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November 08 SEI MSc, Sensors & Sensing - Lecture 5 1
MSc Sensors & Electronic Instrumentation
Sensors & Sensing Principles
Lecture 5: Strain Gauges & Pressure Measurement
Dr Paul W Nutter
Room 113, IT Building (Next to Computer Science)
E-mail: [email protected]
November 08 SEI MSc, Sensors & Sensing - Lecture 5 2
Lecture Aims
The aims of this lecture are:
to discuss the operation of strain gaugesto present forms of gaugesto discuss bridge circuits for signal conversionto present applications of strain gauges (pressure measurement)to discuss pressure sensors
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 3
Strain Gauges
Strain gauges are based on the variation of resistance of a conductor
or semiconductor when subjected to a mechanical stress.
Strain gauges are used to measure the extension or compression of a
body, and have many applications primarily in the measurement ofForce, Pressure and Acceleration.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 4
Resistance of a Wire
The simplest strain gauge can be considered to be constructed from a
single wire having length l, cross sectional areaA, and resistivity, ,as shown below. l
Ar
F
Where the bulk resistance,R, of the wire is given by
Rl
A=
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 5
Stress and Strain
Definitions:
Strain =extension
original length=l
l
Strain is caused by a stressthat is applied to the body, which
using Hookes law is given by
l
l
E
=Area
Force
=Stress
whereEis Youngs Modulus and we are assuming operation in
the elastic region.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 6
Stress on a Wire
If we apply a stress, S, longitudinally to out simple wire (due to a
forceF), then the resistance of the wire will change which is given
by( )
s
Al
sA
l
s
l
Ads
dR
1
+
+
=
which gives
s
A
A
l
sA
l
s
l
Ads
dR
2
+
=
If we divide both sides by the resistance,R, then
s
A
Ass
l
lds
dR
R
1
1
11
+
=
A
A
l
l
R
R
+
=
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 7
Poissons Ratio
If our simple wire is of diameter, d, then we have
( )
A
A
d d d
d
d
dd d=
+
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 9
Gauge Factor
The gauge factor, G, is defined as,
G
RR
ll
=
G
ll
= + +1 2
The second term, 2n, is entirely due to dimensional changes,
whereas the third, Dr/r /Dl/l, is known as the piezoresistive term
and is the change in actual resistivity due to applied strain.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 10
Materials
In metals, the Dl/lterm dominates and the gauge factor is given byG +1 2
Typically, 00.5 and therefore G 12 for common
copper-nickel alloy strain gauges. So for a 1% change in length, the
resistance of a metal strain gauge changes by 12%.
In semiconductors (i.e. silicon) the / term dominates (large
piezoresistivity) and G values of 100 or more can be achieved. This
gives a very large sensitivity.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 11
Measurement of Force
Thus we have shown that there is a relationship between the
change in electric resistance of a material and the strain it
experiences, and that the change depends upon the type of strain
gauge employed - metal or semiconductor.
If the relationship between the strain and force causing it is
known, then from the measurement of resistance change it is
possible to determine the applied force.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 12
Unbonded Strain Guages
The unbonded strain gauge consists of a wire stretchedbetween two points in an insulating medium such as air.
These are typically used for pressure, force and accelerationmeasurement.
An unbonded strain gauge is usually employed in a bridgecircuit and is arranged so that two gauges are lengthened and
two shortened by the displacement of a movable part relative
to a fixed part.
Typical displacements which can be measured - 50mm on aforce lever or diaphragm.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 13
Bonded Strain Gauges
Bonded strain gauges are typically wires cemented onto a suitable
backing or more likely thin film resistors deposited onto a
suitable substrate (often epoxy resin). The gauge is then cemented
onto the test structure from which strain is to be measured.
Bonded strain gauges come in a variety of forms,
linear gauge - measure strain in a single axisrosette gauge - measure strain in three directions
torque gauge for measuring shear strain due to torsion
radial gauge - for attaching to pressure diaphragms
November 08 SEI MSc, Sensors & Sensing - Lecture 5 14
Forms of Bonded Gauge
a) Linear
Foil
Backing
Gauge
Length
c) Torque
b) Rosette
d) Radial
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 15
Typical Characteristics
Gauge Factor 1.8 - 2.35 50-60
Gauge Resistance
(W)
120, 350, 600,
1000>500
Linearity 0.1% 1%
Breaking Strain 25,000 me 5,000 me
Fatigue Life10 million
reversals
10 million
reversals
Metal Semiconductor
1 = 10-6m/m - strain
November 08 SEI MSc, Sensors & Sensing - Lecture 5 16
Limitations
The measurement will only be correct if all the stress istransmitted to the gauge; this is achieved by bonding the
strain gauge with an elastic adhesive, which is stable with
temperature and time.
