A piezoelectric sensor global analytical modeling and integrated design method

Abstract

The application discloses a piezoelectric sensor global analysis modeling and integrated design method, and belongs to the technical field of piezoelectric sensors. The method constructs a global wave differential equation of a multilayer structure containing a backing layer, a piezoelectric layer, a matching layer and a measured medium, and integrates the sound wave propagation law and the piezoelectric layer electromechanical coupling characteristics into a unified mathematical framework; based on the interlayer interface continuity, the electrical boundary and the outer boundary acoustic impedance constraint, the analytical solution is obtained, and the sound intensity transmittance analytical solution is obtained; the parameter reduction equivalent verification is carried out on the double matching layer structure, the analytical solution is mapped and calibrated with the finite element simulation result, and then the structure parameters of each layer are determined according to the target resonance frequency, so that the deterministic integrated design is realized. The application discards the traditional discrete design and the experience trial and error mode, can accurately predict the acoustic behavior of the sensor, realizes the customized forward design, greatly improves the design precision and the research and development efficiency, and is suitable for the rapid research and development and engineering application of the multilayer structure piezoelectric sensor.

Description

Technical Field

[0001] This invention belongs to the field of piezoelectric sensor technology, specifically relating to a global analytical modeling and integrated design method for piezoelectric sensors. Background Technology

[0002] Piezoelectric sensors, based on the direct and inverse piezoelectric effects of piezoelectric materials, achieve efficient bidirectional conversion between electrical energy and mechanical / acoustic energy. They possess core advantages such as fast response speed, excellent electromechanical coupling performance, high detection sensitivity, wide frequency response range, and strong environmental adaptability, playing an irreplaceable role in industrial and scientific research scenarios such as dynamic physical quantity sensing, precision ultrasonic actuation, and vibration energy recovery. Especially in the field of health monitoring of major equipment and engineering structures, piezoelectric ultrasonic sensors, as the core sensing unit for structural damage identification and quantitative defect detection, directly determine the accuracy and reliability of structural damage identification through their resonant frequency control precision, acoustic-to-electric conversion efficiency, and controllability of complex acoustic behavior. They are crucial fundamental components ensuring the service safety of aerospace equipment, rail transit facilities, and large building structures.

[0003] Currently, such as Figure 1 As shown, the design and fabrication of existing piezoelectric ultrasonic sensors generally adopt the mainstream technical route of component-level discrete design and mechanical assembly integration. This technical route treats functional components such as the piezoelectric layer, matching layer, and backing layer as independent acoustic units, and conducts material selection, structural design, and performance simulation for each. Then, the discrete components are combined to form the sensor through processes such as bonding and mechanical assembly. In the supporting design theory system, the existing technology uses discrete theoretical models for analysis of each functional component: the design of the piezoelectric layer is based on the piezoelectric constitutive equation and the lumped parameter equivalent model; the design of the matching layer is based on the acoustic transmission line theory and acoustic impedance matching criteria; and the design of the backing layer is based on the damping sound absorption theory and sound wave attenuation characteristic analysis. The design parameters of each component can only achieve local performance optimization, and there is always a lack of a global unified theoretical model that can deeply integrate the overall geometry of the sensor, piezoelectric constitutive characteristics, sound propagation laws, and vibration dynamics characteristics.

[0004] Existing discrete design theories and technical approaches suffer from numerous insurmountable technical bottlenecks, specifically:

[0005] First, existing technologies cannot achieve high-precision quantitative analysis and prediction of the complex acoustic behavior of sensors. Existing discrete design models cannot incorporate the electromechanical coupling effect of the piezoelectric layer, the acoustic impedance characteristics of the matching layer, the damping and sound absorption characteristics of the backing, and the entire chain of acoustic behavior—including sound wave propagation, interface reflection, and multi-physics coupling within the sensor—into a single global differential equation system for unified solution. Most existing solutions roughly estimate the sensor's resonant characteristics through lumped parameter equivalence and boundary condition simplification. This is particularly problematic for piezoelectric sensors with complex structures such as double-matching layers, where reduced-order equivalence and global co-optimization of multiple parameters are impossible. This leads to significant discrepancies between theoretical design parameters and the actual performance of the sensor, failing to provide accurate theoretical analytical solutions for sensor design.

