Applying Nyquist’s Stability Analysis to Bode Plots With Wrapped Phase Behavior

This brief describes the stability conditions of Nyquist in terms of the data related to a Bode plot of the loop transmission function
$\mathbf {A\beta }(j\boldsymbol{\omega }\mathbf {)}$
. Specifically, it will be shown that the number of encirclements of
$\mathbf {A\beta }(j\boldsymbol{\omega }\mathbf {)}$
around the critical point–1+j0 in the complex plane
$N_{e}$
can be deduced directly from twice the directional sum of the number ±360°-phase jumps that appear in the wrapped phase portion of the Bode plot, together with the phase behavior at DC and infinity, provided the magnitude at each phase jump is greater than unity. An example will be provided to demonstrate the simplicity of the proposed theory.