There are a few limitations of strain gauges, which must be
considered:
The applied stress should not exceed the elastic limit or elseHookes law is no longer valid.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 17
Limitations cont.
Temperature is a source of interference in strain gauges, sincechanges in temperature affects the dimensions and resistivity
of the strain gauge. Temperature effects are very pronounced
in semiconductor strain gauges. The effects of temperature
may be elevated by the use of dummy gauges, which have the
same temperature characteristics as the active gauge.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 18
Limitations cont.
Resistance is measured by passing a current through the straingauge, the resulting power dissipation may cause heating -
typical maximum current is 25mA for metal strain gauges ifthe base material is a good heat conductor, and 5mA if it is a
poor heat conductor. In semiconductor strain gauges the
maximum power dissipation is approx. 250mW.
In spite of these limitations, strain gauges are some of the most
popular sensors because of their small size, high linearity and low
impedance.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 19
Bridge Circuits
vs vo
R1 R2
R3 R4
v1 v2
v1= v
s
R3
R1+ R
3
, v2 = vs
R4
R2+ R
4
The bridge is in balance whenR1/R3= R2/R4, such that vo=0.
vo= v
1 v
2= v
s
R3
R3+ R
1
R
4
R4+ R
2
or vo = vs
1
1+R
1
R3
1
1+R
2
R4
November 08 SEI MSc, Sensors & Sensing - Lecture 5 20
Quarter Bridge
vs vo
R2
R4
Dummy GaugeR
Active Gauge
R+DR
vo= v
s
R + R( )R+ R + R( )
R
R + R
= vs
R + R( )2R + R
1
2
vo= v
s
2R + 2R( ) 2R + R( )4R+ 2R
NBDR can be +ve or -ve
if R
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 21
Temperature Effects
IfR1
is the dummy gauge and R3
is the active gauge, and when
balancedR1= R
2= R
3= R
4=R, then if the resistance ofR
3changes by
a fractionx due to an applied stress, i.e. R3=R+DR and bothR
1and
R3
undergo the same temperature change, y, i.e. R1=R(1+y), and
R3=(R+DR)(1+y) then
vo=
vs
R + R( ) 1+ y( )R 1+ y( ) + R + R( ) 1+ y( )
R
R + R
=
vsR4R
provided DR
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 23
The Half Bridge
The two active, two dummy gauge arrangement is often used in
load cells for measuring weight.
Active Gauges
Dummy Gauges
November 08 SEI MSc, Sensors & Sensing - Lecture 5 24
Half Bridge Output
vo= v
s
R + R( )R + R + R( )
R
R + R + R( )
vo= v
s
R + R( ) R( )2R + 2R
We have twice the sensitivity of the quarter bridge
NBDR can be +ve or -ve
if R
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 25
Full BridgeIn this arrangement two gauges experience compression and two
equal but opposite tension.
vs vo
Active Gauge(compression)
R - DR
Active Gauge(tension)
R+DRActive Gauge
(compression)
R - DR
Active Gauge(tension)
R+DR
November 08 SEI MSc, Sensors & Sensing - Lecture 5 26
Full Bridge
Active Gauges (Tension)
Active Gauges (Compression)
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 27
Full Bridge Output
vo= v
s
R + R( )R + R + R( )
R R( )
R + R R( )
vo= v
s
R + R( ) R + R R( )( ) R R( ) R + R + R( )( )R + R + R( )( ) R + R R( )( )
We have twice the sensitivity of the half bridge
NBDR can be +ve or -ve
if R
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 29
Example Applications
Load
Active Gauges
Dummy Gauges
Cylindrical Load Cell Connected as a half bridge.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 30
Pressure Transducer
Active Gauges
Dummy Gauges
Pressure inlet
Atmospheric pressure
Diaphragm
The dummy gauges sit in the less stressed area of
the diaphragm near the edges. The active gauges
are in the centre and experience a stress as the
diaphragm deforms due to pressure. A half bridge
connection would be used for read out.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 31
Measuring Torque
4545 12
3 4
The shear stress caused by torsion causes strains to appear at 45o
to the shaft axis. The strain gauges must be placed accurately at
45o otherwise they become sensitive to bending and axial stresses
in addition to those caused by torsion.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 32
Effects Due to Loading
In the previous analysis we
have assumed that there are
no loading effects at the
output of the bridge circuit.However, if a finite loading
resistanceRl, exists as
illustrated
vs
R3
R1
R2
R4
vo
Rl
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 33
Equivalent Circuit
vo
31
31
RR
RR
+
42
42
RR
RR
+
Rl
where the series resistance is
calculated by short circuiting
the source.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 34
Calculation of Loading Effect
The output voltage, vl, is then given by
IfRl is infinite, then vl=vo, as expected. IfRl is not infinite, there will
be a reduction in output signal.
