# Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering
## 2 Spectral Considerations of Sierpinski Resonators  
A complete analytical approach to determining the Sierpinski geometry and designing triangles is available in several books, papers, or mathematical packages [13; 14]. Nevertheless, it is a complex task to determine the frequency of resonance when the same triangles are utilized as electromagnetic resonating structures. Only the main resonance frequency \(F_{resonance}\) for equilateral full triangles (or simple geometries such as isosceles and right-angled triangles) was exactly calculated, considering the TM (Transverse Magnetic) mode configuration for the equivalent cavity model. The following formula gives the resonance frequency \(F_{resonance}\) for the main mode of the equilateral triangle [12; 15; 16; 17], as it was also used in [9]:  
\[F_{resonance}=\frac{2c}{3a_{effective}\sqrt{\epsilon_{effective}}} \tag{1}\]where \(c\) is the light speed, and the effective values of the triangle edge \(a\) (_a_,_effective_) and the relative dielectric constant \(\varepsilon\) (_e_,_effective_) might be introduced, as in [12], to account for the needed phenomenological corrections typically adopted for planar high-frequency geometries. It must be remembered that, according to the results in [12], only the effective value of the side length (_a_,_effective_) of the triangle is necessary to have an agreement between the experimental value of the frequency of resonance and the expected one for the main mode given by Equation (1). It has already been discussed in [9] why triangular geometry is advantageous in terms of higher frequencies, array configuration, and footprint compared to circles and squares. For completeness, we recall that the main advantages of using triangles compared to squares are the smaller footprint and a higher resonance frequency. Using basic formulas to determine \(F_{resonance}\), it is expected to be in a ratio of 4/3 using a triangle with the same edge length as a square. At the same time, the area reduction is in the order of 0.4, which is a clear advantage when integrating the resonators into a planar array. Circles are easier to design but difficult to integrate using a side-by-side coupling. The disadvantage with triangles is the necessity for an accurate design accounting for the discontinuities on the corners. An efficient excitation of the resonator's modes implies the proper coupling between the resonator and the signal launcher. Not all the resonance modes can be efficiently excited because the maximum of some modes could be far from the excitation point. How excited should the resonance modes of a triangle be? Many authors have studied triangular antennas with a via hole close to the vertex of the patch triangle, fed by a microstrip lying on the backside of the substrate. The via hole is at a distance \(d\) to the vertex. We underline that there is no published general criterion to precisely determine where the via hole should be positioned depending on the resonance mode to be excited. For this reason, the frequency of resonance of the first few modes has been analytically calculated and experimentally measured in the literature, but the spectrum is rarely shown, probably because the modes are not evidenced due to poor electrical matching. A simple way to feed the triangular resonator by a microstrip on the same substrate where the triangle has been manufactured, instead of using a via hole, is given in Figure 1, where a 50-ohm short line is connected to the edge of the resonator. The conventional labels used in this paper to define different Sierpinski resonators are C0 for the full patch. At the same time, C1, C2, and C3 are concerned with sub-divisions up to the third level of internal complexity. Figure 1: Full equilateral triangle (C0) excited by a microstrip placed in the center of the side and its meshing used for the simulation. The number “1” is the reference port of the simulation, while the external small triangle indicates that the port is grounded.