BASIC MECHANICAL SENSORS

ANDSENSOR PRINCIPLES

Definitions

• Transducer: a device that converts one form of energy into another.

• Sensor: a device that converts a physical parameter to an electrical output.

• Actuator: a device that converts an electrical signal to a physical output.

Sensors :definition and principles

Sensors : taxonomies• Measurand

– physical sensor– chemical sensor– biological sensor(cf : biosensor)

• Invasiveness– invasive(contact) sensor– noninvasive(noncontact) sensor

• Usage type– multiple-use(continuous monitoring) sensor– disposable sensor

• Power requirement– passive sensor– active sensor

Potentiometers

Translational Single turn

Helical

𝑣0=𝑅 𝑖

𝑅 𝑣𝑠=𝑥 𝑖

𝑙 𝑣𝑠

The Wheatstone bridge

Eb A

B

C

D

R1 R2

R3 R4

Rg

Ig

Circuit Configuration

+ VAC -

+ VC -

Eb A

B

C

D

R1 R2

R3 R4 + VA -

E0 = VAC

VA = EbxR4/(R1+R4)

VC = EbxR3/(R2+R3)

E0 = VAC = VA – VC =

))(()(

3241

3142

32

3

41

4

RRRRRRRRE

RRR

RRRE bb

Null-mode of Operation

Eb= 10 V A

B

C

D

R1= 1000

R2= 600

R3 R4= Rx

0 + -

Ig

At balance:

R2R4 = R1R3 or R1/R4 = R2/R3 and the output voltage is zero

9

Example 1

Eb= 10 V A

B

C

D

R1= 1000

R2= 600

R3 R4= Rx

0 + -

Ig

Assume that the bridge shown is used to determine the resistance of an unknown resistance Rx. The variable resistance is the resistance box that allows selection of several resistors in series to obtain the total resistance and it is set until null position in the meter observed. Calculate the unknown resistance if the variable resistance setting indicates 625.4.

The bridge will be balanced if R1/R4 = R2/R3 . Hence, R4 = Rx = R1/(R2/R3) = 1000x625.4/600 = 1042.3 .

Deflection-mode of OperationAll resistors can very around their nominal values as R1 + R1, R2 + R2, R3 + R3 and R4 + R4. Sensitivity of the output voltage to either one of the resistances can be found using the sensitivity analysis as follows

241

4

232

241

31423232413

1

0

)(

)()())((())((

1

RRRE

RRRRRRRRRRRRRRRE

RES

b

bR

232

3

2

0

)(2 RRR

ERE

S bR

232

2

3

0

)(3 RRRE

RES bR

241

1

4

0

)(4 RRRE

RES bR

+ Eg -

ETh = E0

A

B

C

D

RTh

Rg R3 R4

R1 R2

RTh

Ig

ETh = E0 = VAC (open circuit)

RTh = R1//R4 + R2//R3

Ig = E0/(RTh + Rg)

Eg = E0Rg/(RTh + Rg) In case of open-circuit (Rg) Eg = E0

The equivalent circuit

Stress and strain

L

T

A

Tension: A bar of metal is subjected to a force (T) that will elongate its dimension along the long axis that is called the axial direction. Compression: the force acts in opposite direction and shortens the lengthA metal bar

Stress: the force per unit area a = T/A (N/m2)

Bar with tension

L

T

L+dL

dL

StrainStrain: The fractional change in lengtha = dL/L (m/m)

D

L T

Hooke’s lawStress is linearly related to strain for elastic materials

a = a /Ey = (T/A)/Ey

Ey : modulus of elasticity ( Young’s modulus)

Elastic Region

Plastic Region

Strain (a)

Stress (a)

Elastic Limit

Breaking point

The stress-strain relationship

Transverse strainThe tension that produces a strain in the axial direction causes another strain along the transverse axis (perpendicular to the axial axis) as

t = dD/D

This is related to the axial strain through a coefficient known as the Poisson’s ratio as

dD/D = - dL/L

The negative sign indicates that the action is in reverse direction, that is, as the length increases, the diameter decreases and vice versa. For most metals is around 0.3 in the elastic region and 0.5 in the plastic region

