Self-Sustaining Oscillator Based upon Vibrating Nanomechanical Resonator
3.7 Phase Noise of the UHF NEMS Oscillator
feedback amplification gain is changed and thus the RF power driving the device is changed accordingly, the NEMS oscillator output spectrum changes both its peak power and peak frequency (in each case, the loop phase change is tuned to optimize the oscillation to happen at the resonance peak frequency, as addressed in the previous section). The oscillation frequency is pulled upward, as a result of the resonator’s frequency stiffening effect, when the loop gain is increased. Shown in the inset of Fig.
3.15 is the relation of oscillator output versus oscillation frequency, which follows the tendency as depicted by the backbone curve of the NEMS resonator in Fig. 3.14.
The nature of the frequency pulling effect of the NEMS oscillator is based on the transition from linear to nonlinear regime of a doubly-clamped beam Duffing-type resonator, as the driving force is increased. Spontaneously, it becomes very intriguing to drive the NEMS resonator into the nonlinear regime, and to realize and then characterize the nonlinear oscillation with NEMS. Moreover, once self-oscillation is realized with a nonlinear NEMS resonator, the noise-induced switching of self-oscillations between the bistable states of the same frequency-determining resonator could be very interesting and important for both fundamental research and technological applications [29].
We have conducted systematic studies on the NEMS oscillator’s phase noise and frequency stability performance, since noise and sensitivity are crucial for both communications and sensors applications, and the trade-off is that the smaller and more sensitive the device, the more susceptible it is to noise. Of practical importance, low phase noise is desirable for UHF signal processing and communications [4,12-18,30], and sensitivity is most essential for transducer applications such as mass sensors [31,32].
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-140 -120 -100 -80 -60 -40 -20 0 20
Present Measurement Theoretical Prediction Ultimate Limit
Phase Noise, Sφ (dBc/Hz)
Offset Frequency (Hz)
1/f 2
Fig. 3.16 Phase noise performance of the NEMS oscillator for offset frequency from 10Hz to 1MHz, measured by a specialized phase noise analyzer (RDL NTS-1000B Phase Noise Analyzer). Shown are the measured data, the theoretical prediction based on oscillator phase noise theory [33] and the calculated ultimate phase noise performance limited only by the NEMS resonator device itself.
Measured phase noise performance of the oscillator system is shown in Fig. 3.16.
The measured data suggests the present system performance is thermal noise limited (i.e., following the 1/f 2 power law on the phase noise plot). Given the detection noise floor analysis in Section 3.4.2, the electronic thermal noise (Johnson) plus the amplifier noise overwhelms the thermomechanical noise of the NEMS device. Hence the thermal noise induced phase noise limits the oscillator performance, leaving the device’s thermomechanical noise induced phase noise still a fundamental limit to be approached.
The measured phase noise is readily modeled based on the intuitive understanding of phase noise in terms of phase diffusion [33]. As the extrinsic electronic noise at the input of the preamplifier dominates, the measured phase noise can then be predicted by
( )
22 2D S D
= + ω ω
φ , (3-14)
where D is the phase diffusion constant and ω the offset frequency. Phase diffusion constant can be determined by electrical domain measurement according to
2 2 0
Q P
T D k
s
B ⋅ω
≈ , (3-15)
where PS is the power dissipated in the equivalent resistive element that contributes the same amount of electronic noise. As shown in Fig. 3.16, the theory predicts 1/f 2 behavior and matches pretty well with the measured data. The ultimate phase noise performance set by the NEMS device’s intrinsic thermomechanical noise is also shown in Fig. 3.16 (theoretical discussions and equations are addressed in Chapter 2). This limit could be achieved if the transduction and feedback electronics are ideally noise-matched to the NEMS device. The comparison of measured data and calculations suggests important guidelines for further optimization and engineering of the NEMS oscillators with improved phase noise performance. By approaching the thermomechanical noise floor of the NEMS resonators, it will become possible for NEMS oscillators to compete with or even win over the phase noise performance of conventional bulky crystal oscillators.
Fig. 3.17 shows the measured phase noise when the UHF NEMS device is driven at different levels to assume different vibrating amplitudes in the self-oscillating mode.
Note in this set of measurements, we still keep the device operating well in the linear regime, which we assure by our careful calibrations as discussed in the previous sections.
The measured data demonstrates decreased phase noise as the NEMS resonator is driven to larger amplitudes within its mechanical dynamic range. Here a small increment is used in increasing the drive in order to make sure reliable phase noise measurement can be performed by the analyzer at all the levels in this range. Our observed phase noise dependence on resonator amplitude is more reasonable compared to that in another study reported in [34], because better phase noise performance should be expected with increasing power handled by the resonator.
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a = 1.50 nm a = 1.45 nm a = 1.40 nm a = 1.35 nm
Phase Noise, Sφ (dBc/Hz)
Offset Frequency (Hz)
Phase Noise Dependency on the Amplitude of NEMS Device
Fig. 3.17 Phase noise measured at different drive levels, showing that within the dynamic range of the NEMS device, the higher the drive, the lower the phase noise.