A State-Space Phase-Noise Model for Nonlinear MEMS Oscillators ...

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 1, JANUARY 2010

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A State-Space Phase-Noise Model for Nonlinear MEMS Oscillators Employing Automatic Amplitude Control Lin He, Member, IEEE, Yong Ping Xu, Senior Member, IEEE, and Moorthi Palaniapan, Member, IEEE

Abstract—This paper presents a new phase-noise model for nonlinear microelectromechanical-system (MEMS) oscillators. Two widely recognized existing phase-noise models, namely, the linear time-invariant and time-variant models, are first reviewed, and their limitations on nonlinear MEMS oscillators are examined. A new phase-noise model for nonlinear MEMS oscillators is proposed according to the state-space theory. From this model, a closed-form phase-noise expression that relates the circuit and device parameters with the oscillator phase noise is derived and, hence, can be used to guide the oscillator design. The analysis also shows that, despite the nonlinearity in the MEMS resonator, the phase noise is still governed by its linear transfer function. This finding encourages the designers to operate the MEMS resonator far beyond its Duffing bifurcation point to maximize the oscillation signal power and put more emphasis on the low-noise automatic-amplitude-control-loop design to minimize the noise aliasing through amplitude-stiffening effect without concerning the nonlinear chaotic behavior. Index Terms—microelectromechanical systems (MEMS), oscillators, phase noise, state space.

I. INTRODUCTION

A

TIMING reference (or clock) is almost a must for practically every electronic system. Currently, the timing references for commercial products are unanimously provided by crystal oscillators. Although a crystal oscillator is able to provide a stable and clean oscillation, it is not compatible with any mainstream IC process. The timing reference is thus the bottleneck for a more compact and cheaper system. On the other hand, crystal oscillators dissipate significant power in portable devices. The drawbacks of crystal oscillators could be avoided by microelectromechanical-system (MEMS) oscillators in the near future, as it has the potential to be fully integrated with other building blocks into a single chip and the MEMS resonator itself consumes negligible power.

Manuscript received July 25, 2008; revised December 16, 2008 and February 16, 2009. First published March 24, 2009; current version published January 27, 2010. The work of L. He was supported by the National University of Singapore through a Postgraduate Scholarship. This paper was recommended by Associate Editor H. Schmid. L. He is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 Singapore, and also with the Institute of Microsystem and Information Technology, Chinese Academy of Science, Shanghai, China. (e-mail: [email protected]). Y. P. Xu and M. Palaniapan are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 Singapore (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCSI.2009.2018932

The cleanness of a reference oscillation is the defining factor for the fidelity of a communication system, which determines the maximum bit rate and the minimum power consumption. The cleanness of oscillation is usually characterized by its phase noise. Phase noise near the carrier is particularly important in communication systems with narrow channel spacing. In fact, the allowable channel spacing is frequently constrained by the achievable phase noise. From a linear model [1]–[3], the phase noise of an oscillator is given as (1) where is the quality factor, is the signal power, is is the frequency the noise density injected around , and offset. To minimize the phase noise, it is therefore straightforward to maximize the quality factor, to increase the signal power, or to minimize the noise injected. An overwhelmingly high quality factor can be achieved in a MEMS resonator, ranging from tens of thousands to over a million [4], which is comparable to that of crystal resonators. MEMS resonators with power-handling capacity comparable with that of macroscopic crystal resonators have been demonstrated [5]–[7]. At first glance, it seems that MEMS oscillators should be able to achieve extremely low phase noise. In practice, however, a phase noise MEMS oscillator always exhibits a strong which is much worse than that predicted by (1). Attempts have been made to find out the sources of this unexpected phase noise. Roessig et al. [8] suggested that the nonlinearities in the mechanical component caused the noise through noise aliasing. Lee and Nguyen [9] claimed noise in the sustaining circuitry nor the that neither the noise-aliasing mechanism should be responsible for the meaphase noise. Instead, they speculated a link between sured the Duffing behavior and the unexpected phase noise and hence suggested that the oscillation amplitude should not exceed the Duffing bifurcation point to fulfill a low-noise design. However, this link has not been soundly proven yet. It is questionable to apply the bifurcation point derived from open-loop resonators to closed-loop oscillators. Nevertheless, both of the works highlighted a concrete link between the undesirable phase noise and the nonlinearities. Although there are ways to cancel the nonlinearities in a MEMS resonator, either through mechanical compensation, like folded suspension [10], or through nonlinear electrostatic stiffness [11], both of them work well only for low-frequency resonators below hundreds of kilohertz. For MEMS resonators

