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Methods for Stability and Noise Analysis of Coupled Oscillating Systems Downloaded from orbit.dtu.dk on: Dec 17, 2017 Methods for Stability and Noise Analysis of Coupled Oscillating Systems Djurhuus, Torsten Publication date: 2008 Document Version Early version, also known as pre-print Link back to DTU Orbit Citation (APA): Djurhuus, T. (2008). Methods for Stability and Noise Analysis of Coupled Oscillating Systems. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Methods for Stability and Noise Analysis of Coupled Oscillating Systems. Torsten Djurhuus September 28, 2007 Contents Introduction 6 1 Basic Theory 13 1.1EquivalenceTheory............................... 13 1.1.1 Single Oscillators - The Andronov-Hopf Normal-Form ....... 14 1.1.2 Coupled Oscillators - The Synchronized State . .......... 23 1.2AveragingTheory................................ 28 1.3 Symmetry Considerations ........................... 31 2 Single Oscillators Perturbed by White Noise - Inhomogeneous Phase Diffusion 36 2.1 A Linear Response Theory for Noise Forced Limit Cycle Solutions .... 37 2.1.1 Deriving the Oscillator Phase Stochastic Differential Equation . 38 2.1.2 Deriving the Oscillator Phase Asymptotic Statistics - the Fokker- PlanckEquation............................ 43 2.1.3 Calculating the Oscillator Spectrum ................. 46 2.1.4 An Alternative Model of the Asymptotic Phase Statistics of a Free- Running Oscillator. ........................... 48 2.2 Review of 3 Popular Oscillator Phase Noise Methodologies ......... 50 2.2.1 Demir’sPhaseMacroModel...................... 52 2.2.2 Hajimiri’s Impulse Sensitivity Function ................ 54 2.2.3 Kurokawa’s Model - Harmonic Oscillators .............. 54 3 A Phase Macro Model for Coupled Oscillator Systems 57 3.1TheGeneralModelFormulation....................... 59 3.2 A Phase Macro Model for the Sub-Harmonic Injection Locked Oscillator (S-ILO)..................................... 63 3.2.1 Deriving the Torus Projection Operators for the S-ILO ....... 64 3.2.2 TheStochasticDifferentialEquations................ 69 3.2.3 CharacterizingtheSelf-ReferencedS-OSCPhase.......... 71 3.2.4 The Spectrum of a Noise Perturbed Injection locked oscillator . 73 3.2.5 VerificationoftheDevelopedModel................. 76 4 n Unilaterally Ring-Coupled Harmonic Oscillators Perturbed by White Noise 80 4.1 Modes of n Unidirectional Ring Coupled Identical Harmonic Oscillators . 82 4.1.1 Ashwins’s Symmetry Approach .................... 82 4.1.2 Rogge’s Method ............................ 85 4.1.3 Discussion................................ 88 4.2PracticalApplicationsoftheUnidirectionalRing.............. 89 1 CONTENTS CONTENTS 4.3AQuasi-SinusoidalModeloftheUnilateralRing.............. 90 4.4 Linear Start-Up and Stability Analysis .................... 95 4.4.1 Linear Stability of In-Phase Mode for the Unidirectional Ring with β =0 .................................. 99 4.4.2 n=2, β = π : The Cross Coupled Quadrature Oscillator ......100 4.5 Linear Response Noise Analysis - Coupling Induced AM to PM Noise Con- version......................................103 4.5.1 Deriving the Dynamics of the Diagonal Flow . ..........105 4.5.2 Calculating D¯φφ (Method #1) : Rotating the Averaged Phase Equation................................107 4.5.3 Calculating D¯φφ (Method #2) : A PLTV Impulse Response Char- acterization ...............................108 4.5.4 Calculating D¯φφ (Method #3) : the Stochastic Integration Approach110 4.6 A new Definition of Oscillator Q .......................113 4.7 The Cross-Coupled Harmonic Quadrature Oscillator . ..........113 4.7.1 Deriving the CCQO Amplitude/Phase Equations ..........114 4.7.2 Asymmetry Considerations ......................116 4.7.3 LinearResponseNoiseAnalysis....................120 Conclusion 126 A The Noise Appendix : Narrow-band Noise / Stochastic Integration / The Fokker-Planck Equation 130 A.1NarowbandNoise................................131 A.1.1 The Noise Admittance Yn = Gn + jBn ................134 A.1.2APulseTrainNoiseModel......................135 A.2StochasticIntegration.............................137 A.3TheFokker-PlanckEquation.........................140 B Deriving the Averaged Stochastic Differential Equations for a General Class of Second Order Oscillators 142 B.1 A van der Pol Oscillator with a Varactor ...................143 B.1.1APhasorApproach..........................146 C Various Derivations 148 C.1Section1.1.1:Normal-FormCalculations..................148 