Parameters of PZT materials and piezoelectric equations (2）

Second, the piezoelectric parameters

3. There is a complicated relationship between the piezoelectric parameters of piezoelectric materials, such as e = dE and E = -he as described above. Comparing them seems to give d = -1 / h, but it is not true in practice. Because the former is given under the condition of τ = 0, and the latter is given under the condition of I = 0, such a simple comparison cannot generally be made. In addition, piezoelectric materials are anisotropic piezo crystals, and their electrical, mechanical, and electromechanical properties vary with the direction of the electrical or mechanical excitation source. Therefore, there are actually many mechanical parameters (τ, e, c, s), electrical parameters (E, D, ε, β) and piezoelectric parameters (d, g, i, h) connected to the force and electricity. A tensor of components. τ and e each have six independent components, then c and s have 36 components; E and D each have three independent components, then ε and β have 9 components. For example, each e component is related to three E components: the relative elongation e1 (△ l / l) in the X direction is related to the components E1, E2, and E3 of the field strength vector in the three directions of X, Y, and Z. . Therefore, the original relation e = dE is actually: e1 = d11E1 + d21E2 + d31E3
The three normal axis strains (e1, e2, e3) and three independent shear strains (e4, e5, e6) are all related to E in this form, so the d coefficient has 3x6 = 18 components, so Also e2 = d12E1 + d22E2 + d32E3, e3 = d13E1 + d23E2 + d33E3, e4 = d14E1 + d24E2 + d34E3, e5 = d15E1 + d25E2 + d35E3, e6 = d16E1 + d26E2 + d36E3.
This means that each of the four piezoelectric constants of PZT material piezo ring is associated with three electrical and six mechanical components, so they each have 18 components. In the expression method, it is usually indicated in the subscript of the parameter symbol, such as dij, i indicates the direction of the electrical quantity (electric field or electrical displacement) component (there are three directions); j represents the mechanical quantity (stress or strain) component. However, because piezoelectric materials each have a certain symmetry, these components may not all exist independently, some may be zero, and some may be equal to each other or related in a certain relationship, so there are actually much fewer independent components. A specific piezo crystal always involves only a few components and is not complicated to calculate in practice. The number of independent components can usually be reduced to one elastic tensor, one dielectric tensor, and one piezoelectric tensor to determine the properties of the piezoelectric material. In practical applications, there are several components such as "d31", "d33" and "d15". The main application in ultrasonic detection technology is the thickness vibration in the polarization direction of the piezoelectric body (defined as the third direction or the Z direction). Therefore, the parameter of the excitation and change parameters in this polarization direction is "d33 ", Such as d33, g33, etc. The other two directions perpendicular to the polarization direction are designated as "1" (or "X") and "2" (or "Y") directions.

We determine the physical meaning of the relevant piezoelectric parameters as follows:

(1) Strain electric field constant d33 = e / E = W / U (meters / volts), in a mechanical free state (τ = 0), the application of an electric field along the polarization direction causes relative strain along the polarization direction, or Characterize the magnitude of strain generated by a unit voltage in the thickness direction; where W is the simple extension (meters) and U is the applied voltage (volts). (2) Electric field stress constant g33 = -E / τ = -U / P (voltmeter / newton), in the state of electrical open circuit (I = 0), applying stress along the polarization direction causes a relatively open circuit along the polarization direction elegant, or characterize the strength of the open-circuit electric field generated by unit stress in the thickness direction; where U is the open-circuit voltage and P is the sound pressure. The above two parameters (d33, g33) are the main application parameters in electroacoustic transducers. (3) The stress electric field constant i33 = -τ / E (Newton / volt meter) represents the magnitude of the stress generated by the unit electric field strength in the polarization direction (thickness direction). (4) The electric field strain constant h33 = E / e = U / △ t (volts / meter). Characterizes the relative open circuit voltage generated by unit strain along the polarization direction (thickness direction). In the formula, Δt is the thickness change amount, and U is the open circuit voltage. In addition to the above-mentioned piezoelectric parameters, the important parameters that characterize the properties of the piezoelectric body (5), the dielectric constant ε, the dielectric constant of piezoceramic ring componnets are an important macroscopic physical quantity that comprehensively reflects the dielectric behavior of the dielectric. The dielectric constant measurement under an electrostatic field is called a static dielectric constant, and the dielectric constant measurement under an alternating electric field is called a dynamic dielectric constant. The two are different. The magnitude of the dynamic dielectric constant is related to the measurement frequency. (6) Elastic modulus, the strain generated by the piezoelectric effect is in the category of elastic strain, and obviously the state of the strain will be closely related to the elastic modulus of the material.

