What’s RC Discharging Circuit: How to Calculate

RC discharging circuits use what is called the RC time constant which is an inherent property exhibited by an resistor-capacitor combination.

This time constant determines how fast the capacitor discharges and it does so in an exponential way, meaning the voltage does not just drop steadily; it decreases more and more quickly as time gradually passes.

In our earlier RC Charging Circuit tutorial, we talked about how a capacitor charges up through a resistor.

The charging happens until the capacitor reaches a point called “5T”—this is when it has been charging for a time equal to five RC time constants.

Once it reaches that 5T mark, the capacitor is considered fully charged and it will stay that way as long as the power supply continues to provide a constant voltage.

Now if you take this fully charged capacitor and disconnect it from the battery then the energy that was stored during charging would just sit there on the capacitor’s plates indefinitely.

This is assuming we are talking about an ideal capacitor, meaning it has no internal losses, so the voltage across its terminals stays at the same constant value.

But if instead of just disconnecting the battery you replace it with a short circuit (so there is a path for the current to flow) and then close the switch, things change.

The capacitor will start discharging through the resistor. This is what we call an RC discharging circuit.

As the capacitor releases its stored energy by pushing current through the resistor then the voltage across the capacitor (V_C) gradually falls, slowly decaying down to zero, as can be witnessed in the following diagram:

Analyzing an RC Discharging Circuit

In the last tutorial we talked about how in an RC discharging circuit the time constant (τ) still has that vital 63% of the value. That means in a fully charged RC discharging circuit, after one time constant which is called 1T, the voltage across the capacitor will have dropped by 63% from its initial value.

That means the voltage left after one time constant is 1 – 0.63 which is 0.37 or 37% of what it was at the beginning.

Basically the time constant tells you how long it takes for the capacitor’s voltage to drop down to 63% of its fully charged state.

In other words after one time constant in an RC discharge, the voltage on the capacitor’s plates is down to 37% of its final value. Since the final value is zero volts (fully discharged), on a voltage-time curve this is represented as 0.37Vs.

Now the thing about capacitors discharging is that they do not discharge in a steady and linear way. It is not like the voltage just drops at the same speed throughout.

At the start of the discharge process (at time t = 0), the conditions are like this: t = 0, i = 0, and the charge q = Q (which is the full charge). The voltage across the capacitor’s plates is the same as the supply voltage, which means V_C = V_S.

Since the voltage is at its maximum at t = 0, means that the discharge current is also at its maximum and it starts flowing through the RC circuit immediately.

Analyzing the Curves of an RC Discharging Circuit

As soon as we close the switch, the capacitor starts discharging right away, and the process can be clearly witnessed on the above curve. At first the discharging happens rather quickly, so the slope of the curve is very steep.

The discharge rate is at its highest at the beginning but then it slows down as the capacitor loses charge, causing the curve to slowly level off over time.

As the V_C (voltage across the capacitor) drops, the discharge current also decreases.

In the earlier RC charging circuit we observed that the voltage across the capacitor is equal to 0.5V_C at around 0.7T.

By the time the capacitor reaches 5 time constants (5T) it is considered fully discharged and reaches the steady state.

For an RC discharging circuit the voltage across the capacitor V_C, changes over time and can be described by the formula:

V_C = V_S * e^(-t/RC).

Here is what each part means:

V_C represents the voltage across the capacitor
V_S indicates the supply voltage
t shows the time since the supply voltage was removed
RC denotes the time constant of the circuit

Just like in our earlier RC charging circuit, in the RC discharging circuit also the time it takes for the capacitor to discharge down to one time constant (τ) can be still given by the formula τ = R * C wherein R is the resistance in ohms (Ω) and C is the capacitance in farads.

We can use this formula to create a table showing the percentage values for voltage and current at different time intervals during the discharge, based on how much time has passed in terms of the time constants.

Since the RC discharging curve is exponential, the voltage across the capacitor drops off really fast at first but it keeps getting smaller and smaller over time.

After five time constants (5T), the voltage left on the capacitor’s plates is so tiny—less than 1% of what it started with—that, for all practical applications we say the capacitor is fully discharged by that point.

Basically the time constant in an RC circuit is a quick way to measure how fast the capacitor charges up or discharges.

Whether it is charging or discharging the time constant gives you a good idea of how quickly the process happens.

Solving an RC Discharging Circuit Problem

Let us say we have got a capacitor which is fully charged up to 12 volts, and we want to calculate the RC time constant τ for the circuit when it starts discharging.

In this case we are given that the resistor R is 68 kΩ and the capacitor C is 100 μF.

First, to find the RC time constant we use the formula:

τ = R * C

= 68000 * 0.0001

= 6.8 Seconds

Next let us calculate what the voltage across the capacitor will be at 0.7 time constants.

At 0.7 time constants or 0.7T, V_C = 0.5V_C.

∴ V_C = 0.5 x 12 V = 6 V

Now we will calculate the capacitor voltage after 1 full time constant (1T).

At 1 time constant or 1T, V_C = 0.37V_C.

∴ V_C = 0.37 x 12 V = 4.44 V

Finally we calculate when the capacitor is basically “fully discharged.” For this we look at 5 time constants (5T), after which the voltage across the capacitor is less than 1% of the starting value which is considered to be fully discharged.

1 time constant or 1T = 6.8 seconds

∴ 5T = 5 * 6.8 = 34 seconds

References:

Analysis of RC circuits. Charging and discharging

Capacitor Discharge Calculation