Fundamentals of
Measurement Technology
(6)
Prof,Wang Boxiong
For dynamic measurement,the measuring system
must be a linear one,
We can only process linear systems mathematically,
It is rather difficult to perform nonlinear corrections in
situations of dynamic measurement,
Practical systems may be considered as linear systems
within a certain range of operation and permissible
error limits,
It is of general significance to study linear
systems.
3.4 Dynamic characteristics of measuring systems
The input-output relationship of a linear system:
where
x(t)= input of the system
y(t)= output of the system
an,a1,a0,and bm,b1,b0 are system’s parameters.
A linear constant-coefficient system or linear
time-invariant (LTI) system,the parameters are
constants.
3.4.1 Mathematical representation of linear systems
txb
dt
tdx
b
dt
txd
b
dt
txd
b
tya
dt
tdy
a
dt
tyd
a
dt
tyd
a
m
m
mm
m
m
n
n
nn
n
n
011
1
1
011
1
1
(3.3)
Properties:
1,Superposition property (superposability):
If for
then
2,Proportionality
If
then
Where a is a constant.
3.4.1 Mathematical representation of linear systems
tytx 11?
tytx 22?
tytytxtx 2121 (3.4)
tytx?
taytax? (3.5)
3,Differentiation
If
then
4,Integration
If
and for a zero initial condition of the system,then
3.4.1 Mathematical representation of linear systems
tytx?
dt
tdy
dt
tdx? (3.6)
tytx?
tt dttydttx 00
(3.7)
5,Frequency preservability
If
and for
then the output
3.4.1 Mathematical representation of linear systems
tytx?
tjextx?0?
tjeyty 0
Proof:
According to the proportionality property
According to the differentiation property
Since
3.4.1 Mathematical representation of linear systems
tytx 22 (3.8)
2
2
2
2
dt
tdy
dt
txd? (3.9)
2
2
2
2
2
2
dt
tdyty
dt
txdtx (3.10)
tjextx?0?
tx
ex
exj
dt
txd
tj
tj
2
0
2
0
2
2
2
Letting the left-hand side of Eq,(3.10) be zero,
then the right-hand side of Eq,(3.10) must also be
zero,
Solving the equation yields,
where φ is the phase shift,
3.4.1 Mathematical representation of linear systems
0222 dt txdtx?
0222 dt tydty?
tjeyty 0
1,Transfer function
Definition:
For t?0,y(t)=0,the Laplace transform Y(s)
of y(t) is defined as
where s is the Laplace operator,s=a+jb for
a>0.
3.4.2 Representation of system’s characteristics in terms of transfer
function or frequency response
0 dtetysY st
(3.11)
If all the system’s initial conditions are
zero,making Laplace transform of Eq,(3.3)
gives then the expression
The transfer function H(s):
The transfer function H(s) represents the
transfer characteristics of a system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
0111
01
1
1
bsbsbsbsX
asasasasY
m
m
m
m
n
n
n
n
01
1
1
01
1
1
asasasa
bsbsbsb
sX
sYsH
n
n
n
n
m
m
m
m
(3.12)
Properties:
1) The transfer function H(s) won’t change with
input x(t),It represents only the system’s
characteristics,
2) The system described in terms of its transfer
function will provide clearly an output y(t)
for any concrete input x(t).
3) All the coefficients in the equation,an,… a1,
a0,and bm,…,b1,b0,are the constants
determined uniquely by the configuration of
system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
2,Frequency response function
For a steady-state LTI system,let s=jω,
that is,substitute a=0 and b=ωin s=a+jb,
then Eq,(3.11) changes to
Eq,(3.16) is an equivalent of the one-sided
Fourier transform expression discussed in
Chapter 2,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
0 )()( dtetyjY tj (3.16)
In the same manner,we have
H(jω) is called the frequency response or
frequency response function of a measuring
system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
)(
)(
)(
01
1
1
01
1
1
jX
jY
ajajaja
bjbjbjb
jH
n
n
n
n
m
m
m
m
(3.17)
The frequency response function is a special
case of the transfer function,
Although both the transfer function and the
frequency response can be used to represent
the dynamic characteristics of systems,their
significances are different,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.7 Representation of system’s output under different inputs by use of (a)
transfer function,and (b) frequency response function
The frequency response function intuitively
reflects responses of system to input signals of
different frequencies,
To get better measurement results,it is often to
do measurement when a system reaches its steady
state,
We often use the frequency response function in
measurements to describe system’s dynamic
characteristics,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
To control a system,since it is often required
to study its responses to typical disturbances
and the whole features of a process from its
initial transient stage to its final steady-state
stage,the transfer function must be then used,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Usually,H(jω) is a complex function:
where A(ω) is the modulus of the complex
function H(jω),
which is called the amplitude-frequency
characteristic of the system,
φ(ω)is the phase angle of H(jω)
which is called the phase-frequency
characteristic of the system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
)()()()( AeAjH j (3.18)
jH
X
YA)( (3.19)
xyjH a r g (3.20)
Also
P(ω) and Q(ω) are real functions of ω.
