- Email: [email protected]
A measurements in fluids
Ultrasonics 38 (2000) 284–291 www.elsevier.nl/locate/ultras
A novel phase locked cavity resonator for B/A measurements in fluids J.R. Davies a, *, J. Tapson a, B.J.P. Mortimer b a Department of Electrical Engineering, University of Cape Town, Rondebosch 7701, South Africa b School of Electrical Engineering, Cape Technikon, Box 652, Zonnebloem 8000, South Africa
Abstract A new technique for the measurement in fluids of the acoustic non-linearity parameter B/A is presented, together with measured B/A values for several fluids. The non-linearity parameter is measured by phase locking radial modes within a PZT cylinder. The system, which implements the isentropic phase technique, uses continuous wave phase locking to measure the change in sound velocity that is typically associated with a change in ambient pressure under constant entropy. The method provides a means of measuring B/A in vitro both accurately and simply without the typical problems involved in time-of-flight systems. Fluid samples can remain small due to the nature of the cavity resonator, so the system is well suited to small volume, biological samples. © 2000 Elsevier Science B.V. All rights reserved. Keywords: B/A; Cavity resonator; Non-linearity parameter; Phase locking
1. Introduction The acoustic non-linearity parameter B/A is a measure of the non-linearity of the equation of state for a fluid. Advancements in the fields of shockwaves [1] and biomedical ultrasonics have used B/A as a means of modelling finite amplitude propagation in fluids, as a prediction of propagation in biological tissue compositions [2,3]. Much research has focused on empirically determining B/A, with two prevalent methods standing out: the finite amplitude method [4] and the thermodynamic method [5]. The finite amplitude method uses the harmonic distortion of a propagating wave as a measure of B/A. The thermodynamic technique relies on an indirect approach by measuring the gross change in sound speed associated with changes in ambient pressure and temperature. The precision with which B/A can be determined using these techniques is typically 10% for the finite amplitude method and 5% for the thermodynamic method. Recently, the necessity has grown for a simple and accurate technique that can be used to measure B/A for small volume, biological preparations. It is the purpose of this paper to introduce such a technique based on a continuous wave, phase locked, cylindrical cavity resonator. * Corresponding author. Fax: +27-21-6503465. E-mail address: [email protected] (J.R. Davies)
The method prescribed for this technique is a variation of the thermodynamic method, known as the isentropic phase method [6 ]. This method is based upon the initial derivation of the non-linearity parameter which was formulated by Beyer [7] as follows:
A B
∂c =2r c 0 0 ∂p A
B
(1)
s where r is the ambient density for the medium and c 0 o is the infinitesimal sound velocity. The subscript s denotes the derivative of velocity with respect to pressure taken at constant entropy. Using the above expression, B/A can be measured by means of a pressure jump method [8], which enables an approximation of the isentropic process. This pressure jump method is usually taken over a small time frame (typically 2 s), necessitating a fast and precise measurement technique for determining the change in sound velocity. It is the purpose of this paper to describe a phase locked, continuous wave technique for accomplishing this without the need for complex interferometers and expensive electronic systems.
2. Continuous wave measurements Many velocimetric systems operate using a pulsed sound waveform. This introduces several common prob-
0041-624X/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 13 9 - 0
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
285
Fig. 1. The resonator cell, showing the isothermal jacket and the piezoelectric cylinder. The jacket acts as a pressure vessel during measurement. The cylinder and jacket form a cylindrical recess in which the liquid rests. The liquid temperature is measured by means of a platinum resistance thermometer, which protrudes down into the liquid (with no measurable effect on the resonant modes).
lems of pulse detection, most of which arise from the difficulty of detecting the onset of a pulse. This has been overcome previously by using phase detection of repetitive pulses, but many of the threshold uncertainties remain. These systems are particularly complicated by the existence of non-linearities in propagation, which mean that system parameters have to be recalibrated when different fluids are analysed. We have previously made use of phase lock techniques in continuous wave systems to eliminate threshold uncertainties, and to eliminate unwanted system non-linearities [9]. The basis of this technique is that time-of-flight is represented in terms of a continuous signal’s phase and frequency, rather than on the arrival of a pulse. Continuous phase and frequency measurements do not require any threshold detection, allowing time-of-flight measurements to be made to arbitrarily small levels of accuracy. The circuits required to perform phase locking have been widely used in telecommunications (as phase locked loops) and are well understood, simple in design, and usually available as integrated circuit modules. However, we have previously encountered near-field problems (as described by Zhang et al. [10]) when using phase locking in double-disk interferometers. In this paper we make use of a simple continuous wave resonant cavity, which avoids these problems.
