## 3 Filter Design  
With the purpose of designing a band-pass filter having the same size as the unit-cell resonator, the propagation mode paths along the R-X-M triangle and predicted by the dispersion diagram could be followed. The frequency of the modes propagating along the edges of the Brillouin triangle have to be taken into account to define the filter passband, whereas the band-gap can be used to achieve the filter stopband. The computed dispersion diagram shows the propagation of the first mode in the frequency range from 5 to 6.6 GHz. In the present work, only the (M-R) region and part of the (X-M) region from the dispersion diagram were considered due to the fact that the authors' goal was to obtain a selective band-pass filter. The distance between the metallization edge and the unit-cell edge influences the band-gap position. More precisely when the mentioned distance increases, the band-gap shifts to a higher frequency band whereas its width increases. The variation of the \(a2\) and \(a3\) parameters (see Figure 2) has the following effect: as \(a2\) and \(a3\) increase the band-gap shifts to a lower frequency band whereas the width of the band-gap increases. When \(a1\) and \(a4\) parameters decrease, meaning that the whole unit-cell size decreases, the band-gap shifts to a higher frequency band and its width increases. The thickness of the dielectric substrate also has an influence on the band-gap position. Increasing the thickness, the band-gap shifts to a lower frequency band and in the same time its width increases. The filter consists of a unit-cell resonator and two narrow lines coupled to the resonator. Each narrow line has a width of 0.1 mm and a length of 11.52 mm. The two narrow lines are placed symmetrically with respect to the unit-cell resonator. The gap between the coupling narrow line and unit-cell resonator is 0.1 mm. The filter is excited by a pair of non-orthogonal input/output 50 \(\Omega\) microstrip feeding lines of 1.8 mm width and is printed on an ARLON25N substrate with a thickness  
Figure 1: (**a**) Unit-cell resonator. The applied boundary conditions with the irreducible Brillouin triangle; and (**b**) Dispersion diagram of the periodic structure. of 30 mils, relative dielectric permittivity of 3.28, cooper thickness 18 um and loss tangent 0.0025. The layout of the filter in 6.4 GHz frequency band is presented in Figure 2.  
In order to determine the characteristics of the BPF, two approaches could be followed. The first one is to model the BPF using its equivalent circuit (or at least a simplified version) as can be seen in Figure 2. The resonator unit-cell can be model based on transmission line theory [25] providing that the unit-cell size is electrically small enough. The presented unit-cell is \(\lambda_{0}\)/5 so it can be considered not so small and being at the limit of application of such a model. In any case the model would comprise a series connection (L\({}_{2}\) and C\({}_{2}\)) in parallel with the equivalent impedance of the parallel L\({}_{3}\) and C\({}_{3}\) components. In series with the mentioned circuit the capacitance C\({}_{1}\) (formed between the resonator and each narrow coupling line) and L\({}_{l}\) and R components (modeling the transmission line) are placed. The second approach consists in explaining the physical phenomena under the filter behavior. To achieve this aim the modes propagation in the structure should be explained based on dispersion diagram so the second approach is followed. The dimensions of the filter are 29.8 \(\times\) 11.52 \(\times\) 0.762 mm\({}^{3}\) (0.64\(\lambda_{0}\)\(\times\)0.25\(\lambda_{0}\)\(\times\)0.016\(\lambda_{0}\), \(\lambda_{0}\) is the free space wavelength, \(\lambda_{0}\) = 47 mm at 6.4 GHz) considering the input/output feeding lines and 11.92 \(\times\) 11.52 \(\times\) 0.762 mm\({}^{3}\) (0.25\(\lambda_{0}\)\(\times\)0.25\(\lambda_{0}\)\(\times\)0.016\(\lambda_{0}\)) without the input/output feeding lines.