Modern Control Engineering

Section 7–2 / Bode Diagrams 415

40

20

0

• 20

dB

• 40

Exact curve

0.2 0.4 0.6 0.8 1 2 4 6 8 10

v

• 270 °


• 180 °
  
  
  • 90 °
  
  
  
  

0 °

90 °

0.2 0.4 0.6 0.8 1 2 4 6 8 10

v

f

G(jv)

2

2

5

5

4

4

3

1

G(jv)

3

1

Figure 7–11
Bode diagram of the
system considered in
Example 7–3.

Minimum-Phase Systems and Nonminimum-Phase Systems. Transfer func-

tions having neither poles nor zeros in the right-half splane are minimum-phase trans-

fer functions, whereas those having poles and/or zeros in the right-half splane are

nonminimum-phase transfer functions. Systems with minimum-phase transfer functions

are called minimum-phasesystems, whereas those with nonminimum-phase transfer

functions are called nonminimum-phasesystems.

For systems with the same magnitude characteristic, the range in phase angle of the

minimum-phase transfer function is minimum among all such systems, while the range in

phase angle of any nonminimum-phase transfer function is greater than this minimum.

It is noted that for a minimum-phase system, the transfer function can be uniquely

determined from the magnitude curve alone. For a nonminimum-phase system, this is

not the case. Multiplying any transfer function by all-pass filters does not alter the

magnitude curve, but the phase curve is changed.

Consider as an example the two systems whose sinusoidal transfer functions are,

respectively,

G 1 (jv)=

1 +jvT

1 +jvT 1

, G 2 (jv)=

1 - jvT

1 +jvT 1

, 06 T 6 T 1