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A Phase-Noise Model for Nonlinear Piezoelectrically-Actuated MEMS Oscillators Mauricio Pardo1,2, Logan Sorenson1, Farrokh Ayazi1 1
Georgia Institute of Technology, Atlanta, Georgia, USA Fundación Universidad del Norte, Barranquilla, Colombia Email: [email protected]
2
Abstract—This paper presents a new empirical exponential-series-based model that describes the measured phase-noise (PN) of a nonlinear piezoelectric MEMS oscillator. Since the order of terms in the Leeson’s model can only provide at maximum -30 dB/dec slope, a model with higher order terms is proposed to determine an effective resonator quality factor (Qeff) and reflect the reduced contribution of the amplifier flicker-noise. The average fitting error of the proposed model to a 23 MHz in-plane shear (IPS) mode oscillator is found to be less than 1%. Nonlinear operation produces an oscillator with a PN of -130 dBc/Hz at 1kHz offset-frequency, which corresponds to an improvement of about 15 dB compared to linear operation.
I.
INTRODUCTION
High-performance, low phase-noise oscillators form a fundamental component of portable communication systems, and therefore their miniaturization is critical to support multiple wireless standards in a single device. Extensive research has focused on building resonators that can effectively replace large off-chip crystals. Microelectromechanical-systems (MEMS)-based solutions provide an alternative to traditional quartz oscillators in their ability to be integrated into a single chip or compact package [1]. Previous work has shown that capacitively-transduced devices can meet tight phase noise (PN) specifications such as the GSM standard when operated linearly [2]. However, the use of excessively large polarization voltages makes a compact, low-power solution with capacitive resonators challenging. On the other hand, piezoelectrically-transduced resonators exhibit lower motional impedances compared to capacitive devices and do not need a polarization voltage, simplifying the electronic interface design while reducing power and die area. Nevertheless, because of their relatively lower quality factors (Q), piezoelectric devices usually do not exhibit comparable PN performance to capacitive devices. Nonlinear operation of resonators has been discovered to improve the PN by suppressing the flicker noise of the interface circuitry [3]. While piezoelectric devices lack the transducer-induced nonlinearity found in capacitive devices [4], they can still be forced into the nonlinear elastic regime if
978-1-4244-9474-3/11/$26.00 ©2011 IEEE
sufficiently high power is applied to the electrodes [5, 6]. In this work, we demonstrate that a sustaining amplifier with high transimpedance gain is capable of exploiting a beneficial nonlinearity of piezoelectric resonators to improve PN through an adequate selection of the operating phase; a new empirical model is proposed to describe the performance of oscillators working in this regime. The model employs an effective quality factor that better captures the oscillator PN compared to the linear definition using the Leeson’s model. An overview of the operation of the piezoelectric MEMS resonators and characterization results over increasing power levels is given in Section II. Section III details the characteristics that allow the sustaining amplifier to take advantage of the resonator nonlinearity. Section IV presents the PN measurement results of the configured piezoelectric oscillator, so that the proposed PN model can be discussed in Section V. Finally, conclusions are drawn from the application of the model and the accuracy of the PN fitting. II.
PIEZOELECTRIC MEMS RESONATOR NONLINEARITY
A composite aluminum-nitride-on-silicon (AlN-on-Si) resonator is used in this work. The device is fabricated from a 10 μm thick silicon-on-insulator (SOI) substrate. The resonator and stack composition is shown in Figure 1, and it can be described as a two-finger electrode design on top of a plate with lateral dimensions of 156×250 μm. The device is designed to have a longitudinal extensional (LE) resonant frequency of 26 MHz and an in-plane shear (IPS) mode at 23 MHz. The IPS and the LE modes are electrically 180° out of phase with respect to one another. Figure 2 shows an SEM image of the final device; the fabrication process details can be found in [7]. When an AC signal is applied to the drive electrode, an oscillating electric field is produced between top and bottom molybdenum (Mo) layers across the piezoelectric AlN layer, which is converted into mechanical vibration via the inverse piezoelectric effect. Near resonance, the vibration is recaptured by the piezoelectric layer into a net charge, inducing a displacement current in the sense electrode. As the source power is increased, the device enters the nonlinear
221
Figure 3: Visualization of spring-softening and hardening effects in MEMS resonators.
