2103-390 ME Exp and Lab I

1

2103-390 ME Exp and Lab I

Basics for Physical Quantities and Measurement

Physical Quantity

Measured Quantity VS Derived Quantity

Some Terminology

Physical Principle/Relation of An Instrument and Sensor and Sensing Function fs

Measurement System Model

Input-Output relation: y = f ( x ; …)

Important: Identify MS, input x, and output y clearly

2

Theoretical Input-Output Relation and Theoretical Sensitivity

Linear Instrument VS Non-Linear Instrument

Some Common Mechanical Measurements

Calibration

Static Calibration

Calibration Points and Calibration Curve

Calibration Process VS Measurement Process

Some Basic Instrument Parameters

Range and Span

Static Sensitivity K

Resolution

Some Common Practice in Indicating Instrument Errors

3

Basics for Physical Quantities and Measurement

4

Physical QuantityDescribing A Physical Quantity

Physical quantity

A quantifiable/measurable attribute we assign to a particular characteristic of

nature that we observe.

)(][,2.1.3.2

LengthLlmlDimensionQmeasureofunitQunitwrtvaluenumerical

][q

Q

Q

Describing a physical quantity q

1. Dimension

2. Numerical value with respect to the unit of measure

3. Unit of measure

5

Measured Quantity VS Derived Quantity

The Determination of The Numerical Value of A Physical Quantity q

must be either through

Measurement with an instrument Measured quantity

or

Derived through a physical relation Derived quantity(and by no other means)

Because of existing physical relations/laws, we don’t want anybody to make up any

number for a physical quantity.

6

Some Terminology

Measurement / Measure

The process of

quantifying, or

assigning a specific numerical value corresponding to a specific unit (of

measure) to

a physical quantity q of interest in a physical system.

Measurand / Measured Variable

The physical quantity q that we want to measure, e.g., velocity, pressure, etc.

Instrument / Measuring Instrument / Measurement System

The physical tool that we use for quantifying the measurand, e.g., thermometer,

manometer, etc.

7

Often, our desired physical quantity x – measurand – cannot be measured

directly (in its own dimension and unit).

Physical Principle/Relation of An Instrument ( fs )

and

Sensor and Sensing Function fs

8

What is the pressure difference (pa – pb)?

Do we measure the pressure difference (pa – pb) directly in

the unit of pa with a U-tube manometer?

pa

pressure at surface a

pb

pressure at surface b

h

m

Class Discussion

9

What is the pressure difference (pa – pb)?

We do not measure (pa – pb) directly.

Instead, we measure h.

Then, determine the desired measurand (pa – pb)

from the physical principle/relation (from static

fluid)

pa


pb


h

m

...);(1

)(

output

s

measurandinput

yfx

youtputhgm

xmeasurandinputbpap

Measurement System (MS)

y = fs ( x ; …) (sensor stage)

Input measurand x Output y

(pa – pb) h

10

Often, our desired physical quantity x – the measurand – cannot be measured directly (in its own

dimension and unit).

We need to determine/derive its numerical value from

another physical quantity y, which is more easily measured, and

a physical relation/principle.

Measurement System (MS)

(sensor stage)

Input measurand x

(pa – pb)

Output y

h

x

(pa – pb)

y

h

youtput

hgmxmeasurandinput

bpap

ysfx

xmeasurandinputbpap

gmyoutput

h

xfy s

)(

...);(1

)(1

;...)(

11

Physical Principle/Relation of An Instrument [and Sensing

Function fs]

Physical Principle of An Instrument [and Sensing Function fs]

The physical principle that allows us to determine the desired measurand x with

dimension [x] in terms of another physical quantity ys with different dimension [ys].

We refer to the underlying physical relation as sensing function fs.

pa


pb


h

m

youtput

hgmxmeasurandinput

bpap

ysfx

xmeasurandinputbpap

gmyoutput

h

xfy s

)(

...);(1

)(1

;...)(

The physical principle of a U-tube manometer

is static fluid (fluid in static equilibrium).

