What the RC time constant tells you
The time constant τ = RC is the single number that sets how fast a capacitor charges or discharges through a resistor. Multiply ohms by farads and the units genuinely come out in seconds: a volt per amp times a coulomb per volt leaves coulombs per amp, which is seconds. Ten kilohms and one hundred microfarads give exactly one second.
The reason a capacitor charges on a curve rather than a ramp is feedback. Current through the resistor is the voltage across it divided by R, and the voltage across the resistor is whatever the supply has left over after the capacitor's own voltage. So the fuller the capacitor gets, the smaller the driving voltage, the smaller the current, and the slower it fills. Every instant the capacitor closes a fixed fraction of the remaining gap, which is the definition of exponential behaviour.
That fraction is what makes τ so useful. In one time constant the capacitor always covers 63.2% of whatever gap remains, no matter where it started or which direction it is moving. In two it has covered 86.5%, in three 95.0%, and in five 99.3%. Those percentages depend only on t/τ, never on the supply voltage, so the same table serves a 3.3 V logic reset and a 4,160 V capacitor bleed-down.
The four equations, and where each one comes from
All of first-order RC transient response comes from one differential equation, RC · dVC/dt + VC = VS, and four rearrangements of its solution cover everything you will need.
1. The time constant. τ = R × C, with resistance in ohms and capacitance in farads. Raising either one slows the circuit in exact proportion — doubling the resistor and doubling the capacitor gives four times the delay.
2. Voltage while charging. VC(t) = VS(1 − e−t/τ). At t = 0 the exponential is 1 and the capacitor voltage is zero; as t grows the exponential shrinks toward zero and the capacitor voltage approaches the supply. It never arrives, which matters when you specify a target.
3. Voltage while discharging. VC(t) = VS e−t/τ, where VS is now the voltage the capacitor started at. This is the same curve turned upside down; the two always add to the starting voltage at any given t/τ.
4. Time to a target. Take logarithms of the voltage equations. Charging, t = −τ ln(1 − VC/VS); discharging, t = −τ ln(VC/VS). Both require the target to lie strictly between zero and the supply, because the logarithm of zero is undefined — which is the mathematical form of the physical statement that an exponential approach never completes.
Current follows the same exponential in both directions: I(t) = (VS/R) e−t/τ. The magnitude at a given t/τ is identical whether you are charging or discharging; only the direction of flow reverses, and this calculator reports the magnitude. At the first instant the capacitor has no voltage to oppose the source, so it behaves as a short circuit and the current is set entirely by the resistor — plain Ohm's law. That first-instant current is the number that stresses switch contacts and drives inrush limiting.
Worked example: 10 kΩ charging 100 µF from 12 V
Take a 100 µF capacitor charging through a 10 kΩ resistor from a 12 V supply. Every line below can be checked with a scientific calculator.
- Convert to base units. R = 10 kΩ = 1.0 × 104 Ω; C = 100 µF = 1.0 × 10−4 F.
- Multiply for the time constant. τ = 1.0 × 104 × 1.0 × 10−4 = 1.00 s.
- Find the first-instant current. I0 = 12 ÷ 10,000 = 1.20 mA. The resistor dissipates 12 × 0.0012 = 14.4 mW at that moment, falling from there.
- Evaluate the voltage after one second. t/τ = 1, e−1 = 0.367879, so VC = 12 × (1 − 0.367879) = 12 × 0.632121 = 7.585 V, which is 63.21% of the supply.
- Find the current at that same moment. I = 1.20 mA × 0.367879 = 0.4415 mA. Note the voltage across the resistor, 12 − 7.585 = 4.415 V, divided by 10 kΩ, gives the same answer.
- Find how long it takes to reach 6 V. Half the supply, so t = −1.00 × ln(1 − 0.5) = ln 2 = 0.693 s. Every RC halving takes 0.693τ, exactly as every radioactive half-life is 0.693 of its mean lifetime.
- Find the settling time. 5τ = 5.00 s, at which point VC = 12 × 0.993262 = 11.92 V, within 0.08 V of the supply.
- Check the stored energy. Once fully charged, E = ½CV2 = 0.5 × 1.0 × 10−4 × 144 = 7.2 mJ.
Now discharge the same capacitor through the same resistor. τ is unchanged at 1.00 s, so after one second the capacitor holds 12 × 0.367879 = 4.415 V, and it falls to 6 V after the same 0.693 s. Charging and discharging are symmetric whenever the resistance in both paths is the same — and in a real circuit it very often is not, because the discharge path may be a different resistor entirely.
How to read the result
Start with τ against the timescale you care about. If τ is much shorter than the event you are timing, the capacitor is fully settled and behaves as a DC voltage source; if τ is much longer, the capacitor barely moves and behaves as a short-term open circuit. The interesting design region is where τ is within about a decade of your event, and that is where this calculator earns its keep.
Use 5τ as the settling number and 0.693τ as the halving number. Five time constants leaves 0.67% of the gap, which is below the resolution of most measurements and comfortably inside a logic threshold; that is why power-on reset circuits, soft-start ramps and scope-probe compensation are all specified in multiples of τ. If you need tighter than 1%, 7τ leaves 0.09% and 10τ leaves 0.0045%.
