The RC Time Constant

Finds the time constant as τ = RC, the time for the capacitor to reach 63.2% of its final voltage. After 5τ it is charged for all practical purposes.

Explanation

Charge a capacitor through a resistor and the voltage does not leap to its final value; it creeps towards it. The time constant sets the pace.

τ=RC\tau = RC

With RR in ohms and CC in farads, τ\tau comes out in seconds.

That odd 63.2%

After one time constant, the capacitor has reached 63.2% of its final voltage. The strange figure has a reason.

1−1e=1−0.368=0.6321 - \dfrac{1}{e} = 1 - 0.368 = 0.632

Charging follows the exponential V(t)=V0(1−e−t/τ)V(t) = V_0 (1 - e^{-t/\tau}), and putting t=τt = \tau into it leaves exactly 1−e−11 - e^{-1}.

Example

With R=10R = 10 kΩ and C=100C = 100 μF,

τ=10000×100×10−6=1 second\tau = 10000 \times 100 \times 10^{-6} = 1\ \text{second}

So 63.2% after one second, and full for all practical purposes after five.

The same circuit as a filter

The very same RCRC gives a cutoff frequency.

fc=12πRCf_c = \dfrac{1}{2\pi RC}

Above it, signals are increasingly attenuated: a low-pass filter. A circuit with a long time constant is a sluggish one, and a sluggish circuit cannot follow rapid changes — which is to say, high frequencies. The time picture and the frequency picture are one fact seen from two sides.

Where it is used