In this topic, you study Discharging a Capacitor – Derivation, Diagram, Formula & Theory.
Consider the circuit shown in Fig. 1. If the switch Sw is thrown to Position-2 after charging the capacitor C to V volts, the capacitor discharges through the resistor R with the initial current of V/R amperes (as per Ohm’s law). This current is in the opposite direction to that on charge. Therefore, it is considered as negative. As time passes, the charge, the internal p.d. across the capacitor and hence its discharge current gradually decreases exponentially from maximum to zero as illustrated in Fig. 1. Theoretically, these quantities become zero only after an infinite time, but actually, they become zero in a relatively short time.
Fig. 1. Variation Of charge, capacitor p.d. and current during discharge.
Mathematical Expressions for Capacitor-Voltage, Charge and Current at any Instant during Discharge.
Let the pd. across the discharging capacitor C in Fig. 3.14, t seconds after the switch Sw is closed in Position-2 be v volts and the corresponding current i amperes. Then, in this case, the applied voltage V being zero, Equation (3.35) gives
Integrating both sides, we get
Now, when
Hence,
or,
Further,
Substituting these values in Equation (3.44), we have
Now, from Equation (3.42),
But, from Equation (3.44),
where, as already mentioned previously, R is the in•tial discharge current. This may be verified by putting t O in the above equation or v V in Equation (3.42).
I (say). Then Equation (3.46) can be rewritten as
I e-t/CR
Exponential equations (3.44), (3.45) and (3.47) give respectively the instantaneous values of pd. across the capacitor, charge and corresponding current during discharge. The curves in Fig. 3.16 showing variation of these quant•ties with respect to time can be easily derived from these mathematical equations.
Time Constant: At the instant of closing the sw•tch Sw in Position-2, i.e. when t O, p.d. across the capacitor, v V. Therefore, Equation (3.43) gives
• This is the initial rate of change of p.d. across the capacitor. If the p.d. continues to fal at
this initia rate, then the time taken to fall to zero
CR seconds Time constant.
Hence, time constant of R-C series circuit can also be defined as the time (in seconds) during which the p.d. across the capacitor would fall from its initial maximum value to zero on discharge its rate of change were maintained constant at its initial value.
In fact, when t CR, from Equation (3.44),
Also, from Equation (3.47),
These results provide following two more alternative definitions of time constant.
(i) Time constant of R-C series circuit is the time required (in seconds) for the p. d. across the capacitor to fall to 0.368 of its initial maximum value on discharge. Or, (ii) It is the time (in seconds) during which the capacitor discharge current falls to 0.368 of its initial maximum value.