Electrical Resistance of Gage Wire

D

L T

R=L/A

dAARdL

LRdRdR

dAALdL

Ad

ALdR 2

AdA

LdLd

RdR

A = r2 = (/4)D2 and dA/A = 2 dD/D yields dD/D = - dL/L

)21(

L

dLdR

dR

Piezoresistive effect Dimensional effect

Principles of strain measurement dR/R

dL/L

metals semiconductors

Gage factor - K

K = (dR/R)/(dL/L) = (dR/R)/a

For wire type strain gages the dimensional effect will be dominant yielding K 2

For heavily doped semiconductor type gages the piezoreziztive effect is dominant yielding K that ranges between 50 and 200

dR can be replaced by the incremental change R in this linear region yielding R/R = Ka

Bonded Strain-Gages

Backing Resistive Wires

Direction of strain

Con

nect

ing w

ires

T

Strain Gage

Beam

Solid (fixed) platform

A bonded gage Fixing the gage

Examples of bonded gages

Resistance-wire type Foil type Helical-wire type

K 2.0

R0 = 120 or 350 . 600 and 700 gages are also available

Semiconductor strain-gage units

Unbonded, uniformly doped

Diffused p-type gage

20

Fixing the gage

T

Strain Gage

Beam

Solid (fixed) platform

21

22

Strain gage on a specimen

23

The unbonded gage

Poles

Prestrained resistive wire

Unbonded strain-gage pressure sensor

25

Example 2A strain gage has a gage factor 2 and exposed to an axial strain of 300 m/m. The unstrained resistance is 350 . Find the percentage and absolute changes in the resistance.

a = 300 m/m = 0.3x10-3; R/R = Ka = 0.6x10-3 yielding %age change = 0.06% and R = 350x0.6x10-3 = 0.21 .

26

Example 3A strain gage has an unstrained resistance of 1000 and gage factor of 80. The change in the resistance is 1 when it is exposed to a strain. Find the percentage change in the resistance, the percentage change in the length and the external strain (m/m).

R/R (%) = 0.1 %; L/L (%) = [R/R (%)]/K = 1.25x10-3%, and a = [L/L (%)]/100 = 1.25x10-5 = 12.5 m/m

Wheatstone bridge for the pressure sensor

Integrated pressure sensor

Integrated cantilever-beam force sensor

Elastic strain-gageMercury-in-rubber strain-gage plethysmography (volume-measuring) using a four-lead gage applied to human calf.

Venous-occlusion plethysmography

Arterial-pulse plethysmography

Effect of Temperature and Strain in other Directions

)](1[ 00 TTRR R0 is the resistance at T0 and is the temperature coefficient

This is very much pronounced in case of semiconductor gages due to high temperature coefficient.

Effects of wanted strain (sw), unwanted strain (su) and temperature (T) add up in the change in resistance as

R = Rsw + Rsu + RT

The effect of unwanted strain and temperature must be eliminated before the resistance change is used to indicate the strain

Bridge Configurations For Strain Gage Measurements

Solid platform

Cantilever

Strain gage

Q

W

Eb A

B

C

D

R1 R2

R3 R4 = Rx

Rg

Ig

The cantilever beam with a single strain-gage element

A quarter bridge

Analysis of quarter-bridge circuit

)2(2))(()(

))((

2

3241

31420 RR

RE

RRRRRRRRR

ERRRR

RRRREE bbb

Let R1 = R2 = R3 = R and R4 = Rx = R + R = R(1 + R/R), and let x = R/R. The open circuit voltage E0 = 0 at balance (R = 0). At slight unbalance (R 0)

Let x = R/R )

21(4)2(20 x

xE

xx

EE bb

...42

1)2

1(2

1 xxx...)

