Obtaining Frequency Sensitivities to Variations Analytically from Parameterized Nonlinear Oscillator Phase Macromodels

System-level variability analysis and design centering for oscillators relies on fast and accurate methods for obtaining the parametric sensitivities of higher-level performances (such as center frequency) directly from phase macromodels. We present an efficient and elegant method, involving no numerical simulation, for finding parametric sensitivities of oscillator frequencies directly from nonlinear phase domain macromodels. We validate the method, termed FS-PPV, on numerically extracted ring and LC oscillator PPV macromodels, as well as on a purely analytical exact PPV macromodel for idealized ring oscillators. We apply FS-PPV to find statistical distributions of oscillator center frequencies and validate these distributions against Monte-Carlo simulations. FS-PPV achieves speedups of more than 3000times over brute-force Monte-Carlo based parameter variation analysis.