# Root Locus for 4 Real Poles and 1 Real Zero

Cuthbert Nyack
The inclusion of a zero to the 4 real poles means that the asymptotic behaviour is equivalent to that of a 3rd order system with asymptotes at ±60° and -180°.

For parameters (0.0, 1.0, 1.0, 1.0, 1.0, 1.5) show that the addition of a zero at -1.5 increases the K for instability to 7.43.
When the transfer function is expressed this way then the value of K for instability reduces for z < 1.

The properties of the locus for different pole zero locations are illustrated by the following parameters.
(0.0, 3.0, 2.0, 1.0, 0.5, 3.54) z-p-p-p-p
(0.0, 3.0, 2.0, 1.0, 0.5, 2.46) p-z-p-p-p
(0.0, 3.0, 2.0, 1.0, 0.5, 1.5) p-p-z-p-p
(0.0, 3.0, 2.0, 1.0, 0.5, 0.72) p-p-p-z-p
(0.0, 3.0, 2.0, 1.0, 0.5, 0.18) p-p-p-p-z

For parameters (0.0, 1.0, 1.0, 1.0, 1.0, 1.5) in the following applet the system is unstable for K > 4.95(ie 7.43/1.5) because of the way the transfer function is expressed.

Gif image illustrating how applet should appear is shown below. 