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

$$T\left( s \right) = {{KG\left( s \right)} \over {1 + KG\left( s \right)H\left( s \right)}}$$

So the characteristic equation is

$$1 + KG\left( s \right)H\left( s \right) = 1 + K{{N\left( s \right)} \over {D\left( s \right)}} = 0$$

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 $\sigma = {{\sum\limits_{i = 1}^n {{p_i}} - \sum\limits_{i = 1}^m {{z_i}} } \over q}$, and radiate out with angles $\theta = \pm r{{180} \over q}$, 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 $N(s)D'(s)-N'(s)D(s)=0$.
Angle of Departure from Complex Pole Angle of departure from pole pj is $${\theta _{depart,{p_j}}} = 180^\circ + \sum\limits_{i = 1}^m {\angle \left( {{p_j} - {z_i}} \right) - } \sum\limits_{i = 1,\;i \ne j}^n {\angle \left( {{p_j} - {p_i}} \right)}$$
Angle of Arrival at Complex Zero Angle of arrival at zero zj is $${\theta _{arrive,{z_j}}} = 180^\circ - \sum\limits_{i = 1,\;i \ne j}^m {\angle \left( {{z_j} - {z_i}} \right) + } \sum\limits_{i = 1}^n {\angle \left( {{z_j} - {p_i}} \right)}$$
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 $K = - {{D(s)} \over {N(s)}}$, 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