So
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
jQPjH
(3.21)
22 QPA
(3.22)
If the argument ω is expressed in logarithmic
scales and the amplitude A(ω) is expressed in
decibels (dB),then the two graphs for both
amplitude and phase angle of H(jω) are called
Bode plots.
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.8 Bode plots of a first-order system
The polar (or Nyquist) plot is a way of presenting
the frequency response of a system G(jω) on one
graph,as opposed to the separate amplitude and
phase angle Bode plots,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig.3.9 Nyquist plot of the first-order system in Fig,3.8
3,Transfer characteristics of first-
order and second-order systems
Eq,(3.12) can be written as
where αi,βi,piand qi are constants,
Any system can be considered as a parallel
combination or a series combination of a
number of first-order and second-order
elements,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
2
1
22
1 2
rn
i ninii
ii
r
i i
i
ss
s
ps
qsH
(3.23)
Similarly,we have
Conclusion:
The transfer characteristics of first-order and
second-order systems form the basis for the
study of the transfer characteristics of higher-
order systems.
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
2
1
22
1
2
1
22
1
2
2
rn
i
niini
ii
r
i i
i
rn
i ninii
ii
r
i i
i
j
j
pj
q
jj
j
pj
q
jH
1) First-order systems
Any system that follows this relation
is,by definition,a first-order system.
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
txbtya
dt
tdya
001
(3.25)
where we define:
Making Laplace transform yields
The transfer function of the first-order system is
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
tx
a
bty
dt
tdy
a
a
0
0
0
1
(3.26)
ys e n s it iv it s t a t ic ss y s t e m '
0
0
a
bK
c o n s t a n t t im ess y s t e m '
0
1
a
a?
sKXsYs 1? (3.27)
1 s KsX sYsH? (3.28)
Example,the liquid-in-glass thermometer
The relationship between the input Ti(t) and output
To(t) is
where
R -- thermal resistance of thermometer fluid
C -- specific heat of thermometer fluid
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
dt
tdTC
R
tTtT oio (3.29)
Making Laplace transform of Eq,(3.29) and
letting τ=RC,we get
The frequency response
Amplitude,
Phase:
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
sTsTssT ioo
11 ssT sTsH
i
o
(3.30)
11 jjH (3.31)
21
1
jHA (3.32)
a r c t a n jH (3.33)
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.11 Frequency response of first-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.12 First-order systems
(a) Massless single-degree-of-freedom vibration system
(b) RC low-pass filter circuit
2) Second-order systems
A second-order system is one that follows
the equation
Let
Then
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
txbtya
dt
tdya
dt
tyda
0012
2
2
(3.34)
ys e n s it iv it s t a t ic
0
0 —
a
bK?
( r a d / s )f r e q u e n c y n a t u r a l u n d a m p e d
2
0 —
a
a
n
e s sd i m e n s io n l r a t io,d a m p i n g 2
20
1 —
aa
a
sKXsYss
nn
12
2
2
(3.35)
The transfer function is thus
The frequency response of the system is then
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
1222
nn
ss
K
sX
sYsH
(3.36)
nn
nn
j
K
jj
K
X
Y
jH
21
1
2
2
2
2
(3.37)
Example:
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.13 Force measuring spring scale
Using the Newton’s second Law:
where
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
maF
2
2
dt
xdMxK
dt
dxBf o
os
o
i
(3.38)
Nf i f o r c e a p p lie d?
p o in t e r t h eofn t d is p la c e m e?ox
smNB //c o n s t a n t d a m p in g?
mNK s c o n s t a n t s p r in g?
Define
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
sFsXKsBsMs2 (3.39)
sr a dMK sn /
MK
B
s2
)/(1 NmKK
s
The transfer function of the force-measuring
spring scale
Amplitude:
Phase:
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
1222
nn
ss
K
sF
sXsH
(3.41)
2
2
22
41
1
nn
KjHA
(3.42)
2
1
2
a r c t a n
n
n
(3.43)
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.14 Frequency response of second-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.15 Bode plots of second-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.16 Nyquist plot of second-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.17 Examples of second-order system
(a) Spring-damper-mass system
(b) RLC circuit
Measurement Technology
(6)
Prof,Wang Boxiong
For dynamic measurement,the measuring system
must be a linear one,
We can only process linear systems mathematically,
It is rather difficult to perform nonlinear corrections in
situations of dynamic measurement,
Practical systems may be considered as linear systems
within a certain range of operation and permissible
error limits,
It is of general significance to study linear
systems.
