Introduction to Modeling - Rose

```Piezoresistive sensors
 Perform a basic bridge analysis, specifically,
 Explain the difference between longitudinal, transverse,
 find output voltage as a function of input voltage and
center, and boundary stress/strain.
the various resistances, and
 For a given piezoresistive sensor, use the above concepts
 find the relationship between output voltage and
and objectives to find
changes in resistance.
 the sensor’s bridge equation,
 Find changes in resistance of a piezoresistive material
 values of ΔR/R for each resistor,
undergoing deformation due to
 value of Δeo/ei, and
 changes in geometry and/or
 the transducer’s sensitivity
 changes in resistivity
 Calculate the gage factor for a piezoresitor using
piezoresistance coefficients and/or elastoresistance
coefficients where appropriate
 Use the idea of gage factor to explain how the placement
and locations of piezoresitors can affect a device’s
sensitivity
Piezoresistance
Circuit with measurable
output voltage
proportional to
the electrical resistance
Electrical resistance changes with mechanical
deformation (strain)
constant DC voltage
input to bridge
mechanical
input
piezoresitive
sensor
resistance
voltage
output
Wheatstone bridge analysis
Te toca a ti
Find the relationship between the input and
output voltages for the bridge shown. Assume
im is zero.

R1
R4
e o  

R3  R4
 R1  R 2

 ei


Find the relationship between
R1, R2, R3, and R4 for eo = 0.
R1 R 3  R 2 R 4
We would like this output to be zero
when there is no mechanical input
to the transducer.
Bridge balancing
Full bridge:
All four resistors identical piezoresistors with
same value of R, o sea, R1 = R2 = R3 = R4
Half bridge:
R1 and R2 identical piezoresistors with R3 and
R4 identical fixed resistors or vice versa
What is the relation between a change in resistance ΔR and the
output voltage eo?
 eo 
 eo
R1 R 3  R 2 R 4

R1
R4
e o  

R3  R4
 R1  R 2
ei

 ei



eo
 R1
 R1 
R2
 R1  R 2 
2
eo
R2
 R1 
 eo
ei

R2 
eo
R3
R1
 R1  R 2 
2
R3 
R2 
eo
R4
 R 4  higher order term s.
R4
R 3  R 4 
2
R3 
R3
1   R1
R2
R4 





4 R
R
R
R 
R3
R 3  R 4 
2
R4
Gage factors and the piezoresistive effect
What is the relation between deformation and
resistance?
R 
Gage factor:
F 
R R
R

 
R
L
L 
R
A
 A  higher order term s
L
R


R
R 


L
L

A
A
Metals
Semiconductors
Changes in geometry
dominate
Changes in resistivity
dominate
L
R
A
R

L
L

A
R
A
R

* Strain causes differences in atomic spacing, which in turn causes changes in band gaps and thus ρ.


*
Gage factors for strain gages
Piezoresistors made of metals are usually
used in strain gages responding to
uniaxial strain.
Te toca a ti
Find the gage factor for a strain gage subject to
uniaxial strain applied to an isotropic material.
(Hint: you can neglect products of length changes;
i.e., ΔhΔw ≈ 0.)
F  1  2
Gage factors for semiconductors
Recall that in semiconductors changes in
resistivity dominate resistance changes
upon deformation.
Stress formulation:


R
R



R
R

L
L


A
A
  L L   T  T
πL and πT and are the piezoresistance coefficients
• L – longitudinal: in the direction of current
• T – transverse: perpendicular to the direction of current


Strain formulation:


  L L   T  T
γL and γ T and are the elastoresistance coefficients
Gage factors for semiconductors
Te toca a ti
Find the gage factor for a typical semiconductor device. Use the elastoresistance coefficients in your
formulation. (Hints: Recall that in semiconductors changes in resistivity dominate resistance changes
upon deformation. How will you model the stress/strain?)
Gage factor:
F 
R R
R
L
R


R
R
F  L  T
F   L  



  L L   T  T
  L L   T  T
T
L
T
Most likely not isotropic
Gage factor as figure of merit
Gage factor is a figure of merit that
• helps us decide which material to use for a piezoresistor
• helps us decide where to put the piezoresistors
Material
Gage factor, F
Metals
2-5
Cermets (Ceramic-metal mixtures)
5-50
Silicon and germanium
70-135
Physical placement and orientation of piezoresistors
1
2
3
 eo
ei
4
R3
1   R1
R2
R4 
 




4 R
R
R
R 
F 
R R
L

F
4
 1   2   3   4 
=≠ 00 !
