MEMS Oscillator Research Articles

Frequency locking in a nonlinear MEMS oscillator driven by harmonic force and time delay

Vibrations of a nonlinear self- and parametrically excited MEMS device driven by external excitation and time delay inputs are analysed in the paper. The model of MEMS resonator includes a nonlinear van der Pol function producing self-excitation, a periodically varied coefficient which represents Mathieu type of parametric excitation and furthermore, periodic force acting on the resonator. Analysis of frequency locking zones is presented with suggestions for a strategy of a closed loop control. Interactions between self- and parametric excitation lead to quasi-periodic oscillations but under specific conditions the motion becomes harmonic. The so called frequency locking, near the resonance zones is observed. This is caused by the second kind Hopf bifurcation (Neimark–Sacker bifurcation). The amplitudes of periodic oscillations are determined analytically by the multiple time scale method (MS) in the second order perturbation. The effect of external force has been observed by the internal loop occurring inside the frequency locking zone. The localisation of the zones and existence of the internal loop can be controlled by a selection of gains and time delay of displacement or velocity feedbacks.

An analytical formulation for phase noise in MEMS oscillators.

In recent years, there has been much interest in the design of low-noise MEMS oscillators. This paper presents a new analytical formulation for noise in a MEMS oscillator encompassing essential resonator and amplifier nonlinearities. The analytical expression for oscillator noise is derived by solving a second-order nonlinear stochastic differential equation. This approach is applied to noise modeling of an electrostatically addressed MEMS resonator-based square-wave oscillator in which the resonator and oscillator circuit nonlinearities are integrated into a single modeling framework. By considering the resulting amplitude and phase relations, we derive additional noise terms resulting from resonator nonlinearities. The phase diffusion of an oscillator is studied and the phase diffusion coefficient is proposed as a metric for noise optimization. The proposed nonlinear phase noise model provides analytical insight into the underlying physics and a pathway toward the design optimization for low-noise MEMS oscillators.

Reliability for systems of degrading components with distinct component shock sets

This paper studies reliability for multi-component systems subject to dependent competing risks of degradation wear and random shocks, with distinct shock sets. In practice, many systems are exposed to distinct and different types of shocks that can be categorized according to their sizes, function, affected components, etc. Previous research primarily focuses on simple systems with independent failure processes, systems with independent component time-to-failure, or components that share the same shock set or type of shocks. In our new model, we classify random shocks into different sets based on their sizes or function. Shocks with specific sizes or function can selectively affect one or more components in the system but not necessarily all components. Additionally the shocks from the different shock sets can arrive at different rates and have different relative magnitudes. Preventive maintenance (PM) optimization is conducted for the system with different component shock sets. Decision variables for two different maintenance scheduling problems, the PM replacement time interval, and the PM inspection time interval, are determined by minimizing a defined system cost rate. Sensitivity analysis is performed to provide insight into the behavior of the proposed maintenance policies. These models can be applied directly or customized for many complex systems that experience dependent competing failure processes with different component shock sets. A MEMS (Micro-electro mechanical systems) oscillator is a typical system subject to dependent and competing failure processes, and it is used as a numerical example to illustrate our new reliability and maintenance models.

Synchronization in a coupled architecture of microelectromechanical oscillators

There has been much recent interest in engineering the phenomenon of synchronization in coupled micro-/nano-scale oscillators for applications ranging from precision time and frequency references to new approaches to information processing. This paper presents descriptive modelling detail and further experimental validation of the phenomenon of mutual synchronization in coupled MEMS oscillators building upon recent experimental validation of this concept by the present authors. In particular, the underlying dependence of the observation of synchronization on system parameters is studied through numerical and analytical modelling while considering essential nonlinearities in both the resonator and circuit domain. Experimental results demonstrating synchronized response are elaborated based on the realization of electrically coupled MEMS resonator based square-wave oscillators. The experimental results on frequency entrainment are found to be in general agreement with results obtained through analytical modeling and numerical simulation. The concept presented here is scalable and could be used to investigate the dynamics of large-arrays of coupled MEMS oscillators.

Modeling nonlinearities in MEMS oscillators.

We present a mathematical model of a microelectromechanical system (MEMS) oscillator that integrates the nonlinearities of the MEMS resonator and the oscillator circuitry in a single numerical modeling environment. This is achieved by transforming the conventional nonlinear mechanical model into the electrical domain while simultaneously considering the prominent nonlinearities of the resonator. The proposed nonlinear electrical model is validated by comparing the simulated amplitude-frequency response with measurements on an open-loop electrically addressed flexural silicon MEMS resonator driven to large motional amplitudes. Next, the essential nonlinearities in the oscillator circuit are investigated and a mathematical model of a MEMS oscillator is proposed that integrates the nonlinearities of the resonator. The concept is illustrated for MEMS transimpedance-amplifier- based square-wave and sine-wave oscillators. Closed-form expressions of steady-state output power and output frequency are derived for both oscillator models and compared with experimental and simulation results, with a good match in the predicted trends in all three cases.

