Capacitor charge time

When a DC supply is switched onto a series resistor and an initially uncharged capacitor, the capacitor voltage climbs toward the rail as a first-order exponential and never quite reaches the supply in finite time. Engineers, students, and firmware authors often need the inverse of that curve: how many seconds until the voltage hits a chosen threshold such as a logic-high, a brown-out release, or half of the rail. This capacitor charge time calculator accepts series resistance in ohms, capacitance in farads, the DC supply voltage, and the target voltage, then returns t equal to minus R times C times the natural log of one minus V over Vs.

How charging time follows from the RC exponential

Start from Kirchhoff's voltage law on a series loop that contains a constant supply Vs, a resistor R, and a capacitor C that is initially uncharged. The loop equation is R C dVc over dt plus Vc equals Vs. With Vc of zero equal to zero at the switching instant, the unique solution is Vc of t equals Vs times one minus e to the minus t over tau, where the time constant tau equals R times C.

Inverting that closed form for the elapsed time at which Vc equals a chosen target V is a single logarithm: t equals minus tau times ln of one minus V over Vs, which is the same as t equals minus R C ln(1 − V/Vs). The formula is defined only while V and Vs have the same sign and the magnitude of V is strictly smaller than the magnitude of Vs. At V equals zero the capacitor is already at the initial condition, so the time is zero.

At V equal to Vs the argument of the logarithm is zero and the time diverges, matching the physical fact that an exponential approaches its asymptote but never arrives. After one time constant the voltage is about sixty-three point two percent of the supply; after ln(2) times tau it is fifty percent; after three tau roughly ninety-five percent; and after five tau about ninety-nine point three percent, the classic settling rule of thumb.

What to enter and which targets are impossible

Numeric values and numeric strings are both accepted so form widgets and spreadsheet paste stay convenient; SI prefix strings such as one microfarad written as text are not parsed, so pass one times ten to the minus six for a microfarad part and one thousand for a one-kilohm resistor. Resistance and capacitance must be strictly greater than zero. The target must lie strictly between zero and the supply with the same polarity.

Equality with the supply is also rejected because there is no finite charge time. Opposite polarity is unreachable from an initially uncharged state. A classroom check is one kilohm and one microfarad at five volts aiming for two point five volts, which is fifty percent and must return t equal to tau times ln(2).

Where capacitor charge time shows up in real work

Reset supervisors, power-on delay networks, and RC soft-start blobs are sized so a capacitor voltage crosses a comparator or logic threshold after a known delay; inverting V(t) for t is the direct design step, not an afterthought you get by sweeping time. Embedded bring-up teams convert a chosen rail fraction into a firmware timeout, then add margin so brown-out detectors and PLL lock windows are not racing the analog exponential.

Analog and mixed-signal labs ask students for the time to fifty percent, sixty-three percent, and ninety-five percent of a supply so they can place scope cursors before they move on to discharge and multi-stage filters. Keep the forward V(t) charging calculator, the bare tau equals R C landing, energy one half C V squared, and discharge-from-initial-voltage tools on separate pages so each search intent has one clear tool rather than an overloaded multi-formula page that confuses both students and search engines.