An Analytic Circuit-Based Model for White and Flicker Phase Noise in <emphasis emphasistype="italic">LC</emphasis> Oscillators

Abstract

A general circuit-based model of LC oscillator phase noise applicable to both white noise and 1/f noise is presented. Using the Kurokawa theory, differential equations governing the relationship between amplitude and phase noise at the tank are derived and solved. Closed form equations are obtained for the IEEE oscillator phase noise for both white and 1/f noise. These solutions introduce new parameters which take into account the correlation between the amplitude noise and phase noise and link them to the oscillator circuit operating point. These relations are then used to obtain the final expression for voltage noise power density across the output oscillator terminals assuming the noise can be modeled by stationary Gaussian processes. For white noise, general conditions under which the phase noise relaxes to closed-form Lorentzian spectra are derived for two practical limiting cases. Further, the buffer noise in oscillators is examined. The forward contribution of the buffer to the white noise floor for large offset frequency is expressed in terms of the buffer noise parameters. The backward contribution of the buffer to the 1/Delta f2 oscillator noise is also quantified. To model flicker noise, the Kurokawa theory is extended by modeling each 1 /f noise perturbation in the oscillator as a small-signal dc perturbation of the oscillator operating point. A trap-level model of flicker noise is used for the analysis. Conditions under which the resulting flicker noise relaxes to an 1/Delta f3 phase noise distribution are derived. The proposed model is then applied to a practical differential oscillator. A novel method of analysis, splitting the noise contribution of the various transistors into modes is introduced to calculate the Kurokawa noise parameters. The modes that contribute the most to white noise and flicker noise are identified. Further, the tail noise contribution is analyzed and shown to be mostly up-converted noise. The combined white and flicker noise model exhibits the presence of a number of corner frequencies whose values depend upon the relative strengths of the various noise components. The proposed model is compared with a popular harmonic balance simulator and a reasonable agreement is obtained in the respective range of validity of the simulator and theory. The analytical theory presented which relies on measurable circuit parameters provides valuable insight for oscillator performance optimization as is discussed in the paper.