In this topic, you study Charging a Capacitor – Derivation, Diagram, Formula & Theory.
Consider a circuit consisting of an uncharged capacitor of capacitance C farads and a resistor of R ohms connected in series as shown in Fig. 3.14.
Fig. 3.14: Charging and discharging a capacitor through a resistor
When switch Sw is thrown to Position-I, this series circuit is connected to a d.c. source of V volts.
At the instant of closing the switch, there being no initial charge in the capacitor, its internal p.d. (or counter e.m.f.) is zero. As the capacitor starts acquiring more and more charge, this pd. which is proportional to charge, rises at first quickly and then more slow y with the charge in an exponential manner as illustrated in Fig. 3.15 till it becomes equal to the source voltage V. Theoretically speaking, the charge and the p.d. across the capacitor achieve their steady-state values after infinitely long period.
However, in practice, this period is always comparatively small.
Fig. 3.15: Variation of charge, capacitor p.d. and current during charging
At the instant of closing the switch, the p.d. across the capacitor being zero, the entire applied voltage V acts across the resistor R. Hence, the initial charging current I as given by Ohm’s law is
As the p.d. across the capacitor increases, the value Of the charging current reduces. Finally, when the p.d. across the capacitor becomes equal to the source voltage (V), the net voltage acting round the circuit becomes zero and therefore the charging current also reduces to zero. Theoretically, the current becomes zero only after an infinite time. However, in actual practice, it becomes zero in a relatively short time. Fig. 3.15 shows the charging current as a function of time.
Mathematical Expressions for Capacitor-Voltage, Charge and Current at any Instant during Charging
At any instant t seconds from the time Of closing the switch Sw (Fig. 3.14) in Position-I, let
v Voltage across the capacitor, in volts
i Charging current, in amperes
q Charge on the capacitor, in coulombs
Then, by the application of Kirchhoff’s voltage law to the circuit, it follows that
Applied voltage Voltage drop in the resistor + Voltage across the capacitor
But
or,
Hence,
Integrating both sides, we get
— — loge (V —v) + K
CR
where K is the constant Of integration.
• Now, at the instant of closing the switch i.e. when t O, v 0,
— loge (V —v) + loge V loge V —v
et/CR
e-t/CR)
Further, if Q is the total charge on the capacitor when it is fully charged, then obvious
Similarly,
Therefore, from Equation (3.37),
(1 — e-t/CR)
q Q (1— e-t/CR)
Now, from Equation (3.35),
Also from Equation (3.37),
Hence,
V —V V e-t/CR
iR V e-t/CR
— e-t,’CR
where, as already mentioned previously, R is the initial charging current.
This can be verified by putting t O in the above equation, or v O in Equation (3.35).
If
I (say), then Equation (3.39) can be rewritten as
i 1 e-t/CR
Equations (3.37), (3.38) and (3.40) giving respectively the instantaneous values of the pd. across the capacitor, the charge and the corresponding current during charging are all exponential equations. The curves in Fig. 3.15 showing variation of the above mentioned quantities with respect to time during charging can be easily derived from these mathematical equations.
Time Constant:
At the instant of closing the switch Sw in Positlon-l, i.e. when t O, p.d. across the capacitor, v O. Therefore, from Equation (3.36), we have
This is the initial rate Of change of p.d. across the capacitor. If the pd. continues to rise at this initial rate, then time taken to reach its final steady value
CR seconds.
V/CR
This time CR is called the time constant of the RC series circuit.
Therefore, time constant of a RC series circuit may be defined as the time (in seconds) during which the p.d. across the capacitor, starting from zero, would reach its final steady value if its rate of change were maintained constant at its initial value throughout the changing period.
In fact, when t —
– CR from Equation (3.37),
v V (1 — e-CR/CR) — e-1) V 1
Hence alternatively, time constant of R-C series circuit may also be defined as the time required (in seconds) for the p.d. across the capacitor to rise from zero to 0.632 Of its final stead value during charging.
- Also, from Equation (3.40), when t CR,
This result provides one more way Of defining time constant.
It is the time (in seconds) during which the charging current of the capacitor falls to 0.368 of its initial maximum value.