15.2 Simple ac circuits  (Page 2/7)

Capacitor

Now let’s consider a capacitor    connected across an ac voltage source. From Kirchhoff’s loop rule, the instantaneous voltage across the capacitor of [link] (a) is

Recall that the charge in a capacitor is given by $Q=CV.$ This is true at any time measured in the ac cycle of voltage. Consequently, the instantaneous charge on the capacitor is

Since the current in the circuit is the rate at which charge enters (or leaves) the capacitor,

where $I_0=\omega C{V}_0$ is the current amplitude. Using the trigonometric relationship $\text{cos}\omega t=\text{sin}\left(\omega t+\pi \text{/}2\right),$ we may express the instantaneous current as

Dividing $V_0$ by $I_0$ , we obtain an equation that looks similar to Ohm’s law:

The quantity $X_C$ is analogous to resistance in a dc circuit in the sense that both quantities are a ratio of a voltage to a current. As a result, they have the same unit, the ohm. Keep in mind, however, that a capacitor stores and discharges electric energy, whereas a resistor dissipates it. The quantity $X_C$ is known as the capacitive reactance    of the capacitor, or the opposition of a capacitor to a change in current. It depends inversely on the frequency of the ac source—high frequency leads to low capacitive reactance.

A comparison of the expressions for $v_C\left(t\right)$ and $i_C\left(t\right)$ shows that there is a phase difference of $\pi \text{/}2\text{rad}$ between them. When these two quantities are plotted together, the current peaks a quarter cycle (or $\pi \text{/}2\text{rad}$ ) ahead of the voltage, as illustrated in [link] (b). The current through a capacitor leads the voltage across a capacitor by $\pi \text{/}2\text{rad},$ or a quarter of a cycle.

The corresponding phasor diagram is shown in [link] . Here, the relationship between $i_C\left(t\right)$ and $v_C\left(t\right)$ is represented by having their phasors rotate at the same angular frequency, with the current phasor leading by $\pi \text{/}2\text{rad}.$

To this point, we have exclusively been using peak values of the current or voltage in our discussion, namely, $I_0$ and $V_0.$ However, if we average out the values of current or voltage, these values are zero. Therefore, we often use a second convention called the root mean square value, or rms value, in discussions of current and voltage. The rms operates in reverse of the terminology. First, you square the function, next, you take the mean, and then, you find the square root. As a result, the rms values of current and voltage are not zero. Appliances and devices are commonly quoted with rms values for their operations, rather than peak values. We indicate rms values with a subscript attached to a capital letter (such as $I_{\text{rms}}$ ).

Although a capacitor is basically an open circuit, an rms current    , or the root mean square of the current, appears in a circuit with an ac voltage applied to a capacitor. Consider that