## 3 Filter Design
From simulation results in Figure 3a, the 3 dB passband of BPF without harmonic suppression goes from 6.25 to 6.62 GHz, meaning 5.75% fractional bandwidth at the center frequency 6.44 GHz. The minimum insertion loss is 1.7 dB whereas the maximum return loss value is greater than 17 dB (see Figure 3b). Figure 3: **(a)** Simulation _vs_. measurement: Transmission coefficient of the BPF; **(b)** Simulation _vs_. measurement: Reflection coefficient of the BPF. Figure 2: Layout of the band-pass filter (BPF) with its characteristic circuit model. Taking into account the position of the feeding lines with respect to the unit-cell geometry, the propagation follows a quasi-diagonal path, which coincides with the (M-R) region from the dispersion diagram as being together with part of the (X-M) region. Furthermore, from the (M-R) region, as well as part of the (X-M) region of the dispersion diagram resulted in the first mode propagating in a frequency spectrum of between 6 and 6.6 GHz. This band corresponds to the electromagnetic wave propagation through the BPF geometry whereas from 6.94 to 7.9 GHz the structure does not allow any mode propagation which corresponds to the stopband frequency region of the BPF. In Table 1, the 3 dB bandwidth together with the quality factor are presented. In a band-pass filter, the overall width of the passband between the upper and lower 3 dB levels of the filter determines the quality factor Q. The lower the value of the Q factor, the wider the bandwidth and, consequently, the higher the Q factor, the narrower and more selective the filter. According to the analysis shown in Table 1, if three resonators are cascaded in the OY direction (henceforth referred to as 3-cells-oy BPF) the 3 dB passband goes from 6.45 to 6.61 GHz, meaning 2.45% fractional bandwidth at the center frequency 6.52 GHz, whereas in the OX direction (filter henceforth referred to as 3-cells-ox BPF) the 3 dB passband goes from 6.55 to 6.67 GHz, meaning 1.8% fractional bandwidth at the center frequency 6.62 GHz. Moreover, using three cells in the direction of the current flow, the filter becomes more selective (see Figure 4a). The increment in the number of unit-cells in the OY direction has an influence only in the passband of the filter, which becomes narrower; meanwhile the slope in the stop bands remains the same. The benefit of the novel BPF over the square shaped resonator BPF is a quality factor three times greater. Figure 4b shows that the return loss increases with the number of unit-cells. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{Prototype} & \multicolumn{2}{c}{3 dB} & \multirow{2}{*}{\(f_{\mathrm{c}}\)} & \multicolumn{2}{c}{BW(3 dB)} & \multirow{2}{*}{Q} & \multicolumn{2}{c}{Size} & \multirow{2}{*}{Size} \\ \cline{2-2} \cline{5-6}  & \(f_{\mathrm{low}}\)(GHz) & & \(f_{\mathrm{high}}\)(GHz) & & & GHz & \% & & (mm \(\times\)mm) & (mm \(\times\)mm) \\ \hline BPF & 6.25 & 6.62 & 6.44 & 0.37 & 5.57 & 17.40 & 29.80 \(\times\)11.52 * & 11.92 \(\times\)11.52 \\
3-cells-oy BPF & 6.45 & 6.61 & 6.52 & 0.16 & 2.45 & 40.75 & 29.80 \(\times\)34.56 * & 11.92 \(\times\)34.56 \\
3-cells-ox BPF & 6.55 & 6.67 & 6.62 & 0.12 & 1.81 & 55.16 & 52.84 \(\times\)11.52* & 34.96 \(\times\)11.52 \\ Square BPF & 6.13 & 7.30 & 6.70 & 1.17 & 17.4 & 5.72 & 29.80 \(\times\)11.52* & 11.92 \(\times\)11.52 \\ \hline \hline \end{tabular}
* Considering the input/output feeding lines.  
\end{table} 
Table 1: Simulated band-pass filter quality factor. Figure 4: (a) S21 simulation results; and (b) S11 simulation results. In order to suppress the second harmonic, several techniques have been reported in literature. In [26] two quarter wavelength open ended stubs are attached at the edges of the resonators in the main coupling path in order to have harmonic suppression, but this implies additional circuit elements against compact filter design.