lecture notes(sensor)

8/10/2019 Lecture Notes(Sensor)

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SIE1004

MEASUREMENT & SENSOR TECHNOLOGY

WEEK 2

STATIC CHARACTERISTICS OFMEASUREMENT

8/9/2014 SIE1004 Measurement & SensorTechnology

1


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LEARNING OUTCOMES


2

Characteristics of Measurement:

Accuracy & Error;

Precision & Reproducibility;

Resolution;

Sensitivity;

Non-linearity;

type of measurement errors; Statistics analysis.


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STATIC CHARACTERISTICS OF MEASUREMENT


3

Resolution, Precision, Accuracy and Errors;

Hysteresis

Sensitivity

Nonlinearity


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RESOLUTION, PRECISION & ACCURACY


4

Resolution: the smallest increment can bediscerned in a measurement.

Precision:

Repeatability: the variation for a set ofmeasurement in a short period;

Reproducibility: the variation for a set of

measurement in a long period. Accuracy is the closeness of measurement to

the true value.


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STATISTIC ANALYSIS: AVERAGE


5

Average: Mean & Median values

nnxxx

meanx

21

order)ascending(in2/)1(

n

xmedian

x


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STATISTIC ANALYSIS: SPREAD


6

Deviation:

Standard Derivation (s, ):

Population standard deviation:

Sample standard deviation:

Variance (V)

Population variance:

Sample variance:

meanxixid

N

dN

i

i 1

2)(

1

)(1

2

n

dn

i

i

s

2V

2sV


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STATISTICAL ANALYSIS: TEMPERATURE


7

60 TEMPERATURE READING [ref. 3]


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HISTOGRAM: TEMPERATURE DATA


8

60 TEMPERATURE READING [ref. 3]


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STATISTICAL ANALYSIS: TEMPERATURE


9

Mean: = 1103; Median: = 1104; Standard deviation: S=5.79

Variance: =33.49


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CONFIDENCE WITH NORMAL DISTRIBUTION


10

It is only possible to estimate, with someconfidence, the expected limits of errors.

The most common method is to use the normal

distribution.

The average 2 would be with 95%confidence while 3 would be with 99.75%.

= 1 2 ()


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CALCULATION BY SOFTWARE (MATLAB)


11

temperature=[1089,1092,1094*ones(1,2),1095*ones(1,4),1098*ones(1,8),1100*ones(1,9),

1104*ones(1,12),1105*ones(1,4),1107*ones(

1,5),1108*ones(1,5),1110*ones(1,4),1112*ones(1,3),1115*ones(1,2)]

mean(temperature);

std(temperature); var(temperature); std(temperature,1); var(temperature, 1)

(unbiased estimation for sample)

(population calculation)


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CALCULATION BY SOFTWARE (EXCEL)


12

=AVERAGE( : );

=STDEV.P( : ) ; =VAR.P( : )

=STDEV.S( : ) ; =VAR.S( : )(unbiased estimation for sample)

(population calculation)


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ERRORs


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Error is defined as the difference between themeasured value and the true value of the

measurand: Error=(measured)-(true)

Type of errors: Systematic (bias) errors

Random (noise) errors

Target analogy of measurement accuracy (ref. 1)Distinction between systematic and random errors [ref. 3]


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FACTORS OF SYSTEMSTIC ERRORs


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Systematic error sources: Miscalibration: change the input-output

response; aging, damage or abuse of the

sensors; Invasiveness: measurement process change the

intended measurand;

Error introduced in the signal path (friction &

resistance in signal transmission);

Error introduced by human observers.


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ATTENTION TO ACCURACY DESCRIPTION


16

Product A: measurement range 0-100 withaccuracy 0.5%;

Product B:

Which one is more accurate in measuring body

temperature?


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PRECISION


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Precision characterizes the random error.

A highly precise measuring system gives good

repeatability but may not be accurate;

In general, the accuracy of a measurementcannot be better than the precision;

Accuracy and precision are overall

characteristics describe the validity ofmeasurement.


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HYSTERESIS


18

Hysteresis: varying signaldirection of the

movement will have

different result.

Generalized graph of output/input relationship

where hysteresis is present [ref. 1].


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STATIC SENSITIVITY


19

Sensitivity is the ratio of the magnitude ofresponse (output signal) to the magnitude of

quantity measured (input signal)

Static sensitivity,i

o

i

o

qq

dqdqK


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NON-LINEARITY


20

Nonlinear amplification can give rise to unwanted output distortion [ref. 1]

For the input-output relationship, we prefer tohave linear relationship

However, practical units always have somedegree of nonconformity/nonlinearity.


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NON-LINEARITY


21

For ideal linear input-output relationship:= +

Where = is the slope and = -K is the intercept (bias)

Non-linearity function: = ( + )


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EXPRESSION OF ERROR OF NONLINEARITY


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Error of nonlinearity can be expressed in four different ways: (a) best fit line (based on

selected method used to decide this); (b) best fit line through zero; (c) line joining 0% and

100% points; and (d) theoretical line. (ref [1])


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LEAST SQUARED LINEAR FIT


23

Linearized: = + Non-linear error: = Sum of squared errors:

=

=(+ )

Minimize:= 2+ =0;= 2+ =0

http://mathworld.wolfram.com/LeastSquaresFitting.html

http://www.mathworks.com/help/curvefit/least-squares-fitting.html
http://mathworld.wolfram.com/LeastSquaresFitting.htmlhttp://mathworld.wolfram.com/LeastSquaresFitting.htmlhttp://www.mathworks.com/help/curvefit/least-squares-fitting.htmlhttp://www.mathworks.com/help/curvefit/least-squares-fitting.htmlhttp://www.mathworks.com/help/curvefit/least-squares-fitting.htmlhttp://mathworld.wolfram.com/LeastSquaresFitting.html


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SOFTWARE FUNCTION


24

Excel:Slope:

=INDEX(LINEST(known_y's,known_x's),1)

Y-intercept:=INDEX(LINEST(known_y's,known_x's),2)

Matlab:

mdl = fitlm(X,y) returns a linear model of theresponses y, fit to the data matrix X.


