## Capacitors and Capacitive Reactance

Consider the capacitor connected directly to an AC voltage source as shown in this figure. The resistance of a circuit like this can be made so small that it has a negligible effect compared with the capacitor, and so we can assume negligible resistance. Voltage across the capacitor and current are graphed as functions of time in the figure.

The graph in this figure starts with voltage across the capacitor at a maximum. The current is zero at this point, because the capacitor is fully charged and halts the flow. Then voltage drops and the current becomes negative as the capacitor discharges. At point a, the capacitor has fully discharged (*\(Q=0\)* on it) and the voltage across it is zero. The current remains negative between points a and b, causing the voltage on the capacitor to reverse.

This is complete at point b, where the current is zero and the voltage has its most negative value. The current becomes positive after point b, neutralizing the charge on the capacitor and bringing the voltage to zero at point c, which allows the current to reach its maximum. Between points c and d, the current drops to zero as the voltage rises to its peak, and the process starts to repeat. Throughout the cycle, the voltage follows what the current is doing by one-fourth of a cycle:

### AC Voltage in a Capacitor

When a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a \(\text{90º}\) phase angle.

The capacitor is affecting the current, having the ability to stop it altogether when fully charged. Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor. This is considered to be an effective resistance of the capacitor to AC, and so the rms current \(I\) in the circuit containing only a capacitor \(C\) is given by another version of Ohm’s law to be

\(I=\cfrac{V}{{X}_{C}}\text{,}\)

where \(V\) is the rms voltage and \({X}_{C}\) is defined (As with \({X}_{L}\), this expression for \({X}_{C}\) results from an analysis of the circuit using Kirchhoff’s rules and calculus) to be

\({X}_{C}=\cfrac{1}{2\pi \text{fC}}\text{,}\)

where \({X}_{C}\) is called the **capacitive reactance**, because the capacitor reacts to impede the current. \({X}_{C}\) has units of ohms (verification left as an exercise for the reader). \({X}_{C}\) is inversely proportional to the capacitance \(C\); the larger the capacitor, the greater the charge it can store and the greater the current that can flow. It is also inversely proportional to the frequency \(f\); the greater the frequency, the less time there is to fully charge the capacitor, and so it impedes current less.

Although a capacitor is basically an open circuit, there is an rms current in a circuit with an AC voltage applied to a capacitor. This is because the voltage is continually reversing, charging and discharging the capacitor. If the frequency goes to zero (DC), \({X}_{C}\) tends to infinity, and the current is zero once the capacitor is charged. At very high frequencies, the capacitor’s reactance tends to zero—it has a negligible reactance and does not impede the current (it acts like a simple wire). *Capacitors have the opposite effect on AC circuits that inductors have*.