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
§ 9.1 Dynamic Signal in Frequency Domain (1)
• Laplace Transform and Fourier Transform :
Laplace transform for one-sided function x(t)
x(t),t  0
x(t)  
0, t  0

L( x( t ))    x( t )e st dt
0
one-sided Laplace Transf orm
s    j
Fourier transform for one-sided function x(t)
Restrict s    j to p  j

F0 ( x( t ))    x( t )ept dt
0
Extension to two-sided Fourier Transform

F0 ( x( t ))   x( t )e jt dt

Laplace Transform is a one-sided generalized Fourier Transform with
 t
weighted convergent factor e ,   0.
For one-sided function x(t), the Laplace transform is X(s). Then the Fourier
Transform of x(t) is X( j).
§ 9.1 Dynamic Signal in Frequency Domain (2)
• Signal Decomposition and Representation :
(1) Periodic Signal --Fourier Series Representation
x(t)
t
T
x(t  T)  x(t ),    t  , x(t)satisfiesthe Dirichlet conditions
a0 
a0 
x(t) 
  an cosn0 t  bn sinn0 t 
  c n sin(n0 t  n )
2 n1
2 n1
Mean of x(t)
1 T0
(  x(t)dt)
T0 0
DC-component
0  2 fundamental frequency
T
AC-components
c n  an  bn
2
2
n  tan1
an
bn
Periodic signal is represented as a combination of discrete sinusoidal signals.
(2) Nonperiodic Signal --- Fourier Integral Representation
1 
x(t)
j t
x(t) 
C(s)e
d

2  
C(s)  


x(t)e  st dt
, Fourier Transform
s  j
t
Nonperiodic signal includes continuous frequency components with
amplitude as spectral density.
§ 9.1 Dynamic Signal in Frequency Domain (3)
• Frequency-Domain Representation :
x1(t)  Asin0t
Frequency Domain
Time Domain
A
A
t=0
0 t
t
A
2

0
0
0
0


x2 (t)  Asin(0t  0 ), x2 (t ) leads x1(t) by 0, 0  0
A
t=0
A
0
0 t
t
A
2
0

0

0
0
0
0
0

x3 (t)  Asin(0t  0 ), x3 (t) is laged behind x1(t) by 0, 0  0
A
0 t
A
0
t=0
t
2
-A
0
0
0

0

0
0

0
§ 9.1 Dynamic Signal in Frequency Domain (4)
• Dynamic Signal and Measurement :
Arbitrary Function
Generator
Dynamic Signal
Source
t=0
Ideal Signal Flow
Oscilloscope
Spectrum Analyzer
Time Domain
Frequency Domain
1. 0
2
t

(sec)
1.0
1
t
(sec)

Modern Spectrum Analyzer utilizes FFT (Fast Fourier Transform) algorithm for
Real-time Fourier Transform.
§ 9.1 Dynamic Signal in Frequency Domain (5)
• Classification of Dynamic Signal :
Dynamic Signal
Chaotic
Deterministic
Periodic
Sinusoidal
Stochastic
Nonperiodic
Complex Almost
Periodic Periodic
Transient
Most Simpliest
form
Deterministic dynamic signal can be considered as a combination of different
sinusoidal signals in discrete and/or continuous frequency spectrum.
§ 9.2 Transfer Function and Frequency Response (1)
• Steady-state Sinusoidal Response :
Y(s)
X(s)
G(s)
t
0
2
0
Input : x(t)  X 0 sin 0 t, X(s) 
0 X 0
s 2  0
K(s  z 1 )    (s  z m )
, stable system
System: G(s) 
(s  p1 )    (s  p n )
Output : Y(s) X(s) G(s)
n
bi
a
a



s  j0 s  j0 i1 s  p i
2
y s.s.  lim y( t )  lim L1 ( Y(s))  ae  j0 t  a e  j0 t
t 
t 
 y s.s.  Y0 sin(0 t  )
Y0  X 0 G( j0 )

   G(j0 )
§ 9.2 Transfer Function and Frequency Response (2)
• Frequency Transfer Function (Frequency Response Function, FRF):
Def:
A sin t
G(s)
B sin(t  )
Static system   0, G(j ) 
Dynamic system
B
A
B
A
Amplitude ratio of the output sinusoid signal to the input sinusoid signal
Im( G( j))
  G(j)  tan1
, Time Delay  

