Parameters of piezoelectric materials and piezoelectric equations (3)

(11)Mechanical quality factor Qm

When the PZT material piezo ceramic is used for resonance vibration, it is necessary to overcome the internal mechanical friction loss (internal consumption), and when there is a load, it is necessary to overcome the external load loss. The mechanical quality factor Qmo (no-load mechanical Q value) is related to these mechanical losses. And Qm (mechanical Q value under load). It is defined as: Qm = mechanical energy stored by the piezoelectric vibrator at resonance / mechanical energy lost during the resonance period. It reflects the amount of energy consumed by the piezoelectric body to overcome the mechanical loss when is vibrating. A larger Qm means less mechanical energy loss. The existence of Qm also indicates that it is impossible for any piezoelectric material to use all the input mechanical energy for output. At resonance: Qm = (π / 2) [ZC / (Zl-Zb)], where ZC is the acoustic impedance of the piezoelectric vibrator; Zl is the acoustic impedance of the load; Zb is the damping block in the piezoelectric transducer acoustic impedance. For a piezoelectric transducer, its Qm and Qe are not constant. They are related to the operating frequency, frequency bandwidth, manufacturing process, structure, and radiation medium (load) of the piezoelectric transducer. On the piezoelectric transducer used in ultrasonic detection technology, when Qm is too high, it is easy to make the vibration waveform generated by the vibrator too long (ringing phenomenon), resulting in waveform distortion and lower resolution. Similarly, Qe is not larger and larger. The choice and determination of Qm and Qe should be decided according to the actual needs. A large Q value means that the energy consumption is small during the piezoelectric effect. It can reduce the amount of heat generated in the case of high-power and high-frequency applications or pure transmission power applications, which is an advantage. However, for a transducer of detection purposes, a large Q value is disadvantageous for broadening the frequency band, improving the waveform, and increasing the resolution. In addition, since the Q value also changes with the nature of the load (for example, the load medium faced by the water immersion probe and the contacting method probe is different), the influence of the load medium must also be considered when is designing the transducer (radiation impedance ).

(12)Electromechanical coupling coefficient K
This is an important parameter for examining piezoelectric materials from the perspective of energy. Its definition is during positive piezoelectric effect, the external voltage E = 0, and there are: K2 = electric energy stored in the piezoelectric body under the ideal conditions ideal .The total mechanical energy input into the piezoelectric body under the conditions, or in other words: K2 = the converted mechanical energy that causes the charge to move between the connected electrodes / the input mechanical energy that follows the applied stress, the external stress τ during the inverse piezoelectric effect = 0, yes: K2 = mechanical energy stored in the piezoelectric body under ideal conditions / total electrical energy input into the piezoelectric body under ideal conditions or: K2 = converted electrical energy causing mechanical strain / input electrical energy under pressure transistors have elasticity, dielectricity and piezoelectricity at the same time, and they work together. For this reason, it is necessary to introduce this physical quantity to view these characteristics in a unified manner, which indicates the degree of coupling strength between mechanical energy and electrical energy. In a physical sense, it only describes conversion and it is not equal efficiency, and the converted energy may not be completely converted into radiated or output energy (including internal consumption and feedback, etc.). Of course, in a sense, it can also be said that the electromechanical coupling coefficient K represents the "efficiency" of the piezoelectric body converting electrical energy into elastic energy, or converting elastic energy into electrical energy. It is mainly determined by the type of piezoelectric material. It also depends on the vibration mode of the piezoelectric body, but has nothing to do with the value of the resonant frequency of the transducer. In addition, the K value also depends on the structure of the piezoelectric transducer, the operating conditions, and the electrode size and position of the piezoelectric body. We can divide the energy density U (energy in a unit volume) of piezoelectric materials into three parts, one is elastic energy density , one is electric field energy density (dielectric energy density), and one is piezoelectric interchange energy density Um (omit thermal and magnetic energy items).

