# MEMS-Switched Triangular and U-Shaped Band-Stop Resonators for K-Band Operation
## 3 Sierpinski Resonators  
Triangular Sierpinski resonators were considered for their advantages in tuning capability by fixing the footprint of the resonator and increasing the internal complexity, i.e., changing the number of sub-triangles starting from the original full metal patch. In addition, it is easy to combine them in multi-resonant structures, coupling a few resonators in non-conventional geometries. As anticipated in the introduction, it is also interesting to study this geometry from a fundamental point of view, because a detailed theory of the resonant frequencies considering the CPW feeding of the resonators is not yet available for microwave signal processing. The resonators were fully embedded in a CPW boundary, and it is worth noting that this structure, as well as the U-shaped resonators discussed in the next section, is part of the more general family of metamaterial or fractal geometries corresponding to the additional properties in signal propagation [29; 30]. They are characterized by the exhibition of an effective dielectric constant and a magnetic permeability that are both negative, contributing to backward radiation for antennas and significant changes in the propagation properties for planar components. Results and comments about these properties in the studied geometries can be found in [24], or in more general literature such as [25; 31; 32]. In our work, we preliminarily studied a few triangular Sierpinski resonators and selected the configurations promising for filtering adjacent frequencies for fine-tuning. Starting from the full triangular patch, we arrived at the third level of internal complexity determined by the number of sub-triangles, and we defined them as C0, C1, C2, and C3. Single resonator performances are not necessarily the best choice when combining two or more geometries into a complex one. For this reason, after considering the results of the single resonators, we manufactured and tested, for immediate purposes, a structure including the C0 and C1 triangles, i.e., the entire patch and the first level of complexity for the Sierpinski structure working at 20 GHz. At the same time, the C3 and the C2C3 mixed configurations were studied for the 26 GHz device. When we say \"C2C3\", one of the triangles followed the C2 complexity, while the mirrored triangle was designed with the C3 geometry. The mirrored resonators in CPW configurations must have symmetric structures along the central conductor of the CPW and boundaries close to the ground to improve the electrical matching conditions. Primary considerations about the coupling issues and the spectrum reconstruction for CPW-fed Sierpinski structures were discussed in [24]. Figure 5: Experimental response of the RF MEMS SPDT. Reflection (S\({}_{11}\)) and transmission (S\({}_{21}\)) response, with port 3 in isolation. All the used configurations, including the mixed one, namely C2C3 for 26 GHz, are represented in Figure 6. The difference in size was due to the two different working frequencies for the devices studied. The individual response of the resonators was already studied in [24]. The approximated value of the wanted resonance frequency was obtained, accounting for the electromagnetic and experimental trials. Analytical tools are inadequate for predicting resonance frequency, because the theory found in the literature was developed for via-hole or microstrip-fed antenna structures. The CPW environment embedded the resonators completely, and this required a more complicated approach to derive the expected frequency of  
Figure 6: Resonator configurations are used for the 20 and 26 GHz operations. The length of the edge of the individual triangles is 4 mm for the 26 GHz and 6 mm for the 20 GHz operation, respectively. It is worth noting that the configuration C2C3 for 26 GHz is mixed, with the upper part obtained with the internal complexity “C2”, while the lower triangle follows the “C3” complexity. the resonance analytically for the triangular patch and Sierpinski triangles obtained with empty sub-triangles.