3.5 Capacitor Behavior

3.5 Capacitor Behavior

This section reviews the charging progress of a capacitor connected in series to a resistor in a closed circuit. We call this an R–C circuit because it includes a resistor and a capacitor (Fig. 3.15). Closing switch S results in a zero potential difference across the capacitor and the resistor immediately sees the generator voltage. Then, the capacitor starts charging and distributes voltage between both elements for a period of time.

U0​ and a switch $S$. On the right, a graph illustrates the charging and discharging curves of the capacitor. The voltage across the capacitor $U_c(t)$ is shown as $U_0(1-e^{-t/RC})$, and the voltage across the resistor $U_R(t)$ is shown as $U_0{e}^{-t/RC}$. The x-axis represents time $t$, and the y-axis represents voltage. The graph highlights the exponential behavior of the voltages over time.">

Fig. 3.15 Charging and discharging a capacitor via a resistor. Source Own elaboration

What determines the charging time of a circuit with a capacitor is the moment at which the fully charged capacitor cancels the current flowing through the circuit. At this point, the potential difference across the capacitor equals that of the generator emf, while the voltage drop across the resistor is zero. The values of the resistor and the capacitor determine the time it takes to charge the capacitor. U expresses the charging voltage of the capacitor (Eq. 3.12). We can graph the voltage at the capacitor's terminals that this equation shows (Fig. 3.15):

u = U0·(1 - e^(-1/RC)) (3.12)

where U0 is the Generator voltage, R is the resistance, C is the capacitance, and e is the base of the napierian logarithms whose value is 2.71828.

Unlike other mentioned quantities, whose value remains constant and written with a capital letter (E, I, R, etc.), we write the potential difference of the capacitor u in lowercase because it is a variable quantity.

A similar process to the charging process occurs when a charged capacitor with a voltage of E connects to a resistor through a switch and discharges. The discharge time relies on the same factors as its charging counterpart. In this case, we find the capacitor's voltage during discharge using the following equation:

u = E·e^(-τ/RC)

where we define:

τ = RC (3.13)

The time constant is the product of resistance, and capacitance (RC) as we see in the equations above.

The time that the capacitor takes to gain 63% of the complete charge during the process, or to release 63% of the full charged capacitor is the same. We also see that the maximum charge and discharge level are attainable only when time reaches infinite (Eq. 3.12). After five-time constants, the voltage between the capacitor's ends is close to the final value (E or 0). Depending on the process, we consider the charging or discharging process. Capacitors store energy as voltage, because they accumulate charge between its plates, and release it when a consumption circuit is connected, resulting in a net zero-energy consumption process.