INTRODUCTION TO SENSORS

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 1. INTRODUCTION TO SENSORS Content: Sensors and measurement instruments Sensors and electronics Sensors parameters Electronics for sensors. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 2. Measurement instruments Human 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..
. Measurement instruments need calibration.
noteworth exception: color
a procedure assigining to each value of the primary quantity the. corresponding value of the quantity under measurement.. Photochemical reaction: pHcolore Thermal expansion: TL. Equilibrium between the gravity and spring elastic force: M.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 3. 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.
. Mechanical energy Electromagnetical energy Thermal energy Electrical energy. SENSOR. Electrical energy. Magnetical energy Chemical energy. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 4. What is electronics?.
 








 




    
 
 
   


 
   
    

 
 

 




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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 5. Sensors,Transducers,and Actuators a practical definition Any quantity transducer. sensor. Any quantity.
. Electric quantity. actuator. Sensor: device that converts a physical, a chemical or a biological quantity into an electric signal.
Clinical thermometer: temperature
length= transducer
Thermistor: 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. measurand (physical, chemical or biological) circuit element. v = f (t, M ) electronic circuit. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. Classification of sensors: as electronic devices
Resistors





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Thermistor, photoconductor, magnetoresistance, strain gauge, gas sensor… Inductors
Position, fluxgate magnetometer,… Capacitors
Position, pressure, … Diodes
Photodiode, magnetic field sensors,… MOSFET
Magnetic field, ions in solution… Voltage
Hall probe (magnetic field sensors), ion selective electrodes,… Electromotive force (emf )
Thermocouple, Photovoltaic, electrochemical cells (ions and gas sensors)…. 6.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 7. Classification of sensors: sensed quantity and measurement principle
the measured quantity
Sensors of physical quantities.
.
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. Temperature Electromagnetic radiation.
. Magnetic field.
. Mechanical quantities.
. antennas, infrared sensors, visible light, UV, X,
.
. compasses, position sensors.
. Position, strain, acceleration, pressure, flux. Sensors of chemical quantities
. Concentration of chemicals in air.
. Concentration of chemicals in solutions.

. gases and vapours. ions and neutral species. Sensors of biological quantities
. Concentrations of compounds in corporeal fluids
. ions, antibodies, proteins, DNA analysis, viruses and bacterias.
the sensing principle
Physical sensors
.
. Sensors based on physical principles (mechanic or electric) to measure any quantity even non physical
.
. Sensors that uses the properties of molecules to measure any quantity even non chemical
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. e.g. oxygen sensing with magnetic field.. Chemical sensors. e.g. molecular thermometer. Biosensors
. 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. this course “sensori chimici e biosensori” Laurea Specialistica in Ingegneria Elettronica (6 CFU). C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 8. The sensorial paradigm: the living systems Living beings interact with their environment through sensorial receptors.
Two main kinds of receptors:
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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). Jan Bruegel the elder & Peter Paul Rubens Allegory of the five senses (1617-1618) Museo del Prado, Madrid.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 9. The natural senses “Nihilest in intellectunisi prius fueritin sensu”(St.ThomasAquinas) There is nothing in the mind which was not first in the senses Perception, association, memory light Photons with λ=400-700 nm Sound Pressure wave with f
10-103 Hz Equilibrium (gravity) Smell Some volatile compounds Pressure Temperature Electric charge. Communication of experience. Association, memory. Taste some molecules in solution (sweet, salty, bitter, acid, umami). Acquisition of the experiences of others.. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 10. Beyond humans senses Perception, association, memory. DISPLAY. Radiation IR, UV, radio X Light and images Sound Magnetic field, Pressure Force Temperature Voltage Smell …. Measurement instrument. Measurement instruments make visible the imperceptible. model to interpret the data. Communication of experience.. Association, memory. Acquisition of the experiences of others..

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 11. Electronics instrumentation The measure is independent from the observer, the experience is automatically communicated. Processing, storage SENSOR. Radiation IR, UV, radio X Light and images Sound Pressure waves Magnetic field, Pressure Force Temperatura Voltage Smell …. Digital communication of sensorial experience. Perception, associations, memory. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 12. Beyond the senses interface. Radiation IR, UV, radio X Light and images Sound Pressure waves Magnetic field, Pressure Force Temperatura Voltage Smell …. brain-machine interface. SENSOR. neurons-electronics interface. retinal prosthesis: Argus II.        


