Operation of a high quality-factor gyroscope in electromechanical nonlinearities regime

Abstract

This paper describes the operation of a high quality factor gyroscope in various regimes where electromechanical nonlinearities introduce different forms of amplitude-frequency (A-f ) dependence. Experiments are conducted using an epitaxially-encapsulated 2 × 2 mm 2 quad-mass gyroscope (QMG) with a quality factor of 85 000. The device exhibits thirdorder Duffing nonlinearity at low bias voltages (15 V) due to the mechanical nonlinearity in the flexures and at high bias voltages (35 V) due to third-order electrostatic nonlinearity. At intermediate voltages (~26 V), these third-order nonlinearities cancel and the amplitudefrequency dependence is greatly reduced. A model is developed to demonstrate the connection between the electromechanical nonlinearities and the amplitude-frequency dependence, also known as the backbone curve. Gyroscope operation is demonstrated in each nonlinear operating regime and the key performance measures of the gyroscope's performance, angle random walk (ARW) and bias instability, are measured as a function of drive-mode vibration amplitude. We find that low ARW can be achieved even though the gyroscope's drive mode exhibits large amplitude-frequency dependence, and that bias instability is largely independent of the operating regime.

Synchronization of a micromechanical oscillator in different regimes of electromechanical nonlinearity

“…39 . By further increasing the bias voltage to V b ¼ 27 V (green curve), the 3rd-order mechanical and electrostatic nonlinearities cancel (for this reason, this bias point has been called the optimum voltage 3,8 ) and the 5th-order nonlinearity dominates the frequency response (a ¼ 0). Complete analysis of r n is presented elsewhere; 3 here, we focus on studying the synchronization range, which is the difference between the maximum and minimum values of the detuning parameter, r s .…”

“…Due to their small size, low power consumption, waferscale manufacture, and compatibility with integrated circuit fabrication processes, silicon-based micromachined oscillators have begun to replace conventional quartz oscillators in many applications. 1 In applications such as inertial sensors 2,3 and clocks, 4 oscillators based on high quality-factor resonators have advantages of lower thermomechanical noise and lower phase noise. However, when operated at the large vibration amplitudes necessary to achieve high signal-to-noise ratio, high quality-factor resonators are subject to nonlinear phenomena that conventional oscillator designers typically try to avoid.…”

“…The device is operated at a vibration amplitude of up to 580 nm that is large relative to the 3 lm flexure width and 1.5 lm electrode gap, causing it to exhibit nonlinear behavior, documented in detail elsewhere. 3 Synchronization experiments were conducted using a resonator mounted on a printed circuit board (PCB) containing input and output amplifiers and connected to a lock-in amplifier (LIA, Zurich Instruments HF2LI). Using a phase-locked loop (PLL) implemented inside the LIA, the resonator operates as a self-sustained oscillator (e.g., a van der Pol self-sustained oscillator 29,30 ).…”

“…[31][32][33] While complete analysis of electromechanical nonlinearity can be found in the literature (e.g., Refs. 34 and 35), assuming stable oscillation at large amplitudes and using a Taylor series expansion for the electrostatic and mechanical forces, a simplified, forced equation of motion, also used elsewhere, 36,37 including only odd-power stiffness nonlinearities due to the symmetry of the structure, 3 can be written as…”

Mutual 3:1 subharmonic synchronization in a micromachined silicon disk resonator

“…As a result of cubic stiffnesssoftening introduced by the parallel-plate electrodes, 33 the two modes exhibit Duffing nonlinear amplitude-frequency (A-f) dependence. 32,34 To effectively study the synchronization range and the interactions of the two oscillators within this range, it is essential to be able to change the frequency of one oscillator independent from that of the other oscillator. Figure 1(c) presents measurements to show that the frequency of the 3h oscillator can be controlled independently from the frequency of the 9h oscillator using voltage applied to a tuning electrode.…”