An Empirical Phase-Noise Model for MEMS ... - Semantic Scholar
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 5, MAY 2012
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An Empirical Phase-Noise Model for MEMS Oscillators Operating in Nonlinear Regime Mauricio Pardo, Student Member, IEEE, Logan Sorenson, Student Member, IEEE, and Farrokh Ayazi, Senior Member, IEEE
Abstract—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. Index Terms—Detuning, frequency-phase transfer function, MEMS resonator, nonlinearity, power-series-based PN model.
I. INTRODUCTION
O
SCILLATORS are a fundamental building block of synchronous microsystems. Microelectromechanical system (MEMS)-based solutions can provide an alternative to quartz oscillators due to their compatibility with CMOS technology, facilitating the integration into a single chip or a compact package [1]. When a MEMS resonator is used in an oscillator, its quality factor ( ) impacts the phase-noise (PN) performance of the oscillator. A sustaining amplifier with high transimpedance gain is capable of exploiting nonlinear effects in a MEMS resonator to improve the PN of the oscillator [2]. The measurement of such an oscillator shows that the PN performance is progressively improved when the phase-shift provided by the sustaining amplifier forces the operating point away from the mechanical Manuscript received September 06, 2011; revised December 17, 2011; accepted March 14, 2012. Date of publication April 24, 2012; date of current version May 09, 2012. This work was supported by Integrated Device Technology, San Jose, CA. This paper was recommended by Associate Editor T. S. Lande. M. Pardo is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA, and also with the School of Electrical and Electronics Engineering, Universidad del Norte, Barranquilla, Colombia. (e-mail: [email protected]; [email protected]). L. Sorenson and F. Ayazi are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. 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/TCSI.2012.2195129
resonance [2]. Detuning the oscillator allows the selection of a higher phase-frequency slope that emulates a higher-order filter, minimizing the effect of phase variations on the oscillator frequency [3]. In order to deliver high enough power to the resonator, automatic level control is disabled, allowing the signal amplitude to reach supply rails. In addition, high oscillation amplitude saturates the sustaining amplifier, becoming an effective limiting mechanism that minimizes the amplitude variation contribution to the PN. As presented in [2] and [4], an oscillator exhibiting beneficial PN shaping due to nonlinearity can be described by a higher effective compared to when the resonator is operated in its linear regime. However, at frequencies very close-to-carrier, the PN is dominated by flicker noise effects [5] and approaches the same PN limit as in linear operation. As the offset frequency moves away from the carrier, the PN then approaches the high effective limit. Therefore, a PN formulation must include higher-order terms to describe this transition. Available PN models [6]–[8] cannot be applied directly to capture the progressive improvement observed in this work, because the formulations are based on approximations to explain the presence of as the highest order term. In particular, the work from [8] uses the state-space mathematics to explain this term when an automatic amplitude control circuit is used to avoid the nonlinearity of the MEMS resonator. On the other hand, the recent model proposed by [9] acknowledges the possibility of improved PN due to proper phase tuning of a nonlinear resonator, where the expected results are determined by the phase-frequency slope at the selected operating phase, and PN term and additive noise contributions. The captures the experimental results reported in [10] demonstrate this characteristic under nonlinear operation with amplitude limiting enabled. This paper presents an empirical PN model for resonatorbased oscillators that encompasses both linear and nonlinear operation of the resonator, as demonstrated in [2] and [4], where the improvement to the PN in nonlinearity derives from the high-order phase filter action [3]. The paper is organized as follows: first, a brief description of MEMS resonators and their nonlinear behavior is provided. Next, dominant noise sources are discussed, and the contribution of nonlinearity to favor a higher-order phase-filter transfer function is analyzed [11]. A proper description of this filter can be obtained using a powerseries approximation that provides additional details about PN shaping. A description of the testing set-up for three nonlinear piezoelectric resonators is provided, whose results are used to draw conclusions about the practical utility of the described formulation.
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II. OPERATING PRINCIPLES OF MEMS RESONATORS MEMS resonators are vibrating mechanical devices that can use IC processes to reduce size and enable integration. Acoustic waves can be generated in silicon micromechanical devices by using electrostatic transducers or thin films of piezoelectric materials such as aluminum nitride (AlN). Microresonators can be designed to operate in a bulk acoustic wave (BAW) mode, which is more suitable for higher frequency applications. In this mode, the resonance frequency is primarily defined by a single geometrical dimension of the device (for example, the piezoelectric film thickness [12] or the width of the resonator body [13]). A thin film of piezoelectric material can be deposited on the surface of a silicon resonator to excite a lateral BAW mode through the transverse piezoelectric coefficient [14]. The thickness of the silicon layer is typically chosen to be much thicker than that of the piezoelectric layer, so that the resonator frequency is predominantly determined by silicon [13]. The superior acoustic properties of single-crystalline silicon translate to very high and large power handling [15]. A MEMS resonator can be modeled electrically as a two-port admittance network comprising three components: the input transduction, the mechanical response, and the output transduction. The admittance of a piezoelectric device has essentially the same form as that of a series RLC resonator. The large electromechanical coupling of the piezoelectric material results in low motional impedance, relaxing the gain requirements on amplifiers to build oscillators. III. NONLINEARITY OF MEMS RESONATORS The concept of nonlinearity refers to the relation between the amplitude of vibration and the applied force (i.e., Hooke’s law). In a real spring, this relationship is generally nonlinear and given by (1) where is the restoring force, is the displacement, is the linear term, and are the second- and third-order corrections, and aggregates higher-order terms. The displacement-dependent spring constant can be defined as . The solution of the equation of motion describing a nonlinear resonator, of which the Duffing resonator is a particular case ( ), (2) can be solved considering two cases: unforced-undamped, and forced vibrations. An unforced-undamped resonator is expected to oscillate with constant amplitude at the resonance frequency . The nonlinear terms in the restoring force will control the amplitude, which changes the oscillation frequency to according to the relationship (3)
Fig. 1. Amplitude and phase responses of a Duffing-like resonator under different levels of the amplitude-frequency effect.
where corresponds to the magnitude of the vibration-amplitude distribution [16]. Considering the forcing function as , the vibration amplitude near resonance is given by (4)
defined by (3) [16]. From this equation, it can be deterwith mined that for less than , the resonance peak will shift to lower frequencies. This case is known as spring softening, since it can be viewed as being produced by a spring with decreased stiffness. On the other hand, when is greater than , the resonance peak can be moved to higher frequencies becoming a case of spring hardening [16]. Since the stiffness of the nonlinear system is amplitude-dependent, the vibration-amplitude distribution will also exhibit bending. A measure of the maximum vibration amplitude is obtained by calculating the bifurcation point (critical drive) beyond which the amplitude-frequency relationship is no longer a single-valued function [3], [16]. Fig. 1 summarizes the effect and the corresponding modification of the phase response. More detailed equations for the nonlinear resonator can be employed to take into account the mode shape of the resonant structure [17]. MEMS resonators can be forced into the nonlinear elastic regime when sufficiently high power is applied to the device structure, creating regions of high energy density [1], [10], [18], [19]. Intrinsic nonlinearities are dependent on effective acoustic velocity, stiffness coefficients, and propagation of the acoustic wave within the device, while induced nonlinearities may arise due to temperature, force, pressure, acceleration, and vibration, among other sources [5]. IV. OSCILLATOR CONFIGURATION USING RESONATOR
A
NONLINEAR
A MEMS-based oscillator consists of a sustaining amplifier and a micromechanical resonator connected in feedback configuration satisfying Barkhausen criterion [20]. The resonator can be viewed as a band-pass filter, whose selectivity is measured by . A shunt-shunt amplifier is typically used to interface a MEMS resonator, because the input and output impedances are decreased to minimize degradation with loading.
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Fig. 2. Progression of the phase-frequency slope with nonlinearity.
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Fig. 3. A detuned resonator can operate with increased phase-frequency slope. Gain and phase tuning are required to satisfy Barkhausen criterion.
