sensor characteristic

Intelligent Sensor SystemsRicardo Gutierrez-OsunaWright State University

1

Lecture 2: Sensor characteristicsg Transducers, sensors and measurementsg Calibration, interfering and modifying inputsg Static sensor characteristicsg Dynamic sensor characteristics


2

Transducers: sensors and actuatorsg Transducer

n A device that converts a signal from one physical form to a corresponding signal having a different physical form

g Physical form: mechanical, thermal, magnetic, electric, optical, chemical…

n Transducers are ENERGY CONVERTERS or MODIFIERS

g Sensor n A device that receives and responds to a signal or stimulus

g This is a broader concept that includes the extension of our perception capabilities to acquire information about physical quantities

g Transducers: sensors and actuatorsn Sensor: an input transducer (i.e., a microphone)n Actuator: an output transducer (i.e., a loudspeaker)

Sensor Processor Actuator

Inputtransducer

Outputtransducer

Inputsignal

(measurand)

Outputsignal

(measurand)


3

Measurementsg A simple instrument model

n A observable variable X is obtained from the measurandg X is related to the measurand in some KNOWN way (i.e., measuring mass)

n The sensor generates a signal variable that can be manipulated:g Processed, transmitted or displayed

n In the example above the signal is passed to a display, where a measurement can be taken

g Measurementn The process of comparing an unknown quantity with a standard of the

same quantity (measuring length) or standards of two or more related quantities (measuring velocity)

Sensor

Physical measurement

variable

X

Signalvariable

S

Measurement

MMeasurand

PHYSICALPROCESS

DisplaySensor

Physical measurement

variable

X

Signalvariable

S

Measurement

MMeasurand

PHYSICALPROCESS

Measurand

PHYSICALPROCESS

Display


4

Calibrationg The relationship between the physical measurement variable

(X) and the signal variable (S)n A sensor or instrument is calibrated by applying a number of KNOWN

physical inputs and recording the response of the system

Physical input (X)

Sig

nal o

utpu

t (Y

)

Physical input (X)

Sig

nal o

utpu

t (Y

)


5

Additional inputsg Interfering inputs (Y)

n Those that the sensor to respond as the linear superposition with the measurand variable X

g Linear superposition assumption: S(aX+bY)=aS(X)+bS(Y)

g Modifying inputs (Z)n Those that change the behavior of the

sensor and, hence, the calibration curveg Temperature is a typical modifying input

Sig

nal o

utpu

t (Y

)Physical input (X)

Z=Z1

Z=Z2

Sensor

Physical variable X Signalvariable

SMeasurand Interfering input Y

Modifying input Z

Sensor

Physical variable X Signalvariable

SMeasurandMeasurand Interfering input Y

Modifying input Z


6

Sensor characteristics [PAW91, Web99]

g Static characteristicsn The properties of the system after all transient effects have settled to

their final or steady stateg Accuracyg Discriminationg Precisiong Errorsg Drift

g Sensitivityg Linearityg Hystheresis (backslash)

g Dynamic characteristicsn The properties of the system transient response to an input

g Zero order systemsg First order systems

g Second order systems


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Accuracy, discrimination and precisiong Accuracy is the capacity of a measuring instrument to give

RESULTS close to the TRUE VALUE of the measured quantityn Accuracy is related to the bias of a set of measurements

n (IN)Accuracy is measured by the absolute and relative errors

g More on errors in a later slide

g Discrimination is the minimal change of the input necessary to produce a detectable change at the outputn Discrimination is also known as RESOLUTION n When the increment is from zero, it is called THRESHOLD

VALUETRUEERRORABSOLUTE

ERRORRELATIVE

VALUETRUE-RESULTERRORABSOLUTE

=

=


8

Precisiong The capacity of a measuring instrument to give the same

reading when repetitively measuring the same quantity under the same prescribed conditionsn Precision implies agreement between successive readings, NOT

closeness to the true valueg Precision is related to the variance of a set of measurements

n Precision is a necessary but not sufficient condition for accuracy

g Two terms closely related to precisionn Repeatability

g The precision of a set of measurements taken over a short time interval

n Reproducibilityg The precision of a set of measurements BUT

n taken over a long time interval or

n Performed by different operators or

n with different instruments or

n in different laboratories


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Example

gShooting dartsn Discrimination

g The size of the hole produced by a dart

n Which shooter is more accurate? n Which shooter is more precise?

