Modern Control Engineering

460 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

Im

v = 0.8

v = 0.7

j1.5

G

KPlane

v = 0.6

v = 0.9 j^1

G(jv)

K

v = (^1) j0.5 v = 0.4
v = 1.4 v = 1.5
v = 3
v = 0
v =
v = 0.2
v = 0.1

• 1 – 0.5 0 0.5 1 Re
  v = 2

v =–

• j0.5


• j 1

Figure 7–61 –j1.5

Polar plot of

G(jv)/K.

Our problem is to determine the range of the gain Kfor stability. Hence, instead of plotting

Nyquist loci of G(jv)for various values of K, we plot the Nyquist locus of G(jv)/K. Figure 7–61

shows the Nyquist plot or polar plot of G(jv)/K.

SinceG(s)has two poles in the right-half splane, we have Noting that

Z=N+P

for stability, we require Z=0orN=–2. That is, the Nyquist locus of G(jv)must encircle the

–1+j0point twice counterclockwise. From Figure 7–61, we see that, if the critical point lies

between 0 and –0.5, then the G(jv)/Klocus encircles the critical point twice counterclockwise.

Therefore, we require

–0.5K<–1

The range of the gain Kfor stability is

2<K

P=2.

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