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Harmonics of Ring Oscillators

Ring oscillators are probably the most prevalent oscillators these days. They are the oscillator of choice for most micro-controllers and are used more and more often in RF systems as well, due to their low space requirements and easy integration. Having such a wide use, you would think, they are pretty well described in literature, but quite the contrary is true.

The literature on designing ring oscillators is quite weak. I guess most people consider it an easy problem, though the number of papers published on "novel" ideas how to design and build them is quite astonishing. The best resource on design I could find is the book "High-Frequency Oscillator Design for Integrated Transceivers" by van der Tang, Kasperkovitz and van Roermund. The book covers a quite broad range of how to design oscillators, both LC and ring, for integrated RF systems. Even though it is a hands down book, the authors do not shy away from including all the formulas you will need and where they come from, in the description of the design process.

That said, when you start asking, about what harmonics ring oscillators generate, then the results are very bleak. So far, I could only find 2 papers, and one comment to a paper, that explicitly deal with harmonics generation in ring oscillators. The first is "Higher Harmonic Generation in CMOS/SOS Ring Oscillators" by Nobuo Sasaki from 1982. He argues, that the different speed of positive and negative edges will lead to the collapse of all higher modes in short ring oscillators. Thus harmonics should only be visible in longer rings. T.W. Huston argues in "Comments on 'Higher harmonic generation in CMOS/SOS ring oscillators'" that this is not necessarily the case and that even short ring oscillators can show stable higher harmonic modes. The paper "Generation, Elimination and Utilization of Harmonics in Ring Oscillators" by Bhushan and Ketschen analysis higher harmonics as a phenomena, caused by improper start-up. I.e. if the ring oscillator is started with one edge only, then there will be no harmonics. If three or more edges are injected upon startup, then these will be sustained.