Root Loci For Systems With Transport Lag

Root loci for systems with transport lag

Since the right-hand side of the angle condition given by the equation of the angle condition

 has an infinite number of values, there are an infinite number of root loci, as the value of k (k = 0,1,2, ... ) goes from zero to infinity. For example, if k = 1, the angle condition becomes

The construction of the root loci for k = 1 is the same as that for k = 0. A plot of root loci for k = 0,1, and 2 when T = 1 sec is shown in Figure 4.

The magnitude condition states that

Fig: 4 Root-locus plot for the system

The root loci shown in Figure 4 are graduated in terms of K when T = 1 sec. Although there are an infinite number of root-locus branches, the primary branch that lies between -jπ and jπ is most important. Referring to Figure 4, the critical value of K at the primary branch is equal to 2, while the critical values of K at other branches are much higher (8, 14, ...). Therefore, the critical value K = 2 on the primary branch is most significant from the stability viewpoint. The transient response of the system is determined by the roots located closest to the jω axis and lie on the primary branch. In summary, the root-locus branch corresponding to k =0 is the dominant one; other branches corresponding to k = 1, 2, 3, ... are not so important and may be neglected.

This example illustrates the fact that dead time can cause instability even in the first-order system because the root loci enter the right-half s plane for large values of K. Therefore, although the gain K of the first-order system can be set at a high value in the absence of dead time, it cannot be set too high if dead time is present. (For the system considered here, the value of gain K must be considerably less than 2 for a satisfactory operation.)