Abstract
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.
Highlights
COUPLED-OSCILLATOR systems are used in a variety of applications, including power combination, beam steering, wireless distribution of synchronization signals and sensors [1]–[12]
In order to cope with the mentioned analysis problems, several previous works [15], [16], [20] propose the use of oscillator models extracted from a harmonic balance (HB) simulation of the standalone oscillator circuit
A discussion on the initial conditions is given later in this subsection. Note that using this formulation, the system of MM coupled oscillators is modeled with system (11) of MM nonlinear complex equations, which reduces noticeably the computational cost when compared with the circuit-level envelope transient system, especially for a high number MM of oscillator elements
Summary
COUPLED-OSCILLATOR systems are used in a variety of applications, including power combination, beam steering, wireless distribution of synchronization signals and sensors [1]–[12]. Time-domain analyses might not be applicable at microwave frequencies and harmonic balance (HB) requires the individual fulfillment of the oscillation condition in each oscillator circuit This can be achieved by introducing one auxiliary generator (AG) [13], [14] in each oscillator element, so, in a system of MM coupled oscillators, MM oscillation conditions must be satisfied [15], given by the zero value of the current-to-voltage ratio at each of the introduced AGs, YYAAAA,ii = 0, where ii = 1 to MM. In order to cope with the mentioned analysis problems, several previous works [15], [16], [20] propose the use of oscillator models extracted from a HB simulation of the standalone oscillator circuit These models are based on the linearization of the admittance function YY of each oscillator about its standalone free-running oscillation, fulfilling YY(VVoo, ωωoo) = 0.
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