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Parametric excitation can occur when the value of a parameter of an oscillator is modulated at twice the natural frequency of the oscillator. The response grows exponentially and is only limited by a nonlinearity of the system, so large response amplitudes typically occur. However, there is no response unless the parametric drive amplitude is above a threshold value that is dictated by the damping. We investigate parametric excitation in three physical systems. The first involves an acoustic standing wave in a pipe that is driven by a piston at one end. An analysis shows that parametric excitation is not feasible in this system unless one uses a very large-excursion piston (for example, from an aircraft engine). The second system is an inductor-capacitor circuit which can undergo oscillations of the current. An analysis of capacitance modulation with a bank of alternate rotating and stationary parallel plates shows that parametric excitation would be very difficult to achieve. Finally, we describe the construction of a torsional oscillator whose length is modulated. Parametric excitation is successfully demonstrated in this system. A comparison of data to predictions of the standard theory of parametric excitation reveals significant deviations.

by Chiu, Shek Fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves [121]-[125]). / DEDICATION / ACKNOWLEDGE / ABSTRACT / Chapter Chapter 1 --- Introduction --- p.1-1 / Chapter Chapter 2 --- Background --- p.2-1 / Chapter 2.1 --- Basic Oscillator --- p.2-2 / Chapter 2.1.1 --- Introduction --- p.2-2 / Chapter 2.1.2 --- The basic feedback oscillator --- p.2-2 / Chapter 2.1.3 --- The basic negative resistance oscillator --- p.2-3 / Chapter 2.1.4 --- Implementation of an oscillator --- p.2-3 / Chapter 2.1.5 --- The phase noise of an oscillator --- p.2-4 / Chapter a) --- Lesson's model --- p.2-4 / Chapter 2.2 --- Basic Mixer --- p.2-6 / Chapter 2.2.1 --- Introduction --- p.2-6 / Chapter 2.2.2 --- Non-linear resistance mixer --- p.2-6 / Chapter 2.2.3 --- Y-parameter representation --- p.2-7 / Chapter 2.2.4 --- Figure of merit --- p.2-9 / Chapter 2.3 --- Negative Resistance Amplifier --- p.2-11 / Chapter 2.3.1 --- Introdutction --- p.2-12 / Chapter 2.3.2 --- Type of reflection amplifier --- p.2-12 / Chapter 2.3.3 --- The noise figure --- p.2-13 / Chapter 2.4 --- Fundamental Injection-locked Oscillator --- p.2-15 / Chapter 2.4.1 --- Introduction --- p.2-15 / Chapter 2.4.2 --- Injection-locked oscillator --- p.2-15 / Chapter 2.4.3. --- Locking range --- p.2-15 / Chapter 2.4.4 --- Noise behaviour --- p.2-16 / Chapter 2.4.5 --- Applications of ILO --- p.2-17 / Chapter 2.5 --- Quasi-static analysis --- p.2-18 / Chapter 2.5.1 --- Introduction --- p.2-18 / Chapter 2.5.2 --- free running oscillation --- p.2-18 / Chapter 2.5.3 --- Conditions for injection locking --- p.2-22 / Chapter a) --- Stability --- p.2-24 / Chapter 2.5.4 --- Conditions for Two signal injection --- p.2-25 / Chapter a) --- Stability --- p.2-26 / Chapter Chapter 3 --- Frequency conversion of Injection-locked oscillator --- p.3-1 / Chapter 3.1 --- Circuit Description --- p.3 -2 / Chapter 3.1.1 --- One port equivalent circuit --- p.3-5 / Chapter 3.1.2 --- Two port equivalent circuit --- p.3-6 / Chapter 3.2 --- Injection Control Resistance --- p.3-7 / Chapter 3.2.1 --- Introduction --- p.3-7 / Chapter 3.2.2 --- Measurement Setup --- p.3-8 / Chapter 3.2.3 --- Measurement and Experimental