Sensors and Actuators
Introduction to sensors
Sander Stuijk
(s.stuijk@tue.nl)
Department of Electrical Engineering
Electronic Systems
2
BRIDGES AND
RESISTIVE STRAIN SENSORS
(Chapter 5.9-5.12, 9.1)
3
Applications of resistive strain sensors
4
Strain, force, pressure sensors
quantity
excitation
physical effect
sensor
strain
active
resistive effect
strain gauge
acceleration
active
capacitance
capacitive accelerometer
acceleration
active
inductance
inductive accelerometer
acceleration
active
resistive effect
piezoresistive accelerometer
pressure
active
capacitance
capacitive pressure sensor
pressure
active
resistive effect
piezoresistive sensor
resistive effect in strain gauge plays an important role in many
sensors for different quantities
5
Strain gauge
 resistance of a wire
R
l
m l
 2
a ne  a
 changing temperature affects resistance (thermoresistive effect)
 changing dimensions affects resistance (piezoresistive effect)
 strain gauges use piezoresistive effect to sense mechanical stress
 sensor based on strain gauges convert mechanical energy to
electrical energy
 thermoresistive effect is an error source
6
Force
 force leads to deformation of object
Θ
 measuring deformation provides opportunity to sense mechanical
force, which in turn is related to
 torque
 pressure
 acceleration
 mass
 ...
7
Stress and strain
 force leads to deformation of object
Θ
 deformation depends on force per area which is called stress (σ)

F
a
 deformation also depends on
 material properties
 length or volume of object
 deformation per unit length (or volume) is called strain (ε)

dl
l
8
Stress and strain
 object deform under force and
restores to original state when
force is removed (elasticity)
 materials resist deformation (rigidity)
 change in length due to force F given by Hooke’s law

F
dl
 E  E
a
l
stress,
σ = F/a
 E – Young’s modulus, which depends on
 material
 temperature
 ε – strain (unit deformation, dimensionless)
 strain and stress are proportional in elastic zone
rupture
elastic limit
elastic
zone
strain, ε
9
t - Δt t
Strain gauge
F
F
 resistance of a wire
R
l
l
a
l + Δl
 stretching wire longitudinally changes resistance
dR d dl d 1 a  d dl da






R

l
a

l
a
 change in length due to force F given by Hooke’s law

F
dl
 E  E
a
l
stress,
σ = F/a
 E – Young’s modulus, which depends on
 material
 temperature
 ε – strain (unit deformation, dimensionless)
 strain and stress are proportional in elastic zone
rupture
elastic limit
elastic
zone
strain, ε
10
t - Δt t
Strain gauge
F
 longitudinal stress changes
 length of wire (l)
 thickness of wire (t)
F
l
l + Δl
 Poisson ratio gives relation between change in length and thickness
 
dt t
dl l
 Poisson ratio of perfectly compressible material: 0.0 (e.g. cork)
 deformation in one direction does not change other direction
 Poisson ratio of incompressible material: 0.5 (e.g. rubber)
 volume of this material is constant when stress is applied
11
t - Δt t
Strain gauge
F
 longitudinal stress changes
 length of wire (l)
 thickness of wire (t)
F
l
l + Δl
 Poisson ratio gives relation between change in length and thickness
 
dt t
dl l
 Poisson ratio of metals: 0 < ν < 0.5
 volume of metal changes when deformed
 consider a circular wire with diameter t (and thus radius: t/2)
 change in volume per unit volume is then equal to
2
dV dl 2dt
t
V     l 
 
V
l
t
2
dt t
dt
dl
 
  
dl l
t
l

dl dl
 dV dl
  2
 1  2 

V
l
l
l


12
t - Δt t
Strain gauge
F
 longitudinal stress changes
 length of wire (l)
 thickness of wire (t)
F
l
l + Δl
 Poisson ratio gives relation between change in length and thickness
 
dt t
dl l
 Poisson ratio of metals: 0 < ν < 0.5
 volume of metal changes when deformed
 cross-sectional area changes when metals are deformed
 using same approach as used for volume we can show
2
t
da 2dt
a     


