Phase noise is a topic of theoretical and practical interest in electronic circuits, as well as in other fields, such as optics. Although progress has been made in understanding the phenomenon, there still remain significant gaps, both in its fundamental theory and in numerical techniques for its characterization. In this paper, we develop a solid foundation for phase noise that is valid for any oscillator, regardless of operating mechanism. We establish novel results about the dynamics of stable nonlinear oscillators in the presence of perturbations, both deterministic and random. We obtain an exact nonlinear equation for phase error, which we solve without approximations for random perturbations. This leads us to a precise characterization of timing jitter and spectral dispersion, for computing of which we have developed efficient numerical methods. We demonstrate our techniques on a variety of practical electrical oscillators and obtain good matches with measurements, even at frequencies close to the carrier, where previous techniques break down. Our methods are more than three orders of magnitude faster than the brute-force Monte Carlo approach, which is the only previously available technique that can predict phase noise correctly.

Phase noise in oscillators: a unifying theory and numerical methods for characterization

A harmonic oscillator topology displaying an improved phase noise performance is introduced in this paper. Exploiting the advantages yielded by operating the core transistors in class-C, a theoretical 3.9 dB phase noise improvement compared to the standard differential-pair LC-tank oscillator is achieved for the same current consumption. Further benefits derive from the natural rejection of the tail bias current noise, and from the absence of parasitic nodes sensitive to stray capacitances. Closed-form phase-noise equations obtained from a rigorous time-variant circuit analysis are presented, as well as a time-variant study of the stability of the oscillation amplitude, resulting in simple guidelines for a reliable design. Furthermore, the analysis of phase noise is extended to encompass a general harmonic oscillator, showing that all phase noise relations previously obtained for specific LC oscillator topologies are special cases of a very general and remarkably simple result.

Class-C Harmonic CMOS VCOs, With a General Result on Phase Noise

This paper investigates the timing jitter of single-ended and differential CMOS ring oscillators due to supply and substrate noise. We calculate the jitter resulting from supply and substrate noise, show that the concept of frequency modulation can be applied, and derive relationships that express different types of jitter in terms of the sensitivity of the oscillation frequency to the supply or substrate voltage. Using examples based on measured results, we show that thermal jitter is typically negligible compared to supply- and substrate-induced jitter in high-speed digital systems. We also discuss the dependence of the jitter of differential CMOS ring oscillators on transistor gate width, power consumption, and the number of stages.

A study of oscillator jitter due to supply and substrate noise

Radio-frequency (RF) circuits exhibit several distinguishing characteristics that make them difficult to simulate using traditional SPICE transient analysis. The various extensions to the harmonic balance and shooting method simulation algorithms are able to exploit these characteristics to provide rapid and accurate simulation for these circuits. This paper is an introduction to RF simulation methods and how they are applied to make common RF measurements. It describes the unique characteristics of RF circuits, the methods developed to simulate these circuits, and the application of these methods.

Introduction to RF simulation and its application

The voltage controlled oscillator (VCO) is a critical sub-block in communications transceivers. The role of the VCO in a transceiver and the VCO requirements are first reviewed. The necessity of GHz VCOs and the driving factors towards the monolithic integration of the VCO are examined. VCO design techniques are outlined and design trade-offs are explored. The performance of VCOs in different implementation styles is compared to evaluate when and if VCO integration is desirable.

https://www.ee.columbia.edu/%7Ekinget/papers_files/aacd99_book.pdf

Integrated GHz Voltage Controlled Oscillators

In this paper the general correlation spectrum of an oscillator with white and f -α noise sources is derived from the Langevin equations describing the deterministic and stochastic behaviour of an oscillator by perturbation methods. The treatment of f -α noise and the influence of the finite measuring time on the noise spectra are included in a time domain calculation. The theory is applied to a Van der Pol oscillator exhibiting parametric sideband amplification and to a realized oscillator demonstrating the applicability of the theory to technically relevant circuits

Franz X. Kaertner

Papers

Analysis of white and f−α noise in oscillators