SAMPLED-DATA SYSTEMS 103 
continuity in the pulse transfer locus if the contour being mapped were 
exactly the one shown in Fig. 5.9. 
In order to establish a connection between the segments of the pulse 
transfer locus obtained from the portions of the contour on either side of 
the pole (1,0), it is necessary to generate a continuous curve around the 
pole. This is done by taking a small semicircular detour, as shown in 
Fig. 5.12, so oriented that the pole is included definitely on one side or the 

Fig. 5.12. Contour used for pulse transfer Fic. 5.13. Reduced contour used to map 
functions having poles at (1,0). practical pulse transfer functions. 
other. It is conventional to orient the detour to include this pole inside 
the unit circle, as shown in the figure. The same procedure is followed 
for any complex poles whose magnitude is exactly equal to unity, although 
the occurrence of such poles is rare. Since practical pulse transfer func- 
tions vanish for infinite values of z and since the portion of the contour 
along the real axis is self-canceling in the limit, it is customary to map 
only the unit circle and its detours as shown in Fig. 5.13. 
EXAMPLE 
A unity feedback sampled-data control system has a feedforward 
pulse transfer function G(z) given by 
2-1(0.2642-! + 0.368) 
( — 2) — 0.36827!) 
It is seen that the loop pulse transfer function contains a pole at (1,0), 
necessitating the use of the contour shown in Fig. 5.13. For purposes 
of illustration, however, the full contour of Fig. 5.12 will be used. 
Before mapping, the function G(z) will be expressed in positive powers 
of z by multiplying both numerator and denominator by 2z?: 
_ 0.264 + 0.3682 
G@) = @ = 1) = 01368) 

CZ) = 

