Self GMD of 4-strand conductor, 5 mm strands

The self-GMD for inductance of an $n$-strand conductor is the $n^2$-th root of the product of all inter-strand distances, with self-distance $r' = 0.7788r$.

     (1)   (2)
     (3)   (4)
  side of square = 2r, diagonal = 2(sqrt2)r

Data: $r = 5$ mm, $2r = 10$ mm, $r' = 0.7788 \times 5 = 3.894$ mm.

Distances

Each strand: two distances of $2r$ and one of $2\sqrt2 r$.

Self-GMD

$$\begin{aligned} D_s &= \left[(r')^4 \times (2r)^8 \times (2\sqrt2 r)^4\right]^{1/16} \\ &= \left[r' \times (2r)^2 \times 2\sqrt2 r\right]^{1/4} \\ &= \left[3.894 \times 10^2 \times 14.142\right]^{1/4} \\ &= (5507.0)^{1/4} = 8.614\ \text{mm} \end{aligned}$$

In terms of $r$: $D_s = 1.7229r$.

Answer: self-GMD $= 8.61$ mm $= 0.861$ cm.