results --- p.3-9 / Chapter 3.2.4 --- Discussion --- p.3-11 / Chapter 3.2.5 --- Conclusion --- p.3-11 / Chapter 3.3 --- Q Multiplication --- p.3-12 / Chapter 3.3.1 --- Introduction --- p.3-12 / Chapter 3.3.1 --- Measurement setup of reflection gain/loss --- p.3-16 / Chapter a) --- Theory of measurement --- p.3-16 / Chapter 3.3.3 --- Measurement and Experiment results --- p.3-17 / Chapter 3.3.4 --- Discussion --- p.3-17 / Chapter 3.3.5 --- Conclusion --- p.3-19 / Chapter 3.4 --- Impedance Conversion --- p.3-20 / Chapter 3.4.1 --- Introduction --- p.3-20 / Chapter 3.4.2 --- Measurement and Experimental results --- p.3-22 / Chapter 3.4.3 --- Discussion --- p.3-26 / Chapter 3.4.5 --- Conclusion --- p.3-26 / Chapter 3.5 --- Negative Resistance amplification --- p.3-27 / Chapter 3.5.1 --- Introduction --- p.3-27 / Chapter a) --- Small signal response --- p.3-27 / Chapter 3.5.2 --- Measurement and Experimental Results --- p.3-31 / Chapter 3.5.3 --- Discussion --- p.3-32 / Chapter 3.5.4 --- Conclusion --- p.3-35 / Chapter 3.6 --- Frequency Conversion and Noise performance --- p.3-36 / Chapter 3.6.1 --- Frequency Conversion --- p.3-36 / Chapter 3.6.2 --- Noise performance --- p.3-37 / Chapter 3.6.3 --- Measurement setup --- p.3-41 / Chapter 3.6.4 --- Measurement and Experimental results --- p.3-43 / Chapter a) --- Results of the sensitivity measurement --- p.3-43 / Chapter b) --- Results of 3 dB operation bandwidth measurement --- p.3-44 / Chapter c) --- Results of the testing setup in figure 3.6.2 --- p.3-44 / Chapter 3.6.5 --- Discussion --- p.3-46 / Chapter 3.6.6 --- Conclusion --- p.3-48 / Chapter 3.7 --- Large Signal Response --- p.3-49 / Chapter 3.7.1 --- Introduction --- p.3-49 / Chapter 3.7.2 --- Measurement and Experimental results --- p.3-51 / Chapter a) --- The reflection characteristics of ILO at high RF signal level / Chapter a1) --- Gain bandwidth characteristics --- p.3-51 / Chapter a2) --- Gain compression characteristics --- p.3-52 / Chapter b) --- The reflection characteristics of ILO at high IF signal level / Chapter b1) --- Gain bandwidth characteristics --- p.3-54 / Chapter b2) --- Gain compression characteristics --- p.3-55 / Chapter c) --- The conversion properties of ILO / Chapter c1) --- Gain compression characteristics --- p.3-57 / Chapter 3.7.3 --- Discussion --- p.3-60 / Chapter 3.7.4 --- Conclusion --- p.3-61 / Chapter 3.8 --- Image Signal Response --- p.3-62 / Chapter 3.8.1 --- Introduction --- p.3-62 / Chapter 3.8.2 --- Measurement and Experimental results --- p.3-63 / Chapter 3.8.3 --- Discussion --- p.3-65 / Chapter 3.8.4 --- Conclusion --- p.3-66 / Chapter 3.9 --- Conclusion --- p.3-67 / Chapter Chapter 4 --- ILO Regenerative Mixer --- p.4-1 / Chapter 4.1 --- Introduction --- p.4-1 / Chapter 4.2 --- Block diagram representation --- p.4-1 / Chapter 4.3 --- Linear Regenerative Mixer Model --- p.4-2 / Chapter 4.3.1 --- Y-parameter representation --- p.4-2 / Chapter 4.3.2 --- Stability --- p.4-4 / Chapter 4.3.3 --- Linear circuit model --- p.4-5 / Chapter 4.4 --- Design Example and Circuit Description --- p.4-6 / Chapter 4.5 --- Measurement Results --- p.4-8 / Chapter 4.6 --- Conclusion --- p.4-11 / Chapter Chapter 5 --- Conclusion --- p.5-1 / REFERENCE --- p.R-1

