In an RLC circuit, the impedance is minimized when the inductive and capacitive reactances cancel. What condition achieves this?

In an RLC circuit, the impedance is minimized when the inductive and capacitive reactances cancel. What condition achieves this?

• A. X_L = 0
• B. X_L = X_C
• C. X_C = 0
• D. R = 0

Answer: (B)

This question targets resonance in a series RLC circuit, where the inductive and capacitive reactances balance so the impedance is purely resistive. The inductive reactance is XL = ωL and the capacitive reactance is XC = 1/(ωC). The cancellation happens when XL equals XC, i.e., ωL = 1/(ωC). This gives the resonance frequency ω0 = 1/√(LC). At that frequency, the imaginary part of the impedance Z = R + j(XL − XC) is zero, so Z = R. Since the reactive part has vanished, the impedance magnitude reaches its minimum for that circuit configuration. The underlying idea is that the energy sloshes between the inductor and the capacitor without net reactive opposition to the current. The other scenarios don’t produce this balance: removing one reactance or the resistance doesn’t create the canceling effect that yields a purely resistive, minimized impedance.

This question targets resonance in a series RLC circuit, where the inductive and capacitive reactances balance so the impedance is purely resistive. The inductive reactance is XL = ωL and the capacitive reactance is XC = 1/(ωC). The cancellation happens when XL equals XC, i.e., ωL = 1/(ωC). This gives the resonance frequency ω0 = 1/√(LC). At that frequency, the imaginary part of the impedance Z = R + j(XL − XC) is zero, so Z = R. Since the reactive part has vanished, the impedance magnitude reaches its minimum for that circuit configuration. The underlying idea is that the energy sloshes between the inductor and the capacitor without net reactive opposition to the current. The other scenarios don’t produce this balance: removing one reactance or the resistance doesn’t create the canceling effect that yields a purely resistive, minimized impedance.