A rigorous Wiener-Hopf approach is used to investigate the band stop filter characteristics of a coaxial waveguide with finite-length impedance loading. The representation of the solution to the boundary-value problem in terms of Fourier integrals leads to two simultaneous modified Wiener-Hopf equations whose formal solution is obtained by using the factorization and decomposition procedures. The solution involves 16 infinite sets of unknown coefficients satisfying 16 infinite systems of linear algebraic equations. These systems are solved numerically and some graphical results showing the influence of the spacing between the coaxial cylinders, the surface impedances, and the length of the impedance loadings on the reflection coefficient are presented.

Coaxial discontinuity structures are widely used as an element of microwave devices, and in the permeability and permittivity measurement for materials [

Coaxial cable with a finite-length impedance loading in the inner and outer conductors.

A problem similar to the one considered in this work has been recently treated by the authors in the simpler case where the impedance loading is present on the outer conductor only [

The method adopted here is similar to that described in [

Consider a coaxial waveguide whose inner cylinder is of radius

Let the incident TEM mode with angular frequency

The total field

The infinite-range Fourier transform of (

Branch-cuts and integration lines in the complex plane.

Owing to the analytical properties of Fourier integrals,

The solution of (

Now, let us rearrange the equations in (

Now, consider the Wiener-Hopf factorization of the kernel functions

Multiplying both sides of (

The above integrals can be evaluated by using Jordan’s Lemma and the residue theorem. The result is

The formal solution of the coupled system of Wiener-Hopf equations given by (

Reflected field amplitude versus the truncation number.

For

Similarly, the transmitted field in the region

In order to observe the influence of the different parameters such as the surface impedances

Figures

(a) Amplitude of the reflection coefficient versus the frequency, for different values of the impedance loading on the inner conductor (the case where

Figure

Amplitude of the reflection coefficient versus the frequency for different values of the outer cylinder radius

The effect of the width for the impedance loadings on the reflection coefficient is shown in Figure

Amplitude of the reflection coefficient versus the frequency for different impedance loadings length

When we let

Junction of perfectly conducting and impedance loaded coaxial waveguides (

Equations (

Finally, for

Amplitude of the reflection coefficient versus the frequency for different values of

In the present work the propagation of TEM wave in a coaxial waveguide with finite-length impedance loading is investigated rigorously through the Wiener-Hopf technique. In order to obtain the explicit expressions of the reflection coefficient, the problem is first reduced into two coupled modified Wiener-Hopf equations and then solved exactly in a formal sense by using the factorization and decomposition procedures. The formal solution involves infinite series with 8 sets of unknown coefficients satisfying 8 infinite sets of algebraic equations which are solved numerically. The advantage of the present method is that the solution obtained here is valid for all frequencies and impedance lengths. Furthermore, it is observed that for certain values of the surface impedances full reflection occurs, showing that this configuration may be used as a band-stop filter.

Finally, it is noteworthy that the Weiner-Hopf solution provided here could be extended to treat the case where the lengths of the impedances on the inner and outer conductors are different. Other future work could lie in the investigation of wave propagation in coaxial waveguides with successive finite-length impedance loadings.