Displacement amplification in curved piezoelectric diaphragm transducers

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Sensors and Actuators A 141 (2008) 262–265

Letter to the Editor Displacement amplification in curved piezoelectric diaphragm transducers

Abstract Experimental measurements of the displacement amplitude of a special curved piezoelectric diaphragm-type transducer are presented. The diaphragm consisted of a laminated PZT structure, square in shape, 4 mm side length, and 4.1 ␮m total thickness. The transducer was formed by applying a static pressure to one side of the diaphragm that deformed it into a nominally spherical shape. This transducer was similar to curved transducers previously described in the technical literature except that the diaphragm vibrated under tension and static pressure. Scanning laser vibrometer and cursory acoustic measurements were performed to characterize the performance of the transducer. The amplitude of diaphragm vibrations per supplied volt varied by 39 dB and the first natural frequency gradually changed as the static pressure was increased. The displacements at the first natural frequency were observed to be in-phase across the diaphragm surface, as contrasted to the phase variation observed in the first flexural mode of the curved transducers described in the literature. The implication is that a curved transducer formed by the application of external static pressure offers high-displacement amplitude and a low-frequency mode that would couple efficiently to a surrounding acoustic medium. © 2007 Elsevier B.V. All rights reserved. Keywords: Piezoelectric transducers; Curved transducers; Static pressure

1. Introduction In an effort to improve the performance of piezoelectric transducers, researchers have explored the properties of a device consisting of a curved piezoelectric diaphragm [1–7]. The essence of this approach is to exploit the curvature of the diaphragm to convert extensional strains into radial motions, rather than relying solely upon flexure. This vibration is often referred to as the uniform mode. For this mode, it has been found that the resonance frequency [1] of the curved diaphragm is inversely proportional to the radius of curvature, and displacement amplitude can be magnified [6] over that of a flat transducer. The investigations [4–6] have also shown that clamped boundaries and extensional stresses cause additional flexural vibrations. The first natural frequency of the flexural mode is lower than that for the uniform mode, making it potentially useful for some applications. Experimental measurements [5] and numerical computations [6] have shown that the magnitude of the flexural vibrations is on the same order as those of the uniform mode. However, the flexural mode does not vibrate in phase across the diaphragm surface, leading to poor acoustic coupling to the surrounding medium [5]. In this paper, experimental measurements of the displacement amplitude of a special type of curved piezoelectric diaphragmtype transducer are presented. The transducer was formed by applying a static pressure to one side of the diaphragm that deformed it into a nominally spherical shape. A harmonic timevarying voltage was then applied to the diaphragm, causing harmonic displacements. As such, this transducer was simi0924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2007.09.013

lar to the curved transducers described in Ref. [5,6], except that the diaphragm vibrated under tension and static pressure. The effects of tension on small-amplitude linear motions of flat plates are well documented [8–10], while little is known about the effects of tension on curved plate/membrane structures. The experimental observations were surprising. As the magnitude of the static pressure was increased and the transducer assumed a static spherical deformation, the amplitude of the vibrations varied by 39 dB. The displacements at the first natural frequency, thought to be flexural in origin, were observed to be in-phase across the diaphragm surface, as contrasted to the phase variation observed in the first flexural mode of the curved transducers described in Ref. [5]. Efficient coupling of these vibrations to a surrounding acoustic medium would be expected because the displacements were in-phase across the diaphragm surface. 2. Diaphragm transducer fabrication The 4 mm × 4 mm square piezoelectric diaphragm used in this work was formed from standard silicon micromachining and thin-film processes, described elsewhere [11,12]. The individual laminae consisted of Au, PZT, Pt, SiO2 and p+ Si layers of thicknesses 0.3, 1.5, 0.2, 0.1, and 2.0 ␮m, respectively. The Au electrode on top of the 1.5 ␮m-thick lead-zirconate-titanate (PZT) was 2.6 mm × 2.6 mm and centered on the diaphragm. A PZT chemistry of 40% Zr, 60% Ti by atom fraction was chosen for this study. Capacitance was measured as 29 nF.

