A systematic, mathematically and physically sound approach to the energy balance in three-wire, three-phase systems

This paper proposes a systematic approach to the analysis of the three-wire, three-phase systems. The approach is based on the modal analysis of the electric circuits, that, in the case of the three-phase systems, leads to the well known transformations of Fortescue, Lyon, and Clarke-Park. The paper shows how these transformations are closely related to each other, and how the analysis in terms of these same transformations gives evidence of some peculiar characteristics of the three-phase systems that the analysis in terms of the line quantities does not show in such an immediate and clear way. At last, the energy balance is taken into account, showing how the so-called imaginary power is a typical quantity of the three-phase system, and not only a mathematical trick obtained only by a formal extension of the phasorial notation used under sinusoidal conditions. The physical meaning of the imaginary power is also discussed.