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Fig. 1. Illustration of phase noise in a nonlinear MEMS oscillator. The dotted line shows the changes of 1=f and 1=f phase noises at an increased amplitude.

at higher frequency, their mechanical stiffness are so high that the cancellation strategies aforementioned are no longer effective due to the limited electrostatic actuation. Fortunately, the nonlinearities will reduce as the oscillation amplitude decreases. It is therefore a common practice to introduce automatic amplitude control (AAC) to prevent a resonator from entering the strong nonlinear region. The reduction of the oscillation amplitude, however, will cause the linear phase noise to increase, according to (1). A tradeoff has to be made to reach the optimum phase noise at a specific frequency offset, as shown in Fig. 1. It is therefore desirable to maximize the oscillation amplitude without causing the nonlinear phase noise to dominate. However, it is widely believed that the maximum allowable oscillation amplitude is limited by the Duffing bifurcation point; otherwise, a chaotic behavior will be triggered, which abruptly deteriorates the phase noise [9]. Due to the high quality factor, the bifurcation limit of a MEMS resonator is extremely small [7], far smaller than the intrinsic material limit, which seems to doom the future of MEMS reference oscillators. It is thus crucial to understand the details of the interaction between the various noise sources and the nonlinearities in the MEMS oscillator and establish a reliable phase-noise model for nonlinear MEMS oscillators, which may serve as a guide to the low-phase-noise design. The rest of this paper is organized as follows. Section II reviews the amplitude-frequency response of a nonlinear resonator and the existing oscillation structures of MEMS resonators including the structures employing AAC. Section III reviews two of the most recognized phase-noise models, the linear time-invariant (LTI) model and the linear time-variant (LTV) model, with their limitations in analyzing a mechanical oscillator. In Section IV, a state-space analysis is performed on the nonlinear oscillator with the given AAC loop. Numerical simulation results are presented in Section V, respectively. Section VI discusses the limitations of the model and the practical concerns of designing a MEMS oscillator. Section VII concludes this paper. II. MEMS OSCILLATOR A. Nonlinear Resonators Assuming that the nonlinearity mainly comes from the nonlinear spring with both the first- and second-order nonlinear

Fig. 2. Change of open-loop frequency-amplitude response for nonlinear resonator with increasing harmonic force under (a) < 0 and (b) > 0.

terms and , the motion equation for a forced oscillation of a nonlinear MEMS resonator is given by (2) where is the time, is the displacement of the lumped mass , is the damping coefficient, is the linear spring conis the applied harmonic force. Although two stant, and nonlinear terms are enough to model nonlinearity at small vibration amplitudes, the following analysis can still be extended to higher order nonlinearities without affecting the main conclusion. The nonlinear terms from the spring constant cause a peak-frequency shift with vibration-amplitude increases [13] (3) where is the vibration amplitude, is the resonant freis the shifted resonance quency without nonlinear terms, and frequency at , known as amplitude-stiffening (A-S) effect. In practice, the peak frequency can shift to either a higher or lower frequency, depending on the sign of , as shown in Fig. 2. At large vibration amplitudes, the response curve shows hysteresis. The bifurcation point is defined as the critical amplitude that triggers hysteresis, as conceptually shown in Fig. 3. Beyond this point, the amplitude-frequency curve is no longer a single-value function, which means an amplitude uncertainty in open-loop configuration. The amplitude uncertainty, in turn,

HE et al.: STATE-SPACE PHASE-NOISE MODEL

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Fig. 3. Frequency-amplitude response of a nonlinear resonator shows hysteresis and the bifurcation point.

causes a frequency ambiguity, which was believed to be the source of chaotic dynamic behavior [7], [10]. B. MEMS Oscillator and AAC A simple MEMS oscillator can be constructed using the circuit shown in Fig. 4(a) [14]. The resonant beam needs to be polarized at . During vibration, the resonant beam produces a motion current at one electrode. This current is sensed and fed back to the other electrode or through a comparator [14] to generate an electrostatic force. If the gain and phase criterion is satisfied, the resonant beam will oscillate until its amplitude is eventually stabilized by the nonlinearities in the mechanical structure or the circuit in the loop. The amplitude of this oscillator suffers from temperature and supply-voltage variations and, thus, is limited to the low-accuracy arena due to the amplitude-caused frequency drift. An AAC loop is introduced to avoid strong nonlinearity. The most widely used AAC structure [Fig. 4(b)] [8], [9], [12] adjusts the gain of the sense amplifier by replacing the feedback resistor with a MOSFET transistor operating in the linear region. The amplitude of the oscillation signal at the sense-amplifier output is detected and compared with a preset value. The amplified error signal is used to adjust the resistance of the MOSFET transistor. Once a stable oscillation is established, the feedback resistor will precisely match the effective resistance of the MEMS resonator. Therefore, the temperature drift of the effective resistance will cause the same drift in oscillation amplitude and, in turn, cause a frequency drift. A more temperature-robust AAC structure [16] features a fixed-gain amplifier and a linear variable-gain amplifier (VGA) following it, as shown in Fig. 4(c). The gain of the linear VGA is controlled by the difference between the oscillator amplitude through a loop filter . and the present value The MEMS oscillators with AAC loop contain nonlinear building blocks such as an amplitude detector, a VGA, and particularly, a nonlinear resonator, which prevent it from any ordinary linear analysis. Before presenting the proposed state-space model, we shall briefly review the existing LTI and LTV phase-noise models in the next section.

