## 2 Theoretical Model of a Quartz Tuning Fork
### Electrical Model  
The resonance properties of a QTF can be obtained by analyzing its response to a sinusoidal electrical excitation with amplitude \(V_{0}\). By inverse piezoelectric effect, when voltage is applied to the prongs, a strain field and thus surface charges are generated. The electrical response of a QTF can be determined by solving its equivalent circuit [23; 24] represented in Figure 1.  
\(R\), \(C\), and \(L\) are the equivalent electrical resistance, capacitance, and inductance of the QTF, respectively. The current through the RLC circuit \(i(t)\) is associated with the prong's vibration, whereas \(C_{p}\) is the stray capacitance and \(i_{c}(t)\) is the stray current generated by \(C_{p}\). The impedance \(Z\) of this circuit is given by:  
\[\frac{1}{Z(\omega)}=i\omega C_{p}+\frac{1}{i\omega L+R+\frac{1}{i\omega L}} \tag{8}\]  
The output current is given by the sum of \(i(t)\) and \(i_{c}(t)\) and is converted into a voltage signal by a transimpedance amplifier with a feedback resistor \(R_{F}\). By solving the circuit represented in Figure 1, the output signal amplitude \(V_{out}\) of the amplifier is retrieved as a function of the angular frequency of the excitation \(\omega\) = \(2\pi f\):  
\[V_{out}=\frac{A\omega}{Q\,\omega_{o}}\frac{\sqrt{1+2C^{\prime}\left(1-\frac{ \omega^{2}}{\omega_{o}^{2}}\right)+C^{\prime 2}\left(1-\frac{\omega^{2}}{ \omega_{o}^{2}}\right)^{2}+C^{\prime 2}\left(\frac{\omega}{\omega_{o}^{2}Q} \right)^{2}}}{\left(1-\frac{\omega^{2}}{\omega_{o}^{2}}\right)^{2}+\left( \frac{\omega}{\omega_{o}^{2}Q}\right)^{2}} \tag{9}\]  
where \(A\) = \(V_{0}\cdot R_{F}/R\); \(Q\) = \(L\cdot\omega_{0}/R\); \(\omega_{0}\) = \(2\pi f_{0}\) = \(1/\sqrt{LC}\); and \(C^{\prime}\) = \(C_{P}/C\).  
The output signal amplitude calculated by using Equation (9) is plotted in Figure 2a,b at different \(C^{\prime}\) values while keeping \(A\), \(Q\) and \(f_{0}\) fixed, and (b) at different \(Q\) values while keeping \(A\), \(C^{\prime}\) and \(f_{0}\) fixed.  
Figure 1: Equivalent circuit for a QTF. \(R\), \(C\), \(L\) and \(C_{p}\) are the electrical resistance, capacitance, inductance, and stray capacitance of the QTF, respectively.  
Figure 2: (a) Resonance curve of a QTF for different values of \(C^{\prime}\), with \(A\) = \(2\times 10^{-8}\), \(f_{0}\) = 15,820 Hz, \(Q\) = 10,000. (b) Resonance curve of a QTF for different values of \(Q\) with \(A\) = \(2\times 10^{-8}\), \(f_{0}\) = 15,820 Hz, \(C^{\prime}\) = 0.  
All curves are characterized by a peak at the resonance frequency \(f_{0}\) of the QTF. As \(C^{\prime}\), i.e., the stray capacitance, increases, the stray current creates a right-left asymmetry with respect to the peak value and a local minimum appears on the right side, as in the blue curve in Figure 2a. In the case in which the stray capacitance can be neglected (black curve in Figure 2a), the electrical response of the QTF can be represented by a Lorentzian line-shape. As the \(Q\)-factor increases (see Figure 2b) the equivalent electrical resistance of the QTF at the resonance decreases, leading to a higher resonance peak value, and the full width at half maximum (FWHM) of the resonance curve decreases.