Re( G( j))
Phase shift of the output sinusoid signal w .r.t.the input sinusoid signal
G(j) 
Ex: S.S. sinusoidal response and transmission of a mechanical system
y=x
F(s)
C
K
M
f (t)  a sin0 t
G( j)
s.s.
a
t
a
Y(s)
t
§ 9.2 Transfer Function and Frequency Response (3)
f (t)
y ( t )s . s .
f (t)
v ( t ) s. s.
a
b
t
 / 0
y(t)s.s. = bsin(ω0 t + f) b = G(jω0 ) a
Y(jω)
= G(jω) : Dynamic Compliance
F(jω)
= 1
Dynamic Stiffness
a
c
t
 / 0   / 20
v(t)s.s. = y(t)s.s. = ccos(ω0 t + φ) c = G(jω0 ) aω0
= ω0 × y(t +
π
)s.s.
2ω0
v(jω)
= (jω)G(jω) : Mobility
F(jω)
= 1
Impedance
§ 9.2 Transfer Function and Frequency Response (4)
• Frequency Transfer Function and Pole-Zero Diagram :
j
Im
jt
G( jt )
=0

G( jt )
-jt
2
 p2
2
3
 z1
t
G(s)
Gain=K
4

3
 p3
j
1
Re
G(s) s j
(1) G(j) G( j)

 Polar coordinate

(2) Re(G(j))  jIm(G(j ))
 Rectangular coordinate

G(jt ) 
K
4
1 2 3
G(jt )  (  1  2  3 )
1
 p1
G(s) 
K(s  z1 )
(s  p1 )(s  p2 )(s  p3 )
Angular frequencyfrom  0 to    can be realized by using slow ly- sw ept
sinusoidal w ave.
§ 9.2 Transfer Function and Frequency Response (5)
1
1  s
1
G(j) 
 G( j) G( j)
1  j
Ex : G(s) 

G(s) 
1
1  
2 2
  tan1 
1
1
1
 G' (s)
 (s  1 ) 
G' (j) 
1
j  1 

G(jω)
0
10
1
1
1
1
2
2
  45
G(j)
G(jω)
0
45 
.
.
.
.
.
.
.
.

090
0
90
§ 9.2 Transfer Function and Frequency Response (6)
j
1
'
G ( j)
t
1

Magnitude
t  0
1


Rectangular Plot
Static gain  1
G ( j)
'
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
Frequency
7
8
9
10

0
1
2
3
4
5
6
Frequency
7
8
9
10

Polar Plot
Phase(degree)
0
Im
0.5

1
0
45
1
2

1

Re
-20
-40
-60
-80
-100
§ 9.3 Representation of Frequency Transfer Function (1)
• Definitions :
Bode Plot – The plots of magnitude G( j) versus  in log-log rectangular
coordinate and phase G( j) versus  in semi-log rectangular
coordinates, especially through corner plot or asymptotic plot.
G( j) (dB)  20 log G( j) ,
1dB  0.1 bel,
G( j) (bel)  log10 G( j)
2
Power ratio
10
:
1
2
1 octave  frequency ratio :
1
1 decade  frequency ratio
 (log scale)
0.1
1
10 100
 (log scale)
0.5
1
2
4
Nyquist Plot – The plots of vectors G( j) G( j) in polar plot as  is varied
from zero to infinity.
§ 9.3 Representations of Frequency Transfer Function (2)
• Formulation of Bode Plot G(j) :
G( j)  G1( j)  G2 ( j)    GN ( j)
Magnitude in dB:
20log G(j)  20log G1(j)  20log G2(j)    20log GN(j)
Phase:
G( j)  G1( j)  G2 ( j)      GN ( j)
Bode’s Gain-Phase Theorem:
For any stable minimum-phase system, the phase of G( j) is uniquely
related to the magnitude of G( j).
(1) The slope of the G( j) versus  on a log-log scale is weighted most
heavily for the phase shift G( j) of a desired frequency.
(2) The log-log scale G( j) versus  in one portion of the frequency
spectrum and the phase G( j) in the remainder of the spectrum
may be chosen independently.
§ 9.3 Representations of Frequency Transfer Function (3)
Features of Bode’s Plot
1. G( j) can be constructed by the addition and subtraction of fundamental
building blocks in magnitude and phase, respectively.
Five fundamental building blocks:
(1) Constant gain K b
(2) Poles (zeros) at the origin j
(3) Poles (zeros) on the real axis 1  j
2
(4) Complex poles (zeros) 1 j2ξ  n  2 n
 jT
(5) Pure time delay (lead) e d
2. Same types of poles and zeros are mutual mirror images w.r.t. real axis.
§ 9.3 Representations of Frequency Transfer Function (4)
• Bode Plot of Fundamental Building Blocks :
1. Constant Gain, G(s)=Kb
G( j)  K b
Mag : 20logKb (dB), constant
Phase : 0, No delay (static element)
1
2. Pure Integrator, G(s) 
s
1
G( j) 
j
1
Mag : 20log
 20 log (dB)
j
Phase :  90 (lag 90)
Mag ~  (log scale)
  1  0dB
  10  20dB
Mag
( dB )
20 log K b
0
 0.1
(deg)
1
10
100
 (log scale)

