C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
INTRODUCTION TO SENSORSContent:
Sensors and measurement instrumentsSensors and electronicsSensors parameters
Electronics for sensors
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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Measurement instrumentsHuman senses as primary measurement instruments.Humans cannot quantify but they can count.The length is the most simple measurable quantity
Ruler length -> number of ticks.
Other quantities can be indirectly measured through suitable instruments that convert them into a measure of length.
noteworth exception: color
Measurement instruments need calibrationa procedure assigining to each value of the primary quantity the corresponding value of the quantity under measurement.
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Thermal expansion: T L
Photochemical reaction: pH colore
Equilibrium between the gravity and spring elastic force: M
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Transducers, Sensors, and Electronics
Measurement instruments are based on transducers: devices transforming a quantity from a form of energy into something measurable.To measure means to acquire a quantitative information. This information is stored, transmitted and/or processed
Processing: to merge measures with models generating novel informations.Currently, electronics is the technology where all these operations are optimally performed.Sensors are a class of transducers that transform the quantity of interest into a measurable electric quantity.The measurable electric quantities are: magnitude, frequency and phase of voltage.
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Mechanical energy
Electromagnetical energy
Thermal energy
Magnetical energy
Chemical energy
Electrical energyElectrical energySENSOR
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
What is electronics?
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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors, Transducers, and Actuatorsa practical definition
Sensor: device that converts a physical, a chemical or a biological quantity into an electric signal.
Clinical thermometer: temperature length= transducerThermistor: temperature electric resistance= sensor
Sensors are components of electronic circuits. Due to sensors, the quantities of the circuit (current and voltage) become function of environmental quantities.
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Any quantity
Any quantityElectricquantity
transducer sensor
actuator
electronic circuit
circuit element
measurand (physical, chemical or biological)
v = f (t,M)
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Classification of sensors:as electronic devices
ResistorsThermistor, photoconductor, magnetoresistance, strain gauge, gas sensor…
InductorsPosition, fluxgate magnetometer,…
CapacitorsPosition, pressure, …
DiodesPhotodiode, magnetic field sensors,…
MOSFETMagnetic field, ions in solution…
VoltageHall probe (magnetic field sensors), ion selective electrodes,…
Electromotive force (emf) Thermocouple, Photovoltaic , electrochemical cells (ions and gas sensors)…
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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Classification of sensors:sensed quantity and measurement principlethe measured quantity
Sensors of physical quantitiesTemperatureElectromagnetic radiation
antennas, infrared sensors, visible light, UV, X, Magnetic field
compasses, position sensorsMechanical quantities
Position, strain, acceleration, pressure, flux Sensors of chemical quantities
Concentration of chemicals in airgases and vapours
Concentration of chemicals in solutionsions and neutral species
Sensors of biological quantitiesConcentrations of compounds in corporeal fluids
ions, antibodies, proteins, DNA analysis, viruses and bacterias
the sensing principlePhysical sensors
Sensors based on physical principles (mechanic or electric) to measure any quantity even non physical e.g. oxygen sensing with magnetic field.
Chemical sensorsSensors that uses the properties of molecules to measure any quantity even non chemical
e.g. molecular thermometerBiosensors
Sensors that uses the properties of biomolecules (e.g. enzymes, peptides, proteins, nucleobases, DNA, cells…) to measure any quantity even non biological
e.g. immunosensors to detect pesticides
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this course“sensori chimici e biosensori” Laurea Specialistica in Ingegneria Elettronica (6 CFU)
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
The sensorial paradigm: the living systemsLiving beings interact with their environment through sensorial receptors.
