# Complex Power – Definition, Explanation & Significance

## Definition:

Complex Power is basically the representation of electrical power in the form of complex number. Like a complex number, it consists of real and imaginary part. Real part represents the active power whereas the imaginary part represents the reactive power. It is generally represented by symbol S.

If active and reactive power be P & Q respectively, then complex power for an inductive load is written as

S = P + jQ

Complex power for a capacitive load is given as below.

S = P – jQ

## Explanation:

To better understand, let us consider the voltage and current of a load to be V and I respectively. Assume the load to be capacitive. Mind that load is not purely capacitive rather capacitance is dominating in load. Therefore, the current will lead the voltage by some angle ø.

Let voltage across the load V = Ve and I = Iej(Ɵ+ø). Now, Complex Power S can be find by multiplying the voltage (V) with the conjugate of current (I*).

S = VI*

= Vex Ie-j(Ɵ+ø)

= Vie-jø

= VI (cosø – jsinø)

= VIcosø – VIsinø  ………(1)

As we know that, active power (P) and reactive power (Q) is given as

P = VIcosø

Q = VIsinø

Therefore, complex power can be written form (1) as shown below.

S = P – jQ

Carefully observe the above expression. The imaginary part is negative. This means, reactive power is negative. As load was assumed capacitive, the reactive power is due to the capacitance in the load. Since reactive power is negative, this implies that a capacitor is a generator of reactive power. It does not consume reactive power in a circuit rather it produces reactive power.

Complex power S may also be found in the similar way for inductive load. Let voltage across the load V = Ve and I = Iej(Ɵ-ø). Now, the S can be found by multiplying the voltage (V) with the conjugate of current (I*).

S = VI*

= Vex Ie-j(Ɵ-ø)

= Vie

= VI (cosø + jsinø)

= VIcosø + VIsinø  ………(1)

As we know that, active power (P) and reactive power (Q) is given as

P = VIcosø

Q = VIsinø

Therefore, complex power can be written form (1) as shown below.

S = P + jQ

The above expression reveals that reactive power is positive. This simply means that reactive power is being consumed by an inductor.

## Significance of Complex Power:

• Power triangle can easily be constructed if the value of complex power is known. Q will represent the perpendicular, P the base and S is the hypotenuse of power triangle.
• Real part gives the value active power whereas the imaginary part gives the value of reactive power.
• Magnitude of S gives the value of Apparent Power. Thus, apparent power = √(P2+Q2).