Structure design and sensitivity analysis of flexible ultrasonic transducer array

2.1. Piezoelectric effect

The piezoelectric effect was first discovered in 1880 [13]. Under the influence of external forces and electric fields, this effect demonstrates the special electromechanical coupling properties of crystalline materials: when mechanical stress is applied along the crystal in a particular direction, the internal ions will experience relative displacement, changing the electric dipole moment. This is followed by the formation of equal and opposite sign charges on the crystal surface, creating a potential difference. Surface charge density has a linearly positive correlation with applied stress, according to experimental research, and this polarization effect is reversible.

The direct and inverse piezoelectric effects are the two types of piezoelectric effects that can be distinguished. The inverse piezoelectric effect is the property where applying an alternating electric field to a piezoelectric material causes mechanical deformation; the direct piezoelectric effect is the phenomenon where a material experiences internal electrical polarization under mechanical stress, resulting in induced charges on its surface. There is a reversible link between these two phenomena. Based on this idea, piezoelectric transducer technology is now widely used in electromechanical conversion devices like sonar systems and ultrasound imaging.

2.2. Piezoelectric equation

Piezoelectric materials remain electrically neutral when no external force is present. A potential shift is created when mechanical stress is applied because of the positive piezoelectric effect, which causes the material’s surface charges to accumulate. Eq. (1) illustrates the constitutive link between the potential shift $D$ and stress, which is linearly positively connected. The material maintains a zero strain condition in the absence of an external electric field; the inverse piezoelectric effect causes structural strain when an electric field $E$ is applied. Eq. (2) illustrates the constitutive relationship between the strain and electric field strength $E$, which satisfy a linear relationship:

where, $d$ represents the piezoelectric constant of the piezoelectric material, reflecting the coupling between electric displacement and stress.

Furthermore, as Eq. (3) illustrates, the tension and strain produced by piezoelectric materials adhere to Hooke’s law:

where $k$ is the Young’s modulus of piezoelectric material.

Eq. (4) illustrate the connection between the applied electric field $E$ and the potential shift $D$ produced by piezoelectric materials:

The dielectric constant of piezoelectric materials is denoted by $\epsilon$. Its dielectric constant also fluctuates under certain mechanical circumstances.

The dielectric constant, elasticity, and piezoelectricity of piezoelectric materials are represented in matrix form in three dimensions. Eq. (5) displays the dielectric constant matrix of piezoelectric materials:

Eq. (6) displays the matrix of piezoelectric constants:

Eq. (7) displays the matrix of elastic constants:

Both mechanical and electrical boundary constraints govern the behavior of piezoelectric ultrasonic transducers, and four different types of piezoelectric equations can be used to characterize their constitutive relationship. Piezoelectric materials' potential shift response and strain response can be caused by mechanical loads as well as external electric fields within the elastic limit. Two common operating circumstances are included in the electrical boundary conditions: open circuit and short circuit. Accordingly, mechanical freedom and mechanical clamping are the two states into which the mechanical boundary conditions can be separated [14-16].

The first type of boundary condition is mechanical freedom $\left(\sigma =0,\omega \ne 0\right)$, electrical short circuit $\left(E=0,D\ne 0\right)$:

where, $d$ is the piezoelectric strain constant, $d^t$ is the transpose of $d$, $T$ is the stress, $S$ is the strain, ${\epsilon }^T$ is the dielectric constant under constant stress, and $S^E$ is the flexibility matrix.

The second type of boundary condition is mechanical clamping $\left(\sigma \ne 0,\omega =0\right)$, electrical short circuit $\left(E=0,D\ne 0\right)$:

where, ${\epsilon }^S$ is the dielectric constant under constant strain, $e$ is the piezoelectric stress constant, $e^t$ is the transpose of $e$, and $c^E$ is the elastic stiffness under a constant electric field.

The third type of boundary condition is mechanical freedom $\left(\sigma =0,\omega \ne 0\right)$, electrical open circuit $\left(E\ne 0,D=0\right)$:

where, $g$ is the dielectric constant under constant strain, $g^t$ is the transpose of $g$, ${\beta }^T$ is the dielectric isolation ratio under constant stress, and $s^D$ is the flexibility coefficient under constant displacement.

The fourth type of boundary condition is mechanical clamping $\left(\sigma \ne 0,\omega =0\right)$, electrical open circuit $\left(E\ne 0,D=0\right)$:

where, $h$ is the piezoelectric constant, $h^t$ is the transpose of $h$, ${\beta }^S$ is the dielectric isolation ratio under constant strain,, and $c^D$ is the elastic stiffness under constant displacement.

