Almudena Suárez, PhD, is a Full Professor at the University of Cantabria, Spain, and a member of its Communications Engineering Department since 1993. She coauthored the book Stability Analysis of Nonlinear Microwave Circuits and contributed two articles to the Encyclopedia of RF and Microwave Engineering (Wiley). Professor Suárez has published dozens of papers in international journals and has been the leading researcher in several R&D projects. Her areas of interest include the nonlinear design of microwave circuits and, especially, stability and phase-noise analysis. She is a Distinguished Microwave Lecturer of IEEE.

Analysis and Design of Autonomous Microwave Circuits 1. Edition

By Almudena Suarez

John Wiley & Sons

Copyright © 2009 John Wiley & Sons, Inc.
All right reserved.
ISBN: 978-0-470-05074-3

Chapter One

Oscillator Dynamics

1.1 INTRODUCTION

A well-designed free-running oscillator provides a periodic signal of constant amplitude and frequency [f.sub.o] from the energy delivered by direct-current (dc) sources. This has an immediate application for the realization of local oscillators used in the frequency-conversion stages of communication systems. In receivers, the modulated signal at radio-frequency (RF) [f.sub.RF] is mixed with the output of a local oscillator at [f.sub.o], selecting the intermodulation product that corresponds to the frequency difference [f.sub.IF] = [f.sub.RF] - [f.sub.o], This allows down-conversion of the carrier frequency from [f.sub.RF] to [f.sub.IF]. An analogous procedure is followed in transmitters. The intermediate frequency [f.sub.IF] is mixed with the output of the local oscillator, selecting the intermodulation product [f.sub.RF] = [f.sub.IF] + [f.sub.o]. This allows up-conversion of the carrier frequency. The free-running oscillator is usually inserted into a phase-locked loop for this application.

A single oscillator having dc sources only is said to operate in free-running mode. However, other forms of behavior are possible. In injection-locked operation, the oscillation is synchronized with an independent periodic source, which means that the oscillation frequency, influenced by the input source, becomes equal to the input frequency [f.sub.o] = [f.sub.in], with a constant phase shift between the oscillation and the input signal. The injection-locked mode is used for phase noise reduction, frequency division, or phase shifting. In coupled operation, several oscillators are interconnected by means of linear coupling networks and oscillate in a synchronous manner. Coupled-oscillator systems can be used for power combination or beam steering. In this chapter only the free-running mode of an oscillator circuit is considered. Familiarity with the behavior and properties of free-running oscillation is essential for an understanding of any other form of operation (e.g., injection-locked, coupled) treated in subsequent chapters.

Free-running oscillators have essential differences from other RF circuits, such as amplifiers, mixers, and frequency multipliers. The operation frequency or frequencies (in the case of a mixer) of these circuits are determined by the input sources. In contrast, the fundamental frequency of an oscillator is self-generated or autonomous and depends on the values of the circuit elements. Thus, the circuit must be designed accurately to obtain the value desired for the oscillation frequency [f.sub.o]. Due to the absence of time-varying sources, any free-running oscillator can be solved for a mathematical dc solution. The oscillation starts up from any small perturbation of this dc solution and must grow from noise level to a steady-state oscillatory solution with constant amplitude and period. As will be shown, the self-sustained oscillation is only possible in nonlinear, nonconservative systems. Stability concepts are also essential to the understanding of the oscillator behavior. The oscillation startup and the physical observation of the periodic solution are explained from the different stability properties of dc and the steady-state oscillation. Because of the absence of an input periodic source establishing a time reference, arbitrary translations of the periodic waveform along the time axis give other solutions. There is an "irrelevance" with respect to time translations, or in the frequency domain, with respect to the phase origin. Thus, any phase-shifted solution constitutes a valid solution of the oscillator circuit. The absence of a restoring mechanism in the phase value gives rise to the phase noise problem in oscillator circuits.

In this chapter we deal with the main aspects of oscillator behavior. Oscillators are studied in the time domain and in the frequency domain, using impedance-admittance descriptions, which are very helpful for oscillator design, and the describing function approach, which allows nonlinear analysis at the fundamental frequency only. This one-harmonic approach will set the conceptual basis for harmonic balance analysis, covered in detail in Chapter 5. We relate various analysis techniques and unify concepts and properties, derived in the literature from very different viewpoints. Chapter 1 provides a general background for Chapter 2, which is devoted to phase noise analysis; Chapter 3, devoted to global stability analysis; and Chapter 4, devoted to an analysis of injection-locked oscillators and frequency dividers. The chapter is organized as follows. Section 1.2 provides intuitive explanations for oscillation startup and for the mechanism of self-sustained oscillation. In Section 1.3 we present the frequency-domain formulation based on the use of impedance or admittance functions, covering steady-state analysis and the stability of dc and periodic solutions. In Section 1.4 we extend the previous formulation to multiple harmonic components, for conceptual purposes, as this will be necessary for accurate stability analysis of oscillator circuit without limiting assumptions. In Section 1.5 we deal with oscillator circuits from the viewpoint of nonlinear dynamics, with the circuit described by a system of nonlinear differential equations. The main types of steady-state solutions and their properties are presented. In Section 1.6 we introduce formal mathematical procedures for the stability analysis of dc and periodic regimes and provide the necessary background for global stability analysis (i.e., versus variation in a circuit parameter), which is covered in Chapter 3. Finally, in Section 1.7 we emphasize the irrelevance of the oscillator solution versus time translations and show examples of phase shift response versus impulse perturbations. We establish the necessary background for Chapter 2, dealing with stochastic characterization of the spectrum of a noisy oscillator. Two different circuits are considered in this chapter: a parallel resonance oscillator with a two-terminal active element, and a FET-based oscillator at [f.sub.o] = 4.36 GHz. The simplicity of the first circuit makes possible the derivation of meaningful analytical expressions. Comparison with a FET-based oscillator clarifies our understanding of deviations from ideal behavior in practical circuits.

1.2 OPERATIONAL PRINCIPLE OF FREE-RUNNING OSCILLATORS

An ideal circuit given by the parallel connection of an inductor L and a capacitor C, without resistance, will under any initial condition exhibit oscillation at the frequency [[omega].sub.o] = [square root of LC], at which the average energies stored in the magnetic and electric fields are equal, so the sum of the inductor and capacitor susceptances is equal to zero. The total energy in the circuit remains constant during the entire oscillation period, so it is a conservative system. When the electrical energy stored in the capacitor is maximal, the magnetic energy stored in the inductor is zero, and vice versa. The energy displacement from one element to another gives rise to the oscillation observed in the node voltage and branch currents. By Kirchhoff's laws, the sum of the inductor plus capacitor current must be equal to zero, [i.sub.C] + [i.sub.L] = 0, which after some simple manipulations provides the linear...