Figure 1: Schematic of the composite AlN-on-Si MEMS resonator used in this work illustrating the material layers of the composite stack.
operation regime due to material and geometrical effects. The nonlinearity induces the peak of the frequency response to bend and shift toward lower frequencies (i.e. spring-softening), or higher frequencies (i.e. spring-hardening). This is depicted schematically in Figure 3. The mechanically-induced nonlinearity found in piezoelectric resonators can fall into either category depending on the configuration of the device [5]. Figure 4 shows the resonator characterization measurements for power levels between -35 dBm and +30 dBm taken with an Agilent E5071C vector network analyzer (VNA). The resonance mode of interest is the 23 MHz in-plane-shear (IPS) mode as it is able to sustain nonlinear operation at power levels of +30 dBm (the device fractures at power levels of about +20 dBm for the 26 MHz resonance mode). Pronounced nonlinearity in both magnitude and phase appears as sharp vertical transitions in the measurements due to sweeping characteristic of the test equipment.
Figure 2: SEM view of the MEMS resonator used in this work.
Figure 4: Magnitude and phase frequency plots of the in-plane-shear (IPS) mode of the MEMS resonator at 23 MHz showing spring-softening behavior.
III.
TRANSIMPEDANCE AMPLIFIER DESIGN
An inverter-based transimpedance amplifier (TIA) with active feedback resistor is interfaced with the MEMS resonator to analyze the nonlinearity effects on the oscillator PN. The TIA provides interface ports to operate with either 0° or 180° phase-shift resonance modes (see Fig. 5). The input-stage feedback resistor is implemented with a MOS transistor whose operation is controlled by the gate voltage, VCTRL. At one extreme, when VCTRL is minimum, the feedback transistor enters cut-off and the transimpedance gain is set by the inverter at the input. On the other hand, when VCTRL is maximum, the feedback transistor turns on, bypassing the inverter and setting the transimpedance gain to its lowest value equal to the transistor on-resistance. A transimpedance gain close to 100 dBΩ at 23 MHz is able to induce nonlinear effects in the piezoelectric-based resonator. Increasing the gain at the input stage simultaneously moves the associated dominant pole to lower frequencies. This strategy can be used to finely modify the TIA phase-shift, providing a means to move along the resonance curve (Fig. 4). The optimum quiescent point is where the operating phase slope tends to infinity. Here, an infinitesimal change in the phase will result in no frequency shift, leading to long-term phase stability [3] and a beneficial effect in PN. When the resonator is pushed deep into nonlinearity, the phase-frequency plot becomes multivalued with two points of infinite slope. The TIA can be tuned to operate at either point due to the fact that the frequency, which is the variable of interest, remains a single-valued function of the phase.
222
Figure 5: Transimpedance Amplifier Schematic. The generic TIA provides 0º and 180º ports to interface several MEMS devices.
A process-independent threshold-voltage inverter-based comparator is placed at the oscillator output to maintain the noise-floor during the PN measurements. Threshold-voltage insensitivity is achieved by negative feedback when the resistance levels of M13 and M16 are used as control variables [8]. Figure 6 presents the micrograph of the IC fabricated in a 0.5 μm 2P3M CMOS process and interfaced with the 23 MHz AlN-on-Si resonator through minimum-length aluminum bondwires. IV.
OSCILLATOR ARCHITECTURE AND PN PERFORMANCE
The 23 MHz IPS mode is excited using the TIA 180° resonator port. The oscillations were started up properly with low transimpedance gain. As VCTRL is finely reduced, the gain is sufficient to reach the power-supply rails, after which further decrease of VCTRL only affects the operating phase of the TIA. The PN performance is measured with an Agilent E5500 PN test set producing the PN plot set of Figure 7. Figure 7 includes the -30 dB/dec slope line observed under linear operation when VCTRL is set to the minimum gain. This line corresponds to the best fitting with the Leeson’s formula. From this procedure, the Qloaded and the corner flicker frequency are estimated. The quality factor is in close agreement with the value reported by the VNA (Q ~ 4K), and the flicker corner frequency is determined to be 50 kHz. As VCTRL is reduced, nonlinearity causes the PN plots to bend under the linear reference, implying that the oscillator is operating with a higher quality factor compared to the Qloaded. This beneficial nonlinearity causes the 23 MHz IPS mode
oscillator to exhibit a PN of -130 dBc/Hz at 1kHz offset-frequency operating in air. Such performance corresponds to an improvement of about 15 dB compared to linear operation. With a measured Qloaded of 4K, the PN reported in this work is comparable to architectures utilizing very-high-Q piezoelectric devices [6] and vacuum-packaged capacitive resonators at lower frequencies [2]. V.