2.3. Electromechanical coupling coefficient

The Electromechanical Coupling Coefficient (ECC) is a key parameter that characterizes the electromechanical energy conversion efficiency of flexible ultrasonic transducers. Its value directly determines the energy conversion efficiency of the device in the electric vibration field coupling, and thus affects the three key performance indicators of the transducer: sensitivity, operating bandwidth, and conversion efficiency [17]. This coefficient is not an independent parameter, but a composite parameter determined by the piezoelectric constant $d$, elastic constant $s$, and dielectric constant $\epsilon$ together. Therefore, the numerical value of the electromechanical coupling coefficient can serve as a quantitative standard for evaluating the performance of transducers.

The impedance ($Z$) of a piezoelectric vibrator near its resonant frequency can be expressed as:

where, $C_0$ represents the static capacitance, $l$ denotes the thickness, and $v$ signifies the speed of sound.

Under resonance conditions, impedance is minimal; under antiresonance conditions, impedance is maximal. Solving for $f_a$ and $f_r$ yields, and thus obtain:

where, $f_a$ denotes the resonant frequency, $f_r$ denotes the anti-resonant frequency, and $K_t$ denotes the electromechanical coupling coefficient.

Considering the nonlinearity of the tangent function, a more precise formula is obtained as:

The final Eq. (15) allows the calculation of the electromagnetic coupling coefficient based on the resonance frequency $f_a$ and anti-resonance frequency $f_r$:

2.4. Ultrasonic phased control technology

The approach known as Phased Array Ultrasonic Technology (PAUT) carefully regulates the phase connection between the sent and received signals of each member in a transducer array to accomplish beam focusing, dynamic deflection, and scanning. The foundation of ultrasonic phased array technology is Huygens’ principle, which determines the delay relationship of each array element based on the sound path difference of each array element and the sound path depending on the distance between the array element and the focus point. Sound waves’ focusing properties and direction of propagation can be altered by varying the time delay of each array element [18]. Fig. 1 illustrates the ultrasonic phased array technology's focusing concept.

The physical basis of ultrasonic phased array technology includes the principle of wave superposition, coherent interference effect, and Huygens Fresnel principle. Phased array multi element probes provide major advantages in defect detection accuracy and imaging resolution over conventional single element probes because they can electronically deflect and dynamically focus sound beams through time delay control [19, 20].

3.1. Overview of finite element simulation analysis method

Finite Element Analysis (FEA) is a numerical analysis method based on the variational principle, which is suitable for solving multi physics coupling problems, including but not limited to structural mechanics, heat conduction, electromagnetic fields, and fluid dynamics. Its core lies in discretizing the continuous domain into a finite number of units with interpolation functions, and simulating the behavior of actual systems through mathematical approximations and numerical calculations [21, 22].

Fig. 1Ultrasonic phased array focusing schematic diagram

The standard analysis process consists of three key stages:

1) Preprocessing stage: Complete geometric modeling, mesh generation, material parameter definition, and boundary condition setting.

2) Solution stage: Based on the theory of multiphysics coupling, solve nonlinear algebraic equations, focusing on complex working conditions such as material nonlinearity, geometric nonlinearity, and transient response.

3) Post processing stage: Verify results and design iterations through visualization of field variables (such as stress cloud maps and displacement vector maps).

FEA is widely used in structural stress analysis, thermal management of electronic components, fluid flow simulation, and electromagnetic equipment design, especially in multi physics field coupling (such as thermal structural, acoustic solid coupling) where it has significant advantages.

This article uses finite element multi physics simulation software to model and simulate the single element and array form of flexible ultrasonic transducers. The specific simulation analysis process is shown in Fig. 2.

3.2. Structural design of single element flexible ultrasonic transducers

The flexible ultrasonic transducer array is composed of many elements, and the physical characteristics of a single element transducer largely determine the ability of the flexible ultrasonic transducer array to achieve sound beam focusing. In order to better study the performance of the flexible ultrasonic transducer array, this paper first simulates and analyzes the structural parameters of a single element flexible ultrasonic transducer.

This article uses finite element analysis software to simulate and analyze a single element flexible ultrasonic transducer, mainly exploring the influence of the width and thickness of the piezoelectric material on its performance in the single element flexible ultrasonic transducer. The structural design of the ultrasonic transducer is shown in Fig. 3.

Table 1 presents some material and dimensional parameters of the ultrasonic transducer.

