Calculate the time constant of a resistor-capacitor (RC) circuit, and the voltage during charging or discharging at any given time.
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An RC circuit is a simple electrical circuit containing a resistor (R) and a capacitor (C) in series, widely used in electrical filters and timing circuits. As the capacitor charges through a resistor, the voltage across it rises gradually according to the exponential relationship V(t) = V × (1 − e^(−t/τ)), and when it discharges, the voltage drops according to V(t) = V × e^(−t/τ), where τ (tau) is the circuit's "time constant," calculated by multiplying resistance by capacitance (τ = R × C) in seconds. The time constant represents the time needed for the voltage to reach about 63.2% of its final value during charging (or drop to 36.8% of its initial value during discharging), and after about 5τ the capacitor is considered essentially fully charged or discharged for practical design purposes, since the remaining voltage difference becomes negligibly small at that point. This same charging behavior underlies timing circuits, camera flash units, and filters used to smooth out voltage fluctuations in electronic devices.
The RC time constant's defining threshold — 63.2% charged after one time constant — looks like an oddly specific number at first glance, but it emerges naturally and inevitably from the exponential mathematics that genuinely governs how capacitors charge through a resistor, rather than being an arbitrary or rounded reference point chosen for convenience.
The charging equation, V(t) = V × (1 − e^(−t/τ)), is an exponential approach to the final voltage, and exponential functions never actually reach their limiting value in finite time — mathematically, the capacitor never becomes 'perfectly' 100% charged, it just gets progressively, asymptotically closer. This is exactly why engineers needed a practical, well-defined reference point to describe circuit charging speed, rather than trying to specify an exact 'fully charged' time that technically never arrives.
Setting t equal to exactly one time constant (t = τ) into the charging equation gives 1 − e^(−1), which calculates to approximately 0.632, or 63.2% — this specific percentage isn't chosen for round-number convenience; it's simply what falls out mathematically from evaluating the exponential function at exactly one time constant, making it the natural, unavoidable reference point for describing RC circuit charging speed.
The practical '5 time constants to fully charge' rule of thumb used throughout circuit design follows the same underlying math: after 5τ, the voltage has reached about 99.3% of its final value (1 − e^(−5) ≈ 0.993), a remaining gap small enough to be considered negligible for essentially all practical engineering purposes, even though the capacitor technically continues approaching, but never mathematically reaching, exactly 100%.
This exponential charging and discharging behavior is exploited deliberately in countless practical circuits: timing circuits use a known RC time constant to generate precise, predictable delays; camera flash circuits use capacitor discharge timing to control flash duration; and basic analog filter circuits use the frequency-dependent effective impedance of a charging/discharging capacitor to selectively pass or block certain signal frequencies — all applications that depend on the exact, predictable mathematical shape of this exponential charging curve rather than treating it as a rough approximation.
It's the time it takes for the capacitor voltage to reach about 63.2% of its final value during charging, or to drop to about 36.8% of its initial value during discharging — a standard reference point in RC circuit analysis.
In practical design, a capacitor is usually considered fully charged or discharged after about 5 time constants (5τ), at which point it reaches roughly 99.3% of its final value.
The time constant formula τ = R × C only gives a result in seconds when R is in ohms and C is in farads; this calculator handles that unit conversion automatically from the kΩ and μF values you enter.