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2.0 IntroductionIn this chapter we discuss system characteristics that typicalmeasurement elements may posses and their effect on theOVERALL PERFORMANCE of the system.
System Characteristics
Systematic CharacteristicsStatistical Characteristic
- Accuracy- Repeatability
- Tolerance
Static Characteristics- Range, Span,
,Sensitivity,Resolution, etc
Dynamic
Characteristics -Rise time, settling time,
MP, etc
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2.1 Systematic CharacteristicsSystematic characteristics are those that can be exactly qu ant i f ied by m athemat ical or graphical means. These aredistinct from statistical characteristics which cannot beexactly quantified.
Static(steady-state) characteristics- are the relationships which occur between the output O and input I of an element when I is either at constant value
or changing slowly(Figure 2.1)
Figure 2.1 Meaning of element Characteristics
ElementOutput O Input I
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Static(steady- state) Characteristics (contd) Assume I(t) is slowly varying quantity and suppose
I min - minimum input
I max - maximum input
O min - minimum outputO max - maximum output
Range:
The input range of an element is specified by the minimumand maximum values of I ,i.e. I min to I max ( the samedefinition hold for O ).
E.g., a pressure transducer may have an input range of 0 to
10 4 Pa and an output range of 4 to 20mA .
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Static(steady- state) Characteristics (contd)
Span:- is the maximum variation in input or output, i.e. input span isI max - I min , and output span is O max O min .
Thus in the previous example the pressure transducer hasinput span of 10 4Pa and output span of 16mA.
Ideal straight line:
An element is said to be linear if corresponding values of Iand O lie on a straight line defined by eqn.(2.1)
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Static(steady- state) Characteristics (contd)
(2.1) where:
and
Thus the ideal straight line for the pressure transducer(discussed before) is:
Note :The ideal straight line defines the ideal characteristics of anelement. Non-ideal characteristics can be then quantified interms of deviations from the ideal straight line.
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Static(steady- state) Characteristics (contd)
Non-linearity:
can be defined(Figure 2.2) in terms of a function N( I) which isthe difference between the actual and ideal straight-linebehavior, i.e.
(2.2)
A(Imin, O min)
B(Imax , O max )
IdealKI+a
ActualO(I)
Imax
Omax
N(I)
I
O
I
N(I)
ImaxImin
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Static(steady- state) Characteristics (contd)
The maximum non-linearity, , is expressed as percentage tofull-scale deflection(f.s.d), i.e. as a percentage of span. Thus:
(2.3) Max. non-linearity aspercentage of span
Note: In many cases O(I ) & therefore N(I ) can be expressedas a polynomial in I:
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Static(steady- state) Characteristics (contd)
Sensitivity :
-Is the s lope or gradient of the output versus inputcharacteristics O(I)
(2.4)
For a non-linear element
Hysteresis :
For a given value of input I , the output O may depend onwhether I is increasing or decreasing. Hysteresis is, thus, thedifference between these two values of the output ( Figure 2.3 )
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Static(steady- state) Characteristics (contd) Hysteresis (contd )
(2.5) Hysteresis
Hysteresis is usually quantified in terms of the max.hysteresis express as a percentage of f.s.d., i.e. span.
F i g u r e
2 . 3
H y s
t e r e s i s
ImaxIminOmin
Omax
I
O
H(I)
ImaxImin
H
I
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Static(steady- state) Characteristics (contd)
Resolution :-Is the largest change in I that can occur without anycorresponding change in O. (2.6) Resolution expressed as
a percentage of f.s.d
Figure 2.4 Resolution & potentiometer example.
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Static(steady- state) Characteristics (contd)
Non-linearity , hysteresis and resolution effects in manymodern sensors and transducers are so small that it isdifficult and not worthwhile to exactly quantify each individualeffect. In these cases the manufacture defines the
performance of the element in terms of error bands (Figure2.5 )
Oideal
IMaxIMinOmin
OMax
I
O
hh
Oideal
1
2h
2 h
F i g u r e
2 . 5
E r r o r
b a n
d
a n
d r e c t a n g u l a
r P D F
P(O)
O
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Environmental effects:
In general, the output depends not only on the input signal I but also on the environment inputs such as ambienttemp erature , atm os ph er ic pressu re , re lat ive hum idi ty,
su pp ly v ol tage , e tc .
There are two main types of environmental input.
