Sarah H. Nitzan

Epitaxially-encapsulated polysilicon disk resonator gyroscope

We present a 0.6 mm diameter, 20 μm thick epitaxially-sealed polysilicon disk resonator gyro (DRG). High Q (50,000) combined with electrostatic mode-matching and closed-loop quadrature null performed by dedicated electrode sets enables a scale-factor of 0.286 mV/(°/s) and Angle Random Walk (ARW) of 0.006 (°/s)/√Hz. Without precise control of temperature, the minimum Allan deviation is 3.29 °/hr.

Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope.

Parametric amplification, resulting from intentionally varying a parameter in a resonator at twice its resonant frequency, has been successfully employed to increase the sensitivity of many micro- and nano-scale sensors. Here, we introduce the concept of self-induced parametric amplification, which arises naturally from nonlinear elastic coupling between the degenerate vibration modes in a micromechanical disk-resonator, and is not externally applied. The device functions as a gyroscope wherein angular rotation is detected from Coriolis coupling of elastic vibration energy from a driven vibration mode into a second degenerate sensing mode. While nonlinear elasticity in silicon resonators is extremely weak, in this high quality-factor device, ppm-level nonlinear elastic effects result in an order-of-magnitude increase in the observed sensitivity to Coriolis force relative to linear theory. Perfect degeneracy of the primary and secondary vibration modes is achieved through electrostatic frequency tuning, which also enables the phase and frequency of the parametric coupling to be varied, and we show that the resulting phase and frequency dependence of the amplification follow the theory of parametric resonance. We expect that this phenomenon will be useful for both fundamental studies of dynamic systems with low dissipation and for increasing signal-to-noise ratio in practical applications such as gyroscopes.

Encapsulated high frequency (235 kHz), high-Q (100 k) disk resonator gyroscope with electrostatic parametric pump

In this paper, we explore the effects of electrostatic parametric amplification on a high quality factor (Q > 100 000) encapsulated disk resonator gyroscope (DRG), fabricated in 〈100〉 silicon. The DRG was operated in the n = 2 degenerate wineglass mode at 235 kHz, and electrostatically tuned so that the frequency split between the two degenerate modes was less than 100 mHz. A parametric pump at twice the resonant frequency is applied to the sense axis of the DRG, resulting in a maximum scale factor of 156.6 μV/(°/s), an 8.8× improvement over the non-amplified performance. When operated with a parametric gain of 5.4, a minimum angle random walk of 0.034°/√h and bias instability of 1.15°/h are achieved, representing an improvement by a factor of 4.3× and 1.5×, respectively.

Silicon MEMS Disk Resonator Gyroscope With an Integrated CMOS Analog Front-End

We present a 2-mm diameter, 35-μm-thick disk resonator gyro (DRG) fabricated in <;111> silicon with integrated 0.35-μm CMOS analog front-end circuits. The device is fabricated in the commercial InvenSense Fabrication MEMSCMOS integrated platform, which incorporates a wafer-level vacuum seal, yielding a quality factor (Q) of 2800 at the DRGs 78-kHz resonant frequency. After performing electrostatic tuning to enable mode-matched operation, this DRG achieves a 55 μV/°/s sensitivity. Resonator vibration in the sense and drive axes is sensed using capacitive transduction, and amplified using a lownoise, on-chip integrated circuit. This allows the DRG to achieve Brownian noise-limited performance. The angle random walk is measured to be 0.008°/s/√(Hz) and the bias instability is 20°/h.

Frequency-Modulated Lorentz Force Magnetometer With Enhanced Sensitivity via Mechanical Amplification

This letter presents a micromachined silicon Lorentz force magnetometer, which consists of a flexural beam resonator coupled to current-carrying silicon beams via a microleverage mechanism. The flexural beam resonator is a force sensor, which measures the magnetic field through resonant frequency shift induced by the Lorentz force, which acts as an axial load. Previous frequency-modulated Lorentz force magnetometers suffer from low sensitivity, limited by both fabrication restrictions and lack of a force amplification mechanism. In this letter, the microleverage mechanism amplifies the Lorentz force, thereby enhancing the sensitivity of the magnetometer by a factor of 42. The device has a measured sensitivity of 6687 ppm/(mA · T), which is two orders of magnitude larger than the prior state-of-the-art. The measured results agree with an analytical model and finite-element analysis. The frequency stability of the sensor is limited by the quality factor (Q) of 540, which can be increased through improved vacuum packaging.

