The average power of DC is different from the average power in an AC circuit. The alternating current uses sinusoidal voltage and current, which change its direction and magnitude every moment. How could one say that the power in an AC circuit has a fixed value?
Suppose a general load is connected with an AC source, as presented in the diagram. Suppose it has a voltage
$v=V_{m}sin \omega t\ldots (1a)$
Which causes it to flow the following sinusoidal current
$i=I_{m}sin \omega t+\theta \ldots (1b)$
The power flow of the source will be
$p=vi=(V_{m}sin \omega t)(I_{m}sin \omega t+\theta )$
$p=V_{m}I_{m} sin \omega t.sin \omega t+\theta \ldots (1c)$
Using the following trigonometric identity
$sin A .sin B=\frac{1}{2}cos (A-B)-\frac{1}{2}cos (A+B)$
so
$p=\frac{V_{m}I_{m}}{2} [cos (\omega t-\omega t-\theta )-cos (\omega
t+\omega t+\theta )]$
As we know from the following trigonometric identity that
$cos \theta =cos -\theta $
Now
$p=\frac{V_{m}I_{m}}{2}cos \theta -\frac{V_{m}I_{m}}{2}cos (2\omega
t+\theta )$
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Now observe the above equation, the power is made up of two cosine terms. The second term frequency is twice ($2 \omega t)$. As that of current or voltage (both have the same frequency of $\omega t$). This term has a component of time in it, which causes the cosine wave to oscillate and its average value over a cycle is zero. That’s why no energy is transferred due to the second term to the generic load attached to the AC source. Energy only floats back and forth between load and source.
Where the first term has no element of time, so it gives a fixed value for a specific value of theta. This is called the average power dissipated in the load. This average power in an AC circuit is mostly called real or active power. This power depends upon the angle $\theta $, which is the phase difference between current and voltage. So the equation for the real power is
$P=\frac{V_{m}I_{m}}{2}cos \theta$
$=\frac{V_{m}}{\sqrt{2}}\frac{I_{m}}{\sqrt{2}}cos \theta $
Where the relationship between RMS and peak value is $V_{RMS}=\frac{V_{m}}{\sqrt{2}}$, putting it into the above equation will give us
$P=VIcos \theta \ldots (1d)$
In a purely resistive AC circuit, the circuit current and voltage are in-phase, or the phase difference angle is zero i.e. $\theta =0 ^{\circ}$. So, the average power in an AC circuit with pure resistive load will
$P=VIcos 0^{\circ}$
$P=VI (1)cos 0^{\circ}=1$
$P=VI\ldots (2a)$
The equation (2a) means that the power delivered to the load is maximum in a purely resistive circuit.
Where in pure inductive AC load, the load current lags by $90^{\circ}$, i.e. $\theta ^{\circ}=-90^{\circ}$. So, the average power in an AC circuit with an inductive load will be
$P=VIcos (-90^{\circ})$
$P=VI (0)cos -90^{\circ}=0$
$P=0\ldots (2b)$
Equation (2b) means that no power is dissipated in a purely inductive load, the power only floats back and forth between the inductive load and the source.
In a pure capacitive AC circuit, the current leads source voltage by $90^{\circ}$ i.e. $theta =90^{\circ}$. In this case, the average power in an AC circuit with pure capacitive load is
$P=VIcos 90^{\circ}$
$P=VI (0)cos 90=0$
$P=0\ldots (2c)$
According to equation (2c), the power dissipated by the pure capacitor circuit is zero. Power only floats back and forth between the source and the capacitive load.
What is the Power Factor and Why is it Important?
We have seen in the above section that power dissipation depends upon the cosine of the phase angle along with the amplitude of voltage and current. For example, if the phase angle is 90 no power will dissipate, and if it is zero or somewhere between themes the load will dissipate power. The cosine term is very important and is called the power factor (PF).
$Power Factor=cos \theta \ldots (3a)$
Wherefrom impedance triangle we can calculate it through the following formula
$cos \theta =\frac{R}{Z}$
And from the power triangle, we can find like
$cos \theta =\frac{P}{S}$
According to equation (3a), the maximum power factor can be ONE and the minimum PF can be ZERO. An AC circuit containing multiple elements like a resistor, inductor, and capacitor can have a power factor between zero and one.
The power factor has no unit but it is usually expressed as a leading or lagging power factor. The leading/lagging depends upon the phase relationship between current and voltage. If the current leads the voltage, the power factor is called leading, and if the current lags, the power factor is called the lagging power factor. The PF of a capacitive circuit is leading and that for an inductive circuit is lagging.
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Why is the Power Factor Important?
The power factor in an AC circuit helps us in finding how much energy is utilized for performing the desired work. For the power factor equal to one or the “Unity power factor“, the total power is utilized.