A structural parameter equivalent modeling method for the induced voltage of a three-coil abrasive sensor

Abstract

This invention relates to the field of sensor and signal processing technology, specifically to a method for equivalent modeling of the structural parameters of the induced voltage of a three-coil abrasive sensor. The method includes: establishing an equivalent geometric model of the coils and a physical model of the magnetic field for the three-coil abrasive sensor; deriving a structural kernel function; decoupling the structural kernel function to obtain a normalized dimensionless structural kernel function that depends only on the geometric characteristics of the coils; deriving an approximate analytical expression of the Gaussian first derivative of the normalized dimensionless structural kernel function; defining an objective function based on the approximate analytical expression of the Gaussian first derivative; iteratively solving the objective function using a nonlinear least squares algorithm to complete the construction of an approximate equivalent model of the induced field distribution of the three-coil abrasive sensor; this invention realizes a clear physical representation of the mapping relationship between the sensor output abrasive induced voltage and structural parameters, reducing computational complexity.

Description

Technical Field

[0001] This invention relates to the field of sensor and signal processing technology, specifically to a method for equivalent modeling of the structural parameters of the induced voltage of a three-coil abrasive sensor. Background Technology

[0002] Inductive abrasive sensors are key components for health monitoring in the lubrication circuits of large rotating machinery (such as aircraft engines and wind turbines). These sensors capture metal abrasive particles in the oil to achieve online monitoring of the wear condition of mechanical equipment. They offer advantages such as non-invasiveness, insensitivity to oil contamination, and real-time detection, and have been widely used in aerospace, marine engineering, and energy production. The sensor's structural parameters, such as the number of coil turns, radius, and relative spacing, have a decisive impact on its core performance characteristics, including detection sensitivity, signal-to-noise ratio, and spatial resolution. Therefore, optimizing its structure during the design phase is crucial.

[0003] Traditional inductive abrasive sensor output voltage calculations are mostly based on integral models of the Biot-Savart law. These models have complex geometries, and structural optimization typically requires repeated finite element simulations or precise numerical calculations to evaluate output characteristics under different structural parameters. However, a single high-precision simulation is time-consuming, often taking hours or even days. Performing multi-parameter, multi-dimensional global optimization requires substantial computational resources, making engineering implementation difficult. Some studies have attempted to use surrogate models or intelligent optimization algorithms (such as genetic algorithms and particle swarm optimization) to accelerate the optimization process, but these methods still require thousands of iterations of approximate models, resulting in high overall computational costs. More importantly, such optimizations are often "black box" operations, making it difficult to reveal the clear physical relationship between sensor performance and structural parameters, lacking intuitive physical insights and reliable design basis. Furthermore, relying solely on experimental testing to select parameter combinations is not only costly and time-consuming but also makes it difficult to systematically cover comparative verification of multiple sets of structural parameters.

[0004] Therefore, there is an urgent need to establish an equivalent model of the induced voltage structural parameters of a three-coil abrasive sensor that is computationally inexpensive, has clear physical meaning, and is easy to derive and simplify, in order to support subsequent parameter optimization and provide effective support for the design and optimization of high-performance sensors. Summary of the Invention

[0005] To address the above problems, this invention provides a method for equivalent modeling of the structural parameters of the induced voltage of a three-coil abrasive sensor, comprising:

[0006] S1. For a three-coil abrasive sensor, establish its coil equivalent geometric model and magnetic field physical model, including:

[0007] S11. The three-coil abrasive sensor includes one induction coil and two differential excitation coils. All coils are multi-layer rectangular cross-section solid coils.

[0008] S12. Based on the winding layout parameters of the three-coil abrasive sensor, the multi-layer rectangular cross-section solid coil is equivalent to a thin-walled cylindrical model through the second-order moment matching principle, thereby obtaining the equivalent radius of the multi-layer rectangular cross-section solid coil;

[0009] S13. Based on the geometric characteristics of the coil, the precise equations of the axial magnetic field of the induction coil and the differential excitation coil are established using the Biot-Savart law, and the structural kernel function is derived.

