GMR of stranded vs solid conductor

The geometric mean radius (GMR) of a conductor is the radius of a fictitious thin tube that has the same internal plus external flux linkage as the real conductor. For a solid round conductor of radius $R$, $GMR = r' = e^{-1/4}R = 0.7788R$. A stranded conductor of the same overall diameter contains air gaps between strands, so its current is spread over less metal and its effective GMR is smaller.

Example: 7-strand conductor

Take a 7-strand conductor, each strand of radius $r$: one central strand and six touching strands around it. Overall radius $R = 3r$.

        (2)   (3)
     (7)   (1)   (4)      1 = centre strand
        (6)   (5)         ring radius = 2r

Distances between strand centres:

With $n = 7$ strands there are $n^2 = 49$ terms:

$$\begin{aligned} D_s &= \left[(0.7788r)^7 (2r)^{24} (2\sqrt{3}r)^{12} (4r)^6\right]^{1/49} \\ &= 2.1767\,r \end{aligned}$$

Solid conductor of the same overall diameter

$$GMR_{solid} = 0.7788 \times 3r = 2.3364\,r$$

Comparison

$$\frac{D_{s,\,stranded}}{GMR_{solid}} = \frac{2.1767r}{2.3364r} = 0.932$$

So the 7-strand conductor has a GMR about 7% smaller than a solid conductor of the same diameter.

Why: the GMR depends on the actual cross-section that carries current. Stranding leaves air gaps, so the current-carrying material sits in a smaller effective area. A smaller GMR means the stranded conductor has a slightly higher inductance ($L = 2\times10^{-7}\ln(D/D_s)$) than a solid conductor of the same outer diameter.

Answer: $D_s = 2.1767r$ (stranded) < $2.3364r$ (solid).