A method for detecting thickness of a bond coat layer of a thermal barrier coating based on a three-winding transformer model

Abstract

The application discloses a thermal barrier coating bonding layer thickness eddy current measurement method based on a three-winding transformer model. Based on the three-winding transformer model, the signal coupling problem of the thermal barrier coating multilayer structure can be solved, and simple, efficient and accurate bonding layer thickness measurement can be realized. First, a thermal barrier coating calibration piece is prepared, and an eddy current detection probe impedance change signal is obtained. Second, an eddy current detection three-winding transformer model is constructed, and a mathematical expression of the phase of the impedance change is derived. Third, a simplified relationship between the phase of the impedance change and the bonding layer thickness and the ceramic layer thickness is established by using frequency regulation and Taylor series. Fourth, a calibration coefficient is obtained. Finally, the phase of the impedance change of the thermal barrier coating sample to be measured is obtained by using the eddy current method, and the thickness of the bonding layer of the thermal barrier coating sample to be measured is calculated. The application solves the problems that the traditional eigenvalue calibration method is difficult to decouple the signal coupling of the bonding layer, the ceramic layer and the substrate and has poor detection precision.

Description

Technical Field

[0001] This invention relates to a method for detecting the thickness of the thermal barrier coating adhesive layer based on a three-winding transformer model, belonging to the field of eddy current nondestructive testing. Background Technology

[0002] Aero engines are the most advanced products in the equipment manufacturing field, representing a nation's technological level and comprehensive national strength. They have always been considered core technologies affecting national air transport, defense security, and maintaining national strategic advantage. High-pressure turbine blades are core components of aero engines, operating under extremely harsh conditions (high temperature, high pressure, and high stress) for extended periods. The high-temperature, high-pressure gas flow from the combustion chamber blows directly onto the high-pressure turbine blades. The turbine inlet temperature in commercial aero engines can reach 1650℃, while the most advanced high-temperature alloys used in high-pressure vortex blades can only withstand temperatures up to about 1150℃, approaching their temperature limit. Even with advanced film cooling technology, the gap between the material's operating temperature and the turbine inlet temperature cannot be completely resolved or satisfied. Therefore, it is necessary to spray a thermal barrier coating onto the surface of the turbine blade's combustion gas flow channel.

[0003] The ceramic insulating layer of the thermal barrier coating, primarily composed of MCrAlY (M: Ni or / and Co) or NiPtAl, serves to resist oxidation and corrosion, buffer internal stresses arising from the significant difference in thermal expansion coefficients between the ceramic layer and the substrate, and improve the bonding strength of the thermal barrier coating. Its thickness and uniformity analysis are crucial for characterizing the coating quality. The adhesive layer thickness ranges from 50 μm to 150 μm; excessively thick or thin adhesive layers will affect the coating's bonding strength and service life. Therefore, conducting research on non-destructive testing and evaluation of thermal barrier coating quality to achieve adhesive layer thickness detection is of significant strategic importance for monitoring the manufacturing quality and in-service condition of aero-engine blades, thereby ensuring the safe operation of aero-engines.

[0004] Common non-destructive testing (NDT) methods include ultrasonic testing, X-ray testing, infrared testing, and terahertz testing. Ultrasonic testing offers high sensitivity but requires a coupling agent, which can contaminate the surface of the thermal barrier coating. X-ray testing provides direct results but involves strong radiation, potentially harming personnel. Infrared testing has a wide range of applications but requires significant thermal excitation energy, and the large size of the camera limits its portability. Terahertz testing offers high resolution, but the equipment is expensive. Therefore, these methods are not ideally suited for measuring the thickness of thermal barrier coating ceramic layers. Eddy current non-destructive testing, based on Faraday's law of electromagnetic induction, utilizes the eddy currents generated in the test piece to alter the electromagnetic field, which in turn affects the electrical signal of the eddy current sensor. This alteration is used to characterize various aspects of the test piece. Eddy current testing offers advantages such as low cost, high efficiency, no contamination, and ease of detection, making it suitable for measuring the thickness of thermal barrier coating ceramic layers.