[0006] Secondly, existing technologies rely on a trial-and-error R&D model, resulting in low R&D efficiency and insufficient customized design capabilities. Due to the lack of a globally unified theoretical model and accurate analytical solutions, the development of existing piezoelectric sensors generally adopts a finite element black-box simulation-experiment verification model. Researchers need to initially set structural parameters based on engineering experience and finite element simulation, prepare prototypes, and obtain the actual performance of the sensor through testing methods such as transceiver methods. Then, based on the test results, they repeatedly adjust the structural and material parameters of the piezoelectric layer, matching layer, and backing layer until the sensor performance approaches the design target. This model has inherent defects such as long R&D cycles, high prototyping costs, and low design accuracy, and cannot meet the rapid R&D and accurate forward design requirements of high signal-to-noise ratio, high sensitivity customized sensors in the cutting-edge structural health monitoring field.

[0007] In summary, existing piezoelectric sensor design technology is limited by a discrete assembly approach and lacks a globally unified theoretical modeling architecture. This results in core technological bottlenecks such as insufficient design accuracy, low R&D efficiency, inability to accurately predict complex acoustic behavior, and difficulty in achieving forward design of customized devices. These bottlenecks have become key challenges restricting the development and engineering application of high-performance piezoelectric sensor technology. Therefore, developing an integrated design and modeling method that can coordinate all functional components of a piezoelectric sensor within a unified theoretical framework and achieve global synergistic optimization of structural parameters and acoustic performance is a core technological problem urgently needing to be solved in this field. Summary of the Invention

[0008] Based on the core defects of existing piezoelectric sensor design technology, such as discrete assembly, discrete approximation, and reliance on empirical trial and error, this invention provides a global analytical modeling and integrated design method for piezoelectric sensors. This method solves the technical problems of existing technologies, such as the lack of a globally unified theoretical model, the inability to accurately predict complex acoustic behavior, insufficient customized forward design capabilities, and large deviations between device performance and theoretical design. It provides a new design paradigm with both rigorous physical logic and engineering universality for the customized development of complex acoustic devices.

[0009] To achieve the above-mentioned objectives, the present invention adopts the following technical solution: a global analytical modeling and integrated design method for piezoelectric sensors, characterized by comprising the following steps:

[0010] S1. Construct a global wave differential equation for a multilayer structure including a backing layer, a piezoelectric layer, a matching layer, and the medium under test, and incorporate the acoustic wave propagation laws of each layer and the electromechanical coupling characteristics of the piezoelectric layer into a unified mathematical framework.

[0011] S2. Based on the interlayer interface continuity constraint, electrical boundary constraint and outer boundary acoustic impedance constraint, the global wave differential equation is solved analytically to obtain the analytical solution of acoustic intensity transmittance.

[0012] S3. Perform parameter reduction and equivalence verification on the dual-matching layer structure, and unify the multi-dimensional independent parameters into single-matching layer parameters.

[0013] S4. Map and calibrate the analytical solution of sound intensity transmittance with the finite element simulation results to achieve high-precision prediction of sensor acoustic behavior.

[0014] S5. Based on the target resonant frequency, determine the structural parameters of each layer of the sensor in a forward manner based on the global analytical solution to complete the deterministic design.

[0015] Further, in step S1, the global fluctuation differential equation includes:

[0016] For piezoelectric layers, based on the piezoelectric constitutive equation and Newton's laws of motion, an electromechanical coupled wave differential equation under one-dimensional longitudinal vibration is established to fully describe the electromechanical energy conversion characteristics and elastic wave propagation law of piezoelectric materials.

[0017] For the matching layer and the backing layer, based on the elastic wave propagation theory of isotropic elastic media, one-dimensional longitudinal wave differential equations are established to describe the acoustic wave propagation and interface transmission characteristics of each layer.

[0018] Based on the spatial relationship of each layer, the global fluctuation differential equation is formed by connecting them in series, as shown in formula (1):

[0019]

[0020] In the formula, Represents the sound waves within each dielectric layer; This represents the amplitude of the reflected sound wave in that layer of sound waves; Represents the amplitude of the transmitted sound wave in that layer; subscript , , , These represent the parameters corresponding to the tested medium, matching layer, piezoelectric layer, and backing layer, respectively. , , , These represent the wavenumbers of the measured medium, matching layer, piezoelectric layer, and backing layer, respectively. Represents the position of the sound wave axis; Represents the angular frequency of sound waves; Represents the time of dissemination; Represents a complex number; Representative index.

