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A new design of automated piezoelectric composite oscillator technique
Materials Science and Engineering A 442 (2006) 532–537
A new design of automated piezoelectric composite oscillator technique S. Kustov a,∗ , S. Golyandin b , A. Ichino c , G. Gremaud c a
Departament de F´ısica, Universitat de les Illes Balears, cra de Valldemossa km. 7.5, 07122 Palma de Mallorca, Spain b A.F. Ioffe Physico-technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia c Institut de Physique de Mati` ere Complexe, Ecole Polytechnique F´ed´erale de Lausanne, CH 1015, Lausanne, Switzerland Received 10 August 2005; received in revised form 24 January 2006; accepted 9 February 2006
Abstract We describe the latest design of the automated piezoelectric ultrasonic composite oscillator technique, which has traditionally been used for the measurements of elastic and anelastic properties of solids at frequencies of 70–140 kHz in a continuous positive feedback mode. The new equipment features several substantial advantages, as compared to previous constructions, owing to the application of up-to-date data acquisition units and microelectronics components. © 2006 Elsevier B.V. All rights reserved. Keywords: Acoustic methods; Internal friction; Young’s modulus
1. Introduction Transducer systems to induce resonant oscillations include piezoelectric, magnetostrictive, capacitive, magnetic, eddy current, and electromotive units [1,2]. Piezoelectric units in turn use piezoceramic or quartz transducers. Taking into account a variety of requirements for an acoustic measuring system, such as time and temperature stability, linearity, precision, strain amplitude range, background damping introduced by transducers, it has been concluded that the quartz resonator composite oscillator technique has many advantages over others [2,3]. Following initial reports on application of quartz resonators to measurements of elastic and anelastic properties of solids [3,4], a detailed theory of the composite oscillator technique has been developed for longitudinal [5], torsional [6] and flexural [7] modes of oscillations. This technique is being used extensively for decades; see [8–18] for some examples. In [19] we reported a brief sketch of an automated experimental setup. In the meantime, using the same principles as in [19], new electronics have been designed which offer several substantial advantages: • adaptive measurement procedure, when the time-resolution depends upon current values of registered absorption of ultrasound; ∗
Corresponding author. Fax: +34 971173426. E-mail address: [email protected] (S. Kustov).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.02.230
• automatic selection of an appropriate measurement range; • combined analog–digital control of drive voltage (strain amplitude), which substantially simplifies the electronics design; • resonant conditions for oscillations, checked and maintained at each measuring point quite rapidly owing to a procedure of polynomial fitting of the resonant curve; • wide application of highly integrated high-precision analog circuits instead of digital ones, resulting in simple and compact design. We present below a more detailed description of this new design together with examples illustrating its potential. 2. Principles of operation Fig. 1 shows schematically the assembled three-component oscillator and the block diagram of the electronics. The oscillator incorporates a sample, glued to two identical 18.5◦ X-cut ␣-quartz crystals of rectangular cross section for drive and gauge purposes, plated with electrodes on two sides and supported in their centers (for excitation of the fundamental/odd harmonics of longitudinal oscillations). The quartz crystals and the specimen are cemented together at their ends, and the assembly is driven by an ac voltage Ud applied to the drive component. The voltage across the gauge component, Ug , is used to monitor the induced oscillatory strain and to maintain
S. Kustov et al. / Materials Science and Engineering A 442 (2006) 532–537
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Fig. 1. Schematic representation of the composite oscillator and the electronic circuit.
continuous resonant oscillations via a positive feedback loop. The length of the specimen must be selected in such a way that its fundamental resonance frequency matches the one for the drive and gauge transducers. For longitudinal oscillations, the expressions for the total decrement of the composite oscillator, Q−1 (t), and peak value of the strain amplitude of its ith element, εmax 11 (i), in terms of voltages across drive and gauge crystals, have been obtained by Robinson and Egdar [5]: Ud 8Zm l2 d311 2 Ud Q−1 (t) = = Kδ (1a) ω(t)m(t) S1111 Ug Ug √ 2π S1111 Ug max ε11 (i)λ(i) = = K Ug (1b) jω l2 d311 Zm where Zm is the electrical impedance across the gauge component (including cables and input impedance of the measuring circuit), λ(i) is the wavelength of the longitudinal wave in component i, m(t) and ω(t) are the total mass and the angular resonant frequency of the oscillator (ω(t) ≈ ω(i)), respectively, S1111 and d311 are the elastic compliance and the piezoelectric modulus of the transducers, and l2 is the width of the plated side of the transducer. The damping and the resonant frequency of the specimen can be determined from that of the total oscillator (provided that the mass, the resonant frequency and the damping of quartz transducers are known) using the following equations [5]: m(t)ω(t)2 = m(i)ω(i)2 (2a) i
m(t)Q−1 (t) =
m(i)Q−1 (i)
(2b)
i
where m(i), ω(i) and Q−1 (i) is the mass, the resonant frequency and the damping of the ith component of the oscillator, respectively. Q−1 of high-quality transducers is usually substantially lower than that of the specimen, therefore Q−1 of the specimen can be derived with high precision from the measured total damping of the oscillator.
