Capacitor charging calculator

When a DC supply is switched onto a series resistor and an initially uncharged capacitor, the capacitor voltage does not jump; it rises smoothly toward the supply as charge accumulates on the plates. The classical first-order solution is V_c of t equals V_s times one minus e to the minus t over R C, where R is series resistance in ohms, C is capacitance in farads, V_s is the step supply in volts, and t is elapsed time in seconds. Designers use that curve to size timing networks, check how close a rail has settled after a power-on delay, and turn RC products into percent-of-supply milestones at one, two, three, and five time constants. This capacitor charging calculator accepts finite non-negative R and C, a finite supply voltage, and a non-negative time, then returns the capacitor voltage, the RC time constant, the charge fraction, stored charge, and the instantaneous series current from a single deterministic evaluation. Negative elapsed time is rejected with a clear validation error because the transient model is defined for t greater than or equal to zero after the switch closes.

How capacitor voltage rises during series RC charging

Start from Kirchhoff's voltage law on a series loop that contains a constant supply V_s, a resistor R, and a capacitor C that is initially uncharged. The current through the resistor equals C times dV_c over dt, and the resistor drop is R times that current, so the capacitor voltage obeys the linear ODE R C dV_c over dt plus V_c equals V_s.

With V_c of zero equal to zero at the switching instant, the unique solution is V_c of t equals V_s times one minus the exponential of minus t over tau, where the time constant tau equals R times C. At t equals zero the voltage is zero; after one tau it has reached about sixty-three point two percent of V_s; after three tau roughly ninety-five percent; and after five tau about ninety-nine point three percent, which is the classic settling rule of thumb.

The complementary current is I of t equals V_s over R times e to the minus t over tau, large at the first instant and decaying toward zero as the capacitor approaches open-circuit at DC. Stored charge on the plates is simply C times the instantaneous voltage, so once V_c is known the coulomb count follows without a second differential equation.

The model assumes ideal linear parts, no dielectric absorption, and a pure step applied at t equals zero; real switches, ESR, and leakage add second-order effects that belong on separate analysis paths. Energy stored is one half C V_c squared and is intentionally not the primary output here, because search intent for charging voltage versus time is different from search intent for energy or for the bare product Q equals C V at a fixed bias.

Where capacitor charging calculators show up in real work

Introductory electronics labs constantly assign RC charging plots so students can mark one-tau and five-tau milestones on a scope before they move on to discharge and multi-stage filters. Embedded and power-electronics bring-up teams estimate how long a soft-start RC or a reset supervisor network needs before a rail has reached a fraction of the final supply, then convert that delay into firmware timeouts and brown-out margins.

Analog designers sizing sample-and-hold windows, peak detectors, and RC timing blobs need V_c of t rather than only the bare time constant, because percent-of-final voltage is the acceptance criterion on the bench. Keep the static Q equals C V product, the tau equals R C time-constant-only landing, energy one half C V squared, and discharge-from-initial-voltage calculators 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.