Root Locus for 2 Real Poles and 1 Real Zero

Cuthbert Nyack

The applet below shows the root locus for a system with 2 open loop poles and one open loop zero. There are 2 loci and one asymptote at -180°.

This configuration occurs in PI control of a first order system and PD control of a second order system.

for (10.0, 1.0, 1.0, 25.0, 1.0) the zero is far to the left and the step response with the zero is only slightly different from the step response without the zero. The CL poles are at -1.2 ± j3.09.

for (10.0, 1.0, 1.0, 3.75, 1.0) the zero is closer to the imaginary axis and the step response with the zero is significantly different from the step response without the zero. The CL poles are at -2.33 ± j2.36.

for (5.0, 0.1, 0.1, 1.7, 1.0) the poles are close to the imaginary axis and the step response without the zero is oscillatory. With the zero at -1.7, the CL poles are at -1.57 ± j1.59.

Gif image below show how applet should appear.

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