State the Inductance formula

The inductance of a three-phase line per unit length is given by the formula, where D is the GMD (Geometric Mean Distance) and r' is the GMR (Geometric Mean Radius) which is $0.7788r$.

$L = 2 \times 10^{-7} \ln \left( \frac{D}{0.7788r} \right)$

Analyze the effect of doubling parameters

When spacing D is doubled (D₂ = 2D) and radius r is doubled (r2 = 2r), the GMR also effectively doubles (GMR₂ = 2 * GMR₁).

$L_2 = 2 \times 10^{-7} \ln \left( \frac{2D}{2 \times 0.7788r} \right)$

Simplify the expression

The factor of 2 in both the numerator (D) and the denominator (GMR) cancels out, leaving the logarithmic term unchanged.

$L_2 = 2 \times 10^{-7} \ln \left( \frac{D}{0.7788r} \right) = L_1$

Conclusion

Since the ratio inside the logarithm remains the same, the inductance value remains unchanged at 1.2 mH/phase/km.

$L_2 = 1.2 \text{ mH/phase/km}$