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Description
of software for measurement of time of flight, phase velocity,
dispersion, attenuation of sound and impedance of materials.
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Agnieszka and Wiesław
Bicz
Mieczysław Pluta
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This software package
allows to measure time of flight, sound velocity,
attenuation and frequency dependence of attenuation,
dispersion, phase velocity and impedance (density) of any
material. The software uses ultrasonic pulses and is based
on comparison of pulses achieved with reference and measured
medium.
For each measurement it is necessary to choose reference
signal and compare it with the signal, coming from the
measured medium (reflected or transmitted through it). This
allows to use this software with almost any kind of samples,
containments etc. For people using this software it is
necessary to have some knowledge about such kind of
measurements, physics of ultrasounds etc. This software
package gives them very good tool for such measurements.
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I. Introduction to the work with the software |
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First
step
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Reference signal should be prepared, the best way to do it
is to use pure (distilled) water. Using markers in the upper
window most important part of the signal should be chosen.
In the bottom window signal between markers from the upper
window can be seen - magnified. See picture 1.
Picture
1.
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Second
step |
Button “Pattern” should be used. After pressing
this button, chosen signal appears in bottom screen in white
color together with information: "Correlation pattern". It means this signal from this moment will be
“reference signal”. See picture 2.
Picture
2
From this moment key called "Measure" should be
used – all subsequent operations will use signal
stored before (pattern) as reference for comparison
with actually measured signal. See picture 3. For
time of flight measurement the display will show 0 –
nothing changed.
Picture
3.
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Third
step |
In
this moment we have to repeat operations described in the
first step. In upper window we choose - using markers the
most important part of signal we are getting from measured
medium. In the bottom window we can see only signal between
markers from the upper window. See picture 4.
Pay attention on
marker position (it is changed now). It means now we
have another signal (with time offset for example)
Picture
4
In
this moment we have all information which is necessary for
calculation of time of flight (and another functions too),
and then the button “Measure” should be used (see
picture 5). On the bottom window we can see two earlier prepared signals (white –
reference signal; red - measure signal) in this case we
receive result different from zero.
Picture
5
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II.
The time of flight and sound velocity measurement method. |
In most cases we can assume,
that the signal will change after propagation - simple
geometrical comparison of signals won’t work properly.
This is the reason, why we are using following algorithm for
comparison of two signals with different time of flight:
a)
FFT with
Hamming window is made.
b)
In frequency
domain, frequency with maximum amplitude is chosen and using
relatively sharp windowing only this frequency and
frequencies from its neighborhood are taken.
c)
Inverse FFT is
done.
d)
Center point of
achieved signal is taken as time mark, telling us the moment
of “coming” of this signal
Time
of flight can be measured from zero point (start of pulse)
or from the time of “coming” of another signal,
stored as pattern – as described above.
If
the path length is known, it is possible to calculate the
sound velocity in the measured material, using comparison
with reference fluid – for example water.
If
the experimental setup have a containment with measured
fluid, where only a part of the sound propagation path is in
the measured fluid, we can wrote following formula:
T=T1+T2
Where
T1 is time of propagation outside of measured
fluid and T2 in this medium.
We
can measure time of flight in the whole system (T) filled
with water (TW, that has velocity CW),
or measured fluid TX (velocity CX). If
we know the path length (L) in measured fluid, we can
calculate the velocity of
sound in this medium:
T2w=L/CW
T1=TW-T2W
This
(T1) can be obtained after measurement with water,
and this measurement must be done only from time to time,
since parameters of system doesn’t change quickly.
CX
(sound velocity in measured medium) = L/(TX-T1)
The
user of the software must know the path length (L), and
choose appropriate signals (not only direct transmission
must be chosen, but also multiple reflections for example).
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| III.
Theoretical basis of material impedance measurement method |
The measurement of impedance
of an unknown medium can be based on evaluation of the wave,
reflected on the interface of this medium with another
medium (i.e. wall of solid material - in most cases),
eventually transmitted wave too, and comparison with
reflection from another medium with known impedance.
Meanings
of used variables:
Z
- impedance of medium (unit
is Ns/m3)
R
-
the relation of reflected and
transmitted wave (their intensities)
r
- density of material (g/cm3)
c
– sound velocity (m/s)
IR
- intensity of reflected
wave
It
- intensity of transmitted
wave
Geometrical
representation:
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Material
1 (i.e. solid wall) |
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Material
2 (measured medium, water or air) |
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Z1=r1c1 |
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Z2=r2c2 |
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Incident
wave with intensity I |
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Transmitted
wave with intensity IT |
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¾® |
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¾® |
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Reflected
wave with Intensity IR |
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¬¾ |
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In
most cases we are only able to measure the reflected wave
and not the transmitted (this will be disturbed during
propagation in unknown medium). This is the reason, why it
is normally necessary to know the impedance of the solid
wall. But it is also possible to measure it using air as reference
medium (material 2). In this case the measurement would need
following steps:
1)
Measurement
of reflection from the solid wall in contact with air -
result is the intensity of incoming wave. If we know the
impedance of the wall material, this measurement is not
necessary – the intensity can be calculated from the
measurement made in the second step.
