Rules for drawing the Complementary Root Locus (K≤0), are similar to those for the Standard Root Locus (K≥0). Derivations are left to the reader, but follow closely the rules for the Standard Root Locus.
The table below gives the rules for both the Standard Root Locus as well as the Complementary Root Locus. Where the rules are identical, they are only stated once.
Rule Name  Standard Root Locus  Complementary Root Locus 

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.  The locus exists on real axis to the left of an even 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…  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 $\theta = \pm p{{180} \over q}$, where p=0, 2, 4… 
BreakAway and In Points on Real Axis  There are breakaway 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 p_{j} 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 departure from pole p_{j} is $${\theta _{depart,{p_j}}} = \sum\limits_{i = 1}^m {\angle \left( {{p_j}  {z_i}} \right)  } \sum\limits_{i = 1,\;i \ne j}^n {\angle \left( {{p_j}  {p_i}} \right)} $$ 
Angle of Arrival at Complex Zero  Angle of arrival at zero z_{j} is $${\theta _{arrive,{z_j}}} = 180^\circ  \sum\limits_{i = 1,\;i \ne j}^m {\angle \left( {{z_j}  {z_i}} \right) + } \sum\limits_{i = 1}^n {\angle \left( {{z_j}  {p_i}} \right)} $$  Angle of arrival at zero z_{j} is $${\theta _{arrive,{z_j}}} = \sum\limits_{i = 1,\;i \ne j}^m {\angle \left( {{z_j}  {z_i}} \right)  } \sum\limits_{i = 1}^n {\angle \left( {{z_j}  {p_i}} \right)} $$ 
Locus Crosses Imaginary Axis  Use 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 $K =  {{D(s)} \over {N(s)}}$, 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. 
Note the derivation of the rules for drawing the Complementary Root Locus (K≤0), are similar to those for the Standard Root Locus (K≥0), except the angle condition becomes
$$\eqalign{
\angle \left( {K{{N(s)} \over {D(s)}}} \right) &= \angle \left( {  1} \right) \cr
\angle \left( K \right) + \angle \left( {{{N(s)} \over {D(s)}}} \right) &= \angle \left( {  1} \right) \cr
180^\circ + \angle \left( {{{N(s)} \over {D(s)}}} \right) &= \pm r180^\circ ,\quad r = 1,\;3,\;5,\; \ldots \cr
\angle \left( {{{N(s)} \over {D(s)}}} \right) &= \pm p180^\circ ,\quad p = 0,\;2,\;4,\; \ldots \cr} $$
© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.
Erik Cheever Department of Engineering Swarthmore College