On the synchronization condition of second-harmonic coupled QVCOs

We present a nonlinear analysis of quadrature Voltage-Controlled Oscillators (QVCOs) made up of two VCOs mutually coupled at their second-harmonic frequency through a direct coupling circuit. The analysis of the two VCOs, each of which is considered separately, as an injection locked oscillator, reveals some relevant aspects of their behavior related to the steep nonlinearity of the cross-coupled devices, as well as to common-mode voltage at the drain terminals. These aspects influence the synchronization condition of the overall system of the two VCOs, whose oscillation frequency significantly differs from the resonant frequencies of the individual tanks.