# Theoretical Modeling of Piezoelectric Cantilever MEMS Loudspeakers
## 2 Theoretical Modeling of Piezoelectric Cantilever MEMS loudspeaker Using ECM
### Transformer for Multiple Pairs of MPCAs  
As MPCAs are symmetrically distributed, it can be considered that the force of each MPCA acting on the middle coupling element is equal, and the motion of each MPCA is, therefore, the same. Then, the analytical solutions of the electromechanical coupling relationship of multiple pairs of MPCAs can be derived from that of a single MPCA. According to the analytical solutions, a special transformer for multiple pairs of MPCAs can be designed. First, a single MPCA is considered. Figure 2 shows a schematic of a \(i\)-layer MPCA with length \(l\), width \(w\), an nth layer upper and lower z-coordinates of \(h_{n}\) and \(h_{n-1}\), respectively. The MPCA is clamped at \(x=0\) and connected to the middle coupling element at \(x=l\). When a voltage \(U\) is applied to the piezoelectric layers, the MPCA bends and vibrates in the z-direction. Figure 1: Schematic of a piezoelectric cantilever MEMS loudspeaker. To simplify the analysis, three assumptions are made:  
1. The bending vibration caused by the deformation in the y-direction is negligible. The fundamental equations of the piezoelectric materials can then be written as [15] \[s_{x}=s_{11}\sigma_{x}+d_{31}E_{3},\] (1) \[D_{3}=d_{31}\sigma_{x}+\epsilon_{33}^{t}E_{3},\] (2) where \(s_{x}\) denotes the strain in the x-direction, \(s_{11}\) is the compliance at a constant electric field, \(\sigma_{x}\) is the stress in the x-direction, \(d_{31}\) is the piezoelectric constant, \(E_{3}\) is the electric field in the z-direction, \(D_{3}\) is the electric displacement, and \(\epsilon_{33}^{t}\) is the dielectric constant at a constant stress. 2. The MPCA vibrates with a small deflection. Then, the expression for \(s_{x}\) is [15] \[s_{x}=-\frac{\partial^{2}u}{\partial x^{2}}z.\] (3)
3. Layers with no piezoelectric properties can be considered as piezoelectric layers with \(d_{31}=0\) and \(\epsilon_{33}^{t}=0\). Based on the above formulae, the steady-state displacement at \(x=l\) of an MPCA \(u(l)\) and the charge \(Q_{3}\) passing through it can be deduced as follows:  
\[u(l)=-\frac{\frac{F}{k^{2}p}(\sinh kl\cos kl-\sin kl\cosh kl)+\frac{Rlmp}{k^{2 }p}\sinh kl\sin kl}{1+\cosh kl\cos kl}, \tag{4}\]  
\[Q_{3}= -Rkw\frac{-\frac{Rlmp}{k^{2}p}(\sinh kl\cos kl+\sin kl\cosh kl)+ \frac{F}{k^{2}p}\sinh kl\sin kl}{1+\cosh kl\cos kl} \tag{5}\] \[+\sum_{n=1}^{i}(\epsilon_{33n}^{t}-\frac{d_{31n}^{2}}{s_{11}^{n}})E_{n}lw,\]  
where \(F\) denotes the force in the z-direction on an MPCA at \(x=l\), \(\epsilon_{33n}^{t}\) is the dielectric constant of the nth layer in the z-direction, \(d_{31n}\) is the piezoelectric constant of the nth layer, and \(s_{11}^{n}\) is the compliance in the x-direction of the nth layer. Detailed derivations of Equations (4) and (5) are presented in Appendix A, where the expressions for \(k\), \(P\), \(Q\), and \(R\) are also given. According to Equations (4) and (5), the velocity of the MPCA \(v\) at \(x=l\) and the current \(I\) in the MPCA can be derived as follows:  
\[v=j\omega a_{vF}F+j\omega a_{vL}U, \tag{6}\]  
\[I=j\omega a_{vL}F+j\omega a_{vL}U, \tag{7}\]  
Figure 2: Schematic of a multilayer piezoelectric cantilever actuator (MPCA). with  
\[s_{x}=s_{11}\sigma_{x}+d_{31}E_{3},\]0  
\[s_{x}=s_{11}\sigma_{x}+d_{31}E_{3},\]1  
\[s_{x}=s_{11}\sigma_{x}+d_{31}E_{3},\]2 \[s_{x}=s_{11}\sigma_{x}+d_{31}E_{3},\]3  
Equations (6) and (7) describe the relationships among velocity, current, force, and voltage, which indicates the electromechanical conversion relationship in an MPCA. Next, based on the analytical solutions of a single MPCA, the characteristics of multiple MPCA pairs in the piezoelectric cantilever MEMS loudspeaker can be deduced. We denote the force exerted by a single MPCA on the middle coupling element as \(F_{0}\)\((F_{0}=-F)\), the force exerted by all MPCAs on the middle coupling element as \(F_{total}\), and the total current passing through all MPCAs as \(I_{total}\).