## 4 Sensitivity
### Electrical Sensitivity  
In addition to the influence of the physical measurand on the sensor's resonance frequency, an additional sensitivity is required to characterize how the frequency detuning changes the sensor signal. Two sensitivities are required for this purpose, one covering the influence on the amplitude and the other the influence on the phase, where the two sensitivities are referred to as electrical sensitivities. The Figure 4a and Figure 4b illustrate the change in amplitude \(\Delta|\tilde{G}(f)|\) and in phase \(\Delta\tilde{\gamma}(f)\) due to a detuned resonance frequency by \(\Delta f\). To determine the optimal excitation frequencies and the resulting sensitivities, the two first derivatives  
\[\frac{\partial|\tilde{G}(f)|}{\partial f}=-\frac{4\;(f-f_{\text{res}})}{f_{ \text{res}}^{2}\;\left(4\;\left(\frac{f}{f_{\text{res}}}-1\right)^{2}+\frac{1} {Q^{2}}\right)^{\frac{3}{2}}} \tag{22}\]  
and  
\[\frac{\partial\tilde{\gamma}(f)}{\partial f}=-\frac{2\;Qf_{\text{res}}}{4Q^{2 }\;(f-f_{\text{res}})^{2}+f_{\text{res}}^{2}} \tag{23}\]  
as well as the two second derivatives  
\[\frac{\partial^{2}|\tilde{G}(f)|}{\partial f^{2}}=\frac{32\;Q^{2}f^{2}-64\;Q^ {2}ff_{\text{res}}+32\;Q^{2}f_{\text{res}}^{2}-4\;f_{\text{res}}^{2}}{Q^{2}f_{ \text{res}}^{4}\;\left(4\;\left(\frac{f}{f_{\text{res}}}-1\right)^{2}+\frac{1} {Q^{2}}\right)^{\frac{5}{2}}} \tag{24}\]  
and  
\[\frac{\partial^{2}\tilde{\gamma}(f)}{\partial f^{2}}=\frac{16\;Q^{3}f_{ \text{res}}\;(f-f_{\text{res}})}{\left(4\;Q^{2}f^{2}-8\;Q^{2}ff_{\text{res}}+4 \;Q^{2}f_{\text{res}}^{2}+f_{\text{res}}^{2}\right)^{2}} \tag{25}\]  
of the amplitude frequency response \(|\tilde{G}(f)|\) (Equation (4)) and the phase frequency response \(\tilde{\gamma}(f)\) (Equation (5)), respectively, are required. These derivatives are depicted in Figure 4c,d and visualize that the maximum influence on the signal, i.e., the highest sensitivity, is given by the corresponding first derivative at those frequencies at which the second derivatives are zero. These frequencies are referred to as the optimum excitation frequencies.  
Determining the roots of the second derivatives by solving \(\partial^{2}|\tilde{G}(f)|/\partial f^{2}=0\) for \(f\) and \(\partial^{2}\tilde{\gamma}(f)/\partial f^{2}=0\) for \(f\), respectively, yields the two optimum excitation frequencies for the amplitude mode  
\[f_{\rm ex}^{\rm AM}=f_{\rm res}\bigg{(}1\mp\frac{1}{\sqrt{8}\;Q}\bigg{)} \tag{26}\]  
that agree with reported results in [28] and the optimum excitation frequency for the phase mode  
\[f_{\rm ex}^{\rm PM}=f_{\rm res}. \tag{27}\]  
At these optimum excitation frequencies, the values of the electrical sensitivities are given by  
\[S_{\rm elec}^{\rm AM}=\underbrace{\frac{\partial|\tilde{G}(f)|}{|\tilde{G}(f_ {\rm ex}^{\rm AM})|\;\partial f}}_{\frac{\sqrt{8}\;Q}{3}}\Bigg{|}_{f=f_{\rm ex }^{\rm AM}}=\pm\frac{4\sqrt{3}}{9}\frac{Q^{2}}{f_{\rm res}}\cdot\frac{3}{\sqrt {6}\;Q}=\pm\frac{2\sqrt{2}}{3}\frac{Q}{f_{\rm res}} \tag{28}\]  
and  
\[S_{\rm elec}^{\rm PM}=\frac{\partial\tilde{\gamma}(f)}{\partial f}\Bigg{|}_{ f=f_{\rm ex}^{\rm PM}}=-\frac{2Q}{f_{\rm res}}. \tag{29}\]  
Figure 4: Illustration of the relations between shifted amplitude (**a**) and phase (**b**) frequency responses due to a detuned resonance frequency and the corresponding electrical sensitivities. Highest sensitivities are given by the corresponding first derivative at those frequencies at which the second derivatives are zero (**c**,**d**). These frequencies are referred to as the optimum excitation frequencies. The plots are based on the sensor parameters specified in Table 1.