MEMS oscillators – great growth potential and superior performance

MEMS oscillators – great growth potential and superior performance

The low-power potential of oven-controlled mems oscillators

It is shown that oven-controlled micro electromechanical systems (MEMS) oscillators have the potential of attaining a higher frequency stability, with a lower power consumption, than temperature-compensated crystal oscillators (TCXOs) and the currently manufactured MEMS oscillators.

Does greater piezo-resistive transduction give rise to higher anchor loss in a square-extensional mode micromechanical resonator?

The square-extensional (SE) mode resonator was used as the timing element in one of the first demonstrations of a MEMS oscillator meeting the GSM specification for phase noise. Crucial to its high Q was the use of T-shaped tethers in the anchoring design, which could also be used as piezo-resistors that could provide higher transduction. In this work, we examine the relationship between Q and corresponding piezo-resistive transduction efficiency, affected by changes in the compliance of the T-shaped tether anchoring design. Does a high Q necessarily mean a lower piezo-resistive transduction factor as an unavoidable performance tradeoff? Our measurement results, corroborated by finite element simulations, suggest that Q and the related piezo-resistive transduction factor are not opposed to each order in the SE mode resonator. In fact, the two indicative metrics of performance exhibit similar trends to the other with respect to changing the compliance of the anchoring design.

Comb-drive micro-electro-mechanical systems oscillators for low temperature experiments

We have designed and characterized micro-electro-mechanical systems (MEMS) for applications at low temperatures. The mechanical resonators were fabricated using a surface micromachining process. The devices consist of a pair of parallel plates with a well defined gap. The top plate can be actuated for shear motion relative to the bottom fixed plate through a set of comb-drive electrodes. Details on the operation and fabrication of the devices are discussed. The geometry was chosen to study the transport properties of the fluid entrained in the gap. An atomic force microscopy study was performed in order to characterize the surface. A full characterization of their resonance properties in air and at room temperature was conducted as a function of pressure, from 10 mTorr to 760 Torr, ranging from a highly rarefied gas to a hydrodynamic regime. We demonstrate the operation of our resonator at low temperatures immersed in superfluid (4)He and in the normal and superfluid states of (3)He down to 0.3 mK. These MEMS oscillators show potential for use in a wide range of low temperature experiments, in particular, to probe novel phenomena in quantum fluids.

Latest Technology, Technological Challenges, and Market Trends for Frequency Generating and Timing Devices [Application Notes

The second and third generation of MEMS oscillators will target a larger market for compact oscillators in order to meet TCXO and OCXO performance requirements. While silicon-based timing devices are still not as capable as crystal oscillators of undertaking sophisticated tasks , they are getting better, and will eventually replace crystals in many contexts. That will produce further synergies for the industry as mass production becomes cheaper and easier. These days, designers require higher frequencies and low jitter in oscillators, while buyers demand low cost and quick delivery. Timely oscillator options that can deliver the highest desired performance, while minimally compensating design steps are the key to cost-effective solutions. Fortunately, in the ongoing battle to push the limits of technology and lower component costs, oscillator manufacturers continue to close the gap between highlevel performance and cost-effective purchasing, with conventional crystal technology paired with configurable oscillator technology. Like every exciting new technology targeting mass markets and driven by start-ups, confusion or exaggeration are present, but all in all, we believe that MEMS oscillators will follow the successful bulk acoustic wave devices as the second RF MEMS mass product.

An Empirical Phase-Noise Model for MEMS Oscillators Operating in Nonlinear Regime

Nonlinearity of a silicon resonator can lead to improved phase-noise performance in an oscillator when the phase shift of the sustaining amplifier forces the operating point to a steeper phase-frequency slope. As a result, phase modulation on the oscillator frequency is minimized because the resonator behaves as a high-order phase filter. The effect of the increased filtering translates into phase-noise shaping that reflects superior overall performance. Nonlinear effects in MEMS oscillators can be induced via sufficient driving power, generating low-frequency nonwhite noise processes that need to be considered in a phase-noise description. Since the phase-frequency response is not symmetric for a nonlinear detuned resonator, an empirical model based on power series is proposed to describe its effect in the noise sources and to account for the observed higher effective quality factor of the oscillator, the reduction in the corner frequency, and elevated levels of flicker noise very close-to-carrier. The applicability of the presented phase-noise model is shown for three piezoelectric MEMS oscillators, producing a relative fitting error below 1% in all cases.

Behavioural modelling of MEMS oscillators and phase noise simulation

MEMS oscillators offer interesting prospects in terms of performance. Behavioural modelling in Verilog-A and phase noise simulation can help improving the current performances. The proposed macro-model permits multi-physic simulations including mechanical, piezoelectric and electrical analytic descriptions. Phase noise analysis are then performed with this model and compared to the standard Leeson phase noise calculation.

