IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,
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Low Phase-Noise Autonomous Parametric Oscillator Based on a 226.7 MHz AlN Contour-Mode Resonator Cristian Cassella, Student Member, IEEE, and Gianluca Piazza, Member, IEEE Abstract—We present the first parametric oscillator based on the use of a 226.7 MHz aluminum nitride contour-mode resonator. This topology enables an improvement in the phase noise of 16 dB at 1 kHz offset with respect to a conventional feedback-loop oscillator based on the same device. The recorded phase noise is −106 dBc/Hz at 1 kHz offset.
I. Introduction
M
icroelectromechanical systems (MEMS)-based resonators represent good candidates to replace surface acoustic wave (SAW) and quartz resonators used for building frequency references. MEMS resonators are particularly of interest because the fabrication processes used for making MEMS and ICs are similar, hence enabling single-chip oscillator solutions. However, the phase noise performance of MEMS-based oscillators is still worse than what can be achieved with quartz and SAW devices. MEMS resonators can achieve Qs on par with SAW or quartz components, but their small volume limits their power handling and hence the ultimate oscillator phase noise. Although increasing the resonator Q is generally beneficial, this strategy might have several drawbacks in MEMS-based oscillators. In fact, increasing the resonator Q generally reduces the power handling for a given device area and, consequently, limits the oscillator output power [1]. This results in a larger noise floor level. As described in [2], the degradation of the power handling for higher Q systems can be surpassed when resonator arrays are used. However, this approach comes at the expense of a larger device area. The use of non-linearities arising at lower power levels (given the high Q) can be used to reduce the closein phase noise [3], but generally leads to larger coupling between amplitude and phase noise in the circuit. In this work we show a new oscillator topology that permits to attain a lower sensitivity to phase noise sources affecting the circuit without recurring to higher Q resonators. This topology consists of a parametric oscillator whose frequency is set by a 226.7 MHz aluminum nitride (AlN) contour-mode resonator [4]. The use of this new circuit ar-
Manuscript received November 22, 2014; accepted January 22, 2015. The authors are with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.006753
chitecture enabled ~16 dB reduction in the close-in phase noise at 1 kHz offset from the carrier frequency, with respect to a conventional feedback-loop oscillator based on the same resonator. Additionally, different from all parametric oscillators based on MEMS resonators demonstrated to date (e.g., [5]), no external source is used to start the oscillation. This approach drastically reduces the complexity associated with the use of parametric circuits. Although parametric oscillators that did not require external sources to start the oscillation were previously demonstrated with quartz resonators [6], [7], this is the first time this technique was applied to a MEMS resonator and resulted in the effective improvement of the oscillator phase noise. II. Parametric Oscillators Based on 226.7 MHz AlN CMRs Previous work [3], [5], [6], [7] has investigated non-linear oscillator topologies to reduce the phase fluctuations coming from the circuit components. In particular, in [3] the non-linear dynamics of a non-autonomous parametric oscillator were exploited as a powerful tool to enable phase noise reduction in a MEMS-based oscillator. When the resonator employed in an oscillator circuit is parametrically driven (i.e., its stiffness is modulated at twice the oscillation frequency), the oscillator steady state solution is less sensitive to phase fluctuations affecting the circuit components. Parametric modulation has been used in this work to enable phase noise reduction in a MEMS piezoelectric-based oscillator operating in the microwave frequency range. A. Oscillator Topology The oscillator presented in this work (Fig. 1) is based on the use of a capacitive degenerate parametric amplifier [8] working in its division range. A 226.7 MHz aluminum nitride contour-mode resonator, having Q = 2200, Rm = 169 Ω, and kt2 = 1.7% [4] is connected at the so-called signal terminal of the parametric amplifier (Fig. 2). A lowpass filter (Minicircuits SLP 300+, Mini-Circuits, Brooklyn, NY, USA, fc = 270 MHz and IL = 0.5 dB) and an external amplifier (Minicircuits ZKL-2+, labeled here as amplifier 1, and providing 40 dB of gain) follow the reso-
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Fig. 1. Schematic representation of the oscillator topology. A parametric amplifier and a frequency doubler are inserted in a conventional feedback loop oscillator based on a 226.7 MHz AlN contour-mode resonator.
nator. The output of amplifier 1 is plugged into the input port of a resistive 3 dB power splitter (Minicircuits ZFRSC-42-S+). One of the two output ports of the power splitter is sent to a signal source analyzer for measurement. The other port of the power splitter is sent to a frequency doubler (Minicircuits FK-3000+), which has a minimum conversion loss of 13 dB at the frequency of interest. The output of the frequency doubler is sent to a second external amplifier (Minicircuits ZKL-1R5, labeled here as amplifier 2, which can provide a max gain of 35 dB). The dc bias of amplifier 2 is labeled as Va and will be varied to study its impact on the parametric oscillator response. The signal from amplifier 2 is connected to a phase shifter (model no. ATM P1213) and a low-pass filter (Crystek CLPFL-0600, Crystek Corporation, Fort Myers, FL, USA; fc = 600 MHz and IL = 0.5 dB). The output of the filter feeds the input of the parametric amplifier through the RF-port of a bias-T that is also used to reverse bias the variable capacitor, in the parametric amplifier, with voltage, Vbias. B. AlN Contour-Mode Resonator The MEMS device we used in this work as main frequency reference is a 226.7 MHz lateral field excited AlN contour-mode resonator [4]. The device is formed by a piezoelectric layer of AlN sandwiched between two Pt metal plates (Fig. 1). The top metal is an interdigitated metal structure where adjacent metal strips are connected to opposite voltage polarities. The bottom metal is a floating plate that better confines the electric field in the AlN. The CMR was electrically modeled, in ADS, through the use of its modified Butterworth-Van-Dyke (MBVD) model [4]. Each component of the MBVD model was found by fitting the resonator response. In particular, the motional capacitance, Cm, motional inductance, Lm, motional resistance, Rm, and static capacitance, C0, were extracted to be 1.92 fF, 0.256 mH, 169 Ω, and 0.25 pF, respectively. C. Parametric Amplifier Design The parametric oscillator based on the AlN CMR uses a parametric amplifier built in-house and made to work in its instability region. The existence of instability regions in parametric amplifiers has been deeply investigated in
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Fig. 2. Parametric amplifier designed and built for this work. The varactor’s C(V) characteristic is also reported.
