Spectral analysis of instantaneous powers in single-phase and three-phase systems with use of p-q-r theory

Instantaneous active power p, q-axis instantaneous reactive power q_q, and r-axis instantaneous reactive power q_r have been defined as linearly independent of each others in the time domain in three-phase four-wire systems with the use of p-q-r theory. This paper renames the instantaneous active/reactive powers as instantaneous real/imaginary powers to distinguish them from the traditional definition of active/reactive powers. The instantaneous real/imaginary powers are analyzed at various circuit conditions in single-phase and three-phase systems. The q-axis instantaneous imaginary power q_q, is orthogonal to the other two instantaneous powers p and q_r in the frequency domain. By analyzing the spectral distribution of the instantaneous powers p, q_q, q_r, this paper proposes unified definitions for some power quality factors which can evaluate and identify the various circuit conditions both for single-phase and three-phase systems.