In a series RLC circuit, at resonance which statement is true?

In a series RLC circuit, at resonance which statement is true?

• A. X_L ≠ X_C
• B. Z ≈ ∞
• C. X_L = X_C and Z ≈ R
• D. ω0 = 0

Answer: (C)

In a series RLC circuit, the key idea at resonance is that the inductive and capacitive reactances cancel each other out. The total impedance is Z = R + j(X_L − X_C). At the resonance frequency, X_L = X_C, so the imaginary part is zero and the impedance becomes purely real, equal to R (for ideal components). In other words, X_L = X_C and Z is essentially R. The resonance frequency is set by ω0 such that ω0 L = 1/(ω0 C), which gives ω0 = 1/√(LC). This is why you get the maximum current for a given source voltage: the circuit presents only the resistance to the source, minimizing impedance. If non-idealities are present, Z may be very close to R rather than exactly R, hence the ≈ notation. The other statements don’t hold: the impedance isn’t infinite at resonance, and the resonance occurs at a finite frequency, not at zero.

In a series RLC circuit, the key idea at resonance is that the inductive and capacitive reactances cancel each other out. The total impedance is Z = R + j(X_L − X_C). At the resonance frequency, X_L = X_C, so the imaginary part is zero and the impedance becomes purely real, equal to R (for ideal components). In other words, X_L = X_C and Z is essentially R. The resonance frequency is set by ω0 such that ω0 L = 1/(ω0 C), which gives ω0 = 1/√(LC). This is why you get the maximum current for a given source voltage: the circuit presents only the resistance to the source, minimizing impedance. If non-idealities are present, Z may be very close to R rather than exactly R, hence the ≈ notation. The other statements don’t hold: the impedance isn’t infinite at resonance, and the resonance occurs at a finite frequency, not at zero.