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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors

INTRODUCTION TO SENSORSContent:

Sensors and measurement instrumentsSensors and electronicsSensors parameters

Electronics for sensors

1


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.

2

Thermal expansion: T L

Photochemical reaction: pH colore

Equilibrium between the gravity and spring elastic force: M


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.

3

Mechanical energy

Electromagnetical energy

Thermal energy

Magnetical energy

Chemical energy

Electrical energyElectrical energySENSOR


What is electronics?

4


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.

5

Any quantity

Any quantityElectricquantity

transducer sensor

actuator

electronic circuit

circuit element

measurand (physical, chemical or biological)

v = f (t,M)


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)…

6


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

7

this course“sensori chimici e biosensori” Laurea Specialistica in Ingegneria Elettronica (6 CFU)


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


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

9

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


Beyond humans sensesMeasurement instruments make visible the imperceptible

10

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


Electronics instrumentationThe measure is independent from the observer, the experience is automatically communicated

11

SEN

SOR

Processing, storage

Digital communication of sensorial experience.

Radiation IR, UV, radio XLight and imagesSoundPressure wavesMagnetic field,PressureForceTemperatura VoltageSmell…

Perception, associations, memory


Beyond the senses interfaceneurons-electronics interface

12

SEN

SOR

Radiation IR, UV, radio XLight and imagesSoundPressure wavesMagnetic field,PressureForceTemperatura VoltageSmell…

retinal prosthesis: Argus II brain-machine interface


Artificial reality13

Generator of synthetic sensorial signals

science fiction science


From measurement instruments to sensors

atmospheric pressure:barometer

gravitational acceleration:pendulum

angular velocity:gyroscope

light sensor:camera

Sound:microphone


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)

15

9600 dpi = 2 μm

Ink-jet printerhead


Novel technologies

Flexible

16

wearable

Electronic skin


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


Artificial intelligence18

SENSORS

Ifmindacquiresknowledge from sensory experience can sensory experience creates knowledgewithoutthe mind?


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


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)


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


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


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


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


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.


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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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.


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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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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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


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.


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.


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∑


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


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


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


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.

37

sensor

interface circuit

Environmentalquantity M(measurand)

Power supplyV, I

sensitive signalS(M) A

Measurablesignal


Total sensitivity

Each block is characterized by a proper sensitivity

The global sensitivtity is the combination combining all blocks

38

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


Example:temperature sensitive resistor

39

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


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


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.

41

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


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.

42

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


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.

43

iR =vload − vin( )− vload

R= −

vin

R


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

44

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 )


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.

45

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


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.

46

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


Behaviouraround δ=0

47

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


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.

48

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.


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.

49

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)


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.

50

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

+

-


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

+

-


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.

+

-


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.

+

-


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( )

+

-


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


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( )


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.

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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.

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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.


Instrumentation amplifierINA116

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