1. Piezoresistive sensors
Perform a basic bridge analysis, specifically,
Explain the difference between longitudinal, transverse, center, and boundary stress/strain.
find output voltage as a function of input voltage and the various resistances, and
For a given piezoresistive sensor, use the above concepts and objectives to find
find the relationship between output voltage and changes in resistance.
the sensor’s bridge equation,
Find changes in resistance of a piezoresistive material undergoing deformation due to
values of ΔR/R for each resistor,
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

2. Electrical resistance changes with mechanical deformation (strain)
Piezoresistance
Circuit with measurable output voltage proportional to
the electrical resistance
Electrical resistance changes with mechanical deformation (strain)
constant DC voltage
input to bridge
piezoresitive sensor
mechanical input
resistance output
voltage output

3. Wheatstone bridge analysis
Te toca a ti
Find the relationship between the input and output voltages for the bridge shown. Assume im is zero.
Find the relationship between R1, R2, R3, and R4 for eo = 0.
We would like this output to be zero when there is no mechanical input to the transducer.

4. Bridge balancing Full bridge: Half bridge:
All four resistors identical piezoresistors with same value of R, o sea, R1 = R2 = R3 = R4
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?

5. Gage factors and the piezoresistive effect
What is the relation between deformation and resistance?
Gage factor:
Metals
Semiconductors
Changes in geometry dominate
Changes in resistivity dominate *
* Strain causes differences in atomic spacing, which in turn causes changes in band gaps and thus ρ.

6. 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.)

7. Gage factors for semiconductors
Recall that in semiconductors changes in resistivity dominate resistance changes upon deformation.
Stress formulation:
πL and πT and are the piezoresistance coefficients
L – longitudinal:
T – transverse:
in the direction of current
perpendicular to the direction of current
Strain formulation:
γL and γ T and are the elastoresistance coefficients

8. 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:
Most likely not isotropic

9. 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

10. Physical placement and orientation of piezoresistors2 1 3 4
= 0 !
≠ 0

11. Physical placement and orientation of piezoresistors
The sensitivity will decrease.

12. Device case study: Omega PX409 pressure transducer

13. Device case study: Omega PX409 pressure transducer

14. Some definitions

15. 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.

16. 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?

17. i i i i Location of piezoresistors
Where should we place the piezoresistors on the diaphragm?
σC –
σB –
towards the center
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

18. Designing for a given sensitivity
Next, let’s find the change in output voltage corresponding to resistors in these locations. σC σB
= γL εC + γT εB
= (120)(152×10-6) + (-54)(-17×10-6)
= 19.1×10-3
Typical values of stress, strain, and elastoresistance coefficients:
σC = 45.0 MPa, σB = 22.5 MPa γL = 120 <110>
εC = 152 × 10-6, εB = -17 × 10-6 γT = -54 <110>

19. 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. σC
= γL εB + γT εC
= (120)(-17×10-6) + (-54)(152×10-6)
= -10.2×10-3 σB
Typical values of stress, strain, and elastoresistance coefficients:
σC = 45.0 MPa, σB = 22.5 MPa γL = 120 <110>
εC = 152 × 10-6, εB = -17 × 10-6 γT = -54 <110>

20. 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
= (147 – 0) mV/(1 – 0) MPa
= 147 mV/MPa
With ei = 10 V
Need to attenuate the signal.

21. Attenuation of output The gain is K = 100 mV/147 mV = 0.682
inverting amplifier
KIA = -Rf/R0
The gain is
K = 100 mV/147 mV =
K = (KIA) (KIA)
= (-Rf/R0)(-Rf/R0)
= (Rf/R0)2

22. Readily-obtained resistors for use in an op-amp circuit (Ohms)
Te toca a ti
Select two resistors (es decir, Rf y R0) from the available resistors below to achieve K = (Rf/R0)2 =
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
6.2 16 43
110
300
750
6.8 18 47
120
330
820
7.5 20 51
130
360
910 R0 Rf
With Rf = 7.2 Ω and R0 = 6.2 Ω
K = 0.683

23. Attenuation of output
The amplifier goes into the signal conditioning compartment of the transducer

24. 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
the new bridge equation (i.e., eo in terms of ei and the various resistances),
ΔR/R for each resistor,
Δeo/ei, and
the transducer sensitivity (with no amplifier circuit)
Assume the same dimensions, sensor requirements, materials, and stress/strain values as in the case study. 1b
second resistor in series 1a 3a 3b