Virtual damping and Einstein relation in oscillators

This paper presents a new physical theory of oscillator phase noise. Built around the concept of phase diffusion, this work bridges the fundamental physics of noise and existing oscillator phase-noise theories. The virtual damping of an ensemble of oscillators is introduced as a measure of phase noise. The explanation of linewidth compression through virtual damping provides a unified view of resonators and oscillators. The direct correspondence between phase noise and the Einstein relation is demonstrated, which reveals the underlying physics of phase noise. The validity of the new approach is confirmed by consistent experimental agreement.     

Phase-Noise Analysis of Injection-Locked Oscillators and Analog Frequency Dividers

In-depth investigation of the phase-noise behavior of injection-locked oscillators and analog frequency dividers is presented. An analytical formulation has been obtained, which allows a better understanding of the shape of the output phase-noise spectrum of these circuits. The simplicity of this formulation is also helpful for circuit design. Approximate expressions for the corner frequencies of the spectrum are determined, identifying the most influential magnitudes and deriving design criteria. In particular, a technique has been developed to shift the frequency of the first corner of the phase-noise spectrum, up to which the output phase noise follows the input one. The expressions for the corner frequencies can be introduced in either in-house or commercial harmonic-balance software, thus allowing an agile design, as no separate phase-noise analysis is required. The validity of the analytical techniques is verified with the conversion-matrix approach and with measurements using two field-effect-transistor-based circuits: a 4.9-GHz injection-locked oscillator and a frequency divider by 2 with 9.8-GHz input frequency.     

Nonlinear synthesis of phase shifters, based on synchronized oscillators

The phase shifters based on the synchronization of an oscillator to the input signal enable low consumption, low cost, and small size. However, the difficulties for an accurate design, associated to the complexity of the operation regime, limit their application to communication systems. Here, a technique is presented for the nonlinear synthesis of oscillator-based phase shifters. The technique provides the circuit-element values for a specified phase shift. The tuning voltage required to maintain a constant phase-shift value along a certain frequency band is also determined. Using these techniques, an injection-locked voltage-controlled oscillator, covering the 1.17-1.3GHz band has been designed and manufactured. The validity of the design is verified through nonlinear simulations and measurements. These include the phase-noise variation along the phase-shift interval and the circuit response versus modulated inputs. Good agreement has been obtained between measurements and simulations.     

A new approach to nonlinear analysis of noise behavior ofsynchronized oscillators and analog-frequency dividers

An original theory of phase noise in synchronized oscillators is outlined through the phase-locked loop (PLL) approach. The phase-noise spectrum obtained first by the analytical PLL theory and then by a simulator have been compared with very good accuracy. This new approach permits the best understanding of noise conversion in synchronized devices     

Optimal LC-VCO design through evolutionary algorithms

The need for implementing low cost, fully integrated RF wireless transceivers has motivated the widespread use CMOS technology. However, in the particular case for voltage-controlled oscillators (VCO) where ever more stringent specifications in terms of phase-noise must be attained, the design of the on-chip LC tank is a challenging task, where fully advantage of the actual technologies characteristics must be pushed to nearly its limits. To overcome phase-noise limitations arising from the low quality factor of integrated inductors, optimization design methodologies are usually used. In this paper a model-based optimization approach is proposed. In this work the characterization of the oscillator behaviour is guaranteed by a set of analytical models describing each circuit element performance. A set of working examples for UMC130 technology, aiming the minimization of both VCO phase noise and power consumption, is addressed. The results presented, illustrate the potential of a GA optimization procedure design methodology yielding accurate and timely efficient oscillator designs. The validity of the results is checked against HSPICE/RF simulations.     

A varactor configuration minimizing the amplitude-to-phase noise conversion in VCOs

Amplitude-to-phase-noise conversion due to varactors can severely limit the close-in phase noise performance in LC-tuned oscillators. This work proposes a rigorous analysis of this phenomenon, which highlights the fundamental limitations of single-ended tuned and differentially tuned diode varactor configurations. The back-to-back varactor topology is identified as a suitable solution to linearize the tank capacitance. The amplitude to phase noise conversion is greatly attenuated and the 1/f/sup 3/ phase noise is drastically reduced, without impairing the achievable tuning range. These results are validated through circuit simulations in an existing 0.35-/spl mu/m CMOS technology.     