The sustaining-amplifier phase determines the operating point of the system by matching the resonator phase at the desired oscillation frequency. The noise of the sustaining amplifier propagates through the closed-loop system with amplitude and phase components perturbing the oscillator signal, broadening its spectrum. Since the amplitude component can be neglected by means of a limiting mechanism, the phase noise is considered to be the main contributor to these sidebands. Careful circuit design is required to maintain the assumption of PM-only effect in the PN. The advantage of the nonlinear regime is the ability to select a higher phase-frequency slope to desensitize the oscillator frequency with respect to induced phase variations. Once the operating phase is selected and the noise sources are considered as small signals [11], [21], a narrowband approximation allows the definition of a slope in the vicinity of that point, linearizing the phase-frequency relationship locally. Determining this slope corresponds to the definition of open-loop , which is a measure of how the closed-loop system opposes variations in the frequency of oscillation [22]. Thus, it is observed that in nonlinearity, this slope can be higher than the linear case, reducing the effect of phase-modulation variations on oscillator frequency (Fig. 2) [3]. Even though this is a convenient approach to the analysis of nonlinear operation, the lack of symmetry around the operating point will limit the amplitude of the noise sources to an infinitesimal value, since the definition of the slope will not be the same for both sides around the selected phase. Hence, a proper formulation for the PN will require a representation with higher-order terms, and Fig. 2 implies that those terms are necessary when PN results are analyzed as the operating point is tuned.
In addition, the sustaining amplifier also contains nonwhite effects attributed to flicker noise. Transistor-based circuits, whose transconductance is dependent on the bias point, can inject phase shifts resulting in a low-frequency phase modulation [9]. The PN contribution due to the flicker noise is equal to (6) where is a fitting constant dependent on the active devices of the sustaining electronics [24]. Considering that white- and flicker-noise sources are uncorrelated, the total noise of the amplifier can be expressed as (7) or equivalently (8) with defined as the corner frequency where the flicker- and white-noise components become equal. The estimation of is empirical and almost independent of the oscillation power. Thus, when the oscillation power is lowered, the thermal noise is increased, which creates the effect of a reduced value of . [11]. B. Noise of the Nonlinear Resonator The noise spectrum due to Brownian-motion noise in the resonator can be approximated as [25] (9)
V. NOISE SPECTRUM OF THE OSCILLATOR COMPONENTS A. Noise of the Sustaining Amplifier Thermal noise of the amplifier is directly related to the noise factor defined as the ratio of the output noise of the actual amplifier to that of an equivalent noiseless version [23]. Using and dividing by the available signal power , its noise powerdensity is defined as (5) where and are the Boltzmann constant and temperature, respectively. The thermal-noise power density has a white spectrum characteristic.
is the resonator series equivalent impedance at where the oscillation frequency . If matches the resonance frequency, the motional capacitance ( ) and inductance ( ) from the RLC model cancel out, and equals the motional resistance ( ). However, in the nonlinear regime, the resonator operates with a small detuning that increases and includes a phase shift that needs to be compensated by the sustaining amplifier to satisfy the Barkhausen criterion (Fig. 3). Thus, the white-noise contribution of the resonator scales with the motional impedance. The net effect of detuning will increase the noise floor of the oscillator, unless the dominant noise sources stem from the electronics.
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Microresonators can have several sources of flicker noise when nonlinearity is attributed to elevated drive levels. The resonance frequency can be affected by stress in the deposited thin metal films in piezoelectric devices related to the third-order elastic coefficients of the structural material [18], [26], [27]. Dielectric materials present in the resonator (e.g., AlN) can exhibit hysteretic damping [28]. A resonator exhibits flicker noise given by (10) where is a coefficient capturing the hysteresis in the film due to dielectric loss, and is the capacitance of the piezoelectric stack [28]. For the case of AlN, is related to the dielectric loss tangent, which is approximately constant down to a few hundred Hertz [29], contributing additional flicker noise. Additionally, a form of Johnson noise can be observed since self-heating becomes significant for microresonators. Elevated power generates a current associated with random motion of charge carriers in the body of the MEMS device that can be classified as white frequency noise [5]. Design of amplitude control circuitry could mitigate some of the nonwhite processes induced in the resonator [8], [30]; however, for piezoelectrically-transduced resonators studied in this work, the drive level is the mechanism employed to induce beneficial nonlinearity, and the amplitude limiting mechanism utilized is waveform clipping by the sustaining amplifier. VI. PHASE NOISE MODEL FOR AN OSCILLATOR WITH NONLINEAR RESONATOR
A
When the phase is the variable of interest, a resonant device with resonance frequency and quality factor can be expressed as a first-order low-pass filter with a pole equal to defining the Leeson corner frequency, [31]. Considering steady state, the output noise spectral density can be found by multiplying the input spectral noise density and the squared magnitude of . Therefore, the Leeson effect is [11]. defined as the multiplication by of the PN below Equation (11) shows the expression for the output noise spectral density for this case as (11) corresponds to the power spectral denwhere sity of the flicker- and white-noise noise sources for both amplifier and resonator. Equation (11) is the well-known Leeson’s model [4]. Fig. 4(a) shows the squared magnitude of the resonator, and Fig. 4(b) depicts how the PN at the output will be configured. Fig. 4 illustrates the particular case of high- resonators, where the flicker noise corner is much higher that the Leeson corner frequency ( ), which is the case of the MEMS devices considered in this work. For a nonlinear resonator, the associated low-pass transfer function will be equivalent to a higher-order filter, with order proportional to the strength of the induced nonlinearity. Therefore, the Leeson effect is more pronounced due to
Fig. 4. (a) Squared magnitude of resonator as phase filter. (b) The Leeson effect for a first-order equivalent low-pass resonator [4].
Fig. 5. A higher-order phase filter reduces the frequency of the Leeson effect and produces a steeper slope close to the carrier.
an elevated number of roots in the resonator transfer function that increases the slope roll-off (Fig. 5). Mathematically, the transfer function of the higher-order filter can be described by using a truncated power series of the exponential function with a number of terms equal to the order of the filter. This selection also allows a description of how the noise sources are modified by the asymmetry in the nonlinear resonator transfer function. When the power series is used, a parameter is defined as the ratio between the slopes in nonlinear regime at the detuned frequency and in linear operation at resonance. This power-series approximation uses a higher effective equal to , where denotes the loaded quality factor in linear operation. The parameter in the associated input spectral density will be affected by several factors. From a resonator point of view, the detuning that selects the steeper phase-frequency slope also increases the motional impedance of the MEMS device, and therefore its white noise contribution. However, if the elevated drive power generates a signal that forces the sustaining amplifier to operate as a hard limiter, then the noise folding produced for this rail-to-rail operation can dominate the far-from-carrier contribution. In any case, it is expected that the nonlinear oscillator will have an increased noise floor that will reduce the parameter , resulting in what is known as a noise-corrected oscillator [11]. Starting from linear operation, (11) can be modified to give the form of the power series of the exponential function as
(12) reflecting the Leeson effect and The second-order term containing noise processes that vary with and can be viewed as the first three terms of the series expansion for the first-order low-pass equivalent of the resonator.
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Fig. 7. SEM images of the AlN-on-Si resonators used in this work. (a) Twofinger electrode design and (b) Five-finger electrode design.
Fig. 6. Schematic of the two-finger design composite AlN-on-Si MEMS resonator illustrating the material layers of the composite stack.
Hence, when operated in nonlinear regime, (12) can be viewed as a particular case of the general term
(13)
Fig. 8. Representation of the IPS and LE resonance-mode shapes. The deformation of the plate is exaggerated to reveal details.
which generates the required higher-order components to provide an approximation of the expression for the phase filter of order . Notice that the power spectral density of the whiteand flicker- noise processes becomes because their contribution is determined for the specific operating conditions (power and detuning). Thus, (13) is a general expression that considers different levels of beneficial nonlinearity and reduces to Leeson’s model for equal to one (linear operation of the resonator). It can be concluded that for theoretical PN models that consider the transfer function of the resonator, the power series expansion can be an alternative starting point to capture the effect of an increased oscillator .
(LE mode presents 0 at resonance). The shapes of the excited modes are presented in Fig. 8. The outlines correspond to the peak displacements of each half cycle of oscillation. Device characterization is performed using an Agilent E5071C vector network analyzer (VNA) with 50 terminations producing estimates for and motional resistance as presented in Fig. 9. Power-induced nonlinearity is verified using a Mini-Circuits ZFL-1000VH power amplifier. The onset of nonlinearity in the 23 MHz IPS mode occurs close to , and the device can withstand power levels of at least . A similar onset is observed for the 26 MHz LE mode in both devices, but the resonator fractures at about , failing at the corners of the support tethers where the stress concentration is high. The IPS mode is more susceptible to nonlinearity due to the controlling combination of third-order stiffness components of silicon, which is the main structural material of the resonators. For illustration, Fig. 10 shows the progression of the nonlinear effect as the power supplied to the IPS mode is increased.