[0021] Further, in step S2, the interlayer interface constraint is: boundary conditions of continuous mass displacement and continuous normal stress are set at the interface between two adjacent layers to ensure the physical continuity of the multilayer system; the electrical boundary constraint is: grounding and open circuit boundary conditions matching the actual working conditions of the piezoelectric layer are set for the upper and lower electrode surfaces to completely restore the electrical working state of the piezoelectric element; the outer boundary constraint is: acoustic impedance radiation boundary matching the measured medium is set for the radiation end face of the matching layer, and semi-infinite assumed impedance boundary corresponding to the sound absorption characteristics is set for the rear end face of the backing layer.

[0022] Furthermore, in step S2, the global wave differential equation is solved analytically using the separation of variables method and the matrix transfer method to obtain the acoustic intensity transmittance under the single-matching layer parameters of the global wave differential equation. As shown in formula (2), this represents a fundamental leap in piezoelectric sensor design from discrete components to continuous systems.

[0023]

[0024] In the formula, the parameters , , , , , , , , , , , , They are defined as follows:

[0025]

[0026] In the formula, , , , These represent the acoustic impedance of the measured medium, matching layer, piezoelectric layer, and backing layer, respectively. , , These represent the X-axis coordinate, matching layer thickness, and piezoelectric layer thickness at the junction of the measured medium and the matching layer, respectively.

[0027] Furthermore, step S3 specifically includes:

[0028] The global wave differential equation under the double matching layer parameters is established, and the sound intensity transmittance is solved by combining the boundary conditions, as shown in formula (3).

[0029] Acoustic transmittance under dual-matching layer conditions By performing analysis, the parameters of the thickness and acoustic impedance of the double matching layer are unified and reduced to an equivalent analytical solution of the acoustic intensity transmittance of a single matching layer. That is, after the reduction of the order of equation (3), it becomes equation (2).

[0030]

[0031] In the formula, the parameters , , , , , They are defined as follows:

[0032]

[0033] parameter , , , , , , , , They are defined as follows:

[0034]

[0035] parameter , , , , , , , , They are defined as follows:

[0036]

[0037] parameter , , , , , , , , They are defined as follows:

[0038]

[0039] parameter , , They are defined as follows:

[0040]

[0041] In the formula, Represents the wavenumber of the second matching layer. Represents the thickness of the second matching layer. This represents the acoustic impedance of the second matching layer.

[0042] Further, in step S4, a finite element model of the piezoelectric sensor is established, and the proposed global differential equation is simulated and verified. The device simulation results are compared with the analytical solution of the differential equation to verify the model's high-precision prediction capability for the complex acoustic behavior of the sensor and to complete the calibration of the consistency between theory and simulation.

[0043] Furthermore, based on the first type of piezoelectric constitutive equation, a global analytical expression for the sensor output voltage response is constructed as shown in equation (4), yielding the voltage frequency response characteristics at the target resonant frequency.

[0044]

[0045] In the formula, This refers to the output voltage. This is the equivalent electrical admittance; Sound pressure level; The polarization surface area of ​​the piezoelectric layer; The piezoelectric strain constant under short-circuit conditions; The acoustic intensity transmittance is used in the final design of the sensor.

[0046] The present invention also provides a piezoelectric sensor, which is fabricated using a global analytical modeling and integrated design method for piezoelectric sensors as described above. The sensor includes a backing layer, a piezoelectric layer, and at least one matching layer. The thickness and acoustic impedance of each layer are determined by the analytical solution of the global wave differential equation, and the sensor has a preset resonant frequency and acoustic intensity transmittance characteristics.

[0047] Furthermore, in the piezoelectric sensor described above, the matching layer is a single-layer or double-layer structure.

[0048] Compared with the prior art, the present invention has the following significant technical advantages and beneficial effects:

[0049] 1. At the theoretical level, this invention breaks through the core limitations of traditional discrete models and constructs a rigorous, globally unified mathematical architecture. Returning to the vibrational physical essence of elastic wave propagation, this invention completely abandons the simplifications of traditional lumped parameter equivalence and discrete unit approximation. It integrates the piezoelectric layer, matching layer, and backing layer into a single global differential equation system. By analyzing the complex boundary conditions of the multi-layered structure, it achieves closed-loop solution for the entire system. This allows for precise quantitative description of the complex acoustic behavior throughout the sensor, including electromechanical coupling, sound wave propagation, and interface reflection, solving the industry challenge of high-precision analytical prediction of multi-layered piezoelectric acoustic devices in existing technologies.

