This article presents a lumped-parameter model (LPM) providing a deeper understanding of the compliant backplate in capacitive micro-electromechanical systems (MEMS) microphones. Some previous models simplify the backplate as stationary, whereas others treat it as vibrating. This work not only models the backplate as vibrating but also considers the coupling effect between the mechanical and electrical domains. The extended model allows for a more detailed analysis of how the microphone converts sound into an electrical signal. Specifically, the theoretical derivations using Lagrange equations show how backplate motion can impact the microphone's performance. The analysis of the LPM aligns well with the results of finite element analysis (FEA) when the frequency is below the high-order resonance, validating the theoretical concepts. In particular, the model with electrical coupling of the vibrating backplate effectively captures the sensitivity dip resulting from the backplate resonance, unlike models lacking this coupling. The theoretical framework is also extended to the phenomenon of pull-in. A backplate that is overly compliant can narrow the operating frequency range and increase the likelihood of experiencing pull-in. Thus, there is a tradeoff between optimizing the microphone's acoustic performance and ensuring its mechanical robustness. This work provides valuable insights into navigating these tradeoffs.