[0005] Conventional methods for eddy current thickness measurement mainly include model inversion, feature calibration, and machine learning. Specifically:

[0006] First, there's the model inversion method, whose accuracy depends on the consistency between simulation and experiment, requiring high precision matching. Furthermore, due to the coupling relationship between the ceramic layer thickness and the adhesive layer and substrate, multi-frequency excitation is needed, making this method inefficient. Second, there's the feature calibration method. Traditional feature calibration methods are mostly used for single-layer coating thickness measurement. The ceramic layer of a thermal barrier coating is non-conductive, while the adhesive layer and substrate are weakly conductive. Existing feature calibration methods struggle to decouple the relationship between the ceramic layer thickness, adhesive layer, and substrate. Third, there's machine learning. Machine learning-based thickness measurement methods can overcome the challenge of understanding the coupling relationship between different layers of a thermal barrier coating under varying material parameters, while maintaining high computational efficiency. However, this method requires a large amount of sample data to train the neural network.

[0007] Therefore, for those skilled in the art, how to use eddy current testing technology to measure the thickness of the thermal barrier coating adhesive layer has become a technical problem that urgently needs to be solved in this field. Summary of the Invention

[0008] To address the above problems, this invention proposes a method for detecting the thickness of the adhesive layer of a thermal barrier coating based on a three-winding transformer model. By constructing a three-winding transformer model, the method can achieve simple, efficient, and accurate measurement of the adhesive layer thickness, addressing the signal coupling problem in the multi-layer structure of the thermal barrier coating.

[0009] The technical solution of this invention is as follows: It is carried out according to the following steps:

[0010] Step 1: Prepare a thermal barrier coating calibration part, select the test position, use an eddy current detection system to measure the test position of the calibration part, obtain the impedance change signal of the eddy current detection probe, and then obtain the true thickness values ​​of the ceramic layer and the adhesive layer of the thermal barrier coating calibration part through metallographic experiments.

[0011] The ceramic layer thickness of calibration component 1 is h. t1 The adhesive layer thickness is c b1 The phase of the impedance change is Wherein, ΔX1 represents the change in reactance of calibration component 1, and ΔR1 represents the change in resistance of calibration component 1;

[0012] The ceramic layer thickness of calibration component 2 is h. t2 The adhesive layer thickness is c b2 The phase of the impedance change is Wherein, ΔX2 represents the change in reactance of calibration component 2, and ΔR2 represents the change in resistance of calibration component 2;

[0013] The ceramic layer thickness of calibration component 3 is h. t3 The adhesive layer thickness is c b3 The phase of the impedance change is Wherein, ΔX3 represents the change in reactance of calibration component 3, and ΔR3 represents the change in resistance of calibration component 3;

[0014] Step 2: Based on the equivalent eddy current ring and Kirchhoff's laws, construct a three-winding transformer model for eddy current detection and derive the mathematical expression for the phase of the impedance change.

[0015] Step 3: Using frequency modulation and Taylor series, establish a simplified expression for the relationship between the phase of the impedance change and the thickness of the adhesive layer and the ceramic layer.

[0016] Step 4: Use the finite element model to obtain the correction coefficients; using three calibration components, establish a set of equations for the phase of the impedance change with respect to the thickness of the adhesive layer and the ceramic layer based on Step 3, solve the set of equations, obtain the calibration coefficients, and obtain the characteristic curve expression of the adhesive layer thickness.

[0017] Step 5: Use an eddy current detection system to measure the phase of the impedance change of the thermal barrier coating specimen under test, use terahertz technology to obtain the ceramic layer thickness of the thermal barrier coating specimen under test, and substitute it into the characteristic curve expression established in step 4 to obtain the adhesive layer thickness value of the specimen under test.

[0018] Step 2 specifically includes:

[0019] Equivalent eddy currents were constructed for the adhesive layer and the substrate, respectively. Using Kirchhoff's laws, a mathematical expression for the phase P of the impedance change in a three-winding transformer model with thermal barrier coating was established.

[0020]

[0021] Where ΔX represents the change in reactance, and ΔR represents the change in resistance.

[0022]

[0023] Where ω is the angular frequency of the excitation signal, R0 and L0 are the original resistance and inductance of the probe, respectively, R1 and L1 are the resistance and inductance of the equivalent eddy current ring of the adhesive layer, respectively, R2 and L2 are the resistance and inductance of the equivalent eddy current ring of the substrate, and the mutual inductance between the probe and the equivalent eddy current ring of the adhesive layer is M. 01 The mutual inductance between the probe and the equivalent vortex ring of the substrate is M. 02 The mutual inductance between the bonding layer vortex ring and the matrix vortex ring is M. 12 .

