# Relationship between [PERSON]'s Apparent Power and Instantaneous Power in Three-Phase Systems
## 1 Introduction  
Apparent power is used to design generators and transformers in power systems and can be applied to electrical network line calculations, as well as the selection of the size of equipment, such as electronic converters and measurement instruments. In our opinion, apparent power should also measure the combined effects of all phenomena that manifest in power systems. The above information exhibits the importance of apparent power in electrical grids. However, the definition of apparent power in three-phase systems is currently an open issue. Since [PERSON] [1] formulated apparent power in single-phase sinusoidal circuits in the late 19 th century, many definitions of apparent power were proposed in single- and three-phase systems. [PERSON]'s apparent power is related to instantaneous power, and this relationship also extends to the apparent power components, namely, active and reactive power. Instantaneous power is the quantity (along with the active power) directly related to the energy, or the rate with which the energy is transferred in a circuit; hence, the relationship between the apparent and instantaneous power confer a certain physical meaning to the apparent power. For this reason, [PERSON] [2] used instantaneous power to formulate apparent power and its components in single-phase non-sinusoidal systems. However, some of [PERSON]'s apparent power components, such as the reactive and distortion power, did not achieve the unanimous consensus of the scientific community, as specified by [PERSON] [3]; thus, other approaches considered that a physical interpretation was not necessary and definedapparent power in regard to the voltage and current waveforms. [PERSON] [4], for three-phase power systems, and [PERSON] [5], for non-sinusoidal single-phase power systems, pioneered these approaches, and their influence extended to researchers such as [PERSON] in the 1980s [6] and even until now [7; 8]. In spite of the errors regarding [PERSON]'s approach, the search for the physical meaning of apparent power was not abandoned and was recently studied in three-phase systems through different approaches, including the IEEE Standard 1459-2010 [9] and DIN 40110-2 [10]. The IEEE Standard 1459, which was based in part on research developed by [PERSON] [11; 12; 13], and the DIN 40110-Teil 2, which was supported by [PERSON] [14] and [PERSON] [15], were strongly influenced by the definitions of apparent power suggested by [PERSON] in 1920 [16] and by [PERSON] and [PERSON] in 1935 [17], who established that, in single-phase systems, \"Apparent power is numerically equal to the maximum active power that can exist at given points of entry with the given effective value of the sinusoidal current and potential difference and hence is directly related to the size of the required equipment and to the generating and transmitting losses\". In our opinion, the relationships between apparent and instantaneous power established by [PERSON] can also be generalized to three-phase sinusoidal and non-sinusoidal, balanced and unbalanced power systems and can be extended to their components. These relationships are described in Section 2 in which several formal similarities between Buchholz's apparent power and instantaneous power are identified in three-phase, four-wire power systems [18]. According to these similarities, an original procedure to obtain the expressions for [PERSON]'s apparent power and a set of power components (i.e., fundamental, positive-sequence active and reactive power, as well as unbalanced and non-fundamental power) are described in Section 3 for three-phase, wye-configured systems. These types of power are expressed as functions of well-known voltage and current components that obey [PERSON]'s laws, [PERSON]'s theorem, and the Fourier series.