Finite element electromagnetic numerical method for port cross-section transmission matching of waveguide design
By constructing a hybrid basis function form of scalar and vector basis functions, and combining functional discretization and boundary conditions, the difficulty of optimizing and simulating complex waveguide models in the prior art is solved, and fast and accurate generalized eigenvalue solving and simulation design are realized.
Patent Information
- Application Number
- CN202310351445.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-04-04
- Publication Date
- 2026-08-25
- Estimated Expiration
- 2043-04-04
AI Technical Summary
Existing port technologies cannot effectively handle the optimization simulation of complex waveguide models, especially in the presence of losses, resulting in low computational efficiency and inaccurate results.
By employing a hybrid basis function form of scalar and vector basis functions, combined with functional discretization and various boundary conditions, the generalized eigenvalue equation is derived to handle waveguide port matching problems with complex geometry and materials, especially lossy physical problems.
It enables fast and accurate solving of generalized eigenvalue equations, improves the accuracy and efficiency of simulation design, effectively handles the transmission problem of complex waveguide models, and obtains better S-parameters.
Smart Images
Figure CN116306163B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of frequency domain electromagnetic numerical solution technology, specifically a finite element electromagnetic numerical method for port cross-section transmission matching in waveguide design. For waveguide transmission problems, a hybrid basis function of scalar basis functions and vector basis functions is used to solve port transmission problems with discontinuities such as different media and different boundaries. Background Technology
[0002] Waveguides are widely used devices in the microwave field for guiding electromagnetic wave propagation. For example, the design and analysis of integrated electronic and optical circuits often require waveguide structures made of different materials, such as metals, conductive media, and dielectrics. The mode fields of electromagnetic wave propagation differ depending on the waveguide cross-section. For regular waveguides, the cross-sectional shape, size, and dielectric material distribution are constant along the axis or longitudinal direction. In this case, waveguide eigenvalue analysis can be used to solve for the various modes present in the regular waveguide. Especially for regular waveguide problems with particularly simple boundaries and simple media, analytical methods can be used to obtain very accurate calculation results. However, as the complexity of the computational model increases, multilayer media often exhibit anisotropic properties, making analytical methods no longer applicable. Instead, various numerical algorithms (mode matching, multipole methods, spectral methods, etc.) and improved techniques combining mathematics and physics have emerged, aiming to obtain efficient numerical solutions to the original problems. For example, when RCMcphedran analyzed the boundary value problem of a two-dimensional periodic medium with dual sinusoidal modulation, he used coordinate transformation to convert the rough medium surface to a flat plane for analysis. Although this simplified the boundary conditions and made it easier to match the electromagnetic field quantities, the equations of Maxwell's equations were very complex, which increased the difficulty of solving the problem.
[0003] Currently, the classical finite element method is widely used in solving various waveguide and resonant cavity problems. Especially for complex discontinuous structures, the vector basis functions proposed by JCNedelec can avoid spurious solutions. Therefore, the vector finite element method is a significant breakthrough in computational electromagnetics. However, when the waveguide structure undergoes irregular changes, such as the filling medium no longer being uniform, or the waveguide junctions becoming non-uniform with complex structures and materials, traditional two-dimensional port methods face numerous problems such as boundary matching and energy loss.
[0004] In practice, lossless operation is often an ideal state. When the physical structure of a waveguide becomes very complex, not only dielectric losses but also conductor losses may exist. For example, in the high-frequency circuits of microwave tubes, the accompanying losses increase with the increase of the operating frequency band, severely affecting the computational efficiency of beam-wave interaction in high-frequency circuits. In particular, as the number of model meshes increases, the dimension of the generalized matrix equations of the intrinsic waveguide becomes very large, and even ill-conditioned matrices may appear, making the solution of eigenvalues extremely difficult.
[0005] As can be seen, these shortcomings make it impossible to optimize and simulate complex electromagnetic waveguide models using existing port technologies, which can no longer meet the requirements of designers. Therefore, it is necessary to construct stable and reliable numerical methods to obtain high-precision electromagnetic parameters. Summary of the Invention
[0006] To address the aforementioned problems and shortcomings, this invention, starting from the vector wave equation, appropriately constructs a hybrid basis function form of scalar and vector basis functions, providing a finite element electromagnetic numerical method for transmission matching at the port cross-section of waveguide design. This method includes functional discretization and the application of various boundaries. Compared to traditional waveguide solution methods, this invention provides a waveguide port matching technique that can effectively handle complex (anisotropic, non-uniform, etc.) geometries and materials. Especially for lossy physical problems, it can quickly and accurately solve the generalized eigenvalue equations, facilitating the analysis of corresponding electromagnetic parameters and effectively improving simulation design.
