# Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering
## 3 Design of the Single 20 GHz Resonators
The same footprint, reshaping the internal geometry, i.e., changing the level of complexity for the resonator, allows for tuning of the frequency around a chosen value obtained using an electromagnetic simulation. The predicted value for the frequency of resonance \(F_{resonance}\), from the simulations, is given in Table 3, from which, as well as from Figure 7, it is not clear to obtain a law for \(F_{resonance}\) as a function of the internal complexity. Figure 6: Symmetric (**a**) and asymmetric (**b**) configurations of a triangular patch resonator in the CPW configuration, and comparison (**c**) between the expected performances of the transmission S\({}_{21}\) parameter of both structures in dB scale. It is worth noting that, despite the coplanar configuration, and the impossibility of directly using Equation (1), the expected value for \(F_{resonance}\) is now in agreement with the aim of our work to have resonance frequencies close to 20 GHz. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Frequency of Resonance (\(F_{resonance}\)) [GHz]** & \\ \hline
**C0** & **C1** & **C2** & **C3** \\ \hline
20.160 & 20.950 & 19.670 & 19.610 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Predicted frequency of resonance, in GHz, for the 20 GHz Sierpinski resonators. Figure 7: The Sierpinski structures ((**a–c**) above) and the simulated transmission parameters ((**d**), \(S_{21}\) in dB scale) for the C1, C2, and C3 resonators (below), compared with the response of the reference structure C0. Based on the previous considerations, the correct frequency of resonance can be determined by calibrating the design procedure in conjunction with the experimentally measured value of resonance for the full triangle C0 and the Sierpinski ones. Other resonances in the spectrum will also depend, in terms of position and amplitude, on the feeding system and the coupling mechanism. In principle, a connection between different resonators allows for the selection of adjacent resonance frequencies. A non-conventional way to use the internal complexity is obtained using different triangles symmetrically placed on the two sides of the CPW, as shown in Figure 8, where the structure C0C1 is designed with a C0 triangle on the upper side and a C1 one on the lower side. This arrangement provides an intermediate resonance frequency between C0 and C1. Owing to the small difference in the resonance frequencies for C0 and C1, one of the natural consequences is the bandwidth enlargement of the filter. The same occurs if the complexity increases and the bandwidth is enlarged by increasing the number of internal triangles. Such a finding is clearly expressed in Table 4, where the 3-dB bandwidth (from the negative peak value) is shown together with the frequency of resonance for the two simulated structures; i.e., C0 and C0C1. It is worth noting that several resonance modes are expected to be excited, not only the main one, as it occurs for every resonator, but it is not easy to predict at what frequency it will occur nor the amplitude or the electrical matching for each of them. For this reason, it could be necessary to redesign the coupling network more efficiently if high-order modes are required. Then, despite the prediction of the resonance frequency of the main mode, which is not far from the expected value, the question of the effective excitation of the secondary modes and the prediction of the resonance frequency for the Sierpinski triangles with higher internal complexity remains open. Even the main mode is naturally affected by a shift with respect to the predicted one, but this is understandable considering the presence of a coupling network and its contribution to the equivalent circuit of the complete structure. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **C0** & **C0C1** \\ \hline \(F_{resonance}\) [GHz] & 20.16 & 20.95 \\ \hline Bandwidth [GHz] & 0.096 & 0.178 \\ \hline \hline \end{tabular}
\end{table}