Chapter 3
Sensing Elements
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Sensing Elements
Sensing Elements
8.1 Resistive sensing elements
8.2 Capacitive sensing elements
8.3 Inductive sensing elements
8.4 Electromagnetic sensing elements
8.5 Thermoelectric sensing elements
8.6 Elastic sensing elements
8.7 Piezoelectric sensing elements
8.8 Piezoresistive sensing elements
8.10 Hall effect sensors
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149
160
165
170
172
177
182
188
196
Overview of measurement system components
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Resistive sensing elements
Potentiometers(Potentiometric displacement Transdcuer)
Metal and Semiconductor resistive temperature sensors
Thick-film polymer resistive sensors
Metal and semiconductor resistive strain gauges
Semiconductor resistive gas sensors
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Potentiometers
Linear Potentiometer displacement sensors.
linear (rectilinear)
Therefore the open circuit voltage for a linear displacement potentiometer is:
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angular displacement potentiometer
angular (rotary)
the open circuit voltage for an angular displacement potentiometer is:
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Main parameters
Maximum travel dT, θT
Depends on range of displacement to be measured, e.g. 0 to 5 cm, 0 to 300°.
Supply voltage VS
Set by required output range, e.g. for a range of 0 to 5 V d.c., we need VS =5 V d.c.
Resistance RP
For a given load RL , choose RP to be sufficiently small compared with RL
so that maximum non-linearity is acceptable
Power rating Wmax
Wmax should be greater than actual power
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produced in RP.
Calculation of RTh for potentiometer.
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Voltage–displacement relationship for a loaded potentiometer
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Calculating non-linearity for POT transducer
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Calculation of RTh for potentiometer
x = d/dT is the fractional displacement
the corresponding resistance is RPx,
The Thévinin voltage ETh is the open circuit voltage across the output terminals
AB.
The ratio between ETh and supply voltage VS is equal to the ratio of
fractional resistance RP x to total resistance
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The Thévenin impedance ZTh is found by setting supply voltage VS = 0,
The load voltage is thus:
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Classification of potentiometers
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conductive plastic film potentiometers
Continuous track
zero resolution error
higher temperature coefficient of resistance
hybrid track
a conductive film on a precision wire wound track
the best features
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wire wound
- resistive track, of total length dT or θT
- n discrete turns of wire
- resolution error is
dT /n, for rectilinear
θT /n , for rotary
In General (100/n)%
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Resistive metal sensors for temperature measurement
- Idea
The resistance of most metals increases reasonably linearly with
temperature in the range −100 to +800 °C.
The general relationship between the resistance RT Ω of a metal element and
temperature T °C is a power series of the form
where
R0 Ω is the resistance at 0 °C
α, β, γ are temperature coefficients of resistance.
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the variation in the ratio RT /R0 with temperature for the metals
platinum, copper and nickel
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Features of Platinum
- chemically inert
- linear
- repeatable resistance/temperature characteristics
- a wide temperature range (−200 to +800 °C)
- can be used in many types of environments.
- a high degree of purity,
- a typical power 10 mW
R0 = 100.0 Ω
R100 = 138.50 Ω
fundamental interval
R200 = 175.83 Ω,
α = 3.91 × 10−3 °C−1
β = −5.85 × 10−7 °C−2
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Typical construction of platinum element probe
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 Resistive semiconductor sensors for temperature measurement
- They are called Thermistors
- It is made from oxides of the iron group of transition metal elements such as
chromium, manganese, iron, cobalt and nickel.
- NTC and PTC type
Typical thermistor resistance–temperature relationship
Where
Rθ is the resistance at temperature θ kelvin;
K and β are constants for the thermistor.
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Thermistor resistance–temperature characteristics
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 Thick-film polymers
-They can be used as resistive sensors for temperature and humidity
measurement.
-These are pastes consisting of a polymer matrix (usually
epoxy, silicone or phenolic resin) that binds together the filler particles.
