In a certain series RLC circuit, Irms = 9.00 A, ΔVrms = 245 V, and the current leads the voltage by 30.0°.(a)What is the total resistance (in Ω) of the circuit? Ω(b)Calculate the reactance (in Ω) of the circuit (XL − XC). Ω(c)What If? If the voltage in the circuit is maintained by an AC source operating at f = 60.0 Hz and the resonant frequency of the circuit is f0 = 90.0 Hz, what are the values of C (in µF) and L (in mH) in the circuit?
Question
In a certain series RLC circuit, Irms = 9.00 A, ΔVrms = 245 V, and the current leads the voltage by 30.0°.(a)What is the total resistance (in Ω) of the circuit? Ω(b)Calculate the reactance (in Ω) of the circuit (XL − XC). Ω(c)What If? If the voltage in the circuit is maintained by an AC source operating at f = 60.0 Hz and the resonant frequency of the circuit is f0 = 90.0 Hz, what are the values of C (in µF) and L (in mH) in the circuit?
Solution
(a) The total resistance (R) of the circuit can be calculated using Ohm's law, which states that the voltage (V) is equal to the current (I) times the resistance (R). In this case, we are given the rms values for current (Irms) and voltage (ΔVrms), so we can use the formula:
R = ΔVrms / Irms R = 245 V / 9.00 A R = 27.22 Ω
(b) The reactance (X) of the circuit can be calculated using the formula:
X = ΔVrms / (Irms * sin(θ))
where θ is the phase angle. In this case, the current leads the voltage by 30.0°, so θ = 30.0°. Therefore:
X = 245 V / (9.00 A * sin(30.0°)) X = 245 V / (9.00 A * 0.5) X = 54.44 Ω
(c) The values of C (capacitance) and L (inductance) in the circuit can be calculated using the formulas:
C = 1 / (2 * π * f * Xc)
and
L = XL / (2 * π * f)
where f is the frequency. In this case, the voltage in the circuit is maintained by an AC source operating at f = 60.0 Hz and the resonant frequency of the circuit is f0 = 90.0 Hz. Therefore:
C = 1 / (2 * π * 60.0 Hz * 54.44 Ω) C = 4.89 µF
and
L = 54.44 Ω / (2 * π * 60.0 Hz) L = 144.35 mH
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