Computing Timing Jitter From Phase Noise Spectra for Oscillators and Phase-Locked Loops With White and$1/f$Noise

Phase noise and timing jitter in oscillators and phase-locked loops (PLLs) are of major concern in wireless and optical communications. In this paper, a unified analysis of the relationships between time-domain jitter and various spectral characterizations of phase noise is first presented. Several notions of phase noise spectra are considered, in particular, the power-spectral density (PSD) of the excess phase noise, the PSD of the signal generated by a noisy oscillator/PLL, and the so-called single-sideband (SSB) phase noise spectrum. We investigate the origins of these phase noise spectra and discuss their mathematical soundness. A simple equation relating the variance of timing jitter to the phase noise spectrum is derived and its mathematical validity is analyzed. Then, practical results on computing jitter from spectral phase noise characteristics for oscillators and PLLs with both white (thermal, shot) and$bf 1/f$noise are presented. We are able to obtain analytical timing jitter results for free-running oscillators and first-order PLLs. A numerical procedure is used for higher order PLLs. The phase noise spectrum needed for computing jitter may be obtained from analytical phase noise models, oscillator or PLL noise analysis in a circuit simulator, or from actual measurements.     

A study of phase noise in colpitts and LC-tank CMOS oscillators

This paper presents a study of phase noise in CMOS Colpitts and LC-tank oscillators. Closed-form symbolic formulas for the 1/f/sup 2/ phase-noise region are derived for both the Colpitts oscillator (either single-ended or differential) and the LC-tank oscillator, yielding highly accurate results under very general assumptions. A comparison between the differential Colpitts and the LC-tank oscillator is also carried out, which shows that the latter is capable of a 2-dB lower phase-noise figure-of-merit (FoM) when simplified oscillator designs and ideal MOS models are adopted. Several prototypes of both Colpitts and LC-tank oscillators have been implemented in a 0.35-/spl mu/m CMOS process. The best performance of the LC-tank oscillators shows a phase noise of -142dBc/Hz at 3-MHz offset frequency from a 2.9-GHz carrier with a 16-mW power consumption, resulting in an excellent FoM of /spl sim/189 dBc/Hz. For the same oscillation frequency, the FoM displayed by the differential Colpitts oscillators is /spl sim/5 dB lower.     

LC振荡器相位噪声的非线性摄动分析技术

相位噪声是振荡器的重要性能指标,通过对振荡器的非线性微分方程的推导,从而引入新的参数来考虑振幅噪声和相位噪声之间的相关性同时把这些参数和振荡器的工作点联系起来。并且这些公式推导也考虑了LC谐振回路中振幅噪声和相位噪声之间的相关性。在此基础上这里研究了如何在时域状态方程下将噪声看成是对振荡输出信号的小扰动,并建立引入噪声后的随机微分方程,实现LC振荡器的相位噪声分析技术。     

Jitter and phase noise in ring oscillators

A companion analysis of clock jitter and phase noise of single-ended and differential ring oscillators is presented. The impulse sensitivity functions are used to derive expressions for the jitter and phase noise of ring oscillators. The effect of the number of stages, power dissipation, frequency of oscillation, and short-channel effects on the jitter and phase noise of ring oscillators is analyzed. Jitter and phase noise due to substrate and supply noise is discussed, and the effect of symmetry on the upconversion of 1/f noise is demonstrated. Several new design insights are given for low jitter/phase-noise design. Good agreement between theory and measurements is observed     

振荡器时变相位噪声分析技术研究进展

介绍了两种射频振荡器的时变相位噪声模型:Demir&Mehrotra的非线性扰动模型和Hajimiri&Lee的时变相位噪声模型。对一个简单的非线性电导LC振荡器,分别建立了这两种模型,并指出了两种模型的联系和区别。讨论了两种模型的计算方法,介绍了在EDA软件HSpice和SpectreRF下,以及频域中相位噪声的计算问题。最后,进一步讨论了时变相位噪声,指出相位噪声分析的难点和未来的发展方向。     

A theoretical and experimental study of the noise behavior ofsubharmonically injection locked local oscillators

A method for the phase-noise characterization of optically controlled subharmonically injection-locked oscillators that is based on a nonlinear model of synchronized oscillators is presented. It allows FM noise degradation at large-signal levels to be predicted easily and accurately. The theoretical analysis shows that (1) the nth-order subharmonic injection locking oscillator is primarily locked by the nth harmonic output of an injected signal, which is generated by the nonlinearity of the active device; (2) the minimum FM noise degradation factor of the nth-order subharmonically locked oscillator is n² when the injection power is sufficiently strong; and (3) a subharmonic injection locking LO with low injection power, good FM noise degradation, and large locking range can be designed by determining the optimum injection power level, by selecting the optimal nonlinear multiplication factor, and by decreasing the intrinsic noise level of the active device. The experimental results confirm the accuracy of the analysis     

Frequency dependence on bias current in 5 GHz CMOS VCOs: impact on tuning range and flicker noise upconversion

