RC Time Constant & Cut-off Frequency Calculator

RC time constant τ, charge and discharge times (63 %, 99 %), first-order low-pass cut-off frequency, attenuation and phase, and R or C for a target cut-off.

Resistor R Capacitor C

Frequency f (Hz) — attenuation and phaseElapsed time t (ms) — chargeTarget cut-off fc (Hz) — solve R or C

Results
Time constant τ = R × C
Cut-off frequency fc
10 → 90 % rise time
To 99 % (≈ 4.6 τ)
Charged / left after discharge at t
Attenuation at f
Phase at f
C for fc with this R
R for fc with this C

The defaults (10 kΩ, 100 nF) are an example. Capacitor tolerance and temperature behaviour, and the next stage's input resistance, change the real result.

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Formulas

τ = R × C (Ω × F = s)
Charge: Vc(t) = V × (1 − e^(−t/τ)), discharge: Vc(t) = V × e^(−t/τ)
Time to fraction x: t = −τ × ln(1 − x), 10→90 % rise = τ × ln 9 ≈ 2.2 τ
fc = 1 ÷ (2π × R × C)
|H(f)| = 1 ÷ √(1 + (f/fc)²), dB = −10 × log10(1 + (f/fc)²), phase = −atan(f/fc)

Multiples of τ and charge

Time	Charged (%)	Left after discharge (%)
0.5 τ	39.3	60.7
1 τ	63.2	36.8
2 τ	86.5	13.5
3 τ	95.0	5.0
4 τ	98.2	1.8
5 τ	99.3	0.7

Reading a first-order low-pass filter

At the cut-off frequency the output is 1/√2 of the input (−3.01 dB) and the phase is −45°. Ten times higher it is about −20 dB, a hundred times about −40 dB: 20 dB per decade.
Below the cut-off the attenuation is not zero: at half of fc it is about −0.97 dB (89 %), so keep the signal band well below fc.
The output lags by τ. Following a step to 99 % takes about 4.6 τ, so do not push fc too low on inputs that must respond quickly.

Where it is used

Noise filter in front of an ADC: the ATmega328P datasheet (§23.6.1) advises removing components above the Nyquist frequency (f_ADC/2) with a low-pass filter. The same section asks for a source impedance of about 10 kΩ or less, so do not make R too large.
Debouncing switch inputs or delaying a reset is about time, so look at τ and the charge fraction rather than the cut-off.
When filtering PWM into an analog voltage, the attenuation at the PWM frequency sets the remaining ripple (a first-order filter leaves |H| of it).

FAQ

What are τ and fc for 10 kΩ and 100 nF?

τ = 10,000 × 100 × 10⁻⁹ = 1 ms and fc = 1 ÷ (2π × 1 ms) ≈ 159 Hz.

What percentage is one time constant?

Charging reaches 1 − e⁻¹ ≈ 63.2 %; discharging leaves e⁻¹ ≈ 36.8 %. 95.0 % at 3τ and 99.3 % at 5τ.

What fc cuts 50 Hz noise to 1/10?

A first-order filter has |H| = 0.1 at f/fc = √99 ≈ 9.95, so fc ≈ 5 Hz. The response slows to match (τ ≈ 32 ms), so decide together with the signal speed.

What changes when I connect a load after the filter?

The next stage's input resistance in parallel with C divides the output and shifts fc. Choose an input resistance much larger than R or add a buffer.

For ADC resolution and engineering units, use the ADC Resolution & Scaling Calculator. For standard resistor values, use the Voltage Divider Calculator. For capacitor reactance at a frequency and LC resonance, use the Reactance & LC Resonance Calculator.

Basis

τ, the charge curve, cut-off, attenuation and phase are computed from circuit definitions. The ADC input filter advice follows Microchip's ATmega328P datasheet (7810D) §23.6.1 Analog Input Circuitry. Check component tolerances in their datasheets.