A method for extracting electromagnetic parameters based on frequency-independent material response

Abstract

The application discloses a kind of electromagnetic parameter extraction methods based on frequency and material independent reaction, characterized by first to target conductor surface triangulation, then discrete electric field integral equation, obtain the integral expression of partial resistance, partial full-wave inductance and partial full-wave potential coefficient matrix;Recursive expansion is carried out to the full-wave Green function kernel, and is decomposed into frequency-independent geometric integral term of each order and frequency-dependent wave number coefficient term of each order;Pre-calculate the geometric integral term to form a matrix library;In the frequency scanning range, calculate the wave number coefficient term of each order and linearly superimpose the matrix library for each frequency point, reconstruct the full-wave partial inductance matrix and the partial full-wave potential coefficient matrix;Combined with partial resistance matrix, port excitation and load, based on equivalent charge variable replacement to construct modified node analysis matrix equation and solve, extract electromagnetic parameter;The advantage is that frequency-dependent and independent parts are separated, geometric integral term is pre-calculated once, and multiple frequency points are reused, which significantly reduces the wideband scanning calculation amount.

Description

Technical Field

[0001] This invention belongs to the field of computational electromagnetics technology, and in particular to a method for extracting electromagnetic parameters based on frequency- and material-independent reactions. Background Technology

[0002] The Partial Element Equivalent Circuit (PEEC) method is a numerical method that transforms electromagnetic field problems into equivalent circuit network problems. By constructing the resistance, inductance, potential coefficients, and their coupling relationships, this method converts the electromagnetic response of a conductor structure into a form solvable by circuit equations. It has wide applications in parameter extraction for interconnect structures, encapsulation structures, transmission line structures, and complex conductor systems. Under full-wave conditions, the kernel functions in the partial inductance matrix and the partial potential coefficient matrix become: This kernel function contains both near-field singularities and frequency-dependent oscillation characteristics, causing the computational cost of direct double integration of matrix elements to increase linearly with the number of frequency points. This constitutes the main performance bottleneck in wideband multi-frequency sweep analysis for full-wave PEEC engineering applications.

[0003] Existing full-wave PEEC methods typically require recalculating the full-wave kernel function integral for each frequency point during multi-frequency sweep analysis, resulting in a large amount of redundant computation. This is especially true when the mesh size is large and the number of frequency points is large, as the matrix construction time increases significantly, becoming a major factor limiting the engineering application of full-wave PEEC.

[0004] To address the low efficiency of multi-frequency sweeping, various broadband full-wave PEEC optimization schemes have been proposed in the industry, but each has its limitations and can be mainly divided into two categories: one category still calculates the full-wave kernel function integral independently at each frequency point, with the computational load increasing linearly with the number of frequency points; the other category uses model order reduction methods such as asymptotic waveform estimation, rational function fitting, and moment matching to approximate the frequency response, which may lead to decreased accuracy or unstable extrapolation results in broadband and strong resonant structures. Both of these methods have not completely separated the frequency-independent geometric part from the frequency-dependent part of the full-wave integral at the kernel function processing level, making it difficult to simultaneously achieve both broadband sweeping efficiency and full-band computational accuracy.

[0005] Furthermore, when constructing a partial inductance matrix, if the original double surface integral is processed directly, the integration of singular and near-singular terms becomes difficult, which is not conducive to the combination of analytical and numerical integration.

[0006] On the other hand, the frequency and material independent reactions (FMIR) technique was originally proposed in the field of the method of moments. By expanding the Green's function into a Taylor series, the impedance matrix elements are decomposed into a real-valued matrix that is only related to geometry and a scalar product that is related to the propagation constant. It shows a significant computational efficiency advantage in broadband sweeps of scattering analysis. Therefore, how to use FMIR technology to achieve broadband and efficient calculations in the full-wave surface PEEC framework is a problem that needs to be solved. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to provide an electromagnetic parameter extraction method based on frequency-independent material response, which can accurately and efficiently extract the electromagnetic parameters of conductor structures in a wide frequency band and significantly reduce the computational load of matrix construction for multi-frequency sweep.