3. The cavity resonator The heart of this system lies within an acoustic cell, which houses the cylindrical cavity resonator. The geometry of this arrangement can be seen in Fig. 1. Here, the cavity resonator is a piezoelectric ceramic (PZT-5A) cylinder, polarised in wall thickness and attached at one end to an isothermal jacket to form a cup. The cylinder is affixed with an isobutylene compound which provides a watertight seal, whilst maintaining a weak acoustic coupling to the thermal jacket (which is constructed from solid brass). The fixing is deliberately kept weak in order to reduce the effects of the damping typically associated with coupled structures. The cylinder used in this system had a resonant frequency of approximately 700 kHz; a typical admittance and phase plot of the empty cylinder can be seen in Figs. 2 and 3. Included in these figures, for comparison purposes, are the admittance and phase spectra of the cylinder when filled with water. The radial acoustic resonances are much stronger when coupled with water rather than air. We have developed a finite-element model of the system which suggests that these resonant peaks result from radial coupling of wall thickness modes [11]. This phenomenon was observed and characterised in a range of PZT cylinders of varying dimensions. The frequency interval of the radial modes was
286
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
Fig. 2. An admittance plot of the PZT cylinder (38 mm diameter) when empty and when full of water. The overall structure represents the fundamental wall thickness resonance. The satellite peaks represent the wall thickness resonance coupling to the radial cavity modes. Measurements are usually conducted away from this fundamental to avoid any residual effects such as frequency pulling [11].
Fig. 3. A phase plot of the PZT cylinder (38 mm diameter) when empty and when full of water. It can be seen that the satellite resonance phase changes have a very high gradient, making them ideal for automatic phase detection and measurement.
found to be constant, providing a simple method of calculating the infinitesimal sound velocity as follows: df c = D 0 dn
(2)
where df/dn is the change in frequency with mode
number, c is the velocity of sound and D is the inside o diameter of the cylinder. A typical graph displaying the velocities of sound for methanol, glycerine and water is displayed in Fig. 4. The thermal jacket in which the piezoelectric cylinder is held was designed, with the aid of finite element
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
287
Fig. 4. A graph displaying frequency vs. mode number for various liquids. The gradient of the graphs (df/dn) indicates their relative sound velocities.
analysis, to provide an isothermal enclosure for the fluid. The jacket was constructed to withstand an excess pressure of 1000 kPa and special attention was given to reduce any spurious rear-face radiation.
4. Experimental methodology A schematic of the complete system is shown in Fig. 5. This measurement system was designed to provide an uncertainty of less than 1% for B/A, using excess pressures no greater than 200 kPa. This meant that short-term temperature fluctuations within the sample fluid had to be reduced to less than 0.001°C. In order to achieve this requirement, the acoustic cell was immersed within a temperature controlled bath, capable of maintaining the prescribed temperature to within 0.01°C (external to the thermal jacket). In conjunction with the thermal jacket, which was designed to act as a low-pass filter with a time constant of 10 s, this provided the necessary isothermal stability. The temperature of the liquid was monitored using a calibrated platinum 1/10th DIN resistance thermometer. The temperature was measured in terms of resistance, using a six-digit (HP34401A) multimeter connected in a four-point configuration. The pressure system was designed to allow a pressure sweep from 0 to 200 kPa (gauge pressure). Two digital valves operated by a laboratory computer provided the necessary pressurisation and depressurisation of the cell. Pressurisation was obtained from a tank of compressed nitrogen, which was regulated to 200 kPa. The pressure
rise time of 2 s was necessary for the data collection and was introduced using a finely adjustable needle valve. A temperature compensated Motorola MPX2200AP silicon integrated transducer was used for determining the pressure. This transducer was calibrated before each measurement cycle and was located in close proximity to the cell. The phase locked resonance tracking system was designed to maintain a constant phase difference between the driving current and the voltage supplied to the piezoelectric cylinder. This electronic system provides a real time measurement of the change in sound velocity as a function of frequency. An HP53131A 12-digit frequency counter was used to measure the frequency, and was interfaced to the computer. With frequency, temperature and pressure logged by laboratory computer, a simple HPVEE program was written to automate the B/A measurement task. By manipulating Eq. (1), the non-linearity parameter may be expressed as a change in frequency associated with a change in pressure:
A B
Df =2r c2 0 0 A f Dp s 0
B
(3)
where f is the initial frequency and Df is the change in o frequency associated with a change in pressure. The value of p is measured separately before the measureo ment process using picnometry. The measurement procedure begins by filling the piezoelectric cavity completely with the sample fluid. The liquid is typically degassed and allowed to enter the
288
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
Fig. 5. A schematic of the complete measurement system.
