Root-locus Plots

Root-locus plots:

Angle and magnitude conditions:

 

Fig: 1 Control system

 

Consider the system shown in Figure 1. The closed-loop transfer function is

 

 

The characteristic equation for this closed-loop system is obtained by setting the denominator of the right-hand side of the above equation equal to zero. That is,

 

 

Here we assume that G(s)H(s) is a ratio of polynomials in s. Since G(s)H(s) is a complex quantity, the above equation can be split into two equations by equating the angles and magnitudes of both sides, respectively, to obtain the following:

 

 

The values of s that fulfill both the angle and magnitude conditions are the roots of the characteristic equation, or the closed-loop poles. A plot of the points in the complex plane satisfying the angle condition alone is the root locus. The roots of the characteristic equation (the closed-loop poles) corresponding to a given value of the gain can be determined from the magnitude condition.

 

In many cases, G(s)H(s) involves a gain parameter K, and the characteristic equation may be written as

 

Then the root loci for the system are the loci of the closed-loop poles as the gain K is varied from zero to infinity.