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2145-391 Aerospace Engineering Laboratory I Basics for Physical Quantities and Measurement
Physical Quantity
Measured Quantity VS Derived Quantity
Some Terminology
Measurement / Measure
Measurand / Measured Variable
Instrument / Measuring Instrument / Measurement System
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
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Theoretical Input-Output Relation and Theoretical Sensitivity
Linear Instrument VS Non-Linear Instrument
Examples:
Example 1: Define MS, Input x, Output y Clearly
Example 2: Redefine our measurement system for
convenience
Example 3: Find the theoretical input-output relation and the
theoretical sensitivity
Example 4: Theoretical sensitivity and how to increase
sensitivity in the design of the instrument
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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
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Basics for Physical Quantities and Measurement
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Physical QuantityDescribing A Physical Quantity
In an experiment, we want to determine the numerical values of various physical
quantities.
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
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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.
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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 (real) 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.
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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
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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
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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
pressure at surface a
pb
pressure at surface b
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
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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
;...)(
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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
pressure at surface a
pb
pressure at surface b
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).
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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.
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Measurement System Model
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Measurement System Model
....);(xfy
Measurement System Model
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 .
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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
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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
Measurement System (MS)Input measurand x Output y
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.
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Theoretical Input-Output Relation
and
Theoretical Sensitivity
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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
pressure at surface a
pb
pressure at surface b
h
m
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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
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Sensitivity K
Sensitivity
x
y
inputd
outputd
dx
dyK
)(
)(:
Measurement System (MS)Input measurand x Output y
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:
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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.
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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
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Example 1 Determine the theoretical input-output relations for the two measurement systems[See Appendix A for the derivation]
U-Tube Manometer
Input measurand x ? Output y ?
pa – pb [pressure] h [Length]
+bp
+ap
Measurement System 1 (MS1)
U-Tube Manometer
Input measurand x ? Output y ?
p1 – p2 [pressure] h [Length]
+
2p+
1p
Measurement System 2 (MS2)
youtput
m
xmeasurandinput
hgpp 21:MS2
)()()(:MS1 21 hhghhghgpp bbaayoutput
m
xmeasurandinput
ba
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Example 2:
Redefine our measurement system for convenience
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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
Measurement System 1 (MS1)
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.
Measurement System 2 (MS2)
m
+bp+ ap
1h
2h
h
1
2
ah bh
fEquilibrium position
U-Tube Manometerpa – pb [pressure] h [Length]
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Example 3:
Find the theoretical input-output relation and the theoretical sensitivity
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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
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Example 4:
Theoretical sensitivity and how to increase sensitivity in the design of
an instrument
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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
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Some Common Mechanical Measurements
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Some Common Mechanical Measurements
Temperature
Pressure
Velocity
Volume Flowrate
Displacement
Velocity
Acceleration
Force
Torque
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Example 5: Differential Pressure MeasurementInclined Manometer
From Dwyer http://www.dwyer-inst.com/PDF_files/Priced/424_cat.pdf
Principle: Fluid Static
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34From Dwyer http://www.dwyer-inst.com/PDF_files/Priced/424_cat.pdf
Inclined Manometer
Principle: Fluid Static
Input measurand x ? Output y ?
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
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Example 6: Differential Pressure MeasurementPressure Transducer: Capacitance
From Omega: http://www.omega.com/ppt/pptsc.asp?ref=PX653&ttID=PX653&Nav=
Principle: Capacitance
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ap bp
Pressure Transducer (alone)
From Omega: http://www.omega.com/ppt/pptsc.asp?ref=PX653&ttID=PX653&Nav=
Pressure Transducer
Principle: Capacitance
Input measurand x ? Output y ?
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
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Pressure Transducer + Output Indicator
From Omega: http://www.omega.com/ppt/pptsc.asp?ref=PX653&ttID=PX653&Nav=
Pressure Transducer +
Output Indicator
Principle: Capacitance
Input measurand x ? Output y ?
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
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Output Indicator (alone)
From Omega http://www.omega.com/ppt/pptsc.asp?ref=DP24-E&Nav=
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Differential Pressure Measurement
Inclined Manometer
Principle: Fluid Static (Fluid in Equilibrium)
Pressure Transducer
Principle: Capacitance
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Calibration Static Calibration
Calibration Points and Calibration Curve
Some Basic Instrument Parameters
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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.
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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 and Calibration Curve
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.
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Calibration Process VS Measurement Process
Input x [unit of x ]
Out
put y
(x)
[uni
t of y
]
Reference: Known value
Calibration process Measurement process
Static calibration curve (fit): yC(x)
Input x [unit of x ]
Out
put y
(x)
[uni
t of y
]
Reference: Known value
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Range:
Input range:
Output range:
Span:
Input span:
Output span:
Some Basic Instrument ParametersRange and Span
Input x [unit of x ]
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
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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
Input x [unit of x ]
Out
put y
(x)
[uni
t of y
]
1
)()( 1
xx
c
dx
xdyxKK
Calibration points: yM(x)
Static calibration curve (fit): yC(x)
If it is linear, yC(x) = Kx + b.
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[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)
Input x [unit of x ]
Out
put y
(x)
[uni
t of y
]
Ry
Indicated output
displays are finite/digit
Continuous t
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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)
Input x [unit of 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.
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Some Common Practice in Indicating Instrument Errors
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Some Common Practice in Indicating Instrument Errors
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
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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
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U-Tube Manometer Relation
Principle: Static fluid (fluid in static equilibrium)
)2()()()(:)1()(
)1()()()()(
)()()()(:)()(
.........
)()(:21
)()(:2
)()(:1
21
2121
2121
2121
22
11
hhghhghgppC
hhghhgpppp
hhghhgppppBA
Chghhgpp
Bhhgppb
Ahhgppa
bbaamba
bbaaba
bbaaba
mm
bbb
aaa
m
+bp
+ap
1h
2h
h
1
2ah
bh
a
b
U-Tube ManometerInput measurand x ?
pa – pb [pressure]Output y ?
h [Length]
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m
+bp
+ap
1h
2h
h
1
2
ah bhf
)3()(,
)()(
)(
)2()()()(:)1()( 21
gKhK
hgpp
hghgpp
hhghhghgppC
fm
fmba
fmba
bbaamba
If b = b = f
ha = hb
Then Eq. (2) becomes
If the fluid is gas (while manometer fluid is liquid),
)4()( hgpp mba
mf