Series RLC Impedance
|Z| and phase of a series R-L-C at frequency, plus where it resonates.
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The engineering
In series, the inductor and capacitor fight: their reactances subtract, and whichever wins sets the sign of the phase. Below resonance the capacitor dominates (current leads, φ negative); above, the inductor (current lags). At f₀ they cancel exactly and the whole network collapses to bare R — minimum impedance, maximum current, which is why a series-resonant circuit is a frequency-selective *short*.
The resonant-frequency row is the sanity check: if your operating frequency is near f₀, |Z| swings violently with small component shifts and the R you specified is doing all the work. That collapse is also a feature — crystal filters, trap antennas, and spot-frequency shunts all exploit the series-resonant dip on purpose.
Where this math comes from
Kelvin predicted LC oscillation in 1853, but adding the resistor and the driving frequency stayed calculus until 1893, when — at the same Chicago Electrical Congress — Arthur Kennelly published 'Impedance,' putting √(R²+X²) and the complex plane on record, and Charles Steinmetz demonstrated the full phasor method that made AC networks algebra.
The word 'impedance' itself is Heaviside's (1886). Within a decade the series-RLC solution was in every engineering curriculum, and it has opened the AC-circuits chapter of essentially every textbook since.
- 1853Lord KelvinLC oscillation predicted — the reactive tug-of-war.
- 1886Oliver HeavisideCoins 'impedance' for AC opposition.
- 1893Arthur Kennelly'Impedance' paper — complex Z formalized.
- 1893Charles SteinmetzPhasor method makes RLC arithmetic routine.
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