## 2 Analysis Methods
### Acoustic Modeling  
There are three types of common acoustic modeling in a microspeker module: a tube with two open ends [12], a closed cavity [13], and a Helmholtz resonator (HR) [14], as shown in Figure 3.  
A tube open at both ends with rigid walls behaves as an acoustic mass if it is sufficiently short, such that the air inside it moves without appreciable compression. The acoustic mass \(M_{ac}\) of the two open-end tubes is given by Equation (1).  
\[M_{ac}=\frac{\rho L}{A}\left[\mathrm{kg}/\mathrm{m}^{4}\right], \tag{1}\]  
Figure 3: Acoustic modeling: (**a**) A tube with two open ends, (**b**) a closed cavity, and (**c**) a Helmholtz resonator.  
Figure 2: Structure of microspeker module: (**a**) exploded view; (**b**) cross-sectional view.  
where \(\rho\), \(L\), and \(A\) denote the air density, tube length, and cross-sectional area, respectively. In the low-frequency limit, an applied external pressure compresses the enclosed air in the cavity, which then acts as a spring because of elasticity [15]. The acoustic spring constant of the closed cavity \(K_{ac}\) is given by Equation (2).  
\[K_{ac}=\frac{\rho c^{2}}{V}\Big{[}\text{N}/\text{m}^{5}\Big{]}, \tag{2}\]  
The direct analogy relationship between acoustic and mechanical analyses is shown in Equation (3).  
\[M_{me}=M_{ac}A^{2}\,\ K_{me}=K_{ac}A^{2}, \tag{3}\]  
where \(V\), \(M_{me}\), and \(K_{me}\) denote cavity volume, lumped mechanical mass, and lumped mechanical stiffness, respectively.  
A Helmholtz resonator is type of resonator that consists of a cavity with a tube (narrow neck). A microspekar can be thought of as a type of Helmholtz resonator because it has a cavity (the enclosure in which the speaker is mounted) and a tube (the front chamber of the speaker). When the speaker diaphragm moves back and forth, it causes pressure fluctuations in the air inside the enclosure, which can produce resonances in the cavity.  
The geometrical correspondence between a microspekar and a Helmholtz resonator is that both have a cavity and a tube and that both can produce resonances at certain frequencies. However, a microspekar is more complex than a simple Helmholtz resonator because it also has a coil, magnets, and other mechanical parts that are responsible for converting electrical signals into mechanical motion. These parts also contribute to the resonant behavior of the speaker and affect its frequency response.  
The Helmholtz resonator can be treated as a mechanical mass-spring system with a natural frequency \(f_{HR}\), as shown in Equation (4). \(c\) is the speed of sound.  
\[f_{HR}=\frac{1}{2\pi}\sqrt{\frac{K_{me}}{M_{me}}}=\frac{1}{2\pi}\sqrt{\frac{A^ {2}\rho_{0}c^{2}}{\rho_{0}LAV}}=\frac{c}{2\pi}\sqrt{\frac{A}{LV}}, \tag{4}\]  
In acoustics [16], end correction [17; 18] is a short distance applied or added to the actual length of a resonance tube to calculate the precise resonant frequency.  
Hydraulic diameter [19] is a commonly used term for handling flow in noncircular tubes and channels. Using this term, one can calculate flow for a noncircular tube in the same way as for a circular tube. If the cross section has a rectangular shape with width \(b\) and height \(a\), the hydraulic diameter is given by Equation (5).  
\[D_{h}=\frac{2 ab}{a+b}\,\ b\gg a, \tag{5}\]  
The Helmholtz resonator considering an end correction (HREC) is given by Equation (6):  
\[f_{HREC}=\frac{c}{2\pi}\sqrt{\frac{A}{(L+0.85D_{h})V}}, \tag{6}\]  
The resonance frequencies of the two open-end tubes considering an end correction (TEC) are shown in Equation (7).  
\[f_{TEC}=\frac{c}{2(L+0.8D_{h})}, \tag{7}\]  
The air volume of the cavity is 156.34 mm\({}^{3}\). The peak frequencies calculated using Equations (6) and (7) are defined as method #1. The 3D E-M-A coupling method [20] is used to obtain the SPL curve, which is regarded as method #2.