Study of injection-locked non-harmonic oscillators using Volterra series

This study presents a Volterra series approach to analyse injection-locked non-harmonic oscillators. We show that by depicting the voltage transfer characteristics of comparators using hyperbolic tangent functions, non-harmonic oscillators can be analysed analytically using a set of Volterra circuits that are linear, have the same topology and element values but different inputs. We further show that the larger lock range of non-harmonic oscillators as compared with that of their harmonic counterparts is because of the harsher non-linear characteristics of these oscillators and the lower-order attenuation of the high-order frequency components of the oscillators. The reduced non-linear characteristics of ring oscillators because of the absence of positive feedback also gives rise to a smaller lock range as compared with relaxation oscillators. These theoretical findings are validated using both the simulation results of relaxation oscillators and ring oscillators designed in IBM 130 nm complementary metal oxide semiconductor technology and the measurement results of ring oscillators implemented using commercial ICs.