chapter 9 frequency response and transfer function
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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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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• 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
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• Signal Decomposition and Representation : (1) Periodic Signal --- Fourier Series Representation
§ 9.1 Dynamic Signal in Frequency Domain (2)
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
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• Frequency-Domain Representation : tAsin(t)x 01
§ 9.1 Dynamic Signal in Frequency Domain (3)
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
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• 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
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Deterministic dynamic signal can be considered as a combination of different sinusoidal signals in discrete and/or continuous frequency spectrum.
• Classification of Dynamic Signal :
§ 9.1 Dynamic Signal in Frequency Domain (5)
Dynamic Signal
Deterministic Chaotic Stochastic
Periodic Nonperiodic
Sinusoidal ComplexPeriodic
AlmostPeriodic
Transient
Most Simpliestform
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• 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
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• Frequency Transfer Function (Frequency Response Function, FRF):
A
B)G(j 0, system Static
§ 9.2 Transfer Function and Frequency Response (2)
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
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§ 9.2 Transfer Function and Frequency Response (3)
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
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• 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
§ 9.2 Transfer Function and Frequency Response (4)
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)
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§ 9.2 Transfer Function and Frequency Response (5)
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
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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)
§ 9.2 Transfer Function and Frequency Response (6)
'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
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§ 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
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• 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)
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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
§ 9.3 Representations of Frequency Transfer Function (3)
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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
§ 9.3 Representations of Frequency Transfer Function (4)
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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
§ 9.3 Representations of Frequency Transfer Function (5)
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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
§ 9.3 Representations of Frequency Transfer Function (6)
081:Phase
decade/dB40 slope ith w
0dB) ,( through line Straight:Mag
)egratorsintDouble()j(
1)G(j
n
2
n
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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
§ 9.3 Representations of Frequency Transfer Function (7)
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-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
§ 9.3 Representations of Frequency Transfer Function (8)
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
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• 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
§ 9.3 Representations of Frequency Transfer Function (9)
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• 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
§ 9.3 Representations of Frequency Transfer Function (10)
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• 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
§ 9.3 Representations of Frequency Transfer Function (11)
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:
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• 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
§ 9.3 Representations of Frequency Transfer Function (12)
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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
§ 9.3 Representations of Frequency Transfer Function (13)
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• 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
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• 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
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• 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
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§ 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
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§ 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
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§ 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