2141-375
Measurement and Instrumentation
Measurement System
Behavior
Dynamic Characteristics
Dynamic characteristics tell us about how well a sensor
responds to changes in its input. For dynamic signals, the sensor or
the measurement system must be able to respond fast enough to keep
up with the input signals.
Input signal
x(t)
Sensor
or
system
Output signal
y(t)
In many situations, we must use y(t) to infer x(t), therefore a
qualitative understanding of the operation that the sensor or
measurement system performs is imperative to understanding the
input signal correctly.
General Model For A Measurement System
nth Order ordinary linear differential equation with constant coefficient
d n y (t )
d n 1 y (t )
dy (t )
d m x(t )
d m1 x(t )
dx(t )
an

a



a

a
y
(
t
)

b

b



b
 b0 x(t )
n

1
1
0
m
m

1
1
n
n 1
m
m 1
dt
dt
dt
dt
dt
dt
m≤n
y(t)
x(t)
t
a’s and b’s
F(t) = forcing function
Where
= output from the system
= input to the system
= time
= system physical parameters, assumed constant
y(0)
The solution
x(t)
Measurement
system
y(t)
y (t )  yocf  yopi
Where yocf = complementary-function part of solutio
yopi = particular-integral part of solution
Complementary-Function Solution
The solution yocf is obtained by calculating the n roots of the algebraic characteristic
equation
Characteristic equation
an D n  an 1 D n 1  ...  a1D  a0  0
Roots of the characteristic equation:
D  s1 , s2 ,..., sn
Complementary-function solution:
Ce st
1. Real roots, unrepeated:
2. Real roots, repeated:
each root s which appear p times
3. Complex roots, unrepeated:
the complex form: a  ib
C
2
p 1
st

C
t

C
t

...

C
t
e

0
1
2
p 1
Ceat sin(bt   )
[C0 sin(bt  0 )  C1t sin(bt  1 )  C2t 2 sin(bt  2 )
4. Complex roots, repeated:
each pair of complex root which appear p times
 ...  C p 1t p 1 sin(bt   p 1 )]e at
Particular Solution
Method of undetermined coefficients:
yopi  Af (t )  Bf (t )  Cf (t )  ...
Where f(t)
= the function that describes input quantity
A, B, C = constant which can be found by substituting yopi into ODEs
Important Notes
 •After a certain-order derivative, all higher derivatives are zero.
 •After a certain-order derivative, all higher derivatives have the same
functional form as some lower-order derivatives.
 •Upon repeated differentiation, new functional forms continue to arise.
Zero-order Systems
All the a’s and b’s other than a0 and b0 are zero.
y (t )  Kx(t )
a0 y(t )  b0 x(t )
where K = static sensitivity = b0/a0
The behavior is characterized by its static sensitivity, K and remains
constant regardless of input frequency (ideal dynamic characteristic).
xm
Vr
+
x=0
y=V
-
A linear potentiometer used as position
sensor is a zero-order sensor.
x
V  Vr 
here, K  Vr / xm
xm
Where 0  x  xm and Vr is a reference voltage
First-Order Systems
All the a’s and b’s other than a1, a0 and b0 are zero.
a1
dy (t )
 a0  b0 x(t )
dt

dy (t )
 y (t )  Kx(t )
dt
y
K
( D) 
x
D  1
Where K = b0/a0 is the static sensitivity
 = a1/a0 is the system’s time constant (dimension of time)
First-Order Systems: Step Response
Assume for t < 0, y = y0 , at time = 0 the input quantity, x increases instantly
by an amount A. Therefore t > 0
0 t  0
x(t )  AU (t )  
A t  0

dy (t )
 y (t )  KAU (t )
dt
y(t )  Cet /  KA
The complete solution:
2
U(t)
yocf
Transient
response
yopi
Steady state
response
1
Applying the initial condition, we get C = y0-KA, thus
gives
y (t )  KA  ( y0  KA)e t /
0
-1
0
1
2
Time, t
3
4
5
First-Order Systems: Step Response
Here, we define the term error fraction as
y (t )  KA y (t )  y ()

 e t / 
y0  KA
y (0)  y ()
1.0
1.0
.8
.8
Error fraction, em
Output Signal, (y(t)-y0)/(KA-y0)
em (t ) 
0.632
.6
y (t )  y0
 1  e t / 
KA  y0
.4
.2
y (t )  KA
 e t / 
y (0)  KA
.6
.4
0.368
.2
0.0
0.0
0
1
2
3
t/
4
5
0
1
3
2
t/
Non-dimensional step response of first-order instrument
4
5
Determination of Time constant
y (t )  KA
 e t / 
y (0)  KA
ln em  2.3 log em  
t

