RC time constant
τ = RC. The characteristic time for an RC circuit to charge to 1−1/e ≈ 63% of its final voltage, or discharge to 1/e ≈ 37% of its initial voltage. In seconds when R is in ohms and C in farads.
Definition
The RC time constant τ = RC sets the timescale for every exponential process in an RC circuit. When a capacitor C is charged through a resistor R by a step voltage V₀, the capacitor voltage rises as V(t) = V₀(1 − e^(−t/τ)): at t = τ it has reached about 63%, at 3τ about 95%, at 5τ about 99.3%. The discharge through R with the source removed goes the other way: V(t) = V₀e^(−t/τ), dropping to 37% at τ, 5% at 3τ, 0.7% at 5τ.
The derivation comes from Kirchhoff's voltage law around the charging loop: V₀ = IR + Q/C, combined with I = dQ/dt. This gives the first-order linear ODE R dQ/dt + Q/C = V₀, whose solution is Q(t) = CV₀(1 − e^(−t/RC)). The exponential shape is a general feature of any first-order linear system with a single conserved quantity (the capacitor charge); the specific timescale RC is set entirely by the two component values.
Practical consequences are everywhere. In digital logic, RC delays limit how fast a signal line can be driven — double C (by adding load or driving longer traces), double the settling time. In audio circuits, RC pairs form first-order filters with the "corner frequency" f_c = 1/(2πRC) where the response rolls off: above f_c the capacitor shorts the signal to ground for a low-pass filter, or vice versa for high-pass. In power-supply bypass, large electrolytic capacitors with small ESR resistors give τ ≈ 1 ms to smooth mains-frequency ripple. In timer ICs like the 555, an RC external network directly sets the oscillation period. In DRAM, the RC product of the cell capacitor and the bitline resistance determines how often the cells must be refreshed (typically milliseconds).