VII. EXPERIMENTAL VERIFICATION OF THE NONLINEAR PHASE-NOISE MODEL A. Piezoelectric Micromechanical Resonators Composite aluminum-nitride-on-silicon (AlN-on-Si) lateral microresonators are employed to demonstrate the applicability of the empirical PN model proposed in Section VI. The devices are fabricated from a 10 thick silicon-on-insulator (SOI) substrate. The resonator stack composition and thicknesses are 9/2/0.1/1/0.1 of Si/ /Mo/AlN/Mo, respectively. Two designs are utilized: two-finger and five-finger electrode configurations, both on top of a plate with lateral dimensions of 156 250 . Details of the composite material stack are shown in Fig. 6, using the two-finger design for illustration purposes. Fig. 7 shows SEM images of the devices, whose fabrication process is described in [32]. The two-finger electrode configuration excites and senses an in-plane shear (IPS) mode at 23 MHz and a longitudinal extensional (LE) mode at 26 MHz, while the five-finger design produces only a LE mode at 26 MHz. The IPS and LE modes are electrically 180 out-of-phase with respect to one another
B. Transimpedance Amplifier Design An inverter-based transimpedance amplifier (TIA) with active feedback resistor is interfaced with the MEMS resonators [2]. The TIA provides interface ports to operate with either 0 or 180 phase-shift resonance modes (see Fig. 11). The input-stage feedback resistor is implemented with a NMOS transistor whose operation is controlled by the gate voltage, . At one extreme, when is minimum, the feedback transistor enters subthreshold region and the resistance is maximum giving the highest possible gain for the TIA. On the other hand, when is increased to the power-supply rail, the feedback transistor turns on and becomes the preferred path for the current to the output of the inverting stage. The transimpedance gain approximates the transistor on-resistance.
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verter. Thus, when the resistance is increased, the time constant changes proportionally, and the pole moves to lower frequencies. This strategy is used to finely modify the TIA phase-shift, providing a method to move along the phase-frequency curve. A comparator is placed at the oscillator output to work as a digital buffer and to maintain constant amplitude prior to the PN measurement equipment. Thus, the objective is to make the comparator as transparent as possible for the PN measurement. The process-independent threshold-voltage compensated comparator available in [33] is used as the digital buffer. C. Modeling of a Nonlinear Resonator
Fig. 9. Characterization of the AlN-on-Si resonators used in this work. (a) Twofinger electrode design (IPS). (b) Two-finger electrode design (LE). (c) Fivefinger electrode design (LE).
The nonlinear nature of silicon becomes apparent for resonators driven at high amplitudes, and the nonlinearities are assumed to be stiffness-induced for the cases considered. The silicon third-order elastic coefficients are negative for the modes of interest, and an adequate description of the resonator behavior can be obtained using a second-order displacement-dependent spring constant ( ). Nonzero and negative are identified [34] according to the nonlinear strain-dependent Young’s modulus of the main structural material [35]. Equation (2) can be implemented with a large-signal -parameter simulator [e.g., Advance Design System (ADS)] to generate an estimate of can be modeled with an independent current source, and spring and damping forces with voltage-controlled current sources [36]. Nonlinearity is taking into account by using a polynomial voltage-controlled current source for the spring force. The parameters for the RLC series tank can be determined from the -parameter measurements of the considered MEMS resonators, which include a calculation for . Estimation of and can be done by fitting the measured parameter at a specific power level by evaluation of the correlation factor between measurements and simulations. The factor is calculated by the ratio of the maxima of the phase-response derivatives for both nonlinear and linear operation. D. Calculating the Power Delivered to the Resonators
Fig. 10. Magnitude and phase-frequency plots of the resonator 23 MHz-IPS mode for increased power levels.
Changing the setting of simultaneously modifies the phase-shift provided by the TIA because the location of the dominant pole is varied accordingly. The associated time constant can be determined with the value of the active resistor and the gate capacitances of the transistors that configure the in-
From Fig. 9, the worst-case scenario for resonator losses corresponds to a motional resistance equal to 2700 or about 69 . The sustaining amplifier of Fig. 11 provides a transimpedance gain close to 100 , which is sufficient to overcome such level of resonator losses. The design of the sustaining amplifier does not include an amplitude control circuit, which allows the waveform to grow until the large signal saturates the amplifier, hard clipping the output signal. Amplifier saturation becomes an effective form of amplitude control and still allows increased power to drive the resonator. Due to clipping of the generated waveform, amplitude variations are minimized, reducing the AM-to-PM conversion [37] due to the effect. Power delivered to the resonator will remain constant if the power-supply variation is negligible. A 5 V supply voltage is utilized to bias the electronics fabricated on a 0.5 2P3M CMOS process. To find the equivalent power that the VNA would deliver to the resonator during the characterization stage, the transfer characteristic of the sustaining amplifier can be extracted. There are two transfer characteristics, one for each phase-shift version of the TIA. For the
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Fig. 11. Transimpedance amplifier schematic. The generic TIA provides 0 and 180 ports to interface several MEMS devices and resonance modes.
Fig. 12. IC-resonator configuration via bondwires.
180 case, the TIA is only the shunt-shunt feedback stage, and by lowering the gain is increased producing a transfer characteristic similar to that of an inverter. In this situation, the amplitude at the output can reach the rails, producing nearly a square waveform. On the other hand, for the 0 version, the CS stage exhibits a different transfer characteristic due to its fixed load. The passive resistor employed has to be selected such that the symmetry of the transfer characteristic is maintained while at the same time lowering the output voltage at transistor M4 and the power consumption. Thus, as a conservative estimate, selecting about at the resonator input for both TIA versions and given the 50 termination of the measuring equipment, the equivalent power at the VNA source is approximately 13 dBm for the worst-case motional resistance considered. This drive level is above the onset of nonlinearity for all the test resonators. For this estimated power level, equals five for the 23 MHz resonance mode and three for the 26 MHz resonance modes following the procedure of Section VII-C. E. Determining the Parameters
and
Fig. 12 presents a schematic of the IC interfaced with a piezoelectric resonator through minimum-length aluminum bondwires. Three oscillators are configured using a similar set-
Fig. 13. PN results from piezoelectric-based oscillator operating in air. The trendline is added for comparison with Leeson’s model.
up: 23 MHz IPS and 26 MHz LE oscillators with a two-finger electrode resonator, and a 26 MHz LE oscillator using a fivefinger electrode resonator. In all cases, the oscillations start up properly with low transimpedance gain. As is reduced, the gain increases and
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the waveform reaches the saturation levels. Further decrease of only affects the TIA phase-shift to optimize the operating phase-frequency slope. Thus, the PN performance is measured for different settings of with an Agilent E5500 PN test set. The measurement summary includes a slope line, which describes the flicker-noise processes using Leeson’s formula at the limit of linear operation (Fig. 13). The best Leeson’s fitting estimates a noise figure NF ( ) of 45 dB and close to 50 kHz. Notice that the elevated NF stems from the noise folding due to the TIA behavior as hard limiter causing noise components to be folded within the bandwidth of the resonator, due to the rail-to-rail operation of the sustaining amplifier [22], [38]. Since the worst-case for testing is 2700 , and , it is concluded that the white-noise contribution of the oscillator PN is dominated by the electronics.
Fig. 14. PN model for the 23 MHz IPS piezoelectric resonator (2-finger).
F. Application of the Nonlinear Phase-Noise Model Once the parameters have been determined, (13) is used to describe the PN performance for all the oscillators. The standard definition for PN uses the single-sideband expression of the output spectral power density [39] in decibels per Hertz relative to the carrier power. The calculated factor corresponds to the best-measured performance for each case. As (13) uses to include white and flicker noises for amplifier and resonator, the flicker noise term will be presented explicitly to show the reduction in the parameter , and a factor labeled as will be used to represent the white noise of the complete oscillator. It is interesting to note that the best fitting for (13) is given for an effective equal to , the same factor that increases the . Thus, the model for the 23 MHz two-finger electrode design oscillator takes the form:
while for the 26 MHz two-finger and five-finger electrode designs, the models become
and
Fig. 15. PN model for the 26 MHz LE piezoelectric resonator (2-finger).
Fig. 16. PN model for the 26 MHz LE piezoelectric resonator (5-finger).
respectively. Figs. 14–16 show the model description of the measured results. In each case, the linear operation described by (11) has been included for comparison. It can be observed that the induced nonlinearity in the test MEMS resonators produces a maximum performance improvement of 20 dB and 15 dB at an offset frequency of about 200 Hz for the IPS and LE modes, respectively. Note that the nonlinear IPS 23 MHz oscillator, with an absolute figure of at 1 kHz offset frequency, is comparable to architectures with highcapacitive resonators [40]. In addition, when compared with state-of-the-art piezoelectric devices at higher frequencies [41], the IPS mode oscillator can also satisfy the GSM standard (after frequency up-conversion) using a resonator with a of only 4000.
