bridge measure

AE3145 Spring 2000 1

Resistance Strain Gage Circuits

• How do you measure strain in an electrical resistance straingage using electronic instrumentation?

• Topics– basic gage characteristics– Wheatstone Bridge circuitry– Bridge completion, balancing, calibration, switching

• Course Notes provide complete coverage.

• See reference text for further details.


Resistance in Metallic Conductor

L ∆L

Wire before and after strain is applied

ALR ρ=Resistance Equation:

AALL

ALR

∆+∆+∆+−=∆ )( ρρρChange in R:

��

�=≅∆ALddRR ρDifferential change in R:

or: AdA

LdLd

RdR −+=

ρρ

ευρρ )21( ++= d

RdRAfter a bit of manipulation:

DEFINE Gage Factor (GF): εRdRGFFACTORGAGE /==

Ferry alloy

1 2 30

1

2

3

4

% Strain

% ∆R/R

Nickel

10%rhodium/platinum

Constantan alloy

40% gold/palladium

GF is the slope ofthese curves


ExamplesTable 1. Gage Factors for Various Grid Materials

Gage Factor (GF)MaterialLow Strain High Strain

Ultimate Elongation(%)

Copper 2.6 2.2 0.5Constantan* 2.1 1.9 1.0Nickel -12 2.7 --Platinum 6.1 2.4 0.4Silver 2.9 2.4 0.840% gold/palladium 0.9 1.9 0.8Semiconductor** ~100 ~600 --

* similar to “Ferry” and “Advance” and “Copel” alloys.** semiconductor gage factors depend highly on the level and kind of doping used.

Example 1Assume a gage with GF = 2.0 and resistance 120 Ohms. It is subjected to a strain of 5

microstrain (equivalent to about 50 psi in aluminum). Then

!%001.00012.0

)120)(65(2

changeOhms

eRGFR

==

−==∆ ε

Example 2Now assume the same gage is subjected to 5000 microstrain or about 50,000 psi in

aluminum:

changeOhms

eRGFR

%12.1

)120)(65000(2

==

−==∆ ε


Resistance Measuring Circuits

ConstantCurrentSource e R

i

Current Injection

e Rg

Rb

ballast

E

Ballast Circuit

bg

g

RRR

e+

=

( )

( )

( ) εGFRR

ERR

RdR

RRERR

ERR

dRRRR

dRde

gb

gb

g

g

gb

gb

bg

gg

bg

g

2

2

2)

+=

+=

�

��

�

+−

+=

Impractical resolution problems

Output:

Small changes:

EGFde ε4

=

Optimal output (Rb=Rg):

e+de=E/2 + GF/4 ε E

Output includes E/2 plusan incremental, de, which

is VERY small!

Output includes E/2 plusan incremental, de, which

is VERY small!


Wheatstone Bridge Circuit

DC

e

R1 R2

R3R4

e

DC

R2R1

R4 R3

Wheatstone Bridge Back-to-back Ballast Circuits

ERRRR

RRRRERR

RRR

Re))(( 4321

3142

43

3

21

2

++−=

��

�

+−

+=

R2 R4 = R1 R3

Output:

Balance Condition:


Linearized Bridge Equations

ER

dRR

dRRR

RRR

dRR

dRRR

RRde�

��

��

��

−

++��

�

��

−

+=

4

4

3

32

43

43

2

2

1

12

21

21

)()(

[ ]EGFe 43214εεεε −+−=

• The equation identifies the first order (differential) effects only, and so this is the“linearized” form. It is valid only for small (infinitesimal) resistance changes. Largeresistance changes produce nonlinear effects and these are shown in Figure 3 wherefinite changes in R (∆R) in a single arm are considered for an initially balancedbridge.

• Output is directly proportional to the excitation voltage and to the Gage Factor.Increasing either will improve measurement sensitivity.

• Equal strain in gages in adjacent arms in the circuit produce no output. Equal strain inall gages produces no output either.

• Fixed resistors rather than strain gages may be used as bridge arms. In this case thestrain contribution is zero and the element is referred to as a “dummy” element orgage.

