# Piezoelectric ceramics principles and applications apc pdf

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- Piezoelectric Ceramics By Ltd Apc International
- electric field limit for piezoelectric materials
- Piezoelectric Ceramics: Principles and Applications
- The Effects of Piezoelectricity Smart Material

## Piezoelectric Ceramics By Ltd Apc International

Ardid, I. Felis, J. Circular piezoelectric transducers with axial polarization are proposed as low frequency acoustic sensors for dark matter bubble chamber detectors. The axial vibration behaviour of the transducer is studied by three different methods: analytical models, FEM simulation, and experimental setup. To optimize disk geometry for this application, the dependence of the vibrational modes in function of the diameter-to-thickness ratio from 0. Resonant and antiresonant frequencies for each of the lowest modes are determined and electromechanical coupling coefficients are calculated.

From this analysis, due to the requirements of radiopurity and little volume, optimal diameter-to-thickness ratios for good transducer performance are discussed. The first discovery of the piezoelectric effect was found in by Jacques and Pierre Curie. They discovered that a quartz crystal submitted to mechanical action will produce an electrical potential that is proportional to the force applied.

This effect is called direct piezoelectric effect. In the next year, Gabriel Lippmann described an analytical method for the reverse piezoelectric effect and the Curie brothers demonstrated this theory in the laboratory.

Since its discovery, the piezoelectricity has been used in different fields such as defence, medical diagnosis, particle detection, ultrasonics, motors, and echolocation [ 1 , 2 ]. Due to the different vibrations modes in a piezoelectric material, the applications of this phenomenon can be different in each field. The first piezoelectric material used was the quartz; however, this material is not very good to transform the electrical energy into mechanic energy and its sensitivity is small.

This material has a better sensitivity and frequency response. This material is produced using a very intense electrical field that polarizes the ceramic in a specific direction. Piezoceramic transducers have usually a regular geometry and the knowledge of the behaviour of piezoceramic disks is important for transducer design and applications [ 3 , 4 ].

With these transducers, it is possible to generate vibrations from a few to several hundreds of kHz, demonstrating its feasibility to be used as ultrasonic sensors and actuators in this frequency range. For example, recent applications use piezoelectric sensors in order to detect the acoustic signal emitted by the interaction of elementary particles in a fluid target. In this sense, the dark matter detectors PICO bubble chambers use a superheated fluid target filled in a glass vessel [ 5 , 6 ].

Under some thermodynamic conditions, the particle interaction produces a bubble nucleation within the fluid and acoustic waves are emitted during the bubble growth. These sensors are glued to the external walls of the vessel that contains the metastable fluid. The acoustic discernment of the different acoustic events depends forcefully on the complete sensor properties and there are constraints as well in the little size and radiopurity of the piezoelectric ceramics [ 7 ].

These show the influence of bonding the ceramic to the vessel in its acoustic response. Among others, it is extracted that there is an increase of the sensors sensitivity, mainly in low frequencies. This is due to a better adaptation of acoustic impedances between the medium and the ceramic, through the glass of the vessel [ 7 ]. However, the first step in the design of the final transducer is the choice of the type and size of the ceramic taking into account its final use.

For this, and since low radioactivity is a must, the amount of ceramic used in the transducer is a factor of great importance. This is because the ceramic material contains lead, which is usually accompanied with heavy radioisotopes that are alpha emitters and thus a source of background through alpha-neutron reactions. In this paper, we explain a proven methodology to study the optimization of this type of sensors.

For this, several circular section disks or cylinders PIC piezoceramics with different width and height are studied with analytical and numerical methods and the results obtained are contrasted by experimental measurements.

In previous works with circular PZT, some authors studied the natural vibration modes of axial symmetric piezoceramics using Finite Element Methods FEM [ 3 ] focusing on the efficiency and transducer design.

