Charging and discharging a capacitor

Charging a capacitor

When an initially discharged capacitor is connected to a voltage generator E through a resistor of resistance R, a transient current flows: the plates charge progressively until the voltage u_C across its terminals reaches the value E. This regime is called the transient regime.

Law of evolution

The voltage across the capacitor follows an exponential law:

u_C(t) = E * (1 - exp(-t/tau))

where tau = R * C is called the time constant of the circuit, expressed in seconds (with R in ohms and C in farads).

The time constant tau

tau represents the charging speed: the smaller tau is, the faster the capacitor charges. In practice, charging is considered complete after a duration of 5*tau (the capacitor is then charged to more than 99%).

Discharge

If the generator is disconnected and the charged capacitor is connected to a simple resistor R, it discharges according to:

u_C(t) = E * exp(-t/tau)

The voltage decreases exponentially until it reaches zero.

Example

For R = 1000 ohms and C = 470x10^-6 F, tau = RC = 0.47 s. The capacitor will be practically charged after 5tau = 2.35 s.

Common pitfall

Do not confuse the time constant tau (characteristic duration) with the total charging time: the capacitor never charges instantaneously, even if tau is very small, and it never reaches exactly 100% of E in a finite time.