SAMPLED-DATA SYSTEMS 105 
—x/2 to 0 in a counterclockwise direction, as shown in Fig. 5.14b. 
In mapping from ¢ to d, the variable z takes on larger and larger posi- 
tive values, and G(z) becomes smaller and smaller until it vanishes 
when z approaches infinite values. The behavior about the region of 
zero is only of academic interest since it has no significance in the 
determination of the enclosure of the point (—1,0) in this case. Thus, 
the variation of z from d to e is mapped into a point at the origin in Fig. 
5.146. The remainder of the map is obtained by using the conjugate 
values to corresponding points previously plotted. This part of 
the map is shown as efgha. It is seen that the contour does not 
enclose the point (—1,0), so that the system is stable. Had the feed- 
forward gain constant been raised to 2.48 times higher than that used 
in the example, the contour would have enclosed the point (—1,0) and 
the system would be unstable. 
The foregoing example illustrates how the pulse transfer locus is 
plotted or sketched. The only portion of this locus which is significant 
is the map of the unit circle and the detours taken about poles of G(z) 
which lie on the unit circle. Such poles are usually located at the point 
(1,0) since they arise from integration in the feedforward line. In con- 
tinuous systems, the margin by which the critical point (—1,0) is avoided 
is estimated by the use of gain and phase margins or by constant gain loci 
generally referred to as M circles. While analogous margins can be 
employed in the design of sampled-data systems, it is not customary to do 
so. The reason for this is that there is less correlation between the 
margins and the transient response than in continuous systems. Never- 
theless, the larger the margin of avoidance of the critical point, the higher 
is the degree of damping of the system. Also, a substantial margin is 
generally accompanied by relatively docile transient response. In view 
of the fact that it is simpler to invert pulse transfer functions and to 
obtain time-domain performance characteristics, the use of the pulse 
transfer locus as a design tool is not as widespread as in the case of con- 
tinuous systems. 
5.6 Root Loci for Sampled-data Systems 
The response of a system whose over-all pulse transfer function is G(z) 
is determined by the poles and zeros of G(z). For instance, if the poles of 
G(z) are 21, 22, . . . , én, the transient component of the response, g:,(kT), 
is given by 
gir( kT) Ta Ciewe =o Co(z2)* SF Rs Ute Cicax (5.55) 
This relationship was derived in Sec. 5.3 and is shown in Eqs. 5.44 and 
5.45. While straightforward, there is some difficulty in higher-order 