by Wong, Kwok-wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 90-93). / DEDICATION / ACKNOWLEDGEMENTS / ABSTRACT / Chapter CHAPTER 1 --- INTRODUCTION --- p.1 / Chapter CHAPTER 2 --- BASIC OSCILLATOR DESIGN --- p.5 / Chapter CHAPTER 3 --- FUNDAMENTAL INJECTION LOCKING --- p.12 / Chapter 3.1 --- INTRODUCTION --- p.12 / Chapter 3.2 --- NONLINEAR OSCILLATOR MODELS --- p.13 / Chapter 3.3 --- TYPES OF INJECTION LOCKED OSCILLATOR --- p.24 / Chapter 3.4 --- INJECTION LOCKING CHARACTERISTICS --- p.26 / Chapter 3.5 --- CONCLUSION --- p.31 / Chapter CHAPTER 4 --- SUBHARMONIC INJECTION LOCKING --- p.32 / Chapter 4.1 --- INTRODUCTION --- p.32 / Chapter 4.2 --- SUBHARMONIC INJECTION LOCKING --- p.32 / Chapter 4.3 --- SUBHARMONIC INJECTION LOCKING CHARACTERISTICS --- p.36 / Chapter 4.4 --- CONCLUSION --- p.40 / Chapter CHAPTER 5 --- EXPERIMENTAL INVESTIGATIONS ON INJECTION LOCKING --- p.41 / Chapter 5.1 --- INTRODUCTION --- p.41 / Chapter 5.2 --- EXPERIMENTAL CHARACTERISTICS --- p.43 / Chapter 5.3 --- NON-INTEGRAL SUBHARMONIC LOCKING --- p.53 / Chapter 5.3.1 --- Nonlinear feedback model --- p.53 / Chapter 5.3.2 --- Circuit description --- p.55 / Chapter 5.3.3 --- Experimental results --- p.59 / Chapter 5.3.4 --- Summary --- p.64 / Chapter 5.4 --- SELECTIVE SUBHARMONIC LOCKING RANGE ENHANCEMENT --- p.65 / Chapter 5.4.1 --- Mulit-feedback nonlinear model --- p.65 / Chapter 5.4.2 --- Circuit description --- p.65 / Chapter 5.4.3 --- Experimental results --- p.69 / Chapter 5.4.4 --- Summary --- p.71 / Chapter 5.5 --- FEEDBACK TYPE INJECTION LOCKED OSCILLATOR --- p.72 / Chapter 5.5.1 --- Feedback type injection locked oscillator model with different injection points --- p.72 / Chapter 5.5.2 --- Circuit description --- p.73 / Chapter 5.5.3 --- Experimental results --- p.76 / Chapter 5.5.4 --- Summary --- p.76 / Chapter 5.6 --- PHASE TUNING BEYOUND 180 DEGREES BY INJECTION LOCKING --- p.79 / Chapter 5.6.1 --- Phase change by single injection locking --- p.79 / Chapter 5.6.2 --- Phase change by cascaded injection locking --- p.80 / Chapter 5.6.3 --- Experimental results --- p.85 / Chapter 5.6.4 --- Summary --- p.88 / Chapter 5.7 --- CONCLUSION --- p.88 / Chapter CHAPTER 6 --- CONCLUSION --- p.89 / REFERENCES --- p.90 / LIST OF ACCEPTED AND SUBMITTED / PUBLICATIONS DURING THE PERIOD OF STUDY

The limits of current micro-scale technology is approaching rapidly. As the technology is going toward nano-scale devices, physical phenomena involved are fundamentally different from micro-scale ones [1], [2]. Principles in classical physics are no longer powerful enough to explicate the phenomena involved in nano-scale devices. At this stage, quantum mechanic sheds some light on those topics which cannot be described by classical physics. The primary focus of this research work is the development of an analysis technique for understanding the behavior of strongly perturbed harmonic oscillators. Developing ``auxiliary'' boundary value problems we solve monomially perturbed harmonic oscillators. Thereby, we assume monomial terms of arbitrary degree and any finite coefficient desired. The corresponding eigenvalues and eigenvectors can be utilized to solve more complex anharmonic oscillators with non polynomial anharmonicity or numerically defined anharmonicity. A large number of numerical calculations demonstrate the robustness and feasibility of our technique. Particular attention has been paid to the details as have implemented the underlying formula. We have developed iterative expressions for the involved integrals and the introduced ``Universal Functions.'' The latter are applications and adaptations of a concept which was developed in 1990's to accelerate computations in the Boundary Element Method.

