## 4 Conclusions  
This investigation establishes resonator width optimization as a critical design parameter for wideband RF MEMS filters. Through systematic evaluation of three resonator width configurations--0% (L1), 60% (L2), and 100% (L3) matching between cap and bottom wafers--this study demonstrates that complete width matching (L3) provides optimal electrical performance and mechanical reliability. The L3 configuration achieved superior RF performance with 3.34 dB insertion loss across a 4.5 GHz bandwidth (25% fractional bandwidth) and optimal loaded Q-factor (Q\({}_{\text{L}}\) = 3.31). While L1 demonstrated the lowest insertion loss (3.10 dB), L3 provided enhanced electromagnetic wave coupling and minimized contact resistance at the bonding interface. Mechanical reliability testing revealed that L3 withstood forces up to 7.14 Kgf, significantly outperforming L2 (2.24 Kgf) and L1 (4.22 Kgf), representing a 218% improvement over L2. SEM analysis confirmed uniform bonding interfaces with minimal void formation (~180 nm) and tolerance to \(\pm\)10 \(\upmu\)m misalignments. Comprehensive environmental testing per MIL-STD-810E standards, including thermal cycling (~50 \({}^{\circ}\)C to +145 \({}^{\circ}\)C) and humidity exposure, validated long-term stability across all configurations. The L3 design maintained superior performance throughout environmental stress testing. Comparative analysis demonstrates exceptional 25% fractional bandwidth significantly exceeding competing designs (4.3-10%) while maintaining competitive insertion loss.  
It is important to note that while the L3 configuration provides superior overall performance, its reliance on a large, perfectly matched bonding area increases its sensitivity to manufacturing tolerances, especially wafer-to-wafer alignment. This presents a classic engineering trade-off between achieving optimal performance and ensuring a wider process window for manufacturing. This fabrication approach, while specialized, enables performance levels required for demanding applications. The resulting filter is intended for use in advanced hybrid RF modules, where it can be integrated alongside other components to create a compact, high-performance system-in-package.  
This research establishes that complete resonator width matching between cap and bottom wafers simultaneously optimizes Q-factor performance and mechanical bonding reliability, providing a validated framework for developing high-performance, reliable RF MEMS devices for next-generation wireless communication and sensing applications.  
Conceptualization, G.J.; Methodology, M.J.; Validation, S.L.; Formal analysis, S.L. and N.-S.K.; Data curation, Y.J. and N.-S.K.; Writing--original draft, N.-S.K. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by a project for Collaborative R&D between Industry, University, and Research Institute funded by Korea Ministry of SMEs and Startups in 2023 (RS-2023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-023-0233-023-0233-023-0233-0233-0233-0233-023-0233-0233-0233-0233-0233-0233-0233-0233-0233-0233-0233-000223221); in part by the National Research Foundation of Korea (NRF) funded by the Ministry of Science and Information and Communications Technology (MSIT) of Korean government under Grant RS-2023-00252505; in part by the research grant of the Chungbuk National University in 2021. The EDA tool was supported by the IC Design Education Center (IDEC), Korea.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
Authors [PERSON], [PERSON], [PERSON], [PERSON] were employed by the company MISOTECH. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.