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 

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... 
BreakAway and In Points on Real Axis  There are breakaway or in points of the locus on the axis wherever . 
Angle of Departure from Complex Pole  Angle of departure from pole p_{j} is 
Angle of Arrival at Complex Zero  Angle of arrival at zero z_{j} is 
Locus Crosses Imaginary Axis  Use RouthHorwitz 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. 
© Copyright 2005 to 2015 Erik Cheever This page may be freely used for educational purposes.
Erik Cheever Department of Engineering Swarthmore College