Resistance Strain Gage Circuits
• How do you measure strain in an electrical resistance strain
gage 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.
AE3145 Spring 2000
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Resistance in Metallic Conductor
Resistance Equation:
Change in R:
L
A
∆R = ρ
L
L + ∆L
− ( ρ + ∆ρ )
A
A + ∆A
∆L
L
Wire before and after strain is applied
4
R=ρ
10%
rhodium/
platinum
3
% ∆R/R
Differential change in R:
æ Lö
∆R ≅ dR = d ç ρ
è A
or:
dR dρ dL dA
=
+
−
R
ρ
L
A
After a bit of manipulation:
dR dρ
=
+ (1 + 2υ ) ε
R
ρ
Ferry alloy
2
Constantan alloy
1
DEFINE Gage Factor (GF):
GAGE FACTOR = GF =
dR / R
ε
40% gold/palladium
GF is the slope of
these curves
Nickel
0
2
1
3
% Strain
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Examples
Table 1. Gage Factors for Various Grid Materials
Material
Copper
Constantan*
Nickel
Platinum
Silver
40% gold/palladium
Semiconductor**
Gage Factor (GF)
Low Strain
High Strain
2.6
2.2
2.1
1.9
-12
2.7
6.1
2.4
2.9
2.4
0.9
1.9
~100
~600
Ultimate Elongation
(%)
0.5
1.0
-0.4
0.8
0.8
--
* similar to “Ferry” and “Advance” and “Copel” alloys.
** semiconductor gage factors depend highly on the level and kind of doping used.
Example 1
Assume 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
∆R = GF ε R
= 2(5e − 6)(120)
= 0.0012 Ohms
= 0.001% change!
Example 2
Now assume the same gage is subjected to 5000 microstrain or about 50,000 psi in
aluminum:
∆R = GF ε R
= 2(5000e − 6)(120)
= 1.2 Ohms
= 1% change
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Resistance Measuring Circuits
Ballast Circuit
Current Injection
ballast
Constant
Current
Source
i
R
e
Rb
E
Rg
e
Output:
Impractical resolution problems
e=
Rg
Rg + Rb
Small changes:
é dRg
Rg dRg ù
de = ê
E
−
2
êë Rg + Rb Rg + Rb )
Rb Rg E dRg
=
(Rb + Rg )2 Rg
(
=
Rb Rg E
(R
b
Output
Output includes
includes E/2
E/2 plus
plus
an
an incremental,
incremental, de,
de, which
which
is
VERY
small!
is VERY small!
+ Rg )
2
)
GF ε
Optimal output (Rb=Rg):
de =
GF
ε E
4
e+de=E/2 + GF/4 ε E
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Wheatstone Bridge Circuit
R1
R2
R1
e
R2
e
R4
R3
R4
DC
R3
DC
Wheatstone Bridge
Back-to-back Ballast Circuits
Output:
é R2
R3 ù
R2 R4 − R1 R3
e=ê
−
E=
E
( R1 + R2 )( R3 + R4 )
ë R1 + R2 R3 + R4
Balance Condition:
R2 R4 = R1 R3
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Linearized Bridge Equations
0.2
Differential ouput:
0
e/E
é R1 R2 æ dR1 dR2 ö
R3 R4 æ dR3 dR4 öù
ç
÷÷ +
ç
÷ E
−
−
de = ê
2 ç
R2 ø ( R3 + R4 ) 2 çè R3
R4 ÷ø
ë ( R1 + R2 ) è R1
-0.2
Assuming initially balanced bridge:
é dR dR dR dR ù
de = 14 ê 1 − 2 + 3 − 4 E
R2
R3
R4
ë R1
-0.4
Linearized result
-1
-0.5
0
0.5
1
∆R/ R
Nonlinear effects
Using definition of GF
and e+de=de:
e=
•
GF
[ε1 − ε 2 + ε 3 − ε 4 ]E
4
•
•
•
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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. Large
resistance changes produce nonlinear effects and these are shown in Figure 3 where
finite changes in R (∆R) in a single arm are considered for an initially balanced
bridge.
