Charging capacitors

  The capacitor (C) in the circuit diagram is being charged from a supply voltage (Vs) with the current passing through a resistor (R). The voltage across the capacitor (Vc) is initially zero but it increases as the capacitor charges. The capacitor is fully charged when Vc = Vs. The charging current (I) is determined by the voltage across the resistor (Vs - Vc):

Charging current,  I = (Vs - Vc) / R   (note that Vc is increasing)

At first Vc = 0V so the initial current, Io = Vs / R

Vc increases as soon as charge (Q) starts to build up (Vc = Q/C), this reduces the voltage across the resistor and therefore reduces the charging current. This means that the rate of charging becomes progressively slower.

time constant  = R × C  where:   time constant is in seconds (s)

R = resistance in ohms (ohm)

C = capacitance in farads (F)

For example:

If R = 47kohm and C = 22µF, then the time constant,

RC = 47kohm × 22µF = 1.0s.

If R = 33kohm and C = 1µF, then the time constant, 

RC = 33kohm × 1µF = 33ms.

A large time constant means the capacitor charges slowly. Note that the time constant is a property of the circuit containing the capacitance and resistance, it is not a property of a capacitor alone.

Graphs showing the current and

 voltage for a capacitor charging

time constant = RC

charging current

capacitor charging voltage

  The time constant is the time taken for the charging (or discharging) current (I) to fall to 1/e of its initial value (Io). 'e' is the base of natural logarithms, an important number in mathematics (like pi). e = 2.71828 (to 6 significant figures) so we can roughly say that the time constant is the time taken for the current to fall to 1/3 of its initial value.

  After each time constant the current falls by 1/e (about 1/3). After 5 time constants (5RC) the current has fallen to less than 1% of its initial value and we can reasonably say that the capacitor is fully charged, but in fact the capacitor takes for ever to charge fully!

  The bottom graph shows how the voltage (V) increases as the capacitor charges. At first the voltage changes rapidly because the current is large; but as the current decreases, the charge builds up more slowly and the voltage increases more slowly.

   After 5 time constants (5RC) the capacitor is almost fully charged with its voltage almost equal to the supply voltage. We can reasonably say that the capacitor is fully charged after 5RC, although really charging continues for ever (or until the circuit is changed).