Capacitance in AC Circuits

Capacitance in AC Circuits results in a time-dependent current which is shifted in phase by 90^o with respect to the supply voltage producing an effect known as capacitive reactance.

When capacitors are connected across a direct current DC supply voltage, their plates charge-up until the voltage value across the capacitor is equal to that of the externally applied voltage. The capacitor will hold this charge indefinitely, acting like a temporary storage device as long as the applied voltage is maintained.

During this charging process, an electric current ( i ) flows into the capacitor which results in its plates beginning to hold an electrostatic charge. This charging process is not instantaneous or linear as the strength of the charging current is at its maximum when the capacitors plates are uncharged, decreasing exponentially over time until the capacitor is fully-charged.

This is because the electrostaic field between the plates opposes any changes to the potential difference across the plates at a rate that is equal to the rate of change of the electrical charge on the plates. The property of a capacitor to store a charge on its plates is called its capacitance, (C).

Thus a capacitors charging current can be defined as: i = CdV/dt. Once the capacitor is “fully-charged” the capacitor blocks the flow of any more electrons onto its plates as they have become saturated. However, if we apply an alternating current or AC supply, the capacitor will alternately charge and discharge at a rate determined by the frequency of the supply. Then the Capacitance in AC circuits varies with frequency as the capacitor is being constantly charged and discharged.

We know that the flow of electrons onto the plates of a capacitor is directly proportional to the rate of change of the voltage across those plates. Then, we can see that for capacitance in AC circuits they like to pass current when the voltage across its plates is constantly changing with respect to time such as in AC signals.

However, they do not like to pass current when the applied voltage is of a constant steady state value such as in DC signals. Consider the circuit below.

AC Capacitor Circuit

In the purely capacitive circuit above, the capacitor is connected directly across the AC supply voltage. As the supply voltage increases and decreases, the capacitor charges and discharges with respect to this change. We know that the charging current is directly proportional to the rate of change of the voltage across the plates with this rate of change at its greatest as the supply voltage crosses over from its positive half cycle to its negative half cycle or vice versa at points, 0^o and 180^o along the sine wave.

Consequently, the least voltage rate-of-change occurs when the AC sine wave crosses over at its maximum positive peak ( +V_MAX ) and its minimum negative peak, ( -V_MAX ). At these two positions within the cycle, the sinusoidal voltage is constant, therefore its rate-of-change is zero, so dv/dt is zero, resulting in zero current change within the capacitor. Thus when dv/dt = 0, the capacitor acts as an open circuit, so i = 0 and this is shown below.

AC Capacitor Phasor Diagram

At 0^o the rate of change of the supply voltage is increasing in a positive direction resulting in a maximum charging current at that instant in time. As the applied voltage reaches its maximum peak value at 90^o for a very brief instant in time the supply voltage is neither increasing or decreasing so there is no current flowing through the circuit.

As the applied voltage begins to decrease to zero at 180^o, the slope of the voltage is negative so the capacitor discharges in the negative direction. At the 180^o point along the line the rate of change of the voltage is at its maximum again so maximum current flows at that instant and so on.

Then we can say that for capacitors in AC circuits the instantaneous current is at its minimum or zero whenever the applied voltage is at its maximum and likewise the instantaneous value of the current is at its maximum or peak value when the applied voltage is at its minimum or zero.

From the waveform above, we can see that the current is leading the voltage by 1/4 cycle or 90^o as shown by the vector diagram. Then we can say that in a purely capacitive circuit the alternating voltage lags the current by 90^o.

We know that the current flowing through the capacitance in AC circuits is in opposition to the rate of change of the applied voltage. But just like resistors, capacitors also offer some form of resistance against the flow of current. For capacitors in AC circuits opposition is known as Reactance, and as we are dealing with capacitor circuits, it is therefore known as Capacitive Reactance. Thus capacitance in AC circuits suffer from Capacitive Reactance.

Capacitance in AC Circuits – Reactance

Capacitive Reactance in a purely capacitive circuit is the opposition to current flow in AC circuits only. Like resistance, reactance is also measured in Ohm’s but is given the symbol X to distinguish it from a purely resistive value. As reactance is a quantity that can also be applied to Inductors as well as Capacitors, when used with capacitors it is more commonly known as Capacitive Reactance.

For capacitors in AC circuits, capacitive reactance is given the symbol Xc. Then we can actually say that Capacitive Reactance is a capacitors resistive value that varies with frequency. Also, 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 given as:

Capacitive Reactance

Where: F is in Hertz and C is in Farads. 2πƒ can also be expressed collectively as the Greek letter Omega, ω to denote an angular frequency.

From the capacitive reactance formula above, it can be seen that if either of the Frequency or Capacitance where to be increased the overall capacitive reactance would decrease. As the frequency approaches infinity the capacitors reactance would reduce to zero acting like a perfect conductor.

However, as the frequency approaches zero or DC, the capacitors reactance would increase up to infinity, acting like a very large resistance. This means then that capacitive reactance is “Inversely proportional” to frequency for any given value of Capacitance and this shown below:

Capacitive Reactance against Frequency

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

The opposition to current flow, the electrostatic charge on the plates (its AC capacitance value) remains constant as it becomes easier for the capacitor to fully absorb the change in charge on its plates during each half cycle.

Also as the frequency increases the current flowing through the capacitor increases in value because the rate of voltage change across its plates increases.

Then we can see that at DC a capacitor has infinite reactance (open-circuit), at very high frequencies a capacitor has zero reactance (short-circuit).

Capacitance in AC Circuits Example No1

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

In AC circuits, the sinusoidal current through a capacitor, which leads the voltage by 90^o, varies with frequency as the capacitor is being constantly charged and discharged by the applied voltage. The AC impedance of a capacitor is known as Reactance and as we are dealing with capacitor circuits, more commonly called Capacitive Reactance, X_C

Capacitance in AC Circuits Example No2.

When a parallel plate capacitor was connected to a 60Hz AC supply, it was found to have a reactance of 390 ohms. Calculate the value of the capacitor in micro-farads.

This capacitive reactance is inversely proportional to frequency and produces the opposition to current flow around a capacitive AC circuit as we looked at in the AC Capacitance tutorial in the AC Theory section.