the past [8] because it represents a limitation to the use of such amplification mechanism in RF circuits. Parametric amplification [8], [9] is known as a low noise amplification approach that is based on the use of a variable capacitor modulated by a signal (pump or input) at a frequency related to the output frequency. The amplifier topology that we used in this work is described in depth in [10] and it is shown here in Fig. 2. It consists of a 2:1 degenerate configuration, and therefore requires a pump signal having frequency doubled with respect to the output frequency, Ω. Its topology includes two ports: 1) the pump port (input at 2 Ω), which is used to provide the signal needed for modulating the varactor capacitance; 2) the signal port (output at Ω), which is used to deliver the output signal to the next stage. The use of a pump signal permits generation of an equivalent negative resistance, Rneg, which is proportional to the peak-to-peak modulation of the varactor capacitance, C1, induced by the pump signal [8]. C1 is dependent on the varactor working point set by the varactor dc bias (Vbias). When the pump power, Ppump, is larger than a certain threshold, Pth, Rneg exceeds the losses in the parametric circuit and a phase-locking mechanism synchronizes the phase of a mode of the circuit to the phase of the modulation signal [11]. When this happens a parametric oscillation at Ω is activated through a bifurcation phenomenon and the parametric amplifier starts behaving as a frequency divider. This oscillation mechanism is used here to build an autonomous parametric oscillator, based on an AlN CMR, as shown in Fig. 1. The parametric amplifier circuit (Fig. 2) includes two notch filters, placed at the pump and signal ports, a variable capacitor and a stabilization inductor placed in series to the variable capacitor. The notch filters permit reduction of the portion of pump and output energy, respectively, leaking through signal and output ports. In line with [10], the notch filter placed at the pump port was conveniently designed to resonate at a slightly larger frequency than the resonance frequency of the CMR that is employed in the oscillator circuit (Fig. 1). This choice permits reduction of the sensitivity of Pth to un-modeled parasitics. Similarly, a second notch filter was placed at the signal port. The resonance frequency of this filter was set to be approximately two times the resonance frequency of the notch filter connected at the pump port. The variable capacitor was selected to achieve a high parametric gain at the frequency of interest. A hyperabrupt varactor
cassella and piazza: low phase-noise autonomous parametric oscillator
Fig. 3. Circuit schematic used to simulate the parametrically generated sub-harmonic frequency through a commercial HB simulator.
(Skyworks SMV1248) was used because of its wide tuning range and moderate capacitance value (≈5 pF) with respect to other available varactor technologies. As shown in [12], the proper selection of the stabilization inductor is crucial to achieve low Pth. To find its optimum value we recurred to frequency-domain simulations run in a commercial circuit simulator (Agilent ADS, Agilent Technologies Inc., Santa Clara, CA, USA). As described by previous work [10], commercial harmonic balance (HB) algorithms do not permit the detection of parametric instabilities. This is due to the structure of commercial HB solution matrices, which do not include sub-harmonics of the input signal in the list of the analyzed frequencies. To overcome this limitation, Suarez [10] developed the auxiliary generator (AG) technique to detect the existence of parametric instabilities in a parametric amplifier. This method is based on the use of an additional voltage source at the sub-harmonic frequency connected to the circuit. The AG technique can also be employed to predict the parametric amplifier response. However, this technique requires finding, through numerical optimization, the AG’s amplitude and phase that do not alter the impedance seen by the varactor (non-perturbation condition). In this work we developed an alternative continuation method that relies on an auxiliary power generator (pAGs) (Fig. 3) instead of the auxiliary voltage generator reported in [10]. In particular, we used a pAG with output impedance equal to 50 Ω instead of the 50 Ω output load of the circuit shown in Fig. 2. By doing so, we introduced Ω in the list of output frequencies investigated by the HB simulator without changing the impedance seen by the varactor. Therefore, no numerical optimization was needed to satisfy the non-perturbation condition, hence reducing the simulation complexity. Moreover, the power released by the pAG was set to −80 dBm. Such a low power level permitted to neglect any change in the varactor behavior due to the use of the pAG. The complete circuit schematic we adopted in the simulation is shown in Fig. 3. It is well known [8] that the performance of parametric circuits depends largely on the output load connected at the signal port. Because our ultimate goal was to optimize the performance of the parametric amplifier when used in the oscillator circuit (Fig. 1), we connected the signal port of the amplifier to an impedance equivalent to the one seen in the oscillator circuit. This was done by including the CMR and the low-
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Fig. 4. Black dots: measured Pout at 226.7 MHz for different pump power levels and Vbias equal to 4 V. This dc value was chosen to ensure that the varactor operates far enough from its forward conduction, but still exhibits low Pth. In green (dashed line): simulated Pout at 226.7 MHz through HB simulation run in Agilent ADS. Inset: output voltage waveform measured across the 50 Ω output load when Ppump was set to be 16 dBm. It is evident that the output voltage shows an almost pure sinusoidal signal with a frequency equal to 226.7 MHz.
pass filter electrical models in series to the 50 Ω matched input impedance of amplifier 1 in the circuit model. The simulation permitted finding the stabilization inductance value (~135 nH) that minimizes Pth. The same analysis was used to predict the parametric amplifier response for different Ppump values (Fig. 4). The existence of a parametric bifurcation in the circuit was verified by direct measurement. In particular, the pump port of the parametric circuit was connected to a commercial signal generator (model no. Agilent E8257D), which was programmed to release power, Ppump, at twice the resonance frequency of the MEMS resonator. The CMR was then connected to the parametric amplifier signal port, in series to the low-pass filter and the 50 Ω input port of a spectrum analyzer (model no. Agilent 8562EC). We report in Fig. 4 both simulated and measured output power at Ω, Pout, for different Ppump values (Fig. 2). It is evident that Pout shows a bifurcation at Ppump close to 10 dBm. This power threshold, whose value matches closely what was predicted by the circuit simulation, represents the minimum pump power, Pth, that permits activation of the sub-harmonic oscillation in the system. Moreover, as Ppump is increased beyond Pth, an increase in the output power was observed due to a larger modulation of the varactor capacitance, which implies a larger value of Rneg. III. Oscillator Start-Up We showed that a large pump power at the input of a parametric amplifier can lead to the generation of a parametric oscillation at half of the pump frequency. To build an autonomous parametric oscillator, the generation of a self-sustained pump signal is required to maintain the system into oscillation. In most parametric oscillators, this signal is initially provided by an external source [5]. In our case, we use the intrinsic power-dependent non-linearity of the frequency doubler to simultaneously start the oscillations and provide the parametric gain. Microwave frequency doublers are non-linear circuits, characterized by a conversion loss (CL) and by an input rejection (RJ). CL is defined as the
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maximum ratio between input and output powers, within its operational bandwidth. RJ is defined as the ratio between the input power and the maximum amount thereof leaking to the output port. Ideally, these devices should present CL equal to 1 and RJ tending to infinity. However, because of the intrinsic non-linear behavior of any diode frequency doubler, both CL and RJ are related to the input power. In particular, for the frequency doubler of this work, CL is about 50 dB and RJ is about 23 dB when the input power is smaller than −10 dBm. Hence, at low power levels (during oscillator start-up), the frequency doubler can be modeled as a simple linear impedance having real part related to its loss and imaginary part related to the phase difference between input and output ports. The same consideration applies to the behavior of the parametric circuit when it is driven by low Ppump levels. Therefore, the oscillator start-up conditions are the same as a regular feedback loop. In the following sections we refer to this operative regimen as the conventional regimen. As oscillations build up and the power into the doubler increases, the doubler reaches a different operating state (CL = 13 dB and RJ = 48 dB), which ensures the presence of a pump signal at the pump port of the parametric circuit (i.e., Ppump ≠ 0). By controlling Ppump through the external amplifier dc bias, Va (Fig. 1), it is possible to reach the pump power threshold, Pth, that activates the parametric oscillation in the parametric circuit. In the following sections we refer to this operative region as the parametric regimen.