12

Measurement System (MS)Input measurand x Output y

(pa – pb) h

Principle: Fluid Static

pa

pb

h

m

Physical Principle: Fluid Statics

...);()(1

...);()( 1

xfyppg

h

yfxhgpp

s

xmeasurandinput

bamyoutput

syoutput

m

xmeasurandinput

ba

fs is the sensing function.

13


14

Measurement System / Instrument

Control stage

Process/System

SensorstageInput x Output y

Signal path

Transducerstage

Signal Conditioning

stage

Outputstage

Sensor (or sensor-transducer) Output ys: ys = f (x;…)

Sensor/Transducer

• employs physical phenomena

• to sense the desired physical quantity x

• in terms of another more easily measured quantity ys .

Thermal expansion

Temperature T

Length L (scale)

Thermometer

15


....);(xfy


Input x

Measurand q

in a physical system

Output y(Numerical value )

OutputStage

....);( mo yfy

of

Sensor stage(Sensing element)

....);(xfy ss

sf

Signal Modification Stage

....);( smm yfy

mf

Physical Principle of The Instrument and Sensing Function ( fs )

We shall refer to

the physical principle that allows the sensor to sense the desired measurand x with

dimension [x] in terms of another physical quantity ys with different dimension [ys] as the

physical principle of the instrument, and

the corresponding underlying physical relation as the sensing function fs .

16

Input - Output Relation:

We are then interested in the input-output relation

....);(....);( 1

outputinputinputoutput

yfxxfy

....);(xfy

....);(xfy

Measurement System (MS)Input measurand x

(physical quantity)

Output y

(physical quantity)

How to find the output-input relation

- Theoretical

- Actual Static Calibration

17

Important: Identify MS, input x, and output y clearly

When considering measurement system (or subsystem) characteristics

1. Measurement System: Identify the measurement system (MS) clearly

(physically as well as functionally), from input

x to output y.

2. Input Measurand x: Identify the physical quantity that is the input measurand x and its

dimension/unit.

3. Output y: Identify the physical quantity that is the output y and its

dimension/unit.

4. Calibration Curve: Find and draw the calibration curve ( y VS x ) for the system


Note:

1. It helps to identify the dimensions of the input and output physical quantities clearly. Is it length, pressure, velocity, or voltage, etc?

2. Recognize that if there is no output indicator, we cannot yet know the numerical value.

For example, the output of the pressure transducer is voltage output, but without a voltmeter or an output indicator, we cannot yet know the numerical

value of this voltage output.

In this regard, e.g., when perform uncertainty analysis, the output indicator must be accounted for as part of the measurement system.

18

Measurement System Model: First-Order System

The ODE has the solution

where the complementary solution is given by

and is the time constant.

The particular solution depends on the input forcing function.

Input )(tx Measurement System ))(( txyOutput

)(1 tFyadt

dya o First-Order System:

)()()( tytyty pc

/1)( t

c ecty

oaa /1

19

First-Order System: Step Forcing Input

o

nt

o

onn

aa

tteyKA

ytyt n

/

/;1)(

)(

1

= Normalized time

0 1 2 3 4 5 60

0.10.20.30.40.50.60.70.80.9

1

tn

tntn

o

on

yKA

yty

)( t n

n

0 00.368 0.3080.5 0.3931 0.6322 0.8653 0.9504 0.9825 0.9936 0.998

1/e =

20

Theoretical Input-Output Relation

and

Theoretical Sensitivity

21

The Theoretical Input-Output Relation

and Sensitivity

pa – pb

(input measurand)

h(output)

The input-output relation:

gKppK

ppg

h

xfy

mmeasurandinput

ba

measurandinput

bamoutput

inputoutput

1,)(

)(1

....);(

The Theoretical Input-Output Relation and Theoretical Sensitivity for U-Tube Manometer

MS: U-Tube

Manometer

Input measurand x ? Output y ?

