Locking time of oscillators under external frequency injection

In this work, time domain analysis is used to solve Adler´s equation to obtain the phase shift between a free-running oscillator and an externally injected oscillator. In addition, the time required to acquire locking between the two oscillators is also deduced. The dependency of the locking time versus characteristics of the oscillator such as the quality factor, Q, and its free-running frequency are also discussed. Finally, the analyses are verified by simulations of a typical LC oscillator under injection. The results of this analysis enable designers to evaluate the exact timing budget required to achieve injection locking.