[0010] S2. Decouple the structure kernel function to obtain a normalized dimensionless structure kernel function that depends only on the geometric characteristics of the coil;

[0011] S3. Derive the approximate analytical expression of the Gaussian first derivative of the normalized dimensionless structure kernel function;

[0012] S4. Define the objective function based on the approximate analytical expression of Gauss's first derivative, and use the nonlinear least squares algorithm to iteratively solve it, thus completing the construction of an approximate equivalent model of the induction field distribution of the three-coil abrasive sensor.

[0013] The beneficial effects of this invention are:

[0014] This invention transforms the complex physical model of a sensor into a concise analytical model based on Gaussian first-order derivatives through series expansion and approximate derivation. This model achieves a high degree of decoupling and refinement of the structural response characteristics, specifically by constructing a structural kernel function and introducing an amplitude coefficient. The complex geometric dependence of the coils, along with the scaling factor c, is transformed into two parameters with clear physical meaning. The Levenberg-Marquardt optimization algorithm is used in the middle region to effectively suppress the impact of numerical errors on model accuracy, ensuring that the equivalent model achieves a fitting accuracy of over 97% in the core signal response region. The structural parameter equivalent modeling method for the induced voltage of a three-coil abrasive sensor proposed in this invention is not only applicable to differential three-coil structures, but can also be widely extended to the modeling and signal processing of various inductive and non-inductive sensors by adjusting the mapping logic of the characteristic parameters. The concise analytical form greatly facilitates the optimization design of model parameters. Attached Figure Description

[0015] Figure 1 This is a flowchart of the structural parameter equivalent modeling method in the induced voltage of the three-coil abrasive sensor of the present invention;

[0016] Figure 2 This is a schematic diagram of the structure of the three-coil abrasive sensor of the present invention;

[0017] Figure 3 This is a magnetic induction intensity distribution diagram of the two excitation coils of the three-coil abrasive sensor of the present invention;

[0018] Figure 4 This is a comparison diagram of the differential magnetic field and structural kernel function distribution of the three-coil abrasive sensor of the present invention;

[0019] Figure 5 This is a diagram showing the Maxwell simulation signal filtering and demodulation results of an embodiment of the present invention;

[0020] Figure 6 This is a schematic diagram comparing the analytical model, the equivalent model, and the Maxwell simulation model of this invention. Detailed Implementation

[0021] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0022] Please see Figures 1-2 This invention provides a method for equivalent modeling of the structural parameters of the induced voltage of a three-coil abrasive sensor, comprising the following steps:

[0023] S1. For a three-coil abrasive sensor, establish its coil equivalent geometric model and magnetic field physical model, including:

[0024] S11. The three-coil abrasive sensor includes one induction coil and two differential excitation coils. All coils are multi-layer rectangular cross-section solid coils.

[0025] A multi-layer rectangular cross-section solid coil is a precision electromagnetic coil constructed by winding a single solid conductor (solid wire) and arranging it in multiple layers. This coil has a regular overall structure and rectangular conductor cross-section, making it a typical winding configuration used in devices such as abrasive sensors and inductive sensors.

[0026] In some embodiments, such as Figure 2 The three-coil abrasive sensor shown has a differential excitation coil arranged on the left and right sides of the induction coil, respectively.

[0027] S12. Based on the winding layout parameters of the three-coil abrasive sensor, the multi-layer rectangular cross-section solid coil is equivalent to a thin-walled cylindrical model through the second-order moment matching principle, thereby obtaining the equivalent radius of the multi-layer rectangular cross-section solid coil.

