Applet below show the root locus for a system with a 3rd order polynomial in the denominator of the transfer function. If a pole exists for a given si then the magnitude and angle condition are satisfied (cyan and light magenta lines crossing on the real axis at the real coordinate of the pole).

For parameters (0.0, 3.0, 2.75, 0.75, 0.0), open loop poles are at -0.5, -1.0, -1.5 for K = 0.

(1.0, 3.0, 2.75, 0.75, 0.8 and 0.0), closed loop poles are at -2.07, -0.47 ± j0.8 for K = 1.0.

(7.6, 3.0, 2.75, 0.75, 0.8 and 0.0), closed loop poles are at -3.04, -0.0± j1.66 and system is oscillatory for K = 7.6.

System is unstable for K > 7.6.

For parameters (0.0, 2.5, 3.0, 1.0, 0.0 and 1.0), open loop poles are at -0.5, -1.0 ± j1.0 for K = 0.

(6.6, 2.5, 3.0, 1.0, 1.73 and 0.0), closed loop poles are at -2.5, -0.0± j1.73 and system is oscillatory for K = 6.6.

System is unstable for K > 6.6.

Gif image showing how applet should appear is shown below.

COPYRIGHT © 2006 Cuthbert Nyack.