[0008] The technical solution adopted by this invention to solve the above-mentioned technical problems is as follows: a method for extracting electromagnetic parameters based on frequency- and material-independent reactions, comprising the following steps: Step 1: Perform surface triangular meshing on the target conductor structure. Define RWG basis functions on the common edges inside the mesh as unknown edge currents, and define surface pulse basis functions on the mesh cells as unknown surface charges. Step 2: Based on the material parameters of the target conductor and the relationship between the mesh topology, the electric field integral equation is discretized using the Galerkin discretization method to obtain the integral expressions of a portion of the resistance matrix, a portion of the full-wave inductance matrix, and a portion of the full-wave potential coefficient matrix. Step 3: For the full-wave Green's function kernels of a portion of the full-wave inductance matrix and a portion of the full-wave potential coefficient matrix, respectively, recursively expand them using the frequency-independent reaction method to decompose them into frequency-independent geometric integral terms of each order of inductance and each order of potential, as well as frequency-dependent inductance wavenumber coefficient terms and each order of potential wavenumber coefficient terms. Step 4: Calculate the geometric integral terms of inductance and potential at each order through the recursive scheme of natural separation of singularities, and obtain the matrix library of geometric integral terms of inductance and the matrix library of geometric integral terms of potential. Step 5: For each frequency point within the preset sweep range, calculate the inductance wavenumber coefficients and potential wavenumber coefficients of each order based on the wavenumber corresponding to the current frequency point, and linearly superimpose them with the corresponding geometric integral term matrix library to reconstruct the full-wave inductance matrix and full-wave potential coefficient matrix of the current frequency point. Step 6: Combine the partial resistance matrix with the full-wave partial inductance matrix and the full-wave partial potential coefficient matrix, introduce port excitation and port load, construct and solve the compact modified node analysis matrix equation to obtain the edge current, surface charge and surface potential at each frequency point. Step 7: Extract electromagnetic parameters based on the edge current, surface charge, and surface potential at each frequency point.

[0009] Compared with the prior art, the advantages of this invention are that, in step 3, frequency-independent material-independent reaction recursive expansion is introduced into the full-wave surface PEEC framework. Taylor expansion is performed on the full-wave Green's function kernel of part of the full-wave inductance matrix and part of the full-wave potential coefficient matrix, decomposing each matrix element into the sum of products of geometric integral terms of each order (i.e., frequency-independent) that are only related to geometry and mesh, and wavenumber coefficient terms of each order (i.e., frequency-dependent) that are only related to wavenumber, thus achieving complete separation of the frequency-dependent and frequency-independent parts. In step 4, a recursive scheme based on singularity natural separation is used to calculate all frequency-independent inductance geometric integral terms and potential geometric integral terms of each order sequentially, establishing a matrix library of inductance geometric integral terms and a matrix library of potential geometric integral terms. This pre-calculation process only needs to be performed once, fundamentally avoiding the huge overhead of repeatedly calculating the double integral of the kernel function containing frequency oscillation characteristics at each frequency point in traditional methods. In step 5, for the pre- For each frequency point within the sweep range, it is only necessary to quickly calculate the inductance wavenumber coefficients and potential wavenumber coefficients of each order based on the current wavenumber. Then, these are linearly superimposed with the stored geometric integral term matrix library to efficiently reconstruct the full-wave partial inductance matrix and full-wave partial potential coefficient matrix of the current frequency point. This reduces the computational cost of single-frequency matrix reconstruction from the traditional O(N²×Ng²) to O(N²×P), ​​and the acceleration effect is more significant with more frequency points. In step 6, the partial resistance matrix is ​​further combined with the reconstructed full-wave partial inductance matrix and full-wave partial potential coefficient matrix, and a compact modified node analysis matrix equation is constructed. This causes the dense frequency-dependent part of the matrix equation related to the partial potential coefficient matrix to degenerate into a scalar diagonal structure. This, combined with the geometric integral term matrix library pre-established in step 4 and the linear superposition reconstruction method in step 5, improves the numerical stability and overall efficiency of multi-frequency point solutions. In the verification example of the signal line-ground plane interconnect structure, under the condition of 200 frequency points sweep, the total time for constructing part of the inductor matrix was reduced from 389 s to 234 s, achieving a speedup of 1.66×. The matrix construction method was changed from repeatedly performing double integration point by point to pre-calculating once and then rapidly reconstructing multiple frequency points. As the number of sweep frequency points further increases, the acceleration effect will continue to expand, while the storage overhead of the geometric integral term remains unchanged. The peak memory increased from 48 MB to 105 MB, demonstrating the technical feature of achieving computational acceleration by pre-calculating and storing the geometric integral term. This invention maintains accuracy comparable to point-by-point full-wave calculation across the entire frequency band (especially wideband strong resonant structures), avoiding the accuracy degradation or extrapolation instability problems that may occur in existing model order reduction methods. At the same time, it significantly reduces the amount of repeated integration calculations during wideband sweep, achieving efficient and accurate extraction of wideband electromagnetic parameters of the full-wave surface PEEC.

