A Step Forward in Application of Dynamic Harmonic Domain: Phase Shifting Property of Harmonics

In this paper, a novel analytical approach is put forward which leads to phase-shifting property of harmonics in a periodic system. Based on the illustrations, it is shown that if all of the input sources of the periodic system are shifted by
$\boldsymbol{\alpha }$
, the entire outputs will be shifted by
$\boldsymbol{\alpha }$
as well in both transient and steady states. There is no limitation for such a system as it can be either single or three phase, linear or nonlinear, or supplied by periodic balanced or unbalanced sinusoidal/nonsinusoidal sources. According to this concept, the source angle affects only the phase angle of harmonics linearly but not their magnitude. However, related time-domain response may be greatly affected. These characteristics of the harmonics are observed using the dynamic harmonic domain analysis which calculates exact variations of harmonics. To such aim, a phase-shifting matrix is defined which provides exact phase-shift calculation considering each harmonic order. The introduced concept significantly reduces the required simulation time and represents the effects of the source phase angle on the harmonic content and time-domain response without performing extra simulations. Finally, the novel concept is successfully applied to two test cases, followed by analysis and discussion.