Modern Control Engineering

Section 7–7 / Relative Stability Analysis 465

|G

| in dB

|G

| in dB

0

+

• 
  

0

+

• 
  

|G

| in dB

+

• 
  

|G

| in dB

+

• 
  


• 90 °


• 180 °


• 270 °


• 90 °


• 180 °


• 270 °

G G

Logv

Logv

Logv

Logv

Positive

gain margin

Positive

phase margin

Negative

gain margin

Negative

phase margin

Stable system

Im Im

Re Re

Unstable system

Stable system Unstable system

Stable system Unstable system

(a)

(b)

(c)

Negative

phase margin

Negative

phase

margin

Positive

gain

margin

Positive

gain

margin

Positive

phase

margin

Positive

phase

margin

Negative

gain

margin

Negative

gain

margin

1

Kg

f

g

G(jv)

Gplane Gplane

1

Kg

f

g

G(jv)

1 1

• 1 – 1

0 0

• 270 ° – 180 ° – 90 ° –^270 ° –^180 ° –^90 °
  G G

Figure 7–67
Phase and gain
margins of stable and
unstable systems.
(a) Bode diagrams;
(b) polar plots;
(c) log-magnitude-
versus-phase plots.

Figures 7–67(a), (b), and (c) illustrate the phase margin of both a stable system and

an unstable system in Bode diagrams, polar plots, and log-magnitude-versus-phase plots.

In the polar plot, a line may be drawn from the origin to the point at which the unit cir-

cle crosses the G(jv)locus. If this line lies below (above) the negative real axis, then the