Derivation of Floquet Eigenvectors Displacement for Optimal Design of LC Tank Pulsed Bias Oscillators

The paper presents an approximated and compact derivation of the mutual displacement of Floquet eigenvectors in a class of LC tank oscillators with time varying bias. In particular it refers to parallel tank oscillators of which the energy restoring can be modeled through a train of current pulses. Since Floquet eigenvectors are acknowledged to give a correct decomposition of noise perturbations along the stable orbit in oscillator's space state, an analytical and compact model of their displacement can provide useful criteria for designers. The goal is to show, in a simplified case, the achievement of oscillators design oriented by eigenvectors. To this aim, minimization conditions of the effect of stationary and time varying noise as well as the contribution of jitter noise introduced by driving electronics are deduced from analytical expression of eigenvectors displacement.

Floquet Eigenvectors Theory of Pulsed Bias Phase and Quadrature Harmonic Oscillators

The paper presents an analytical derivation of Floquet eigenvalues and eigenvectors for a class of harmonic phase and quadrature oscillators. The derivation refers in particular to systems modeled by two parallel RLC resonators with pulsed energy restoring. Pulsed energy restoring is obtained through parallel current generators with an impulsive characteristic triggered by the resonators voltages. In performing calculation the initial hypothesis of the existence of stable oscillation is only made, then it is verified when both oscillation amplitude and eigenvalues/eigenvectors are deduced from symmetry conditions on oscillator space state. A detailed determination of the first eigenvector is obtained. Remaining eigenvectors are hence calculated with realistic approximations. Since Floquet eigenvectors are acknowledged to give the correct decomposition of noise perturbations superimposed to the oscillator space state along its limit cycle, an analytical and compact model of their behavior highlights the unique phase noise properties of this class of oscillators.

Phase and Quadrature Pulsed Bias LC-CMOS VCO

Pulsed bias is an attempt to improve the performance of oscillators in integrated circuits as a result of architectural innovation. Given the relatively low value of resonator quality factor achievable on-chip, for a specified bias voltage level, pulsed bias may result in a lower power consumption and in an improvement of the spectral purity of the oscillation. The main drawback of this approach is the need to introduce a certain time delay in order to properly position pulses with respect to oscillation waveform. Delay accumulation requires further energy dissipation and introduce additional jitter. In this paper we present a new architecture capable to avoid unnecessary delay, based on the idea to apply the pulsed bias approach to a quadrature oscillator. A ?rst circuit-level implementation of this concept is presented with simulation results.

CMOS Phase and Quadrature Pulsed Differential Oscillators Coupled through Microstrip Delay-Lines

An innovative solution to design phase and quadrature pulsed coupled oscillators systems through electromagnetic waveguides is described in this paper. Each oscillator is constituted by an LC differential resonator refilled through a couple of current pulse generator circuits. The phase and quadrature coupling between the two differential oscillators is achieved using delayed replicas of generated fundamentals from a resonator as driving signal of pulse generator injecting in the other resonator. The delayed replicas are obtained by microstrip-based delay-lines. A 2.4 - 2.5 GHz VCO has been implemented in a 150 nm RF CMOS process. Simulations showed at 1 MHz offset a phase noise of -139.9 dBc/Hz and a FOM of -189.1 dB.