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 m^{th} order polynomial;
D(s) is n^{th} order.
 N(s) has zeros at z_{i} (i=1..m); D(s)
has them at p_{i} (i=1..n).
 The difference between n and m is q, so q=nm.

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... 
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 Arrival at Complex Zero 
Angle of arrival at zero z_{j} is $${\theta _{arrive,{z_j}}} = 180^\circ  \sum\limits_{i = 1,\;i \ne j}^m {\angle \left( {{z_j}  {z_i}} \right) + } \sum\limits_{i = 1}^n {\angle \left( {{z_j}  {p_i}} \right)} $$ 
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. 