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First, the piezoelectric equation.

For the performance of piezoelectric materials, we have the following four considerations: 1. Piezoelectric materials are elastomers, which obey Hooke's law in terms of mechanical effects, that is, the elastic relationship between stress τ and strain e: τ = ce Or e = sτ where c is the elastic modulus, also known as the elastic stiffness constant or elastic stiffness constant, which represents the force required for the unit to produce a unit strain; s is the elastic compliance coefficient, it is also known as the elastic compliance constant, which represents the stress and the relationship between strains and s = 1 / c. The physical meaning of the above relationship is within the limit of elasticity, the stress of the elastomer is proportional to the stress.

Piezoelectric materials are ferroelectrics. In the electrical effect, the electrical parameters,the electric field strength E and the electric displacement strength D follow a dielectric relationship: E = βD or D = εE, where ε is the permittivity, and It is called the dielectric constant (unit: law / meter), it reflects the dielectric properties of the material, and reflects the polarization properties of the piezoelectric body, which is related to the capacitance formed by the electrodes attached to the piezoelectric body, that is, the capacitance C = εA / t, where A is the relative area of the two pole plates, t is the distance between the two poles or the thickness of the piezoelectric chip, and is therefore related to the electrical impedance of the piezoelectric ceramics transducer. Dielectric constant ε is usually expressed by relative dielectric constantεr, and its value is equal to the ratio of dielectric capacitance to vacuum capacitance under the same electrode: εr = C dielectric / C vacuum = ε dielectric / ε vacuum M), β is the dielectric induction coefficient, also known as the dielectric isolation rate, it indicates how fast the electric field of the dielectric changes with the electric displacement vector, and β = 1 / ε, but this coefficient is generally less used. The physical meaning of the above dielectric relationship expression is that when a dielectric is in the electric field E, the electric field inside the dielectric can be expressed by the electric displacement D.

The magnetic effects of piezoelectric materials piezo ceramic disc crystal B = μH, where B is the magnetic induction strength, H is the magnetic field strength, and μ is the magnetic permeability. Among the thermal effects of piezoelectric materials, Q = φσ / ρc, where Q Is heat; φ is temperature; σ is entropy; ρ is medium density; c is material specific heat. For piezoelectric bodies, we usually don't consider the magnetic effect and think that there is no heat exchange during the piezoelectric effect ( this is not true, but these two aspects are omitted when simplifying the analysis). Therefore, only the mechanical and electrical effects described above are generally considered, and the interactions between them must also be considered at the same time. The two mechanical quantities, stress τ and strain e, and the two electrical quantities, electric field strength E and electric displacement strength D, are related. The expression describing the interaction between them is the so-called piezoelectric equation. In the working state of a piezoelectric body, its mechanical boundary conditions can be mechanical freedom and mechanical clamping, while electrical boundary conditions can be electrical short circuit and electrical open circuit. According to different boundary conditions, choose different Independent and dependent variables, different types of piezoelectric equations can be obtained.

(1) Suppose the stress τ is applied to the piezoelectric body under the condition that the electrical output is short-circuited, that is, the electric field strength E = 0, which is: D = dτ | E = 0, where d is called the piezoelectric constant and reflects the piezoelectric material .The coupling relationship between elastic properties and dielectric properties is not only related to stress and strain, but also to the strength of electric field and electric displacement. It is also called piezoelectric strain electric field constant, piezoelectric modulus, piezoelectric strain constant, piezoelectric emission coefficient, etc. Similarly, when the piezoelectric body generates strain e under the action of the stress τ, there are: D = ie, where the proportionality coefficient i is also the piezoelectric constant, which is called the piezoelectric stress electric field constant, which is also called the piezoelectric stress constant and piezoelectric emission. Assuming that the stress τ is applied to the piezoelectric body under the condition of an electrical open circuit, that is, the output current I = 0, E = -gτ | I = 0, and the piezoelectric constant g in the formula is called piezoelectric strain electrical induction. Constants are also referred to as electric field stress constants, piezoelectric strain constants, piezoelectric voltage constants, and piezoelectric acceptance coefficients. when the strain e is generated by the piezoelectric body under the stress τ, there are: E = -he. The piezoelectric constant h in the formula is called the piezoelectric stress electrical induction constant, which is also called the piezoelectric strain constant and piezoelectric stiffness. piezoelectric acceptance coefficient, etc. The above four equations actually reflect the case of the positive piezoelectric effect.

