Reactance, resonance and time constants

capacitive reactanceinductive reactanceRC time constantresonant frequency formulaRMS voltage

How capacitors and inductors oppose alternating current, the frequency at which they cancel, and the RC and RL constants that set switching speed.

Inductive Reactance (X_L = 2πfL)

X_L = 2\pi f L

An inductor's opposition to AC current, rising in proportion to frequency.

Capacitive Reactance (X_C = 1/2πfC)

X_C = \frac{1}{2\pi f C}

A capacitor's opposition to AC current, falling as frequency rises.

LC Resonant Frequency

f = \frac{1}{2\pi\sqrt{LC}}

The natural oscillation frequency of an inductor-capacitor pair.

RC Time Constant

\tau = R C

Characteristic charging/discharging time of a resistor-capacitor circuit.

RL Time Constant (τ = L/R)

\tau = \frac{L}{R}

How quickly current builds or decays in an inductor-resistor circuit.

RC Capacitor Discharge

V = V_{0} \, e^{-t/\tau}

Exponential decay of the voltage on a capacitor discharging through a resistor.

RMS and Peak Voltage

V_{rms} = \frac{V_{peak}}{\sqrt{2}}

For a sine wave, the effective (heating-equivalent) voltage is the peak divided by √2.

How they fit together

Capacitors and inductors oppose current without dissipating energy, and they do it in opposite directions with frequency: inductive reactance rises with f, capacitive reactance falls with it. Where the two are equal they cancel, and that is resonance — the basis of every radio tuner since Marconi. The time constants are the same components viewed in the time domain rather than the frequency domain: τ = RC and τ = L/R are the same physics answering a different question.

Use reactance when the circuit is driven by a steady sine wave and you want current or a filter's corner; use the time constant when something is switched and you want to know how long the settling takes. One τ gets you 63% of the way, and the practical rule is five τ for fully charged or discharged — 99.3%, close enough for any real design. Two persistent errors: reactances do not add to resistance arithmetically, they add in quadrature to give impedance, so a 30 Ω resistor with 40 Ω of reactance gives 50 Ω and not 70 Ω. And an AC voltage quoted as 120 V is RMS, while the peak is about 170 V — insulation and diode ratings have to survive the peak, not the RMS.