- The time constant of a capacitor discharging through a resistor is a measure of how long it takes for the capacitor to discharge
- The definition of the time constant is:
The time taken for the charge, current or voltage of a discharging capacitor to decrease to 37% of its original value
- Alternatively, for a charging capacitor:
The time taken for the charge or voltage of a charging capacitor to rise to 63% of its maximum value
- 37% is 0.37 or (where e is the exponential function) multiplied by the original value (I0, Q0 or V0)
- This is represented by the Greek letter tau, , and measured in units of seconds (s)
- The time constant provides an easy way to compare the rate of change of similar quantities eg. charge, current and p.d.
- It is defined by the equation:
- = time constant (s)
- R = resistance of the resistor (Ω)
- C = capacitance of the capacitor (F)
The graph of voltage-time for a discharging capacitor showing the positions of the first three time constants
- The time to half, t1/2 (half-life) for a discharging capacitor is:
The time taken for the charge, current or voltage of a discharging capacitor to reach half of its initial value
- This can also be written in terms of the time constant, :
t1/2 = ln(2) ≈ 0.69 = 0.69 RC
Note that the time constant is not the same as half-life. Half-life is how long it takes for the current, charge or voltage to halve whilst the time constant is to 37% of its original value (not 50%).
Although the time constant is given on the datasheet, you will be expected to remember the half-life equation t1/2 = 0.69RC
- Problems involving the time constant tend to involve
- Determining the time constant from a graph
- To find the time constant from a voltage-time graph, calculate 0.37V0 and determine the corresponding time for that value
The time constant shown on a charging and discharging capacitor
A capacitor of 7 nF is discharged through a resistor of resistance R. The time constant of the discharge is 5.6 × 10-3 s.
Calculate the value of R.