Two main kinds of receptors:Physical: tactile sense, temperature, optic (sight), acustic (hearing)……
Chemical: olfaction (smell), taste (flavour)
The signals of receptors (sensations) are processed and integrated to form the knowledge (perception).On the other hand, living beings act in the environment through the actuators
Towards outside: mechanical (muscles), acoustic (sounds),…Towards inside: e.g. The organs influence each other with biochemical actuators (hormones)
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Jan Bruegel the elder & Peter Paul RubensAllegory of the five senses (1617-1618)Museo del Prado, Madrid
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
The natural senses“Nihilest in intellectunisi prius fueritin sensu” (St. Thomas Aquinas)There is nothing in the mind which was not first in the senses
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SoundPressure wave with f 10-103 HzEquilibrium (gravity)
lightPhotons with λ=400-700 nm
SmellSome volatile compounds
PressureTemperatureElectric charge
Tastesome molecules in solution(sweet, salty, bitter, acid, umami)
Communication of experience.
Perception, association, memory
Acquisition of the experiences of others.
Association, memory
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Beyond humans sensesMeasurement instruments make visible the imperceptible
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Radiation IR, UV, radio XLight and imagesSoundMagnetic field,PressureForceTemperature VoltageSmell… M
easu
rem
ent i
nstr
umen
t
DIS
PLAY
Communication of experience.
Perception, association, memory
Acquisition of the experiences of others.
Association, memory
model to interpret the data
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Electronics instrumentationThe measure is independent from the observer, the experience is automatically communicated
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SEN
SOR
Processing, storage
Digital communication of sensorial experience.
Radiation IR, UV, radio XLight and imagesSoundPressure wavesMagnetic field,PressureForceTemperatura VoltageSmell…
Perception, associations, memory
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Beyond the senses interfaceneurons-electronics interface
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SEN
SOR
Radiation IR, UV, radio XLight and imagesSoundPressure wavesMagnetic field,PressureForceTemperatura VoltageSmell…
retinal prosthesis: Argus II brain-machine interface
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Artificial reality13
Generator of synthetic sensorial signals
science fiction science
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
From measurement instruments to sensors
atmospheric pressure:barometer
gravitational acceleration:pendulum
angular velocity:gyroscope
light sensor:camera
Sound:microphone
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Microsensors
Integrated sensorsSilicon technology (microelectronics) enables the fabrication of integrated systems where both the sensitive element and the electronics are integrated in the same chip. MEMS (Micro Electro Mechanical Systems)
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9600 dpi = 2 μm
Ink-jet printerhead
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Novel technologies
Flexible
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wearable
Electronic skin
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Ubiquitous sensing
I computer sono sempre più potenti, più piccoli e pervasivi.La rete rende il dato disponibileovunque e a chiunque.1012 sensori connessi.Big Data
microphone2 image sensors3D accelerometer3D gyroscopePressure sensor3D compassAmbient light sensorProximity sensor
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Artificial intelligence18
SENSORS
Ifmindacquiresknowledge from sensory experience can sensory experience creates knowledgewithoutthe mind?
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensor systems
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sensorY=f(M)
electronic interface amplifier filter
A/D conversion
μP
storage communication
actuation
The worldmeasurand
ener
gy
N
v, iv, iv, iY
display
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
General parameters of sensors
Sensor signalA measurable quantity that depends on the sensor itselfExcept few cases, Most sensors are passive devices: a circuit is necessary to generate the signal.
The actual sensor is RS(M), but its value can be evaluated only from the measure of V0(M).
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V0
RSIA
RS(M) is the sensor : a resistance whose value depends on some “ambiental” quantity (e.g. Temperature, light, magnetic field, gas, mechanical stress,…)V0(M) is the sensor signal whose value depends on RS and the circuit used to generate the signal.The circuit is designed to optimize some property of V0(M)
sensor
interface circuit
Environmentalquantity M(measurand)
Power supplyV, I
sensitive signalS(M)
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensor response and feature extraction
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The sensors adaptation to a new value of the measurand follows a proper dynamic. Then the signal always evolves in time.
In practice, a sensor is instantaneous when the response time is much shorter than the typical rate of variation of the measurand.
Synthetic descriptors (features) are extracted from the time evolution of the sensor signal.
Absolute steady-state signal (Veq)Differential or relative signal (with respect to a baseline V0 (reference measurand condition)
for delayed sensors:Response time (T90)
Interval of time necessary to achieve 90% of the steady-state signal.