2)
Measurement of
reflection from solid wall in contact with reference fluid
with known impedance (for example water) – this is
only necessary for obtaining value of wall material
impedance. If we know this parameter, we can calculate R for
water and do not need the measurement with air (we are able
to calculate the intensity of incoming wave).
3)
Measurement of
signal reflected from measured fluid.
If
the setup doesn’t change, measurements from steps 1
and 2 must be done only from time to time (relatively seldom).
The formulas used here are as follow:
R=IR/IT=
(Z2-Z1)/(Z2+Z1)
I=IR
+ IT=const
I
is the same in each case - with any material, because it
depends only on used transducer and pulser, and can be also
directly measured in comparison with air (which has almost 0
impedance and thus 100% reflection – IT»0)
or calculated if we know both impedances (Z2 and
Z1 – this is the case, if we know the
impedance of the wall material and are using known fluid,
for example water).
I=IR+IR/R
(if we know R and IR)
We
can measure IR (reflected wave) or IT
(transmitted) wave. If we know only IR, and the
intensity of the incoming wave I, we can calculate R:
R=IR/(I-IR)
After
comparison with another substance we are able to calculate
the impedance of the measured substance:
Z1=Z2(1-R)/(1+R)
Z2=Z1(R+1)/(1-R)
The
first equation is necessary to calculate the impedance of
wall material (assuming, that Z2 is the impedance
of known material – i.e. water) and the second for
calculation of impedance of measured material (Z1
is impedance of wall material).
In
the case of impedance measurement only with reflected signal
it is not necessary to measure the signal using FFT
filtration, because it doesn't change the shape (only in
special cases, that are not realistic here). But this is
also not worse, if we do it.
If
we know the impedance and sound velocity of a material, we
are able to calculate the density (r=c/Z).
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IV. Description of method for measurement of phase velocity,
dispersion and attenuation of sound |
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A.
Introduction |
The idea of the measurement is
based on sending and receiving signals through measured
medium. This operation is repeated many times, after some
time period or after some changes was made (different
material, temperature etc.). Each received signal can be
described by
 where
n=1,2,...N. Each signal can be also described using its
Fourier transform:
 (1a)
after
inverse Fourier transform is applied, signal can be restored
from his spectrum   (1b)
The distribution of harmonics of this spectrum shows the transfer function
of the whole unit, where  is
the function of electronics (generator, transducer,
amplifier...), wave propagation  , including diffraction effects  and
influence of the medium  .
 (2)
In
evaluation of material properties the most interesting thing
are changes of medium properties  . Thus it is necessary to design the system in the
way, that all other elements are negligible (at least in
frequency region of interest). This means, that it is
necessary to use:
a – broadband devices (capable of
transmitting and receiving short pulses),
b – transducer with small or
controlled diffraction effects.
In all cases of relative measurements, where measured signal is compared
with reference (index zero), it is allowed to assume, that
the characteristic of electronics  and geometry of the system
 do
not change. Thus:
 (3)
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B.
Measurement with changing distance: |
Measurements
made in two distances  and
 (4a)
 (4b)
Compound factor responsible for propagation
 (5)
we are writing it as
where
 ,
 -
slowness
 - attenuation
coefficient.
For the case without attenuation and
dispersion
(6a)
We are writing it as the superposition of
propagating waves
(6b)
Careful comparison of this both formulas
allows to see, that propagation of wave packet can be
treated as a superposition of harmonic waves with different
slowness (6b).
From (6a) follows, that with the distance compound
amplitudes of harmonics are changing:
Imaginary part of the coefficient 
is responsible for attenuation. Inclusion of this
effect for chosen harmonics allows us to write compound
amplitude after propagation:
Calculation of compound coefficient
is possible after logarithm of formula (5)
 (7)
where
Difference of compound amplitudes
In large distances this difference changes
quickly with frequency and if it is larger than 
we get non continuous phase readings. In this moment,
for the purpose of doing the phase continuous, it is
necessary to be able to estimate the properties of the
medium and try to make a model, that follows the reality. To
calculate the dispersion we describe slowness as containing
two parts – constant and changing:
where first two parts are describing the
propagation in a medium without dispersion (see (6)). Phase
velocity is calculated from phase difference:
 ,
and from this equation
(8)
The calculation of attenuation using this
method gives realistic results in the central region of
transfer bandwidth, under the condition, that propagation is
going only in one mode (only one path). If many paths are
existing, interference effects occur, that are causing waves
in the attenuation diagram. The existence of such effects
can be shown after measurements for some distances and
comparison of results. In the border region of transfer
function noise occurs, resulting from division of two small
noise functions.
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C. Measurement with changing
medium parameters.
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We are going back to describing individual measurements using 
and reference signal with
 .
For relative measurements it is necessary to store
reference measurement, made with known medium. For water
solutions the best way could be to use water as reference
medium. Treating attenuation of water as negligible and
taking slowness 
of water as reference, for calculation of attenuation
and dispersion of measured medium we can use modified
formulas (7)
and (8):
 (9)
 (10)
Similar formulas can be used for calculating changes of attenuation and
dispersion of medium. As reference one of the previous done
measurements can be taken.
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| Legends |
Formulas are given for continuous variable t,
but in computing we are using discrete variable iT,
where T is the sampling period.
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| For
download |
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material parameter
measurement.pdf
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