Understanding Relaxation Oscillator Circuits Using Fast-Slow System Representations

Abstract

We demonstrate how to utilize a fundamental principle of nonlinear dynamics, namely Liénard-type representations of ordinary differential equations, also referred to as fast-slow systems, to describe and understand relaxation oscillations in electronic circuits. Relaxation oscillations are characterized by periods of slow signal changes followed by fast, sudden transitions. They are generated either intentionally by means of usually simple circuits or occur often unintentionally where they would not have been expected, such as in circuits where there is only one dominant energy storage device. The second energy storage required to promote oscillatory solutions of the governing equations can also be provided by spurious elements or mechanisms. Conditions distinguishing harmonic from (anharmonic) relaxation oscillations are discussed by considering the underlying eigenvalues of the system. Subsequently, we shown how to intuitively understand the relaxation oscillations through analyses in the phase diagram based on the fast-slow system representation of the nonlinear differential equation. Practical examples of oscillators, including RC and LR op-amp circuits and the so-called "Joule thief" circuit, are discussed to illustrate the principle. The method's applicability is not limited to electrical circuits but extends to a variety of disciplines, such as chemistry, biology, geology, meteorology, and the social sciences.