Method for calculating skin depth of conductor with electrical orthotropic property

Abstract

The invention provides a skin depth calculation method for a conductor with electrical orthotropic property, which comprises the following steps of: S1, establishing a physical model of a measurement area according to a measured object and a coil; establishing an anisotropic wave equation based on a Maxwell equation and a physical model; s2, solving an anisotropic wave equation of the conductivity tensor to obtain an exponential decay general solution of the magnetic induction intensity; s3, utilizing a finite element method to solve the magnetic induction intensity of the junction of the measured object and the air, and obtaining a magnetic field distribution analytical solution of the measured object according to a magnetic index attenuation general solution; s4, determining the depth direction of eddy current attenuation according to the spatial relationship between the measured object and the coil; and S5, determining the skin depth of the measured object according to the Ampere's law, the depth direction of eddy current attenuation and the magnetic field distribution analytic solution of the measured object. According to the relation between the magnetic induction intensity and the current density, the magnetic induction intensity in each direction is calculated, the current density of different points is obtained, and therefore the accurate skin depth can be obtained.

Description

TECHNICAL FIELD

[0001] The present application relates to a skin depth calculation method, in particular to a skin depth calculation method of an electrically orthotropic conductor. BACKGROUND

[0002] Orthotropicity refers to the fact that an object exhibits different physical properties in three mutually perpendicular principal axis directions (i.e., x, y, and z axes) and is symmetric in properties in a plane perpendicular to the principal axes. Materials with orthotropicity include fiber-reinforced composites and wood. Orthotropic materials are widely used in the fields of aerospace and civil engineering and construction. By regularly detecting and tracking the expansion of defects in anisotropic materials, maintenance strategies can be optimized to avoid premature replacement or sudden failure. Various non-destructive testing techniques have been reported for online monitoring of orthotropic materials. Among them, electromagnetic testing technology can detect various types of fiber damage at high detection speed and high signal-to-noise ratio. It is widely used in damage detection and safety evaluation of composite materials due to its non-contact, fast, and good safety advantages. Eddy current testing is a non-destructive testing technology based on electromagnetic induction principle. Defects (such as cracks, holes, and corrosion) in materials will interfere with the distribution and strength of eddy currents, thereby changing the impedance of the coil.

[0003] However, due to the skin effect, eddy current testing can only be used to detect surface and near-surface defects. Therefore, it is necessary to study the true skin depth of composite structures under different excitation frequencies to determine the effective detection range. When calculating the skin depth of materials with orthotropic properties using a finite element model, high-density meshing is required. High-density meshing generates a large-scale linear equation system, which requires a large amount of memory and computing time during the solving process. SUMMARY

[0004] The purpose of the present application is to provide a skin depth calculation method for an electrically orthotropic conductor to solve the problem of consuming a large amount of memory and computing time in calculating the skin depth of an orthotropic object.

[0005] The purpose of the present application is achieved as follows:

[0006] A skin depth calculation method for an electrically orthotropic conductor, comprising the following steps:

[0007] S1. A physical model of the measurement region is established according to the measured object and the coil, and an anisotropic wave equation of the electrical conductivity tensor is established based on Maxwell's equations and the physical model;

[0008] S2. The anisotropic wave equation of the electrical conductivity tensor is solved to obtain an exponential decay general solution of the magnetic induction intensity of the measured object in each direction;

[0009] S3. Solving the magnetic induction intensity at the interface between the measured object and air under non-planar wave excitation by using finite element method, and obtaining the analytical solution of the magnetic field distribution of the measured object by using the exponential decay general solution of the magnetic induction intensity of the measured object in each direction;

[0010] S4. Determining the depth direction of eddy current decay according to the spatial relationship between the measured object and the coil;

[0011] S5. Determining the skin depth of the measured object according to Ampere's law, the depth direction of eddy current decay and the analytical solution of the magnetic field distribution of the measured object.

[0012] Further, the specific way of establishing the physical model in step S1 is:

[0013] The position where the central axis of the coil is extended to intersect with the measured object is taken as the origin, and the direction of the extended central axis of the coil is taken as the y-axis; the direction perpendicular to the y-axis in the cross section of the measured object detected by the coil is taken as the x-axis, and the axial direction of the measured object is taken as the z-axis; the lift-off height between the coil and the surface of the measured object is h.

