How do X_L and X_C vary with frequency, and what is the behavior of a series LC circuit as frequency changes?

• A. X_L decreases with frequency; X_C increases; at resonance X ≈ 0; inductive at high frequencies; series LC tends to infinite impedance away from resonance
• B. X_L increases with frequency; X_C decreases with frequency; near resonance X ≈ 0; at low frequencies capacitive; at high frequencies inductive; series LC tends to zero reactance at ω0 and high impedance away from ω0
• C. X_L constant; X_C constant; resonance occurs when L equals C
• D. X_L increases with frequency; X_C increases with frequency; resonance has X ≈ X

Answer: (B)

In a series LC circuit, the two reactive parts respond opposite to frequency: the inductor’s reactance increases with frequency (X_L = ωL), while the capacitor’s reactance decreases with frequency (X_C = 1/(ωC)). The total reactance is X = X_L − X_C = ωL − 1/(ωC).

At low frequencies, the capacitor dominates, so the circuit looks capacitive (negative reactance). At high frequencies, the inductor dominates, so it looks inductive (positive reactance). There is a particular frequency, ω0 = 1/√(LC), where the two reactive parts cancel each other out, so X ≈ 0 and the impedance is purely resistive (equal to the series resistance, or zero in an ideal lossless case).

Thus, the impedance is smallest and purely resistive at resonance, and it grows large as you move away from that frequency, approaching infinity in the ideal case as ω → 0 or ω → ∞.