Physical placement and orientation of piezoresistors
 eo
ei

F
4
 1   2   3   4 
The sensitivity will decrease.
Device case study: Omega PX409 pressure transducer
Device case study: Omega PX409 pressure transducer
Some definitions
Schematics of the sensor
Piezoresistors detect stress and
strain of the wafer using a bridge
configuration. Electrical output
signal is proportional to the inlet
pressure.
Designing for a given sensitivity
We would like to design a pressure sensor with a 0–1 MPa (145 psi) span, 0–100 mV full scale
output, a 10 VDC excitation, and p-Si piezoresistors.
What is the sensitivity?
η = FSO/Span
= (100 – 0) mV/(1 – 0) MPa
= 100 mV/MPa
How do we design the senor
to achieve this sensitivity?
Location of piezoresistors
Where should we place the piezoresistors on the
diaphragm?
σC – towards the center
σB – along the boundary
Typical values at max deflection:
σC = 45.0 MPa, σB = 22.5 MPa
εC = 152 × 10-6, εB = -17 × 10-6
i
i
i
i
 eo
ei

F
4
 1   2   3   4 
Designing for a given sensitivity
Next, let’s find the change in output voltage corresponding to resistors in these locations.
 eo
ei
1F   R1  R 2  R 3  R 4 
  1  2   3   4  

44  R
R
R
R 
 R1 , 3
σC
R
σB
  L L   T  T
= γL εC + γT εB
= (120)(152×10-6) + (-54)(-17×10-6)
Typical values of stress, strain, and elastoresistance coefficients:
σC = 45.0 MPa, σB = 22.5 MPa
εC = 152 × 10-6, εB = -17 × 10-6
γL = 120 <110>
γT = -54 <110>
= 19.1×10-3
Designing for a given sensitivity
Te toca a ti
Find the values of ΔR2/R and
ΔR4/R using the same assumed
values of stress, strain, and
elastoresistance.
 R2,4
σC
R
  L L   T  T
= γL εB + γT εC
σB
= (120)(-17×10-6) + (-54)(152×10-6)
Typical values of stress, strain, and elastoresistance coefficients:
σC = 45.0 MPa, σB = 22.5 MPa
εC = 152 × 10-6, εB = -17 × 10-6
γL = 120 <110>
γT = -54 <110>
= -10.2×10-3
Designing for a given sensitivity
Requirements:
•
•
•
•
0–1 MPa (145 psi) span
0–100 mV full scale output
10 VDC excitation
η = 100 mV/MPa
η = FSO/Span
 eo

ei

1   R1  R 2  R 3  R 4 





4 R
R
R
R 
1
4
= (147 – 0) mV/(1 – 0) MPa
19 . 1    10 . 2   19 . 1    10 . 2   10  3
 14 . 7 mV/V.
With ei = 10 V
 e o  147 mV
= 147 mV/MPa
Need to attenuate the signal.
Attenuation of output
inverting amplifier
KIA = -Rf/R0
Rf
The gain is
K = 100 mV/147 mV
= 0.682
K = (KIA) (KIA)
= (-Rf/R0)(-Rf/R0)
= (Rf/R0)2
voltage signal
from bridge
Ro
+
Rf
Ro
+
transducer
output
Te toca a ti
Select two resistors (es decir, Rf y R0) from the available resistors below
to achieve K = (Rf/R0)2 = 0.682.
Readily-obtained resistors for use in an op-amp circuit (Ohms)
1
8.2
22
56
150
390
1.5
9.1
24
62
160
430
2.7
10
27
68
180
470
4.3
11
30
75
200
510
4.7
12
33
82
220
560
5.1
13
36
91
240
620
5.6
15
39
100
270
680
R0
6.2
16
43
110
300
750
6.8
18
47
120
330
820
7.5
20
51
130
360
910
Rf
With Rf = 7.2 Ω and R0 = 6.2 Ω
K = 0.683
Attenuation of output
The amplifier goes into the signal conditioning
compartment of the transducer
voltage signal
from bridge
Ro
Rf
+
Rf
Ro
+
transducer
output
Te toca a ti
A configuration of p-Si resistors on a
square diaphragm is suggested in
which two resistors in series are used
to sense the maximum stress, σC, as
shown in the figure. Find
second resistor
in series
1b
1a
3a
3b
a. the new bridge equation (i.e., eo
in terms of ei and the various
resistances),
b. ΔR/R for each resistor,
c. Δeo/ei, and
d. the transducer sensitivity (with
no amplifier circuit)
Assume the same dimensions,
sensor requirements, materials, and
stress/strain values as in the case
study.
```