[0014] Further, the specific way of determining the depth direction of eddy current decay in step S4 is:

[0015] S4-1. Calculating the point where the maximum current density is located according to Ampere's law;

[0016] S4-2. Determining the curvature radius r of the cross section of the composition unit of the measured object at the origin;

[0017] S4-3. The depth direction of eddy current decay is the straight line between the point where the maximum current density is located and (0, r); when the curvature radius is -∞ or +∞, the depth direction of eddy current decay is the straight line parallel to the y-axis and passing through the point where the maximum current density is located.

[0018] Further, the exponential decay general solution of the magnetic induction intensity of the measured object in each direction is:

[0019]

[0020] Where i=x, y, z, C 1i , C 2i , C 3i , C 4i , C 5i and C 6i are undetermined constants.

[0021] Further, the specific way of determining the skin depth of the measured object is:

[0022] The point at which the current density decays to the surface current density 1 / e in the depth direction of eddy current attenuation is calculated, and the distance between the point at which the current density decays to the surface current density 1 / e and the point at which the current density maximum is located is taken as the skin depth of the measured object.

[0023] The application deeply analyzes the electromagnetic field distribution characteristics in the measured object with orthogonal anisotropy, and establishes a skin depth representation method closer to the actual working condition. The application solves the electromagnetic field boundary conditions by establishing a finite element model, and shows good adaptability when dealing with composite materials with complex geometric boundaries and orthogonal anisotropic conductive characteristics; the magnetic field distribution and current distribution expressions in the measured object and the edge are directly obtained, the spatial distribution characteristics of the electromagnetic field in the anisotropic conductor are obtained, and the change trend of the current density in different structure regions is effectively revealed. In the electrically orthogonal anisotropic material, the conductivity tensor is in diagonal form in the orthogonal coordinate system, so that the Maxwell equation set is decoupled in three spatial dimensions, and the separation of variables is used in the general solution, which not only simplifies the solving process of the equation, but also directly reflects the separability of the spatial mode of the electromagnetic field caused by the orthotropy of the material. Based on the traditional skin depth definition, the application realizes the quantification of the current concentration degree under special shape by introducing the curvature radius parameter, so that the definition of the skin depth direction is closer to the actual current distribution.

[0024] The application can be applied to the generalized skin depth solving of different composite material structures under the action of different structure probes and different excitation frequencies, and has good expansibility and adaptability. Under the condition of multi-frequency excitation, the attenuation law of the electromagnetic field in the anisotropic composite material under different frequencies can be quantitatively analyzed, so that the effective detection depth can be dynamically regulated.

[0025] The application does not need to perform high-density grid division on the finite element model, only needs to obtain the magnetic induction intensity of the coil and the air boundary in the finite element model, and can obtain the analytical solution of the magnetic induction intensity according to the magnetic induction intensity constraint equation, thereby saving the memory; and the skin depth is calculated according to the analytical solution of the magnetic induction intensity, thereby saving the calculation time. According to the relationship between the magnetic induction intensity and the current density, the quantitative law of the spatial field quantity decaying with the depth can be obtained, and the finite element method only gives the specific value of the skin depth. BRIEF DESCRIPTION OF DRAWINGS

[0026] Figure 1 is the flow chart of the application.

[0027] Figure 2 is the physical model of the excitation-composite material structure coupling system, a is the physical model on the YOZ section, and b is the physical model on the XOY section.

[0028] Figure 3 is a solving domain schematic diagram.

[0029] Figure 4 It is a schematic diagram of the depth direction of the carbon fiber composite cable, a is a schematic diagram of the eddy current detection of the carbon fiber composite cable by the coil, and b is a schematic diagram of the depth direction of the carbon fiber composite cable on the XOY section.

[0030] Figure 5 This is the comparison between the theoretical calculation results and simulation data of the magnetic induction intensity distribution at an excitation frequency of 1 MHz. DETAILED DESCRIPTION

[0031] The present invention will be further described below in conjunction with the accompanying drawings.

[0032] like Figure 1 As shown, the skin depth calculation method of the electrically orthogonal anisotropic conductor of the present invention includes the following steps:

[0033] S1. Establish a physical model of the measurement area based on the object to be measured and the coil; and establish an anisotropic wave equation of the conductivity tensor based on Maxwell's equations and the physical model.

[0034] The object being measured is a conductor, and its electromagnetic properties (such as electrical conductivity and magnetic permeability) directly determine the calculated value of the skin depth.

