The RC time constant
When a capacitor charges through a resistor, the voltage rises exponentially. After one time constant τ = R × C it reaches about 63.2% of the final value; after 5τ it is above 99% and is considered fully charged. Discharging works the same way in reverse.
τ = R × C
V(t) = V × (1 − e^(−t/τ))
fc = 1 ÷ (2π × R × C)
Worked example
10 kΩ and 100 nF: τ = 10,000 × 0.0000001 = 1 ms, fully charged after about 5 ms. Used as a filter, the cutoff frequency is 1 ÷ (2π × 0.001) = 159 Hz.
Low-pass and high-pass filters
- Low-pass: R in series, C to ground. Frequencies above fc are attenuated at 20 dB per decade, handy for smoothing a PWM output or debouncing a button.
- High-pass: C in series, R to ground. Blocks DC and passes frequencies above fc, e.g. audio coupling.
- At fc the output is 70.7% (−3 dB) of the input in both cases.
- An LC circuit resonates at f0 = 1 ÷ (2π√(L × C)).
Frequently asked questions
What is 5 time constants?
After 5τ a charging capacitor is at 99.3% of the supply voltage, which is usually treated as fully charged.
What is the cutoff frequency of 1k and 100nF?
1 ÷ (2π × 1,000 × 0.0000001) ≈ 1.59 kHz, and τ = 0.1 ms.
Is the formula the same for high-pass?
Yes. Low-pass and high-pass RC filters both have fc = 1 ÷ (2πRC); only the positions of R and C swap.