Phase margin revisited: phase-root locus, bode plots, and phase shifters

In learning undergraduate controls, one of the most abstract and confusing concepts is that of phase margin (PM). One frustration is the difficulty in conceptualizing what physical process could produce the pure phase shift referred to in the definition of PM. Another frustration occurs when simple second-order formulas given for PM do not work in practice. Finally, when resonances occur, there may be multiple gain-crossover frequencies, and which one to use to compute the PM may be unclear. This paper offers visualizations and explanations that students have found helpful in learning about PM-both its evaluation and its application. The phase-root locus (PRL) plot reveals the effect of adding phase in the same way that conventional gain root locus shows the effects of adding gain. The PRL ultimately leads to a definition of PM involving phase shifting that results in physical (real-coefficient) systems, unlike the usual abstract Nyquist plot rotation. This definition of PM suggests a simple, effective compensator design method for improving PM via phase shifters, a solution illustrated by numerical examples. These materials can either be presented in lectures or assigned as supplementary readings and may inspire student projects.