Page 546 - Mechanical Engineers' Handbook (Volume 2)

6 Pole Locations in the z-Domain 537

Damping Ratio
In the s-plane a constant damping ratio may be represented by a radial line from the origin.
A constant damping ratio locus (for 0
1) in the z-plane is a logarithmic spiral. Figure
31 shows constant
loci in both the s-plane and the z-plane.
If all the poles in the s-plane are specified as having a damping ratio not less than a
specified value
, then the poles must lie to the left of the constant-damping-ratio line in
1
the s-plane (shaded region). In the z-plane, the poles must lie in the region bounded by
logarithmic spirals corresponding to

(shaded region).
1
Damped Natural Frequency d
The rise time or the speed of response depends on the damped natural frequency and the
d
damping ratio
of the dominant complex-conjugate closed-loop poles. In the s-plane the
constant loci are horizontal lines, while in the z-plane they are radial lines emanating
d
from the origin.
Settling Time t s
The settling time is determined by the value of attenuation
of the dominant closed-loop
poles
j . If the settling time is specified, it is possible to draw a line
,in
d
1
the s-plane corresponding to a given settling time. The region to the left of the line

in the s-plane corresponds to the interior of a circle with radius e
T in the z-plane, as
1
1
shown in Fig. 32.
Remark
To transform s-plane pole locations to the z-domain, the transformation z e , where T is
sT
the sampling time, is employed.

6.3 Root Locus in the z-Domain
The root locus method for continuous-time systems can be extended to discrete-time systems
without modifications, except that the stability boundary is changed from the j axis in the
s-plane to the unit circle in the z-plane. The reason for being able to extend the root-locus
method is that the characteristic equation for the discrete-time system is of the same form
as that for the root loci in the s-plane. For the discrete-time case the CLCE is

Figure 31 (a) Constant
loci in the s-plane; (b) constant
loci in the z-plane.