Fig. 4. Schematic diagrams of (a) a direct feedback MEMS oscillator, (b) a MEMS oscillator with AAC loop and gain control performed on the sense amplifier, and (c) a MEMS oscillator with fixed gain sense amplifier and AAC performed on a linear VGA.

III. EXISTING PHASE-NOISE THEORY A. Phase-Noise Basics Ideally, the output of an oscillator may be expressed as , where is the amplitude and is the frequency. In practice, however, the amplitude and the phase offset are functions of time. Ignoring the waveform distortion, the output can be thus given by

(4)

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Fig. 5. Linear oscillator system.

which further decomposes into

Fig. 6.

LC oscillator showing charge injection.

of the oscillator. In other words, the noise power spectral density is shaped by (5) (9) The aforementioned equation indicates that the amplitude varia, tion is actually caused by the in-phase component while the phase variation is actually caused by the quadrature , with respect to the ideal oscillation component . The marking of noise in the time domain referred to the oscillation waveform is the key to understand the dynamics of an oscillator. The amplitude and phase noises produce noise sidebands around the oscillation frequency. In a practical oscillator, will be reduced by an amplitude-limiting the noise in mechanism like saturation or eliminated by the application of will a limiter to the output signal, whereas the noise in not be affected and will accumulate over time. Thus, the noise sideband is dominated by the phase portion, known as phase noise.

, and hence (10)

When

,

, leading (9), which can be written as

(11) With the quality factor generally defined as [3] (12) (11) is reduced to

B. LTI Model tank oscillator The LTI model was first derived from an [1], [2] and extended to a general oscillator with a linear frequency selection tank [3]. Take a simple feedback system shown represents a noise source, and in Fig. 5 as an example. is the frequency selection tank. If, at frequency , the precisely matches 360 and the loop phase shift across matches unity, the closed-loop transfer funcgain across tion (6) will build up until it is goes to infinity; thus, an oscillation at limited by some nonlinear mechanism. Suppose that the oscilla, tion frequency slightly deviates from , i.e., the phase shift deviates from 360 or the loop gain deviates from the unity; thus, its noise transfer function can be approximated as

(7) Since

To gain more insight, let

and, for most practical cases, , (7) reduces to (8)

This equation indicates that a noise component at is multiplied by when it appears at the output

(13) The output noise can be equally divided into phase and amplitude fluctuations with reference to the output oscillation. Taking only phase fluctuation into account yields (14) Because of the simplification, the LTI phase-noise model provides important qualitative design insights but shows significant discrepancies from the measurement result. This model, along with the LTV model, predicts boundless growth of power spectrum density at a very small offset frequency compared with that of the measured Lorentzian shape [17]. To answer this discrepancy, [18]–[20] use an analogy between the phase noise and physical diffusion process, while [21] models a feedback gain that is slightly less than unity. In addition, the LTI model cannot phase noise observed in practical oscillators. explain the C. LTV Model The LTV model was proposed by Hajimiri and Lee [22] to include the omnipresent nonlinearity in a linear model. This model oscillator shown in Fig. 6 as an can be explained using an example. Assume that a charge injection occurs when the oscillator is oscillating at a steady state. It can be seen from Fig. 7 that the resultant amplitude and phase change is time dependent. In particular, if the charge is applied at the peak of the voltage across the capacitor, the oscillation amplitude will increase, as shown in Fig. 7(a). Assuming that an amplitude change does not


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Fig. 9. Equivalent block diagram of the LTV model.