The tuning curve of an LC-tuned voltage-controlled oscillator (VCO) substantially deviates from the ideal curve 1//spl radic/(LC(V)) when a varactor with an abrupt C(V) characteristic is adopted and the full oscillator swing is applied directly across the varactor. The tuning curve becomes strongly dependent on the oscillator bias current. As a result, the practical tuning range is reduced and the upconverted flicker noise of the bias current dominates the 1/f/sup 3/ close-in phase noise, even if the waveform symmetry has been assured. A first-order estimation of the tuning curve for MOS-varactor-tuned VCOs is provided. Based on this result, a simplified phase-noise model for double cross-coupled VCOs is derived. This model can be easily adapted to cover other LC-tuned oscillator topologies. The theoretical analyses are experimentally validated with a 0.25 /spl mu/m CMOS fully integrated VCO for 5 GHz wireless LAN receivers. By eliminating the bias current generator in a second oscillator, the close-in phase noise improves by 10 dB and features -70 dBc/Hz at 10 kHz offset. The 1/f/sup 2/ noise is -132 dBc/Hz at 3 MHz offset. The tuning range spans from 4.6 to 5.7 GHz (21%) and the current consumption is 2.9 mA.     

High-Q active resonators using amplifiers and their applications to low phase-noise free-running and voltage-controlled oscillators

This paper presents a new technique to design high-Q active resonators. The active resonators are then used in the design of low phase-noise oscillators. The proposed new technique uses an amplifier to generate a negative resistance, which compensates for the resonator losses and increases the Q factor. The active resonator using this technique shows a high loaded Q factor of 548.62 from measurement at the fixed 10-GHz resonant frequency. Considerations to design a voltage tunable active resonator is given and measurements show that the loaded Q factors exceed 500 with a 120-MHz tuning range. A low phase-noise free-running and voltage-controlled oscillator (VCO) were designed as an application of the proposed active resonators. The phase noise of the free-running oscillator using the active resonator is -114.36 dBc/Hz at 100-kHz offset, which is 14 dB lower than the phase noise of the passive resonator oscillator. In the case of a VCO using the active resonator, the phase-noise performance is below -110 dBc/Hz over the whole tuning range, which is lower 13 dB compared to the passive resonator VCO. The total dc power consumptions are approximately 500 mW.     

Noise in IMPATT Diode Amplifiers and Oscillators

The results of experimental and theoretical investigations of the noise characteristics of lMPATT diode amplifiers and oscillators are presented. The oscillator noise is shown to consist of three different contributions: modulation noise, selectively amplified primary noise, and conversion noise. The influence of the active device nonlinearity and load circuit parameters is discussed in detail. The experimental results are in good agreement with the theoretical predictions. It is especially pointed out that the large correlation between AM and FM fluctuations, usually measured in IMPATT oscillators, indicates nonoptimum AM noise performance. Experimental techniques for achieving optimum AM noise performance are demonstrated (orthogonal noise tuning). By a simple extension of the model, the noise behavior of an injection phase-locked oscillator can be described. The calculated AM and FM noise power spectra for the synchronized oscillator are also shown to be in good agreement with experimental results. Finally the signal-to-noise ratio for current modulated IMPATT oscillators is investigated and optimization is demonstrated.     

Transmission-line stabilized monolithic oscillators

A novel technique for frequency stabilization and phase-noise reduction of monolithic oscillators is presented in this paper. It employs simple transmission-line resonators, which are many wavelengths long to increase the oscillator quality factor. Monolithic oscillators at 20 and 40 GHz are realized for the application of this technique. Phase noise reduction of more than 20 dB was achieved for both oscillators. The single-sideband phase noise obtained was -100 dBc/Hz at 100-kHz offset for the 20-GHz oscillator and -90 dBc/Hz at 1-MHz offset for the 40-GHz oscillator. The approach is implemented by using readily available transmission lines, which are open- or short-circuited at one end and connected to the monolithic-microwave integrated-circuit (MMIC) oscillator at the other end. Thus, it presents significant potential in the development of low-cost MMIC oscillators with enhanced noise performance     

Analysis of stabilization circuits for phase-noise reduction in microwave oscillators

Two configurations for oscillator phase-noise reduction using stabilization circuits have been demonstrated in the literature. One of them is based on the self-injection of the oscillator signal, after it passes through a long delay line or a high-quality-factor resonator. The second one is a stabilization loop, containing a frequency discriminator. In this paper, an in-depth analytical comparison of these two configurations, respectively based on injection locking and phase-locking principles, is presented. Analytical expressions are provided for the variation of the steady-state solution and its phase noise versus the parameters of the feedback network. The expressions are rigorously validated with harmonic balance. Instabilities reported by other authors are investigated through bifurcation analysis. The new expressions enable a good understanding of the amplitude and frequency jumps and sharp phase-noise maxima obtained simulations and measurements versus the feedback parameters. A practical 5-GHz voltage-controlled oscillator has also been implemented, for validation purposes.     

An Analysis of 1/f2 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.