Description of the piezoelectric equation
For the properties of piezoelectric materials, we have the following four aspects to consider:
1 piezoelectric material is elastic, which obeys Hooke's law in the mechanical effect on the stress and strain, namely tau obeying the elastic relationship between E: =ce or e=s tau tau, type C is also known as the elastic modulus, elastic stiffness constants or elastic stiffness constants, representation of an object is the force required to produce unit strain; s elastic compliance coefficient, also called the elastic compliance constant material, said the relationship between stress and strain and s=1/c.
The physical significance of the relation is: in the limit of elasticity, elastic stress and strain are proportional.
2 piezoelectric materials is ferroelectrics, it in the electrical effect, obey the dielectric relation between the electrical parameter - E electric field intensity and electric displacement intensity of D: E= beta D or D= E, type E as a capacitance ratio, also known as dielectric constant (unit: method / M), which reflect the dielectric the electrical properties of materials, the piezoelectric body is the reflection of the polarization properties of the piezoelectric body, and an electrode attached capacitance.
That is, the capacitor C= epsilon A/t, type A is two plate relative area, t is the distance between the two poles or piezoelectric wafer thickness, and associated with the resistance of the piezoelectric body resistance.
The dielectric constant of the commonly used relative dielectric constant epsilon r said, its value is equal to the ratio of dielectric capacitor and the vacuum capacitor electrode under the condition of same: epsilon r=C mediated /C vacuum = epsilon / epsilon (epsilon mediated vacuum vacuum =8.85x10-2 / M)
Beta for the dielectric coefficient of induction, also known as the dielectric isolation rate, it said dielectric electric field along with the electric displacement vector change speed, and beta =1/ epsilon, but this coefficient is generally used less. The physical meaning of the dielectric relation is: when a dielectric in electric field E, the electric field inside dielectric can use the electric displacement D said.
3 piezoelectric materials in magnetic effect: B= mu H, type B for the magnetic induction intensity for magnetic field strength, H, Mu is the permeability
4 piezoelectric materials in thermal effect in Q=: Phi sigma / P C, type Q is heat; Phi sigma temperature; entropy; P as medium density; C is the specific heat.
For the piezoelectric body, we usually do not consider the magnetic effect and that the piezoelectric effect in the process without heat exchange (of course this is not true, but simply omitted in the simplified analysis, these two aspects). Therefore, the general only consider mechanical effect and electrical effects mentioned above, but also have to consider the interaction between them.
The two mechanical quantity -- stress tau and strain E and two electrical quantities -- E electric field intensity and electric displacement intensity D together, expression describing the interaction between them is called the piezoelectric equation.
In the piezoelectric body under the working state, the mechanical boundary conditions can be a mechanical freedom and mechanical clamping two kind of situations, and the electric boundary conditions are electrical short circuit and electrical open two cases, according to the different boundary conditions, the choice of different independent variables and dependent variables, you can get the piezoelectric equations of different types.
1) assumed in the electrical output short circuit, i.e. E=0 electric field intensity under the conditions of the piezoelectric body applied stress tau, tau, are: D=d E=0, type in D is called the piezoelectric constant, reflecting the coupling between the piezoelectric material elastic properties and dielectric properties, it is not only related with the stress, strain, and the strength is related to electric field strength, electric displacement, it is also called the piezoelectric strain field constants, piezoelectric modulus, piezoelectric strain constants, piezoelectric emission coefficient.
Similarly, the piezoelectric stress in the tau generated under the action of strain e,: D=ie, type of proportional coefficient I is the piezoelectric constant, called piezoelectric stress constant electric field, also known as the piezoelectric stress constant, piezoelectric emission coefficient.
Assume that the open circuit state in power, namely the output current of I=0 under the conditions of the piezoelectric body applied stress tau, tau, are: E=-g I=0, type in the piezoelectric constant G is called the piezoelectric strain constant electric induction, also known as the electric field stress constants, piezoelectric strain constants, piezoelectric voltage constant, pressure electric receiving coefficient.
