RC Circuit Fundamentals: Time Constant, Charging, and Discharging Explained

Combining what we have learnt in previous articles opens up a whole new set of ideas to explore. The combination of a resistor and Capacitors results in an RC Circuit. One of the very important types of electronic circuits to be studied is due to their wide range of applications.

Table Of Contents

1. Basics of RC Circuit:
2. Time Constant and its importance:
Conclusion:

Related Articles:

1. Basics of RC Circuit:

1.1 Charging without a Resistor

Let’s consider a circuit with a battery of emf ‘E’ and a capacitor (capacitance ‘C’) connected in series to it via a switch ‘S’. Note that the capacitor is uncharged initially.

We know, charging will start as soon as the Switch ‘S’ is closed, and it is also seen that the capacitor gets charged very quickly. This is shown in Fig.2, where we plot the charge v/s time graph to keep a track of the charge appearing on the capacitor plates w.r.t time.

When we say that a capacitor is fully charged, it means that the capacitor plates have the maximum charge they can hold. In this case, it will be ‘CE’. Also, this state where the charge doesn’t change anymore is called ‘Steady State.‘
As the charge keeps on developing on the capacitor, the current in the circuit keeps on decreasing till it becomes zero at steady state. This whole situation, where every parameter(charge, current) is going through a change, is called ‘Transient state.‘

Fig.2 With reference to Fig.1 (a) and (b)

What happens when Resistance is added?

1.2 Charging with a Resistor

Now, we add a resistor with resistance R in series with the capacitor. The capacitor is uncharged initially. We have the most basic RC Circuit possible for analysis.

Fig.3 (a) Resistor is attached in series

We can sum up the role of a resistor here in simple words. It is mainly used to increase the charging time of the capacitor. This makes the transient state last a little longer. How long ?, can be decided and pre-calculated on the basis of R and C values chosen to build the resultant RC circuit.

Charging in an RC Circuit with a resistor and a capacitor together — Fig. 4 Charging in an RC Circuit

In case of Charging, you can imagine a resistor as a cunning friend who doesn’t want you(the capacitor) to grow in his/her life. This friend somehow tries to delay the work that you would have achieved very easily earlier.

1.3 Discharging without Resistor:

Consider we have a capacitor, fully charged with the help of a battery of emf ‘E’, and we need to discharge it.

This can be done by simply disconnecting the battery and then allowing the two plates of the capacitor to be connected.

Fig.5 Before closing the switch (Capacitor is fully charged with battery E)

Fig.6 Battery is removed, and Switch S is closed.

Similar to the charging process, the discharging of a capacitor also happens instantly in the absence of a resistor. This can be seen in the plot in Fig.7

Fig. 7 Plot of q v/s t for situations in Fig. 5 and 6

1.4 Discharging with a Resistor:

In case of discharging, the resistor increases the discharging time. We can say that the resistor here acts as a good friend who doesn’t allow you(the capacitor) to fall(discharge) immediately.

Discharging still happens, but at a slower rate. This can be observed from the plot between q (charge on capacitor plate) v/s time ‘t’ as shown below.

So we can clearly see from Fig.8 that it takes time to reach 0 charge (for practical purposes). But how much time it takes is decided by the R and C values

2. Time Constant and its importance:

We represent ‘Time Constant’ by the Greek letter ‘Tau’ $(\tau)$

Time Constant is mainly formulated as the product of the Resistance and the Capacitance. It’s an important representation mainly used in RC(resistance-capacitance) and RL (resistance-inductance) circuits.

$\tau = R \times C$

Now that we know how it is formulated, replace R X C with the time constant.

Let’s start with the charging case :

$q = q_{0}\left(1 – e^{-t/(RC)}\right)$

$q = q_{0}\left(1 – e^{-t/\tau}\right)$

Inserting t = $\tau$

$q = q_{0}\left(1 – e^{-1}\right)$

$q= q_{0}\left(1 – \frac{1}{e}\right)$

$q = 0.63\, q_{0}$

On plotting the same, we get a graph shown below:

From the above analysis, we can define time constant (For charging) as the time required to charge the capacitor to 63% of the total charge it can achieve, i.e., 63 % of the steady state charge

Let’s discuss the Discharging case :

$q = q_{0}\left(e^{-t/(RC)}\right)$

$q = q_{0}\left( e^{-t/\tau}\right)$

Inserting t = $\tau$

$q = q_{0}\left(e^{-1}\right)$

$q = \frac{q_{0}}{e}$

$q = 0.37\, q_{0}$

Again, from the above analysis, we can define time constant (For discharging) as the time required to discharge the capacitor to 37% of the total charge it had initially, i.e., 37 % of the steady state charge

We can keep on inserting such time values and keep a track of the charge on the capacitor.

Important: Note that q is the charge on the capacitor plates at time instant ‘t.’

Charging case:

It is seen that approximately after a minimum (t = 5 * time constant), the capacitor is 99% charged. We consider that as approx fully charged in reality while doing some projects and all!

Inserting $t= 5\tau$

$q = q_{0}\left(1 – e^{-5\tau/\tau}\right)$

$q = q_{0}\left(1 – e^{-5}\right)$

$q = 0.99q_{0}$

Discharging case:

It can be seen again from the calculations below that, –> It takes about (t = 5 * time constant) to discharge the capacitor to an extent where only 1% of the initial charge remains on the plates.
We consider this as approx. fully discharged case for project purposes

Inserting $t= 5\tau$

$q = q_{0}\left(e^{-5\tau/\tau}\right)$

$q = q_{0}\left( e^{-5}\right)$

$q=0.01q_{0}$

Conclusion:

So, based on our previous articles’ knowledge (Resistors and Capacitors), we have learnt about the basics of the RC Circuit. There will be another article to go into further depth in the case of RC Circuits.

Have a look at the derivations, as those will provide a proper insight and reasoning of why exactly things happen in that way. It’s always good to have a look at the mathematical analysis of the physics concept you study!

Keep Learning!