104 SAMPLED-DATA CONTROL SYSTEMS 
This function plane will map the contour shown in Fig. 5.14a. Starting 
with the point a, which is the value z = —1.0, the value of G(z) is seen 
to be —0.038. As the various values of z are substituted in the range 
from a to b, a map is generated as shown in Fig. 5.14b. At a phase 
angle just short of 7/2, the locus again is real and crosses the real axis 
at about —0.4. At the point b on the contour, the function G(z) is 
dominated by the behavior of the factor (2 — 1) in the denominator. 
G(z)-plane \ 

(a) 
Fic. 5.14. Pulse transfer locus used in example. (a) Contour used in z plane. (6) 
Corresponding points on pulse transfer locus. 
As the radius of the detour from b to c becomes increasingly small, this 
behavior becomes more and more dominant. Thus, to study the 
transfer locus in this region, G(z) can be approximated by 
1 
G(z) = ag 
To study the behavior of the map in the region b-c on the contour, a 
detail is shown in Fig. 5.15. The 
complex number (zg — 1) is repre- 
sented by a vector extending from 
the point (1,0) to the contour. The 
angle of this vector at b is 7/2 and, 
c when substituted into G(z) as approx- 
5 anne imated in this region, yields, 
(z-1) 7 
G(z) = Gale 


Fic. 5.15. Detail of contour © on z As the contour is traced from b toc, 
plane. the angle changes from 7/2 to 0 in 
a clockwise direction, and G(z) has a phase angle which changes from 