2
dl
a
t
 
 da
 2

dt t
a
l
dt
dl 
 
  

dl l
t
l
13
t - Δt t
Strain gauge
F
 longitudinal stress changes
 length of wire (l)
 thickness of wire (t)
F
l
l + Δl
 Poisson ratio gives relation between change in length and thickness
 
dt t
dl l
 Poisson ratio of metals: 0 < ν < 0.5
 volume of metal changes when deformed
 because of volume change
 amplitude of vibrations in metal lattice changes
 results in change of specific resistivity (for metals)
d

C
dV
V
 C – Bridgman’s constant
14
t - Δt t
Strain gauge
F
 longitudinal stress changes
 length of wire (l)
 thickness of wire (t)
F
l
l + Δl
 stretching wire longitudinally changes resistance
dR d dl d 1 a  d dl da






R

l
a

l
a
 using results found so far we find
dR d dl da




R

l
a  dR
dV dl da

C



 dR dl
R
V
l
a
d
dV 
C 1  2   1  2 


C

R
l
dV dl
da
dl 

V
 1  2 ,
 2 
V
l
a
l
dR
dl

G
R
l
 G – gauge factor
15
t - Δt t
Strain gauge
F
F
 change in resistance related
to change in length
dR
dl
G
R
l
l
l + Δl
 remember Hooke’s law (relates stress σ to strain ε)

F
dl
 E  E
a
l
 change in resistance related to force (per unit area) and strain
dR
F
 G  G
R
aE
 strain gauge can be used to sensor force and its derived quantities
 gauge factor is constant for metals, hence
 dR 
  R0 1  G   R0 1  x 
R  R0  dR  R0 1 
R
0 

 typically x < 0.002
16
Strain gauge
example – strain gauge attached to aluminum strut
 strain gauge with R = 350 Ω and G = 2.1
 aluminum strut has E = 73 GPa
 outer diameter of the strut: D = 50 mm
 inner diameter of the strut: d = 47.5 mm
what is the change in resistance when the strut supports a load of
1000 kg?
 area supporting the force
D d
a

 D 2  d 2   50mm2  47.5mm2
4

4
  191mm
2
 change in resistance
R  RG  RG
F
9800 N
 3502.1
aE
191106 m2  73 109 Pa   0.5
 change in resistance is less than 0.15% of the initial resistance
17
Metal strain gauge
 change in resistance due to stress depends on
dR d dl da d
dl




 1 2 
R

l
a

l
 change in conductivity
 deformation of the conductor
 metals
 change in conductivity related to deformation
d

C
dV
V
 change in conductivity small compared to dimensional change
d

 1 2 
dl
l
 metals have linear relation between ΔR and Δl
18
Silicon strain gauge
 change in resistance due to stress depends on
dR d dl da d
dl




 1 2 
R

l
a

l
 change in conductivity
 deformation of the conductor
 silicon
 change in conductivity dominates dimensional change
d

 1 2 
dl
l
 stress influences
 lattice spacing between atoms
 changes band gap energy
 influences number of carriers and average mobility of carriers
 silicon has a larger non-linearity compared to metal
19
Typical characteristics
Gauge factor
Base resistance (Ω)
Non-linearity
Sensitivity
Deformation
Breaking strain
Fatigue life
TCR (10-6/°C)
Metal
Semiconductor
2.0
-100 to 150
100 - 1000
>200
0.1%
1%
5mΩ/Ω
5mΩ/Ω
2-3μm/m
2-3μm/m
25000
5000
10 million reversals
10 million reversals
10 - 2000
90000
 semiconductor strain gauges have
 high gauge factor
 strong temperature dependency
 magnitude and sign of gauge factor depends on doping
20
Construction
 bonded strain gauges
 wires cemented onto a backing or
 thin film resistor deposited on a substrate
 resistor forms a long, meandering wire
force
D
d
 strain gauge connected to test object
force
21
Interface circuit
 strain gauge in resistive divider
 stress applied to gauge (resistive change ‘x’)
 what is the output voltage vo of the circuit?
 R0 1  x  
 R2 
1 x 
Vr  
Vr  
vo  
Vr
 k 1 x 
 R1  R2 
 kR0  R0 1  x  
non-linearity
 for strain gauges holds that x << k (typically x < 0.002)
1
x
 1 x 
vo  
Vr 
Vr
Vr 
k