[0010] Furthermore, in step 4, the geometric integral terms of inductance and potential at each order are calculated sequentially using a recursive scheme based on the natural separation of singularities. These calculations include both zero-order and higher-order geometric integral terms. The zero-order geometric integral term is calculated using a two-layer semi-analytical scheme with an inner-layer analytical approach and an outer-layer numerical approach. Higher-order geometric integral terms are obtained from the previous order geometric integral term through a single-step linear recursion. Each recursive process involves only addition, subtraction, and multiplication operations, without introducing any additional approximation or truncation errors. Thus, while ensuring the accuracy of singular term processing, the one-time pre-calculation and multi-frequency reuse of frequency-independent geometric integral terms are achieved.

[0011] Furthermore, in step 5, when reconstructing the full-wave partial inductance matrix, the inductance wavenumber coefficients of each order are multiplied by the inductance geometric integral term matrix library according to their corresponding orders and then summed; when reconstructing the full-wave partial potential coefficient matrix, the potential wavenumber coefficients of each order are multiplied by the potential geometric integral term matrix library according to their corresponding orders and then summed. This ensures that the matrix construction at each frequency point only requires low-order polynomial summation and scalar multiplication, avoiding the need to recalculate the double integral containing the oscillation kernel at each frequency point in traditional methods, thus reducing the computational load per frequency point from... Down to Furthermore, the truncation order P in the reconstruction formula can be independently controlled according to accuracy requirements, achieving a flexible balance between computational efficiency and accuracy under wideband sweep frequency.

[0012] Furthermore, in step 6, an equivalent charge variable substitution is used to construct a compact modified nodal analysis matrix equation. The expression for the compact modified nodal analysis matrix equation is as follows: ,in, For the port excitation vector, Represents angular frequency. This represents the full-wave inductance matrix. Represents a partial resistance matrix. This represents the edge-face correlation matrix constructed from the mesh topology. This represents the full-wave partial potential coefficient matrix. It is the identity matrix. Represents the edge current vector. Represents the equivalent charge vector. , Represented as a surface potential vector; The compact modified node analysis matrix equations at each frequency point are solved using a dense complex linear equation solver to obtain the edge currents of each inner edge. and the equivalent charge on each triangular face This reduces the surface potential. , This degenerates the lower right block of the matrix equation into a scalar diagonal matrix, significantly reducing the numerical rigidity of the equation system and improving the convergence speed and stability of the solver for dense complex linear equation systems.

[0013] Furthermore, step 6 also includes multiplying by a frequency correlation correction coefficient when constructing the compact corrected node analysis matrix equation at each frequency point. The DC resistance is corrected to a frequency-dependent broadband loss model, with frequency-dependent correction coefficients. Surface impedance in Based on skin depth With conductor thickness The relative relationships are uniformly expressed as: ,in The conductivity of a conductor. For conductor thickness, For skin depth; when When it degenerates into a DC loss model, when In some cases, the model degenerates into a high-frequency surface impedance model; in others, a broadband thickness correction model is used. This achieves continuous, seamless conductor loss modeling from DC to microwave frequencies. The frequency-dependent correction coefficient is directly embedded in the matrix equation of step 6, without changing the FMIR recursion and reconstruction framework of steps 3 to 5. This allows the invention to accurately simulate the frequency-dependent loss characteristics of conductors of different thicknesses during the skin effect evolution process while maintaining the advantages of broadband and efficient parameter extraction, thus expanding the applicability of the method to practical engineering structures.