1
y (t )  KA
 e t / 
y (0)  KA
0.368
Error fraction, em
em 
.1
Slope = -1/
.01
.001
0
1
2
3
t
4
5
First-Order Systems: Ramp Response
Assume that at initial condition, both y and x = 0, at time = 0, the input quantity
start to change at a constant rate qis Thus, we have
Therefore
The complete solution:
t0
0
x(t )  
qist t  0
dy (t )

 y (t )  KqistU (t )
dt
y (t )  Ce t /  Kqis (t   )
Transient Steady state
response response
Applying the initial condition, gives
Measurement error
em  x(t ) 
y (t )  Kqis (e t /  t   )
y (t )
 qise t /  qis
K
Transient
error
Steady
state error
First-Order Instrument: Ramp Response
10
Output signal, y/K
8
6
Steady state
time lag = 
4
Steady state
error = qis
2
0
0
2
4
6
8
10
t/
Non-dimensional ramp response of first-order instrument
First-Order Systems: Frequency Response
From the response of first-order system to sinusoidal inputs, x(t )  A sin t
we have
dy

 y  KA sin t
D 1y(t )  KAsin t
dt
The complete solution:
y (t )  Ce t / 
KA
1  ( )
Transient
response
2

sin t  tan 1 
Steady state
response
=

Frequency
response
If we do interest in only steady state response of the system, we can write the
equation in general form
y(t )  Cet /  B( ) sin t   ( )
B( ) 
KA
1  ( ) 
2 1/ 2
 ( )   tan 1 
Where B() = amplitude of the steady state response and () = phase shift
First-Order Instrument: Frequency Response
M ( ) 
The amplitude ratio
1
M ( ) 
B
1

KA 1   2 1/ 2

( ) 2  1

The phase angle is
Dynamic error
1.2
 ( )   tan 1 ( )
0
.8
-2
-3 dB
0.707
.6
-4
.4
-6
-8
-10
.2
Cutoff frequency
-20
0.0
.01
.1
1

10
100
-20
Phase shift, ()
0
Decibels (dB)
Amplitude ratio
-10
1.0
-30
-40
-50
-60
-70
-80
-90
.01
.1
1
10
100

Frequency response of the first order system
Dynamic error, () = M(): a measure of an inability of a system to
adequately reconstruct the amplitude of the input for a particular frequency
Dynamic Characteristics
Frequency Response describe how the ratio of output and input changes
with the input frequency. (sinusoidal input)
Dynamic error, () = 1- M() a measure of the inability of a system or sensor to
adequately reconstruct the amplitude of the input for a particular frequency
Bandwidth the frequency band over which M()  0.707 (-3 dB in decibel unit)
Cutoff frequency: the frequency at which the system response has fallen to
0.707 (-3 dB) of the stable low frequency.
tr 
0.35
fc
First-Order Systems: Frequency Response
Ex: Inadequate frequency response
Suppose we want to measure
x(t )  sin 2t  0.3 sin 20t
x(t)
With a first-order instrument whose  is 0.2 s and
static sensitivity K
Superposition concept:
For  = 2 rad/s: B(2 rad/s ) 
y(t)/K
For  = 20 rad/s: B(20 rad/s ) 
K
  21.8o  0.93K  21.8o
0.16  1
K
  76o  0.24 K  76o
16  1
Therefore, we can write y(t) as
y(t )  (1)(0.93K ) sin( 2t  21.8o )  (0.3)(0.24K ) sin( 20t  76o )
y(t )  0.93K sin( 2t  21.8o )  0.072K sin( 20t  76o )
Dynamic Characteristics
Example: A first order instrument is to measure signals with frequency content up to 100 Hz with
an inaccuracy of 5%. What is the maximum allowable time constant? What will be the phase shift
at 50 and 100 Hz?
Solution: Define
Qo (i )
K
 M ( ) 
Qi (i )
 2 2  1


