motion of the cantilever's tip using a laser beam and a photosensitive detector ([17] p. 196). Concerning the signal-to-noise ratio (SNR), the optical beam deflection method is equivalent to the Michelson interferometer [18]. However, to eliminate the optical displacement sensor that occupies most of the space in such systems, piezoresistive [19], capacitive [20], and piezoelectric [21] readout schemes have been presented [22].  
Micromechanical cantilevers combined with a piezoelectric layer offer the great advantage that the piezoelectric material can be used to drive the resonator, as well as to detect its state. Although more sophisticated structures have been presented [23], most piezoelectric cantilevers have two electrodes, so the cantilever can be interpreted as a simple electrical two-pole (one-port) with a lumped-element equivalent electrical circuit in accordance with Figure 1. Based on the analogies between mechanical and electromechanical systems [24], the electromechanical resonator can be modeled as an electrical series resonance circuit with the motional resistance \(R_{\mathrm{m}}\), the motional capacitance \(C_{\mathrm{m}}\), and the motional inductance \(L_{\mathrm{m}}\)[25]. The capacitance \(C_{0}\) results from the two electrodes that typically form a plate capacitor around the piezoelectric layer. Its dielectric losses are covered by the resistance \(R_{0}\).  
Apart from pulse mode operation, the two common continuous operation modes are often referred to as impedance-based measurement and oscillator-based measurement, respectively [25]. A closed-loop oscillator-based system contains a positive feedback network and generates an oscillating signal that is frequency-modulated (FM) by the measurand, while the signal's amplitude is usually constant. On the contrary, open-loop impedance-based systems are characterized by amplitude-modulated (AM) and phase-modulated (PM) signals. Apart from practical differences between closed-loop and open-loop operation, FM and PM sensor systems are fully equivalent in terms of the minimum reachable limit of detection [26; 27]. For open-loop operation, from previous studies it is known that the maximum sensitivity in AM mode is achieved by driving the resonator at a certain frequency [28] that is not equal to the frequency for which the sensitivity in PM mode is at its maximum [29]. However, a comprehensive analysis of the differences between AM and PM modes, especially with respect to the reachable frequency-dependent limits of detection, has not previously been performed and is presented in this article.  
This article is organized as follows: Section 2 presents some fundamental characteristics of electromechanical resonators and introduces the properties of a magnetoelastic cantilever-type magnetic field sensor that is exemplarily utilized to illustrate the deduced relations. Section 3 presents the considered readout system utilized to reconstruct the time-varying physical quantity being measured. Both expressions for the frequency-dependent sensitivities, as well as frequency-dependent amplitude noise and phase noise, are deduced in Section 4 and Section 5, respectively. Based on these, Section 6 presents the resulting limits of detection. This article finishes with a summary of the findings in Section 7.  
All expressions are given as a function of the two basic parameters of a resonator, the resonance frequency \(f_{\mathrm{res}}\) and the quality factor \(Q\). However, please note that several simplifying assumptions are made. It is assumed that the quality factor is independent of the measurand, and \(Q>100\), which allows us to neglect a resonance detuning due to mechanical losses. All given results apply not only to cantilever-type sensors but to all types of micromechanical resonators that can be described by a damped harmonic oscillator.  
Figure 1: Lumped-element equivalent electrical circuit of a piezoelectric resonator.