ 0.0031 W
f. The resonant frequency of the circuit can be calculated using the formula:
f0 = 1 / (2 * π * sqrt(L * C))
f0 = 1 / (2 * π * sqrt(10 * 200 * 10^-6))
f0 = 1 / (2 * 3.1416 * sqrt(0.002))
f0 = 1 / (2 * 3.1416 * 0.0447)
f0 ≈ 7.1 Hz
Therefore, the capacitive reactance is approximately 26.50 ohms, the inductive reactance is about 3770.80 ohms, the impedance is around 4813.18 ohms, the rms consumed in the circuit is roughly 0.00104 A, the voltage across the resistor is about 3.12 V, the voltage across the inductor is approximately 3.92 V, the voltage across the capacitor is approximately 0.03 V, the power consumed in the circuit is approximately 0.0031 W, and the resonant frequency of the circuit is around 7.1 Hz.