1G 2G 3G nG

R c

Bode diagram of the open loop system

Plotting methods of the Bode diagram of the open-loop system

sGsGsGsGsG n321

s replaced by j

n

i

iGjn

i

i eGjG1

1

n

i

iGjG1

n

iiGjG

1

magnitude characteristic

phase characteristic

open-loop transfer function

n

ii

n

i

i GGG11

lg20lg20lg20Logarithmic magnitude characteristic

That is, Bode diagram of a open loop system is the superposition of the

Bode diagrams of the typical elements.

Example

Constant gain K =10 (20lg10=20dB)

A pole at origin

First break frequency: 2 rad/s

)15.0(

10)(

sssG 10(s)G1

ssG

1)(2

15.0

1)(3

ssG

-20dB/dec -40dB/dec

Example

)15.0(

10)(

sssG

Facility method to plot the magnitude response

of the Bode diagram

Summarizing example, we have:

1) Mark all break frequencies in the -axis of the Bode diagram.

2) Determine the slope of the L() of the lowest frequency band (before the first break frequency) according to the number of the integrating

elements:

[-20] dB/dec for 1 integrating element

[-40] dB/dec for 2 integrating elements

3) Continue the L() of the lowest frequency band until to the first break frequency, afterwards change the the slope of the L()

The slope of the L() should be increased 20dB/dec for the break frequency of the 1th-order differentiating element .

The slope of the L() should be decreased 20dB/dec for the break frequency of the Inertial element

Example: Plot the L() of the G(s)

)101.001.0)(11.0(

)1(10)(

22

ssss

ssG

2

111

)01.0(1

01.0 )1.0(90)(

tgtgtgo

100 9.174

10 5.56

1 3.51

)(

o

o

o

The Bode diagram

Facility method to plot the magnitude response

of the Bode diagram

0dB, 0o

100 10 1 0.1

)( ),( L

20dB

-45o

-90o

40dB

-180o

-135o

-225o

-270o

60dB/dec

20dB/dec

1.25dB

r

20dB/dec

There is a resonant peak Mr at:

7.705.021100 21 22 nr

dBMr 25.1154.1

12

1

2

Minimum-phase transfer function

A transfer function is called a minimum phase transfer function if its zeros and poles all lie in the left hand of s-plane.

A transfer function is called a non-minimum phase transfer function if it has any zero or pole lie in the right hand of s-plane.

For the minimum phase systems we can affirmatively determine the relevant transfer function only from the magnitude

response of the Bode diagram.

For the non-minimum phase system we must combine the magnitude response and phase response together to determine

the transfer function.

Minimum phase transfer function versus

Non-minimum phase transfer function

phase minimum-non 1

1 )2(

phase minimum 1

1 )1(

Ts

Ts

[-20]

)(L

1/T

)(

1/T

090)1(

)2(

)2)(1(

phase minimum-non 1 )4(

phase minimum 1 )3(

s

s

[+20]

)(L

)(

090

/1

/1

)4)(3(

)3(

)4(

The magnitude responses are the same.

But the phase responses are different when vary from 0 to infinite.

2 20 200 0dB

100 10 1 0.1

)(L

40dB/dec

40dB/dec

20dB/dec

)1005.0(

)15.0()(

:

2

ss

sKsG

diagramthe Bodefrom the G(s) we can get

40

0)5.0log(20log20log20)(20

2

K

KL

Example 1#

Determine the transfer function of the minimum phase

systems from the magnitude response of the Bode diagram

0dB 100 10 1 0.1

)(L

0.5 200

20dB/dec 20dB/dec 20dB

05.0 0)log(20)2.0log(20log202log20)(

2.0 20log202log20)(

2 0log20log20)(

:

22002

1/1

5.0

1

TTL

TdBL

KKL

and

T

Determine the transfer function of the minimum phase

systems from the magnitude response of the Bode diagram

)1)(1()(

:

21

sTsT

KssG

diagramodefrom the B the G(s) we can get

Example 2#

2.0 136.812

1log20

100 20log20log20)(

0.1T 10T

1

2

10

KdBKL

0dB

100 10 1 0.1

)(L

20dB/dec

20dB/dec 60dB/dec

8.136 dB

20 dB 1)0.04ss(0.01s

1)100(0.01sG(s) :

2

then

)12(

)101.0()( :

2

2

TssTs

sKsGdiagramodefrom the B the G(s) we can get

Determine the transfer function of the minimum phase

systems from the magnitude response of the Bode diagram

Example 3#