Modern Control Engineering

Section 7–8 / Closed-Loop Frequency Response of Unity-Feedback Systems 485

|G

| in dB

G

15

10

5

0

• 5


• 10


• 15


• 180 ° – 150 ° – 120 ° – 90 °

Mr= 1.4

20 log K= 4

G(jv)

G(jv)

K

Figure 7–86
Determination of the
gainKusing the
Nichols chart.

andNloci differently, resulting in a different closed-loop frequency-response curve. For

a small value of the gain K, the G(jv)locus will not be tangent to any of the Mloci, which

means that there is no resonance in the closed-loop frequency response.

EXAMPLE 7–24 Consider the unity-feedback control system whose open-loop transfer function is

Determine the value of the gain Kso that Mr=1.4.

The first step in the determination of the gain Kis to sketch the polar plot of

Figure 7–86 shows the Mr=1.4locus and the G(jv)/Klocus. Changing the gain has no effect on

the phase angle, but merely moves the curve vertically up for K>1and down for K<1.

In Figure 7–86, the G(jv)/Klocus must be raised by 4 dB in order that it be tangent to the

desiredMrlocus and that the entire G(jv)/Klocus be outside the Mr=1.4locus. The amount of

vertical shift of the G(jv)/Klocus determines the gain necessary to yield the desired value of

Mr. Thus, by solving

we obtain

K=1.59

20 logK= 4

G(jv)

K

=

1

jv(1+jv)

G(jv)=

K

jv(1+jv)