In this paper, the Reynolds equation of air damping of MEMS wine-glass resonator is solved by numerical calculation method considering the air slip boundary conditions. The corresponding pressure distribution between internal gaps is obtained by MATLAB software. And then size effects of resonator on air damping and quality factor are analyzed. It is found that MEMS disk radius and thickness have slight effects on the change of the pressure distribution. But the effects of spacing between disk body and electrodes are very significant. The designed ratio value of disk amplitude and spacing should not be over 70%, otherwise the changes of air pressure between gaps will be very remarkable and increased exponentially, the air damping will begin to increase doubly, and the quality factor will be reduced greatly. The analyzed results are very meaningful for optimizing design size of MEMS disk resonator.

Ideally, a mechanically coupled array-composite resonator can improve linearity and motional resistance by a factor equal to the number of constituent resonators used in the array. Owing to fabrication nonidealities, however, random variations in the dimensions of resonators and coupling beams often compromise the actual increase in output current. As a result, previous array composites have fallen short of their expected impedances and linearity improvements, particularly when the number of constituent resonators used in the array is large. Therefore, in this paper, we present a detailed theoretical analysis on the centroid offset of beams caused by process variations and some related substantive issues. Specifically, an equivalent circuit model for a coupling trapezoid beam is first deduced in the case of an offset centroid. Then, a model for a coupled double-disk resonator including a beam with an offset centroid is established by combining the equivalent circuit model with an existing circuit model of a radial-contour mode disk resonator. Finally, numerical results are obtained by simulations using ANSYS and PSpice with the mechanical and electrical models, respectively. It is shown that the resonant frequency varies by 45.7, 110, and 250 ppm when the beam centroid shifts by 6, 12, and 18‰, respectively. When the centroid shifts towards one disk by 18‰, the output current amplitude of this disk decreases by 38.7‰ to 1.657 μA (compared with 1.724 μA at zero offset), while the output current amplitude of the other disk increases by 35.9‰ to reach 1.785 μA. In addition, when there is an offset, the Q of the resonator decreases. In particular, when the centroid has an 18‰ offset, Q decreases by nearly 19‰.

Capacitive micro-electromechanical system (MEMS) disk resonators fabricated by deep reactive-ion etching (DRIE) have large sensitive capacitances and low motional resistances. However, for the MEMS disk structure with a high aspect ratio, the cross section takes on a trapezoidal profile, which will affect the performance of the resonator. In this study, we firstly analyzed the electrostatic tuning mechanism of electrical stiffness produced by the electrostatic force, and the dependence of resonance frequency variation on the inclination angle and dc bias voltage is obtained and the electromechanical coupling strength was changed owing to the inclination angle. Secondly, after analyzing the feasibility of overcoming the inclination effect by introducing a tuning disk array, the optimal tuning voltage of the inclined disk resonator array and the scale of the array with a small motional resistance can be obtained. The results show that, for an array with the same inclination angle of 0.1° and biased at 10 V, when the tuning voltage is 20 V, the relative error of the resonance frequency can be reduced to 12.4 ppm. In addition, the optimal tuning voltage increases as the inclination angle increases, and when the inclination angle is 0.3°, the optimal tuning voltages are 28.68 and 57.35 V for the dc bias voltages of 10 and 30 V, respectively. If the motional resistance needs be reduced to 50 Ω, the integrated number of disk resonators will increase to 1048, and the optimal tuning voltage can reach 37.5 V. These results can provide some theoretical basis for the large-scale integration of the disk resonator array in the future.