Floquet theory and non-linear perturbation analysis for oscillators with differential-algebraic equations

Abstract

Oscillators are key components of electronic systems. In RF communication systems, they are used for frequency translation of information signals and for channel selection, and in digital electronic systems, they are used as a time reference, i.e. a clock signal, in order to synchronize operations. Undesired perturbations in practical electronic systems adversely affect the spectral and timing properties of oscillators, which is a key performance limiting factor, being a major contributor to bit‐error‐rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled‐data systems. Characterizing how perturbations affect oscillators is therefore crucial for practical applications. The traditional approach to analysing perturbed nonlinear systems (i.e. linearization) is not valid for oscillators. In this paper, we present a theory and efficient numerical methods, for non‐linear perturbation and noise analysis of oscillators described by a system of differential‐algebraic equations (DAEs). Our techniques can be used in characterizing phase noise and timing jitter due to intrinsic noise in IC devices, and evaluating the effect of substrate and supply noise on the timing properties of practical oscillators. In this paper, we also establish novel results for periodically time‐varying systems of linear DAEs, which we rely on in developing the above theory and the numerical methods. Copyright © 2000 John Wiley & Sons, Ltd.

Floquet Eigenvectors Theory of Pulsed Bias Phase and Quadrature Harmonic Oscillators

“…Floquet eigenvectors approach is indeed acknowledged as a correct analytical methodology for the description of noise perturbations [4].…”

“…the eigenvalues for the pulsed bias phase and quadrature oscillator are reported in both calculated (continuous traces) and simulated (dotted traces) cases as a function of resonators quality factor. It can be observed a very good match between calculated and simulated 2,3 0.66 12 e A  s  (error always <1%) whereas calculated 4  is underestimated with respect to the simulated one (error <5% for ). Underestimation is due to Q  10 0   assumption which leads to higher error in determination of expression (32).…”

“…This property is independent from resonators losses. The oscillation period can indeed be written as 1 4 2…”

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

“…Although electronic simulators offer results in good agreement with effective measurements, they cannot be used to infer general properties of the underlying system. Despite the great amount of work presented even recently in literature [3,4], since the pioneeristic work of Kaertner [5] and the latter efforts accomplished by Hajimiri-Lee [6], Demir [7,8] and Buonomo [9] not many significant theoretical contributions have really extended the capability to develop innovative techniques in the design of oscillators systems.…”

Purely Imaginary Eigenvalues (PIE) in Extended Circuits with Mem-Devices

“…Qualitative results are related e.g. to stability properties of equilibria and operating points [3,4,5,6,7], oscillations [8,9,10,11], bifurcations [12,13,14] or chaotic phenomena [3,15,16,17,18,19,20,21,22]. These references are just a sample of the huge literature addressing qualitative aspects in electrical and electronic engineering (cf.…”