Calculator of Alexandria

RC time constant calculator

Time constant τ = RC, capacitor voltage at any time while charging or discharging, and the time to reach 99%, with the exponential curve derived.

Capacitor is

Supply voltage when charging, starting voltage when discharging.

Time constantτ\tau
Capacitor voltage at that time (V)V(t)V(t)
Share of full charge
Time to 99% (s)4.6τ4.6\tau

Formula

τ=RC,V=V0 (1−e−t/τ)\tau = RC, \quad V = V_0\,(1 - e^{-t/\tau})
How it's derived

The formula

τ=RC\tau = RC

Charging from empty toward the supply voltage V0V_0:

V(t)=V0(1−e−t/τ)V(t) = V_0\left(1 - e^{-t/\tau}\right)

Discharging from V0V_0:

V(t)=V0 e−t/τV(t) = V_0\,e^{-t/\tau}

RR is the resistance in ohms, CC the capacitance in farads, τ\tau the time constant in seconds and V(t)V(t) the voltage across the capacitor after time tt.

Where it comes from

A capacitor stores charge qq, and its voltage is q/Cq/C. When it charges through a resistor from a supply V0V_0, the resistor gets whatever voltage the capacitor hasn’t yet taken, so by Ohm’s law the current is:

I=V0−q/CRI = \frac{V_0 - q/C}{R}

Current is the rate charge arrives, I=dq/dtI = dq/dt. As the capacitor fills, the gap V0−q/CV_0 - q/C shrinks, so the charging slows. The rate is always proportional to how far is left to go:

ddt(V0−V)=−1RC(V0−V)\frac{d}{dt}\left(V_0 - V\right) = -\frac{1}{RC}\left(V_0 - V\right)

Anything that shrinks at a rate proportional to its own size decays exponentially. The gap starts at V0V_0, so V0−V=V0e−t/RCV_0 - V = V_0 e^{-t/RC}, which rearranges to the charging equation. Discharging is the same idea with the gap being the voltage itself.

The product RCRC has units of seconds (ohms times farads), and it sets the speed. After one τ\tau the capacitor is 1−e−1=63.2%1 - e^{-1} = 63.2\% charged. To reach 99%, solve e−t/τ=0.01e^{-t/\tau} = 0.01:

t=τln⁡100≈4.6 τt = \tau \ln 100 \approx 4.6\,\tau

Worked example

A 10 kΩ resistor and a 100 µF capacitor on a 5 V supply:

τ=10,000×0.0001=1 s\tau = 10{,}000 \times 0.0001 = 1\ \text{s} V(1 s)=5(1−e−1)=3.16 VV(1\ \text{s}) = 5\left(1 - e^{-1}\right) = 3.16\ \text{V}

It reaches 99% (4.95 V) after about 4.6 s.

Watch out for

  • “Fully charged after 5τ” is a rule of thumb. At 5τ5\tau the capacitor is at 99.3%. Mathematically it never quite gets to 100%.
  • Include every resistance. The source’s internal resistance and any load across the capacitor change the effective RR. A load in parallel also lowers the final voltage.
  • Electrolytic tolerance. Electrolytic capacitors are often ±20%, so timing from an RC circuit is only as good as that.

Sources

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