M ( )  M (0)
1

Dynamic error 
100%  
 1 100%
2
2
M (0)
   1 
From the condition |Dynamic error| < 5%, it implies that 0.95 
1
  1
2 2
 1.05
But for the first order system, the term 1 /  2 2  1 can not be greater than 1 so that the
constrain becomes
1
0.95 
Solve this inequality give the range
  1
2 2
1
0    0.33
The largest allowable time constant for the input frequency 100 Hz is
The phase shift at 50 and 100 Hz can be found from
   arctan 
This give  = -9.33o and = -18.19o at 50 and 100 Hz respectively

0.33
 0.52 ms
2 100 Hz
Second-Order Systems
In general, a second-order measurement system subjected to arbitrary input, x(t)
2
a2
d y (t )
dy (t )

a
 a0 y (t )  b0 x(t )
1
2
dt
dt
 D 2 2

 2 
D  1 y(t )  Kx(t )
 n n

1 d 2 y (t ) 2 dy (t )

 y (t )  Kx(t )
2
2
n dt
n dt
The essential parameters
K

b0
a0
a1
2 a0 a2
n 
a0
a2
= the static sensitivity
= the damping ratio, dimensionless
= the natural angular frequency
Second-Order Systems
Consider the characteristic equation
1 2 2
D 
D 1  0
2
n
n
This quadratic equation has two roots:
S1, 2  n  n  2 1
Depending on the value of , three forms of complementary solutions are possible
    2 1  t

 n
Overdamped ( > 1):
yoc (t )  C1e
Critically damped ( = 1):
yoc (t )  C1e nt  C2tent
Underdamped (< 1): :
yoc (t )  Ce nt sin n 1   2 t  

 C2 e
    2 1  t

 n

Second-Order Systems
Case I Underdamped (< 1):
Case 2 Overdamped ( > 1):


S1, 2   n  n  2  1
S1, 2      2  1 n
   j d
Case 3 Critically damped ( = 1):
yt
Ae
S1, 2  n
 t
t
sin( d t   )
yt
 1
 1
t
Second-order Systems
Example: The force-measuring spring
consider a spring with spring constant Ks under applied force fi
and the total mass M. At start, the scale is adjusted so that xo = 0
when fi = 0;
forces=(mass)(acceleration)
dxo
d 2 xo
fi  B
 K s xo  M
dt
dt 2
( MD 2  BD  K s ) xo  f i
the second-order model:
K
1
Ks
m/N
Ks
rad/s
M
B

2 Ks M
n 
Second-order Systems: Step Response
For a step input x(t)
 D 2 2

 2 
D  1 y(t )  KAU (t )
 n n

1 d 2 y 2 dy

 y  KAU (t )
2
2
n dt
n dt
With the initial conditions: y = 0 at t = 0+, dy/dt = 0 at t = 0+
The complete solution:
Overdamped ( > 1):
   2  1   
y (t )

e
2
KA
2  1
Critically damped ( = 1):
y (t )
 (1  nt )e nt  1
KA
Underdamped (< 1): :
y (t )
e  nt

sin 1   2 nt    1
KA
1  2

 2 1 nt


   2 1
2  1
2

e
    2 1  t

 n

1
  sin 1 1   2

Second-order Instrument: Step Response
Ringing frequency:
Output signal, y(t)/KA
2.0
Td 
2
d
d  n 1   2
=0
Ringing frequency:
0.25
Rise time decreases  with but
increases ringing
1.5
0.5
1.0
Optimum settling time can be obtained
from  ~ 0.7
.5
1.0
Practical systems use 0.6<  <0.8
2.0
0.0
0
2
4
6
8
10
nt
Non-dimensional step response of second-order instrument
Dynamic Characteristics
1.4
overshoot
Output signal, y(t)/KA
1.2
100%  5%
1.0
.8
.6
.4
settling
time
.2
rise time
0.0
0
5
10
15
Time, t (s)
Typical response of the 2nd order system
20
Second-order System: Ramp Response
For a ramp input x(t )  qistU (t )
1 d 2 y 2 dy

 y  KqistU (t )
2
2
n dt
n dt
 nt 

nt
1

e
(1

)

