chapter 9 frequency response and transfer function § 9.1 dynamic signal in frequency domain § 9.2...

Chapter 9 Frequency Response and Transfer Function

§ 9.1 Dynamic Signal in Frequency Domain

§ 9.2 Transfer Function and Frequency Response

§ 9.3 Representations of Frequency Transfer Function

§ 9.4 Frequency Domain Specifications of System Performance

• Laplace Transform and Fourier Transform :

§ 9.1 Dynamic Signal in Frequency Domain (1)

Laplace transform for one-sided function x(t)

js

ansformLaplace Trone-sided dte)t(x))t(x(L

0t 0,

0t x(t),x(t)

0

st

Fourier transform for one-sided function x(t)

Extension to two-sided Fourier Transform

Laplace Transform is a one-sided generalized Fourier Transform with weighted convergent factor

For one-sided function x(t), the Laplace transform is X(s). Then the Fourier Transform of x(t) is

0

pt0 dte)t(x))t(x(F

jp to js Restrict

tj0 dte)t(x))t(x(F

0. ,e t

).j(X

• Signal Decomposition and Representation : (1) Periodic Signal --- Fourier Series Representation


Periodic signal is represented as a combination of discrete sinusoidal signals.(2) Nonperiodic Signal --- Fourier Integral Representation

conditions Dirichlet the satisfies x(t),t ),t(x)Tt(x

j t

st

s j

1x(t) C(s)e d

2

C(s) x(t)e dt , Fourier Transform

Nonperiodic signal includes continuous frequency components with amplitude as spectral density.

n

n1n b

atan2

n2

nn bac frequency lfundamenta T2

0

)x(t)dtT

1(

x(t)of Mean

0T

0 0

1n

n0n0

1n0n0n

0 )tnsin(c2

atnsinbtncosa

2

ax(t)

DC-component

AC-components

t

x(t)

t

x(t)

T

• Frequency-Domain Representation : tAsin(t)x 01


0 , by )t( xleads )t( x),tAsin((t)x 0012002

0 , by )t( xbehind laged is )t( x),tAsin((t)x 0013003

t=0 t

A

-A

t0

A

0

Time Domain Frequency Domain

0

2

0

0

t=0

t

A

-A

t0

A

0

0

00

2

0

0

0

0

t=0

t

A

-A

t0

A

0

0

0

0

2 0

0

00

• Dynamic Signal and Measurement : § 9.1 Dynamic Signal in Frequency Domain (4)

Modern Spectrum Analyzer utilizes FFT (Fast Fourier Transform) algorithm for

Real-time Fourier Transform.

Arbitrary FunctionGenerator

t=0

Dynamic SignalSource

Ideal Signal Flow

Oscilloscope Spectrum Analyzer

Time Domain Frequency Domain

(sec)

(sec)

t

t

0.1

0.1

1

2

Deterministic dynamic signal can be considered as a combination of different sinusoidal signals in discrete and/or continuous frequency spectrum.

• Classification of Dynamic Signal :


Dynamic Signal

Deterministic Chaotic Stochastic

Periodic Nonperiodic

Sinusoidal ComplexPeriodic

AlmostPeriodic

Transient

Most Simpliestform

• Steady-state Sinusoidal Response :

)G(j

)j(GXY

)tsin(Yy

eaae))s(Y(Llim)t(ylim y

ps

b

js

a

js

a

G(s)X(s) Y(s):Output

system stable ,)ps()ps(

)zs()zK(sG(s) :System

s

XX(s) ,tsinX x(t):Input

0

000

00.s.s

tjtj1

tts.s.

n

1i i

i

00

n1

m1

20

2

0000

00

§ 9.2 Transfer Function and Frequency Response (1)

Y(s)X(s)G(s)

t0

0

2

• Frequency Transfer Function (Frequency Response Function, FRF):

A

B)G(j 0, system Static


G(s)tsinA )tsin(B

Def:

signal sinusoid input the w.r.t.signal sinusoid output the of shift Phase

Delay Time ,))j(GRe(

))j(GIm(tan )G(j

signal sinusoid input the to signal sinusoid output the of ratio mplitude AA

B)G(j

1

Ex: S.S. sinusoidal response and transmission of a mechanical system

C

K M

y=x

tsina)t(f 0

Y(s)F(s))j(G

a

a

t t

s.s.

system Dynamic


s.s. 0 0y(t) = bsin(ω t + f) b = G(jω ) a

Y(jω)= G(jω) : Dynamic Compliance

F(jω)

1 = Dynamic Stiffness

s.s. s.s. 0 0 0

0 s.s.0

v(t) = y(t) = ccos(ω t + φ) c = G(jω ) aω

π = ω × y(t + )

2ω

v(jω)= (jω)G(jω) : Mobility

F(jω)