The Young’s modulus of the sample is conventionally determined from the fundamental angular frequency of longitudinal oscillations of a rod-shaped sample: E = ρω2 l12 /π2 , where ρ and l1 are the density and the length of the sample. Eqs. (1) and (2) show that measurements of the decrement of the sample, oscillatory strain amplitude and Young’s modulus are reduced to measurements of two electrical voltages and resonant frequency of the oscillator, provided that (i) the system is at resonance; (ii) the mass, resonant frequency and Q−1 of the oscillator without the sample are known; (iii) the coefficients K␦ and K are known (these coefficients depend on the geometry of the transducers, fundamental characteristics of quartz crystals and the electrical impedance Zm of the measuring circuit [5]). The electronics is controlled via a general-purpose data acquisition PCI card. The electronic circuitry performs two basic functions: (i) maintaining resonant conditions over a wide range of oscillatory strain amplitudes and (ii) conditioning Ud and Ug signals. Therefore, the setup includes two (nearly) identical channels, Ud - and Ug -channels. The Ug channel forms a part of the positive feedback loop. Both channels use symmetrical input/output circuits to minimize the electric pickup between drive and gauge voltage signals. This is especially important for high damping values when the Ud /Ug ratio becomes high (see Eq. (1)). Apart from the symmetric preamplifier/divider connected directly to the quartz crystals, each measuring channel includes • a line of three digitally-controlled amplifiers, which is supplied with a circuitry for automatic range selection (four-step automatic gain control units); • a one-chip hybrid eight-pole octave Butterworth filter with the center frequency of 100 kHz to improve signal-to-noise ratio and thus to increase the dynamic range of measurements; • a highly linear integrated RMS converter. The control of each measuring channel requires three TTL lines (inputs of subrange status) and one A/D converter (for the analogue voltage of RMS value of the Ud (Ug ) signal).
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A key element of the positive feedback loop is the automatic gain control amplifier and phase-shifter units. The AGC amplifier provides a constant ac voltage at the output, independent of the gauge voltage across the transducer, which serves afterwards in control of the drive voltage level. The analog phase-shifter is used to control the overall phase shift in the positive feedback loop and, thus, to maintain the oscillator at resonance. The latter condition corresponds to a purely resistive impedance of the entire electromechanical circuit (including drive and gauge components and measuring circuit) and, hence, to a maximum voltage across the gauge quartz transducer. The phase of the feedback loop is controlled by one analog line. Before each measurement, the resonant condition (i.e. the maximum response signal) is verified and maintained. To minimize the time for this procedure, the following algorithm is adopted. Initially and only once, before a measurement is started, the “global search” of the resonance is performed with 1000 steps over the total phase shift range from 0 to π rad. Afterwards, only a “fine tuning” procedure is performed in the close proximity to the resonance. To further minimize the tuning time, only a few points of the resonant curve are measured (typically, five points are sufficient). Afterwards, the position of the maximum response is found from a polynomial fitting to these few points. The “fine tuning” procedure is repeated, if necessary, until the resonant maximum is close to the middle of the tuning range and only afterwards a measurement is carried out. An essential ingredient of the entire system is the drive voltage control circuitry, which includes a digitally-controlled four-step resistive divider and an analog gain control module based on a high-precision analog multiplier. These are key elements which control the drive voltage level and, thus, the oscillatory strain amplitude over the range exceeding 80 dB with a resolution of better than 0.5%. The last figure gives an estimate of the strain amplitude resolution, when the strain amplitude should be controlled or, as a particular case, kept constant. In a single measurement of the strain amplitude dependence, the set-up controls the excitation voltage in the range from ∼20 to 280 VRMS . For a typical mass (m ∼ 0.5 g) and damping of a specimen (Q−1 ∼ 10−4 to 10−2 ) these values ensure oscillations with strain amplitudes ranging typically from 10−7 to 2 × 10−4 , sufficient usually for the majority of materials science problems. The low limit for strain amplitudes of the setup is restricted by the noise level of the output power amplifier. The gauge signal contains less noise since it is picked up from the quartz transducer and this procedure is equivalent to rather narrow band filtering of the drive voltage. In the present set-up, the output noise of the amplifier (with the open positive feedback loop) is compensated in the software, assuming that this noise is not correlated with the essential signal. The upper limit of the strain amplitudes is determined either by the mechanical strength of the quartz transducers (for the case of low total damping), or by the maximum drive voltage provided by the power amplifier (for the case of high damping). According to Robinson and Edgar [5], the practical range of strain amplitudes can be extended from 10−12 to 5 × 10−4 .
