106 SAMPLED-DATA CONTROL SYSTEMS 
sampled-data feedback systems in obtaining the roots of the character- 
istic equation which determines the poles of the over-all pulse transfer 
function. This problem is analogous to that encountered in continuous 
feedback systems where the poles of the feedforward and feedback trans- 
fer functions are readily available but the poles of the over-all closed-loop 
transfer function are not. The poles of the over-all response pulse trans- 
fer function K(z) are the roots of a characteristic equation whose form is 
generally 
1+ GH(z) = 0 (5.56) 
where GH(z) is the loop pulse transfer function. It is recalled from a 
previous section that the form of the loop pulse transfer function may 
differ from that employed in (5.56), depending on the number and loca- 
tions of the samplers in the system. 
The problem is to find the roots of (5.56), knowing readily only the roots 
of GH(z). Asin the continuous-system problem, a root locus is employed 
to either estimate or obtain exactly the root locations of (5.56) in the 
complex plane. Stated differently, it is necessary to find all those values 
of z in the complex z plane which satisfy the condition 
GH(z) = 1/7 + n2an (5:57) 
As in all relationships between complex variables, (5.57) implies two 
separate equalities, one stating that the amplitudes are equal and the 
other that the angles are equal. In view of the fact that most practical 
systems either contain or are designed with gain constants which are 
adjustable, the amplitude relationship is considered relatively unimpor- 
tant. On the other hand, the phase relationship is a property of the 
system which reflects the organic characteristics of the components com- 
prising the system and is therefore of prime importance. Thus, a root 
locus is desired in which all the points on that locus satisfy the phase rela- 
tion implicit in (5.57), using the gain as a parameter. The relationship 
describing the root locus is 
Ang [GH(z)] = 7 + n2a (5.58) 
It is seen that except for a change in variable from s to z and the use of 
the z plane instead of the s plane, the problem posed by (5.58) is identical 
to that of plotting a root locus of a closed-loop continuous system. The 
transfer functions involved, namely, G(s) and G(z), respectively, are 
ratios of polynomials in their respective variables, so that the rules 
governing the plotting or sketching of the root loci are identical. Since 
these rules are well known and available! they will not be repeated here. 
The only significant difference to be found is that the behavior of the poles 
is observed in relation to the unit circle as contrasted to the imaginary 
axis. If poles lie inside the unit circle, they represent a stable system, 