(pa – pb)

[pressure]h [Length]

Define The Measurement System MS

(Define the input and the output quantities clearly.)

pa


pb


h

m

22

Theoretical Input-Output Relation and Graph, andLinear VS Non-Linear Instrument

xmeasurandinput

ba pp )(

youtput

h

gK

dx

dy

m1

slope

Linear Instrument

Output y is a linear function of input

measurand x.

The slope K is constant throughout

the range

xmeasurandinput

youtput

Kdx

dy

General Non-Linear Instrument

Output y is not a linear function of

input measurand x.

The slope K is not constant

throughout the range

23

Sensitivity K

Sensitivity

x

y

inputd

outputd

dx

dyK

)(

)(:


If K is large, small change in input produces large change in output.

The instrument can detect small change in input measurand more easily.

xmeasurandinput

youtput

Kdx

dy:

24

Example 1: Define MS, Input x, Output y Clearly

Note: Here, we define a measurement system in a more general term, based on the

interested functional relation.

25

m = density of manometer fluid

a = density of fluid a

b = density of fluid b

pa = static pressure at center a

pb = static pressure at center b

p1 = static pressure at 1

p2 = static pressure at 2

ha = elevation at center a

hb = elevation at center b

h1 = elevation at free surface 1

h2 = elevation at free surface 2

h = h2 – h1

g

m

+bp

+ap

1h

2h

h

1

2ah

bh

a

b

26

Example 1 Determine the theoretical input-output relations for the two measurement systems[See Appendix A for the derivation]

U-Tube Manometer


pa – pb [pressure] h [Length]

+bp

+ap

Measurement System 1 (MS1)

U-Tube Manometer


p1 – p2 [pressure] h [Length]

+

2p+

1p


youtput

m

xmeasurandinput

hgpp 21:MS2

)()()(:MS1 21 hhghhghgpp bbaayoutput

m

xmeasurandinput

ba

27

Example 2:

Redefine our measurement system for convenience

28

Example 2 Redefine our measurement system for convenience

U-Tube Manometerpa – pb [pressure] h [Length]

m

+bp+ ap

1h

2h

h1

2

ah bh

f


MS1: It is not convenient to measure the change from the two

interfaces.

We may redefine our output/system.

MS2: Here, it is more convenient to measure the change with

respect to one stationary reference point.


m

+bp+ ap

1h

2h

h

1

2

ah bh

fEquilibrium position

U-Tube Manometerpa – pb [pressure] h [Length]

29

Example 3:

Find the theoretical input-output relation and the theoretical sensitivity

30

Example 3Find the theoretical input-output relation and the theoretical sensitivity

m

+bp

+ap

1h

2h

h

1

2

ah bhf

The theoretical input-output relation is

gKppKKppf

ppg

h

hgpp

fmxmeasurandinput

ba

xmeasurandinput

bas

xmeasurandinput

bafmyoutput

youtputfm

xmeasurandinput

ba

)(

1,)();)((

)()(

1

,)()(

Measurement System (MS)(pa - pb) h

31

Example 4:

Theoretical sensitivity and how to increase sensitivity in the design of

an instrument

32

Example 4 How can we increase the sensitivity of the manometer?Differential Pressure Measurement - Inclined Manometer

Fox et al, 2010, Example Problem 3.2 pp. 59-61.

1p

2p

sin)/(

1,)(

)(sin)/(

1

sin)/()(

sin)/(:

sin:

)/(44

:

)()(

221

212

221

221

2

21

21

2

2121

DdgKppKL

ppDdg

L

LDdgpp

LDdhh

Lh

LDdhLdhD

hhgpp

mxmeasurandinputyoutput

xmeasurandinputmyoutput

youtputm

xmeasurandinput

youtput

youtput

youtputyoutput

m

xmeasurandinput

Principle: Static fluid

Appropriate sensitivity K can be chosen by changing d/D and sin , e.g.,

Smaller Higher K

33


34


Temperature

Pressure

Velocity

Volume Flowrate

Displacement

Velocity

Acceleration

Force

Torque

35

Example 5: Differential Pressure MeasurementInclined Manometer

From Dwyer http://www.dwyer-inst.com/PDF_files/Priced/424_cat.pdf


36From Dwyer http://www.dwyer-inst.com/PDF_files/Priced/424_cat.pdf

Inclined Manometer



pa – pb [pressure](at the free surfaces)

