Phase Noise Reduction in an MEMS Oscillator Using ...

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Phase Noise Reduction in an MEMS Oscillator Using a Nonlinearly Enhanced Synchronization Domain Oriel Shoshani, Daniel Heywood, Yushi Yang, Thomas W. Kenny, and Steven W. Shaw Abstract— We investigate the phase dynamics of a closed-loop MEMS-based oscillator and demonstrate how one can exploit nonlinear behavior to improve oscillator phase noise characteristics by synchronizing the oscillator with a weak harmonic drive. Analytical predictions are based on an oscillator model that incorporates a resonator element with a weak cubic (i.e., Duffing type) nonlinearity, weak coupling to a clean external harmonic drive, and both thermal and frequency noise terms. The method of stochastic averaging is used to derive an expression for the rate of phase diffusion induced by the noises, and the results predict a remarkable phase noise reduction of three order of magnitude when the oscillator is synchronized to the weak external field. The predictions are experimentally demonstrated using a closed-loop oscillator with a double-anchored doubleended-tuning-fork MEMS resonator with coupling to a small external sinusoidal signal from a signal generator, demonstrating a 30 dB/Hz drop in phase noise. The results show how one can generate a clean high-power signal from a relative noisy oscillator by synchronizing it with a low-power external signal. The results also confirm recent studies showing that the parameter range of synchronization is significantly expanded when the resonator operates in its nonlinear regime. [2016-0067] Index Terms— Frequency stability, nonlinear oscillators, oscillator noise, phase noise, synchronization.

I. I NTRODUCTION

D

UE TO their inherent compatibility with semiconductor technology and the ability to fulfill device miniaturization requirements, MEMS-based oscillators are an attractive replacement for quartz crystals in time-keeping applications [4]. They offer fast-responding, low powerconsumption elements that are readily integrable with electronic circuits in fabrication [5]–[7]. However, as the size of MEMS devices are reduced, resonator frequency becomes more highly dependent upon fabrication variances and more susceptible to environmental effects, both of which result

Manuscript received March 31, 2016; revised June 23, 2016; accepted July 10, 2016. This work was supported in part by the U.S. Army Research Office under Grant W911NF- 12-1-0235, in part by the Defense Advanced Research Projects Agency through the Dynamic Enabled Frequency Sources Program under Grant FA8650-13-1-7301, and in part by the National Science Foundation under Grant CMMI-1234067. Subject Editor L. Buchaillot. O. Shoshani and S. W. Shaw are with the Department of Mechanical and Aerospace Engineering, Florida Institute of Technology, Melbourne, FL 32901 USA (e-mail: [email protected]; [email protected]). D. Heywood, Y. Yang, and T. W. Kenny are with the Department of Mechanical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JMEMS.2016.2590881

in degradation of frequency stability [8], [9]. Moreover, in order to produce large signal-to-noise ratio (SNR), these MEMS devices may need to operate nonlinearly [10], which leads to further degradation in the frequency precision due to amplitude-modulation (AM) to phase-modulation (PM) conversion [11]. In this paper, we describe an approach to reducing phase noise in a relatively noisy oscillator using coupling of a weak signal from a cleaner oscillator, essentially using synchronization to achieve clean signal amplification. The synchronization of oscillators, a concept that goes back to Huygens [12], provides an approach for reducing phase noise or, equivalently, increasing frequency stability. This approach has been refined in other applications, e.g., synchronization of triode oscillators for radio communication systems [13], [14], synchronization of optical oscillators in laser systems [15], [16] and synchronization of MEMS oscillators for ultra-sensitive resonant sensing [17], [18] and timekeeping [1], [19]. The practical significance of phase noise reduction due to synchronization has led to a large number of studies. Kurokawa [20] showed that the phase noise of an oscillator can be considerably improved by synchronization with an external signal, while the amplitude noise is slightly degraded. Chang et al. [21] showed the 1/N phase noise reduction in N mutually synchronized oscillators for a case where the amplitude noise and AM to PM conversion are negligible. Razavi [22] performed a theoretical and experimental investigation of injection locking\pulling, phase-locked oscillators, and phase noise reduction, in which he derived an identity for the requisite oscillator nonlinearity in order to obtain injection locking. Li et al. [23] presented discrete-time and frequency domain analyses of injection pulling and provided a detailed discussion on the benefits and deficiencies of each approach. Amro et al. [24] presented analysis for a pair of coupled oscillators with unequal noises. They showed that if the noisy oscillator is sufficiently noisy such that it is no longer synchronized with the clean oscillator, then it can actually improve the performance of the clean oscillator as the noisy oscillator becomes more noisy. Recent studies have experimentally demonstrated that nonlinearity in the frequency selective element (the resonator) of the oscillator can be exploited to manifest a regime in which the synchronization domain expands with increasing oscillation amplitude, thus providing a new strategy to enhance the synchronization of oscillators [25], [26].

