On the Synchronization Condition for Superharmonic Coupled QVCOs

A steady-state nonlinear analysis of quadrature voltage-controlled oscillators (QVCOs) comprising two VCOs mutually coupled at their second-harmonic frequency through a direct coupling circuit is presented. The analysis is based on an accurate prediction of the behavior of each VCO, which is analyzed separately as an injection locked oscillator taking into account both higher order harmonics of the differential tank voltage and the effect of the common-mode voltage at the drain terminals. We show that synchronization of the two VCOs is possible only at a frequency, derived in closed-form, which differs appreciably from the tank's resonant frequency.