0
Mag
( dB )
20
0
 0.1
(deg)
0
 90
1
10
 (log scale)

Mag  20 (log )
 Straight line with slope  20dB / decade
Output amplitude is reduced as input frequency is increased.
§ 9.3 Representations of Frequency Transfer Function (5)
1
3. First order pole, G(s) 
1  s
1
G( j) 
1  j
1
Mag : 20log
 10 log(1  2 2 )
1  j
Mag
(dB)
Asymptotes
1


0
-3

Im(G( j))
1
Phase :   tan
  tan ( )
(deg)
Re(G( j))
Asymptotes:
1
0
(i) Low frequency   (  1)

Mag : 0dB -45
G(j)  1 (pure gain)  
Phase : 0-90
 (log scale)
1
1
(ii) High frequency   (  1)

G(j) 

1

 (log scale)
Asymptotes
Mag : 20log (slope  20dB / decade)
1
(pure integrator)  
j
Phase :  90 (lag 90)
(iii) Corner frequency  
1


1
1
1

, 20log
 3dB
1
Mag :
G(j )  
1 j
1 j
2

Phase : 45

§ 9.3 Representations of Frequency Transfer Function (6)
Mag
n
(dB)
(  1.0) -34dB
4. Complex Poles G(s)  2
2
s  2ns  n
1
0
G( j) 
-3dB
2
j 2
1  ( )( j)  ( )
n
n

(deg)

1
set u 
 G( j) 
0
n
1  2uj  ( ju)2
2
Mag : 20log G(j )  10 log (1  u )  4 u
2u
Phase :    tan 1
1  u2
2 2
2
2
Mag : 0dB
G(j)  1 (Pure gain)  
Phase : 0
(ii) High frequency  n (u  1)
  0.01
  0 .5
1

  0.707
n
(log scale)
n
(log scale)
  0.01
1
  0 .5

-90
  0.707
-180
Asymptotes
Asymptotes:
(i) Low frequency   n (u  1)
Asymptotes
  n

Mag (dB)
phase slope
0.01
0.5
34
0
-3
100
0.707
2
2
Mag : Straight line through (n , 0dB)
1

G(j) 
(Double int egrators)  
with slope  40dB / decade
 2
Phase : 180
(j )

n
(iii ) Corner frequency  n
Mag : 20log 2 (dB)
G(j n )  
Phase : 90
§ 9.3 Representations of Frequency Transfer Function (7)
5. Pure Time Delay G(s)  eTds
G( j)  e  jTd
Mag : 0dB
Phase :   Td (rad) (Linear functionof )
180

Td (deg)