Fig. 7. Impulse response of an LC oscillator. Fig. 10. Noise folding due to ISF decomposition.

is the phase of the th harmonic of the ISF. Fig. 10 where shows how the noise folding happens in an oscillator. Note that noise is caused by the upconversion of the noise the through the coefficient . Since is the dc value of the ISF, the phase noise can be greatly reduced by the proper ISF design, through the adjustment of the rise- and fall-time symmetry. However, the LTV model is derived from an implicit assumption that the frequency does not change with amplitude variation, which shows significant disagreement for pedagogical oscillators [23].

Fig. 8. Waveform and ISF for an LC oscillator.

affect the oscillation frequency, the timing of the zero crossings (or phase) would not be affected in this case. On the other hand, if this impulse is applied at the zero crossing, it affects the zero crossing but has negligible effect on the amplitude, as shown in Fig. 7(b). Based on the aforementioned observation, an impulse , as a function of when the injecsensitivity function (ISF) tion takes place, can be derived. The phase change with response to an impulse input is thus given by (15)

IV. STATE-SPACE MODEL Analyzing phase noise in the state space is not a new idea. To the author’s knowledge, a unifying phase-noise model proposed by Kaertner and Demir et al. [24]–[27] belongs to this category. In this model, the noise perturbation is decomposed into orbital and phasal components along the Floquet vectors in the state space and models the phasal perturbation as a diffusion problem. The soundness of this model is verified in numerical simulators [28], [29]. In addition, the oscillation envelop can also be predicted by employing this model [30]. However, due to the complexity in mathematics, this model fails to arrive at a closed-form solution based on the circuit or device parameters. To guide the oscillator design process, we combine the mathematical soundness of the state-space approach and the understandability of Hajimiri’s approach in our model. Our model is different from the model of Kaertner and Demir et al. in that the orbital component still influences the phase.

is the unit step function, is the maximum where is the charge displacement across the capacitor, and ISF and is a dimensionless periodic function with a period of . Fig. 8 shows the ISF for an oscillator, which has its maximum value near the zero crossings of the oscillation and a zero value at the peak of the oscillation waveform. The phase is shift of the oscillator caused by a current perturbation thus given in the block diagram shown in Fig. 9. can be expressed as a Fourier series

A. Unperturbed Oscillator Trajectory in State Space Taking only the first- and second-order nonlinearities and into account for simplicity, the motion of a second-order spring–mass–damping resonator is given by its state-space equation

(16)

(17)

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Fig. 11. Unperturbed decomposition.


oscillation

trajectory

y (t)

and

perturbation Fig. 12. Macro model of second-order nonlinear oscillator with noise in the observed velocity and feedback gain.

where is the excitation force or feedback force applied to the resonator. A feedback mechanism observes the velocity and applies a force which is proportional to the observed so that the velocity. The feedback gain is finely adjusted to is exactly balanced by the feedback force mechanical loss (similar to the assumption in the LTI model), as given by

(18)

is a monotonic function of time (within a period). vector Therefore, a phase can be uniquely defined by the vector for any unperturbed trajectory with a different orbit. Therefore, if the trajectory is perturbed by a component proportional to , the phase will remain fixed but the orbit will change. On the contrary, a perturbation in parallel with is not going to change the motion trajectory (or orbit), but will change the speed of cycling and hence affect the phase [refer to (18)]. Therefore, the perturbation can be decomposed into an orbital perturbation that is parallel to and a phase perturbation , as shown in Fig. 11 that is in parallel to

Equation (18) has a periodic solution plotted in the state-space plane (Fig. 11) where the position is plotted against velocity . The solution forms a closed curve or unperturbed trajectory. Since the mechanical loss is fully compensated, the unperturbed trajectory follows an orbit that has a fixed energy given by (21) (19)

where

B. Velocity Observation Noise and Gain Variation The noise introduced in the sense amplifier and the driving . The buffer can be simplified as a velocity observation noise feedback gain may not precisely match all the time, which is . The velocity observamodeled as a feedback gain variation tion noise and the gain variation are included in the macro model of a second-order nonlinear oscillator, as shown in Fig. 12. The describing function of the oscillator shown in Fig. 12 is thus given by (20)

where can be viewed as the impulse-phase-sensitive funccan be viewed as the impulse-orbit-sensitive tion while function. Here, an analogy to LTV model is applied. Combining (20) and (21) yields

C. Orbital and Phase Perturbations Let us use Fig. 11 to study the influence of over the motion trajectory. We noticed that, in an unperturbed trajectory, the

(22)


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Comparing (22) with (18), the seeming timing shift caused by the phase perturbation is given by

(23.a)

and the orbital shift caused by the orbital perturbation is given by

Fig. 13. Equivalent models of a perturbed free-running nonlinear oscillator described by (20).

