BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 207 
of the error pulse sequence is 
R*(s) 
from which it follows that 
_ G(s) R*(s) 
The problem is to reduce (8.19) to a form which will be amenable to ready 
inversion into the time domain. The exact procedure for doing this 
depends somewhat on the form of the data hold. 
If it is assumed that the data hold is a simple clamp, or zero-order 
hold, the transfer function G(s) is given by 
il = aS 
Eo) = —— (8.20) 
and the approach is to include all factors containing rational powers of s 
in the plant transfer function G,(s). For the situation assumed, the 

Fria. 8.6. Model of system used for determination of ripple at output. 
model used to develop the desired relationships is given by Fig. 8.6, where 
it is seen that the numerator of (8.20) is a pulse-to-pulse relationship and 
is contained in a separate box. ‘The addition of the extra sampler in the 
error line is redundant since the operation 1 — e~7* represents the differ- 
ence of two samples. The modified plant pulse transfer function is now 
G,(s)/s and is assumed to be the ratio of rational polynomials in s. 
If G(s)/s is the ratio of rational polynomials in s, it may be expanded 
into partial fractions as follows: 
G(s) a Ay A; AG 
= pee 3 ae soe 
get sta aap nes (8.21) 




As 
s gti ar ‘gs? a 
where n is the order of the pole of G,(s) at the origin. For purposes of 
discussion it is assumed that all other poles are simple. This decomposi- 
tion into partial fractions makes possible the rearrangement of the model 
into the form shown in Fig. 8.7. The output of each elementary path 
contributes to the continuous output. At the same time, it is possible to 
connect fictitious samplers at the output of each elementary path to 
obtain expressions describing the pulse sequences of these paths. For 
instance, the output pulse sequence of the gth path, c}(¢), is described by 