Veq −V0; Veq
V0
; Veq −V0
V0
time
Veq
V0
V0+0.9 (Veq-V0)
T90
(Veq-V0)
time
mea
sura
nd
time
(Veq-V0)
Instantaneous
delayed
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parameters:Reversibility
Reversibility is the capability of a sensor to follow, with its own dynamics, the variations of the measurand.
A sensor is reversible if, when the stimulus stops, the signal comes back to the pristine value.
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M
t
V
t
ReversibleIntegral
“dosimeter”single use
“disposable”
t
Reversible with memory (hysteresis)
V
t t
t
t
M M M
t
V V
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parameters:the response curve
Each sensor defines a mapping from the space of the measurand to the space of the signals.If both these spaces are one-dimensional, the sensor is represented by a function V=f(M) called the response curve. This function allows utilizing the sensor as a measurement instrument: from the measurand signal, the measurand is estimated.Response curves are almost always made by a linear region, a non-linear region and a saturation region.The response curve is obtained measuring the sensor signal correspondent to known values of the measurand (calibration)
These conditions are provided by:Standards: reference samples (e.g. masses) or known experimental conditions (e.g. melting point of ice)Reference instruments: certified instruments used to measure the “true” values of the measurand
The performance of the sensor is limited by the goodness of the calibration.
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M
V
linearity
non-linearity
saturation
m1
m2
m3
tV
v1
v2
v3
t
measurand
sensor signal
m1 m2 m3
v1
v2
v3
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parametersSensitivity
The sensitivity (S) describes the capability of the sensor to follow the variations of the measurand.Analytically, it is the derivative of the response curve
In case of a non linear response curve, S is a function of the measurand.Largest values of S are found in the linear region close to the origin.
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S = dV
dM
V
M
M
S
linearregion non-linear region saturation
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Instrumental errors and intrinsic fluctuations
Every physical measure contains a part of uncertainty.This uncertainty comes from the limits of the measurement instrument and from the intrinsic statistical nature of the measured quantity.In case of electric quantities, the intrinsic statistical nature is the electronic noise.Which of the two sources prevail depends on the characteristics of the measurement instrument.
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Example: repeated measurements of the length of a rod performed with instruments with different measurement errors
lmin=0.1 cmL=(2.3 ÷ 2.4) cm 2.35 ± 0.05 cm
118
118,5
119
119,5
120
120,5
121
121,5
122
0 1 2 3 4 5 6
Errore strumentale 10 mm
Lung
hezz
a (c
m)
MISURE
Δ=10 mm
119,6
119,8
120
120,2
120,4
0 1 2 3 4 5 6
Errore strumentale 1 mm
Lung
hezz
a (c
m)
MISURE
Δ=2 mm
instrumental error=1 mm
repeated measurements repeated measurements
leng
th (c
m)
leng
th (c
m)
Measurement error hides the intrinsic fluctuation.
Intrinsic fluctuation are observable.
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
The actual response curve
Due to the fluctuation of the signal and the limited accuracy of the signal measurement, the response curve is modified in a deterministic part function of the average of the measurand [f(M)] and a stochastic part due to the fluctations ( v).
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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parameters: ResolutionDue to the fluctations, the estimation of M is affected by an uncertainty. This uncertainty is quantified by the resolution.
The resolution is the smallest change of M that produces a observable change of signals.The signal change is larger than the fluctuation
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Vmeas
Mestim
Vmeas
Resolution and sensitivity
The resolution is inversely proportional to the sensitivity.
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Resolution and measurement errors
Fluctuations are sometimes small, in these cases the resolution is dominated by the measurement error of the instrument used to acquire the signal.
In practice, the accuracy is defined by the number of bits of the Analog to Digital converter.Example: the measurement error due to a 10 bits ADC in a range of 10 V is
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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parametersLimit of detection
The limit of detection (LOD) is the resolution evaluated as the signal approaches zero.
The origin of the response curve is not accessible by a measurement but by the analytical extension of the response curve.When the resolution is determined by the noise, the LOD is the smallest theoretical measurable quantity.
