Comparison of the calculated method to the driving voltage applied across the lay in single and double layers of piezoelectric material of active sound absorption

2.1. Theoretical method

Several authors have presented work on the following topics: the derivation of the relation between normal forces on opposite faces of a thin piezoelectric material, the velocities of those faces, the voltage across and current through the piezoelectric material. Consider a piezoelectric material, which consists of a non-conducting piezoelectric layer of thickness $t$ much smaller than the dimensions of its faces, each of area $A$. Denoting rectangular coordinates by $(x_1,x_2,x_3)$, we define the $x_3$ axis normal to the faces of the piezoelectric material so that its thickness extends from $x_3=\text{0}$ (face 1) to $x_3=t$ (face 2).

For harmonic electric and/or acoustic excitation at angular frequency $\omega$, we represent (1) all time-dependent quantities by the usual product of constant complex amplitude and (2) a time-dependent exponential. For example, at time $t$ the force $f_1$ on face 1 of the piezoelectric material is $f_1\left(t\right)=F_1{e}^{j\omega t}$, where $F_1$ is the complex amplitude of that force. Thus we define $F_i$ and $U_i$ ($i=\text{1,}\text{2}$) as complex amplitudes of the normal forces and velocities of the piezoelectric material faces. We also define $V$ and $I$ as the complex amplitudes of the voltage across and current into the piezoelectric material. The sign conventions of these quantities are indicated in Fig. 1.

The relation between the complex amplitudes described above can be found using the fact that for $0<x_3<t$. The elastic stress and strain, and the electric displacement and field intensity must satisfy the following: the usual mechanical laws of motion, Maxwell's equations, and the constitutive piezoelectric equations for the piezoelectric material. The interaction between these fields depends on the orientation of the symmetry directions of the material relative to the piezoelectric material faces. Let us assume that the material is isotropic in the $x_1-x_2$ direction and is polarized in the $x_3$ direction. If field disturbances are limited to the $x_3$ direction, they may be related using only the elements $c_t^D$, $e_{33}$, and ${\epsilon }_{33}^S$ of the elastic stiffness, piezoelectric stress constant, and dielectric impermissibility matrices. As a result of these assumptions, one obtains [6]:

In the above, $v_x=\sqrt{\frac{c_t^D}{\rho }}$ is the velocity of longitudinal elastic (acoustic) waves traveling in the thickness direction of the piezoelectric material. $\rho$ is the density of the piezoelectric material, and $k=\frac{\omega }{v_x}$ is the acoustic number.

If we set $F_1=F_2=\text{0}$, the electrical impedance $Z_{el}=\frac{V}{I}$ of the air-loaded piezoelectric material may be obtained from Eqs. (1)-(3). The result is:

The material parameters $c_t^D$, $e_{33}$, and ${\epsilon }_{33}^S$ can thus be obtained by fitting Eq. (4) to measurements of electrical impedance versus frequency.

Suppose the above piezoelectric material separates two semi-infinite media of specific acoustic impedance $R_1$ and $R_2$ bounded by face 1 and 2, respectively. If a plane acoustic sound wave is normally incident on face 1, the net force $F_1$ results from the superposition of the incident and reflected sound waves, while $F_2$ is due to the transmitted sound wave. Letting $P_i$, $P_r$ and $P_t$ be the complex pressure amplitudes of the incident, reflected and transmitted sound waves, respectively, we have:

Insertion of Eq. (5)-(8) into Eq. (1)-(3) yields:

In the above, $\theta =kd$, and $R_x=\rho v_x$ is the specific acoustic impedance of the piezoelectric material.

In Eq. (9)-(11) $J=\frac{I}{A_0}$ is the complex amplitude of the current density into face 1 of the piezoelectric material.

Eq. (9)-(11) are simultaneous equations relating the complex amplitudes $P_i$, $P_r$, $P_t$, $V$ and $J$ of five sigmoidally varying quantities. If two of these amplitudes are specified, the equations may be solved for the remaining three. If $P_i$ is set to desired incident sound wave amplitude and $P_r=\text{0}$, thus we can have:

where $Z_{E1}=j\rho C_t^{\mathrm{D}}\mathrm{t}\mathrm{a}\mathrm{n}\frac{kd}{2}$, $Z_{E1}=\frac{-j\rho c_t^D}{\mathrm{s}\mathrm{i}\mathrm{n}kd}$, $C_0=\frac{{\epsilon }_{33}^S}{d}$, $N=\frac{e_{33}}{t}$, $d$, $\rho$, $C_t^D$, ${\epsilon }_{33}^S$, $e_{33}$ are thickness, density, elastic coefficient, dielectric constant, and coupling constant of the piezoelectric material, respectively.

2.2. Electro-acoustic analogy method

The structure of the piezoelectric material used in this paper is shown in Fig. 2.

