Piezoresistive sensors

Perform a basic bridge analysis, specifically, find output voltage as a function of input

voltage and the various resistances, and find the relationship between output

voltage and changes in resistance. Find changes in resistance of a

piezoresistive material undergoing deformation due to

changes in geometry and/or 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

Explain the difference between longitudinal, transverse, center, and boundary stress/strain.

For a given piezoresistive sensor, use the above concepts and objectives to find

the sensor’s bridge equation, values of ΔR/R for each resistor, value of Δeo/ei, and the transducer’s sensitivity

Piezoresistance

Electrical resistance changes with mechanical deformation (strain)

mechanical input

resistance outputvoltage output

constant DC voltageinput to bridge

Circuit with measurable output voltage proportional tothe electrical resistance

piezoresitive sensor

Wheatstone bridge analysis

Find the relationship between the input and output voltages for the bridge shown. Assume im is zero.

io eRR

RRR

Re

43

4

21

1

Te toca a ti

We would like this output to be zero when there is no

mechanical input to the transducer.

Find the relationship between R1, R2, R3, and R4 for eo = 0.

4231 RRRR

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

4231 RRRR

What is the relation between a change in resistance ΔR and the output voltage eo?

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RRR

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s.order termhigher 44

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RReR

ReR

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Gage factors and the piezoresistive effect

What is the relation between deformation and resistance?

Gage factor:L

RRF

ALR

AA

LL

RR

Metals

Changes in geometry dominate

Semiconductors

Changes in resistivity dominate

AA

LL

RR

RR

sorder termhigher

AARL

LRRR

*

* 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.

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.)

Te toca a ti

LL + L

original area A,height h and width w

appliedstress

new area,height h h andwidth w w

21F

Gage factors for semiconductors

Recall that in semiconductors changes in resistivity dominate resistance changes upon deformation.

AA

LL

RR

RR

TTLL

TTLL

πL and πT and are the piezoresistance coefficients

• L – longitudinal: • T – transverse:

in the direction of currentperpendicular to the direction of current

γL and γ T and are the elastoresistance coefficients

Stress formulation:

Strain formulation:

Gage factors for semiconductors

RR

TTLL

TTLLRR

L

TTLF

Gage factor: L

RRF

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?)

Te toca a ti

TLF 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

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L

RRF

43214

F

1

4

2

3

= 0 !≠ 0

Physical placement and orientation of piezoresistors

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o 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? towards the center

along the boundary

Typical values at max deflection:

σC = 45.0 MPa, σB = 22.5 MPaεC = 152 × 10-6, εB = -17 × 10-6

43214

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σC –σB –

i

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Designing for a given sensitivity

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>

RR

RR

RR

RR

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41 43214

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o

TTLLRR

3,1σC

σB = γL εC + γT εB

= (120)(152×10-6) + (-54)(-17×10-6)

= 19.1×10-3

Next, let’s find the change in output voltage corresponding to resistors in these locations.

Designing for a given sensitivity

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>

σC

σB

TTLLRR

4,2

= γL εB + γT εC

= (120)(-17×10-6) + (-54)(152×10-6)

= -10.2×10-3

Te toca a ti

Find the values of ΔR2/R and ΔR4/R using the same assumed values of stress, strain, and elastoresistance.

Designing for a given sensitivity

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3102.101.192.101.1941

Requirements:

• 0–1 MPa (145 psi) span• 0–100 mV full scale

output• 10 VDC excitation• η = 100 mV/MPa

mV 147 oeWith ei = 10 V

η = FSO/Span

= (147 – 0) mV/(1 – 0) MPa

= 147 mV/MPa

mV/V. 7.14 Need to attenuate the signal.

Attenuation of output

The gain is

K = 100 mV/147 mV = 0.682

Ro

voltage signalfrom bridge transducer

output

Rf

Ro

Rf

K = (KIA) (KIA)

= (-Rf/R0)(-Rf/R0)

= (Rf/R0)2

inverting amplifier

KIA = -Rf/R0

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.

1 8.2 22 56 150 3901.5 9.1 24 62 160 4302.7 10 27 68 180 4704.3 11 30 75 200 5104.7 12 33 82 220 5605.1 13 36 91 240 6205.6 15 39 100 270 6806.2 16 43 110 300 7506.8 18 47 120 330 8207.5 20 51 130 360 910

Readily-obtained resistors for use in an op-amp circuit (Ohms)

R0

Rf

With Rf = 7.2 Ω and R0 = 6.2 ΩK = 0.683

Attenuation of output

The amplifier goes into the signal conditioning compartment of the transducer

Ro

voltage signalfrom bridge transducer

output

Rf

Ro

Rf

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

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.

1a1b

3b3a

second resistor in series