# Root Locus for 3 Real Poles and 1 Real Zero

Cuthbert Nyack
Applet shows the root locus for 3 real poles and 1 real zero. The presence of the zero means that one of the locus ends on the zero and there are now 2 asymptotes at ±90°. The asympotes appear to intersect the real axis at point p = (sum of poles - sum of zeros)/(number of poles - number of zeros).

Parameters (0.0, 1.0, 0.75, 0.25, 10.0, 1.0) show that when the zero is far to the left, then the locus around the origin appears similar to that of 3 poles with asymptotes at ±90° appearing to intersect the real axis at 4.0. The locus crosses the imaginary axis and the system is unstable for K > 2.8.

Parameters (0.0, 1.0, 0.75, 0.25, 2.0, 1.0) show that the asymptotes now correspond to the imaginary axis and the system is always stable.

Parameters (0.0, 1.0, 0.75, 0.25, 1.0, 1.0) show that the zero cancels the pole at -1 and the locus now has the characteristics of a 2 pole system.

Parameters (8.0, 1.0, 0.75, 0.25, 1.5, 1.0) show that the step response has transient oscillations at w = ~2.4 and the frequency response is peaked at that frequency. Increasing the gain increases the transient oscillation frequency but it does not go away.

Root locus for different pole zero locations are illustrated by the following paramaters:-
(0.0, 3.0, 2.0, 1.0, 3.5, 1.0) z-p-p-p
(0.0, 3.0, 2.0, 1.0, 2.5, 1.0) p-z-p-p
(0.0, 3.0, 2.0, 1.0, 1.5, 1.0) p-p-z-p
(0.0, 3.0, 2.0, 1.0, 0.5, 1.0) p-p-p-z

Gif image below show how applet should appear.