RLC Series Impedance

RLC impedance: |Z| = √(R² + (X_L − X_C)²)

Feed a series RLC circuit with a sinusoidal signal of angular frequency ω = 2πf and each element responds with its own reactance. The inductor gives X_L = ωL, the capacitor gives X_C = 1/(ωC). Combine them with the resistance and you get the impedance magnitude, |Z| = √(R² + (X_L − X_C)²), measured in ohms. Something special happens at the resonant frequency f₀ = 1/(2π√LC): the two reactances are equal (X_L = X_C) and cancel out, leaving |Z| = R. In a series RLC that's the minimum; flip to parallel and it becomes the maximum. Example: take R = 100 Ω, L = 10 mH, C = 1 µF at f = 1 kHz. Then ω ≈ 6283 rad/s, X_L ≈ 62.8 Ω, X_C ≈ 159 Ω, and |Z| ≈ √(100² + 96.2²) ≈ 139 Ω.

Applications: filters, radio tuning, RF matching, power factor

This is what analog filters are built on — low-pass, high-pass, band-pass, notch. It also drives the radio tuning circuits that pick out a station, the 50 Ω impedance matching in RF amplifiers and antennas, and EMI suppression networks. On the factory floor it shows up as power-factor correction, where capacitor banks offset inductive loads to cut reactive current and the penalties the utility charges for it.

FAQ

What is the Q factor? Q = ω₀L/R tells you how sharp the resonance is. A high Q gives a narrow bandwidth, which is what you want when a radio has to be selective. A low Q spreads things out into a flatter response.

Series vs parallel? At resonance a series RLC drops to its minimum impedance (R) and behaves like a band-pass. A parallel RLC does the opposite, rising to maximum impedance and acting as a band-stop or tank circuit.

What is the phase angle? φ = arctan((X_L − X_C)/R). It comes out positive when the circuit looks inductive, with current lagging voltage, and negative when it looks capacitive, with current running ahead.

Why 50 Ω? It's an old RF compromise. Coaxial cable handles the least loss around 77 Ω and the most power around 30 Ω, and 50 Ω splits the difference. That's why you see it on transmitters, antennas and nearly every piece of lab gear.