The Analytic Determination of the PPV for Second-Order Oscillators Via Time-Varying Eigenvalues

Time-varying eigenvalues may be used to formulate a set of linearly independent solutions for an arbitrary dynamical linear time-varying system. In this brief, it is shown how these quantities are used to determine analytically the perturbation projection vector (PPV) associated to a given oscillator. The PPV can be further used to estimate the spectral and timing properties of the oscillator output subject to the presence of noise sources. For this purpose, a complete set of solutions of the variational system associated to the oscillator under consideration is generated in terms of time-varying eigenvalues. To compute these quantities, a solution for a particular form of the Riccati equation must be found. It is shown how to obtain a solution for this equation from the steady-state behavior of the oscillator. A simple example demonstrating the application of the concepts mentioned above is also provided.