Complex power is a way of representing power in an AC circuit where voltage and current may not be in phase. It combines real power, which performs useful work, with reactive power, which is associated with energy storage in magnetic and electric fields.
The concept is useful in cable design because load current, power factor and reactive power all affect current flow, voltage drop and network loading.
Real, reactive and apparent power
| Quantity | Meaning | Typical unit |
|---|---|---|
| Real power, P | Power converted into useful work, heat, light or motion | W or kW |
| Reactive power, Q | Power associated with stored and released energy in inductance and capacitance | var or kvar |
| Apparent power, S | Combined effect of voltage and current magnitude, before separating real and reactive components | VA or kVA |
Complex power is written as a complex number:
The real component is real power, and the imaginary component is reactive power. The magnitude of S is the apparent power, and the angle of the complex power vector is the phase angle between voltage and current.
Complex power definition
Mathematically, complex power is defined as the voltage multiplied by the complex conjugate of the current:
The complex conjugate is used so the sign of the reactive power follows the usual convention for lagging and leading power factor. Complex power is commonly expressed in VA or kVA, while its real component is expressed in W or kW.
Voltage and current as complex quantities
In sinusoidal steady-state AC analysis, voltage and current can be represented as complex quantities:
The complex conjugate of the current is:
Substituting these into the complex power expression gives:
The product VmIm gives the apparent power magnitude, while θV – θI gives the phase angle of the complex power.
Why complex power matters in cable calculations
For a given real power, a lower power factor requires more current. Higher current increases cable losses and voltage drop, and it may require a larger conductor or a different supply arrangement. Complex power therefore provides a compact way to connect load demand, power factor, current and voltage-drop calculations.
For related cable calculations, see Voltage Drop Calculations for Cable Sizing and Cable Sizing.
For cable-specific application of complex power to conductor losses, see Cable Power Loss.