Injection signaling in relaxation oscillators

This paper investigates relation between injection signaling and the lock range of relaxation oscillators. We show that lock range is determined by the effective injection signals of Volterra circuits contributed by both external injection signals and the nonlinearity of oscillators. The larger the harmonic tones of injection signals and the higher the degree of the nonlinearity of oscillators, the larger the effective injection signals subsequently the larger the lock range. We further show in order to maximize the contribution of external injection signals, injection phase needs to be π / 2 , valid for oscillators with either single or multi-tone injections. Moreover, we show to maximize lock range, the duty cycle of injection signals needs to be 50%. Finally, we show the phase noise of relaxation oscillators has a similar profile as that of harmonic oscillators. The quality factor of relaxation oscillators is smaller as compared with that of harmonic oscillators. The higher the degree of the nonlinearity of the relaxation oscillator, the smaller the quality factor of the relaxation oscillator subsequently the higher the phase noise. The theoretical findings on the lock range of relaxation oscillators are validated using the simulation results of a dual-comparator relaxation oscillator designed in TSMC 180 nm 1.8 V CMOS technology.