The basic differences of the present set-up from the previous design [19] are as follows. (a) Extensive application of analog integrated circuits with analog control instead of digital control. This makes the circuitry simple, compact and reliable. For instance, only four analog (two D/A and two A/D channels) and 10 digital TTL lines are required for a complete control of the equipment. Therefore, the control can be performed by means of a standard data-acquisition PCI card. (b) The present system is adaptive: temporal resolution of the equipment depends on the current values of measured damping. In the composite oscillator technique, for not too high values of damping, the properties of the oscillator (specimen) limit the temporal resolution, because each step-like change of the drive voltage leads to a transient in the time dependence of strain amplitude. If the step is small and the change of the decrement during this transient is neglected, the time constant τ of the transient is simply given as τ = T/(πQ−1 (t)), where T is the period of oscillation. Thus, the higher is the damping, the faster can the measurements be performed. This feature of the equipment is especially important if the decrement values change orders of magnitudes and if the decrement is time-dependent. (c) Resonant conditions are checked and maintained at each measuring point. This procedure is necessary since the subrange switch introduces certain unwanted phase shifts, but can be done rapidly owing to the polynomial fitting of the resonant curve. The software developed uses LabVIEW. It includes rather simple measurements of time and strain amplitude dependence as well as more complicated software which allows for simultaneous measurements of temperature spectra of the decrement and Young’s modulus for two values of strain amplitudes and of the strain amplitude dependence of the same parameters during single heating/cooling scan. The two stabilized strain amplitudes usually fall within strain–amplitude-independent and strain–amplitude-dependent ranges. They are stabilized in turn for each temperature step and the measurements are taken for each value of the strain amplitudes. Since the measurements are fast (as compared to the temperature variation rate), such two measurements can be regarded as referring to the same temperature. The spectra registered for the low strain amplitude value yields directly the strain–amplitude-independent damping, whereas the difference between the spectra at high and low strain amplitudes yields the temperature spectra of the strain–amplitude-dependent damping. Similar procedure can be applied to measurements of the Young’s modulus for low and high strain amplitudes, yielding temperature spectra of the amplitude-dependent modulus defect. In addition to the temperature spectra measurements, the strain amplitude dependence of decrement and modulus defect can be registered at any temperature, interrupting the temperature spectra measurement. The time to measure the strain amplitude dependence (containing about 100 experimental points) is rather short, typically about 1 min, which corresponds to the
S. Kustov et al. / Materials Science and Engineering A 442 (2006) 532–537
temperature variation for typical heating–cooling rates of only 1–2 K. This allows one to measure, in the majority of applications, the strain amplitude dependence at a certain temperature with a precision of about 1 K. After measurements of the strain amplitude dependence, the software returns to the registration of the temperature spectra. Problems which can be encountered in assembling the oscillator and in measurements of temperature spectra are mostly related to the selection of adhesive and purity of quartz transducers, as discussed earlier [19,20]. The basic technical characteristics of the setup are as follows: • the frequency range 70–140 kHz; • the maximum strain amplitude typically not less than 2 × 10−4 ; • the noise level for strain amplitude channel (strain amplitude value referring to quartz transducers)—3 × 10−9 ;
Fig. 2. Strain amplitude dependence of the total decrement, δ (a) and its strain–amplitude-dependent part, δh (b), for martensitic samples of polycrystalline binary NiTi, and of Cu-based alloys subjected to different heat treatments: quenching from homogenization temperatures into the martensitic state (Q) and step-quenching with subsequent -phase ageing (SQ).
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• the maximum drive voltage 280 VRMS ; • the noise level for the drive voltage channel—5 mV with the booster amplifier ON, 0.5 mV for the measuring channel alone.
3. Examples of application Fig. 2(a) shows examples of strain amplitude dependence of the total logarithmic decrement δ registered for some selected martensitic alloys at room temperature. We notice here that the strain amplitude dependence can be registered over a wide range of strain amplitudes even for the background damping levels differing nearly two orders of magnitude (NiTi and Cu–Zn–Al–B alloy in the quenched state). It is interesting to remark that the strain amplitude dependence of the decrement is rather weak in NiTi, as compared to the level of background damping: the amplitude-dependent part of the damping δh is only around 50% of the background amplitude-independent damping δi , Fig. 2(a). Nevertheless, the high precision and linearity of the measuring
Fig. 3. Effect of time of martensite ageing elapsed after the heat treatment of step-quenched Cu–21.3Zn–5.4Al–0.05B polycrystalline alloy on the strain–amplitude-dependent decrement, δh (a) and modulus defect, ( E/E)h (b).
