200 SAMPLED-DATA CONTROL SYSTEMS 
system has a Laplace transform given by R*(s). The output of the 
continuous system is C(s), whose inverse gives the time function which 
contains the ripple component being sought. If this output is sampled 
by a synchronous switch, the output pulse sequence is obtained from 
C(z). An approach to obtaining C(s) is to express the pulse sequence at 
the input of the system by the Laplace transform R*(s). 
The Laplace transform of the continuous output C(s) is given by 
C(s) = R*(s)G(s) (8.1) 
Recalling that R*(s) is given by the summation of displaced transforms 
as follows: 
+ 2 
Rs) =), RG + nie (8.2) 
where wo is 27/T, the output C(s) is then given by 
1 
C(s) = 7 ) R(s + njwo)G(s) (8.3) 
It is recognized that the central term of this series, R(s)G(s), is the output 
that would be obtained had the system not been sampled or sampled at a 
very high rate. Thus, (8.3) can be written as follows: 
565 
Gio 7 G(s) R(s) ie 7 G(s) R(e + njou) (8.4) 
where the first term in (8.4) represents the ‘‘smooth”’ output and the 
summation represents the ripple 
Ris) R*(s) Gis) LCS)-~___, component. The summation is the 
T Riz) T Cl) result of sampling and may be ex- 
Linear system pressed as a percentage of the smooth 
Fia. 8.1. System used for evaluation of output at the various sampling inter- 
Tupple. vals. While the inversion of (8.4) is 
possible in principle, it is evident that consideration of an infinite number 
of terms is not practical. As a result, only the first few terms of the 
summation are employed in practical situations. 
EXAMPLE 
To illustrate the application of this representation of ripple effects, 
it is assumed that G(s) in Fig. 8.1 is 
G(s) = 

1 
sta 
