Capacitance in AC Circuits

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

During this charging process, a charging current, ( i ) will flow into the capacitor opposing any changes to the voltage at a rate that is equal to the rate of change of the electrical charge on the plates.

This 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 capacitors in AC circuits like to pass current when the voltage across its plates is constantly changing with respect to time such as in AC signals, but it does not like to pass current when the applied voltage is of a constant 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, 0o and 180o along the sine wave.

Consequently, the least voltage change occurs when the AC sine wave crosses over at its maximum or minimum peak voltage level, ( Vm ). At these positions in the cycle the maximum or minimum currents are flowing through the capacitor circuit and this is shown below.

AC Capacitor Phasor Diagram

At 0o 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 90o 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 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 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 90o as shown by the vector diagram. Then we can say that in a purely capacitive circuit the alternating voltage lags the current by 90o.

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 through the circuit, but with capacitors in AC circuits this AC resistance is known as Reactance or more commonly in capacitor circuits, Capacitive Reactance, so capacitance in AC circuits suffers from Capacitive Reactance.

Capacitive 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. 2πF 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).

AC Capacitance Example No1.

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

In AC circuits, the sinusoidal current through a capacitor, which leads the voltage by 90o, 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, XC

AC Capacitance 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.

• R
RB

Hey it’s good

• V
Vipin PVarghese

Hai,
I am insect killer machine manufacture. In my unite we using one step up transformer(2.4kv AC) and one 7.5 K DC out put capacitor. transformer’s two end we connected to a power grid for swamping insects , and the 7.5 K DC capacitor we connected in between the transformer and electric power grid.
My question
1) for what purpose we connected the capacitor?
2) it is what kind of capacitor?
3)if capacitor got damage the circuit is not working, what is the reason?

Anybody can help me.

• J
Jyrki

Hi,
Great tutorials!
A note about units in graphs. It soul be easier to read the graphs if naming the coordinate axis ie time would implicate time division ticks tau rather than degrees.
Same consistency is desired in formulas like now reads (2pif-90 degrees) should read (2pif-pi/2).
Small things in otherwise great work.

• Wayne Storr

The x-axis of a sinusoidal waveform is time, (t) as it is a rotating vector and its corresponding sine function is periodic in time.

Tau, (T) is the rate at which a capacitor charges as expressed in terms of its time constant and is not related to a sinusoid.

• Byron Brubaker

Nothing goes “through” a capacitor unless you damage the dielectric. The first drawing is wrong.

• j
john kinnear

I am interested in this theory as I have a motor on a saw that fails to run . can it be a break in the capacitor?

• Wayne Storr

Could be, single-phase capacitor run motors have two windings, one of them is in series with a low value capacitor (less than 20uF). If you have supply to the motor and it fails to start, it is possible that one of the two windings is gone (open-circuit).

• p
prince of the seas

hi please can u help me with theory of capacitance in a.c circuit

• H
Harish Zambani

Hi Wayne,

How the high frequency transients are removed from DC circuits using capacitors?

Can you throw some knowledge of yours on this.

• trsk

You can use a ferrite bead for that.

• Wayne Storr

When a capacitor is placed across a voltage source, the impedance of the capacitor forms a voltage divider with the impedance of the source. If the source is steady and constant the impedance of the capacitor is very high and the capacitor acts like an open-circuit. If a high frequency transient or voltage spike occurs on the voltage source then the impedance of the capacitor becomes very low and the capacitor acts as a short-circuit resulting in attenuation of the transient.

• g
giussepe

i have this high volatge transformer which input is 220v/60hz and secondary core output is 2500v 7~9mA. i need to conect a high voltage capacitor on the output wiring,what will be the high voltage capacitor? v= ? capacitance= ?
the output of this transformer is conected to a electrical shock grid to kill insect.thanks alot

• i
ilkay

Hi Wane, Could you explain what exactly you mean by ”a charging current, ( i ) will flow into the capacitor opposing any changes to the voltage at a rate that is equal to the rate of change of the electrical charge on the plates.”

I thought the capacitor opposes the current flow when it has charged but I don’t get what you mean by ” current opposes the change in voltage” please explain..

• Wayne Storr

The basic property of a capacitor is its ability to store electrical energy in the form of an electric charge on it plates with the ability of a capacitor to store a charge being its capacitance. The flow of electrons onto the plates is known as a charging current with the amount of charge on the plates calculated by multiplying the amount of current that flows by the time for which it flows. However, this charging current is not instant or constant but will flow at a rate that starts fast at the instant the voltage source is connected but gradually slows down as the plates charge up.

This charging current is being pushed onto the plates by the voltage source, but equally the capacitor opposes a change in voltage across it at a rate that is equal to the rate of change slowing down the charging current and this continues until the voltage across the plates is equal to the supply voltage. The capacitor is then said to be fully charged but in reality the charging is never totally complete. The amount of charge is given as: Q = C x V and is generally fairly constant for a particular size and arrangement of plates.

• k