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circuits allows one to follow small variations of the non-linear part of the decrement, δh = δ − δi , over more than two orders of magnitude, Fig. 2(b). Fig. 2(a) exemplifies also the absence of a strain amplitude hysteresis in NiTi and step-quenched Cu–Al–Ni alloys immediately after the heat treatment, but its existence in Cu–Zn–Al–B samples prepared otherwise. The presence of a strain amplitude hysteresis in the last case is a perfect indication of the mobility of point-like defects. Finally, Fig. 2(b) shows that the useful range of studying the amplitude-dependent decrement covers the range of around five orders of magnitude. Fig. 3 depicts the effect of ageing after the heat treatment on the strain amplitude-dependent components of the decrement and the modulus defect for a step-quenched Cu–Zn–Al–B martensitic alloy. We emphasize first that the initial measurement was made only 10 min after the heat treatment. This time is found to be sufficient to assemble the oscillator and to perform
the measurement. Second, we mention the dynamic range of measurements: the values of the decrement and modulus defect cover typically four orders of magnitude, allowing one to analyse various stages in the strain amplitude dependence and their variations with ageing. Finally, Fig. 4 gives an example of a further analysis of the data from Fig. 3. Fig. 4(a) shows the stress amplitude dependence of the anelastic strain amplitude, εan (σ 0 ), derived from the stress amplitude dependence of the amplitude-dependent modulus defect, ( E/E)h (σ 0 ): εan (σ 0 ) = ε0 × ( E/E)h (σ 0 ). The dynamic range of the anelastic strain amplitude reaches nearly seven orders of magnitude. The effect of ageing on anelastic strain amplitude, Fig. 4(a), has been interpreted as due to the shifts to higher strain amplitudes (i) of the entire strain amplitude dependences and (ii) of the stage of the rapid increase of the anelasticity [21]. The origins of these two shifts were supposed to be homogeneous atomic reordering, and growth of the pinning atmospheres around partial dislocations in faulted martensites [21]. Fig. 4(b) represents the result of scaling of the curves from Fig. 4(a) along the stress axis, with the scaling coefficient for each curve yielding the kinetics of the reordering. The shift of the second stage of the strain amplitude dependence, Fig. 4(b), yields, in turn, the kinetics of the pinning process [21]. 4. Conclusion We have reported a design of automated resonant composite oscillator technique, which fully uses the advantages inherent in this acoustic method: • measurements cover a wide range of damping and strain amplitude: no mechanical parts are involved, resulting in a low level of background damping; • measurements are fast: this property is advantageous for various in situ experiments and is important for studying transient phenomena; • measurements are simple and accurate: in practice, only two voltages across the transducers need to be measured; • measurements are detailed: practically continuous curves of damping or modulus versus strain amplitude can be registered; • resonant conditions can be easily maintained: positive feedback in a closed-loop configuration holds the oscillator at resonance. Acknowledgement SK acknowledges Ram´on y Cajal Program (Spain). References
Fig. 4. Stress amplitude dependence of anelastic strain amplitude εan for stepquenched samples of Cu–21.3Zn–5.4Al–0.05B alloy subjected to different martensite ageing periods after the heat treatment (a), and the same stress amplitude dependence reduced to the reference curve corresponding to the sample subjected to 0.6 ks of martensite ageing (b). Coefficients for scaling the curves in (a) along stress axis are 0.71, 0.56, 0.43, 0.29 for martensite ageing periods of 1.8, 3.6, 9.6 and 166 ks, respectively; data from [21].
[1] [2] [3] [4] [5]
A.S. Nowick, Prog. Met. Phys. 4 (1953) 1–70. S.D. Devine, W.H. Robinson, Adv. Magn. Reson. 10 (1982) 53–117. J. Marx, Rev. Sci. Instrum. 22 (1951) 503–509. S.L. Quimby, Phys. Rev. 25 (1925) 558–573. W.H. Robinson, A. Edgar, IEEE Trans. Sonics Ultrasonics SU-21 (1974) 98–105.
S. Kustov et al. / Materials Science and Engineering A 442 (2006) 532–537 [6] W.H. Robinson, S.H. Carpenter, J.L. Tallon, J. Appl. Phys. 45 (1978) 1975–1981. [7] S.D. Devine, W.H. Robinson, IEEE Trans. Ultrason. Ferroelectricity Frequency Contr. 45 (1998) 11–22. [8] T.A. Read, Trans. AIME 143 (1941) 30–41. [9] A.S. Nowick, Phys. Rev. 80 (1950) 249–257. [10] J. Weertman, E.I. Salkovitz, Acta Metall. 3 (1955) 1–9. [11] I. Hiki, J. Phys. Soc. Jpn. 13 (1958) 1138–1144. [12] G.S. Baker, S.H. Carpenter, Rev. Sci. Instrum. 36 (1965) 29–31. [13] B.K. Kardashev, S.P. Nikanorov, Sov. Phys. Solid State 13 (1971) 128–131. [14] R.B. Schwarz, Rev. Sci. Instrum. 48 (1977) 111–115.
