A series RLC circuit consists of
a 75Ω resistor, a 5.0 μf capacitor
and a 75mH inductor. They are
connected across a generator of
frequency 250Hz with a rms
voltage of 12V. Determine the
impedance of the circuit.
A. 127.32 Ω
B. 117.8 Ω
C. 84.51Ω
D. 74.39Ω
E. 75.6Ω
To find the impedance of the series RLC circuit, we need to calculate the total resistance (R), total inductive reactance (XL), and total capacitive reactance (XC), and then combine them using the impedance formula:
Impedance (Z) = √(R^2 + (XL - XC)^2)
Given:
R = 75 Ω
C = 5.0 μF
L = 75 mH
f = 250 Hz
Vrms = 12 V
First, let's calculate XL and XC using the formulas:
XL = 2πfL
XC = 1/(2πfC)
XL = 2π(250)(75 * 10^(-3)) = 0.942 Ω
XC = 1/(2π(250)(5 * 10^(-6))) = 127.323 Ω (rounded to 3 decimal places)
Next, let's calculate the impedance:
Z = √(R^2 + (XL - XC)^2)
Z = √((75)^2 + (0.942 - 127.323)^2)
Z = √(5625 + (-126.381)^2)
Z = √(5625 + 15982.854761)
Z = √(21607.854761)
Z = 146.951 Ω (rounded to 3 decimal places)
Therefore, the impedance of the circuit is approximately 146.951 Ω, which is not one of the given answer choices.
To determine the impedance of the circuit, we need to calculate the total opposition to the flow of current, taking into account the resistance (R), capacitance (C), and inductance (L).
The formula to calculate the impedance (Z) of an RLC circuit is:
Z = √(R² + (Xl - Xc)²)
Where:
R = resistance = 75Ω
Xl = inductive reactance = 2πfL
Xc = capacitive reactance = 1/(2πfC)
f = frequency = 250Hz
L = inductance = 75mH = 0.075H
C = capacitance = 5.0 μF = 5.0 x 10⁻⁶F
Calculating Xl:
Xl = 2πfL = 2π(250)(0.075) = 37.5π
Calculating Xc:
Xc = 1/(2πfC) = 1/(2π(250)(5.0 x 10⁻⁶)) = 1/(0.03141 x 10⁻³) = 31830.58007
Now we can substitute the values into the impedance formula:
Z = √(75² + (37.5π - 31830.58007)²)
Calculating the equation:
Z = √(5625 + (117.809724 - 31830.58007)²)
Z = √(5625 + 31692.87748²)
Z = √(5625 + 1005004065.886948)
Taking the square root:
Z ≈ √1005009689.88695
Z ≈ 31743.78159
Therefore, the impedance of the circuit is approximately 31743.78159Ω.
None of the given options is a close approximation to the calculated value, so there might be a mistake in the question or the choices.