CHARGING AND DISCHARGING A CAPACITOR
CHARGING AND DISCHARGING A CAPACITOR
I - Goal :
· Study of the charge and discharge voltages of a polarized capacitor through a resistor and their variations as a function of time.
· Verification of the association laws of polarized capacitors.
II - Theoretical reminders :
II-1- Definition of a capacitor :
A capacitor is made up of two conductive plates carrying opposite electrical charges +q and –q, separated by an insulator. The relationships between the quantities u(t), i(t) and q(t) are algebraic: they depend on the chosen orientation conventions.
II-2- Principle :
The capacitor of capacity C is charged by a direct voltage generator E (the switch K is in position A). It is discharged through resistor R (switch K is in position B) (see Figure 1).
II-3- Case of charging a capacitor :
The capacitor being initially discharged, at time t = 0 we flip the switch K to position A and the capacitor charges via a resistor R. By application of the law of meshes and taking into account the sign conventions at the terminals of the different elements present in the circuit, it is easy to obtain the following relationships :
By deriving this last equation with respect to time we obtain :
Hence the differential equation governing the evolution of the charging voltage as a function of time is:
The solution of this first order differential equation with constant coefficients and constant second member is as follows:
The quantity τ corresponds to the time constant which characterizes the evolution of the state of charge of the capacitor in this circuit. Let E be the voltage applied across the circuit. Flipping the switch K to position A causes the voltage UC to increase and the intensity i to decrease exponentially, as the Capacitor charges (see Figure 2).
II-4- Case of discharge of a capacitor :
The capacitor being initially charged under the potential difference E, we switch K to position B at time t = 0, the capacitor discharges through the resistance R. By proceeding in the same way as that presented in the previous part, the voltage and current intensity in the circuit now follow the laws:
In the RC portion of the circuit the discharge current flows in the opposite direction to the charge current, its absolute value is maximum at t = 0 and decreases exponentially as the capacitor discharges (see figure 3).
III - Case of charging two capacitors :
We want to study the behavior of the charge of two capacitors placed on the one hand in series (see figure 4) and on the other hand in parallel (see figure 5). The experiment in this part consists of measuring the voltage difference across the equivalent capacitor and from the charge curve we deduce the value of the equivalent capacity Ceq in the two types of association.