NONLINEAR PN MODEL FORMULATION
Since the TIA circuit does not include automatic gain control, Leeson’s model should predict three sections on the PN plot for a high Q resonator: two linear segments with slopes equal to -30 dB/dec (modulated flicker noise) and -10 dB/dec (flicker noise), and the noise floor. Increased close-to-carrier slopes shown in Figure 7 indicate that the Leeson’s formula cannot describe this behavior, and therefore an alternative model with higher order terms is needed. Initially conceived as an empirical model, a mathematical derivation for the Leeson’s formula is found in [11], and it can be used as starting point for a more complete formulation. If |H(jω)| is the equivalent low-pass expression for the resonator and SΦi(ω) and SΦo(ω) are the one-side PN densities at the input and output of the TIA [11], the general form of a PN model can be expressed as
Figure 7: PN results from 23 MHz in-plane shear oscillator operating in air. The –30 dB/dec trend line is added for comparison with Leeson’s model. Figure 6: IC-resonator configuration via bondwires.
223
Sφi (ω ) = Sφo (ω ) ⋅ M (ω )
[
(1)
]
−1
M (ω ) = (H( jω ) −1)⋅ (H * ( jω ) − 1) .
(2)
For the case where the nonlinearity is introduced by the resonator only, SΦo(ω) remains dependent on the flicker and thermal noise of the TIA, which implies that higher order terms are needed in |H(jω)| to capture the nonlinear effects. Good quantitative description of the PN results is obtained by expanding M(ω) as an exponential series. However, this approach requires an adjustment to Q reflecting the strength of the nonlinearity. A multiplicative constant, N, is defined for that purpose. When the resonator is operated linearly, H(jω) is expressed by a first-order low-pass filter. For nonlinear operation, the equivalent low-pass transfer function can be modeled as cascaded first-order sections. The number of stages, which is equal to N, can be estimated using the phase-frequency plot for the MEMS resonator (Fig. 4). N is found as the ratio of the maxima of the derivatives of the phase responses for both linear and nonlinear regimes. Alternatively, it can also be obtained directly from the PN results as ∂ (3) N= (L( f )) 10 −1, ∂f max which allows the definition of Qeff and fceff as
Qeff = N⋅ Q,
f ceff = f c N
Figure 8: Fitting of the PN result using the exponential-series model for the 23MHz IPS mode oscillator. Qeff of 20K is used in the model.
and special packaging techniques. Additionally, a reduction in the flicker corner frequency value has been observed and quantified as that the same factor that increases the Qloaded. Under these conditions, the average relative error between PN measurements and the proposed PN model is calculated to be less than 1%. ACKNOWLEDGMENTS The authors wish to thank the MOSIS Fabrication Service for IC fabrication and Dr. Wanling Pan for device fabrication. REFERENCES
(4)
[1]
producing the proposed PN model in (5). For the present demonstration, the 23 MHz-oscillator best PN exhibits a maximum slope of –60 dB/dec, which produces N equal to 5. Therefore, using the parameters estimated in Section IV, a close fitting using (5) is attained with Qeff and fceff equal to 20K and 10 kHz, respectively. Figure 8 shows these results compared with the best fit using Leeson’s model. CONCLUSIONS A new empirical model based on a truncated exponential-series expansion has been proposed. The model includes higher order terms that allow a close fit of the PN measurement results when the resonator is operated in the nonlinear regime. It is found that the nonlinearity exhibited by a 23 MHz in-plane shear resonator leads to a beneficial effect on the oscillator PN when an appropriate operating point is chosen. Beneficial nonlinearity means that the PN response is superior to the equivalent oscillator working in the linear regime; that is, the effective Q exhibited by the oscillator is higher than the Qloaded. Operating Qs as high as 20K can be obtained from relatively low Q piezoelectrically-transduced resonators, which do not require large polarization voltages