mean


10

Accuracy and errorsg Systematic errors

n Result from a variety of factorsg Interfering or modifying variables (i.e., temperature)

g Drift (i.e., changes in chemical structure or mechanical stresses)

g The measurement process changes the measurand (i.e., loading errors)

g The transmission process changes the signal (i.e., attenuation)

g Human observers (i.e., parallax errors)

n Systematic errors can be corrected with COMPENSATION methods (i.e., feedback, filtering)

g Random errorsn Also called NOISE: a signal that carries no informationn True random errors (white noise) follow a Gaussian distribution

n Sources of randomness:g Repeatability of the measurand itself (i.e., height of a rough surface)

g Environmental noise (i.e., background noise picked by a microphone)g Transmission noise (i.e., 60Hz hum)

n Signal to noise ratio (SNR) should be >>1g With knowledge of the signal characteristics it may be possible to interpret a signal with

a low SNR (i.e., understanding speech in a loud environment)


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Example: systematic and random errors

Systematicerror(Bias)

Randomerror

(Precision)

Systematicerror(Bias)

Randomerror

(Precision)


12

More static characteristicsg Input range

n The maximum and minimum value of the physical variable that can be measured (i.e., -40F/100F in a thermometer)

n Output range can be defined similarly

g Sensitivityn The slope of the calibration curve y=f(x)

g An ideal sensor will have a large and constant sensitivity

n Sensitivity-related errors: saturation and “dead-bands”

g Linearityn The closeness of the calibration curve to a specified straight line (i.e., theoretical

behavior, least-squares fit)

g Monotonicityn A monotonic curve is one in which the dependent variable always increases or

decreases as the independent variable increases

g Hystheresisn The difference between two output values that correspond to the same input

depending on the trajectory followed by the sensor (i.e., magnetization in ferromagnetic materials)

g Backslash: hystheresis caused by looseness in a mechanical joint


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Dynamic characteristicsg The sensor response to a variable input is different from that

exhibited when the input signals are constant (the latter is described by the static characteristics)

g The reason for dynamic characteristics is the presence of energy-storing elementsn Inertial: masses, inductances

n Capacitances: electrical, thermal

g Dynamic characteristics are determined by analyzing the response of the sensor to a family of variable input waveforms:n Impulse, step, ramp, sinusoidal, white noise…

time

Am

plitu

de

time

Am

plitu

de

1A

1/A time

Am

plitu

de

time

Am

plitu

de

time

Am

plitu

de


14

Dynamic modelsg The dynamic response of the sensor is (typically) assumed to

be linear n Therefore, it can be modeled by a constant-coefficient linear differential

equation

n In practice, these models are confined to zero, first and second order. Higher order models are rarely applied

g These dynamic models are typically analyzed with the Laplace transform, which converts the differential equation into a polynomial expressionn Think of the Laplace domain as an extension of the Fourier transform

g Fourier analysis is restricted to sinusoidal signals n x(t) = sin(ωt) = e-jωt

g Laplace analysis can also handle exponential behaviorn x(t) = e-σtsin(ωt) = e-(σ +jω)t

x(t)y(t)adt

dy(t)a

dty(t)d

adty(t)d

a 012

2

2k

k

k =+++�


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The Laplace Transform (review)g The Laplace transform of a time signal y(t) is denoted by

n L[y(t)] = Y(s)g The s variable is a complex number s=σ+jω

n The real component σ defines the real exponential behaviorn The imaginary component defines the frequency of oscillatory behavior

g The fundamental relationship is the one that concerns the transformation of differentiation

g Other useful relationships are

( )0f-sY(s)y(t)dtd

L =

[ ]

[ ]

[ ]2s

1r(t)L:Ramp

s1

u(t)L:Step

1��L:Impulse

=

=

= ( )[ ] ( )

( )[ ]

( )[ ]22

22

1-

ss

�cosL:Cosine

s�sinL:Sine

a-satexpL:Decay

+=

+=

=


16

The Laplace Transform (review)g Applying the Laplace transform to the sensor model yields

n G(s) is called the transfer function of the sensor

g The position of the poles of G(s) -zeros of the denominator- in the s-plane determines the dynamic behavior of the sensor such asn Oscillating componentsn Exponential decays

n Instability

( )

o12

2k

k

o12

2k

k

o12

2

2k

k

k

asasasa1

X(s)Y(s)

G(s)