cylinder under laminar flow, ensuring no microbubble accumulation. The cell is then immersed in a temperature controlled bath, and the entire system is left to reach a state of equilibrium at a prescribed temperature. The pressure sensor is zeroed and the pressure is increased to 180 kPa (gauge). Once thermal equilibrium is reached, the pressure is vented through the needle valve (see Fig. 6). During this venting process, the frequency is recorded (via GPIB Bus) by a computer (Fig. 7). Each venting process lasts on average 2 s, which is sufficient to accumulate 100 data points. The experiment is then repeated 50 times, over different modes, so that a weighted least square regression can be performed for the final determination of B/A.
5. Electronic system The electronic system used in this method is well known in electrical engineering and has been used extensively in microwave systems. The exceptional ‘Q’, or sharpness, of the radial acoustic modes (typically 5000) provides an excellent subject for mode locking, resulting in resolutions of typically 1 part in 107. A
circuit diagram for the electronic system is shown in Fig. 8. The general approach for this system was to lock the system drive frequency to a radial acoustic mode within the piezoelectric cylinder. To accomplish this, the current and voltage of the PZT transducer have to be phase locked to each other. The current supplied to the PZT cylinder was measured using a 1.8 V series resistance. The phase difference between the driving voltage and the current was then measured using an analogue multiplier (AD633), together with an active low pass filter. A stable voltage controlled oscillator (HP8656) was used to generate the driving sinusoid signal. The stability of this VCO was critical in determining the overall stability of the system. The phase measurement serves as input to a feedback loop controller, which amplifies it and adds an offset. The offset provides the ability to lock off quadrature, which was often required. This variable phase locking scheme allows the system to lock on a resonant mode with a phase setpoint anywhere between 20° and 160°. Since the acoustic resonance modes of the cylinder had a typical impedance of 1 V, a low impedance driving source was used (a 200 W, 1 MHz amplifier based on an Apex PA19 opamp). Low
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
289
Fig. 6. A typical frequency sweep of pressure vs. time. The full-scale horizontal axis is approximately 4 s.
Fig. 7. Capture of frequency vs. time associated with the depressurisation of the sample fluid.
voltages — generally less than 1 V rms — were used in order to reduce the unwanted effect of self-heating of the sample liquid. The system operates by maintaining an exact integer mode number within the cylindrical cavity, regardless of any phase velocity changes. Any changes in sound velocity are generated as a change in frequency, which is measured and displayed using the frequency counter. The bandwidth of the VCO was typically 5 kHz, with the centre frequency set to the resonant frequency of the selected radial mode at ambient pressure. In order
to jump from one mode to another, the centre frequency of the VCO was merely adjusted to that mode. This type of phase locking scheme is simple in construction and provides the required resolution of typically 0.1 Hz in 700 kHz.
6. Results Table 1 shows the measured results for four liquids, together with B/A values taken from the literature [12].
290
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
Fig. 8. A complete circuit diagram for the phase locked loop. The analogue multiplier (AD633) acts as the phase detector, with an offset which provides the ability to lock off quadrature. The feedback is provided with variable gain and an integrator time constant of 1 ms. Table 1 Measurements of the non-linearity parameter B/A. The table shows a comparison of measured B/A values with published values [12] Fluid
Temperature (°C )
Density r (kg/m3)
Velocity (m/s)
B/A±DB/A
B/A literature
Water FC43 FC75 CCl 4
30 30 30 25
995.7 1861 1768 1002
1508.8 638 567 1519
5.13±0.02 13.19±0.023 12.83±0.021 8.34±0.04
5.22, 5.14 12.85 (20°C ) 12.19 (20°C ) 8.70 (25°C )
The results of this experiment generally concur with those found in the literature, and an estimated measurement error for the system is set as 0.48%. Each sample run acquired 100 measurement readings of pressure vs. frequency. The average linearity of the slope df/dp is estimated to have r2=0.9997. The two slow sound speed fluorocarbons which have large B/A values provided a means of testing the ability of the phase lock system to track large changes in velocity. The B/A results obtained for FC43 and FC75 were both satisfactory and had a standard deviation of ±0.02.
