APPLICATION OF CONVENTIONAL TECHNIQUES 129 
tion is given by 

sin [(w -+ nw) wl 
(w =e Nw) 2, 
G(jo + jnwo) (6.10) 
H,G* (jw) = > (1 ao joT + Jno) EWeT neo 
If all harmonics are ignored and the (sin v/x)? term approximated by 
unity, 
HG* (jw) & A + joT)e—“'G (Jo) 
., A + joT 
GG.) 2” 
~ G(ju) (6.11) 
From (6.11) it is evident that the first-order hold and the sampler may 
be replaced by unity to the same degree of approximation that requires 
a delay to represent the clamp. Although this approximation is crude, it 

(14+ 7s)(1—e-75 2 
Ts2 
(a) 
Fia. 6.11. Block diagram of approximately equivalent systems. 

is useful if the transmission of G(jw) is very small at frequencies above 
a/T radians/sec. As an example of the nature of the approximation, 
consider the system shown in Fig. 6.11, where the continuous system of 
Fig. 6.116 results if the sampler and hold of the system of Fig. 6.1la are 
replaced by a through connection. The step responses of these systems 
are plotted in Fig. 6.12, where it is obvious that the equivalence between 
the two systems is not impressive. Actually, the sampled-data-system 
response closely resembles the response of the continuous system with a 
gain of 4, rather than a gain of 1. It is not evident from the expression 
for the first-order hold, (6.10), however, that this should be so. 
Only the simplest possible approximation to the zero- and first-order- 
hold circuits has been considered in this outline of one method for the 
design of sampled-data control systems. The extension of the method to 
establish more sophisticated approximations suitable to a wider range of 
devices is obviously possible but has limited value. When the simplest 
approximation derived above is not valid, then the system designer 
would be best advised to abandon the method altogether rather than try 
to refine 1t with more complicated approximations. 
