Phase Noise Derivation

/ edmundsj
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Here I derive the linear phase noise model developed by Behzad Razavi in his 1996 paper on the subject, which gives a closed-form expression for phase noise in an oscillator. I also describe a conceptual basis for phase noise in terms of the transfer function of the oscillator and the noise shaping properties of the oscillator.
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[@XiaolongHuang-y1b]

Very helpful brief concept. Thank you.

[@xlai]

Thank you for the explanation. A helpful supplement to Razavi's book.

[@santhoshk8703]

Beautiful explanation

[@milkarton]

Perfectly explained, thank you.

[@javidmusayev1092]

I see two issues with this derivation. First, A(w0) need not be -1 as in Barkhausen criterion, because in real oscillators it would be the nonlinearity (voltage limitation) that would suppress the large swing oscillation. It does not mean that A(w0) becomes one (e.g. in ring oscillator example it is not 1). Second and even more important point is that the phase noise conceptually is a sampled noise. Therefore, input noise sources at low frequencies up-convert to frequencies around the carrier when phase noise is observed. This phenomenon cannot be described by such linear model where Vin is passed through H(w), because none of the low-frequency components of Vin would appear around w0 according to the given derivation.