Microstrip ring circuits have been extensively studied in the past three decades. A magnetic-wall model has been commonly used to analyze these circuits. Unlike the conventional magnetic-wall model, a simple transmission-line model, unaffected by boundary conditions, is developed to calculate the frequency modes of ring resonators of any general shape such as annular, square, or meander ring resonators. The new model can be used to extract equivalent lumped element circuits and unloaded Qs for both closed- and open-loop ring resonators. Several new bandpass filter structures, such as enhanced coupling, slow-wave, asymmetric-fed with two transmission zeros, and orthogonal direct-fed, have been proposed. These new proposed filters provide advantages of compact size, low insertion loss, and high selectivity. Also, an analytical technique is used to analyze the performance of the filters. The measured results show good agreement with the simulated results. A compact elliptic-function lowpass filter using microstrip stepped impedance hairpin resonators has been developed. The prototype filters are synthesized from the equivalent circuit model using available element-value tables. The filters are evaluated by experiment and simulation with good agreement. This simple equivalent circuit model provides a useful method to design and understand this type of filters and other relative circuits.Finally, a tunable feedback ring resonator oscillator using a voltage controlled piezoelectric transducer is introduced. The new oscillator is constructed by a ring resonator using a pair of orthogonal feed lines as a feedback structure. The ring resonator with two orthogonal feed lines can suppress odd modes and operate at even modes. A voltage controlled piezoelectric transducer is used to vary the resonant frequency of the ring resonator. This tuned oscillator operating at high oscillation frequency can be used in many wireless and sensor systems.

Parametric excitation is a fundamental feature of dynamical systems arising across the applied sciences. In this thesis we study the structure of parametric res- onance and its in uence of the global nonlinear dynamics in a number of oscillating systems which arise in engineering contexts. The parametrically excited Helmholtz oscillator and the elliptically excited pen- dulum are two systems where the interaction of regular and parametric excitation are important for a complete understanding of the dynamics. We examine the resonance structure of the Helmholtz oscillator and use the Melnikov function to demonstrate the e ect that the parametric excitation has on the nonlinear dynam- ics. The estimates produced in this analysis are then compared to a numerical study of the engineering integrity. For the elliptically excited pendulum we discuss the quantitative e ects of introducing ellipticity to the pro le of excitation. We go on to examine the e ect of periodic time varying mass in the Helmholtz oscillator and demonstrate that the resonance structure exhibits the phenomenon of coexistence. The evolution of the systems engineering integrity is examined and compared to the purely parametrically excited case. Finally we examine a system incorporating two pendulums on a rigid rig modelled by two linear springs. The parametric resonance in this case is mapped using numerical Floquet theory and the structure of the linear resonance is shown to organise solution space for the nonlinear system.

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