Lecture 2: Sensor characteristics
g
g
g
g
Transducers, sensors and measurements
Calibration, interfering and modifying inputs
Static sensor characteristics
Dynamic sensor characteristics
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
Wright State University
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Transducers: sensors and actuators
g 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 actuators
n Sensor: an input transducer (i.e., a microphone)
n Actuator: an output transducer (i.e., a loudspeaker)
Input
signal
(measurand)
Sensor
Input
transducer
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
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Processor
Actuator
Output
transducer
Output
signal
(measurand)
2
Measurements
g A simple instrument model
Physical
measurement
variable
Measurand
X
Signal
variable
Sensor
Display
Measurement
S
M
PHYSICAL
PROCESS
n A observable variable X is obtained from the measurand
g 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 Measurement
n 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)
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Calibration
g The relationship between the physical measurement variable
(X) and the signal variable (S)
Signal output (Y)
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)
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Additional inputs
g 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)
Modifying
input Z
Physical variable X
Interfering input Y
Sensor
g Modifying inputs (Z)
n Those that change the behavior of the
sensor and, hence, the calibration curve
S
Signal output (Y)
Measurand
Signal
variable
Z=Z1
Z=Z2
g Temperature is a typical modifying input
Physical input (X)
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Sensor characteristics [PAW91, Web99]
g Static characteristics
n The properties of the system after all transient effects have settled to
their final or steady state
g
g
g
g
g
g
g
g
Accuracy
Discrimination
Precision
Errors
Drift
Sensitivity
Linearity
Hystheresis (backslash)
g Dynamic characteristics
n The properties of the system transient response to an input
g Zero order systems
g First order systems
g Second order systems
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Accuracy, discrimination and precision
g Accuracy is the capacity of a measuring instrument to give
RESULTS close to the TRUE VALUE of the measured quantity
n Accuracy is related to the bias of a set of measurements
n (IN)Accuracy is measured by the absolute and relative errors
ABSOLUTE ERROR = RESULT - TRUE VALUE
RELATIVE ERROR =
ABSOLUTE ERROR
TRUE VALUE
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 output
n Discrimination is also known as RESOLUTION
n When the increment is from zero, it is called THRESHOLD
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Precision
g The capacity of a measuring instrument to give the same
reading when repetitively measuring the same quantity under
the same prescribed conditions
n Precision implies agreement between successive readings, NOT
closeness to the true value
g 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 precision
n Repeatability
g The precision of a set of measurements taken over a short time interval
n Reproducibility
g The precision of a set of measurements BUT
n
n
n
n
taken over a long time interval or
Performed by different operators or
with different instruments or
in different laboratories
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Example
g Shooting darts
n Discrimination
g The size of the hole produced by a dart
n Which shooter is more accurate?
n Which shooter is more precise?
mean
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Wright State University
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Accuracy and errors
g Systematic errors
n Result from a variety of factors
g
g
g
g
g
Interfering or modifying variables (i.e., temperature)
Drift (i.e., changes in chemical structure or mechanical stresses)
The measurement process changes the measurand (i.e., loading errors)
The transmission process changes the signal (i.e., attenuation)
Human observers (i.e., parallax errors)
n Systematic errors can be corrected with COMPENSATION methods (i.e.,
feedback, filtering)
g Random errors
n Also called NOISE: a signal that carries no information
n 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 >>1
g 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
Systematic
error
(Bias)
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Ricardo Gutierrez-Osuna
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Random
error
(Precision)
11
More static characteristics
g 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 Sensitivity
n 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 Linearity
n The closeness of the calibration curve to a specified straight line (i.e., theoretical
behavior, least-squares fit)
g Monotonicity
n A monotonic curve is one in which the dependent variable always increases or
decreases as the independent variable increases
g Hystheresis
n 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 characteristics
g 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 elements
n 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:
1/A
time
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time
time
Amplitude
1
Amplitude
Amplitude
Amplitude
A
Amplitude
n Impulse, step, ramp, sinusoidal, white noise…
time
time
13
Dynamic models
g 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
dk y(t)
d2 y(t)
dy(t)
+L
a
+
a
+ a0 y(t) = x(t)
ak
1
2
2
k
dt
dt
dt
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 expression
n 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 behavior
n x(t) = e-σtsin(ωt) = e-(σ +jω)t
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Wright State University
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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 behavior
n The imaginary component defines the frequency of oscillatory behavior
g The fundamental relationship is the one that concerns the
transformation of differentiation
d

