Analysis of a Coaxial Waveguide with Finite-Length Impedance Loadings in the Inner and Outer Conductors

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 WienerHopf 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.


Introduction
Coaxial discontinuity structures are widely used as an element of microwave devices, and in the permeability and permittivity measurement for materials 1-3 .Hence, the diffraction at discontinuities in coaxial waveguides is a very important topic in microwave theory and have been subjected to numerous past investigations.Most simple types of discontinuities such as steps in inner or outer conductors see, e.g., 4-7 and wall impedance discontinuities 8, 9 were analyzed and characterized.For example, in 8, 9 the scattering of a shielded surface wave in a coaxial waveguide by a wall impedance discontinuity in the inner cylinder has been analyzed.These classical results are related mostly with isolated discontinuities, and fail when there are several of them close enough to interfere with each other.The aim of the present work is to consider a new canonical scattering problem consisting of the propagation of the dominant TEM mode at the finite-length impedance discontinuities in the inner and outer conductors of a coaxial waveguide see Figure 1 .The contributions from the successive

Analysis
Consider a coaxial waveguide whose inner cylinder is of radius ρ a, while the radius of the outer cylinder is ρ b with ρ, φ, z being the usual cylindrical coordinates.The part 0 < z < l of the inner and outer conductors are characterized by different constant surface impedances denoted by Z 1 η 1 Z 0 and Z 2 η 2 Z 0 , respectively, with Z 0 being the characteristic impedance of the free space.
Let the incident TEM mode with angular frequency ω and propagating in the positive z direction be given by H i ρ 0, H i φ u i , H i z 0, with where an exp −iωt time factor is assumed and suppressed.k is the propagation constant which is assumed to have a small imaginary part corresponding to a medium with damping.
The lossless case can be obtained by letting Im k → 0 at the end of the analysis.

Formulation of the Problem
The total field u T ρ, z can be written as u 1 ρ, z appearing in 2.1b is an unknown function which satisfies the Helmholtz equation and the following boundary conditions and continuity relations: where

2.3e
To obtain the unique solution to the mixed boundary value problem stated by 2.2 and 2.3a -2.3d one has to take into account the following edge and radiation conditions 17 :

2.3f
The infinite-range Fourier transform of 2.2 yields

2.4e
In 2.4a , K α stands for

2.5
The square-root function is defined in the complex α-plane, cut along α k to α k i∞ and α −k to α −k − i∞, such that K 0 k see Figure 2 .
Branch-cuts and integration lines in the complex plane.
Owing to the analytical properties of Fourier integrals, F ρ, α and F − ρ, α are yet unknown functions which are regular in the half-planes Im α > Im −k and Im α < Im k , respectively.The function F 1 ρ, α defined by 2.4d is an unknown entire function.
The solution of 2.4a reads Here A α and B α are spectral coefficients to be determined.By using the Fourier transform of the boundary conditions in 2.3a and 2.3b , one can easily show that they are related to P 1 a, α and P 1 b, α through with P 1 ρ, α being given by

2.8b
Multiplying both sides of 2.3c and 2.3d by e iαz and then integrating from 0 to l we obtain 2.9 Putting ρ b and then ρ a in 2.8a and its derivative with respect to ρ, and then using the relations 2.9 we end up with the following coupled systems of modified Wiener-Hopf equations valid in the strip Im −k < Im α < Im k :

2.11b
Mathematical Problems in Engineering 7

Solution of the Wiener-Hopf Equations
Now, let us rearrange the equations in 2.10 as

2.13
Notice that R − 1 * α and R − 2 * α are regular functions of α in the lower half-plane Im α < Im k except at the pole singularity occurring at α −k, while S 1 α and S 2 α are regular in the upper half-plane Im α > Im −k .Now, consider the Wiener-Hopf factorization of the kernel functions where V 1,2 α and V − 1,2 α denote certain functions which are regular and free of zeros in the half-planes Im α > Im −k and Im α < Im k , respectively.Their explicit expressions can easily be found by using the method described in 18

2.17
The positions of the integration lines L ± are shown in Figure 2.
The above integrals can be evaluated by using Jordan's Lemma and the residue theorem.The result is 18b Mathematical Problems in Engineering

