A negative-capacitance equivalent circuit model for parallel-plate capacitive-gap-transduced micromechanical resonators

Abstract

A small-signal equivalent circuit for parallel-plate capacitive-gap-transduced micromechanical resonators is introduced that employs negative capacitance to model the dependence of resonance frequency on electrical stiffness in a way that facilitates circuit analysis, that better elucidates the mechanisms behind certain potentially puzzling measured phenomena, and that inspires circuit topologies that maximize performance in specific applications. For this work, a micromechanical disk resonator serves as the vehicle with which to derive the equivalent circuits for both radial-contour and wine-glass modes, which are then used in circuit simulations (via simulation) to match measurements on actual fabricated devices. The new circuit model not only correctly predicts the dependence of electrical stiffness on the impedances loading the input and output electrodes of parallel-plate capacitive- gap-transduced micromechanical device, but does so in a visually intuitive way that identifies current drive as most appropriate for applications that must be stable against environmental perturbations, such as acceleration or power supply variations. Measurements on fabricated devices confirm predictions by the new model of up to 4× improvement in frequency stability against dc-bias voltage variations for contour- mode disk resonators as the resistance loading their ports increases. By enhancing circuit visualization, this circuit model makes more obvious the circuit design procedures and topologies most beneficial for certain mechanical circuits, e.g., filters and oscillators.

Low-Power MEMS-Based Pierce Oscillator Using a 61-MHz Capacitive-Gap Disk Resonator

“…The Pierce oscillator topology used here combines a twoport frequency-selective vibrating MEMS wine-glass disk resonator [20] wired in closed-loop positive feedback with a single transconducting gain device, as shown in Fig. 1.…”

“…1(b), which comprises expansion and contraction of the disk along the orthogonal axes. The frequencies of the modes derive from the transcendental equations summarized in Table I, with resultant operating frequency for the (2, 1) mode used in the present work taking the form [20], [23] where ω nom is the angular resonance frequency and R, E, σ , and ρ are the disk radius, Young's modulus, Poisson ratio, and density, respectively, and K is a material-dependent parameter equal to 0.373 [20]. Equation (3) specifically gives the nominal frequency of the isolated disk, with no outside interactions, e.g., no applied voltages that can shift the frequency.…”

Development of Nonlinear Electromechanical Coupled Macro Model for Electrostatic MEMS Cantilever Beam

“…The stiffness expression is derived at the position of the maximum deflection y L of cantilever i.e. at its tip [18]. The stiffness k(Nm −1 ) is defined as fL e /y L , where f is distributed transverse load per unit length over partial length L e originating from the tip of the cantilever beam as shown in Fig.…”

“…The surface leakage current flows due to the generation of electric field between the contact pads of cantilever and bottom electrode and increases with voltage. The spring softening of beam is a coupled field effect and is modelled mathematically as if it was a negative capacitance [18] by series capacitance, -C 1 referred to electrical side.…”

Micromechanical disk array for enhanced frequency stability against bias voltage fluctuations

“…Pursuant to suppress the electrical stiffness and enhance frequency stability, the work in this report develops a new negative capacitance based equivalent circuit model [8], which reveals that capacitive-gap transduced micromechanical resonators can offer better frequency stability against environment fluctuations when used in large mechanicallycoupled arrays [9]. The key to enhanced frequency stability is the electrode-to-resonator capacitance (Co) generated by the parallel combination of input/output electrodes overlapping each resonator in the array that in turn reduces the efficacy of the bias voltagecontrolled electrical stiffness.…”