(23.b) Here, can be viewed as a fractional orbital shift, and hence, can be approximated as , the amplitude change is the unperturbed oscillation amplitude. where D. Additional Timing Shift by A-S Effect Aside from the phase perturbation, the orbital change will cause an additional seeming timing shift due to the A-S effect, as is indicated in (3). The resonance frequency is actually a . Within a narrow band function of the oscillation amplitude of the orbit

(24) is the amplitude of unperturbed oscillation, is the where , is the perturbed frequency at unperturbed frequency at , and is the fracamplitude , equals tional orbital shift. This frequency shift results in an additional seeming timing shift (25)

Therefore, the equivalent model for a perturbed free-running nonlinear oscillator is shown in Fig. 13. The perturbed oscillator , where is the unhas an output is the perturbed oscillation from (18) and total timing shift. is transformed by The observed velocity noise into a normalized orbital perturbation and by into a . This transformation is simnormalized phase perturbation ilar to the phase and orbital decompositions in the LTI model.

Fig. 14. Illustration of 0(y ), H (y ), x_ 0(y ), and xH _ (y).

If the nonlinear spring terms and do not exist, and will degenerate to sinusoidal functions, and and will be equally split, which agrees well with the LTI model. , , , and as a function of Fig. 14 shows time. E. Amplitude-Limiting AAC If the oscillator is free of any amplitude-limiting mechanism, the orbital perturbation will be integrated and result in an unbounded orbital shift. Such an undesirable unbounded shift is avoided by employing the AAC. The following analysis is performed on the oscillator with an ideal position sensor, an ideal velocity sensor, and an AAC loop, as shown in Fig. 15(a). The detected oscillation amplitude is compared with a preset . Their difference is fed to the AAC loop filter to value . produce the gain perturbation signal Inside the AAC loop, the amplitude detection introduces extra , and the VGA has an input-renoise which is modeled as at the gain-control node [Fig. 15(a)]. Asferred noise suming that the amplitude control loop responds much slower and can be rethan the oscillation itself, placed with their average values for simplicity, which is given as (26)

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Three phase-noise sources are identified in (29). is caused by the normalized phase perturbation, corresponding indito the linear phase noise in the LTI model. cates an extra phase noise caused by the feedback gain variation through the asymmetry of the oscillation waveform. It looks like a novel phase-noise mechanism that was never paid attention to. However, we found through numerical simulation that is so small that it can be practically ignored even though an asymmetrical waveform is produced by the first nonlinear correction . The last term is caused by the orbital term noise fluctuation through the A-S effect, which aliases the into phase noise. component in So far, we have proved that, as long as the unperturbed trajectory forms a closed trajectory, the amplitude noise as well as the phase noise from a nonlinear mechanical oscillator can be still studied using a linear model. Therefore, the oscillation amplitude should only be constrained by the hard limits such as material, structure, or supply voltage, provided that the circuit design ensures a noiseless AAC loop. At a frequency close to dc, the orbital fluctuation (30)

Fig. 15. (a) Nonlinear MEMS oscillator with ideal position sensor and AAC loop and (b) its equivalent linear model.

Fig. 15(b) shows the derived linear equivalent model for this ideal nonlinear MEMS oscillator. The transfer function of the fractional orbital fluctuation is given by

(27) In the derived linear model, it is interesting to note that the AAC loop features the same integrator as in the derived oscillator [18], which reminds us of the model for a linear similarity between the AAC loop and the phase-locked loop (PLL). Without the negative feedback of the AAC loop, the orbit shift will build up exactly in the same way as the phase difference between two frequencies does. Therefore, the same loop filter design strategy used in PLL can be applied directly to the AAC loop. A second-order loop filter borrowed from the charge-pump PLL [19] can satisfy both the noise shaping and stability requirement, as given in the following: (28)

F. Total Phase Noise The transfer function of the phase shift can be derived from the total timing shift using

(29)

