This chapter addresses the application of symbolic techniques on the characterization of ring Voltage Controlled Oscillators (VCO). The symbolic characterization of the VCOs comprises both the frequency-control voltage response and a simplified formula for evaluating the phase-noise in the oscillator. Two methodologies are shown for generating the frequency-control voltage response. The first one is based on the evaluation of the delay introduced by each VCO delay cell. In the second an approximate expression for the equivalent resistance of each VCO delay cell load is considered, and used for deriving the VCO model. The adoption of the proposed methodologies to submicron transistor sizes is illustrated. The application of the VCO characterization into an optimization based design is described.
Keywords: Ring VCO, VCO model, symbolic characterization, optimization-based design, deepsubmicron VCO characterization, semi-symbolic methodology, phase-noise characterization, Maneatis delay cell, equivalent resistance, phase-noise minimization.