A Wiener-Hopf analysis of the parallel plate waveguide with finite length impedance loading

Authors


Abstract

[1] In the present work a rigorous Wiener-Hopf approach is used to investigate the band-stop filter characteristics of a parallel plate 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 which are uncoupled by using the pole removal technique. The solution involves four infinite sets of unknown coefficients satisfying four infinite systems of linear algebraic equations. These systems are solved numerically and some graphical results showing the influence of the waveguide spacing, surface impedances and the length of the impedance loading on the reflection coefficient are presented.

1. Introduction

[2] In the present work we consider the propagation of a TEM mode, in a parallel plates waveguide with a finite length impedance loading as depicted in Figure 1. The parallel plates at y = 0 and y = b are perfectly conducting for x < 0 and x > l, and are characterized by constant surface impedances for 0 < x < l. For the sake of generality we assume that the surface impedances of the lower and upper plates are different from each other and denoted by Z1 = η1Z0 and Z2 = η2Z0, respectively, with Z0 being the characteristic impedance of the free space. It is assumed that ReZ1,2 > 0. The scattering coefficients for wall impedance change in parallel-plate waveguides have been investigated by several authors. Among them one can cite, for example, Johansen [1962] who considered the case where the part x < 0 of the parallel plates are perfectly conducting while the part x > 0 has the same surface impedance. Heins and Feshbach [1947] provided a Wiener-Hopf solution to the problem of coupling of two ducts. Karajala and Mittra [1965] have considered the scattering at the junction of two semi-infinite parallel plate waveguides with impedance walls by mode matching method. Arora and Vijayaraghavan [1970] used the Wiener-Hopf technique to compute the scattering of shielded surface wave in a parallel-plate waveguide consisting of inductively reactive guiding surfaces and characterized by an abrupt wall reactance discontinuity. Finally Büyükaksoy et al. [2006] reconsidered the problem treated in the work of Johansen [1962] in the more general case where the impedances of the upper and lower plates of the semi-infinite parallel plate waveguide are different from each other.

Figure 1.

Geometry of the problem.

Table 1. Some Typical Values of βm and νm for Two Different Values of k
βmνm
k = 1 + i0.01
β1 = 0.90634 + i0.01006ν1 = 0.80580 + i0.01031
β2 = 0.00144 + i5.65590ν2 = 0.00112 + i5.68768
β3 = 0.00071 + i11.3960ν3 = 0.00055 + i11.4119
β4 = 0.00047 + i17.1173ν4 = 0.00037 + i17.1279
β5 = 0.00035 + i22.8340ν5 = 0.00027 + i22.8419
 
k = 2 + i0.01
β1 = 1.91031 + i0.01002ν1 = 1.82292 + i0.01010
β2 = 0.00336 + i5.41755ν2 = 0.00300 + i5.48303
β3 = 0.00161 + i11.2797ν3 = 0.00144 + i11.3118
β4 = 0.00106 + i17.0401ν4 = 0.00095 + i17.0614
β5 = 0.00079 + i22.7762ν5 = 0.00071 + i22.7921

[3] The aim of this work is to compute the reflection and transmission coefficients of a TEM wave in a parallel plate waveguide with finite length impedance loading. One has to notice that this configuration can be used as a band-stop filter at microwave frequencies. Hence, the solution of this problem is of importance for the band-stop filter design using the rectangular waveguide with finite length impedance loading. This work may also serve as a first-order approximation to the grooves or periodic gratings inside a parallel-plate waveguide [Lee et al., 1994; Hwang and Eom, 2005; Tayyar et al., 2008]. It is well known that the grooved parallel plate waveguide exhibit also band-pass and band-stop filter behavior at microwave frequencies.

[4] The representation of the solution to the three-part mixed boundary-value problem in terms of Fourier integrals leads to two simultaneous modified Wiener-Hopf equations which are uncoupled by using the pole removal technique. It consists of considering the analytical properties of the functions that occur and introducing some infinite sums over certain poles with unknown expansion coefficients [Büyükaksoy et al., 2006; Idemen, 1976; Abrahams, 1987]. The solution involves four infinite sets of unknown coefficients satisfying four infinite systems of linear algebraic equations. These systems are solved numerically and some graphical results showing the influence of the waveguide spacing, surface impedances and the length of the impedance loading on the band-stop filter characteristics are presented.

[5] The results are compared with the results obtained by Tayyar et al. [2008] for dielectric filled grooved waveguide problem. It is shown that the results are very satisfactory especially when the grooves are shallow and filled with low loss and high contrast dielectric materials.

2. Analysis

[6] Let the incident TEM mode propagating in the positive x direction be given by

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where k is the propagation constant which is assumed to have a small imaginary part corresponding to slightly lossy medium. The lossless case can be obtained by letting Im (k) → 0 at the end of the analysis.

