# Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering
## 2 Spectral Considerations of Sierpinski Resonators
\begin{table}
\begin{tabular}{c c c} \hline \hline \begin{tabular}{c} **Resonance** \\ **Frequency for C0** \\ **(MWOffice)** \\ **[GHz]** \\ \end{tabular} & \
\begin{tabular}{c} **Resonance** \\ **Frequency for C0 from Ref.** \\ **[12]** \\ **[GHz]** \\ \end{tabular} &
\begin{tabular}{c} **\% Error on the C0 Frequency** \\ **of Resonance** \\ \end{tabular} \\ \hline
1.290 & 1.280 & 0.008 \\ \hline
2.255 & 2.242 & 0.006 \\ \hline
2.615 & 2.550 & 0.025 \\ \hline
3.440 & 3.400 & 0.012 \\ \hline
3.850 & 3.824 & 0.007 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequencies of resonance, in GHz, for the C0 configuration, comparing the experimental results of [12] with the simulations obtained using CADENCE Microwave Office (MWOffice) release 22.1. Using a CPW to obtain a triangular resonator implies an excitation obtained by side-coupling the triangle with the central conductor and a ground plane surrounding the geometrical figure. A CPW allows for an in-plane electric field, and it is convenient for filters to be manufactured on a stack of non-interacting planes. A preliminary evaluation of the C0 structure using the same RO5870 substrate was attempted with a CPW designed to have a 50-ohm transmission line exciting the resonator again. For this purpose, the CPW structure was composed of a central conductor 1200 \(\upmu\)m wide with a gap 50 \(\upmu\)m wide. The  
Figure 4: Excitation by microstrip of the C0 configuration using a corner of the triangle (**a**), and its spectral response using the reflection parameter \(\mathrm{S}_{11}\) (**b**). The number “1” in (**a**) is the port number for the simulation and the triangle is the ground reference, respectively. The meshing used for the simulation is also shown in (**a**). gap size was selected to be equal to the structures that were successively manufactured on silicon, placing the distance between the triangle side and the lateral ground as the CPW gap. At the same time, the width of the central conductor was scaled to account for a 50-ohm line. The resonator was symmetrically placed in the CPW to improve the efficiency of the electrical coupling, as discussed later. In Figure 5, the designed structure and its spectral response to the reflection scattering parameter S\({}_{11}\) are plotted. Analyzing Figure 5, it turns out that the excited spectrum is similar to that obtained with microstrip feeding, except for two major differences: (i) the frequencies of resonance are shifted up or down, and (ii) a good value for the depth of the peak is obtained, but an impedance transformation is needed at the I/O ports to improve the zero-level response of the device. The experienced changes in the frequencies of resonance in comparison with the microstrip configuration, using data from Table 1 and the following Table 2, are difficult to categorize as a simple up-shift or down-shift, because the excitation configuration can have an important role in the manifestation of specific resonance modes. A more detailed theoretical electromagnetic analysis, eventually followed by purposely designed prototypes, is reserved for further contributions to the same topic, studying from theoretical and experimental perspectives the critical role of the feeding in the effective excitation of the CPW Sierpinski resonators. For now, we designed and manufactured devices only with the choice of side coupling according to the typical CPW layout. In fact, except for the first two modes, downshifted by approximately 320 MHz compared to those excited by a microstrip, the following three modes are not comparable in an easy way. This finding suggests that an equivalent circuit and an electromagnetic formulation of the triangular patches must be re-calculated to have a reliable theory to predict the frequencies of resonance when a CPW configuration feeds the triangular resonator. Figure 5: CPW excited C0 resonator (**a**) and its microwave response in reflection (S\({}_{11}\) parameter) (**b**).