## 7 Practical Application  
In this section, a practical case study to verify all of the concepts discussed in the previous sections is developed. Figure 7 shows a three-phase, four-wire electrical system with three unbalanced three-phase linear loads. The loads are modelled at a constant impedance. The impedance values of loads are listed in Tables 1-3. The voltages are unbalanced and sinusoidal in bus 1 (slack bus), in which  
\[V_{an}=230\;e^{j0},\;V_{bn}=205\;e^{-j115},\;V_{cn}=220\;e^{j135}\]  
Using the 'PSPICE' analysis software, the line-to-neutral voltages in the buses (1, 2, and 3) and the currents circulating in the loads are obtained. These magnitudes are displayed in Tables 4 and 5. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(R\) & \(X\) \\ \hline
**Impedances** & **(\(\Omega\))** & **(\(\Omega\))** \\ \hline \(Zab\) & \(12\) & \(3\) \\ \(Zbc\) & \(8\) & \(4\) \\ \(Zca\) & \(5\) & \(2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Load-1 Impedances. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(R\) & \(X\) \\ \hline
**Impedances** & **(\(\Omega\))** & **(\(\Omega\))** \\ \hline \(Za\) & \(12\) & \(3\) \\ \(Zb\) & \(8\) & \(4\) \\ \(Zc\) & \(5\) & \(2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Line-to-neutral voltage. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(R\) & \(X\) \\ \hline
**Impedances** & **(\(\Omega\))** & **(\(\Omega\))** \\ \hline \(Za\) & \(12\) & \(3\) \\ \(Zb\) & \(8\) & \(4\) \\ \(Zc\) & \(5\) & \(2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Load-3 Impedances. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(V_{an}\) (V) & \multicolumn{3}{c}{\(V_{bn}\) (V)} & \multicolumn{3}{c}{\(V_{cn}\) (V)} \\ \hline  & **Modulus** & **Modulus** & **Modulus** & **Modulus** & **Angle** \\ \hline Bus 2 & \(187.88\) & \(0.49\) & \(161.97\) & \(-116.90\) & \(171.72\) & \(132.68\) \\ Bus 3 & \(180.50\) & -0.09 & \(143.48\) & \(-120.47\) & \(161.03\) & \(130.36\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Line-to-neutral voltage. Figure 7: Electric system with unbalanced three-phase load. Table 6 shows the voltages expressed in symmetric components for each of the nodes of the system. These values are referred to as phase A. Table 7 shows the values of the components that form the vectors of the apparent power and unbalanced power, considering the positive-sequence voltage. To calculate \(P_{z+}^{z}\) and \(Q_{z+}^{z}\), expressions (16) and (17) have been used. For \(P_{+}\) and \(Q_{+}\), expressions (33) and (34) have been used. Table 8 shows the values of the components that form the vectors of apparent power and unbalanced power, considering the negative-sequence voltage. To calculate \(P_{z-}^{z}\) and \(Q_{z-}^{z}\), expressions (24) and (25) have been used. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**\(V_{a+}\) (V)**} & \multicolumn{2}{c}{\(V_{a-}\) (V)} & \multicolumn{2}{c}{\(V_{a\oplus}\) (V)} \\ \hline  & **Modulus** & **Modulus** & **Modulus** & **Modulus** & **Modulus** & **Angle** \\ \hline Bus 1 & 217.012 & 6.598 & 23.722 & -38.783 & 10.866 & \(-\)111.978 \\ Bus 2 & 173.122 & 5.308 & 18.389 & \(-\)28.891 & 5.568 & \(-\)96.214 \\ Bus 3 & 161.054 & 3.260 & 20.632 & \(-\)25.965 & 1.223 & \(-\)19.535 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Positive-, negative-, and zero-sequence line-to-neutral voltage.