This paper presents a variation-tolerant quick-start-up 39.25 MHz crystal oscillator and a theoretical analysis of a proposed chirp injector (CI) and negative resistance booster (NRB) for the quick start up. A new analysis of the theoretical minimum start-up time provides a new criterion for evaluating the start-up time in crystal oscillators. The calculated theoretical minimum start-up time at 39.25 MHz is   ${11}.{1}\ {\upmu }{s}$  . A new analysis of the proposed CI and NRB clarifies the validity of the universal optimum control method. It explains the reason why the proposed CI and NRB reduce the start-up time and its variation. The proposed crystal oscillator with the proposed CI and NRB exhibits a start-up time of 158 μs at 39.25 MHz. The variation of the start-up time variations is   ${\pm}{13}\% $   over the supply voltage range of 1.2–1.8 V and   ${\pm }{7}\% $   over the temperature range of   $- {30}\text{ to } {125}\;^\circ {\rm C}$  . The power consumption of the crystal oscillator in the steady state is 181 μW with a phase noise of   $- {147}\;{\bf{dBc}}/{\bf{Hz}}$   at 1 kHz offset frequency, which corresponds to a figure of merit (FoM) of 276 dB.
['Shunta Iguchi', 'Hiroshi Fuketa', 'Takayasu Sakurai', 'Makoto Takamiya']
Variation-Tolerant Quick-Start-Up CMOS Crystal Oscillator With Chirp Injection and Negative Resistance Booster