Necessary and sufficient conditions for frequency entrainment of quasi-sinusoidal injection-synchronized oscillators

A method is presented which permits the first-approximation exact analysis of the dynamical stability of fundamental-mode injectionsynchronized oscillators (FISO´s) characterized by a quasi-sinusoidal quasi-static behavior. By combining small parameter and stroboscopic transformation techniques, the phase-lock stability investigation of an nth-order system is reduced to the simple Hurwitz test on an nth degree polynomial easily obtainable from steady state describing quantities. On this basis, equations for critical locking are also derived, which demonstrate the existence of a pair of limit curves (Locus and Boundary) already conjectured and looked for in the past, but only with partial success. As an example of application, a sixth-order oscillator is finally investigated. The wide class of treatable circuits (which may include several resistive and reactive nonlinear elements though dependent on the same control variable, as well as multiple-tuned resonators), together with the simplicity of the calculations required, make the proposed method suitable for use in FISO design through computer-aided techniques.