An electronic oscillator in a phase-locked loop

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Thermal, Statistical & Complex Systems · Nonlinear dynamics and chaos

Many Parts, One Behaviour, Ch Synchronisation, cascades and the resilience of networks

Problem

An electronic oscillator in a phase-locked loop, free-running at angular frequency ω\omega, is injection-locked to an external reference signal of fixed angular frequency ωref\omega_{\rm ref} (the reference is unaffected by the oscillator). Its phase obeys θ˙=ω+Ksin⁡(θref−θ)\dot\theta = \omega + K\sin(\theta_{\rm ref}-\theta) with θref=ωreft\theta_{\rm ref}=\omega_{\rm ref}t, where KK (in rad s−1\mathrm{rad\,s^{-1}}) is the injection strength. The detuning is Δω=ω−ωref=1.4 rad s−1\Delta\omega = \omega-\omega_{\rm ref} = 1.4\,\mathrm{rad\,s^{-1}}. ((a)) Derive Adler's equation for the phase difference ϕ=θ−θref\phi=\theta-\theta_{\rm ref} and hence find the smallest injection strength KcK_{\rm c} at which the loop can lock. ((b)) At K=KcK=K_{\rm c} exactly, find the fixed point ϕ∗\phi^* and show that the usual linear stability test gives no verdict there. Say in one sentence what this signals about the fixed point. ((c)) At K=2.0 rad s−1K=2.0\,\mathrm{rad\,s^{-1}}, find the stable locked phase difference and confirm its stability.

Browse the problem library