Root Locus for 3 Real Poles and 2 Real Zeros

Cuthbert Nyack

Applet below show the angle condition for 3 real poles and 2 real zeros. To find the root locus for a system with poles at -0.5, -1.0, -1.75 and zeros at -2.0 and -2.15 set the following parameters:
(0.0, 1.75, 1.0, 0.5, 2.15, 2.0). This shows 3 loci and 1 asymptote at -180°. One loci goes from the pole at -1.75 to the zero at -2.0. A second starts from the pole at -0.5 and moves left, while the third starts at -1 and moves right. When these 2 meet at ~-1.75 they break away from the real axis eventually meeting at the break-in point ~-3.85. From there one moves to the left towards ∞ and the second moves to the right ending at the zero at -2.15.

Changing K from 0 to ~4.5 gives a satisfactory step and frequency response because the poles move upwards(reduce risetime) and to the left(reduce transients).

It is still possible for this system to be unstable when the poles are close to the imaginary axis and the zeros are far to the left. eg (0.0, 0.25, 0.25, 0.25, 3.0, 3.0). System is stable for K < ~0.02 and K > ~ 0.68 and unstable in between.

Root locus for different pole zero locations are illustrated by the following paramaters:-
(0.0, 2.0, 1.0, 0.5, 2.5, 3.0) z-z-p-p-p
(0.0, 2.0, 1.0, 0.5, 1.5, 2.5) z-p-z-p-p
(0.0, 2.0, 1.2, 0.5, 0.8, 2.75) z-p-p-z-p
(0.0, 2.0, 1.2, 0.5, 0.1, 2.75) z-p-p-p-z
(0.0, 3.0, 1.2, 0.25, 2.0, 2.5) p-z-z-p-p
(0.0, 3.0, 2.5, 0.5, 2.0, 1.5) p-p-z-z-p
(0.0, 3.0, 2.5, 1.5, 2.0, 0.5) p-p-z-p-z
(0.0, 3.5, 3.0, 2.5, 1.0, 0.5) p-p-p-z-z

Gif image showing how applet should appear is shown below.

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