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 in
all gages produces no output either.
Fixed resistors rather than strain gages may be used as bridge arms. In this case the
strain contribution is zero and the element is referred to as a “dummy” element or
gage.
6
Temperature Effects
• Gage material can respond as much to temperature as to strain:
Table 2. Properties of Various Strain Gage Grid Materials
Material
Constantan
Isoelastic
Manganin
Nichrome
IridiumPlatinum
Monel
Nickel
Karma
Composition
45% NI, 55%
Cu
36% Ni, 8%
Cr, 0.5% Mo,
55.5% Fe
84% Cu, 12%
Mn, 4% Ni
80% Ni, 20%
Cu
95% Pt, 5% Ir
Use
Strain
Gage
Strain
gage
(dynamic)
Strain
gage
(shock)
Thermometer
Thermometer
67% Ni, 33%
Cu
74% Ni, 20%
Cr, 3% Al, 3%
Fe
Strain
Gage
(hi temp)
GF
Resistivity
(Ohm/mil-ft)
Temp.
Coef. of
Resistance
(ppm/F)
Temp.
Coef. of
Expansion
(ppm/F)
Max
Operating
Temp. (F)
2.0
290
6
8
900
3.5
680
260
0.5
260
6
2.0
640
220
5
2000
5.1
135
700
5
2000
1.9
240
1100
-12
45
2400
2.4
800
10
800
8
1500
• For “isoelastic” material:
dR / R 260
ε=
=
= 74 microstrain / 0 F
GF
3.5
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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
ε = 13 − 8 = 5 microstrain / 0 F
• Combining with direct temperature effects in constantan:
ε = 3 + 5 = 8 microstrain / 0 F
• Gages are cold-worked to match thermal properties of selected
substrate materials:
Table 3. Materials for which Strain Gages can be Compensated (typical)
PPM/°F
3
6
9
13
15
40
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Material
Molybdenum
Steel; titanium
Stainless steel, Copper
Aluminum
Magnesium
Plastics
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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 Different
Substrates
Accuracy
High
High
High
High
Average
Average
Average
Average
Substrate
Conductivity
Good
Good
Poor
Poor
Good
Good
Poor
Poor
Substrate Thickness
thick
thin
thick
thin
thick
thin
thick
thin
Power Density (r)
(Watts/sq. in.)
2-5
1-2
O.5-1
0.05-0.2
5-10
2-5
1-2
0.1-O.5
Table Notes:
Good = aluminum, copper
Poor = steels, plastics, fiberglass
Thick = thickness greater than gage element length
Thin = thickness less than gage element length
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Computations
• We can compute excitation limits…
• Power dissipated in a gage:
E2
Pg =
4 Rg
• Power density in the gage (power per unit area) from previous:
ρ=
Pg
Ag
=
Pg
LgWg
• Maximum excitation depends on:
– maximum power density allowed
– gage area
– gage resistance
Emax = 2 ρ Lg Wg Rg
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Wheatstone Bridge Compensation
• A desirable characteristic of the Wheatstone Bridge is its ability
to compensate for certain kinds of problems.
• Consider temperature induced changes in strain: ει=ει+ειΤ=:
GF
GF T
e=
[ε1 − ε 2 + ε 3 − ε 4 ] E + éëε1 − ε 2T + ε 3T − ε 4T E
4
4
• If we are careful how these extra terms are combined in the
bridge equation, it is possible to cancel them out.
• This is called bridge compensation.
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Application:
• Half-bridge compensation:
1
2
e=
(
)
GF
(ε 1 − ε 2 )E + GF ε 1T − ε 2 T E
4
4
e
4
3
since gages #3 and #4 are dummy resistances
and therefor experience no strain or
temperature changes (if located together).