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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Generalized sensitivity and resolution
The concepts of sensitivity and resolution are valid for any input/output system
e.g. clinical thermometer
e.g. amplifier
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AVoutVin
ADC
S = 10ticks◦C
Ntick, err =1
2ticks
ΔTris =Ntick, err
S=
1
2 · 10 = 0.05◦C
S = A =Vout
Vin
Vin,res =ΔVout
A
example:A = 100; Vout = [−12; +12] V ; ADCres = 10 bit
ΔVout =24
210= 23 mV ;
Vin,res =23
100= 230 μV
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
SAMPLE0 1000 2000 3000 4000
V o
ut [V
]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SAMPLE0 1000 2000 3000 4000
R 1
[KO
hm]
0
5
10
15
20
25
Resistance measurement with a voltage divider31
Vin=5 V R1 V0
R0=100 KΩ
V0 = ViR1
R0 +R1
R1 =R0
Vi
V0− 1
Limit of detection and resolution:smallest measurable resistanceArduino UNO ADC: 10 bit
ΔV0,err =Vi
210= 4 mV
S =dV0
dR= Vi
R0
(R0 +R1)2;
SR1=0 =Vi
R0=
5
100 · 103 = 49 μA
SR1=20KΩ = 5100
1202= 35 μA
R1,LOD =ΔV0,err
SR1=0=
4 · 10−3
49 · 10−6= 81 Ω;
R1,res20KΩ =ΔV0,err
SR1=20KΩ=
4 · 10−3
35 · 10−6= 114 Ω;
Response curve V0=f(R1) is not linear, then the resolution depends on R1.
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parametersAccuracy and Reproducibility
Sistematic and random errors define the accuracy and the reproducibility of a sensor.Accuracy is the difference between estimated and “true” value of measurands.
the true value is not accessible because each measurement is always affected by an error. Accuracy is defined respect to a “reference” measurement system
Reproducibility (or precision) is the dispersion of repeated measurements performed in the same conditions.Both terms are statistical quantities: given N measures, the accuracy is associated to the average and the reproducibility to the variance.
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••
• •
• •
•
•
••
•
•
•
••• • •• ••
•• •
••
•• ••
• •
• •
•
•
••
•
•
•
•
•• • •• ••
•• •
••
••
Yes AccuracyNo Reproduc.
Yes AccuracyYes Reproduc.
No AccuracyYes Reproduc.
No AccuracyNo Reproduc.
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parametersselectivity and cross-sensitivity
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quantities
sensitivity
SELECTIVEsensor
quantities
sensitivity
NON-SELECTIVEsensor
environment
quantities
magnitude
V = S j ⋅ M j + err V = Sk ⋅ M k
k∑
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensors parameters:drift
Progressive deterioration of the sensitivity
Sensitivity changes with the time, then drift is an additional systematic error that affects the estimation of the measurandDrift affects the calibration lifetime.
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M0 =va
k t0( )
<M1 =va
k t1( )
<M2 =va
k t2( )
A is the response curve calibrated at t=t0. B, and C are the actual, but unknown, curves at t=t1
and t=t2. respectively.The signal corresponding to a constant stimulus MA
becomes progressively smaller.The obsolete response curve results in a measurand underestimation.
MM0
A t=t0B t=t1
va
V
C t=t2
time
M1 M3
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Examples of parameters of a real sensor:accelerometer ADXL50A
response curve
Sensitivity
Dynamic range=±50 gThe sensitivity is constant throughout the dynamic range
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V a( )=V0 +a ⋅K V0 =1.8 V ; K =0.019 Vg
a=V −1.80.019
g
S = dVda
=K S =19 mVg
V a( )
a V0
+50 g-50 g
the practical unit for acceleration is the gravitational acceleration 1 g = 9.8 m/s2
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Parameters of a real sensor:accelerometer ADXL50A
NoiseSpectral density of thermal noise
If the signal is filtered by a low-pass filter with a corner frequency of 10 Hz, the rms value of the noise is:
Resolution
With a bandwidth of 10 Hz the resolution is 20 mg.Airbag control needs a quick measurement of the acceleration, T=0.1 ms B=10 KHz ares=0.6 g
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125 μVHz
Vnoise , rms =125 μVHz
⋅ 10Hz =395μV
ares =Vn
S=
125 μVHz
19 mVg
=6.6 mgHz
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Some issues about electronics for sensors
Most sensors are passive devices able to condition the networks at which they are connected.