1
k

1
k

1


 maximal sensitivity when k = 1
1
x
vo  Vr  Vr
2
2
offset voltage
R1=kR0
Vr
R2=R0(1+x)
vo
22
Interface circuit
 strain gauge in resistive divider
 stress applied to gauge (resistive change ‘x’)
 two problems when measuring output voltage
 non-linearity in response
 offset voltage present
vo when ignoring non-linearity
|vo/Vr|
vo measured (includes non-linearity)
offset voltage
R1=kR0
Vr
R2=R0(1+x)
x
vo
23
Interface circuit
 remove offset voltage by placing strain gauge in bridge
 stress applied to gauge (resistive change ‘x’)
 what is the output voltage vo? (assume k = R1/R4 = R2/R0 = 1)
 R4
R3 
Vr   R0  R0 1  x  V   1  1  x V    x V   x Vr
vo  

r
r
 2 R R 2  x   r
4
4

2
x
2
2

x
 R1  R4 R2  R3 




0
 0

 intermezzo: two sources of non-linearity
non-linearity
 strain gauge itself does not adhere to R = R0(1+x)
 interface circuit causes non-linear resistance – voltage relation
R1
R2
vo
Vr
R4
R3=R0(1+x)
24
Interface circuit
 remove offset voltage by placing strain gauge in bridge
 stress applied to gauge (resistive change ‘x’)
 what is the output voltage vo? (assume k = R1/R4 = R2/R0 = 1)
 R4
R3 
Vr   R0  R0 1  x  V   1  1  x V    x V   x Vr
vo  

r
r
 2 R R 2  x   r
4
4

2
x
2
2

x
 R1  R4 R2  R3 




0
 0

 compare output voltage bridge and divider
non-linearity
1
x
vo,divider  Vr  Vr
2
2
 bridge removes offset
 bridge reduces sensitivity
R1
R2
vo
Vr
R4
R3=R0(1+x)
25
Interface circuit
 increase sensitivity by adding extra strain gauge to bridge
 stress applied to gauge (resistive change ‘x’)
 what is the output voltage vo? (assume k = R0/R4 = R2/R0 = 1)
 R4
 R0
R3 
R 1  x  
x 
x
Vr  
Vr  
vo  

 0
Vr   Vr
2
2 x
 R1  R4 R2  R3 
 R0 2  x  R0 2  x  
non-linearity
 compare output voltage to single sensor solution
vo,sin gle 
x
x
Vr 
Vr
4  2x
4
 extra sensor increases sensitivity
R1=R0(1+x)
R2
vo
Vr
R4
R3=R0(1+x)
26
Interface circuit
 remove non-linearity by adding applying opposing signal to gauge
 stress applied to gauge (resistive change ‘x’)
 what is the output voltage vo? (assume k = R1/R4 = R2/R0 = 1)
 R4
 R

 R
R3 
R0 1  x 
R 1  x  
Vr   0 
Vr   0  0
Vr   x Vr
vo  

2 R0 
2
 R1  R4 R2  R3 
 2 R0 R0 1  x   R0 1  x  
 2 R0
 sensitivity equal to previous arrangement
 non-linearity removed
active gauge (tension)
R1
vo
Vr
active gauge
(compression)
R2=R0(1-x)
R4
R3=R0(1+x)
27
Interface circuit
 add more strain gauges to increase sensitivity
 stress applied to gauge (resistive change ‘x’)
 what is the output voltage vo?
 R4
R3 

R0 1  x 
R0 1  x 
Vr  
vo  

Vr


R

R
R

R








R
1

x

R
1

x
R
1

x

R
1

x
4
2
3 
 1
0
0
0
 0

 2x
1 x 1 x 


Vr   xV
Vr 
2 
2
 2
active gauge (tension)
R1=R0(1+x)
vo
Vr
active gauge
(compression)
R2=R0(1-x)
R4=R0(1-x)
R3=R0(1+x)