[0014] Furthermore, in step 7, the electromagnetic parameters include one or more of the following: port voltage, port current, input impedance, input admittance, parasitic parameter RLC, power distribution, power consumption, and equivalent S-parameters. Attached Figure Description

[0015] Figure 1 This is a model diagram of a transmission line instance; Figure 2 This is an overall flowchart of the method of the present invention; Figure 3 This is a schematic diagram showing the result of input impedance changing with frequency; Figure 4 for A schematic diagram showing the results as a function of frequency. Detailed Implementation

[0016] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0017] This embodiment uses a three-dimensional transmission line structure with a copper signal line suspended above a finite ground plane as the verification object to demonstrate the entire process of the method of the present invention, from geometric modeling to broadband extraction of full-wave electromagnetic parameters. The structure is as follows: Figure 1As shown, the verification structure is a three-dimensional transmission line model with a copper signal line suspended above a finite ground plane. The specific parameters of the verification structure are as follows: the copper signal line is 20mm long and 1mm wide, suspended 0.5mm above the finite ground plane; the ground plane is 20mm long and 10mm wide; the conductor conductivity is... A horizontal cut is made along the center of the signal line, with the excitation port and load port connected to either side. An 86Ω current source is applied to the excitation port, and an 86Ω matching resistor is connected to the load port. A wideband scan is performed within the frequency range of 1GHz to 10GHz to extract the input impedance. and equivalent parameters The results were compared and verified with the simulation reference curves of the reference simulation software FEKO.

[0018] like Figure 2 As shown, a method for extracting electromagnetic parameters based on frequency- and material-independent reactions includes the following steps: Step 1: Perform surface triangular meshing on the target conductor structure to obtain multiple triangular face elements, and obtain the mesh topology relationship based on the adjacency relationship between the mesh elements. Define RWG basis functions as edge current unknowns on the common edges inside each triangular mesh, and define surface pulse basis functions as surface charge unknowns on each triangular mesh element. In this embodiment, the conductor surfaces of the signal line and the ground plane are respectively divided into triangular meshes. The surface of the signal line (including the top and bottom surfaces and four side surfaces) and the top surface of the ground plane are both divided into triangular mesh elements. The internal common edge is the unknown edge current, and the triangular mesh element is the element where the unknown surface charge is located. An RWG basis function is defined on each internal common edge, supported on a pair of adjacent triangular mesh elements on both sides of the edge. The surface charge is uniformly spread on each triangular mesh element using the surface pulse basis function.

[0019] Step 2: Based on the material parameters of the target conductor and the mesh topology, the electric field integral equation is discretized using the Galerkin discretization method to obtain integral expressions for a partial resistance matrix, a partial full-wave inductance matrix, and a partial full-wave potential coefficient matrix. The material parameters of the target conductor include conductivity, permeability, and conductor thickness. The partial resistance matrix is ​​determined by the conductor resistance on each support edge. The matrix elements of the partial inductance matrix and the partial potential coefficient matrix are represented by double surface integrals, with the integrand being the full-wave Green's function. The core operation of Galerkin discretization is to expand the unknowns (edge ​​current J and surface charge ρ) using basis functions, and then use the test function to perform an inner product on both sides of the integral equation, thereby transforming the continuous integral equation into a system of linear equations.