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[15] H.J. Kaufmann, P.P. Pal-Val, Phys. Status Solidi A 62 (1980) 569–575. [16] A. Wolfenden, M.R. Harmouche, J.T. Hartman Jr., J. Phys. 46 (1985) C10391–C10-394. [17] S. Kustov, G. Gremaud, W. Benoit, S. Golyandin, S. Sapozhnikov, Y. Nishino, S. Asano, J. Appl. Phys. 85 (1999) 1444–1455. [18] T. Kosugi, D. McKay, A.V. Granato, J. Alloy Compd. 310 (2000) 111–114. [19] G. Gremaud, S. Kustov, Ø. Bremnes, Mater. Sci. Forum 366–368 (2001) 652–666. [20] T. Kosugi, Jpn. J. Appl. Phys. 33 (1994) 2862–2866. [21] S. Kustov, M. Corr`o, J. Pons, E. Cesari, J. Van Humbeeck, Proceedings of the ICOMAT 2005, Shanghai, June 14–17, 2005.
A new design of automated piezoelectric composite oscillator technique S. Kustov a,∗ , S. Golyandin b , A. Ichino c , G. Gremaud c a
Departament de F´ısica, Universitat de les Illes Balears, cra de Valldemossa km. 7.5, 07122 Palma de Mallorca, Spain b A.F. Ioffe Physico-technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia c Institut de Physique de Mati` ere Complexe, Ecole Polytechnique F´ed´erale de Lausanne, CH 1015, Lausanne, Switzerland Received 10 August 2005; received in revised form 24 January 2006; accepted 9 February 2006
Abstract We describe the latest design of the automated piezoelectric ultrasonic composite oscillator technique, which has traditionally been used for the measurements of elastic and anelastic properties of solids at frequencies of 70–140 kHz in a continuous positive feedback mode. The new equipment features several substantial advantages, as compared to previous constructions, owing to the application of up-to-date data acquisition units and microelectronics components. © 2006 Elsevier B.V. All rights reserved. Keywords: Acoustic methods; Internal friction; Young’s modulus
1. Introduction Transducer systems to induce resonant oscillations include piezoelectric, magnetostrictive, capacitive, magnetic, eddy current, and electromotive units [1,2]. Piezoelectric units in turn use piezoceramic or quartz transducers. Taking into account a variety of requirements for an acoustic measuring system, such as time and temperature stability, linearity, precision, strain amplitude range, background damping introduced by transducers, it has been concluded that the quartz resonator composite oscillator technique has many advantages over others [2,3]. Following initial reports on application of quartz resonators to measurements of elastic and anelastic properties of solids [3,4], a detailed theory of the composite oscillator technique has been developed for longitudinal [5], torsional [6] and flexural [7] modes of oscillations. This technique is being used extensively for decades; see [8–18] for some examples. In [19] we reported a brief sketch of an automated experimental setup. In the meantime, using the same principles as in [19], new electronics have been designed which offer several substantial advantages: • adaptive measurement procedure, when the time-resolution depends upon current values of registered absorption of ultrasound; ∗
Corresponding author. Fax: +34 971173426. E-mail address: [email protected] (S. Kustov).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.02.230
• automatic selection of an appropriate measurement range; • combined analog–digital control of drive voltage (strain amplitude), which substantially simplifies the electronics design; • resonant conditions for oscillations, checked and maintained at each measuring point quite rapidly owing to a procedure of polynomial fitting of the resonant curve; • wide application of highly integrated high-precision analog circuits instead of digital ones, resulting in simple and compact design. We present below a more detailed description of this new design together with examples illustrating its potential. 2. Principles of operation Fig. 1 shows schematically the assembled three-component oscillator and the block diagram of the electronics. The oscillator incorporates a sample, glued to two identical 18.5◦ X-cut ␣-quartz crystals of rectangular cross section for drive and gauge purposes, plated with electrodes on two sides and supported in their centers (for excitation of the fundamental/odd harmonics of longitudinal oscillations). The quartz crystals and the specimen are cemented together at their ends, and the assembly is driven by an ac voltage Ud applied to the drive component. The voltage across the gauge component, Ug , is used to monitor the induced oscillatory strain and to maintain
S. Kustov et al. / Materials Science and Engineering A 442 (2006) 532–537
533
Fig. 1. Schematic representation of the composite oscillator and the electronic circuit.
