# Root Loci For Systems With Transport Lag

**Root loci for systems with transport lag**

*Fig: 1 Thermal system Fig: 2 Block diagram of the system*

Figure 1 shows a thermal system in which hot air is circulated to keep the temperature of a chamber constant. In this system, the measuring element is placed downstream a distance L ft from the furnace, the air velocity is v ft/sec, and T = L/v sec would elapse before any change in the furnace temperature is sensed by the thermometer. Such a delay in measuring, delay in controller action, or delay in actuator operation, and the like, is called transport lag or dead time. Dead time is present in most process control systems.

The input x(t) and the output y(t) of a transport lag or dead time element are related by

Suppose that the feedforward transfer function of this thermal system can be approximated by

as shown in Figure 2. Let us construct a root-locus plot for this system. The characteristic equation for this closed-loop system is

It is noted that for systems with transport lag the rules of construction presented earlier need to be modified. For example, the number of the root-locus branches is infinite, since the characteristic equation has an infinite number of roots. The number of asymptotes is infinite. They are all parallel to the real axis of the s plane.

From the above equation we obtain,

Thus, the angle condition becomes