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Fig. 1. Simplified circuit schematic of the experimental setup. The DADETF resonator, shown in an SEM micrograph and a finite element computational model on the right, is driven in closed loop at a constant phase shift by a Zurich Instruments HF2LI lock-in amplifier, with a transimpedance amplifier and bandpass filter for signal conditioning. The external signal is produced from an Agilent 33120A function generator and is weakly injected to the oscillator.

Different aspects of frequency stability, including sensitivity to fluctuations of parameters, have been considered in numerous studies [27]–[29]. Rubiola and Giordano [30] and Sansa et al. [31] highlighted the importance of flicker noise frequency fluctuations as a limiting factor of the oscillator frequency short-term stability at relatively large time scales. The goal of this paper is to examine phase noise characteristics at small time scales of an externally-synchronized MEMS oscillator with a weakly nonlinear, lightly damped resonator element. While this synchronization is known to result in an expanded synchronization domain, it also provides a dramatic reduction in the phase noise. To this end, we augment the investigations described in [25] and [26] by including a stochastic analysis of the model and experimental tests with the following specific objectives: (i) to obtain a systematic and consistent derivation of a Langevin model for the oscillator phase; (ii) to use the model to provide theoretical predictions of the effective synchronization domain and the attendant phase noise reduction; and (iii) to demonstrate experimental validation of the theoretical predictions using a MEMS oscillator. The paper is organized as follows: We start with a description of the hardware setup in Section II, so that experimental comparisons can be made directly as analytical predictions are derived. Section III presents the derivation of a simplified Langevin equation (a noisy Adler equation) for the biased phase dynamics of a general noisy oscillator with a Duffing type resonator that is coupled to a weak external field. Section IV provides the specific predictions of interest from the model, namely the enhanced synchronization regime and the phase noise, along with a comparison of experimental results. Section V summarizes the main conclusions of the investigation and presents potential future research topics along similar lines. II. E XPERIMENTAL S ETUP AND D EVICE P ROPERTIES The device of interest is a closed-loop oscillator with a micro-mechanical frequency selective element, specifically, a double-anchored double-ended-tuning-fork (DADETF) resonator, as depicted in Fig. 1. The resonator consists of two beams 200μm long and 6μm thick connected on both ends to perforated masses, which are further anchored to a support

