Real, reactive power and power factor can be explained using different concepts.  Sometimes it is convenient to use the concept of complex power.  In an electrical system, if the voltage and current are treated as vectors, these can be expressed in complex numbers.  The complex power S, of the system is then given by:

where:

• P is the complex power in VA
• V is the voltage in complex form in V
• I^* is the conjugate of the current in complex form (the imaginary part is multiplied by -1) in A

Complex power S, is a vector and is commonly represented as:

where

• S is the complex power in VA
• P is the real power transmitted in W
• Q is the reactive power transmitted in VAr
• j is the complex operator 

The quantity P, represents the real power transmitted in the system.  This is the power that actually does work (i.e. rotating motors, heating homes, etc.).  The reactive power Q, does not do actual real work, but is responsible for maintaining the magnetic and electric fields of the electrical system.  These fields provided the energy storage mechanisms necessary for the distribution system to work. The magnitude of the complex power S, is often referred to as apparent power.

Complex power as a vector is illustrated in the diagram above.  The ratio between the real and apparent power is the power factor of a system:

giving

If the current is lagging the voltage in phase (for example in an inductive circuit) the power factor is said to be lagging.  Conversely if the current leads the voltage the power factor is said to be leading.  For a power factor of 1, the real and apparent power will be the same, with the reactive power being equal to zero.  As the power factor is reduce, more reactive power in the system is required.  Generation of reactive power required additional equipment,  increasing transmission system costs and operating energy loss.  By ensuring high power factors these costs and loses are reduced.

References

• William D.Stevenson, "Elements of Power System Analysis", McGraw-Hill, 1982
• This post is reproduced and expanded in our Wiki page on complex power.