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VIII. CONCLUSIONS A power series expansion provides a convenient and practical way to model the PN behavior of a nonlinear oscillator, because it provides the higher-order terms needed to reflect the relationship between the noise processes and the nonlinear phase-frequency response when the resonator is operated with elevated drive levels. This formulation reflects the higher effective and the reduction of the corner frequency for beneficial nonlinearity, and simplifies to the Leeson’s model for linear operation. Beneficial nonlinearity means that the PN response is superior to the equivalent oscillator working in linear regime; that is, the effective exhibited by the oscillator is higher than . The presented model, restricted to linear operation, sets two limiting cases bounding the true PN of a nonlinear oscillator. In the first case, the close-to-carrier PN approaches a linear oscillator with equal to . In the second case, the far-from-carrier performance is better described by a linear oscillator with equal to and reduced corner frequency. Thus, the nonlinear oscillator outperforms the linear oscillator with in terms of the integrated PN. This formulation is able to account for these two features using the same factor , which can be found using the equivalent power delivered to the resonator and a large-signal -parameter simulator, enabling simple predictive estimates of the nonlinear PN when transfer characteristics of the resonator are available over increasing power levels. For the cases considered, the average relative error between PN measurements and the proposed PN model is calculated to be less than 1%. ACKNOWLEDGMENT The authors wish to thank the MOSIS Fabrication Service for IC fabrication and Dr. W. Pan for device fabrication. REFERENCES [1] F. Ayazi, “MEMS for integrated timing and spectral processing (invited paper),” in Proc. IEEE CICC 2009, pp. 65–72. [2] M. Pardo, L. Sorenson, and F. Ayazi, “A phase-noise model for nonlinear piezoelectrically-actuated MEMS oscillators,” in Proc. ISCAS 2011, pp. 221–224. [3] B. Yurke et al., “Theory of amplifier noise evasion in an oscillator employing a nonlinear resonator,” Phys. Rev. A, vol. 51, pp. 4211–4229, 1995. [4] M. Pardo, L. Sorenson, and F. Ayazi, “Phase noise shaping via forced nonlinearity in piezoelectrically-actuated silicon micromechanical oscillators,” in Proc. IEEE MEMS 2011, pp. 780–783. [5] F. Walls and J. Gagnepain, “Environmental sensitivities of quartz oscillators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 39, no. 2, pp. 241–249, 1992. [6] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, 1966. [7] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, pp. 179–194, Feb. 1998. [8] L. He, Y. P. Xua, and M. Palaniapan, “State-space phase-noise model for nonlinear MEMS oscillators employing automatic amplitude control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 1, pp. 189–199, Jan. 2010. [9] P. Ward and A. Duwel, “Oscillator phase noise: Systematic construction of an analytical model encompassing nonlinearity,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 58, no. 1, pp. 195–205, 2011. [10] H. K. Lee et al., “Verification of the phase noise model for MEMS oscillators operating in the non-linear regime,” in Proc. Transducers, Jun. 2011, pp. 510–513.
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[11] E. Rubiola, The Leeson Effect: Phase Noise in Quasilinear Oscillators. Nancy, France: Universite Henri Poincare, 2005. [12] K. M. Lakin et al., “Thin film resonator technology,” in Proc. IEEE Freq. Contr. Symp., May 1987, pp. 371–381. [13] G. K. Ho et al., “Piezoelectric-on-silicon lateral bulk acoustic wave micromechanical resonators,” J. Microelecmech. Syst., vol. 17, no. 2, pp. 512–520, Apr. 2008. [14] S. Humad et al., “High frequency micromechanical piezo-on-silicon block resonators,” in Proc. IEDM, Dec. 2003, pp. 39.3.1–39.3.4. [15] R. Abdolvand and F. Ayazi, “Enhanced power handling and quality factor in thin-film piezoelectric-on-substrate resonators,” in Proc. IEEE Int. Ultrason. Symp., New York, Oct. 2007, pp. 608–611. [16] 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. [17] W. Weaver Jr., S. Timoshenko, and D. Young, Vibration Problems in Engineering, 5th ed. New York: Wiley, 1990. [18] H. Li et al., “Nonlinear free and forced oscillations of piezoelectric microresonators,” J. Micromech. Microeng., vol. 16, no. 2, p. 356, 2006. [19] M. Shahmohammadi et al., “Low jitter thin-film piezoelectric-on-substrate oscillators,” in Proc. IEEE FCS 2010, pp. 613–617. [20] F. He, R. Ribas, C. Lahuec, and M. Jezequel, “Discussion on the general oscillation startup condition and the Barkhausen criterion,” Analog Integr. Circuits Signal Process., vol. 59, no. 2, pp. 215–221, 2009. [21] W. P. Robins, Phase Noise in Signal Sources: Theory and Applications. London, U.K.: , 1982, vol. 9, IEE Telecommunication Series. [22] B. Razavi, “A study of phase noise in CMOS oscillators,” IEEE J. Solid-State Circuits, vol. 31, no. 3, pp. 331–343, Mar. 1996. [23] U. Rohde, Microwave and Wireless Synthesizers: Theory and Design. New York: Wiley, 1997. [24] P. Gray, P. Hurst, S. Lewis, and R. Meyer, Analysis and Design of Analog Integrated Circuits. New York: Wiley, 2001. [25] T. Gabrielson, “Mechanical-thermal noise in micromachined acoustic and vibration sensors,” IEEE Trans. Electron. Dev., vol. 40, no. 5, pp. 903–909, 1993. [26] J. Gagnepain, “Nonlinear properties of quartz crystal and quartz resonators: A review,” in Proc. 35th Annu. Symp. Freq. Control, 1981, pp. 14–30. [27] E. P. EerNisse, “Quartz resonator frequency shifts arising from electrode stress,” in Proc. 29th Annu. Symp. Freq. Control, 1975, pp. 1–4. noise in polycrystalline silicon thin films,” Phys. [28] L. Michelutti, “ Rev. B, vol. 57, no. 19, pp. 12 360–12 363, 1998. [29] X. Li and T. L. Tansley, “Laser-induced chemical vapor deposition of AlN films,” J. App. Phys., vol. 68, pp. 5369–5371, 1990. [30] 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., 2003, pp. 341–349. [31] E. Rubiola and R. Brendel, A Generalization of the Leeson Effect. Besançon, France: FEMTO-ST Institute, 2010. [32] W. Pan et al., “Thin-film piezoelectric-on-substrate resonators with enhancement and TCF reduction,” in Proc. IEEE MEMS 2010, pp. 104–107. [33] K. Sundaresan et al., “Process and temperature compensation in a 7 MHz CMOS clock oscillator,” Proc. IEEE J. Solid-State Circuits, vol. 41, no. 2, pp. 433–442, Feb. 2006. [34] K. Y. Kim and W. Sachse, “Nonlinear elastic equation of state of solids subjected to uniaxial homogeneous loading,” J. Mat. Sci., vol. 35, pp. 3197–3205, 2000. [35] V. Kaajakari et al., “Nonlinear mechanical effects in silicon longitudinal mode beam resonators,” Sensors Actuators, vol. 120, pp. 64–70, 2004. [36] T. Veijola and T. Mattila, “Modeling of nonlinear micromechanical resonators and their simulation with the harmonic-balance method,” Int. J. RF Microw. Comput.-Aided Eng., vol. 11, no. 5, pp. 310–321, Sep. 2001. [37] S. Levantino, C. Samori, A. Zanchi, and A. L. Lacaita, “AM-to-PM conversion in varactor-tuned oscillators,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 49, no. 7, pp. 509–513, Jul. 2002. [38] C. Samori, A. L. Lacaita, F. Villa, and F. Zappa, “Spectrum folding and phase noise in LC tuned oscillators,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 45, pp. 781–790, Jul. 1998. [39] IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology – Random Instabilities, IEEE Std. 1139-2008. [40] H. Miri Lavasani, A. K. Samarao, G. Casinovi, and F. Ayazi, “A 145 MHz low phase noise capacitive silicon micromechanical resonator,” in Proc. IEEE IEDM 2008, pp. 675–678.
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[41] H. M. Lavasani, W. Pan, B. Harrington, R. Abdolvand, and F. Ayazi, 1.7 GHz 0.18 CMOS tunable TIA using broadband “A 76 current pre-amplifier for high frequency lateral MEMS oscillators,” IEEE J. Solid-State Circuits, vol. 46, pp. 224–235, 2011.