In the Fig. 2, $d$ is the thickness of the piezoelectric material. $U_1$, $U_2$ are the vibration velocity in the direction of thickness. $F_1$, $F_2$ are force press on the piezoelectric material. $V$ is voltage. According to the structure of the piezoelectric material, the equivalent circuit figuration is attained and shown in Fig. 3.

Fig. 2Structure of the piezoelectric ceramic

Fig. 3The equivalent circuit figuration

In the Fig. 3, $Z_{E1}=j\rho C_t^{\mathrm{D}}\mathrm{t}\mathrm{a}\mathrm{n}\frac{kt}{2}$, $Z_{E1}=\frac{-j\rho C_t^D}{\mathrm{s}\mathrm{i}\mathrm{n}kt}$, $C_0=\frac{{\epsilon }_{33}^S}{t}$, $N=\frac{e_{33}}{t}$, $t$, $\rho$, $C_t^D$, ${\epsilon }_{33}^S$, $e_{33}$ are thickness, density, elastic coefficient, dielectric constant, and coupling constant of the piezoelectric material, respectively. The following equations can be derived according to the electro circuit theory:

According to the Eq. (5)-(8) and the Eq. (15)-(17), the following equations yield:

In the Eq. (18)-(20), there are five complex amplitudes $p_i\text{,}{p}_r\text{,}{p}_t\text{,}V\text{,}I$. If two of these amplitudes are specified, the equations may be solved for the remaining three. For example, if $p_i$ is set to desired incident wave amplitude and $p_r$ is set to zero, the transformed equations read:

From the Eq. (21)-(23), the voltage $V$ of the piezoelectric material is attained:

According to the Eq. (24), the voltages applied on the piezoelectric material can make the reflected wave be zero and the aim of active absorption is gained.

According to the Eq. (12)-(14) and Eq. (24), the calculated results are the same, thus denote that the methods are correct and applicable.

The density of the air is $\rho =\text{1.21}\text{kg/}{\text{m}}^{\text{3}}$. The propagation velocity of sound wave in the air is $c=\text{343}\text{m}\text{/}{\text{s}}^{\text{-1}}$. The density, the propagation velocity of sound wave in the piezoelectric material, and the dielectric constant of the piezoelectric material, respectively, are $\rho =\text{5400}\text{kg/}{\text{m}}^{\text{-3}}$, $c_t=\text{4540}\text{m/s}$, ${\epsilon }_{33}^S=\text{1200}{\epsilon }_0$. The diameter of the piezoelectric material is $d=\text{29}\text{mm}$ and the thickness is $t=\text{1.5}\text{mm}$. If the amplitude of the incidence wave is 1.00 Pa, the theoretical calculated amplitude and phase of the piezoelectric material, according to the Eq. (24), are shown in Fig. 4.

Fig. 4The theoretical calculated amplitude and phase of the piezoelectric ceramic

For the same piezoelectric material, the thickness is $t=\text{2.5}\text{mm}$. If the amplitude of the incidence wave is 1.00 Pa, the theoretical calculated amplitude and phase of the piezoelectric material, according to the Eq. (24), are shown in Fig. (5).

Fig. 5The theoretical calculated amplitude and phase of the piezoelectric ceramic

For the same piezoelectric material, the thickness is $t=\text{3.5}\text{mm}$. If the amplitude of the incidence wave is 1.00 Pa, the theoretical calculated amplitude and phase of the piezoelectric material according to the Eq. (24) are shown in Fig. (6).

Fig. 6The theoretical calculated amplitude and phase of the piezoelectric ceramic

According to the Fig. 4-6, the conclusion can be drawn:

(1) For the piezoelectric material of the same thickness, the amplitude and the phase of the voltage are diminished when the incident frequency is added.

(2) For the same frequency of the incident wave, the amplitude of the voltage is added and the phase is diminished when the thickness of the piezoelectric material is added.

3.1. Theoretical method

We now consider the piezoelectric ceramic, which consists a sandwich of two piezoelectric layers of the type described above, with face 2 of one layer fixed to face 1 of the other layer. For each layer a set of equations equivalent to Eq. (1)-(3) may be written. And yet, in order to allow for different material properties and thicknesses, we subscript the characteristic quantities of the layer “1” and “2”. We now have the additional boundary conditions and the forces and velocities of the two surfaces in contact are equal. The result of this boundary condition is a set of equations relating seven complex amplitudes, i.e.:

Expressions for the coefficients $D_{ij}$ are given in the Appendix.

