High School

An RLC circuit is driven by an AC generator. The voltage of the generator is [tex]V_{\text{RMS}} = 97.9 \, \text{V}[/tex]. The figure shows the RMS current through the circuit as a function of the driving frequency.

1. What is the resonant frequency of this circuit? Please notice that the resonance curve passes through a grid intersection point.
- [tex]4.00 \times 10^2 \, \text{Hz}[/tex]

2. If the inductance of the inductor is [tex]L = 273.0 \, \text{mH}[/tex], then what is the capacitance [tex]C[/tex] of the capacitor?
- (Provide the capacitance calculation)

3. What is the ohmic resistance of the RLC circuit?
- [tex]122.4 \, \Omega[/tex]

4. What is the power of the circuit when the circuit is at resonance?
- (Provide the power calculation)

Answer :

Therefore, the power of the circuit when the circuit is at resonance is 77.8 W.

An RLC circuit is driven by an AC generator. The voltage of the generator is V_RMS = 97.9 V. The figure shows the RMS current through the circuit as a function of the driving frequency. The resonant frequency of this circuit is given by 4.00×10^2 Hz.

The inductance of the inductor is L = 273.0 mH.The capacitive reactance X_c of the capacitor in the RLC circuit can be calculated using the formula:$$X_C=\frac{1}{2\pi fC}$$where f is the frequency of the AC voltage source and C is the capacitance of the capacitor.

The resonant frequency of the circuit occurs when the capacitive and inductive reactances are equal and opposite. Therefore,X_L = X_CwhereX_L = 2πfL and X_C = 1/2πfCTherefore,2πfL = 1/2πfCwhere f is the resonant frequency of the circuit.Substituting the values of f and L, we get:2π × 4.00×10^2 × 273.0×10^-3 = 1/2π × CTherefore, C = 1/(2π × 4.00×10^2 × 273.0×10^-3) = 0.296 × 10^-6 FThe ohmic resistance of the RLC circuit is 122.4 ohm.

The power of the circuit when the circuit is at resonance can be calculated using the formula:P = V_RMS^2/Rwhere R is the resistance of the circuit.Substituting the values of V_RMS and R, we get:P = (97.9)^2/122.4 = 77.8 W

Therefore, the power of the circuit when the circuit is at resonance is 77.8 W.

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