## 3 Sensor System  
For operating the cantilever-type sensor, an electrical system as depicted in Figure 3 is considered [29]. The basic principle is based on driving both the sensor and a capacitive branch to neutralize the parallel resonance due to the cantilever's static capacitance \(C_{0}\). For this, a differential input transimpedance/charge amplifier is utilized. Please note that the figure depicts a simplified illustration of such an amplifier. Refer to [39] for further details on differential input charge amplifiers. The excitation signal \(v_{\text{ex}}(t)=\hat{V}_{\text{ex}}\cos(2\pi f_{\text{ex}}t)\) with the excitation amplitude \(\hat{V}_{\text{ex}}\) and the excitation frequency \(f_{\text{ex}}\) leads to currents through the sensor and through the neutralization branch that, in the frequency domain, can be described by \(I_{\text{sensor}}(f)=V_{\text{ex}}(f)\cdot\hat{Y}_{\text{sensor}}(f)\) and \(I_{\text{n}}(f)=V_{\text{ex}}(f)\cdot\hat{j}2\pi fC_{\text{n}}\) where \(V_{\text{ex}}(f)\) is the amplitude spectrum of \(v_{\text{ex}}(t)\). By means of the transimpedance \(T(f)=-(j2\pi fC_{\text{f}})^{-1}\) both currents are transformed into the two voltage spectra  
\[V_{\text{sensor}}(f)=T(f)I_{\text{sensor}}(f)=T(f)V_{\text{ex}}(f)\cdot\left( \frac{1}{R_{0}(f_{\text{res}})}+j\left(2\pi fC_{0}+\frac{\tilde{C}(f)}{QR_{ \text{m}}}\right)\right) \tag{10}\]  
and  
\[V_{\text{n}}(f)=T(f)I_{\text{n}}(f)=T(f)V_{\text{ex}}(f)\cdot j2\pi fC_{\text{ n}} \tag{11}\]  
yielding a differential output voltage spectrum of  
\[V_{\text{d}}(f)=V_{\text{sensor}}(f)-V_{\text{n}}(f)=T(f)V_{\text{ex}}(f) \left(\frac{1}{R_{0}(f_{\text{res}})}+j\frac{\tilde{C}(f)}{QR_{\text{m}}}\right) \tag{12}\]  
Figure 3: Electrical readout system for the measurement of a physical quantity (here, a magnetic flux density) that affects the resonance frequency of an electromechanical resonator. The sensor is driven in the vicinity of its resonance frequency by an electrical excitation signal \(v_{\text{ex}}(t)\), leading to an amplitude-modulated and phase-modulated signal \(v_{\text{d}}(t)\) at the output of a differential input transimpedance amplifier. A subsequent quadrature detector performs both the amplitude demodulation and the phase demodulation. if the neutralization capacitance \(C_{\rm n}\) is matched to the sensor's static capacitance \(C_{0}\). The dielectric losses taken into account by \(R_{0}\) are important for the noise analysis in Section 5 below. However, due to its high resistance (compare Table 1), the impact of \(R_{0}\) can be neglected here, yielding  
\[V_{\rm d}(f)=j\ T(f)V_{\rm ex}(f)\ \frac{\tilde{G}(f)}{QR_{\rm m}}=-\frac{V_{\rm ex }(f)\cdot\tilde{G}(f)}{2\pi f_{\rm ex}C_{\rm f}\cdot QR_{\rm m}}. \tag{13}\]  
This spectrum corresponds with the time domain signal  
\[v_{\rm d}(t)=-\frac{\tilde{V}_{\rm ex}}{2\pi f_{\rm ex}C_{\rm f}\cdot QR_{\rm m }}\cdot|\tilde{G}(f_{\rm ex},B(t))|\cdot\cos\left(2\pi f_{\rm ex}t+\tilde{\gamma }(f_{\rm ex},B(t))\right) \tag{14}\]  
when considering that the resonator, i.e., \(\tilde{G}(f)\) (Equation (3)), is affected by an ambient magnetic flux density \(B(t)=B_{\rm bias}+B_{\rm x}(t)\) that consists of the constant bias to set the sensor's magnetic operating point (initial state) and a time-dependent magnetic signal \(B_{\rm x}(t)\) to be measured. Both the magnitude  
\[|\tilde{G}(f_{\rm ex},B(t))|=|\tilde{G}(f_{\rm ex},B_{\rm bias})|\cdot\left[1 +\underbrace{\frac{\partial|\tilde{G}(f)|}{|\tilde{G}(f_{\rm ex},B_{\rm bias} )|\,\partial B_{\rm x}}}_{S_{\rm AM}}\cdot B_{\rm x}(t)\right] \tag{15}\]  
and the phase  
\[\tilde{\gamma}(f_{\rm ex},B(t))=\tilde{\gamma}(f_{\rm ex},B_{\rm bias})+ \underbrace{\frac{\partial\tilde{\gamma}(f)}{\partial B_{\rm x}}}_{S_{\rm PM}} \cdot B_{\rm x}(t). \tag{16}\]  
of the resonator's frequency response \(\tilde{G}(f)\) can be described by a sum of a static term, defined by \(B_{\rm bias}\), and a (fractional) dynamic term that changes with \(B_{\rm x}(t)\).