A. Frequency Noise in AlN Contour-Mode Resonators It has been observed [13] that AlN CMRs suffers from high level of frequency fluctuations, which limit the minimum phase noise attainable by AlN contour-mode oscillators. We measured the spectrum of these fluctuations in the resonator we used for the parametric oscillator. This was done via an open-loop noise measurement [13]. The measurement revealed that the resonator frequency fluctuations have a 1/f spectrum, indicative of flicker frequency noise. The measured frequency noise was used to analytically predict the phase noise we would measure in a closed-loop oscillator, if the phase fluctuations coming from the circuit were to be evaded. This noise was compared with the best measured phase noise of the feedback-loop oscillator obtained by removing the frequency doubler and parametric amplifier from the oscillator circuit shown in Fig. 1 (Fig. 5). As evident from Fig. 6, the measured phase noise is larger than the analytical prediction which considers noise coming only from the resonator. This experiment proves that circuit noise dominates over the frequency noise of the resonator and justifies using parametric amplification to reduce the oscillator noise.
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Fig. 5. Conventional feedback-loop oscillator formed by removing the parametric amplifier and the frequency doubler from the circuit shown in the figure.
B. Phase Noise Evasion in Parametric Oscillators Previous work has investigated parametric oscillators to reduce the phase noise coming from the circuit components. In fact, diode-based parametric amplifiers exhibit a lower flicker noise than any transistor-based amplifier. In this work we study the phase noise in a parametric oscillator for values of Ppump larger than Pth. To do so we conducted a symbolic analysis using a simplified model that describes parametric modulation of the resonator stiffness (or equivalently Cm) in a negative resistance oscillator. The simplification is based on the idea of viewing the modulation of the external variable capacitance (the varactor) connected to the MEMS resonator as a modulation of the equivalent resonator motional capacitance (Cm) [4], which directly relates to its stiffness.
IV. Phase Noise of Parametric Oscillators
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x + 2Γx + (ω 0 2 + δ cos(2Ωt + 2θ(t) − π/2))x (1) = f cos(Ωt + θ(t)).
In (1), 2Γ is equal to the ratio between the resonator resonance frequency, ω0, and its quality factor, Q. Ω is the oscillation frequency, δ is equal to A∙ω02/Q, in which A is the peak-to-peak amplitude of the stiffness modulation. Finally, f is the amplitude of the forcing term at Ω. The presence of phase noise sources is considered by using a slow time-dependent phase term, θ(t), in the forcing term. No resonator noise is initially considered. θ(t) has spectral density Sθ and represents the intrinsic noise of the circuit
Fig. 6. In black: closed-loop phase noise predicted by open-loop measurement of the resonator frequency fluctuations. Part of this curve was measured (continuous line) and the remainder (>100 Hz) is extrapolated (dashed line). In green: best phase noise measured for the configuration shown in Fig. 5. Va was set to 12 V and the output power was 14 dBm. In red: time-domain output voltage for Va = 12 V. The output frequency is 226.7 MHz.
cassella and piazza: low phase-noise autonomous parametric oscillator
components. Because the signal at 2Ω is obtained by frequency doubling the oscillation term at Ω, it carries the phase fluctuation 2θ(t). An additional phase shift, equal to −π/2, is also considered in the phase of this signal. This shift was added as in a parametric oscillator the stiffness modulation has to be phase-locked to the resonator displacement at Ω. By using perturbation theory [9], [11], we first analyzed the stability of the oscillator fixed point. The system was found to be stable for 0 ≤ δ < 4ΓΩ. Next, we calculated the phase noise affecting the fixed point, Sϕ(foff), after being processed by the oscillator noise transfer function (foff represents the frequency offset at which the phase noise is evaluated). The expression of Sϕ(foff), is given in (2).
S φ(f off ) = S θ ⋅
(4ΓΩ − δ)2 , (2) 2 16Ω 2f off
where, by assuming low damping in the resonator, we could neglect the frequency detuning. It is important to point out that (2) remains valid only within the stability domain of (1), hence for 0 ≤ δ < 4ΓΩ. When no additional parametric drive is applied (δ = 0), Sϕ(foff) is equal to
S φ(f off ) = S θ ⋅
1 2 f off
2
⋅
( 2ΩQ ) , (3)
where ω0/2Q is the Leeson’s corner frequency [14]. Therefore, by comparing (2) with (3) it can be easily verified that when additional parametric drive is applied (i.e., δ ≠ 0) the phase noise contribution due to circuit noise can be reduced. This reduction always leads to a lower phase noise as long as the resonator frequency fluctuations do not dominate the phase fluctuations coming from the circuit. The phase noise performance of the parametric oscillator presented in this work was also investigated through ADS. However, because of the limitations in the HB simulator that were discussed in Section II, a closed-loop simulation of the circuit shown in Fig. 1 is not possible. Therefore, an ad hoc technique was developed to compute the phase noise evasion attained by the oscillator operating in the parametric regimen. This is based on the use of a simplified circuit topology, which includes only the parametric amplifier (Fig. 7) and two power generators. One generator is a pAG connected at the 50 Ω output load of the parametric amplifier. The second power generator is used to represent the pump signal coming from amplifier 2 (Fig. 1). The recorded phase fluctuations affecting the oscillator response in the close-in region, when operating in the conventional regimen, were included in ADS through the use of a phase modulator (Fig. 7). By doing so, we could capture the effect of all the circuit noise sources affecting the oscillator circuit shown in Fig. 1. The proposed simplified circuit for studying phase noise is similar to the circuit we analyzed to determine the per-
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Fig. 7. Circuit schematic adopted to simulate the phase noise of the parametric oscillator shown in Fig. 1 through a commercial simulator (Agilent ADS).
