Watch s travel round the D-contour while its image L(s) is traced in step, the same colour for the same stretch in both planes. An arrow from −1 spins as the plot goes round; count its turns, apply Z = P − N, and check the answer against the closed-loop poles. Drag K and watch stability flip.
Loading the lab…
Open-loop transfer function L(s) = G(s)H(s)
Open-loop stable, so the closed loop is stable exactly when the plot stays clear of −1. Unstable for K > 60.
s-plane: the D-contour, clockwise round the right half plane
L-plane: where each s lands. The arrow from −1 counts the turns.
Step 1. As rises from to , climbs the imaginary axis and draws the polar plot, the curve you sketch in the exam. It starts on the real axis at and ends at the origin, arriving at angle because has 3 more poles than zeros.
Stable for: 0 < K < 60
Drag K while the animation runs and watch the plot swell or shrink past −1.