L [Length, mmW]

ab pp

Balance position: apply pa = pb

apL = 0

ab pp

Measure position: apply pa > pb

L = 0 L (mmW)

ap

37

Example 6: Differential Pressure MeasurementPressure Transducer: Capacitance

From Omega: http://www.omega.com/ppt/pptsc.asp?ref=PX653&ttID=PX653&Nav=

Principle: Capacitance

38

ap bp

Pressure Transducer (alone)


Pressure Transducer



pa – pb [pressure](at the ports)

V [Voltage, Vdc]

Without output stage such as voltmeter or

output indicator, however, we cannot yet

know the numerical value of the output

(voltage).

This is not yet a complete measurement

system – no output stage.

of the pressure transducer alone

39

Pressure Transducer + Output Indicator


Pressure Transducer +

Output Indicator



pa – pb [pressure](at the ports)

V [Voltage, Vdc]

Specification/Characteristics (of the measurement system, e.g.,

accuracy, etc.)

Need to take into account the characteristics (e.g., accuracy,

etc.) of the output stage – i.e., output indicator – also.

ap bp

40

Output Indicator (alone)

From Omega http://www.omega.com/ppt/pptsc.asp?ref=DP24-E&Nav=

42

Calibration Static Calibration


Some Basic Instrument Parameters

43

CalibrationStatic Calibration

Calibration is the act of applying a known/reference value of input to a

measurement system.

Objectives of Calibration Process

1. Determine the actual input-output relation of the instrument.

2. Quantify various performance parameters of the instrument, e.g.,

range, span, linearity, accuracy, etc.

The known value used for the calibration is called the reference/standard.

Static Calibration:

• A calibration procedure in which the values of the variables involved remain constant.

• That is, they do not change with time.

44

Calibration points: yM(x)

Static calibration curve (fit): yC(x)

If it is linear, yC(x) = Kx + b.

Input x [unit of x ]

Out

put y

(x)

[uni

t of y

]

Reference: Known value


Calibration Points: We first have a set of calibration points.

Calibration Curve: For convenience in usage, we fit the curve through calibration

points and use the fitted equation in measurement.

45

Calibration Process VS Measurement Process


Out

put y

(x)

[uni

t of y

]


Calibration process Measurement process



Out

put y

(x)

[uni

t of y

]


46

Range:

Input range:

Output range:

Span:

Input span:

Output span:

Some Basic Instrument ParametersRange and Span


Out

put y

(x)

[uni

t of y

]

ro =

ym

ax -

ym

in

ymax

ymin

xmin xmax

ri = xmax - xmin

Calibration

maxmin to xx

minmax xxri

maxmin to yy

minmax yyro

Full - scale operating range (FSO)

= Output span

= minmax yyro

47

Static Sensitivity:

• is the slope of the static calibration curve yC(x) at that point.

• In general, K = K(x).

• If the calibration curve is linear, K = constant over the range.

Some Basic Instrument ParametersStatic Sensitivity K

1

)()( 1

xx

c

dx

xdyxKK


Out

put y

(x)

[uni

t of y

]

1

)()( 1

xx

c

dx

xdyxKK

Calibration points: yM(x)


If it is linear, yC(x) = Kx + b.

48

[Output] Resolution (Ry) is the smallest physically indicated

output division that the instrument displays or is marked.

Note that

• while the input may be continuous (e.g., temperature in the room),

• the indicated output displays are finite/digit (e.g., ‘digital’ or numbered scale on

the output display).

Some Basic Instrument ParametersResolution (Ry)


Out

put y

(x)

[uni

t of y

]

Ry

Indicated output

displays are finite/digit

Continuous t

49

Some Basic Instrument ParametersResolution and Static Sensitivity KThe smallest change in input that an instrument can indicate.