42(

4

32

0 xxxEE b

Since x<<1, higher order terms can be neglected yielding R

REx

EE bb

440

Sensitivity analysis can also be used

241

1

4

0

)(4 RRRE

RES bR

RRER

RRRESRE b

bR

4)( 240 4

Sensitivity analysis

Effect of Temperature and Tensile Strain

• R = RQ + RW + RT • The effect of unwanted strain and temperature must be

eliminated. • The circuit as it is provides no compensation.• Using a second strain gage of the same type for R1 can

compensate effect of temperature. • This second gage can be placed at a silent location

within the sensor housing, hence kept at the same temperature as the first one.

• As a result, both R1 and R4 have the same amount of changes due to temperature that cancel each other in the equation yielding perfect temperature compensation

36

Wheatstone Bridge with Strain Gages and Temperature Compensation

Bridge with Two Active Elements

Cantilever

Strain gages

Q

W

The cantilever beam with two opposing strain gages

Eb A

B

C

D

R1 R-R R2

R3 R4

R+R

Rg

Ig

Circuit for the half-bridge

Circuit analysis

Let R2 = R3 = R; R1 = R - R; R4 = R + R, the open circuit voltage E0 = 0 at balance (R = 0). At slight unbalance (R 0)

))(( 3241

31420 RRRR

RRRREE b

))(()()(RRRRRR

RRRRRREb

RRE

RRE b

b

24

2

Eb A

B

C

D

R1 R-R R2

R3 R4

R+R

Rg

Ig

Insensitivity of half-bridge Wanted

strain Unwanted

strain Temperature

R4

R1

Effects of wanted and unwanted strains and temperature on measuring gages

Bridge with Four Active Elements (Full Bridge)

The force, when applied in the direction shown, causes tension on gages at the top surface (R + RQ) and compression on gages at the bottom surface (R - RQ).

The tensile force W causes (R + RW) on all gages.

The temperature also produces (R + RT) on all gages.

Q

W R1

R2 R4

R3

The cantilever beam with four strain gages (full bridge)

Eb A

B

C

D

R1 R-R

R4 R+R

Rg

Ig

R3 R-R

R2 R+R

• The strain gages that are working together are placed into opposite (non-neighboring) arms of the bridge.

• The strain gage resistors are manufactured for a perfect match to have the open circuit voltage E0 = 0 at balance (R = 0).

• At slight unbalance (R 0) with R1 = R3 = R - R; R2 = R4 = R + R

))(( 3241

31420 RRRR

RRRREE b

RRE

RRRRRRRRRRRRRRRRE bb

))(())(())((

L = n2G, where

n= number of turns of coil

G = geometric form factor

= effective permeability

Self-inductance

Inductive sensors

Mutual inductance

Differential transformer

LVDT transducer

(a)electric diagram and

(b)cross-section view

LVDT

Capacitive sensors +QQ

x Area = A

xAC r0

dv/dt

i

1

C +

Cv

i

(a ) (b )

Capacitive displacement transducer

(a)single capacitance and (b)differentialcapacitance

20 xA

xCKysensitivit r

xdx

CdC

orxC

dxdC

0

0

00

1

)(

)()(

xARRCwhere

j

jxE

jXjV

r

I

Capacitive sensor for measuring dynamic displacement changes

Piezoelectric sensors

kfq k is piezoelectric constant C/N

Akfx

Ckfv

r 0

Kxq

K is proportionality constant C/m

RCs iidtdxK

dtdqi

Rv

xtdxK

dtdvCii

dtiC

vv

Rs

CC

00

0

)(

)1(

1)()(0

jjK

jXjV S

KS=K/C, V/m; = RC, s

Response to step displacement

High-frequency response

High-frequency circuit model for piezoelectric sensor. RS is the sensor leakage resistance and CS the capacitance. Lm, Cm and Rm represent the mechanical system.

Piezoelectric sensor frequency response.

51

Quantum Tunneling Composites

(a) Structure (b) Effect of pressure

Structure and effect of pressure for QTC

52

Effect of Pressure on a QTC Pill

53

QTC as a Pressure Sensor