[0050] 2. At the design level, this invention breaks through the technical barriers of traditional trial-and-error methods, achieving forward-oriented, precise, and customized design. It establishes a quantitative mapping relationship between structural parameters and device performance through a global analytical solution, and combines this with a double-matching layer parameter order reduction equivalence method to achieve order reduction equivalence verification of the theoretical model. The optimal combination of structural parameters can be directly solved forward based on the target resonant frequency and application scenario requirements, significantly shortening the R&D cycle, reducing prototyping costs, and demonstrating strong customized design capabilities.

[0051] 3. At the engineering level, it provides a new design paradigm that combines physical rigor and universality. The global analytical modeling architecture of this invention can be universally extended to the forward design of piezoelectric ultrasonic transducers and multilayer acoustic devices with different resonant frequencies and layer counts. It can be widely used in structural health monitoring in aerospace, rail transportation, civil engineering and other fields, as well as in non-destructive testing, ultrasonic imaging, precision drive and other scenarios, and has great engineering promotion value. Attached Figure Description

[0052] Figure 1 This is a schematic diagram of a piezoelectric sensor structure;

[0053] Figure 2 This is a schematic diagram of the propagation of sound waves inside a piezoelectric sensor, where a is a schematic diagram of the sound wave reflection and transmission at the interfaces of various media in the sensor, and b is a schematic diagram of the sound wave propagation inside each medium.

[0054] Figure 3 Theoretical analytical diagram of sound intensity transmittance (variable matching layer thickness);

[0055] Figure 4 Theoretical analytical diagram of sound intensity transmittance (variable resonant frequency);

[0056] Figure 5 The simulation diagram of sound intensity transmittance (variable matching layer thickness);

[0057] Figure 6 The simulation diagram of sound intensity transmittance (variable resonant frequency);

[0058] Figure 7 For sound intensity transmittance simulation - analytical absolute error diagram (variable matching layer thickness);

[0059] Figure 8 Simulation of sound intensity transmittance - Analysis of absolute error diagram (variable resonant frequency);

[0060] Figure 9 The sensor simulation-theoretical voltage frequency response diagram (resonant frequency is 80kHz) is shown, where a is the actual voltage frequency response diagram and b is the voltage frequency response diagram after normalization.

[0061] Figure 10 The sensor simulation-theoretical voltage frequency response diagram (resonant frequency is 150kHz) is shown in Figure a, where a is the actual voltage frequency response diagram and b is the voltage frequency response diagram after normalization. Detailed Implementation

[0062] The present invention will be further described in detail below with reference to specific embodiments. These embodiments are only used to explain the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

[0063] Example 1

[0064] 1. Global analytical modeling and integrated design method for piezoelectric sensors

[0065] This method abandons the traditional discrete lumped parameter approximation and returns to the vibrational physical essence of elastic wave propagation. It incorporates the entire structure of the piezoelectric sensor into a unified mathematical framework for forward design, specifically including the following steps:

[0066] Step 1: Construct the global wave differential equation for the multilayer structure

[0067] By treating the backing layer, piezoelectric layer, and matching layer of the piezoelectric sensor as a continuous longitudinal propagation system of elastic waves, and combining the boundary conditions between each layer, a global differential equation covering the entire structure is constructed:

[0068] For piezoelectric layers, based on the piezoelectric constitutive equation and Newton's laws of motion, an electromechanical coupled wave differential equation under one-dimensional longitudinal vibration is established to fully describe the electromechanical energy conversion characteristics and elastic wave propagation law of piezoelectric materials.

[0069] For the matching layer and the backing layer, based on the elastic wave propagation theory of isotropic elastic media, one-dimensional longitudinal wave differential equations are established to describe the acoustic wave propagation and interface transmission characteristics of each layer.

[0070] like Figure 2As shown, according to the spatial relationship of each layer, the above equations are connected in series to form a global fluctuation differential equation as shown in formula (1), so as to realize the continuous and integrated mathematical description of the entire sensor structure.

