Modern Control Engineering

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

and

Figure 7–43 compares frequency-response curves of

in three different representations. In the log-magnitude-versus-phase plot, the vertical

distance between the points v=0andv=vr, where vris the resonant frequency, is the

peak value of G(jv)in decibels.

Since log-magnitude and phase-angle characteristics of basic transfer functions have

been discussed in detail in Sections 7–2 and 7–3, it will be sufficient here to give exam-

ples of some log-magnitude-versus-phase plots. Table 7–2 shows such examples. (How-

ever, more on Nichols charts will be discussed in Section 7–6.)

G(jv)=

1

1 + 2 zaj

v

vn

b + aj

v

vn

b

2

n

1

G(jv)

=-/G(jv)

0

5

5

10

0 °

90 °

180 °

|G

| in dB

G

Mr Mr

0.2vn 0.5vn vn 2 vn

vr

(a)

v = 0

v =`

∞

v

Im

Re

vn vr

vr vn

Mr

(b) (c)

12

15

6

3

0 1

6

3

9

180 ° – 90 ° 0 °

|G

| in dB

G

Figure 7–43

Three representations of the frequency response of for z>0.

(a) Bode diagram; (b) polar plot; (c) log-magnitude-versus-phase plot.

1

1 + 2 zaj

v vn

b+aj