[0035] like Figure 2 As shown, the physical model is: the central axis of the coil is extended to intersect with the object to be measured, the intersection position is the origin, the extension direction of the central axis of the coil is the y-axis, the direction perpendicular to the y-axis on the cross section of the object to be measured is the x-axis, the axial direction of the object to be measured is the z-axis, and the lift-off height is h.

[0036] like Figure 3 As shown in Figure 1, the solution domain consists of the source region (Ω1), the non-conductive region (Ω2), and the conductive medium (Ω3), satisfying Ω = Ω1 ∪ Ω2 ∪ Ω3. The magnetic induction intensity constraint equation of the object under test with orthogonal anisotropy is:

[0037] ▽ 2 B i +γ i 2 B i =0

[0038] like Figure 2 As shown, for the magnetic induction intensity at the interface between the object being measured and the air, according to Figure 2 From the physical model in , we can know that each coordinate has magnetic induction intensity in the x, y and z directions respectively.

[0039] In the rectangular coordinate system, the anisotropic wave equation of the conductivity tensor can be obtained:

[0040]

[0041] wherein B i is the magnetic induction intensity of the corresponding direction, i = x, y, z; γ i 2 is the propagation constant of the medium, γ i 2 = -jωμσ i , wherein ω is the angular frequency of the electromagnetic wave oscillation, μ is the magnetic permeability of the measured object, σ i is the electrical conductivity of the measured object.

[0042] S2. Solve the anisotropic wave equation of the electrical conductivity tensor to obtain the exponential decay general solution of the magnetic induction intensity of the measured object in each direction.

[0043] The exponential decay general solution of the magnetic induction intensity of the measured object in each direction is obtained by using the separation of variables method as follows:

[0044]

[0045] wherein k i , m i and n i are separation constants, γ i 2 = k i 2 + m i 2 + n i 2 ; C 1i , C 2i , C 3i , C 4i , C 5i and C 6i are undetermined constants.

[0046] Define k i = a i + jb i , m i = c i + jd i and n i = e i +jf i . Therefore:

[0047]

[0048] S3. Solve the magnetic induction intensity at the air interface of the measured object under the excitation of non-planar waves by using the finite element method, and obtain the analytical solution of the magnetic field distribution of the measured object by using the exponential decay general solution of the magnetic induction intensity of the measured object in each direction.

[0049] When the interface is between two conductive media with limited conductivity, there is no free charge on the surface of the medium, and the relative magnetic permeability μ r,j = μ r,j+1 = 1, the boundary condition is:

[0050]

[0051] wherein, is a normal vector, Γ j,j+1 is the intersection of Ω j and Ω j+1 , j = 1 is a coil, j = 2 is air, j = 3 is a measured object, and B is a magnetic induction intensity. When at Γ 1,2 , B1 is the magnetic induction intensity of the coil at the intersection of the coil and air, and B2 is the magnetic induction intensity of the air at the intersection of the coil and air; when at Γ 2,3 , B2 is the magnetic induction intensity of the air at the intersection of the measured object and air, and B3 is the magnetic induction intensity of the measured object at the intersection of the measured object and air, that is, the magnetic induction intensity of the outer surface of the measured object.

[0052] According to the boundary condition, at the intersection Γ 1,2 of the coil and air, the magnetic induction intensity B1 of the coil is equal to the magnetic induction intensity B2 of the air, and at the intersection Γ 2,3 of the measured object and air, the magnetic induction intensity B3 of the measured object is equal to the magnetic induction intensity B2 of the air.

[0053] To obtain the parameters of the measured object, when the measured object is a cylinder, the radius of the measured object needs to be obtained; when the measured object is a square, the side length of the measured object needs to be obtained; and the inner diameter, outer diameter, number of turns and lift-off height of the coil need to be obtained.

[0054] For a complex conductor structure, a single coil-cable numerical simulation is realized by using a frequency domain finite element method and a magnetic field interface in an AC / DC module of COMSOL software, and a finite element model is established. Taking a carbon fiber composite cable as an example, according to the coil parameters, the inner diameter R1 is 5 mm, the outer diameter R2 is 10 mm, the number of turns N is 350, and the lift-off height is 0.5 mm. The radius of each single carbon fiber composite cable rod is r = 4 mm. An excitation current of 1 A is applied, and the excitation frequency is 1 MHz.

[0055] The lift-off height in the finite element model is the same as the lift-off height when the measured object is actually detected by eddy current.

