大三上学期课件

15.2

Elemental

Bode

DiagramsWe

will learn

how

to

plot

Bode

Diagrams

of

simple(elemental)

TFs

and

then

learn

how

to

combine

them.In

general

an

open-loop

TF

will

have

the

following

form:一阶微分环节二阶微分环节m1m2

][1

2

i

s2s(1T

s)KG(s)

i1l1nl2

l

nl比例环节2

][1

2n1k1n2r1nrnr

r

k

s2(1T

s)s

s<0微分环节>0积分环节惯性环节振荡环节n1

n2

2

i1

l

1

nl

nl

K

m1(1

jTi

)m2

[122

j

2

]lG(

j)

G(s)

|s

j]k

1r

1nrr2nrk

[1

j2(1

jT

)(

j)1)

Proportional

elementG(s)

KG(

j

)

K4030K=10020K=10M

K10090o

45o0o-45o-90oM

db

20

lg

K

(

)

0·增益K为常数，对数

幅频特性为一水平直线，改变K，会上升或下降,

相应的相角不变，为0；·

K>1,分贝值为正，K<1，分贝值为负。10-11001011022)

Integral

element1jG(

s

)

1

G(

j

)

s211

j

G(

j

)

j

eM

()

1M

20lg

1

20lgdb()

9040200

20dB

/

dec-20=1时，对数幅频特性为0

-90o

45o0o分40贝，斜率为-20dB/dec;相角为常数-90o-45o-90o传递函数包含(j)-n时，对数幅频特性为-n20lg

，相频特性为-90on。10-11001011023) Differential

elementG(s)

sG(

j

)

j4020020dB

/

dec-20-4090o

45o-45o0ojG(

j

)

j

e

2M(

)

Mdb

20lg

(

)

90·与积分环节的对数频率特性以横轴互为镜像;n·传递函数包含(j)时，对数幅频特性为n20lg

，相频特性为90on。-90o10-11001011024)

Inertial

element11

20

lg12T

22T

240200lo ss

filterstraight-line

approximation2

3,()

45M

db

()

20

lg

1

时，TMdb

()

20lg

1

时,

20dB

/

decthe

true

curveTM

db

(

)

0,

(

)

0

时,1T-20-4090o

45o0o-45o-90o0.01/T

0.1/T1/T

10/T100/TM

db

( )

20

lg

T

,

( )

90

The

low-frequency

asymptoteis

0db,

the

high-frequencyasymptote

is

–20db/dec.break

frequency

=1/T；Phase

is

central

symmetryabout

(1/T,

-45º).·

For

(1+jT)-n,

The

low-frequencyasymptote

is

0db,

the

high-frequencyasymptote

is n*20db/dec;

Phase

iscentral

symmetry

about

(1/T,

-n*45º).5)-order

differential

element4020the

exact

curve-20020dB

/

decstraight-line

approximationM

db

()

20

lg

2T

2

1( )

arctan

T1

时，TMdb

()

20lg 2

3,()

45-4090o

45o0o

1

时,-45o-90o0.01/T

0.1/T1/T

10/T100/TTMdb

()

0,(

)

0

1

时,TMdb

()

20

lgT

,(

)

901This

plots

are

symmetry

about

the

frequency

axis

with

those

ofFor

(1+jT)n,

the

low-frequency

asymptote

is

0db,

the

high-1

Tsfre

uenc

as

m tote

is

n*20db/dec;

Phase

is

central

s

mmetr

about(1/T,

n*45º).6)

Second-order

oscillatory

element

(振荡环节)4020Mdb

()

20

lg0straight-line

approximation

40dB

/

decnn2n

(

)

arctan-20-40-602n

21

(Depend

on

and

/

n)

180o90o

时,0onnMdb

( )

0,

( )

0

时,0.1

1

10

100n-90o-180o),()

180nMdb

()

40lg(nwhen

(break

frequency)dbM

()

20

lg

2

,

()

90The

asymptote

is

independent

of

.The

discrepancy

depends

upon

.Phase

is

central

symmetry

about(ωn,

-90º).=0==3.0=1.0

,

M

()

1，G（j

）趋近于半圆=0.2=0.1=n令dM

()

