## 4 Design and Electrical Performance Characterization
### Mode Selection  
This section evaluates multiple characterization charts to establish an optimal design compromise between the quality factor, frequency stability, and mode separation.  
The eigen-mode analysis conducted in Ansoft HFSS (Table 2) yields resonant frequencies and the corresponding unloaded quality factors \(Q_{u}\). Lower \(Q_{u}\) values associated with eigen-modes 1, 2, 3, and 9 are excluded from the analysis, as these modes are not dominated by the dielectric resonator's characteristic behavior. The fourth mode, designated as TE\({}_{01\delta}\) in cylindrical coordinates (\(r\),\(\phi\),\(z\)), represents the fundamental mode of the dielectric resonator, fully determined by its geometric dimensions and aspect ratio. Conversely, as evidenced by Table 2, higher-order modes are undesirable as their closer spacing reduces the Spurious Free Dynamic Range (SFDR) of the system.  
This mode exhibits no radial variation, but features azimuthal \(\phi\) and axial \(z\) variations. This mode in dielectric resonators exhibits a horizontal magnetic dipole behavior [33], which is essential for achieving high unloaded Q and stable coupling in microwave filter applications [34]. Figure 9 illustrates both the electric and magnetic field distributions.  
The \(Q_{u}\) effectively corresponds to the intrinsic quality factor \(Q_{i}\) in this setup, comprising the following three principal loss components:  
* Dielectric losses from the resonator substrate.
* Losses originating from the bonding material.
* Non-radiative contributions independent of external loading.  
\begin{table}
\begin{tabular}{c c c} \hline
**Eigen-Mode Number** & **Unloaded Quality Factor \(Q_{u}\)** & **Eigen-Mode Frequency (GHz)** \\ \hline Mode 1 & 86.69 & 10.291 \\ Mode 2 & 651.54 & 10.610 \\ Mode 3 & 651.60 & 10.617 \\ Mode 4 & 2639.34 & 11.401 \\ Mode 5 & 1819.19 & 14.569 \\ Mode 6 & 1830.09 & 14.572 \\ Mode 7 & 2162.55 & 15.105 \\ Mode 8 & 1987.70 & 15.121 \\ Mode 9 & 302.31 & 15.304 \\ Mode 10 & 1741.95 & 16.274 \\ \hline \end{tabular}
\end{table}
Table 2: First ten eigen-modes.  
Figure 9: Electric and magnetic field—\(TE_{01\delta}\) mode.  
Quality Factor Enhancement  
Following the operating mode selection, the stand-alone DR design optimization under representative conditions focuses on maximizing the unloaded quality factor \(Q_{u}\) while minimizing the physical housing dimensions to ensure stable resonant frequency behavior under extreme conditions, including temperature drifts and fabrication tolerances. A parametric study evaluated the following two key variables: the DR aspect ratio ranging from 0.4 to 0.55 (achieved through height variation at fixed diameter) and the housing dimensions where cavity height remains constant while lateral (X-Y) dimensions scale proportionally from an initial 0.89 \(\lambda_{\text{eff}}\)\(\times\) 0.89 \(\lambda_{\text{eff}}\) baseline.  
As shown in Figure 10a,b, the resonance frequency stabilizes when the enclosure size increases by 40% relative to the initial dimensions, with corresponding stabilization in \(Q_{u}\). The maximum \(Q_{u}\) occurs at AR = 0.55 (Figure 10b), which was selected as the optimal point despite the continuing upward trend in quality factor. This conservative choice prevents the excitation of adjacent modes near the target frequency while meeting the design constraints. The optimal configuration combines an aspect ratio of 0.55 with 40% increased lateral enclosure dimensions (1.25 \(\lambda_{\text{eff}}\)\(\times\) 1.25 \(\lambda_{\text{eff}}\)), achieving optimal balance between robustness and performance.