Section 7–7 / Relative Stability Analysis 463
s Plane
jv
G(jv)
0 s
jv 4
jv 3
jv 2
jv 1
GPlane
Im
– (^10) Re
Constants
curves
Constantv
curves
v 4
v 3
v 2
v 1
- s 4
- s 3
- s 2
Figure 7–63 –s 1
Conformal mapping
ofs-plane grids into
theG(s)plane.
s Plane
jv
0 s
(a) (b)
s Plane
jv
0 s
Figure 7–64
Two systems with
two closed-loop
poles each.
Consider the conformal mapping of constant-slines (lines s=s+jv, where sis
constant and vvaries) and constant-vlines (lines s=s+jv, where vis constant and
svaries) in the splane. The s=0line (the jvaxis) in the splane maps into the Nyquist
plot in the G(s)plane. The constant-slines in the splane map into curves that are sim-
ilar to the Nyquist plot and are in a sense parallel to the Nyquist plot, as shown in Fig-
ure 7–63. The constant-vlines in the splane map into curves, also shown in Figure 7–63.
Although the shapes of constant-sand constant-vloci in the G(s)plane and the
closeness of approach of the G(jv)locus to the –1+j0point depend on a particular
G(s), the closeness of approach of the G(jv)locus to the –1+j0point is an indication
of the relative stability of a stable system. In general, we may expect that the closer the
G(jv)locus is to the –1+j0point, the larger the maximum overshoot is in the step
transient response and the longer it takes to damp out.
Consider the two systems shown in Figures 7–64(a) and (b). (In Figure 7–64, the :’s
indicate closed-loop poles.) System (a) is obviously more stable than system (b) because
the closed-loop poles of system (a) are located farther left than those of system (b).
Figures 7–65(a) and (b) show the conformal mapping of s-plane grids into the G(s)
plane. The closer the closed-loop poles are located to the jvaxis, the closer the G(jv)
locus is to the –1+j0point.