Watch the first-instant current whenever the resistance is small. It equals the full supply divided by R alone, so a 12 V rail charging a bank through 0.1 Ω sees 120 A at the moment of connection. That surge is what welds relay contacts and blows soft-start resistors, and no capacitance value changes it — only more resistance, or active inrush limiting, does.
Finally, treat a long discharge τ as a safety figure, not a convenience. A bleeder resistor is sized so a capacitor bank falls to a safe voltage in a defined time, and the time you need is set by the target you accept: 4.6τ to fall to 1% of the starting voltage, 6.9τ to fall to 0.1%. Enter your own safe-touch voltage as the target and read the time directly. NFPA 70E requires stored electrical energy to be released before you can establish an electrically safe work condition, and equipment standards set the discharge time the design must meet; take the target voltage and the allowed time from whichever document governs your installation rather than from a habit.
Percentage charged and remaining at each time constant
| Time constants (n = t/τ) | Charged toward supply | Remaining while discharging | Typical use of this point |
|---|---|---|---|
| 0 | 0.00% | 100.00% | The switching instant; current is at its peak |
| 0.5 | 39.35% | 60.65% | |
| 0.693 | 50.00% | 50.00% | The RC half-time, ln 2 |
| 1 | 63.21% | 36.79% | The definition of one time constant |
| 2 | 86.47% | 13.53% | |
| 2.303 | 90.00% | 10.00% | One decade, ln 10 |
| 3 | 95.02% | 4.98% | Often adequate for analogue settling |
| 4 | 98.17% | 1.83% | |
| 4.605 | 99.00% | 1.00% | Common bleed-down specification |
| 5 | 99.33% | 0.67% | The conventional “fully charged” point |
| 7 | 99.91% | 0.09% | Ten-bit settling |
| 10 | 99.9955% | 0.0045% | Better than 14-bit settling |
Percentages are rounded from the exponential; e⁻¹ = 0.367879, e⁻² = 0.135335, e⁻⁵ = 0.0067379. The 0.693 and 2.303 rows are ln 2 and ln 10, the two logarithms worth memorising.
Assumptions and pitfalls
- The source must be stiff. The formula assumes the supply holds its voltage regardless of current. If the source has meaningful output resistance, add it to R — that is why the input on this page asks for the total series resistance, not just the visible resistor.
- One R and one C only. A network of several resistors and capacitors is not a single exponential. Reduce it to a Thévenin equivalent first, or use the capacitor series and parallel calculator to combine capacitors before you start.
- Leakage sets a ceiling on long time constants. Above about a megohm the capacitor's own insulation resistance and the board surface begin to compete with your resistor, and an electrolytic can leak enough to prevent full charge entirely. Long timers use film or ceramic capacitors for this reason.
- Charge and discharge paths are usually different. A 555 timer charges through two resistors and discharges through one, so its two time constants differ; the 555 timer calculator handles that asymmetry explicitly.
- Tolerance stacks multiplicatively. A 5% resistor with a 20% electrolytic gives a time constant good to roughly ±25%. If timing matters, use a 1% resistor and a film capacitor, or trim.
- A target of exactly zero or exactly the supply has no answer. The exponential approaches but never reaches its limit, so the calculator reports no finite time rather than a misleading large one. Specify a real threshold instead.
The frequency-domain twin, and when to use it instead
The same RC product describes this circuit's behaviour with sine waves, and the translation is one equation: fc = 1 ÷ (2πRC) = 1 ÷ (2πτ). A one-second time constant is a 0.159 Hz corner frequency. Use the time domain when your signal is a step, a pulse or a switch closure; use the RC filter cutoff frequency calculator when it is a continuous waveform whose amplitude and phase you care about. Two constants link the views. The 10% to 90% rise time of a step is τ ln 9 = 2.197τ, usually quoted as 2.2τ; multiply that by the bandwidth 1 ÷ (2πτ) and the τ cancels, leaving 2.197 ÷ 2π = 0.35. That 0.35 figure holds for this single-pole response and is only an approximation for a multi-pole one.
If you want the capacitor's opposition at a single frequency rather than its behaviour over time, that is reactance, and the capacitive reactance calculator gives it. The two views describe the same component: at the corner frequency, the capacitor's reactance is exactly equal to the resistance.
This page models an ideal capacitor and an ideal resistor. It does not include dielectric absorption, which lets a capacitor recover some voltage after being shorted and matters in sample-and-hold and high-accuracy integrator circuits; it does not include the capacitor's series inductance, which only shows up at nanosecond timescales; and it does not model semiconductor switch resistance, which is frequently the dominant resistance in a switched-capacitor circuit.
Key terms
- Time constant (τ)
- The product RC, in seconds. The time in which the capacitor covers 63.2% of the remaining gap between its present voltage and its final voltage.
- Transient
- The temporary behaviour of a circuit between one steady state and the next. In an RC circuit the transient is the exponential curve; the steady state is what remains after it has decayed.
- Settling time
- The time to come within a stated tolerance of the final value. Conventionally 5τ for 1%, but state the tolerance because the multiplier depends on it.
- Bleeder resistor
- A resistor permanently connected across a capacitor so that stored energy drains to a safe level within a specified time after the supply is removed.