Due to the sensor, the electric quantities (current and voltage) become function of outer world quantities.
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sensor
interface circuit
Environmentalquantity M(measurand)
Power supplyV, I
sensitive signalS(M) A
Measurablesignal
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Total sensitivity
Each block is characterized by a proper sensitivity
The global sensitivtity is the combination combining all blocks
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sensorinterfacecircuit
amplifier filter A/D convM NY v0 va vf
N = fAD v f( )v f = fF va( )va = fA v0( )v0 = fC Y( )Y = fS M( )
S = dNdM
=dNdv f
⋅dv f
dva
⋅dva
dv0
⋅dv0
dY⋅
dYdM
=dfAD
dv f
⋅dfF
dva
⋅dfA
dv0
⋅dfC
dY⋅dfS
dM
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Example:temperature sensitive resistor
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T [°C]
RT
R0
R0α =Δ R
Δ T∆R∆T
Apparently: arbitrary values of sensitivity can be obtained combining the sensor with circuit parameters.Signal limitations restrict the dynamic range
V2 is confined in [-V +V] range
ΔT ΔR ΔV1 ΔV2
RT (T ) = R0(1 + αT );V1 = RT (T ) · I0V2 = A · V1 = A · I0 ·R0(1 + αT )
S =∂V2
∂T=
∂V2
∂V1· ∂V1
∂R· ∂R∂T
S = A · I0 · α ·R0
S =∏
j
Sj
T [°C]
V2
+V
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Example:resolution of a temperature sensitive resistance
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S =∂V2
∂T=∂V2
∂V1
⋅∂Vi
∂R⋅∂R∂T
= Si ⋅ ST ⋅ SA
S = A ⋅ I ⋅α ⋅ Ro
ΔTres = limΔV2 →Vnoise
ΔV2
Stot= lim
ΔV2 →Vnoise
ΔV2
A ⋅ I ⋅α ⋅ Ro
ΔTres = limΔV2 →Vnoise
ΔV2
S j
j
∏
Vnoise2 = A2Vn1
2 +VnA2 = A2 ⋅ Vjohnson
2 +RT2 ⋅ In
2( )+VnA2
Vnoise2 = A2 ⋅ 4kTRT B+RT
2 ⋅ In2( )+VnA
2
∝ I ∝A
ΔT ΔR ΔV1 ΔV2
Arbitrary increase of sensitivity does not improve the resolution
The amplifier is applied to both signal and noise, then it doesn not increase the performance.Noise increases with the current level and the resistance
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Current source with op-amps
The current delivered by an ideal current source is independent from the impedance of the load.
In feedback configuration, op-amps delivers the output voltage necessary to maintain at zero the differential voltage across the input terminals.
In a simple circuit, such as the inverting amplifier, the current in the feedback network does not depend on the feedback impedance, then the the circuit acts as a current source for the resistor RF
The non-ideality of the op-amp limits the ideality of the current source.
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Iin =Vin
Rin
= IF ; Vin −0Rin
=0−Vout
RF
Vout = −RF
Rin
⋅Vin = −Iin ⋅RF
A=− Vout
V+ −V−
A=∞ e Vout finite⇒V+ =V−
Inverting amplifier: equivalent circuit
iF =vin
Rin
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Howland current source 1
In Howland current sources the lack of ideality of a current source is complemented by the additional current provided by an op-amp.A simple implementation of the idea is achieved with a non-inverting amplifier.
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R1 =R2 ; R3 =R4 =R
V0 =VS ⋅ 1+ R2
R1
⎛
⎝⎜
⎞
⎠⎟=2⋅VS
IS = I1+ I2 =Vi −VS
R4
+2⋅VS −VS
R3
=Vi −VS
R+
VS
R=
Vi
R
-+
RS
R1
R2
R3R4
VS
Vo
ViISI1
I2basic current source
correction of the lack of ideality
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Howland current source 2Another implementation of the concept: the output of the differential amplifier (ideal op-amp) is Vload-Vin
The current in the resistor R is
Since the input impedance of the ideal op-amp is infinite, the current IR is totally injected in the load, and then it does not depend on the load.
