Antenna equivalent radius

Part of a series on
Antennas
Common types Dipole Fractal Loop Monopole Satellite dish Television Whip
Components Balun Block upconverter Coaxial cable Counterpoise (ground system) Feed Feed line Low-noise block downconverter Passive radiator Receiver Rotator Stub Transmitter Tuner Twin-lead
Systems Antenna farm Amateur radio Cellular network Hotspot Municipal wireless network Radio Radio masts and towers Wi-Fi Wireless
Safety and regulation Wireless device radiation and health Wireless electronic devices and health International Telecommunication Union (Radio Regulations) World Radiocommunication Conference
Radiation sources / regions Boresight Focal cloud Ground plane Main lobe Near and far field Side lobe Vertical plane
Characteristics Array gain Directivity Efficiency Electrical length Equivalent radius Factor Friis transmission equation Gain Height Radiation pattern Radiation resistance Radio propagation Radio spectrum Signal-to-noise ratio Spurious emission
Techniques Beam steering Beam tilt Beamforming Small cell Bell Laboratories Layered Space-Time (BLAST) Massive Multiple-input multiple-output (MIMO) Reconfiguration Spread spectrum Wideband Space Division Multiple Access (WSDMA)
v t e

The equivalent radius of an antenna conductor is defined as:^[1][2]

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

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 ${\displaystyle \scriptstyle {\lambda }}$.