[0007] A finite element electromagnetic numerical method for transmission matching of port cross-sections in waveguide design includes the following steps:
[0008] Step A: Based on the physical structure of the target electronic device, and in conjunction with the working environment and boundary conditions, perform simulation modeling.
[0009] First, the electric field vector waveguide equation derived from the time-harmonic Maxwell equations in a dielectric waveguide structure is given:
[0010]
[0011] Where E is the electric field intensity vector, ε r μ is the relative permittivity of the medium. r Let be the relative permeability of the medium, k0 be the wave number in vacuum, D be the electric displacement vector, and B be the magnetic flux density. In practical waveguide physics problems, three commonly used boundary conditions are as follows:
[0012]
[0013] Where H is the magnetic field strength vector. It is the ideal conductor boundary condition (PEC). It is an ideal magnetic wall boundary condition (PMC). This is the boundary condition for applying impedance, where n represents the outward normal unit vector on the surface, and Z... s It represents the surface impedance of a conductor material.
[0014] Step B: Use tetrahedral elements to divide the three-dimensional solution domain. Surface discretization and volume discretization must be compatible.
[0015] Step C: Combining finite element scalar basis functions and vector basis functions, derive the eigenvalue equations for waveguide structures under general boundaries;
[0016] To solve equation (1), in the three directions x, y, and z of the spatial rectangular coordinate system, it is assumed that the geometry of the waveguide cross-section remains constant along the longitudinal z-axis (e.g., ...). Figure 2 The electric and magnetic fields are divided into transverse and longitudinal fields, as detailed below:
[0017]
[0018] Where the subscript t represents the transverse field (including the x and y axes), z represents the longitudinal field, γ is the propagation constant, z represents the z-direction vector, and E, E t H and H t It is a vector, E z With H z It is a scalar. Further definition:
[0019]
[0020] in Substituting equations (3) and (4) into the first equation of equation (1), we can decompose it into a transverse field and a longitudinal field, as follows:
[0021] Lateral field:
[0022] Longitudinal field:
[0023] Since the waveguide cross-section is considered as a two-dimensional region, step B can discretize the two-dimensional cross-sectional region into multiple small triangular units. Based on the forms in (5) and (6), the vector electric field E is discretized using the vector stacked basis function N. t The scalar electric field E is discretized using the scalar basis function L. z .
[0024] To clearly describe the implementation process, a simple first-order basis function is used below. Note that this method is also applicable to second-order and higher-order basis functions.
[0025] Consider a triangular surface element u with 3 nodes i, i∈{0,1,2}. Its transverse electric field can be expressed as:
[0026]
[0027] The longitudinal electric field is represented as:
[0028]
[0029] in It is a transverse vector electric field The expansion coefficients, expressed as vectors express; It is a longitudinal scalar electric field The expansion coefficients, expressed as vectors Substituting equations (7) and (8) into equations (5) and (6), we obtain the following matrix equation:
[0030]
[0031]
[0032] definition Equations (9) and (10) can then be transformed into the following system of generalized eigenvalue equations:
[0033]
[0034] Where the matrix and All are related to the vector basis function N, matrix It is simultaneously dependent on both vector basis functions N and scalar basis functions L, and the matrix It is related to the scalar basis function L.
[0035] Since the finite element method is based on mathematical functional variation, it is a numerical algorithm for effectively solving partial differential equations by constructing a complete basis function space. Therefore, different functional variation expressions are used for different physical problems. For example, for the eigenmodes of dielectric-filled waveguides, it is necessary to establish longitudinal electric and magnetic fields simultaneously, and also combine transverse electric and magnetic fields, as well as various different functional variation expressions such as vector potential and scalar potential. In most existing literature, waveguide problems with general boundary conditions (ideal electric walls and ideal magnetic walls) are mainly considered. This is because the tangential electric field is 0 on the ideal electric wall and the tangential magnetic field is 0 on the ideal magnetic wall, and the resulting functional expression is well-known and easy to solve. In order to make a comparative analysis with the present invention, the element matrix expression of the triangular surface element in formula (11) under general boundary conditions is given below:
[0036]
[0037]
[0038]
[0039]
[0040] Where the subscripts i∈{0,1,2} and j∈{0,1,2} represent the three nodes on the triangular facet element, and i and j can be the same or different. Based on the node and edge information of each facet element, each facet element can be integrated into an element matrix using the well-known finite element method to obtain the matrix form of the entire computational domain. As can be seen from the element matrix expression above, the final matrix of the entire computational domain is... and Both are sparse symmetric matrices. By combining the boundary conditions and solving the generalized eigenvalue equation (11), the eigenvalues and eigenvectors can be obtained.