Hollow Square- and Ring-Plate MEMS Oscillators Embedded in a Phase-Locked Loop for Low Limit of Detection in Liquid

This letter demonstrates hollow plate microelectromechanical-system oscillators implemented into a digital phase-locked loop (PLL). The resonators feature innovative geometries: hollow square and ring plates, operating in Lamé or Kirkhope modes, respectively, with resonance frequencies comprised between 24 and 78 MHz. Advantageously, the ring-plate resonator presents better frequency stability when compared with its counterpart square plate. The PLL provides a real-time monitoring of the oscillators' frequency shift, which was tested by changing the electrostatic actuation amplitude. Ring-plate resonators show sensitivity between 38 and 232 Hz/V. Finally, the feedback loop was operated with a square-plate device filled with an aqueous solution demonstrating a sensitivity of up to 284 Hz/V.

Multitest MT9928xm deployed for calibration of MEMS oscillators

Multitest MT9928xm deployed for calibration of MEMS oscillators

Use of Electromechanical Feedback in MEMS for Suppressing Electronics Noise

Abstract At the pull-in point, a capacitive MEMS sensor becomes infinitely sensitive to applied force as the effective spring constant goes to zero because of electromechanical feedback We show that this phenomenon can be used to fully eliminate the noise contribution of readout electronics. Experimentally, we show that the electronics noise and interference contribution to system resolution could be suppressed by an order of magnitude, reaching the intrinsic resolution of the MEMS microphone. Experiments are in good agreement with a theory based on a small signal model of a harmonic MEMS oscillator. The technique allows the use of standard integrated electronics with noise-critical MEMS sensors, such as microphones, pressure sensors and accelerometers.

System-level circuit simulation of nonlinearity in micromechanical resonators

We here present an equivalent circuit model capable of capturing the effects of nonlinearity in micromechanical resonators, particularly those driven and sensed by capacitive transducers. A charge controlled capacitor and a variable capacitor are developed to capture the nonlinear nature of the mechanical restoring force and the capacitive transducer. Distinctive features of the frequency response (like spring softening, hysteresis) induced by mechanical and capacitive nonlinearity are highlighted by illustrating the simulation results of a well-studied longitudinal beam resonator, which are verified against analytical solution of the Duffing equation. The circuit simulation approach is capable of capturing prominent parametric excitation effects in narrow-gap capacitive transducers which are commonly excluded in the approximations of the analytical model for simplicity. Finally, we constructed the full equivalent circuit for a fully differential Lamé bulk mode microresonator. The comparison between experimental and simulation results shows the proposed model can well capture the nonlinear behavior of this resonator which illustrates an approach to realize full system-level circuit simulation of microresonator based devices, such as precision resonant sensors, mechanical RF filters and MEMS oscillators.

Stable Operation of MEMS Oscillators Far Above the Critical Vibration Amplitude in the Nonlinear Regime

In microelectromechanical systems resonators, nonlinear operation is feasible, but instabilities can arise if open-loop resonators operate above the critical vibration amplitude. This fact has led to a reluctance to operate resonator-based oscillators above this amplitude. This study experimentally demonstrates stable operation of these oscillators far beyond the critical vibration amplitude.

A design methodology for fully integrated MEMS and NEMS Pierce oscillators

This work proposes a generic methodology to evaluate the feasibility of designing a Pierce oscillator based on a micro or nanoresonator fabricated with any given CMOS technology and with the constraint of using only one single active transistor. Both the design and the transimpedance gain of the electronics are analytically determined for resonators with capacitive or resistive detection. This approach is applied and validated in the case of (a) an electrostatically actuated clamped-clamped beam resonator using a capacitive transduction, and (b) a piezoresistive crossbeam, both based on a standard 0.13μm CMOS technology.

D-1-5 Coupled Vibrating-Body Field Effect Transistorの特性解析(D-1 マイクロアクチュエータ,口頭発表:マイクロナノメカトロニクス)

In this study, we evaluated properties of Vibrating-Body Field Effect Transistor (VB-FET), which integrated MOSFET into MEMS oscillator, taking into account electro-mechanical interaction between gate and channel. Our modeling was started to express VB-FET as an equivalent circuit and then derived Lagrange's function and dissipation function of VB-FET from the equivalent circuit. From the modeling we calculated a transconductance and an output resistance of Coupled VB-FET and found that the value of the transconductance peak is increased up to 6 times and the output resistance of VB-FET is reduced by half at the resonance frequency. As described here, VB-FET has unique properties which offer new class of functions in MEMS.

Chaotic Phenomena in MEMS with Duffing Equation

Recently, there are many difficult for maintenance in the power in established sensor networks. In order to solve this problems, the power development has been interested using vibration in MEMS that insert the MEMS oscillator. In this paper, we propose the MEMS system with Duffing equation to generate vibration signal that can be use power signal in MEMS and confirm and verify the chaotic behaviors in vibration signal of MEMS by computer simulation. As a verification methods, we confirm the existence of period motion and chaotic motion by parameter variation through the time series, phase portrait, power spectrum and poincare map.