Circuit design / transient response

RC time constant

Compute the time constant, corner frequency and charge curve of a resistor–capacitor network. The 63.2% per τ progression is tabulated explicitly, which is what most timing and debounce calculations actually need.

Full charge (5τ)5.00 msInitial current0.5000 mACharge at full0.500 µC

Time constant τ

1.000 ms

Time to reach 63.2% of the step

Cut-off frequency

159.15 Hz

−3 dB point of the low-pass response

Voltage at 1.00 ms

3.1606 V

63.21% of the input

Stored energy

0.0012 mJ

At full charge, ½CV²

Charge curve at time constants

The classic 63.2% per τ progression, with the residual gap to the supply rail.

τ = 1.000 ms

Charge curve at time constants
Time	V across C	Share of V in	Remaining gap
0.5 τ (0.50 ms)	1.9673 V	39.35%	3.0327 V
1 τ (1.00 ms)	3.1606 V	63.21%	1.8394 V
2 τ (2.00 ms)	4.3233 V	86.47%	0.6767 V
3 τ (3.00 ms)	4.7511 V	95.02%	0.2489 V
4 τ (4.00 ms)	4.9084 V	98.17%	0.0916 V
5 τ (5.00 ms)	4.9663 V	99.33%	0.0337 V

After 5τ the capacitor is within 0.7% of the supply — the usual definition of “fully charged” in switching analysis.

Charge curve

Capacitor voltage against time, measured in time constants.

Voltage at each time constant

The 63 / 86 / 95 / 98 / 99% progression.

Canonical equations

The exact relationships implemented by this engine.

τ = R·C · seconds, with C in farads

V_C = V_in(1 − e^(−t/τ)) · from an uncharged state

V_C = V₀·e^(−t/τ) · decay toward zero

f_c = 1/(2πRC) · the −3 dB frequency of the divider

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

An RC network responds to a step with a decaying exponential governed by a single number: τ = RC. Everything else — the 63.2% at one time constant, the 99.3% at five, the corner frequency — follows from it. This engine computes τ and evaluates the curve wherever you point it.

Step-by-step methodWorked exampleAssumptions explained

How to use

Enter the series resistance.

Enter the capacitance in microfarads.

Set the step amplitude — the voltage the input swings to.

Choose an evaluation time to sample the curve.

Read the time constant and the corner frequency.

Check the table: five time constants is the usual definition of “fully charged”.

The formula

\tau = RC, \qquad V_C(t) = V_{\text{in}}\left(1 - e^{-t/\tau}\right), \qquad f_c = \frac{1}{2\pi\tau}, \qquad E = \tfrac{1}{2}CV^2

Reference: Standard first-order linear network solution; IEC 60617 for RC filter conventions.

Worked example

10 kΩ and 0.1 µF driven by a 5 V step (default inputs)

Inputs

R = 10 kΩ.
C = 0.1 µF.
Step amplitude 5 V.
Evaluate at 1 ms.

Calculation path

1
Time constant: 10,000 × 0.1×10⁻⁶ = 1 ms.
2
Corner frequency: 1 / (2π × 0.001) = 159.15 Hz.
3
At t = τ, the capacitor reaches 63.2% of 5 V = 3.16 V.
4
Stored energy at full charge: ½ × 0.1 µF × 25 = 1.25 µJ.

Result

τ = 1 ms · corner 159.15 Hz · 3.16 V at one time constant · 1.25 µJ stored at full charge.

Interpreting the result

The 63.2% figure is not arbitrary — it is 1 − e⁻¹. At two time constants you are at 86.5%, at three 95.0%, at five 99.3%. Most digital timing specifications use 5τ for “settled”. The corner frequency matters separately: above it the network attenuates at 20 dB per decade, which is why an RC pair is the simplest possible low-pass filter.

Assumptions & limitations

Ideal first-order model: no equivalent series resistance, no leakage, no dielectric absorption.

The source has zero output impedance and switches instantaneously.

The capacitor starts fully discharged for the charging curve.

Component values do not drift with temperature or voltage; ceramic capacitors in particular lose capacitance under DC bias.

No load is connected across the capacitor — a load adds a parallel resistance and changes τ.

Frequently asked questions

What exactly does the time constant mean?

It is the time to reach 63.2% of the final value, and also the time for a discharging capacitor to fall to 36.8% of its starting voltage. It is the reciprocal of the exponential decay rate.

How do I use this for a debounce circuit?

Choose τ so the capacitor voltage stays below the logic threshold for the duration of the contact bounce, typically 5–20 ms. Then check that the rise time does not violate the input’s slew requirement.

Is the corner frequency the same as the bandwidth?

For a single-pole RC low-pass filter, yes — the −3 dB point and the half-power bandwidth are the same number, 1/(2πRC).

Why five time constants for “fully charged”?

At 5τ the capacitor is within 0.7% of the supply. Beyond that the remaining change is below the noise floor of most measurements, so further waiting buys nothing observable.

What happens if I add a load resistor?

It sits in parallel with the capacitor, so the effective resistance is R ‖ R_load and τ shrinks. The final voltage also drops, because the network becomes a divider.

Where does the energy go during charging?

Half the energy supplied by the source is stored in the capacitor and half is dissipated in the series resistance, regardless of the resistor value. That 50% efficiency is a fundamental property of charging a capacitor through a resistor.