formance of the parametric amplifier (Fig. 3). However, when the parametric amplifier operates as the core element of a parametric oscillator, its phase noise transfer function is substantially different from the one of a frequency divider. In particular, in parametric frequency dividers, the phase noise of the pump signal is fixed and sets the phase noise of the output signal. In contrast, in parametric oscillators, the pump signal is generated from the output signal at Ω, and therefore its close-in phase noise is dependent on the parametric amplifier working point. To use ADS to evaluate the phase noise of a parametric oscillator, no initial phase noise condition at 2Ω has to be included in the simulation. This is done to prevent the simulator from converging to the trivial solution of a frequency divider. It is important to emphasize that the use of the simplified circuit shown in Fig. 7 is suitable to model the close-in phase noise evasion attained by the oscillator shown in Fig. 1. This is conceptually explained by the fact that the close-in phase noise attained by the oscillator after migrating to the parametric regimen is exclusively dependent on the phase noise transfer function of the parametric amplifier, where the parametric oscillation is generated and sustained [3]. Furthermore, as it will be shown in the next section, the simulation results are supported by the experimental data. We report in Fig. 8 the simulated phase noise at 0.1, 1, and 10 kHz offsets for different pump power levels. In addition to the circuit phase noise introduced through the phase modulator, the simulation also included the CMR frequency noise that was separately measured in an openloop configuration (Fig. 6). Similarly to what shown in [13], the CMR’s resonance frequency fluctuations were modeled by using a 1/f noise source modulating the resonator Cm, whose magnitude was calibrated to produce the same noise that was measured experimentally. It is evident that the simulated phase noise values vary depending on which operating regimens we are in (function of Ppump value). For Ppump < Pth (region I), the oscillator phase noise is not dependent on the pump power level. When Ppump > Pth (region III), a parametric oscillation at Ω is active. Consequently, the noise transfer function depends on the parametric amplifier’s operational point. The activation of the parametric oscillation leads to lower phase noise with respect to the case in which Ppump < Pth. Moreover, as predicted by (3), phase noise lowers
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Fig. 8. Simulated phase noise at 0.1, 1, and 10 kHz offsets from the carrier for the cases including the CMR frequency noise (continuous lines) or considering only circuit noise (dotted lines).
with larger values of Ppump. This reduction would be even larger if the CMR were not to suffer from frequency noise (Fig. 8, dotted lines). When Ppump ≈ Pth (region II), a local increase of the phase noise is obtained. This increase is likely due to operation close to the parametric instability. We suspect that when the oscillator operates close to such an unstable point, its steady-state solution can be more sensitive to the phase fluctuations affecting the circuit, hence resulting in larger phase noise values. Such increase in the phase noise is also confirmed experimentally as it will be shown in the next section. V. Experimental Results and Discussion The best phase noise we measured at the output of the parametric oscillator is compared, in Fig. 9 (blue curve), to the best phase noise of the conventional feedback loop oscillator shown in Fig. 5. As evident, this curve matches the phase noise analytically predicted (Fig. 9, black curve) by considering the measured frequency noise of the CMR as the main noise source in the loop. This fact proves that the oscillator phase noise is now limited by the resonator noise and that cancellation of the circuit noise was achieved. Moreover, the measured phase noise in the close-in region shows excellent agreement with the simulated phase noise (Fig. 8) relative to the same pump power (~17 dBm) driving the parametric circuit in the actual experiment. An improvement of more than 16 dB at 1 kHz offset was obtained by using the new oscillator topology (Fig. 1) rather than the conventional configuration shown in Fig. 5. The effect of the parametric circuit on output power and phase noise of the MEMS oscillator was measured for different Va, and for a fixed Vbias (4 V). In particular, we report in Fig. 10 the measured output power for each Va we investigated. Similarly, we report in Fig. 11 the recorded phase noise at 0.1, 1, and 10 kHz offsets, relative to the same Va values. As suggested by Fig. 10, increasing Va makes the pump signal at the input of the parametric amplifier larger. Intuitively, this increase leads
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Fig. 9. In black: predicted closed-loop phase noise by open-loop measurement of the resonator frequency fluctuations. Part of this curve was measured (continuous line) and the remainder (>100 Hz) is extrapolated (dashed line). In green: best phase noise measured for the configuration shown in Fig. 5. Va was set to 12 V and the output power was 14 dBm. In red: simulated phase noise, in the close-in region, through Agilent ADS. In blue: best phase noise we measured in the parametric oscillator. Va is set to 12 V and the output power is 12.5 dBm. Both the measured curves were evaluated at one output port of the power splitter following amplifier 1.
to a larger modulation of the varactor capacitance and, consequently, to an increase of the output power at the main output frequency. This fact was confirmed by an open loop measurement of the parametric circuit. After disconnecting the frequency doubler from the power splitter (Fig. 1), we drove it at the same frequency and output power measured at the oscillator output in the closedloop experiment. By doing so, we were able to measure the pump power, Ppump, at the pump port (Fig. 2) of the parametric circuit for each Va we investigated in the closed-loop experiment. Similarly, for the same Va values, we measured the amplifier 1 (Fig. 1) output power, Pout, at the corresponding oscillator output frequency. This probing location was preferred to the parametric circuit signal port so as to preserve the same impedance seen by the varactor in the closed-loop experiment. The variation of Pout with respect to Ppump is shown in Fig. 12. It is evident that the oscillator migrates from the conventional regimen to the parametric regimen when the pump power is about 5 dBm. This power level, attained at Va equal to 6.2 V, is slightly lower than the power threshold we found through our circuit simulation (Figs. 3–8). This fact remains under investigation, but it is likely due to unmodeled aspects, such as the dependence of the amplifier 2 output impedance on Va or the presence of non-linearities in the amplifiers. Regarding the measured phase noise performance (Fig. 11), the activation of the parametric oscillation permits
Fig. 10. Oscillator output power for different dc biases, Va, of amplifier 2.
cassella and piazza: low phase-noise autonomous parametric oscillator
Fig. 11. Measured phase noise at 0.1, 1, and 10 kHz offsets at different amplifier dc biases (Va). As evident, a generally larger phase noise evasion is attained at larger Va values and, consequently, at higher Ppump (Fig. 12). However, the measured phase noise shows a local maximum when the oscillator operates in the proximity of the parametric instability (Va ~ 6.2 V). This local degradation of the phase noise performance is determined by operation close to the parametric instability and follows what was predicted by the circuit simulation (Fig. 8).
improvement of the phase noise with respect to what is measured when the oscillator operates as a conventional feedback-loop oscillator. Moreover, as predicted by both the circuit simulation (Fig. 8, Region III) and the analytical model, we observed a larger reduction of the phase noise at larger Va and, therefore, at higher Ppump values (Fig. 11). Furthermore, we found a local maximum in the phase noise when the oscillator was made to operate close to its bifurcation. These experimental data follow the trend predicted by the circuit simulation (Fig. 8, Region II) and further validate the use of ADS for predicting phase noise performance in the parametric oscillator. It is important to note that, despite the high power level in the resonator, the measured improvement in the close-in phase noise cannot be attributed to the resonator non-linearity given the time constant (
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Low Phase-Noise Autonomous Parametric Oscillator Based on a 226.7 MHz AlN Contour-Mode Resonator Cristian Cassella, Student Member, IEEE, and Gianluca Piazza, Member, IEEE Abstract—We present the first parametric oscillator based on the use of a 226.7 MHz aluminum nitride contour-mode resonator. This topology enables an improvement in the phase noise of 16 dB at 1 kHz offset with respect to a conventional feedback-loop oscillator based on the same device. The recorded phase noise is −106 dBc/Hz at 1 kHz offset.
