Charging and Discharging of Capacitor with Examples

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Charging of Capacitor

Charging and Discharging of Capacitor with Examples- When a capacitor is connected to a DC source, it gets charged. As has been illustrated in figure 6.47. In figure (a), an uncharged capacitor has been illustrated, because the same number of free electrons exists on plates A and B. When a switch is closed, as has been shown in figure (b), then the source, moves electrons towards B via the circuit. In this way, the flow of electrons starts from plate A, and electrons start to store on plate B. As a result, plate A becomes positive with respect to plate B. So long as this process of charging continues, voltages across plates keep increasing very rapidly, until their value equates to applied voltage V. However, their polarity remains inverse, as has been depicted vide figure (c). When a capacitor gets fully charged, the value of the current then becomes zero.

Figure 6.47; Charging a capacitor

When a charged capacitor is dissociated from the DC charge, as has been shown in figure (d), then it remains charged for a very long period of time (depending on the leakage resistance), and one feels an intense shock if touched. From a practical point of view, the capacitance of any capacitor installed in a circuit cannot be restored until resistance has been installed in the circuit. Because, resistance introduces an element of time during the charging or discharging of a capacitor (that’s by means of resistance, a charged capacitor will require a certain amount of time for getting discharged). When a capacitor is either charged or discharged through resistance, it requires a specific amount of time to get fully charged or fully discharged. That’s the reason, voltages found across a capacitor do not change immediately (because charge requires a specific time for movement from one point to another point). The rate at which a capacitor charges or discharges, is determined through the time constant of a circuit. The charge available on a capacitor can be determined with the help of the following equation;

Q = CV

We can differentiate and change this equation into the following equation;

i = dQ / dt = d / dt (CV) = C dv / dt

The following conclusions can be inferred from the above–mentioned equation;

(i). As Q = CV, it means that voltages found across any capacitor, are directly proportional to charge, instead of current.

(ii). Voltages parallel to a capacitor may also be found when there is no flow of current.

(iii). A capacitor has a capacity to store charge.

(iv). It has become clear from i = C dv / dt that a current in a capacitor exists at a time when voltages found parallel to it, change with the time. If dv = dt = 0, that’s when its voltages are constant, then i = 0. As such, the capacitor functions as an open circuit.

(v). i = C dv / dt can also be written as; dv / dt = i/ C

It is obvious from this equation that in the situation of a charge or discharge, the rate of change in voltage is directly proportional to capacitance, on any given value of current i. The higher the value of C, the lower the ratio of change in capacitive voltage. Moreover, capacitor voltages do not change forthwith.

Charging a Capacitor Through a Resistor

Let us assume that a capacitor having a capacitance C, has been provided DC supply by connecting it to a non-inductive resistor R. This has been shown in figure 6.48. On closing the switch, voltages across the capacitor do not proceed instantaneously to their final steady value.

Figure 6.48; Charging a capacitor through a resistor

Let us assume that;

V = Instantaneous voltages across capacitor

i = Instantaneous current

q = Instantaneous charge

Applied voltages are equal to the following;

(a). Voltage drops taking place in resistor

(b). Voltage drops found parallel to a capacitor,

i.e., V = i R + ʋ … (1)

We know that i = C d ʋ / dt

∴ V = CR d ʋ / dt + ʋ + … (2)

On closing the switch,

ʋ = 0

∴ V = CR dʋ/dt or, dʋ/dt = V/ CR

This is the initial rate of change in voltage found parallel to a capacitor. After solving equation (2), we get;

ʋ = V (1 – e^-t/CR) … (3)

Through this equation, changes in voltages across a capacitor can be determined; As ʋ = q / c, and V = Q/ C, therefore, equation (3) can be written as follows;

q/C = Q/C (1 – e^-t/CR) … (4)

Through equation (4), changes in the charge stored in a capacitor can be monitored. Equation 91) can also be arranged in the following manner;

i R = V – ʋ

∴ i = 1 / R (V – ʋ)

Equation (3) can also be written in the following form;

ʋ = V – V e^-t/CR

V – ʋ = V e^-t/CR

∴ i R = V e^-t/CR

∴ i = V e^-t/CR/ R … (5)

V/R means the initial current i, value of which can be acquired by placing ʋ = 0 in the equation (1).

Discharging of Capacitor

When a wire is connected across a charged capacitor, as has been illustrated in fig. 6,49, the capacitor discharges. For doing so, a very low resistance path (i.e., wire) is connected to a switch parallel to the capacitor, as can be seen in fig. (b). When the switch is closed, as shown in fig.(b), then electrons existing on plate B start moving towards plate A via circuit (Remember that during charging and discharging, the flow of current does not occur in dielectrics, because dielectric is an insulating material), as a result of which stored energy in the capacitor begins to consume as a result of a flow of current via the low resistance wire route. When the number of free electrons on both the plates becomes equal, then the charge becomes neutral. At that moment, voltages found parallel to a capacitor become zero, and the capacitor discharges completely. This has been shown in figure (C).

Figure 6.49; Discharging a Capacitor

Discharging a Capacitor Through a Resistor

Have a look at the circuit shown in figure 6.50. When a switch is pushed up and closed, the capacitor charges via a resistor. Now, if the switch is pushed down, then the capacitor installed in the resistance series, becomes short-circuited. As such, the value of V becomes zero. By putting the value of V in equation (1) expressed above;

Figure 6.50; Discharging a capacitor through a resistor