BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 213 
between zero and unity, it is seen that the value of the continuous output 
is explored at any and all sampling intervals. 
If G(z) is the pulse transfer function of the system shown in Fig. 8.11a, 
oe ale) 
Yr Cle) 
| 
t 2 Pts | 
R(s) Riz) C(s) 
[——_ --70 
! C(z,A) 
r(t r*(t) ATs 
C\s) 5s O 
Ris) = 5 | ea : 
(b) 
Fia. 8.11. Application of modified z transform to determine ripple. 
then the output of the system at sampling instants is obtained by inver- 
sion of the output 2 transform C(z), given by 
C(z) = G(z) R(z) (8.28) 
With the fictitious time advance inserted as shown in Fig. 8.116, the z 
transform of the output is a function of A and is given by 
C(z,A) = G(z,A) R(z) (8.29) 
Inversion of C(z,A) gives the output as a function of the sampling instant 
and A. Hence, 
c(mT,A) = Z-1C(z,A) (8.30) 
By leaving A as a parametric variable, it is possible to determine the out- 
put anywhere within a sampling interval. 
EXAMPLE 
If the transfer function of the continuous element G(s) in Fig. 8.11 is 
taken as 
1 
G(s) = pags 
From Appendix II, the advanced pulse transfer function G(z,A) is given 
by 
e@AT 
GGA) = a 
If the input to the system is a unit step function, then the advanced 
output C(z,A) is given by 
Ces). — 
e7aAT 
