Capacitance in AC Circuits





	

Capacitance in AC Circuits


Capacitors that are connected to a sinusoidal supply produce reactance from
the effects of supply frequency and capacitor size


Capacitance in AC Circuits results in a time-dependent current which is
shifted in phase by 90o 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

	
	capacitance in ac circuits

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 rate-of-change occurs when the AC sine wave
crosses over at its maximum positive peak ( +VMAX ) and its minimum negative
peak, ( -VMAX ). 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

	
	capacitance in ac circuits 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. 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
	
	capacitance in ac circuits 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





	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.

	
Answer Example No1

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



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.
	
	capacitance of a capacitor

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.
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