Consider that there is a mutual coupling of ZM = j0.4 between the branch elements 4 and 5 as shown in Figure 3.4. Recalculate the bus admittance matrix.

Question

Consider that there is a mutual coupling of [latex]Z_{M}[/latex] = j0.4 between the branch elements 4 and 5 as shown in Figure 3.4. Recalculate the bus admittance matrix.

Accepted Answer

The primitive impedance matrix is

[latex]\overset{ \ ightharpoonup \ }{ Z}_{P} =\left|\begin{matrix} 1 & 0 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 & 0 \ 0& 0& 3 & 0 &0 \ 0 &0&0& j0.2 & j0.4 \ 0 & 0& 0 & j0.4 & j0.3\end{matrix} ight| [/latex]

For the coupled branch,

[latex] \overline{Y}_{jj} = \frac{1}{\overline{Z}_{jj}}[/latex]

Therefore,

[latex]\left|\begin{matrix} j0.2 & j0.4 \ j0.4 & j0.3 \end{matrix} ight|^{-1} =\left|\begin{matrix} -j3 & j4 \ j4 & -j2 \end{matrix} ight| [/latex]

Then the primitive Y matrix is

[latex]\overline{Y}_{P}=\left|\begin{matrix} 1 & 0 & 0 & 0 & 0 \ 0 & 0.5 & 0 & 0 & 0 \ 0 & 0 & 0.333 & 0 & 0 \ 0 & 0 & 0 & j3 & -j4 \ 0 & 0 & 0 & -j4 & j2 \end{matrix} ight| [/latex]

Then the Y bus matrix is

[latex]\overline{Y}_{B}=\left|\begin{matrix} 1+j3 & j & -4j \ j & 0.5-j3 & j2 \ -j4 & 2j & 0.333+j2 \end{matrix} ight| [/latex]

If similar currents as in Example 3.1 are injected at buses 1 and 3, then

[latex] \left|\begin{matrix} V_{1} \ V_{2} \ V_{3} \end{matrix} ight|= \left|\begin{matrix} 0.549-j0.015 & 0.54-j0.033 & 0.543-j0.093 \ 0.54-j0.033 & 0.563+j0.147 & 0.537-j0.123 \ 0.543-j0.093 & 0.537- j0.123 & 0.565-j0.096 \end{matrix} ight| \left|\begin{matrix}1 \ 0 \ 1 \end{matrix} ight| =\left|\begin{matrix} 1.092-j0.079 \ 1.077-j0.155 \ 1.108+j0.003 \end{matrix} ight| [/latex]

Question 3.7: Consider that there is a mutual coupling of ZM = j0.4 betwee......

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