## 4 Analysis of Numerical Results
### Glycerin Resonator  
In this subsection, we explore the integration of glycerin resonators into the patch antenna by evaluating the performance of different geometric shapes. The \(|S_{11}|\) parameter for cap-based, rectangular prism-based, cone-based, half-cylinder-based, and trapezoidal prism-based geometries are shown in Figure 8a, Figure 8b, Figure 8c, Figure 9a, and Figure 9b, respectively. Furthermore, the corresponding radiation patterns for cap-based, rectangular prism-based, cone-based, half-cylinder-based, and trapezoidal prism-based geometries are shown in Figure 10a, Figure 10b, Figure 10c, Figure 11a, and Figure 11b, respectively. Moreover, Table 5 lists key parameter values obtained at the resonant frequency. Figure 8: Patch antenna with a glycerin-made resonator: magnitude of the scattering parameter (\(|S_{11}|\)). (**a**) cap-based; (**b**) rectangular prism-based; and (**c**) cone-based. Figure 9: Patch antenna with a glycerin-made resonator: magnitude of the scattering parameter (\(|S_{11}|\)). (**a**) half-cylinder-based; and (**b**) trapezoidal prism-based. Figure 10: Patch antenna with a glycerin-made resonator: the radiation pattern plot of the 1D far-field realized gain (\(\phi=0\)). (**a**) cap-based; (**b**) rectangular prism-based; and (**c**) cone-based. The evaluation of the cap-based geometries reveals distinct characteristics in both \(|S_{11}|\) and radiation pattern as illustrated in Figure 8a and Figure 10a, respectively. The cylindrical hemisphere geometry reduces the resonant frequency, showing its potential for reducing  
Figure 11: Patch antenna with a glycerin-made resonator: the radiation pattern plot of the 1D far-field realized gain (\(\phi=0\)). (**a**) half-cylinder-based; and (**b**) trapezoidal prism-based. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multicolumn{1}{c}{**Geometry**} & \multicolumn{1}{c}{\(|S_{11}|\)} & \multicolumn{1}{c}{**BW**} & \multicolumn{1}{c}{**MLM**} & \multicolumn{1}{c}{**HPBW**} & \multicolumn{1}{c}{**SLL**} \\  & **(dB)** & **(MHz)** & **(dBi)** & **(\({}^{\circ}\))** & **(dB)** \\ \hline Patch antenna & \(-34.30\) & \(60\) & \(2.79\) & \(148.2\) & w/o \\ \hline Rectangular prism & \(-7.24\) & w/o & \(3.95\) & \(117.2\) & \(-14.9\) \\ \hline Hollow rectangular prism & \(-1.21\) & w/o & \(-3.82\) & \(140.5\) & w/o \\ \hline Cylindrical hemisphere & \(-3.72\) & w/o & \(2.04\) & \(120.2\) & \(-8.4\) \\ \hline Concew hemisphere & \(-11.48\) & \(54\) & \(5.46\) & \(101.8\) & \(-9.2\) \\ \hline Hollow concave hemisphere & \(-6.19\) & \(11\) & \(3.08\) & \(127.3\) & \(-8.4\) \\ \hline Hemisphere & \(-3.27\) & w/o & \(0.89\) & \(91\) & w/o \\ \hline Inverted cone & \(-21.08\) & \(47\) & \(4.24\) & \(125.2\) & \(-12.4\) \\ \hline Hollow inverted cone & \(-4.68\) & w/o & \(2.69\) & \(139.1\) & w/o \\ \hline Cone & \(-10.57\) & \(27\) & \(3.42\) & \(146.9\) & w/o \\ \hline LPA Inverted half-cylinder & \(-4.39\) & w/o & \(2.18\) & \(139.2\) & w/o \\ \hline LPA hollow inverted half-cylinder & \(-2.34\) & w/o & \(-1.54\) & \(140.7\) & w/o \\ \hline LPA half-cylinder & \(-0.66\) & w/o & \(-4.65\) & \(152.4\) & \(-6.4\) \\ \hline WPA inverted half-cylinder & \(-6.30\) & \(32\) & \(3.19\) & \(120.5\) & \(-10.7\) \\ \hline WPA hollow inverted half-cylinder & \(-13.56\) & \(54\) & \(2.47\) & \(143.5\) & w/o \\ \hline WPA half-cylinder & \(-8.95\) & \(8\) & \(3.28\) & \(144\) & w/o \\ \hline LPA inverted trapezoidal prism & \(-21.20\) & \(45\) & \(5.33\) & \(104.1\) & \(-11.9\) \\ \hline LPA hollow inverted trapezoidal prism & \(-15.98\) & \(31\) & \(4.28\) & \(117.7\) & \(-6.2\) \\ \hline LPA trapezoidal prism & \(-22.81\) & \(57\) & \(2.74\) & \(145.6\) & w/o \\ \hline WPA inverted trapezoidal prism & \(-10.30\) & \(3\) & \(3.37\) & \(120.9\) & \(-7.9\) \\ \hline WPA hollow inverted trapezoidal prism & \(-8.44\) & \(9\) & \(3.63\) & \(119.8\) & w/o \\ \hline WPA trapezoidal prism & \(-9.33\) & \(17\) & \(3.99\) & \(118.6\) & \(-9.5\) \\ \hline \hline \end{tabular}
\end{table}