Question

Determine the equivalent circuit for the line in Problem 5.26.

Short answer

The equivalent circuit of the line is shown below.

Step-by-step solution

Write the given data from the question.        

The length of the line $l=350\text{\hspace{0.33em}km}$.

The power rating $V=500\text{\hspace{0.33em}kV}$.

The frequency of the system $f=60\text{\hspace{0.33em}Hz}$.

The positive sequence series reactance $z=j0.34\frac{\Omega }{\text{km}}$.

The positive sequence shunt admittance $y=j4.5\times {10}^{-6}\frac{\text{S}}{\text{km}}$.

Determine the equation to calculate the value of the parameters of the equivalent circuit.

The equation to calculate the characteristics impedance is given as follows.

$Z_c=\sqrt{\frac{z}{y}}$…… (1)

The equation to calculate the propagation constant is given as follows,

$\gamma l=l\sqrt{zy}$ …… (2)

The equation to calculate the$Z\text{'}$ is given as follows.

$Z\text{'}=j{Z}_{c}sin\left(\beta l\right)$…… (3)

Here $\beta l$is the electrical length.

The equation to calculate the $\frac{Y\text{'}}{2}$is given as,

$\frac{Y\text{'}}{2}=\frac{yl}{2}\frac{jtanh\left(\frac{\beta l}{2}\right)}{\frac{j\beta l}{2}}$…… (4)

Determine the equivalent circuit.

Calculate the characteristics impedance.

Substitute $j0.34\frac{\Omega }{\text{km}}$ for$z$ and $j4.5\times {10}^{-6}\frac{\text{S}}{\text{km}}$ into equation (1).

$\begin{array}{c}{Z}_c=\sqrt{\frac{j0.34}{j4.5\times {10}^{-6}}}\\ =\sqrt{75.55\times {10}^3}\\ =274.9\Omega \end{array}$

Calculate the value of $\gamma l$

Substitute $300\text{\hspace{0.33em}km}$for $l$ $j0.34\frac{\Omega }{\text{km}}$ For $z$and$j4.5\times {10}^{-6}\frac{\text{S}}{\text{km}}$ for $y$into equation (2).

$\begin{array}{l}\gamma l=300\sqrt{j0.34\times j4.5\times {10}^{-6}}\\ \gamma l=300\times j1.237\times {10}^{-3}\\ \gamma l=j0.3711\text{\hspace{0.33em}pu}\end{array}$

Consider the relationship,

$\gamma l=\alpha l+j\beta l$

Therefore,$\beta l=j0.3711\text{\hspace{0.33em}pu}$

Calculate the value of$Z\text{'}$

Substitute $274.9\Omega$ for $Z_c$and $0.3711\text{\hspace{0.33em}pu}$for$\beta l$ into equation (3).

$\begin{array}{c}Z\text{'}=j\left(274.9\sin \left(0.3711\right)\right)\\ =j\left(274.9\right)\left(0.3626\right)\\ =j99.68\Omega \end{array}$

Calculate the value of$\frac{Y\text{'}}{2}$

Substitute $0.3711\text{\hspace{0.33em}pu}$for$\beta l$ , $j4.5\times {10}^{-6}\frac{\text{S}}{\text{km}}$for $300\text{\hspace{0.33em}km}$for $l$into equation (4).

The equivalent circuit of the line is shown below.

$\begin{array}{l}\frac{Y\text{'}}{2}=\frac{j4.5\times {10}^{-6}\times 300}{2}\frac{j\tanh \left(\frac{0.3711}{2}\right)}{\frac{j0.3711}{2}}\\ \frac{Y\text{'}}{2}=0.675\times {10}^{-3}\frac{j\tan \left(0.1855\right)}{0.1855}\\ \frac{Y\text{'}}{2}=0.675\times {10}^{-3}\times j1.011\\ \frac{Y\text{'}}{2}=j6.828\times {10}^{-4}\text{\hspace{0.33em}S}\end{array}$