# Root Locus for 4th Order Polynomial

Cuthbert Nyack
Applet shows the root locus of a system with a 4th order characteristic equation. There are 4 asymptotes at ±45° and ±135°

Parameters (0.0, 5.0, 8.75, 6.25, 1.5, 0.0) show a system with open loop denominator s4 + 5.0s3 + 8.75s2 + 6.25s + 1.5.

Applet shows that this system has open loop poles(starting point of loci for K = 0) at -0.5, -1.0, -1.5, -2.0 and is stable for K < 7.96.

To find the closed loop poles for any given K eg K = 2.
1. Set K = 2.0. 1. Change si until the magnitude and angle conditions are satisfied( crossing of cyan and light magenta lines on the real axis). In this case si =0.75 and the crossing occurs at -2.17 and -0.33. The closed loop poles for K = 2 are then -0.33 ± j0.75 and -2.17 ± j0.75.

Parameters (0.0, 4.8, 8.75, 6.25, 1.5, 0.0) show a system with open loop denominator s4 + 4.8s3 + 8.75s2 + 6.25s + 1.5.

Applet shows that this system has open loop poles(starting point of loci for K = 0) at ~-0.55, ~-0.65, ~-1.75 ± ~j0.85 and is stable for K < 8.17.

To find the closed loop poles for any given K eg K = 4.
1. Set K = 4.0. 1. Change si until the magnitude and angle conditions are satisfied( crossing of cyan and light magenta lines on the real axis). In this case si =0.93, 1.15 and the crossing occurs at -0.17 and -2.17. The closed loop poles for K = 4 are then -0.17 ± j0.93 and -2.17 ± j1.15.

Parameters (0.0, 5.0, 8.0, 6.0, 2.0, 0.0) show a system with open loop denominator s4 + 5s3 + 8s2 + 6s + 2.

Applet shows that this system has open loop poles(starting point of loci for K = 0) at ~-1.0, ~-2.8, ~-0.55 ± ~j0.6 and is stable for K < 6.18.

To find the closed loop poles for any given K eg K = 1.
1. Set K = 1.0. 1. Change si until the magnitude and angle conditions are satisfied( crossing of cyan and light magenta lines on the real axis). In this case si =0.0, 0.76 and the crossing occurs at -2.72, -1.56 and -0.34. The closed loop poles for K = 1 are then -2.72, -1.56 and -0.34 ± j0.76.

Gif image illustrating how applet should appear is shown below.