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C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 13. Artificial reality. Generator of synthetic sensorial signals. science. science fiction.  


  
. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. From measurement instruments to sensors atmospheric pressure: barometer. light sensor: camera. gravitational acceleration: pendulum. angular velocity: gyroscope. Sound: microphone.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 15. Microsensors
. Integrated sensors
Silicon 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). Ink-jet printerhead. 9600 dpi = 2 μm. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 16. Novel technologies
.
. Flexible. Electronic skin.
. wearable.

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 disponibile ovunque e a chiunque. 1012 sensori connessi. Big Data.







. microphone 2 image sensors 3D accelerometer 3D gyroscope Pressure sensor 3D compass Ambient light sensor Proximity sensor. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. Artificial intelligence Ifmind acquiresknowledge from sensoryexperience can sensoryexperiencecreates knowledge withoutthe mind?. SENSORS. 18.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 19. Sensor systems The world. sensor Y=f(M). measurand. Y. electronic interface. v, i. amplifier. v, i. v, i. filter. A/D conversion. actuation. energy. N. μP. communication. storage. display. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 20. General parameters of sensors
. Sensor signal
A measurable quantity that depends on the sensor itself
Except few cases, Most sensors are passive devices: a circuit is necessary to generate the signal.. Power supply V, I. Environmental quantity M (measurand). sensor. sensitive signal S(M). interface circuit. V0 IA. RS. 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).
The actual sensor is RS (M), but its value can be evaluated only from the measure of V 0(M)..

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 21. Sensor response and feature extraction The sensors adaptation to a new value of the measurand follows a proper dynamic. Then the signal always evolves in time.. measurand.
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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.. Veq V0.
Differential or relative signal (with respect to a baseline V0 (reference measurand condition). Veq ; V0. Instantaneous (Veq-V0).
Absolute steady-state signal (Veq). Veq − V0 ;. time. time. Veq − V0 V0. delayed.
for delayed sensors:
Response time (T90). V0+0.9 (Veq-V0).
Interval of time necessary to achieve 90% of the steadystate signal.. (Veq-V0). time. T90. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 22. 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.
. Reversible with memory (hysteresis). Reversible. M. M. M. t V. M. t. t V. t. Integral “dosimeter”. V. t. single use “disposable”. t V. t. t.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. Sensors parameters: the response curve

