The time constant
When a capacitor charges through a resistor, its voltage does not jump instantly; it rises along an exponential curve. The speed of that curve is set by the time constant:
τ = R × C
with R in ohms and C in farads, τ is in seconds. 1 kΩ × 1 µF = 1 ms.
Charging and discharging
- Charging: V(t) = V₀ × (1 − e^(−t/τ))
- Discharging: V(t) = V₀ × e^(−t/τ)
| Time | Charge reached | Remaining while discharging |
|---|---|---|
| 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% |
Engineers usually consider a capacitor fully charged or discharged after 5τ.
Worked example
A 10 kΩ resistor charges a 100 µF capacitor from a 5 V supply.
- Time constant: 10,000 × 0.0001 = 1 second.
- After 1 second, the capacitor voltage is 5 × (1 − e⁻¹) = 3.16 V, or 63.2% of full charge.
- It is practically full after 5 seconds.
- Used as a low-pass filter, this RC pair has a cutoff frequency of 1 ÷ (2π × 1) = 0.159 Hz.
RC filters
A resistor and capacitor make the simplest filter:
- Low-pass (output across the capacitor): passes slow signals and blocks fast ones.
- High-pass (output across the resistor): blocks DC and slow changes, passes fast ones.
- At the cutoff frequency fc = 1 ÷ (2πRC), the output is 3 dB down, about 70.7% of the input amplitude, and the phase shift is 45°.
Common uses
- Timers and delays, such as the classic 555 timer, whose period depends on RC.
- Debouncing buttons so a single press is not read as several.
- Smoothing power supplies and sensor readings.
- Coupling audio stages while blocking DC.
- Reset circuits that hold a microcontroller in reset for a few milliseconds at power-up.
Practical notes
Real capacitors have tolerances of ±10 to 20% and leakage, especially electrolytic types, so timing circuits that must be precise use film capacitors or crystals. Always respect the capacitor’s voltage rating and polarity.