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)
Key Notes
- Time constant: τ = RC: Where R is in ohms and C is in farads, giving τ in seconds. During discharge: V(t) = V₀ × e^(−t/τ). During charging: V(t) = V_supply × (1 − e^(−t/τ)). The exponential shape is governed entirely by τ.
- 63.2% rule: After one time constant τ, a discharging capacitor reaches 1/e ≈ 36.8% of its initial voltage — equivalently, it has discharged 63.2%. After 5τ: less than 1% remains — considered fully discharged or charged in practice. The 5τ rule is the engineering standard for settling time.
- Corner frequency: f_c = 1/(2πRC): The RC circuit acts as a low-pass filter. Signals below f_c pass through with little attenuation; signals above f_c are increasingly attenuated (slope of −20 dB/decade). This is the most common first-order filter in electronics.
- High-pass RC filter: Taking V_out across R (instead of C) gives a high-pass filter with the same corner frequency f_c. Below f_c: signals are attenuated. Above f_c: signals pass. Used for blocking DC and passing AC in audio coupling circuits.
- Applications: RC time constants appear in power supply ripple filtering, pulse shaping in digital circuits, debouncing mechanical switches, sensor signal conditioning (averaging), timing circuits (555 timer), camera flash charging circuits, and audio equalization.