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. 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. 23. measurand. m3 m2 m1. t V. sensor signal. v3 v2 v1. t. V saturation. v3 non-linearity. v2. v1. of the calibration.. linearity m1. m2. M. m3. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 24. Sensors parameters Sensitivity 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
. S=. linear region. dV dM. 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.
. V. non-linear region. saturation. M S. M.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 25. 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.
. instrumental error=1 mm. lmin=0.1 cm L=(2.3 ÷ 2.4) cm
2.35 ± 0.05 cm. Example: repeated measurements of the length of a rod performed with instruments with different measurement errors 122 Errore strumentale 10 mm. Errore strumentale 1 mm 120,4. 121,5. 120,2. length Lunghezza(cm) (cm). length Lunghezza (cm) (cm). 121 120,5 120 119,5 Δ=10 mm. 120. Δ=2 mm 119,8. 119 118,5. 119,6 118 0. 1. 2. 3. 4. 5. 6. MISURE. repeated measurements. Measurement error hides the intrinsic fluctuation.. 0. 1. 2. 3. 4. 5. 6. MISURE. repeated measurements. 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).. 26.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 27. Sensors parameters:Resolution Due 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. V meas. V meas. M estim. 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
. 28.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 29. Sensors parameters Limit 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.
. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 30. Generalized sensitivity and resolution
. The concepts of sensitivity and resolution are valid for any input/output system
e.g. clinical thermometer. ticks ◦C 1 Ntick, err = ticks 2 1 Ntick, err = = 0.05◦ C ΔTris = S 2 · 10 S = 10.
e.g. amplifier. Vout Vin ΔVout = A. S=A= Vin,res. example: A = 100; Vout = [−12; +12] V ; ADCres = 10 bit 24 = 23 mV ; 210 23 = 230 μV = 100. ΔVout = Vin,res. Vin. A. Vout. A D C.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 31. Resistance measurement with a voltage divider R0=100 KΩ. 25. 1. R1 V0 = Vi R0 + R1. 0.9 0.8. V0. 0.6. R 1 [KOhm]. R1. 20. R0 −1. Vi V0. 0.7 V out [V]. Vin=5 V. R1 =. 0.5 0.4. 15. 10. 0.3 0.2. Response curve V0=f(R1) is not linear, then the resolution depends on R1.. 5. 0.1 0. 0. 1000. 2000 SAMPLE. 3000. 4000. 0. 0. 1000. 2000 SAMPLE. 3000. 4000. Vi = 4 mV 210 R0 dV0 = Vi ; S= dR (R0 + R1 )2 Vi 5 = = 49 μA SR1 =0 = R0 100 · 103 100 SR1 =20KΩ = 5 = 35 μA 1202 ΔV0,err 4 · 10−3 R1,LOD = = = 81 Ω; SR1 =0 49 · 10−6 ΔV0,err 4 · 10−3 R1,res20KΩ = = = 114 Ω; SR1 =20KΩ 35 · 10−6 ΔV0,err =. Limit of detection and resolution: smallest measurable resistance Arduino UNO ADC: 10 bit. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 32. Sensors parameters Accuracy 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.
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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.
. • •• •• • • •••••••. Yes Accuracy Yes Reproduc.. • •. • •. • • •. • • •. • •. •. • •. Yes Accuracy No Reproduc.. •. • •. • • •. • • •. • •• •• • • •• •••• •. • • • •. No Accuracy No Reproduc.. No Accuracy Yes Reproduc..

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 33. Sensors parameters selectivity and cross-sensitivity. magnitude. environment. SELECTIVE sensor. sensitivity. sensitivity. quantities NON-SELECTIVE sensor. quantities. quantities. V = ∑ Sk ⋅ M k. V = S j ⋅ M j + err. k. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 34. 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 measurand
Drift affects the calibration lifetime.
.

. V. time. A t=t0.
B t=t1.
. C t=t2. va. M0 M1. M3. M. 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.. M0 =. va v v < M1 = a < M 2 = a k(t0 ) k(t1 ) k( t 2 ).