The first part here is the mechanical part of the material-mechanical elastic energy, the second part of piezoceramic ring componnets is the electrical part-electric field energy, and the third part is the energy density of the interaction between elastic energy and dielectric energy. The total internal energy is: U = Ue + Ud + 2Um. Considering that piezoelectric energy is interchangeable energy, it is doubled. Therefore, we can define the electromechanical coupling coefficient in another way: K = Um / ( UeUd) 1/2. Or: K = Geometric mean value of piezoelectric energy / elastic energy and dielectric energy. The reason for choosing the geometric mean value of elastic energy and dielectric energy is to consider the uneven energy distribution of each tiny part of the piezoelectric crystal . In this way, we can say that the ratio of the energy that can be piezoelectrically converted in a unit volume of piezoelectric material is the electromechanical coupling coefficient. For example, Ud and Ue cannot be piezoelectrically converted, But it is not energy loss. For specific materials, such as quartz, the energy loss is small and the conversion efficiency is very high, but its electromechanical coupling coefficient is lower than that of piezoelectric ceramics, while the conversion efficiency of piezoelectric ceramics is not high. A large part can be piezoelectrically converted, which means that its electromechanical coupling coefficient is high. From here we can recognize the difference between electromechanical coupling coefficient and efficiency. The electromechanical coupling coefficient is a ratio of energy, dimensionless, and its maximum value is 1, when K = 0, it means that no piezoelectric effect occurs. The common electromechanical coupling coefficients are as follows:

(1) Electromechanical coupling coefficient Kp for radial vibration (also known as planar electromechanical coupling coefficient): Reflects the electromechanical coupling effect of a thin disc-shaped piezoelectric crystal when it is subjected to radial telescopic vibration, provided that the wafer diameter is ≥3 times the wafer thickness t , Its thickness direction is the polarization direction and the direction of the applied electric field.

(2) Transverse vibration (transverse length vibration) electromechanical coupling coefficient K31 reflects the electromechanical coupling effect when the long sheet-shaped piezoelectric crystal with the thickness direction as the polarization direction stretches and contracts in the length direction, provided that the length of the sheet is l≥3 times. The width and thickness of the flakes.

(3) Electromechanical coupling coefficient K33 of longitudinal vibration (longitudinal length vibration): reflects the electromechanical coupling effect of telescopic vibration along the length direction when the slender rod-shaped piezoelectric crystal is polarized in the thickness direction, and the electric field direction is the same as the polarization direction. The condition is a rod width and thickness or diameter with a length l≥3 times.

(4) Electromechanical coupling coefficient Kt of thickness vibration: reflects the electromechanical coupling effect of sheet-shaped piezoelectric crystals polarized in the thickness direction and the electric field direction is also in the thickness direction. The condition is that the thickness of the wafer is smaller than the side length or diameter of the wafer.

(5) Electromechanical coupling coefficient of thickness shear vibration K15: It reflects the electromechanical coupling effect of thickness shear vibration of piezoelectric crystal.

In summary, we can conclude that the main selection principles when selecting piezoelectric materials to make piezoelectric transducers in practical applications of ultrasonic testing are as follows: (1) The larger the value of d33--d33, the better the emission performance. . Obviously, when is making a transmitting transducer, it is better to choose a material with a d33 value as large as possible; (2) The larger the value of g33--g33, the better the receiving performance. Obviously, if you want to make a receiving transducer, you should choose a material with a large value of g33 as much as possible; when you need to make a transducer that combines both transmission and reception, as a comprehensive consideration, you should choose a value close to and as large as d33 and g33 as well. (3) Acoustic impedance Z (Z = ρc)-Considering that the reflectance and transmittance of ultrasonic waves are related to the difference in acoustic impedance between the medium. the smaller difference in acoustic impedance is the higher ultrasonic transmittance. In order to make as many ultrasonic waves as possible from the piezoelectric transducer enter the test medium, a piezoelectric material whose acoustic impedance is as close as possible to the acoustic impedance of the contact medium should be selected. It should be noted that the existence of the electric field will affect the apparent sound velocity in the piezoelectric material, and even the acoustic impedance of the piezoelectric material will change in the working state. (4) Electromechanical coupling coefficient Kt of thickness vibration-In the ultrasonic detection technology, the most important application is the thickness vibration type piezoelectric chip, so the larger the value of Kt, the better the electromechanical conversion performance, which  the sensitivity of the transducer is higher. (5) Electromechanical coupling coefficient Kp of radial vibration--When the piezoelectric chip is performing thickness vibration, there is also radial vibration at the same time, which will interfere with the thickness vibration and cause waveform distortion, noise increase or increase, etc. It is hoped that the Kp value should be as small as possible. In general consideration, the larger the Kt / Kp value, the better.

