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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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