[0056] When the finite element model is established, the same spatial coordinate system as the physical model is established, and the surface of the measured object is the surface detected by the coil. Therefore, the coordinates of each point in the region of the finite element model can be obtained.

[0057] By applying frequency domain current or voltage excitation to the coil, the coil generates a magnetic field, thereby exerting electromagnetic excitation on the measured object. The magnetic induction intensity attenuates when propagating in the air. By using the finite element method, the material of the measured object is equivalent to the air medium, and the magnetic induction intensity Γ 1,2 on the boundary is obtained. According to the continuity of the boundary condition, the magnetic induction intensity of each point at the interface between the measured object and the air is obtained, that is, the magnetic induction intensity of the outer surface of the measured object on Γ 2,3 .

[0058] In the finite element model, the frequency of the current or voltage excitation applied to the coil needs to be consistent with the frequency of the current or voltage excitation applied to the coil in actual detection.

[0059] The magnetic induction intensity of each point includes the magnetic induction intensity in the x direction, the y direction, and the z direction. For example, at the coordinate (1.33, 0.23, 0.14), the magnetic induction intensity in the x direction is 0.0038 T, the magnetic induction intensity in the y direction is 0.016 T, and the magnetic induction intensity in the z direction is 0.0023 T.

[0060] When the main magnetic field of the measured object is symmetric about the xoy plane or the yoz plane, the exponential decay general solution of the magnetic induction intensity of the measured object in each direction can be simplified. When the main magnetic field is odd symmetric about the xoy plane, C5=-C6 in the near coil action area; when the main magnetic field is even symmetric about the xoy plane, C5=C6 in the near coil action area; when the main magnetic field is odd symmetric about the yoz plane, C1=-C2 in the near coil action area; and when the main magnetic field is even symmetric about the yoz plane, C1=C2 in the near coil action area. The simplified general solution is easier to calculate.

[0061] By using Matlab programming, the magnetic induction intensity of each point in the x direction at the interface between the measured object and the air is substituted into the exponential decay general solution of the magnetic induction intensity of the measured object in the x direction, and the values of C 1x , C 2x , C 3x , C 4x , C 5x , C 6x , a x , b x , c x , d x , e x , and f x are calculated; similarly, the magnetic induction intensity of each point in the y direction at the interface between the measured object and the air is substituted into the exponential decay general solution of the magnetic induction intensity of the measured object in the y direction, and the values of C 1y , C 2y , C 3y , C 4y , C 5y , C 6y , ay , b y , c y , d y , e y , and f y ; the magnetic induction intensity of all points z direction of air interface is substituted into the exponential decay general solution of the magnetic induction intensity of the measured object in z direction, and the values of C 1z , C 2z , C 3z , C 4z , C 5z , C 6z , a z , b z , c z , d z , e z , and f z are calculated. Then the values are substituted into the exponential decay general solution of the magnetic induction intensity of the measured object in each direction to obtain the analytical solution of the magnetic field distribution of the measured object, B=B x i+B y j+B z k.

[0062] S4. According to the spatial relationship between the measured object and the coil, the depth direction of the eddy current decay is determined.

[0063] According to the specific magnetic field expression on the conductor cross section (denoted as cross section S) under the dominant action of the main magnetic field, the maximum current density |J| eddy on the cross section S is calculated based on the Ampere law ▽×B=μJ max , and the point coordinates (x0, y0) where the maximum value is located are obtained.

[0064] As shown in Figure 4 , the physical model has been established before, and the spatial relationship between the measured object and the coil can be obtained through the physical model of the measured object and the coil. The curvature radius of the cross section of the constituent unit of the measured object at the origin is calculated. When the measured object is composed of repeated or independent constituent units, the constituent units form the measured object by rigid connection (such as welding and bonding), such as the carbon fiber rod in the carbon fiber composite cable in Figure 4 ; when the measured object is not formed by rigid connection, but a whole, such as a steel plate, the constituent unit of the measured object is itself. The straight line where the point where the maximum current density is located and (0, r) are located together is the depth direction of the eddy current decay, that is, (y0-r)x-x0y+rx0=0.

[0065] When the curvature radius is -∞ or +∞, the depth direction is x=x0.

[0066] S5. According to the Ampere law, the analytical solution of the magnetic field distribution of the measured object, and the depth direction of the eddy current decay, the skin depth of the measured object is determined.