[1] Modifying input (I M)- causes change of the linearsensitivity of the element. If I M is the deviation in the
modifying input from the standard conditions( I M =0 ), then thiscause the linear sensitivity to shift from
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Environmental effects(contd) :
Figure 2.6 Modifying & InterferingInputs
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Environmental effects(contd) :
[2] Interfering input (I I )- causes the straight line intercept (or zero bias) of the element to change from
If both modifying & interfering input differs from the standardvalue ( i.e., and ) , then eqn.[2.2] becomes
(2.7)
A numerical examples on determining the value of Km , KI ,a
& K associated with the general model equation will follow inthe tutorial section.
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2.2 Generalized model of a system elementFigure 2.7 shows eqn.(2.7) in block diagram form to represent the
static characteristics of an element. For completeness the diagramalso show the transfer function G(s) , which represent dynamiccharacteristic of an element.
Static Dynamic
K mImI
KIIinput
K m
X
K
N()
K I
G(s)
I m (Modifying) I I (Interfering)
N(I)
O
a
O
F i g u r e
2 . 7
G e n e r a
l M o
d e
l o
f e
l e m e
n t
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2.3 Dynamic Characteristics:
If the input signal I to an element is changed suddenly ,from one value to another, then the output signal O will notinstantaneously change to its new value.
The ways in which an element responds to sudden input
changes are termed its dyn amic charac te r is t ics , and theseare most conveniently summarized using a t ransfer fun c t ion G(s).
For example, if the temperature input to a t he rmocoup le issuddenly changed from 25 0C to 100 0C, some time will elapsebefore the e.m.f. output completes the change from say 1mVto 4mV.
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2.3.1 Transfer function G(s) for typical system elements
As an example , let find G(s) of thermocouple inserted in fluidenvironment.
preliminaries Heat transfer take place as a result of one or more ofthree possible types of mechanism; conduct ion ,co nv ect ion*, radia t ion
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Preliminaries(contd)
The temperature of a sensing element @ any instant oftime depends on the ra te of t ransfer of heat both to andfrom the sensor.
Returning back to the example, we have from Newtons law of cooling the convective heat flow W watts between asensor @ T 0C and fluid @ T F0C is given by
(2.8)
Where;U : is the convection heat transfer coefficient and [Wm -2 0 C -1 ]
A : is the heat transfer area[m 2 ]
2.3.1 Transfer function G(s) for typical system elements
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Thermocouple example (contd)
i.e.
(2.11)
The quantity MC/UA has the dimensions of time:
And is referred to as the time constant ( ) of the system. Thedifferential equation become;
(2.12)
2.3.1 Transfer function G(s) for typical system elements
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Thermocouple example (contd)
Taking LT on both sides of eqn.(2.12) gives
(2.13)
Where; T(0-) : is the temperature deviation @ initial condition prior to t=0and by assumption , T(0-)=0
Therefore, eqn.(2.13) becomes
(2.14)Or , equivalently
(2.15)
2.3.1 Transfer function G(s) for typical system elements
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Thermocouple example (contd)
The TF in eqn.(2.15) only relates changes in sensortemperature to changes in fluid temperature. The overallrelationship between changes in sensor output signal O and
fluid temperature is; (2.16)
Where; O/ T is the steady-state sensitivity of the temperature sensor. (foran ideal element O/ T=K).
2.3.1 Transfer function G(s) for typical system elements
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Example 2.1
For a copper-cons tan tan (Type-T ) t he rmocoup le junction,the first four terms in the polynomial relating e.m.f. E(T),expressed in V, and junction temperature T 0C are
for the range 0 to 400 0C.
Calculate:(a) E/ T for small fluctuations in temperature around 100 0 C.(b) Assuming =10s, find G(s) relating e.m.f & fluid temp 0
2.3.1 Transfer function G(s) for typical system elements
h
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Example 2.1 (contd)
Solution :
(a)
Evaluation @ 100 0C yields
(b) Using eqn.(2.16) , we get
Exercise2.1: For the same thermocouple find (a) E IDEAL(T) & (b) N(T)
2.3.1 Transfer function G(s) for typical system elements
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In the general case of an element with static characteristics givenby eqn[2.7] and dynamic characteristics defined by G(s), the effectof small, rapid changes in I is evaluated using Figure 2.8, in whichsteady-state sensitivity (O/ I )I 0 =K+K MIM+ (dN/d I )I 0 , and I 0 is thesteady-state value of I around which the fluctuations are takingplace.