base. Resonator parameters were estimated based on openloop measurements [32], as follows: The natural frequency and quality factor are found to be: fr = 1.188MHz and Q = 7.013 × 103 , and the dimensionless Duffing parameter, which represents a stiffness term that is cubic in the displacement and arises from a combination of mechanical (hardening, arising from mid-line stretching of the beams [33]) and electrostatic (softening) is found to be: α = 4.2 × 10−3 . Note that α results in an amplitude-dependent frequency of free vibration [33], [34]. Oscillation was achieved by using a Zurich Instruments HF2LI lock-in amplifier to electrostatically drive the resonator at a constant phase shift. The output of the lock-in amplifier was maintained at a constant amplitude (R). A transimpedance amplifier and bandpass filter were used to condition the resonator output prior to measurement by the lock-in amplifier. The external signal was obtained from Agilent 33120A function generator and its amplitude (E) was kept at fixed ratio with respect to the oscillator amplitude (E/R = 8.33 × 10−2 ). This setup provides self-sustained oscillation at a given amplitude that is then coupled to a weak external harmonic drive. The system is subject to unmodeled dynamics from a variety of sources that result in fluctuations in the response. The spectra of fluctuating quantities were measured using HP 89410A vector signal analyzer (VSA). In this work we do not consider the sources of these noises, but only their affect on the phase noise performance of the system, which is intimately related to frequency stability as the latter is simply time derivative of the former. We now turn to a model that describes these dynamics. III. M ODEL F ORMULATION We consider a model for a closed-loop oscillator with a high-Q weakly nonlinear (Duffing type) resonator element, thermal (additive) and frequency (multiplicative) noise sources, and an external harmonic drive. When expressed in terms of a dimensionless resonator displacement (x) the model is ˙ E cos(ωt) + ξ1 (t) + ξ2 (t)x x¨ + Q −1 x˙ + x + αx 3 = S(x, x)+ (1) where overdots denote derivatives with respect to the dimensionless time t = 2π fr td ( fr is the resonator natural frequency and td is dimensional time), S represents the saturated closedloop feedback input (gains, phase shifts, etc., for a specific model; see Agrawal et al. [1]), α is the Duffing nonlinearity parameter, Q is the quality factor (Q  1), E and ω are the drive amplitude (E  1) and drive angular frequency (which is close to the natural frequency, ω = 1 − ω, ω  1), and ξ1,2 are wide-sense stationary, zero-mean, additive and multiplicative noises, respectively, which are also assumed to be weak (||ξ1,2 ||  1). The resonator considered in this work is assumed to operate in an amplitude range for which a weakly nonlinear model holds, and while the resonator in the experimental device has a hardening nonlinearity (α > 0), the analysis presented is also applicable for a resonator with a softening nonlinearity (α < 0).

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In many cases the additive and multiplicative noises are used to model thermal and frequency fluctuations, respectively. Furthermore, these different noise sources can be associated with different correlation times and as a result be important at different time scales. For example, if the thermal and frequency fluctuations are white and flicker frequency noises, respectively, then the short term stability of the oscillator will be governed by thermal fluctuations (additive noise) at short times and by frequency fluctuations (multiplicative noise) at longer times. Here, however, we do not associate the additive and multiplicative noises with specific sources, but rather use them in a generic way to account for the combined effect of a variety of noise sources (fluctuation of temperature and frequency, feedback phase, quality factor, etc.) and consider only small time scales of the short term stability. Thus, we assume that the correlation times (τc1,2 ) of ξ1,2 are sufficiently small when compared to the system relaxation time, τr = Q/(π fr ), i.e., τc1,2 /τr  1, or, equivalently, that the noises are wide-band compared to the generated signal. We then apply the method of stochastic averaging to obtain expressions that approximate the slow evolution of the oscillator amplitude a(t) and phase θ (t), where x(t) = a(t) cos(ωt + θ (t)). These are given by 3Sξ2 (2ω) Sξ (ω) a s − − E sin θ + a+ 12 2 2 2Q 16ω 4ω a   Sξ2 (2ω) 2 1/2 1 Sξ1 (ω) + a η1 , + ω 2 8 3α 2 E

ξ (2ω) θ˙ = ω + a − cos θ − 2 2 8ω a 8ω   Sξ1 (ω) 1/2 1 Sξ2 (2ω) + 2Sξ2 (0) + + η2 ω 8 a2

a˙ =

where



Sξn (ω) + i ξn (ω) = 2

∞

(2)

3 S˜ξ2 (2ω) S˜ξ (ω) r − E˜ sin θ + R + 12 2 2Q 16ω 4ω R  1/2 S˜ξ (2ω) 2 1 S˜ξ1 (ω) + 2 R η1 . + ω 2 8

r˙ = −

(6)

