202 SAMPLED-DATA CONTROL SYSTEMS 
This steady-state ripple is constant and dependent on the sampling 
frequency and system time constant. 
The procedure outlined in this section and illustrated in the example is 
not readily applicable to the problem of obtaining an exact expression for 
the ripple. Other procedures to be 
described in later sections can be 
used to better advantage for this 
problem. However, forsteady-state 
conditions where ripple is present, 
Fic. 8.2. Feedback sampled-data sys- the approach outlined in this section 
tem used for evaluation of ripple. i Aa a 
as considerable usefulness 
The method outlined here has more limited application in the case of 
feedback systems, such as those shown in Fig. 8.2. Here the Laplace 
transform of the continuous output C(s) is related to the Laplace trans- 
form of the sampled error sequence H*(s) by the expression 



Cs) — 1G (yas) (8.5) 
and it has been shown that H*(s) is given by 
iL es 
7 R(s + njwo) 
E*(s) = -S=5 (8.6) 
1+ A » G(s + njwo) 
Dy Lays 
Then C(s) is given by n 
- D RCo ate) 
CGC) = amet G(s) (8.7) 
ee ~ Dy G(s + njwo) 
The inversion of (8.7) is a difficult, if not impossible, procedure. It is 
possible, however, to obtain approximations by taking only a few of the 
terms of the infinite summations in both numerator and denominator. 
If the system is low-pass relative to the sampling frequency, only the 
first term or two is significant. In view of the fact that other methods 
are available to obtain exact expressions for the ripple in any given 
sampling interval, the approximation techniques described here are of 
limited value. 
8.2 The Multiple-rate Sampling Technique 
An approach to the evaluation of the ripple involves the use of a 
fictitious sampler at the output of the system whose period is a fraction 
