Genetic algorithm design for E-plane waveguide filters

Abstract

In this paper, a new approach has been proposed to design any arbitrary E-plane filters, including band-pass, and band-stop filters, with a desired frequency response. The proposed method is based on replacing all conventional resonators with a new form of a resonator, which is made of a patterned plane. The patterned plane is a metal plane with infinitesimal thickness that some of its parts, are removed. The introduced patterned plane is supported by a thin and low permittivity dielectric slab, and is located longitudinally in the middle of a rectangular waveguide, parallel to the E-plane. To design the proposed filters, the scattering parameters of the structure should be calculated. For this purpose, a coupled set of electric field integral equations have been derived, and solved by method of moments. Then, a suitable cost function was defined, and optimized using genetic algorithm. MATLAB GA tool has been used to optimize the cost function. The proposed method facilitates and accelerates the optimization process in comparison to full wave simulator software. As examples, both band-pass, and band-stop E-plane filters have been designed. The results show that the proposed filter, in comparison to the conventional filters, has some advantages, such as frequency band selectivity, compactness, and the ability of adjustment to any desired frequency response. The results of designed structures are validated by HFSS simulator software.

Appendix

The electric dyadic Green’s functions derived as follows.

$$G_{yy} = \mathop \sum \limits_{m = 0} \mathop \sum \limits_{n = 0} \frac{{\varepsilon_{m} \varepsilon_{n} }}{{2j\omega \varepsilon_{0} ab\varGamma_{mn} }}\left\{ {\left( {k_{0}^{2} - k_{y}^{2} } \right)\sin \left( {k_{x} x} \right)\cos \left( {k_{y} y} \right)\sin \left( {k_{x} x'} \right)\cos \left( {k_{y} y'} \right)\exp \left( { - \varGamma_{mn} \left| {z - z'} \right|} \right)} \right\}$$

(22)

$$G_{xz} = \mp \mathop \sum \limits_{m = 0} \mathop \sum \limits_{n = 0} \frac{{\varepsilon_{m} \varepsilon_{n} }}{{2j\omega \varepsilon_{0} ab\varGamma_{mn} }}\left\{ {\varGamma_{mn} k_{y} \sin \left( {k_{x} x} \right)\cos \left( {k_{y} y} \right)\sin \left( {k_{x} x'} \right)\sin \left( {k_{y} y'} \right)\exp \left( { - \varGamma_{mn} \left| {z - z'} \right|} \right)} \right\}$$

(23)

$$G_{zx} = \pm \mathop \sum \limits_{m = 0} \mathop \sum \limits_{n = 0} \frac{{\varepsilon_{m} \varepsilon_{n} }}{{2j\omega \varepsilon_{0} ab\varGamma_{mn} }}\left\{ {\varGamma_{mn} k_{y} \sin \left( {k_{x} x} \right)\sin \left( {k_{y} y} \right)\sin \left( {k_{x} x'} \right)\cos \left( {k_{y} y'} \right)\exp \left( { - \varGamma_{mn} \left| {z - z'} \right|} \right)} \right\}$$

(24)

$$G_{zz} = \mathop \sum \limits_{m = 0} \mathop \sum \limits_{n = 0} \frac{{\varepsilon_{m} \varepsilon_{n} }}{{2j\omega \varepsilon_{0} ab\varGamma_{mn} }}\left\{ {\sin \left( {k_{x} x} \right)\sin \left( {k_{y} y} \right)\sin \left( {k_{x} x'} \right)\sin \left( {k_{y} y'} \right)\left[ {\left( {\varGamma_{mn}^{2} + k_{0}^{2} } \right) - \,2\varGamma_{mn} \delta \left( {z - z'} \right)} \right]\exp \left( { - \varGamma_{mn} \left| {z - z'} \right|} \right)} \right\}$$

(25)

where $\varGamma_{mn} = \sqrt {\left( {m\pi /a} \right)^{2} + \left( {n\pi /b} \right)^{2} - k_{0}^{2} }$ is the propagation constant, $k_{0}^{2} = \omega^{2} \mu_{0} \varepsilon_{0}$, $k_{x} = m\pi /a$, and $k_{y} = n\pi /b$. In Eqs. 22, 23, upper and lower sign corresponds to $z \ge z',$ and $z < z'$ respectively and the Neumann factor $\varepsilon_{n}$ is given by:

$$\varepsilon_{n} = \left\{ {\begin{array}{*{20}c} {1, n = 0} \\ {2, n \ne 0} \\ \end{array} } \right.$$