## 2 Analysis Methods
### E-M-A Coupling Model  
The E-M-A coupling method of the microspeaker uses the coupled equations of the voltage, mechanical vibration, and acoustic wave equations, which are shown in Equations (11)-(13), respectively.  
\[V_{e}=IR_{e}+L_{e}\frac{dI}{dt}+Bl\dot{z}, \tag{11}\]  
where \(V_{e}\), \(I\), \(R_{e}\), \(Bl\), and \(\dot{z}\) are the input voltage, electrical current, direct current resistance of the voice coil, inductance, force factor, and mechanical velocity, respectively.  
\[[M]\big{\{}\tilde{z}\big{\}}+[R]\big{\{}\dot{z}\big{\}}+[K]\big{\{}z\big{\}}= \{F_{L}\}+\{F_{P}\}, \tag{12}\]  
where \([M]\), \(\{\tilde{z}\}\), \([R]\), \(\{\dot{z}\}\), \([K]\), \(\{z\}\), \(\{F_{L}\}\), and \(\{F_{P}\}\) are the mass, acceleration, damping, velocity, stiffness, displacement, Lorentz force, and pressure force vectors, respectively.  
\[\
abla^{2}P+k^{2}P=0, \tag{13}\]  
where \(\
abla^{2}\), \(P\), and \(k\) are the Laplace operator, acoustic pressure, and wave number, respectively.  
Equations (14)-(17) indicate the coupled relationships between the electromagnetic, mechanical, and acoustic domains.  
\[V_{Backmf}=Bl\dot{z}, \tag{14}\]  
\[F_{L}=BII, \tag{15}\]  
\[F_{P}=S_{d}P, \tag{16}\]  
\[U=S_{d}\dot{z}, \tag{17}\]  
where \(V_{Backmf}\) and \(U\) represent the back electromotive force and volume velocity, respectively.  
The coil information, magnetic flux density on the coil, the flux density distribution in cross-section, and the force factor of the microspeaker are shown in Figure 6. The material of the magnets is neodymium with magnet grade N52H. The voice coil is the Daikoku high-tension wire (DHT), which can provide a high tensile strength. SDP is an elastomer compound, while the CDP is an aluminum compound. The top plates and yoke are made of steel plate cold commercial (SPCC).The material of frame and protector are polycarbonate and stainless steel (SUS 301), respectively. The front and back chamber are made by a 3D printer with printing material Nylon \(11\). The mechanical material properties are shown in Table 1.  
Figure 5: Piston source model.  
E-M-A coupling analysis is performed to derive the SPL of a microspeaker. For the accurate analysis of complex acoustic structures, FEM is used instead of the lumped parameter method (LPM) [23]. The boundary condition should be well defined, simulating the real experimental environment, where it is in an anechoic chamber. A spherical air region is formed, and the boundary condition of outward spherical wave radiation is applied at the exterior surface. The special wave radiation boundary condition can provide minimal reflections to the acoustic wave. The SPL comparison between analysis results and experiment results shows that the spherical wave radiation boundary condition is good enough for the acoustic analysis of the microspeaker. The E-M-A coupling model is illustrated in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Part**} & \multirow{2}{*}{**Material**} & **Density** & **Young’s Modulus** & \multirow{2}{*}{**Poisson’s Ratio**} \\  & & [kg/m\({}^{3}\)] & [GPa] & & \\ \hline Magnet & N52H & 7790.0 & 41.4 & 0.29 \\ Voice coil & DHT & 5320.0 & 110 & 0.35 \\ Suspension & Polyimide Film & 2110.0 & 2.50 & 0.34 \\  & Elastomer & & & \\ SDP & Compound & 1230.0 & 0.13 & 0.40 \\  & Aluminum & & & \\ CDP & Compound & 856.1 & 69.3 & 0.28 \\  & SPCC & 7308.0 & 190 & 0.29 \\ Frame & Polycarbonate & 1165.0 & 2.4 & 0.37 \\ Protector & SUS 301 1/4H & 7551.0 & 193 & 0.28 \\ Chambers & Nylon 11 & 1040.0 & 0.335 & 0.32 \\ \hline \end{tabular}
\end{table}
Table 1: Mechanical material properties.  
Figure 6: Electromagnetic analysis results.