Question

Identical series capacitors are installed at both ends of the line in Problem 5.14, providing $\mathbf{40}\mathbf{\%}$ total series compensation. Determine the equivalent ABCDparameters of this compensated line. Also calculate the impedance of each series capacitor.

Short answer

Therefore, the equivalent ABCD parameters are $$\begin{gathered} \left[0.9248\angle 0.64°86.66\angle 82.31° \\ 1.688\angle 90.2°\&0.9248\angle 0.64°\right] \end{gathered}$$ and reactance of each capacitance is$-\mathrm{j}26.86\mathrm{\Omega }.$.

Step-by-step solution

Given data.

From problems 5.23,

The ABCD parameters are,

$$\begin{gathered} \mathrm{A}=0.8794\angle 0.{66}^0\mathrm{pu} \\ \mathrm{D}=0.8794\angle 0.{66}^0\mathrm{pu} \\ \mathrm{C}=1.688\times {10}^{-3}\angle 90.2^0\mathrm{S} \\ \mathrm{B}=134.8\angle 85.3^0\mathrm{\Omega } \end{gathered}$$

The$40\%$series compensation is mentioned.

The impedance from problem 5.23 is,

$\mathrm{Z}=(11+\mathrm{j}134.3)\mathrm{\Omega }$

Determine the formula of ABCD parameters.

Write the formula of equivalent ABCD parameters.

$\mathbf{\left[}\begin{array}{c}{\mathbf{A}}_{\mathbf{eq}}\mathbf{}\mathbf{}\mathbf{}{\mathbf{B}}_{\mathbf{eq}}\\ {\mathbf{C}}_{\mathbf{eq}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}}{\mathbf{D}}_{\mathbf{eq}}\end{array}\mathbf{\right]}\mathbf{=}\mathbf{\left[}\begin{array}{c}{\mathbf{A}}_{\mathbf{S}}\mathbf{}\mathbf{}\mathbf{}{\mathbf{B}}_{\mathbf{S}}\\ {\mathbf{C}}_{\mathbf{S}}\mathbf{}\mathbf{}{\mathbf{D}}_{\mathbf{S}}\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{c}\mathbf{A}\mathbf{}\mathbf{}\mathbf{}\mathbf{B}\\ \mathbf{C}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{D}\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{c}{\mathbf{A}}_{\mathbf{R}\mathbf{}}\mathbf{}\mathbf{}{\mathbf{B}}_{\mathbf{R}}\\ {\mathbf{C}}_{\mathbf{R}}\mathbf{}\mathbf{}\mathbf{}{\mathbf{D}}_{\mathbf{R}}\end{array}\mathbf{\right]}$ ..................(1)

Here, eq is representing equivalent, S is representing sending end, and R is representing receiving end.

Determine the formula of ABCD parameters and capacitance reactance.

The reactance of each series capacitor with compensation is,

$$\begin{gathered} \mathrm{X}=-\left(\mathrm{j}\frac{134.3}{2}\right)\left(0.4\right)\mathrm{\Omega } \\ =\mathrm{j}26.86\mathrm{\Omega } \end{gathered}$$

The ABCD parameters at sending and receiving end will be,

$$\begin{gathered} \left[\begin{array}{c}{\mathrm{A}}_{\mathrm{S}}{\mathrm{B}}_{\mathrm{S}}\\ {\mathrm{C}}_{\mathrm{S}}{\mathrm{D}}_{\mathrm{S}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{A}}_{\mathrm{R}}{\mathrm{B}}_{\mathrm{R}}\\ {\mathrm{C}}_{\mathrm{R}}{\mathrm{D}}_{\mathrm{R}}\end{array}\right] \\ =\left[\begin{array}{c}1-\mathrm{j}26.86\\ 01\end{array}\right] \end{gathered}$$

Substitute $\left[\begin{array}{c}1-\mathrm{j}26.86\\ 01\end{array}\right]\mathrm{for}\left[\begin{array}{c}{\mathrm{A}}_{\mathrm{R}}{\mathrm{B}}_{\mathrm{R}}\\ {\mathrm{C}}_{\mathrm{R}}{\mathrm{D}}_{\mathrm{R}}\end{array}\right],\left[\begin{array}{c}{\mathrm{A}}_{\mathrm{S}}{\mathrm{B}}_{\mathrm{S}}\\ {\mathrm{C}}_{\mathrm{S}}{\mathrm{D}}_{\mathrm{S}}\end{array}\right]$and $\left[\begin{array}{c}0.8794\angle 0.66°134.8\angle 85.3°\\ 1.688\times 10^(-3)\angle 90.2°0.8794\angle 0.{66}^{\circ }\end{array}\right]\mathrm{for}\left[\begin{array}{c}\mathrm{A}\mathrm{B}\\ \mathrm{C}\mathrm{D}\end{array}\right]\mathrm{in}\mathrm{equation}\left(1\right)$role="math" localid="1655372052655" $$\begin{gathered} \left[\begin{array}{c}{\mathrm{A}}_{\mathrm{eq}}{\mathrm{B}}_{\mathrm{eq}}\\ {\mathrm{C}}_{\mathrm{eq}}{\mathrm{D}}_{\mathrm{eq}}\end{array}\right]=\left[\begin{array}{c}1-\mathrm{j}26.86\\ 01\end{array}\right]\left[\begin{array}{c}0.8794\angle 0.{66}^{\circ }134.8\angle 85.3^{\circ }\\ 1.688\times {10}^{-3}\angle 90.20.8794\angle 0.{66}^{\circ }\end{array}\right]\mathrm{for}\left[\begin{array}{c}\mathrm{A}\mathrm{B}\\ \mathrm{C}\mathrm{D}\end{array}\right]\left[\begin{array}{c}1-\mathrm{j}26.86\\ 01\end{array}\right] \\ =\left[0.9248\angle 0.64°86.66\angle 82.31° \\ 1.688\angle 90.2°0.9248\angle 0.64°\right] \end{gathered}$$

Therefore, the equivalent ABCD parameters are $$\begin{gathered} \left[0.9248\angle 0.64°86.66\angle 82.31° \\ 1.688\angle 90.2°0.9248\angle 0.64°\right] \end{gathered}$$ and reactance of each capacitance is $-\mathrm{j}26.86\mathrm{\Omega }.$.