Strain Gages

• Electrical resistance in material changes when the material is deformed

RA

R – Resistance

ρ – Resistivityl – LengthA – Cross-sectional area

log log logR A

dR

R

d d A

A

Taking the differential

Change in resistance is from change in shape as

well as change in resistivity

For linear deformations

R

RSs

ε – strainSs – sensitivity or gage factor(2-6 for metals and 40 – 200 for

semiconductor)

• The change in resistance is measured using an electrical circuit

• Many variables can be measured – displacement, acceleration, pressure, temperature, liquid level, stress, force and torque

• Some variables (stress, force, torque) can be determined by measuring the strain directly

• Other variables can be measured by converting the measurand into stress using a front-end device

Outputvo

Direction ofSensitivity

(Acceleration)

Strain Gage

Housing

SeismicMass

m

Base MountingThreads

Strain Member Cantilever

Strain gage accelerometer

Direction ofSensitivity

FoilGrid

BackingFilm

Solder Tabs(For Leads)

Single ElementTwo-Element Rosette

Three-Element Rosettes

Nickle-PlatedCopper Ribbons

WeldedGold Leads

Doped SiliconCrystal

(P or N Type)

PhenolicGlass

BackingPlate

Strain gages are manufactured as metallic foil (copper-nickel alloy – constantan)

Semiconductor (silicon with impurity)

Potentiometer or Ballast Circuit

vo

Output vref

(Supply)

Strain Gage

+

-

Rc

R

• Ambient temperature changes will introduce error

• Variations in supply voltage will affect the output

• Electrical loading effect will be significant

• Change in voltage due to strain is a very small percentage of the output

refvRR

Rv

co

Question: Show that errors due to ambient temperature changes will cancel if the temperature coefficients of R and Rc are the same

Wheatstone Bridge Circuit

vref

(Constant Voltage)

-

+ R1

A

R2

R3

R4

B

RL

vo

- +

Load (High)

Small i

1 3 1 4 2 3

1 2 3 4 1 2 3 4

( )

( ) ( ) ( )( )ref ref

o ref

R v R v R R R Rv v

R R R R R R R R

R

R

R

R1

2

3

4

When the bridge is balanced

True for any RL

Null Balance Method

• When the stain gage in the bridge deforms, the balance is upset.

• Balance is restored by changing a variable resistor

• The amount of change corresponds to the change in stain

• Time consuming – servo balancing can be used

Direct Measurement of Output Voltage

• Measure the output voltage resulting from the imbalance

• Determine the calibration constant

• Bridge sensitivity

v

v

R R R R

R R

R R R R

R R

o

ref

2 1 1 2

1 22

4 3 3 4

3 42

To compensate for temperature changes, temperature coefficients of adjacent pairs should be the same

The Bridge Constant

• More than one resistor in the bridge can be active

• If all four resistors are active, best sensitivity can be obtained

• R1 and R4 in tension and R2 and R3 in compression gives the largest sensitivity

• The bridge sensitivity can be expressed as

4o

ref

v Rk

v R

k bridge output in the general case

bridge output if only one strain gage is activeBridge Constant

Example 4.4

A strain gage load cell (force sensor) consists of four identical strain gages, forming a Wheatstone bridge, that are mounted on a rod that has square cross-section.  One opposite pair of strain gages is mounted axially and the other pair is mounted in the transverse direction, as shown below.  To maximize the bridge sensitivity, the strain gages are connected to the bridge as shown.  Determine the bridge constant k in terms of Poisson’s ratio v of the rod material.

vref

+−

+

−

vo

1 2

3 4

1

Axial Gage

2 Transverse Gage

Cross SectionOf SensingMember

3

4

Transverse strain = (-v) x longitudinal strain

Calibration Constant

v

vCo

ref

Ck

Ss4 4

o

ref

v Rk

v R

R

RSs

k – Bridge Constant

Ss – Sensitivity or gage factor

Example 4.5A schematic diagram of a strain gage accelerometer is shown below.  A point mass of weight W is used as the acceleration sensing element, and a light cantilever with rectangular cross-section, mounted inside the accelerometer casing, converts the inertia force of the mass into a strain.  The maximum bending strain at the root of the cantilever is measured using four identical active semiconductor strain gages. Two of the strain gages (A and B) are mounted axially on the top surface of the cantilever, and the remaining two (C and D) are mounted on the bottom surface. In order to maximize the sensitivity of the accelerometer, indicate the manner in which the four strain gages A, B, C, and D should be connected to a Wheatstone bridge circuit.  What is the bridge constant of the resulting circuit?

vref

+−

+

−

δvo

A

B

C

D

W

Strain Gages A, B

C, D

l

b

h

AB

CD

Obtain an expression relating applied acceleration a (in units of g) to bridge output (bridge balanced at zero acceleration) in terms of the following parameters:

W = Mg = weight of the seismic mass at the free end of the cantilever elementE = Young’s modulus of the cantileverl = length of the cantileverb = cross-section width of the cantileverh = cross-section height of the cantileverSs = gage factor (sensitivity) of each strain gagevref = supply voltage to the bridge.