Single-Structure Micromachined Three-Axis Gyroscope With Reduced Drive-Force Coupling

This letter presents a micromachined silicon three-axis gyroscope based on a triple tuning-fork structure utilizing a single vibrating element. The mechanical approach proposed in this letter uses a secondary “auxiliary” mass rather than a major “proof” mass to induce motion in the proof mass frame for Coriolis force coupling to the sense mode. These auxiliary masses reduce the unwanted mechanical coupling of force and motion from the drive mode to the three sense modes. The experimental data show that the bias error due to coupling is reduced by a factor up to 10, and the bias instability of each sense axis is reduced by a factor of up to 3 when the gyroscope is actuated using the auxiliary masses rather than the major masses. The gyroscope exhibits a bias instability of 0.016°/s, 0.004°/s, and 0.043°/s for the x-, y-, and z-sense modes, respectively. Furthermore, initial temperature characterization results show that the gyroscope actuated by the auxiliary masses ensures a better bias instability performance in each sense axis over a temperature range from 10 °C to 50 °C in comparison with the gyroscope actuated by the major masses.

Impact of gyroscope operation above the critical bifurcation threshold on scale factor and bias instability

This paper investigates the impact of operating a vibratory rate gyro (VRG) at large oscillation amplitude where the VRGs driven axis behaves like a nonlinear oscillator, described by the Duffing equation. Although open-loop resonators operating above a critical amplitude exhibit catastrophic jump instabilities, we demonstrate that through closed-loop operation, the drive axis can be stably operated at an amplitude above this threshold without impacting drive-axis stability or bias instability, resulting in decreased Angle Random Walk (ARW).

Impact of Synchronization in Micromechanical Gyroscopes

In this paper, we study the occurrence of synchronization between the two degenerate resonance modes of a microdisk resonator gyroscope. Recently, schemes involving the simultaneous actuation of the two vibration modes of the gyroscope have been implemented as a promising new method to increase their performance. However, this strategy might result in synchronization between the two modes, which would maintain frequency mode-matching but also may produce problems, such as degrading stability and sensitivity. Here, we demonstrate for the first time synchronization between the degenerate modes of a microgyroscope and show that synchronization arising from mutual coupling dramatically reduces frequency instability at the cost of increased amplitude instability. We present an alternative synchronization scheme that suppresses this drawback while still taking advantage of a passive frequency mode-match operation. [DOI: 10.1115/1.4036397]

Predicting the closed-loop stability and oscillation amplitude of nonlinear parametrically amplified oscillators

This work investigates the closed-loop operation of microelectromechanical oscillators in the presence of both cubic (Duffing) nonlinearities and parametric amplification. We present a theoretical model for this system that enables us to predict oscillation amplitude and instability and experimentally verify it using a silicon disk resonator with a quality factor (Q) of 85 000 and a natural frequency of 251 kHz. We determine that, contrary to previous understanding gained from analyzing the open-loop system, the presence of cubic nonlinearities does not limit the maximum stable oscillation amplitude if the resonator is operated in a closed loop. In addition, the stability and amplitude behavior predicted by our theoretical model are independent of the presence or severity of cubic nonlinearities, or on drive amplitude.

Countering the Effects of Nonlinearity in Rate-Integrating Gyroscopes

This paper addresses the impact of cubic nonlinearity on the operation of a rate-integrating gyroscope (RIG). It is demonstrated that below the bifurcation threshold, cubic nonlinearity results in angle-dependent frequency split between the two resonant modes of the gyroscope, which impacts angle-dependent bias, quadrature error, and controller efficacy in addition to distorting the scale factor due to off-resonant excitation. These errors are experimentally demonstrated using a high-Q disk resonator gyroscope, which are shown to be in close agreement with theory. A method of compensating for angle-dependent frequency error is proposed and experimentally validated. It is demonstrated that mode mismatch can be experimentally reduced to the level of thermal noise, effectively cancelling the effects of nonlinearity and eliminating the distortion of readout angle.