Note that in contrast to the phase dynamics, dissipation is explicitly present in the amplitude dynamics. Hence, perturbations of the amplitude decay relatively rapidly when considered on the time scale of the phase evolution. This point can also be seen from the free-running oscillator equations (Eqs. (2)-(3) with Sξn = ξn = E = 0), where the phase dynamics are associated with a zero eigenvalue and thus subject to slow evolution. Consequently, we assume that the amplitude is adiabatically following the phase and noises (˙r ≈ 0), resulting in the perturbed amplitude  3Q S˜ξ2 (2ω) Q S˜ξ1 (ω) − 2Q E˜ sin θ a = R+ R + 8ω2 2ω2 R 1/2  Q S˜ξ2 (2ω) 2 ˜ + 2 Sξ1 (ω) + R η1 + O( 2 ). ω 2 (7)

(3)

ξn (t)ξn (t + τ ) exp(−i ωτ )dτ,

0

η1,2 (t) = 0, ηn (t)ηm (t + τ ) = δnm δ(τ ),

˜ S˜ξn = O(1) and  1. with E˜ = E/ , S˜ξn = Sξn / , E, Note that is an artificial book-keeping parameter introduced in order to emphasize that the external signal and noises are relatively weak compared to the deterministic part of the free-running oscillator. Thus, assuming that the perturbed amplitude is given by a = R + r (t) + O( 2 ) and substituting this form into Eq. (5), we obtain the following governing equation for the amplitude perturbation away from R,

We define the biased oscillator phase as φ = θ + θ0 , tan θ0 = 2ω/(3α Q R 2 ), which represents the phase difference between the external signal and the oscillator. Substitution of Eq. (7) into Eq. (3) and retaining terms up to O( ), yields an approximated Langevin equation for φ given by

(4)

are the spectral values of ξ1,2 at angular frequency ω, according to the Wiener–Khinchin theorem and η1,2 are simple delta-correlated Gaussian noises, according to the limit theorem of Stratonovich [2]. Note that in the absence of the external signal (E = 0) and noises (Sξn = ξn = 0), the closed-loop oscillator has a stable limit-cycle with an amplitude a ∗ = s Q ≡ R which is the ratio of the amplifier saturation level (s) to the system dissipation (Q −1 ), which describes the condition for which the loop dynamics sustain the isolated oscillator. When the external signal and noises are small in comparison with to the closed-loop gain (E, Sξ1 /R, Sξ2 R  s), we can rewrite Eq. (2) as ⎡ a 3 S˜ξ2 (2ω) S˜ξ (ω) s − ⎣ E˜ sin θ − a− 12 a˙ = − 2 2 2Q 16ω 4ω a 1/2 ⎤  S˜ξ2 (2ω) 2 1 S˜ξ1 (ω) + a − η1 ⎦ (5) ω 2 8

φ˙ ≈ ω˜ − A sin φ + ν(t)

(8)

where 3α R 2

ξ (2ω) − 2 2 8ω 8ω 3α Q + (4Sξ1 (ω) + 3R 2 Sξ2 (2ω)), 32ω3 is the total frequency offset from all effects,   2 1/2 E 3α Q R 2 A= , 1+ R 2ω ω˜ = ω +

is the mean phase modulation amplitude, and  1/2 R 2 Sξ2 (2ω) 3α Q R Sξ1 (ω) + η1 (t) ν(t) = 2ω2 2 8   1 Sξ2 (2ω) + 2Sξ2 (0) Sξ1 (ω) 1/2 + + η2 (t). ω 8 R2

(9)

(10)

captures the net effects of the noises acting on the phase φ. Note that while Eq. (8) takes into account AM to PM conversion due to the Duffing nonlinearity, it solely describes