Equation (30) indicates that and will be shaped by , will stay untouched. The suggested AAC loop filter has but a high gain at dc to minimize the orbital fluctuation so that the phase noise can be minimized. will appear unattenuated in the orbital fluctuation Since noise component is the only source at low frequencies, its phase noise in the close-in region. Thus, that gives rise to attention must be paid to design a low-noise amplitude detector as well as a low-noise error amplifier. To use this model in a practical MEMS oscillator, proper and converscaling is required to accommodate the , one should divide the output sion. For example, to obtain noise figure in the sense interface by the velocity-to-voltage transduction ratio. V. NUMERICAL SIMULATION Simulink is used to numerically simulate the ideal nonlinear oscillator shown in Fig. 15(a). A time-domain Monte Carlo simulation is performed to capture the power spectrum density around the oscillation frequency. The core Simulink setup is shown in Fig. 16. Two nonlinear resonators with normalized stiffness and resonant frequency are used in the simulation, as given in Table I. The quality factors are set to 1000 instead of hundreds of thousands in a practical case, due to the accuracy achievable with the simulation tool. The peak-frequency shifts and are of the same amount, except that, in caused by resonator 1, both of them tilt to the same direction while, in resonator 2, they completely cancel each other, as shown in Fig. 17. For resonator 1, the oscillation amplitude is already far beyond the Duffing bifurcation point. Fig. 18 shows the state-space trajectories of the stable oscillations for resonator 1, resonator 2, and a linear resonator. Although the trajectories for oscillators 1 and 2 show asymmetry against the oscillation amplitude due to the first-order nonlinear correction term, the waveforms have symmetrical rising and falling times.


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Fig. 16. Simulink setup for the Monte Carlo simulation of nonlinear oscillator.

Fig. 17. Amplitude-frequency responses of resonators 1 and 2 in Table I.

Therefore, the LTV model should yield a close result as that of the LTI model. A two-tone test is employed to verify the linearity between a control signal at the AAC loop and the resulted phase change. The testing tones and are injected at the amplitude comparison node C (Fig. 16). To remove the amplitude noise, the oscillation output is fed to an amplitude limiter. The limiter output is multiplied with an ideal sinusoid signal and filtered with a low-pass filter. This treatment greatly reduces the number of data collected without affecting the simulated phase noise. The simulated sideband spectra after the amplitude limiter are shown in Fig. 19. For resonator 1, only two tones are clearly seen in the phase plot. The absence of an intermodulation tone indicates a perfect linear relationship. For resonator 2, no tone is seen at the desired frequency, indicating that neither the A-S-induced phase shift nor the asymmetry-related phase shift exists. To verify the quantitative prediction power of the proposed phase-noise model, the same oscillator based on resonator 1 was simulated with flicker-noise sources inserted to nodes C and G (Fig. 16). The power spectrum densities of the injected noise

Fig. 18. State-space trajectories of the oscillator built from resonator 1, resonator 2, and an ideal linear resonator.

TABLE I PARAMETERS FOR THE SIMULATION SETUP

sources are also given in Table I. Fig. 20 shows the comparison of the simulated sideband spectrum with the predicted phase-

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be kept as low as possible to allow the increase of the oscillation amplitude. In practice, however, the concern of temperature stability prevents the oscillation amplitude from further growing; otherwise, the amplitude control has to be extremely stable and precise. VII. CONCLUSION

Fig. 19. Simulated two-tone spectrum sidebands for resonators 1 and 2.

A linear deterministic phase-noise model based on state-space theory has been derived for a nonlinear MEMS oscillator with AAC loop. It shows that the oscillation amplitude can vibrate far beyond the bifurcation point without triggering chaotic behavior. To further improve the phase-noise performance, a low-noise AAC loop should be employed. This finding paves the way to the design of ultralow noise MEMS oscillators that may one day replace the well-known crystal oscillator. APPENDIX

(A.1)

(A.2) Fig. 20. Simulated phase-noise spectrum sideband versus the LTI (and LTV) model and state-space model.

noise curve from the LTI (or LTV) model and the state-space model. The state-space model agrees well with the simulated spectrum, but the LTI model underestimates the resultant phase noise.

(A.3)

ACKNOWLEDGMENT VI. DISCUSSION We would like to remind the reader again that our model is design oriented, which agrees well with the Monte Carlo simulation results only at the “rolling-off” region but fails to predict the plateau of Lorentzian shape due to the perfect loss compensation assumption. In a practical MEMS oscillator, other phase-noise mechanisms may be triggered, such as the noise aliasing due to nonlinear capacitive transduction [33] and electrical gain stage [16]. This paper also highlights a different design strategy from that oscillator where AAC is typically not employed to of an avoid the downconversion of the noise component at double the oscillation frequency [34]. The noise in the key building blocks in the AAC loop, such as the rectifier and error amplifier, must

L. He would like to thank L. Shao for the inspiring discussions, F. Michel and A. Taschwer for the proof reading, M. Keller for the simulation setup, and the anonymous reviewers for their time and efforts to make this work publishable. REFERENCES [1] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, Feb. 1966. [2] L. S. Cutler and C. L. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards,” Proc. IEEE, vol. 54, no. 2, pp. 136–154, Feb. 1966. [3] B. Razavi, “A study of phase noise in CMOS oscillators,” IEEE J. Solid-State Circuits, vol. 31, no. 3, pp. 331–343, Mar. 1996. [4] M. Palaniapan and L. Khine, “Micromechanical resonator with ultrahigh quality factor,” Electron. Lett., vol. 43, no. 20, pp. 1090–1092, Sep. 2007.