[0067] According to Ampere's law, the point where the current density decays to 1 / e of the surface current density is calculated, that is, the point where the current density J0 / e is located. The current density values ​​at different points may be the same. It is necessary to find the point (x1, y1) in the depth direction of eddy current decay and where the current density is J0 / e.

[0068] Then the skin depth δ of the object being measured is: δ=[(x1-x0) 2 +(y1-y0) 2 ] 1 / 2 .

[0069] like Figure 5 As shown, the theoretical calculation results of the magnetic field distribution of the carbon fiber composite parallel cable structure under the excitation field generated by the circular coil are compared with the numerical calculation data. The left column shows the influence of the change of x on the magnetic induction intensity in each direction when y and z are fixed; the middle column shows the influence of the change of y on the magnetic induction intensity in each direction when x and z are fixed; the right column shows the influence of the change of z on the magnetic induction intensity in each direction when x and y are fixed. The first row shows the change of magnetic induction intensity in the x direction, the second row shows the change of magnetic induction intensity in the y direction, and the third row shows the change of magnetic induction intensity in the z direction. The determination coefficient R of each component magnetic field intensity 2 They are all higher than 0.819, indicating that the model has significant predictive ability, the electromagnetic field prediction results can effectively reflect the physical laws, and the overall prediction accuracy is high.

[0070] When calculating the skin depth of objects of the same shape and size but made of different materials, the present invention simply changes the conductivity in the formula. Even when calculating the skin depth of objects of different shapes and sizes, or when changing the excitation frequency, the present invention does not require meshing. This is unlike methods that use finite element models to calculate skin depth, which require re-meshing. For example, higher frequencies require finer meshes, resulting in a cumbersome process and limited flexibility. Furthermore, after each modification, the finite element model must iteratively solve the large-scale matrix equation, often requiring extended calculation times.

Claims

1. A method for calculating the skin depth of an electrically orthogonal anisotropic conductor, characterized in that: The steps include: S1. Establish a physical model of the measurement area based on the object being measured and the coil; and establish an anisotropic wave equation for the conductivity tensor based on Maxwell's equations and the physical model. S2. Solve the anisotropic wave equation of the conductivity tensor to obtain the exponential decay general solution of the magnetic induction intensity of the object under test in all directions; S3. Use the finite element method to solve the magnetic induction intensity at the interface between the measured object and the air under non-plane wave excitation. Utilize the general solution of the exponential decay of the measured object's magnetic induction intensity in all directions to obtain an analytical solution for the magnetic field distribution of the measured object. S4. Determine the depth direction of eddy current attenuation based on the spatial relationship between the object being measured and the coil; S5. Determine the skin depth of the object under test based on Ampere's law, the depth direction of eddy current attenuation, and the analytical solution of the magnetic field distribution of the object under test.

2. The skin depth calculation method according to claim 1, wherein the step The specific way to establish the physical model in S1 is: Extend the central axis of the coil to the position where it intersects with the object being measured as the origin, and the direction in which the central axis of the coil is extended is the y-axis; the direction perpendicular to the y-axis in the cross-section of the object being measured detected by the coil is the x-axis, and the axial direction of the object being measured is the z-axis; the lift-off height between the coil and the surface of the object being measured is h.

3. The skin depth calculation method according to claim 2, wherein: The specific method of determining the depth direction of eddy current attenuation in step S4 is: S4-1. Calculate the point of maximum current density according to Ampere's law; S4-2. Determine the radius of curvature r of the cross section of the component unit of the object being measured at the origin; S4-3. The point with the maximum current density and the straight line (0, r) are the depth direction of eddy current attenuation. When the radius of curvature is -∞ or +∞, the depth direction of eddy current attenuation is the straight line parallel to the y-axis and passing through the point with the maximum current density.

4. The skin depth calculation method according to claim 1, wherein: The general solution of the exponential decay of the magnetic induction intensity of the object being measured in all directions is: B i =(C 1i e jkix +C 2i e jkix )(C 3i e jmiy +C 4i e jmiy )(C 5i e jniz +C 6i e jniz ) Where i = x, y, z, C 1i 、C 2i 、C 3i 、C 4i 、C 5i and C 6i is an undetermined constant.

5. The skin depth calculation method according to claim 1, wherein: The specific method for determining the skin depth of the object being measured is: The point where the current density decays to 1 / e of the surface current density in the depth direction of eddy current decay is calculated, and the distance between the point where the current density decays to 1 / e of the surface current density and the point where the current density is maximum is taken as the skin depth of the object under test.

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