 If the filler material is metallic, e.g. silver flakes or copper particles, then the
paste has similar resistive properties to a metal.
 If carbon particles are used as the filler material, then the paste has the
resistive properties of a semiconductor.
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Equations for measuring temperature with reference
•Thermoresistor (RTD)
for T between 25 and 200 °C is
Thermistor
Rθ1 Ω is the resistance at reference temperature θ1 K, usually θ1 =
25 °C = 298 K.
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Metal and semiconductor resistive strain gauges
Understanding Stress and Strain
What do we mean by Stress ?
F

A
Stress is defined by force/area
There are two types of Stress effect
- Tensile stress effect
- Compressive stress effect
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Effect of tensile stress
Tensile stress
F

A
which tends to increase the length of the body.
eL- longitudinal strain
eT- transverse strain
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Effect of compressive stress.
F
Compressive stress 
A
which tends to reduce the length of the body.
eL- longitudinal strain
eT- transverse strain
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What do we mean by Strain?
Strain in the body which is defined as
(change in length)
Strain 
(original unstressed length)
-l
Compressiv e strain 
l
 Δl
Tensil strain 
l
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Strain Gauge (SG)
A strain gauge is a metal or semiconductor element whose resistance
changes when under strain.
How does the change of the length affect on the change
of resistance?
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Basic Relationships
The relationship between strain and stress is
[8.8]
The relation between longitudinal strain eL and accompanying transverse
strain eT is:
[8.9]
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The relationship between strain and Resistance of element
The resistance of an element of length l, cross-sectional area A and resistivity ρ
[8.10]
If the element is strained, so that the change in resistance ΔR is given by:
[8.11]
i.e.
[8.12]
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Dividing throughout by R = ρl/A yields
The ratio Δl/l is the longitudinal strain eL in the element. Since cross-sectional
area A = wt
From [8.9] and [8.12]
[8.13]
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Gauge Factor G
It is a ratio which equals to the fractional change in resistance divide by (strain)
or
where R0 is the unstrained resistance of the gauge.
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Gauge factor of a strain gauge
(1/e) (Δρ/ρ) - Changes in resistivity piezoresistive effect) is small (around 0.4)
For most metals ν ≈ 0.3,
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Features of metal type strain gauge :
• Gauge factor 2.0 to 2.2
• Unstrained resistance 120 ± 1 Ω
• Linearity within ±0.3%
• Maximum tensile strain +2 × 10−2
• Maximum compressive strain −1 × 10−2
• Maximum operating temperature 150 °C
• A maximum gauge current between 15 mA and 100 mA, depending on area, is
specified in order to avoid self-heating effects.
•The change in resistance
-at maximum tensile strain is ΔR = +4.8 Ω,
-at maximum compressive strain ΔR = −2.4 Ω.
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Features of semiconductor type strain gauge :
• the piezoresistive term (1/e) (Δρ/ρ) can be large, giving large gauge factors.
•The most common material is silicon doped with small amounts of p-type or ntype material.
• Greater sensitivity to temperature changes
Values of Gauge factors
• for p-type silicon of between +100 and +175
• for n-type silicon between −100 and −140.
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Semiconductor resistive gas sensors
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Gas to be detected
Material
Idea
carbon monoxide (CO)
and hydrocarbons
chromium titanium
oxide
oxygen atoms near the surface
react with reducing gas
molecules; this reaction takes up
conduction electrons so that
fewer are available for
conduction. This causes a
decrease in electrical
conductivity and a corresponding
increase in resistance
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Gas to be detected
Material
Idea
Oxidising gases such as
oxides of nitrogen
(NOx ) and ozone
tungsten oxide
The atoms near the surface react
with oxidising gas molecules;
this reaction takes up
conduction electrons, again
causing a decrease in electrical
conductivity and an increase in
resistance with gas
concentration.