2 
qo
2q
 qis t  is
K
n
With the initial conditions: y = dy/dt = 0 at t = 0+
The possible solutions:
Overdamped:
2qis  2 2  1  2  2  1   
y (t )
 qist 
1
e
2
K
n 
4   1

Critically damped:
Underdamped:
2q
y(t )
 qist  is
K
n
2qis
y (t )
 qist 
K
n
 2 2  1  2  2  1
4   1
2
e
 2 1  n t

    2 1  t

 n




nt nt 

1

(
1

)e 

1




e  nt
sin 1   2 nt  
1 
 2 1   2




2 1   2
  tan
2 2  1
1
Second-order Instrument: Step Response
Steady state error =
Output signal, y(t)/K
10
8
2qis
n
Steady state 2

time lag =
Ramp input
n
6
 = 0.3
4
0.6
1.0
2.0
2
0
0
2
4
6
8
10
Time, t (s)
Typical ramp response of second-order instrument
Second-order Instrument: Frequency Response
The response of a second-order to a sinusoidal input of the form x(t) = Asint
KA
y (t )  yoc (t ) 
sin t   ( )
1/ 2
2 2
2
1   / n   2 / n 

where


 ( )   tan 1
2
 / n  n / 
The steady state response of a second-order to a sinusoidal input
ysteady (t )  B( ) sin t   ( )
B( ) 
KA
1   /     2 /   
2 2
n
2
1/ 2
 ( )   tan 1
n
2
 / n  n / 
Where B() = amplitude of the steady state response and () = phase shift
M ( ) 
B
1

KA 1   /  2 2  2 /  2 1/ 2
n
n



Second-order Instrument: Frequency Response
The phase angle
The amplitude ratio
M ( ) 
1
 ( )   tan 1
1   /     2 /   
2 2
2
n
1/ 2
n
2
 / n  n / 
0
6
0.3
1.5
3
0.5
1.0
0
-3
1.0
-6
-10
-15
.5
2.0
0.0
.01
.1
1
n
10
100
-20
Phase shift, 
 = 0.1
Decibel (dB)
Amplitude ratio
2.0
0 = 0.1
0.3
-40
-60
0.5
1.0
-80
2.0
-100
-120
-140
-160
-180
.01
.1
1
n
Magnitude and Phase plot of second-order Instrument
10
100
Second-order Systems
For overdamped ( >1) or critical damped ( = 1), there is neither overshoot nor steadystate dynamic error in the response.
In an underdameped system ( < 1) the steady-state dynamic error is zero, but the speed
and overshoot in the transient are related.
arctan( d /  )
Maximum
overshoot:
M p  exp  / 1   2
Peak time:
tp 
Resonance
amplitude:
d


d
 r   n 1  2 2
Mr 
1
2 1  
Td
overshoot

1.2
1.0
o
tr 
Output signal, q (t)/Kqis
Rise time:
Resonance
frequency:
1.4
.8
peak
time
.6
.4
settling
time
.2
2
rise time
0.0
0
where  = n ,  d   n 1   2 , and   arcsin( 1   2 )
5
10
Time, t (s)
15
20
Dynamic Characteristics
Speed of response: indicates how fast the sensor (measurement system) reacts
to changes in the input variable. (Step input)
Rise time: the length of time it takes the output to reach 10 to 90% of full response
when a step is applied to the input
Time constant: (1st order system) the time for the output to change by 63.2% of its
maximum possible change.
Settling time: the time it takes from the application of the input step until the output
has settled within a specific band of the final value.
Dead time: the length of time from the application of a step change at the input of
the sensor until the output begins to change