Expressions For Phase Noise Research Articles

A Cryo-CMOS Oscillator With an Automatic Common-Mode Resonance Calibration for Quantum Computing Applications

This article presents a 4-to-5GHz LC oscillator operating at 4.2K for quantum computing applications. The phase noise (PN) specification of the oscillator is derived based on the control fidelity for a single-qubit operation. To reveal the substantial gap between the theoretical predictions and measurement results at cryogenic temperatures, a new PN expression for an oscillator is derived by considering the shot-noise effect. To reach the optimum performance of an LC oscillator, a common-mode (CM) resonance technique is implemented. Additionally, this work presents a digital calibration loop to adjust the CM frequency automatically at 4.2K, reducing the oscillator’s PN and thus improving the control fidelity. The calibration technique reduces the flicker corner of the oscillator over a wide temperature range (10 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> and 8 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> reduction at 300K and 4.2K, respectively). At 4.2K, our 0.15-mm2 oscillator consumes a 5-mW power and achieves a PN of −153.8dBc/Hz at a 10MHz offset, corresponding to a 200-dB FOM. The calibration circuits consume only a 0.4-mW power and 0.01-mm2 area.

A 19.5-GHz 28-nm Class-C CMOS VCO, With a Reasonably Rigorous Result on 1/f Noise Upconversion Caused by Short-Channel Effects

Class-C operation is leveraged to implement a $K$ -band CMOS voltage-controlled oscillator (VCO) where the upconversion of $1/f$ current noise from the cross-coupled transistors in the oscillator core is robustly contained at a very low level. Implemented in a bulk 28-nm CMOS technology, the 12%-tuning-range VCO shows a phase noise as low as −112 dBc/Hz at 1-MHz offset (−86 dBc/Hz at 100 kHz offset) from a 19.5 GHz carrier while consuming 20.7 mW, achieving a figure of merit (FoM) of −185 dBc/Hz. The design is complemented by a theoretical investigation of $1/f$ noise upconversion caused by short-channel effects in the cross-coupled transistors, obtaining the first instance of a closed-form phase noise expression in the $1/f^{3}$ region.

An Accurate Analysis of Phase Noise in CMOS Ring Oscillators

We present an accurate analysis of phase noise in ring oscillators, where each stage in the ring is a CMOS inverter loaded by a capacitance. Closed-form phase noise expressions are obtained in both 1/ ${f} ^{{2}}$ and 1/ ${f} ^{{3}}$ offset frequency regions, displaying an excellent agreement with numerical simulations using either ideal Verilog-A MOS models (allowing theoretical predictions to be tested with exactly the same assumptions under which they were derived), or commercial BSIM4 MOS models.

A low phase noise InGaP–GaAs HBT Colpitts VCO with a high quality differential inductor

A balanced Colpitts voltage controlled oscillator (VCO) topology with its two implementations in a GaAs heterojunction bipolar transistor (HBT) process is introduced in this paper. A detailed analysis of phase noise, originating from the work by Hajimiri and Lee, is given, which depicts a specific outline of the phase noise generation mechanisms in GaAs HBT based harmonic oscillators including LC cross coupled/Colpitts/class C VCOs. In addition, phase noise expressions for HBT based VCOs are derived, compared and act as an optimization tool in oscillator designs. The theoretical analysis of phase noise is authenticated with two balanced Colpitts VCOs with their respective center oscillation frequencies of 850 MHz and 1670 MHz, which are fabricated in a 2 µm GaAs HBT H20HL process. Measurement results show a phase noise of − 139 dBc/Hz at 1 MHz offset from an 850 MHz carrier and a phase noise of − 134 dBc/Hz or less at 1 MHz offset from a 1700 MHz carrier. The output power variation is less than 0.5 dBm over the tuning range. A relatively good match between numerical simulation and measurements is observed.

Phase Noise Analysis of Bipolar Class-C VCOs With Delay in Oscillator Loop

The expression for phase noise of a bipolar class-C voltage controlled oscillator (VCO) with an ideal delay in the oscillator loop is derived in this paper. The impact of internal device parasitics is then quantified and supported with circuit simulation results. A bipolar class-C VCO is fabricated in a 130-nm BiCMOS technology to validate the analytical predictions. It exhibits a phase noise performance of −112.2 dBc/Hz at 1-MHz offset when running at 19.2 GHz. The tuning range is 7%, and the overall FoM and FoMT are 180.4 dBc/Hz and 177.3 dBc/Hz, respectively.

