RC Time Constant Calculator

The time constant tau = R x C sets the pace of every exponential transient in a resistor capacitor network, and the product of ohms and farads works out directly in seconds. Charging follows V(t) = V0 x (1 - e to the minus t over tau), so at one time constant the capacitor has reached 63.2 percent of the supply, which is simply 1 - 1/e; the sequence continues 86.5, 95.0, 98.2, and 99.3 percent at two through five time constants, which is why 5 tau is treated as practically complete. Discharge is the mirror image, falling to 36.8 percent at one tau. The same network viewed in the frequency domain has a corner at fc = 1 / (2 pi R C), the point where capacitive reactance equals the resistance and response is 3 dB down. A 10 kilohm resistor with a 1 microfarad capacitor gives tau of 10 ms and a cutoff near 15.9 Hz. Several practical factors move the real answer away from the ideal. Source impedance adds to R, so a circuit driven through a logic output or an op amp with finite output resistance is slower than the nominal calculation. Component tolerance dominates in electrolytics, which are commonly specified at minus 20 to plus 80 percent and lose capacitance as they age and dry out; film and C0G ceramic parts hold much tighter values. Class II ceramics such as X7R also lose substantial capacitance under DC bias. When RC networks are used for timing rather than filtering, the switching threshold matters more than tau itself, since the delay to a threshold Vth is minus tau times the natural log of 1 minus Vth over V0; the classic 555 timer switches at one third and two thirds of the supply, which yields intervals of 0.693 R C.

The capacitor is older than the circuit theory that describes it. The Leyden jar was discovered independently by Ewald Georg von Kleist in 1745 and by Pieter van Musschenbroek at Leiden in 1746, giving experimenters their first means of storing appreciable charge, though its behavior was described qualitatively for the next century. Michael Faraday's investigations of dielectrics and specific inductive capacity in the 1830s supplied the physical understanding, and the unit of capacitance was later named the farad in his honor. The exponential RC law became commercially urgent with submarine telegraphy. Long cables behave as distributed resistance and capacitance, and pulses sent into them emerged smeared and delayed. William Thomson, later Lord Kelvin, analyzed this retardation in 1855 and showed that signalling speed was governed by the product of the cable's resistance and capacitance, scaling with the square of its length, a result that shaped the design of the Atlantic telegraph cables and made the RC product a design quantity rather than a curiosity. Oliver Heaviside extended the analysis through the 1870s and 1880s with the telegrapher's equations and his operational calculus, giving engineers systematic methods for transient response. In the twentieth century the RC network moved from long cables to nearly every circuit board, and the introduction of the 555 timer by Signetics in 1971, designed by Hans Camenzind, made a resistor and capacitor pair the default way to generate a delay or an oscillation.