Question

The three-phase impedance load shown in Figure 8.7 has the following phase impedance matrix:

$Z_P=\left[\begin{array}{ccc}5+j10& 0& 0\\ 0& 5+j10& 0\\ 0& 0& 5+j10\end{array}\right]$

Determine the sequence impedance matrix $Z_S$for this load. Is the load symmetrical?

Short answer

The sequence impedance matrix is $\left[\begin{array}{ccc}5+j10& 0& 0\\ 0& 5+j10& 0\\ 0& 0& 5+j10\end{array}\right]$and the load is symmetric.

Step-by-step solution

Write the given data from the question.

The phase impedance matrix.

$Z_P=\left[\begin{array}{ccc}5+j10& 0& 0\\ 0& 5+j10& 0\\ 0& 0& 5+j10\end{array}\right]$

Determine the equation to calculate the sequence impedance matrix.

From the equation 8.2.19 from the text book.

${\mathit{Z}}_{\mathbf{s}}\mathbf{=}{\mathit{A}}^{\mathbf{-}\mathbf{1}}{\mathit{Z}}_{\mathbf{p}}\mathit{A}$ …… (1)

Here,

The matrix are,

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]$

The inverse of the matrix A,

role="math" localid="1656765203880" $A^{-1}=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]$

Consider the identities.

$$\begin{gathered} {\mathit{a}}^{\mathbf{4}}\mathbf{=}\mathit{a} \\ {\mathit{a}}^{\mathbf{3}}\mathbf{=}\mathbf{1} \\ \mathbf{1}\mathbf{+}\mathit{a}\mathbf{+}{\mathit{a}}^{\mathbf{2}}\mathbf{=}\mathbf{0} \end{gathered}$$

Calculate the sequence impedance matrix.

Calculate the sequence impedance matrix.

Substitute$\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]$for$A^{-1}$,role="math" localid="1656763746879" $\left[\begin{array}{ccc}5+j10& 0& 0\\ 0& 5+j10& 0\\ 0& 0& 5+j10\end{array}\right]$for $Z_p$and $\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]$for into equation (1).

role="math" localid="1656764313695" $$\begin{gathered} Z_S=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]\left[\begin{array}{ccc}5+j10& 0& 0\\ 0& 5+j10& 0\\ 0& 0& 5+j10\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right] \\ {Z}_S=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]\left[\begin{array}{ccc}5+j10& 5+j10& 5+j10\\ 5+j10& a^2\left(5+j10\right)& a\left(5+j10\right)\\ 5+j10& a\left(5+j10\right)& a^2\left(5+j10\right)\end{array}\right] \\ {Z}_s=\frac{1}{3}\left[\begin{array}{ccc}\left(5+j10\right)+\left(5+j10\right)+\left(5+j10\right)& \left(5+j10\right)\left(1+a^2+a\right)& \left(5+j10\right)\left(1+a+a^2\right)\\ \left(5+j10\right)\left(1+a+a^2\right)& \left(5+j10\right)\left(1+a^3+a^3\right)& \left(5+j10\right)\left(1+a^2+a^4\right)\\ \left(5+j10\right)\left(1+a^2+a\right)& \left(5+j10\right)\left(1+a^4+a^2\right)& \left(5+j10\right)\left(1+a^3+a^3\right)\end{array}\right] \end{gathered}$$

Substitute for $a^4$and for $a^3$into above matrix.

$$\begin{gathered} Z_s=\frac{1}{3}\left[\begin{array}{ccc}\left(15+j30\right)& \left(5+j10\right)\left(1+a^2+a\right)& \left(5+j10\right)\left(1+a+a^2\right)\\ \left(5+j10\right)\left(1+a+a^2\right)& \left(5+j10\right)\left(1+1+1\right)& \left(5+j10\right)\left(1+a^2+a\right)\\ \left(5+j10\right)\left(1+a^2+a\right)& \left(5+j10\right)\left(1+a+a^2\right)& \left(5+j10\right)\left(1+1+1\right)\end{array}\right] \\ {Z}_s=\frac{{\displaystyle 1}}{{\displaystyle 3}}\left[\begin{array}{ccc}\left(15+j30\right)& \left(5+j10\right)\left(1+a^2+a\right)& \left(5+j10\right)\left(1+a+a^2\right)\\ \left(5+j10\right)\left(1+a+a^2\right)& \left(15+j30\right)& \left(5+j10\right)\left(1+a^2+a\right)\\ \left(5+j10\right)\left(1+a^2+a\right)& \left(5+j10\right)\left(1+a+a^2\right)& \left(15+j30\right)\end{array}\right] \end{gathered}$$

Substitute 0 for $1+a+a^2$into above matrix.

$$\begin{gathered} Z_s=\frac{1}{3}\left[\begin{array}{ccc}\left(15+j30\right)& 0& 0\\ 0& \left(15+j30\right)& 0\\ 0& 0& \left(15+j30\right)\end{array}\right] \\ {Z}_s=\left[\begin{array}{ccc}5+j10& 0& 0\\ 0& 5+j10& 0\\ 0& 0& 5+j10\end{array}\right] \end{gathered}$$

Hence, the sequence impedance matrix is $\left[\begin{array}{ccc}5+j10& 0& 0\\ 0& 5+j10& 0\\ 0& 0& 5+j10\end{array}\right]$and the load is symmetric.