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the noisy phase dynamics and does not explicitly depend on the amplitude. A similar phase equation can be obtained using other methods, such as the perturbation projection vector (PPV) method [35], [36], orthogonal decomposition [28], and the method of isochrons and phase response curves [37]. However, in contrast to these alternative methods, our derivation yields the well-known noisy Adler equation (Eq. (8)) [38] that appears in many other applications, including overdamped Josephson junctions [39], the theory of charge density waves [40], dithered-ring-laser gyroscopes [41], and vortex diffusion in layered superconductors [42]. IV. R ESULTS We now consider how this model is used to predict synchronization and phase noise performance in this class of systems and compare these predictions with results from the experimental device. A. Nonlinear Synchronization Enhancement We start the synchronization analysis with an investigation of the deterministic synchronized solution and the enhanced synchronization domain that stems from the resonator nonlinearity. The noise-free synchronization characteristics are determined from the deterministic part (Sξn = ξn = 0) of the Adler equation, Eq. (8) [43], which models the dynamics of an overdamped mass particle moving on a washboard potential. By setting φ˙ = 0 in the deterministic part of Eq. (8), we obtain an equation for the system fixed points (φ ∗ ) which correspond to a phase-locked/synchronized solutions sin φ ∗ =

ω˜ A

(11)

2

where ω˜ = ω + 3α8ωR is the first order approximation of the frequency mismatch between the self-sustained oscillator 2 angular-frequency ωo = 1 + 3α8ωR (i.e., the free vibration frequency of Duffing resonator) and the external signal frequency ω = 1 − ω. For a frequency mismatch satisfying −A < ω˜ < A, the washboard potential is sufficiently flat such that the system has ∗ . Note a pair of fixed points, one stable φs∗ and one unstable φus that the condition |ω| ˜ c = A defines the range of existence of these solutions, the stable of which represents synchronized response. This condition yields the following relation between the system parameters   2 1/2 E 3α Q R 2 , (12) |ω| ˜ c= 1+ R 2ω a result that is in agreement with those derived in [25] and [26]. Following [26], two limiting cases of Eq. (12) are of interest, which have distinct scaling behavior with the amplitude R, as follows: (i) A linear resonator element (α = 0) which yields 2|ω| ˜ c= 2E/R = 2E/(s Q), so that, as expected, the frequency range of synchronization is proportional to the strength of the interaction with the external perturbation E and the bandwidth of the resonant response (Q −1 ). (ii) A nonlinear resonator dominated by the Duffing term ˜ c ≈ 3α Q E R/ω. (3α Q R 2 /(2ω)  1) which yields 2|ω|

Fig. 2. Nonlinear synchronization domain enhancement; frequency mismatch bandwidth (2|ω| ˜ c ) versus oscillator amplitude (R). Analytical prediction (black line), linear limiting case (green line), limiting nonlinear case (blue line), and experimental measurements (red circles).

Fig. 2 shows the analytical predictions and experimental measurements of the synchronization region width (2|ω| ˜ c) for a fixed ratio of external perturbation amplitude to oscillator amplitude (E/R = 8.33 × 10−2 ), along with the analytical prediction calculated from Eq. (12), including the two limiting cases noted above. Note that all experimental measurements are in the range where the synchronization domains are significantly larger than their linear counterparts, that is, 3α Q R 2 /(2ω)  1. Also note that in this range the biased phase (φ) and the oscillator phase (θ ) coincide for all practical purposes, since φ = θ + θ0 , tan θ0 = 2ω/(3α Q R 2 ). That is, there there is no time delay between signals of the external perturbation and the synchronized oscillator. B. Effects of Noise on Synchronization ˙ = limt →∞ φ(t ) ) and The average phase velocity (φ t

  diffusion coefficient (D = limt →∞ 2t1 [φ(t) − φ(t)]2 ) of the noisy Adler equation Eq. (8) are the two quantities that characterize the dynamics, and these can be calculated analytically [2], [3]. The solution of the Adler equation can be interpreted as a random walk of an overdamped mass on the washboard potential. We consider the effective noise ν(t) to be delta-correlated ν(t)ν(t + τ ) = 2Dν δ(τ ) with intensity Dν that is proportional to the spectral values (Sξ1,2 , ξ2 ) of the (non-white) additive noise at the fundamental frequency and the (non-white) multiplicative noise at the second harmonic and at DC (see Eq. (10)). Fig. 3 shows the theoretical time ˙ versus the normalized derivative of the averaged phase φ detuning parameter for different levels of noise intensity Dν . ˙ captures the phase noise in Note that the mean phase rate φ the system and can be considered as the frequency deviation from the synchronized solution due to the presence of noise. The curves reveal that for sufficiently small values of the ratio of noise intensity to external-field coupling, specifically, for 0 < Dν /A < 0.005, the noise has a negligible effect on the region of synchronization, resulting in phase locking in a noise-dependent range over the Shapiro step, that is, ˙ = 0 where phase-locking occurs (this the segment of φ behavior is well known for Josephson junctions [44]); these

