The Resonance Condition for Slow Wave Antennas: a Lagrangian Approach

TL;DR

This paper introduces a Lagrangian formalism to analyze the resonance properties of slow wave antennas with subwavelength elements, providing a more accurate physical model and demonstrating how periodic loading affects resonant length.

Contribution

It develops a novel Lagrangian approach for modeling periodically loaded antennas, improving upon traditional transmission line models and validating with simulations and measurements.

Findings

Periodic loading shortens the resonant length without changing the pattern.

Resonant length is proportional to the product of the half-wavelength and the velocity ratio.

The Lagrangian model accurately predicts loaded resonant length.

Abstract

A proof of the resonant property of linear periodically loaded antennas with subwavelength elements is obtained by applying a Lagrangian formalism. A Lagrangian is developed by modeling the antenna with lumped inductance and capacitance elements on a single line, thereby physically similar to the antenna and thus avoiding the inaccurate two parallel conductor transmission line model. An equation for the antenna current driven by an incident electromagnetic field is obtained via vector and scalar potentials. It is shown that periodic loading provides a means to shorten the resonant length while the antenna pattern remains unchanged. The Lagrangian model is validated through a calculation showing the loaded resonant length is determined by a product of a resonant half-wavelength dipole with the ratio of the free space velocity and the longitudinal traveling wave velocity. A periodically loaded disk-on-rod antenna example with simulations and measurements provides further validation of the mathematics.