## 2 Problem Formulation  
Patch antennas are essential in wireless and satellite communication systems, particularly in microwave and millimeter-wave RF-based front-end architectures. Patch antenna design continuously refines these systems to enhance energy efficiency, coverage, flexibility, and reliability, driving advancements in front-end engineering. Achieving these improvements requires meticulous selection of substrates and embedding materials, as both directly influence key antenna performance parameters such as impedance matching, radiation efficiency, and BW. Embedding materials, classified as dielectric or conductive, serve multiple critical functions in antenna design. High-permittivity materials contribute to miniaturization by reducing the antenna's physical footprint without compromising performance. Proper material selection enhances gain and directivity by mitigating surface wave losses and optimizing radiation characteristics. Additionally, embedding materials support impedance matching, minimizing reflection losses and improving signal transmission efficiency. In harsh environments, encapsulating materials protect antennas from moisture, temperature variations, and mechanical stress, ensuring long-term reliability. The literature has pointed out that the behavior of a patch antenna is strongly influenced by the electrical and mechanical characteristics of the used substrate, which directly impact key evaluation parameters obtained from the radiation pattern and \(S_{11}\) scattering parameter (e.g., BW, directionality, and resonant frequency). For example, Figure 1 illustrates the radiation patterns of a single rectangular patch antenna with substrates of different dielectric constants and loss tangents. The difference between radiation patterns is attributed to the fact that dielectric constant and loss tangent are inversely proportional to the radiation efficiency of the patch antenna, which is expressed as [17]  
\[\eta_{\text{rad}}^{\text{patch}}=\frac{1}{1+\frac{\tan\delta+\tan\delta_{H}}{ 240\text{C}^{\text{patch}}}\frac{W}{h}\sqrt{\frac{\varepsilon_{\text{r}}}{ \mu_{\text{r}}}}}, \tag{1}\]  
where \(\varepsilon_{\text{r}}\) is the dielectric constant, \(\mu_{\text{r}}\) is the magnetic constant, \(\tan\delta\) is the dielectric loss tangent, \(\tan\delta_{H}\) is the magnetic loss tangent, \(G_{\text{r}}^{\text{patch}}\) is the radiation conductance, \(W\) is the width of the radiation element, and \(h\) is the height of the substrate. Figure 1: Radiation pattern plot of the far-field realized gain at 2.45 GHz, cut in (\(\phi=0\)). Moreover, fringing fields can significantly impair the performance of patch antennas if appropriate countermeasures are not implemented. These fields increase the effective dimensions of the patch, thereby altering its impedance characteristics and leading to energy dissipation within the substrate instead of efficient radiation into the surrounding environment [4; 17]. This issue becomes more pronounced in substrates with a high loss tangent, where a more significant portion of the electromagnetic energy dissipates as heat rather than being radiated as an electromagnetic field. As a result, the radiation efficiency and overall performance of the antenna experience substantial reductions [4; 17]. Furthermore, this problem intensifies when the substrate quality is poor (e.g., FR-4 material). Given the limitations regarding substrates with high \(\varepsilon_{\mathrm{r}}\) and tangent loss (\(\tan\delta\)), the challenge of designing antennas with suitable performance highlights the need to explore alternative approaches, including the integration of resonators. Indeed, embedding patch antennas in dielectric and conductive resonators is promising, mainly when poor substrates (i.e., those with high dielectric constants) are used [17]. The resonator works as an impedance-matching component, confining electromagnetic waves within a specific geometry that defines the resonant modes.