Noise Model of Relaxation Oscillators Due to Feedback Regeneration Based   on Physical Phase Change
  A new approach to investigate noise spikes due to regeneration in a relaxation oscillator is proposed. Noise spikes have not been satisfactorily accounted for in traditional phase noise models. This paper attempts to explain noise spikes/jump phenomenon by viewing it as phase change in the thermodynamic system(for example, from gas to liquid or magnetization of ferromagnet). Both are due to regeneration (positive feedback in oscillator as well as alignment of spin due to positive feedback in ferromagnet). The mathematical tool used is the partition function in thermodynamics, and the results mapped between thermodynamic system and relaxation oscillator. Theory is developed and formula derived to predict the magnitude of the jump, as a function of design parameter such as regeneration parameter or loop gain. Formulas show that noise increases sharply as regeneration parameter/loop gain approaches one, in much the same way when temperature approaches critical temperature in phase change. Simulations on circuits (Eldo) using CMOS as well as Monte Carlo simulations (Metropolis) on ferromagnet (Ising model) were performed and both show jump behaviour consistent with formula. Measurements on relaxation oscillators fabricated in 0.13um CMOS technology verify such behaviour, where the sharp increase in noise when regeneration parameter/loop gain is close to one, matches closely with the theoretical formula. Using the formula the designer can quantify the variation of noise spikes dependency on design parameters such as gm (device transconductance), R, I0, via their influence on regeneration parameter/loop gain. 
Bosco Leung
cond-mat.stat-mech