Formula
Tau is resistance times capacitance. Charging from zero toward the supply is Vs times (1 − e^(−t/τ)). Discharging from Vs toward zero is Vs times e^(−t/τ). The chart draws both from 0 to 5τ. The solid line is the charge. The dashed line is the discharge. The table repeats the voltages so the colors are not the only record.
τ = R C
Vc = Vs (1 − e^(−t/τ))
Vd = Vs e^(−t/τ)
One time constant reaches 63.2% on the way up and leaves 36.8% on the way down. The cutoff of this R and C is on the RC filter calculator. A 555 monostable is a different fraction of the same product, on the 555 timer calculator.
Worked examples
10 kΩ and 100 µF. Tau is 1.000 s.
Charging toward 5 V, at t = τ. 3.1606 V.
Discharging from 5 V, at t = τ. 1.8394 V.
Reference
At 5τ the remaining gap on a charge is e⁻⁵ of the supply, about 0.67%.
Questions
What is tau for 10 kΩ and 100 µF?
1 second. Tau is R times C.
What voltage is that after one time constant?
Charging from 0 toward 5 V, 5 × (1 − 1/e) = 3.161 V. Discharging from 5 V, 5/e = 1.839 V.
When is the capacitor finished?
The ideal curve never finishes. At five time constants the charge is 99.3% of the way there. The table shows 0, 0.5, 1, 2, 3, and 5 tau.
Sources
- Series RC step response. τ = RC. Charge and discharge are the two exponentials above.