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Typical construction of a metal oxide sensor
- It consists of an alumina substrate with a film of oxide printed on one
side and a platinum heater grid on the other.
-A typical NOX sensor has an ambient temperature range of −20 °C to +60 °C
and operating power of 650 mW.
-The resistance is typically 6 kΩ in air, 39 kΩ in 1.5 ppm NO2 and 68 kΩ in 5.0
ppm NO2.
- A typical CO sensor has an ambient temperature range of −20 °C to +60 °C
and an operating power of 650 mW. The resistance is typically 53 kΩ in air, 85
kΩ in 100 ppm CO and 120 kΩ in 400 ppm CO
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Capacitive sensing elements
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Capacitance of parallel plate capacitor
The capacitance of this parallel plate capacitor is given by:
where
ε0 - the permittivity of free space (vacuum) of magnitude 8.85 pF m−1,
ε - the relative permittivity or dielectric constant of the insulating material,
A - m2 is the area of overlap of the plates
d - m is their separation
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Examples of Capacitive SE
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Types of capacitive displacement sensors
Variable separation displacement sensor
If the displacement x causes the plate separation to increase to d + x
the capacitance of the sensor is:
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Case study 1: Differential capacitive displacement sensor
It is also, called push-pull displacement sensor
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Case study 2: Pressure measurement
-one plate is a fixed metal disc, the other is a flexible flat circular
diaphragm, clamped around its circumference;
- the dielectric material is air (ε ≈ 1).
-The diaphragm is an elastic sensing element which is bent into a
curve by the applied pressure P.
The deflection y at any radius r is given by:
a = radius of diaphragm
t = thickness of diaphragm
E = Young’s modulus
ν = Poisson’s ratio.
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The deformation of the diaphragm means that the average separation of the
plates is reduced. The resulting increase in capacitance ΔC is given by
where:
d is the initial separation of the plates and
at zero pressure.
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the capacitance
Variable area displacement sensor
In the variable area type, the displacement x causes the overlap area to decrease
by ΔA = wx, where w is the width of the plates, giving:
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Variable dielectric displacement sensor
The total capacitance of the sensor is the sum of two capacitances, one with
area A1 and dielectric constant ε1, and one with area A2 and dielectric constant
ε 2,
Since A1 = wx, A2 = w(l − x), when w is the width of the plates,
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Case Study 3: Capacitive level sensor
-Construction
-Type of liquid
-Total Resistance
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 Thin-film capacitive humidity sensor
-Construction
-Dielectric change
-Plates are a layer of tantalum and a thin layer of chromium
The capacitance–humidity relation is therefore the linear equation:
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Capacitive pressure sensor
flat circular diaphragm
fixed metal disc
The deflection y at any radius r is given by:
where
a = radius of diaphragm
t = thickness of diaphragm
E = Young’s modulus
ν = Poisson’s ratio.
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The deformation of the diaphragm means that the average separation of the
plates is reduced. The resulting increase in capacitance ΔC is given by
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Application of Capacitive SE:
Ceramic liquid-filled differential pressure sensor
Why does silicon is used as dielectric liquids ? Page 163
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Inductive sensing elements
Variable inductance (variable reluctance) displacement sensors
Linear Variable Differential Transformer (LVDT) displacement sensor
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Review of magnetic circuit
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Simple magnetic circuit
In electrical circuit
In magnetic circuit
For this figure
the flux in the magnetic circuit is
the total flux N linked by the entire coil of n turns is
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The reluctance of a magnetic circuit
The reluctance of a magnetic circuit is given by:
where
l -the total length of the flux path
μ -the relative permeability of the circuit material
μ0- the permeability of free space = 4π × 10−7Hm−1 A cross-sectional area of the flux path
Self-inductance of a coil
By definition the self-inductance L of the coil is the total flux per unit current
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inductive displacement sensor (Variable reluctance elements)
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Calculation for typical variable reluctance displacement sensor
The elements of a typical variable reluctance displacement sensor are:
- a ferromagnetic core in the shape of a semitoroid (semicircular ring)
- a variable air gap
-a ferromagnetic plate or armature
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The Inductance of reluctance displacement sensor
First of all we have to calculate the total Reluctance Calculation
The total reluctance of the magnetic circuit is the sum of the individual reluctances
- The reluctance of the core
- The reluctance of air gap
- The reluctance of armature
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The total reluctance can be written as
The Inductance of reluctance displacement sensor is
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Differential reluctance displacement sensor
Note: The relationship between L1, L2 and displacement x is non-linear, but if
the sensor is incorporated into the a.c. deflection bridge of then the overall
relationship between bridge out of balance voltage and x is linear
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Linear Variable Differential Transformer (LVDT)
- The primary winding is energised by an a.c. voltage of amplitude
and frequency f Hz
-The two secondaries are connected in series opposition
- The output voltage is the difference (V1 − V2 ) of the voltages induced in the secondaries.
