Analytical expressions for phase noise eigenfunctions of LC oscillators

We obtain analytical expressions for eigenfunctions that characterize the phase noise performance of generic LC oscillator structures. Using these, we also obtain analytical expressions for the timing jitter and spectrum of such oscillators. Our approach is based on identifying three fundamental parameters, derived from the oscillator's steady state, that characterize these eigenfunctions. Our analysis accounts for the nonlinear mechanism that stabilizes oscillator amplitudes. It also lays out, quantitatively and in analytical form, how symmetry in an LC oscillator's negative resistance mechanism impacts the oscillator's eigenfunctions and its phase noise/jitter characteristics. We show that symmetry results in particularly simple forms for the PPV and resultant phase noise. We compare our expressions with existing LC oscillator design formulae and show that the expressions match for symmetric nonlinearities. We validate our analytical results against simulation on practical CMOS LC oscillator circuits. Our expressions and symmetry results are expected to be useful tools for optimizing phase noise performance during the design of LC oscillators.