[7] The total field uT(x, y) can be written as

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In (2), u(x, y) is the unknown function which satisfies the Helmholtz equation

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It is appropriate to use the following Fourier integral representation for the solution of (3):

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Here, K(α) denotes the square-root function

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which is defined in the complex α -plane, cut along α = k to α = k + i∞ and α = −k to α = −ki∞, such that K(0) = k. The spectral coefficients A(α) and B(α) in (4a) will be determined with the aid of the following boundary conditions:

equation image
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To obtain the unique solution to the mixed boundary value problem stated by (3) and (5a)(5d), one has to take into account the following edge and radiation conditions [Idemen, 2000]:

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Notice that dos Santos and Teixeira [1989] have shown that the edge conditions in (5e) are equivalent to specifying the space of solutions to the Wiener-Hopf equation as the Sobolev space H1/2; 2+ (equation image). The existence of a unique solution and continuous dependence on the known data may be discussed by formulating the boundary value problem in a Sobolev space setting by looking for solutions in the finite energy norm space and using the operator-theoretic methods. But this fairly difficult task is beyond the scope of this paper and for practical purposes we have chosen the classical formulation here. Hence, in this paper we provide a formal solution to the problem based on the assumption of existence and uniqueness.

[8] Incorporating (4a) into (5a)(5d) and then inverting the resulting integral equations, we get

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with

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and

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Owing to the analytical properties of Fourier integrals, Φ1,2+ (α) and Φ1,2 (α) are yet unknown functions which are regular in the half-planes Im(α) > Im(−k) and Im(α) < Im(k), respectively. The function F1,2(α) defined by (7e)(7f) are an unknown entire functions.

[9] From equations (6a) and (6b) we have

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Substituting (8a) and (8b) into (6c) and (6d), one obtains the following two coupled modified Wiener-Hopf equations of the third kind valid in the strip Im(−k) < Im(α) < Im(k)

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and

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with

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and

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Notice that P(*)(α) and Q(*)(α) are regular in the lower half plane except at the pole singularity occurring at α = −k.

3. Solution of the Simultaneous Modified Wiener-Hopf Equations

[10] It is well-known that the most important step to solve the functional equation of the Wiener-Hopf type is the factorization of the kernel functions M1,2(α) and N(α) appearing in (10a) and (10b) as follows:

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and

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where M1,2+(α), N+(α) and M1,2(α), N(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.

[11] The explicit expressions of M1,2+(α) and N+(α) can be obtained by following the procedure outlined by Mittra and Lee [1971]

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here βm′s, vm′s and αm′s are the roots of the functions M1,2(α) and N(α ), respectively (Figure 2):

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with

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In (13a)(13c), C is the Euler's constant given by C = 0.57721….

Figure 2.

Location of the roots βm and vm of M1(α) and M2(α) in the complex α-plane.

[12] For the case b = 0.55, η1 = 0.1i and η1 = 0.2i the first five zeros of the scalar kernels M1,2(α) in (10a) are obtained as follows:

[13] It can be easily shown that one has

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when |α∣ → ∞ in the respective region of regularity of the functions M1,2+(α), N+(α) and M1,2(α), N(α).

[14] Multiplying (9a) from the left by (kα)N(α)/M1(α) and eiαl+(α)/M1+(α) respectively, one obtains

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and

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Now, after multiplying (9b) by (kα)N(α)/M2(α) and eiαl+(α)/M2+(α) respectively one gets

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and

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The first term appearing in the left hand side of (14a) is evidently regular in the upper half-plane. The third term and the right hand side of the same equation have singularities in both half-planes. Hence, one has necessarily to apply the Wiener-Hopf decomposition procedure to these terms. After performing the Wiener-Hopf decomposition procedure, (14a) can be rearranged as

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where M1(α) denotes the derivative of M1(α) with respect to α. The regularity of the left hand side of the equation (15) in the upper half-plane may be violated by the simple poles occurring at the zeros of M1(α) lying in the upper half-plane, namely at α = βm, m = 1,2,…. If the infinite series of poles equation imageam/(αβm) are subtracted from both sides of equation (15), we obtain

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To cancel out the unwanted poles, the expansion coefficients am should satisfy the following equation:

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In equation (16b) the dash (′) stands for the derivation with respect to α. All functions on the left hand side of the equation (16a) now are regular in the upper half-plane (Im(α) > Im(−k)) while those on the right hand side are regular in the lower half-plane (Im(α) < Im(k)). The application of the analytical continuation principle together with the Liouville's theorem to the equation (16a) yields:

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A similar treatment of the equations (14b)(14d) enables us to obtain the following results:

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and

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where the unknown constants bm, cm and dm are defined by

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and

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The constants vm appearing in the equations (17c) and (17d) are the zeros of M2(α) (see (13d)). By using (17a) and (17b) in (14a), (17c) and (17d) in (14c), respectively one obtains the solutions of the simultaneous modified Wiener-Hopf equations in (9a) and (9b) as:

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and

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The expressions of F1(α) and F2(α) in (19a) and (19b) involve the unknown constants am, bm, cm and dm which can be determined through the 4 systems of infinitely many algebraic equations. These linear systems are obtained from the expressions (19a) and (19b) and (17a)(17d) themselves by substituting there α = ±βm and α = ±vm. The result is as follows:

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and

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In equations (20a)(20d), P(*)(−βm), R+(βm), Q(*)(−vm), S+(vm) stand for:

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and

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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 [1978]. By taking into account the edge conditions in (5e) and the orders of βm and vm for sufficiently large m

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[see, e.g., Heins and Feshbach, 1947], we have:

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This shows 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 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.

