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2. 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
exponential function y = a ekx
power function
y = ao + a1xc
(3.2)
(3.3)
(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
dy
dx
x X
Sx 
a
b
y
Measurement error Δx of
quantity X for a given Δy can
be small enough for a high
sensitivity.
 y
 xa
 xb
x
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 = Δy/Δx
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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 ± Δ.
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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 – Δ.
This results in the error in intercept (new
intercept a1, real a)
δa = a1 – a
and in the error of the slope
δb = Δ /(x2-x1)
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Hysteresis
 - Max. difference at
output for specified
input
y

x
This is a deviation of the sensor output, when it is
approached from different directions.
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Nonlinearity
y
±% FSO
Best straight line
x
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
y
Cycle 1
Cycle 2

Δ - max. difference
between output readings
for the same direction
x
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.
δr = Δ / FSO
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Resolution and threshold
y
 resolution
threshold
x
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

F ( s)  L f (t )   e  st f (t )dt
(2)
0
where s = σ + jω
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Integrating (2) by parts it is easy to show that
 df (t ) 
L
  sL f (t )  f (0)
 dt 
(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
1  B1 s  B2 s 2  ...Bm s m
Y ( s)
K ( s) 
k
X ( s)
1  A1 s  A2 s 2  ... An s n
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
x(t)
x(t) = 1(t)
0 for t < 0
1
1(t) =
1 for t  0
t
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 analogue
L{1(t)} = X(s) = 1/s
Y (s)  K (s)
1
k
1


s 1  s s
Inertial element of the 1-st order
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Inertial element of the 1-st order,
calculation of operator transmittance

f(s) e
 st

f ( t ) dt   e  stk ( 1  e t /  ) dt
0
0

f(s)/ k  e
 st
0
dt   e
( st  t /  )
0
1
1

s s 1/ 
k
f(s)
1  s


dt  1 / s e
 st 
0
1
( s 1 /  )t 

e
0
s 1/ 
1/ s
1  s
1

s

Therefore
K( s ) 
Y( s )
k

1 / s 1  s
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Time response to the step function of inertial element of
the 1-st order
y ( t )  L1Y ( s )  k (1  e t /  )
k
y ' ( t )  e t / 

y(t)
k


τ – time constant, a measure of
sensor inertia;
For electric analogue τ = RC
t
y(    k ( 1  1 / e )  0.6321 k
what for t = τ means 63% of a steady value.
For t = 3τ one gets 95% of a steady value.
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Response to the step function of an inertial element
of higher order
y(t)
k
95%
90%
t95 - 95% response time
Δt = t90 – t10 – rise time
τ = t63 – time constant
63%
Electric analogue of
the inertial element of
2nd order.
10%
0
t10

t90 t95
t
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
y(t)
1 – oscillations
2 – critical damping
3 – overdamping
K
2
0
3
t
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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RS
L
kS
u1(t)
u(t)
y
C
q = CU1(t) = kU1(t)
R
m
F(t)
q  2q  02 q  02Cu(t )
y 
Rs
1
F( t )
y 
y
m
mks
m
Mechanic analogue
1
LC
R

2L
02 
Electric analogue
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Transmitance of the oscillating system
k 02
K ( s)  2
s  2s   02
The denominator of expression for transmittance can have:
1) Two real roots
s1, 2     2  02
2)
overdamping
One real root
s  
3)
critical damping
Two complex roots
s1, 2    j 02   2    j
After inverse Laplace
transformation one gets:

underdamping
(oscillations)
y(t )  k 1  e t sin( t   )

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