Differential ouput:

Assuming initially balanced bridge:

Using definition of GF and e+de=de:

ER

dRR

dRR

dRR

dRde�

��

�−+−=

4

4

3

3

2

2

1

141

-0.4

-0.2

0

0.2

-1 -0.5 0 0.5 1∆R/R

e/E

Nonlinear effects

Linearized result


Temperature Effects

• Gage material can respond as much to temperature as to strain:

• For “isoelastic” material:0/ 260 74 /

3.5dR R microstrain FGF

ε = = =

Table 2. Properties of Various Strain Gage Grid MaterialsMaterial Composition Use GF Resistivity

(Ohm/mil-ft)Temp.

Coef. ofResistance

(ppm/F)

Temp.Coef. of

Expansion(ppm/F)

MaxOperatingTemp. (F)

Constantan 45% NI, 55%Cu

StrainGage 2.0 290 6 8 900

Isoelastic 36% Ni, 8%Cr, 0.5% Mo,55.5% Fe

Straingage(dynamic)

3.5 680 260 800

Manganin 84% Cu, 12%Mn, 4% Ni

Straingage(shock)

0.5 260 6

Nichrome 80% Ni, 20%Cu

Ther-mometer 2.0 640 220 5 2000

Iridium-Platinum

95% Pt, 5% Ir Ther-mometer 5.1 135 700 5 2000

Monel 67% Ni, 33%Cu 1.9 240 1100

Nickel -12 45 2400 8Karma 74% Ni, 20%

Cr, 3% Al, 3%Fe

StrainGage(hi temp)

2.4 800 10 1500


Temperature Effects - cont’d

• Strains can appear due to differential expansion also:– Consider a “constantan” gage on aluminum– Constantan=8 µstrain /F and aluminum=13 µstrain/F

• Combining with direct temperature effects in constantan:

• Gages are cold-worked to match thermal properties of selectedsubstrate materials:

013 8 5 /microstrain Fε = − =

Fnmicrostrai 0/853 =+=ε

Table 3. Materials for which Strain Gages can be Compensated (typical)PPM/°F Material 3 Molybdenum 6 Steel; titanium 9 Stainless steel, Copper13 Aluminum15 Magnesium40 Plastics


Gage Heating: Excitation Voltage Limits

• Gage output can be increased by increasing excitation– this also increases current flow through gage– creates direct resistance heating (P=E2/R)

Table 4. Strain Gage Power Densities for Specified Accuracies on DifferentSubstrates

Accuracy SubstrateConductivity

Substrate Thickness Power Density (r)(Watts/sq. in.)

High Good thick 2-5High Good thin 1-2High Poor thick O.5-1High Poor thin 0.05-0.2

Average Good thick 5-10Average Good thin 2-5Average Poor thick 1-2Average Poor thin 0.1-O.5

Table Notes:Good = aluminum, copperPoor = steels, plastics, fiberglassThick = thickness greater than gage element lengthThin = thickness less than gage element length

AE3145 Spring 2000 10

• We can compute excitation limits…• Power dissipated in a gage:

• Power density in the gage (power per unit area) from previous:

• Maximum excitation depends on:– maximum power density allowed– gage area– gage resistance

2

4gg

EPR

=

g g

g g g

P PA L W

ρ = =

max 2 g g gE L W Rρ=

Computations

AE3145 Spring 2000 11

Wheatstone Bridge Compensation

• A desirable characteristic of the Wheatstone Bridge is its abilityto compensate for certain kinds of problems.

• Consider temperature induced changes in strain: ει=ει+ειΤ=:

• If we are careful how these extra terms are combined in thebridge equation, it is possible to cancel them out.

• This is called bridge compensation.

[ ]1 2 3 4 1 2 3 44 4T T T TGF GFe E Eε ε ε ε ε ε ε ε�= − + − + − + −�

AE3145 Spring 2000 12

Application:

• Half-bridge compensation:

• If gages experience same temperature change, then theeffects are automatically cancelled in the bridge.

• RULE #1: equal changes in adjacent arms will cancel out.

