Bifurcation in classical bipolar transistor oscillator circuits

The results of a systematic asymptotic analysis of two classical oscillator circuits are presented in this paper. Both circuits are composed of passive components and a single bipolar transistor. They are each described by a third-order system of nonlinear, ordinary differential equations that is singularly perturbed in the limit $\varepsilon \to 0$. Here $\varepsilon $ is the inverse of the circuit's “Q.” The two-timing method is used to construct the asymptotic behavior of the solutions as $\varepsilon \to 0$. These solutions are shown to depend upon amplitudes that evolve, on a slow time scale, according to amplitude equations. These amplitude equations are analyzed and the results are physically interpreted.