3.4 Dynamic characteristics of measuring systems
The input-output relationship of a linear system:
where
x(t)= input of the system
y(t)= output of the system
an,a1,a0,and bm,b1,b0 are system’s parameters.
A linear constant-coefficient system or linear
time-invariant (LTI) system,the parameters are
constants.
3.4.1 Mathematical representation of linear systems
txb
dt
tdx
b
dt
txd
b
dt
txd
b
tya
dt
tdy
a
dt
tyd
a
dt
tyd
a
m
m
mm
m
m
n
n
nn
n
n
011
1
1
011
1
1
(3.3)
Properties:
1,Superposition property (superposability):
If for
then
2,Proportionality
If
then
Where a is a constant.
3.4.1 Mathematical representation of linear systems
tytx 11?
tytx 22?
tytytxtx 2121 (3.4)
tytx?
taytax? (3.5)
3,Differentiation
If
then
4,Integration
If
and for a zero initial condition of the system,then
3.4.1 Mathematical representation of linear systems
tytx?
dt
tdy
dt
tdx? (3.6)
tytx?
tt dttydttx 00
(3.7)
5,Frequency preservability
If
and for
then the output
3.4.1 Mathematical representation of linear systems
tytx?
tjextx?0?
tjeyty 0
Proof:
According to the proportionality property
According to the differentiation property
Since
3.4.1 Mathematical representation of linear systems
tytx 22 (3.8)
2
2
2
2
dt
tdy
dt
txd? (3.9)
2
2
2
2
2
2
dt
tdyty
dt
txdtx (3.10)
tjextx?0?
tx
ex
exj
dt
txd
tj
tj
2
0
2
0
2
2
2
Letting the left-hand side of Eq,(3.10) be zero,
then the right-hand side of Eq,(3.10) must also be
zero,
Solving the equation yields,
where φ is the phase shift,
3.4.1 Mathematical representation of linear systems
0222 dt txdtx?
0222 dt tydty?
tjeyty 0
1,Transfer function
Definition:
For t?0,y(t)=0,the Laplace transform Y(s)
of y(t) is defined as
where s is the Laplace operator,s=a+jb for
a>0.
3.4.2 Representation of system’s characteristics in terms of transfer
function or frequency response
0 dtetysY st
(3.11)
If all the system’s initial conditions are
zero,making Laplace transform of Eq,(3.3)
gives then the expression
The transfer function H(s):
The transfer function H(s) represents the
transfer characteristics of a system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
0111
01
1
1
bsbsbsbsX
asasasasY
m
m
m
m
n
n
n
n
01
1
1
01
1
1
asasasa
bsbsbsb
sX
sYsH
n
n
n
n
m
m
m
m
(3.12)
Properties:
1) The transfer function H(s) won’t change with
input x(t),It represents only the system’s
characteristics,
2) The system described in terms of its transfer
function will provide clearly an output y(t)
for any concrete input x(t).
3) All the coefficients in the equation,an,… a1,
a0,and bm,…,b1,b0,are the constants
determined uniquely by the configuration of
system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
2,Frequency response function
For a steady-state LTI system,let s=jω,
that is,substitute a=0 and b=ωin s=a+jb,
then Eq,(3.11) changes to
Eq,(3.16) is an equivalent of the one-sided
Fourier transform expression discussed in
Chapter 2,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
0 )()( dtetyjY tj (3.16)
In the same manner,we have
H(jω) is called the frequency response or
frequency response function of a measuring
system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
)(
)(
)(
01
1
1
01
1
1
jX
jY
ajajaja
bjbjbjb
jH
n
n
n
n
m
m
m
m
(3.17)
The frequency response function is a special
case of the transfer function,
Although both the transfer function and the
frequency response can be used to represent
the dynamic characteristics of systems,their
significances are different,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.7 Representation of system’s output under different inputs by use of (a)
transfer function,and (b) frequency response function
The frequency response function intuitively
reflects responses of system to input signals of
different frequencies,
To get better measurement results,it is often to
do measurement when a system reaches its steady
state,
We often use the frequency response function in
measurements to describe system’s dynamic
characteristics,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
To control a system,since it is often required
to study its responses to typical disturbances
and the whole features of a process from its
initial transient stage to its final steady-state
stage,the transfer function must be then used,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Usually,H(jω) is a complex function:
where A(ω) is the modulus of the complex
function H(jω),
which is called the amplitude-frequency
characteristic of the system,
φ(ω)is the phase angle of H(jω)
which is called the phase-frequency
characteristic of the system,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
)()()()( AeAjH j (3.18)
jH
X
YA)( (3.19)
xyjH a r g (3.20)
Also
P(ω) and Q(ω) are real functions of ω.