[0024] Step 3 specifically includes:

[0025] Step 3.1: Divide both the numerator and denominator of the mathematical expression for the phase P of the impedance change by A2A4. Under high-frequency excitation, the phase P of the impedance change is:

[0026]

[0027] in,

[0028]

[0029] in,

[0030]

[0031] Among them, a 01 a represents the mutual inductance coupling coefficient between the coil and the equivalent eddy current ring of the bonding layer. 12 a represents the mutual inductance coupling coefficient between the equivalent eddy current in the bonding layer and the equivalent eddy current ring in the matrix. 02 This represents the mutual inductance coupling coefficient between the coil and the equivalent eddy current ring of the substrate. m and n are exponential fitting coefficients closely related to the geometric parameters of the coil. h t c represents the thickness of the ceramic layer. b h represents the thickness of the adhesive layer. b This represents the equivalent vortex ring thickness of the adhesive layer. k b δ is a correction factor for the penetration depth of the adhesive layer. b denoted as the penetration depth of the eddy currents in the adhesive layer. Wherein, f is the excitation frequency, σ b The conductivity of the adhesive layer is μ. b denoted as ρ, where ρ is the magnetic permeability of the adhesive layer.

[0032] Under high-frequency excitation, Approximately zero. Based on the expressions for mutual inductance and mutual inductance coupling coefficient, the phase P of the impedance change can be approximated as...

[0033]

[0034] Where G≈L1L2 / (L1R2+L2R1).

[0035] Step 3.2: Expand the Taylor series at the zero point.

[0036]

[0037] The phase P of the impedance change and the thickness c of the adhesive layer are obtained. b and ceramic layer thickness h t Simplified expressions between

[0038]

[0039] Where, k b ' is a correction factor (related to the excitation frequency and the conductivity of the adhesive layer), k b =-2δ b k b / n can be obtained through finite element simulation. α1, α2, and β2 are calibration coefficients.

[0040] Step 4 specifically includes:

[0041] Step 4.1: Apply the finite element model and optimization algorithm to obtain the correction coefficient k. b '.

[0042] Step 4.2: Using the three calibration components, establish a system of equations for solving the coefficients.

[0043]

[0044] Solve the system of equations to obtain the calibration coefficients α1, α2 and β2.

[0045] Step 4.3: Obtain the characteristic curve expression for adhesive layer thickness detection:

[0046] This invention solves the problem that traditional eigenvalue calibration methods struggle to decouple the relationship between the adhesive layer thickness, ceramic layer, and substrate, and proposes a method for detecting the adhesive layer thickness of thermal barrier coatings based on a three-winding transformer model. Compared with existing detection methods, this invention has the following advantages: First, compared to electron microscopy, this invention uses the conventional eddy current method as the signal acquisition means, which has advantages such as fast detection speed, low cost, and ease of automation; Second, compared to the eddy current method based on inversion and machine learning, this invention, by utilizing the conversion of the phase of coil impedance change and related parameters, can accurately obtain the adhesive layer thickness, which cannot be directly measured.

[0047] Therefore, the beneficial effects of the present invention are as follows:

[0048] I. This invention starts from the basic principle of eddy current detection, introduces the concept of equivalent eddy current ring, constructs equivalent eddy currents in the adhesive layer and the substrate respectively, and uses Kirchhoff's law to establish a mathematical expression for the phase P of the impedance change of the three-winding transformer model with thermal barrier coating.

[0049] Second, this invention utilizes frequency modulation and Taylor series to establish a simplified expression between the phase of impedance change and the thickness of the adhesive layer and the ceramic layer, thereby improving computational efficiency.

[0050] Third, this invention uses a finite element model to obtain correction coefficients; and uses three calibration components to obtain calibration coefficients, successfully decoupling the relationship between the adhesive layer thickness and the ceramic layer and the substrate.

[0051] Fourth, the entire testing process of this invention is non-destructive and highly efficient, avoiding the need to prepare a large number of thermal barrier coating calibration parts and conduct destructive experimental verification, which greatly improves the economy and efficiency of adhesive layer thickness testing. Attached Figure Description

[0052] Figure 1 This is a schematic diagram illustrating the principle of eddy current bonding layer thickness detection based on a three-winding transformer model.

[0053] Figure 2 This is a flowchart for measuring the thickness of the eddy current bond layer.

[0054] Figure 3 This is an equivalent eddy current model diagram.

[0055] Figure 4 This is a flowchart of the optimization algorithm for obtaining the correction coefficients.

[0056] Figure 5 This is a graph showing the measurement error of the adhesive layer thickness. Detailed Implementation

[0057] To clearly illustrate the technical features of this patent, the following detailed description is provided through specific embodiments and in conjunction with the accompanying drawings.

