Question

Reconsider Problem 4.28 with still another alternate phase placement shown below.

	Physical Positions
	1	2	3	1’	2’	3’
Phase Placement	C	A	B	B	A	C

Find the inductive reactance of the line in $\mathit{\Omega }\mathbf{/}\mathit{m}\mathit{i}\mathbf{/}\mathit{P}\mathit{h}\mathit{a}\mathit{s}\mathit{e}$.

Short answer

The inductive reactance is $0.2511\mathrm{\Omega }/\mathrm{mi}\mathrm{per}\mathrm{phase}$.

Step-by-step solution

Given data

Consider a double circuit configuration of three phase line with bundle conductors of three conductors.

Determine three bundled conductor GMR

Write the expression of the bundled spacing conductor.

$\begin{array}{c}\mathrm{d}=21\mathrm{in}\\ =21\mathrm{in}\times \frac{1\mathrm{ft}}{12\mathrm{in}}\\ =1.750\mathrm{in}\end{array}$

GMR of each sub conductors,${\mathrm{D}}_{\mathrm{s}}=0.0485\mathrm{ft}$

Write the expression for three bundles GMR.

$\begin{array}{c}{\mathrm{D}}_{\mathrm{SL}}=\sqrt[3]{{\mathrm{D}}_{\mathrm{s}}\times {\mathrm{d}}^2}\\ =\sqrt[3]{0.0485\times 1{.75}^2}\\ =0.5296\mathrm{ft}\end{array}$

Determine distance between sub conductors

Write the expression for the total GMR of three phase double circuit bundled conductor.

$\mathrm{GMR}=\sqrt[3]{{\left(\mathrm{GMR}\right)}_{\mathrm{A}}{\left(\mathrm{GMR}\right)}_{\mathrm{B}}{\left(\mathrm{GMR}\right)}_{\mathrm{C}}}$

Write the expression of the distance between the sub conductors.

$\begin{array}{c}{\mathrm{D}}_{{11}^{\text{'}}}=64\mathrm{ft}\\ {\mathrm{D}}_{{22}^{\text{'}}}=\sqrt{{36}^2+{32}^2}\\ =48.166\mathrm{ft}\\ {\mathrm{D}}_{{33}^{\text{'}}}=\sqrt{{36}^2+{64}^2}\\ =73.430\mathrm{ft}\end{array}$

Write the expression for the GMR of conductor A.

$\begin{array}{c}{\left(\mathrm{GMR}\right)}_{\mathrm{A}}=\sqrt{{\mathrm{r}}^{\text{'}}{\mathrm{D}}_{{11}^{\text{'}}}}\\ =\sqrt{0.5296\times 64}\\ =5.822\mathrm{ft}\end{array}$

Write the expression for the GMR of conductor B.

$\begin{array}{c}{\left(\mathrm{GMR}\right)}_{\mathrm{B}}=\sqrt{{\mathrm{r}}^{\text{'}}{\mathrm{D}}_{{22}^{\text{'}}}}\\ =\sqrt{0.5296\times 48.166}\\ =5.051\mathrm{ft}\end{array}$

Write the expression for the GMR of conductor C.

$\begin{array}{c}{\left(\mathrm{GMR}\right)}_{\mathrm{C}}=\sqrt{{\mathrm{r}}^{\text{'}}{\mathrm{D}}_{{33}^{\text{'}}}}\\ =\sqrt{0.5296\times 73.43}\\ =6.236\mathrm{ft}\end{array}$

Write the expression for the total GMR.

$\begin{array}{c}\mathrm{GMR}=\sqrt[3]{{\left(\mathrm{GMR}\right)}_{\mathrm{A}}{\left(\mathrm{GMR}\right)}_{\mathrm{B}}{\left(\mathrm{GMR}\right)}_{\mathrm{C}}}\\ =\sqrt[3]{5.822\times 5.051\times 6.236}\\ =5.681\mathrm{ft}\end{array}$

Determine distance between sub conductors

Write the expression for the distance between the sub conductors of the given double circuit configuration.