[0028] In some embodiments, the core of the second-order moment matching principle lies in modeling based on the principle of equivalent average magnetic flux area of ​​the coil. For a multilayer rectangular cross-section solid coil, the number of turns is uniformly distributed in the radial direction (from the inner radius to the outer radius). This is because the induced electromotive force and the magnetic flux area passing through the coil... They are positively correlated. In order to ensure that the equivalent geometric model (i.e. the equivalent thin-walled cylindrical model) can accurately reproduce the electromagnetic induction characteristics of the original multi-layer rectangular cross-section solid coil, the square of the radius of the equivalent geometric model needs to be equal to the second-order origin moment of the radius distribution of the original multi-layer rectangular cross-section solid coil (i.e., the expected value of the uniform distribution of the square of the radius).

[0029] Assuming the number of coil turns is in the interval [ , The distribution is uniformly distributed on the surface, and the second moment of each turn radius is used to replace the actual distribution:

[0030] ,

[0031] The formula for calculating the equivalent radius R of a multilayer rectangular cross-section solid coil is:

[0032] ,

[0033] In the formula, R O R represents the outer radius of the coil. I This indicates the inner radius of the coil.

[0034] S13. Based on the geometric characteristics of the coil, the precise equations of the axial magnetic field of the induction coil and the differential excitation coil are established using the Biot-Savart law, and the structural kernel function is derived.

[0035] In some embodiments, based on Figure 2 The three-coil abrasive sensor structure shown has the following precise equation for the axial magnetic field of the induction coil:

[0036] ,

[0037] ,

[0038] In the formula, B s (z) represents the axial magnetic field of the induction coil, μ0 = 4π × 10 -7 H represents the free permeability. N represents the density of the induction coil. s L represents the number of turns of the induction coil. s g represents the length of the induction coil. s ( ) represents the kernel function of the induction coil, u represents the axial position variable of the induction coil end face, and R s represents the equivalent radius of the induction coil; z represents the axial position coordinate of the sensor.

[0039] The exact equation for the axial magnetic field of the differential excitation coil is expressed as:

[0040] ,

[0041] ,

[0042] ,

[0043] In the formula, B1(z) and B2(z) represent the axial magnetic fields of the differential excitation coils on the left and right sides, respectively. N represents the differential excitation coil density. D L represents the number of turns in the differential excitation coil. D L represents the length of the differential excitation coil. G The distance between the inner end faces of the two differential excitation coils is represented by I, which represents the excitation current, and g is the excitation current. D ( ) represents the kernel function of the differential excitation coil, v represents the axial position variable of the differential excitation coil end face, and R D This represents the equivalent radius of the differential excitation coil. The magnetic induction intensity of the two differential excitation coils is as follows: Figure 3 As shown.

[0044] The structure kernel function S(z) is expressed as:

[0045] ,

[0046] ,

[0047] In the formula, B0(z) represents the differential axial magnetic field. Figure 4 This is a comparison diagram of the differential magnetic field and structural kernel function distribution of a three-coil abrasive sensor.

[0048] S2. Decouple the structure kernel function to obtain a normalized dimensionless structure kernel function that depends only on the geometric characteristics of the coil:

[0049] S21. Analyze the contribution of abrasive particles to the eddy current of a three-coil abrasive sensor.

[0050] In some embodiments, a conductive spherical particle with conductivity σ and radius a is taken as an example, located on the central axis (i.e., the z-axis) of the sensor and subjected to a differential axial magnetic field B0(z). It is assumed that the size of the conductive spherical particle is much smaller than the coil and the frequency is sufficiently low, such that the magnetic flux density within the conductive spherical particle can be approximated as spatially uniform along the z-axis and equal to B0(z). Under this uniform field limit, the induced electric field and eddy currents within the conductive spherical particle are both pure azimuth fields. Applying Faraday's law to a circular loop with radius rsinθ within the conductive spherical particle, the expression for the azimuth electric field can be obtained. r represents the radial distance from any point inside the conductive spherical particle to its center, θ represents the angle between the position vector and the sensor axis, and ω represents the angular frequency of the excitation current; the azimuth current density can be obtained according to Ohm's law. Substituting this current distribution into the standard dipole moment definition... Furthermore, by utilizing spherical symmetry, the volume integral can be simplified to a single radial moment, ultimately yielding the eddy current contribution to the sensor. for:

[0051] .