[0058] The schematic diagram of the eddy current adhesive layer thickness detection principle based on a three-winding transformer model in this case is as follows: Figure 1 As shown, the flowchart of the eddy current measurement method for the thermal barrier coating adhesive layer thickness is as follows: Figure 2 As shown.

[0059] Specifically, the following steps are included:

[0060] Step 1: Prepare a thermal barrier coating calibration part, select the test position, use an eddy current detection system to measure the test position of the calibration part, obtain the impedance change signal of the eddy current detection probe, and then obtain the true thickness values ​​of the ceramic layer and the adhesive layer of the thermal barrier coating calibration part through metallographic experiments.

[0061] The ceramic layer thickness of calibration component 1 is h. t1 The adhesive layer thickness is c b1 The phase of the impedance change is Wherein, ΔX1 represents the change in reactance of calibration component 1, and ΔR1 represents the change in resistance of calibration component 1;

[0062] The ceramic layer thickness of calibration component 2 is h. t2 The adhesive layer thickness is c b2 The phase of the impedance change is Wherein, ΔX2 represents the change in reactance of calibration component 2, and ΔR2 represents the change in resistance of calibration component 2;

[0063] The ceramic layer thickness of calibration component 3 is h. t3 The adhesive layer thickness is c b3 The phase of the impedance change is Where ΔX3 represents the change in reactance of calibration component 3, and ΔR3 represents the change in resistance of calibration component 3.

[0064] Step 2: Based on the equivalent eddy current loop and Kirchhoff's laws, construct a three-winding transformer model for eddy current detection, and derive the mathematical expression for the phase of the impedance change. Specifically, this includes:

[0065] Equivalent eddy currents were constructed for the adhesive layer and the substrate, respectively. Using Kirchhoff's laws, a mathematical expression for the phase P of the impedance change in a three-winding transformer model with thermal barrier coating was established.

[0066]

[0067] Where ΔX represents the change in reactance, and ΔR represents the change in resistance.

[0068]

[0069] Where ω is the angular frequency of the excitation signal, R0 and L0 are the original resistance and inductance of the probe, respectively, R1 and L1 are the resistance and inductance of the equivalent eddy current ring of the adhesive layer, respectively, R2 and L2 are the resistance and inductance of the equivalent eddy current ring of the substrate, and the mutual inductance between the probe and the equivalent eddy current ring of the adhesive layer is M. 01 The mutual inductance between the probe and the equivalent vortex ring of the substrate is M. 02 The mutual inductance between the bonding layer vortex ring and the matrix vortex ring is M. 12 .

[0070] A three-winding transformer model is as follows: Figure 1 As shown in the figure, U is the excitation voltage, R0 is the original resistance of the probe, R1 is the equivalent eddy current ring resistance of the adhesive layer, R2 is the equivalent eddy current ring resistance of the substrate, L0 is the original inductance of the probe, L1 is the equivalent eddy current ring inductance of the adhesive layer, L2 is the equivalent eddy current ring inductance of the substrate, i0 is the excitation current, i1 is the induced current of the adhesive layer, and i2 is the induced current of the substrate.

[0071] Equivalent eddy current model such as Figure 3 As shown in the figure, h is the probe thickness, and c b c represents the thickness of the adhesive layer. s h is the matrix thickness. b h is the equivalent vortex ring thickness of the adhesive layer. s r is the equivalent vortex ring thickness of the matrix. i r is the inner radius of the probe. o r is the outer radius of the probe. 11 r is the inner radius of the equivalent vortex ring of the bonding layer. 12r is the outer radius of the equivalent vortex ring of the bonding layer. 21 r is the inner radius of the equivalent vortex ring of the matrix. 22 c is the outer radius of the equivalent vortex ring of the matrix. b -h b This is the distance between the equivalent vortex ring of the adhesive layer and the equivalent vortex ring of the substrate.

