Question

A , three-phase completely transposed overhead line has one ACSR conductor per phase and flat horizontal phase spacing, With $7m$between adjacent conductors. Determine the inductance in $\frac{H}{m}$and the inductive reactance in $\frac{\Omega }{km}$.

Short answer

The inductance and inductive reactance of completely transposed overhead line are $1.31\times {10}^{-7}\frac{\text{H}}{\text{m}}$and $0.493\frac{\Omega }{\text{km}}$respectively.

Step-by-step solution

Write the given data from the question.

The voltage of the line,$V=230\text{\hspace{0.17em}kV}$

The frequency of the system, $f=60\text{\hspace{0.17em}Hz}$

Spacing between the conductor is $7\text{\hspace{0.17em}m}$

Refer to table A4, the geometric mean radius of the overhead conductor for $954\text{\hspace{0.17em}kcmil}$,

$\begin{array}{l}{D}_s=0.0403\text{\hspace{0.17em}ft}\\ {\text{D}}_s=0.0403\times 0.305\text{\hspace{0.17em}m}\\ D_s=0.0123\text{\hspace{0.17em}m}\end{array}$

.

Determine the equation to calculate the inductance and inductive reactance.

The equation to calculate the geometric mean distance of completely transposed overhead line is given as follows.

$D_{eq}=\sqrt[3]{D_{12}{D}_{23}{D}_{31}}$ …… (1)

Here, $D_{12}$is the distance between conductor $1$and $2$, $D_{23}$is the distance between conductor $2$and$3$, $D_{31}$is distance between $3$and $1$.

The equation to calculate the inductance transposed overhead line inductance is given as follows.

$L=2\times {10}^{-7}ln\left(\frac{D_{eq}}{D_s}\right)\frac{H}{m}$…… (2)

The equation to calculate inductive reactanceof transposed overhead lineis given as follows,

$X=2\pi fL$…… (3)

Conversion from foot to metres.

$1ft=0.305m$

Calculate the inductance and inductive reactance of the transposed overhead line.

Consider the completely transposed overhead line schematic diagram as follows.

Calculate the geometric mean distance.

Substitute $7\text{\hspace{0.17em}m}$for$D_{12}$, $7m$for $D_{23}$and localid="1656756061748" $14\text{\hspace{0.17em}m}$
for $D_{31}$into equation (1).

$\begin{array}{l}{D}_{eq}=\sqrt[3]{7\times 7\times 14}\\ D_{eq}=8.819\text{\hspace{0.17em}m}\end{array}$

Calculate the inductance of the transposed overhead conductors.

Substitute for , for into equation (2).

$\begin{array}{l}L=2\times {10}^{-7}\ln \left(\frac{8.82}{0.0123}\right)\\ L=2\times {10}^{-7}\ln \left(717.073\right)\\ L=2\times {10}^{-7}\times 6.575\\ L=1.31\times {10}^{-6}\frac{\text{H}}{\text{m}}\end{array}$

Calculate the inductive reactance of the transposed overhead conductors.

Substitute $1.31\times {10}^{-6}\frac{\text{H}}{\text{m}}$for $L$and$60\text{\hspace{0.17em}Hz}$for $f$into equation (3).

$\begin{array}{l}X=2\pi \times 60\times 1.31\times {10}^{-6}\\ X=493.85\times {10}^{-6}\\ X=0.493\frac{\Omega }{\text{km}}\end{array}$

Hence the inductance and inductive reactance of completely transposed overhead line are $1.31\times {10}^{-7}\frac{\text{H}}{\text{m}}$and r$0.493\frac{\Omega }{\text{km}}$espectively.