F. Ayazi, “MEMS for Integrated Timing and Spectral Processing,” Invited Paper, Proc. IEEE CICC 2009, pp. 65-72, 2009. [2] V. Kaajakari et al., "Square-extensional mode single-crystal silicon micromechanical resonator for low-phase-noise oscillator applications," IEEE Electron Device Letters, vol. 25, no. 4, pp. 173-175, Apr. 2004. [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. Agarwal et al., "Nonlinear Characterization of Electrostatic MEMS Resonators," 2006 IEEE IFCS Exposition, pp. 209-212, 2006. [5] H. Li et al., "Nonlinear free and forced oscillations of piezoelectric microresonators," J. Micromech. Microeng., vol. 16, no. 2, pp. 356, 2006. [6] M. Shahmohammadi et al., “Low jitter thin-film piezoelectric-onsubstrate oscillators,” IEEE FCS 2010, pp. 613-617, 2010. [7] W. Pan et al., “Thin-Film Piezoelectric-on-Substrate Resonators with Q Enhancement and TCF Reduction,” IEEE MEMS 2010, pp. 104-107, 2010. [8] K. Sundaresan et al., “Process and temperature compensation in a 7MHz CMOS clock oscillator,” IEEE JSSC, vol. 41, no. 2, pp. 433-441, 2006. [9] H.M. Lavasani et al., “A 500MHz Low Phase-Noise AlN-on-Silicon Reference Oscillator”, IEEE CICC 2007, pp. 599-602, 2007. [10] H.M. Lavasani et al., “A 76dBOhm, 1.7 GHz, 0.18um CMOS Tunable Transimpedance Amplifier Using Broadband Current Pre-Amplifier for High Frequency Lateral Micromechanical Oscillators,” IEEE ISSCC 2010, pp. 318-320, 2010. [11] G. Sauvage, “Phase noise in oscillators: A mathematical analysis of Leeson’s model,” IEEE Trans. Instrum. Meas., vol. 26, no. 4, pp. 408–410, Dec. 1977.
224
Georgia Institute of Technology, Atlanta, Georgia, USA Fundación Universidad del Norte, Barranquilla, Colombia Email: [email protected]
2
Abstract—This paper presents a new empirical exponential-series-based model that describes the measured phase-noise (PN) of a nonlinear piezoelectric MEMS oscillator. Since the order of terms in the Leeson’s model can only provide at maximum -30 dB/dec slope, a model with higher order terms is proposed to determine an effective resonator quality factor (Qeff) and reflect the reduced contribution of the amplifier flicker-noise. The average fitting error of the proposed model to a 23 MHz in-plane shear (IPS) mode oscillator is found to be less than 1%. Nonlinear operation produces an oscillator with a PN of -130 dBc/Hz at 1kHz offset-frequency, which corresponds to an improvement of about 15 dB compared to linear operation.
I.
INTRODUCTION
High-performance, low phase-noise oscillators form a fundamental component of portable communication systems, and therefore their miniaturization is critical to support multiple wireless standards in a single device. Extensive research has focused on building resonators that can effectively replace large off-chip crystals. Microelectromechanical-systems (MEMS)-based solutions provide an alternative to traditional quartz oscillators in their ability to be integrated into a single chip or compact package [1]. Previous work has shown that capacitively-transduced devices can meet tight phase noise (PN) specifications such as the GSM standard when operated linearly [2]. However, the use of excessively large polarization voltages makes a compact, low-power solution with capacitive resonators challenging. On the other hand, piezoelectrically-transduced resonators exhibit lower motional impedances compared to capacitive devices and do not need a polarization voltage, simplifying the electronic interface design while reducing power and die area. Nevertheless, because of their relatively lower quality factors (Q), piezoelectric devices usually do not exhibit comparable PN performance to capacitive devices. Nonlinear operation of resonators has been discovered to improve the PN by suppressing the flicker noise of the interface circuitry [3]. While piezoelectric devices lack the transducer-induced nonlinearity found in capacitive devices [4], they can still be forced into the nonlinear elastic regime if
978-1-4244-9474-3/11/$26.00 ©2011 IEEE
sufficiently high power is applied to the electrodes [5, 6]. In this work, we demonstrate that a sustaining amplifier with high transimpedance gain is capable of exploiting a beneficial nonlinearity of piezoelectric resonators to improve PN through an adequate selection of the operating phase; a new empirical model is proposed to describe the performance of oscillators working in this regime. The model employs an effective quality factor that better captures the oscillator PN compared to the linear definition using the Leeson’s model. An overview of the operation of the piezoelectric MEMS resonators and characterization results over increasing power levels is given in Section II. Section III details the characteristics that allow the sustaining amplifier to take advantage of the resonator nonlinearity. Section IV presents the PN measurement results of the configured piezoelectric oscillator, so that the proposed PN model can be discussed in Section V. Finally, conclusions are drawn from the application of the model and the accuracy of the PN fitting. II.