[0020] Step 3: For the full-wave Green's function kernel of a portion of the full-wave inductance matrix and the full-wave Green's function kernel of a portion of the full-wave potential coefficient matrix, respectively, a recursive expansion is performed using the frequency-independent reaction method to decompose and obtain frequency-independent geometric integral terms of inductance and potential, as well as frequency-dependent wavenumber coefficient terms of inductance and potential. Specifically: the full-wave Green's function kernel of part of the full-wave inductance matrix and the full-wave Green's function kernel of part of the full-wave potential coefficient matrix are both The full-wave Green's function kernel is expanded around the center. Recursive expansion of reaction methods that are independent of frequency and material: Among them, the development center Selection of distances: For mutual terms, the distance between the geometric centers of the two triangular faces is taken; for independent terms, the weighted average of the distances between multiple pairs of Gaussian integral points is taken to avoid... To prevent instability in the unfolding process caused by the value approaching zero, the unfolding truncation order P is set to 3 in this embodiment. For wavenumber coefficients, This is a geometric integral term.

[0021] Step 4: Calculate the geometric integral terms of inductance and potential at each order using a recursive scheme based on the natural separation of singularities, thereby obtaining the inductance geometric integral term matrix library and the potential geometric integral term matrix library. The specific operation process is as follows: Among them, the zeroth-order geometric integral term corresponds to the original static integral with a 1 / R singular kernel. It is accurately calculated using a two-layer semi-analytical scheme with an inner layer of analysis and an outer layer of numerical analysis: the inner layer integral (the integral over the source triangle) uses the analytical integral formula on the triangle element to accurately handle the 1 / R singularity; the outer layer integral (the integral over the observed triangle) uses numerical Gaussian integration; the combination of the two can accurately handle the self-term and the nearest neighbor term with a 1 / R singular kernel without truncation or regularization. The specific operation of calculating the geometric integral terms of inductance at each order through the recursive scheme of natural separation of singularities is as follows: Step 4-1-1, Calculate The zeroth-order geometric integral term at time: ,in, This represents the positive / negative observation triangle of the RWG basis function corresponding to edge m. This represents the positive / negative source triangle of the RWG basis function corresponding to edge n. Represents the observation triangular unit The RWG basis functions of m above, Represents source triangular face elements The RWG basis functions of n above, Indicates the observation point With the source The distance between them This represents the integral element on the observed triangular unit. Represents the integral surface element on the source triangular facet; Step 4-1-2, for Higher-order geometric integral terms, calculating intermediate quantities: ,in, Indicates the observation triangle The unfolding center and the source triangle The distance between the centers of expansion, Indicates the current recursive order. ; Step 4-1-3, based on the intermediate quantity obtained in step 4-1-2 and the first-order geometric integral term The recurrence relation for the corresponding higher-order geometric integral term, obtained through single-step linear recursion, is as follows: ; Step 4-1-4: Repeat steps 4-1-1 to 4-1-3, calculating sequentially up to the truncation order P, and then calculate all the resulting geometric integral terms of each order. Stored as a matrix of inductance geometric integral terms; The specific operation of calculating the geometric integral terms of each potential step by step using the recursive scheme of natural separation of singularities is as follows: Step 4-2-1, Calculate The zeroth-order geometric integral term at time: , Let these represent the areas of the triangular face elements containing face s and face t, respectively. Represents the observation triangular unit scalar impulse basis functions on Represents source triangular face elements scalar impulse basis functions on This represents the integral element on the observed triangular unit. Represents the integral surface element on the source triangular facet; Step 4-2-2, for Higher-order geometric integral terms, calculating intermediate quantities: ,in, Represents the observed triangle surface The unfolding center to the source triangle face The distance from the center of the unfolding, Indicates the current recursive order. ; Step 4-2-3, based on the intermediate quantity obtained in step 4-2-2 and the first-order geometric integral term The recurrence relation for the corresponding higher-order geometric integral term, obtained through single-step linear recursion, is as follows: ; Step 4-2-4: Repeat steps 4-2-1 to 4-2-3, calculating sequentially up to the truncation order P, and then calculate all the resulting geometric integral terms of each order. Stored as a matrix of potential geometric integral terms.

