Modern Control Engineering

In designing a lag–lead compensator, we frequently chose g=b. (This is not

necessary. We can, of course, choose gZb.) In what follows, we shall consider the

case where g=b. The polar plot of the lag–lead compensator with Kc=1andg=b

becomes as shown in Figure 7–109. It can be seen that, for 0<v<v 1 ,the

compensator acts as a lag compensator, while for v 1 <v<qit acts as a lead

compensator. The frequency v 1 is the frequency at which the phase angle is zero. It

is given by

(To derive this equation, see Problem A–7–21.)

Figure 7–110 shows the Bode diagram of a lag–lead compensator when Kc=1,

g=b=10,and Notice that the magnitude curve has the value 0 dB at the

low- and high-frequency regions.

T 2 =10T 1.

v 1 =

1

1 T 1 T 2

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

Im

01 Re

v =v 1 v =^0

Figure 7–109 v =`

Polar plot of a

lag–lead

compensator given

by Equation (7–27),

withKc=1and

g=b.

10

0

• 10


• 20


• 30

90 °

0 °

• 90 °

v in rad/sec

dB

0.01

T 1

1

T 1

10

T 1

0.1

T 1

0.001

T 1

100

T 1

Figure 7–110

Bode diagram of a

lag–lead

compensator given

by Equation (7–27)

withKc=1,

g=b=10,and

T 2 =10T 1.

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