Inductance of 4-conductor bundle line, 6 m spacing

Each phase is a square 4-conductor bundle. For inductance, its self-GMD is $D_s = (r' \cdot d \cdot d \cdot \sqrt2 d)^{1/4} = 1.09\sqrt[4]{r'd^3}$. The line is taken as transposed.

   o o        o o        o o
   o o        o o        o o
    A    6 m   B    6 m   C
   d = 45 cm, r = 12 mm

Data: $r = 0.012$ m, $r' = 0.7788 \times 0.012 = 0.009346$ m, $d = 0.45$ m, $D = 6, 6, 12$ m, 200 km.

Mutual GMD

$$D_m = \sqrt[3]{6 \times 6 \times 12} = 7.5595\ \text{m}$$

Self GMD of bundle

$$D_s = 1.09\sqrt[4]{0.009346 \times 0.45^3} = 0.18629\ \text{m}$$

Inductance

$$\begin{aligned} L &= 2\times10^{-7}\ln\frac{7.5595}{0.18629} = 2\times10^{-7}\times3.7033 \\ &= 7.4065\times10^{-7}\ \text{H/m} = 0.7407\ \text{mH/km} \\ L_{200} &= 0.7407 \times 200 = 148.13\ \text{mH} \end{aligned}$$

Answer: $L = 148.1$ mH per phase for 200 km (0.7407 mH/km).