Modern Control Engineering

the lead compensator. The minimum value of ais usually taken to be about 0.05. (This

means that the maximum phase lead that may be produced by a lead compensator is

about 65°.) [See Equation (7–25).]

Figure 7–91 shows the polar plot of

withKc=1.For a given value of a, the angle between the positive real axis and the tan-

gent line drawn from the origin to the semicircle gives the maximum phase-lead angle,

fm.We shall call the frequency at the tangent point vm.From Figure 7–91 the phase

angle at v=vmisfm,where

(7–25)

Equation (7–25) relates the maximum phase-lead angle and the value of a.

Figure 7–92 shows the Bode diagram of a lead compensator when Kc=1anda=0.1.

The corner frequencies for the lead compensator are v=1/Tandv=1/(aT)=10/T.

By examining Figure 7–92, we see that vmis the geometric mean of the two corner fre-

quencies, or

logvm=

1

2

alog

1

T

+log

1

aT

b

sinfm=

1 - a

2

1 +a

2

=

1 - a

1 +a

Kc a

jvT+ 1

jvaT+ 1

(0 6 a 6 1)

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

Im

Re

vm

01

v = 0 v =`

fm

1

2 (1+a)

a

1

Figure 7–91^2 (1–a)

Polar plot of a lead

compensator

a(jvT+1)/(jvaT+1),

where0<a<1.

10

0

v in rad/sec

• 10


• 20

90 °

0 °

dB

0.1

T

1

T

10

T

100

T

(^10)
T
Figure 7–92 fm
Bode diagram of a
lead compensator
a(jvT+1)/(jvaT+1),
wherea=0.1.
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