the transient behavior of lc and ring oscillators under external frequency injection

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 and ring oscillators considering their time-varying nature. then, the analysis is verified by simulations of a ring as well as a typical rf-lc oscillator. likewise, the effect of initial phase difference of injection signal on locking time and phase noise is theoretically studied. for ring oscillators, a delay-based time–domain and perturbation analysis are used to reveal the dependency of circuit parameters to the locking time. finally, the design insights are deduced which enable the 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 there is no transient behavior while ring oscillator signal propagates from a stage to another, or why the initial phase shift of injection signal has no effect on the phase noise of oscillator.