[5] V. Kaajakari, T. Mattila, A. Oja, J. Kiihamäki, H. Kattelus, M. Koskenvuori, P. Rantakari, I. Tittonen, and H. Seppä, “Square extensional mode single-crystal silicon micromechanical RF resonator,” in Proc. Transducers, 2003, pp. 951–954. [6] V. Kaajakari, T. Mattila, J. Kiihamäki, H. Kattelus, A. Oja, and H. Seppä, “Nonlinearities in single-crystal silicon micromechanical resonators,” in Proc. Transducers, 2003, pp. 1574–1577. [7] V. Kaajakari, T. Mattila, A. Oja, and H. Seppä, “Nonlinear limits for single-crystal silicon microresonators,” J. Microelectromech. Syst., vol. 13, no. 5, pp. 715–724, Oct. 2004. [8] T. A. Roessig, R. T. Howe, and A. P. Pisano, “Nonlinear mixing in surface micromachined tuning fork oscillators,” in Proc. IEEE Int. Freq. Control Symp., May 1997, pp. 778–782. [9] S. Lee and C. T.-C. Nguyen, “Influence of automatic level control on micromechanical resonator oscillator phase noise,” in Proc. IEEE Int. Freq. Control Symp., May 2003, pp. 341–349. [10] J. Cao and C. T.-C. Nguyen, “Driving amplitude dependence of micromechanical resonator series motional resistance,” in Proc. Transducers, 1999, pp. 1826–1829. [11] M. Agarwal, K. Park, R. Candler, M. Hopcroft, C. Jha, R. Melamud, B. Kim, B. Murmann, and T. W. Kenny, “Non-linearity cancellation in MEMS resonators for improved power handling,” in IEDM Tech. Dig., Dec. 2005, pp. 286–289. [12] Y.-W. Lin, S. Lee, S.-S. Li, Y. Xie, Z. Ren, and C. T.-C. Nguyen, “Series-resonant VHF micromechanical resonator reference oscillators,” IEEE J. Solid-State Circuits, vol. 39, no. 12, pp. 2477–2491, Dec. 2004. [13] L. D. Landau and E. M. Lifshitz, “Resonance in nonlinear oscillations,” in Mechanics, 3 ed. Reading, MA: Butterworth-Heinemann, 1982, vol. 1, Course of Theoretical physics, pp. 87–92. [14] S. S. Bedair and G. K. Fedder, “CMOS MEMS oscillator for gas chemical detection,” in Proc. IEEE Sens., Oct. 2004, pp. 955–958. [15] J. A. Geen, S. J. Sherman, J. F. Chang, and S. R. Lewis, “Single-chip surface micromachined integrated gyroscope with 50 /h Allan deviation,” IEEE J. Solid-State Circuits, vol. 37, no. 12, pp. 1860–1866, Dec. 2002. [16] L. He, Y. Xu, and M. Palaniapan, “A CMOS readout circuit for SOI resonant accelerometer with 4-g bias stability and 20-g/rt Hz resolution,” IEEE J. Solid-State Circuits, vol. 43, no. 6, pp. 1480–1490, Jun. 2008. [17] W. A. Edson, “Noise in oscillators,” Proc. IRE, vol. 48, no. 8, pp. 1454–1466, Aug. 1960. [18] R. Kubo, “Stochastic Liouville equations,” J. Math. Phys., vol. 4, no. 2, pp. 174–183, Feb. 1963. [19] M. Lax, “Classical noise. V. noise in self-sustained oscillators,” Phys. Rev., vol. 160, no. 2, pp. 290–307, Aug. 1967. [20] D. Ham and A. Hajimiri, “Virtual damping and Einstein relations in oscillators,” IEEE J. Solid-State Circuits, vol. 38, no. 3, pp. 407–418, Mar. 2003. [21] F. Herzel, “An analytical model for the power spectral density of a voltage-controlled oscillator and its analogy to the laser linewidth theory,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 9, pp. 904–908, Sep. 1998. [22] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [23] G. J. Coram, “A simple 2-D oscillator to determine the correct decomposition of perturbations into amplitude and phase noise,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 48, no. 7, pp. 896–898, Jul. 2001. [24] F. X. Kaertner, “Determination of the correlation spectrum of oscillators with low noise,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 1, pp. 90–101, Jan. 1989. noise in oscillators,” Int. [25] F. X. Kaertner, “Analysis of white and f J. Circuit Theory Appl., vol. 18, no. 5, pp. 485–519, Sep./Oct. 1990. [26] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp. 655–674, May 2000. [27] A. Demir, “Phase noise and timing jitter in oscillators with colorednoise sources,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 12, pp. 1782–1791, Dec. 2002. [28] V. Vasudevan, “A time-domain technique for computation of noisespectral density in linear and nonlinear time-varying circuits,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 2, pp. 422–433, Feb. 2004. [29] B. De Smedt and G. Gielen, “Accurate simulation of phase noise in oscillators,” in Proc. Eur. Solid-State Circuits Conf., 1997, pp. 208–211.