[0052] S22. Analyze the magnetization contribution of abrasive particles to a three-coil abrasive sensor.

[0053] In some embodiments, taking a ferromagnetic spherical particle with magnetic susceptibility χ and radius a as an example, when it is located on the central axis of the sensor (i.e., the z-axis), its magnetization response to the differential axial magnetic field B0(z) is expressed using a scalar demagnetization factor N. d =1 / 3 for modeling:

[0054] ,

[0055] ,

[0056] ,

[0057] M represents the uniform magnetization intensity within the ferromagnetic spherical particle, H int H represents the internal magnetic field, and H0 represents the unmagnetized magnetic field. This represents the unit vector along the positive direction of the sensor axis. (Further details are needed for a complete translation.) To obtain effective magnetic susceptibility The magnetic dipole moment is given by the formula It is concluded that This represents the volume of a ferromagnetic spherical particle. This is the component of the magnetic moment along the z-axis. Ultimately, the magnetization contribution of the ferromagnetic spherical particle to the equivalent magnetic dipole on the z-axis of the sensor is obtained. :

[0058] .

[0059] S23. Calculate the induced voltage based on the electromagnetic reciprocity theorem and the magnetic dipole theory.

[0060] In some embodiments, the total magnetic flux generated by the magnetic dipole in the induction coil Numerically equal to the unit harmonic current flowing through the induction coil. The dot product of the sensitivity field generated by time and the original magnetic dipole moment, i.e. ,in, = Let Bs(z) represent the total contribution of the abrasive particles, and Bs(z) represent the axial magnetic field of the induction coil. This is based on Faraday's law. The induced voltage can be obtained. .

[0061] By separating the eddy current factor and magnetization factor of the abrasive particles from the geometric characteristics of the coil, the linear mapping relationship between the induced voltage and the structural kernel function is obtained as follows:

[0062] ,

[0063] In the formula, U(z) represents the induced voltage of the three-coil abrasive sensor, ω represents the angular frequency of the excitation current, j represents the complex form, and α e =σωa 2 V / 10 represents the eddy current factor of the abrasive grains, α m =χV / μ0(1+χ / 3) represents the magnetization factor of the abrasive grain, and S(z) represents the structural kernel function.

[0064] S24. Construct a normalized dimensionless structure kernel function that depends only on the geometry of the coil. :

[0065] ,

[0066] ,

[0067] ,

[0068] In the formula, μ0 represents the free permeability. N represents the density of the induction coil. s L represents the number of turns of the induction coil. s Indicates the length of the induction coil. N represents the differential excitation coil density. D L represents the number of turns in the differential excitation coil. D I represents the length of the differential excitation coil. M Indicates the amplitude of the excitation current. This represents the core structure of the differential excitation coil. This represents the core structure of the induction coil; , , , , R s R represents the equivalent radius of the induction coil. D The value represents the equivalent radius of the differential excitation coil, and z represents the axial position coordinate of the sensor. This represents the sensor's normalized axial coordinates.

[0069] S3. Derive the approximate analytical expression of the Gaussian first derivative of the normalized dimensionless structure kernel function.

[0070] In some embodiments, step S3 includes:

[0071] S31. Perform series expansion and limit analysis on the normalized dimensionless structure kernel function.

[0072] The embodiments of this invention employ an additive power series representation. This method exhibits better numerical stability over a wide axial range and can naturally achieve attenuation characteristics that conform to physical laws. The expression is as follows:

[0073] ,

[0074] ,

[0075] In the formula, B represents the central slope of the normalized dimensionless structure kernel function at the origin, S'(0) represents the first derivative of the normalized dimensionless structure kernel function at the origin, and s1 represents the coefficient of the first term in the expansion of the normalized dimensionless structure kernel function. This represents the normalized dimensionless structure kernel function. Represents the normalized odd-power expansion coefficients. Approximated by the inverse polynomial. The symmetric shape term leads to the infinite series form within the effective sensing region:

[0076] ,

[0077] a k >0 indicates the coefficient of even-degree terms, and k=1,2,3...n indicates the index of the terms in the series summation.