DC
• If gages experience same temperature change, then the
effects are automatically cancelled in the bridge.
• RULE #1: equal changes in adjacent arms will cancel out.
• NOTE: Single gages cannot be compensated...
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Leadwire Effects
• When gages are located long distances from the instrumentation,
the effect of the added leadwire resistance can unbalance the
bridge:
• 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:
E
A'
A
This
This looks
looks like
like aa
duplicated
duplicated wire
wire but
but
itit really
really puts
puts aa
leadwire
leadwire in
in AA’E
AA’E
and
another
in
and another in ABC
ABC
DC
e
D
B
C
since each leadwire resistance appears in adjacent arms (RULE #1)
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Leadwire Desensitization
• Added leadwire resistance in an arm will also desensitize the gage.
Consider a single arm:
R
g
R
RS
e
e=
1
4
GF
εE =
1+ β
1
4
GF *ε E
R
R
DC
• We can correct by defining a “new” gage factor:
GF
GF * =
≈ GF (1 − β )
1+ β
• Example:
64.9
= 38.9 Ohms
1000
β = 38.9 /120 = 0.324
Rs = 2 × 300 ×
GF * =
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Two
Two 300
300 ftft lengths
lengths of
of
28
28 AWG
AWG wire
wire for
for aa 120
120
Ohm
Ohm gage
gage with
with initial
initial
GF=2.0
GF=2.0
2.0
GF
=
= 1.51
1 + β 1.324
New
New GF*
GF*
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Other Issues
• Bridge balancing: how do you get rid of any initial unbalance in
the bridge due to unequal initial gage resistances?
• Bridge switching: how can you use a single Wheatstone Bridge
instrument in order to read outputs from several different strain
gages?
– Interbridge switching
– Intrabridge switching
AE3145 Spring 2000
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Wheatstone Bridge Applications
• Gages can be arranged in a Wheatstone Bridge to cancel some
strain components and to amplify others:
Table 5. Bridge Configurations for Uniaxial Members
No.
K
Configuration
Notes
1
A-1
Must use dummy gage in an adjacent arm (2
or 4) to achieve temperature compensation
1
1
A-2
Rejects bending strain but not temperature
compensated; must add dummy gages in
arms 2 & 4 to compensate for temperature.
2
3
1
A-3
Temperature compensated but sensitive to
bending strains
(1+ν)
2
1
A-4
2
Best: compensates for temperature and
rejects bending strain.
2(1+ν)
3
4
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Wheatstone Bridge Applications - cont’d
• Bending applications:
Table 6. Bridge Configurations for Flexural Members
No.
K
Configuration
Notes
1
F-1
Also responds equally to axial strains; must
use dummy gage in an adjacent arm (2 or 4)
to achieve temperature compensation
1
1
F-2
Half-bridge; rejects axial strain and is
temperature compensated; dummy resistors
in arms 3 & 4 can be in strain indicator.
2
2
1
F-3
2
Best: Max sensitivity to bending; rejects axial
strains; temperature compensated.
4
3
4
1
F-4
2(1+ν)
4
2
Adequate, but not as good as F-3;
compensates for temperature and rejects
axial strain.
3
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Wheatstone Bridge Applications - cont’d
• Torsion applications:
Table 7. Bridge Configuration for Torsion Members
No.
T-1
K
Configuration
Notes
1
Half Bridge: Gages at ±45° to centerline
sense principal strains which are equal &
opposite for pure torsion; bending or axial
force induces equal strains and is rejected;
arms are temperature compensated.
2
2
3
T-2
4
Best: full-bridge version of T-1; rejects axial
and bending strain and is temperature
compensated.
4
2
1
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Wheatstone Bridge Applications - cont’d
• Rings loaded diametrically are very popular as a means to
transduce load by converting it into a proportional strain:
3
2
1
4
• How would one wire the gages to best advantage in a
Wheatstone Bridge?
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