About RC Time Constant Calculator — τ, Charge Time & Voltage Curve
Every resistor-capacitor pair has one number that sets its rhythm: the time constant τ = R·C. Charge a capacitor through a resistor and the voltage climbs the exponential Vc = V(1 − e^(−t/τ)) — 63.2% of the way there after one τ, 86.5% after two, 99.3% after five. This calculator computes τ from your R and C (kilohms × microfarads give milliseconds directly), evaluates both the charging and discharging curves at any elapsed time, and inverts the curve to answer the practical question: how long until the capacitor reaches x%?
The inverse solve t = −τ·ln(1 − x/100) is the working formula behind reset delays, debounce circuits, sawtooth timers, and camera-flash charge indicators. It also explains why the tool rejects a 100% target: the exponential only approaches the supply asymptotically, so "fully charged" is a convention, not an event — engineering practice calls 5τ (99.3%) fully charged, and this calculator makes that ladder explicit instead of leaving it folk knowledge.
How It Works
- Enter the resistance in kilohms and the capacitance in microfarads. Their product is the time constant in milliseconds (kΩ × µF = ms): 10 kΩ × 100 µF = 1,000 ms = 1.0 s. The headline shows τ in the natural unit — microseconds for logic-level RCs, seconds for power-supply bleeders.
- Enter the supply (or initial) voltage and an elapsed time to read the state of both curves at that moment: how far a charging capacitor has climbed toward the supply, and how much a discharging one still holds, in percent and in volts.
- Enter a target percentage for the inverse solve. The tool returns the exact time as t = −τ·ln(1 − x/100), reported in seconds and in multiples of τ — 50% takes 0.693τ (the ln 2 that RC oscillator formulas are built from), 95% takes 3τ, 99% takes 4.6τ.
- Use the 1τ–5τ chart below the tool as the mental model: the curve covers most of its ground early (63% in the first τ) and crawls at the end — each additional τ closes 63.2% of whatever gap remains. That geometric tail is why doubling a delay from 95% to 99% costs 1.6 more time constants, not a fraction.
- Remember the same τ governs discharge: a capacitor bleeding through a resistor falls to 36.8% at 1τ and under 1% at 5τ. That is the sizing logic for bleeder resistors on power-supply capacitors — pick R so that 5τ passes before fingers can reach the terminals.
Worked Example
Take R = 10 kΩ and C = 100 µF: τ = 10,000 Ω × 0.0001 F = 1.0 s. Charging from a 5 V supply, after one time constant (t = 1 s) the capacitor sits at 1 − e⁻¹ = 63.21% of the supply — 3.16 V. How long to reach 99%? t = −τ·ln(1 − 0.99) = −1.0 × ln(0.01) = 4.605 s, the "4.6τ to 99%" rule seen in datasheets. The same pair discharging from 5 V holds 5 × e⁻² = 0.677 V (13.5%) after 2 s. And asking the tool for 100% correctly fails: ln(0) is undefined because the exponential never quite arrives — at 5τ (5 s) the capacitor is at 99.33%, which engineering convention rounds to "fully charged."
RC charge time chart: how charged is a capacitor after 1τ to 5τ?
The canonical time-constant ladder, computed from the exponential itself (not transcribed): percentage charged after each whole multiple of τ, and the mirror image — what a discharging capacitor still holds. The numbers are universal: they depend only on t/τ, never on the particular R, C, or voltage.
| Elapsed time | Charged to (% of supply) | Remaining on discharge (%) |
|---|---|---|
| 1τ (one time constant) | 63.2 | 36.8 |
| 2τ | 86.5 | 13.5 |
| 3τ | 95.0 | 5.0 |
| 4τ | 98.2 | 1.8 |
| 5τ ("fully charged" convention) | 99.3 | 0.7 |
Formulas
- Time constant
τ = R × C- Charging curve
Vc(t) = V × (1 − e^(−t/τ))- Discharging curve
Vc(t) = V × e^(−t/τ)- Time to a target percentage
t = −τ × ln(1 − x/100)
Standards & References
- First-order linear step response — the solution of RC·dVc/dt + Vc = V, standard in every circuits text (e.g. Horowitz & Hill, The Art of Electronics; Nilsson & Riedel, Electric Circuits)
- The 5τ ≈ 99.3% "fully charged" convention is engineering practice, not a physical event — the exponential approaches the supply asymptotically
Frequently Asked Questions
What does the time constant actually mean physically?
τ is the time the circuit would take to finish charging if it kept its INITIAL charging rate — instead the rate decays, so after one τ it has covered only 63.2% of the gap. τ is also the memory of the circuit: after each additional τ, 63.2% of whatever error remains disappears. Big R means small charging current, big C means more charge needed; their product is the natural timescale, independent of the voltage applied.
How long does a capacitor take to charge fully?
Strictly forever — the curve approaches the supply asymptotically and never touches it, which is why this calculator rejects a 100% target (the math would be ln(0)). The working convention is 5 time constants: 99.33% charged, indistinguishable from full for almost every purpose. If your application defines "charged" differently (an ADC threshold, a POR trip point), use the inverse solve with that exact percentage instead of the convention.
Why is 63.2% the magic number instead of something round?
Because 1 − 1/e = 0.63212…, and e is baked into the physics: the charging current is proportional to the remaining voltage gap, and any quantity whose rate of change is proportional to itself follows an exponential. The same 63.2% appears in thermal time constants, motor speed responses, and low-pass filter step responses — all first-order systems share the number, which is exactly what makes τ such a portable concept.
How do I pick R and C for a specific delay?
Work backwards from the inverse formula. Decide the delay t and the threshold your circuit trips at (say a logic input reading high at 50% of supply): then R·C = t / (−ln(1 − x/100)) — for 50% that is t/0.693. Prefer adjusting R rather than C (resistor values are cheap and fine-grained), keep R below a megohm or so to dwarf leakage currents, and remember electrolytic capacitor tolerance is ±20% — precision delays belong to timer ICs, not bare RCs.
Does the same τ apply to discharging?
Yes — the discharge through the same resistor is the mirror curve V·e^(−t/τ) with the identical time constant: 36.8% remains after 1τ, under 1% after 5τ. This symmetry is the basis of bleeder-resistor sizing on power supplies: pick R so 5τ elapses within your safety window, then check the resistor's continuous dissipation V²/R while the supply runs. Note the discharge path is often different from the charge path (a different R), so compute each direction with its own resistance.
What if my capacitor charges through the resistor but discharges through a diode or switch?
Then each direction has its own time constant — τ_charge = R_charge·C and τ_discharge = R_discharge·C — and you run this calculator once per direction. That asymmetry is deliberate in many circuits: a slow charge through 100 kΩ with a fast dump through a transistor makes a sawtooth; a fast charge through a diode with a slow bleed makes a peak detector. The math in each phase is still the single-τ exponential this tool computes.