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# About the piezoelectric property of a material characteristics table.

The piezoelectric property has shapes and the conditions of a direction. This condition is expressed in the vector and a symbol of tensor quantity. The superscript and subscript of this symbol have a meaning. Those outlines are summarized.

### Equivalent circuit

The piezoelectric element is expressed by the equivalent circuit of FIG near the resonance frequency.

The above chart is a characteristic of the impedance and phase of near the resonance frequency.
And $C_d$ is the current component flowing in the piezo element. $L_1$ and $C_1$ are the vibration mode of the piezo element. It represents the mechanical vibrations decided by the element size and elastic constant and piezoelectric constant.
$R_1$ represents the mechanical vibration loss.

### Coupling factors $k_p、k_t、k_{31}、k_{33}、k_{15}$

Electromechanical coupling factor $k$ is a factor represents the electrical and mechanical conversion capacity. It is defined as the square root of the ratio of “Arisen mechanical energy” and “Given electrical energy”, or “Given mechanical energy” and “Arisen electrical energy”. It is one of an amount representing the piezoelectric effect. The following is a practical formula to calculate the coupling factor $k$ from the resonant frequency and the anti-resonance frequency.

### Frequency constants $N_p、N_t、N_{31}、N_{33}、N_{15}$

A frequency constant $N$ serves as length ℓ and the product with resonance frequency $f_r$ of the corresponding direction. It’s used to determine the size and the resonance frequency. The frequency constant according to vibration mode is shown by the following formula.

### Dielectric constant ${\varepsilon_{11}}^T、{\varepsilon_{33}}^T$ & Capacitance $C^T$

The dielectric constant $\varepsilon^T$ is defined by the electric displacement arising when given an electric field. Used for the analysis of the piezoelectric constant by measuring the capacitance $C^T$ at a sufficiently lower frequency than the resonant frequency. The ratio of the permittivity $\varepsilon_0$ in vacuum is the relative permittivity $\varepsilon^T/\varepsilon_0$.
These relationships are expressed by the following equation.

$C^T = \varepsilon^T\cdot\displaystyle\frac{A}{t}$

The material properties table becomes the following formula to describe the relative permittivity.

$C^T = \displaystyle\frac{\varepsilon^T}{\varepsilon{_0}}\cdot\varepsilon{_0}\cdot\displaystyle\frac{A}{t}$

$A$：Electrode area [m2]　$t$：Interelectrode distance [m]　$\varepsilon{_0}\ = 8.854\times10^{-12}$ [F/m]

### Piezoelectric constants $d_{31}、d_{33}、d_{15}、g_{31}、g_{33}、g_{15}$

Piezoelectric constant is a constant that represents the largeness of the piezoelectric effect along with the coupling factor. In the piezoelectric constants there are four constants of the piezoelectric strain constant $d$, piezoelectric voltage constant $g$ and the piezoelectric stress constant $e$ and $h$. Usually, $d$ constant and $g$ constant are used. These are defined as follows.

In the definition of the previous formula, the amount of displacement with respect to the applied voltage from the $d$ constant can be calculated. Also from the $g$ constant, can be calculated output voltage with respect to the applied force.
$e$ constant and $h$ constant is in a reciprocal relation with the $g$ constant and $d$ constant.

### Elastic constants ${Y_{11}}^E、{s_{11}}^E、{Y_{33}}^E、{s_{33}}^E、{Y_{55}}^E、{s_{55}}^E$

A ratio with the longitudinal strain of the same direction as perpendicular stress is Young’s modulus $Y$. Without considering the other direction, if a particular direction the target, handled also as elastic stiffness $c$. The elastic compliance $s$ is in a reciprocal relation with the Young’s modulus $Y$.

$Y = c = \displaystyle \frac{1}{s}$

In the piezoelectric ceramics it is directly related to the frequency constant that determines the resonance frequency. And is also the amount related to the generated force.

### Poisson’s ratio $\sigma$

A poisson’s ratio is defined by the ratio of the transversal strain and a longitudinal strain, which arisen in perpendicular stress.

$\sigma = \displaystyle \frac{\alpha}{\beta} = -\displaystyle \frac{{s_{12}}^E}{{s_{11}}^E}$

Poisson’s ratio is particularly the amount related to the resonant frequency of the binding region.

### Dielectric loss $\tan \delta$

When in the piezoelectric body of lossless applying a sine wave AC electric field $E$ of angular frequency $\omega$, electric displacement $\dot{D}$ vibrates in $\pi/2$ phase advanced for the electric field $\dot{E}$. In practice, electric displacement $\dot{D}$ is $\delta$ only delayed, and arisen the loss of the phase difference. This loss will be action such as to be transformed to the dielectric heat generation. (Please refer to the following FIG.) There is a relation, as shown in the following formula to between $C_1$ and $R_1$ and $\tan \delta$ of the equivalent circuit of FIG.

### Mechanical quality factor (Mechanical$Q$) $Q_m$

The piezoelectric body has elastic loss like the dielectric loss. Against stress by AC electric field, arising a phase difference of $\delta_m$ to the strain.

The magnitude of the $Q_m$, it acting on the sharpness of the mechanical vibration in the resonance frequency.

### Curie point $T_c$

The dielectric constant $\varepsilon$ of the piezoelectric body, will increase to infinity with increasing temperature $T$. As a result, the crystal becomes unstable and will change rapidly at the temperature $\theta_0$ at which there is a crystal system. This temperature $\theta_0$ is the Curie point. It is the critical temperature at which completely depolarized. Piezoelectricity, will be lost in this temperature. Changes in the dielectric constant $\varepsilon$ in a high temperature range will be the relationship, such as the following formula.