Finding the Tuning Curve of a CMOS—LC VCO

A nonlinear perturbation model of a complementary LC-tuned voltage-controlled oscillator is derived, which consists of two mutually-coupled second-order differential equations. The first-order approximate periodic solution of the describing equations is found, obtaining closed-form expressions for both the amplitude and the harmonics of oscillation, as well as for the correction of the oscillation frequency due to the nonlinear effect of varactors. This allows us to find the tuning curve in explicit form. The accuracy of presented formulas was validated by circuit simulations.