PIEZOELECTRIC MEMS RESONATOR NONLINEARITY
A composite aluminum-nitride-on-silicon (AlN-on-Si) resonator is used in this work. The device is fabricated from a 10 μm thick silicon-on-insulator (SOI) substrate. The resonator and stack composition is shown in Figure 1, and it can be described as a two-finger electrode design on top of a plate with lateral dimensions of 156×250 μm. The device is designed to have a longitudinal extensional (LE) resonant frequency of 26 MHz and an in-plane shear (IPS) mode at 23 MHz. The IPS and the LE modes are electrically 180° out of phase with respect to one another. Figure 2 shows an SEM image of the final device; the fabrication process details can be found in [7]. When an AC signal is applied to the drive electrode, an oscillating electric field is produced between top and bottom molybdenum (Mo) layers across the piezoelectric AlN layer, which is converted into mechanical vibration via the inverse piezoelectric effect. Near resonance, the vibration is recaptured by the piezoelectric layer into a net charge, inducing a displacement current in the sense electrode. As the source power is increased, the device enters the nonlinear
221
Figure 3: Visualization of spring-softening and hardening effects in MEMS resonators.
Figure 1: Schematic of the composite AlN-on-Si MEMS resonator used in this work illustrating the material layers of the composite stack.
operation regime due to material and geometrical effects. The nonlinearity induces the peak of the frequency response to bend and shift toward lower frequencies (i.e. spring-softening), or higher frequencies (i.e. spring-hardening). This is depicted schematically in Figure 3. The mechanically-induced nonlinearity found in piezoelectric resonators can fall into either category depending on the configuration of the device [5]. Figure 4 shows the resonator characterization measurements for power levels between -35 dBm and +30 dBm taken with an Agilent E5071C vector network analyzer (VNA). The resonance mode of interest is the 23 MHz in-plane-shear (IPS) mode as it is able to sustain nonlinear operation at power levels of +30 dBm (the device fractures at power levels of about +20 dBm for the 26 MHz resonance mode). Pronounced nonlinearity in both magnitude and phase appears as sharp vertical transitions in the measurements due to sweeping characteristic of the test equipment.
Figure 2: SEM view of the MEMS resonator used in this work.
Figure 4: Magnitude and phase frequency plots of the in-plane-shear (IPS) mode of the MEMS resonator at 23 MHz showing spring-softening behavior.
III.
TRANSIMPEDANCE AMPLIFIER DESIGN
An inverter-based transimpedance amplifier (TIA) with active feedback resistor is interfaced with the MEMS resonator to analyze the nonlinearity effects on the oscillator PN. The TIA provides interface ports to operate with either 0° or 180° phase-shift resonance modes (see Fig. 5). The input-stage feedback resistor is implemented with a MOS transistor whose operation is controlled by the gate voltage, VCTRL. At one extreme, when VCTRL is minimum, the feedback transistor enters cut-off and the transimpedance gain is set by the inverter at the input. On the other hand, when VCTRL is maximum, the feedback transistor turns on, bypassing the inverter and setting the transimpedance gain to its lowest value equal to the transistor on-resistance. A transimpedance gain close to 100 dBΩ at 23 MHz is able to induce nonlinear effects in the piezoelectric-based resonator. Increasing the gain at the input stage simultaneously moves the associated dominant pole to lower frequencies. This strategy can be used to finely modify the TIA phase-shift, providing a means to move along the resonance curve (Fig. 4). The optimum quiescent point is where the operating phase slope tends to infinity. Here, an infinitesimal change in the phase will result in no frequency shift, leading to long-term phase stability [3] and a beneficial effect in PN. When the resonator is pushed deep into nonlinearity, the phase-frequency plot becomes multivalued with two points of infinite slope. The TIA can be tuned to operate at either point due to the fact that the frequency, which is the variable of interest, remains a single-valued function of the phase.
