rtitioning the overall oscillator circuit into a core system and a set of perturbation components. The core system is solved by a multitude of different steady-state solutions. Each solution is called a state of the system. The perturbation components disturb the core, driving it from one state to another. This state-changing behavior captures the essence of the oscillator's dynamic behavior. Detailed observation shows that the changes in the state of an oscillator are dominated by a slow-varying component. Typically, this component is the one of main interest. Using averaging techniques, it is possible to identify the part of the state-changing equations that generates this slow-varying behavior. These averaged equations can be solved very efficiently. They can be used as a starting point for oscillator phase-noise analysis and for the generation of compact behavioral models. In what follows, section 5.1 introduces the basic concepts underlying the oscillator analysis elaborated in this chapter. Hereby, we use the earth's motion around the sun as an example. The mathematics underlying these concepts is addressed in sections 5.3 5.1 THE STORY BEHIND THE MATH to 5.6. The theory is applied to oscillator phase noise analysis in section 5.7 and to the behavioral modeling of harmonic oscillators in section 5.8. 5.1 The story behind the math Constructing of compact oscillator models requires a clear understanding of the mech- anisms that govern oscillator dynamics. Oscillators exhibit a mixture of fast- and slow- varying behavior. Rapid variations are caused by, for instance, energy that oscillates between a set of energy storing elements, e.g capacitors and inductors. Energy ex- changes with the surroundings cause slow-varying modulations. Understanding the interplay between fast- and slow-varying dynamics is key to understanding an oscilla- tor's behavior. In order to clarify the fundamental mechanisms that govern an oscillator's behavior, we consider one of the first oscillations ever studied: the motion of the eart