So
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
jQPjH
(3.21)
22 QPA
(3.22)
If the argument ω is expressed in logarithmic
scales and the amplitude A(ω) is expressed in
decibels (dB),then the two graphs for both
amplitude and phase angle of H(jω) are called
Bode plots.
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.8 Bode plots of a first-order system
The polar (or Nyquist) plot is a way of presenting
the frequency response of a system G(jω) on one
graph,as opposed to the separate amplitude and
phase angle Bode plots,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig.3.9 Nyquist plot of the first-order system in Fig,3.8
3,Transfer characteristics of first-
order and second-order systems
Eq,(3.12) can be written as
where αi,βi,piand qi are constants,
Any system can be considered as a parallel
combination or a series combination of a
number of first-order and second-order
elements,
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
2
1
22
1 2
rn
i ninii
ii
r
i i
i
ss
s
ps
qsH
(3.23)
Similarly,we have
Conclusion:
The transfer characteristics of first-order and
second-order systems form the basis for the
study of the transfer characteristics of higher-
order systems.
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
2
1
22
1
2
1
22
1
2
2
rn
i
niini
ii
r
i i
i
rn
i ninii
ii
r
i i
i
j
j
pj
q
jj
j
pj
q
jH
1) First-order systems
Any system that follows this relation
is,by definition,a first-order system.
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
txbtya
dt
tdya
001
(3.25)
where we define:
Making Laplace transform yields
The transfer function of the first-order system is
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
tx
a
bty
dt
tdy
a
a
0
0
0
1
(3.26)
ys e n s it iv it s t a t ic ss y s t e m '
0
0
a
bK
c o n s t a n t t im ess y s t e m '
0
1
a
a?
sKXsYs 1? (3.27)
1 s KsX sYsH? (3.28)
Example,the liquid-in-glass thermometer
The relationship between the input Ti(t) and output
To(t) is
where
R -- thermal resistance of thermometer fluid
C -- specific heat of thermometer fluid
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
dt
tdTC
R
tTtT oio (3.29)
Making Laplace transform of Eq,(3.29) and
letting τ=RC,we get
The frequency response
Amplitude,
Phase:
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
sTsTssT ioo
11 ssT sTsH
i
o
(3.30)
11 jjH (3.31)
21
1
jHA (3.32)
a r c t a n jH (3.33)
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.11 Frequency response of first-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.12 First-order systems
(a) Massless single-degree-of-freedom vibration system
(b) RC low-pass filter circuit
2) Second-order systems
A second-order system is one that follows
the equation
Let
Then
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
txbtya
dt
tdya
dt
tyda
0012
2
2
(3.34)
ys e n s it iv it s t a t ic
0
0 —
a
bK?
( r a d / s )f r e q u e n c y n a t u r a l u n d a m p e d
2
0 —
a
a
n
e s sd i m e n s io n l r a t io,d a m p i n g 2
20
1 —
aa
a
sKXsYss
nn
12
2
2
(3.35)
The transfer function is thus
The frequency response of the system is then
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
1222
nn
ss
K
sX
sYsH
(3.36)
nn
nn
j
K
jj
K
X
Y
jH
21
1
2
2
2
2
(3.37)
Example:
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.13 Force measuring spring scale
Using the Newton’s second Law:
where
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
maF
2
2
dt
xdMxK
dt
dxBf o
os
o
i
(3.38)
Nf i f o r c e a p p lie d?
p o in t e r t h eofn t d is p la c e m e?ox
smNB //c o n s t a n t d a m p in g?
mNK s c o n s t a n t s p r in g?
Define
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
sFsXKsBsMs2 (3.39)
sr a dMK sn /
MK
B
s2
)/(1 NmKK
s
The transfer function of the force-measuring
spring scale
Amplitude:
Phase:
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
1222
nn
ss
K
sF
sXsH
(3.41)
2
2
22
41
1
nn
KjHA
(3.42)
2
1
2
a r c t a n
n
n
(3.43)
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.14 Frequency response of second-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.15 Bode plots of second-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.16 Nyquist plot of second-order system
3.4.2 Representation of system’s characteristics in terms of transfer function or frequency
response
Fig,3.17 Examples of second-order system
(a) Spring-damper-mass system
(b) RLC circuit