[0078] S32. Higher-order truncation of the denominator M defines the upper limit of error.

[0079] Infinite series form The infinite denominator series is truncated to order M:

[0080] ,

[0081] The remainder term of the denominator series is ignored. Within the given middle (valid) interval Inside, This represents the boundary value of the normalized effective sensing interval, assuming that both the truncated denominator and the complete denominator are uniformly away from zero, i.e., there exists... Make , .

[0082] Utilizing the spatial distribution characteristics of the structural kernel in the intermediate region, the error caused by truncation can be limited to... Therefore, for For any given tolerance, M can be chosen such that Small enough, and Always below the tolerance. It can replace the infinite series form in the effective sensing region described in S31. This is used for subsequent parameter modeling and structural optimization.

[0083] S33. In The positive definiteness condition, in the common case where the roots are real and negative, involves truncating the denominator and factoring into... get:

[0084] ,

[0085] in, As an effective scale, the representative value is located in the middle range. When aggregated, the product can be approximated by a single-scale power function, where h represents the exponential decay coefficient and p represents the power exponent. This yields a compact power function approximation. For larger p values, this power function form in standard scaling... The following can be transformed into the exponential limit form. Therefore, it can be concluded that when hour, Therefore, under the large p-value limit, the sensor's structural kernel approaches the product of a linear factor and a Gaussian envelope:

[0086] .

[0087] Where A represents the amplitude coefficient. denoted by , where is the normalized axial coordinate of the sensor, and c represents the scale coefficient.

[0088] S4. Define the objective function according to the approximate analytical expression of Gauss's first derivative, and solve it iteratively using the nonlinear least squares algorithm to finally complete the construction of the equivalent model of the sensor structure.

[0089] In some embodiments, N discrete sampling points are selected within the central region of the sensor. The objective function is defined as follows:

[0090] ,

[0091] Among them, y i This indicates that the i-th discrete sampling point is determined according to the normalized dimensionless structure kernel function. Calculated sample values, This represents the normalized axial coordinate of the sensor corresponding to the i-th discrete sampling point.

[0092] The Levenberg-Marquardt (LM) algorithm is used to iteratively minimize the objective function until the parameters converge, and the optimal feature parameters A and c are output.

[0093] In one specific embodiment, the differential excitation coil and the induction coil are defined as having an inner diameter of 4mm, an outer diameter of 8mm, an axial length of 10mm, and a spacing of 10mm between the two differential excitation coils. Both the differential excitation coil and the induction coil have 320 turns. Within the sensor's observation area... ,in This represents the normalized dimensionless axial displacement. This explains the solution process for the parameters A and c of the Gaussian first-order derivative equivalent model for the approximate distribution of the induced field in this invention.

[0094] First, based on the equivalent radius calculation formula in S1, the equivalent radii of the differential excitation coil and the induction coil are obtained. ≈6.1101mm, then the normalized dimensionless structure kernel function in step S24 was used. Definition, calculation of dimensionless parameters , , , .

[0095] Within the intermediate region, the number of discrete sampling points N = 1630, and the sampling step size is approximately 0.01. This intermediate region corresponds to... Within this intermediate region, the y-values ​​of each discrete sampling point are calculated. i Numerical values. The LM algorithm was used for fitting, with a function tolerance set to 10. −8 The step size tolerance is also set to 10. −8 During the optimization process, the initial value of the amplitude coefficient was set to... =-0.5; Attenuation coefficient set to =1. The algorithm generally converges to the optimal solution within 10 iterations, yielding the optimal solution A≈-0.4897, c≈0.8852. Based on the above fitting results, an approximate equivalent model of the induced field distribution of the three-coil abrasive sensor is established:

[0096] .

[0097] Furthermore, the accuracy of the approximate equivalent model of the sensor is verified.