1 = Impedance

ab

t

)t(f

.s.s)t(y

0/

ac

t

)t(f

.s.s)t(v

0 0/ / 2

• Frequency Transfer Function and Pole-Zero Diagram :

coordinate rRectangula

))jIm(G(j))Re(G(j (2)

coordinate Polar

)j(G)G(j (1)

)s(Gjs


4t

1 2 3

t 1 2 3

KG(j )

G(j ) ( )

wave.sinusoidal swept-slowly usingby realized be can to 0 fromfrequency Angular

)ps)(ps)(ps(

)zs(K)s(G

321

1

3

j

34

2

13p 1z

2p

1p

1

2t Gain=K

j

tj

=0

-jt

Re

Im

)j(G t

)j(G t

G(s)


1

22tan

1

1

)j(G)j(Gj1

1)G(j

s1

1G(s) :Ex

)s('G1

)1(s

11G(s)

1j

1)(jG'

0

G(j )

1 0

0 90

.

.

.

.

G(jω) G(jω)

1

.

.

.

.

1 0

2

145

0 90

145

2

0 1 2 3 4 5 6 7 8 9 100.2

0.4

0.6

0.8

1

Frequency

Mag

nitu

de

0 1 2 3 4 5 6 7 8 9 10-100

-80

-60

-40

-20

0

Frequency

Pha

se(d

egre

e)


'G (j )

j

1

t

0 t

1

'G (j )

1Static gain

1

Im

Re0.5

1

0 45

2

1

Polar Plot

Rectangular Plot

§ 9.3 Representation of Frequency Transfer Function (1)• Definitions :

Bode Plot – The plots of magnitude versus in log-log rectangular

coordinate and phase versus in semi-log rectangular

coordinates, especially through corner plot or asymptotic plot.

)j(G

)j(G

2

10

20

G(j ) (dB) log G(j ) ,

1dB 0.1 bel,

G( j ) (bel) log G(j )

Nyquist Plot – The plots of vectors in polar plot as is varied

from zero to infinity.

)()( jGjG

:1

2 ratiofrequency octave 1

:1

10 ratiofrequency decade 1

0.1 1 10 100

scale) (log

4210.5scale) (log

Power ratio

• Formulation of Bode Plot G(j) : )j(G)j(G)j(G)j(G N21

Magnitude in dB:

1 220 20 20 20 Nlog G(j ) log G (j ) log G (j ) log G (j )

Phase:

)j(G)j(G)j(G)j(G N21

Bode’s Gain-Phase Theorem:

For any stable minimum-phase system, the phase of is uniquely

related to the magnitude of .

(1) The slope of the versus on a log-log scale is weighted most

heavily for the phase shift of a desired frequency.

(2) The log-log scale versus in one portion of the frequency

spectrum and the phase in the remainder of the spectrum

may be chosen independently.

)j(G

)j(G

)j(G

)j(G

)j(G )j(G

§ 9.3 Representations of Frequency Transfer Function (2)

1. can be constructed by the addition and subtraction of fundamental

building blocks in magnitude and phase, respectively.

Five fundamental building blocks:

(1) Constant gain

(2) Poles (zeros) at the origin

(3) Poles (zeros) on the real axis

(4) Complex poles (zeros)

(5) Pure time delay (lead)

2. Same types of poles and zeros are mutual mirror images w.r.t. real axis.

Features of Bode’s Plot

bK

j

)j(G

j12

n2

nj2ξ1 dTje


1. Constant Gain, G(s)=Kb

2. Pure Integrator,

• Bode Plot of Fundamental Building Blocks :

bK)j(G

element)(static delay No ,0 :Phase

constant ),dB(20logK :Mag b

s

1)s(G

j

1)j(G

)dB(log20j

120log :Mag

Output amplitude is reduced as input frequency is increased.decade/dB20 slope withline Straight

)(log 20Mag

20dB10

0dB1

scale) (log ~Mag

)90 (lag 90 :Phase

scale) (log

01 10 100

)dB(

Mag

bKlog20

(deg)

0

0.1

scale) (log

01 10

)dB(

Mag

20

(deg)

0

90

0.1


3. First order pole,

s1

1)s(G

j1

1)j(G

)1log(10j1

120log :Mag 22

)1( 1

frequency Low )i(

0:Phase

0dB:Mag gain) (pure 1)G(j

)(tan))j(GRe(

))j(GIm(tan :Phase 11

:Asymptotes

)1( 1

frequency High )ii(

)90 (lag 90 :Phase

)decade/dB20 (slope 20log:Mag )integrator pure(

j

1)G(j

1

frequency Corner )iii(

45:Phase

dB3j1

120log ,

2

1

j1

1:Mag

)1

G(j

Mag(dB)

0

-45

0

-3

-90

(deg)