2.18h
Here the dash denotes the derivative with respect to α : These coupled systems of algebraic equations will be solved numerically.The approach used in solving the infinite system of algebraic equations is similar to that employed by Rawlins 20 .
By using the edge conditions and the asymptotic behavior of β 1 n and β 2 n , one can show that the convergence of the infinite series appearing in these equations is rapid enough to allow truncation at, say N. Consequently the infinite systems are replaced by the corresponding finite systems of 8N × 8N algebraic equations and then solved by standard numerical algorithms.The value of N was increased until the reflected field amplitude being calculated did not change in a given number of decimal places.A typical result is provided by Figure 3.It can be seen that the reflected field amplitude becomes insensitive to the increase of the truncation number for N > 6.

Scattered Field
For ρ ∈ a, b and z < 0, the reflected wave propagating backward is obtained by taking the inverse Fourier transform of F − ρ, α , namely, , α e −iαz dα.

3.1
The above integral is calculated by closing the contour in the upper half plane and evaluating the residue contributions from the simple poles occurring at the zeros of T 1 a, b, α lying in the upper α-half-plane.The reflection coefficient R of the fundamental mode is defined as the ratio of reflected and incident waves amplitudes.It is computed from the contribution of the first pole at α k.The result is Similarly, the transmitted field in the region ρ ∈ a, b and z > l is obtained by inverting F ρ, α :

Mathematical Problems in Engineering
The transmission coefficient T of the fundamental mode, which is defined as the ratio of the amplitudes of the transmitted and incident waves, is obtained by evaluating the integral in 3.3 for z > l.This integral is now computed by closing the contour in the lower half of the complex α-plane.The pole of interest is at α −k whose contribution gives

3.4c
The first term in 3.4a or in 3.4b cancels out the incident fundamental mode as expected.The transmission coefficient is then given by

Computational Results
In order to observe the influence of the different parameters such as the surface impedances η 1 and η 2 , the width l of the impedance loadings, and the distance b − a between the two coaxial cylinders on the reflection coefficient, some numerical results are presented in this section.In what follows the impedances are assumed to be purely reactive.
Mathematical Problems in Engineering decreasing values of the outer cylinder radius, the band-stop frequencies of the configuration are shifted to the right.This means that total reflections occur at higher frequencies if the radius of the outer cylinder decreases.The effect of the width for the impedance loadings on the reflection coefficient is shown in Figure 6.The number of resonances corresponding to the variation of the reflected field increases when the value of l increases.But the amplitude related to the reflected field is not affected too much by the width of the impedance surfaces.It is also seen that the curves related to reflected field amplitude approaches to the one calculated from 5.2a , when kl 1.

Discussion
When we let l → ∞, the Wiener-Hopf equations in 2.10 reduce to Then, the reflection coefficient in 3.2 takes the following form:  or which is nothing but the results related to the junction of perfectly conducting and impedance coaxial waveguides shown in Figure 7. Equations 5.2a and 5.2b are obtained by using the Wiener-Hopf equations in 5.1a and 5.1b , respectively.It can be checked easily that the two expressions are equivalent.Finally, for |η 2 | → 0, the results obtained in the present work to that previously obtained in 3 .Indeed, from Figure 8 we can see that for decreasing values of |η 2 |, the curves approach the result related to the finite-length impedance loading in the outer conductor of a coaxial waveguide obtained in 3 solid line in Figure 8 .This can be considered as a check of the analysis made in this paper.

Conclusion
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.

Figure 1 :
Figure 1: Coaxial cable with a finite-length impedance loading in the inner and outer conductors.

Figure 4 : 9 Figure 5 :
Figure4: a Amplitude of the reflection coefficient versus the frequency, for different values of the impedance loading on the inner conductor the case where η 1 and η 2 are capacitive .b Amplitude of the reflection coefficient versus the frequency, for different values of the impedance loading on the inner conductor the case where η 1 is capacitive and η 2 is inductive .c Amplitude of the reflection coefficient versus the frequency, for different values of the impedance loading on the inner conductor the case where η 1 and η 2 are inductive .

Figure 6 :Figure 7 :
Figure 6: Amplitude of the reflection coefficient versus the frequency for different impedance loadings length l.

MathematicalFigure 8 :
Figure 8: Amplitude of the reflection coefficient versus the frequency for different values of η 2 .