Chapter 10

# 10.3 Evans Root Locus Construction Rule # 1: Beginning, End and Symmetry

This Rule deals with the beginning and end of the Root Locus plot and its symmetry. We begin by writing the closed loop system characteristic equation in the form:

 $1+KG(s)=0$ Equation 10-3

Secondly, we factor G(s) and rewrite the polynomial equation as:

 $1+KG(s)=1+K\frac{N(s)}{D(s)}=1+K\frac{\prod_{i=1}^{m}(s-z_i)}{\prod_{i=1}^{n}(s-p_i)}=0$ Equation 10-4

Therefore:

 $\prod_{i=1}^{n}(s-p_i) + K\prod_{i=1}^{m}(s-z_i) = 0$ Equation 10-5

Notice that when K = 0, the roots of the characteristic equation are simply the roots of D(s). Notice that as the gain approaches infinity ($K\rightarrow \infty$) the roots of the characteristic equation are given by the roots of N(s).

We observe that the root loci start at the poles, and finish at the zeros. So, if there is an excess of finite poles over finite zeros, as is the case in strictly proper systems, where do these extra loci go as $K\rightarrow \infty$? The only way to satisfy the criterion is to conclude that the excess branches must tend to zeros located at infinity as $K\rightarrow \infty$. Observations lead us to form our first rule for root locus construction:

Rule 1: Each branch of the root locus is a continuous curve that begins at a pole of G(s), and ends at a zero of G(s). If the open loop system has m zeros and n poles, with $m \leq n$, then m of the root locus branches will begin at a pole and end at a zero. The n-m remaining branches of root loci will go to infinity $(\infty)$. The number of branches leaving a pole is equal to the multiplicity of the pole, r. The root loci are symmetrical with respect to the Real axis.

As an example, consider RL shown in Figure 10‑4. There are three RL segments, starting at -2, -5 and -10, which are open loop pole locations. Since there are no open loop zeros, three segments go to infinity. The plot is symmetrical about the Real axis.