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92Electronic Journal «Technical Acoustics» http://www .ejta.org
2007, 13
Ahmed Chitnalah*1, Djilali Kourtiche2, Hicham Jakjoud1,
Mustapha Nadi2
1 Electrical Systems and Telecommunication Laboratory (LSET), Cadi Ayyad University, BP549, 40 000 Marrakech, Morocco
2 Nancy Electronic Instrumentation Laboratory (LIEN), Henri Poincare University, BP239, 54506 Vandoeuvre-les-nancy, France
Pulse echo method for nonlinear ultrasound parameter measurement
Received 28.05.2007, published 04.06.2007
Pulse echo method for nonlinear ultrasound measurement attracts the interest of the scientific community because it can be used for non destructive evaluation and in vivo measurement. Also, it can be useful in quantitative harmonic imaging. The technique is based on the measurement of the reflected harmonics from an interface separating two mediums. The use of this method for nonlinear ultrasound parameter measurement was suggested. However the mathematical model derived was based on the plane wave theory ignoring diffraction phenomenon. An analytical model taking into account diffraction is proposed in this paper. Using a decomposition of the source function in terms of Gaussian beams, analytical expressions are obtained for the fundamental and second harmonic reflected. For the experimental study a ring transducer is used for transmission allowing machining a tube to insert the hydrophone. The hydrophone is a PZT with 0.4 mm in diameter. The harmonics reflected from two interfaces (ethanol/water and water/glycerol) are measured. Comparing the measured values to the theoretical results allow to deduce the nonlinear parameter for water and ethanol. The results obtained agree with the value find in the literature and show the validity of the model.
INTRODUCTION
Two main methods are usually used in nonlinear ultrasound parameter measurement. Thermodynamic method gives accurate measurements but it requires complex apparatus. Finite amplitude are more useful but one needs an analytical expression relating the second harmonic to the nonlinear parameter /?0=1+B/A. Some approximations are necessary to
derive such expressions. Many studies dealt with this problem considering the plane wave theory [1, 2]. The method is improved by including absorption and diffraction into the mathematical model [3-7].
Pulse echo method is suitable for in vivo measurement because it allows using echographic devices for harmonic imaging. This technique for nonlinear ultrasound parameter measurement [8, 9] was suggested, using the plane wave theory to calculate the reflected
*
Corresponding author, e-mail: [email protected]
fields. A theoretical model taking into account diffraction and absorption based on the quasilinear theory and a set of Gaussian beams was derived [10]. The measurement [11, 12] based on this model are first made by using a perfect reflector (total reflection).
In his paper we present the values of the nonlinear parameter deduced from the measurement of the second harmonic reflected from a plane interface between two mediums water/glycerol and ethanol/water. The measurements agree well with the value available in the literature and show the validity of the theoretical model and the validity of this method.
1. B/A DETERMINATION
Let us consider, Vr2 and Vr1, the electrical voltages received by the hydrophone for the second harmonic and the fundamental reflected from the interface. These voltages are linked to the ultrasound fields q1r (r, z) and q2r (r, z)) by the following equations:
Vr1 =n« qir(r = 0z = 2 zo), (1)
where rjh\ is the hydrophone sensitivity (V/Bar) at the fundamental frequency f0, z0 is the
interface position and z is the coordinate along the axis propagation and r is the radial coordinate;
Vr 2 = n h2 qir (r = 0 z = 2 Z0), (2)
where nh2 is the hydrophone sensitivity (V/Bar) at the second harmonic frequency 2f0.