Equations (25)-(28) play a similar role for the double piezoelectric ceramic as Eqs. (9)-(11) for the single layer, i.e., they relate the complex amplitudes $P_i$, $P_r$, $P_t$, $V_1$, $J_1$, $V_2$ and $J_2$ to the complex coefficients $D_{ij}$. In this case, for example, if we set $P_i$ to a desired incident wave amplitude and $P_r=P_t=\text{0}$, according to the Equation (1)-(3) and the Equation (5)-(8), the following formula can be obtained:

where $Z_{E11}=j{\rho }_1{C}_{1t}^{\mathrm{D}}\mathrm{t}\mathrm{a}\mathrm{n}\frac{k_1{t}_1}{2}$, $Z_{E21}=\frac{-j{\rho }_1{C}_{1t}^D}{\mathrm{s}\mathrm{i}\mathrm{n}{k}_1{t}_1}$, $C_1=\frac{{\epsilon }_{133}^S}{t_1}$, $N_1=\frac{e_{133}}{t_1}$, $Z_{E12}=j{\rho }_2{C}_{2t}^{\mathrm{D}}\mathrm{t}\mathrm{a}\mathrm{n}\frac{k_2{t}_2}{2}$, $Z_{E22}=\frac{-j{\rho }_2{C}_{2t}^D}{\mathrm{s}\mathrm{i}\mathrm{n}{k}_2{t}_2}$, $Z_{E22}=\frac{-j{\rho }_2{C}_{2t}^D}{\mathrm{s}\mathrm{i}\mathrm{n}{k}_2{t}_2}$, $C_2=\frac{{\epsilon }_{233}^S}{t_2}$, $N_2=\frac{e_{233}}{t_2}$.

3.2. Electro-acoustic analogy method

According to the structure of the piezoelectric ceramic, the equivalent circuit figuration of the front piezoelectric ceramic is attained and shown in Fig. 8.

Fig. 8The equivalent circuit figuration of the front piezoelectric ceramic

In the Fig. 8, $Z_{E11}=j{\rho }_1{C}_{1t}^{\mathrm{D}}\mathrm{t}\mathrm{a}\mathrm{n}\frac{k_1{t}_1}{2}$, $Z_{E21}=\frac{-j{\rho }_1{C}_{1t}^D}{\mathrm{s}\mathrm{i}\mathrm{n}{k}_1{t}_1}$, $C_1=\frac{{\epsilon }_{133}^S}{t_1}$, $N_1=\frac{e_{133}}{t_1}$, $t_1$, ${\rho }_1$, $C_{1t}^D$, ${\epsilon }_{133}^S$, $e_{133}$ are thickness, density, elastic coefficient, dielectric constant, and coupling constant of the front piezoelectric ceramic, respectively. The following equations can be derived according to the electro circuit theory:

The equivalent circuit figuration of the back piezoelectric ceramic is shown in the Fig. 9^.

Fig. 9The equivalent circuit figuration of the back piezoelectric ceramic

In the Fig. 9, $Z_{E12}=j{\rho }_2{C}_{2t}^{\mathrm{D}}\mathrm{t}\mathrm{a}\mathrm{n}\frac{k_2{t}_2}{2}$, $Z_{E22}=\frac{-j{\rho }_2{C}_{2t}^D}{\mathrm{s}\mathrm{i}\mathrm{n}{k}_2{t}_2}$, $C_2=\frac{{\epsilon }_{233}^S}{t_2}$, $N_1=\frac{e_{233}}{t_2}$, $t_2$, ${\rho }_2$, $C_{2t}^D$, ${\epsilon }_{233}^S$, $e_{233}$ are thickness, density, elastic coefficient, dielectric constant, and coupling constant of the back piezoelectric ceramic, respectively. The following equations can be derived according to the electro circuit theory:

According to the Equations (36)-(38), the following equations yield:

According to the Eq. (39)-(42), the voltages applied on the piezoelectric ceramics can make the reflected wave and transmission wave be zero. The aim of active absorption and active isolation is gained.

According to the Eq. (29)-(32) and Eq. (39)-(42), the calculated results are the same, thus denote that the methods are correct and applicable.

Fig. 10The theoretical calculated amplitude and phase of the front piezoelectric ceramic

The front and back of piezoelectric ceramic are the same. The density, the acoustic impedance, the dielectric constant, and coupling constant of the piezoelectric ceramics, respectively, are $\rho =\text{2430}\text{kg}\text{/}{\text{m}}^{\text{-3}}$, $\rho C_t^D=\text{7.6×}{\text{10}}^{\text{6}}\text{kg}{\text{(}{\text{m}}^{\text{2}}\text{/}\text{s)}}^{\text{-1}}$, ${\epsilon }_{33}^S=\text{174.3}{\epsilon }_0$, $e_{33}=\text{3.95}\text{C/}{\text{m}}^{\text{-2}}$. The diameter of the piezoelectric ceramic is $d=\text{100}\text{mm}$ and the thickness is $t=\text{1.5}\text{mm}$. If the amplitude of the incidence wave is 1.00 Pa, the theoretical calculated amplitude and phase of the front piezoelectric ceramic according to the Eq. (41)-(44) are shown in Fig. 10.

The theoretical calculated amplitude and phase of the back piezoelectric ceramic according to the Eq. (41)-(42) are shown in Fig. 11.

Fig. 11The theoretical calculated amplitude and phase of the back piezoelectric ceramic