# Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering
## 3 Design of the Single 20 GHz Resonators  
Single Sierpinski microwave resonators were designed in a coplanar waveguide (CPW) configuration for band-stop filtering to take advantage of the confinement of the electromagnetic field mainly in the plane of the wafer, which allows the studied structures to be also suitable for realization on multiple planes without mutual interference. The simulations were performed with the CADENCE-AWR Microwave Office Software release 22.1. The CPW symmetry, with one central conductor and two lateral ground planes, imposes a symmetrical arrangement of the resonating structure, which is the main difference for the same resonators excited by microstrip transducers. The above consideration is demonstrated in Figure 6, where single and doubled patch triangles were simulated for the 20 GHz operation. The best impedance matching (with respect to the 50 \(\Omega\) I/O lines) is displayed by the red curve, related to the transmission parameter S\({}_{21}\) of the symmetric structure; i.e., when the peak of the band-stop filter is deeper. The CPW was designed as a wideband transmission line with the central conductor 80 \(\upmu\)m wide and the slot 50 \(\upmu\)m wide. The choice for the side length of the triangle was initially made in agreement with Equation (1), accounting for the necessary recalculation of the geometrical side length \(a\) from the effective one, in order to obtain the envisaged frequencies. On the other hand, this equation was unsuccessful in having a good prediction of \(F_{resonance}\) for the case of the CPW-fed resonators, even considering that the CPW has been manufactured onto an oxidized high resistivity Silicon wafer, and accounting for a dielectric constant value of approximately \(\varepsilon=5.5\) for a wide frequency range, as obtained in [21]. For this reason, we tried values of \(a\) busing electromagnetic simulations to obtain operating frequencies close to the desired ones, choosing \(a(20\ \mathrm{GHz})=6\ \mathrm{mm}\) and \(a(26\ \mathrm{GHz})=4\ \mathrm{mm}\). As a result, the C0 configuration was simulated almost in agreement with the desired value of \(F_{resonance}\), while we saw that higher frequencies needed further elaboration to improve both the prediction of the resonance and the selectivity of the resonating structure. An additional comment must be made regarding the symmetry of the boundary conditions. As has been shown in [9], the separation between the resonator and the central conductor of the CPW, as well as the distance between the edges of the triangle and the ground plane, must be equal to provide a symmetric boundary for the resonator, favoring a simpler design procedure. An additional consideration is related to the fact that symmetry must also be used for the boundaries, as demonstrated in [9], where the peak of resonance for a 20 GHz resonator is improved by 5 dB when passing from 25 \(\upmu\)m to 50 \(\upmu\)m of separation between the triangle edge and the ground plane of the CPW. Following these preliminary considerations, conventional and non-conventional Sierpinski structures have been designed by increasing the internal complexity or combining different levels of complexity. Figure 7 shows the structures and the expected responses for C1, C2, and C3 compared with the reference C0 around 20 GHz. \begin{table}
\begin{tabular}{c c} \hline \hline
**Frequency of Resonance for C0** & **Frequency of Resonance for C0** \\ (Microstrip) & **(CPW)** \\
**[GHz]** & **[GHz]** \\ \hline
1.290 & 0.970 \\ \hline
2.255 & 1.930 \\ \hline
2.615 & 2.915 \\ \hline
3.440 &? \\ \hline
3.850 & 3.925 \\ \hline
4.660 & 4.870 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Frequencies of resonance, in GHz, simulated by Microwave Office for the C0 configuration, and the corresponding resonances obtained for the CPW-fed structure. Only an approximate agreement can be obtained. Analysis of the expected performance of the above resonators allows for important conclusions.