[0072] Step 3: Using frequency modulation and Taylor series, establish a simplified expression for the relationship between the phase of the impedance change and the thicknesses of the adhesive layer and the ceramic layer. Specifically, this includes:

[0073] Step 3.1: Divide both the numerator and denominator of the mathematical expression for the phase P of the impedance change by A2A4. Under high-frequency excitation, the phase P of the impedance change is:

[0074]

[0075] in,

[0076]

[0077] in,

[0078]

[0079] Among them, a 01 a represents the mutual inductance coupling coefficient between the coil and the equivalent eddy current ring of the bonding layer. 12 a represents the mutual inductance coupling coefficient between the equivalent eddy current in the bonding layer and the equivalent eddy current ring in the matrix. 02 This represents the mutual inductance coupling coefficient between the coil and the equivalent eddy current ring of the substrate. m and n are exponential fitting coefficients closely related to the geometric parameters of the coil. h t c represents the thickness of the ceramic layer. b h represents the thickness of the adhesive layer. b This represents the equivalent vortex ring thickness of the adhesive layer. k b δ is a correction factor for the penetration depth of the adhesive layer. b denoted as the penetration depth of the eddy currents in the adhesive layer. Wherein, f is the excitation frequency, σ b The conductivity of the adhesive layer is μ. b denoted as ρ, where ρ is the magnetic permeability of the adhesive layer.

[0080] Under high-frequency excitation, Approximately zero. Based on the expressions for mutual inductance and mutual inductance coupling coefficient, the phase P of the impedance change can be approximated as...

[0081]

[0082] Where G≈L1L2 / (L1R2+L2R1).

[0083] Step 3.2: Expand the Taylor series at the zero point.

[0084]

[0085] The phase P of the impedance change and the thickness c of the adhesive layer are obtained. b and ceramic layer thickness h t Simplified expressions between

[0086]

[0087] Where, k b ' is a correction factor (related to the excitation frequency and the conductivity of the adhesive layer), k b =-2δ b k b / n can be obtained through finite element simulation. α1, α2, and β2 are calibration coefficients.

[0088] Step 4: Using a finite element model, obtain the correction coefficients; using three calibration components, establish a system of equations based on Step 3 regarding the phase of the impedance change and the thickness of the adhesive layer and the ceramic layer, solve the system of equations, obtain the calibration coefficients, and derive the characteristic curve expression for the adhesive layer thickness. Specifically, this includes:

[0089] Step 4.1: Apply the finite element model and optimization algorithm to obtain the correction coefficient k. b '.

[0090] Step 4.2: Using the three calibration components, establish a system of equations for solving the coefficients.

[0091]

[0092] Solve the system of equations to obtain the calibration coefficients α1, α2 and β2.

[0093] Step 4.3: Obtain the characteristic curve expression for adhesive layer thickness detection:

[0094] The optimization algorithm for obtaining the correction coefficients based on the finite element model is as follows: Figure 4 As shown in the figure, f is the excitation frequency, σ b σ is the electrical conductivity of the adhesive layer. s h is the conductivity of the matrix. t c is the thickness of the ceramic layer. bn and c bm It is the adhesive layer thickness of the two calibration parts, P i For the phase characteristics of the test piece i, c bi For the adhesive layer thickness of specimen i calculated using the adhesive layer thickness characteristic curve in step 4, c btiThe actual adhesive layer thickness of specimen i is set in the simulation. The adhesive layer thickness and conductivity of the tested specimen are the same as those of the calibration specimen.

[0095] Step 5: Use an eddy current detection system to measure the phase of the impedance change of the thermal barrier coating specimen under test, use terahertz technology to obtain the ceramic layer thickness of the thermal barrier coating specimen under test, and substitute it into the characteristic curve expression established in step 4 to obtain the adhesive layer thickness value of the specimen under test.

[0096] The relative error of adhesive layer thickness measurement is as follows Figure 5 As shown in the figure, TC represents the thickness of the ceramic layer.

[0097] This invention starts from the basic principles of eddy current detection, introduces the concept of equivalent eddy current loops, and constructs equivalent eddy currents for the adhesive layer and the substrate respectively. Using Kirchhoff's laws, a mathematical expression for the phase P of the impedance change in a three-winding transformer model of a thermal barrier coating is established. By utilizing frequency modulation and Taylor series, a simplified expression is established between the phase of the impedance change and the thicknesses of the adhesive layer and the ceramic layer, improving computational efficiency. A finite element model is used to obtain correction coefficients; and three calibration components are used to obtain calibration coefficients, successfully decoupling the relationship between the adhesive layer thickness, the ceramic layer, and the substrate. This invention solves the difficulties of complex decoupling and low efficiency in eddy current thermal barrier coating detection. Addressing the signal coupling problem in multi-layered thermal barrier coating structures, it avoids the need for extensive fabrication of thermal barrier coating calibration components and destructive experimental verification, greatly improving the economy and efficiency of adhesive layer thickness detection. It has significant engineering implications and value for efficient quantitative detection of thermal barrier coating adhesive layer thickness.