[0098] Calculate its position in the middle region ( The coefficient of determination (R) of the region (which covers the core part of the signal amplitude and induced voltage) 2 Mean absolute error (MAE), root mean square error (RMSE), and peak relative error (RE) peakThese metrics are used to evaluate the predictive accuracy and stability of the model. Its expression is:

[0099] ,

[0100] ,

[0101] ,

[0102] ,

[0103] In the formula, Indicates the i-th discrete sampling point according to Calculated sample values, express peak value express The peak value.

[0104] The calculated value is: R 2 =0.9999, RE peak The results show that the coefficient of determination of the approximate equivalent model in the intermediate region is as high as 0.9999, the peak amplitude error is much less than 1%, and the RMSE is much lower than 0.5%, which meets the accuracy requirement of more than 97% of the preset threshold.

[0105] The Gaussian first derivative approximation analytical formula proposed in this invention can approximate the shape, amplitude, and peak position of the original induced voltage signal with high accuracy in the middle region of the signal, meeting the accuracy requirements of practical applications such as metal particle detection and defect location.

[0106] In some embodiments, simulation verification is performed based on ANSYS Maxwell, with the following specific parameters:

[0107] Excitation coils: Two sets of identical coils, inner diameter 4mm outer diameter It is 8mm, axial length 5mm, spacing 10mm, number of turns The value is 320. A sinusoidal excitation current with an amplitude of 1A and a frequency of 20kHz is applied.

[0108] Induction coil: located at the center of the structure, inner diameter 4mm outer diameter It is 8mm, axial length 5mm, number of turns It is 320.

[0109] Abrasive particles: speed set to 3 m / s, magnetic susceptibility set to 4000.

[0110] Materials: The coil is a copper conductor, and the surrounding medium is air.

[0111] Mesh generation: Adaptive mesh refinement is adopted, and the axial path mesh size is controlled within 0.1 to 0.5 mm to ensure calculation accuracy;

[0112] Solution settings: A transient solver is used.

[0113] Extract the induced voltage signal along the axial centerline as a reference signal. The extracted signal is then filtered and demodulated, such as... Figure 5 As shown. To align the simulation data with the mathematical model, the peak value of the induced voltage waveform is identified. With negative peak time Calculation center time The time axis is shifted to a zero point with the waveform center. Normalization is then performed by dividing the simulated voltage amplitude by the maximum absolute value to obtain the normalized amplitude. .

[0114] Because Maxwell simulation outputs time-domain data, while analytical models... Equivalent Model For spatial domain data, i.e., in S23 Equivalent to .

[0115] in, For abrasive particle parameter coefficients, a mapping relationship needs to be established. :

[0116]

[0117] in, For abrasive particle velocity, The equivalent radius of the induction coil. The zero point of the time is the center of the abrasive grain signal. This is obtained after mapping. Similarly, divide the simulated voltage amplitude by its maximum absolute value, i.e. The error caused by abrasive particles in the signal was removed, resulting in a synchronized analytical model sequence. and equivalent model sequence Ultimately, as Figure 6 As shown, a comprehensive comparison chart of ferromagnetic wear particle signals under the analytical model, equivalent model, and Maxwell simulation model is obtained.

[0118] The equivalent model described in step S4 is adopted. For the normalized simulation signal Calculate the error index within the intermediate region.

[0119] The typical simulation and equivalent model calculation results of this embodiment are as follows:

[0120] Determination coefficient of the intermediate region Peak relative error Mean absolute error Root mean square error .

