Series RLC Impedance Calculator

Series RLC impedance

Wire R, L and C in series and the same current runs through all three, while the source voltage splits up among them. The magnitude of the total impedance comes out to Z = sqrt(R^2 + (Xl - Xc)^2). Here the inductive reactance is Xl = 2πfL and the capacitive reactance is Xc = 1/(2πfC). Plug in R = 100 Ω, L = 10 mH, C = 1 µF and f = 1 kHz and you get Z ≈ 124 Ω with a capacitive phase, because at that frequency Xc outweighs Xl.

Resonance happens when Xl equals Xc. The reactive part of Z drops to zero and the impedance bottoms out at R. Current peaks at this point, and the voltages across L and C cancel out against each other even though each one on its own can run far higher than the source voltage (that's the Q-factor amplification). Stay below resonance and the circuit acts capacitive; go above it and it turns inductive.

Applications

Series RLC circuits show up early in any Electrical Engineering program (Sedra/Smith - Microelectronics, Boylestad - Circuit Analysis, Nilsson/Riedel). In practice you find them inside band-pass filters, crystal radio receivers, impedance matching networks, AC sensors and the snubber circuits that protect switching converters. Standards like IEEE 519, IEC 61000-4 and ABNT NBR 5410 point to series tuned filters when the goal is harmonic mitigation or EMC compliance.

FAQ

Why is the impedance minimum at resonance? Xl and Xc sit 180° apart, so they cancel and all that's left in the expression is R.

Can the voltage across L or C exceed the source voltage? It can. At resonance each one reaches Q times the input voltage, which is exactly what makes crystal receivers work.

How do I know if the circuit is capacitive or inductive? Just compare Xl and Xc. If Xl > Xc the circuit is inductive and the current lags; if Xc > Xl it's capacitive and the current leads.