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3. Sensor characteristicsStatic sensor characteristics

Relationships between output and input signals of the sensor in conditionsof very slow changes of the input signal determine the static sensorcharacteristics.

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

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

y

x

a b

xa xb

y

XxdxdyS x

Measurement error Δx of quantity X for a given Δy can be small enough for a high sensitivity.

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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 powers1dB = 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 = Δ /(x2-x1)

and in the error of the slope

δb = Δ /(x2-x1)

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Hysteresis

This is a deviation of the sensor output, when it is approached from different directions.

y

x

- Max. difference atoutput for specified input

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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.

y

x

FSO

Best straight line

Nonlinearity

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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

y

x

Cycle 1 Cycle 2

Δ - max. difference between output readings for the same direction

Repeatability

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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.

y

x

threshold

resolution

Rsolution and threshold

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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

signaly 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. timek – static sensitivity of a transducerm ≤ n

Eq. (1) can be transformed by the Laplace integral transformation

(2)

where s = σ + jω

0

)()()( dttfetfLsF st

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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

)0()()( ftfsLdt

tdfL

nn

mm

sAsAsAsBsBsB

ksXsYsK

...1...1

)()()( 2

21

221

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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x(t)

t

1

x(t) = 1(t) 0 for t < 0 1(t) = 1 for t

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

Inertial element of the 1-st order

R

C u1(t) u2(t)

L{1(t)} = X(s) = 1/s

ssk

ssKsY 1

11)()(

Resistance thermometerimmersed in the liquidof elevated temperature

Electric analog

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Therefore

ssk)s(f

ss/

/ss

e/s

es/dtedtek/)s(f

dt)e(kedt)t(fe)s(f

t)/s(st)/tst(st

/tstst

11

11

111

111

1

0

1

000

00

sk

s/)s(Y)s(K

11

Inertial element of the 1-st order,calculation of operator transmittance

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Time response to the step function of inertial element of the 1-st order

t

y(t) k

)1()()( /1 teksYLty

k.)e/(k(y 6321011

τ – time constant, a measure of sensor thermal inertia;

For electric analogue τ = RC

what means for t = τ 63% of a steady value.For t = 3τ one gets 95% of a steady value.

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Insertion of a thermometer into an insulating sheath transforms it into an inertial element of higher order.

y(t)

k 95% 90%

63%

10%

0 t10 t90 t95 t

Response to the step function of an inertial element of higher order

t95 - 95% response timeΔt = t90 – t10 – rise timeτ = t63 – time constant

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Response of an oscillating system to the step function

y(t)

t

K

0

2 3

1

1 – oscillations2 – critical damping3 – 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

kS

m

F(t)

y

RS

R

C

u(t)

u1(t)

L

q = CU1(t) = kU1(t)

m)t(Fy

mky

mRy

s

s 1

LRLC

tCuqqq

2

1)(2

20

20

20

Electric analogue

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The denominator of expression for transmittance can have:1) Two real roots

overdamping

2) One real root

critical damping

3) Two complex roots

underdamping(oscillations)

Transmitance of the oscillating system

20

2

20

2)(

ssk

sK

jjs 2202,1

s

After inverse Laplace transformation one gets:

20

22,1 s

)sin(1)( tekty t