## 2 Problem Formulation
The electromagnetic field distribution inside the resonator depends on its shape, configuration, dimensions, and the particular resonant mode being excited. These modes are classified into the transverse electric (TE) and transverse magnetic (TM) types [5]. Furthermore, the materials used in the composition of the resonator exhibit electromagnetic properties that significantly affect wave propagation and, consequently, impact the overall performance of patch antennas [18; 19; 20]. For example, [PERSON] et al. [21] proposed a SPIDA antenna embedded in dielectric waveguides, achieving size reduction, multi-band operation, and radiation pattern modification to enable beamforming through frequency tuning. Furthermore, [PERSON] and [PERSON] [22] explored the application of resonators to enhance antenna performance, demonstrating that combining tapered dielectric resonators with band notches effectively improves ultra-wideband antenna characteristics. They also showed that an inverted conical resonator significantly enhances antenna gain and impedance bandwidth. Moreover, [PERSON] and [PERSON] [23] proposed a compact water patch antenna that incorporates an annular water ring beneath the patch, resulting in a reduced center frequency and overall antenna size. Their results confirmed a stable radiation pattern and wide bandwidth. The authors of [24] introduced a dielectric waveguide patch antenna fed by an L-shaped metallic probe and enclosed in a transparent Plexiglass container. This design achieved a wide impedance bandwidth, high gain, excellent radiation efficiency, and a broadside radiation pattern across the entire bandwidth. [PERSON] et al. [25] introduced a hybrid resonator utilizing distilled and saline water, achieving size reduction, wide bandwidth, and high efficiency. [PERSON] et al. [26] developed a wideband circularly polarized antenna incorporating liquid ionic resonators. Their experimental validation showed a significant increase in bandwidth while maintaining a compact design. Additionally, [PERSON] et al. [27] introduced a single-fed, dual-band, circularly polarized reconfigurable liquid dielectric resonator antenna fabricated using 3 D printing technology. The measured results validated its dual-band reconfigurability and wide bandwidths. Furthermore, [PERSON] et al. [28] proposed a trapezoidal dielectric resonator antenna for millimeter-wave applications, which was designed to generate multiple frequency bands while maintaining a wide BW. The literature has shown that introducing resonators with a specific dielectric constant and geometry around a patch antenna can bring some benefits. Typically, resonators with higher dielectric constants reduce the resonant frequency by modifying the wavelength of nearby electromagnetic waves. Furthermore, resonators can broaden or narrow the BW, depending on its influence on impedance matching and radiation efficiency. High-loss dielectrics introduce additional losses and decrease BW, while low-loss materialsenhance impedance matching and may increase BW [18; 19; 20]. Moreover, embedding a patch antenna within a resonator with a specific combination of material and geometry can significantly modify \(S_{11}\) and the radiation pattern (e.g., BW, effective gain, directionality, and resonant frequency). Looking at all the papers discussed above and references therein on patch antennas embedded in resonators made of dielectric and conductive materials, we see that most studies focus on specific geometric shapes and materials under distinct design specifications and constraints, making it difficult to assess their benefits based on well-established performance criteria comparatively. Given the advantages that resonators offer to patch antennas, there is a clear necessity to provide a comprehensive analysis of how the combination of dielectric and conductive materials with different geometries to form a resonator influences the performance of a patch antenna.