Abstract
A semianalytical method for the global prediction and understanding of the transient dynamics of oscillator circuits is presented. It covers both the linear and nonlinear transient stages, which are related with the circuit generalized eigenvalues, here introduced for the first time. The transient model relies on the application of the implicit function theorem to the harmonic-balance (HB) system, in order to derive a reduced-order nonlinear differential equation from a given observation node. This requires the extraction of a nonlinear admittance function, depending on the voltage excitation and oscillation frequency, which is done with a forcing auxiliary generator (AG). The linearization of this admittance function for each excitation amplitude provides a sequence of linear ordinary differential equations (ODEs), describing the system dynamics in the vicinity of each point of the transient trajectory, which can be reconstructed from the expression of the solution increment at each time step. The sequence of differential equations provides a set of generalized eigenvalues, responsible for the acceleration or deceleration of the oscillation growth and capable to detect spurious transient frequencies. The concept of escape time, or time required by the transient trajectory to go through a certain interval of amplitude values, is also introduced, for the first time to our knowledge. The method has been successfully applied to analyze the transient dynamics of several FET oscillators, including dual-frequency oscillators and switched oscillators.
Highlights
T transient dynamics is a key characteristic of the oscillator behavior [1]–[7] that highly influences the performance of the system in which it is embedded
The envelope-transient approach [8]–[12] relies on a Fourier-series representation of the circuit variables with timevarying harmonic terms, which enables the derivation of a nonlinear differential equation system in these slowly varying terms
The general equation (5) will be considered, which exhibits an implicit dependence on both the amplitude V and frequency
Summary
T transient dynamics is a key characteristic of the oscillator behavior [1]–[7] that highly influences the performance of the system in which it is embedded. The first-order assumption [19], [20] will fail if the transient effects cannot be accurately described through a firstorder Taylor series expansion in j To cope with this limitation, this work considers a general dependence on j. As in all the LL methods [22]–[29], this resolution is exponential, in the sense that, in each time interval (tn, tn+1), the transient depends on the eigenvalues of the system Jacobian matrix, evaluated at tn These eigenvalues are calculated in an indirect manner, applying, under V variations, a series of pole-zero identification to a frequency-domain function, derived from the characteristic system.
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