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STATIC CALIBRATION


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Calibration shall refer and follow thestandards.

Measurement are taken by cycles and usually

done from low end to top end and thenreduced to the low end of the range.

It effectively combines the errors due to

nonlinearity, hysteresis and non-repeatability.


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CALIBRATION of a WEIGHTING SCALE


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A low cost, 0-5lb spring weighting scalehas been calibrated by placing accurate

weights on the platform.

Several cycles were completed beforedata recording started.

Fit a straight line to the data and

determine the accuracy, hysteresis andlinearity errors. Estimate the maximum

systematic and random errors.


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SCALE CALIBRATION DATA [ref. 3]


27


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CALIBRATION CURVE [ref. 3]


28


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HYSTERESIS IN MEASUREMENT


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-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6

Hysteresis

Cycle 3 Upwards

Cycle 3 Downwards

Cycle 4 Upwards

Cycle 4 Downwards


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DEVIATION DATA [ref. 3]


30


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PLOT OF DEVIATION DATA [ref. 3]


31


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NON-LINEAR CASE STUDY


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We measured a thermocouple junctionvoltage as 645 at the steam point(100), 3375 at the zinc point (420)and 11476 at the silver point (962).

Determine the sensitivity function if it is

assumed to be 3rd order polynomial: =

+ + and study the non-linearity.


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CASE DISCUSSION


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Based on the 3 calibration points, we canestablish equations as:

645 =100+ 100+ 100,3375 =420+ 420+ 420,11476 =962+ 962+ 962,

http://www.wolframalpha.com/input/?i=645%3D100x%2B100^2y%2B100^3z%2C+3375+%3D420x%2B420^2+y%2B420^3z%2C+114

76+%3D962x%2B962^2+y%2B962^3z

Key in 645=100x+100^2y+100^3z, 3375 =420x+420^2 y+420^3z, 11476 =962x+962^2 y+962^3z at http://www.wolframalpha.com/

= 6.06 + 3.61 10+ 2.59 10Plot by MATLAB>> t=0:10:1000;

>> plot(t, 6.06*t+3.61*10^(-3)*t.^2+2.59*10^(-6)*t.^3)

http://www.wolframalpha.com/input/?i=plot+V%3D+6.06T%2B3.61%C3%9710^%28

-3%29+T^2%2B2.59%C3%9710^%28-6%29+T^3+
http://www.wolframalpha.com/input/?i=645%3D100x%2B100^2y%2B100^3z,+3375+%3D420x%2B420^2+y%2B420^3z,+11476+%3D962x%2B962^2+y%2B962^3zhttp://www.wolframalpha.com/input/?i=645%3D100x%2B100^2y%2B100^3z,+3375+%3D420x%2B420^2+y%2B420^3z,+11476+%3D962x%2B962^2+y%2B962^3zhttp://www.wolframalpha.com/http://www.wolframalpha.com/input/?i=plot+V%3D+6.06T%2B3.61%C3%9710^(-3)+T^2%2B2.59%C3%9710^(-6)+T^3http://www.wolframalpha.com/input/?i=plot+V%3D+6.06T%2B3.61%C3%9710^(-3)+T^2%2B2.59%C3%9710^(-6)+T^3http://www.wolframalpha.com/input/?i=plot+V%3D+6.06T%2B3.61%C3%9710^(-3)+T^2%2B2.59%C3%9710^(-6)+T^3http://www.wolframalpha.com/input/?i=plot+V%3D+6.06T%2B3.61%C3%9710^(-3)+T^2%2B2.59%C3%9710^(-6)+T^3http://www.wolframalpha.com/http://www.wolframalpha.com/input/?i=645%3D100x%2B100^2y%2B100^3z,+3375+%3D420x%2B420^2+y%2B420^3z,+11476+%3D962x%2B962^2+y%2B962^3z


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MATH TOOLS FROM WOLFRAM & MATLAB


34

0 100 200 300 400 500 600 700 800 900 10000

2000

4000

6000

8000

10000

12000

14000


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LEARNING OUTCOMES


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Characteristics of Measurement: Resolution, Accuracy & Error (systematic &

random);

Precision & Reproducibility (statistical analysis); Sensitivity (linear & non-linear);

Calibration of measuring system;


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REFERENCES

8/9/2014 SIE1004 Measurement & Sensor 36

1. Measurement, Instrumentation and Sensors Handbook, Edited by John G.Webster, CRC Press & IEEE Press, 1999

2. Principles of measurement systems, by John P. Bentley, Pearson Publication,

2005.

3. Introduction to Engineering Experimentation, by Anthony J. Wheeler & Ahmad R.

Ganji, Pearson, 2010.

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