[0121] Keeping the abrasive particle velocity, magnetic susceptibility, excitation current magnitude, and frequency constant, the sensor structural parameters were changed, and multiple sets of comparisons were performed, as shown in Table 1 below:

[0122] Table 1 Examples of Multiple Sets of Structural Parameters

[0123]

[0124] Table 2 Comparison results of equivalent model, analytical model, and Maxwell simulation model

[0125]

[0126] To comprehensively verify the accuracy and universality of the equivalent model proposed in this invention under different sensor structures, this embodiment selected 12 sets of sensor structure parameters with significant differences for comparative testing (as shown in Table 1). The selected parameter range not only covers conventional geometric dimensions, but also specifically includes extreme cases such as asymmetric number of turns (Group 10) and special aspect ratios (Groups 11 and 12) to test the boundary performance of the model. The experimental results are shown in Table 2. In the comparison between the equivalent model and the analytical model, the coefficient of determination R for all 12 sets of parameters is... 2 All are above 0.995, and the peak relative error RE peak The root mean square error (RMSE) was generally below 0.4% (except for a few extreme groups), indicating that the equivalent model had a very high degree of consistency in its mathematical derivation. Furthermore, in the comparison between the equivalent model and the Maxwell finite element simulation model, despite a wide range of changes in structural parameters, the RMS error remained consistent across the vast majority of groups. 2 It remains above 0.99. Even for the more structurally unique Group 11 (narrow and thick structure), R... 2 The accuracy remains above 0.965, with a root mean square error (RMSE) of only 0.0958. This result fully demonstrates that the equivalent model proposed in this invention has excellent robustness, can adapt to the design requirements of sensors with different sizes, numbers of turns, and spacings, and can be directly used to guide the parameter optimization and performance prediction of sensors. It also shows that the equivalent model proposed in this invention maintains high accuracy even in the complex three-dimensional field environment of electromagnetic simulation verification. This approximation method has good robustness to the effects of actual multi-layer windings, finite sizes, and boundary effects, and the computational complexity is significantly reduced after model simplification.

[0127] In summary, within the field of three-coil abrasive sensor technology as described in this invention, existing technologies largely rely on empirical design and repeated experiments, lacking systematic theoretical guidance and quantitative optimization pathways in their design process. Specifically, traditional design methods struggle to establish a clear mathematical relationship between the sensor's core structural parameters (such as coil spacing, number of turns, and coil radius) and the final output signal characteristics, resulting in long design cycles, low optimization efficiency, and difficulty in accurately achieving the predetermined detection performance indicators.

[0128] Compared to these traditional methods, the structural parameter equivalent modeling method for the induced voltage of the three-coil abrasive sensor proposed in this invention exhibits significant advantages. This method is stable and robust throughout the entire optimization design system, providing a reliable optimal solution regardless of the complexity or simplicity of the parameter combination to be optimized.

[0129] This method excels in handling complex optimization problems with multiple parameters and strong coupling. By utilizing the established explicit mathematical mapping relationships, it significantly improves the directionality of the design process and the optimality of the results. Employing electromagnetic field theory and data mapping, it effectively addresses the design challenges of complex structures. The results demonstrate that this method exhibits good accuracy and generalization ability in sensor performance tuning, providing a new paradigm for the intelligent design of high-performance sensors. Future research can further explore the applicability of this method in the design of different types of sensors (such as capacitive and piezoelectric sensors), and its application in more complex scenarios such as multi-physics coupling, to continuously improve the robustness and universality of this method.

[0130] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for equivalent modeling of structural parameters of the induced voltage of a three-coil abrasive sensor, characterized in that, include: S1. For a three-coil abrasive sensor, establish its coil equivalent geometric model and magnetic field physical model, including: S11. The three-coil abrasive sensor includes one induction coil and two differential excitation coils. All coils are multi-layer rectangular cross-section solid coils. S12. Based on the winding layout parameters of the three-coil abrasive sensor, the multi-layer rectangular cross-section solid coil is equivalent to a thin-walled cylindrical model through the second-order moment matching principle, thereby obtaining the equivalent radius of the multi-layer rectangular cross-section solid coil; S13. Based on the geometric characteristics of the coil, the precise equations of the axial magnetic field of the induction coil and the differential excitation coil are established using the Biot-Savart law, and the structural kernel function is derived. S2. Decouple the structure kernel function to obtain a normalized dimensionless structure kernel function that depends only on the geometric characteristics of the coil; S3. Derive the approximate analytical expression of the Gaussian first derivative of the normalized dimensionless structure kernel function; S4. Define the objective function based on the approximate analytical expression of Gauss's first derivative, and use the nonlinear least squares algorithm to iteratively solve it, thus completing the construction of an approximate equivalent model of the induction field distribution of the three-coil abrasive sensor.

