RC Time Constant Formula
Calculate the RC time constant using τ = RC.
Understand how resistors and capacitors control charging and discharging rates in circuits.
The Formula
The RC time constant tells you how quickly a capacitor charges or discharges through a resistor.
After one time constant (1τ), the capacitor reaches about 63.2% of its final voltage.
After five time constants (5τ), the capacitor is considered fully charged (99.3%).
Variables
| Symbol | Meaning |
|---|---|
| τ | Time constant (seconds, s) |
| R | Resistance (Ohms, Ω) |
| C | Capacitance (Farads, F) |
Charging and Discharging Percentages
| Time | Charging (% of V_max) | Discharging (% remaining) |
|---|---|---|
| 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% |
Example 1
A 10 kΩ resistor is connected to a 47 μF capacitor. What is the time constant? How long to fully charge?
τ = R × C
τ = 10,000 Ω × 0.000047 F
τ = 0.47 s
Full charge ≈ 5τ = 5 × 0.47
τ = 0.47 s, full charge ≈ 2.35 s
Example 2
A camera flash uses a 100 μF capacitor with a 500 Ω resistor to charge from a 300 V supply. What is the time constant, and what voltage is reached after 1τ?
τ = R × C = 500 × 0.0001 = 0.05 s
After 1τ, voltage = 63.2% of 300 V
V(1τ) = 0.632 × 300
τ = 0.05 s (50 ms), V after 1τ ≈ 189.6 V
When to Use It
Use the RC time constant when you need to:
- Design timing circuits and oscillators
- Calculate how long a capacitor takes to charge or discharge
- Build low-pass or high-pass filters (cutoff frequency = 1/(2πRC))
- Analyse transient response in power supply smoothing circuits
The full voltage and current equations during charging are:
- V(t) = V_max × (1 - e^(-t/τ))
- I(t) = (V_max/R) × e^(-t/τ)