## 2 Analysis Methods
### Back Chamber Effect  
A microspeaker can be assumed as an element of a mass, a spring, and a damping (MCK) system with a forced vibration. It is inserted into a back chamber to prevent the leakage of sound having an opposite phase from the backside of the diaphragm. Owing to space limitation in smartphones, the back volume in the speaker box is always limited. The back volume is a small, closed air volume under the diaphragms that includes the center diaphragm (CDP) and the side diaphragm (SDP). The back volume adds additional stiffness to the mechanical system of the microspeaker (unit stiffness), resulting in an increased fundamental resonance frequency.  
The fundamental resonance frequency is determined by the moving mass and the total stiffness of the mechanical system. The cross-sectional view of the microspeaker unit is shown in Figure 3a. The material of SDP is an elastomer compound, and it has the lowest Young's modulus. The CDP is a flat plate made by aluminum compound to keep the coil vibrating uniformly. The unit stiffness is controlled by the material and structure of SDP; the vibration mode of diaphragms and voice coil is shown in Figure 4; the other parts have been hidden.  
The mechanical parts work together to convert electrical signals into mechanical motion, which creates sound waves in the surrounding air. The stiffness and the mass of moving parts affect the speed at which the diaphragms can move and the mount of force it can generate. This, in turn, affects the frequency response and the volume of the sound produced by the microspeaker. A heavier coil or more flexible diaphragm will result in a lower resonance frequency, which means the microspeaker will be less capable of producing high-frequency SPL. A lighter coil or a more rigid diaphragm will result in a higher resonance frequency, which means the microspeaker will be more capable of producing high-frequency SPL.  
The microspeaker can also ideally be represented as a rigid piston at low frequencies because the movement of the diaphragm is similar to the movement of a piston in an engine, as shown in Figure 5. The added stiffness [21] can be calculated by Equation (8). The relationship between total stiffness, unit stiffness, and added stiffness is shown in Equation (9). The unit stiffness is only determined by the mechanical structure and material of the diaphragms; thus, it can be obtained by mechanical analysis. The total stiffness is the sum of unit stiffness and added stiffness, which determined the fundamental resonance of microspeaker. The fundamental resonance frequency can be expressed as Equation (10).  
\[K_{added}=\frac{\rho_{0}c^{2}{S_{d}}^{2}}{V_{back}}, \tag{8}\]  
\[K_{Total}=K_{Ushift}+K_{added}, \tag{9}\]  
\[f_{0}=\frac{1}{2\pi}\sqrt{\frac{K_{Total}}{M}}, \tag{10}\]  
Figure 4: Vibration mode: (**a**) cross-section view of microspeaker unit and (**b**) the vibration mode of diaphragms and voice coil.  
where \(\rho_{0}\) is a density of air. \(S_{d}\) is the effective radiation area [22]. \(V_{back}\) is the back volume. \(f_{0}\) is the fundamental resonance frequency, which the first peak shows in the SPL curve around 1 kHz. \(M\) is the moving mass of the mechanical system that mainly comes from the voice coil and center diaphragm.