continuous resonant oscillations via a positive feedback loop. The length of the specimen must be selected in such a way that its fundamental resonance frequency matches the one for the drive and gauge transducers. For longitudinal oscillations, the expressions for the total decrement of the composite oscillator, Q−1 (t), and peak value of the strain amplitude of its ith element, εmax 11 (i), in terms of voltages across drive and gauge crystals, have been obtained by Robinson and Egdar [5]: Ud 8Zm l2 d311 2 Ud Q−1 (t) = = Kδ (1a) ω(t)m(t) S1111 Ug Ug √ 2π S1111 Ug max ε11 (i)λ(i) = = K Ug (1b) jω l2 d311 Zm where Zm is the electrical impedance across the gauge component (including cables and input impedance of the measuring circuit), λ(i) is the wavelength of the longitudinal wave in component i, m(t) and ω(t) are the total mass and the angular resonant frequency of the oscillator (ω(t) ≈ ω(i)), respectively, S1111 and d311 are the elastic compliance and the piezoelectric modulus of the transducers, and l2 is the width of the plated side of the transducer. The damping and the resonant frequency of the specimen can be determined from that of the total oscillator (provided that the mass, the resonant frequency and the damping of quartz transducers are known) using the following equations [5]: m(t)ω(t)2 = m(i)ω(i)2 (2a) i
m(t)Q−1 (t) =
m(i)Q−1 (i)
(2b)
i
where m(i), ω(i) and Q−1 (i) is the mass, the resonant frequency and the damping of the ith component of the oscillator, respectively. Q−1 of high-quality transducers is usually substantially lower than that of the specimen, therefore Q−1 of the specimen can be derived with high precision from the measured total damping of the oscillator.
The Young’s modulus of the sample is conventionally determined from the fundamental angular frequency of longitudinal oscillations of a rod-shaped sample: E = ρω2 l12 /π2 , where ρ and l1 are the density and the length of the sample. Eqs. (1) and (2) show that measurements of the decrement of the sample, oscillatory strain amplitude and Young’s modulus are reduced to measurements of two electrical voltages and resonant frequency of the oscillator, provided that (i) the system is at resonance; (ii) the mass, resonant frequency and Q−1 of the oscillator without the sample are known; (iii) the coefficients K␦ and K are known (these coefficients depend on the geometry of the transducers, fundamental characteristics of quartz crystals and the electrical impedance Zm of the measuring circuit [5]). The electronics is controlled via a general-purpose data acquisition PCI card. The electronic circuitry performs two basic functions: (i) maintaining resonant conditions over a wide range of oscillatory strain amplitudes and (ii) conditioning Ud and Ug signals. Therefore, the setup includes two (nearly) identical channels, Ud - and Ug -channels. The Ug channel forms a part of the positive feedback loop. Both channels use symmetrical input/output circuits to minimize the electric pickup between drive and gauge voltage signals. This is especially important for high damping values when the Ud /Ug ratio becomes high (see Eq. (1)). Apart from the symmetric preamplifier/divider connected directly to the quartz crystals, each measuring channel includes • a line of three digitally-controlled amplifiers, which is supplied with a circuitry for automatic range selection (four-step automatic gain control units); • a one-chip hybrid eight-pole octave Butterworth filter with the center frequency of 100 kHz to improve signal-to-noise ratio and thus to increase the dynamic range of measurements; • a highly linear integrated RMS converter. The control of each measuring channel requires three TTL lines (inputs of subrange status) and one A/D converter (for the analogue voltage of RMS value of the Ud (Ug ) signal).