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iR =vload − vin( )− vload
R= −
vin
R
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Measure of a resistive sensor with a voltage divider
The voltage divider is the simplest circuit available to extract a signal from a resistive sensor
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ΔM→ ΔR→ ΔV R = f ( M ); V0 = f ( R )
Let us consider a linear sensor. The quantity δ depends from the measurand.
The relationship between the output and the variation of RS is non-linear.
Then, even with a linear sensor, the voltage divider gives rise to a non linear relationship between the signal and the measurand.Ri can be chosen to optimize either the sensitivity or the linearity
Vo =Vi ⋅Ro ⋅ 1+δ( )
Ri + Ro ⋅ 1+δ( )
RS = Ro +ΔRo( M )= Ro ⋅ 1+ ΔRo( M )Ro
⎛
⎝⎜
⎞
⎠⎟= Ro ⋅ 1+δ( M )( )
(e.g. δ =α ⋅T )
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Sensitivity optimizationThe total sensitivity is the product of the sensitivity of the sensor and the sensitivity of the circuit.
The sensitivity of the circuit is independent from the nature of the sensor.To optimize the contribution of the circuit, let us maximize dV/dδ.
Since the relationship V=f(δ) is non linear, the sensitivity depends on δ.Let us choose δ=0. In this way, the LOD is optimized.
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Vo =V ⋅Ro ⋅ 1+δ( )
Ri +Ro ⋅ 1+δ( )
S = dVdδ
=VRo ⋅ Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦−Ro ⋅ Ro ⋅ 1+δ( )⎡⎣ ⎤⎦
Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦2 =V Ro ⋅Ri
Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦2
max S⇒ dSdRi
=0
dSdR
=VRo ⋅ Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦
2−Ro ⋅Ri ⋅2⋅ Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦
Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦4
dSdR
=0⇒ Ro ⋅ Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦2−Ro ⋅Ri ⋅2⋅ Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦=0
Ro ⋅ Ri +Ro ⋅ 1+δ( )⎡⎣ ⎤⎦=Ro ⋅Ri ⋅2; Ri +Ro ⋅ 1+δ( )=Ri ⋅2; Ro +Ro ⋅δ =Ri
the maximum sensitivity around δ=0 is obtained when: Ri =Ro
S = dVdM
=dVdδ
⋅dδdM
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Linearity and sensitivity
The ratio Ri/R0 affects the sensitivity and the linearity of the signal respect to the measurandSensitivity and linearity are inversely correlated.
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0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
DELTA
V/V0
Vo =V ⋅R0 ⋅ 1+δ( )
Ri + Ro ⋅ 1+δ( )⇒
VoV=
1+δ( )RiRo+ 1+δ( )
RiRo
curve parameter:
current source
voltage source
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Behaviouraround δ=0
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Vo =V ⋅Ro ⋅ 1+δ( )
Ro +Ro ⋅ 1+δ( )=V ⋅
1+δ2+δ
linearization around δ=0
when δ <<1⇒Vo δ( )=Vo δ =0( )+dVo
dδ δ=0⋅δ
Vo δ( )=V2+V 1
2+δ( )2
δ=0
⋅δ
Vo δ( )=V2+
V4⋅δ
S = dVo
dδ=
V4
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Wheatstone bridge
Circuit used to measure an unknown resistor through a null measurement.The null conditition ( V=0) is achieved changing the value of three known resistors.
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R1
R2
R3
Rx
ΔVV2
V1
V1 =V Rx
R3 +Rx
V2 =V R2
R1+R2
ΔV =V1−V2 =V ⋅ Rx
R3 +Rx
−R2
R1+R2
⎛
⎝⎜
⎞
⎠⎟=0
Rx
R3 +Rx
−R2
R1+R2
=1
R3
Rx
+1−
1R1
R2
+1=0⇒ R3
Rx
=R1
R2
⇒ Rx =R3 ⋅R1
R2
old box of resistors
modern box of resistors with R variable in the range 0-12 K , with 6 decades of steps of 10 m each.