Fundamental noise dynamics in cascaded-order Brillouin lasers

The dynamics of cascaded-order Brillouin lasers make them ideal for applications such as rotation sensing, highly coherent optical communications, and low-noise microwave signal synthesis. Remarkably, when implemented at the chip scale, recent experimental studies have revealed that Brillouin lasers can operate in the fundamental linewidth regime where optomechanical and quantum noise sources dominate. To explore new opportunities for enhanced performance, we formulate a simple model to describe the physics of cascaded Brillouin lasers based on the coupled mode dynamics governed by electrostriction and the fluctuation-dissipation theorem. From this model, we obtain analytical formulas describing the steady-state power evolution and accompanying noise properties, including expressions for phase noise, relative-intensity noise, and power spectra for beat notes of cascaded laser orders. Our analysis reveals that cascading can enhance laser noise, resulting in a broader emission linewidth and larger intensity fluctuations with increased power. Consequently, higher-coherence laser emission can be achieved if indefinite cascading can be prevented. In addition, we derive a simple analytical expression that enables the Stokes linewidth to be obtained from spectra of beat notes between distinct cascaded laser orders and their relative powers. We validate our results using stochastic numerical simulations of the cascaded laser dynamics.

Phase noise of electrical oscillators due to multi‐Lorentzian noise sources based on the LTV model

Using the general expression of the phase variance expressed by the linear time variant (LTV) phase noise model, phase variances due to multi-Lorentzian and white noise sources of electrical oscillators have been extracted and phase noise expressions due to white and Flicker noises have been calculated. Despite the basic LTV model in which the expressed phase noise due to white noise is valid for offset frequencies larger than the −3 dB corner frequency of the output and it tends to infinity for small offset frequencies, in the proposed model, the extracted phase noises due to white and Flicker noises are valid for the entire spectrum and become limited for small offset frequencies. Comparison between the results of the proposed model by those of the harmonic balance technique shows the validity of the model. Also, the effects of the physical parameters of the oscillator circuit on the extracted phase noise can be observed in the expressions. The proposed model is applicable to calculate the phase noise of the oscillators with single perturbation source applied to the intended output node.

Analysis and Design of a Multi-Core Oscillator for Ultra-Low Phase Noise

In this paper, we exploit an idea of coupling multiple oscillators to reduce phase noise (PN) to beyond the limit of what has been practically achievable so far in a bulk CMOS technology. We then apply it to demonstrate for the first time an RF oscillator that meets the most stringent PN requirements of cellular basestation receivers while abiding by the process technology reliability rules. The oscillator is realized in digital 65-nm CMOS as a dual-core LC-tank oscillator based on a high-swing class-C topology. It is tunable within 4.07–4.91 GHz, while drawing 39–59 mA from a 2.15 V power supply. The measured PN is $-$ 146.7 dBc/Hz and $-$ 163.1 dBc/Hz at 3 MHz and 20 MHz offset, respectively, from a 4.07 GHz carrier, which makes it the lowest reported normalized PN of an integrated CMOS oscillator. Straightforward expressions for PN and interconnect resistance between the cores are derived and verified against circuit simulations and measurements. Analysis and simulations show that the interconnect resistance is not critical even with a 1% mismatch between the cores. This approach can be extended to a higher number of cores and achieve an arbitrary reduction in PN at the cost of the power and area.

On the Phase Noise Performance of Transformer-Based CMOS Differential-Pair Harmonic Oscillators

The white noise to phase noise conversion of one- and two-port CMOS differential-pair harmonic oscillators with transformer-based resonators is addressed in this paper. First, the operation of double-tuned transformer resonators is reviewed and design guidelines are proposed to maximize the quality factor. A rigorous approach is then employed to approximate the transformer network with a second order RLC model near the resonances, greatly simplifying the problem of handling the complex equations of a higher order resonator. The results are applied to phase noise calculations, leading to simple, yet accurate, closed-form $1/f^{2}$ phase noise expressions in excellent agreement with the simulation results. It is formally proved, in a general case, that high-order resonators do not provide any fundamental advantage in comparison with simple LC-tanks. The two-port transformer based oscillator may be exploited to limit the phase noise contribution of the core transistors through optimization of the bias point and by leveraging the transformer voltage gain.

The Analysis of the Phase Noise of the Closed-Loop Driver Circuit in Micromechanical Gyroscope Based on the Phase-Locked Principle

Compared with the closed-loop drive circuit in micromechanical gyroscope of AGC principle, the closed-loop driver circuit based on the phase locked principle can overcome its shortcomings which cannot control the frequency characteristics of the output signal, reducing the frequency deviation and the frequency jitter of the output signal, and shortening the settling time of the closed-loop drive system. In this paper, the noise of the closed-loop phase drive circuit based on phase locked principle is analyzed and deduced its phase noise expressions.

A Time-Variant Analysis of Phase Noise in Series Quadrature Oscillators

This paper presents a systemic analysis for phase noise performances of the series quadrature oscillator (QOSC) by using the time-variant impulse sensitivity function (ISF) model. The effective ISF for each noise source in the oscillator is derived mathematically. According to these effective ISFs, the explicit closed-form expression for phase noise due to the total thermal noise in the series QOSC is derived, and the phase noise contribution from the flicker noise in the regenerative and coupling transistors is also figured out. The phase noise contributions from the thermal noise and the flicker noise are verified by SpectreRF simulations.

