Question

Perform the indicated matrix multiplications in (8.2.21) and verify the sequence impedances given by (8.2.22) through (8.2.27).

Short answer

Answer

The diagonal sequence impedances is,

$$\begin{gathered} Z_1=Z_2 \\ {Z}_1\frac{1}{3}\left(Z_{aa}+Z_{bb}+Z_{cc}-Z_{ab}-Z_{ac}-Zbc\right) \\ {Z}_0=\frac{1}{3}\left(Z_{aa}+Z_{bb}+Z_{cc}+2Z_{bc}+2Z_{zc}+2Z_{bc}\right) \end{gathered}$$

The off diagonal sequence impedances. Is,

$$\begin{gathered} Z_{01}=Z_{20} \\ {Z}_{01}=\frac{1}{3}\left(Z_{aa}+a{Z}_{bb}+z^2{Z}_{cc}-a^2{Z}_{ab}-a{Z}_{bc}-Z_{bc}\right) \\ {Z}_{02}=Z_{10} \\ {Z}_{02}=\frac{1}{3}\left(Z_{aa}+a{Z}_{bb}+a^2{Z}_{cc}-a^2{Z}_{ab}-a{Z}_{bc}-Z_{bc}\right) \end{gathered}$$

Further sequence impedances are,

$$\begin{gathered} Z_{12}=\frac{1}{3}\left(Z_{aa}+z^2{Z}_{bb}+2a{Z}_{ab}+2a^2{Z}_{ac}+2Z_{bc}\right) \\ {Z}_{21}=\frac{1}{3}\left(Z_{aa}+a{Z}_{bb}+a^2{Z}_{cc}+2a^2{Z}_{ab}+2a{Z}_{ac}+2Z_{bc}\right) \end{gathered}$$

The sequence impedances are verified.

Step-by-step solution

Write the given data from the question:

The sequence Impedance matrix is,

$Z_s=\left[\begin{array}{ccc}{Z}_0& Z_{01}& Z_{02}\\ Z_{10}& Z_1& Z_{12}\\ Z_{20}& Z_{21}& Z_2\end{array}\right]$

The phase impedance matrix is,

$Z_p=\left[\begin{array}{ccc}{Z}_{aa}& Z_{ab}& Z_{ac}\\ Z_{ab}& Z_{bb}& Z_{bc}\\ Z_{ac}& Z_{cb}& Z_{cc}\end{array}\right]$

Consider the matrixA.

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

The inverse of the matrix A,

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

Write the equation that is need to be verified.

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)

Perform the matrix multiplication.

$$\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}{Z}_{aa}& Z_{ab}& Z_{ac}\\ Z_{ab}& Z_{bb}& Z_{bc}\\ Z_{ac}& Z_{cb}& Z_{cc}\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}{Z}_{aa}+Z_{ab}+Z_{ac}& Z_{aa}+a^2{Z}_{ab}+a{Z}_{zc}& Z_{aa}+a{Z}_{zb}+a^2{Z}_{ac}\\ Z_{ab}+Z_{bb}+Z_{bc}& Z_{ab}+a^2{Z}_{bb}+a{Z}_{bc}& A_{zb}+a{Z}_{bb}+a^2{Z}_{bc}\\ Z_{ac}+Z_{cb}+Z_{cc}& Z_{ac}+a^2{Z}_{cb}+a{Z}_{cc}& Z_{ac}+a{Z}_{zb}+a^2{Z}_{cc}\end{array}\right] \\ {Z}_s=\frac{1}{3}\left[\begin{array}{ccc}\left\{\begin{array}{c}{Z}_{aa}+Z_{ab}+Z_{ac}\\ +Z_{ab}+Z_{bb}+Z_{bc}\\ +Z_{ac}+Z_{bc}+Z_{cc}\end{array}\right\}& \left\{\begin{array}{c}{Z}_{aa}+a^2{Z}_{ab}+a{Z}_{ac}\\ -Z_{ab}+a^2{Z}_{bb}+a{Z}_{bc}\\ -Z_{ac}+a^2{Z}_{ab}-a{Z}_{cc}\end{array}\right\}& \left\{\begin{array}{c}{Z}_{aa}+a{Z}_{ab}+a^2{Z}_{ac}\\ +Z_{ab}+a{Z}_{bb}+a^2{Z}_{bc}\\ +Z_{ac}+a{Z}_{cb}+a^2{Z}_{cc}\end{array}\right\}\\ \left\{\begin{array}{c}{Z}_{aa}+Z_{ab}+Z_{ac}\\ +a\left(Z_{ab}+Z_{bb}+Z_{bc}\right)\\ +a^2\left(a{c}_{+Z_{cb}+Z_{cc}}\right)\end{array}\right\}& \left\{\begin{array}{c}{Z}_{aa}-a^2{Z}_{ab}-a{Z}_{zc}\\ -a\left(Z_{ac}-a^2{Z}_{ab}-a{Z}_{ac}\right)\\ -a^2\left(Z_{ac}+a^2{Z}_{cb}-a{Z}_{cc}\right)\end{array}\right\}& \left\{\begin{array}{c}{Z}_{aa}+a{Z}_{ab}+a^2{Z}_{ac}\\ +a\left(Z_{ab}+a{Z}_{bb}+a^2{Z}_{bc}\right)\\ +a^2\left(Z_{ac}+a{Z}_{cb}+a^2{Z}_{cc}\right)\end{array}\right\}\\ \left\{\begin{array}{c}{Z}_{aa}+Z_{ab}+Z_{ac}\\ +a^2\left(Z_{ab}+Z_{bb}+Z_{bc}\right)\\ +a\left(Z_{ac}+Z_{cb}+Z_{cc}\right)\end{array}\right\}& \left\{\begin{array}{c}{Z}_{aa}+a^2{Z}_{ab}+a{Z}_{ac}\\ -a^2\left(Z_{ab}-a^2{Z}_{bb}-a{Z}_{bc}\right)\\ -a\left(Z_{ac}-a^2{Z}_{cb}+a{Z}_{cc}\right)\end{array}\right\}& \left\{\begin{array}{c}{Z}_{aa}+a{Z}_{ab}+a^2{Z}_{ac}\\ +a^2\left(Z_{ab}+a{Z}_{bb}+a^2{Z}_{bc}\right)\\ +a\left(Z_{ac}+a{Z}_{cb}+a^2{Z}_{cc}\right)\end{array}\right\}\end{array}\right] \end{gathered}$$uncaught exception: Invalid chunk

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Substitute $\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]$for $A^{-1},\left[\begin{array}{ccc}{Z}_{aa}& Z_{ab}& Z_{ac}\\ Z_{ab}& Z_{bb}& Z_{bc}\\ Z_{ac}& Z_{cb}& Z_{cc}\end{array}\right]$for $Z_p$and $\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]$for A into equation (1).

Substitute 0 for into above matrix.

By comparing the above matrix with the sequence impedance matrix. The diagonal sequence impedances.

The off diagonal sequence impedances are,

Further sequence impedances are,