APPLICATION OF CONVENTIONAL TECHNIQUES 131 
following facts. First, G*(jw) is periodic, with period wo, so that the 
approximate plot need be constructed over a range of width wo only. 
Secondly, G*(—jw) is the conjugate of G*(jw), which permits the designer 
to construct the entire plot from the half obtained for positive frequencies. 
In other words, one need consider G*(jw) only over the range 0 Sw S 
wo/2. The number of terms required to give a reasonable approximation 
depends upon the accuracy required and the speed with which the series 
(6.12) converges. For the method to be effective, it is necessary that 
G(jw) approach zero at least as rapidly as 1/w? for frequencies above wo, 
and it is preferable if the cutoff is 
even more rapid. As an example of 
the limitation of the method, consider 
the system shown in Fig. 6.13. In 
this example, the plant is of first order 
only, and the series (6.12) does not Fic. 6.13. Block diagram for example 
converge sufficiently rapidly to make illustrating limitation on use of method 
the approximate method useful. 
The plot of G(yjw) for this example is shown in Fig. 6.14, along with the 
exact G*(jw). The construction of three terms of the series for G(j1) is 
shown, and it is obvious from this construction that an accurate approxi- 
mation cannot be made in this case. Although the higher-sideband terms 
are very small in amplitude, they converge so slowly that poor accuracy 
results. The difficulty is obvious in the simple example used here for 


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Fic. 6.14. Polar plots showing construction of approximate plots. 