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 35. Examples of parameters of a real sensor: accelerometer ADXL50A the practical unit for acceleration is the gravitational acceleration 1 g = 9.8 m/s2.
. response curve V ( a ) =V0 +a⋅ K V0 =1.8 V ; K = 0.019 . a=.
. V g. V −1.8 g 0.019 V (a). Sensitivity. S=. dV mV = K S =19  da g. V0 -50 g.
. +50 g. a. Dynamic range=±50 g
The sensitivity is constant throughout the dynamic range. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 36. Parameters of a real sensor: accelerometer ADXL50A
. Noise
Spectral density of thermal noise. 125. μV Hz.
If the signal is filtered by a low-pass filter with a corner frequency of 10 Hz, the rms value of the noise is:. Vnoise , rms =125
. μV ⋅ 10Hz = 395μV Hz. Resolution ares =. μV Vn 125 Hz mg = = 6.6 mV Hz S 19 g.
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.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 37. 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.
. Power supply V, I. Environmental quantity M (measurand). sensor. sensitive signal S(M). Measurable signal. A. interface circuit. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 38. Total sensitivity v0 = f C (Y ). Y = fS (M ). M. sensor. Y. interface circuit. va = f A ( v0 ) v0. amplifier. N = f AD ( v f ). v f = f F ( va ) va. filter. vf. A/D conv. Each block is characterized by a proper sensitivity
The global sensitivtity is the combination combining all blocks
. S=. dN dN dv f dva dv0 dY df AD df F df A df C df S = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ dM dv f dva dv0 dY dM dv f dva dv0 dY dM. N.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 39. Example: temperature sensitive resistor RT. ∆T ∆R. R0α =. ΔR ΔT. RT (T ) = R0 (1 + αT ); V1 = RT (T ) · I0 V2 = A · V1 = A · I0 · R0 (1 + αT ). R0 T [°C]. ∂V2 ∂V1 ∂R ∂V2 = · · ∂T ∂V1 ∂R ∂T S = A · I 0 · α · R0  S= Sj S=. ΔT
ΔR
ΔV1
ΔV2. j. V2 +V. 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. T [°C]. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 40. Example: resolution of a temperature sensitive resistance ∂V2 ∂V2 ∂Vi ∂R = ⋅ ⋅ = Si ⋅ ST ⋅ SA ∂T ∂V1 ∂R ∂T S = A ⋅ I ⋅ α ⋅ Ro S= ΔT
ΔR
ΔV1
ΔV2. ΔTres =. ΔV2 ΔV2 = lim ΔV2 →Vnoise S ΔV2 →Vnoise A ⋅ I ⋅ α ⋅ R tot o. ΔTres =. ΔV2 ΔV2 →Vnoise ∏ Sj. lim lim. j.
. 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. 2 2 Vnoise = A2Vn12 +VnA2 = A2 ⋅ (V johnson + RT2 ⋅ I n2 ) +VnA2 2 Vnoise = A2 ⋅ (4kTRT B+ RT2 ⋅ I n2 ) +VnA2. ∝I. ∝A.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 41. 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.
. . iF =. . vin Rin.  
. Inverting amplifier: equivalent circuit 
. . V A= − out V+ −V− A= ∞ e Vout finite ⇒V+ =V−. . . . . I in =  . Vin V − 0 0 −Vout = I F ; in = Rin Rin RF. Vout = −. RF ⋅V = −I in ⋅ RF Rin in. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 42. Howland current source 1 correction of the lack of ideality. R2. R1 Vo R3. basic current source Vi. I2. R4. I1. 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.
. +. VS. RS. IS. R1 = R2  ; R3 = R4 = R ⎛ R ⎞ V0 =VS ⋅ ⎜1+ 2 ⎟ = 2⋅VS ⎝ R1 ⎠ V −V 2⋅VS −VS Vi −VS VS Vi I S = I1 + I 2 = i S + = + = R4 R3 R R R.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 43. Howland current source 2 Another implementation of the concept: the output of the differential amplifier (ideal opamp) is Vload-Vin
The current in the resistor R is
. iR =
. (v. load. ). − vin − vload R. =−. vin R. 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..  



. . . . .  
.  . 



. 
. .
.  
.   .  . C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 44. 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. R = f ( M ); V0 = f ( R ). ΔM → ΔR → ΔV
. Let us consider a linear sensor.The quantity δ depends from the measurand.. ⎛ ΔR ( M RS = Ro + ΔRo ( M ) = Ro ⋅ ⎜1+ o Ro ⎝
 .
. .
.
. (. ). The relationship between the output and the variation of RS is non-linear.. Vo = Vi ⋅
. )⎞ ⎟ = Ro ⋅ 1+ δ( M ) ⎠ ( e.g.   δ = α ⋅T ). (. Ro ⋅ 1+ δ. (. ). Ri + Ro ⋅ 1+ δ. ). 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.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 45. Sensitivity optimization




. The 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. Vo =V ⋅ S=. S=. dV dV dδ = ⋅ dM dδ dM. Ro ⋅ (1+ δ ) Ri + Ro ⋅ (1+ δ ). R ⋅⎡⎣ R + R ⋅ (1+ δ )⎤⎦ − Ro ⋅⎡⎣ Ro ⋅ (1+ δ )⎤⎦ dV Ro ⋅ Ri =V o i o =V 2 2 ⎡⎣ Ri + Ro ⋅ (1+ δ )⎤⎦ ⎡⎣ Ri + Ro ⋅ (1+ δ )⎤⎦ dδ. max S ⇒. dS =0 dRi 2. R ⋅⎡⎣ R + R ⋅ (1+ δ )⎤⎦ − Ro ⋅ Ri ⋅2⋅⎡⎣ Ri + Ro ⋅ (1+ δ )⎤⎦ dS =V o i o 4 ⎡⎣ Ri + Ro ⋅ (1+ δ )⎤⎦ dR 2 dS = 0 ⇒ Ro ⋅⎡⎣ Ri + Ro ⋅ (1+ δ )⎤⎦ − Ro ⋅ Ri ⋅2⋅⎡⎣ Ri + Ro ⋅ (1+ δ )⎤⎦ = 0 dR 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. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 46. Linearity and sensitivity curve parameter:. voltage source. Ri Ro. 1. The ratio Ri/R0 affects the sensitivity and the linearity of the signal respect to the measurand
Sensitivity and linearity are inversely correlated.
. Vo = V ⋅. R0 ⋅ (1+ δ ) V (1+ δ ) ⇒ o= Ri + Ro ⋅ (1+ δ ) V Ri + 1+ δ ( ) Ro. 
. 0.9. 
 0.8.
. 0.7. . V/V0. 0.6 0.5. . 0.4 0.3 0.2 0.1 0.  0. 2. 4. 6. 8. 10 DELTA. current source 12. 14. 16. 18. 20.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 47. Behaviour around δ=0

