Antenna equivalent radius

Lore & Background

The equivalent radius is derived by considering a conductor whose cross-section dimensions are small compared to the wavelength, with current flowing only axially and varying slowly along the length. Under these conditions, the current is approximately uniformly distributed around the circumference due to the skin effect, and only current in a local neighborhood contributes significantly to the magnetic vector potential at a point. This reduces the three-dimensional problem to a two-dimensional one of an infinitely long conductor with constant surface current density.

The derivation begins by dividing the circumference of the arbitrary cross-section into differential segments, each approximated as a vertical line current. The magnetic vector potential at a fixed point on the circumference is the sum of potentials from all such line currents. The average potential over the circumference is then computed. For a cylindrical conductor with the same linear current density, the potential at any point on its surface equals its average potential. Equating the two average potentials and exponentiating yields the formula for the equivalent radius.

Formulas for specific cross-sections exist: those for square and triangular cross-sections follow from numerical evaluation of the double integral, while all other formulas are exact. The equivalent radius is consistent under scaling—if the cross-section dimensions are multiplied by a factor, the equivalent radius scales by the same factor—and for a cylindrical conductor, the equivalent radius equals the actual radius.

Reader's Guide

The antenna equivalent radius is significant because it bridges the gap between theoretical and practical antenna design. Many analytical formulas and computational tools are built around the simple geometry of a cylindrical conductor, but real antennas often use conductors with square, rectangular, triangular, or other non-circular cross-sections. By providing a single effective radius that preserves the magnetic vector potential behavior, the equivalent radius allows engineers to apply existing cylindrical-conductor results directly to these more complex shapes, provided the cross-section dimensions are small relative to the wavelength.

Its legacy lies in enabling accurate modeling without requiring full three-dimensional electromagnetic simulation for every non-circular geometry. The derivation relies on quasi-static conditions and the assumption that current distribution is uniform around the circumference, which holds when skin effect dominates and the conductor is electrically small. The formulas for square and triangular cross-sections, derived from numerical evaluation of the double integral, extend the concept to common practical shapes. The scaling property ensures that the equivalent radius behaves intuitively when dimensions are changed. This concept remains a standard tool in antenna theory, allowing the vast body of work on cylindrical antennas to be leveraged for a wide variety of conductor shapes.