124 SAMPLED-DATA CONTROL SYSTEMS 
If T is sufficiently small, or, equivalently, if wo is high, and G(jw) is low- 
pass, all harmonics can be neglected, and the transfer function in (6.5) 
reduces to 
HG*(jo) 2 e-iTo”? we G (jo) (6.6) 
Actually, for low frequencies, ee is approximately unity, so that 
finally 
HG* (jw) & e-#T’?E (jw) (6.7) 
In words, the loop transfer function is approximately the equivalent of 
replacing the sample and hold operation by a pure time delay of half a 

Fia. 6.4. Sketch showing the delay of 7'/2 sec introduced by the zero-order hold. 
sampling period. This approximation may also be quickly obtained 
by consideration of time-domain plots of a signal and its sampled and 
clamped resultant as shown in Fig. 6.4. The boxcar output of the zero- 
order hold seems to follow the input curve delayed by 7/2 sec. In terms 
of block diagrams the system of Fig. 6.5a is replaced or approximated by 
the continuous system of Fig. 6.5b. The applications of standard fre- 
quency design methods to the approximate system are straightforward, as 
will be illustrated by an example. 
EXAMPLE Method 1 
As an illustrative example of the application of the approximating 
technique outlined above, the compensation of the system shown in 
Fig. 6.6 will be considered. For the purposes of this example the 
sampling rate will be taken to be 1 per second, and the Bode diagram or 
logarithmic frequency plots used to suggest the compensation required. 
A polar plot or Nichols chart could be as easily used, of course. The 
separate logarithmic plots of gain and phase are chosen arbitrarily to 
illustrate the use of a conventional technique with this particular 
