NETWORKED SYSTEM of CRYSTAL OSCILLATORS a Thesis

NETWORKED SYSTEM OF CRYSTAL OSCILLATORS A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Applied Mathematics with a Concentration in Dynamical Systems by Steven Isaac Reeves Spring 2016 iii Copyright c 2016 by Steven Isaac Reeves iv In solving a problem of this sort, the grand thing is to be able to reason backward. – Sherlock Holmes v ABSTRACT OF THE THESIS NETWORKED SYSTEM OF CRYSTAL OSCILLATORS by Steven Isaac Reeves Master of Science in Applied Mathematics with a Concentration in Dynamical Systems San Diego State University, 2016 The dynamics of a system of coupled crystal oscillators (CCOST) is examined with the aim of developing a stable precision timing device. Symmetry is used to establish the existence and stability of collective patterns of oscillations in the CCOST device. We investigate N identical crystal oscillators, where each is described by a two-mode nonlinear oscillator circuit that exhibits SO(2) × SO(2)-symmetry. The coupling is assumed to be identical, and two different topologies, unidirectional and bidirectional, are considered. The unidirectional topology leads to a network with SO(2) × SO(2) × ZN -symmetry. On the other hand, the bidirectional topology yields a network with SO(2) × SO(2) × DN -symmetry. The possible patterns of oscillation are classified using these symmetries and their respective isotropy subgroups. The effects of noise on the unidirectional CCOST device will be investigated with respect to phase error reduction. Phase error reduction of certain patterns will be tested against an uncoupled-averaged control group. This work will instruct on the design rules for the proposed precision timing device. vi TABLE OF CONTENTS PAGE ABSTRACT ....................................................................................v LIST OF TABLES.............................................................................. vii LIST OF FIGURES ............................................................................ viii ACKNOWLEDGMENTS ..................................................................... xi CHAPTER 1 Introduction ............................................................................1 2 Crystal Oscillators .....................................................................4 2.1 Two-Mode Oscillator Model ...................................................5 2.2 Averaging ........................................................................6 2.3 Stability ..........................................................................8 3 Coupled Crystal Oscillator System.................................................... 12 3.1 Averaged Equations ............................................................. 12 3.2 Symmetries of Collective Patterns ............................................. 16 3.2.1 The unidirectional case .................................................... 17 3.2.2 The bidirectional case ..................................................... 21 3.3 Isotypic Decomposition......................................................... 28 3.4 Spectrum of Eigenvalues and Linear Stability ................................. 31 4 Numerical Continuation ............................................................... 38 4.1 Unidirectionally Coupled Ring ................................................. 38 4.2 Bidirectionally Coupled Ring .................................................. 44 5 Phase Error ............................................................................. 48 5.1 Uncoupled Control Group ...................................................... 50 5.2 Unidirectional Coupling ........................................................ 51 6 Discussion and Future Work........................................................... 54 BIBLIOGRAPHY .............................................................................. 57 APPENDICES A Averaging............................................................................... 60 vii LIST OF TABLES PAGE 2.1 Classification of solutions of Eq. (2.6) based on isotropy subgroups. ...............9 3.1 One-dimensional irreducible representations and spatial subgroups of the bifurcating periodic solutions. ..................................................... 22 3.2 Isotropy subgroups of periodic solutions bifurcating from DN Hopf bifurcation at a Γ-simple subspace consisting of the sum of isomorphic two-dimensional absolutely irreducible representations of DN ...................... 23 3.3 Spatial subgroups of the rotating waves bifurcating simultaneously from the symmetric Hopf bifurcation point. ................................................ 