L  y(t) = sY(s) - f (0 )
 dt

g Other useful relationships are
Impulse : L[ W] = 1
Step :
Ramp :
L[u(t)] =
1
s
1
L[r(t)] = 2
s
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
Wright State University
Decay :
L[exp(at )] = (s - a )
Sine :
L[sin( W )] =
Cosine :
L[cos(
-1
s2 +
s
W )] = 2
s +
2
2
15
The Laplace Transform (review)
g Applying the Laplace transform to the sensor model yields
 dk y

d2 y
dy
+ ao y(t) = x(t)
L ak k + L a 2 2 + a1
dt
dt
 dt

⇓
(a s
k
k
)
+ L a 2s2 + a1s + a o Y(s) = X(s)
⇓
G(s) =
Y(s)
1
=
X(s) ak sk + L a 2s2 + a1s + ao
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 as
n Oscillating components
n Exponential decays
n Instability
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Wright State University
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Pole location and dynamic behavior
jω
jω
-σ1
σ
-σ2
e − 1t sin(
+jω1
e−
e−
2t
1t
t
t)
1
σ
-σ1
t
-jω1
jω
jω
+jω1
σ1
σ
t
sin(
+jω2
σ
+jω1
e
-σ1 σ
+
sin(
t)
1
jω
+jω1
1t
+σ1 σ
t
-jω1
Intelligent Sensor Systems
Ricardo Gutierrez-Osuna
Wright State University
t)
t
+jω2
jω
2
t
e + 1t sin(
t)
1
17
Zero-order sensors
g Input and output are related by an equation of the type
Y(s)
y(t) = k ⋅ x(t) ⇒
=k
X(s)
n Zero-order is the desirable response of a sensor
g No delays
g Infinite bandwidth
g The sensor only changes the amplitude of the input signal
n Zero-order systems do not include energy-storing elements
n Example of a zero-order sensor
g A potentiometer used to measure linear and rotary displacements
n This model would not work for fast-varying displacements
VCC
X
Y
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First-order sensors
g Inputs and outputs related by a first-order differential equation
dy
Y(s)
1
k
a1
+ a 0 y(t) = x(t) ⇒
=
=
dt
X(s) a1s + a0 τ s + 1
n First-order sensors have one element that stores energy and one that
dissipates it
n Step response
g 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 response
g Better described by the amplitude and phase shift plots
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Wright State University
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First-order sensor response
g Step response
g Frequency response
n Corner frequency ωc=1/τ
n Bandwidth
From [PAW91]
g Ramp response
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Example of a first-order sensor
g A mercury thermometer immersed into a fluid
n What type of input was applied to the sensor?
n Parameters
g
g
g
g
θF
C: thermal capacitance of the mercury
R: thermal resistance of the glass to heat transfer
θF: temperature of the fluid
θ(t): temperature of the thermometer
θF
R
θ
C
n The equivalent circuit is an RC network
g Derivation
n Heat flow through the glass ( F − W)/R
n Temperature of the thermometer rises as
n Taking the Laplace transform
d
W
dt
=
F
− W
RC
(s) − V
⇒ (RCs + 1) V = F (s) ⇒
RC
F (s)
⇒ V =
⇒ W = F 1− e − t/RC
(RCs + 1)
s
V =
F
(
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)
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Second-order sensors
g Inputs and outputs are related by a second-order differential
equation
d2 y
dy
Y(s)
1
a 2 2 + a1
+ a0 y(t) = x(t) ⇒
=
dt
dt
X(s) a 2s 2 + a1s + a0
n We can express this second-order transfer function as
Y(s)
k
= 2
X(s) s + 2
with k =
n Where
2
n
n
s+
1
,
a0
2
n
=
a1
,
2 a0a1
n
=
a0
a2
g k is the static gain
g ζ is known as the damping coefficient
g ωn is known as the natural frequency
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Second-order step response
g Response types
n Underdamped (ζ<1)
n Critically damped (ζ=1)
n Overdamped (ζ>1)
g Response parameters
n
n
n
n
Rise time (tr)
Peak overshoot (Mp)
Time to peak (tp)
Settling time (ts)
From [PAW91]
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Second-order response (cont)
g Ramp response
g Frequency response
From [PAW91]
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Example of second-order sensors
g 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 accelerometer
n The armature suffers an acceleration
g We will assume that this acceleration is
orthogonal to the direction of gravity
x0
M
n x0 is the displacement of the mass M with
respect to the armature
n The equilibrium equation is:
K
B
&x&
i
M(&x&i − &x&0 ) = Kx 0 + Bx& 0
⇓
Ms2 Xi (s) = X0 (s) K + Bs + Ms2
⇓
X0 (s) M
K/M
=
s 2 Xi (s) K s 2 + s(B/M) + K/M
[
Intelligent Sensor Systems
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Wright State University
]
25
References
[PAW91]
[Web99]
[Tay97]
[Fdn97]
[BW96]
R. Pallas-Areny and J. G. Webster, 1991, Sensors and
Signal Conditioning, Wiley, New York
J. G. Webster, 1999, The Measurement, Instrumentation
and Sensors Handbook, CRC/IEEE Press , Boca Raton, FL.
H. R. Taylor, 1997, Data Acquisition for Sensor Systems,
Chapman and Hall, London, UK.
J. Fraden, 1997, Handbook of Modern Sensors. Physics,
Designs and Applications, AIP, Woodbury, NY
J. Brignell and N. White, 1996, Intelligent Sensor Systems,
2nd Ed., IOP, Bristol, UK
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Wright State University
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