Antenna equivalent radius

The equivalent radius of an antenna conductor is defined as:

${\displaystyle r_{e}=\exp \left\{{1 \over L^{2}}\oint _{\ell }\oint _{\ell }\ln \vert {\boldsymbol {x}}-{\boldsymbol {y}}\vert \;dx\;dy\right\}}$

where ${\displaystyle \scriptstyle \ell }$ denotes the conductor's circumference, ${\displaystyle \scriptstyle {L}}$ is the length of the circumference, ${\displaystyle \scriptstyle {\boldsymbol {x}}}$ and ${\displaystyle \scriptstyle {\boldsymbol {y}}}$ are vectors locating points along the circumference, and ${\displaystyle \scriptstyle {dx}}$ and ${\displaystyle \scriptstyle {dy}}$ are differentials segments along it. The equivalent radius allows the use of analytical formulas or computational or experimental data derived for antennas constructed from small conductors with uniform, circular cross-sections to be applied in the analysis of antennas constructed from small conductors with uniform, non-circular cross-sections. Here "small" means the largest dimension of the cross-section is much less than the wavelength ${\displaystyle \scriptstyle {\lambda }}$.