## 7 Conclusions  
In this article, the performance of quasi-resonantly driven piezoelectric sensors operated in amplitude mode and phase mode has been investigated. Regardless of the physical quantity that a micromechanical sensor with a piezoelectric layer is designed to detect, changes in its resonance frequency result in both amplitude (AM) and phase modulation (PM) of the sensor signal if the sensor is operated in an open-loop configuration. The degree of the respective modulation is essentially determined by the frequency of the signal used to electrically excite the sensor. The objective of the analysis presented here was to evaluate the respective performance in terms of the minimum reachable limit of detection (LOD) as a function of the frequency of the physical signal to be measured. Thus, based on the fundamental properties of micromechanical resonators, comprehensive analyses of the frequency-dependent sensitivities, as well as the frequency-dependent amplitude noise and phase noise, were performed and respective analytical expressions were derived. Different optimal excitation frequencies result for the two types of operation, and these also lead to different limits of detection. It has been shown that the low-frequency LOD in phase mode is always about 6 dB better than the LOD in amplitude mode. In addition, the bandwidth in terms of detectivity is generally significantly larger, and never worse, in phase mode compared with amplitude mode. Accordingly, an evaluation of the sensor signal's phase with appropriate excitation of the sensor is generally preferable to an evaluation of the sensor signal's amplitude. To illustrate all the analytical relationships in this article, an exemplary piezoelectric magnetic field sensor was considered. In Table 2, all electromechanical parameters of this sensor, and further parameters of the employed sensor system, as well as various numerical values for the derived equations, are summarized. Even if the exemplary sensor is a magnetic field sensor, all derived equations have general validity for all micromechanical sensors which can be described by the differential equation of motion of a damped driven harmonic oscillator (Equation (1)). Several directions exist for the future continuation of the work presented here. First, confirmation of the theory developed here on the basis of measurements is advisable. Provided that the parasitic capacitance of the sensor is compensated as presented here, PM operation would always be preferred. However, it would also be interesting to compare AM and PM operation modes without neutralization, because the effective impedance leading to the electrical sensitivity can take a significantly different form. Due to the asymmetry caused by the additional parallel resonance, however, the calculations are considerably complicated, meaning that numerical simulation would possibly be more appropriate in this case. Figure 8: Illustrations of the limits of detection in AM mode (**a**) and in PM mode (**b**). For both types of operation, the low-frequency detectivity is limited by the thermal mechanical noise of the resonator. The bandwidth in which the detectivity is constant with frequency is significantly higher in PM mode. In general, the bandwidth can never be higher in AM mode than in PM mode. In addition, independent of the dominant type of noise, the detectivity in PM mode is always more than 6 dB better compared with an operation in AM mode. The plots are based on the sensor parameters given in Table 1.  
**Author Contributions:** Conceptualization: P.D., Methodology: P.D., Software: P.D., Validation, P.D. and M.H., Formal analysis: P.D. and M.H., Investigation: P.D., Writing--original draft: P.D., Writing--review & editing: P.D. and M.H., Visualization: P.D. and M.H., Supervision: M.H., Project administration: M.H., Funding acquisition: M.H. All authors have read and agreed to the published version of the manuscript.