Question

Consider a symmetrical with sub conductors arrange in a Circle of radius$\mathbf{A}$. The inductance of a single-phase symmetrical bundle-conductor line is given by
$\mathbf{L}\mathbf{=}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{In}\frac{\mathbf{GMD}}{\mathbf{GMR}}\frac{\mathbf{H}}{\mathbf{m}}$

Where is given by${\mathbf{[}\mathbf{Nr}\mathbf{\text{'}}{\left(A\right)}^{\mathbf{N}\mathbf{-}\mathbf{1}}\mathbf{]}}^{\frac{\mathbf{1}}{\mathbf{N}}}\mathbf{,}\mathbf{r}\mathbf{\text{'}}\mathbf{=}\mathbf{\left(}{\mathbf{e}}^{\frac{\mathbf{1}}{\mathbf{4}}}\mathbf{r}\mathbf{\right)}\mathbf{,}\mathbf{r}$conductor radius, an$\mathbf{GMD}$is approximately the distance$\mathbf{D}$between the bundle centers. Note that$\mathbf{A}$is related to the sub conductor spacing$\mathbf{S}$in the bundle circle by$\mathbf{S}\mathbf{=}\mathbf{2}\mathbf{Asin}\mathbf{\left(}\frac{\mathbf{\pi }}{\mathbf{N}}\mathbf{\right)}$

Now consider a$\mathbf{965}\mathbf{kV}$, single-phase, bundle-conductor line with eight sub conductors per phase, phase spacing$\mathbf{D}\mathbf{=}\mathbf{20}\mathbf{m}$, and the sub conductor spacing$\mathbf{S}\mathbf{=}\mathbf{45}\mathbf{.}\mathbf{72}\mathbf{cm}$. Each sub conductor has a diameter of$\mathbf{4}\mathbf{.}\mathbf{572}\mathbf{cm}$. Determine the line inductance in$\frac{\mathbf{H}}{\mathbf{m}}$.

Short answer

The inductance of single-phase symmetrical bundle conductor line$7.38\times {10}^{-7}\frac{\mathrm{H}}{\mathrm{m}}$

Step-by-step solution

Step: write given data from the question.

The voltage,$\mathrm{V}=965\mathrm{kV}$

Phase spacing between the conductors,$GMD=20\mathrm{m}$

Sub conductor spacing,$S=45.72\mathrm{cm}$

The number of the sub conductor in symmetrical bundle,$N=8$

The diameter of each sub conductor,$d=4.572\mathrm{cm}$

Determine the formulas to calculate the inductance.

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

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

The equation of sub conductor spacing in bundle conductor is given as follows.

$\mathbf{S}\mathbf{=}\mathbf{2}\mathbf{Asin}\mathbf{\left(}\frac{\mathbf{\pi }}{\mathbf{N}}\mathbf{\right)}$ …… (2)

Here, is related to the sub conductor spacingin the bundle circle.

The equation of GMR of the conductor is given as follows.

$\mathbf{GMR}\mathbf{=}{\mathbf{[}\mathbf{Nr}\mathbf{\text{'}}{\left(A\right)}^{\mathbf{N}\mathbf{-}\mathbf{1}}\mathbf{]}}^{\frac{\mathbf{1}}{\mathbf{N}}}\mathbf{r}\mathbf{\text{'}}$ …… (3)

The inductance of single-phase symmetrical bundle conductor line is given as follows.

$\mathbf{L}\mathbf{=}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{In}\frac{\mathbf{GMD}}{\mathbf{GMR}}\frac{\mathbf{H}}{\mathbf{m}}$ …… (4)

Calculate inductance of single-phase symmetrical bundle conductor line.

Calculate the radius of the conductor.

Substitut$4.572\mathrm{cm}$for$d$into equation (1).

$$\begin{gathered} r=\frac{4.572\times {10}^{-2}}{2} \\ r=0.02286\mathrm{m} \end{gathered}$$

Calculate the value of .

Substitute$45.72\mathrm{cm}$for$S$and $8$for$N$into equation (2).

$$\begin{gathered} 45.72\times {10}^{-2}=2\times A\times \sin \left(\frac{180}{8}\right) \\ A=\frac{45.72\times {10}^{-2}}{2\times \sin \left(\frac{180}{8}\right)} \\ A=\frac{0.4572}{0.5973} \\ A=0.5973 \end{gathered}$$

Calculate the GMR of the conductor.

Substitute $8$for$N,0.5973$for$A$into equation (3).

$$\begin{gathered} \mathrm{GMR}={\left[8\times 0.7788\times \times 0.02286\times 0.{5974}^{8-1}\right]}^{\frac{1}{8}} \\ \mathrm{GMR}={\left[8\times 0.0178\times 0.{5974}^7\right]}^{\frac{1}{8}} \\ \mathrm{GMR}=0.4994\mathrm{m} \end{gathered}$$

Calculate the$\mathrm{GMR}$inductance of the conductors.

Substitute $20\mathrm{m}$for $\mathrm{GMD}$and $0.4994\mathrm{m}$for$\mathrm{GMR}$into equation (4).

$$\begin{gathered} L=2\times {10}^{-7}In\left(\frac{20}{0.4994}\right) \\ L=2\times {10}^{-7}\times 3.69 \\ L=7.38\times {10}^{-7}\frac{\mathrm{H}}{\mathrm{m}} \end{gathered}$$

Hence the inductance of single-phase symmetrical bundle conductor line$7.38\times {10}^{-7}\frac{\mathrm{H}}{\mathrm{m}}$