RC Time Constant Calculator

The RC time constant (τ, tau) describes how quickly a capacitor charges or discharges through a resistor. It is one of the most fundamental concepts in electronics, governing everything from filter design to timer circuits to signal smoothing.

Formula: τ = R × C where:

• τ (tau): the time constant in seconds
• R: resistance in ohms (Ω)
• C: capacitance in farads (F)

Charging voltage over time: V(t) = V_supply × (1 − e^(−t/τ))

Discharging voltage over time: V(t) = V_initial × e^(−t/τ)

What each variable means:

• e: Euler’s number ≈ 2.71828 (base of natural logarithm)
• t: elapsed time in seconds
• V_supply: source voltage applied during charging
• V_initial: voltage across capacitor at start of discharge

Key milestones during charging (% of V_supply reached):

• 1τ: 63.2% charged
• 2τ: 86.5% charged
• 3τ: 95.0% charged
• 4τ: 98.2% charged
• 5τ: 99.3% charged ← considered “fully charged” in practice

Worked example: A circuit uses R = 10,000 Ω (10 kΩ) and C = 100 μF (0.0001 F). τ = 10,000 × 0.0001 = 1 second After 2 seconds (2τ): V = 5V × (1 − e^(−2)) = 5 × 0.865 = 4.32V (86.5% charged) Full charge reached in ≈ 5 seconds.

Common RC circuit applications:

• Low-pass filters: attenuate high-frequency signals, pass low frequencies. Cutoff frequency: f_c = 1 ÷ (2π × R × C)
• High-pass filters: opposite behavior
• 555 timer circuits: RC determines pulse timing
• Debounce circuits: smooth mechanical switch bounce
• Camera flash: capacitor charges slowly, discharges instantly for the flash

Cutoff frequency example: Same circuit above. f_c = 1 ÷ (2π × 10,000 × 0.0001) = 1 ÷ 6.283 = 0.159 Hz — very low cutoff, suitable for DC smoothing only.