## 3 Performance Characterization and Reliability Assessment
### RF Performance Measurement
The S-parameter measurements shown here directly support the quantitative performance metrics presented elsewhere in the paper, including the superior Q-factor and lower insertion loss achieved by the L3 configuration. The loaded Q-factor (\(\mathrm{Q_{L}}\)) was determined from the measured frequency response using the 3 dB bandwidth method, calculated as \(\mathrm{Q_{L}}=\mathrm{f_{c}}/\Delta\mathrm{f_{3 dB}}\), where \(\mathrm{f_{c}}\) is the center frequency and \(\Delta\mathrm{f_{3 dB}}\) is the bandwidth between the points where the insertion loss is  
Figure 8: Comparison of simulated and measured frequency response graphs for L1, L2, and L3 structures. The red lines represent the simulation results, while the blue lines indicate the measured data. Figure 9: Comparison of measured frequency responses for five distinct samples (S1–S5) for each of the L1, L2, and L3 filter configurations. The plots show the overlaid S21 (insertion loss) and S11 (return loss) for each sample, demonstrating the consistency of the fabrication process. 3 dB higher than its minimum value in the passband. The unloaded Q-factor QU is then calculated from the relationship in Equation (5). The unloaded Q-factor QU is calculated from the relationship in Equation (5) [21]. \[\text{Q}_{\text{U}}=\frac{\text{Q}_{\text{L}}}{1{-}10^{(\frac{-1}{20})}} \tag{5}\]  
Figure 10 presents a comprehensive comparative analysis of the RF performance measurements across the three resonator width configurations (L1, L2, and L3), organized in four distinct bar charts that highlight critical performance metrics. The return loss (RL) chart (a) shows L3 demonstrating favorable performance (10.94 dB) compared to L2 (10.68 dB) and L1 (12.23 dB), indicating superior impedance matching and reduced signal reflections. Most importantly, the insertion loss (IL) chart (b) highlights the critical signal transmission efficiency, with L3 demonstrating the lowest insertion loss (3.34 dB) compared to L2 (3.56 dB) and L1 (3.10 dB), confirming that 100% resonator width matching provides optimal signal transmission efficiency while minimizing energy losses. The bandwidth chart (c) demonstrates that all three configurations maintain an identical 4.5 GHz bandwidth, indicating that resonator width matching preserves the filter's frequency range capabilities while enhancing other performance aspects. The Q-factor measurements (d) reveal the performance characteristics for each configuration, showing both loaded Q-factor (Q\({}_{\text{L}}\)) and unloaded Q-factor (Q\({}_{\text{U}}\)) values. The L3 configuration achieves Q\({}_{\text{L}}\) = 3.31 and Q\({}_{\text{U}}\) = 10.38, compared to L2 (Q\({}_{\text{L}}\) = 3.61, Q\({}_{\text{U}}\) = 10.74) and L1 (Q\({}_{\text{L}}\) = 3.69, Q\({}_{\text{U}}\) = 12.28). \(\text{Q}_{\text{U}}\) represents the intrinsic quality factor of the resonator without external loading effects. Collectively, these performance measurements establish a clear correlation between  
Figure 10: Comparative analysis of RF performance metrics across three resonator width configurations (L1, L2, and L3): (**a**) return loss performance in dB, (**b**) insertion loss measurements in dB, (**c**) bandwidth characteristics in GHz, and (**d**) Q-factor analysis showing both loaded (QL) and unloaded (QU) quality factors for each configuration. resonator width matching and enhanced RF performance, validating the study's central hypothesis that optimizing the geometric relationship between cap wafer and device substrate resonators yields both electrical performance and mechanical reliability benefits.