2. The method for equivalent modeling of structural parameters of the induced voltage of a three-coil abrasive sensor according to claim 1, characterized in that, The formula for calculating the equivalent radius R of a multilayer rectangular cross-section solid coil is: ， In the formula, R O R represents the outer radius of the coil. I This indicates the inner radius of the coil.

3. The method for equivalent modeling of structural parameters of the induced voltage of a three-coil abrasive sensor according to claim 1, characterized in that, The exact equation for the axial magnetic field of the induction coil is expressed as: ， ， In the formula, B s (z) represents the axial magnetic field of the induction coil, and μ0 represents the free permeability. N represents the density of the induction coil. s L represents the number of turns of the induction coil. s g represents the length of the induction coil. s ( ) represents the kernel function of the induction coil, u represents the axial position variable of the induction coil end face, and R s The value represents the equivalent radius of the induction coil; z represents the axial position coordinate of the sensor. The exact equation for the axial magnetic field of the differential excitation coil is expressed as: ， ， ， In the formula, B1(z) and B2(z) represent the axial magnetic fields of the two differential excitation coils. N represents the differential excitation coil density. D L represents the number of turns in the differential excitation coil. D L represents the length of the differential excitation coil. G The distance between the inner end faces of the two differential excitation coils is represented by I, which represents the excitation current, and g is the excitation current. D ( ) represents the kernel function of the differential excitation coil, v represents the axial position variable of the differential excitation coil end face, and R D This represents the equivalent radius of the differential excitation coil; The structure kernel function S(z) is expressed as: ， ， In the formula, B0(z) represents the differential axial magnetic field.

4. The method for equivalent modeling of structural parameters of the induced voltage of a three-coil abrasive sensor according to claim 1, characterized in that, Step S2 includes: Based on the electromagnetic reciprocity theorem and magnetic dipole theory, the eddy current factor and magnetization factor of the abrasive particles are separated from the geometric characteristics of the coil, and the linear mapping relationship between the induced voltage and the structural kernel function is obtained as follows: ， In the formula, U(z) represents the induced voltage, ω represents the angular frequency of the excitation current, j represents the complex form, and α e The eddy current factor, α, represents the abrasive grains. m S(z) represents the magnetization factor of the abrasive grains, and S(z) represents the structural kernel function. Constructing a normalized dimensionless structure kernel function that depends only on the geometry of the coil : ， ， ， In the formula, μ0 represents the free permeability. N represents the density of the induction coil. s L represents the number of turns of the induction coil. s Indicates the length of the induction coil. N represents the differential excitation coil density. D L represents the number of turns in the differential excitation coil. D I represents the length of the differential excitation coil. M Indicates the amplitude of the excitation current. This represents the core structure of the differential excitation coil. This represents the core structure of the induction coil; , , , , R s R represents the equivalent radius of the induction coil. D denoted by z, which represents the equivalent radius of the differential excitation coil, and z represents the axial position coordinate of the sensor.

5. The method for equivalent modeling of structural parameters of the induced voltage of a three-coil abrasive sensor according to claim 1, characterized in that, Gaussian first derivative approximate analytical expression for: ， In the formula, A represents the amplitude coefficient. denoted by , where is the normalized axial coordinate of the sensor, and c represents the scale coefficient.

6. The method for equivalent modeling of structural parameters of the induced voltage of a three-coil abrasive sensor according to claim 1, characterized in that, Select N discrete sampling points within the middle region of the sensor. The objective function is defined as follows: ， Among them, y i This indicates that the i-th discrete sampling point is determined according to the normalized dimensionless structure kernel function. Calculated sample values, This represents the normalized axial coordinate of the sensor corresponding to the i-th discrete sampling point.

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