[0071]

[0072] In the formula, Represents the sound waves within each dielectric layer; This represents the amplitude of the reflected sound wave in that layer of sound waves; Represents the amplitude of the transmitted sound wave in that layer; subscript , , , These represent the parameters corresponding to the tested medium, matching layer, piezoelectric layer, and backing layer, respectively. , , , These represent the wavenumbers of the measured medium, matching layer, piezoelectric layer, and backing layer, respectively. Represents the position of the sound wave axis; Represents the angular frequency of sound waves; Represents the time of dissemination; Represents a complex number; Representative index.

[0073] Step 2: Solving the global wave differential equation

[0074] For the global differential equation established in step 1, continuous constraints are established for the entire boundary and interlayer interfaces. The closed-loop solution of the system is then obtained through analytical methods, specifically including:

[0075] Interlayer interface constraints: Boundary conditions for continuous mass displacement and continuous normal stress are set at the interface between adjacent layers to ensure the physical continuity of the multilayer system.

[0076] Electrical boundary constraints: For the upper and lower electrode surfaces of the piezoelectric layer, grounding and open circuit boundary conditions are set to match the actual working conditions of the project, so as to fully restore the electrical working state of the piezoelectric element;

[0077] External boundary constraints: For the radiating end face of the matching layer, set an acoustic impedance radiation boundary that matches the medium under test; for the rear end face of the backing layer, set a semi-infinite assumed impedance boundary corresponding to the sound absorption characteristics.

[0078] Based on the above boundary conditions, the acoustic intensity transmittance under single-matching layer parameters of the global differential equation is obtained analytically using the separation of variables method and matrix transfer method. As shown in formula (2), this represents a fundamental leap in piezoelectric sensor design from discrete components to continuous systems.

[0079]

[0080] In the formula, the parameters , , , , , , , , , , , , They are defined as follows:

[0081]

[0082] In the formula, , , , These represent the acoustic impedance of the measured medium, matching layer, piezoelectric layer, and backing layer, respectively. , , These represent the X-axis coordinate, matching layer thickness, and piezoelectric layer thickness at the junction of the measured medium and the matching layer, respectively.

[0083] Step 3: Validation of the reduced order of the differential equation's double-matched layer parameters.

[0084] Based on the expansion of the global fluctuation differential equation, a double-matched layer mathematical model is constructed, and the mathematical equations are verified after parameter reduction and equivalence. Specifically, this includes:

[0085] Establish the global differential equation under the double matching layer parameters, and solve the sound intensity transmittance by combining the boundary conditions, as shown in formula (3);

[0086] Acoustic transmittance under dual-matching layer conditions Analysis was performed to unify the parameters of the thickness and acoustic impedance of the double matching layer, reducing the order to an equivalent analytical solution of the acoustic intensity transmittance of a single matching layer. Verification showed that formula (3) became formula (2) after the order reduction.

[0087]

[0088] In the formula, the parameters , , , , , They are defined as follows:

[0089]

[0090] parameter , , , , , , , , They are defined as follows:

[0091]

[0092] parameter , , , , , , , , They are defined as follows:

[0093]

[0094] parameter , , , , , , , , They are defined as follows:

[0095]

[0096] parameter , , They are defined as follows:

[0097]

[0098] In the formula, Represents the wavenumber of the second matching layer. Represents the thickness of the second matching layer. This represents the acoustic impedance of the second matching layer;

[0099] After order reduction verification, this theoretical formula can realize the integrated design and analysis of piezoelectric sensors. The theoretical analytical values ​​of acoustic intensity transmittance for different matching layer thicknesses and different piezoelectric sensor resonant frequencies are plotted as follows: Figure 3-4 As shown.

[0100] Step 4: Finite element simulation mapping verification of the theoretical model

[0101] A finite element model of the piezoelectric sensor was established, and the proposed global differential equation was verified through simulation. The results of sound intensity transmittance are as follows: Figure 5-6 As shown, the device simulation results are compared with the analytical solutions of the differential equations to verify the model's high-precision prediction capability for the complex acoustic behavior of the sensor, and to complete the consistency calibration of the mapping between theory and simulation.

[0102] like Figure 7-8 As shown, regardless of the thickness of the matching layer or across an extremely wide range of sensor resonant frequency domains, the maximum absolute deviation between the theoretical analytical solution and the finite element numerical solution is strictly suppressed to an extremely small order of magnitude (Δ < 0.003). This high fidelity across different topologies strongly demonstrates that the mathematical architecture method proposed in this invention possesses excellent positive prediction capabilities when dealing with complex acoustic boundary conditions.