7. Conclusions The measurement systems together with the phase locking technique described in this paper have been used to adequately determine the non-linearity parameter B/A in several fluids. The system not only provides an easier approach to previous time-of-flight techniques,
but also has an average measurement accuracy of less than 1%. The phase locking scheme set out provides a simple, generic technique for phase locking continuous radial waves within a piezoelectric cylinder. The use of the cylindrical cavity is described as a novel technique for determining B/A in small sample volumes. Furthermore, the cylindrical cavity provides a means of determining the infinitesimal sound velocity of a liquid, with the only calibration being that of the diameter. This system should be well suited for small volume biological measurements of B/A in vitro.
References [1] B. Mortimer, L. Felthun, B. Skews, Shock Waves 8 (1998) 63–69. [2] X. Gong, R. Feng, C. Zhu, T. Shi, J. Acoust. Soc. Am. 76 (1984) 949–950. [3] P. Jiang, E. Everbach, R. Apfel, Ultrasound Med. Biol. 17 (1991) 829–838.
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291 [4] W. Cobb, J. Acoust. Soc. Am. 73 (1983) 1525–1531. [5] W. Law, L. Frizzell, F. Dunn, J. Acoust. Soc. Am. 74 (1983) 1295–1297. [6 ] E. Everbach, R. Apfel, J. Acoust. Soc. Am. 98 (1995) 3428–3438. [7] R. Beyer, J. Acoust. Soc. Am. 32 (1960) 719–721. [8] C. Sehgal, R. Bahn, J. Greenleaf, J. Acoust. Soc. Am. 76 (1984) 1023–1029.
291
[9] J. Davies, J. Tapson, B. Mortimer, J. Acoust. Soc. Am. 103 (5:2) (1998) 3080. [10] J. Zhang, F. Dunn, J. Acoust. Soc. Am. 89 (1991) 73–79. [11] J. Bell et al., in preparation. [12] C. Everbach, Tissue composition determination via measurement of the acoustic non-linearity parameter. Ph.D. dissertation, Yale University, 1989, pp. 13–16.
A novel phase locked cavity resonator for B/A measurements in fluids J.R. Davies a, *, J. Tapson a, B.J.P. Mortimer b a Department of Electrical Engineering, University of Cape Town, Rondebosch 7701, South Africa b School of Electrical Engineering, Cape Technikon, Box 652, Zonnebloem 8000, South Africa
Abstract A new technique for the measurement in fluids of the acoustic non-linearity parameter B/A is presented, together with measured B/A values for several fluids. The non-linearity parameter is measured by phase locking radial modes within a PZT cylinder. The system, which implements the isentropic phase technique, uses continuous wave phase locking to measure the change in sound velocity that is typically associated with a change in ambient pressure under constant entropy. The method provides a means of measuring B/A in vitro both accurately and simply without the typical problems involved in time-of-flight systems. Fluid samples can remain small due to the nature of the cavity resonator, so the system is well suited to small volume, biological samples. © 2000 Elsevier Science B.V. All rights reserved. Keywords: B/A; Cavity resonator; Non-linearity parameter; Phase locking
1. Introduction The acoustic non-linearity parameter B/A is a measure of the non-linearity of the equation of state for a fluid. Advancements in the fields of shockwaves [1] and biomedical ultrasonics have used B/A as a means of modelling finite amplitude propagation in fluids, as a prediction of propagation in biological tissue compositions [2,3]. Much research has focused on empirically determining B/A, with two prevalent methods standing out: the finite amplitude method [4] and the thermodynamic method [5]. The finite amplitude method uses the harmonic distortion of a propagating wave as a measure of B/A. The thermodynamic technique relies on an indirect approach by measuring the gross change in sound speed associated with changes in ambient pressure and temperature. The precision with which B/A can be determined using these techniques is typically 10% for the finite amplitude method and 5% for the thermodynamic method. Recently, the necessity has grown for a simple and accurate technique that can be used to measure B/A for small volume, biological preparations. It is the purpose of this paper to introduce such a technique based on a continuous wave, phase locked, cylindrical cavity resonator. * Corresponding author. Fax: +27-21-6503465. E-mail address: [email protected] (J.R. Davies)
The method prescribed for this technique is a variation of the thermodynamic method, known as the isentropic phase method [6 ]. This method is based upon the initial derivation of the non-linearity parameter which was formulated by Beyer [7] as follows:
A B
∂c =2r c 0 0 ∂p A
B
(1)
s where r is the ambient density for the medium and c 0 o is the infinitesimal sound velocity. The subscript s denotes the derivative of velocity with respect to pressure taken at constant entropy. Using the above expression, B/A can be measured by means of a pressure jump method [8], which enables an approximation of the isentropic process. This pressure jump method is usually taken over a small time frame (typically 2 s), necessitating a fast and precise measurement technique for determining the change in sound velocity. It is the purpose of this paper to describe a phase locked, continuous wave technique for accomplishing this without the need for complex interferometers and expensive electronic systems.