A Time Variant Analysis of Phase Noise in Differential Cross-Coupled LC Oscillators

This paper presents a systemic analysis for phase noise performances of differential cross-coupled LC oscillators by using Hajimiri and Lee's model. The effective impulse sensitivity functions (ISF) for each noise source in the oscillator is mathematically derived. According to these effective ISFs, the phase noise contribution from each device is figured out, and phase noise contributions from the device noise in the vicinity of the integer multiples of the resonant frequency, weighted by the Fourier coefficients of the effective ISF, are also calculated. The explicit closed-form expression for phase noise of the oscillator is definitely determined. The validity of the phase noise analysis is verified by good simulation agreement.

A State-Space Phase-Noise Model for Nonlinear MEMS Oscillators Employing Automatic Amplitude Control

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

Phase-induced intensity noise in Bragg-grating-pair delay line signal processors

Investigation of the phase-induced intensity noise generated by fibre Bragg-grating-pair delay line signal processors is presented. The phase noise expressions of the transmissive and reflective Bragg-grating-pair structures are derived for the first time. They show that the transmissive and reflective structures have different phase noise spectra even though they have the same transfer characteristic. Experimental results are presented to verify the theory.

Comparison of phase noise models on a class of ring oscillators using low voltage swing fully differential delay cells

This paper compares the phase noise models on a class of differential saturated ring oscillator with low voltage swing. Variance parameter/single stage jitter is used as the metric for comparison. When first passage time (fpt) is employed, first for the simple case of the RC representation of the differential pair, an exact expression involving the parabolic cylinder function is used in this comparison. Then for the general case, the expression using time scaling is used for comparison. On the last passage time (lpt) model, the probability distribution function (pdf) of lpt is used to further differentiate the models. The fpt part in the complete lpt pdf is identified. Comparison of the phase noise's power spectral density expression is then presented. The phase noise expression of this author does not go to infinity as offset frequency approaches zero. Finally simulation results on example designs are presented, where it is shown that the model by this author, when compared to other models, can improve accuracy by 7-10 dBc/Hz.

Concurrent Dual-Frequency Oscillators and Phase-Locked Loops

A dual-frequency oscillator employing a fourth-order tank is shown to have the ability to generate simultaneous oscillations at two frequencies. A nonlinear analysis to determine the steady-state and transient behavior of this oscillator is presented. Further, the phase-noise expression for the dual-frequency oscillator is derived and compared with that of a single-frequency oscillator. A dual-loop phase-locked loop (PLL) is designed to lock the frequencies of the dual-frequency oscillator to two external references independently. A prototype of a dual-frequency voltage-controlled oscillator (VCO) along with the dual-loop PLL is implemented in a BiCMOS SiGe technology. The dual-frequency VCO oscillates simultaneously at 2.33 and 4.98 GHz with 12.5% and 10.3% tuning ranges, respectively. The PLL has locking ranges of 4.2% and 3.6% for 2.33 and 4.98 GHz, respectively. Potential applications of concurrent multifrequency oscillators in multifunctional communication systems and multiband beam forming are discussed.

An Analysis of $1/f^{2}$ Phase Noise in Bipolar Colpitts Oscillators (With a Digression on Bipolar Differential-Pair LC Oscillators)

This work presents an analysis of phase noise in the 1/f2 region displayed by both single-ended and differential bipolar Colpitts oscillators. Very accurate and rigorous symbolic phase noise expressions are derived, enabling a deeper insight into the major mechanisms of phase noise generation, and providing new tools for design optimization. Phase noise expressions for the cross-coupled differential-pair LC-tank oscillator are derived as well. The theoretical analysis is validated on a 3 GHz differential bipolar Colpitts VCO implemented in a 0.35 mum SiGe process. Measurements show a phase noise of -123 dBc/Hz or less at 1MHz offset frequency from the 2.8-3.1 GHz carrier, for a phase noise figure-of-merit of at least 183 dBc/Hz across the tuning range. A very good agreement between theory, numerical simulations, and measurements is observed

Multiphase LC oscillators

This paper proposes an extensive analysis of multiphase LC oscillators. N oscillators coupled in a ring topology can generate multiphase outputs. For their nature, these oscillators operate off-resonance and consequently their phase noise worsens at increasing coupling strength. Since the coupling transistors raise the total power consumption, the noise-power product degrades even more with respect to a stand-alone oscillator. On the other hand, at high coupling factors, component mismatches affect less the phase accuracy. Closed-form expressions for phase noise and phase accuracy are derived which are verified against circuit simulations.