_2. Performance Characteristics of Measurement System

8/13/2019 _2. Performance Characteristics of Measurement System

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ENGG3201-Measurement Technique

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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(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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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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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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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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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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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 ]



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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)



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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)



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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).



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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


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)


ENGG3201 M T h i


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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


ENGG3201 M T h i


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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

ENGG3201 M t T h i


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Second-order elements(contd)

(2.19)

(2.20)

Where , ,

And this gives


ENGG3201 M t T h iq


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(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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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


0

ENGG3201 Measurement Technique


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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:


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) :


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) :


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)


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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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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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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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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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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(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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(b) Current transmitter(contd )

Partial derivatives

Overall mean value

Overall standarddeviation



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(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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Opposing environmental inputs(contd )

Figure 2.15 Compensation for interfering inputs

(b) Using a differential system



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Isolation*

Zero environmental sensitivity High-gain negative feedback*

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_2. Performance Characteristics of Measurement System