An AC voltage of the form Δv = (65.0 V)sin(340t) is applied to a series RLC circuit. If R = 46.0Ω, C = 27.0 μF, and L = 0.300 H, find the following.
(a) impedance of the circuit
Ω
(b) rms current in the circuit
A
(c) average power delivered to the circuit
W
a. Z = R + jXl + jXc = 46 + j340*0.3 - j1/340*27*10^-6 = 46 + j102 - j109,
Z = 46 - j7 = 46.5 Ohms[-8.65o].
b. Irms = Erms/Z = (65*0.707)/46.5[-8.65] = 0.988A[8.65o].
c. P = (Irms)^2 * R = (0.988)^2 * 46 = 44.9 Watts.
To find the impedance of the circuit, we need to use the formula for impedance in a series RLC circuit:
Z = √(R^2 + (Xl - Xc)^2)
where R is the resistance, Xl is the reactance due to inductance, and Xc is the reactance due to capacitance.
(a) Finding the impedance of the circuit:
Given:
R = 46.0 Ω
C = 27.0 μF = 27.0 × 10^-6 F
L = 0.300 H
First, we need to find the reactance due to inductance (Xl) and the reactance due to capacitance (Xc).
For inductance:
Xl = 2πfL
where f is the frequency in hertz, and L is the inductance in henries.
The form of the AC voltage given is Δv = (65.0 V)sin(340t), so we can determine the frequency by comparing it to the general form of sinusoidal function: sin(ωt).
The angular frequency (ω) is related to the frequency (f) by the formula ω = 2πf.
Comparing the given AC voltage form to sin(ωt), we can see that ω = 340.
Now we can calculate Xl:
Xl = 2πfL = 2π(340)(0.300) = 643.96 Ω
For capacitance:
Xc = 1 / (2πfC)
Now we can calculate Xc:
Xc = 1 / (2πfC) = 1 / (2π(340)(27.0 × 10^-6)) = 121.36 Ω
Now we can calculate the impedance:
Z = √(R^2 + (Xl - Xc)^2)
Z = √((46.0)^2 + (643.96 - 121.36)^2)
Z = √(2116 + 430619.68)
Z = √432735.68
Z ≈ 658.40 Ω
So, the impedance of the circuit is approximately 658.40 Ω.
(b) Finding the rms current in the circuit:
The rms current (Irms) is given by the equation:
Irms = Vrms / Z
where Vrms is the rms voltage.
The rms voltage is the peak voltage divided by the square root of 2, since the given voltage is in the form of Δv = (65.0 V)sin(340t):
Vrms = 65.0 / √2
Vrms ≈ 45.96 V
Now we can calculate Irms:
Irms = Vrms / Z = 45.96 / 658.40
Irms ≈ 0.0699 A
So, the rms current in the circuit is approximately 0.0699 A.
(c) Finding the average power delivered to the circuit:
The average power (Pavg) is given by the equation:
Pavg = (Irms^2) * R
where Irms is the rms current.
Now we can calculate Pavg:
Pavg = (Irms^2) * R = (0.0699)^2 * 46.0
Pavg ≈ 0.221 W
So, the average power delivered to the circuit is approximately 0.221 W.