2. Continuous wave measurements Many velocimetric systems operate using a pulsed sound waveform. This introduces several common prob-
0041-624X/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 13 9 - 0
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
285
Fig. 1. The resonator cell, showing the isothermal jacket and the piezoelectric cylinder. The jacket acts as a pressure vessel during measurement. The cylinder and jacket form a cylindrical recess in which the liquid rests. The liquid temperature is measured by means of a platinum resistance thermometer, which protrudes down into the liquid (with no measurable effect on the resonant modes).
lems of pulse detection, most of which arise from the difficulty of detecting the onset of a pulse. This has been overcome previously by using phase detection of repetitive pulses, but many of the threshold uncertainties remain. These systems are particularly complicated by the existence of non-linearities in propagation, which mean that system parameters have to be recalibrated when different fluids are analysed. We have previously made use of phase lock techniques in continuous wave systems to eliminate threshold uncertainties, and to eliminate unwanted system non-linearities [9]. The basis of this technique is that time-of-flight is represented in terms of a continuous signal’s phase and frequency, rather than on the arrival of a pulse. Continuous phase and frequency measurements do not require any threshold detection, allowing time-of-flight measurements to be made to arbitrarily small levels of accuracy. The circuits required to perform phase locking have been widely used in telecommunications (as phase locked loops) and are well understood, simple in design, and usually available as integrated circuit modules. However, we have previously encountered near-field problems (as described by Zhang et al. [10]) when using phase locking in double-disk interferometers. In this paper we make use of a simple continuous wave resonant cavity, which avoids these problems.
3. The cavity resonator The heart of this system lies within an acoustic cell, which houses the cylindrical cavity resonator. The geometry of this arrangement can be seen in Fig. 1. Here, the cavity resonator is a piezoelectric ceramic (PZT-5A) cylinder, polarised in wall thickness and attached at one end to an isothermal jacket to form a cup. The cylinder is affixed with an isobutylene compound which provides a watertight seal, whilst maintaining a weak acoustic coupling to the thermal jacket (which is constructed from solid brass). The fixing is deliberately kept weak in order to reduce the effects of the damping typically associated with coupled structures. The cylinder used in this system had a resonant frequency of approximately 700 kHz; a typical admittance and phase plot of the empty cylinder can be seen in Figs. 2 and 3. Included in these figures, for comparison purposes, are the admittance and phase spectra of the cylinder when filled with water. The radial acoustic resonances are much stronger when coupled with water rather than air. We have developed a finite-element model of the system which suggests that these resonant peaks result from radial coupling of wall thickness modes [11]. This phenomenon was observed and characterised in a range of PZT cylinders of varying dimensions. The frequency interval of the radial modes was
286
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
Fig. 2. An admittance plot of the PZT cylinder (38 mm diameter) when empty and when full of water. The overall structure represents the fundamental wall thickness resonance. The satellite peaks represent the wall thickness resonance coupling to the radial cavity modes. Measurements are usually conducted away from this fundamental to avoid any residual effects such as frequency pulling [11].
