RC Circuit Time Constant Calculator
Calculate the RC time constant (τ = RC) for resistor-capacitor circuits.
Find capacitor charge and discharge voltages, 63% charge time, and 5-tau settling time.
What Is the RC Time Constant? An RC circuit consists of a resistor (R) and a capacitor (C) in series. The time constant τ (tau) = R × C. τ has units of seconds when R is in ohms (Ω) and C is in farads (F). The time constant determines how quickly the capacitor charges or discharges.
Charging (capacitor starts empty) Voltage across capacitor: V(t) = V_supply × (1 − e^(−t/τ)) At t = τ: V = 63.2% of supply voltage At t = 2τ: V = 86.5% At t = 3τ: V = 95.0% At t = 5τ: V = 99.3% (considered “fully charged” in practice) The capacitor never reaches 100% mathematically — it approaches asymptotically.
Discharging (capacitor starts fully charged) Voltage across capacitor: V(t) = V₀ × e^(−t/τ) At t = τ: V = 36.8% of initial voltage At t = 2τ: V = 13.5% At t = 5τ: V = 0.67% (considered “fully discharged”)
Why e (Euler’s Number)? The exponential function e^(−t/τ) emerges from the differential equation governing the circuit: C × dV/dt + V/R = V_supply This is a first-order linear ODE — its solution always involves e ≈ 2.71828. This makes RC circuits the simplest example of exponential decay in engineering.
RC Filters An RC circuit also acts as a frequency filter. Low-pass filter (output across capacitor): passes low frequencies, blocks high frequencies. High-pass filter (output across resistor): passes high frequencies, blocks low frequencies. Cutoff frequency: f_c = 1/(2πRC) Hz At f_c, the output voltage is reduced to 70.7% (−3 dB) of the input. This is the fundamental building block of audio equalizers, anti-aliasing filters, and power supply smoothing.
Common Applications Power supply smoothing: C smooths ripple from rectified AC (large C → more smoothing). Debouncing: RC circuit ignores brief switch contact bounce. Signal integration/differentiation: RC circuits approximate these mathematical operations. Timing circuits: 555 timer IC uses RC time constant to set pulse width. Flash photography: capacitor stores energy, discharges instantly through flash tube.
Practical Notes Capacitor values: pF (10⁻¹² F), nF (10⁻⁹ F), µF (10⁻⁶ F), mF (10⁻³ F), F. Resistor values: Ω, kΩ (10³), MΩ (10⁶). 1 µF + 1 kΩ: τ = 10⁻⁶ × 10³ = 10⁻³ s = 1 ms. Common in audio circuits. 1 µF + 1 MΩ: τ = 10⁻⁶ × 10⁶ = 1 s. Common in timing circuits.