I. Introduction
M
icroelectromechanical systems (MEMS)-based resonators represent good candidates to replace surface acoustic wave (SAW) and quartz resonators used for building frequency references. MEMS resonators are particularly of interest because the fabrication processes used for making MEMS and ICs are similar, hence enabling single-chip oscillator solutions. However, the phase noise performance of MEMS-based oscillators is still worse than what can be achieved with quartz and SAW devices. MEMS resonators can achieve Qs on par with SAW or quartz components, but their small volume limits their power handling and hence the ultimate oscillator phase noise. Although increasing the resonator Q is generally beneficial, this strategy might have several drawbacks in MEMS-based oscillators. In fact, increasing the resonator Q generally reduces the power handling for a given device area and, consequently, limits the oscillator output power [1]. This results in a larger noise floor level. As described in [2], the degradation of the power handling for higher Q systems can be surpassed when resonator arrays are used. However, this approach comes at the expense of a larger device area. The use of non-linearities arising at lower power levels (given the high Q) can be used to reduce the closein phase noise [3], but generally leads to larger coupling between amplitude and phase noise in the circuit. In this work we show a new oscillator topology that permits to attain a lower sensitivity to phase noise sources affecting the circuit without recurring to higher Q resonators. This topology consists of a parametric oscillator whose frequency is set by a 226.7 MHz aluminum nitride (AlN) contour-mode resonator [4]. The use of this new circuit ar-
Manuscript received November 22, 2014; accepted January 22, 2015. The authors are with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.006753
chitecture enabled ~16 dB reduction in the close-in phase noise at 1 kHz offset from the carrier frequency, with respect to a conventional feedback-loop oscillator based on the same resonator. Additionally, different from all parametric oscillators based on MEMS resonators demonstrated to date (e.g., [5]), no external source is used to start the oscillation. This approach drastically reduces the complexity associated with the use of parametric circuits. Although parametric oscillators that did not require external sources to start the oscillation were previously demonstrated with quartz resonators [6], [7], this is the first time this technique was applied to a MEMS resonator and resulted in the effective improvement of the oscillator phase noise. II. Parametric Oscillators Based on 226.7 MHz AlN CMRs Previous work [3], [5], [6], [7] has investigated non-linear oscillator topologies to reduce the phase fluctuations coming from the circuit components. In particular, in [3] the non-linear dynamics of a non-autonomous parametric oscillator were exploited as a powerful tool to enable phase noise reduction in a MEMS-based oscillator. When the resonator employed in an oscillator circuit is parametrically driven (i.e., its stiffness is modulated at twice the oscillation frequency), the oscillator steady state solution is less sensitive to phase fluctuations affecting the circuit components. Parametric modulation has been used in this work to enable phase noise reduction in a MEMS piezoelectric-based oscillator operating in the microwave frequency range. A. Oscillator Topology The oscillator presented in this work (Fig. 1) is based on the use of a capacitive degenerate parametric amplifier [8] working in its division range. A 226.7 MHz aluminum nitride contour-mode resonator, having Q = 2200, Rm = 169 Ω, and kt2 = 1.7% [4] is connected at the so-called signal terminal of the parametric amplifier (Fig. 2). A lowpass filter (Minicircuits SLP 300+, Mini-Circuits, Brooklyn, NY, USA, fc = 270 MHz and IL = 0.5 dB) and an external amplifier (Minicircuits ZKL-2+, labeled here as amplifier 1, and providing 40 dB of gain) follow the reso-
0885–3010 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Fig. 1. Schematic representation of the oscillator topology. A parametric amplifier and a frequency doubler are inserted in a conventional feedback loop oscillator based on a 226.7 MHz AlN contour-mode resonator.
nator. The output of amplifier 1 is plugged into the input port of a resistive 3 dB power splitter (Minicircuits ZFRSC-42-S+). One of the two output ports of the power splitter is sent to a signal source analyzer for measurement. The other port of the power splitter is sent to a frequency doubler (Minicircuits FK-3000+), which has a minimum conversion loss of 13 dB at the frequency of interest. The output of the frequency doubler is sent to a second external amplifier (Minicircuits ZKL-1R5, labeled here as amplifier 2, which can provide a max gain of 35 dB). The dc bias of amplifier 2 is labeled as Va and will be varied to study its impact on the parametric oscillator response. The signal from amplifier 2 is connected to a phase shifter (model no. ATM P1213) and a low-pass filter (Crystek CLPFL-0600, Crystek Corporation, Fort Myers, FL, USA; fc = 600 MHz and IL = 0.5 dB). The output of the filter feeds the input of the parametric amplifier through the RF-port of a bias-T that is also used to reverse bias the variable capacitor, in the parametric amplifier, with voltage, Vbias. B. AlN Contour-Mode Resonator The MEMS device we used in this work as main frequency reference is a 226.7 MHz lateral field excited AlN contour-mode resonator [4]. The device is formed by a piezoelectric layer of AlN sandwiched between two Pt metal plates (Fig. 1). The top metal is an interdigitated metal structure where adjacent metal strips are connected to opposite voltage polarities. The bottom metal is a floating plate that better confines the electric field in the AlN. The CMR was electrically modeled, in ADS, through the use of its modified Butterworth-Van-Dyke (MBVD) model [4]. Each component of the MBVD model was found by fitting the resonator response. In particular, the motional capacitance, Cm, motional inductance, Lm, motional resistance, Rm, and static capacitance, C0, were extracted to be 1.92 fF, 0.256 mH, 169 Ω, and 0.25 pF, respectively. C. Parametric Amplifier Design The parametric oscillator based on the AlN CMR uses a parametric amplifier built in-house and made to work in its instability region. The existence of instability regions in parametric amplifiers has been deeply investigated in
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Fig. 2. Parametric amplifier designed and built for this work. The varactor’s C(V) characteristic is also reported.