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˙ in the presence of noise for Dν / A = 1 (red), Fig. 3. Averaged phase rate φ Dν / A = 10−2 (green), Dν / A = 5 × 10−3 (black), Dν / A = 10−3 (blue), ˜ A = −1. and Dν = 0 (dashed line). The inset shows a blow-up near ω/

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Fig. 4. Phase noise (Lφ ) versus frequency mismatch (2π fr ω) ˜ at three different offset frequencies: f = 0.5(Hz) (blue), f = 1(Hz) (black), f = 2(Hz) (red). Solid lines are theoretical predictions and , ,  are the corresponding experimental measurements.

spectrum cases are shown as the dashed, blue, and black curves in Fig. 3. In contrast, for relatively large noise intensities, e.g., Dν /A ≈ 1, the noise completely eliminates the synchronization region and the time derivative of the averaged phase difference varies linearly with the frequency mismatch (ω), ˜ resulting in continual phase drift of a free runing oscillator, as depicted by the red line in Fig. 3. The transition between these limits is evident in Fig. 3, for example, as depicted by the green curve. 1) Phase Noise Analysis: The noisy phase dynamics can be determined from the model by linearization of Eq. (8) about the appropriate state and subtracting the mean steady-state value. For the synchronized response, which is achieved for Dν /A < 0.005, the model is linearized about the stable fixed point (φs∗ ) of the deterministic model (Sξn = ξn = 0). This results in the following equation for the noisy phase deviation (δφ = φ − φ) δ φ˙ + (A2 − ω˜ 2 )1/2 δφ = ν(t).

(13)

Note that, the phase deviation, δφ, is an Ornstein-Uhlenbeck process. Thus, the variance of the phase (δφ 2  = φ 2 −φ2 ) can be readily calculated from Eq. (13) and is given by    Dν 1 − exp −2(A2 − ω˜ 2 )1/2 t . δφ 2  = √ A2 − ω˜ 2 (14) Thus, as the offset frequency (ω) ˜ increases and the synchronized solution moves from the center (ω/A ˜  1) of the Shapiro step towards the edges (|ω|/A ˜ ≈ 1), both the variance of the locked solution (δφ 2 ss = √ 2Dν 2 ) and the A −ω˜ time it takes to achieve this value (tss ∝ (A2 − ω˜ 2 )−1/2 ) increase up to the boundary points of the synchronization domain, ω/A ˜ = ±1, at which points Eq. (14) recovers the free-running solution behavior, δφ 2  = 2Dν t. By taking the Fourier transform of Eq. (13) we obtain the following phasenoise spectrum  2Dν , (15) Lφ ( f ) = 10 log10 4π 2 f 2 + A2 − ω˜ 2 which again, at the boundary points of the synchronization domain, ω/A ˜ = ±1, recovers the free-running solution

 Lφ ( f ) = 10 log10

Dν . 2π 2 f 2

(16)

