RC Discharge Time Calculator
Calculate the discharge time of a capacitor through a resistor for any target voltage.
Includes the time constant tau, half-life, and 5-tau rule.
RC Discharge Time
When a charged capacitor is connected across a resistor, the stored charge bleeds out through the resistor and the voltage decays exponentially. This is a first-order linear circuit and one of the cornerstone results in electronics.
Formula
V(t) = V₀ × e^(−t/τ)
Solving for t:
t = −τ × ln(V/V₀)
Where:
- V₀ = initial voltage (volts)
- V = target voltage at time t (volts)
- τ = R × C = the time constant (seconds)
- R = resistance in ohms
- C = capacitance in farads
The Time Constant τ
τ is the time it takes the voltage to fall to 1/e ≈ 36.8% of its starting value. It is the single number that characterizes the speed of the entire decay.
Decay Reference Table
| Multiples of τ | V / V₀ | % Discharged |
|---|---|---|
| 1τ | 36.8% | 63.2% |
| 2τ | 13.5% | 86.5% |
| 3τ | 5.0% | 95.0% |
| 4τ | 1.8% | 98.2% |
| 5τ | 0.7% | 99.3% |
The “5-tau rule” is the engineering rule of thumb that capacitors are considered fully discharged after 5 time constants.
Worked Example
A 470 µF capacitor is charged to 12 V and discharged through a 10 kΩ resistor.
- τ = 10 000 × 470 × 10⁻⁶ = 4.7 s
- Time to reach 1 V: t = −4.7 × ln(1/12) = 4.7 × 2.485 ≈ 11.7 s
- Time to reach 0.1 V: t ≈ 22.5 s ≈ 4.8τ
Common Use Cases
| Application | Why RC discharge matters |
|---|---|
| Power supply bleed resistors | Safe handling after disconnect |
| Camera flash | Strobe duration tuning |
| Filter circuits | Cutoff frequency f = 1 / (2πRC) |
| Timing circuits | 555 timer, debounce, delay |
| ESD protection | Charge dissipation paths |
Relationship to Half-Life
The voltage half-life is t₁/₂ = τ × ln 2 ≈ 0.693 × τ. This is mathematically identical to the half-life concept used in radioactive decay — only the variable being tracked changes.