222
Figure 5: Transimpedance Amplifier Schematic. The generic TIA provides 0º and 180º ports to interface several MEMS devices.
A process-independent threshold-voltage inverter-based comparator is placed at the oscillator output to maintain the noise-floor during the PN measurements. Threshold-voltage insensitivity is achieved by negative feedback when the resistance levels of M13 and M16 are used as control variables [8]. Figure 6 presents the micrograph of the IC fabricated in a 0.5 μm 2P3M CMOS process and interfaced with the 23 MHz AlN-on-Si resonator through minimum-length aluminum bondwires. IV.
OSCILLATOR ARCHITECTURE AND PN PERFORMANCE
The 23 MHz IPS mode is excited using the TIA 180° resonator port. The oscillations were started up properly with low transimpedance gain. As VCTRL is finely reduced, the gain is sufficient to reach the power-supply rails, after which further decrease of VCTRL only affects the operating phase of the TIA. The PN performance is measured with an Agilent E5500 PN test set producing the PN plot set of Figure 7. Figure 7 includes the -30 dB/dec slope line observed under linear operation when VCTRL is set to the minimum gain. This line corresponds to the best fitting with the Leeson’s formula. From this procedure, the Qloaded and the corner flicker frequency are estimated. The quality factor is in close agreement with the value reported by the VNA (Q ~ 4K), and the flicker corner frequency is determined to be 50 kHz. As VCTRL is reduced, nonlinearity causes the PN plots to bend under the linear reference, implying that the oscillator is operating with a higher quality factor compared to the Qloaded. This beneficial nonlinearity causes the 23 MHz IPS mode
oscillator to exhibit a PN of -130 dBc/Hz at 1kHz offset-frequency operating in air. Such performance corresponds to an improvement of about 15 dB compared to linear operation. With a measured Qloaded of 4K, the PN reported in this work is comparable to architectures utilizing very-high-Q piezoelectric devices [6] and vacuum-packaged capacitive resonators at lower frequencies [2]. V.
NONLINEAR PN MODEL FORMULATION
Since the TIA circuit does not include automatic gain control, Leeson’s model should predict three sections on the PN plot for a high Q resonator: two linear segments with slopes equal to -30 dB/dec (modulated flicker noise) and -10 dB/dec (flicker noise), and the noise floor. Increased close-to-carrier slopes shown in Figure 7 indicate that the Leeson’s formula cannot describe this behavior, and therefore an alternative model with higher order terms is needed. Initially conceived as an empirical model, a mathematical derivation for the Leeson’s formula is found in [11], and it can be used as starting point for a more complete formulation. If |H(jω)| is the equivalent low-pass expression for the resonator and SΦi(ω) and SΦo(ω) are the one-side PN densities at the input and output of the TIA [11], the general form of a PN model can be expressed as
Figure 7: PN results from 23 MHz in-plane shear oscillator operating in air. The –30 dB/dec trend line is added for comparison with Leeson’s model. Figure 6: IC-resonator configuration via bondwires.
223
Sφi (ω ) = Sφo (ω ) ⋅ M (ω )
[
(1)
]
−1
M (ω ) = (H( jω ) −1)⋅ (H * ( jω ) − 1) .
