A Phasor-Domain Study of Lock Range of Harmonic Oscillators With Multiple Injections

This brief presents a phasor-domain study of the lock range of harmonic oscillators with multiple injections. The intrinsic relation between the lock range of harmonic oscillators with multiple injections and that with a single injection is obtained. We show that harmonic oscillators with multiple injections exhibit a larger lock range as compared with those with single injection if the phase of the injection signals is properly chosen. We further show that the maximum lock range provided by dual injections is twice that provided by a single injection. The findings are verified using $LC$ oscillators with a single injection and dual injections designed in IBM 130-nm 1.2-V CMOS technology.