2104 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 4, OCTOBER 2010 AC Power Theory From Poynting Theorem: Accurate Identification of Instantaneous Power Components in Nonlinear-Switched Circuits Francisco de León, Senior Member, IEEE, and José Cohen Abstract—This paper contributes to narrowing the long- standing theoretical gap with power theory (or “power defini- tions”) for nonlinear ac switching circuits. The true instantaneous energy transformation and storage components of ac circuits are identified from the Poynting Theorem. This paper tackles the problem of power identification from the most general form of energy conservation. Therefore, it is no longer necessary to mathematically “define” powers to fit the engineering solution of a problem. The identification technique does not present problems with physical meaning since it is in full agreement with Maxwell’s Equations. In this paper, the method is applied to the identifica- tion of the power components of single-phase switched circuits. Instantaneous energy is decomposed only into energy transformed (related to active power) and energy stored (related to reactive power). Examples that have caused physical interpretation prob- lems with other power theories are presented for illustration and validation. Index Terms—Active power, alternating current circuits, energy restored, energy stored, energy transformed, instantaneous power, nonlinear circuits, power definitions, power theory, reactive power. I. INTRODUCTION P OWER definitions have been the subject of much research for more than 100 years. At the end of the 19th century, Steinmetz generated and compiled most of the available knowl- edge for the analysis of power in ac circuits in [1]. The problems with the definition of power (and power factor) for unbalanced circuits were identified as early as 1920 [2]; there are more than 70 pages with discussions on the definition of power. A second round of discussions took place in 1933 [3] and focused on the definition of reactive power for nonlinear circuits. Companion papers and discussions extend to almost 60 pages. The discus- sions on what power really is in nonlinear and unbalanced cir- cuits have continued and a great number of papers have been published on the matter. We have compiled more than 200 pa- pers on the subject. In 2000, the IEEE published the standard Manuscript received March 19, 2009; revised May 03, 2010; accepted June 20, 2010. Date of publication August 03, 2010; date of current version September 22, 2010. Paper no. TPWRD-00227-2009. F. de León is with the Polytechnic Institute of New York University, Brooklyn, NY 11201 USA (e-mail: fdeleon@poly.edu). J. Cohen is with the Departamento de Ingeniería Eléctrica, División de Estudios de Posgrado, Universidad Nacional Autónoma de México (UNAM), México City 04510, México. (e-mail: jcohenmex@hotmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRD.2010.2054117 1429-2000 on power definitions that has served to further ignite the discussions [4]. One can learn about the issues and history of power defini- tions through some of the papers of Emanuel [5]–[9] and Czar- necki [10]–[16] and their references. Numerous attempts to establish a power theory that fits some observable phenomena have produced a gamut of power con- cepts lacking physical meaning when applied to other cases. Through mathematical manipulations, many authors have tried to generalize power definitions applicable to a particular case. Some authors, for example, Shepherd and Fang [17], have ex- plicitly admitted the weaknesses of their power theories in re- gards to physical interpretation. In this paper, we propose an instantaneous power theory di- rectly derived from Maxwell’s Equations and specifically from the Poynting Vector Theorem. Accordingly, only two energy (or power) components exist: 1) the energy transformed yielding the active power and 2) the energy that is stored/restored in the electromagnetic fields that gives birth to the reactive power. Note that the power components of this paper are not defined from a particular example, but are accurately computed from the most fundamental conservation of energy principles. Only the instantaneous information on terminal voltage and current are required to fully characterize the power phenomena of a switching load. There are publications in favor of [18] and against [19] the use of the Poynting vector to describe power phenomena in electrical circuits. For us, the Poynting vector is not merely a mathematical tool for calculating energy flow as claimed in [19]. Poynting theorem has been derived from the experimen- tally macroscopically undisputed Maxwell’s Equations. There- fore, it offers the best physical description yet available for the representation of electrical power and energy phenomena. This paper advocates for the time-domain analysis of powers for nonlinear circuits. Note that the commonly used quantities to characterize power such as apparent power , reactive power , power factor , etc., do not exist in instantaneous terms. Those quantities are simply definitions that have shown to be useful and fully meaningful only for linear ac circuits [20]. Ex- tensions to nonlinear circuits have failed to provide the same physical meaning. The contribution of this paper is the proper identification of the instantaneous energy (and power) components for non- linear-switched ac circuits. The results are different from the instantaneous power theories of Fryze [21] and Akagi et al. [22]. Both of those power theories lack sound physical meaning 0885-8977/$26.00 © 2010 IEEE