- A ferromagnetic core or plunger moves inside the former
-With the core removed the secondary voltages are ideally equal so that VOUT = 0.
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LVDT secondary waveforms
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A.C. characteristics of LVDT(Measuring ΔVp-p)
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DC characteristics of LVDT after Rectifier
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Characteristics of LVDT after phase sensitive detector and LPF
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Electromagnetic sensing elements
- The operation of these elements is based on Faraday’s law of electromagnetic
induction
-These elements are used for the measurement of linear and angular velocity
Faraday’s law of electromagnetic induction
if the flux N linked by a conductor is changing with time, then a back e.m.f. is
induced in the conductor with magnitude equal to the rate of change of flux, i.e.
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Variable reluctance tachogenerator
- measuring angular velocity
-It consists of
• a toothed wheel of ferromagnetic material
• a coil wound onto a permanent magnet, extended by a soft iron pole piece.
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The resulting cyclic variation in reluctance
with angular rotation θ
The total flux N linked by a coil of n turns is
We see that a reluctance minimum corresponds to a flux maximum and
vice versa.
This relation may be approximated by
where
a is the mean flux
b is the amplitude of the flux variation
m is the number of teeth.
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Output signal for variable reluctance tachogenerator
a back e.m.f. is induced in the conductor with magnitude equal to the rate of
change of flux
The induced e.m.f. is given by
Thus
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Amplitude and Frequency Output signal for variable reluctance tachogenerator
Amplitude of signal
Frequency of signal
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Thermoelectric sensing elements
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Principle
A
junction potential
B
If two different metals A and B are joined together, there is a difference
in electrical potential across the junction called the junction potential
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The realtionship
This junction potential depends on the metals A and B and the temperature T
°C of the junction, and is given by a power series of the form
Example:
The values of constants a1, a2, etc., depend on the metals A and B. For
example, the first four terms in the power series for the e.m.f. of an iron v.
constantan (Type J ) junction are as follows, expressed in μV:
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Practical circuit
A thermocouple is a closed circuit consisting of two junctions, at
different temperatures T1 and T2 °C.
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Thermocouple laws
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Law 1
Law 1 states that the e.m.f. of a given thermocouple depends only on
the temperatures of the junctions and is independent of the
temperatures of the wires connecting the junctions.
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Law 2
Law 2 states that if a third metal C is introduced into A (or B) then,
provided the two new junctions are at the same temperature (T3), the
e.m.f. is unchanged.
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Law 3
Law 3 states If a third metal C is inserted between A and B at either
junction, then d the two new junctions AC and CB are both at the
same temperature (T1 or T2), then the e.m.f. is unchanged
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Law 4 (law of intermediate metals)
Law 4 (law of intermediate metals) can be used, for example, to deduce the
e.m.f. of a copper–iron (AB) thermocouple, given the e.m.f. values for copper–
constantan (AC) and constantan–iron (CB) thermocouples.