vl
vo
=1
1+ Rb
Rl
where
Rb=
R2R
4
R2+ R
4
+R
1R
3
R1+ R
3
For example, ifRl= 10R
b, then v
l/v
o=1/1.1=0.91, and 9% of the signal
is lost due to loading effects.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 35
Amplification of Bridge Output
We can drive a differential amplifier using the bridge circuit,
s
R3
vsvo
R1 R2
R4
-
+
0V
Rf
Rf
vo
November 08 SEI MSc, Sensors & Sensing - Lecture 5 36
Equivalent Circuit
ov
-
+
0V
Rf
Rf
vo
31
31
1
RR
RRR
i
+
=
42
42
2
RR
RRR
i
+
=
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 37
Gain of Circuit
For a full bridge:R1=R+DR,R2=R-DR,R3=R-DR andR4=R+DR,
Ri1 =
R1R3
R1 + R3
=
R+ R( ) R R( )2R
=R
2R
2
2R
R
2for R
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 39
Pressure Measurements
Divided into three categories:
1. Absolute pressure pressure at a point in a fluid relative to avacuum (absolute zero of pressure)
2. Gauge Pressure pressure relative to local atmosphericpressure.
3. Differential Pressure difference between two unknownpressures, neither of which is atmospheric pressure.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 40
Conversion factors for units of pressure
S.I. Unit of pressure is the Pascal (Pa).
1 Pa = 1 N/m2 = 1.45 x 10-4 lb/in2
1 lb/in2 = 6895 N/m2 = 0.0703 kg/cm2
1 atm = 101,325 N/m2 = 14.7 lb/in2
1 bar = 100,000 N/m2 = 14.5 lb/in2
1 mmHg = 133.3 N/m2 = 1.93 x 10-2 lb/in2
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 41
Manometers
Column of liquid supported produced pressureP = rh, where r is the
density of the liquid and h the height of the column. Thus for case a)
PA = PB = rha
a) Absolute
Unknown
pressure
Vacuum
haAB
Open to atmosphere
Unknown pressure
hg
b) Gauge
hd
c) Differential
November 08 SEI MSc, Sensors & Sensing - Lecture 5 42
Dead Weight Calibration System
Sensor
Under
Test
V1
Screw Press
Piston & Cylinder
Valve Priming Pump
& Reservoir
Weight (m)
The system is pressurised when the valve V1 is opened.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 43
Calibration Procedure
Extend the screw press to its zero position Apply weights representing the required pressure to the
piston (P = m.g/A, where m is the mass applied,A thearea of the piston andgthe acceleration due togravity).
Pressurise the system through valve V1 and then closethe valve.
Operate screw press until piston is just raised thensensor should then read pressureP.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 44
Accuracy of Calibration
Precision of manufacture of piston.The following all affect the accuracy of calibration:
Friction in the piston. Temperature of the gas in the system.
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 45
Sensors using elastic properties.
Three types of device:
1. Bourdon Tubes basis of many mechanical gauges.
2. Bellows low cost barometers.
3. Diaphragms or Membranes most commonly used structures for pressure
sensing.
November 08 SEI MSc, Sensors & Sensing - Lecture 5 46
Bourdon Tubes
Tube cross section
a) C type b) Spiral c) Twisted Tube
Stiff in x-y
Soft Rot.
Free end usually connected to needle dial. C-type used up to 7 x108 N/m2 (100,000
psi). The spiral and twisted versions produce larger displacements and are used below
1 x 106 N/m2. Best accuracy ~ 0.1%
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 47
Bellows
a) Single Bellows Gauge
(Gauge Pressure)
b) Double Bellows Gauge
(Differential Pressure)
Reversible with low hysterisis
Often used in aneroid barometers
November 08 SEI MSc, Sensors & Sensing - Lecture 5 48
Diaphragms and Membranes
D
tr
dmdr
PressurepDiaphragm
Y= Youngs Modulus of
diaphragm
r = density (SI units)u = Poissons ratio
D, t dm in mm
Centre deflection( )
3
42
265
13
Yt
pDdm
=
dm is linearly related to pressurep if tdm 5.0
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November 08 SEI MSc, Sensors & Sensing - Lecture 5 49
The End