# 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
### Instantaneous Power  
Instantaneous power refers to the velocity of the energy delivery in a system, which is directly related to the energy and its components to identify power phenomena that manifest in the system. The instantaneous active power transferred under fundamental-frequency and positive-sequence voltages and currents is the only instantaneous power component delivered in efficient (ideal) power systems. The other instantaneous power components manifest due to the inefficiencies present in the system, such as reactance, imbalance, and distortion. Therefore, a power system is less efficient when the instantaneous power possesses more components. The instantaneous power source in the system illustrated in Figure 1 can be expressed as follows:  
\[p_{S}(t)=\sum_{z=A,B,C}v_{zN}\cdot i_{z} \tag{1}\]  
where \(v_{zN}\) is the source of instantaneous line-to-neutral voltages in each phase (\(z=A,B,C\)) of the point of common coupling (PCC) of the source and the load and \(i_{z}\) is the line current (Figure 1). The instantaneous line-to-neutral voltage sources and the line currents can be decomposed into fundamental-frequency (\(v_{zN1}\), \(i_{z1}\)) and non-fundamental (\(v_{zNH}\), \(i_{zH}\)) line-to-neutral voltages and line currents, respectively, using the Fourier series:  
\[v_{zN}=v_{zN1}+v_{zNH}=v_{zN1}+\sum_{x\
eq 1}^{\infty}v_{zNx}\;i_{z}=i_{z1}+i_{ zH}=i_{z1}+\sum_{y\
eq 1}^{\infty}i_{zy} \tag{2}\]  
If the source fundamental and non-fundamental line-to-neutral voltages and currents are used as determined by the Fourier series, the source instantaneous power can be expressed as follows:  
\[p_{S}(t)=\sum_{z=A,B,C}(v_{zN1}+v_{zNH})\cdot(i_{z1}+i_{zH})=p_{S1}(t)+p_{SH}(t) \tag{3}\]  
Figure 1: Three-phase, four-wire, unbalanced, non-sinusoidal power system. (The head of the arrow voltages indicates the point of greatest potential). where  
\[p_{S1}(t)=\sum_{z=A,B,C}v_{zN1}\cdot i_{z1} \tag{4}\]  
is the source fundamental instantaneous power, and  
\[p_{SH}(t)=\sum_{z=A,B,C}(v_{zN1}\cdot i_{zH}+v_{zN1}\cdot i_{z})=p_{SH}^{I}(t)+p _{SH}^{V}(t) \tag{5}\]  
is the source non-fundamental instantaneous power, which has two components, i.e., \(p_{SH}^{I}(t)\), which represents the source instantaneous power caused exclusively by distorted currents, whereas \(p_{SH}^{V}(t)\) defines the source instantaneous power caused by distorted voltages. The fundamental-frequency source line-to-neutral voltages and line currents can be expressed according to the following components in each phase (\(z=A,B,C\)) by applying [PERSON]'s theorem:  
\[v_{zN1}=v_{zN1}^{+}+v_{zN1}^{-}+v_{zN1}^{0}=v_{zN1}^{+}+v_{zN1}^{U}\;i_{z1}=i_ {z1}^{+}+i_{z1}^{-}+i_{z1}^{0}=i_{z1}^{+}+i_{z1}^{U} \tag{6}\]  
where \(v_{zN1}^{+}\) and \(i_{z1}^{+}\) are the fundamental-frequency positive-sequence source line-to-neutral voltages and line currents, respectively, and  
\[v_{zN1}^{U}=v_{zN1}^{-}+v_{zN1}^{0}\;\text{and}\;i_{z1}^{U}=i_{z1}^{-}+i_{z1}^{0} \tag{7}\]  
are the source unbalanced line-to-neutral voltages and unbalanced line currents, respectively, which are formed by the fundamental-frequency negative- (\(v_{zN1}^{-}\), \(i_{z1}^{-}\)) and zero-sequence (\(v_{zN1}^{0}\), \(i_{z1}^{0}\)) components of the source line-to-neutral voltages and line currents. Hence, the source fundamental-frequency instantaneous power (4) can also be expressed as a function of the symmetrical components of the voltages and currents resulting from [PERSON]'s theorem, as defined in Equation (6).