The voltage produced by two metals A and B is the same as the sum of the
voltages produced by each metal (A and B) relative to a third metal C.
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law 5 (law of intermediate temperatures)
The fifth law (law of intermediate temperatures) is used in interpreting
e.m.f. measurement
Junction pairs at T1 and T3 produce the same voltage as two sets of junction
pairs spanning the same temperature range (T1 to T2 and T2 to T3
where T3 is the intermediate temperature. If T2 = 0 °C, then
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Thermocouple installations
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Case1
The meter located just outside the pipe
 It is completely useless.
The reference junction temperature T2 can vary widely from sub-zero
temperatures in cold weather to possibly +50 °C
if a steam leak occurs; the measured e.m.f. is therefore meaningless.
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Case 2
the meter located in the control room
the thermocouple is connected with copper leads
the reference junction is still located outside the pipe
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Case 3
 the thermocouple is extended to the control room using extension or
compensation leads made of chromel and alumel.
 the reference junction is now in the control room where the variation
in ambient temperature is smaller, possibly 10 °C at most.
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Case 2
 utilizing the law of intermediate .
 The thermocouple e.m.f. is ET1,T2 for a measured junction
 The e.m.f. source producing ET2,0 is known as an automatic reference
junction compensation circuit (ARJCC).
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Thermocouple data and characteristics
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Thermocouple packaging
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Piezoelectric sensing elements
Direct piezoelectric effect (ultrasonic receiver)
F
F
––––
Si +
O–
O–
Si
O
O
Si
Si
Si +
O
++++
++++
++++
Si
O–
Si +
––––
O
O
Si
Si
++++
O
––––
––––
If a force is applied to any crystal, then the crystal atoms are displaced
slightly from their normal positions in the lattice.
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Static and dynamic relations
Static relation
This displacement x is proportional to the applied force F
Dynamic relation
k -The stiffness of the crystal is large, typically 2 × 109 Nm−1.
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net charge calculation
In a piezoelectric crystal, this deformation of the crystal lattice results in the
crystal acquiring a net charge q, proportional to x
We know that
We obtain
where
d = K/k C N−1 is the charge sensitivity to force
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Inverse piezoelectric effect (ultrasonic transmitters)
if a voltage V applied to the crystal it causes a mechanical displacement x
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Properties of piezoelectric materials.
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Application of Piezoelectric SE: Displacement , Velocity, Acceleration
Consider a body, e.g. a part of a machine such as the casing of a pump or a
compressor, which is executing sinusoidal vibrations with displacement amplitude
at frequency f Hz, i.e.
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Photoresistor Sensor
A sensor that changes resistance as light is shined on it is made from Cadmium
Circuit applications
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Characteristics
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photodiode
A reversed-biased photodiode light sensor
• a reverse-biased photodiode has its reverse leakage increased by light shining on it.
• Photons, which are particles of light that are high-frequency electromagnetic waves,
are absorbed in the reverse-biased diode depletion layer.
•They produce free electrons and holes that increase the reverse current
•The more photons, the higher the intensity of light, the more energy is absorbed,
and the larger the reverse current.
• The photodiode is a light sensor with a variable current output.
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Phototransistors
Bipolar transistor operation
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phototransistor
A phototransistor, a transistor designed to be activated by light, has the same
basic operation as the NPN and PNP transistor described except it has no base
connection.
More light intensity produces more collector current
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LED Light Source
LED is not a sensor, it is a very important light source for light sensors.
• An LED is a forward-biased Semiconductor
•LEDs are made from special semiconductor materials other than silicon,
but still have the same type of junction characteristics.
• When a rated amount of current is passed through the forward-biased diode it
emits light
•The amount of current, I, through the diode can be adjusted by choosing the
value of R when a given voltage, V, is used.
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Charge carriers in a Hall-effect
When the charge carriers in a Hall-effect apparatus are negative, the
upper edge of the conductor becomes negatively charged, and c is at a
lower electric potential than a.