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A key element of the positive feedback loop is the automatic gain control amplifier and phase-shifter units. The AGC amplifier provides a constant ac voltage at the output, independent of the gauge voltage across the transducer, which serves afterwards in control of the drive voltage level. The analog phase-shifter is used to control the overall phase shift in the positive feedback loop and, thus, to maintain the oscillator at resonance. The latter condition corresponds to a purely resistive impedance of the entire electromechanical circuit (including drive and gauge components and measuring circuit) and, hence, to a maximum voltage across the gauge quartz transducer. The phase of the feedback loop is controlled by one analog line. Before each measurement, the resonant condition (i.e. the maximum response signal) is verified and maintained. To minimize the time for this procedure, the following algorithm is adopted. Initially and only once, before a measurement is started, the “global search” of the resonance is performed with 1000 steps over the total phase shift range from 0 to π rad. Afterwards, only a “fine tuning” procedure is performed in the close proximity to the resonance. To further minimize the tuning time, only a few points of the resonant curve are measured (typically, five points are sufficient). Afterwards, the position of the maximum response is found from a polynomial fitting to these few points. The “fine tuning” procedure is repeated, if necessary, until the resonant maximum is close to the middle of the tuning range and only afterwards a measurement is carried out. An essential ingredient of the entire system is the drive voltage control circuitry, which includes a digitally-controlled four-step resistive divider and an analog gain control module based on a high-precision analog multiplier. These are key elements which control the drive voltage level and, thus, the oscillatory strain amplitude over the range exceeding 80 dB with a resolution of better than 0.5%. The last figure gives an estimate of the strain amplitude resolution, when the strain amplitude should be controlled or, as a particular case, kept constant. In a single measurement of the strain amplitude dependence, the set-up controls the excitation voltage in the range from ∼20 to 280 VRMS . For a typical mass (m ∼ 0.5 g) and damping of a specimen (Q−1 ∼ 10−4 to 10−2 ) these values ensure oscillations with strain amplitudes ranging typically from 10−7 to 2 × 10−4 , sufficient usually for the majority of materials science problems. The low limit for strain amplitudes of the setup is restricted by the noise level of the output power amplifier. The gauge signal contains less noise since it is picked up from the quartz transducer and this procedure is equivalent to rather narrow band filtering of the drive voltage. In the present set-up, the output noise of the amplifier (with the open positive feedback loop) is compensated in the software, assuming that this noise is not correlated with the essential signal. The upper limit of the strain amplitudes is determined either by the mechanical strength of the quartz transducers (for the case of low total damping), or by the maximum drive voltage provided by the power amplifier (for the case of high damping). According to Robinson and Edgar [5], the practical range of strain amplitudes can be extended from 10−12 to 5 × 10−4 .
The basic differences of the present set-up from the previous design [19] are as follows. (a) Extensive application of analog integrated circuits with analog control instead of digital control. This makes the circuitry simple, compact and reliable. For instance, only four analog (two D/A and two A/D channels) and 10 digital TTL lines are required for a complete control of the equipment. Therefore, the control can be performed by means of a standard data-acquisition PCI card. (b) The present system is adaptive: temporal resolution of the equipment depends on the current values of measured damping. In the composite oscillator technique, for not too high values of damping, the properties of the oscillator (specimen) limit the temporal resolution, because each step-like change of the drive voltage leads to a transient in the time dependence of strain amplitude. If the step is small and the change of the decrement during this transient is neglected, the time constant τ of the transient is simply given as τ = T/(πQ−1 (t)), where T is the period of oscillation. Thus, the higher is the damping, the faster can the measurements be performed. This feature of the equipment is especially important if the decrement values change orders of magnitudes and if the decrement is time-dependent. (c) Resonant conditions are checked and maintained at each measuring point. This procedure is necessary since the subrange switch introduces certain unwanted phase shifts, but can be done rapidly owing to the polynomial fitting of the resonant curve. The software developed uses LabVIEW. It includes rather simple measurements of time and strain amplitude dependence as well as more complicated software which allows for simultaneous measurements of temperature spectra of the decrement and Young’s modulus for two values of strain amplitudes and of the strain amplitude dependence of the same parameters during single heating/cooling scan. The two stabilized strain amplitudes usually fall within strain–amplitude-independent and strain–amplitude-dependent ranges. They are stabilized in turn for each temperature step and the measurements are taken for each value of the strain amplitudes. Since the measurements are fast (as compared to the temperature variation rate), such two measurements can be regarded as referring to the same temperature. The spectra registered for the low strain amplitude value yields directly the strain–amplitude-independent damping, whereas the difference between the spectra at high and low strain amplitudes yields the temperature spectra of the strain–amplitude-dependent damping. Similar procedure can be applied to measurements of the Young’s modulus for low and high strain amplitudes, yielding temperature spectra of the amplitude-dependent modulus defect. In addition to the temperature spectra measurements, the strain amplitude dependence of decrement and modulus defect can be registered at any temperature, interrupting the temperature spectra measurement. The time to measure the strain amplitude dependence (containing about 100 experimental points) is rather short, typically about 1 min, which corresponds to the
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temperature variation for typical heating–cooling rates of only 1–2 K. This allows one to measure, in the majority of applications, the strain amplitude dependence at a certain temperature with a precision of about 1 K. After measurements of the strain amplitude dependence, the software returns to the registration of the temperature spectra. Problems which can be encountered in assembling the oscillator and in measurements of temperature spectra are mostly related to the selection of adhesive and purity of quartz transducers, as discussed earlier [19,20]. The basic technical characteristics of the setup are as follows: • the frequency range 70–140 kHz; • the maximum strain amplitude typically not less than 2 × 10−4 ; • the noise level for strain amplitude channel (strain amplitude value referring to quartz transducers)—3 × 10−9 ;
Fig. 2. Strain amplitude dependence of the total decrement, δ (a) and its strain–amplitude-dependent part, δh (b), for martensitic samples of polycrystalline binary NiTi, and of Cu-based alloys subjected to different heat treatments: quenching from homogenization temperatures into the martensitic state (Q) and step-quenching with subsequent -phase ageing (SQ).