Mauricio Pardo (S’03) received the B.S. degree in electronics engineering from Universidad del Norte, Barranquilla, Colombia, in 2002, and the M.S. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 2007. He is currently working toward the Ph.D. degree at Georgia Institute of Technology. In January 2003, he joined the faculty of Universidad del Norte, Barranquilla, Colombia, where he was an Associate Professor in the School of Electrical and Electronics Engineering. In August 2004, he was the Undergraduate Program Coordinator at the same university. His research interests are in the areas of oscillators and PLLs for clock cleaners and frequency synthesizers. Mr. Pardo is a 2005 recipient of the Fulbright/Colciencias/DNP Scholarship to pursue his doctoral studies in electronic design applications and microelectronics.
Logan Sorenson (S’02) received the B.S. and the M.S degrees in electrical engineering from the Illinois Institute of Technology (IIT), Chicago, in 2004 and 2007, respectively. He is currently working toward the Ph.D. degree at Georgia Institute of Technology, Atlanta. In 2003–2004, he spent one year on exchange in Stockholm, Sweden, at the Royal Institute of Technology (KTH). His research interests are in the areas of design, modeling, and simulation of resonant M/NEMS devices and phononic crystals. Mr. Sorenson was a recipient of a Camras and Grainger scholarship at IIT and the 2005 IEC Everitt Award of Excellence. He also earned the IIT Graduate Outstanding Academic Achievement Award, and, in 2007, the prestigious Georgia Tech Institute Fellowship to pursue doctoral studies in electrical and computer engineering.
Farrokh Ayazi (S’96–M’00–SM’05) received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 1994, and the M.S. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 1997 and 2000, respectively. In December 1999, he joined the faculty of Georgia Institute of Technology, Atlanta, where he is now a Professor in the School of Electrical and Computer Engineering. His research interests are in the areas of integrated micro- and nanoelectromechanical systems, interface IC design for MEMS and sensors, and RF MEMS. Prof. Ayazi is a 2004 recipient of the NSF CAREER Award, the 2004 Richard M. Bass Outstanding Teacher Award (determined by the vote of the ECE senior class), and the Georgia Tech College of Engineering Cutting Edge Research Award for 2001–2002. Dr. Ayazi is an editor for the IEEE Journal of Micro-Electro-Mechanical Systems, and the IEEE TRANSACTIONS ON ELECTRON DEVICES. He and his students won the best paper awards at Transducers 2011, the IEEE International Frequency Control Symposium in 2010, and IEEE Sensors conference in 2007. Dr. Ayazi is the Co-Founder and Chief Technology Officer of Qualtré Inc., a spin-out from his research Laboratory that commercializes multiaxis Bulk Acoustic Wave (BAW) silicon gyroscopes and six-degrees-of-freedom inertial sensors for consumer electronics and personal navigation systems.
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An Empirical Phase-Noise Model for MEMS Oscillators Operating in Nonlinear Regime Mauricio Pardo, Student Member, IEEE, Logan Sorenson, Student Member, IEEE, and Farrokh Ayazi, Senior Member, IEEE
Abstract—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. Index Terms—Detuning, frequency-phase transfer function, MEMS resonator, nonlinearity, power-series-based PN model.
I. INTRODUCTION
O
SCILLATORS are a fundamental building block of synchronous microsystems. Microelectromechanical system (MEMS)-based solutions can provide an alternative to quartz oscillators due to their compatibility with CMOS technology, facilitating the integration into a single chip or a compact package [1]. When a MEMS resonator is used in an oscillator, its quality factor ( ) impacts the phase-noise (PN) performance of the oscillator. A sustaining amplifier with high transimpedance gain is capable of exploiting nonlinear effects in a MEMS resonator to improve the PN of the oscillator [2]. The measurement of such an oscillator shows that the PN performance is progressively improved when the phase-shift provided by the sustaining amplifier forces the operating point away from the mechanical Manuscript received September 06, 2011; revised December 17, 2011; accepted March 14, 2012. Date of publication April 24, 2012; date of current version May 09, 2012. This work was supported by Integrated Device Technology, San Jose, CA. This paper was recommended by Associate Editor T. S. Lande. M. Pardo is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA, and also with the School of Electrical and Electronics Engineering, Universidad del Norte, Barranquilla, Colombia. (e-mail: [email protected]; [email protected]). L. Sorenson and F. Ayazi are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. 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/TCSI.2012.2195129
resonance [2]. Detuning the oscillator allows the selection of a higher phase-frequency slope that emulates a higher-order filter, minimizing the effect of phase variations on the oscillator frequency [3]. In order to deliver high enough power to the resonator, automatic level control is disabled, allowing the signal amplitude to reach supply rails. In addition, high oscillation amplitude saturates the sustaining amplifier, becoming an effective limiting mechanism that minimizes the amplitude variation contribution to the PN. As presented in [2] and [4], an oscillator exhibiting beneficial PN shaping due to nonlinearity can be described by a higher effective compared to when the resonator is operated in its linear regime. However, at frequencies very close-to-carrier, the PN is dominated by flicker noise effects [5] and approaches the same PN limit as in linear operation. As the offset frequency moves away from the carrier, the PN then approaches the high effective limit. Therefore, a PN formulation must include higher-order terms to describe this transition. Available PN models [6]–[8] cannot be applied directly to capture the progressive improvement observed in this work, because the formulations are based on approximations to explain the presence of as the highest order term. In particular, the work from [8] uses the state-space mathematics to explain this term when an automatic amplitude control circuit is used to avoid the nonlinearity of the MEMS resonator. On the other hand, the recent model proposed by [9] acknowledges the possibility of improved PN due to proper phase tuning of a nonlinear resonator, where the expected results are determined by the phase-frequency slope at the selected operating phase, and PN term and additive noise contributions. The captures the experimental results reported in [10] demonstrate this characteristic under nonlinear operation with amplitude limiting enabled. This paper presents an empirical PN model for resonatorbased oscillators that encompasses both linear and nonlinear operation of the resonator, as demonstrated in [2] and [4], where the improvement to the PN in nonlinearity derives from the high-order phase filter action [3]. The paper is organized as follows: first, a brief description of MEMS resonators and their nonlinear behavior is provided. Next, dominant noise sources are discussed, and the contribution of nonlinearity to favor a higher-order phase-filter transfer function is analyzed [11]. A proper description of this filter can be obtained using a powerseries approximation that provides additional details about PN shaping. A description of the testing set-up for three nonlinear piezoelectric resonators is provided, whose results are used to draw conclusions about the practical utility of the described formulation.
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II. OPERATING PRINCIPLES OF MEMS RESONATORS MEMS resonators are vibrating mechanical devices that can use IC processes to reduce size and enable integration. Acoustic waves can be generated in silicon micromechanical devices by using electrostatic transducers or thin films of piezoelectric materials such as aluminum nitride (AlN). Microresonators can be designed to operate in a bulk acoustic wave (BAW) mode, which is more suitable for higher frequency applications. In this mode, the resonance frequency is primarily defined by a single geometrical dimension of the device (for example, the piezoelectric film thickness [12] or the width of the resonator body [13]). A thin film of piezoelectric material can be deposited on the surface of a silicon resonator to excite a lateral BAW mode through the transverse piezoelectric coefficient [14]. The thickness of the silicon layer is typically chosen to be much thicker than that of the piezoelectric layer, so that the resonator frequency is predominantly determined by silicon [13]. The superior acoustic properties of single-crystalline silicon translate to very high and large power handling [15]. A MEMS resonator can be modeled electrically as a two-port admittance network comprising three components: the input transduction, the mechanical response, and the output transduction. The admittance of a piezoelectric device has essentially the same form as that of a series RLC resonator. The large electromechanical coupling of the piezoelectric material results in low motional impedance, relaxing the gain requirements on amplifiers to build oscillators. III. NONLINEARITY OF MEMS RESONATORS The concept of nonlinearity refers to the relation between the amplitude of vibration and the applied force (i.e., Hooke’s law). In a real spring, this relationship is generally nonlinear and given by (1) where is the restoring force, is the displacement, is the linear term, and are the second- and third-order corrections, and aggregates higher-order terms. The displacement-dependent spring constant can be defined as . The solution of the equation of motion describing a nonlinear resonator, of which the Duffing resonator is a particular case ( ), (2) can be solved considering two cases: unforced-undamped, and forced vibrations. An unforced-undamped resonator is expected to oscillate with constant amplitude at the resonance frequency . The nonlinear terms in the restoring force will control the amplitude, which changes the oscillation frequency to according to the relationship (3)
Fig. 1. Amplitude and phase responses of a Duffing-like resonator under different levels of the amplitude-frequency effect.