[0103] 2. Analyze the voltage response of a piezoelectric sensor at a specific resonant frequency based on the above method.

[0104] Based on the accurate characterization of acoustic energy at complex boundaries using wavefield transmission dynamics, the dynamic evolution characteristics of the sensor's macroscopic electrical output were further explored. Based on the first type of piezoelectric constitutive equation, and with strict mechanical freedom and electrical open-circuit boundary constraints, a global analytical model mapping the mechanical stress field to the system-level voltage response was constructed, as shown in equation (4):

[0105]

[0106] In the formula, For output voltage, For equivalent admittance, For sound wave sound pressure, The polarization surface area of ​​the piezoelectric layer. The piezoelectric strain constant is given under short-circuit (constant electric field) conditions. Represents the angular frequency of sound waves. The acoustic intensity transmittance is used for the final design of the sensor.

[0107] Example 2

[0108] Based on the aforementioned global analytical modeling and integrated design method, this embodiment designs two piezoelectric sensors with center resonant frequencies of 80kHz and 150kHz, respectively, and conducts finite element simulation analysis to verify the technical feasibility and performance advantages of the present invention. The specific implementation process is as follows:

[0109] 1. Global Modeling and Parameter Optimization: Based on the target resonant frequency and application scenario, material selection is completed, and the optimal combination of structural parameters is obtained through analytical solution of the global differential equation system and forward optimization.

[0110] 2. Sound intensity transmittance verification: Using finite element analysis software, reasonable boundary conditions are set to carry out sound intensity transmittance simulation analysis, and the results are compared with the analytical solution of the global differential equation to verify the consistency between the theoretical analytical solution and the physical response of the simulated device.

[0111] 3. Voltage frequency response verification: such as Figure 9-10 As shown, the voltage frequency response curve of the piezoelectric sensor is simulated and analyzed by setting reasonable boundary conditions using finite element analysis software. The analytical solution of the voltage frequency response curve is obtained based on the analytical solution of sound intensity transmittance. The voltage simulation and analytical frequency response curve are then compared and analyzed to verify the consistency between the theoretical analytical solution and the simulated device voltage frequency response.

[0112] In summary, the simulation analysis results of sound intensity transmittance and voltage frequency response curves both prove that the global analytical modeling and integrated design architecture proposed in this invention can realize the forward precision design of piezoelectric sensors. The theoretical analytical solution and the actual physical response of the device have extremely high mapping consistency, which completely breaks the limitation of traditional acoustic design relying on trial and error, and provides a brand-new physical design paradigm for the customized development of complex multilayer acoustic devices.

Claims

1. A global analytical modeling and integrated design method for piezoelectric sensors, characterized in that, Includes the following steps: S1. Construct a global wave differential equation for a multilayer structure including a backing layer, a piezoelectric layer, a matching layer, and the medium under test, and incorporate the acoustic wave propagation laws of each layer and the electromechanical coupling characteristics of the piezoelectric layer into a unified mathematical framework. S2. Based on the interlayer interface continuity constraint, electrical boundary constraint and outer boundary acoustic impedance constraint, the global wave differential equation is solved analytically to obtain the analytical solution of acoustic intensity transmittance. S3. Perform parameter reduction and equivalence verification on the dual-matching layer structure, and unify the multi-dimensional independent parameters into single-matching layer parameters. S4. Map and calibrate the analytical solution of sound intensity transmittance with the finite element simulation results to achieve high-precision prediction of sensor acoustic behavior. S5. Based on the target resonant frequency, determine the structural parameters of each layer of the sensor in a forward manner based on the global analytical solution to complete the deterministic design.

2. The global analytical modeling and integrated design method for piezoelectric sensors according to claim 1, characterized in that, Step S1, the global fluctuation differential equation includes: For piezoelectric layers, based on the piezoelectric constitutive equation and Newton's laws of motion, an electromechanical coupled wave differential equation under one-dimensional longitudinal vibration is established to fully describe the electromechanical energy conversion characteristics and elastic wave propagation law of piezoelectric materials. For the matching layer and the backing layer, based on the elastic wave propagation theory of isotropic elastic media, one-dimensional longitudinal wave differential equations are established to describe the acoustic wave propagation and interface transmission characteristics of each layer. Based on the spatial relationship of each layer, the global fluctuation differential equation is formed by connecting them in series, as shown in formula (1): In the formula, Represents the sound waves within each dielectric layer; This represents the amplitude of the reflected sound wave in that layer of sound waves; Represents the amplitude of the transmitted sound wave in that layer; subscript , , , These represent the parameters corresponding to the tested medium, matching layer, piezoelectric layer, and backing layer, respectively. , , , These represent the wavenumbers of the measured medium, matching layer, piezoelectric layer, and backing layer, respectively. Represents the position of the sound wave axis; Represents the angular frequency of sound waves; Represents the time of dissemination; Represents a complex number; Representative index.