$\begin{array}{c}{\mathrm{D}}_{12}=36\mathrm{ft}\\ {\mathrm{D}}_{{12}^{\text{'}}}=32\mathrm{ft}\\ {\mathrm{D}}_{1^{\text{'}}2}=\sqrt{{36}^2+{64}^2}\\ =73.43\mathrm{ft}\\ {\mathrm{D}}_{1^{\text{'}}{2}^{\text{'}}}=32\mathrm{ft}\\ {\mathrm{D}}_{23}=\sqrt{{36}^2+{32}^2}\\ =48.166\mathrm{ft}\\ {\mathrm{D}}_{{23}^{\text{'}}}=32\mathrm{ft}\\ {\mathrm{D}}_{2^{\text{'}}3}=64\mathrm{ft}\\ {\mathrm{D}}_{2^{\text{'}}{3}^{\text{'}}}=36\mathrm{ft}\end{array}$

Determine the distance between sub conductors.

$\begin{array}{c}{\mathrm{D}}_{13}=32\mathrm{ft}\\ {\mathrm{D}}_{{13}^{\text{'}}}=\sqrt{{36}^2+{32}^2}\\ =48.166\mathrm{ft}\\ {\mathrm{D}}_{1^{\text{'}}3}=96\mathrm{ft}\\ {\mathrm{D}}_{1^{\text{'}}{3}^{\text{'}}}=\sqrt{{36}^2+{32}^2}\\ =48.166\mathrm{ft}\end{array}$

Determine equivalent GMD between conductors

Write the expression for the equivalent GMD between conductors A,B.

$\begin{array}{c}{\mathrm{D}}_{\mathrm{ABeq}}=\sqrt[4]{{\mathrm{D}}_{12}{\mathrm{D}}_{1^{\text{'}}{2}^{\text{'}}}{\mathrm{D}}_{{12}^{\text{'}}}{\mathrm{D}}_{1^{\text{'}}2}}\\ =\sqrt[4]{36\times 32\times 73.43\times 32}\\ =40.562\mathrm{ft}\end{array}$

Write the expression for the equivalent GMD between conductors B,C

$\begin{array}{c}{\mathrm{D}}_{\mathrm{BCeq}}=\sqrt[4]{{\mathrm{D}}_{23}{\mathrm{D}}_{2^{\text{'}}{3}^{\text{'}}}{\mathrm{D}}_{2,3}{\mathrm{D}}_{{23}^{\text{'}}}}\\ =\sqrt[4]{48.166\times 32\times 64\times 36}\\ =43.410\mathrm{ft}\end{array}$

Determine equivalent GMD of the bundle

Write the expression for the equivalent GMD between conductors A,C.

$\begin{array}{c}{\mathrm{D}}_{\mathrm{ACeq}}=\sqrt[4]{{\mathrm{D}}_{13}{\mathrm{D}}_{1^{\text{'}}{3}^{\text{'}}}{\mathrm{D}}_{{13}^{\text{'}}}{\mathrm{D}}_{1^{\text{'}}3}}\\ =\sqrt[4]{32\times 48.166\times 96\times 48.166}\\ =51.668\mathrm{ft}\end{array}$

Write the expression for the total GMD of the bundle.

$\begin{array}{c}\mathrm{GMD}=\sqrt[3]{{\mathrm{D}}_{\mathrm{ABeq}}{\mathrm{D}}_{\mathrm{BCeq}}{\mathrm{D}}_{\mathrm{ACeq}}}\\ =\sqrt[3]{40.562\times 43.41\times 51.668}\\ =44.976\mathrm{ft}\end{array}$

Determine total reactance of line in ohms in mile per phase

Write the expression for total reactance of line in ohms in mile per phase.

$\begin{array}{c}{\mathrm{X}}_{\mathrm{L}}=0.2794\log \frac{\mathrm{GMD}}{\mathrm{GMR}}\mathrm{\Omega }/\mathrm{mi}\mathrm{per}\mathrm{phase}\\ =0.2794\log \frac{44.976}{5.681}\\ =0.2511\mathrm{\Omega }/\mathrm{mi}\mathrm{per}\mathrm{phase}\end{array}$

Thus, the inductive reactance is $0.2511\mathrm{\Omega }/\mathrm{mi}\mathrm{per}\mathrm{phase}$.