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Piezoelectric, mechanical and dielectric properties of the PZT film in this study can be found in Ref. [12]. 3. Experimental methods The piezoelectric diaphragm transducer was mounted in a fixture that allowed the application of a static pressure to one side using a cast iron bellows developed for testing pressure transducers, and access to the other side for displacement and acoustic measurements. A photograph of the apparatus is shown in Fig. 1. A voltage source was connected to the electrical contacts of the diaphragm to cause it to undergo AC vibrations. In Fig. 1, the apparatus that applied an external static pressure is not visible, what is shown is the vibrating surface area, electrical contacts, and relative fixture geometry. Behind the vibrating diaphragm was a sealed cavity connected to a manually controlled bellows to control the static pressure. The sealed cavity exposed the entire back surface of the vibrating diaphragm to the static air pressure, ensuring a uniform background pressure across the diaphragm. A sensor was located in the cavity to measure the cavity pressure. Deflections and velocities of the diaphragm were measured with a scanning laser vibrometer (Polytec PI). Static deflections of the diaphragm center were measured as static pressure was applied. AC vibration amplitudes were measured over the diaphragm surface with a grid density of 0.125 mm. These results were then truncated at the diaphragm center, and the grid aligned with coordinate axes. Acoustic measurements were collected at a distance of 10 cm from the diaphragm and recorded with a dynamic signal analyzer (Agilent 35670A). A 1/4 in. diameter free-field measurement microphone, aligned normal to the surface of the diaphragm, was used to measure the acoustic amplitude spectra at this distance.

Fig. 1. Transducer and mounting fixture.

Fig. 2. Static center displacement w caused by external static pressure P, and best fit of Eq. (1).

4. Results Application of a static external pressure to one side of the transducer diaphragm caused the diaphragm to assume a nominally spherical shape. Measurements of the displacement of the center of the diaphragm w versus externally applied static pressure P are shown in Fig. 2. The expected static pressure P center deflection w behavior of a pressurized membrane structure is [13]: P =γ

σt Et w + δ 4 w3 , a2 a

(1)

where σ is the residual stress, t the thickness, a the distance from center to edge, E the effective biaxial elastic modulus, and γ and δ are constants particular to the shape of the membrane—these can be found for a square diaphragm in Ref. [13]. The best-fit line is shown in Fig. 2. From this data, the residual stress σ and modulus E for the diaphragm were estimated to be 71 MPa and 113 GPa, respectively. The curvature of a square pressurized membrane can be estimated as 1.66(w/a2 ) [13]. At a static pressure of P = ±20 kPa, the magnitude of the center deflection was approximately 50 ␮m, and the radius of curvature was estimated to be 48 mm. Consider the laser vibrometer measurements of the displacement amplitude spectrum of the diaphragm center for several externally applied pressures shown in Fig. 3. The displacement amplitudes were normalized to the voltage provided to the diaphragm transducer. When taking these measurements, care was taken to ensure that voltage excitation levels were small enough so that the displacement amplitude spectra were independent of excitation level, thus precluding distortion caused by nonlinear oscillations. In Fig. 3a, the externally applied static pressure P ranged from 0 to −24 kPa, which caused the diaphragm to deform statically inward. Negative static pressures were chosen to facilitate scanning laser vibrometer measurements of displacement amplitude.

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Letter to the Editor / Sensors and Actuators A 141 (2008) 262–265

Fig. 3. Displacement amplitude at diaphragm center. (a) Static pressure ranging from P = 0.0 to 24.0 kPa and (b) static pressure ranging from P = −2.0 to 2.0 kPa. Numeric labels on individual curves refer to static pressure P in units of kPa.

Positive static pressures caused a convex surface that tended to scatter the scanning laser beam. Similar vibration behavior for positive static pressures were observed but not included in this paper. At 4 kHz, the displacement amplitude increased from 0.032 ␮m/V at a static pressure of P = 0.0 kPa to 0.18 ␮m/V at a static pressure of P = −8.0 kPa. As the magnitude of the static pressure was increased, i.e., going below P = −8.0 kPa, the displacement amplitude at 4 kHz gradually decreased from 0.18 ␮m/V. The first resonance peak in displacement amplitude was observed at a frequency near 8 kHz. As the static pressure changed from P = 0 to −8.0 kPa, the frequency for the first resonance peak gradually increased from 8.55 to 9.18 kHz. A detailed examination of the displacement amplitude spectrum as it changed with small static pressure is shown in Fig. 3b. In this figure, the static pressure ranged from P ± 2 kPa, with one measurement at P = −0.7 kPa. Ideally, one might expect to observe the lowest displacement amplitude at a static pressure of P = 0 kPa, but that was not the case. As shown in Fig. 3b, the lowest displacement amplitude was observed at a static pressure of P = −0.7 kPa. At a static pressure of P = −0.7 kPa and frequency of 4 kHz, the center displacement amplitude was only 0.00185 ␮m/V, 39.8 dB smaller than the displacement amplitude at a static pressure of P = −8.0 kPa from Fig. 3a at the same frequency. Scanning laser vibrometer measurements were taken to determine the shape of the mode associated with the first resonance