•If M = 5 gm, E = 5x1010 N/m2, l = 1 cm, b = 1 mm, h = 0.5 mm, Ss = 200, and vref = 20 V, determine the sensitivity of the accelerometer in mV/g.•If the yield strength of the cantilever element is 5xl07 N/m2, what is the maximum acceleration that could be measured using the accelerometer? •If the ADC which reads the strain signal into a process computer has the range 0 to 10 V, how much amplification (bridge amplifier gain) would be needed at the bridge output so that this maximum acceleration corresponds to the upper limit of the ADC (10 V)?•Is the cross-sensitivity (i.e., the sensitivity in the two directions orthogonal to the direction of sensitivity small with this arrangement? Explain.•Hint: For a cantilever subjected to force F at the free end, the maximum stress at the root is given by

6

2

F

bh

Mechanical Structure

Signal Conditioning

MEMS Accelerometer

Applications: Airbag Deployment

Data Acquisition

ACBridge

CalibrationConstant

Oscillator Power Supply

AmplifierDemodulator

And FilterDynamic

StrainStrain

Reading

• Supply frequency ~ 1kHz

• Output Voltage ~ few micro volts – 1 mV

• Advantages – Stability (less drift), low power consumption

• Foil gages - 50Ω – kΩ

• Power consumption decreases with resistance

• Resolutions on the order of 1 m/m

Semiconductor Strain Gages

Single Crystal ofSemiconductor

Gold Leads

ConductorRibbons

Phenolic GlassBacking Plate

• Gage factor – 40 – 200

• Resitivity is higher – reduced power consumption

• Resistance – 5kΩ

• Smaller and lighter

Material Composition Gage Factor(Sensitivity)

Temperature Coefficient of

Resistance (10-6/C)

Constantan 45% Ni, 55% Cu 2.0 15

Isoelastic 36% Ni, 52% Fe, 8% Cr, 4% (Mn, Si, Mo)

3.5 200

Karma 74% Ni, 20% Cr, 3% Fe, 3% Al

2.3 20

Monel 67% Ni, 33% Cu 1.9 2000

Silicon p-type 100 to 170 70 to 700

Silicon n-type -140 to –100 70 to 700

Properties of common strain gage material

Disadvantages of Semiconductor Strain Gages

• The strain-resistance relationship is nonlinear

• They are brittle and difficult to mount on curved surfaces.

• The maximum strain that can be measured is an order of magnitude smaller 0.003 m/m (typically, less than 0.01 m/m)

• They are more costly

• They have a much larger temperature sensitivity.

−3 −2 −1 1 2 3

−0.2

−0.1

0.1

0.2

0.3

0.4

−0.3

Strain

×103

ResistanceChange

= 1 Microstrain = Strain of 1×10-6

−3 −2 −1 1 2 3

−0.2

−0.1

0.1

0.2

0.3

0.4

−0.3

Strain×103

ResistanceChange

R

R

R

R

P-type

N-type

For semiconductor strain gages

R

RS S 1 2

2

• S1 – linear sensitivity

• Positive for p-type gages

• Negative for n-type gages

• Magnitude is larger for p-type

• S2 – nonlinearity

• Positive for both types

• Magnitude is smaller for p-type

Linear Approximation

Strain

Change inResistance

QuadraticCurve

max

−max

LinearApproximation

0

R

R

R

RS

Ls

Error eR

R

R

RS S S

Ls

1 22

21 2sS S S

J e d S S S ds 2

1 22 2

max

max

max

max

Quadratic Error

Minimize Error 0.s

J

S

max

max

2221)2(

dSSS s = 0

sSS 1

Maximum Error

e Smax max 22

Range – change in resistance

R

RS S S S

S

1 22

1 22

12

max max max max

max

Percentage nonlinearity error

22 max

1 max

max error100% 100%

range 2p

SN

S

2 max 150 %pN S S

Temperature Compensation

CompensationFeasible

CompensationNot Feasible

CompensationFeasible

(−β)

Concentration of Trace Material (Atoms/cc)

Tem

pera

ture

coe

ffic

ient

s (p

er °

F)

0

1

2

3α = Temperature Coefficient of Resistanceβ = Temperature Coefficient of Gage Factor

α

1 .oR R T

1 .s soS S T

Resistance change due to temperatureSensitivity

change due to temperature

R4

R1 R2

R3

δvo

+

−

CompensatingResistor

Rc

vref

+−

vi

RR−

vref

+

−

vi

RR

+

Rc

Self Compensation with a Resistor

vR

R Rvi

c

ref

v

v

R

R R

kSo

c

s

ref

4

1 .1 .

1 .oo

so soo c o c

R TRS S T

R R R T R

For self compensation the output after the temperature change must be the same

R R R R To c o c ( )

c oR R

Possible only for certain ranges