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When the charge carriers are positive, the upper edge becomes
positively charged, and c is at a higher potential than a.
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Hall voltage
If d is the width of the conductor, the Hall voltage is
Vd- drift velocity
VH - Hall voltage generated across the conductor
B - uniform magnetic field
EH - the magnitude of the electric field due to the charge separation
(sometimes referred to as the Hall field).
the drift speed is expressed as
n- carrier density
A - the cross-sectional area of the conductor
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we obtain
Because A=t d , where t is the thickness of the conductor, we can also
express Equation as
- the Hall coefficient
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Elastic sensing elements
If a force is applied to a spring, then the amount of extension or compression of
the spring is approximately proportional to the applied force
Principle of elastic sensing elements
Converting an input force into an output displacement
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Main Application of Elastic elements
- measuring torque
torque = force × distance
- measuring pressure
- measuring acceleration
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How to convert displacement (in elastic element ) into electrical signal ?
In a measurement system an elastic element will be followed by a suitable
secondary displacement sensor :
potentiometer
strain gauge
LVDT
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Dynamic model of elastic element for measurement linear accelerometer
x- is the displacement of the mass relative
to the casing
m -mass
a-acceleration
kx -spring force
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Dynamic model of elastic element for measurement pressure
A- area
P -input pressure
AP=f- produced force
- damping force
Kx -spring force
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Dynamic model of elastic element for measurement Angular acceleration
I- moment of inertia
Cθ -spring torque
-damping torque
-angular acceleration
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Dynamic model of elastic element for measurement linear acceleration
T -input torque
Cθ -spring torque
-damping torque
-angular acceleration
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Practical elastic sensing elements using strain gauges
a) Cantilever load cell
Strain Gauges 1 and 3 sense a tensile strain +e so that their resistance increases
by ΔR.
Strain Gauges 2 and 4 sense a compressive strain −e so that their
resistance decreases by an equal amount.
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(b) Pillar load cell
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(c) Torque sensor
- Gauge 1 is mounted with its active axis at +45° to the shaft axis,
- The tensile strain has a maximum value +e
- Gauge 2 at −45° to the shaft axis
- The compressive strain has a maximum value −e.
Gauges 3 and 4 are mounted at similar angles on the other side of
the shaft and experience strains +e and −e respectively
This maximum strain is given by
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Unbonded strain gauge accelerometer
-The space between the seismic mass and casing is filled with liquid to provide
damping.
-The unbonded strain gauges are stretched fine metal wires, which provide the
spring restoring force as well as acting as secondary displacement sensors.
-The gauges are prestressed, so that at zero acceleration each gauge
experiences a tensile strain e0 and has a resistance R0(1 + Ge0).
- If the casing is given an acceleration a, then the resultant displacement of
the seismic mass m relative to the casing is
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- Gauges 1 and 3 increase in length from L to L + x, and gauges 2 and 4
decrease in length from L to L − x.
-The tensile strain in gauges 1 and 3 increases to e0 + e, and that in gauges 2
and 4 decreases to e0 − e, where:
the maximum acceleration induced strain is only one-half of the initial strain
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Elastic pressure sensing elements
( this subject will be introduced in details later )
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Piezoresistive sensing elements
The piezoresistive effect is defined as the change in resistivity  of material
with applied mechanical strain , and is represented by the term (l/) (∆/).
Piezoresistive sensing elements are made from semiconductor materialsusually silicon with boron as the trace impurity for the P –type materials and
arsencic as trace impurity for the N-type, the resistivity  can be expressed .
1

eN
e- the electron charge, which depends on the type of impurity
N- the number of charge carriers, which depends on the concentration of impurity
- the mobility of charge carriers, which depends on strain and its direction
relative to crystal
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Remember , these section were introduced
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.10
-149,
-160,
-165,
-170,
-172
-177,
-182,
-188,
-196
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