[0022] Step 5: For each frequency point within the preset frequency sweep range (e.g., 1GHz to 10GHz, evenly divided into 200 frequency points), calculate the inductance wavenumber coefficients and potential wavenumber coefficients of each order based on the wavenumber corresponding to the current frequency point; linearly superimpose the inductance wavenumber coefficients of each order with the inductance geometric integral term matrix library to reconstruct the full-wave partial inductance matrix of the current frequency point; linearly superimpose the potential wavenumber coefficients of each order with the potential geometric integral term matrix library to reconstruct the full-wave partial potential coefficient matrix of the current frequency point. Let the element of the full-wave partial inductance matrix in the m-th row and n-th column be denoted as... , ,in, Represents the permeability of free space. This represents the length of the inner edge of the RWG basis function corresponding to edge m. This represents the length of the inner edge of the RWG basis function corresponding to edge n. Corresponding to the positive and negative triangular faces of the RWG basis functions. For the symbolic factors determined by the combination of faces, Represents the imaginary unit. This indicates the wave number corresponding to the current frequency. Indicates the observation triangle The unfolding center and the source triangle The distance between the centers of expansion, The truncation order; Let the element of the full-wave partial potential coefficient matrix in the s-th row and t-th column be denoted as... , ,in, Indicates the dielectric constant of the medium. Let S and T represent the areas of the triangular face elements containing face S and face T, respectively. Represents the observed triangle surface The unfolding center to the source triangle face The distance from the center of the unfolding.

[0023] Since the inductance geometric integral term and the potential geometric integral term are pre-stored, the matrix reconstruction at each frequency point only requires the linear superposition of coefficients, without the need to repeatedly perform double integration.

[0024] Step 6: Combine the partial resistance matrix with the full-wave partial inductance matrix and the full-wave partial potential coefficient matrix, introduce port excitation and port load, and construct a compact modified nodal analysis matrix equation based on equivalent charge variable substitution. The partial resistance matrix elements only reflect conductor losses under DC conditions and cannot characterize the increase in losses caused by the high-frequency skin effect. Therefore, when constructing the compact modified nodal analysis matrix equation at each frequency point, a frequency-related correction factor is multiplied. The DC resistance is corrected to a frequency-dependent broadband loss model; the compact corrected node analysis matrix equation is directly solved by calling the dense complex linear equation solver to obtain the edge current, surface charge and surface potential at each frequency point; Specifically, the expression for the compact modified node analysis matrix equation is: ,in, For the port excitation vector, Represents angular frequency. This represents the full-wave inductance matrix. Represents a partial resistance matrix. This represents the edge-face correlation matrix constructed from the mesh topology. This represents the full-wave partial potential coefficient matrix. It is the identity matrix. Represents the edge current vector. Represents the equivalent charge vector. , It is represented as a surface potential vector; in this embodiment, the 86 Ω matching resistor load at both ends is coupled into the equation via admittance. The compact modified node analysis matrix equations at each frequency point are solved using a dense complex linear equation solver to obtain the edge currents of each inner edge. and the equivalent charge on each triangular face And thus through the reduction surface potential Reduced surface potential .

[0025] Convert the surface potential vector φ into an equivalent charge vector. This makes the lower right block of the compact modified node analysis matrix equation consist of a frequency-dependent dense matrix. Exactly degenerates into a scalar diagonal matrix This completely eliminates the construction and storage overhead of frequency-dependent dense inverse matrix terms, fundamentally avoiding... The numerical instability risk introduced by explicit calculation is mitigated; the diagonalization effect and the FMIR geometric pre-computation mechanism form a deep technical synergy, achieving systematic computational acceleration and accuracy assurance in broadband electromagnetic parameter extraction.

[0026] In this embodiment, the frequency correlation correction coefficient Surface impedance in Based on skin depth With conductor thickness The relative relationships are uniformly expressed as: ,in The conductivity of a conductor. For conductor thickness, For skin depth; when When it degenerates into a DC loss model, when In the case of a certain condition, the model degenerates into a high-frequency surface impedance model, while in other cases, a broadband thickness correction model is used. The three cases are naturally unified within a single formula framework, without the need for manual model switching or setting frequency thresholds.