the past [8] because it represents a limitation to the use of such amplification mechanism in RF circuits. Parametric amplification [8], [9] is known as a low noise amplification approach that is based on the use of a variable capacitor modulated by a signal (pump or input) at a frequency related to the output frequency. The amplifier topology that we used in this work is described in depth in [10] and it is shown here in Fig. 2. It consists of a 2:1 degenerate configuration, and therefore requires a pump signal having frequency doubled with respect to the output frequency, Ω. Its topology includes two ports: 1) the pump port (input at 2 Ω), which is used to provide the signal needed for modulating the varactor capacitance; 2) the signal port (output at Ω), which is used to deliver the output signal to the next stage. The use of a pump signal permits generation of an equivalent negative resistance, Rneg, which is proportional to the peak-to-peak modulation of the varactor capacitance, C1, induced by the pump signal [8]. C1 is dependent on the varactor working point set by the varactor dc bias (Vbias). When the pump power, Ppump, is larger than a certain threshold, Pth, Rneg exceeds the losses in the parametric circuit and a phase-locking mechanism synchronizes the phase of a mode of the circuit to the phase of the modulation signal [11]. When this happens a parametric oscillation at Ω is activated through a bifurcation phenomenon and the parametric amplifier starts behaving as a frequency divider. This oscillation mechanism is used here to build an autonomous parametric oscillator, based on an AlN CMR, as shown in Fig. 1. The parametric amplifier circuit (Fig. 2) includes two notch filters, placed at the pump and signal ports, a variable capacitor and a stabilization inductor placed in series to the variable capacitor. The notch filters permit reduction of the portion of pump and output energy, respectively, leaking through signal and output ports. In line with [10], the notch filter placed at the pump port was conveniently designed to resonate at a slightly larger frequency than the resonance frequency of the CMR that is employed in the oscillator circuit (Fig. 1). This choice permits reduction of the sensitivity of Pth to un-modeled parasitics. Similarly, a second notch filter was placed at the signal port. The resonance frequency of this filter was set to be approximately two times the resonance frequency of the notch filter connected at the pump port. The variable capacitor was selected to achieve a high parametric gain at the frequency of interest. A hyperabrupt varactor
cassella and piazza: low phase-noise autonomous parametric oscillator
Fig. 3. Circuit schematic used to simulate the parametrically generated sub-harmonic frequency through a commercial HB simulator.
(Skyworks SMV1248) was used because of its wide tuning range and moderate capacitance value (≈5 pF) with respect to other available varactor technologies. As shown in [12], the proper selection of the stabilization inductor is crucial to achieve low Pth. To find its optimum value we recurred to frequency-domain simulations run in a commercial circuit simulator (Agilent ADS, Agilent Technologies Inc., Santa Clara, CA, USA). As described by previous work [10], commercial harmonic balance (HB) algorithms do not permit the detection of parametric instabilities. This is due to the structure of commercial HB solution matrices, which do not include sub-harmonics of the input signal in the list of the analyzed frequencies. To overcome this limitation, Suarez [10] developed the auxiliary generator (AG) technique to detect the existence of parametric instabilities in a parametric amplifier. This method is based on the use of an additional voltage source at the sub-harmonic frequency connected to the circuit. The AG technique can also be employed to predict the parametric amplifier response. However, this technique requires finding, through numerical optimization, the AG’s amplitude and phase that do not alter the impedance seen by the varactor (non-perturbation condition). In this work we developed an alternative continuation method that relies on an auxiliary power generator (pAGs) (Fig. 3) instead of the auxiliary voltage generator reported in [10]. In particular, we used a pAG with output impedance equal to 50 Ω instead of the 50 Ω output load of the circuit shown in Fig. 2. By doing so, we introduced Ω in the list of output frequencies investigated by the HB simulator without changing the impedance seen by the varactor. Therefore, no numerical optimization was needed to satisfy the non-perturbation condition, hence reducing the simulation complexity. Moreover, the power released by the pAG was set to −80 dBm. Such a low power level permitted to neglect any change in the varactor behavior due to the use of the pAG. The complete circuit schematic we adopted in the simulation is shown in Fig. 3. It is well known [8] that the performance of parametric circuits depends largely on the output load connected at the signal port. Because our ultimate goal was to optimize the performance of the parametric amplifier when used in the oscillator circuit (Fig. 1), we connected the signal port of the amplifier to an impedance equivalent to the one seen in the oscillator circuit. This was done by including the CMR and the low-
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Fig. 4. Black dots: measured Pout at 226.7 MHz for different pump power levels and Vbias equal to 4 V. This dc value was chosen to ensure that the varactor operates far enough from its forward conduction, but still exhibits low Pth. In green (dashed line): simulated Pout at 226.7 MHz through HB simulation run in Agilent ADS. Inset: output voltage waveform measured across the 50 Ω output load when Ppump was set to be 16 dBm. It is evident that the output voltage shows an almost pure sinusoidal signal with a frequency equal to 226.7 MHz.
pass filter electrical models in series to the 50 Ω matched input impedance of amplifier 1 in the circuit model. The simulation permitted finding the stabilization inductance value (~135 nH) that minimizes Pth. The same analysis was used to predict the parametric amplifier response for different Ppump values (Fig. 4). The existence of a parametric bifurcation in the circuit was verified by direct measurement. In particular, the pump port of the parametric circuit was connected to a commercial signal generator (model no. Agilent E8257D), which was programmed to release power, Ppump, at twice the resonance frequency of the MEMS resonator. The CMR was then connected to the parametric amplifier signal port, in series to the low-pass filter and the 50 Ω input port of a spectrum analyzer (model no. Agilent 8562EC). We report in Fig. 4 both simulated and measured output power at Ω, Pout, for different Ppump values (Fig. 2). It is evident that Pout shows a bifurcation at Ppump close to 10 dBm. This power threshold, whose value matches closely what was predicted by the circuit simulation, represents the minimum pump power, Pth, that permits activation of the sub-harmonic oscillation in the system. Moreover, as Ppump is increased beyond Pth, an increase in the output power was observed due to a larger modulation of the varactor capacitance, which implies a larger value of Rneg. III. Oscillator Start-Up We showed that a large pump power at the input of a parametric amplifier can lead to the generation of a parametric oscillation at half of the pump frequency. To build an autonomous parametric oscillator, the generation of a self-sustained pump signal is required to maintain the system into oscillation. In most parametric oscillators, this signal is initially provided by an external source [5]. In our case, we use the intrinsic power-dependent non-linearity of the frequency doubler to simultaneously start the oscillations and provide the parametric gain. Microwave frequency doublers are non-linear circuits, characterized by a conversion loss (CL) and by an input rejection (RJ). CL is defined as the
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maximum ratio between input and output powers, within its operational bandwidth. RJ is defined as the ratio between the input power and the maximum amount thereof leaking to the output port. Ideally, these devices should present CL equal to 1 and RJ tending to infinity. However, because of the intrinsic non-linear behavior of any diode frequency doubler, both CL and RJ are related to the input power. In particular, for the frequency doubler of this work, CL is about 50 dB and RJ is about 23 dB when the input power is smaller than −10 dBm. Hence, at low power levels (during oscillator start-up), the frequency doubler can be modeled as a simple linear impedance having real part related to its loss and imaginary part related to the phase difference between input and output ports. The same consideration applies to the behavior of the parametric circuit when it is driven by low Ppump levels. Therefore, the oscillator start-up conditions are the same as a regular feedback loop. In the following sections we refer to this operative regimen as the conventional regimen. As oscillations build up and the power into the doubler increases, the doubler reaches a different operating state (CL = 13 dB and RJ = 48 dB), which ensures the presence of a pump signal at the pump port of the parametric circuit (i.e., Ppump ≠ 0). By controlling Ppump through the external amplifier dc bias, Va (Fig. 1), it is possible to reach the pump power threshold, Pth, that activates the parametric oscillation in the parametric circuit. In the following sections we refer to this operative region as the parametric regimen.
