A Capacitive Microcantilever: Modelling, Validation, and Estimation Using Current Measurements present a mathematical model for the dynamics of an electrostatically actuated micro-

cantilever. For the common case of cantilevers excited by a periodic voltage, we show that the underlying linearized dynamics are those of a periodic system described by a Mathieu equation. We present experimental results that confirm the validity of the model, and in particular, illustrate that parametric resonance phenomena occur in capacitively actuated micro-cantilevers. We propose a system where the current measured is used as the sensing signal of the cantilever state and position through a dynamical observer. By investigating how the best achievable performance of an optimal observer depends on the excitation frequency, we show that the best such frequency is not necessarily the resonant frequency of the cantilever. @DOI: 10.1115/1.1767851# The recent advances in the field of miniaturization and micro fabrication have paved the way for a new range of applications, bringing along the promise of unprecedented levels of performance. In particular, scanning probe devices have proven to be extremely versatile instruments, with applications that range from surface imaging at the atomic scale @1#, to ultra high density data storage and retrieval @2#, and to biosensors @3,4#, to cite but a few. The working principle for most of these devices is based on a measurement of displacement. As an example, consider imaging in atomic force microscopy: the topography of a sample is reconstructed from the displacement of the cantilever-probe, caused by the interaction forces with the sample @5,6#. In biosensors applications the displacement of a cantilever can be related to the binding of molecules on the ~activated! surface of the cantilever beam, and is therefore used to compute the strength of these bonds, as well as the presence of specific reagents in the solution under consideration @7,8#. It is clear that the sensitivity of these devices strongly depends on the smallest detectable motion, which poses a constraint on the practically vs. theoretically achievable performance. In order to make the gap between the two smaller, while at the same time providing compactness of devices and faster dynamics, much of the research effort has been focused on the design of integrated detection schemes. The most common solutions for integrated detection make use of the piezoresistive, @9,10#, piezoelectric @11‐13#, thermal expansion @14# or capacitive effects @15‐17#. A major advantage of capacitive detection, is the fact that it offers both electrostatic actuation as well as integrated detection, without the need for an additional position sensing device. The common scheme used in capacitive detection is to apply a second AC voltage at a frequency much higher than the mechanical bandwidth of the cantilever. The current output at that frequency is then used to estimate the capacitance, and consequently the cantilever position. This sensing scheme is the simplest position detection scheme available, however, it is widely believed to be less accurate than optical levers or piezoresistive sensing. The device that we propose is an electrostatically actuated microcantilever. More precisely, in our design the microcantilever constitutes the movable plate of a capacitor and its displacement is controlled by the voltage applied across the plates. In order to measure the cantilever displacement, we propose a novel scheme that avoids the use of a high frequency probing signal by the use of a dynamical state observer, whose input is the current through the capacitive cantilever. For the purpose of implementation, this scheme offers significant advantages as it involves simpler circuitry. By using an optimal observer, or by tuning the observers gains, it is conceivable that a high fidelity position measurement can be obtained, thus improving resolution in atomic force microscopy applications. In this paper, we present a model for this electrostatically actuated microcantilever. Using simple parallel plate theory and for the common case of sinusoidal excitation, it turns out that its dynamics are governed by a special second order linear periodic differential equation, called the Mathieu equation. We produce experimental evidence that validates the mathematical model, including a mapping of the first instability region of the Mathieu equation. The optimal observer problem that was formulated also in @18# is solved here following a different and simpler approach. This optimal design is then used as an analysis tool to select the frequency of excitation that corresponds to the best achievable observer performance. In other words, the optimal observer design is used to actually design the system ~rather than the observer! ,b y selecting the excitation frequency that produces the least estimation error. Interestingly, it turns out that this frequency is not necessarily the resonant frequency of the cantilever, and it depends on the statistics of the measurement and process noise.