Question

A single-phase overhead transmission line consists of two solid aluminum conductors having a radius of 3 cm with a spacing 3.5m between centers. (a) Determine the total line inductance in $\frac{\mathbf{m}\mathbf{H}}{\mathbf{m}}$. (b) Given the operating frequency to be 60Hz, find the total inductive reactance of the line in $\frac{\mathbf{\Omega }}{\mathbf{km}}$and in $\frac{\mathbf{\Omega }}{\mathbf{mi}}$. (c) If the spacing is doubled to 7 m, how does the reactance change?

Short answer

Answer

(a) The total inductance of the single-phase overhead transmission line is $2\times \frac{mH}{m}$.

(b) The inductive reactance in per kilometer and per miles are $0.753\frac{\Omega }{km}$and $1.21\frac{\Omega }{mi}$respectively.

(c) The total inductance of the single-phase overhead transmission line is $2.28\times {10}^{-3}\frac{mH}{m}$and the reactance in per kilometer and per miles are $0.753\frac{\Omega }{km}$and $1.21\frac{\Omega }{mi}$respectively.

If the spacing is double, the inductive reactance is increased by 14%.

Step-by-step solution

Write the given data by the question.

The radius of the conductors,$r=3cm$

The spacing between the conductors,$D=3.5m$

The frequency of the system,$f=60Hz$

Determine the formulas to calculate the total inductance, inductive reactance and effect of changing the spacing between the conductors.

The expression to calculate the single-phase overhead transmission line inductance is given by,

$\mathit{L}\mathbf{=}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathit{I}\mathit{n}\left(\frac{D}{0.7788r}\right)$ …… (1)

The expression to calculate single-phase overhead transmission line inductive reactance is given by,

$\mathbf{\pi }\mathbf{=}\mathbf{2}\mathbf{XL}$ …… (2)

The expression to calculate the change in the reactance is given by,

$\mathbf{∆}\mathbf{X}\mathbf{=}\mathbf{\left(}\frac{{\mathbf{X}}_{\mathbf{2}}\mathbf{-}\mathbf{X}}{{\mathbf{X}}_{\mathbf{1}}}\mathbf{\right)}\mathbf{\times }\mathbf{100}$ …… (3)

Here, ${\mathbf{X}}_{\mathbf{1}}$is the reactance when the spacing is $\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{m}$and ${\mathbf{X}}_{\mathbf{2}}$is the reactance when the spacing is $\mathbf{7}\mathbf{m}$.

Calculate the total inductance of the single-phase overhead transmission line conductor.

(a)

Calculate the inductance of the overhead transmission line.

Substitute 3.5m for D and 3cmfor r into equation (1).

$$\begin{gathered} L=4\times {10}^{-7}In\left(\frac{3.5}{0.7788\times 3\times {10}^{-2}}\right) \\ L=4\times {10}^{7}In\left(149.80\right) \\ L=4\times {10}^{-7}\times 5 \\ L=2\times {10}^{-3}\frac{mH}{m} \end{gathered}$$

Hence the total inductance of the single-phase overhead transmission line is $2\times {10}^{-3}\frac{mH}{m}$.

Calculate the inductive reactance of the line in Ωkmand in Ωmi.

(b)

Calculate the inductive reactance of the overhead transmission line.

Substitute $2\times {10}^{-3}\frac{mH}{m}$for $L$and $60Hz$for $f$into equation (2).

$$\begin{gathered} X_1=2\mathrm{\pi }\times 60\times 2\times {10}^{-3} \\ {\mathrm{X}}_1=753.98\times {10}^{-3} \\ {\mathrm{X}}_1=0.753\frac{\mathrm{\Omega }}{\mathrm{km}} \end{gathered}$$

Calculate inductive reactance of the overhead transmission line in ohms per miles.

$$\begin{gathered} {\mathrm{X}}_1=0.753\times 1.609 \\ {\mathrm{X}}_1=1.21\frac{\mathrm{\Omega }}{\mathrm{mi}} \end{gathered}$$

Hence inductive reactance in per kilometer and per miles are $0.753\frac{\mathrm{\Omega }}{\mathrm{km}}$and $1.21\frac{\mathrm{\Omega }}{\mathrm{mi}}$respectively.

Calculate the inductance and inductive reactance when the spacing between the conductor is  7m.

(c)

Calculate the inductance of the overhead transmission line.

Substitute 7m for D and 3cm for r into equation (1).

$$\begin{gathered} L=4\times {10}^{-7}In\left(\frac{7}{0.7788\times 3\times {10}^{-2}}\right) \\ L=4\times {10}^{-7}In\left(299.6\right) \\ 4\times {10}^{-7}In\times 5.702 \\ 4\times {10}^{-3}\frac{mH}{m} \end{gathered}$$

Hence the total inductance of the single-phase overhead transmission line is $2.28\times {10}^{-3}\frac{mH}{m}$.

Calculate the inductive reactance of the overhead transmission line.

Substitute $2.28\times {10}^{-3}\frac{mH}{m}$for L and 60Hz for f into equation (2).

$$\begin{gathered} X_2=2\mathrm{\pi }\times 60\times 2.28\times {10}^{-3} \\ {\mathrm{X}}_{2}258.42\times {10}^{-3} \\ {\mathrm{X}}_2=0.858\frac{\mathrm{\Omega }}{\mathrm{km}} \end{gathered}$$

Calculate inductive reactance of the overhead transmission line in ohms per miles.

$$\begin{gathered} {\mathrm{X}}_2=0.858\times 1.609 \\ {\mathrm{X}}_2=1.38\frac{\mathrm{\Omega }}{\mathrm{mi}} \end{gathered}$$

Calculate the change in inductance due to change in the spacing.

Substitute $1.38\frac{\mathrm{\Omega }}{\mathrm{mi}}$for $X_2$and $1.21\frac{\mathrm{\Omega }}{\mathrm{mi}}{\mathrm{X}}_1$for $X_1$into equation (3).

$$\begin{gathered} ∆\mathrm{X}=\left(\frac{1.38-1.21}{1.21}\right)\times 100 \\ =\frac{0.17}{1.21}\times 100 \\ =0.140\times 100 \\ =14\% \end{gathered}$$

Hence the total inductance of the single-phase overhead transmission line is $2.28\times {10}^{-3}\frac{\mathrm{mH}}{\mathrm{m}}$and the reactance in per kilometer and per miles are $0.753\frac{\mathrm{\Omega }}{\mathrm{km}}$and $1.21\frac{\mathrm{\Omega }}{\mathrm{mi}}$respectively.

If the spacing is double, the inductive reactance is increased by 14%.