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G(s) -. 5(s + 2)/(s +l)(s + 3)
┏━━━┓
┃\ ┃
┃~ ┃
┣━━━┫
┃J"',F ┃
┃'N ┃
┃,. ┃
┗━━━┛
c)
┏━━━━┳━━━━┓
┃ Im / ┃ k ┃
┃C ┃- ~- ┃
┣━━━━╋━━━━┫
┃QA' ┃" jF'-/ ┃
┗━━━━┻━━━━┛
G(s) : 5(s +2)/s(s +l)(s +3)
Flg. 5.19 Nyquist plots for Examples 5.6 and 5.7.
Nyquist plot as shown. However, the reader should keep in mind that, in a given
problem, the locations of closed-loop poles are not known as assumed in this
discussion.
Example 5.6
Draw the Nyquist plot for a unity feedback system with
5(s + 2)
G(s) = (s+l) s+3~
Solution. The first step is to select some points along the Nyquist contourin
the s-plane as shown in Fig. 5.19a Consider an arbitrary point P. Let Vi, 1V2, and
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 479
V3 be the vectors drawn to point P from the zeros and poles as shown. Then,
5{ ViI
{ V211 V3~
/V; = ZVi -/V2 - z V3
If point P coincides wiLh A, I ViI = 2, rv21 : 1, and [ V31 = 3 so that [ vh i = 10/3.
The phase angles /Vi = Z V2 - Z V3 = 0 SO that /Vh = O. Thus the point A in
the s-plane maps to point A' on the real axis in the F-(llane with abscissa equal to
10l3.ln a similar fashion, we find the magnitude and phase angles at other image
points such as IV~I = O, /V/ = -90 deg; iv61 = o, zv~ = o, ancl lv~f = o,
/V5 = 90 deg.
.Based on this information, the Nyquist plot can be sketched as shown in Fig.
5.19b. Observe that the Nyquist plot does not encircle the origin butjust goes around
it in a semicircle of "zero" radius. As we move clockwise in the s-plane starting
at point A, we move in the counterclockwise direction in the F-plane from A'.
In this example we didn't have any poles of the mapping funcXon on the imag-
mary axis. If we did, then we have to draw semicircles of infinitesimally small
radii around each one to prevent a breakdown of the mapping procedure at these
points. We illustrate the procedure of drawing such Nyquist plots with the help of
Example 5.7.
Example 5.7
Draw the Nyquist plot for the system with
5(s + 2)
G(s) = s~s + ) s+ 3~
So/ution. Here, we have a pole at s = 0 at the origin. As said above, we draw
a semicircle ofinfirutesimally small radius around it as shown in Fig. 5.19c. The
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