Question

Find the GMR of a stranded conductor consisting of six outer strands surrounding and touching one central strand, all strands having the same radius.

Short answer

Therefore the value of the GMR is $2.177r$.

Step-by-step solution

Determine the distance between conductors.

Consider the figure for calculation of geometric mean distance.

Let’s assume the radius of each conductor is r.

Calculate distance from conductor $1$to every outer conductor,

The distance from the conductor$1$.

$D_{11}=0.7788r$

The distance of conductorlocalid="1650468369011" $1$from conductor$2$.

$D_{12}=2r$

The distance of conductor $1$from conductor localid="1650468656873" $6$and localid="1650468669940" $7$are equal to distance of conductor localid="1650468683909" $1$from localid="1650468696931" $2$.

$\begin{array}{l}{D}_{16}=D_{17}\\ D_{16}=2r\end{array}$

The distance of conductor $1$from conductors localid="1650468724058" $3$and localid="1650468743916" $5$are same.

$\begin{array}{l}{D}_{13}=D_{15}\\ D_{13}=\sqrt{{\left(4r\right)}^2-{\left(2r\right)}^2}\\ D_{13}=\sqrt{16r^2-4r^2}\\ D_{13}=2\sqrt{3}r\end{array}$

The distance of conductor $1$from conductor .

For inner conductor,

The distance of conductor ,

The distance of conductor from conductors and are the same.

Further calculation,

The distance of conductor from conductor .

For inner conductor,

The distance of conductor,

The distance of conductor from conductors and are the same.

Further calculation,

Determine the GMR of stranded conductor.

$D_s=\sqrt{\left(D_{11}{D}_{12}{D}_{13}{D}_{14}{D}_{15}{D}_{16}{D}_{17}\right)\left(D_{71}{D}_{72}{D}_{73}{D}_{74}{D}_{75}{D}_{76}{D}_7\right)}$

Substitute $0.7788r$for$d11$ , $2r$for$D12$ , $2r$for$D16$ , $2$r for D$17$, $2\sqrt{3r}$for D$13$, $2\sqrt{3}$ for D$15$ , $4$rfor D$14$ , $0.7788r$for D$77$ , $2r$for$D_{72}$ ,$2r$for$D_{73}$ ,$2r$for$D_{75}$ ,$2r$for , $2r$for$D_{71}$ , and $2r$for$D_{74}$ into above equation.$\begin{array}{l}{D}_s=\sqrt[49]{{(0.7788r\times 2r\times 2r\times 2r\times 2\sqrt{3}r\times 2\sqrt{3}r\times 4r)}^6(0.7788r\times 2r\times 2r\times 2r\times 2r\times 2r\times 2r\times )}\\ D_s=\sqrt[49]{{\left(0.7788\right)}^7{\left(r\right)}^{49}{\left(2\right)}^{34}{\left(2\sqrt{3}\right)}^{12}{\left(4\right)}^6}\\ D_s=r\sqrt[49]{{\left(0.7788r\right)}^7{\left(2\right)}^{34}{\left(2\sqrt{3}\right)}^{12}{\left(4\right)}^6}\\ D_s=2.177r\end{array}$

Hence the value of GMR is$2.177r$.