Fig. 2Simulation flowchart

Fig. 3Structure diagram of the ultrasonic transducer

3.3. The influence of piezoelectric material thickness on the performance of ultrasonic transducers

Fig. 4 displays the admittance and impedance curves for various thicknesses. The resonant and antiresonant frequencies of the corresponding structures were extracted, the data recorded, and the corresponding electromechanical coupling coefficients calculated using Eq. (15). ultimately yielding the parameters of the ultrasonic transducer for different piezoelectric material thicknesses.

Table 2 records the values of various parameters for ultrasonic transducers at different piezoelectric material thicknesses. Plotting the data from Table 2 yields the variation curve of the electromechanical coupling coefficient with piezoelectric material thickness.

In this section, under the parameter setting of a piezoelectric material width of 1.8, the influence of the thickness of the piezoelectric material on the performance of the ultrasonic transducer was studied. Fig. 5 shows the variation curves of the piezoelectric coupling coefficient at different piezoelectric material thicknesses. The simulation results show that when the thickness increases from 0.1 to 1.0, the electromechanical coupling coefficient varies between 0.081 and 0.612, and when the thickness is, the electromechanical coupling coefficient of the ultrasonic transducer takes the maximum value of 0.612.

Fig. 4Conductance and impedance curves of piezoelectric material with thicknesses of 0.2 mm and 0.8 mm

a) At a thicknesses of 0.2 mm

b) At a thicknesses of 0.8 mm

Fig. 5ECC for different piezoelectric material thicknesses

3.4. The influence of piezoelectric material width on the performance of ultrasonic transducers

Fig. 6 displays the admittance and impedance curves for partial widths. The resonant and antiresonant frequencies of the corresponding structures were extracted from these curves, and the recorded data were used to calculate the electromechanical coupling coefficient of each structure based on Eq. (15). This process ultimately yielded the parameter values for the ultrasonic transducer at different piezoelectric material widths.

Fig. 6Conductance and impedance curves of piezoelectric materials with widths of 0.1 mm, 0.2 mm, 1.1 mm, and 1.2 mm

a) At a width of 0.1 mm

b) At a width of 0.2 mm

c) At a width of 1.1 mm

d) At a width of 1.2 mm

Table 3 records the values of various parameters for ultrasonic transducers with different piezoelectric material widths. Plotting the data from Table 3 yields the variation curve of the electromechanical coupling coefficient with piezoelectric material width.

In this section, under the parameter setting of a piezoelectric material thickness of 0.4, the influence of the piezoelectric material width on the performance of the ultrasonic transducer was studied. Fig. 7 shows the variation curves of the piezoelectric coupling coefficient at different piezoelectric material widths. The simulation results show that the electromechanical coupling coefficient of the ultrasonic transducer presents a sinusoidal relationship with the width. When the width is, the electromechanical coupling coefficient is very high, reaching 0.770 at 0.4, 0.692 at 0.8, 0.653 at 0.8, and 0.612 at 1.8 respectively. The electromechanical coupling coefficient keeps showing peaks and troughs as the width increases. Although the electromechanical coupling coefficient of the ultrasonic transducer is very high when the width is small, it needs to be considered that in actual preparation, if the width is too small, it will greatly increase the preparation difficulty. Moreover, the wider the ultrasonic transducer is, the narrower the ultrasonic beam it emits and the stronger the directivity. This is because the acoustic wave interference formed by large-sized transducers in space is more concentrated, and the energy is more focused in the direction of the main lobe. Meanwhile, an overly large width will lead to a significant reduction in the flexibility of the flexible ultrasonic transducer.

It can be known from the data in Table 4 that the electromechanical coupling coefficient of PZT-5H is 0.52. In the simulation results of this section: The electromechanical coupling coefficient of the structure reaches 0.770 when the width is 0.4. It reaches 0.692 when the width is 0.8. It reaches 0.653 when the width is and 0.612 when the width is 1.8, which is almost 1.5 times that of the piezoelectric ceramic sheet.

Table 4 shows the main parameters of certain types of piezoelectric materials.

Fig. 7ECC for different piezoelectric material widths

4.1. The influence of ultrasonic transducer spacing on beam focusing

Fig. 8 illustrates the focusing effect of ultrasonic beams at partial spacings. Data was extracted from this figure, ultimately yielding parameter values for different ultrasonic transducer spacings.

Table 5 records the values of various parameters at different ultrasonic transducer spacings. By plotting the data from Table 5, we can obtain a line chart showing the variation of these parameters with different ultrasonic transducer spacings.