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Application of the Wheatstone bridge to the measure of a resistive sensor
3 fixed resistances and one resistive sensor
The fixed resistors are chosen in order to balance the bridge when the measurand is null.
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ΔV =V2−V R0
R0 +R0 ⋅(1+δ)=
V2−
V2+δ
=V2
δ2+δ( )
=V4
δ1+ δ
2
RS =Ro +ΔRo =Ro ⋅ 1+ΔRo
Ro
⎛
⎝⎜
⎞
⎠⎟=Ro ⋅ 1+δ( )
if δ«1⇒ΔV =V4δ
it corresponds to a voltage divider with the offset subtraction.Respect to the voltage divider it exploits the whole range provided by the voltage source.
+
-
R0(1+δ)
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
2 identical sensors exposed to the same measurand
The sensitivity increases using two identical sensors exposed to the same measurand.
The sensors are connected in different legs and in opposite positions.
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Ro
R0(1+δ)
R0(1+δ)
ΔV
Ro
ΔV =Vi ⋅R0 ⋅(1+δ)
R0 +R0 ⋅(1+δ)−V R0
R0 +R0 ⋅(1+δ)=
=V ⋅ 1+δ2+δ
−1
2+δ⎛⎝⎜
⎞⎠⎟=V ⋅ δ
2+δ
δ«1⇒δ«2⇒ ΔV =V
2δ
S =V
2
For small variation of the sensor response
The sensitivity is twice the sensitivity with one sensor
+
-
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
2 identical sensors exposed to opposite changes
The use of identical sensors exposed to opposite variations allows to fully exploit the bridge featuresthis case is appliable to some experimental conditions such as the measurement of the deformation of a beam or a cantilever.
The sensors are connected in the same leg.
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Ro
R0
R0(1+δ)
R0(1-δ)
ΔV
V2V1
ΔV =Vi ⋅R0
2⋅R0
−Vi ⋅R0 ⋅(1−δ)
R0 ⋅(1−δ)+R0 ⋅(1+δ)=
=Vi ⋅2−Vi ⋅
1−δ2
=Vi ⋅2⋅δ
S = dΔVdδ
=Vi
2
linear !
the same of the sensitivity in the origin in the case with 2 sensors exposed to same stimulus
+
-
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
2 pairs of sensors: same and opposite stimuli
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R0(1+δ)
ΔV
R0(1+δ)
R0(1-δ)ΔV =Vi ⋅
R0 ⋅(1+δ)R0 ⋅(1+δ)+R0 ⋅(1−δ)
−Vi ⋅R0 ⋅(1−δ)
R0 ⋅(1+δ)+R0 ⋅(1−δ)=Vi ⋅δ
S = dΔVdδ
=1⋅Vi
R0(1-δ)
linear !
The sensitivity is four times the sensitivity with one sensor and twice the sensitivity with two sensors.
+
-
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
General case with 4 identical sensors
Let us consider 4 sensors with the same null resistance but each undergoing a different relative change of resistance
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Ri =R0 ⋅ 1+δi( ) i=1−4
R3=R0(1+δ3)
V0
R2=R0(1+δ2)
R1=R0(1+δ1)
R4=R0(1+δ4)
VoVi=
R2
R1 + R2
−R4
R3 + R4
VoVi=
R0 1+δ2( )R0 1+δ1( )+ R0 1+δ2( )
−R0 1+δ4( )
R0 1+δ3( )+ R0 1+δ4( )
If δ «1, the second order terms (δ2, δiδj) are negligible, then the following expression is found:
VoVi=
14+δ1 −δ2 −δ3 +δ4( )
This property is useful to compensate cross-interferences.