Td  1, G(j)  e j
 (rad s)
 (deg)
0
0
0.1
5.73
1
2
57.3
114.6

Mag
(dB)
 (log scale)
0

(deg)
0
5.73
57.3

114.6
0 .1
1
2
 (log scale)
§ 9.3 Representations of Frequency Transfer Function (8)
s
8(1  )
2
 corner frequency : A  2, B  8, C  24
Ex: G(s) 
s
s
s(1  )(1 
)
8
24
Bode Diagram
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
40
Magnitude (dB)
AM
-20dB/D
20
-20dB/D
0dB/D
0
asymptotes
BM
CM
-20
-40dB/D
-40
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∘
-60
-80
＊
-45
Phase (deg)
A Pi
P1 C
Pi
BPi
AP
BP
＊
-90
P2
A Pf
CP
＊
-135
P3
BPf
CPf
-180
-1
10
0
10
2
8 101
24
Frequency (rad/sec)
2
10
3
10
Corner Phase:
(1)P1: -45∘
(2)P2: -90∘
(3)P3: -135∘
§ 9.3 Representations of Frequency Transfer Function (9)
• Non-minimum Phase G(j) :
j
t
Gm (s) 
sz
, p, z  0
sp

Nonminimum phase
Gn (s)
90
j
t
sz
Gn (s) 
, p, z  0
sp
180
0
p
z

 (log scale)
Minimum phase Gm (s)
A non-minimum phase all pass network
G( j)
j
1

 (log scale)
G( j)
0
 180
G(s) pole-zero diagram
Symmetric lattice network
 360
n
 (log scale)
§ 9.3 Representations of Frequency Transfer Function (10)
• Phase Lead and Phase Lag Compensator :
Phase Lead:
G(s) 
1  s
1  s
Mag (dB)
20 log10 
1
m 
 

  sin1(

 1
)
 1
Phase Lag:
60 
50 
m
1

1

m
Mag (dB)
1  s
G( s) 
1  s
1
m 
 

1

m
 (log scale)
40
30
20
10
0
1 2 3 4 5 6 7 8 9 10
 (log scale)
1

 (log scale)
 20 log10 

m

 (log scale)
Lead and Lag Compensators are mutual mirror images w.r.t. real axis.

§ 9.3 Representations of Frequency Transfer Function (11)
• General Shape of Nyquist Plot :
K[(j)  b (j)    b ] b (j)  b (j)  
G(j) 

, n  N q
(j) [(j)  (j)    a ]
(j)  a (j)  
nm
Asymptotes:
Low frequency   0
m1
m
m1
N
m1
m
q1
q
m1
m
0
n1
n
n1
0
Im
N  0, lim G( j)  0
0

Type 2
0
Type 0
0
Re
N  1, lim G( j)  90
0
N  2, lim G( j)  180
0
0
Type 1
High frequency   
Im
n  m  1, lim
G( j)  90
 
n-m=3
n  m  2, lim G( j)  180
 
n-m=2
Re
 
n-m=1
n  m  3, lim G( j)  270
 
§ 9.3 Representations of Frequency Transfer Function (12)
• Nyquist Plot of Fundamental Building Blocks :
n
4. Complex Poles, G(s) 
2
Im
s2  2ns  n
1. Constant Gain, G(s)=Kb
j
2
1


 90
  n
  n
Kb
1
2
1
2. Pure Integrator, G(s) 
j
s
  90
j
peak frequency
Approach circle for   1
5. Pure Time Delay, G(s)  e
 Tds
Im
1

0
3. First-order Pole, G(s) 
Im

1
1  s
1
0.5
0
45
Re

2Td
unit circle


Td



1

Re
 decreas


0
3
2Td
0
Re
§ 9.3 Representations of Frequency Transfer Function (13)
e  j T
Ex: Polar plot of G( j) 
(1  j )
Im
d

1
Re
0
Spiral
Ex: Polar plot of G( j) 
Im


0
1
( j)(1  j)
Re
§ 9.4 Frequency Domain Specifications of System Performance (1)
• Frequency Response Test :
Obtain the steady-state frequency response of a system to a sinusoidal
input signal.
Controlled Environment
Function
Generator
t0
x( t )
L-T-I
System

A
y( t )
s.s.
B
t
0
Phase measurement by Lissajou Plot
A
y( t )
B
Recorder
t
0
2
tan
1

same frequency
x( t )
x(t )  A sint
y( t )
B
2
 2

2 1
x( t )
y( t )
A
B
t
y( t )
y(t )  A sin(t  )
t

  0
x( t )
A
For nonlinear system, the output response is not in the same sinusoidal
waveform and frequency as those of input signal.
x( t )
A
  90
§ 9.4 Frequency Domain Specifications of System Performance (2)
• System Identification :
Assume the LTI system:
X( j)
LTI System
Y( j)
Levy's method (1959)
a0  a1s  a2s2  a3s3    
G(s) 
Total no. of unknow n # ai  # bi  N
b0  b1s  b2s2  b3s3    
(a0  a22  a44  )  j(a1  a32  a54  )   j N()
 G(j) 


