Spectral Analysis of Sierpiński Microwave Resonators

Triangular Sierpiński structures have been theoretically studied for microwave applications around the lower edge of the X-band (8-12 GHz). In detail, we present electromagnetic simulations for microstrip-fed resonators on a silicon substrate, encompassing: (i) Equilateral triangles and the spectrum of high-order modes; (ii) Classical and internally stretched Sierpiński triangles; (iii) Cascaded and mixed triangular resonators. We compared analytical results with electromagnetic simulations to reconstruct the spectrum of full, non-subdivided triangles and extended them to predict the resonance modes of Sierpiński structures. For a 6 mm edge length, the analytically predicted fundamental resonance of the full triangle (9.42 GHz) agrees with the simulated one (9.58 GHz) within 2%. The first level of internal subdivision causes a coarse frequency jump of about −17% (to 7.96 GHz), while further subdivisions (C2, C3) provide a fine, nearly linear tuning of about 130 MHz. Cascading two resonators deepens the notch to −27 dB, mixing triangles of different complexity along the feed line produces multiple closely spaced resonances, and stretching the internal triangle splits the C1 resonance into two modes, enabling dual-frequency operation.