(2)
For the case where the nonlinearity is introduced by the resonator only, SΦo(ω) remains dependent on the flicker and thermal noise of the TIA, which implies that higher order terms are needed in |H(jω)| to capture the nonlinear effects. Good quantitative description of the PN results is obtained by expanding M(ω) as an exponential series. However, this approach requires an adjustment to Q reflecting the strength of the nonlinearity. A multiplicative constant, N, is defined for that purpose. When the resonator is operated linearly, H(jω) is expressed by a first-order low-pass filter. For nonlinear operation, the equivalent low-pass transfer function can be modeled as cascaded first-order sections. The number of stages, which is equal to N, can be estimated using the phase-frequency plot for the MEMS resonator (Fig. 4). N is found as the ratio of the maxima of the derivatives of the phase responses for both linear and nonlinear regimes. Alternatively, it can also be obtained directly from the PN results as ∂ (3) N= (L( f )) 10 −1, ∂f max which allows the definition of Qeff and fceff as
Qeff = N⋅ Q,
f ceff = f c N
Figure 8: Fitting of the PN result using the exponential-series model for the 23MHz IPS mode oscillator. Qeff of 20K is used in the model.
and special packaging techniques. Additionally, a reduction in the flicker corner frequency value has been observed and quantified as that the same factor that increases the Qloaded. Under these conditions, the average relative error between PN measurements and the proposed PN model is calculated to be less than 1%. ACKNOWLEDGMENTS The authors wish to thank the MOSIS Fabrication Service for IC fabrication and Dr. Wanling Pan for device fabrication. REFERENCES
(4)
[1]
producing the proposed PN model in (5). For the present demonstration, the 23 MHz-oscillator best PN exhibits a maximum slope of –60 dB/dec, which produces N equal to 5. Therefore, using the parameters estimated in Section IV, a close fitting using (5) is attained with Qeff and fceff equal to 20K and 10 kHz, respectively. Figure 8 shows these results compared with the best fit using Leeson’s model. CONCLUSIONS A new empirical model based on a truncated exponential-series expansion has been proposed. The model includes higher order terms that allow a close fit of the PN measurement results when the resonator is operated in the nonlinear regime. It is found that the nonlinearity exhibited by a 23 MHz in-plane shear resonator leads to a beneficial effect on the oscillator PN when an appropriate operating point is chosen. Beneficial nonlinearity means that the PN response is superior to the equivalent oscillator working in the linear regime; that is, the effective Q exhibited by the oscillator is higher than the Qloaded. Operating Qs as high as 20K can be obtained from relatively low Q piezoelectrically-transduced resonators, which do not require large polarization voltages
F. Ayazi, “MEMS for Integrated Timing and Spectral Processing,” Invited Paper, Proc. IEEE CICC 2009, pp. 65-72, 2009. [2] V. Kaajakari et al., "Square-extensional mode single-crystal silicon micromechanical resonator for low-phase-noise oscillator applications," IEEE Electron Device Letters, vol. 25, no. 4, pp. 173-175, Apr. 2004. [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. Agarwal et al., "Nonlinear Characterization of Electrostatic MEMS Resonators," 2006 IEEE IFCS Exposition, pp. 209-212, 2006. [5] H. Li et al., "Nonlinear free and forced oscillations of piezoelectric microresonators," J. Micromech. Microeng., vol. 16, no. 2, pp. 356, 2006. [6] M. Shahmohammadi et al., “Low jitter thin-film piezoelectric-onsubstrate oscillators,” IEEE FCS 2010, pp. 613-617, 2010. [7] W. Pan et al., “Thin-Film Piezoelectric-on-Substrate Resonators with Q Enhancement and TCF Reduction,” IEEE MEMS 2010, pp. 104-107, 2010. [8] K. Sundaresan et al., “Process and temperature compensation in a 7MHz CMOS clock oscillator,” IEEE JSSC, vol. 41, no. 2, pp. 433-441, 2006. [9] H.M. Lavasani et al., “A 500MHz Low Phase-Noise AlN-on-Silicon Reference Oscillator”, IEEE CICC 2007, pp. 599-602, 2007. [10] H.M. Lavasani et al., “A 76dBOhm, 1.7 GHz, 0.18um CMOS Tunable Transimpedance Amplifier Using Broadband Current Pre-Amplifier for High Frequency Lateral Micromechanical Oscillators,” IEEE ISSCC 2010, pp. 318-320, 2010. [11] G. Sauvage, “Phase noise in oscillators: A mathematical analysis of Leeson’s model,” IEEE Trans. Instrum. Meas., vol. 26, no. 4, pp. 408–410, Dec. 1977.
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