2
4
2
4
(b0  b2  b4  )  j(b1  b3  b5  )   j D()
Error betw eenmeasured curveF(j) and model G(j) :
N()
E( )  F(j)  G( j)  F( j) 
D()
Define w eightederror :
D( )E( )  D( )F(j)  N()  A()  jB()
D()E()  A 2 ()  B2 ()
m
Total error in m  1 frequencypoints K
Et   ( A 2 (K )  B2 (K ))
K 0
Least - squareminimizati on :
MinE t  N equations
a i , bi
Solve N unknow nai, bi by matrix inversion
A simplified low - order model can be directly identified in experimental test.
§ 9.4 Frequency Domain Specifications of System Performance (3)
• Frequency-Domain Specifications :
Cutoff Frequency and Bandw idth:
3dB decay  Half pow erpoint
Gain(dB)
M
p
3dB
0
Slope, Sc.f.
T(0 )
0

B.W.
p
c.f .
 (log scale)
M : Resonance Peak
r
 : Resonance Frequency(   )
r
M : Maximum Peak
r
n
Pow er  F  V  CV 2 (For damper)
Pow er 1
1
2
 CV 2  CV2  V2 
V
2
2
2
(1) Control system
DC to 3dB decay
(2) Narrow - band structuralsystem
Peak to 3dB decay
Gain
(dB)
3dB
p
 : Peak Frequency
p
 : Cutoff Frequency

c.f .
B.W. : Bandwidth (Usually 3dB decay point) 0    
c.f .
s : Cutoff rate
c.f .
B.W.
(3) Others (Ex : EH servo)
Ex : Frequency at  90 phase point
§ 9.4 Frequency Domain Specifications of System Performance (4)
For 1st-order pure dynamics
C(s)
1

R(s) 1  s
No resonance peak
T(s) 
1
1
B.W.  , ( c.f.  )


T(j)  10%,   10c.f.
Sc.f.  20dB / decade
For 2nd-order pure dynamics

C(s)
T(s) 

R(s) s  2 s  
2
n
2
n
2
n
1
M 
,   0.707, M  % o.s.
2 1  
p
   1-2
p
p
2
(Note :      )
2
n
p
d
B.W.   [(1  2 )  4  4  2 ]
4
2
n
T(j)  10%,   3
c.f .
S
c.f .
 40dB / decade
2
n
1
2

B.W.
0.707 n
 1 2n
a
§ 9.4 Frequency Domain Specifications of System Performance (5)
Ex: Identify the structure modal parameters of the experimental FRF given by
dB
F(s)
1
2
ms  cs  k
28
4
0
0
8
200

G(s) 
-52
0
 180

 90
Y(s)
Y(s)
F(s)

1
r
r
[(
)(
)]
2j s  p
sp
p, r are modal parameters (complex number)
1
Sol : G(j) 
m(j )2  c( j)  k
(1)   n
p    j
r  r 
Y(j) 1
1
 , 20log  4dB  k  0.631N
m
F(j) k
k
(2)   n
Y(j)
1
1
rad

,


200
,
20log[
]  52dB  m  0.0099kg
sec
F(j) m2
m(200)2
(3)   n
Y(j)
1
1

,   8 rad
, 20log( )  28dB  c  0.0049N  s
sec
m
F(j) c
8c
§ 9.4 Frequency Domain Specifications of System Performance (6)
102.04
s2  0.48s  64.28
102.04

(s  0.24  j8.01)(s  0.24  j8.01)
G(s) 

Pole  zero diagram
j
Static gain
=1.563
j8.01
1
12.74
12.74
(

)
2j (s  0.24  j8.01) (s  0.24  j8.01)

 0.24
Modal parameters :
  0.24
  8.01
r  12.74
  0
The impulse response functionis
x(t)  r e t sin(t   )
 12.74e 0.24t sin(8.01t )
 j8.01