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To the theory of helical waveguides

A. S. Yurkov

TL;DR

The paper extends the anisotropic-conductivity approximation for tiny- pitch helical waveguides to dielectric-filled windings, deriving a general dispersion framework for slow waves with azimuthal order $n=0$ and providing explicit field expressions. A transcendental dispersion equation for the slowing coefficient $\xi=k/k_0$ is obtained from the solvability of a boundary-value system, with numerical solution via bracketing and reductions that recover the classical $\varepsilon=\mu=1$ case. Closed-form EM-field components, expressed in terms of the winding current, are derived for both inside and outside the dielectric, together with exact power and impedance formulas. These results enable accurate modeling of such waveguides as frequency-dependent equivalent long lines, facilitating practical RF design for antennas and resonators involving dielectric effects.

Abstract

It makes sense to consider a helical waveguide with a fine pitch approximately, replacing the turns with anisotropic conductivity: infinite along the turns and zero across them. This approach has been known for a long time, but calculation formulas within it have only been obtained for the case where the winding does not contain a dielectric core. This paper addresses this gap in the theory: calculation formulas are obtained for the case where the waveguide contains a dielectric with a certain permittivity and magnetic permeability. An equation determining the slowing factor is found, and a method for its numerical solution is proposed. Explicit formulas are obtained for the wave impedance.

To the theory of helical waveguides

TL;DR

The paper extends the anisotropic-conductivity approximation for tiny- pitch helical waveguides to dielectric-filled windings, deriving a general dispersion framework for slow waves with azimuthal order and providing explicit field expressions. A transcendental dispersion equation for the slowing coefficient is obtained from the solvability of a boundary-value system, with numerical solution via bracketing and reductions that recover the classical case. Closed-form EM-field components, expressed in terms of the winding current, are derived for both inside and outside the dielectric, together with exact power and impedance formulas. These results enable accurate modeling of such waveguides as frequency-dependent equivalent long lines, facilitating practical RF design for antennas and resonators involving dielectric effects.

Abstract

It makes sense to consider a helical waveguide with a fine pitch approximately, replacing the turns with anisotropic conductivity: infinite along the turns and zero across them. This approach has been known for a long time, but calculation formulas within it have only been obtained for the case where the winding does not contain a dielectric core. This paper addresses this gap in the theory: calculation formulas are obtained for the case where the waveguide contains a dielectric with a certain permittivity and magnetic permeability. An equation determining the slowing factor is found, and a method for its numerical solution is proposed. Explicit formulas are obtained for the wave impedance.

Paper Structure

This paper contains 6 sections, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Representation of a spiral waveguide wave as a superposition of $H$- and $E$-waves
  3. Dispersion law
  4. Electromagnetic field expressed in terms of the current flowing through the winding
  5. Wave power, wave impedance and equivalent long line
  6. Conclusion

Figures (2)

  • Figure 1: The slowdown coefficient $\xi$ depending on the wavelength $\lambda$ for a waveguide with a diameter of 30 mm, with a winding pitch of 1 mm. 1: $\varepsilon=\mu=1$; 2: $\varepsilon=5$, $\mu=1$; 3: $\varepsilon=1$, $\mu=5$.
  • Figure 2: Wave impedance $W$ depending on wavelength $\lambda$ for a waveguide with a diameter of 30 mm, with a winding pitch of 1 mm. 1: $\varepsilon=\mu=1$; 2: $\varepsilon=5$, $\mu=1$; 3: $\varepsilon=1$, $\mu=5$.