Understanding Relaxation Oscillator Circuits Using Fast-Slow System Representations

We demonstrate the utilization of the fundamental principle of nonlinear dynamics, namely, the 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 often occur unintentionally where they would not have been expected, such as in circuits with 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. The conditions that distinguish harmonic from (anharmonic) relaxation oscillations are discussed by considering the underlying eigenvalues of the system. Subsequently, we show how to intuitively understand relaxation oscillations through analyses of 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 this principle. The applicability of the method is not limited to electrical circuits, but extends to a variety of disciplines, such as chemistry, biology, geology, meteorology, and social sciences.