Figure 2.8 Element model for dynamic calculations
2.3.1 Transfer function G(s) for typical system elements
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Transfer function for standard 2 nd order element is given by
Second-order elements
(2.17)
Unit step response of 2 nd order element is found as:
G(s)Output Input I
(2.18)
Expressing (2.18) in partial fraction we get:
2.3.2 Identification of the dynamics of an element
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Second-order elements(contd)
(2.19)
(2.20)
Where , ,
And this gives
2.3.2 Identification of the dynamics of an element
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Second-order elements(contd)
(2.21)
Three different case result depending on the value of
Case 1: Critically damped (=1 )
Using Trigonometric relationship (refer the triangle above), we get
Case 2: Underdamped (
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Second-order elements(contd)
F i g u r e
2 . 9
R e s p o n s e o
f a
2 n d
o r d e r
e l e m e n
t t o a u n
i t s t e p
2.3.2 Identification of the dynamics of an element
0
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Second-order elements(contd)
The time TP @ which the 1 st oscillation peak occur is given by:
The settling time Ts ( the time for the response to settle out
approximately within 2% of the final steady-state value) is given by:
Maximum Overshoot(MP), which is the difference (T P )-1 (for
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Sinusoidal response of 1 st and 2 nd order element:
2.3.2 Identification of the dynamics of an element
Figure 2.10 Frequency response of an element with linear dynamic
In the steady-sate , the output O satisfies the following 4 rules:
O is also a sine wave
The frequency of O is also The amplitude of O is The phase difference between O & I is
KG(s)Input output
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Sinusoidal response of 1 st and 2 nd order element(contd) :
2.3.2 Identification of the dynamics of an element
Example 2.2 Using the rules on pp 31, find the amplitude ratio& phase relations for a 2 nd order element with:
Solution :
so that
Amplitude ratio =
Phase difference =
(2.26a)
(2.26b)
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Sinusoidal response of 1 st and 2 nd order element(contd) :
2.3.2 Identification of the dynamics of an element
F i g u r e
2 . 1
1 F r e q u
e n c y r e s p o n s e c h a r a c t e r i s t
i c s
o f 2 n d
o r d e r e
l e m e n t w
i t h
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2.4 Statistical Characteristics
Statistical variations in the output of a single element with
time- repeatability
Repeatability- is the ability of an element to give the sameoutput for the same input when repeatedly applied on it.
the most common cause of lack of repeatability in the output isdue to rando m f luc tua t ion of the environmental input( I M,I I ) with time.
If the coupling constants( K M,K I ) are non-zero, then there willbe corresponding time variation in the output.
By making reasonable assumptions about the probability density
functions( PDF s) of the inputs, I ,I M,I I we can find ( or at least
approximate) the probability density function of the output O .
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2.4 Statistical Characteristics(contd)
Statistical variations in the output of a single element with
time- repeatability(contd )
Very often, the PDF of the inputs can be assumed to be theNormal probability distribution or the Gaussian distribution, i.e.,
(2.28)
Where: = the mean ( specified center of the distribution)= standard deviation ( spread of the distribution).
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2.4 Statistical Characteristics(contd)
Statistical variations in the output of a single element with
time- repeatability(contd ) Recall the general equation for the output of a measurementsystem (eqn(2.7))
A small deviation in the output O can be approximated as
Which means O is approximated by a linear combination of thedeviations of the inputs, I ,I M,I I .
It can be shown that, if y is a linear combination of theindependent variables x1, x2, x3, i.e.,
(2.29)
(2.30)
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q
2.4 Statistical Characteristics(contd)
Statistical variations in the output of a single element with
time- repeatability(contd ) And if x1, x2, x3 have normal distributions with standarddeviations 1, 2, 3, respectively, then the output will also havea normal distribution with standard deviation
(2.31)
From eqns[2.29] and [2.31] we see that the standard deviation of O, i.e. of O about mean( ), is given by:
(2.32)
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q
2.4 Statistical Characteristics(contd)
Statistical variations in the output of a single element with
time- repeatability(contd ) The corresponding mean of the element output is given by:
and the corresponding PDF is:
(2.33)
(2.34)
Statistical variations amongst a batch of similar elements-tolerance
Reading Assignment :
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q
2.5 Identification of static characteristics-Calibration
Is an experimental procedure used to determine most of the
static performance parameters.
