# Design and Analysis of a Novel Microspeaker with Enhanced Low-Frequency SPL and Size Reduction
## 2 Analysis Method (Prototype)
### Electromagnetic Analysis  
The governing equation for the electromagnetic analysis is expressed as follows:  
\[V_{e}=IR+L\frac{dI}{dt}+Bl(z)\frac{dz}{dt} \tag{1}\]  
where \(V_{e}\), \(I\), \(R\), \(L\), \(t\), \(l\), and \(z\) are the voltage, current, resistance, inductance, time, coil length, and displacement, respectively. The vibration of the coil generates a back electromotive force (back EMF), which is expressed as \(Bl(z)\frac{dz}{dt}\) in Equation (1). The back EMF is the product of the force factor and the velocity in the \(z\)-direction, derived in the mechanical domain. Equation (1) is called the voltage equation, and it can be described as a circuit shown in Figure 3 in the electromagnetic domain. To solve the voltage equation, we identified the parameters in advance. Figure 4: Flow chart (procedures) of the coupled analysis. Figure 3: Equivalent circuit (electromagnetic and mechanical) and acoustic domain for the coupled analysis. The flux density was obtained by using a finite element method (FEM)-based commercial software (Ansys). The flux density in the coil was obtained by solving the following governing equations:  
\[ \begin{array}{l}B=\
abla\times A\\ \
abla\times H=J\\ B=\mu H\end{array}  \tag{2}\]  
where \(B\) is the magnetic flux density, \(\
abla\times\) is the vector curl operation, \(A\) is the magnetic vector potential, \(J\) is the current density, \(H\) is the magmatic intensity, and \(\mu\) is the magnetic permeability. The flux linkage is defined as follows:  
\[\lambda(z,I)=NSB(z,I) \tag{3}\]  
where \(N\) is the number of coil turns, \(S\) is the coil cross-sectional area, and \(B\) is the average flux density passing through the coil. The inductance is expressed as the derivative of the flux linkage to the displacement expressed in Equation (4):  
\[L=\frac{d\lambda}{dl} \tag{4}\]  
The force that causes coil vibration is the Lorentz force. If the direction of the current flowing in the coil and the direction of the magnetic field are known, the direction of the Lorentz force can be determined. The force factor is derived by dividing the Lorentz force by the current. They are expressed as  
\[ \begin{array}{l}F_{Lorentz}=\oint Idl\times B\\ Force\ factor=F_{Lorentz}/I=Bl\end{array}  \tag{5}\]  
The material properties needed for the electromagnetic analysis are listed in Table 1.  
The top plates and the yoke were made of SPCC (Steel Plate Cold Commercial) and had a nonlinear relationship between \(B\) and \(H\). \(B\) and \(H\) were correlated with a nonlinear parameter called magnetic permeability. This nonlinear value was calculated from the slope of the \(B\)-\(H\) curve. The \(B\)-\(H\) curve of SPCC is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Part** & **Material** & **Property** & **Value** & **Unit** \\ \hline Air & - & Permeability & \(4\pi\times 10^{-7}\) & H/m \\ Magnet (N52H) & NdFeB & Remanence & 1.45 & T \\ Top plate & SPCC & Permeability & \(B\)-\(H\) curve & H/m \\ Yoke & SPCC & Permeability & \(B\)-\(H\) curve & H/m \\ Coil & Copper & Permeability & \(4\pi\times 10^{-7}\) & H/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties used for the electromagnetic analysis. Figure 5: Nonlinear \(B\)-\(H\) curve of SPCC. A sphere shape domain was made around the microspeker to simulate the air region. At the surface of the air region, we applied a magnetic insulation boundary condition, meaning that the flux was tangential to its boundaries. The hexahedral element was used at the coil, magnet, and center diaphragm because of its simple shape. For the rest of the parts, a tetrahedral element was used because of the complexity of its boundary shape. Because the electromagnetic force was calculated from the coil, we used the fine element size in the coil. In addition, the top plates and the yoke that provide paths for the magnetic fluxes were also finely meshed for accurate results.