## 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 *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. It is assumed that Re*Z*_{1,2} > 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.

β_{m} | ν_{m} |
---|---|

k = 1 + i0.01 | |

β_{1} = 0.90634 + i0.01006 | ν_{1} = 0.80580 + i0.01031 |

β_{2} = 0.00144 + i5.65590 | ν_{2} = 0.00112 + i5.68768 |

β_{3} = 0.00071 + i11.3960 | ν_{3} = 0.00055 + i11.4119 |

β_{4} = 0.00047 + i17.1173 | ν_{4} = 0.00037 + i17.1279 |

β_{5} = 0.00035 + i22.8340 | ν_{5} = 0.00027 + i22.8419 |

k = 2 + i0.01 | |

β_{1} = 1.91031 + i0.01002 | ν_{1} = 1.82292 + i0.01010 |

β_{2} = 0.00336 + i5.41755 | ν_{2} = 0.00300 + i5.48303 |

β_{3} = 0.00161 + i11.2797 | ν_{3} = 0.00144 + i11.3118 |

β_{4} = 0.00106 + i17.0401 | ν_{4} = 0.00095 + i17.0614 |

β_{5} = 0.00079 + i22.7762 | ν_{5} = 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.