# Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering
## 2 Spectral Considerations of Sierpinski Resonators
The same arrangement was used for the Sierpinski triangle with complexity C1; the results are shown in Figure 2. In both cases, we used, for the simulation performed using the CADENCE Microwave Office Software release 22.1, the same geometry and materials considered in the 1987 paper regarding the spectrum of an equilateral triangular and the influence of effective values for the dielectric constant and the edge side [12]. In the paper, the conclusion was that the theory matches the experiment when an effective side length is considered without changes in the formulas for the dielectric constant. Then, the substrate was the Rogers product RO5870 with \(\varepsilon=2.32\), and a thickness of \(t=1590\)\(\upmu\)m.  
The 50-ohm microstrip was designed with a width of \(W=4700\)\(\upmu\)m. The substrate is grounded, and the input/output port is referred to that ground. The spectral response obtained for C0 and C1 by the simulation is shown in Figure 3, where the simulated scattering (S) parameter in the reflection, i.e., the matrix element \(S_{11}\), is plotted on a dB scale. Figure 3: Spectral response of the triangles C0 and C1 excited by a short microstrip connected to the edge of the triangles. The dark curve is for C0 and the red one is for C1. Figure 2: First-order Sierpinski triangle excited by a microstrip and the meshing used for the simulation. The number “1” and the downward external small triangle are the port number for the simulation and its ground reference, respectively. From the spectral analysis, we can argue two main facts: (a) the excitation of the modes is much more efficient for the full triangle C0 than the C1 configuration, and (b) the excited modes for the C0 resonator agree very well with the measured ones in [12], as shown in Table 1.  
For an error not exceeding 2%, calculating the quantity \(\mathit{err}=\bigl{|}f_{sim}-f_{exp}\bigr{|}/f_{exp}\) is evaluated, where \(f_{sim}\) is the simulation result and \(f_{exp}\) is the experimentally measured frequency of resonance from [12]. The third mode gives the worst case, but it is the one excited with the lowest efficiency. The interpretation of the resonance modes is more difficult when we analyze the C1 configuration. In the same frequency range used for simulating C0, only two modes are clearly visible for C1, and a third one is depressed by the proximity with a more intense mode, around 4.56 GHz. It should be stressed that this comparison was performed by changing the excitation system of the triangle with respect to [12], but a full agreement was obtained for C0. To complete the analysis, the following Figure 4 shows how the position of the excitation point of the C0 structure, placed on the corner of the triangle, affects the resonance. The spectrum is almost the same but with a less efficient excitation of the resonance modes and broader peaks because of the utilization of a discontinuity point to feed the resonator. There are a wide range of instances in the literature regarding Sierpinski fractals for antennas, but most of the contributions are, as in our case, the result of systematic simulations and experimental efforts to reconstruct the spectrum. A good overview of the guidelines that should be followed for designing a fractal antenna based on Sierpinski geometry can be found in [18; 19], also inspired by the seminal review in [20]. The papers suggested a fitting procedure using electromagnetic simulations and experimental measurements. The authors of [18; 19] were inspired by the necessity to have formulas for practical purposes, and the simulation and experimental efforts produced clear guidelines for antennas in a microstrip-fed structure. As a result, valuable formulas have been published for this specific configuration to provide a reliable numerical approach to the design of triangular antennas. Again, this is a different finding than our efforts because triangular antennas were studied in those papers and fed using a microstrip.