## 4 MEMS AE Sensor Geometries  
Typical MEMS AE sensors are designed using a cantilever beam or plate (i.e., diaphragm). Figures 3 and 5 show common configurations used in the design of MEMS AE sensors. The cantilever beam can be designed as lumped mass or distributed mass models. If the lump mass model is considered, the fundamental frequency is calculated by  
\[f_{1}=\frac{1}{2\pi}\sqrt{\frac{k}{m}} \tag{14}\]  
where \(k\) is transverse stiffness of cantilever beam and \(m\) is lump mass. Assuming the rotation at the cantilever beam and mass is restrained, stiffness is calculated by \(k=12\frac{EI}{L^{3}}\) where \(I\) is the moment of inertia, \(L\) is the cantilever length. If the distributed mass model is considered, the fundamental frequency is calculated by [40]  
\[f_{n}=\frac{\alpha_{n}}{2\pi}(\frac{1}{L})^{2}\sqrt{\frac{EI}{\rho_{d}A}} \tag{15}\]  
where \(\alpha_{n}\) is coefficient depending on mode number (e.g., 3.516 for the first mode, 22.03 for the second mode) and \(A\) is cross sectional area. The diaphragm shapes can be circular, hexagonal, octagonal or square. Analytical solutions of circular and square plates exist. Each model frequency of rectangular plates for simply supported boundary conditions at all sides is calculated by  
\[f_{mn}=\frac{1}{2\pi}\Big{(}\frac{m\pi}{a}\Big{)}^{2}+\Big{(}\frac{n\pi}{b} \Big{)}^{2}\Big{]}\sqrt{\frac{D}{\rho_{d}h}} \tag{16}\]  
where \(D=\frac{EI^{3}}{12(1-\
u^{2})}\), \(m\), \(n\) are integers (e.g., 1 for the first mode), \(a\), \(b\) are two edge distances. The equation for circular plates is  
\[f_{n}={\lambda_{n}}^{2}\frac{1}{2\pi}\sqrt{\frac{D}{\rho_{d}ha^{4}}} \tag{17}\]  
where \(\lambda_{n}\) depends on the mode number. The value is 3.01146 for the first mode, 6.20540 for the second mode etc. [41]. For instance, the circular microstructure made of 10 \(\upmu\)m silicon with the material properties as \(E=156\) GPa, \(\upsilon=0.125\), \(\rho_{d}=2330\) kg/m\({}^{3}\), Figure 6 shows the relationship between fundamental frequency and the diameter of microstructure. The diameter of a 10 \(\upmu\)m-thick silicon microstructure should be between 560-340 \(\upmu\)m to tune the sensor between 100-300 kHz. For complex geometries without analytical formulation, the frequency responses are obtained by finite element models. While conventional bulky AE sensors are calibrated using ASTM standards [42; 43], the properties of MEMS AE sensors are extracted from electromechanical testing. The resonant frequency of MEMS AE sensors can be measured through impedance measurement. Figure 7 shows an example of admittance and phase values of a MEMS AE sensor designed with a mass connected with four beams [39]. The measurement was conducted using an impedance analyzer in atmospheric pressure. The features of resonant frequency and quality factor can be extracted from the impedance measurement. The sensor geometry configured for each transduction principle influences the bandwidth of MEMS AE sensors. As capacitive sensors require two parallel electrodes, they have a lower quality  
Figure 6: The relationship between fundamental frequency and diameter of a 10 \(\upmu\)m silicon microstructure with simply supported boundary condition. Figure 7: An example of admittance and phase plots of a MEMS AE sensor tuned near 170 kHz. factor as compared to piezoresistive and piezoelectric MEMS AE sensors. The quality factor of a capacitive sensor is increased by adding additional etch holes to provide air flow between two parallel plates [25]. The bandwidth can be widened by increasing the damping via adding backing material, which reduces the response magnitude of the vibrating microstructural element.