Fig. 3. A phase plot of the PZT cylinder (38 mm diameter) when empty and when full of water. It can be seen that the satellite resonance phase changes have a very high gradient, making them ideal for automatic phase detection and measurement.
found to be constant, providing a simple method of calculating the infinitesimal sound velocity as follows: df c = D 0 dn
(2)
where df/dn is the change in frequency with mode
number, c is the velocity of sound and D is the inside o diameter of the cylinder. A typical graph displaying the velocities of sound for methanol, glycerine and water is displayed in Fig. 4. The thermal jacket in which the piezoelectric cylinder is held was designed, with the aid of finite element
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
287
Fig. 4. A graph displaying frequency vs. mode number for various liquids. The gradient of the graphs (df/dn) indicates their relative sound velocities.
analysis, to provide an isothermal enclosure for the fluid. The jacket was constructed to withstand an excess pressure of 1000 kPa and special attention was given to reduce any spurious rear-face radiation.
4. Experimental methodology A schematic of the complete system is shown in Fig. 5. This measurement system was designed to provide an uncertainty of less than 1% for B/A, using excess pressures no greater than 200 kPa. This meant that short-term temperature fluctuations within the sample fluid had to be reduced to less than 0.001°C. In order to achieve this requirement, the acoustic cell was immersed within a temperature controlled bath, capable of maintaining the prescribed temperature to within 0.01°C (external to the thermal jacket). In conjunction with the thermal jacket, which was designed to act as a low-pass filter with a time constant of 10 s, this provided the necessary isothermal stability. The temperature of the liquid was monitored using a calibrated platinum 1/10th DIN resistance thermometer. The temperature was measured in terms of resistance, using a six-digit (HP34401A) multimeter connected in a four-point configuration. The pressure system was designed to allow a pressure sweep from 0 to 200 kPa (gauge pressure). Two digital valves operated by a laboratory computer provided the necessary pressurisation and depressurisation of the cell. Pressurisation was obtained from a tank of compressed nitrogen, which was regulated to 200 kPa. The pressure
rise time of 2 s was necessary for the data collection and was introduced using a finely adjustable needle valve. A temperature compensated Motorola MPX2200AP silicon integrated transducer was used for determining the pressure. This transducer was calibrated before each measurement cycle and was located in close proximity to the cell. The phase locked resonance tracking system was designed to maintain a constant phase difference between the driving current and the voltage supplied to the piezoelectric cylinder. This electronic system provides a real time measurement of the change in sound velocity as a function of frequency. An HP53131A 12-digit frequency counter was used to measure the frequency, and was interfaced to the computer. With frequency, temperature and pressure logged by laboratory computer, a simple HPVEE program was written to automate the B/A measurement task. By manipulating Eq. (1), the non-linearity parameter may be expressed as a change in frequency associated with a change in pressure:
A B
Df =2r c2 0 0 A f Dp s 0
B
(3)
where f is the initial frequency and Df is the change in o frequency associated with a change in pressure. The value of p is measured separately before the measureo ment process using picnometry. The measurement procedure begins by filling the piezoelectric cavity completely with the sample fluid. The liquid is typically degassed and allowed to enter the
288
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
Fig. 5. A schematic of the complete measurement system.
cylinder under laminar flow, ensuring no microbubble accumulation. The cell is then immersed in a temperature controlled bath, and the entire system is left to reach a state of equilibrium at a prescribed temperature. The pressure sensor is zeroed and the pressure is increased to 180 kPa (gauge). Once thermal equilibrium is reached, the pressure is vented through the needle valve (see Fig. 6). During this venting process, the frequency is recorded (via GPIB Bus) by a computer (Fig. 7). Each venting process lasts on average 2 s, which is sufficient to accumulate 100 data points. The experiment is then repeated 50 times, over different modes, so that a weighted least square regression can be performed for the final determination of B/A.
5. Electronic system The electronic system used in this method is well known in electrical engineering and has been used extensively in microwave systems. The exceptional ‘Q’, or sharpness, of the radial acoustic modes (typically 5000) provides an excellent subject for mode locking, resulting in resolutions of typically 1 part in 107. A
circuit diagram for the electronic system is shown in Fig. 8. The general approach for this system was to lock the system drive frequency to a radial acoustic mode within the piezoelectric cylinder. To accomplish this, the current and voltage of the PZT transducer have to be phase locked to each other. The current supplied to the PZT cylinder was measured using a 1.8 V series resistance. The phase difference between the driving voltage and the current was then measured using an analogue multiplier (AD633), together with an active low pass filter. A stable voltage controlled oscillator (HP8656) was used to generate the driving sinusoid signal. The stability of this VCO was critical in determining the overall stability of the system. The phase measurement serves as input to a feedback loop controller, which amplifies it and adds an offset. The offset provides the ability to lock off quadrature, which was often required. This variable phase locking scheme allows the system to lock on a resonant mode with a phase setpoint anywhere between 20° and 160°. Since the acoustic resonance modes of the cylinder had a typical impedance of 1 V, a low impedance driving source was used (a 200 W, 1 MHz amplifier based on an Apex PA19 opamp). Low
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
289
Fig. 6. A typical frequency sweep of pressure vs. time. The full-scale horizontal axis is approximately 4 s.
