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).
|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.|