Instantaneous power components definition for three-phase circuits based on the four- dimension complex number

A succinct definition of instantaneous power components based on the concept of four-dimension complex number is proposed in this paper. The instantaneous complex power is defined as the product of complex voltage and the conjugate of complex current, and the instantaneous real (active) power and imaginary (reactive) power are defined as its real part and imaginary part respectively, which are similar to the definitions of conventional power components in frequency domain. This generalized definition with clear physical meanings is valid for sinusoidal or nonsinusoidal, balanced or unbalanced three-phase power systems with or without zero-sequence components.