. . Vo =V ⋅. . Ro ⋅ (1+ δ ) 1+ δ =V ⋅ Ro + Ro ⋅ (1+ δ ) 2+ δ. linearization around δ=0 when δ <<1⇒Vo (δ ) =Vo (δ = 0 ) + Vo (δ ) =. V 1 +V 2 2 ( 2+ δ ). dVo dδ. ⋅δ δ =0. ⋅δ δ =0. V V Vo (δ ) = + ⋅δ 2 4 dV V S= o = dδ 4. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 48. 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.
. V1 =V. R1. R3 V1. V2. ΔV R2. Rx R3 + Rx. V2 =V. R2 R1 + R2. ⎛ Rx R2 ⎞ ΔV =V1 −V2 =V ⋅ ⎜ − ⎟=0 ⎝ R3 + Rx R1 + R2 ⎠ R2 1 1 R R R ⋅R Rx − = − = 0 ⇒ 3 = 1 ⇒ Rx = 3 1 R R R3 + Rx R1 + R2 Rx R2 R2 3 1 +1 +1 Rx R2. Rx 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. 49. 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. . ⎛ ΔR ⎞ RS = Ro +ΔRo = Ro ⋅ ⎜1+ o ⎟ = Ro ⋅ (1+ δ ) Ro ⎠ ⎝ V R0 V V V δ V δ ΔV = −V = − = = 2 R0 + R0 ⋅(1+ δ ) 2 2+ δ 2 ( 2+ δ ) 4 1+ δ2. . R0(1+δ). 
.  
. +. 
. V if   δ«1⇒ ΔV = δ 4 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.. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 50. 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. . R0 ⋅(1+ δ ) R0 −V = R0 + R0 ⋅(1+ δ ) R0 + R0 ⋅(1+ δ ) ⎛ 1+ δ 1 ⎞ δ − =V ⋅ ⎜ ⎟ =V ⋅ ⎝ 2+ δ 2+ δ ⎠ 2+ δ.
 Ro. ΔV =Vi ⋅ R0(1+δ) -. ΔV. R0(1+δ). Ro. +. For small variation of the sensor response. δ «1 ⇒ δ «2 ⇒ ΔV =. S=. V 2. V δ 2. The sensitivity is twice the sensitivity with one sensor.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 51. 2 identical sensors exposed to opposite changes . The use of identical sensors exposed to opposite variations allows to fully exploit the bridge features  this 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..
. R0 R0 ⋅(1− δ ) −Vi ⋅ = R0 ⋅(1− δ )+ R0 ⋅(1+ δ ) 2⋅ R0 1− δ Vi ⋅ V⋅ = ⋅δ = i −Vi ⋅ linear ! 2 2 2. ΔV =Vi ⋅ Ro. R0(1+δ) V1. V2. ΔV R0. +. R0(1-δ). S=. dΔV Vi = dδ 2. 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. 52. 2 pairs of sensors:same and opposite stimuli.
 R0(1-δ). ΔV =Vi ⋅. R0(1+δ). ΔV R0(1+δ). R0(1-δ). R0 ⋅(1+ δ ) R0 ⋅(1− δ ) −Vi ⋅ =Vi ⋅δ R0 ⋅(1+ δ )+ R0 ⋅(1− δ ) R0 ⋅(1+ δ )+ R0 ⋅(1− δ ). linear ! +. S=. dΔV =1⋅Vi dδ. 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. 53. 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. Ri = R0 ⋅ (1+ δi )    i =1−4
 R1=R0(1+δ1). R2 R4 Vo = −
Vi R1 + R2 R3 + R4. R3=R0(1+δ3). R0 (1+ δ2 ) R0 (1+ δ4 ) Vo = − Vi R0 (1+ δ1 ) + R0 (1+ δ2 ) R0 (1+ δ3 ) + R0 (1+ δ4 ) -. V0. If δ «1, the second order terms (δ2, δiδj) are negligible, then the following expression is found:. +. R4=R0(1+δ4). R2=R0(1+δ2). Vo 1 = (+δ1 − δ2 − δ3 + δ4 ) Vi 4 This property is useful to compensate cross-interferences.. C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 54. Proof. (. ). Let s write: Ri = R0 ⋅ 1+ δi = R0 + Δ Ri. R + ΔR4 R + ΔR2 Vo = − Vi R + ΔR3 + R + ΔR4 R + ΔR1 + R + ΔR2.
 R1=R0(1+δ1). ( assumption : ΔR «R e ΔR ΔR i. R3=R0(1+δ3). R2=R0(1+δ2). j. = 0). numerator:. -. V0. i. +. R4=R0(1+δ4). ( 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 ⋅ R 2 + 2 ⋅ R ⋅ ( ΔR1 + ΔR2 + ΔR3 + ΔR4 ) ≅ 4 ⋅ R 2 Vo R ⋅ (+ΔR1 − ΔR2 − ΔR3 + ΔR4 ) 1 ⎛ ΔR1 ΔR2 ΔR3 ΔR4 ⎞ = = ⋅ + − − + = Vi. 4 ⋅ R2. 1