Both δφ 2  and Lφ ( f ), which quantify the phase-noise in the time and frequency domains, respectively, show that the largest reduction in phase noise will occur at the center of the Shapiro step, ω/A ˜  1. This result agrees with the intuition that the “overdamped mass” will remain in the well in spite of the random “kicks,” so long as the washboard potential slope (ω) ˜ is not large compared to A. Fig. 4 shows a comparison between the theoretical predictions from Eqs. (15)-(16) and experimental measurements of the phase noise. The phase noise is computed for the given oscillator and external perturbation amplitudes of Rd = 300 mV, E d = 25 mV and three different offset frequencies, f = 0.50, 1.0, 2.0 Hz. The effective noise intensity Dν is estimated from Eqs. (15)-(16) in order to achieve optimal agreement with the measurements (by means of a nonlinear least squares fit [45]) and yields the dimensional value Dνd ≡ 2π fr Dν = 1.98×10−4 Hz. This is the only parameter obtained from fitting experimental results to the model. Note that the ratio of noise intensity to external field coupling yields Dν /A = 8.22 × 10−6 , which is considerably smaller than the theoretical upper bound of the synchronized solution (Dν /A)max = 5 × 10−3 , thus confirming the validity of the theoretical model. Also, note that there is a considerable drop in the phase noise when the oscillator is synchronized, e.g., for f = 0.5 Hz there is a drop of 32 dB in the phase ˜ > noise at the transition from non-synchronized (|2π fr ω| 24.1 Hz) to synchronized (|2π fr ω| ˜ < 24.1 Hz), that is, the phase noise is reduced by three orders of magnitude in synchronized operation. The slight asymmetry in the experimental measurement may be due to the noise induced frequency drift 2

(2ω)

ξ2 3α Q 2 (ω˜ − ω − 3α8ωR = 32ω ), 3 (4Sξ1 (ω) + 3R Sξ2 (2ω)) − 8ω2 which is not accounted for in our model prediction. Fig. 5 shows the Allan deviation of the oscillator in both free-running and synchronized modes, along with that of the external harmonic signal. As it can be seen, the oscillator frequency stability is improved significantly over both short and long averaging times. The stability on short averaging times is

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R EFERENCES

Fig. 5. Allan deviation of the free-running oscillator (red), the synchronized oscillator with zero frequency mismatch (black), and the external harmonic signal (blue).

governed by noises with small correlation time compared to the relaxation time of the oscillator, and this noise reduction is explained by our model. Of course, the noise reduction is limited by the quality of the external signal, which in the ideal case serves as the noise floor at zero frequency mismatch. V. C LOSING R EMARKS We derived and investigated a model for the phase noise of a MEMS based closed-loop oscillator that is synchronized by an external harmonic field. The case of weak external field and small intensity noise allowed us to reduce the problem to a single stochastic differential equation governing the phase dynamics of the synchronized oscillator. The corresponding Langevin equation for the oscillator phase is shown to be capable of predicting the remarkable phase noise reduction in the synchronized regime, along with the ability to enhance the synchronization region, by choosing a Duffing resonator as the frequency selective element, which allows for operation at relatively large oscillator amplitudes. The analytical predictions are validated by experimental measurements and provide useful insights into methods for reducing phase noise by exploiting system nonlinearities. The most important feature of these results is that the system acts like a “clean” amplifier, specifically, one can use a lowpower signal with good frequency stability to drive a noisy MEMS oscillator to produce a high-power signal that is significantly cleaner than that of the MEMS oscillator; in our device we obtain a twelvefold amplification of the injected signal. In future work the analysis will be generalized to include the transition zones near the bifurcation points where the oscillator switches from non-synchronized to synchronized behavior. These regions are associated with phase slips which increase the phase noise considerably (see Fig. 4) and limit oscillator performance, and are of interest in other applications, such as quantum computing, where the readout of the qubits is done via a Josephson junction bifurcation amplifier that has similar characteristic transition dynamics. ACKNOWLEDGMENT The authors would like to thanks the SNF staff for their help during the fabrication process.

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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SHOSHANI et al.: PHASE NOISE REDUCTION IN AN MEMS OSCILLATOR

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Oriel Shoshani received the B.S. degree in mechanical engineering from the Ben-Gurion University of the Negev, Beer-Sheva, Israel, in 2008, and the Ph.D. degree in mechanical engineering from the Technion-Israel Institute of Technology, Haifa, Israel, in 2014. He is currently a Post-Doctoral Fellow with the Department of Mechanical and Aerospace Engineering, Florida Institute of Technology. His current research interests include dynamics of fluctuating nonlinear vibrational systems and applications to micro/nano-electro-mechanical systems. He has received awards and honors including the Sherman Fellowship for graduate studies achievements (2009), the Irwin and Joan Jacobs Fellowship for excellence in studies (2011, 2013), and the Pnueli prize for Ph.D. thesis (2015).

