Deep Analysis of Capacitors in AC Circuits

    A capacitor connected to a sinusoidal supply will have a reactance that is affected by the frequency of the supply and the size of the capacitor.

    When capacitors are connected across a DC supply voltage, they become charged to the value of the applied voltage, acting like temporary storage devices, and they hold or retain this charge indefinitely as long as the supply voltage is present.

    During this charging process, charging current (I) will flow into the capacitor at a rate equal to the rate of change of charge on the plates opposite to any change in voltage.

    This charging current can be defined as: i = CdV/dt. Once a capacitor is "fully charged," when it is saturated with electrons, the capacitor blocks more electrons from flowing to its plates. However, if an alternating current, or AC power source, is applied, the capacitor will alternately charge and discharge at a rate determined by the frequency of the power source. The capacitance in an AC circuit will then change with frequency as the capacitor is constantly charging and discharging.

    We know that the current that electrons flow across the plates of a capacitor is proportional to the rate of change of the voltage across those plates. We can then see that a capacitor in an AC circuit likes to pass current when the voltage across the plates is constantly changing with respect to time, such as in an AC signal, but it doesn't like to pass current when the applied voltage is a constant value, such as in a DC signal. Consider the circuit below.

    AC capacitor circuit

    In the pure capacitive circuit above, the capacitor is connected directly across the AC supply voltage. As the supply voltage increases and decreases, the capacitor will charge and discharge according to this change. We know that the charging current is proportional to the rate of change of the voltage across the plates, which is greatest as the supply voltage changes from the positive half cycle to the negative half cycle, or vice versa at the points 0o and 180o along the sine wave.

    Therefore, when the AC sine wave crosses at the maximum or minimum peak voltage level (Vm), the minimum voltage change occurs. At these locations in the cycle, the maximum or minimum current flows through the capacitor circuit as shown below.

    AC Capacitor Phase Diagram

    At 0°, the rate of change of the supply voltage increases in the positive direction, resulting in the maximum charging current at that moment. When the applied voltage reaches its maximum peak at 90° for a short period of time, the supply voltage neither increases nor decreases, so no current flows through the circuit.

    When the applied voltage begins to drop to zero at 180o, the slope of the voltage is negative, so the capacitor discharges in the negative direction. At the 180o point along the line, the rate of change of the voltage is again at a maximum, so the maximum current flows at this instant, and so on.

    Then we can say that for a capacitor in an AC circuit, whenever the applied voltage reaches its maximum value, the instantaneous current is at its minimum value or zero; similarly, when the applied voltage reaches its minimum value, the instantaneous value of the current is at its maximum value or peak value. Or zero.

    From the waveform above, we can see that the current leads the voltage by 1/4 cycle or 90o as shown in the vector diagram. We can then say that in a purely capacitive circuit, the AC voltage lags the current by 90o.

    We know that the current flowing through a capacitor in an AC circuit is opposite to the rate of change of the applied voltage, but just like a resistor, a capacitor also provides some form of resistance to oppose the flow of current through the circuit, but capacitors are used in AC. This AC resistance is called reactance, or more commonly in capacitor circuits, capacitive reactance, so capacitors in AC circuits suffer from the effect of capacitive reactance.

    Capacitive reactance

    Capacitive reactance in a purely capacitive circuit simply opposes the flow of current in an AC circuit. Like resistance, reactance is measured in ohms, but the symbol X distinguishes it from a purely resistive value. Since reactance is a quantity that can apply to both inductors and capacitors, it is often referred to as capacitive reactance when used with capacitors.

    For capacitors in an AC circuit, the symbol for capacitive reactance is Xc. So we can actually say that capacitive reactance is the value of the capacitor resistance, which varies with frequency. Again, capacitive reactance depends on the capacitance of the capacitor in farads as well as the frequency of the AC waveform, and the formula used to define capacitive reactance is:

    Capacitive reactance

    Where: F is in Hertz and C is in Farads. 2πƒ can also be denoted collectively as the Greek letter omega, with ω representing the angular frequency.

    From the capacitive reactance formula above, we can see that if we were to increase either the frequency or the capacitance, the total capacitive reactance would decrease. As the frequency approaches infinity, the reactance of the capacitor decreases to zero, just like a perfect conductor.

    However, as the frequency approaches zero or DC, the reactance of the capacitor increases to infinity, acting like a very large resistor. This means that for any given capacitance value, the capacitive reactance is "inversely proportional" to the frequency, as shown below:

    Capacitor impedance frequency

    The capacitive reactance of a capacitor decreases as the frequency across it increases, so the capacitive reactance is inversely proportional to the frequency.

    Opposing the current flow, the static charge on the plates (its AC capacitance value) remains constant because it is easier for the capacitor to fully absorb the change in charge on its plates during each half cycle.

    Likewise, as the frequency increases, the value of the current flowing through the capacitor also increases because the rate of change of voltage across the capacitor plates increases.

    We can then see that in DC a capacitor has infinite reactance (open circuit) and at very high frequencies it has zero reactance (short circuit).

    AC Capacitor Example 1

    Find the rms current that flows in the AC capacitor circuit when a 4 μF capacitor is connected across a 880V, 60Hz supply.

    In an AC circuit, the sinusoidal current through a capacitor causes the voltage to lead by 90o, which varies with frequency as the capacitor is constantly charged and discharged by the applied voltage. The AC impedance of a capacitor is known as reactance and the capacitive circuit we are dealing with is usually called capacitive reactance, XÇ

    AC Capacitor Example 2.

    When a parallel plate capacitor is connected to a 60Hz AC source, the reactance is found to be 390 ohms. Calculate the value of the capacitor in microfarads.

    This capacitive reactance is inversely proportional to frequency and will react against the current flow around the capacitive AC circuit.