where corresponds to the magnitude of the vibration-amplitude distribution [16]. Considering the forcing function as , the vibration amplitude near resonance is given by (4)
defined by (3) [16]. From this equation, it can be deterwith mined that for less than , the resonance peak will shift to lower frequencies. This case is known as spring softening, since it can be viewed as being produced by a spring with decreased stiffness. On the other hand, when is greater than , the resonance peak can be moved to higher frequencies becoming a case of spring hardening [16]. Since the stiffness of the nonlinear system is amplitude-dependent, the vibration-amplitude distribution will also exhibit bending. A measure of the maximum vibration amplitude is obtained by calculating the bifurcation point (critical drive) beyond which the amplitude-frequency relationship is no longer a single-valued function [3], [16]. Fig. 1 summarizes the effect and the corresponding modification of the phase response. More detailed equations for the nonlinear resonator can be employed to take into account the mode shape of the resonant structure [17]. MEMS resonators can be forced into the nonlinear elastic regime when sufficiently high power is applied to the device structure, creating regions of high energy density [1], [10], [18], [19]. Intrinsic nonlinearities are dependent on effective acoustic velocity, stiffness coefficients, and propagation of the acoustic wave within the device, while induced nonlinearities may arise due to temperature, force, pressure, acceleration, and vibration, among other sources [5]. IV. OSCILLATOR CONFIGURATION USING RESONATOR
A
NONLINEAR
A MEMS-based oscillator consists of a sustaining amplifier and a micromechanical resonator connected in feedback configuration satisfying Barkhausen criterion [20]. The resonator can be viewed as a band-pass filter, whose selectivity is measured by . A shunt-shunt amplifier is typically used to interface a MEMS resonator, because the input and output impedances are decreased to minimize degradation with loading.
PARDO et al.: AN EMPIRICAL PHASE-NOISE MODEL FOR MEMS OSCILLATORS OPERATING IN NONLINEAR REGIME
Fig. 2. Progression of the phase-frequency slope with nonlinearity.
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Fig. 3. A detuned resonator can operate with increased phase-frequency slope. Gain and phase tuning are required to satisfy Barkhausen criterion.
The sustaining-amplifier phase determines the operating point of the system by matching the resonator phase at the desired oscillation frequency. The noise of the sustaining amplifier propagates through the closed-loop system with amplitude and phase components perturbing the oscillator signal, broadening its spectrum. Since the amplitude component can be neglected by means of a limiting mechanism, the phase noise is considered to be the main contributor to these sidebands. Careful circuit design is required to maintain the assumption of PM-only effect in the PN. The advantage of the nonlinear regime is the ability to select a higher phase-frequency slope to desensitize the oscillator frequency with respect to induced phase variations. Once the operating phase is selected and the noise sources are considered as small signals [11], [21], a narrowband approximation allows the definition of a slope in the vicinity of that point, linearizing the phase-frequency relationship locally. Determining this slope corresponds to the definition of open-loop , which is a measure of how the closed-loop system opposes variations in the frequency of oscillation [22]. Thus, it is observed that in nonlinearity, this slope can be higher than the linear case, reducing the effect of phase-modulation variations on oscillator frequency (Fig. 2) [3]. Even though this is a convenient approach to the analysis of nonlinear operation, the lack of symmetry around the operating point will limit the amplitude of the noise sources to an infinitesimal value, since the definition of the slope will not be the same for both sides around the selected phase. Hence, a proper formulation for the PN will require a representation with higher-order terms, and Fig. 2 implies that those terms are necessary when PN results are analyzed as the operating point is tuned.
In addition, the sustaining amplifier also contains nonwhite effects attributed to flicker noise. Transistor-based circuits, whose transconductance is dependent on the bias point, can inject phase shifts resulting in a low-frequency phase modulation [9]. The PN contribution due to the flicker noise is equal to (6) where is a fitting constant dependent on the active devices of the sustaining electronics [24]. Considering that white- and flicker-noise sources are uncorrelated, the total noise of the amplifier can be expressed as (7) or equivalently (8) with defined as the corner frequency where the flicker- and white-noise components become equal. The estimation of is empirical and almost independent of the oscillation power. Thus, when the oscillation power is lowered, the thermal noise is increased, which creates the effect of a reduced value of . [11]. B. Noise of the Nonlinear Resonator The noise spectrum due to Brownian-motion noise in the resonator can be approximated as [25] (9)
V. NOISE SPECTRUM OF THE OSCILLATOR COMPONENTS A. Noise of the Sustaining Amplifier Thermal noise of the amplifier is directly related to the noise factor defined as the ratio of the output noise of the actual amplifier to that of an equivalent noiseless version [23]. Using and dividing by the available signal power , its noise powerdensity is defined as (5) where and are the Boltzmann constant and temperature, respectively. The thermal-noise power density has a white spectrum characteristic.
is the resonator series equivalent impedance at where the oscillation frequency . If matches the resonance frequency, the motional capacitance ( ) and inductance ( ) from the RLC model cancel out, and equals the motional resistance ( ). However, in the nonlinear regime, the resonator operates with a small detuning that increases and includes a phase shift that needs to be compensated by the sustaining amplifier to satisfy the Barkhausen criterion (Fig. 3). Thus, the white-noise contribution of the resonator scales with the motional impedance. The net effect of detuning will increase the noise floor of the oscillator, unless the dominant noise sources stem from the electronics.
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Microresonators can have several sources of flicker noise when nonlinearity is attributed to elevated drive levels. The resonance frequency can be affected by stress in the deposited thin metal films in piezoelectric devices related to the third-order elastic coefficients of the structural material [18], [26], [27]. Dielectric materials present in the resonator (e.g., AlN) can exhibit hysteretic damping [28]. A resonator exhibits flicker noise given by (10) where is a coefficient capturing the hysteresis in the film due to dielectric loss, and is the capacitance of the piezoelectric stack [28]. For the case of AlN, is related to the dielectric loss tangent, which is approximately constant down to a few hundred Hertz [29], contributing additional flicker noise. Additionally, a form of Johnson noise can be observed since self-heating becomes significant for microresonators. Elevated power generates a current associated with random motion of charge carriers in the body of the MEMS device that can be classified as white frequency noise [5]. Design of amplitude control circuitry could mitigate some of the nonwhite processes induced in the resonator [8], [30]; however, for piezoelectrically-transduced resonators studied in this work, the drive level is the mechanism employed to induce beneficial nonlinearity, and the amplitude limiting mechanism utilized is waveform clipping by the sustaining amplifier. VI. PHASE NOISE MODEL FOR AN OSCILLATOR WITH NONLINEAR RESONATOR
A
When the phase is the variable of interest, a resonant device with resonance frequency and quality factor can be expressed as a first-order low-pass filter with a pole equal to defining the Leeson corner frequency, [31]. Considering steady state, the output noise spectral density can be found by multiplying the input spectral noise density and the squared magnitude of . Therefore, the Leeson effect is [11]. defined as the multiplication by of the PN below Equation (11) shows the expression for the output noise spectral density for this case as (11) corresponds to the power spectral denwhere sity of the flicker- and white-noise noise sources for both amplifier and resonator. Equation (11) is the well-known Leeson’s model [4]. Fig. 4(a) shows the squared magnitude of the resonator, and Fig. 4(b) depicts how the PN at the output will be configured. Fig. 4 illustrates the particular case of high- resonators, where the flicker noise corner is much higher that the Leeson corner frequency ( ), which is the case of the MEMS devices considered in this work. For a nonlinear resonator, the associated low-pass transfer function will be equivalent to a higher-order filter, with order proportional to the strength of the induced nonlinearity. Therefore, the Leeson effect is more pronounced due to
Fig. 4. (a) Squared magnitude of resonator as phase filter. (b) The Leeson effect for a first-order equivalent low-pass resonator [4].
Fig. 5. A higher-order phase filter reduces the frequency of the Leeson effect and produces a steeper slope close to the carrier.
an elevated number of roots in the resonator transfer function that increases the slope roll-off (Fig. 5). Mathematically, the transfer function of the higher-order filter can be described by using a truncated power series of the exponential function with a number of terms equal to the order of the filter. This selection also allows a description of how the noise sources are modified by the asymmetry in the nonlinear resonator transfer function. When the power series is used, a parameter is defined as the ratio between the slopes in nonlinear regime at the detuned frequency and in linear operation at resonance. This power-series approximation uses a higher effective equal to , where denotes the loaded quality factor in linear operation. The parameter in the associated input spectral density will be affected by several factors. From a resonator point of view, the detuning that selects the steeper phase-frequency slope also increases the motional impedance of the MEMS device, and therefore its white noise contribution. However, if the elevated drive power generates a signal that forces the sustaining amplifier to operate as a hard limiter, then the noise folding produced for this rail-to-rail operation can dominate the far-from-carrier contribution. In any case, it is expected that the nonlinear oscillator will have an increased noise floor that will reduce the parameter , resulting in what is known as a noise-corrected oscillator [11]. Starting from linear operation, (11) can be modified to give the form of the power series of the exponential function as
(12) reflecting the Leeson effect and The second-order term containing noise processes that vary with and can be viewed as the first three terms of the series expansion for the first-order low-pass equivalent of the resonator.
