# Relationship between [PERSON]'s Apparent Power and Instantaneous Power in Three-Phase Systems
## 2 Formulations of Apparent and Instantaneous Power in Three-Phase Wye-Configured Systems
### [PERSON]'s Apparent Power  
[PERSON] defined apparent power in 1922 [4] as a function of RMS line-to-neutral voltages and line currents in three-phase, three-wire power systems, with no physical significance. If we consider the neutral conductor to be a different subsystem to the source and load of the system illustrated in Figure 1, the source and load can be considered as a three-port subsystem to which [PERSON]'s approach can be applied. Hence, the source apparent power can be expressed according to [PERSON]'s approach as follows:  
\[S_{S}=\sqrt{\sum_{z}V_{zN}^{2}}\sqrt{\sum_{z}l_{z}^{2}}=\sqrt{\big{(}V_{AN}^{2 }+V_{BN}^{2}+V_{CN}^{2}\big{)}\big{(}l_{A}^{2}+l_{B}^{2}+l_{C}^{2}\big{)}} \tag{13}\]  
Using the Fourier series, the squares of the RMS values of these voltages and currents can be expressed as follows:  
\[V_{zN}^{2}=V_{zN1}^{2}+V_{zNH}^{2}=V_{zN1}^{2}+\sum_{x\
eq 1}^{\infty}V_{zNx}^{2 }\,l_{z}^{2}=l_{z1}^{2}+l_{zH}^{2}=l_{z1}^{2}+\sum_{y\
eq 1}^{\infty}l_{zy}^{2} \tag{14}\]  
where (\(V_{zN1}\) and \(l_{z1}\)) denote the fundamental-frequency RMS values of each phase (\(z=A,B,C\)) of the source line-to-neutral voltages and line currents, respectively, and (\(V_{zNH}\) and \(l_{zH}\)) represent the RMS values of all harmonics without fundamental phase voltages and currents. Thus, the source apparent power can be expressed as follows:  
\[S_{S}=\sqrt{\sum_{z}(V_{zN1}^{2}+V_{zNH}^{2})}\cdot\sqrt{\sum_{z}\big{(}l_{z1} ^{2}+l_{zH}^{2}\big{)}}=\sqrt{S_{S1}^{2}+S_{SH}^{2}} \tag{15}\]  
where \(S_{S1}\) is the source fundamental-frequency apparent power,  
\[S_{S1}=\sqrt{\sum_{z}V_{zN1}^{2}}\cdot\sqrt{\sum_{z}l_{z1}^{2}}=\sqrt{\big{(} V_{AN1}^{2}+V_{BN1}^{2}+V_{CN1}^{2}\big{)}\big{(}l_{A1}^{2}+l_{B1}^{2}+l_{C1}^{2} \big{)}} \tag{16}\]  
and \(S_{SH}\) represents the source non-fundamental power,  
\[S_{SH}=\sqrt{\sum_{z}V_{zN1}^{2}\cdot\sum_{z}l_{zH}^{2}+\sum_{z}V_{zNH}^{2} \cdot\sum_{z}l_{z}^{2}}=\sqrt{\big{(}S_{SH}^{I}\big{)}^{2}+\big{(}S_{SH}^{V} \big{)}^{2}} \tag{17}\]  
where  
\[S_{SH}^{I}=\sqrt{\sum_{z}V_{zN1}^{2}\cdot\sum_{z}l_{zH}^{2}}=\sqrt{\big{(}V_{ AN1}^{2}+V_{BN1}^{2}+V_{CN1}^{2}\big{)}\big{(}l_{AH}^{2}+l_{BH}^{2}+l_{CH}^{2} \big{)}} \tag{18}\]  
is the non-fundamental power corresponding to the distorted currents, exclusively, and  
\[S_{SH}^{V}=\sqrt{\sum_{z}V_{zNH}^{2}\cdot\sum_{z}l_{z}^{2}}=\sqrt{\big{(}V_{ AH}^{2}+V_{BH}^{2}+V_{CH}^{2}\big{)}\big{(}l_{A}^{2}+l_{B}^{2}+l_{C}^{2}\big{)}} \tag{19}\]  
is the non-fundamental power due to the distorted voltages. On the other hand, the fundamental-frequency source line-to-neutral voltages and line currents of each phase (\(z=A,B,C\)) can be expressed as functions of their positive- (+), negative- (-), and zero-sequence (0) components, according to [PERSON]'s theorem:  
\[ \begin{array}{c}\sum\limits_{z}V_{zN1}^{2}=\sum\limits_{z}\left[\left(V_{zN1}^ {+}\right)^{2}+\left(V_{zN1}^{-}\right)^{2}+\left(V_{zN1}^{0}\right)^{2}\right] =\sum\limits_{z}\left[\left(V_{zN1}^{+}\right)^{2}+\left(V_{zN1}^{U}\right)^{2 }\right]\\ \sum\limits_{z}I_{z1}^{2}=\sum\limits_{z}\left[\left(I_{z1}^{+}\right)^{2}+ \left(I_{z1}^{-}\right)^{2}+\left(I_{z1}^{0}\right)^{2}\right]=\sum\limits_{z }\left[\left(I_{z1}^{+}\right)^{2}+\left(I_{z1}^{U}\right)^{2}\right]\end{array}