A. Frequency Noise in AlN Contour-Mode Resonators It has been observed [13] that AlN CMRs suffers from high level of frequency fluctuations, which limit the minimum phase noise attainable by AlN contour-mode oscillators. We measured the spectrum of these fluctuations in the resonator we used for the parametric oscillator. This was done via an open-loop noise measurement [13]. The measurement revealed that the resonator frequency fluctuations have a 1/f spectrum, indicative of flicker frequency noise. The measured frequency noise was used to analytically predict the phase noise we would measure in a closed-loop oscillator, if the phase fluctuations coming from the circuit were to be evaded. This noise was compared with the best measured phase noise of the feedback-loop oscillator obtained by removing the frequency doubler and parametric amplifier from the oscillator circuit shown in Fig. 1 (Fig. 5). As evident from Fig. 6, the measured phase noise is larger than the analytical prediction which considers noise coming only from the resonator. This experiment proves that circuit noise dominates over the frequency noise of the resonator and justifies using parametric amplification to reduce the oscillator noise.
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Fig. 5. Conventional feedback-loop oscillator formed by removing the parametric amplifier and the frequency doubler from the circuit shown in the figure.
B. Phase Noise Evasion in Parametric Oscillators Previous work has investigated parametric oscillators to reduce the phase noise coming from the circuit components. In fact, diode-based parametric amplifiers exhibit a lower flicker noise than any transistor-based amplifier. In this work we study the phase noise in a parametric oscillator for values of Ppump larger than Pth. To do so we conducted a symbolic analysis using a simplified model that describes parametric modulation of the resonator stiffness (or equivalently Cm) in a negative resistance oscillator. The simplification is based on the idea of viewing the modulation of the external variable capacitance (the varactor) connected to the MEMS resonator as a modulation of the equivalent resonator motional capacitance (Cm) [4], which directly relates to its stiffness.
IV. Phase Noise of Parametric Oscillators
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x + 2Γx + (ω 0 2 + δ cos(2Ωt + 2θ(t) − π/2))x (1) = f cos(Ωt + θ(t)).
In (1), 2Γ is equal to the ratio between the resonator resonance frequency, ω0, and its quality factor, Q. Ω is the oscillation frequency, δ is equal to A∙ω02/Q, in which A is the peak-to-peak amplitude of the stiffness modulation. Finally, f is the amplitude of the forcing term at Ω. The presence of phase noise sources is considered by using a slow time-dependent phase term, θ(t), in the forcing term. No resonator noise is initially considered. θ(t) has spectral density Sθ and represents the intrinsic noise of the circuit
Fig. 6. In black: closed-loop phase noise predicted by open-loop measurement of the resonator frequency fluctuations. Part of this curve was measured (continuous line) and the remainder (>100 Hz) is extrapolated (dashed line). In green: best phase noise measured for the configuration shown in Fig. 5. Va was set to 12 V and the output power was 14 dBm. In red: time-domain output voltage for Va = 12 V. The output frequency is 226.7 MHz.
cassella and piazza: low phase-noise autonomous parametric oscillator
components. Because the signal at 2Ω is obtained by frequency doubling the oscillation term at Ω, it carries the phase fluctuation 2θ(t). An additional phase shift, equal to −π/2, is also considered in the phase of this signal. This shift was added as in a parametric oscillator the stiffness modulation has to be phase-locked to the resonator displacement at Ω. By using perturbation theory [9], [11], we first analyzed the stability of the oscillator fixed point. The system was found to be stable for 0 ≤ δ < 4ΓΩ. Next, we calculated the phase noise affecting the fixed point, Sϕ(foff), after being processed by the oscillator noise transfer function (foff represents the frequency offset at which the phase noise is evaluated). The expression of Sϕ(foff), is given in (2).
S φ(f off ) = S θ ⋅
(4ΓΩ − δ)2 , (2) 2 16Ω 2f off
where, by assuming low damping in the resonator, we could neglect the frequency detuning. It is important to point out that (2) remains valid only within the stability domain of (1), hence for 0 ≤ δ < 4ΓΩ. When no additional parametric drive is applied (δ = 0), Sϕ(foff) is equal to
S φ(f off ) = S θ ⋅
1 2 f off
2
⋅
( 2ΩQ ) , (3)
where ω0/2Q is the Leeson’s corner frequency [14]. Therefore, by comparing (2) with (3) it can be easily verified that when additional parametric drive is applied (i.e., δ ≠ 0) the phase noise contribution due to circuit noise can be reduced. This reduction always leads to a lower phase noise as long as the resonator frequency fluctuations do not dominate the phase fluctuations coming from the circuit. The phase noise performance of the parametric oscillator presented in this work was also investigated through ADS. However, because of the limitations in the HB simulator that were discussed in Section II, a closed-loop simulation of the circuit shown in Fig. 1 is not possible. Therefore, an ad hoc technique was developed to compute the phase noise evasion attained by the oscillator operating in the parametric regimen. This is based on the use of a simplified circuit topology, which includes only the parametric amplifier (Fig. 7) and two power generators. One generator is a pAG connected at the 50 Ω output load of the parametric amplifier. The second power generator is used to represent the pump signal coming from amplifier 2 (Fig. 1). The recorded phase fluctuations affecting the oscillator response in the close-in region, when operating in the conventional regimen, were included in ADS through the use of a phase modulator (Fig. 7). By doing so, we could capture the effect of all the circuit noise sources affecting the oscillator circuit shown in Fig. 1. The proposed simplified circuit for studying phase noise is similar to the circuit we analyzed to determine the per-
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Fig. 7. Circuit schematic adopted to simulate the phase noise of the parametric oscillator shown in Fig. 1 through a commercial simulator (Agilent ADS).