Figure 3.

Refected field amplitude versus the truncation number N.

4. Scattered Field

4.1. Reflected Field

[15] The evaluation of the integral (3) for x < 0 will give us the reflected wave propagating backward as follows

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where equation image is a straight line parallel to the real α-axis, lying in the strip Im(−k) < Im(α) < Im(k). 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 K(α) sin [K(α) b] lying in the upper α-half-plane. The reflection coefficient ℛ of the fundamental mode is defined as the complex coefficient multiplying the travelling wave term exp (−ikx) and is computed from the contribution of the first pole at α = k. The result is

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with

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4.2. Transmitted Field

[16] Similarly, the transmission coefficient T of the fundamental mode which is defined as to be the complex coefficient of exp (ikx), is obtained by evaluating the integral in (23) for x > 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

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with

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Notice that the first term in (25a) cancels out exactly the incident TEM wave and the second term gives the transmission coefficient. This is an expected result since the incident TEM wave does not satisfy the impedance boundary conditions given in (5c) and (5d) for x ∈ (0, l).

5. Numerical Results

[17] In order to observe the influence of the different parameters such as the surface impedances (η1 and η2), the width (l) of the impedance surfaces, and the separation distance b between the two-plates on the reflection coefficient, some numerical results are presented in this section. In what follows the impedances are assumed to be purely reactive, i.e. ηj = iςj, j = 1, 2.

[18] Figure 4 shows the amplitude of the reflection coefficient versus the frequency for different values of ς2 while ς1, b and l are held fixed. If ς1 is positive (capacitive) and ς2 is decreasing by taking negative values (inductive), the band-stop frequency of the configuration remains unchanged, but the quality factor diminish.

Figure 4.

Amplitude of the reflection coefficient versus the frequency for different values of η2 while η1 is held fixed.

[19] In Figure 5, ς2 is negative (inductive) and kept constant, while ς1 increases by taking positive values. In this case the band stop frequency is shifted to the right and we observe a better quality factor.

Figure 5.

Amplitude of the reflection coefficient versus the frequency for different values of η1 while η2 is held fixed.

[20] Figure 6 displays the variation of the reflection coefficient versus ς1. When ς1 (>0) increases, the reflection coefficient tends to unity and no transmitted field is observed. For fixed ς2 and for ς1 < 0, full transmission occurs for a specified frequency and for certain values of ς1.

Figure 6.

Amplitude of the reflected field versus ς1 = Im(η1) for the different values of kl.

[21] The effect of the width for the impedance loadings on the reflection coefficient is shown in Figure 7. 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 (28a), when kl ≫ 1.

Figure 7.

Amplitude of the reflection coefficient versus the frequency for different values of the length of the impedance surface, l.

[22] Finally, Figure 8 depicts the variation of the reflected field amplitude for different values of distance between the parallel-plates. it is observed that the amplitude of the reflected field decreases when the separation distance b increases, as expected.

Figure 8.

Amplitude of the reflection coefficient versus the frequency for different values of the distance between the waveguide plates.

[23] For the special case where η1 = η2 = η, the scalar kernels M1(α) and M2(α) reduce to

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and we have

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In this special case the reflection coefficient reduces to

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[24] In the Figure 9 a comparison is made with the rigorous Wiener-Hopf solution obtained by Tayyar et al. [2008] for a paralel plate waveguide with oppositely placed rectangular grooves filled with dielectric materials. This configuration may be simulated by the following standard impedance boundary condition

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applied at the opening of the groove, where

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is the normalized impedance. ɛr and μr are the relative constitutive parameters of the material filling the groove, while t denotes the groove's depth [Senior and Volakis, 1995].

Figure 9.

Comparison of the results with the first-order impedance approximation for dielectric filled grooves in a parallel plate waveguide.

6. Concluding Remarks

[25] In the present work the band-stop filter characteristics of a parallel plate 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 rigorously by using the pole removal method. It is observed that when one of the impedances is capacitive (ζj > 0j = 1 or j = 2) and takes large values, full reflection occurs. This configuration may be used as a band-stop filter if one of the impedances is inductive.

[26] When we let l → ∞, the reflection coefficient in (24a) and (24b) reduces to

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with

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where F2+(βm) and F1+(vm) are to be solved through

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This is the nothing but exactly the result obtained previously by Büyükaksoy et al. [2006] for the junction of perfectly conducting and impedance parallel plate semi-infinite waveguides (see Figure 7).

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