RC Time Constant Explained: Tau, the Charge Curve, and the 5τ Rule

Every resistor-capacitor pair has a single number that governs how fast it reacts: the time constant, written with the Greek letter tau (τ). It shows up in filter cutoffs, timer periods, button debounce windows, and the rise time of a logic edge. Once you can compute it and read the charge curve in your head, a surprising amount of analog timing stops being mysterious. This guide walks through the formula, the famous 63% and 99% markers, a worked example, and the everyday circuits where τ decides the design.

The Formula: τ = R · C

The time constant is just the product of resistance and capacitance:

τ = R · C

Use ohms for R, farads for C, and you get seconds for τ. That unit bookkeeping is where most beginners trip. A 10 kΩ resistor is 10,000 Ω, and a 100 µF capacitor is 0.0001 F. Multiply them and you get 1 second. Keep the kilo- and micro- prefixes in by mistake and your answer is off by the product of those prefixes, which here would be a factor of a hundred thousand.

The time constant does not tell you when the capacitor is "done." It tells you the natural timescale of the circuit. After one τ the capacitor has moved 63.2% of the way toward its final voltage. After another τ it covers 63.2% of the remaining gap, and so on. The motion is exponential, not linear, which is exactly why a single number captures the whole behavior.

Why One Tau Is 63%, Not 100%

Charging a capacitor from zero toward a supply voltage V0 follows:

V(t) = V0 · (1 − e^(−t/τ))

At t = τ the exponent is −1, and e^(−1) ≈ 0.368. So the voltage is 1 − 0.368 = 0.632 of the supply, or 63.2%. That number is not arbitrary trivia; it falls straight out of the math, and it is the reason engineers measure RC circuits in multiples of τ instead of fixed milliseconds.

Discharging is the mirror image through the same τ:

V(t) = V0 · e^(−t/τ)

At one τ the capacitor has fallen to 36.8% of where it started. Charge climbs to 63.2% while discharge drops to 36.8% over the same interval, because the two curves are reflections of one exponential. A common mistake is assuming the two are symmetric in some simpler way; they share τ, but they head in opposite directions from different starting points.

The 5τ Rule: When a Capacitor Counts as Full

Mathematically the exponential never actually reaches the supply. Each τ closes 63.2% of the remaining distance, so the voltage creeps closer forever without arriving. In practice, engineers draw the line at five time constants:

| Time | Charge (% of supply) | Discharge (% of start) | |------|----------------------|------------------------| | 1τ | 63.2% | 36.8% | | 2τ | 86.5% | 13.5% | | 3τ | 95.0% | 5.0% | | 4τ | 98.2% | 1.8% | | 5τ | 99.3% | 0.7% |

At 5τ the capacitor sits at 99.3% of the supply, which is close enough for almost any design to call it fully charged. So the rule of thumb is simple:

Full charge (or discharge) time ≈ 5 × R × C

For τ = 1 ms that is about 5 ms. For τ = 1 second it is about 5 seconds. The same 5τ window tells you when a bleeder resistor has drained a storage cap down to 0.7% of its original charge, which is the practical "safe to touch" threshold.

A Worked Example: 10 kΩ and 100 µF

Take a 10 kΩ resistor feeding a 100 µF capacitor from a 5 V supply.

• τ = R · C = 10,000 Ω × 0.0001 F = 1 second
• At 1τ (1 s): V = 5 × 0.632 = 3.16 V
• At 5τ (5 s): V = 5 × 0.993 = 4.97 V, effectively full

So after one second the cap has reached 3.16 V, after five seconds it is essentially at the 5 V rail. If you instead need to know when the voltage crosses a specific level, invert the charge formula: t = −τ · ln(1 − Vtarget/V0). To hit 2 V on this circuit, t = −1 × ln(1 − 2/5) ≈ 0.51 seconds. The RC time constant calculator runs both directions and draws the curve, so you can read time-to-voltage or voltage-at-time without reaching for a calculator and the natural log.

I keep a 10 kΩ / 100 µF pair in my head as a reference because it lands on exactly one second. When I am sketching a new timer or filter, I scale from there: drop the cap to 100 nF and τ falls to 1 ms; bump the resistor to 100 kΩ and it climbs to 10 seconds. That anchor has saved me from more than one off-by-a-thousand error when I was staring at a 104-marked ceramic cap and almost typed it in as 100 µF instead of 100 nF.

Where Tau Shows Up: Filters, Timers, and Debounce

The same τ drives several everyday circuits:

• Passive RC filters. The cutoff frequency of an RC low-pass or high-pass is f = 1 / (2π·R·C), tied directly to τ. A 1.6 kΩ resistor with a 100 nF capacitor gives τ = 160 µs and a corner near 1 kHz. Size R and C to put the cutoff where you want it.
• 555 timers and reset circuits. Astable and monostable timers charge a capacitor through a resistor to set their period or pulse width. The 1τ and 5τ markers tell you when the timing cap crosses the threshold.
• Button debounce. A mechanical switch bounces for a few milliseconds. An RC network plus a Schmitt-trigger input smooths the edge; pick R and C so 5τ comfortably exceeds the bounce duration.
• Discharge bleeders. A resistor across a storage cap drains it after power-off. Read the 5τ row to find the wait-before-touch interval.

If you are sizing real parts, you will also want to decode the component markings: a resistor color code calculator turns the bands into a value you can plug into τ = R · C. Mixing up µF and nF is the single most common RC mistake, and it is always a clean factor-of-1000 error in your time constant, so it pays to match the unit to the part marking every time.

Putting It Together

The RC time constant compresses a whole exponential into one number. Compute τ = R · C in ohms and farads, remember that 1τ gets you to 63% and 5τ effectively completes the move, and you can size filters, timers, and debounce circuits by feel. When the numbers matter, let a tool draw the curve and the 0τ–5τ table so you can see exactly where the voltage lands instead of trusting a straight-line guess. The exponential rewards a little intuition, and τ is where that intuition starts.

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Made by Toolora · Updated 2026-06-13