Consider the unity feedback system shown in Fig. 13.10. The system open-loop transfer function is
T_{OL}(s)=\frac{K}{s(s+2)} , (13.37)
where K ≥ 0 is the open-loop gain. The closed-loop transfer function is
T_{CL}(s)=\frac{K}{(s^{2}+2s+K )} . (13.38)
Hence the closed-loop system characteristic equation is
s^{2}+2s+K=0 , (13.39)
and hence the roots are
s_{1} =-1-\sqrt{1-K} , s_{1} =-1-\sqrt{1+K} (13.40)
As the open-loop gain varies from zero to infinity, the roots change as follows:
_ For K = 0: s_{1} = −2 and s_{2} = 0 .
_ For K increasing from 0 to 1: Both roots remain real, with s_{1} moving from −2 to −1 and s_{2} moving from 0 to −1.
_ For K between 1 and infinity: The roots are complex conjugate, s_{1} = -1-j\sqrt{K-1} and s_{2} = -1+j\sqrt{K-1} .
The migration of roots for 0 ≤ K < ∞ is shown in Fig. 13.11. Once the root locus is plotted, it is easy to determine the locations of the characteristic roots necessary for a desired system performance. For instance, if the desired damping ratio for the system considered in this example is, ξ= 0.7, the roots must be located at points A_{1} and A_{2} , which correspond to
s_{1} = -1+j , s_{2} = -1-j .
The characteristic equation for these roots takes the form
(s + 1 − j )(s + 1 + j ) = 0Or
s^{2} + 2s + 2 = 0. . (13.41)
When Eq. (13.41) is compared with the characteristic equation in terms of K [Eq. (13.39)], the required value of the open-loop gain is found to be K = 2.
