3. Sensor characteristics

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3. Sensor characteristics Static sensor
characteristics

• Relationships between output and input signals of
  the sensor in conditions
• of very slow changes of the input signal
  determine the static sensor
• characteristics.
• Some important sensor characteristics and
  properties include
• transfer function, from which a sensitivity can
  be determined
• span (input full scale) and FSO (full scale
  output)
• calibration error
• hysteresis
• nonlinearity
• repeatability
• resolution and threshold

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Transfer function
An ideal relationship between stimulus (input) x
and sensor output y is called transfer function.
The simplest is a linear relationship given by
equation y a bx (3.1) The slope
b is called sensitivity and a (intercept) the
output at zero input. Output signal is mostly of
electrical nature, as voltage, current,
resistance. Other transfer functions are often
approximated by logarithmic function y a
b lnx (3.2) exponential function y a
ekx (3.3) power function y ao
a1xc (3.4) In many cases none of above
approximatios fit sufficiently well and higher
order polynomials can be employed. For nonlinear
transfer function the sensitivity is defined
as S dy/dx (3.5) and depends on the
input value x.
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Sensitivity
Measurement error ?x of quantity X for a given
?y can be small enough for a high sensitivity.
Over a limited range, within specified accuracy
limits, the nonlinear transfer function can be
modeled by straight lines (piece-wise
approximation). For these linear approximations
the sensitivity can be calculated by
S ?y/?x
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Span and full scale output (FSO)
An input full scale or span is determined by a
dynamic range of stimuli which may be converted
by a sensor,without unacceptably high
inaccuracy. For a very broad range of input
stimuli, it can be expressed in decibels, defined
by using the logarithmic scale. By using decibel
scale the signal amplitudes are represented by
much smaller numbers. For power a decibel is
defined as ten times the log of the ratio of
powers 1dB 10 log(P/P0) Similarly for the
case of voltage (current, pressure) one
introduces 1dB 20 log(V2/V1) Full scale
output (FSO) is the difference between ouput
signals for maximum and minimum stimuli
respectively. This must include deviations from
the ideal transfer function, specified by ?.
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Calibration error

• Calibration error is determined by innacuracy
  permitted by a manufacturer after calibration of
  a sensor in the factory.
• To determine the slope and intercept of the
  function one applies two stimuli x1 and x2 and
  the sensor responds with A1 and A2. The higher
  signal is measured with error ?.
• This results in the error in intercept (new
  intercept a1, real a)
• da a1 a ? /(x2-x1)
• and in the error of the slope
• db ? /(x2-x1)

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Hysteresis
This is a deviation of the sensor output, when it
is approached from different directions.
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Nonlinearity
This error is specified for sensors, when the
nonlinear transfer function is approximated by a
straight line. It is a maximum deviation of a
real transfer function from the straight line and
can be specified in of FSO. The approximated
line can be drawn as the so called best
straight line which is a line midway between two
parallel lines envelpoing output values of a real
transfer function. Another method is based on the
least squares procedure.
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Repeatability
This error is caused by sensor instability and
can be expressed as the maximum difference
between output readings as determined by two
calibration cycles, given in of FSO. dr ?
/ FSO
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Rsolution and threshold
Threshold is the smallest increment of stimulus
which gives noticeable change in
output. Resolution is the step change at output
during continuous change of input.
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Dynamic characteristics
When the transducing system consists of linear
elements dissipating and accumulating energy,
then the dependence between stimulus x and output
signal y can be written as equality of two
differential equations A0y A1y(1) A2y(2)
... Any(n) k (B0x B1x(1) B2x(2) ...
Bmx(m)) (1) y(1) 1-st derivative vs.
time k static sensitivity of a
transducer m n Eq. (1) can be transformed
by the Laplace integral transformation
(2) where s s j?
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Integrating (2) by parts it is easy to show
that (3)
Transforming eq. (1) using Laplace
transformation, with the help of property (3) and
with zero initial conditions one obtains the
expression for operator transmittance of the
sensor
In effect we transfer from differential to
algebraic equations. The analysis of an operator
transmittance is particularly useful when the
transducer is built as a measurement chain. The
response y(t) one obtains applying reverse
Laplace transformation.
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Excitation by a step-function
The response of a sensor system depends on its
type. It can be an inercial system, which
consists of accumulation elements of one
type (accumulating kinetic or potencial energy)
and dissipating elements.
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An example of inertial transducer
Resistance thermometer immersed in the liquid of
elevated temperature
Electric analog
L1(t) X(s) 1/s
Inertial element of the 1-st order
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Inertial element of the 1-st order, calculation
of operator transmittance
Therefore
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Time response to the step function of inertial
element of the 1-st order
t time constant, a measure of sensor thermal
inertia For electric analogue t RC
what means for t t 63 of a steady value. For
t 3t one gets 95 of a steady value.
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Response to the step function of an inertial
element of higher order
t95 - 95 response time ?t t90 t10 rise
time t t63 time constant
Insertion of a thermometer into an insulating
sheath transforms it into an inertial element of
higher order.
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Response of an oscillating system to the step
function
1 oscillations 2 critical damping 3
overdamping
Tranducer of the oscillation type consists of
accumulating elements of both types and
dissipating elements. Mechanical analogue is a
damped spring oscillator (the spring accumulates
potential energy, the mass kinetic energy,
energy is dissipated by friction). Electric
analogue is an RLC circuit.
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Mechanic analogue
Electric analogue
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Transmitance of the oscillating system

• The denominator of expression for transmittance
  can have
• Two real roots
• overdamping
• 2) One real root
• critical damping
• 3) Two complex roots
• underdamping
• (oscillations)

After inverse Laplace transformation one gets