In electrical engineering, power is the rate at which energy is transferred or converted in an electrical system. Complex power is a concept used to describe the total power in a system that has both reactive (stored) and active (real) power components.

This concept is crucial in understanding power flow in AC circuits, especially in systems with significant inductive or capacitive loads where the phase difference between voltage and current leads to less efficient power transfer. Understanding complex power helps in designing and operating electrical systems more efficiently, especially in terms of power factor correction and load management.

1. Real Power (P): This is the power consumed by the circuit in performing useful work. It is measured in watts (W). Real power is the power that is converted into heat, light, motion, etc., in electrical devices like motors and lamps.

2. Reactive Power (Q): This power is due to energy storage and release in the circuit's inductors and capacitors. It is measured in reactive volt-amperes (var). Reactive power does not perform any real work but is essential for creating the magnetic and electric fields necessary for the operation of AC equipment.

3. Apparent Power (S): This is a combination of real and reactive power and represents the total power in the circuit. It is measured in volt-amperes (VA). Apparent power is the product of the circuit's voltage and current without considering the phase difference between them.

Complex power (S) is represented as a complex number, with real power as the real component and reactive power as the imaginary component. It is given by the formula:

$S=P+jQ$

where j is the imaginary unit. The magnitude of the complex power gives the apparent power, and the angle of the complex power vector (often denoted as φ) gives the phase difference between the voltage and current.

Mathematically, complex power is defined as the product of the voltage and the complex conjugate of the current:

where I* is the complex conjugate of the current. The reason for using the complex conjugate of the current in this calculation is to account for the phase difference between voltage and current.

Complex power is often expressed in units of volt-amperes (VA) or kilovolt-amperes (kVA), units of apparent power. The real power component of complex power is expressed in units of watts (W) or kilowatts (kW).

Complex Power Theory

Voltage (V) and Current (I) as Complex Numbers

In AC circuits, the voltage and current can be represented as $V=V_m{e}^{j\theta V}$ and $I=I_m{e}^{j\theta I}$ , where V_m​ and _m​ are the magnitudes of the voltage and current, respectively, and θV​ and θI​ are their respective phases.

Complex Conjugate of Current (I^∗)

 The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. So, if $I=I_m{e}^{j\theta I}$, $I^{*}=I_m{e}^{-j\theta I}$​.

Complex Power (S)

The complex power is calculated using $S=VI*$. This calculation results in a complex number, where the real part represents the real power (P) and the imaginary part represents the reactive power (Q).

The formula can also be represented as:

$S=V{I}^{*}=V_m{I}_m{e}^{j(\theta V-\theta I)}$

where V_m​I_m​ gives the magnitude of the apparent power, and θV​−θI​ gives the phase angle of the complex power.