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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]


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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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.


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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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.


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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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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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.


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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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.


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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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


Forms of Bonded Gauge

a) Linear

Foil

Backing

Gauge

Length

c) Torque

b) Rosette

d) Radial

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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


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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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.


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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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


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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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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The Half Bridge

The two active, two dummy gauge arrangement is often used in

load cells for measuring weight.

Active Gauges

Dummy Gauges


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


if R

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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


Full Bridge

Active Gauges (Tension)

Active Gauges (Compression)

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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


if R

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Example Applications

Load

Active Gauges

Dummy Gauges

Cylindrical Load Cell Connected as a half bridge.


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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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.


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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Equivalent Circuit

vo

31

31

RR

RR

+

42

42

RR

RR

+

Rl

where the series resistance is

calculated by short circuiting

the source.


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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Amplification of Bridge Output

We can drive a differential amplifier using the bridge circuit,

s

R3

vsvo

R1 R2

R4

-

+

0V

Rf

Rf

vo


Equivalent Circuit

ov

-

+

0V

Rf

Rf

vo

31

31

1

RR

RRR

i

+

=

42

42

2

RR

RRR

i

+

=

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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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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.


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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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


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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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.


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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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.


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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Bellows

a) Single Bellows Gauge

(Gauge Pressure)

b) Double Bellows Gauge

(Differential Pressure)

Reversible with low hysterisis

Often used in aneroid barometers


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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