The reflected fields are expressed [10]:
™ - BmFm ( z )l-l
q„(r,z)=r,P z X(z)e 1 aJ , (3)
m=1
where P0 is the source level pressure, a is the source radius, a01 is the thermoviscous absorption at f0, r1 is the reflection coefficient at f0, Am and Bm are the well known Wen and Breazeale Gaussian [13] beams’ coefficients,
1 k0 a2
Fm(z)=-------- and 0
1 . z 0m 2 B’ (4)
1 + i------- m v ’
z0 m
where k0 is the wave number,
q2r (r, z) = /° k\ P02 e “““ z X Am Aj r2 f0 Gm,j (r, z / z ') dz' + ^ [ Gm,j (r, z / z' ) dz'
10
,0n’0_ P2 e-«02 z 2 P C2 P0
H0 0 m, j=1
(5)
where c0 is the sound speed, f30 is the nonlinearity parameter, and p0 the ambient density, a02 is the thermoviscous absorption at 2f0, T2 is the reflection coefficient at 2f0,
. ( am j +2iz'bm j («02 - 2 a») z' -0.5 k( - I ,j ,j
Gm](r,z/z')=----------------—— e 1 Cm,j +izdm,j ^, (6)
c .+iz d .
m, j m,j
z0m + z0 j , 7 1
am,j =----------------L and bm, j =------
z0 mz0 j
z0 mz0 j
cmi =1 + 0.5izam. and dm , = 0.5 am , + izbm.,
m, j m, j m, j m, j m, j 5
7 -7
r = ^ 01 ^11
7 +7
01 11
and
7 -7
r = 02 12
j- o '
7 +7
02 12
(7)
(8) (9)
where 701 and Z02 are the acoustic impedances related respectively to the fundamental and second harmonic in the first medium (before the interface). 711 and 712 are the same acoustic impedances in the second medium (after the interface),
7 = 7
01 0
( a ^ 1 + j— k
k0 y
and
7 = 7
02 0
1 + j
. a
02
2k
(10)
where 70 represents the characteristic acoustic impedance in the first medium (before the interface),
7 = 7
11 1
.a
and
7 = 7
12 1
V1 y
a
1 + j— 2k
(11)
1y
where 71 and k1 represent respectively the characteristic acoustic impedance and the wave number in the second medium (after the interface). The absorption coefficients are a11 and a12 for the fundamental and second harmonic.
Assuming that the hydrophone sensitivity is almost the same (nh1 =nh2) for the fundamental (f0) and the second harmonic (2f0) then we can write:
q = Vn = qlr(0,2z0) = Pq k0 P0 e-(«02-aM)zU(z0)
Vr1 q2r (0,2z0) 2 Po C0 r1
V ( zo)
10
V (zo) = X A.F. (2 zo),
m=1
10
U(zo) =X A„Aj
m, j =1
r2 io
e
(a02 - 2a01) z'
0 c ■ + iz'd
m, j m, j
dz'+r12
e
(a02 - 2a01) z'
o c . + iz'd
m, j m, j
■dz'
(12)
(13)
(14)
The term
U ( zo) V ( zq)
takes into account the absorption in the medium and the diffraction. We
need to remind here that this equation is derived using only the quasilinear approximation. Then, the ratio Q can be expressed as:
Q =^Po
and
ko 1
0 2 Po Co2 r1
e
U ( zo)
V ( zo)
(15)
The measurement of the slope s permits to determine the value of the nonlinear parameter
Pq.
0
2. EXPERIMENTAL SET UP
The experimental set up is shown in figure 1. It consists of a tank (30*20*15 cm3) which contains the medium at the ambient temperature, a transmitter module and a system of reception. The ultrasonic probe is made of a ring functioning at 1 MHz with the hydrophone integrated in the same body (figure 1). The hydrophone is PZT of 0.4 mm of diameter. The transmitting transducer is supplied by a generator of signals associated with a power amplifier in pulse echo mode (width of 5 cycles and a repetition rate of 500 Hz).
The received signals are visualized on the oscilloscope and are transferred towards the computer via an IEEE connection. All devices are controlled by a software developed using HPVEE language. The interface is located at the position z0, which is the focal length of the
transmitter. The interface is made of a thin Mylar sheet allowing the ultrasound transmission to the second medium. The characteristics of the mediums studied are given in table 1.
