Self-geometrical mean distance abbreviated as self-GMD plays vital role in simplifying the inductance calculations in the case of multiconductor arrangements in transmission lines. Symbol D with subscript s is used to denote self-GMD (Ds).
The mathematical equation for inductance per conductor per metre is:
Inductance per conductor per meter = 2 * 10-7 * (1/4) + 2 * 10-7 loge (d/r)
Where in the above mathematical equation:
2 * 10-7 * (1/4) implies inductance flux within the solid conductor
The current can be confined to the conductor surface and internal conductor flux linkage would be almost zero, if the original solid conductor is replaced by equivalent hollow cylinder having hollow having very thing walls. As a result the inductance resulting from internal flux would be zero and the term 2 * 10-7 * (1/4) can be removed.
The radius of equivalent hollow cylinder should be quite smaller as compared to physical radius of the conductor, doing so provides sufficient room for additional flux thereby compensating for the absence of internal flux linkage.
In case of solid round conductor having radius r, the the self-GMD or GMR equals 0.7788 r
Now,
Inductance per conductor per meter = 2 * 10-7 loge (d/Ds)
Where Ds = Self-GMD = 0.7788 r
Self GMD of conductor is independent of spacing between conductors.
It depends on size and shape of conductor.