C.2 Calculating the Eigenvalues of the Matrix J11 onpage98.........150 C.3 Calculating the Eigenvalues of the Matrix J22 onpage99.........151 C.4CalculationsUsedintheProofofTheorem3.3onpage68.........153 C.5TheILOMonodromyMatrix.........................156 C.6 Computing the Spectrum of the Noise Perturbed ILO . ..........158 D Floquet Theory 159 D.1 Deriving the Monodromy Matrix Φ(2π, 0) ..................163 2 Abstract In this thesis a study of analytical and numerical models of coupled oscillating systems, perturbed by delta-correlated noise sources, is undertaken. These models are important for the attainment of a qualitative understanding of the complex dynamics seen in various physical, biological, electronic systems and for the derivation of fast and computationally efficient CAD routines. The text concentrates on developing models for coupled electronic oscillators. These circuit blocks find use in RF/microwave and optical communication systems as coherent multi-phase signal generators, power combiners and phase-noise fil- ters; to name but a few of the possible applications areas. Taking outset in the established single-oscillator phase-macro model, a novel numer- ical algorithm for the automated phase-noise characterization of coupled oscillators, per- turbed by noise, is developed. The algorithm, which is based on stochastic integration and Floquet theory and is independent of circuit topology and parameters, proceeds by deriving the invariant manifold projection operators. This formulation is easily integrated into commercial CAD environments, such as SPICE™ and SPECTRE™. The algorithm improves the computational efficiency, compared to brute-force Monte-Carlo techniques, by several orders of magnitude. Unilateral ring-coupled oscillators have proven a reliable and power efficient way to create coherent multi-phase signal generators in the RF/microwave frequency range. A complete and self-contained study of this complex multi-mode system is undertaken. The developed model explains the existence of a so-called dominant mode, ensuring a consistent signal phase pattern following start-up. A linear response model is derived to investigate linear stability and noise properties. It is shown that a linear coupling transconductor will cancel the coupling induced noise contribution in the single-side band phase-noise spectrum. This phenomena was not discussed in any of the previous pub- lications considering this circuit. The model gives a general insight into the qualitative properties of unilateral ring-coupled oscillators, perturbed by white noise. 3 Resumé I denne afhandling vil der blive udført et studie af mulige analytiske og numeriske mod- eller af koblede oscillerende systemer. Disse modeller er vigtige med henblik på at opnå et kvalitativt overblik, mht. den komplekse dynamik der observeres i diverse fy- siske, biologiske, elektroniske systemer og for udviklingen af hurtige, beregnings-effektive simulerings-routiner. Teksten koncentrerer sig om modellering af elektroniske koblede systemer. Disse kredsløbs-blokke finder anvendelse i RF/mikrobølge og optiske kommu- nikations systemer som multi-fase signal-kilder, power combiners og fase-støjs filtre; for bare at nævne nogle få eksempler. Med udgangspunkt i den allerede eksisterende enkelt-oscillator phase-macro model vil der blive udviklet en ny algoritme med henblik på at finde en automatiseret, topologi og parameter uafhængig fase-støjs beskrivelse af koblede oscillatorer. Denne algoritme, som bygger på stokastisk integration og Floquet teori, er baseret på udledningen af ortogonale projektions operatorer, der afbilder støj-responsen ned på den invariante manifold. På denne form er algoritmen meget velegnet til at blive integreret i kommercielle kredsløbs simulations-platforme, så som SPICE™ og SPECTRE™. Algoritmen er flere størrelsesor- dner hurtigere ift. en simple Monte-Carlo fremgangsmåde. Unilateral ring-kobling af oscillatorer har vist sig at være en pålidlig og effektiv måde at frembringe multi-fase signaler i RF/mikrobølge frekvens-området. Et komplet studie af dette komplekse multi-mode system vil blive udført. Den udviklede model forklarer eksistensen af en såkaldt dominant mode, der sikre at et konsistent fase-mønster altid følger efter opstart. En linear respons model er udviklet med henblik på at undersøge linær stabillitet og støj egenskaber. Det vil blive vist, hvordan linær kobling vil resultere i, at
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