K

Linear static calibration curve: yC(x) = Kx + b.

z(x)


Out

put y

(x)

[uni

t of y

]

Rx: Input Resolution

K

yx R

R

Output

Resolution:

RyIndicated output displays are finite/digit,

the input resolution is correspondingly considered finite.

50


51


Error e in % FSO:

Error e in % Reading:

Error e in [unit output]/[unit input]:

%100Error

FSO][% or

ee

%100Error

Reading% y

ee

einput valuCurrent xx

ee

Errorinput]unit output / unit[

spanOutput minmax -y yro

valueReading/Current )( xy

52

Example: the input range of a pressure transducer is 0 – 10 bar,

the output range is 0 - 5 V, and

the current reading is 3 V (or 6 bar), then

Error: 1% FSO

This error is considered fixed and applied to all reading values.

That means the % reading error at lower reading values are more than this value.

Error: 1% Reading

Error: 5 mV / bar

mVVV

re

e o

5005.05100

1100

FSO][%Error

mVVV

ye

e

3003.03100

1100

Reading%Error

mVbar

bar

mV

xee

3065

inputunit output / unit Error

53

Example: Sketch The Estimated Calibration Curve, and Extract Some Input-Output and Uncertainty Information

from The Curve

Given two pressure transducers with the following specifications

Input range: No.1 0 to 10 in WC No.2 0 to 20 in WC.

Output range: 0 to 5 VDC (both)

Accuracy: + 1% FSO (both)

Assume that both have a linear calibration curve.

1. Sketch the estimated linear calibration curve and find the estimated static sensitivity K for each

device.

2. What are the expected errors/uncertainties from these devices?

in unit of output

in unit of input

3. If we are to measure a pressure difference of 5 in WC, what is the expected output from each?

4. If we read the output as 2.5 V, what is the input (measured value) of each?

5. At the current reading (corresponding to 5 in WC input), what is the expected uncertainty in % of

reading for each?

Note: We shall discuss more about error VS uncertainty later.

54

1. Sketch the estimated linear calibration curve and find the estimated static sensitivity K for each device.

Input (x) : Pressure difference (in WC)

Output (y): Voltage DC (Volt)

The estimated linear calibration curve is the straight line joining the two extreme points:

( Inputmin , Outputmin)

( xmax , ymax)

( 0 in WC, 0 V)

( 10 in WC, 5 V) ( 20 in WC, 5 V)

( xmin , ymin)

( Inputmax , Outputmax)

( 0 in WC, 0 V)

( 10 in WC, 5 V)

( xmax , ymax)

( xmin , ymin) ( 0 in WC, 0 V)

( 20 in WC, 5 V)

1

2

( 0 in WC, 0 V)

The estimated static sensitivity K is the slope of the estimated calibration curve.

K = 5 V / 10 in WC

= 0.5 V/in WCK = 5 V / 20 in WC

= 0.25 V/in WC( Inputmin , Outputmin)

( Inputmax , Outputmax)

55

2. What are the expected uncertainties from these devices? 1) in unit of output 2) in unit of input

Accuracy: + 1% FSO (both)

in unit of output:

No. 1 & 2: [since both have the same % full-scale accuracy and the same full-scale output span]

in unit of input:

We can employ the calibration curve by finding the corresponding x for the given y of +

50 mV.

Note that since the two models have different sensitivity, they have different uncertainty in

unit of input.

Or, we can find the corresponding full-scale input span [the calibration curve is linear] and find

No. 1:

No. 2:

mVVV

re

e o

5005.05100

1100

FSO][%error scale-Full

inWCinWCe 1.010100

1error scale-Full

inWCinWCe 2.020100

1error scale-Full

56

3. If we are to measure a pressure difference of 5 in WC, what is the expected output from each?


Output (y): Voltage DC (Volt)

( 0 in WC, 0 V)

( 10 in WC, 5 V) ( 20 in WC, 5 V)

( 0 in WC, 0 V)

We come back again to the calibration curve.

Equation: We can also use the equation for the linear calibration curve:

where (x, y) is any point on the curve.

