Series R-L-C Reactance, Impedance, and Resonant Frequency
The base AC-circuit relations behind filter, coil, and cable-reactance work: inductive reactance XL = 2 pi f L (rises with frequency), capacitive reactance XC = 1/(2 pi f C) (falls with it), series impedance Z = sqrt(R^2 + (XL - XC)^2), and power factor R/Z. At 60 Hz a 10 ohm / 0.05 H / 50 uF branch has XL 18.85, XC 53.05 ohm, so it is capacitive (leading), Z = 35.6 ohm, PF 0.28. The branch resonates where XL = XC, at f0 = 1/(2 pi sqrt(L C)) = 100.7 Hz, where the reactances cancel, Z collapses to just R, and current peaks -- the frequency a passive filter is tuned to and a cap bank must avoid. Single-frequency, linear, lumped-element steady state; a real cable/coil is distributed, and a harmonic or transient study governs a power-system resonance.
Formula and source
XL = 2 pi f L; XC = 1/(2 pi f C); net X = XL - XC; Z = sqrt(R^2 + (XL - XC)^2); power factor = R/Z; resonant frequency f0 = 1/(2 pi sqrt(L C)).
Series R-L-C reactance, impedance, and resonance relations (classic AC circuit theory), by name; a harmonic or transient study governs a real power-system resonance.
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