Rules for Making Root Locus Plots

The table below summarizes how to sketch a root locus plot (K≥0).  This is also available as a Word Document or PDF.

You can also find a page that includes the rules for the Complementary Root Locus (K≤0).


 

The closed loop transfer function of the system shown is

So the characteristic equation is

As K changes, so do locations of closed loop poles (i.e., zeros of characteristic equation). The table below gives rules for sketching the location of these poles as K varies from 0 to infinity (K>0).
 

Rule Name Description
Definitions
  • The loop gain is KG(s)H(s) which can be rewritten as KN(s)/D(s).
  • N(s), the numerator, is an mth order polynomial; D(s) is nth order.
  • N(s) has zeros at zi (i=1..m);  D(s) has them at pi (i=1..n).
  • The difference between n and m is q, so q=n-m.
Symmetry The locus is symmetric about real axis (i.e., complex poles appear as conjugate pairs).
Number of Branches There are n branches of the locus, one for each pole of the loop gain.
Starting and Ending Points The locus starts (when K=0) at poles of the loop gain, and ends (when K∞ ) at the zeros. Note: there are q zeros of the loop gain as |s|∞ .
Locus on Real Axis The locus exists on real axis to the left of an odd number of poles and zeros.
Asymptotes as |s| If q>0 there are asymptotes of the root locus that intersect the real axis at , and radiate out with angles , where r=1, 3, 5...
Break-Away and -In Points on Real Axis There are break-away or -in points of the locus on the axis wherever .
Angle of Departure from Complex Pole Angle of departure from pole pj is
Angle of Arrival at Complex Zero Angle of arrival at zero zj is
Locus Crosses Imaginary Axis Use Routh-Horwitz to determine where the locus crosses the imaginary axis.
Determine Location of Poles, Given Gain "K" Rewrite characteristic equation as D(s)+KN(s)=0. Put value of K into equation, and find roots of characteristic equation.  (This may require a computer)
Determine Value of "K", Given Pole Locations Rewrite characteristic equation as , replace "s" by the desired pole location and solve for K.  Note: if "s" is not exactly on locus, K may be complex, but the imaginary part should be small. Take the real part of K for your answer.

References

© Copyright 2005 to 2015 Erik Cheever    This page may be freely used for educational purposes.

Erik Cheever       Department of Engineering         Swarthmore College