Theoretical analysis contrasted with experimental measurements of resonant vibration with interferometry and laser Doppler vibrometer have been used [ 8 ]. Additional characterization methods and theoretical approach formulas can also be found. Here, the main parameters of piezoceramics will be compared: the resonance, , and antiresonance, , frequencies of the electrical impedance and its amplitude, the product of each frequency resonance and the length thickness, , or diameter, associated with the mode of vibration , and the piezoelectric coupling factor,.

Moreover, the ratio of coupling factors of the lowest piezoelectric modes gives us a quantified estimation of the energy distribution in this frequency range. The conclusions of these studies lead us to have design principles to select a specific piezoceramic circular geometry with a radio-clean piezoelectric material that can be used in the next generation of dark matter bubble chamber detectors PICO L [ 9 ].

As discussed, the optimization of the volume is based on a free bounded circular ceramic such as that shown in Figure 1 that shows the geometrical scheme of a piezoceramic disk with thickness t and radius. The piezoelectric disk is polarized along the thickness -axis , and the two opposite plane faces are covered with complete electrodes. The vibrational behaviour of a circular piece of the characteristics shown in Figure 1 can be obtained from the constitutive equations for a piezoelectric material.

See [ 10 — 12 ] for a comprehensive review in this regard. For the purposes of this text, it is important to note that there are three different modes of vibration of thin piezoceramic disks: tangential, transverse, and radial extensional modes. However, theoretical and experimental analysis of these modes [ 8 ] shows that only the resonance frequencies of the radial vibrations modes can be measured in an impedance analysis. For this reason, we only analyse the vibrational characteristics of the extensional modes.

Assuming that there is a harmonic radial vibration in time with a known frequency , the radial displacement in the median plane along the radius can be expressed as.

The resonance and antiresonance frequencies can be found where the current becomes infinite and becomes zero, respectively. With this, the characteristic equations for both resonance frequencies can be written as where and are the Bessel function of the first kind, for orders 0 and 1, respectively, and. Equation 4 is the well-known result for two-dimensional analysis of the radial modes [ 8 ]. From 4 and 5 , the resulting resonant and antiresonant frequencies of the radial extensional mode for a circular piezoelectric piece with free boundary conditions can be expressed as.

As mentioned, the basis for comparing these analytical results with experimental measurements or numerical simulations of the electrical impedance is that it reaches a local minimum when the sample vibrates at the resonance frequency and a maximum at antiresonance.

With this, it has obtained the impedance curves that have been used in this study. In the sizing of tetrahedral meshing elements, it was taken into account that the smaller wavelength maximum frequency was discretized in almost ten parts.

A mechanical free boundary condition was set in all the contours of the transducer and uniformly electrical open-circuit voltages were set to the electrodes. In Figure 2 , an example of the type of mesh used to discretize the solutions is shown. The number of meshing elements with tetrahedral structure was Table 1 shows the thickness and diameter of the different cylindrical ceramics that have been simulated in order to study different vibrational modes.

The selected dimensions are the typical ones used in PICO detectors. The geometries with a ratio equal to or more than 5 can be considered as disks [ 3 ]. The smaller the ratio, the more the ceramic behaves as a cylinder. With this in mind, the FEM method can be used to study the ceramics with different radius and length in order to obtain the resulting impedance response and the frequencies and amplitudes of each peak. Subsequently, the resulting behaviour will be studied taking into account the volume of each testing piece.

In order to contrast theoretical predictions of models with experimental results, we have measured the bandwidth frequency electrical impedance response of several cylindrical PIC piezoceramics with different diameter and thickness see Table 2. The samples are held in air only by three metal wires at the bottom and another one at the top to avoid loads and constraints for the vibration.

When the ceramic is excited in a resonance frequency of a radial vibration, the value of the impedance reaches a minimum and the admittance is maximum.

The modulus of the electrical impedance obtained is shown. We can distinguish between the low frequency radial modes and the high frequency thickness modes. The local minima and maxima appearing in the impedance curve correspond to resonance and antiresonance frequencies, respectively [ 13 ]. For the first three modes of Figure 4 that correspond to radial modes left square , a good agreement, either in the frequency position of the peaks or in their amplitude, can be seen with a difference smaller than 0.