X(s)Y(s)asasasa

x(t)y(t)adtdy

adt

yda

dtyd

aL

+++==

⇓=+++

⇓

=+++

�

�

�


17

Pole location and dynamic behavior

-σ1

-σ2σ

jω

t1e−t2e−t

σ1σ

jω

t1e+

t-σ1σ

jω

t

-σ1 σ

jω( )tsine 1

t1−

t-jω1

+jω1

σ

jω

t

( )tsin 1

+jω1

+jω2

+jω1

+jω2

( )tsin 2

+σ1 σ

jω

( )tsine 1t1+

t-jω1

+jω1


18

Zero-order sensorsg Input and output are related by an equation of the type

n Zero-order is the desirable response of a sensorg No delaysg Infinite bandwidth

g The sensor only changes the amplitude of the input signal

n Zero-order systems do not include energy-storing elementsn Example of a zero-order sensor

g A potentiometer used to measure linear and rotary displacementsn This model would not work for fast-varying displacements

kX(s)Y(s)

x(t)ky(t) =⇒⋅=

VCC

X

Y


19

First-order sensorsg Inputs and outputs related by a first-order differential equation

n First-order sensors have one element that stores energy and one that dissipates it

n Step responseg y(t) = Ak(1-e-t/τ)

n A is the amplitude of the step

n k (=1/a0) is the static gain, which determines the static response

n τ (=a1/a0) is the time constant, which determines the dynamic response

n Ramp response g y(t) = Akt - Akτu(t) + Akτe-t/τ

n Frequency responseg Better described by the amplitude and phase shift plots

1sk

asa1

X(s)Y(s)

x(t)y(t)adtdy

a01

01 +=

+=⇒=+

τ


20

First-order sensor responseg Step response

g Ramp response

g Frequency responsen Corner frequency ωc=1/τn Bandwidth

From [PAW91]


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Example of a first-order sensorg A mercury thermometer immersed into a fluid

n What type of input was applied to the sensor?n Parameters

g C: thermal capacitance of the mercury

g R: thermal resistance of the glass to heat transfer

g θF: temperature of the fluid

g θ(t): temperature of the thermometer

n The equivalent circuit is an RC network

g Derivationn Heat flow through the glass

n Temperature of the thermometer rises asn Taking the Laplace transform

θF

R

C

θF θ

( )/R�W�F −

RC�W�

dt�W�d F −=

( )

( ) ( )t/RCF

F

FF

e1�W�1RCs

(s)�V�

(s)�V� 1RCsRC

�V�(s)�V� s

−−=⇒+

=⇒

⇒=+⇒−=


22

Second-order sensorsg Inputs and outputs are related by a second-order differential

equation

n We can express this second-order transfer function as

n Whereg k is the static gain

g ζ is known as the damping coefficient

g ωn is known as the natural frequency

012

2012

2

2 asasa1

X(s)Y(s)

x(t)y(t)adtdy

adt

yda

++=⇒=++

2

0n

10

1

0

2nn

2

2n

aa

,aa2

a,

a1

kwith

s2sk

X(s)Y(s)

===

++=


23

Second-order step responseg Response types

n Underdamped (ζ<1)

n Critically damped (ζ=1)n Overdamped (ζ>1)

g Response parametersn Rise time (tr)

n Peak overshoot (Mp)n Time to peak (tp)n Settling time (ts)

From [PAW91]


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Second-order response (cont)g Ramp response g Frequency response

From [PAW91]


25

Example of second-order sensorsg A thermometer covered for protection

n Adding the heat capacity and thermal resistance of the protection yields a second-order system with two real poles (overdamped)

g Spring-mass-dampen accelerometern The armature suffers an acceleration

g We will assume that this acceleration is

orthogonal to the direction of gravity

n x0 is the displacement of the mass M with respect to the armature

n The equilibrium equation is:

( )

[ ]

K/Ms(B/M)sK/M

KM

(s)Xs(s)X

MsBsK(s)X(s)XMs

xBKxxxM

2i

20

20i

2

000i

++=

⇓++=

⇓+=− ��

M

K B

x0

ix&&

M

K B

x0

ix&&

M

K B

x0

ix&&


26

References[PAW91] R. Pallas-Areny and J. G. Webster, 1991, Sensors and

Signal Conditioning, Wiley, New York[Web99] J. G. Webster, 1999, The Measurement, Instrumentation

and Sensors Handbook, CRC/IEEE Press , Boca Raton, FL.[Tay97] H. R. Taylor, 1997, Data Acquisition for Sensor Systems,

Chapman and Hall, London, UK.[Fdn97] J. Fraden, 1997, Handbook of Modern Sensors. Physics,

Designs and Applications, AIP, Woodbury, NY[BW96] J. Brignell and N. White, 1996, Intelligent Sensor Systems,

2nd Ed., IOP, Bristol, UK

sensor characteristic

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