3. The global analytical modeling and integrated design method for piezoelectric sensors according to claim 1, characterized in that, Step S2, the interlayer interface constraint: boundary conditions of continuous mass displacement and continuous normal stress are set at the interface between two adjacent layers to ensure the physical continuity of the multilayer system; the electrical boundary constraint: grounding and open circuit boundary conditions matching the actual working conditions of the engineering are set for the upper and lower electrode surfaces of the piezoelectric layer to completely restore the electrical working state of the piezoelectric element; the outer boundary constraint: acoustic impedance radiation boundary matching the measured medium is set for the radiating end face of the matching layer, and a semi-infinite assumed impedance boundary corresponding to the sound absorption characteristics is set for the rear end face of the backing layer.

4. The global analytical modeling and integrated design method for piezoelectric sensors according to claim 1, characterized in that, Step S2: The global wave differential equation is solved analytically using the separation of variables method and the matrix transfer method to obtain the acoustic intensity transmittance under the single-matching layer parameters of the global wave differential equation. As shown in formula (2): In the formula, the parameters , , , , , , , , , , , , They are defined as follows: In the formula, , , , These represent the acoustic impedance of the measured medium, matching layer, piezoelectric layer, and backing layer, respectively. , , These represent the X-axis coordinate, matching layer thickness, and piezoelectric layer thickness at the junction of the measured medium and the matching layer, respectively.

5. The global analytical modeling and integrated design method for piezoelectric sensors according to claim 4, characterized in that, Step S3 specifically includes: The global wave differential equation under the double matching layer parameters is established, and the sound intensity transmittance is solved by combining the boundary conditions, as shown in formula (3). Acoustic transmittance under dual-matching layer conditions By performing analysis, the parameters of the thickness and acoustic impedance of the double matching layer are unified and reduced to an equivalent analytical solution of the acoustic intensity transmittance of a single matching layer. That is, after the reduction of the order of equation (3), it becomes equation (2). In the formula, the parameters , , , , , They are defined as follows: parameter , , , , , , , , They are defined as follows: parameter , , , , , , , , They are defined as follows: parameter , , , , , , , , They are defined as follows: parameter , , They are defined as follows: In the formula, Represents the wavenumber of the second matching layer. Represents the thickness of the second matching layer. This represents the acoustic impedance of the second matching layer.

6. The global analytical modeling and integrated design method for piezoelectric sensors according to claim 1, characterized in that, Step S4: Establish a finite element model of the piezoelectric sensor, perform simulation verification on the proposed global differential equation, compare the device simulation results with the analytical solution of the differential equation, verify the model's high-precision prediction capability for the complex acoustic behavior of the sensor, and complete the consistency calibration of the mapping between theory and simulation.

7. The global analytical modeling and integrated design method for piezoelectric sensors according to claim 1, characterized in that, It also includes constructing a global analytical formula (4) for the sensor output voltage response based on the first type of piezoelectric constitutive equation, thus obtaining the voltage frequency response characteristics at the target resonant frequency. In the formula, This refers to the output voltage. This is the equivalent electrical admittance; Sound pressure level; The polarization surface area of ​​the piezoelectric layer; The piezoelectric strain constant under short-circuit conditions; The acoustic intensity transmittance is used in the final design of the sensor.

8. A piezoelectric sensor, characterized in that, The sensor is fabricated using a global analytical modeling and integrated design method for piezoelectric sensors as described in any one of claims 1-7, and includes a backing layer, a piezoelectric layer, and at least one matching layer; the thickness and acoustic impedance of each layer are determined by the analytical solution of the global fluctuation differential equation, and the sensor has a preset resonant frequency and acoustic intensity transmittance characteristics.

9. The piezoelectric sensor according to claim 8, characterized in that, The matching layer can be a single-layer or double-layer structure.

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