peak. These measurements are shown in Fig. 4. The displacement amplitude was measured over one-quarter of the transducer area. The remaining area was observed to behave with even symmetry. In Fig. 4a, the displacement amplitude of the diaphragm is presented at a frequency of 8.56 kHz, corresponding to the first resonant peak at a static pressure of P = 0 kPa shown in Fig. 3a. When a static pressure of P = −8 kPa was applied, the displacement amplitude shown in Fig. 4b was observed at a frequency of 9.22 kHz. This situation corresponded with the peak in displacement amplitude at P = −8 kPa static pressure contained in Fig. 3a. In-phase vibration was observed in Fig. 4 for both static pressure conditions. These measurements contrast with the flexural modes observed in curved piezoelectric transducers without static pressure, in which phase variation in the displacement amplitude was observed for the first flexural mode [5]. Consequently, it would be expected that the diaphragm transducer operating with a static pressure on one side would couple more efficiently to the surrounding medium than a curved transducer without static pressure applied to one side of the diaphragm. Acoustic measurements were performed to determine the coupling of the diaphragm transducer with applied static pressure to airborne acoustic waves. Comparison of the measured acoustic amplitude spectra and acoustic amplitude spectra predicted from laser vibrometer displacement amplitude measurements are shown in Fig. 5. The predicted acoustic pressure amplitude spectra were computed using a simple-source model

Fig. 4. Surface displacement amplitude (quarter section of transducer diaphragm). (a) Static pressure 0 kPa; frequency 8.56 kHz and (b) static pressure −8 kPa; frequency 9.22 kHz.

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the application of external pressure amplifies the displacement amplitude, and the displacement amplitude associated with the first resonant peak has little or no phase variation across the vibrating surface. These properties may impact acoustic applications that require high output at low frequencies. References

Fig. 5. Measured acoustic spectra p, and acoustic spectra p predicted from laser vibrometer measurements of displacement amplitude and a simple-source model.

for the transducer [14]: p=

1 Q ρc , 2 λr

(2)

where p is the acoustic pressure amplitude, ρ = 1.10 kg/m3 , c = 343 m/s are the density and sound speed of air at the measurement temperature and altitude, Q the source strength, λ = c/f the wavelength, and r is the distance from the transducer to the microphone. The source strength was based upon the displacement amplitude spectra for the center of the diaphragm taken from Fig. 3a, weighted by a factor of 41% derived from the mode shape measurements at the frequency of the first resonant peak in Fig. 4b to account for the variation in displacement amplitude across the diaphragm surface. The fixture geometry was not ideal for acoustic measurements. The wavelength in air at a frequency of 9 kHz was 3.8 cm, so that the fixture and table would cause a degree of baffle effect (interference) not contained in the simple-source model (2). For the first resonant peak with P = 0 kPa static pressure, the measured acoustic pressure amplitude was 0.0174 Pa/V at a frequency of 8.37 kHz, while the predicted acoustic pressure amplitude was 0.0095 Pa/V at a frequency of 8.64 kHz. For the first resonant peak with P = −8 kPa static pressure, the measured acoustic pressure amplitude was 0.042 Pa/V at a frequency of 9.3 kHz, while the predicted acoustic pressure amplitude was 0.046 Pa/V at a frequency of 9.29 kHz. These measurements indicated that the diaphragm displacements at the first resonant peak did indeed couple efficiently to the acoustic medium, as would be expected by in-phase vibration of the transducer surface. 5. Conclusions Experimental measurements indicate that the vibrations of a piezoelectric diaphragm transducer deformed into a spherical shape by application of a static pressure have special properties. In particular, the curvature and tension associated with

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D.J. Morris D.F. Bahr School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA M.J. Anderson ∗ Department of Mechanical Engineering, University of Idaho, Moscow, ID 83844, USA ∗ Corresponding

author. E-mail addresses: [email protected] (D.J. Morris), [email protected] (M.J. Anderson) 19 July 2007 Available online 29 September 2007