In the previous RC Charging Circuit tutorial, we saw how a Capacitor, C charges up through the resistor until it reaches an amount of time equal to 5 time constants known as 5T, and then remains fully charged as long as a constant supply is applied to it.
If this fully charged capacitor is now disconnected from its DC battery supply voltage, the stored energy built up during the charging process would stay indefinitely on its plates, (assuming an ideal capacitor and ignoring any internal losses), keeping the voltage stored across its connecting terminals at a constant value.
If the battery was replaced by a short circuit, when the switch is closed the capacitor would discharge itself back through the resistor, R as we now have a RC discharging circuit. As the capacitor discharges its current through the series resistor the stored energy inside the capacitor is extracted with the voltage Vc across the capacitor decaying to zero as shown below.
RC Discharging Circuit
As we saw in the previous tutorial, in a RC Discharging Circuit the time constant ( τ ) is still equal to the value of 63%. Then for a RC discharging circuit that is initially fully charged, the voltage across the capacitor after one time constant, 1T, has dropped by 63% of its initial value which is 1 – 0.63 = 0.37 or 37% of its final value.
Thus the time constant of the circuit is given as the time taken for the capacitor to discharge down to within 63% of its fully charged value. So one time constant for an RC discharge circuit is given as the voltage across the plates representing 37% of its final value, with its final value being zero volts (fully discharged), and in our curve this is given as 0.37Vs.
As the capacitor discharges, it does not lose its charge at a constant rate. At the start of the discharging process, the initial conditions of the circuit are: t = 0, i = 0 and q = Q. The voltage across the capacitors plates is equal to the supply voltage and VC = VS. As the voltage at t = 0 across the capacitors plates is at its highest value, maximum discharge current therefore flows around the RC circuit.
RC Discharging Circuit Curves
When the switch is first closed, the capacitor starts to discharge as shown. The rate of decay of the RC discharging curve is steeper at the beginning because the discharging rate is fastest at the start, but then tapers off exponentially as the capacitor looses charge at a slower rate. As the discharge continues, VC reduces resulting in less discharging current.
We saw in the previous RC charging circuit that the voltage across the capacitor, C is equal to 0.5Vc at 0.7T with the steady state fully discharged value being finally reached at 5T.
For a RC discharging circuit, the voltage across the capacitor ( VC ) as a function of time during the discharge period is defined as:
- VC is the voltage across the capacitor
- VS is the supply voltage
- t is the elapsed time since the removal of the supply voltage
- RC is the time constant of the RC discharging circuit
Just like the previous RC Charging circuit, we can say that in a RC Discharging Circuit the time required for a capacitor to discharge itself down to one time constant is given as:
Where, R is in Ω and C in Farads.
Thus we can show in the following table the percentage voltage and current values for the capacitor in a RC discharging circuit for a given time constant.
RC Discharging Table
|Percentage of Maximum
|0.5 time constant
|0.5T = 0.5RC
|0.7 time constant
|0.7T = 0.7RC
|1.0 time constant
|1T = 1RC
|2.0 time constants
|2T = 2RC
|3.0 time constants
|3T = 3RC
|4.0 time constants
|4T = 4RC
|5.0 time constants
|5T = 5RC
Note that as the decaying curve for a RC discharging circuit is exponential, for all practical purposes, after five time constants the voltage across the capacitor’s plates is much less than 1% of its inital starting value, so the capacitor is considered to be fully discharged.
So an RC circuit’s time constant is a measure of how quickly it either charges or discharges.
RC Discharging Circuit Example No1
A capacitor is fully charged to 10 volts. Calculate the RC time constant, τ of the following RC discharging circuit when the switch is first closed.
The time constant, τ is found using the formula T = R*C in seconds.
Therefore the time constant τ is given as: T = R*C = 100k x 22uF = 2.2 Seconds
a) What value will be the voltage across the capacitor at 0.7 time constants?
At 0.7 time constants ( 0.7T ) Vc = 0.5Vc. Therefore, Vc = 0.5 x 10V = 5V
b) What value will be the voltage across the capacitor after 1 time constant?
At 1 time constant ( 1T ) Vc = 0.37Vc. Therefore, Vc = 0.37 x 10V = 3.7V
c) How long will it take for the capacitor to “fully discharge” itself, (equal to 5 time constants)
1 time constant ( 1T ) = 2.2 seconds. Therefore, 5T = 5 x 2.2 = 11 Seconds