1Asymptotes

Asymptotes

scale) (log

scale) (log

1


4. Complex Poles

)0.1(

s2s)s(G

2nn

2

2n

2

nn

)j

()j)(2

(1

1)j(G

2

n )ju(uj21

1)j(Gu set

2222 u4)u1(log10)G(j20log :Mag

2

1

u1

u2tan :Phase

:Asymptotes

)1u( frequency Low )i( n

)1u( frequency High )ii( n

frequency Corner )iii( n

09:Phase

(dB) 220log:Mag)G(j n

)dB( Mag slope phase

0.5

0.707

0.01

0

-3

34

22

100

n

-180

0

-90

(deg)

Asymptotes

1

Mag(dB)

0-3dB

Asymptotes

-34dB

1

scale) (logn

scale) (logn

01.0

01.0

5.0

5.0

707.0

707.0

0:Phase

0dB:Mag gain) (Pure 1)G(j


081:Phase

decade/dB40 slope ith w

0dB) ,( through line Straight:Mag

)egratorsintDouble()j(

1)G(j

n

2

n

5. Pure Time Delay

(deg) T180

) of function (Linear )rad( T :Phase

0dB :Mag

e)j(G

d

d

jTd

jd e)G(j ,1T

)srad( (deg)

0

0.1

1

0

5.73

57.3114.6

2

sTde)s(G

(deg)

0

Mag)dB(

scale) (log

scale) (log 1.0 1 2

6.114

3.5773.50


-80

-60

-40

-20

0

20

40

Ma

gn

itu

de

(d

B)

10-1

100

101

102

103

-180

-135

-90

-45

Ph

as

e (

de

g)

Bode Diagram

Frequency (rad/sec)

MA

MB

MC

PA PB

PCPfA

PiBPiA

PfB

PiC

PfC

-20dB/D

0dB/D

-20dB/D

-40dB/D

P1

P2

P3

＊

＊

＊

2 8 24

A B C

s8(1 )

2G(s) corner frequency : 2, 8, 24s s

s(1 )(1 )8 24


Ex:

Phase:(1)Starting from -90∘(2)From APi (0.1AP) to APf (10AP): increase 90∘(3)From BPi (0.1BP) to BPf (10BP): decrease 90∘(4)From CPi (0.1CP) to CPf (10CP): decrease 90∘

Magnitude:(1) 1st slope: -20 dB/decade(2) 2nd slope: 0 dB/decade(3) 3rd slope: -20 dB/decade(4) 4th slope: -40 dB/decade

Corner Phase:(1)P1: -45∘(2)P2: -90∘(3)P3: -135∘

asymptotes

• Non-minimum Phase G(j) :

j

t

1

)j(G

)j(G

0

180

360n scale) (log

scale) (log

t

j

0z p, ,ps

zs)s(Gm

0z p, ,ps

zs)s(Gn

A non-minimum phase all pass network

G(s) pole-zero diagram Symmetric lattice network

j

180

90

0z p

mMinimum phase G (s)

n

Nonminimum phase

G (s)

scale) (log


• Phase Lead and Phase Lag Compensator :

1

1

s

G(s)s

1

m

)1

1(sin 1

Phase Lead:

Phase Lag:

s1

s1)s(G

1

m

Lead and Lag Compensators are mutual mirror images w.r.t. real axis.

m

m

1

1

)dB( Mag

10log20

scale) (log

scale) (log )dB( Mag

m

10log20

1

1

m

scale) (log

scale) (log

1 2 3 4 5 6 7 8 9 10

0102030405060


• General Shape of Nyquist Plot :1 1

1 0 1

1 1

0 1

m m m m

m m m

N q q n n

n

K[( j ) b ( j ) b ] b ( j ) b ( j )G( j ) , n N q

( j ) [( j ) ( j ) a ] ( j ) a ( j )mn

270)j(Glim 3,mn

180)j(Glim 2,mn

90)j(Glim 1,mn

Low frequency

Im

Re0

0Type 2

0 Type 1

Type 0


180)j(Glim 2,N

90)j(Glim 1,N

0)j(Glim 0,N

0

0

0

High frequency

0

n-m=2

n-m=3

n-m=1

Im

Re

Asymptotes:

• Nyquist Plot of Fundamental Building Blocks : 1. Constant Gain, G(s)=Kb

2. Pure Integrator, s

1)s(G

j

bK

3. First-order Pole,

s1

1)s(G

4. Complex Poles,

s2s)s(G

2nn

2

2n

5. Pure Time Delay,

sTde)s(G

dT2

Im

Re

unit circle

dT2

3

dT

0

j

0

1j 90

0.5 1

Im

Re045

1

Im

Re 0

n

1

21

decreas

90

n

Approach circle for 1

peak frequency


Ex: Polar plot of

Ex: Polar plot of

)j1(

e)j(G

dTj

)j1)(j(

1)j(G

Im

Re

0

1

Im

Re

Spiral

0


• Frequency Response Test : § 9.4 Frequency Domain Specifications of System Performance (1)

Obtain the steady-state frequency response of a system to a sinusoidal

input signal.

