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0066_frame_C27 Page 15 Wednesday, January 9, 2002 7:10 PM

FIGURE 27.17 Nyquist diagram.

FIGURE 27.18 Nyquist diagram and root locus of Example 4.
The Nyquist stability criterion can now be stated as follows:

A necessary and sufficient condition for the closed-loop stability of a system defined by the loop
transfer function L(s) is that
Z = N + P (27.12)

be equal to zero, where N is the net number of encirclements of the −1 point in the L(s)-plane, and
P is the number of unstable poles of the loop transfer function L(s).

Example 4
Consider the system with the loop transfer function
2 2s +
s –
2
KL s() = KG s()Hs() = K---------------------------------- (27.13)
ss + 1) s + 2)
(
(
Let us determine the range of the gain K such that the closed-loop system is stable. Since there is a
pole at s = 0, we need to modify the Nyquist path to detour around the origin. The contour is shown
in Fig. 27.18(a), where the detour is chosen to be a semicircle of radius approaching zero in the limit.
We use the following procedure to sketch the Nyquist plot in Fig. 27.18(b):
+
1. Determine L(jω) as ω → 0 : L(s) is of system type 1 and thus
1
Ljw) ≈ ----- = ∞ – 90°
(
∠
jw
according to Table 27.2.

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