[0027] Step 7: Extract electromagnetic parameters based on the edge current, surface charge, and surface potential at each frequency point. Electromagnetic parameters include one or more of the following: port voltage, port current, input impedance, input admittance, parasitic parameter RLC, power distribution, power consumption, and equivalent S-parameters. Specifically: the port current is obtained by summing the edge current expansion coefficients corresponding to all RWG basis functions at the excitation port; the port voltage is obtained by subtracting the surface potentials on the positive and negative triangular facets of the excitation port; when an external voltage source is applied for excitation, the voltage is a known quantity, and the port current is extracted after solving the problem; when an external current source is applied for excitation, the current is a known quantity, and the port voltage is extracted after solving the problem. The input impedance is obtained by the ratio of the surface potential difference between the positive and negative triangular facets of the excitation port to the excitation current, and the input admittance is obtained by taking its reciprocal. The parasitic parameter RLC is extracted from the real and imaginary parts of the input impedance. The power distribution and power consumption are obtained by combining the current and charge of each element with a loss model. The input impedance is obtained by the ratio of the port voltage to the excitation current. Then, the equivalent S-parameters are calculated: ,in This is the port reference impedance. Its magnitude is then taken. Compare and verify with the reference results.

[0028] Figure 3 The results show a comparison between the real and imaginary parts of the input impedance as a function of frequency in the 1–10 GHz frequency band using the proposed method (i.e., FMIR-PEEC in the figure) and the reference simulation software FEKO. The curves of the two methods are highly consistent. The real part remains stable in the range of 80–100 Ω throughout the entire frequency band, while the imaginary part fluctuates slightly near zero. This verifies the accuracy of the proposed method in calculating the input impedance.

[0029] Figure 4 The modulus of the equivalent S-parameter is given. The comparison results with frequency variation show that the two curves of the method proposed in this invention (i.e., FMIR-PEEC in the figure) and the reference simulation software FEKO have the same trend in the full frequency band from 1 to 10 GHz, the resonant frequency matches well, and the average absolute error is 0.32 dB, which further verifies the accuracy of the method of this invention in calculating the input impedance.

[0030] Table 1

[0031] Table 1 compares the accuracy of the proposed method with the reference simulation software FEKO in a frequency range of 1–10 GHz and with 200 frequency points. As shown in Table 1, the proposed method and the reference simulation software FEKO yield highly consistent results: the real part of the input impedance is... The mean absolute error is 0.45 Ω, and the imaginary part of the input impedance is... The mean absolute error is 0.40 Ω, and the modulus of the equivalent S-parameter is... The mean absolute error was 0.32 dB, which verifies the calculation accuracy of the method of the present invention in a wide frequency range of 1 to 10 GHz.

[0032] Table 2

[0033] Table 2 shows the efficiency comparison with the direct frequency-point integration method. As can be seen from Table 2, the speedup of the method proposed in this invention is 1.66×; the peak memory increases from 48 MB to 105 MB, reflecting the pre-computation reuse strategy of trading space for time; as the number of sweep frequency points further increases, the time advantage of the method proposed in this invention will continue to expand, while the storage overhead of the geometric integral term remains unchanged.

[0034] The above embodiments are for illustrative purposes only and are not intended to limit the scope of protection of this invention. All equivalent substitutions, improvements, and modifications made within the spirit and principles of this invention should fall within the scope of protection of this invention.