For the thickness mode right square , it should be pointed out that there is a large relative deformation and the effects of the losses are larger than those in radial modes. Anyway, we are not interested in very high frequencies, and thus the high frequency thickness modes are not of application in this work.

Even so, when the ratio is smaller and smaller, this mode becomes of lower frequency, and experimental results and the simulations are in better agreement. With these experimental, numerical, and analytical methods, the relationship between the resonance frequencies and geometry parameters of the disk have to be studied. All these approaches will be taken into account and the results will be compared and discussed. Generally, the analytical and numerical results are in good agreement except for higher modes than the third one.

In this range, the frequencies are close to the thickness vibration mode frequencies of the piezoelectric cylinder and strong coupling vibration of thickness and radial direction occurs. In addition, in these regions, the analytical solutions may have some errors due to the small diameter-to-thickness ratio; also the analytical method is based on a membrane theory [ 8 ]. We can see that the resonance frequency increases with decreasing diameter in a nonlinear relationship, especially for small diameters.

Both experimental and simulated values show that frequency decreases with an increase in thickness and it converges to the analytical solution for low thickness.

Since the analytical solution is valid only for ideal disks, it is independent of the thickness. For this, the analytical solutions are constant and close to the FEM solutions for larger diameter-to-thickness ratio, that is, for small thickness. The frequency coefficient describes the relationship between the geometrical dimensions of a body and the corresponding series resonant frequency.

When mechanical free piezoelectric ceramic element is excited with an alternating electric field and there is an impedance minimum, the radial resonance frequency matches with the series resonance frequency. The relationship between the radial mode resonance frequency constant, , and the diameter of the ceramic element, , is expressed by. At higher frequencies, there is another impedance minimum, the thickness resonance frequency.

Thickness mode frequency constant, , is related to the thickness of the ceramic element, , by. Figure 7 shows the relation to different ratio obtained both in FEM methods dashed line and experimentally squares.

It can be seen that when the frequency coefficient decreases significantly. This behaviour offers a geometrical distinction between disks and cylinders. In accordance with [ 1 ], the measures of resonant and antiresonant frequencies of a piezoelectric disk should be made better in terms of ratio than the conventional standards of measurement of piezoelectric coefficients [ 14 , 15 ].

The black solid line indicates the frequency constant in the theoretical model, which supports the above statement. Although high modes present a larger deviation between experimental and predicted results, it is interesting to show the distinction between antiresonant frequency constant in low and high modes. Figure 8 shows the value of for the first four low modes radial modes. As expected, for lower ratio, increases. Both numerical and analytical results agree with experimental values above.

However, there is a clear trend upwards for lower ratios. Additionally, for thickness mode is shown. Since there is no thickness variation of these measures, the frequency constant remains quite constant and it is quite different to the low modes pattern.

This behaviour shows clearly the difference between radial and thickness modes in piezoelectric disks. With this, we can predict that, for low ratios, both radial and thickness modes will be at similar frequencies, being possible that the thickness mode frequency is lower than the radial one.

## electric field limit for piezoelectric materials

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Ardid, I. Felis, J. Circular piezoelectric transducers with axial polarization are proposed as low frequency acoustic sensors for dark matter bubble chamber detectors. The axial vibration behaviour of the transducer is studied by three different methods: analytical models, FEM simulation, and experimental setup. To optimize disk geometry for this application, the dependence of the vibrational modes in function of the diameter-to-thickness ratio from 0. Resonant and antiresonant frequencies for each of the lowest modes are determined and electromechanical coupling coefficients are calculated. From this analysis, due to the requirements of radiopurity and little volume, optimal diameter-to-thickness ratios for good transducer performance are discussed.

[(Piezoelectric Ceramics: Principles and Applications)] [Author: Ltd Apc International] APC International, Ltd.'s textbook on the principles and applications of (December, ) by Ltd Apc International Free PDF d0wnl0ad, audio books.

## Piezoelectric Ceramics: Principles and Applications

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### The Effects of Piezoelectricity Smart Material

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