199

[30] P. Vanassche, G. Gielen, and W. Sansen, “Efficient analysis of slowvarying oscillator dynamics,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 8, pp. 1457–1467, Aug. 2004. [31] B. Linares-Barranco and T. Serrano-Gotarredona, “A loss control feedback loop for VCO stable amplitude tuning of RF integrated filters,” in Proc. IEEE Int. Symp. Circuits Syst., Aug. 2002, pp. 521–524. [32] F. M. Gardner, “Charge-pump phase-locked loops,” IEEE Trans. Commun., vol. COM-28, no. 11, pp. 1849–1858, Nov. 1980. [33] V. Kaajakari, J. K. Koskinen, and T. Mattila, “Phase noise in capacitively coupled micromechanical oscillators,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 52, no. 12, pp. 2322–2331, Dec. 2005. [34] Q. Huang, “Phase noise to carrier ratio in LC oscillators,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 7, pp. 965–980, Jul. 2000. Lin He (S’02–M’08) was born in Hunan, China, on September 1977. He received the B.Sc. degree in material science and the M.Eng. degree in instrumentation engineering both from Southeast University, Nanjing, China, in 1999 and 2002, respectively. He received the Ph.D. degree in electronic engineering at the National University of Singapore, Singapore. From August 2002 to September 2007, he was with the Signal Processing and VLSI Lab, National University of Singapore, working on silicon resonant accelerometers and CMOS readout circuits for aerospace and inertial navigation application. From November 2007 to august 2009, he was with the Institute for Micro System Technique (IMTEK), University of Freiburg, Freiburg, Germany, where he was the Leader for the sensor readout team, supervising the microelectromechanical-system (MEMS) gyroscope projects and the magnetic sensor projects. He is currently with the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Science. His research interests include CMOS mixed-signal circuits, sensor interfaces, MEMS oscillators, and integrated MEMS gyroscopes. Yong Ping Xu (S’90–M’92–SM’01) received the Ph.D. degree in electronics from The University of New South Wales (UNSW), Sydney, Australia, in 1994. From 1978 to 1987, he was with Qingdao Semiconductor Research Institute, China, initially as an IC Designer and later as the Deputy R&D Manager and the Director. From 1993 to 1995, he was with UNSW on an industry collaboration project with GEC Marconi Pty Ltd., Sydney, where he was involved in the design of sigma–delta ADCs. He became a Lecturer with the University of South Australia, Adelaide, Australia, in 1996. Since 1998, he has been with the Department of Electrical and Computer Engineering, National University of Singapore, where he is currently an Associate Professor. His main research interests are in mixed-signal and RF integrated circuits and systems and currently focus on applications in wireless communication, biomedical devices, and MEMS. He has authored and coauthored two book chapters, six granted patents with another eight pending, and over 60 technical journal and conference papers. Dr. Xu is a corecipient of the 2007 DAC/ISSCC Student Design Contest Award and a recipient of the 2004 Excellent Teacher Award from the National University of Singapore. He was the General Cochair of the 2002 IEEE Asia Pacific Circuits and Systems and the Technical Program Committee Cochair of the 2007 IEEE International Workshop on Radio Frequency Integration Technology.

Moorthi Palaniapan received the B.Eng. (with first class honors) and M.Eng. degrees in electrical engineering from the National University of Singapore, Singapore, Singapore, in 1995 and 1997, respectively, and the Ph.D. degree from the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, in 2002. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, National University of Singapore. His research interests include integrated microelectromechanical sensors, actuators, resonator designs, power electronic circuits, and biosensors.