Under the conditions of fixed parameters (the number of array elements = 7, the center frequency = 1 MHz, and the focusing distance = 25), the influence of the spacing of ultrasonic transducers on the beam focusing effect was studied. Fig. 9 is a line graph of the changes in the values of each parameter under different ultrasonic transducer spacings. As shown in Fig. 9, the sound intensity at the focusing point gradually decreases with the increase of the spacing of the ultrasonic transducers. When the spacing is 0.1 to 0.5, the longitudinal focusing distance differs significantly from the specified focusing distance. Other focusing distances do not differ much from the specified one. An increase in the spacing will make the side lobes more obvious and simultaneously lead to a decrease in the sound intensity of the main lobe. Therefore, the spacing of ultrasonic transducers should not be too large. However, the actual manufacturing process accuracy should be taken into account. If the spacing of ultrasonic transducers is too small, it will increase the precision requirements for the actual manufacturing process, which undoubtedly will increase the difficulty of actual manufacturing.

Fig. 8Ultrasonic beam focusing diagrams with spacings of 0.4 mm, 0.8 mm, 1.2 mm, and 1.8 mm

a) At a spacing of 0.4 mm

b) At a spacing of 0.8 mm

c) At a spacing of 1.2 mm

d) At a spacing of 1.8 mm

Fig. 9Line graph of the variation of the values of each parameter for different ultrasonic transducer spacing

4.2. The influence of the number of ultrasonic transducers on beam focusing

Fig. 10 illustrates the focusing effect of partial ultrasonic beams. Data was extracted from this figure, ultimately yielding parameter values for different numbers of ultrasonic transducers.

Table 6 records the values of various parameters under different numbers of ultrasonic transducers. By plotting the data results from Table 6, we can obtain a line chart showing the variation of parameter values with different numbers of ultrasonic transducers.

Under fixed parameter conditions (array element spacing = 0.8, center frequency = 1 MHz, focusing distance = 25), the influence of the number of ultrasonic transducers on the beam focusing effect was studied. Fig. 11 is a line graph of the changes in the values of each parameter under different numbers of ultrasonic transducers. As shown in Fig. 11, when the number is 14, the longitudinal focusing distance differs significantly from the set focusing distance. Other focusing distances do not differ much from the set ones. When the number is 14, the relative sound intensity of the focusing point fluctuates between 1.47 and 3.41. In addition, the sound intensity of the focusing point increases with the increase in the number of ultrasonic transducers. As shown in Fig. 11, an increase in the number of ultrasonic transducers will enhance the acoustic intensity of the main lobe, make the energy of the synthetic beam more concentrated, and achieve a more ideal focusing effect. In actual preparation, the increase in the number of ultrasonic transducer array elements is often accompanied by an increase in the number of channels in the electronic circuit, which increases the manufacturing cost. A larger number will also reduce the flexibility of the array and increase the difficulty of actual manufacturing.

Fig. 10Ultrasonic beam focusing diagrams with quantities of 3, 5, 7 and 9

a) At a quantity of 3

b) At a quantity of 5

c) At a quantity of 7

d) At a quantity of 9

4.3. The influence of the center frequency of the ultrasonic transducer on beam focusing

Fig. 12 illustrates the focusing effect of ultrasonic beams at selected center frequencies. Data were extracted from the figure, ultimately yielding parameter values for different ultrasonic transducer center frequencies.

Fig. 11Line graph of variation of values of each parameter for different number of ultrasonic transducers

Table 7 presents the values of various parameters at different center frequencies of ultrasonic transducers. By plotting the data from Table 7, we can obtain line graphs illustrating the variation of these parameters across different center frequencies.

Under fixed parameter conditions (array element spacing = 0.8, array element number = 7, focusing distance = 25), the influence of the center frequency of the ultrasonic transducer on the beam focusing effect was studied. Fig. 13 is a line graph of the changes in the values of each parameter at different center frequencies of the ultrasonic transducer. As shown in Fig. 13, when the center frequency is 1.0 to 2.5 MHz, the focusing distance is not much different from the setting, and at 1.0 MHz, the relative sound intensity reaches the maximum. When other parameters are constant, although the increase in the center frequency of the ultrasonic transducer can significantly reduce the beam width of the main lobe, it will simultaneously cause an increase in the number of side lobes and an enhancement of the relative sound pressure level. These factors will lead to significant side lobe interference to the main lobe signal.

Fig. 12Ultrasonic beam focusing diagrams with the center frequency of 1 MHz, 2 MHz, and 3 MHz

a) At a center frequency of 1 MHz

b) At a center frequency of 2 MHz

c) At a center frequency of 3 MHz

Fig. 13Line graph of variation of values of each parameter at different ultrasonic transducer center frequencies