[0098] There are many specific ways to implement this invention. The above description is only a preferred embodiment of this invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of this invention, and these improvements should also be considered within the scope of protection of this invention.

Claims

1. A method for detecting the thickness of the thermal barrier coating adhesive layer based on a three-winding transformer model using eddy currents, characterized in that, Includes the following steps: Step 1: Prepare a thermal barrier coating calibration part, select the test position, use an eddy current detection system to measure the test position of the calibration part, obtain the impedance change of the eddy current detection probe, and obtain the true thickness values ​​of the ceramic layer and the adhesive layer of the thermal barrier coating calibration part through metallographic experiments. The ceramic layer thickness of calibration component 1 is The adhesive layer thickness is The phase of the impedance change is , and These represent the changes in reactance and resistance of calibration component 1, respectively. The ceramic layer thickness of calibration component 2 is The adhesive layer thickness is The phase of the impedance change is , and These represent the changes in reactance and resistance of calibration component 2, respectively. The ceramic layer thickness of calibration component 3 is The adhesive layer thickness is The phase of the impedance change is , and These represent the changes in reactance and resistance of calibration component 3, respectively. Step 2: Based on the equivalent eddy current loop and Kirchhoff's laws, construct a three-winding transformer model for eddy current detection and derive the phase of the impedance change. The mathematical expression, ； ； in, It is the change in reactance. It is the change in resistance. It is the angular frequency of the excitation signal. and These are the resistance and inductance of the equivalent eddy current ring in the adhesive layer, respectively. and These are the resistance and inductance of the equivalent eddy current ring in the matrix, respectively, and the mutual inductance between the probe and the equivalent eddy current ring in the adhesive layer is... The mutual inductance between the probe and the equivalent vortex ring of the substrate is The mutual inductance between the equivalent vortex ring of the bonding layer and the equivalent vortex ring of the matrix is ; Step 3: Using frequency modulation and Taylor series, establish expressions for the phase of impedance change, adhesive layer thickness, and ceramic layer thickness; Step 4: Using a finite element model, obtain the correction coefficients; using three calibration components, establish a system of equations based on the expression in Step 3, solve the system of equations, obtain the calibration coefficients, and obtain the characteristic curve expression for the adhesive layer thickness, including: Step 4.1: Apply the finite element model and optimization algorithm to obtain the correction coefficients. , It is related to the excitation frequency, the conductivity of the adhesive layer, and the permeability of the magnetic layer. Step 4.2: Using the three calibration components, establish a system of equations for solving the coefficients. ； in, To determine the penetration depth of eddy currents in the adhesive layer, solve the system of equations to obtain the calibration coefficients. and ; Step 4.3: Obtain the characteristic curve expression for the adhesive layer thickness: In the formula The thickness of the ceramic layer; Step 5: Use an eddy current detection system to measure the phase of the impedance change of the thermal barrier coating specimen under test, use terahertz technology to obtain the ceramic layer thickness of the specimen, and substitute it into the characteristic curve expression established in step 4 to obtain the adhesive layer thickness value of the specimen.

2. The method for detecting the thickness of the thermal barrier coating adhesive layer based on a three-winding transformer model according to claim 1, characterized in that, Step 3 specifically includes: Step 3.1: Phase of the impedance change Dividing both the numerator and denominator of the mathematical expression by A2A4, under high-frequency excitation, the phase of the impedance change... for: ； in, ； in, ； Where L0 is the original inductance of the probe, a 10 a represents the mutual inductance coupling coefficient between the coil and the equivalent eddy current ring of the bonding layer. 12 a represents the mutual inductance coupling coefficient between the equivalent eddy current in the bonding layer and the equivalent eddy current ring in the matrix. 20 represents the mutual inductance coupling coefficient between the coil and the equivalent eddy current ring of the substrate; m and n are exponential fitting coefficients closely related to the geometric parameters of the coil; c b h represents the thickness of the adhesive layer. b This represents the equivalent vortex ring thickness of the adhesive layer; where, k b is a correction factor for the penetration depth of the adhesive layer; where, f is the excitation frequency, σ b The conductivity of the adhesive layer is μ. b The magnetic permeability of the adhesive layer; Under high-frequency excitation, Approximately zero; according to the expressions for mutual inductance and mutual inductance coupling coefficient, the phase of the impedance change... Approximately simplified to: ； in, ; Step 3.2: Expand using Taylor series at zero: ； The phase of the impedance change is obtained Adhesive layer thickness c b Ceramic layer thickness h t The expression: ； in, It was obtained through finite element simulation.

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