conventional E-plane filter has advantages such as low-loss, low-cost, and mass-produc- tion, but it has a drawback of weak performance in the stop-band, also; the size of filter becomes large at low frequencies. Some studies have been done to enhance rejection band performance of such filters (Arndt et al. 1986; Budimir 1997). In one of them, a few of single metal inserts in the conventional filter, are replaced with multiple metal inserts (Arndt et al. 1986). As another work, extracted pole resonator along with metal-insert E-plane, makes transmission zero in the rejection band, so it improves frequency response selectivity. Band-stop cavities, as extracted poles, in addition to the single, or double ridge E-plane resonators have been used for this purpose (Kozakowski and Deleniv2011; Mansour and Fouladian1989). These structures are not in-line, there is a T-junction whose one port becomes short circuited, to make band-stop cavity. It has been shown that transmission zero also could be made by folding the structure and cross coupling between non-adjacent resonators (Ofli et al. 2005).

By now, a few waveguide band-stop filters have been introduced. The conventional one uses half-wavelength band-stop cavities, which are connected to main waveguide by inductive irises, so it is not an in-line structure, also such a resonators is spaced threequarter wavelength (Matthaei et al. 1980). In another filter, band-reject cavities have been used, but the conventional irises have been replaced by asymmetrical slots (Montejo-Garai et al. 2011).

In this paper, an approach is proposed so that E-plane waveguide band-pass and bandstop filter with desired frequency response characteristics can be designed. In the presented structure, longitudinal patterned planes, which are designed using genetic algorithm optimization, act as resonators or anti resonators in band-pass and band-stop waveguide filters, respectively. Moreover, in the presented method, a filter structure with any arbitrary frequency response characteristics can be designed. The proposed method facilitates and accelerates the optimization process in comparison to simulator software. It can be shown, by using the proposed method; one can design waveguide E-plane band-pass and band-stop filters with shorter length compared with the conventional one. Furthermore, the introduced approach could be helpful to design waveguide filters with higher selectivity features. Moreover, provided that the design procedure is feasible, the presented method can be used to design a two port E-plane waveguide device of any desired frequency characteristic.

Basically, the proposed method is based on finding optimum patterned plane using genetic algorithm (GA), according to a defined cost function. Genetic algorithm finds the optimum pattern whose frequency response can be fitted to that of desired one. For this purpose, a coupled set of electric field integral equations (EFIEs) of an arbitrary pattern are derived and solved by method of moments (MoM) to obtain the scattering parameters of the proposed structure.

2 Design procedure

2.1 Electric field integral equation formulation

Figure 1 shows the proposed E-plane waveguide filter compared with the conventional band-pass one. In the proposed E-plane filter, half-wavelength hollow waveguide sections are replaced by longitudinal patterned planes, which act as resonators. The patterned plane is a metal plane with shorter length than a quarter-wavelength that its determined parts are

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removed. Such a patterned plane can be placed on a thin and low relative dielectric permittivity substrate in order to facilitate the fabrication process.

To design the proposed E-plane waveguide band-pass or band-stop filter, one can use the equivalent circuit approach. Figure 2 depicts a typical band-pass filter using N parallel resonators and impedance inverters. Elements of Li, and Ci (i = 1, …, N) for each parallel resonators and the value of impedance inverters, Ki (i = 1,…, N ? 1), can be obtained by type and specifications of desired band-pass filter and the corresponding relations which can be found in many literatures (Collin 1996).

Fig. 1 a Three dimensional view, b side view of the proposed E-plane waveguide filter, c side view of the conventional E-plane waveguide band-pass filter. Metalized parts are shown as black sections

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468 Page 4 of 13 H. Ghorbaninejad, A. Ghajar

It is aimed to model the proposed waveguide band-pass filter shown in Fig. 1a as one shown in Fig. 2. For this purpose, each LC resonator in the equivalent circuit model is replaced by a introduced longitudinal patterned plane, so that the frequency responses of the proposed longitudinal patterned plane can be fitted to that of the given resonator in the equivalent circuit model. As well as, each impedance inverter is replaced by a hollow waveguide section whose length is odd multiples of quarter wavelength in the center frequency of filter. To fit the transmission coefficient (S21) of the proposed longitudinal patterned plane to that of the circuit model resonator, it is necessary to obtain general scattering parameters of the structure. For this purpose, coupled electric field integral equations in terms of electric currents on metal parts of the patterned plane are derived. According to the equivalence theorem, Fig. 3 shows the equivalent structure to obtain electric and magnetic fields in the waveguide region, in which a longitudinal patterned plane is located.

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In Fig. 3, J is unknown induced electric surface current on the metalized parts of the longitudinal patterned plane. By deriving appropriate dyadic electric Green’s function, tangential electric fields due to electric surface current, were expressed in terms of Eigenmodes series expansion and given by,

where Gij, and i, j = y, z are the electric dyadic Green’s function which are provided in the appendix, also Jy, and Jz are y- and z-directed electric surface currents respectively. Sm is metalized parts of the longitudinal patterned plane.

Boundary condition requires the continuity of tangential electric field, so that the total tangential electric field across the metalized parts of the longitudinal patterned plane must

Fig. 2 Typical band-pass filters using parallel resonators and impedance inverters

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Fig. 3 Equivalent structure to obtain electric and magnetic fields in waveguide region

q

Y10TE ¼ C10=ðjxl0eÞ is propagation constant and wave admittance of dominant TE10 incident mode, respectively. Also k0 is free space wave number. A, introduced in Eq. 5, is normalization factor, and is chosen so that satisfy incident unit power normalization

condition,

Z

2.2 Galerkin’s method of moment solution

In former section, tangential electric fields were expressed in terms of unknown electric currents on the metalized parts of longitudinal patterned plane, and a set of coupled integral equations were obtained. To solve the coupled set of integral equations, unknown electric currents expanded in series of y- and z-directed overlapping sub-domain sinusoidal basis functions:

ks parameter determines the smoothness degree of basis function. By applying Galerkin’s method to each one of the integral equations, a matrix system of linear equation is

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