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Daniel Heywood received the B.S. degree in mechanical engineering from the University of California, Berkeley, in 2014. He is currently pursuing the degree with the Department of Mechanical Engineering, Stanford University. His current research focuses on the development of new noninvasive cardiovascular sensing techniques.

Yushi Yang received the bachelor’s degrees in mechanical engineering from Purdue University and Shanghai Jiao Tong University in 2011, and the M.S. and Ph.D. degrees in mechanical engineering from Stanford University in 2013 and 2016, respectively. Her research interests include developing MEMS fabrication techniques, studying the nonlinear behavior of bulk-mode MEMS resonators, and analyzing the coupling behavior of MEMS resonators. Thomas W. Kenny received the B.S. degree in physics from the University of Minnesota, Minneapolis, in 1983, and the M.S. and Ph.D. degrees in physics from the University of California, Berkeley, in 1987 and 1989, respectively. From 1989 to 1993, he was at the Jet Propulsion Laboratory, National Aeronautics and Space Administration, Pasadena, CA, where he was involved in research on the development of electron-tunneling high-resolution microsensors. In 1994, he joined the Department of Mechanical Engineering, Stanford University, Stanford, CA, where he directs microsensor-based research in a variety of areas, including resonators, wafer-scale packaging, cantilever beam force sensors, microfluidics, and novel fabrication techniques for micromechanical structures. He is the Founder and CTO of Cooligy (now a division of Emerson), a microfluidics chip cooling component manufacturer, and the Founder and a Board Member of SiTime Corporation (now a division of MegaChips), a developer of timing references using MEMS resonators. He is Founder and a Board Member of Applaud Medical, developing non-invasive therapies for kidney stones. He is currently the Richard Weiland Professor of Mechanical Engineering and the Senior Associate Dean of Engineering for Student Affairs. He was the General Chairman of the 2006 Hilton Head Solid-State Sensor, Actuator, and Microsystems Workshop, and the General Chair of TRANSDUCERS 2015 meeting in Anchorage. From 2006 to 2010, he was on leave to serve as the Program Manager in the Microsystems Technology Office at the Defense Advanced Research Projects Agency, starting and managing programs in thermal management, nanomanufacturing, manipulation of Casimir forces, and where he received the Young Faculty Award. He has authored or co-authored over 250 scientific papers and holds 50 issued patents, and has been advisor to more than 50 graduated Ph.D. students from Stanford. Steven W. Shaw received the B.A. degree in physics from the University of Michigan–Flint in 1978, the M.S.E. degree in applied mechanics from the University of Michigan–Ann Arbor in 1979, and the Ph.D. degree in theoretical and applied mechanics from Cornell in 1983. He is currently the Harris Professor of Mechanical and Aerospace Engineering with the Florida Institute of Technology, Melbourne, FL, and is the University Distinguished Professor with the Department of Mechanical Engineering and the Adjunct Professor with the Department of Physics and Astronomy with Michigan State University, East Lansing, MI. He has held visiting appointments at Cornell University, the University of Michigan, Caltech, the University of Minnesota, the University of California–Santa Barbara, and McGill University. He currently serves as an Associate Editor for two journals, Nonlinear Dynamics and the SIAM Journal on Applied Dynamical Systems. His research interests are in applied dynamical systems and currently include modeling, analysis, and design of micro-scale resonators for sensing and signal processing applications, and nonlinear vibration absorbers for automotive applications. Dr. Shaw is a member and a fellow of the ASME. He is a recipient of the SAE Arch T. Colwell Merit Award, the Henry Ford Customer Satisfaction Award, the ASME Henry Hess Award, and the ASME N. O. Myklestad Award.