RC Time Constant Calculator (Charging, Discharging and Cutoff Frequency)

How it works

τ = R × C · f_c = 1 ÷ (2π × τ) · charging: V(t) = V_s + (V₀ − V_s) e^(−t/τ) · discharging: V(t) = V₀ e^(−t/τ) · I(t) = (ΔV ÷ R) e^(−t/τ) · time to x% = −τ ln(1 − x) · rise time 10–90% = τ ln 9 ≈ 2.2τ · E = ½CV²

The time constant τ = RC sets the pace of charging and discharging in a resistor–capacitor circuit. After one τ the capacitor reaches about 63.2% of the applied voltage, after five τ it is over 99% charged, which engineers treat as complete; when discharging, 36.8% of the voltage is left after one τ. The same τ fixes the −3 dB cutoff frequency of a simple RC low-pass or high-pass filter, f_c = 1/(2πτ). Enter R and C with the unit menus (Ω, kΩ, MΩ and F down to pF), or solve for the resistor or capacitor that gives a target time constant or cutoff frequency — the calculator then suggests the nearest standard E24 resistor or E12 capacitor and what it would give. Add the supply voltage and a time to see the capacitor voltage and current at that moment, the time to reach any charge level, the 10–90% rise time and the charge and energy stored. A table steps through 0.5τ to 5τ and two charts draw the voltage and current curves.

Worked example

With R = 10 kΩ and C = 47 µF, τ = 10,000 Ω × 0.000047 F = 0.47 s. Charging from 0 to 5 V, the capacitor is at 5 × (1 − e^(−1)) = 3.161 V after 0.47 s, and the current has fallen from 5 V ÷ 10 kΩ = 0.5 mA to 0.1839 mA. It reaches 90% after 0.47 × ln 10 = 1.082 s and is essentially full after 5 × 0.47 = 2.35 s; the rise time is 1.033 s and it stores ½ × 47 µF × 5² = 587.5 µJ. As a filter the cutoff is 1 ÷ (2π × 0.47) = 0.3386 Hz. For a 1 kHz low-pass with a 10 nF capacitor, R = 1 ÷ (2π × 1000 × 10 nF) = 15.92 kΩ; the nearest E24 resistor, 16 kΩ, gives 994.7 Hz.

Sources: RC time constant — Wikipedia · Electronics Tutorials — RC charging circuit

Frequently asked questions