## 5 Conclusions  
This paper provided a comprehensive study on patch antennas embedded in dielectric and conductive material-based resonators, examining their effects on key performance parameters such as impedance matching, bandwidth, radiation efficiency, and gain. The study systematically investigated fifteen distinct geometric resonator shapes composed of distilled water, glycerin, and graphite, providing valuable insights into their influence on antenna behavior.  
The numerical results highlighted that specific resonator geometries, particularly combined with a high dielectric constant material, effectively enhance bandwidth, improve impedance matching, and increase radiation efficiency. We also observed that certain geometries allow multi-band operation due to multiple impedance matching points. Conversely, the study also identified cases where the inclusion of a resonator does not yield a substantial performance enhancement. In a few cases, it even degrades the performance of a patch antenna. Specifically, some configurations exhibited degraded impedance matching, while others led to shifts in the resonant frequency without significant bandwidth or radiation efficiency improvements. These findings emphasize the importance of careful material and geometric selections for achieving meaningful performance gains.  
Overall, this paper contributes to advancing patch antenna design by offering insights and guidance for selecting one of the fifteen geometric shapes and one of the three materials (distilled water, glycerin, and graphite) for embedding a patch antenna into a resonator. The discussions are particularly relevant in scenarios requiring compact designs, enhanced efficiency, or multi-band operation. Future work may focus on other materials, an array of patch antennas, experimental validation, and further resonator-container fabrication combinations to address practical implementation challenges.  
Conceptualization, C.M.S.F.; Data curation, C.M.S.F. and C.M.M.A.; Formal analysis, C.M.S.F., L.G.S.C., B.R.D. and M.V.R.; Investigation, C.M.S.F. and C.M.M.A.; Methodology, C.M.S.F., C.M.M.A., L.G.S.C., B.R.D. and M.V.R.; Supervision, M.V.R.; Validation, B.R.D., M.A.O.S., G.L.R. and M.V.R.; Visualization, C.M.S.F., C.M.M.A. and L.G.S.C.; Writing--original draft, C.M.S.F., C.M.M.A. and L.G.S.C.; Writing--review and editing, B.R.D., M.A.O.S., G.L.R. and M.V.R. All authors have read and agreed to the published version of the manuscript.  
This research was supported in part by Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES) under Grant 001; Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) under grants 404068/2020-0, 314741/2020-8, and 445958/2024-3; Fundacao de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG) under grants APQ-03609-17 and APQ-04623-22, and Instituto Nacional de Energia Eletrica (INERGE).  
Not applicable.  
All data supporting the results reported in this study are provided in the Appendix of this manuscript.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.