The extreme point need not be (0, 0). It is common to find that it is not.

5 in WC

No.1: Output = 2.5 V

5 V

No. 2: Output = 1.25 V

range]n calibratio thebeyond usedbe[cannot,)()(

maxmin

maxmin

minmax

min

minmax

min

yyy

xxx

xx

xx

yy

yy

),( yx

),( minmin yx

Output Input

57

4. If we read the output as 2.5 V, what is the input (measured value) of each?

Again, we come back to the calibration curve.

Equation: Again, we can also use the equation for the linear calibration curve.

2.5 V

No.1: Input = 5 in WC No. 2: Input = 10 in WC


Output (y): Voltage DC (Volt)( 10 in WC, 5 V) ( 20 in WC, 5 V)

( 0 in WC, 0 V)

5 V

range]n calibratio thebeyond usedbe[cannot,)()(

maxmin

maxmin

minmax

min

minmax

min

yyy

xxx

xx

xx

yy

yy

58

5. At the current reading (corresponding to 5 in WC input), what is the expected uncertainty in % of reading for each?

Because the instrument “accuracy” (say in unit of input) is considered a fixed

uncertainty, we have

No.1:

No.2:

reading(current)of%2%1005

1.0yUncertaint

inWC

inWC

reading(current)of%4%1005

2.0yUncertaint

inWC

inWC

59

Instrument Errors (Elemental Errors)

See, e.g.,

Dunn, P. F., 2005, Measurement and data analysis for engineering and

science, McGraw-Hill, New York.

Figliola, R. S., and Beasley, D. E., 2000, Theory and design for mechanical

measurements, 3rd Edition, Wiley, New York.

Doebelin, E. O., 1989, Measurement systems: application and design, 4th

Edition, McGraw-Hill, New York.

60

Instrument Errors: Hysteresis and Repeatability Errors

FSO%100% maxmax

,,

o

hh

downscaleMupscaleMh

r

ee

yye

Hysteresis Error, eh:

Upscale and downscale calibrations give

different results. This may due to, e.g., friction,

residual electrical charge, etc.

Input x [unit x]

Output y [unit y]

Upscale

Downscale

ehmax

FSO%100)2(

%

deviation Standard

maxmax

o

YR

Y

r

Se

S

Repeatability Error, eR:

Measure of the ability of an instrument to

indicate the same output value upon repeated

but independent applications of the same input.

Input x [unit x]

Output y [unit y]

Note that the standard also has fossilized uncertainty. Hence, conceptually there is also ‘horizontal error bar.’

61

Instrument Errors: Linearity / Nonlinearity Error

FSO%100%

)()(

maxmax

o

LL

CML

r

ee

xyxye

Linearity Error, eL: (Independent Linearity)

Measure of maximum deviations of calibration

points from the best fit linear calibration curve.

Input x [unit x]

Output y [unit y]

yC(x)

eLmax

FSO%100%

)()(

maxmax

o

LL

eML

r

ee

xyxye

Linearity Error, eL: (Endpoint Linearity)

Measure of maximum deviations of calibration

points from the straight line passing through the

end points.

(More conservative)

Input x [unit x]

Output y [unit y]

eLmax

ye(x) = yend-to-end(x)

62

Estimate of The Overall Instrument Error/Uncertainty (UI):RSS Method

For all known sources of elemental instrument errors [1] (some we may not know), say i = 1, 2,

…, M sources, the overall instrument error ec can be estimated using the root-sum-square

(RSS) method:

This is often taken as instrument uncertainty UI at the design-stage uncertainty analysis.

Example: If the hysteresis, linearity, sensitivity, and repeatability errors are known, then the

overall instrument error ec can be estimated as

[1] It may be more suitable to refer to these as uncertainties since for our measurement we do

not know the values of the errors for certain and, for us, these are simply ‘possible value

that an error may have.’ However, the term error is still very common.

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21 CUeeee IMc

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2103-390 ME Exp and Lab I

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Transcript of 2103-390 ME Exp and Lab I