Closed-form locking time formula applying time-variant nature of LC Oscillators

In this work, time domain analysis is used to solve Adler´s equation in order to obtain the required time, for an oscillator under external injection, reaching the steady-state condition. Mathematical approach has been applied to fully describe the transient of frequency acquisition in injection-locked LC oscillators considering their time-variant nature. Both perturbation analysis and time domain solutions are used completely describe the locking time dependencies. Also, the effect of initial phase difference of injection signal on locking time and phase noise is theoretically studied Finally, design insights are deduced which enables designers to evaluate and minimize the timing budget required to achieve injection locking in designing a fast locking oscillator. The mathematical consequences in this work explain why the initial phase shift of injection signal has no effect on the phase noise of oscillator.