## 3 Transduction Principles of MEMS AE Sensors
### Capacitive MEMS AE Sensors  
A capacitance is made of two conductors with opposite charges. A parallel plate capacitor is the most common configuration for AE sensing. The diaphragm forming the moving electrode of parallel plate capacitor can be built using a hexagonal [23], circular [24], square [25] or rectangular [26] shape.  
Capacitance change due to dynamic stimulus is detected via change in capacitance (\(C\)), which is related to dielectric constant (\(\epsilon\)), overlapping area (\(A\)) or gap (\(g\)) between two capacitive layers. The capacitance between two parallel plates is measured by  
\[C=\epsilon\frac{A}{g} \tag{8}\]As shown in Figure 3d, one layer is deposited on the substrate and the other layer is free to vibrate under inertial force. The change in capacitance is related to  
\[\frac{dC}{C}=\frac{de}{e}+\frac{dA}{A}-\frac{dg}{g} \tag{9}\]  
If the dominant factor to capacitance change is gap change (\(x(t)\)), the produced displacement current (\(i(t)\)) is calculated as  
\[i(t)=V_{DC}\frac{\epsilon_{o}A}{g^{2}}\frac{dx(t)}{dt} \tag{10}\]  
where \(V_{DC}\) is DC bias voltage, \(t\) is time and \(\epsilon_{o}\) is dielectric constant of air [25]. The operation of capacitive MEMS sensors requires constant DC bias voltage, which may introduce long-term durability challenges [27] in AE applications.  
Figure 4 shows an example of a differential MEMS AE sensor achieved by designing two different gaps between capacitive layers that change due to in-plane excitation. The geometry can be modified to have different mass and spring elements, and multiple individual units can be connected to increase total capacitance and current output, which improves the signal-to-noise ratio. Capacitive MEMS AE sensors have advantages of large bandwidth, sensitivity, efficiency, integrated circuit compatibility and geometry flexibility.  
The important variables for designing capacitive MEMS AE sensors are:  
* Fundamental frequency, \(f_{1}\), which is controlled by stiffness and mass of microstructure. The geometry can be varied to tune to a particular frequency.
* Collapse voltage, which is defined as the bias voltage causing the failure of membrane. The value depends on stiffness, gap and overall area [29]. As the sensitivity increases with higher \(V_{DC}\) as shown in Equation (10), the design with higher collapse voltage improves the signal-to-noise ratio.
* Quality factor controls the bandwidth of a capacitive MEMS sensor. Quality factor depends on squeeze film damping between stationary and moving layers. High quality factor improves signal-to-noise ratio while it reduces the bandwidth. Squeeze film damping occurs due to the pressure of thin viscous fluid between two layers. The resonant frequency is also influenced by the damping level in the atmospheric condition [23; 30].
* Dielectric charging, which is caused by the accumulation of unwanted charges stored and trapped on the dielectric surface [27]. This can increase the stiction of two layers, leading to permanent  
Figure 4: An example of a capacitive MEMS AE sensor for in-plane wave sensing [28].  
device failure. The failure can be prevented by depositing dielectric posts or a layer between two electrodes [21].  
Post-manufacturing packaging of capacitive MEMS AE sensors has limited robustness. Wafer-level packaging can be adapted from capacitive MEMS ultrasonic transducers [31].