Question

The capacitance per phase of a balanced three-phase overhead line is given by

$\mathbf{C}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{0389}}{\mathbf{log}\left({\displaystyle \frac{\mathrm{GMD}}{r}}\right)}\frac{\mathbf{\mu f}}{\mathbf{mi}\mathbf{-}\mathbf{phase}}$

For the line of Problem 4.24, determine the capacitive reactance per phase in$\mathbf{\Omega }\mathbf{.}\mathbf{mi}$.

Short answer

The capacitive reactance per phase is $148.273\times {10}^3\frac{\mathrm{\Omega }}{\mathrm{mi}}.$.

Step-by-step solution

Write the given data by the question.

The frequency of the system,$f=60\mathrm{Hz}$

The horizontal separation between center of C and that of A is ${40}^{"}$.

The separation between A and B is ${40}^{"}$.

The vertical separation of A from the line of C-B is ${16}^{"}$.

From appendix A4 the outer diameter of 336.4kcmil ACSR 26/7 conductor $d=0.721\mathrm{in}$.

Determine the formula to calculate the capacitive reactance per phase.

The equation to calculate the radius of the conductor is given as follows.

$\mathbf{r}\mathbf{=}\frac{\mathbf{d}}{\mathbf{2}}$ …… (1)

The equation to calculate the geometric mean distance is given as follows.

$\mathbf{GMD}\mathbf{=}\sqrt[\mathbf{3}]{{\mathbf{D}}_{\mathbf{AB}}{\mathbf{D}}_{\mathbf{BC}}{\mathbf{D}}_{\mathbf{CA}}}$ …… (2)

Here, ${\mathbf{D}}_{\mathbf{AB}}$ is the distance between conductor A and B, ${\mathbf{D}}_{BC}$is the distance between conductor B and C,${\mathbf{D}}_{\mathit{C}\mathit{A}}$ is distance between C and A .

The equation to calculate the capacitance is given as follows.

$\mathbf{C}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{0389}}{\mathbf{log}\left({\displaystyle \frac{\mathrm{GMD}}{r}}\right)}\frac{\mathbf{\mu f}}{\mathbf{mi}\mathbf{-}\mathbf{phase}}$ .....(3)

The equation to calculate the capacitive reactance is given a follow.

${\mathbf{x}}_{\mathbf{c}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi fC}}$ …… (4)

Calculate the per phase capacitive reactance.

Calculate the radius of the conductor.

Substitute 0.721 in for d into equation (1).

$$\begin{gathered} r=\frac{0.721}{2} \\ =0.361\mathrm{in} \end{gathered}$$

Draw the schematic diagram as,

Calculate the phase distance between $A$and$B$.

$$\begin{gathered} D_{AB}=\sqrt{{16}^2+{40}^2} \\ {D}_{AB}=\sqrt{1856} \\ {D}_{AB}=43.08\mathrm{in} \\ \mathrm{Calculate}\mathrm{the}\mathrm{phase}\mathrm{distance}\mathrm{between}\mathrm{B}\mathrm{and}\mathrm{C}. \\ {\mathrm{D}}_{\mathrm{BC}}=80\mathrm{in} \\ \mathrm{Calculate}\mathrm{the}\mathrm{phase}\mathrm{distance}\mathrm{between}A\mathrm{and}C. \\ {\mathrm{D}}_{\mathrm{AC}}=\sqrt{{16}^2+{40}^2} \\ {\mathrm{D}}_{\mathrm{AC}}=\sqrt{1856} \\ {\mathrm{D}}_{\mathrm{AC}}=43.08\mathrm{in} \end{gathered}$$

Calculate the geometric mean distance.

Substitute 43.08 in forlocalid="1655879766479" $D_{AB}\&D_{AC}$and $D_{BC}$for into equation (2).

$$\begin{gathered} \mathrm{GMD}=\sqrt[3]{43.08\times 80\times 43.08} \\ \mathrm{GMD}=\sqrt[3]{148470.911} \\ \mathrm{GMD}=52.952\mathrm{in} \\ \mathrm{Calculate}\mathrm{the}\mathrm{capacitive}\mathrm{reactance}. \\ \mathrm{Substitute}52.952\mathrm{in}\mathrm{for}\mathrm{GMD},\mathrm{and}0.361\mathrm{in}\mathrm{for}\mathrm{r}\mathrm{into}\mathrm{equation}\left(3\right). \\ \mathrm{C}=\frac{0.0389\times {10}^{-6}}{\log \left({\displaystyle \frac{52.952}{0.361}}\right)} \\ \mathrm{C}=\frac{0.0389\times {10}^{-6}}{2.1663} \\ \mathrm{C}=1.79\times {10}^{-8}\frac{\mathrm{F}}{\mathrm{mi}} \end{gathered}$$

Calculate the capacitive reactance.

$$\begin{gathered} \mathrm{Substitute}1.79\times {10}^{-8}\frac{\mathrm{F}}{\mathrm{mi}}\mathrm{for}\mathrm{C}\mathrm{and}60\mathrm{Hz}\mathrm{for}\mathrm{f}\mathrm{into}\mathrm{equation}\left(4\right). \\ {\mathrm{X}}_{\mathrm{c}}=\frac{1}{2\mathrm{\pi }\times 60\times 1.789\times {10}^{-8}} \\ {\mathrm{X}}_{\mathrm{c}}=\frac{{10}^8}{674.43} \\ {\mathrm{X}}_{\mathrm{c}}=148.273\times {10}^3\frac{\mathrm{\Omega }}{\mathrm{mi}} \\ \mathrm{Hence},\mathrm{the}\mathrm{capacitive}\mathrm{reactance}\mathrm{is}148.273\times {10}^3\frac{\mathrm{\Omega }}{\mathrm{mi}}. \end{gathered}$$