(2) Assuming that the piezoelectric body does not bear external force and the stress is zero, that is, τ = 0, the piezoelectric body can deform freely. Under this condition, an electric field is applied, the relationship between the strain e and the electric field strength E is: e = dE | τ = 0, where d is the piezoelectric strain constant. The relationship between the strain e and the electrical displacement intensity D is: e = gD, where g is the piezoelectric voltage constant. If the piezoelectric body is clamped so that it cannot deform, the strain is zero, that is, e = 0. Under this condition When an electric field is applied.The relationship between the stress τ and the electric field strength E is: τ = -iE ｜ e = 0, where is the piezoelectric stress constant, and the relationship between the stress τ and the electric displacement strength D is: τ = -hD, where h is the piezoelectric strain constant. The above four equations reflect the situation of the inverse piezoelectric effect Pzt material piezoelectric ceramic. In practical applications, mechanical quantities and electrical quantities always exist at the same time, so we can get the following four sets of piezoelectric equations. Pay attention to understand the relationship between the parameters through the piezoelectric equation, and we should mainly understand its physical meaning:

(1) Type d piezoelectric equation: e = sEτ + dE D = dτ + ετE where d is the piezoelectric strain constant; sE = 1 / cE is the elastic compliance coefficient when the electric field strength E is constant (superscript indicates this parameter (constant, the same applies hereinafter); ετ is the dielectric constant when the stress τ is constant.

(2) g-type piezoelectric equation: e = sDτ + gD E = -gτ + βτD where g is the piezoelectric voltage constant; sD = 1 / cD is the elastic compliance coefficient when the electric displacement intensity D is constant; βτ = 1 / ετ is the dielectric induction rate when the stress τ is constant.

(3) Type i piezoelectric equation: τ = cEe-iE D = ie + εeE .where is the piezoelectric stress constant; cE is the elastic modulus when the electric field strength E is constant; εe is the dielectric constant when the strain e is constant constant.

(4) h-type piezoelectric equation: τ = cDe-hD E = -he + βeD where h is the piezoelectric strain constant; cD is the elastic modulus when the electrical displacement strength D is constant; βe = 1 / εe is the strain dielectric induction at constant . The above four sets of piezoelectric equations can be obtained as follows: (1), d = (δe / δE) τ = (δD / δτ) E (meter / volt or Coulomb / Newton) (δ is used to represent the partial differential symbol) This means the relative strain caused by the electric field when the stress is constant or the relative electrical displacement caused by the stress when the electric field strength is constant.

(5)g = (-δE / δτ) D = (δe / δD) τ (volt meter / Newton or meter 2 / Coulomb) This means that the electric field strength change caused by stress (relative open circuit voltage) is unchanged when the electric displacement intensity is unchanged ), Or the relative strain caused by the electrical displacement strength when the stress is constant.

(6) i = (-δτ / δE) e = (δD / δe) E (Newton / volt meter or Coulomb / meter 2) This means the relative stress caused by the electric field when the strain is constant, or Relative electrical displacement caused by strain.

(7) h = (-δE / δe) D = (-δτ / δD) e (Newton / Coulomb or Volt / meter) This means that the electric field strength change caused by the strain (relative open circuit voltage) when the electric displacement strength is constant. , Or the relative stress caused by the electric displacement strength when the strain is constant. d and i represent the strain or stress change caused by the electric field, that is, the inverse piezoelectric effect. In practical applications, they reflect the ability of piezoelectric materials to emit ultrasonic waves, especially with d as the most important and most commonly used. The larger d and i, the larger the sound pressure generated by the same electric field strength, or as long as a smaller alternating voltage is applied, a larger amplitude can be obtained, that is, a larger mechanical output power can be obtained. g and h represent the change in electric field strength caused by stress or strain, that is, the positive piezoelectric effect. In the practical applications, they reflect the ability of piezoelectric materials to receive ultrasonic waves, with g being the most important and most commonly used. The greater g and h, the higher the relative open circuit voltage generated under the same stress or strain condition, or even the weaker ultrasonic wave can generate a larger relative open circuit voltage, that is, the higher the receiving sensitivity. These four parameters have the following conversion relationship: d = ετg = ieE; g = βτd = heD; i = εeh = dcE; h = βei = gcD

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