F i g u r e
2 . 1
2 C a l i
b r a
t i o n o
f a n e l e m e n
t
F i g u r e
2 . 1
3 s i m p
l i f i e d t r a c e a
b i l i t y
l a d d e r
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q
2.6 The Accuracy of measured system in the steady-state
Accuracy is quantified using measurement error E where:
E = Measured value true Value = System output System input
2.6.1 Measurement error of a system of ideal elements
Let
Then the overall output for the above system become:
If the measured system is complete, then
I=I 1 O1=I 2 O2=I 3 InO3 Ii Oi On=O
Measured Value
True Value
K 1 K 2 K 3 K i K n1 2 3 i n
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2.6.1 Measurement error of a system of ideal elements()
Thus if
We have E = 0 and the system is perfectly accurate!
Example 2.3 consider the simple temperature measurementsystem depicted below:
For this example,
Measuredtemperature
Truetemperature
TT 0C E(T) V
e.m.f volts
TM 0CV Thermocouple Amplifier Indicator
K 1=40V/ 0C K 2= 1000V/V K 2= 25 0C/V
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2.6.2 The error PDF of a system of non-ideal elementsObjective : to calculate the system error PDF , p(E) ( using
Eqns(2.32)-(2.34)Table 2.1 : General calculation of system p(E)
Mean values of element outputs
: :
: :
(2.35)
1 2 i n
I=I 1 O1=I 2 O2 = I 3 I i Oi I n On =O
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2.6.2 The error PDF of a system of non- ideal elements()Table 2.1 : General calculation of system p(E) (contd )
Mean value of system error
(2.36)
Standard deviation of element outputs
(2.37)
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2.6.2 The error PDF of a system of non- ideal elements()Table 2.1 : General calculation of system p(E) (contd )
Standard deviation of element outputs(contd )
(2.37)
Standard deviation of system error
(2.38)
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2.6.2 The error PDF of a system of non- ideal elements()Table 2.2 : Model for temp 0 measurement system element(e.g.2.4)
(a) Platinum resistance temp 0 detector
Model eqn.
Individual mean
values (between 100 to 130 0C) Individual stand.deviations
Partial derivatives
Overall mean value
Overall standarddeviation
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2.6.2 The error PDF of a system of non- ideal elements()Table 2.2 : Model for temp 0 measurement system element(e.g.2.4)
(b) Current transmitter
Model eqn. 4 to 20mA output for 138.5 to 149.8 input
( 100 to 130 0C)
Ta = deviations of ambient temperature from 20 0C
Individual mean
values
Individual stand.deviations
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2.6.2 The error PDF of a system of non- ideal elements()Table 2.2 : Model for temp 0 measurement system element(e.g.2.4)
(b) Current transmitter(contd )
Partial derivatives
Overall mean value
Overall standarddeviation
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2.6.2 The error PDF of a system of non- ideal elements()Table 2.2 : Model for temp 0 measurement system element(e.g.2.4)
(b) Current transmitter
Model eqn.
Individual mean
values Individual stand.deviations
Partial derivatives
Overall mean value
Overall standard
deviation
(100 to 1300C record for 4 to 20mA input)
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2.6.2 The error PDF of a system of non- ideal elements()Table 2.3 : Summary of calculation of and for Example 2.4)
Mean
Standard deviation
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2.7 Error reduction techniquesCompensation methods for non-linear and environmental effects.
Compensating non-linear element
Figure 2.14 Compensating non-linear element
U(I) C(U) I U C
UncompensatedNon-linear element
compensated Non-linear element
Thermistor Deflection bridge
ResistanceTemperature Voltage
k R k ETh V
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2.7 Error reduction techniques(contd)
Opposing environmental inputs
e.g. Variations in temperature T 2 of the reference junction of athermocouple.
(a) Using opposing environmental inputs
K I
K
K I+ +
+
-I U C=KI
i f K I =K I
Compensating element Uncompensated element
I I
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2.7 Error reduction techniques(contd)
Opposing environmental inputs(contd )
Figure 2.15 Compensation for interfering inputs
(b) Using a differential system
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2.7 Error reduction techniques(contd)
Isolation*
Zero environmental sensitivity High-gain negative feedback*