formance of the parametric amplifier (Fig. 3). However, when the parametric amplifier operates as the core element of a parametric oscillator, its phase noise transfer function is substantially different from the one of a frequency divider. In particular, in parametric frequency dividers, the phase noise of the pump signal is fixed and sets the phase noise of the output signal. In contrast, in parametric oscillators, the pump signal is generated from the output signal at Ω, and therefore its close-in phase noise is dependent on the parametric amplifier working point. To use ADS to evaluate the phase noise of a parametric oscillator, no initial phase noise condition at 2Ω has to be included in the simulation. This is done to prevent the simulator from converging to the trivial solution of a frequency divider. It is important to emphasize that the use of the simplified circuit shown in Fig. 7 is suitable to model the close-in phase noise evasion attained by the oscillator shown in Fig. 1. This is conceptually explained by the fact that the close-in phase noise attained by the oscillator after migrating to the parametric regimen is exclusively dependent on the phase noise transfer function of the parametric amplifier, where the parametric oscillation is generated and sustained [3]. Furthermore, as it will be shown in the next section, the simulation results are supported by the experimental data. We report in Fig. 8 the simulated phase noise at 0.1, 1, and 10 kHz offsets for different pump power levels. In addition to the circuit phase noise introduced through the phase modulator, the simulation also included the CMR frequency noise that was separately measured in an openloop configuration (Fig. 6). Similarly to what shown in [13], the CMR’s resonance frequency fluctuations were modeled by using a 1/f noise source modulating the resonator Cm, whose magnitude was calibrated to produce the same noise that was measured experimentally. It is evident that the simulated phase noise values vary depending on which operating regimens we are in (function of Ppump value). For Ppump < Pth (region I), the oscillator phase noise is not dependent on the pump power level. When Ppump > Pth (region III), a parametric oscillation at Ω is active. Consequently, the noise transfer function depends on the parametric amplifier’s operational point. The activation of the parametric oscillation leads to lower phase noise with respect to the case in which Ppump < Pth. Moreover, as predicted by (3), phase noise lowers
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Fig. 8. Simulated phase noise at 0.1, 1, and 10 kHz offsets from the carrier for the cases including the CMR frequency noise (continuous lines) or considering only circuit noise (dotted lines).
with larger values of Ppump. This reduction would be even larger if the CMR were not to suffer from frequency noise (Fig. 8, dotted lines). When Ppump ≈ Pth (region II), a local increase of the phase noise is obtained. This increase is likely due to operation close to the parametric instability. We suspect that when the oscillator operates close to such an unstable point, its steady-state solution can be more sensitive to the phase fluctuations affecting the circuit, hence resulting in larger phase noise values. Such increase in the phase noise is also confirmed experimentally as it will be shown in the next section. V. Experimental Results and Discussion The best phase noise we measured at the output of the parametric oscillator is compared, in Fig. 9 (blue curve), to the best phase noise of the conventional feedback loop oscillator shown in Fig. 5. As evident, this curve matches the phase noise analytically predicted (Fig. 9, black curve) by considering the measured frequency noise of the CMR as the main noise source in the loop. This fact proves that the oscillator phase noise is now limited by the resonator noise and that cancellation of the circuit noise was achieved. Moreover, the measured phase noise in the close-in region shows excellent agreement with the simulated phase noise (Fig. 8) relative to the same pump power (~17 dBm) driving the parametric circuit in the actual experiment. An improvement of more than 16 dB at 1 kHz offset was obtained by using the new oscillator topology (Fig. 1) rather than the conventional configuration shown in Fig. 5. The effect of the parametric circuit on output power and phase noise of the MEMS oscillator was measured for different Va, and for a fixed Vbias (4 V). In particular, we report in Fig. 10 the measured output power for each Va we investigated. Similarly, we report in Fig. 11 the recorded phase noise at 0.1, 1, and 10 kHz offsets, relative to the same Va values. As suggested by Fig. 10, increasing Va makes the pump signal at the input of the parametric amplifier larger. Intuitively, this increase leads
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Fig. 9. In black: predicted closed-loop phase noise by open-loop measurement of the resonator frequency fluctuations. Part of this curve was measured (continuous line) and the remainder (>100 Hz) is extrapolated (dashed line). In green: best phase noise measured for the configuration shown in Fig. 5. Va was set to 12 V and the output power was 14 dBm. In red: simulated phase noise, in the close-in region, through Agilent ADS. In blue: best phase noise we measured in the parametric oscillator. Va is set to 12 V and the output power is 12.5 dBm. Both the measured curves were evaluated at one output port of the power splitter following amplifier 1.
to a larger modulation of the varactor capacitance and, consequently, to an increase of the output power at the main output frequency. This fact was confirmed by an open loop measurement of the parametric circuit. After disconnecting the frequency doubler from the power splitter (Fig. 1), we drove it at the same frequency and output power measured at the oscillator output in the closedloop experiment. By doing so, we were able to measure the pump power, Ppump, at the pump port (Fig. 2) of the parametric circuit for each Va we investigated in the closed-loop experiment. Similarly, for the same Va values, we measured the amplifier 1 (Fig. 1) output power, Pout, at the corresponding oscillator output frequency. This probing location was preferred to the parametric circuit signal port so as to preserve the same impedance seen by the varactor in the closed-loop experiment. The variation of Pout with respect to Ppump is shown in Fig. 12. It is evident that the oscillator migrates from the conventional regimen to the parametric regimen when the pump power is about 5 dBm. This power level, attained at Va equal to 6.2 V, is slightly lower than the power threshold we found through our circuit simulation (Figs. 3–8). This fact remains under investigation, but it is likely due to unmodeled aspects, such as the dependence of the amplifier 2 output impedance on Va or the presence of non-linearities in the amplifiers. Regarding the measured phase noise performance (Fig. 11), the activation of the parametric oscillation permits
Fig. 10. Oscillator output power for different dc biases, Va, of amplifier 2.
cassella and piazza: low phase-noise autonomous parametric oscillator
Fig. 11. Measured phase noise at 0.1, 1, and 10 kHz offsets at different amplifier dc biases (Va). As evident, a generally larger phase noise evasion is attained at larger Va values and, consequently, at higher Ppump (Fig. 12). However, the measured phase noise shows a local maximum when the oscillator operates in the proximity of the parametric instability (Va ~ 6.2 V). This local degradation of the phase noise performance is determined by operation close to the parametric instability and follows what was predicted by the circuit simulation (Fig. 8).
improvement of the phase noise with respect to what is measured when the oscillator operates as a conventional feedback-loop oscillator. Moreover, as predicted by both the circuit simulation (Fig. 8, Region III) and the analytical model, we observed a larger reduction of the phase noise at larger Va and, therefore, at higher Ppump values (Fig. 11). Furthermore, we found a local maximum in the phase noise when the oscillator was made to operate close to its bifurcation. These experimental data follow the trend predicted by the circuit simulation (Fig. 8, Region II) and further validate the use of ADS for predicting phase noise performance in the parametric oscillator. It is important to note that, despite the high power level in the resonator, the measured improvement in the close-in phase noise cannot be attributed to the resonator non-linearity given the time constant (