Claims

1. A method for extracting electromagnetic parameters based on frequency- and material-independent reactions, characterized in that... Includes the following steps: Step 1: Perform surface triangular meshing on the target conductor structure. Define RWG basis functions on the common edges inside the mesh as unknown edge currents, and define surface pulse basis functions on the mesh cells as unknown surface charges. Step 2: Based on the material parameters of the target conductor and the relationship between the mesh topology, the electric field integral equation is discretized using the Galerkin discretization method to obtain the integral expressions of a portion of the resistance matrix, a portion of the full-wave inductance matrix, and a portion of the full-wave potential coefficient matrix. Step 3: For the full-wave Green's function kernels of a portion of the full-wave inductance matrix and a portion of the full-wave potential coefficient matrix, respectively, recursively expand them using the frequency-independent reaction method to decompose them into frequency-independent geometric integral terms of each order of inductance and each order of potential, as well as frequency-dependent inductance wavenumber coefficient terms and each order of potential wavenumber coefficient terms. Step 4: Calculate the geometric integral terms of inductance and potential at each order through the recursive scheme of natural separation of singularities, and obtain the matrix library of geometric integral terms of inductance and the matrix library of geometric integral terms of potential. Step 5: For each frequency point within the preset sweep range, calculate the inductance wavenumber coefficients and potential wavenumber coefficients of each order based on the wavenumber corresponding to the current frequency point, and linearly superimpose them with the corresponding geometric integral term matrix library to reconstruct the full-wave inductance matrix and full-wave potential coefficient matrix of the current frequency point. Step 6: Combine the partial resistance matrix with the full-wave partial inductance matrix and the full-wave partial potential coefficient matrix, introduce port excitation and port load, construct and solve the compact modified node analysis matrix equation to obtain the edge current, surface charge and surface potential at each frequency point. Step 7: Extract electromagnetic parameters based on the edge current, surface charge, and surface potential at each frequency point.

2. The method for extracting electromagnetic parameters based on frequency- and material-independent reactions according to claim 1, characterized in that... In step 4, the geometric integral terms of inductance and potential are calculated step by step using a recursive scheme based on the natural separation of singularities. These include both zero-order and higher-order geometric integral terms. The zeroth-order geometric integral term is calculated using a two-layer semi-analytical scheme with an inner-layer analytic approach and an outer-layer numerical approach; higher-order geometric integral terms are obtained from the previous-order geometric integral term through single-step linear recursion.

3. The method for extracting electromagnetic parameters based on frequency- and material-independent reactions according to claim 2, characterized in that... In step 5, when reconstructing the full-wave partial inductance matrix, the inductance wavenumber coefficients of each order are multiplied by the inductance geometric integral term matrix library according to their corresponding orders and then summed; when reconstructing the full-wave partial potential coefficient matrix, the potential wavenumber coefficients of each order are multiplied by the potential geometric integral term matrix library according to their corresponding orders and then summed.

4. The method for extracting electromagnetic parameters based on frequency- and material-independent reactions according to claim 1, characterized in that... In step 6, the equivalent charge variable substitution is used to construct the compact modified nodal analysis matrix equation. The expression of the compact modified nodal analysis matrix equation is as follows: ,in, Represents the imaginary unit. Represents angular frequency. This represents the full-wave inductance matrix. Represents a partial resistance matrix. This represents the edge-face correlation matrix constructed from the mesh topology. This represents the full-wave partial potential coefficient matrix. It is the identity matrix. Represents the edge current vector. For the port excitation vector, Let φ represent the equivalent charge vector, and let φ represent the surface potential vector. The compact modified node analysis matrix equations at each frequency point are solved using a dense complex linear equation solver to obtain the edge currents of each inner edge. and the equivalent charge on each triangular face This reduces the surface potential. , .

5. The method for extracting electromagnetic parameters based on frequency- and material-independent reactions according to claim 1, characterized in that... Step 6 also includes multiplying by a frequency correlation correction factor when constructing the compact corrected node analysis matrix equation at each frequency point. The DC resistance is corrected to a frequency-dependent broadband loss model, with frequency-dependent correction coefficients. Surface impedance in Based on skin depth With conductor thickness The relative relationships are uniformly expressed as: ,in, The conductivity of a conductor. For conductor thickness, To reach skin depth, Represents angular frequency. Indicates permeability, Represents the hyperbolic cotangent function; when When it degenerates into a DC loss model, when In some cases, the model degenerates into a high-frequency surface impedance model; in others, a broadband thickness correction model is used.

6. The method for extracting electromagnetic parameters based on frequency- and material-independent reactions according to claim 1, characterized in that... In step 7, the electromagnetic parameters include one or more of the following: port voltage, port current, input impedance, input admittance, parasitic parameter RLC, power distribution, power consumption, and equivalent S-parameters.

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