Or, the piezoelectric stress in the tau generated under the action of strain e,: E=-he, type in the piezoelectric constant h is called the piezoelectric stress induction constant, also known as the piezoelectric strain constants, piezoelectric stiffness constant, piezoelectric receiving coefficient. The above four equations are actually reflects positive piezoelectric effect of the situation.
2) assumes that the piezoelectric body does not bear the external force, stress is zero, that is =0, the piezoelectric free deformation, under this condition, the applied electric field, then:
The relationship between strain E and electric field strength of E: e=dE| =0, type D piezoelectric strain constant, the relationship between strain E and the electric displacement intensity of D is: e=gD, type G piezoelectric voltage constant. If the piezoelectric body clamping, so that it can not deformation, strain is zero, which is e=0 under this condition, the applied electric field, then:
The relationship of E tau and electric field strength for: tau =-iE ~ e=0, type I for the piezoelectric stress constant, the relationship between stress and electric displacement intensity of D tau: =-hD, type H for the piezoelectric strain constant, the above four equations reflects the inverse piezoelectric effect of the situation.
In practical application, always mechanical quantity and electric quantity at the same time, so we can get the following four groups of piezoelectric equation: pay attention to understand the relation between the parameters through the piezoelectric equation, the main should understand its physical meaning:
1) d type piezoelectric equation: e=sE +dE, D=d tau + epsilon tau E type D as the piezoelectric strain constant; sE=1/cE E electric field intensity is constant elastic compliance coefficient (superscript indicates that the parameters are constant, the following are the same); epsilon tau tau for stress constant dielectric constant.
2) g type piezoelectric equation: e=sD +gD, E=-g tau + beta tau D type G as the piezoelectric voltage constant; sD=1/cD for the electric displacement intensity D constant elastic compliance coefficient; beta tau epsilon tau tau =1/ stress constant dielectric timing induction rate.
3) I type piezoelectric equation: =cEe-iE, D=ie+ e eE type I piezoelectric stress constant; cE E electric field intensity is constant elastic modulus; epsilon e strain e as a constant dielectric constant.
4) H type piezoelectric equation: =cDe-hD, E=-he+ beta eD; type H piezoelectric strain constant; cD for the electric displacement intensity D constant elastic modulus; beta e=1/ epsilon e strain e as a constant dielectric timing induction rate.
Four groups of piezoelectric equations above we can get the following solution:
I. d= (delta e/ Delta E) t = (delta D/ delta tau) E, (M / V or Coulomb / Newton) (here using delta said partial differential symbols, the same below), which express the relative potential relative strain or stress caused by electrical field strength unchanged unchanged by the stress induced shift.
II. g= (delta E/ delta tau) D= (delta e/ delta tau (D), volt meter / Newton or m 2/ Coulomb), which expresses the change of electric field intensity of the electric displacement intensity constant caused by stress (relative to the open circuit voltage), the relative strain or stress remains unchanged by the electric displacement intensity caused by.
IV. i= (delta tau Delta (delta D/ Delta E) e= E) E, (Newton / volt meter or Coulomb / M 2), this indicates strain constant relative stress induced by the electric field, or the relative potential caused by the strain field intensity constant shift.
IIV. h= (delta E/ Delta D= (E) - delta tau delta e (D), the Newton / Coulomb or V / M), which expresses the change of electric field strength caused by the same strain electric displacement intensity when the (relatively open circuit voltage), or strain constant by potential shift caused by the relative stress intensity.
D and I represent the strain caused by the electric field or stress changes, i.e., the inverse piezoelectric effect. In practical application, which reflects the ability of pressure transmitting ultrasonic electric material, in particular to D is the most important and most commonly used. D and I is larger, the means of generating the same electric field intensity of the greater sound pressure, or alternating voltage applying only smaller, can obtain larger amplitude, is also can obtain larger output power machinery.
G and H represent the changes of the electric field strength caused by stress or strain, which is the direct piezoelectric effect. In practical application, they reflect the ability of piezoelectric materials and receiving ultrasonic waves, especially to G is the most important and most commonly used. G and H is more big, mean the same stress or strain conditions relative to the open circuit voltage is high, or even to receive weak ultrasonic can also produce relatively large open circuit voltage, i.e. the higher receiver sensitivity.