+
-
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Proof
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R3=R0(1+δ3)
V0
R2=R0(1+δ2)
R1=R0(1+δ1)
R4=R0(1+δ4)
Let s write: Ri = R0 ⋅ 1+δi( ) = R0 + ΔRi
VoVi=
R+ΔR4
R+ΔR3 + R+ΔR4
− R+ΔR2
R+ΔR1 + R+ΔR2
assumption : ΔRi«R e ΔRiΔRj = 0( )numerator:
R+ΔR4( ) ⋅ R+ΔR1 + R+ΔR2( )− R+ΔR2( ) ⋅ R+ΔR3 + R+ΔR4( )≅ R ⋅ +ΔR1 −ΔR2 −ΔR3 +ΔR4( )
denominator:
R+ΔR3 + R+ΔR4( ) ⋅ R+ΔR1 + R+ΔR2( )= 4 ⋅R2 + 2 ⋅R ⋅ ΔR1 +ΔR2 +ΔR3 +ΔR4( ) ≅ 4 ⋅R2
VoVi=R ⋅ +ΔR1 −ΔR2 −ΔR3 +ΔR4( )
4 ⋅R2 =14⋅ +
ΔR1
R−ΔR2
R−ΔR3
R+ΔR4
R
⎛
⎝⎜
⎞
⎠⎟=
14⋅ +δ1 −δ2 −δ3 +δ4( )
+
-
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Differential measurements
Sensors may be sensitive to more than one measurand.Such sensors may be used in ensembles, for instance all sensors are exposed to a common measurand (not the object of the measurement) and only one to the measurand of interest.
e.g. semiconducting resistors are sensitive to temperature and light
they are arranged so that they are exposed to the same temperature but only one is illuminated by the light
The difference between sensors rejects the common mode . Such a measurement requires a differential amplifier
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R=R0 ⋅ 1+α ⋅T +γ ⋅ Iλ( )
λ
V1 =R0 ⋅i1 ⋅ 1+α ⋅T +γ ⋅ Iλ( )V2 =R0 ⋅i1 ⋅ 1+α ⋅T( )
λ
V0 = AD ⋅ V2 −V1( )= AD ⋅R0 ⋅i1 ⋅γ ⋅ Iλ
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Differential amplifier
A differential amplifier with infinite common mode rejection (CMRR) can be obtained with an ideal op-amp and resistors with accurate values.
The gain depends on the ratio between R2 and R1. To change the gain and to maintain the CMRR it is necessary to change two pairs of resistors of exactly the same quantity.
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V+ =V2R2
R1+R2
I1 = I2 ⇒ V1−V−
R1
=V− −Vout
R2
op. amp. V+=V-
V− =V1R2 +Vout R1
R1+R2
V+ =V− ⇒ V2R2
R1+R2
=V1R2 +Vout R1
R1+R2
Vout =R2
R1
V2 −V1( )
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Instrumentation amplifier
The instrumentation amplifier is a differential amplifier with a input stage.
The gain can be adjusted changing only the resistor (R1). The whole circuit can be integrated (highest CMRR) except R1: the resistor controlling the gain.
57
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
The differential stage amplifies the difference between v01 and v02.
The input stage is the part where the relationship between the resistor R1 and the gain is established.
58
Instrumentation amplifiergain
vout =R4
R3
⋅ v01−v02( )
hypothesis: ideal op-ampsanalysis of the input stage:
i= v1−v2
R1
vo1−v1
R2
= i= v1−v2
R1
; v2 −vo2
R2
= i= v1−v2
R1
vo1 =R2
R1
v1−v2( )+v1
vo2 =−R2
R1
v1−v2( )+v2
vo1−vo2 =R2
R1
v1−v2( )+v1+R2
R1
v1−v2( )−v2 =
=2 R2
R1
v1−v2( )+ v1−v2( )= 1+2 R2
R1
⎛
⎝⎜
⎞
⎠⎟⋅ v1−v2( )
vout =R4
R3
⋅ 1+2 R2
R1
⎛
⎝⎜
⎞
⎠⎟⋅ v1−v2( )
Usually, R4=R3, then when R1= G=1.On the other hand, the case R1=0 is not possible.
C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors
Instrumentation amplifierINA116
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