The Spectrum of a Noisy Free-Running Oscillator Explained by Random Frequency Pulling

We show that the Lorentzian spectrum of a free-running LC oscillator arises from the random pulling of its frequency by circuit noise sources. To prove this, we analyze the oscillator's spectrum using Adler's differential equation, first in response to pulling by a single injected tone and then by two tones mirrored around the free-run frequency. A novel analysis predicts the frequencies and amplitudes of the sidebands caused by the phase modulation that results. Ultimately, we show that frequency pulling by white noise accounts for the Lorentzian broadening of the oscillator spectrum, and that a complete pulling analysis predicts the same linewidth as a simple circuit analysis.