For nonlinear system, the output response is not in the same sinusoidal

waveform and frequency as those of input signal.

FunctionGenerator

)t(x )t(y

Recorder

Controlled Environment

L-T-ISystem

A

0t

Phase measurement by Lissajou Plot

t0

2

A

tsinA)t(x

)tsin(A)t(y

t

t

)t(x

)t(y

same frequency

1

2

B)t(y

A)t(x

1

2

2tan

t0

Bs.s.

B)t(y

A)t(x

90

B)t(y

A)t(x

0

• System Identification : § 9.4 Frequency Domain Specifications of System Performance (2)

(1959) method sevy'L

:)G(j model and )F(j curve measured between Error

:error weightedDefine

K pointsfrequency 1m in error Total

:onminimizati square-Least

inversionmatrix byb ,a unknown N Solve ii

test. alexperiment in identifieddirectly be can model order-low simplified A

)(D

)(N

j

j

)bbb(j)bb(b

)aaa(j)aa(a)G(j

45

231

44

220

45

231

44

220

Nb#a#unknown of no. Total sbsbsbb

sasasaaG(s) ii3

32

210

33

2210

)(D

)(N)j(F)j(G)F(j)E(

)(jB)(A)(N))F(jD())E(D(

))(B)(A(Em

0KK

2K

2t

:system LTI the AssumeLTI System

)j(X )j(Y

)(B)(A)(E)(D 22

equations NMinEii b ,a

t

• Frequency-Domain Specifications : § 9.4 Frequency Domain Specifications of System Performance (3)

damper) (For CVVFPower

point power Half decay 3dB

:Bandwidth andFrequency Cutoff

2

0

r

r r n

p

p

c . f .

c . f .

c . f .

M : Resonance Peak

: Resonance Frequency( )

M : Maximum Peak

: Peak Frequency

: Cutoff Frequency

B.W. : Bandwidth (Usually 3dB decay point)

s : Cutoff rate

V 2

1VCVCV

2

1

2

Power2

22

2

decay 3dB to Peak

system structural band-Narrow (2)

decay 3dB to DC

system Control (1)

point phase 90 atFrequency :Ex

servo) EH :(Ex Others (3)

)dB(Gain

scale) (log

pM

0 p .f.c

c.f.S ,SlopedB3

.W.B

T(0 )

0

)dB(

Gain

dB3

.W.B

§ 9.4 Frequency Domain Specifications of System Performance (4)

2

2 2

2

2 2

2

2 4 4 2

10

40

n

n n

p p

2

p n p d n

14 2

n

c . f .

c . f .

C(s)T(s)

R(s) s s

1M , 0.707, M % o.s.

2 1

1-2 (Note : )

B.W. [(1 ) ]

T(j ) %, 3

S dB / decade

For 1st-order pure dynamics

For 2nd-order pure dynamics

1

1

1 1

10 0

20

c.f.

c.f .

c.f .

C(s)T(s)

R(s) s

No resonance peak

B.W. , ( )

T(j ) %, 1

S dB / decade

.W.B

707.0 n

n2

a

1

§ 9.4 Frequency Domain Specifications of System Performance (5)Ex: Identify the structure modal parameters of the experimental FRF given by

k)j(c)m(j

1)G(j :Sol

2

1

2

Y(s)G(s)

F(s)

r r [( ) ( )]

j s p s p

p, r are modal parameters (complex number)

p j

r r

)s(F Y(s)

kcsms1

2

n (1)

Y(j ) 1 1 N, 20log 4dB k 0.631 mF(j ) k k

n (2)

2 2

Y(j ) 1 1rad , 200 , 20log[ ] 52dB m 0.0099kgsecF(j ) m m(200)

n (3)

28

Y(j ) 1 1rad s , 8 , 20log( ) dB c 0.0049Nsec mF(j ) c 8c

dB

90

1800

4

28

80 2000

-52

§ 9.4 Frequency Domain Specifications of System Performance (6)

))01.8j24.0s(

74.12

)01.8j24.0s(

74.12(

2j

1

)01.8j24.0s)(01.8j24.0s(

04.102

28.64s48.0s

102.04G(s)

2

0

12.74r

8.01

0.24

:parameters Modal

)t01.8sin(12.74e

)tsin(er x(t)

is function response impulse The

0.24t

t

01.8j

01.8j

24.0

jStatic gain

=1.563

diagram zeroPole

chapter 9 frequency response and transfer function § 9.1 dynamic signal in frequency domain § 9.2...

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