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Fig. 7. SEM images of the AlN-on-Si resonators used in this work. (a) Twofinger electrode design and (b) Five-finger electrode design.
Fig. 6. Schematic of the two-finger design composite AlN-on-Si MEMS resonator illustrating the material layers of the composite stack.
Hence, when operated in nonlinear regime, (12) can be viewed as a particular case of the general term
(13)
Fig. 8. Representation of the IPS and LE resonance-mode shapes. The deformation of the plate is exaggerated to reveal details.
which generates the required higher-order components to provide an approximation of the expression for the phase filter of order . Notice that the power spectral density of the whiteand flicker- noise processes becomes because their contribution is determined for the specific operating conditions (power and detuning). Thus, (13) is a general expression that considers different levels of beneficial nonlinearity and reduces to Leeson’s model for equal to one (linear operation of the resonator). It can be concluded that for theoretical PN models that consider the transfer function of the resonator, the power series expansion can be an alternative starting point to capture the effect of an increased oscillator .
(LE mode presents 0 at resonance). The shapes of the excited modes are presented in Fig. 8. The outlines correspond to the peak displacements of each half cycle of oscillation. Device characterization is performed using an Agilent E5071C vector network analyzer (VNA) with 50 terminations producing estimates for and motional resistance as presented in Fig. 9. Power-induced nonlinearity is verified using a Mini-Circuits ZFL-1000VH power amplifier. The onset of nonlinearity in the 23 MHz IPS mode occurs close to , and the device can withstand power levels of at least . A similar onset is observed for the 26 MHz LE mode in both devices, but the resonator fractures at about , failing at the corners of the support tethers where the stress concentration is high. The IPS mode is more susceptible to nonlinearity due to the controlling combination of third-order stiffness components of silicon, which is the main structural material of the resonators. For illustration, Fig. 10 shows the progression of the nonlinear effect as the power supplied to the IPS mode is increased.
VII. EXPERIMENTAL VERIFICATION OF THE NONLINEAR PHASE-NOISE MODEL A. Piezoelectric Micromechanical Resonators Composite aluminum-nitride-on-silicon (AlN-on-Si) lateral microresonators are employed to demonstrate the applicability of the empirical PN model proposed in Section VI. The devices are fabricated from a 10 thick silicon-on-insulator (SOI) substrate. The resonator stack composition and thicknesses are 9/2/0.1/1/0.1 of Si/ /Mo/AlN/Mo, respectively. Two designs are utilized: two-finger and five-finger electrode configurations, both on top of a plate with lateral dimensions of 156 250 . Details of the composite material stack are shown in Fig. 6, using the two-finger design for illustration purposes. Fig. 7 shows SEM images of the devices, whose fabrication process is described in [32]. The two-finger electrode configuration excites and senses an in-plane shear (IPS) mode at 23 MHz and a longitudinal extensional (LE) mode at 26 MHz, while the five-finger design produces only a LE mode at 26 MHz. The IPS and LE modes are electrically 180 out-of-phase with respect to one another
B. Transimpedance Amplifier Design An inverter-based transimpedance amplifier (TIA) with active feedback resistor is interfaced with the MEMS resonators [2]. The TIA provides interface ports to operate with either 0 or 180 phase-shift resonance modes (see Fig. 11). The input-stage feedback resistor is implemented with a NMOS transistor whose operation is controlled by the gate voltage, . At one extreme, when is minimum, the feedback transistor enters subthreshold region and the resistance is maximum giving the highest possible gain for the TIA. On the other hand, when is increased to the power-supply rail, the feedback transistor turns on and becomes the preferred path for the current to the output of the inverting stage. The transimpedance gain approximates the transistor on-resistance.
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verter. Thus, when the resistance is increased, the time constant changes proportionally, and the pole moves to lower frequencies. This strategy is used to finely modify the TIA phase-shift, providing a method to move along the phase-frequency curve. A comparator is placed at the oscillator output to work as a digital buffer and to maintain constant amplitude prior to the PN measurement equipment. Thus, the objective is to make the comparator as transparent as possible for the PN measurement. The process-independent threshold-voltage compensated comparator available in [33] is used as the digital buffer. C. Modeling of a Nonlinear Resonator
Fig. 9. Characterization of the AlN-on-Si resonators used in this work. (a) Twofinger electrode design (IPS). (b) Two-finger electrode design (LE). (c) Fivefinger electrode design (LE).
The nonlinear nature of silicon becomes apparent for resonators driven at high amplitudes, and the nonlinearities are assumed to be stiffness-induced for the cases considered. The silicon third-order elastic coefficients are negative for the modes of interest, and an adequate description of the resonator behavior can be obtained using a second-order displacement-dependent spring constant ( ). Nonzero and negative are identified [34] according to the nonlinear strain-dependent Young’s modulus of the main structural material [35]. Equation (2) can be implemented with a large-signal -parameter simulator [e.g., Advance Design System (ADS)] to generate an estimate of can be modeled with an independent current source, and spring and damping forces with voltage-controlled current sources [36]. Nonlinearity is taking into account by using a polynomial voltage-controlled current source for the spring force. The parameters for the RLC series tank can be determined from the -parameter measurements of the considered MEMS resonators, which include a calculation for . Estimation of and can be done by fitting the measured parameter at a specific power level by evaluation of the correlation factor between measurements and simulations. The factor is calculated by the ratio of the maxima of the phase-response derivatives for both nonlinear and linear operation. D. Calculating the Power Delivered to the Resonators
Fig. 10. Magnitude and phase-frequency plots of the resonator 23 MHz-IPS mode for increased power levels.
Changing the setting of simultaneously modifies the phase-shift provided by the TIA because the location of the dominant pole is varied accordingly. The associated time constant can be determined with the value of the active resistor and the gate capacitances of the transistors that configure the in-
From Fig. 9, the worst-case scenario for resonator losses corresponds to a motional resistance equal to 2700 or about 69 . The sustaining amplifier of Fig. 11 provides a transimpedance gain close to 100 , which is sufficient to overcome such level of resonator losses. The design of the sustaining amplifier does not include an amplitude control circuit, which allows the waveform to grow until the large signal saturates the amplifier, hard clipping the output signal. Amplifier saturation becomes an effective form of amplitude control and still allows increased power to drive the resonator. Due to clipping of the generated waveform, amplitude variations are minimized, reducing the AM-to-PM conversion [37] due to the effect. Power delivered to the resonator will remain constant if the power-supply variation is negligible. A 5 V supply voltage is utilized to bias the electronics fabricated on a 0.5 2P3M CMOS process. To find the equivalent power that the VNA would deliver to the resonator during the characterization stage, the transfer characteristic of the sustaining amplifier can be extracted. There are two transfer characteristics, one for each phase-shift version of the TIA. For the
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Fig. 11. Transimpedance amplifier schematic. The generic TIA provides 0 and 180 ports to interface several MEMS devices and resonance modes.
Fig. 12. IC-resonator configuration via bondwires.
180 case, the TIA is only the shunt-shunt feedback stage, and by lowering the gain is increased producing a transfer characteristic similar to that of an inverter. In this situation, the amplitude at the output can reach the rails, producing nearly a square waveform. On the other hand, for the 0 version, the CS stage exhibits a different transfer characteristic due to its fixed load. The passive resistor employed has to be selected such that the symmetry of the transfer characteristic is maintained while at the same time lowering the output voltage at transistor M4 and the power consumption. Thus, as a conservative estimate, selecting about at the resonator input for both TIA versions and given the 50 termination of the measuring equipment, the equivalent power at the VNA source is approximately 13 dBm for the worst-case motional resistance considered. This drive level is above the onset of nonlinearity for all the test resonators. For this estimated power level, equals five for the 23 MHz resonance mode and three for the 26 MHz resonance modes following the procedure of Section VII-C. E. Determining the Parameters
and
Fig. 12 presents a schematic of the IC interfaced with a piezoelectric resonator through minimum-length aluminum bondwires. Three oscillators are configured using a similar set-
Fig. 13. PN results from piezoelectric-based oscillator operating in air. The trendline is added for comparison with Leeson’s model.
up: 23 MHz IPS and 26 MHz LE oscillators with a two-finger electrode resonator, and a 26 MHz LE oscillator using a fivefinger electrode resonator. In all cases, the oscillations start up properly with low transimpedance gain. As is reduced, the gain increases and
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the waveform reaches the saturation levels. Further decrease of only affects the TIA phase-shift to optimize the operating phase-frequency slope. Thus, the PN performance is measured for different settings of with an Agilent E5500 PN test set. The measurement summary includes a slope line, which describes the flicker-noise processes using Leeson’s formula at the limit of linear operation (Fig. 13). The best Leeson’s fitting estimates a noise figure NF ( ) of 45 dB and close to 50 kHz. Notice that the elevated NF stems from the noise folding due to the TIA behavior as hard limiter causing noise components to be folded within the bandwidth of the resonator, due to the rail-to-rail operation of the sustaining amplifier [22], [38]. Since the worst-case for testing is 2700 , and , it is concluded that the white-noise contribution of the oscillator PN is dominated by the electronics.
