The capacitor is a device used to store energy in the form of electrical charge which can be later utilised to supply charge or energy once the power source is disconnected from it. It is used in the electric circuits of radios, computers, etc. along with these capacitors. It also provides temporary storage of energy in circuits which can be supplied when required. The property of the capacitor to store energy is known as capacitance.
Charging and Discharging of Capacitor Derivation
Charging and diTscharging of capacitors holds importance because it is the ability to control as well as predict the rate at which a capacitor charges and discharges that makes capacitors useful in electronic timing circuits. It happens when the voltage is placed across the capacitor and the potential cannot rise to the applied value instantaneously. As the charge on the terminals gets accumulated to its final value, it tends to repel the addition of further charge accumulation.
Thus following are the factors on which rate at which a capacitor can be charged or discharged depends on:
The capacitance of the capacitor and
The resistance of the circuit through which it is being charged or is discharged.
Charging of a Capacitor
Let us take a capacitor (C) in series with a resistor (R) forming an RC Charging Circuit and is connected across a DC battery supply (Vs) with a switch. Now at any time zero, when the switch is first closed, the capacitor gradually starts charging up through the resistor until the voltage across it reaches the supply voltage of the DC battery supply.
Now at time t=0, the switch is open and the capacitor is fully charged. These are the initial conditions of the circuit, thus at t = 0, i = 0 and q = 0. Now when the switch is closed, the time begins at t = 0 and current begins to flow into the capacitor via the resistor and charge starts accumulating on the capacitor. Since the initial voltage across the capacitor is zero i.e (Vc = 0) at t = 0, the capacitor is in state of short circuit with respect to the external circuit. In this situation, the maximum current flows through the circuit opposed only by the resistor R. Now applying the Kirchhoff’s Voltage Law (KVL), the voltage drops around the circuit are given as:
Thus the current flowing in the circuit is called the Charging Current and is found by using Ohm's law as i = Vs/R.
[Image will be Uploaded Soon]
As the capacitor starts charging up, the potential difference across its plates slowly increases and the actual time is taken for the charge on the capacitor to reach 63% of its maximum possible voltage, in the curve time corresponding to 0.63Vs is known as one Time Constant (tau).
The capacitor continues to charge and corresponding to it, the voltage difference between Vs and Vc reduces. The circuit current also decreases. Now at a condition greater than five-time constants (5T) when the capacitor is said to be fully charged, t = ∞, i = 0, q = Q = CV. At time tending to infinity, the charging current finally diminishes to zero and the capacitor now acts as an open circuit with the supply voltage value entirely across the capacitor as Vc = Vs.
Thus we can say that the time required for a capacitor to charge up to one time constant (1T) is RC (time constant only specifies a rate of charge where R is in Ω and C in Farads).
Now we know that the voltage V is related to charge on a capacitor by the equation, Vc = Q/C, the voltage across the capacitor ( Vc ) at any instant of time during the charging is given as:
Now after a time period equivalent to 4-time Constants (4T), the capacitor in this RC charging circuit is virtually fully charged and the voltage across the capacitor now becomes approx 98% of its maximum value, 0.98Vs. This time taken for the capacitor to reach this 4T point is known as the Transient Period.
After a time of 5T, the capacitor is said to be fully charged with the voltage across the capacitor (Vc ) being equal to the supply voltage( Vs ). As the capacitor becomes fully charged, no more current flows in the circuit. The time period after 5T is called the Steady-State Period.
Discharging of Circuit
At time equals to 5-time constants i.e 5T, the capacitor remains fully charged as long as there is a constant supply applied to it. Now when this fully charged capacitor is disconnected from its DC battery supply, the stored energy accumulated during the charging process will stay indefinitely on its plates, keeping the voltage across its connecting terminals at a constant value.
Now if the battery is replaced by a short circuit when the switch is closed, the capacitor would discharge itself back through the resistor, R as we now have an RC discharging circuit. As the capacitor keeps on discharging, its current through the series resistor the stored energy inside the capacitor is extracted with the voltage Vc across the capacitor that decays to zero finally.
In the RC Circuit Discharging, the time constant (τ) is still equal to the value of 63%. Thus for the 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 thus is 1 – 0.63 = 0.37 or 37% of the final value.
[Image will be Uploaded Soon]
As shown in the graph when the switch is first closed, the capacitor starts to discharge. The rate of decay of the RC discharging curve can be seen to be steeper at the beginning because the discharging rate is fastest at the start and then decreases exponentially as the capacitor loses charge at a slower rate. As the discharging continues, the value of VC reduces resulting in a less discharging current.
We know that from 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.
Now For the RC discharging circuit, the voltage across the capacitor ( VC ) is the function of time during the discharge period and is defined as