0d122

1

2r

nr0

2

时,有谐振频率

1

2

2

,谐振峰值M

=0.1=0.220=0.3=0.50=1.0=0.7Approximate

magnitude-20-40=0.140dB

/

dec=0.2=0.3=0.5=0.7-90=1.0-1807)

Second-order

Differential

elementn

22n22ns22s

1

G(

j)

1

jG(s)

2j

arctann2

n222nn2n2

21)

eG

(

j

)

(1

)

(n

n时,dbM

()

0,

()

0

时,when

n(break

frequency)Mdb

()

20

lg

2

,(

)

90( )

180nMdb

( )

40

lg(

),6222

2M

db

(

)

20

lg

(1

2

)

(

)nn

2604020Approximate

magnitude40dB

/

dec2nn

21

( )

arctan0-20-40180o90o-90o0o()

180ndb(Depend

on

and

/

n)

n时,Mdb

()

0,

()

0

n时,M

()

40

lg(

),-180o0.1110100nwhen

n(break

frequency)Mdb

()

20

lg

2

,

()

90Asymptotes

of

Bode

diagrams4040

dB

/

dec200-20=1.0=0.7=0.5=0.3=0.2=0.118090=1.0=0.7=0.5=0.3=0.2=0.1njs8)

Delay

element

(延迟环节)G

(

s

)

e

j

G

(

j

)

ec(t)1(t

)r(t

)0－11M

(

)

1,

M

db

(

)

0( )

(rad

)

57.3

（oMdb200Magnitude

is

independentof

.Phase

angle

depends

on

and

τ.-200o－９0o－１８0o0.111010015.3 Steps

for

Plotting

Bode

Diagramsnnj

i

(

)i1GH

(

j

)

GH

(s)

|s

j

Mi

()

ei

1ni1Mdb

20

lg

M

()

20

lg

Mi

()nLogarithmic(对数化)

(

)

i

(

)i

1Method

1:

directly

adding

component

parts

(SP

15.2

)GH

(s)

500(s

2)s(s

20)(s2

4s

100)Preparation

Step

–

Writing

the

open-loop

frequency

response

inBode

form

ly:0.5(

1

s

1)GH

(s)

24s21s(

s

1)(

1

s

1)20

100

100Step

1–

Calculate

the

break

frequencies:

1

1;

2

2;

3

10

4

20Step

2–

Determine

the

frequency

range

to

be

plotted:0.1

200Step

3–Plot

the

straight-line

magnitude

approximationStep

4–

Graphically

add

all

element

magnitudes.Step

5–

Plot

Deviations(画出转折频率处的偏差）Step

6–

Complete

the

magnitude

curve.Step

7–

Plot

the

individual

phase

diagram

of

the

components.Step

8–

Graphically

add

all

element

phases.Method

2:•开环传函典型环节分解；•

将各环节的交接频率标在半对数坐标图的

轴上；开环传函典型环节分解；将各环节的交接频率标在半对数坐标图的轴上；•

低频段渐近特性曲线，即

<

bmin频段内，由积分环节和低频段渐近特性曲线，即<bmin频段内，由积分环节和开环增益决定K开环增益决定M( )

20

lg

20

lg

K

20

lg

(

)

90

db直线斜率为-20vdB/dec，还需确定直线上一点：K

K在<bmin范围内，任选一点0，计算Mdb

(0

)

20

lg

K

20v

lg0取频率为0=1，则Mdb

(1)

20

lg

K低频渐近线或其延长线与0分贝线的交点

v

v000

0

1

K

v

or

K

0

：20

lg•

对数幅频渐近曲线为分段折线，每一个交接频率处，曲线斜率发生变化，变化规律取决于该交接频率对应的典型环节的种类；•每两个相邻交接频率之间为直线对数幅频渐近曲线为分段折线，每一个交接频率处，曲线斜率发生变化，变化规律取决于该交接频率对应的典型环节的种类；每两个相邻交接频率之间为直线2

1sM

db

(

2

)

M

db

(1

)

k

(dB

/

dec

)lg

lg

直线方程：TypicalelementsTransferFunctionCrossoverre

uencSlopechange-orderelements11

Ts1T-

20dB/dec1

Ts20dB/decSecond-orderelements1s

2

2

s

1

2

n

nn-

40dB/decs

2

2

s

1

2

n

n40dB/decMinimum-phase

system:

分子m次，分母n次，

＝∞时，幅频渐近线斜率为-(n-m)