Fig. 14. PN model for the 23 MHz IPS piezoelectric resonator (2-finger).
F. Application of the Nonlinear Phase-Noise Model Once the parameters have been determined, (13) is used to describe the PN performance for all the oscillators. The standard definition for PN uses the single-sideband expression of the output spectral power density [39] in decibels per Hertz relative to the carrier power. The calculated factor corresponds to the best-measured performance for each case. As (13) uses to include white and flicker noises for amplifier and resonator, the flicker noise term will be presented explicitly to show the reduction in the parameter , and a factor labeled as will be used to represent the white noise of the complete oscillator. It is interesting to note that the best fitting for (13) is given for an effective equal to , the same factor that increases the . Thus, the model for the 23 MHz two-finger electrode design oscillator takes the form:
while for the 26 MHz two-finger and five-finger electrode designs, the models become
and
Fig. 15. PN model for the 26 MHz LE piezoelectric resonator (2-finger).
Fig. 16. PN model for the 26 MHz LE piezoelectric resonator (5-finger).
respectively. Figs. 14–16 show the model description of the measured results. In each case, the linear operation described by (11) has been included for comparison. It can be observed that the induced nonlinearity in the test MEMS resonators produces a maximum performance improvement of 20 dB and 15 dB at an offset frequency of about 200 Hz for the IPS and LE modes, respectively. Note that the nonlinear IPS 23 MHz oscillator, with an absolute figure of at 1 kHz offset frequency, is comparable to architectures with highcapacitive resonators [40]. In addition, when compared with state-of-the-art piezoelectric devices at higher frequencies [41], the IPS mode oscillator can also satisfy the GSM standard (after frequency up-conversion) using a resonator with a of only 4000.
PARDO et al.: AN EMPIRICAL PHASE-NOISE MODEL FOR MEMS OSCILLATORS OPERATING IN NONLINEAR REGIME
VIII. CONCLUSIONS A power series expansion provides a convenient and practical way to model the PN behavior of a nonlinear oscillator, because it provides the higher-order terms needed to reflect the relationship between the noise processes and the nonlinear phase-frequency response when the resonator is operated with elevated drive levels. This formulation reflects the higher effective and the reduction of the corner frequency for beneficial nonlinearity, and simplifies to the Leeson’s model for linear operation. Beneficial nonlinearity means that the PN response is superior to the equivalent oscillator working in linear regime; that is, the effective exhibited by the oscillator is higher than . The presented model, restricted to linear operation, sets two limiting cases bounding the true PN of a nonlinear oscillator. In the first case, the close-to-carrier PN approaches a linear oscillator with equal to . In the second case, the far-from-carrier performance is better described by a linear oscillator with equal to and reduced corner frequency. Thus, the nonlinear oscillator outperforms the linear oscillator with in terms of the integrated PN. This formulation is able to account for these two features using the same factor , which can be found using the equivalent power delivered to the resonator and a large-signal -parameter simulator, enabling simple predictive estimates of the nonlinear PN when transfer characteristics of the resonator are available over increasing power levels. For the cases considered, the average relative error between PN measurements and the proposed PN model is calculated to be less than 1%. ACKNOWLEDGMENT The authors wish to thank the MOSIS Fabrication Service for IC fabrication and Dr. W. Pan for device fabrication. REFERENCES [1] F. Ayazi, “MEMS for integrated timing and spectral processing (invited paper),” in Proc. IEEE CICC 2009, pp. 65–72. [2] M. Pardo, L. Sorenson, and F. Ayazi, “A phase-noise model for nonlinear piezoelectrically-actuated MEMS oscillators,” in Proc. ISCAS 2011, pp. 221–224. [3] B. Yurke et al., “Theory of amplifier noise evasion in an oscillator employing a nonlinear resonator,” Phys. Rev. A, vol. 51, pp. 4211–4229, 1995. [4] M. Pardo, L. Sorenson, and F. Ayazi, “Phase noise shaping via forced nonlinearity in piezoelectrically-actuated silicon micromechanical oscillators,” in Proc. IEEE MEMS 2011, pp. 780–783. [5] F. Walls and J. Gagnepain, “Environmental sensitivities of quartz oscillators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 39, no. 2, pp. 241–249, 1992. [6] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, 1966. [7] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, pp. 179–194, Feb. 1998. [8] L. He, Y. P. Xua, and M. Palaniapan, “State-space phase-noise model for nonlinear MEMS oscillators employing automatic amplitude control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 57, no. 1, pp. 189–199, Jan. 2010. [9] P. Ward and A. Duwel, “Oscillator phase noise: Systematic construction of an analytical model encompassing nonlinearity,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 58, no. 1, pp. 195–205, 2011. [10] H. K. Lee et al., “Verification of the phase noise model for MEMS oscillators operating in the non-linear regime,” in Proc. Transducers, Jun. 2011, pp. 510–513.
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[41] H. M. Lavasani, W. Pan, B. Harrington, R. Abdolvand, and F. Ayazi, 1.7 GHz 0.18 CMOS tunable TIA using broadband “A 76 current pre-amplifier for high frequency lateral MEMS oscillators,” IEEE J. Solid-State Circuits, vol. 46, pp. 224–235, 2011.
Mauricio Pardo (S’03) received the B.S. degree in electronics engineering from Universidad del Norte, Barranquilla, Colombia, in 2002, and the M.S. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 2007. He is currently working toward the Ph.D. degree at Georgia Institute of Technology. In January 2003, he joined the faculty of Universidad del Norte, Barranquilla, Colombia, where he was an Associate Professor in the School of Electrical and Electronics Engineering. In August 2004, he was the Undergraduate Program Coordinator at the same university. His research interests are in the areas of oscillators and PLLs for clock cleaners and frequency synthesizers. Mr. Pardo is a 2005 recipient of the Fulbright/Colciencias/DNP Scholarship to pursue his doctoral studies in electronic design applications and microelectronics.
Logan Sorenson (S’02) received the B.S. and the M.S degrees in electrical engineering from the Illinois Institute of Technology (IIT), Chicago, in 2004 and 2007, respectively. He is currently working toward the Ph.D. degree at Georgia Institute of Technology, Atlanta. In 2003–2004, he spent one year on exchange in Stockholm, Sweden, at the Royal Institute of Technology (KTH). His research interests are in the areas of design, modeling, and simulation of resonant M/NEMS devices and phononic crystals. Mr. Sorenson was a recipient of a Camras and Grainger scholarship at IIT and the 2005 IEC Everitt Award of Excellence. He also earned the IIT Graduate Outstanding Academic Achievement Award, and, in 2007, the prestigious Georgia Tech Institute Fellowship to pursue doctoral studies in electrical and computer engineering.
Farrokh Ayazi (S’96–M’00–SM’05) received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 1994, and the M.S. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 1997 and 2000, respectively. In December 1999, he joined the faculty of Georgia Institute of Technology, Atlanta, where he is now a Professor in the School of Electrical and Computer Engineering. His research interests are in the areas of integrated micro- and nanoelectromechanical systems, interface IC design for MEMS and sensors, and RF MEMS. Prof. Ayazi is a 2004 recipient of the NSF CAREER Award, the 2004 Richard M. Bass Outstanding Teacher Award (determined by the vote of the ECE senior class), and the Georgia Tech College of Engineering Cutting Edge Research Award for 2001–2002. Dr. Ayazi is an editor for the IEEE Journal of Micro-Electro-Mechanical Systems, and the IEEE TRANSACTIONS ON ELECTRON DEVICES. He and his students won the best paper awards at Transducers 2011, the IEEE International Frequency Control Symposium in 2010, and IEEE Sensors conference in 2007. Dr. Ayazi is the Co-Founder and Chief Technology Officer of Qualtré Inc., a spin-out from his research Laboratory that commercializes multiaxis Bulk Acoustic Wave (BAW) silicon gyroscopes and six-degrees-of-freedom inertial sensors for consumer electronics and personal navigation systems.