⋅ (+δ1 − δ2 − δ3 + δ4 ) 4. ⎜ 4 ⎝. R. R. R. ⎟ R ⎠.

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 55. 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. R = R0 ⋅ (1+ α ⋅T + γ ⋅ I λ ) 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 λ. λ. . . . . V1 = R0 ⋅i1 ⋅ (1+ α ⋅T + γ ⋅ I λ ). V0 = AD ⋅ (V2 −V1 ) = AD ⋅ R0 ⋅i1 ⋅γ ⋅ I λ.
. V2 = R0 ⋅i1 ⋅ (1+ α ⋅T ).  . C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 56. 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.
. . . .  . . op. amp.
V+=V-. . . .
.  . R2 R1 + R2 V −V V −V I 1 = I 2 ⇒ 1 − = − out R1 R2. V+ =V2.  .  
. . . . V1 R2 +Vout R1 R1 + R2 V R +V R R2 V+ =V− ⇒ V2 = 1 2 out 1 R1 + R2 R1 + R2 R Vout = 2 (V2 −V1 ) R1 V− =. .

C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 57. 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.
. .
. .  . . . . .
. . . .
. . . C. Di Natale, University of Rome Tor Vergata: Introduction to Sensors. 58. Instrumentation amplifier gain analysis of the input stage: hypothesis: ideal op-amps .
. 
. .  . . . . . .
. . . .
. .  . . .
. The differential stage amplifies the difference between v01 and v02.. vout =
. R4 ⋅ ( v01 − v02 ) R3. The input stage is the part where the relationship between the resistor R1 and the gain is established..   
. v1 − v2 R1 v −v v −v v −v vo1 − v1 = i = 1 2 ; 2 o2 = i = 1 2 R2 R1 R2 R1 R vo1 = 2 ( v1 − v2 ) + v1 R1 R vo2 = − 2 ( v1 − v2 ) + v2 R1 R R vo1 − vo2 = 2 ( v1 − v2 ) + v1 + 2 ( v1 − v2 ) − v2 = R1 R1 ⎛ R R ⎞ = 2 2 ( v1 − v2 ) + ( v1 − v2 ) = ⎜1+ 2 2 ⎟ ⋅ ( v1 − v2 ) R1 R1 ⎠ ⎝ R ⎛ R ⎞ vout = 4 ⋅ ⎜1+ 2 2 ⎟ ⋅ ( v1 − v2 ) R3 ⎝ R1 ⎠. i=. 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 amplifier INA116. 59.