# Relationship between [PERSON]'s Apparent Power and Instantaneous Power in Three-Phase Systems
## 4 Formulation of the Neutral Conductor Effects According to the Proposed Methodology  
The method described in Section 3 was useful to obtain the expressions for the apparent power and its components in the sources and loads of power systems. These expressions are summarized in Table 1. However, a relevant application of the proposed methodology is the establishment of expressions to measure the power effects caused by neutral conductor operation, which can be expressed by the neutral-displacement power (\(S_{N}\)). In this section, our proposed neutral-displacement power is expressed using the methodology described in Section 3 and the Conservation of Energy Principle in regard to the entire three-phase power system represented in Figure 1. In order to obtain accurate expressions for this power, it was necessary to consider the neutral conductor to be a different subsystem from the sources and loads (Figure 1). When the neutral conductor impedance is not negligible or the neutral conductor does not exist, the source and load instantaneous power in the system of Figure 1 generally do not have the same values; the same occurs with the source and load apparent types of power, even if the subsystems are directly connected. These differences are caused by the phenomena that manifest via the operation of the neutral conductor and are identified by instantaneous power (\(p_{N}(t)\)) in order to satisfy the Conservation of Energy Principle in the system presented in Figure 1.  
As the loads are usually more unbalanced and distorted than the sources, the power effects that manifest via the neutral conductor (or neutral path) should be obtained as the differences between the load and source instantaneous types of power in order to satisfy the Conservation of Energy Principle throughout the entire power system. Hence, the neutral displacement instantaneous power can be expressed as follows:  
\[p_{N}(t)=p_{L}(t)-p_{S}(t)=\sum_{z=A,B,C}(v_{2n}-v_{zN})\cdot i_{z} \tag{35}\]  
This quantity identifies the phenomena associated with the operation of the neutral conductor. One possible expression for the neutral displacement power can be identified by applying steps (2-a), (2-b), and (3) of the methodology established in Section 3 to Equation (35), as follows:  
\[S_{N}=\sqrt{S_{L}^{2}-S_{S}^{2}}=\sqrt{\left[\left(V_{An}^{2}+V_{Bn}^{2}+V_{ Cn}^{2}\right)-\left(V_{AN}^{2}+V_{BN}^{2}+V_{CN}^{2}\right)\right]\left(I_{A} ^{2}+I_{B}^{2}+I_{C}^{2}\right)} \tag{36}\]This power measures the effects of the neutral conductor operation and shows apparent power dimensions as it is defined as the quadratic difference between the apparent power of the load (\(S_{L}\)) and source (\(S_{S}\)). Equation (36) shows that the neutral-displacement power (\(S_{N}\)) is not the neutral apparent power determined by the product of the RMS values of the neutral voltage drop and the neutral current (\(V_{nN}\cdot I_{N}\)). Further, the neutral-displacement power is more suitable than the neutral apparent power (\(V_{nN}\cdot I_{N}\)) for measuring the effects of the neutral conductor due to the following points:  
1. In Expression (36), not only the active and reactive power in the neutral conductor are included, but also the power due to imbalances and distortions in the loads and sources;
2. The neutral-displacement power does not directly depend on the value of the neutral current (\(I_{N}\)) and can also be applied to three-phase systems with no neutral conductor.