1. Strain Gages
Electrical resistance in material changes when the material is deformed
R – Resistance
ρ – Resistivity
l – Length
A – Cross-sectional area
Taking the differential
Change in resistance is from change in shape as well as change in resistivity
For linear deformations
– strain
Ss – sensitivity or gage factor
(2-6 for metals and 40 – 200 for semiconductor)

2. Strain gage accelerometer
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
Output vo
Direction of
Sensitivity
(Acceleration)
Strain
Gage
Housing
Seismic
Mass m
Base
Mounting
Threads
Strain Member
Cantilever
Strain gage accelerometer

3. Semiconductor (silicon with impurity)
Strain gages are manufactured as metallic foil (copper-nickel alloy – constantan)
Direction of
Sensitivity
Foil
Grid
Single Element
Two-Element Rosette
Backing
Film
Solder Tabs
(For Leads)
Three-Element Rosettes
Semiconductor (silicon with impurity)
Doped Silicon
Crystal
(P or N Type)
Welded
Gold Leads
Nickle-Plated
Copper Ribbons
Phenolic
Glass
Backing
Plate

4. Potentiometer or Ballast Circuit+ v
Strain Gage o
Output v R
ref
(Supply) - Rc
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
Question: Show that errors due to ambient temperature changes will cancel if the temperature coefficients of R and Rc are the same

5. Wheatstone Bridge Circuit
Small i + R1 vo R2 RL
Load (High) R4 R3 - B - +
vref
(Constant Voltage)
When the bridge is balanced
True for any RL

6. Direct Measurement of Output Voltage
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
To compensate for temperature changes, temperature coefficients of adjacent pairs should be the same

7. 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
Bridge Constant

8. 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.
Axial Gage 1 2 1 + vo 3 2
Transverse Gage −
Cross Section
Of Sensing
Member 3 4 4 − +
vref
Transverse strain = (-v) x longitudinal strain

9. Calibration Constant k – Bridge Constant
Ss – Sensitivity or gage factor

10. Example 4.5
A 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? A C
Strain Gages A, B + W
δvo −
C, D D B l b A B + C − h
vref D

11. 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 element
E = Young’s modulus of the cantilever
l = length of the cantilever
b = cross-section width of the cantilever
h = cross-section height of the cantilever
Ss = gage factor (sensitivity) of each strain gage
vref = 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

12. MEMS Accelerometer Signal Conditioning Mechanical Structure
Applications: Airbag Deployment

13. Data Acquisition Supply frequency ~ 1kHz
Bridge
Calibration
Constant
Oscillator
Power Supply
Amplifier
Demodulator
And Filter
Dynamic
Strain
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

14. Semiconductor Strain Gages
Ribbons
Single Crystal of
Semiconductor
Gold Leads
Phenolic Glass
Backing Plate
Gage factor – 40 – 200
Resitivity is higher – reduced power consumption
Resistance – 5kΩ
Smaller and lighter

15. Temperature Coefficient of Resistance (10-6/C)
Properties of common strain gage material
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
n-type
-140 to –100

16. 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 m/m (typically, less than 0.01 m/m)
They are more costly
They have a much larger temperature sensitivity.
Resistance
Change
Resistance
Change
P-type
N-type
0.4
0.4
0.3
 = 1 Microstrain
= Strain of 1×10-6
0.3
0.2
0.2
0.1
0.1 −3 −2 −1 1 2 3
×103  −3 −2 −1 1 2 3
×103 
−0.1
−0.1
Strain
Strain
−0.2
−0.2
−0.3
−0.3

17. For semiconductor strain gages
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

18. Linear Approximation Error Quadratic Error Minimize Error = 0
Curve
Change in
Resistance
Linear
Approximation
−max
Error
max 
Strain
Quadratic Error
Minimize Error
= 0
Maximum Error

19. Range – change in resistance
Percentage nonlinearity error

20. Temperature Compensation
α = Temperature Coefficient of Resistance
β = Temperature Coefficient of Gage Factor 3 α
Compensation
Feasible
Temperature coefficients (per °F) 2
(−β)
Compensation
Not Feasible
Compensation
Feasible 1
Concentration of Trace Material (Atoms/cc)
Resistance change due to temperature
Sensitivity change due to temperature

21. Self Compensation with a Resistor+
δvo R R − + − vi Rc − R3 R4 +
Compensating
Resistor Rc − +
vref vi
For self compensation the output after the temperature change must be the same
vref
Possible only for certain ranges