Fractral theory in filter design

Fractals are geometric figures characterized by self-similarity and complexity across scales. Common examples are Koch curve, Sierpinski gasket and Minkowski island. These structures are recursive and can occupy a large effective electrical length in a compact physical space.Fractals are useful  in microstrip and planar filter design  because, Fractal geometries allow for longer current paths in small areas, reducing the overall footprint of filters.Due to their multi-scale nature, fractal filters can inherently support multiple resonances, useful for designing multi-band filters (e.g., WiFi + LTE).Also, the Recursive and distributed features of fractrals  lead to enhanced coupling and broader passbands. The main advanmtages of fractal based designs are Size reduction, Improved performance at multiple frequencies, Better stopband suppression and Design flexibility.

 The structure is built atop an FR4 substrate with a relative permittivity of 4.4 and an adjustable thickness, typically 1.6 mm. A ground plane lies beneath the substrate at the bottom (z = 0), while the resonator and the microstrip feed line are etched on the top surface. The feed line is designed to have a 50-ohm characteristic impedance, calculated using standard empirical microstrip design equations based on the substrate parameters. It is modeled as a rectangular metallic strip that runs from the left edge of the board and connects directly to the base of the Sierpinski triangle. The fractal resonator is generated using a recursive MATLAB function, which subdivides an equilateral triangle into three smaller corner triangles while omitting the center, with the number of iterations controlling the complexity and multi-band behavior. Both the feed and resonator share the same metal layer, ensuring continuity. A lumped port is introduced at the start of the microstrip line to excite the structure and to measure the reflection coefficient (S11), replacing the earlier probe-based approach for improved realism.

The simulation methodology involves defining the full 3D geometry using OpenEMS's CSXCAD system through MATLAB scripting. All material properties and boundary structures are specified, and the mesh is finely controlled using SmoothMeshLines to ensure numerical stability and resolution suitable for high-frequency operation. The maximum mesh size is determined by the highest frequency of interest to accurately capture field variations. A Gaussian pulse centered at the midpoint between the start and stop frequencies is applied using the SetGaussExcite function to excite the system across the desired frequency range. To mimic an open environment and prevent artificial reflections, Perfectly Matched Layer (PML) absorbing boundary conditions are applied on all sides. The simulation is written to an XML file and executed using the OpenEMS engine. Following the time-domain simulation, port voltage data is extracted, and the reflection coefficient is computed and plotted as a function of frequency.

The output of the simulation is a plot of S11 (in dB) versus frequency, where dips in the curve indicate resonant frequencies of the structure. Due to the fractal nature of the Sierpinski geometry, multiple resonances are observed across the frequency range, confirming its utility for multiband applications such as filters or antennas. This setup, with its realistic microstrip feed, provides a closer approximation to practical PCB implementation and allows for easy scaling and tuning of parameters such as substrate properties, resonator size, and iteration depth.