A Comprehensive Analysis of Quadrature Signal Synthesis in Cross-Coupled RF VCOs

This paper presents a linear model for cross-coupled quadrature voltage-controlled oscillators (QVCOs) together with a simple generalized proof for the condition of quadrature oscillation. The analysis provides insight into the oscillation mechanism and the previously observed frequency shift when the magnitude and phase of the coupling signal are deliberately modified by a complex coupling coefficient K_c. It is demonstrated that the steady-state oscillation condition G_MR_p=1 holds only if G_M is replaced by an effective large-signal transconductance G_Meff that is a function of the coupling transconductance G _Mc and the imaginary part of K_c. Closed-form expressions for the phase imbalance and amplitude error in presence of mismatch between the LC tank circuits as well as device transconductances are also derived. We introduce the new concept of quadrature resistance/quadrature conductance (R_quad/G_quad), an incremental element synthesized by the coupling transistors, and show that its magnitude is responsible for the frequency shift and quadrature oscillation. A one-port model of the QVCO is studied and the loading effect of G_quad on the tank is examined. Particularly, it is shown that G_quad degrades the open-loop quality factor and worsens the phase noise. The analysis in this paper can be applied to most QVCO topologies presented in the literature