## 4 Analysis of Numerical Results
### General Comments  
The numerical results analyzed in previous sections reveal that performance can significantly vary with the choice of material and geometry of the resonator when the performance evaluation parameters (e.g., \(|S_{11}|\), BW, and MLM) are considered. Specifically, the graphite resonators exhibit the highest number of geometric shapes with better MLM compared to the patch antenna without a resonator, followed by DW resonators and the glycerin resonators, respectively. Among the combination of material and geometries with MLM exceeding that of the patch antenna without a resonator, the combinations using DW attain the majority of acceptable \(|S_{11}|\) values, followed by those using graphite, and, finally, those using glycerin. Moreover, when combined with the appropriate geometry, each resonator material enhances the radiation pattern. Several of the evaluated resonators achieved MLM values exceeding 5 dBi, indicating improved radiation efficiency. Moreover, certain resonators show reductions in HPBW and SLL, improving antenna directionality. For instance, the inverted cone resonator with DW material achieves the highest improvements in terms of MLM and interesting reductions in HPBW and SLL, compared to the patch antenna without a resonator. Furthermore, this resonator maintains an acceptable \(|S_{11}|\) at the resonant frequency of 2.45 GHz. Similarly, the DW material in the LPA hollow inverted prism and inverted prism geometric shapes also achieves MLM values greater than 5 dBi, provide significant reductions in terms of HPBW and SLL, resulting in improved directionality while maintaining acceptable \(|S_{11}|\). Resonators based on glycerin material also achieve MLM values exceeding 5 dBi, indicating enhanced radiation efficiency. Among these, the concave hemisphere exhibits the highest MLM, followed by the LPA inverted trapezoidal prism. Furthermore, use of this material improves antenna directionality due to the attained reductions in HPBW and SLL. Moreover, these resonators maintain acceptable \(|S_{11}|\), with the LPA inverted trapezoidal prism showing superior performance. Using graphite results in resonators that are slightly outperformed by DW and glycerin resonators. When the hollow inverted cone is considered, its use achieves an MLM greater than 5 dBi. This resonator also achieves reductions in HPBW and SLL while maintaining an acceptable \(|S_{11}|\). As a result of these characteristics, the antenna performance is enhanced. Using resonators is an effective approach to enhancing some performance parameters of a patch antenna; however, it can also degrade certain others. For instance, a hollow rectangular prism can influence the performance of a patch antenna differently depending on the material used, and when made of DW, it improves the SLL but negatively affects other parameters. Conversely, when filled with glycerin, it enhances only the HPBW, whereas, with graphite, it preserves the SLL and MLM while also improving the HPBW. Similarly, depending on the material, the WPA hollow inverted half-cylinder resonator exhibits different effects. When constructed from DW or glycerin, it improves only the HPBW. However, when made from graphite, it enhances the MLM, HPBW, and SLL