Nonlinear oscillations in single transistor circuits

In this thesis, several basic modern communication circuits are analyzed using nonlinear analysis and bifurcation theory. Each circuit is characterized by a single bipolar transistor described mathematically by the Ebers-Moll model. The analyses are divided into three parts. In the first, a tuned collector oscillator is analyzed to resolve a vertical secondary Hopf bifurcation and the associated loss of stable circuit oscillation. Introducing an imperfection in the current source bends the secondary solution branch subcritically. However, using a simple RC circuit to bias the transistor bends the bifurcation supercritically, thus predicting a periodically modulated amplitude of oscillation, or squegging. In the second part, collector saturation effects are included in the transistor's mathematical model as a term that becomes large rapidly as a voltage passes a saturation value. This term is similar to the reaction term of large activation energy combustion theory. A model problem based on the van der Pol oscillator is used to demonstrate two methods of analysis. The first is a phase plane method using boundary layer theory and matched asymptotic expansions, while the second is a dominant balance method. The methods are supported with a numerical experiment. Three circuits are analyzed to reveal three different amplitude limiting effects. For a Colpitts oscillator, the amplitude response is clipped at the saturation voltage. For a simplified tuned collector oscillator, saturation effects are required to set up stable oscillations. And for the tuned collector of the first part, saturation effects bend the secondary solution branch supercritically again establishing squegging. In the third part, we analyze two frequency converter circuits used in radio reception. For a tuned collector converter, the circuit analysis decouples to yield a forced oscillator which drives a linear filter. The analysis of the forced oscillator reveals interesting solution response structure, including limit points and isola birth. The circuit for a Colpitts converter does not decouple as neatly, but the associated forced oscillator behaves identically to the tuned collector oscillator with a parameter restriction. The results for the forced oscillators are interpreted for their function in converters.