(6) Dielectric constant ε—The piezoelectric wafer forms a capacitor after the electrodes are coated, and its capacitance conforms to C = εA / t, that is, the dielectric constant ε, the relative area A of the electrodes, and the electrode spacing (wafer Thickness) t related. In the circuit, a small capacitance means a large capacitive reactance, which is suitable for use as a high-frequency piezoelectric element. In particular, ultrasonic detection transducer mostly works in the megahertz frequency range, so it is required that the ε of the piezoelectric material be smaller. Conversely, when used to make low-frequency piezoelectric components (such as speakers and microphones in the audio range), a material with a large ε should be selected to meet the matching requirements of large capacity and low capacitive reactance. It should be noted that the value of ε is also related to the mechanical freedom of the transducer, that is, the dielectric constants of the mechanical clamping state and the mechanical free state are different, so there are differences between εe and ετ. In addition, the relationship between ε and frequency is also more sensitive, so the ε value should be actually measured on the condition of the specific operating frequency. It means that piezoelectric wafers of the same thickness have a higher resonance frequency, or the thickness of the wafer is larger at the same resonance frequency, which is convenient for processing and manufacturing high-frequency components. Therefore, a material with a larger Nt value should be selected.

(8) Ferroelectric Curie point Tc--The ferroelectric crystal only has ferroelectricity within a certain temperature range. When the temperature reaches the ferroelectric curie point, the crystal will lose ferroelectricity, and the dielectric, piezoelectric, optical, elastic, and thermal properties are all abnormal. Most ferroelectrics have only one curie point, but a few ferroelectrics have upper and lower curie points, and they have ferroelectricity only in the temperature range between the upper and lower curie points. For example, the upper curie point of lead zirconate titanate is 115-120 ° C and the lower curie point is -5 ° C. If 5% calcium titanate is added to barium titanate, the lower curie point can reach -40 ° C. . In addition, some ferroelectrics have no curie point, such as some special polymer piezoelectric materials (because they have melted or even burned when they reach a certain temperature).

It should be noted that when the actual temperature has not reached the curie point, the  performance of many piezoelectric transducers (such as Kt, etc.) has significantly decreased or deteriorated (for example, the barium titanate probe deteriorates at 60-70 ° C) Moreover, the highest temperature at which it can work is not equal to being able to withstand sudden temperature changes, which is caused by the existence of anisotropy including the coefficient of thermal expansion. Therefore, in the case of higher temperatures such as welding electrode leads and heating during pouring of the absorption block during the actual use of the transducer and the process of making the transducer, When selecting a piezoelectric material, specific consideration should be given to the operating conditions of the transducer.

(9) Mechanical quality factor Qm and electrical quality factor Qe-In practical applications, if the Qm and Qe values are large, there will be a "ringing" phenomenon, resulting in waveform distortion and reduced resolution, which are not conducive to detection. The situation arises. Therefore, starting from the needs of detection technology, in order to truly reflect the characteristics of the echo signal and ensure that the detection resolution meets the detection requirements, Qm and Qe are generally not expected to be too large. In addition to taking into consideration when selecting materials, when is designing and manufacturing transducers, Frequence, Qm and Qe values need to be appropriately reduced by increasing the damping on the structure and changing the impedance on the circuit. Of course, reducing Qm and Qe values comes at the expense of sensitivity (reduced output power). Therefore, the appropriate Q value should be selected and adjusted according to the needs of the actual application (according to experience, the actual Q value of the ultrasonic detection transducer should not be greater than 10).

(10) Aging performance of piezoelectric materials piezoceramic cylinder tube-The piezoelectric properties of polarized piezoelectric materials will have irreversible changes with time. This phenomenon is called "aging", such as dielectric constant, dielectric Losses, piezoelectric constants, electromechanical coupling coefficients, and elasticity usually decrease with time, and frequency constants and mechanical Q values increase with time. The change of these parameters is basically linear with the logarithmic value of time. It is generally considered as a unit of ten years, which is called "ten-year aging". Obviously, this index reflects the time stability of piezoelectric materials. When making piezoelectric transducers, due consideration should also be given to selecting materials with better time stability. On a specific ultrasonic transducer, this aging phenomenon will be specifically manifested in sensitivity, initial wave occupation, and electrical noise level. Therefore, attention should also be paid to the effect of aging on the purchase and storage of the transducer.

(11) Thermal stability of piezoelectric materials-This refers to the piezoelectric properties of piezoelectric materials that are constant or non-degraded after a period of continuous operation in a certain temperature range below the curie point, especially for high temperature environments .The working transducer should be selected from materials with good thermal stability.
The above 11 items are the main considerations and selection principles when we choose piezoelectric materials to make ultrasonic testing transducers. We should comprehensively consider and select appropriately according to the specific application and needs.