Thermodynamical Phase Noise in Oscillators Based on <i>L-C</i> Resonators

Abstract

Using a new Admittance-based model for electrical noise able to handle Fluctuations and Dissipations of electrical energy, we explain the phase noise of oscillators that use feedback around L-C resonators. We show that Fluctuations produce the Line Broadening of their output spectrum around its mean frequency f<sub>0</sub> and that the Pedestal of phase noise far from f<sub>0</sub> comes from Dissipations modified by the feedback electronics. The charge noise power 4FkT/R C<sup>2</sup>/s that disturbs the otherwise periodic fluctuation of charge these oscillators aim to sustain in their L-C-R resonator, is what creates their phase noise proportional to Leeson’s noise figure F and to the charge noise power 4kT/R C<sup>2</sup>/s of their capacitance C that today’s modelling would consider as the current noise density in A<sup>2</sup>/Hz of their resistance R. Linked with this (A<sup>2</sup>/Hz?C<sup>2</sup>/s) equivalence, R becomes a random series in time of discrete chances to Dissipate energy in Thermal Equilibrium (TE) giving a similar series of discrete Conversions of electrical energy into heat when the resonator is out of TE due to the Signal power it handles. Therefore, phase noise reflects the way oscillators sense thermal exchanges of energy with their environment

Electrical Noise, Brownian Motion and the Arrow of Time

“…Thus, a resistor in TE would collect an average power P C by its C acting as a receiving antenna for TAs and would release an equal amount (on average) of electrical power converted into heat by its R. Note this picture for resistance R as a random set of chances to convert electrical energy into another form: usually phonons, but it could be photons as well (think of a radiation resistance). This discrete nature of R, or better said: of the electrical noise itself, allows a direct explanation for the phase noise of electronic oscillators (their line broadening for example) as due to the λ TAs associated with the resistance R of their resonators by Equation (2) [6] [7].…”