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

Lee, Hyung Kyu; Melamud, R.; Chandorkar, Saurabh A.; Salvia, J.; Yoneoka, S.; Kenny, Thomas W.

doi:10.1109/jmems.2011.2170821

Cited by 52 publications

(22 citation statements)

References 19 publications

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“…However, in this latter case, the resonator-drive phase difference is itself determined by the feedback, and both amplitude and frequency are single-valued functions of this phase. Therefore, all three operating conditions at the same frequency might be stable [18], and this is indeed confirmed by a stability analysis using Eq. (3), and by our measurements [Fig.…”

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confidence: 66%

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Surpassing Fundamental Limits of Oscillators Using Nonlinear Resonators

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In its most basic form an oscillator consists of a resonator driven on resonance, through feedback, to create a periodic signal sustained by a static energy source. The generation of a stable frequency, the basic function of oscillators, is typically achieved by increasing the amplitude of motion of the resonator while remaining within its linear, harmonic regime. Contrary to this conventional paradigm, in this Letter we show that by operating the oscillator at special points in the resonator’s anharmonic regime we can overcome fundamental limitations of oscillator performance due to thermodynamic noise as well as practical limitations due to noise from the sustaining circuit. We develop a comprehensive model that accounts for the major contributions to the phase noise of the nonlinear oscillator. Using a nano-electromechanical system based oscillator, we experimentally verify the existence of a special region in the operational parameter space that enables suppressing the most significant contributions to the oscillator’s phase noise, as predicted by our model.

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confidence: 66%

“…This scheme is also known as a phase feedback oscillator [13,18], which is commonly used to suppress one quadrature of the amplifier noise. It also provides, in principle, a quantum nondemolition method [19] to track the resonator phase.…”

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Surpassing Fundamental Limits of Oscillators Using Nonlinear Resonators

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“…Two recent papers have presented experimental results showing parameter scans of oscillators based on high-Q resonators driven into their nonlinear regime [10,11]. A major interest of these works is to trace out the characteristic driven resonator Duffing curve, showing multiple solutions for the amplitude and phase of the driven oscillations for a given driving frequency, over some frequency range and for sufficiently large drive amplitudes.…”

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confidence: 99%

Phase noise of oscillators with unsaturated amplifiers

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We study the role of amplifier saturation in eliminating feedback noise in self-sustained oscillators. We extend previous works that use a saturated amplifier to quench fluctuations in the feedback magnitude, while simultaneously tuning the oscillator to an operating point at which the resonator nonlinearity cancels fluctuations in the feedback phase. We consider a generalized model which features an amplitude-dependent amplifier gain function. This allows us to determine the total oscillator phase noise in realistic configurations due to noise in both quadratures of the feedback, and to show that it is not necessary to drive the resonator to large oscillation amplitudes in order to eliminate noise in the phase of the feedback.

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“…Corrections to the spring constant should then be taken into account, which give rise to a stiffening of the resonator and a higher resonance frequency and bistability when driven beyond the critical amplitude. 14,15 Throughout this work, the resonator is driven below this amplitude, and although the nonlinearity is clearly present, the open-loop amplitude and phase response remain single-valued functions of the driving force.…”

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confidence: 99%

Nonlinear dynamics of a microelectromechanical oscillator with delayed feedback

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We study the dynamics of a nonlinear electromechanical oscillator with delayed feedback. Compared to their linear counterparts, we find that the dynamics is dramatically different. The well-known Barkhausen stability criterion ceases to exist, and two modes of operation emerge: one characterized by hysteresis in combination with a bistable frequency and amplitude; the other, by self-stabilization of the oscillation frequency and amplitude. The observed features are captured by a model based on a Duffing equation with delayed force feedback. Nonlinear oscillators with delayed force feedback are exemplary for a large class of dynamic systems. Oscillators are ubiquitous in nature and engineering, with implementations in physical, life, and social sciences. [1][2][3] Typical components of an oscillator are a resonant system and a positive feedback loop. Oscillators are self-sustained and produce an ac output signal from a dc input. Compared to a resonator, which is driven by an ac signal, an oscillator exhibits a reduced line width and improved phase-noise performance. These properties make them interesting as sensitive detectors and as timing references. A common example is the quartz crystal oscillator, with widespread application in electronic circuits.In current applications the oscillating element is typically linear, implying that the displacement is proportional to the driving force as in Hooke's law. In a linear system, displacement-proportional feedback modifies the spring constant, whereas velocity-proportional feedback modifies the damping. Stability for the latter systems is bounded by the well-known Barkhausen criterion. 4,5 It states that a selfsustained oscillation occurs when the phase shift around the loop is an integer multiple of 2π , and the loop gain exceeds the value k/Q, where k is the spring constant and Q the quality factor without feedback. Although the linear regime has been studied in great detail, much less is known about nonlinear oscillating elements. In a nonlinear oscillator, the oscillating frequency depends on the amplitude of the motion. Micro-and nanoscale electromechanical devices exhibit strong nonlinearity, which makes them ideal devices to study such systems.6-8 We demonstrate that in the presence of a delayed force feedback, the nonlinearity gives rise to intricate dynamic behavior near the oscillation threshold.To construct a strongly nonlinear oscillator, we use a doubly clamped micromechanical beam with a high aspect ratio as the oscillating element. Figure 1(a) shows the device, with dimensions L × w × h = 750 × 8 × 0.5 μm 3 , fabricated from silicon nitride using standard microfabrication processes.9 An integrated piezoelectric actuator enables periodic driving of the beam, as well as static tuning of its resonance frequency. The motion of the beam is probed with pm/ √ Hz sensitivity using an optical deflection technique, by, is shown in Fig. 1(b), reflecting a laser beam off the surface of the device onto a position-sensitive photodetector. 10 The experiments are condu...

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

Cited by 52 publications

References 19 publications

Surpassing Fundamental Limits of Oscillators Using Nonlinear Resonators

Surpassing Fundamental Limits of Oscillators Using Nonlinear Resonators

Phase noise of oscillators with unsaturated amplifiers

Nonlinear dynamics of a microelectromechanical oscillator with delayed feedback

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