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• the maximum drive voltage 280 VRMS ; • the noise level for the drive voltage channel—5 mV with the booster amplifier ON, 0.5 mV for the measuring channel alone.
3. Examples of application Fig. 2(a) shows examples of strain amplitude dependence of the total logarithmic decrement δ registered for some selected martensitic alloys at room temperature. We notice here that the strain amplitude dependence can be registered over a wide range of strain amplitudes even for the background damping levels differing nearly two orders of magnitude (NiTi and Cu–Zn–Al–B alloy in the quenched state). It is interesting to remark that the strain amplitude dependence of the decrement is rather weak in NiTi, as compared to the level of background damping: the amplitude-dependent part of the damping δh is only around 50% of the background amplitude-independent damping δi , Fig. 2(a). Nevertheless, the high precision and linearity of the measuring
Fig. 3. Effect of time of martensite ageing elapsed after the heat treatment of step-quenched Cu–21.3Zn–5.4Al–0.05B polycrystalline alloy on the strain–amplitude-dependent decrement, δh (a) and modulus defect, ( E/E)h (b).
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circuits allows one to follow small variations of the non-linear part of the decrement, δh = δ − δi , over more than two orders of magnitude, Fig. 2(b). Fig. 2(a) exemplifies also the absence of a strain amplitude hysteresis in NiTi and step-quenched Cu–Al–Ni alloys immediately after the heat treatment, but its existence in Cu–Zn–Al–B samples prepared otherwise. The presence of a strain amplitude hysteresis in the last case is a perfect indication of the mobility of point-like defects. Finally, Fig. 2(b) shows that the useful range of studying the amplitude-dependent decrement covers the range of around five orders of magnitude. Fig. 3 depicts the effect of ageing after the heat treatment on the strain amplitude-dependent components of the decrement and the modulus defect for a step-quenched Cu–Zn–Al–B martensitic alloy. We emphasize first that the initial measurement was made only 10 min after the heat treatment. This time is found to be sufficient to assemble the oscillator and to perform
the measurement. Second, we mention the dynamic range of measurements: the values of the decrement and modulus defect cover typically four orders of magnitude, allowing one to analyse various stages in the strain amplitude dependence and their variations with ageing. Finally, Fig. 4 gives an example of a further analysis of the data from Fig. 3. Fig. 4(a) shows the stress amplitude dependence of the anelastic strain amplitude, εan (σ 0 ), derived from the stress amplitude dependence of the amplitude-dependent modulus defect, ( E/E)h (σ 0 ): εan (σ 0 ) = ε0 × ( E/E)h (σ 0 ). The dynamic range of the anelastic strain amplitude reaches nearly seven orders of magnitude. The effect of ageing on anelastic strain amplitude, Fig. 4(a), has been interpreted as due to the shifts to higher strain amplitudes (i) of the entire strain amplitude dependences and (ii) of the stage of the rapid increase of the anelasticity [21]. The origins of these two shifts were supposed to be homogeneous atomic reordering, and growth of the pinning atmospheres around partial dislocations in faulted martensites [21]. Fig. 4(b) represents the result of scaling of the curves from Fig. 4(a) along the stress axis, with the scaling coefficient for each curve yielding the kinetics of the reordering. The shift of the second stage of the strain amplitude dependence, Fig. 4(b), yields, in turn, the kinetics of the pinning process [21]. 4. Conclusion We have reported a design of automated resonant composite oscillator technique, which fully uses the advantages inherent in this acoustic method: • measurements cover a wide range of damping and strain amplitude: no mechanical parts are involved, resulting in a low level of background damping; • measurements are fast: this property is advantageous for various in situ experiments and is important for studying transient phenomena; • measurements are simple and accurate: in practice, only two voltages across the transducers need to be measured; • measurements are detailed: practically continuous curves of damping or modulus versus strain amplitude can be registered; • resonant conditions can be easily maintained: positive feedback in a closed-loop configuration holds the oscillator at resonance. Acknowledgement SK acknowledges Ram´on y Cajal Program (Spain). References
Fig. 4. Stress amplitude dependence of anelastic strain amplitude εan for stepquenched samples of Cu–21.3Zn–5.4Al–0.05B alloy subjected to different martensite ageing periods after the heat treatment (a), and the same stress amplitude dependence reduced to the reference curve corresponding to the sample subjected to 0.6 ks of martensite ageing (b). Coefficients for scaling the curves in (a) along stress axis are 0.71, 0.56, 0.43, 0.29 for martensite ageing periods of 1.8, 3.6, 9.6 and 166 ks, respectively; data from [21].
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