Envelope Domain Formulation for the Analysis of the Nonlinear Transient Dynamics of Coupled Oscillators

A reduced-order envelope-domain formulation of coupled-oscillator systems based on realistic nonlinear models of the oscillator elements is presented for the first time to our knowledge. The formulation, based on numerical models of the transistor-based oscillators, enables an accurate prediction of the nonlinear dynamics of the coupled system, including the oscillation build-up, the locked and unlocked states, and the oscillator ON-OFF switching. To increase the applicability of the method, both admittance- and impedance-type models are extracted through harmonic balance simulations, under a voltage and current excitation, respectively, at the node/branch where the oscillator is connected to the coupled system. They are used to derive a nonlinear differential-equation system able to describe the transient dynamics of the entire structure. Because the oscillators are coupled through current injection, the impedance-based formulation is formally different from the admittance one, so it requires a dedicated derivation. For illustration, the method has been applied to exhaustively investigate the nonlinear dynamics of a system of three FET-based oscillators at 5 GHz. Very good agreement has been obtained with both circuit-level envelope transient (when applicable) and with measurements.