**5. Electrical Part**

This section focuses on maximizing the conversion between mechanical deflection and electrical signal.

The nominal capacitance *C*<sup>0</sup> = *Lbε*0*εr*/*d* is the capacitance value without any displacement, where *εr*,*ε*<sup>0</sup> are the relative permittivity of the media (equal to unity in air) and vacuum, respectively. For the different geometries considered, *C*<sup>0</sup> may take values between 10−<sup>4</sup> and 100 pF. The expression of nominal capacitance indicates that the change of the distance between two electrodes will cause a capacitance variation.

The dynamic capacitance caused by deflection of the cantilever is given by [38]:

$$\mathcal{C}(t) = \int\_0^L \frac{b\varepsilon\_0 \varepsilon\_r}{d + \mathcal{W}\_\text{tr}(\mathbf{x}, \omega)} \, d\mathbf{x} \exp(i\omega t) \tag{23}$$

The model applies a method called DC bias sensing [39] (Chapter 5). Figure 8 presents the sensing scheme. In an electromechanical system, a polarization voltage *Vdc* on the electrodes is required to generate an electrical signal related to the mechanical behavior of the moving electrode.

**Figure 8.** Sensor conditioning circuit, where: *C*0, *Cp*, *Rf* , *Vdc*, *Vout* are the capacitance of the cantilever, parasitic capacitance, resistance, polarization voltage, and output voltage, respectively.

The application of the force generated by the photoacoustic effect sets the movable electrode in motion. This movement causes the changes of the capacitance from the maximal value *Cmax* to the minimal *Cmin*. *Cmax* and *Cmin* correspond to the minimal and maximal distances between the cantilever and support, respectively.

The capacitance variation can take place at a constant charge or a constant voltage [40]. If the time constant *Rf C*<sup>0</sup> >> 1/*ωn*, the electric charge stored in the capacitor remains constant. *Rf* 100 *G*Ω − 10 *T*Ω is the value of the resistor placed between the cantilever and the polarization voltage *Vdc*. In the constant charge regime, the voltage of the measured signal is given by *Vout*(*t*) = *C*0*Vdc*/*C*(*t*) while its amplitude is given by:

$$V\_{out} = V\_{dc} \int\_{0}^{L} \frac{\mathcal{W}\_{n}(\mathbf{x}, \omega)}{Ld} \, d\mathbf{x} \tag{24}$$

#### *Results and Discussion*

Figure 9 presents the results obtained for a bias *Vdc* = 1 V. The maximum values follow the tendencies given by the displacement. The values change with the gap *d* and frequency *fn*. Indeed, according to Equation (22), Equation (24) can be simplified as follows:

$$V\_{out} \simeq 0.39 V\_{dc} \frac{w\_n(\omega)}{d} \tag{25}$$

Decreasing the gap *d* between the two electrodes should lead to an output signal amplitude increase. However, simultaneously it increases the squeeze film damping and reduces the cantilever displacement. The optimization of this parameter will be discussed in the last section.

Equation (25) indicates that the signal output does not depend on the area of the capacitor as it would be expected based on Equation (23). The output signal amplitude is given for an open circuit, without any read-out circuit which can modify the signal. In a complete system, the signal is attenuated by a parasitic capacitance *Cp*, which is the sum of the parasitic capacitance of the resonator itself and the one which comes from read-out circuit. The output signal attenuation can be estimated with the ratio *C*0/(*C*<sup>0</sup> + *Cp*) [41].

**Figure 9.** Amplitude of the output voltage versus width and thickness for a cantilever of fundamental resonance frequency equal to 11 kHz (**a**) and 60 kHz (**b**) for a gap between the support and the cantilever equal to *d* = 10 μm. The polarization voltage is *Vdc* = 1 V.
