# Theoretical Modeling of Piezoelectric Cantilever MEMS Loudspeakers
## 1 Introduction  
Compared to conventional electrodynamic loudspeakers, microelectromechanical system (MEMS) loudspeakers are considered to be more suitable for use in portable equipment applications owing to their higher energy efficiency [1] and smaller thickness. Among the different types of MEMS loudspeakers [1; 2; 3; 4; 5], piezoelectric MEMS loudspeakers have attracted attention because they require a low actuation voltage to provide a high actuation force [6] and are insensitive to dust [7].  
Many previous studies have been devoted to analyzing the performances of piezoelectric MEMS loudspeakers. Early works focused on conducting measurements for the analysis of different types of piezoelectric MEMS loudspeakers. [PERSON] et al. measured the displacement, sound pressure response, and sensitivity of a piezoelectric membrane, which works as both a microphone and an MEMS loudspeaker [8]. [PERSON] et al. measured the sound pressure response of a piezoelectric MEMS loudspeaker fabricated with a ZnO thin film [9]. With the increasing capacity of numerical computations, the finite element method (FEM) has become a capable tool for the simulation of MEMS loudspeakers. [PERSON] et al. utilized FEM simulations to study a novel type of MEMS loudspeaker featuring membrane-less two-way sound generation [10]. [PERSON] et al. employed an FEM simulation to predict and compare the performance of their proposed suspension-spring and dual-electrode designs [11]. Although FEM can predict the performance of MEMS loudspeakers precisely, it requires a large amount of mesh generation to ensure the accuracy of the simulation, which substantially increases the computational complexity and time consumption. In comparison with FEM, the equivalent circuit method (ECM) is more efficient. [PERSON] et al.  
analyzed a new type of MEMS loudspeaker accurately using ECM [12]. [PERSON] et al. established an ECM model for MEMS loudspeakers using a piezoelectric circular plate [13], and the model facilitated fast numerical analysis via analytical solutions.  
Recently, piezoelectric cantilever MEMS loudspeakers [14] have attracted increasing attention for portable audio applications because of their low cost and easy manufacturing procedure. They are driven by multilayer piezoelectric cantilever actuators (MPCAs) and exhibit piston vibration behaviors over a large surface. However, the existing ECM models cannot be applied to piezoelectric cantilever MEMS loudspeakers owing to the driving complexity of MPCAs. This significantly increases the difficulty of modeling and analyzing piezoelectric cantilever MEMS loudspeakers.  
In this study, an efficient ECM model of the piezoelectric cantilever MEMS loudspeaker with a special transformer suitable for MPCAs is presented. Based on theoretical analysis and derivation, a new transformer for MPCAs to achieve electromechanical conversion is proposed, which is required to establish the ECM model. FEM models with varying levels of complexity are used to validate the ECM model in three cases. Case 1 verifies the analytical solutions of a single MPCA, while cases 2 and 3 corroborate the feasibility of using the ECM model with the special transformer for different structures using paired MPCAs.  
This paper is organized as follows: Section 2 introduces the piezoelectric cantilever MEMS loudspeaker, proposes a special transformer, and presents an ECM model. Section 3 employs the FEM models in three aforementioned cases for validation. Section 4 concludes the paper.