• NOTE: Single gages cannot be compensated...

DC

e

1 2

34

( ) ( )EGFEGFe TT2121 44

εεεε −+−=

since gages #3 and #4 are dummy resistancesand therefor experience no strain ortemperature changes (if located together).

AE3145 Spring 2000 13

Leadwire Effects

• When gages are located long distances from the instrumentation,the effect of the added leadwire resistance can unbalance thebridge:

• For 100 ft of 26 AWG wire in two leadwires:R=40.8*100/1000*2 = 8.16 Ohms

compared to a 120 Ohm gage.• A 3-wire hookup can eliminate the unbalance problem:

since each leadwire resistance appears in adjacent arms (RULE #1)

DCeA

E

C

B D

A'

This looks like aduplicated wire but

it really puts aleadwire in AA’E

and another in ABC

This looks like aduplicated wire but

it really puts aleadwire in AA’E

and another in ABC

AE3145 Spring 2000 14

Leadwire Desensitization

• Added leadwire resistance in an arm will also desensitize the gage.Consider a single arm:

• We can correct by defining a “new” gage factor:

• Example:

* (1 )1GFGF GF β

β= ≈ −

+

*

64.92 300 38.91000

38.9 /120 0.3242.0 1.51

1 1.324

sR Ohms

GFGF

β

β

= × × =

= =

= = =+

DC

e

R

RR

Rg

RS

*1 14 41

GFe E GF Eε εβ

= =+

Two 300 ft lengths of28 AWG wire for a 120Ohm gage with initial

GF=2.0

Two 300 ft lengths of28 AWG wire for a 120Ohm gage with initial

GF=2.0

New GF*New GF*

AE3145 Spring 2000 15

Other Issues

• Bridge balancing: how do you get rid of any initial unbalance inthe bridge due to unequal initial gage resistances?

• Bridge switching: how can you use a single Wheatstone Bridgeinstrument in order to read outputs from several different straingages?– Interbridge switching– Intrabridge switching

AE3145 Spring 2000 16

Wheatstone Bridge Applications

• Gages can be arranged in a Wheatstone Bridge to cancel somestrain components and to amplify others:

Table 5. Bridge Configurations for Uniaxial MembersNo. K Configuration Notes

A-1 1

1

Must use dummy gage in an adjacent arm (2or 4) to achieve temperature compensation

A-2 2

1

3

Rejects bending strain but not temperaturecompensated; must add dummy gages inarms 2 & 4 to compensate for temperature.

A-3 (1+ν)

1

2

Temperature compensated but sensitive tobending strains

A-4 2(1+ν)

1

4

2

3

Best: compensates for temperature andrejects bending strain.

AE3145 Spring 2000 17

Wheatstone Bridge Applications - cont’d

• Bending applications:Table 6. Bridge Configurations for Flexural Members

No. K Configuration Notes

F-1 1

1Also responds equally to axial strains; mustuse dummy gage in an adjacent arm (2 or 4)to achieve temperature compensation

F-2 2

1

2

Half-bridge; rejects axial strain and istemperature compensated; dummy resistorsin arms 3 & 4 can be in strain indicator.

F-3 4

1

3

2

4

Best: Max sensitivity to bending; rejects axialstrains; temperature compensated.

F-4 2(1+ν)

1

4

2

3

Adequate, but not as good as F-3;compensates for temperature and rejectsaxial strain.

AE3145 Spring 2000 18


• Torsion applications:

Table 7. Bridge Configuration for Torsion MembersNo. K Configuration Notes

T-1 2

12

Half Bridge: Gages at ±45° to centerlinesense principal strains which are equal &opposite for pure torsion; bending or axialforce induces equal strains and is rejected;arms are temperature compensated.

T-2 4

12

43Best: full-bridge version of T-1; rejects axialand bending strain and is temperaturecompensated.

AE3145 Spring 2000 19


• Rings loaded diametrically are very popular as a means totransduce load by converting it into a proportional strain:

• How would one wire the gages to best advantage in aWheatstone Bridge?

1

2

4

3

bridge measure

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