E TEP z An zyxw Approach to the Non-Active Power Concept z in Terms of the Poynting-ParkVector A. Ferrero, S. Leva, A. P. Morando Abstract zyxwvutsrq This paper reconsiders the approach to the electric systems in terms zyxwv of Maxwell equations, and reconsiders in particular the energy transfer involved in an electric system in terms of the flux of the Poynting vector. This approach is extended to the three-phase systems, and the Park transformation is reconsidered by applying it to the Poynting vector. In this way a correct physical meaning can be assigned to the non-active components of the Park instantaneous power by tracing them back to the components of the Poynting vector. 1 Introduction zyxwvuts There is no doubt that the modern theory of the elec- tromagnetic phenomena is totally based on the well known Maxwell equations. These equations fully ex- plain the interactions between the electric and magnetic fields and therefore explain every electromagnetic phe- nomenon, including the energy transfer in an electric circuit. There is also no doubt that, under most practical sit- uations, an electric circuit can be studied by means of mathematical tools that are much simpler to handle than the Maxwell equations. This is due to the fact that the electric circuits considered for almost all practical ap- plications in the electric power systems can be regarded as a very particular case of an electromagneticfield and therefore suitableconstraintscan be considered when in- tegrating the Maxwell equations. This approach allows from moving from the field theory (always ruled by the Maxwell equations) to the circuit theory, ruled by sim- pler equations based on the relationships between volt- ages, currents and circuit element properties [l]. The foundation of the circuit theory goes back to the end of the 1 91h and the beginning of the zyxwvu 20th century and struck so deeply root into the mind of the modern elec- trical engineers that the constraints considered to derive the circuit equationsfrom the Maxwell ones are too often forgotten. Clear evidence of such forgetfulnessis given by the way the electric systems under non-sinusoidal conditions have been analysed: at the Authors’ knowl- edge, all known attempts started from the equations em- ployed in the sinusoidal conditions and modified them in the attempt (not always successful)to extend their va- lidity to the non-sinusoidal situations. The result of such a way of approaching the problem is that the physical meaning assigned to the defined (or re-defined) quantities cannot usually be framed into the general theory of the electric systems. When the three-phase circuits are considered under non-sinusoidal unbalanced situations, things become even more difficult,due to the increasing number of phe- nomena involved. Their analysis in terms of the zyxwvu ap [2] ordq [3- 101componentsleads to interesting results from the practical point of view, especially when compensa- tion and measurement techniques are considered, but has not yet found a physically sound explanation for the defined power components other than the instantaneous one, that appear to be derived only by means of mere mathematical considerations. The above considerations lead to conclude that a dangerous gap is arising between the practical ap- proaches that are being proposed in the literature and more and more used by the practitioners, and a physi- cally sound approach that describes the physical phe- nomena coherently with the fundamental Maxwell equations. In order to fill this gap, and find a physically sound theoreticalbasis to the practical approaches,the Poynting vector zyxwv [ 1 11 should be reconsidered, since it represents, in the interpretationgiven by Slepian [ 12,131,the bridge be- tween the field theory and the circuit theory [ 141. The explanationof the energy flow and conservation in single-phase circuits comes quite immediately, as it will be briefly recalled in the following section, by the analysis of the flux of the Poynting vector through a given surface,as clearly shown also by [ 151.When three- phase circuits are concerned, the approach is no longer so immediate, since the resultant electric and magnetic field produced by the considered three-phase element should be considered in order to evaluate the Poynting vector. This paper is aimed to show how an electric and a magnetic field can be assumed as associated with the Park vectors of the voltages and currents appertaining to a three-phase element showing physical symmetry.Tak- ing into account these fields leads to the definition of a “Park-transformed” Poynting vector able to explain the energy flow and conservation for the three-phase ele- ment. The physical meaning for the non-active Park power componentsis also found, thus fixing an unsolved point in the analysis of the three-phase systems in terms of the Park componentsand giving evidence of the phys- ical, and not only mathematical, correctness of the Park approach. ETEP Vol. 1 I, No. 5, September/October 2001 29 I