24 0 3.4 Possible spatial subgroups K depending on which element of K fixes the kernel (dg)(0,λ)...................................................................... 27 viii LIST OF FIGURES PAGE 2.1 (Left) Schematic of a quartz crystal. (Right) A quartz crystal behaves like circuit composed of an inductor, capacitor and resistor, so it oscillates with a precise resonant frequency when it is subjected to an electric field. .......................4 2.2 Two-mode crystal oscillator circuit. A second set of spurious RLC components (R2;L2;C2) are introduced by parasitic elements. [37] [33] ..............................5 2.3 Time series solutions of a single two-mode crystal oscillator model (2.4). (Top left) Two parameter bifurcation diagram indicating the regions of existence of the time-series solutions as a function of parameters R1=a and R2=a. (Top right) Unstable Mixed Mode (i1(t); i2(t)), (bottom left) Stable Mode 1 (i1(t); 0) , (bottom right) Stable Mode 2 (0; i2(t)). Parameter values are: −4 −4 R1 = 30:9Ω, R2 = 181:1Ω, L1 = 5:2 × 10 H, L2 = 2:6 × 10 H, −13 −14 8 C1 = 1:0 × 10 F , C2 = 2:5 × 10 F , a = 939, b = 3 × 10 . ....................... 11 3.1 CCOST concept with unidirectionally coupled crystal oscillators. ........................ 12 3.2 Rings of N = 16 oscillator units representing a 2π periodic so- 8 lution with spatial subgroups K = D2(κ, (π; γ )) (left) and K = 4 D2(κ, (π; γ )) (right) corresponding respectively to the cases m = 1 and m = 2. The various colors represent different waveforms and the numeric labels the relative phases between cells of same color, with 02 meaning oscillation at twice the frequency of the other units. ....................... 26 3.3 Rings of N = 16 units representing a 2π-periodic solution with spatial 0 8 0 4 subgroups K = Z2(π; γ ) (left) and K = Z2(π; κγ ) (right) corresponding to the cases m = 1 and m = 2 respectively. The various colors represent different waveforms and the numeric labels the relative phases between cells of same color. Phase shifts between different colors are uncorrelated and 02 means oscillation at twice the frequency of the other units........................................................................... 29 4.1 Bifurcation diagrams for a ring of N = 3 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z3-symmetry, i.e., cyclic permutations of three crystal oscillators. Parameter values are: R1 = 30:9Ω, −4 −4 −13 R2 = 181:1Ω, L1 = 5:2 × 10 H, L2 = 2:6 × 10 H, C1 = 1:0 × 10 F , −14 C2 = 2:5 × 10 F , a = 939, b = 3E08................................................. 39 4.2 Bifurcation diagrams for a ring of N = 4 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z4-symmetry, i.e., cyclic permutations of four crystal oscillators. Parameter values: same as in Fig. 4.1 ............ 41 ix 4.3 Bifurcation diagrams for a ring of N = 5 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z5-symmetry, i.e., cyclic permutations of five crystal oscillators. Parameter values: same as in Fig. 4.1 ............ 42 4.4 Bifurcation diagrams for a ring of N = 6 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z6-symmetry, i.e., cyclic permutations of six crystals. Parameter values: same as in Fig. 4.1........................ 44 4.5 Bifurcation diagrams for a ring of N crystal oscillators coupled biidirectionally in a ring configuration. The diagrams depict the existence and stability of various branches of collective oscillations that appear, mainly via Hopf bifur- cations, as a function of coupling strength λ. The ring exhibits DN -symmetry, i.e., cyclic permutations of three crystal oscillators. Parameter values are: −4 −4 R1 = 30:9Ω, R2 = 181:1Ω, L1 = 5:2 × 10 H, L2 = 2:6 × 10 H, −13 −14 8 C1 = 1:0 × 10 F , C2 = 2:5 × 10 F , a = 939, b = 3 × 10 . ....................... 45 4.6 Two-parameter bifurcation diagrams for a ring of N crystal oscillators coupled biidirectionally in a ring configuration. Parameter values are: R1 = 30:9Ω, −4 −4 −13 R2 = 181:1Ω, L1 = 5:2 × 10 H, L2 = 2:6 × 10 H, C1 = 1:0 × 10 F , −14 8 C2 = 2:5 × 10 F , a = 939, b = 3 × 10 . ............................................. 47 1 5.1 This figure displays 22MHz rotating wave solution, RW1 , of the unidirectional CCOST model, with 3 nodes and coupling strength λ = 0:99. Above, the −7 current, Xi, is plotted over a period of 1:0 × 10 seconds. Below, one noise function is displayed over the same duration. ............................................. 49 5.2 The uncoupled-averaged (λ = 0) phase error as a function of array size. The top section displays the average,p maximum, and minimum phase errors of 50 samples for each N.A 1= N scaling is observed. Below, the range of phase error values are displayed as a

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