Transcript Analog Sensors for Motion Measurement

Strain Gages
• Electrical resistance in material changes when the material is deformed

R
A
R – Resistance
ρ – Resistivity
l – Length
A – Cross-sectional area
log R  log   log A
Taking the differential
dR d d  A


R

A
For linear deformations
R
 Ss 
R
Change in resistance is
from change in shape as
well as change in
resistivity
ε – strain
Ss – 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
Housing
Output
vo
Strain
Gage
m
Seismic
Mass
Strain Member
Cantilever
Base
Mounting
Threads
Strain gage accelerometer
Direction of
Sensitivity
(Acceleration)
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)
Phenolic
Glass
Backing
Plate
Welded
Gold Leads
Nickle-Plated
Copper Ribbons
Potentiometer or Ballast Circuit
+
vo
Output
Strain Gage
vref
R
-
(Supply)
Rc
R
vo 
vref
R  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
Wheatstone Bridge Circuit
A
Small i
+
R1
R2
RL
R4
R3
vo
Load
(High)
-
B
-
vref
+
(Constant Voltage)
vo 
R1vref
(R1  R2 )

R3vref
(R3  R4 )

(R1R4  R2 R3 )
vref
(R1  R2 )(R3  R4 )
When the bridge is balanced
R1 R3

R2 R 4
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
vo  R2 R1  R1R2   R4 R3  R3 R4 


2
2
v ref
R

R
 R1  R2 
 3 4
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
 vo
vref
Bridge Constant k 
k
R
4R
bridge output in the general case
bridge output if only one strain gage is active
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 crosssection. 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
2
1
1
2
3
Cross Section
Of Sensing
Member
+
vo
Transverse
Gage
4
−
3
4
−
Transverse strain = (-v) x longitudinal strain
vref
+
Calibration Constant
v o
 C
v ref
k
C  Ss
4
k – Bridge Constant
Ss – Sensitivity or gage factor
R
 Ss 
R
 vo
vref
k
R
4R
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
+
δvo
W
−
C, D
D
B
l
b
h
A
B
C
D
−
vref
+
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

6 F
bh 2
MEMS Accelerometer
Signal Conditioning
Mechanical Structure
Applications: Airbag Deployment
Data Acquisition
Dynamic
Strain
AC
Bridge
Amplifier
Oscillator
Power Supply
Demodulator
And Filter
Calibration
Constant
• 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
Strain
Reading
Semiconductor Strain Gages
Conductor
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
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
Silicon
n-type
-140 to –100
70 to 700
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.
Resistance
Change
R
R
Resistance
Change
P-type
R 0.4
0.4
 = 1 Microstrain
0.3
= Strain of
0.2
0.3
1×10-6
0.2
0.1
0.1
−3
−2
N-type
R
−1
1
−0.1
2
Strain
3 ×103 
−3
−2
−1
1
−0.1
−0.2
−0.2
−0.3
−0.3
2
3 ×103 
Strain
For semiconductor strain gages
R
 S1  S2  2
R
• 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
 R 
 R   Ss 
L
R
R
Change in
Resistance
Quadratic
Curve
Linear
Approximation
−max
0
Strain
max
Error e 

R  R 
    S1   S2  2  Ss 
R  R L
  S1  Ss    S2 2
Quadratic Error
J
J
 0.
Minimize Error
 Ss
 max




(2 ) S1  Ss   S2 2 d = 0
 max 2

e d 
max
 S1  Ss   S2  
 max
max
2
Maximum Error
max
S1  S s
emax  S2  2max
2 2
d
Range – change in resistance
R
 S1 max  S2  2max   S1 max  S2  2max
R
 2S1 max

 
Percentage nonlinearity error
2
S2 max
max error
Np 
100% 
100%
range
2S1 max
N p  50S2max S1 %

Temperature coefficients (per °F)
Temperature Compensation
α = Temperature Coefficient of Resistance
β = Temperature Coefficient of Gage Factor
3
α
2
Compensation
Feasible
(−β)
1
Compensation
Not Feasible
Compensation
Feasible
0
Concentration of Trace Material (Atoms/cc)
Sensitivity
change due to
temperature
R  Ro 1  .T 
Ss  Sso 1   .T 
Resistance
change due to
temperature
Self Compensation with a Resistor
R1
R
R2
R
+
δvo
−
R
R
+
−
−
vi
R4
R3
vi
+
Compensating
Resistor
Rc
vref
vi 
R
vref
 R  Rc 
v o
kSs
R


v ref  R  Rc  4
Possible only for certain ranges
−
vref
Rc
+
For self compensation the output after the
temperature change must be the same
Ro 1   .T 
Ro
Sso 
Sso 1   .T 
 Ro 1   .T   Rc 
 Ro  Rc 
Ro  Rc (  )   Ro  Rc T
  
Rc   
 Ro




