GMR for inductance vs capacitance

The GMR (self-GMD) used in line parameter calculations is different for inductance and capacitance because inductance depends on the magnetic flux, which exists inside the conductor too, while capacitance depends on the electric field, which exists only outside the conductor.

GMR for inductance

• Current flows through the whole cross-section, so there is magnetic flux inside the conductor as well as outside.
• The internal flux linkage ($\mu_0 I/8\pi$ per metre) is included by replacing the actual radius $r$ with a smaller fictitious radius:

$$r' = r\,e^{-1/4} = 0.7788\,r$$

• For a stranded or bundled conductor, the self-GMD uses $r'$ for each strand: $D_{sL} = (r' \cdot D_{12} \cdots D_{1n} \cdots)^{1/n^2}$; for a 2-bundle, $D_{sL} = \sqrt{r'd}$.

GMR for capacitance

• Charge resides on the surface of a conductor; the electric field inside a conductor is zero.
• So there is no "internal" term and the actual outer radius $r$ is used: $D_{sC} = (r \cdot D_{12} \cdots)^{1/n^2}$; for a 2-bundle, $D_{sC} = \sqrt{rd}$.

Comparison

Example: for $r = 1$ cm, GMR for inductance $= 0.7788$ cm, while for capacitance it is 1 cm. The mutual GMD between phases is the same for both calculations.