Fig. 7. Capture of frequency vs. time associated with the depressurisation of the sample fluid.
voltages — generally less than 1 V rms — were used in order to reduce the unwanted effect of self-heating of the sample liquid. The system operates by maintaining an exact integer mode number within the cylindrical cavity, regardless of any phase velocity changes. Any changes in sound velocity are generated as a change in frequency, which is measured and displayed using the frequency counter. The bandwidth of the VCO was typically 5 kHz, with the centre frequency set to the resonant frequency of the selected radial mode at ambient pressure. In order
to jump from one mode to another, the centre frequency of the VCO was merely adjusted to that mode. This type of phase locking scheme is simple in construction and provides the required resolution of typically 0.1 Hz in 700 kHz.
6. Results Table 1 shows the measured results for four liquids, together with B/A values taken from the literature [12].
290
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291
Fig. 8. A complete circuit diagram for the phase locked loop. The analogue multiplier (AD633) acts as the phase detector, with an offset which provides the ability to lock off quadrature. The feedback is provided with variable gain and an integrator time constant of 1 ms. Table 1 Measurements of the non-linearity parameter B/A. The table shows a comparison of measured B/A values with published values [12] Fluid
Temperature (°C )
Density r (kg/m3)
Velocity (m/s)
B/A±DB/A
B/A literature
Water FC43 FC75 CCl 4
30 30 30 25
995.7 1861 1768 1002
1508.8 638 567 1519
5.13±0.02 13.19±0.023 12.83±0.021 8.34±0.04
5.22, 5.14 12.85 (20°C ) 12.19 (20°C ) 8.70 (25°C )
The results of this experiment generally concur with those found in the literature, and an estimated measurement error for the system is set as 0.48%. Each sample run acquired 100 measurement readings of pressure vs. frequency. The average linearity of the slope df/dp is estimated to have r2=0.9997. The two slow sound speed fluorocarbons which have large B/A values provided a means of testing the ability of the phase lock system to track large changes in velocity. The B/A results obtained for FC43 and FC75 were both satisfactory and had a standard deviation of ±0.02.
7. Conclusions The measurement systems together with the phase locking technique described in this paper have been used to adequately determine the non-linearity parameter B/A in several fluids. The system not only provides an easier approach to previous time-of-flight techniques,
but also has an average measurement accuracy of less than 1%. The phase locking scheme set out provides a simple, generic technique for phase locking continuous radial waves within a piezoelectric cylinder. The use of the cylindrical cavity is described as a novel technique for determining B/A in small sample volumes. Furthermore, the cylindrical cavity provides a means of determining the infinitesimal sound velocity of a liquid, with the only calibration being that of the diameter. This system should be well suited for small volume biological measurements of B/A in vitro.
References [1] B. Mortimer, L. Felthun, B. Skews, Shock Waves 8 (1998) 63–69. [2] X. Gong, R. Feng, C. Zhu, T. Shi, J. Acoust. Soc. Am. 76 (1984) 949–950. [3] P. Jiang, E. Everbach, R. Apfel, Ultrasound Med. Biol. 17 (1991) 829–838.
J.R. Davies et al. / Ultrasonics 38 (2000) 284–291 [4] W. Cobb, J. Acoust. Soc. Am. 73 (1983) 1525–1531. [5] W. Law, L. Frizzell, F. Dunn, J. Acoust. Soc. Am. 74 (1983) 1295–1297. [6 ] E. Everbach, R. Apfel, J. Acoust. Soc. Am. 98 (1995) 3428–3438. [7] R. Beyer, J. Acoust. Soc. Am. 32 (1960) 719–721. [8] C. Sehgal, R. Bahn, J. Greenleaf, J. Acoust. Soc. Am. 76 (1984) 1023–1029.
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