Fig. 1. Experimental set up and ultrasound probe description
Table 1. Acoustical parameters in standard conditions
P (kg/m3) C0 (m/s) & a01 (Np/m/MHz)
water 992 1480 3.5 0.0265
ethanol 789 1229 5.3 0.09
glycerol 1239 1851 6 2.4
3. RESULTS
3.1. Transmitter calibration
Electrical characterization was carried thanks to the network analyzer HP4195. This allows us to measure the electrical impedance around the resonance frequency f0. Using a wattmeter
we can measure the real part of the supply power given to the transducer. Only one part of this energy is converted into the mechanical one and the challenge is to deduce it. Thus, we
measure the transverse distribution of the sound field in the near zone of the transmitter (z = zv = 2cm) . Figure 2 shows the beam pattern at this location for three different excitation
levels.
We assume that only the fundamental is produced in the near zone (z = 2 cm), the higher harmonics are still very lowers. So the axial incident pressure field can be developed using the linear theory as
where, 2a is the inner diameter and 2b is the external diameter of the ring (figure 1).
Let us consider VXi (r, zp) the electrical voltage measured by the hydrophone at the location
rj (xj, yj). The incident pressure field at this point can be expressed as
(16)
%(r, zp) = %i Vu(r, zp K
Then the initial pressure P0 can be obtained using equation
(17)
(18)
where, S is the effective surface of he transmitter S = n(b2 - a2). Using eqs. (13) and (14) gives
(19)
0,04
—*— P0=0,1825 Bar
P0=0,365 Bar
—a—P0=0,5475 Bar
0,03
! 0,02 pH
0
-30 -20 -10 0 10 20 30
Radial coordinate r, mm
Fig. 2. Beam pattern at z = 2 cm for different excitation level P0: P0=0.1825, P0=0.365, P0=0.5475 Bar
The results obtained are summarized in the table 2, for different excitation voltage Vt. The slope of the variation of P0 vs. Vt gives the sensitivity of the transducer (^e=0.0184 Bar/V).
Table 2. Initial pressure P0 vs. excitation voltage Vt
Vt (V) 12.5 25 37.5 50 72
P0 (Bar) 0.17 0.34 0.5 0.63 0.9
3.2. B/A Measurement
The variation of the ratio Q vs. P0 is shown in the figure 3, for the case of the reflection from the interface ethanol/water. We note a linear variation provided that the initial pressure level is less than 0.6 Bar. For higher than this limit we observe a saturation which is related the strong nonlinear parameter of ethanol. Limiting the pressure level to 0.6 Bar, the slope allows to deduce the nonlinear parameter /30 =6.3 for ethanol.
Fig. 3. The ratio Q vs. P0 in the case of ethanol/water interface
The results obtained for a reflection from water/glycerol interface are drawn in the figure
4. For P0 less than 0.2 Bar the signal is not enough to produce significant distortion. For P0 exceeding this limit the ratio Q shows a perfect linear variation. The slope gives j30 =3.33 for water.
The discrepancies in those results should be explained by:
• Measurement errors coming from the non alignment of the probe with the interface on the axis of propagation and the interface which is not plane because of ultrasound pressure on its surface.
• The theoretical expressions of the harmonics give the punctual fields however the hydrophone measures the average pressure on its surface.
Fig. 4. The ratio Q vs. P0 in the case of water/glycerol interface
CONCLUSIONS
An analytical model is presented for the fundamental and second harmonic generation associated with reflection. The model is based on the quasilinear theory and the superposition of Gaussian beams. Measurement of reflected harmonics from two interfaces; water/glycerol and ethanol/water; are performed and compared to the theoretical results. The ratio of the second harmonic and fundamental reflected allows deducing the nonlinear parameter of the first medium located before the interface. The experimental results demonstrate the validity of the proposed method for nonlinear ultrasound parameter measurement.
ACKNOWLEDGEMENTS
This work is supported by the Moroccan Ministry of education and scientific research (Grant program Protars II: P21/01).
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