The charging and discharging of a capacitor is never instant. When a sudden step of voltage is applied to one plate of a capacitor, the other plate voltage will step in voltage by the same amount. If a resistor is present that connects the second plate to a different voltage level the capacitor will then charge or discharge to this other voltage level. The time needed for this change is about 4 time constants, as shown in Figure – theoretically the time is infinite but a time of four times the time constant allows the charge to reach 98% of the final amount. A figure of 3 times the time constant is sometimes used, representing 95% charging.
The quantity called time constant, T, is measured by R × C where R is the resistance of the charge or discharge resistor and C is the capacitance. For C in farads and R in ohms, the time constant T is in seconds. For the more practical units of μF and kW, T is in milliseconds (ms); and for C in nF and R in kW, T is in microseconds (μs).
These calculations of charging and discharging times are important in determining the shape of the output when a step voltage is applied to a capacitor–resistor combination.
The bases of these time constant calculations are the exponential charging and discharging formulae for a capacitor. For a capacitor discharging from an initial voltage V0 through a time constant RC, the voltage V across the capacitor changes in time t is described by the equation:
For a capacitor being charged to a voltage V0 through a time constant, the equation becomes:
If we assume that t = 4RC, and rearrange the first equation, we get:
V/V0 = exp(−4)
which gives V/V0 = 0.0183, so the voltage has reached 1.8% of the initial voltage, well discharged. For the charging of a capacitor the four time constants give (1 − 0.0183) = 0.9817, around 98% of final voltage.
We can also rearrange the equations to find the time required to charge or discharge a capacitor to a required level. For a discharge:
with the symbol meanings as before, and ln meaning natural logarithm. For example, if you want to find that time is needed to discharge from 10 V to 4 V with time constant 10 μs, the formula becomes:
The negative sign is needed because of the negative values of the natural logarithm (ln).
For charging, the formula becomes T = −RC ln(1− V/V0)