(7) Frequency constant N: Units Hz · m, MHz · mm, and KHz · mm. We know that the resonance frequency of a piezoelectric body is not only related to the characteristics of the material itself, but also to the external dimensions of the material, so the evaluation of it inconvenience. The purpose of introducing the parameter of frequency constant is to avoid the influence of the external dimensions of the material, and only as a piezoelectric performance parameter is related to the material properties for easy evaluation. According to the different vibration modes of the piezoelectric body, it can be divided into: (a) thickness vibration frequency constant Nt = ft, (b) length extension vibration frequency constant Nl = fl, (c) radial extension vibration frequency constant Nd = fd, f is the resonance frequency; t is the thickness of the vibrator; l is the vibrator length; d is the vibrator diameter. The main application of ultrasonic testing technology is the thickness vibration mode, with Nt as an important parameter commonly used, and its resonance frequency: f = (K / 4π2M) 1/2 fundamental frequency resonance f = (1 / 2t) (c / ρ) 1/2 = C / 2t where: K = n2 (π2 / 2) (cA / t); M = ρtA / 2; W = K / M = 2πf (circular frequency) where A is the area of the piezoelectric chip; t is the thickness of the piezoelectric wafer; n is a multiple of the frequency doubling vibration; when the fundamental frequency vibration is taken, n = 1; ρ is the density of the piezoelectric body; c is the elastic constant of the piezoelectric body along the axis of the vibration direction; C is the piezoelectric crystal The speed of sound in the case of thickness vibration mode is the longitudinal wave velocity CL in the crystal. According to C = λf (λ is the wavelength), it can be known that the thickness of the piezoelectric crystal. when the fundamental frequency is used as the thickness resonance is t = λ / 2. This can determine the thickness of a piezoelectric chip that resonates at a certain fundamental frequency. Example 1: Given that barium titanate Nt = 2520Hz·m, what is the thickness of the chip if a piezoelectric chip with a center frequency of 2.5MHz is to be made?

It is known that CLZ = 3780m / s for lead zirconate titanate (PZT-5A). If you want to make a piezoelectric chip with a center frequency of 5MHz, what is the thickness of the chip (8) dielectric loss. When a dielectric crystal is suddenly exposed to an electric field, the polarization intensity does not reach the final value at once, because although the orientation of molecules (electric domains) will try to follow the direction of the electric field, when they do, they will obstructed by the viscosity of the piezo ceramic ring, it is necessary to absorb energy from the electric field, which manifests itself as a relaxation time, that is, polarization is a relaxation phenomenon (polarization relaxation). If the medium is subjected to an alternating electric field and the alternating frequency is relatively high, it will cause the polarization to follow in a timely manner and lag, which will cause the so-called dielectric loss and cause the dynamic dielectric constant to differ from the static dielectric constant. Part of the energy supplied to the dielectric is consumed by forcing the rotation of the inherent electric moment and converted into thermal energy to be consumed. Another cause of dielectric loss is the leakage of the dielectric, especially under the action of high temperature and strong electric field. Due to leakage, electrical energy is converted into heat and consumed (conductance loss). We can use a parallel loss resistance Rn to represent the consumption of electrical energy in the medium. The current through the medium can be divided into a part of the IR that consumes energy and a part of the IC that does not consume energy through the pure capacitance of the medium. We use the dielectric loss tangent to represent: tgδ = IR / IC = 1 / ωC0Rn where ω is the circular frequency of the alternating electric field; C0 is the electrostatic capacitance value of the dielectric sample with the electrodes; δ is the hysteresis of current versus voltage Angular dielectric loss tangent is also called dielectric loss, dielectric loss factor, and it is related to electric field strength, temperature and frequency.

(9)Electrical quality factor Qe

(10)The inverse of the dielectric loss tangent is the electrical quality factor: Qe = 1 / tgδ = ωcorn at resonance: Qe = (π / 4K2) (Zl / ZC), where K is the electromechanical coupling coefficient; Zl is the acoustic impedance of the load ; ZC is the acoustic impedance of the piezoelectric body. The electrical quality factor Qe is defined as: Qe = electric energy stored by the piezoelectric vibrator at resonance / electric energy lost during the resonance cycle. It reflects the amount of electrical energy (converted into thermal energy) consumed by the piezoelectric body under the action of an alternating electric field. A larger Qe means less power loss. The existence of Qe shows that it is impossible for any piezoelectric material to completely convert electrical energy into mechanical energy, and the reason for its energy loss is the above-mentioned dielectric loss.