This section covers basic concepts about apparent, active (real) and reactive power which is important ingredients in the analysis of a power system.

Consider the general single-phase circuit with a sinusoidal voltage $v={{V}_{m}}sin\left( wt \right)$ applied. A current $i={{I}_{m}}sin(wt\pm \theta )$ results and is leading (θ is positive) for a capacitive type circuit and is lagging (θ is negative) for an inductive type circuit.

The instantaneous power is

$p=vi={{V}_{m}}{{I}_{m}}Sin\omega tSin\left( wt+\theta \right)$

Using the trigonometric identity,

$Sin\alpha Sin\beta =\frac{1}{2}\left( Cos\left( \alpha -\beta \right)-Cos\left( \alpha +\beta \right) \right)$

So, the power expression is;

$P=\frac{{{V}_{m}}{{I}_{m}}}{2}\left( \cos \left( wt-wt-\theta \right)-\cos \left( wt+wt+\theta \right) \right)$

$P=\frac{{{V}_{m}}{{I}_{m}}}{\sqrt{2}}\left( \cos \left( -\theta \right)-\cos \left( 2wt+\theta \right) \right)$

Because

$V=\frac{{{V}_{m}}}{2}$

$I=\frac{{{I}_{m}}}{2}$

So,

$P=VI\left( \cos \left( \theta \right)-VI\cos \left( 2wt+\theta \right) \right)~~~~~~~~\cdots ~~~~~~\left( 1 \right)$

The second term of equation (1) represents a cosine wave of twice the frequency of the applied voltage; since the average value of a cosine wave is zero, this term contributes nothing to the average power. However, the first term is of particular importance because of the terms V, I and Cosθ are all constant and do not change with time. Thus, the average value of power, P, is given by

$P=VICos\left( \theta \right)~~\text{ }\cdots \text{ }~~~\left( 2 \right)$

Where V and I are the effective RMSrms values of voltage and current and θ is the phase angle between the voltage and current. Since the phase angle for a single-phase circuit is always between ±90^{o}, so

$Cos\left( \theta \right)\geqq 0$

And

$P\geqq 0$

The term Cosθ is called the power factor and the angle θ is referred to as the power factor angle. In the inductive circuit, where the current lags the voltage, the power factor is described as a lagging power factor. In the capacitive circuit, where the current leads the voltage, the power factor is considered as leading power factor.

**Apparent Power**

As the product VI in equation (2) does not represent either average power in watts or reactive power in vars, it is defined by a new term, apparent power. The product VI, called apparent power, has the unit of volt-ampere (VA) and is indicated by the letter S. Thus;

$S=VI$

So,

$P=S*Cos\theta ~~~\text{ }\cdots \text{ }~~~~\left( 3 \right)$

**Power Triangle**

Equation (3) suggests a relationship between active Power P and apparent power S using triangle known as Triangle.

Where the real power P is along the horizontal axis, the reactive power Q is along the vertical axis, and the apparent power S is the hypotenuse.Then,

$Q=S*Sin\theta ~~~~~\text{ }\cdots \text{ }~~\left( 4 \right)$

And

$S=\sqrt{{{P}^{2}}+{{Q}^{2}}}$

**Reactive Power**

Reactive power Q is defined by equation (4)

$Q=VI~Sin\theta $

If a circuit has inductive element only, Q is inductive reactive Power, Q_{L }expressed in terms of VARS.

If the circuit has capacitive element only, Q is capacitive reactive power,Q_{C} expressed by the same unit as Q_{L}.

If the circuit contains both inductance and capacitance, the net reactive power,Q_{t} is the difference between the capacitive reactive power and inductive reactive power. In such a case, capacitor returns energy to the circuit while inductor takes energy from the circuit.

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