New Modalities in Performing the “abc-$\alpha\beta$” and “$\alpha\beta$-abc” Coordinates Conversion using Hilbert Transform

The paper presents new modalities to transform the relations from the “abc” coordinates to the “ αβ” coordinates and vice versa using the Hilbert transform. The new analysis of the unbalanced three-phase system is achieved by symmetrical components method. The method is based on the properties of the Hilbert transform. At the beginning, relations are developed for calculating the symmetrical components in “abc” natural coordinates of a three-phase system. In the end, similar relationships in “ αβ” coordinates are developed. There are also presented the conversion relations between different quantities expressed in both coordinate systems. The analysis is done for the general case of an unbalanced, non-sinusoidal, distorted three-phase system. Symmetrical components will be implicitly determined for each group of harmonics of the same frequency. The main conclusion is that the Hilbert transform based method represents an important analytical and computational tool regarding the unbalanced three-phase systems.