[0041] Step D: Consider the lossy boundary conditions—impedance boundary conditions—and derive the generalized eigenvalue equations for the waveguide structure.
[0042] In practice, various loss scenarios often occur, including not only dielectric losses but also conductor losses. To further improve the computational efficiency of finite element waveguide eigenvalue analysis for lossy problems, in step D, this invention details the key techniques for handling impedance boundary conditions in the two-dimensional port problem with losses. Through theoretical derivation, this boundary condition is added to the vector wave equation, thereby obtaining the specific form of the generalized eigenvalue equation.
[0043] The third equation in formula (2) is the expression for the impedance boundary, which contains a magnetic field vector H. However, the generalized eigenvalue equation set (11) does not contain a magnetic field vector, so the magnetic field vector needs to be eliminated. Considering that the magnetic field vector can be derived from the electric field vector, that is:
[0044]
[0045] Where ω is the angular frequency, and μ is the permeability, the value of which is determined by the permeability μ0 in vacuum and the relative permeability μ of the medium. r Multiplying them together. Substituting formulas (3) and (4) into the above formula (16), we get:
[0046]
[0047] make Then H = zH z +H t z represents the longitudinal vector along the z-axis, E z It is the longitudinal electric field, E t It is a transverse electric field, H zIt is a longitudinal magnetic field, H t It is a transverse magnetic field. Substituting equation (3) into the expression for the impedance boundary, we get:
[0048]
[0049] Substituting equation (17) into equation (18), the impedance boundary condition can be further derived as follows:
[0050]
[0051] Where η0 is the wave impedance in vacuum.
[0052] Because of the introduction of impedance boundary conditions, unlike the previous PEC and PMC boundaries, the impedance boundary conditions contribute to the surface integral calculation, thus affecting the matrix in step C. and This will change; through derivation, we have:
[0053]
[0054] in and It is an intermediate variable in the derivation, specifically in the form of:
[0055]
[0056] Similarly, the generalized eigenvalue equations (11) can be obtained. Once the transmission coefficient γ is solved, other electromagnetic parameters can be obtained.
[0057] Step E: Using hybrid basis functions, the generalized eigenvalue equations of the discrete waveguide structure are derived.
[0058] This step is the core step of the generalized eigenvalue equation and is also the innovation of this invention. For different physical models, it is necessary to discretize the final mathematical model to facilitate programming implementation. In fact, selecting appropriate basis functions often plays a crucial role in the convergence and stability of the entire numerical solution.
[0059] From the specific form of the electromagnetic field, equation (3), it can be seen that the transverse field is a vector and the longitudinal field is a scalar. Considering that using vector basis functions can effectively avoid spurious solutions in the problem of discontinuity in complex structures, this invention uses vector stacked basis functions for the vector field and general scalar basis functions for the longitudinal field. The following will use this hybrid basis function to perform the generalized eigenvalue equation (11) of the discrete waveguide structure.
[0060] To illustrate the structure of stacked basis functions more vividly, Figure 3A schematic diagram of the transverse field t in this invention is given. In the following step, the discretization form of each matrix in (11) will be derived in conjunction with the two-dimensional mixed basis function.
[0061] The nodes and edges of the triangular face element are numbered starting from 0. First, define the three edges l on the triangular face element. i The local index numbers are shown in Table 1 below:
[0062] Table 1: Local edge numbering on triangular face units
[0063] 0 1 2 1 0 2 2 0 1
[0064] Define the three nodal scalar basis functions of the triangular face element as L i (i = 0, 1, 2), define the vector basis functions of the three edges as follows:
[0065]
[0066] For the gradient term in the above equation The present invention adopts the following form, which avoids the tedious calculation of traditional scalar basis function differentiation.
[0067]
[0068] Where t i The magnitude of is the length of the i-th edge, and the vector direction is perpendicular to the i-th edge and points towards the interior of the face triangle, specifically as follows: Figure 3 As shown. S is the area of the triangular facet unit.
[0069] First, derive equation (12). In the matrix According to the formula
[0070]
[0071] but:
[0072]
[0073] Similarly, for In the matrix Then we have:
[0074]
[0075] Combining with the existing formula (27):
[0076]
[0077] Equation (26) can be solved, and then equation (13) can be obtained.
[0078] The following derivation of equation (14) is based on equation (23). In the matrix
[0079]
[0080] According to the formula
[0081]
[0082] Then equation (28) becomes:
[0083]
[0084] The following derivation of equation (15) is based on equation (23). Matrix. Where Combining formula (27), we get:
[0085]
[0086] The discrete forms of all general boundary condition generalized eigenvalue equations have now been discussed.
[0087] Next, consider step D, where the matrix in equation (21) of the eigengeneralized equation system includes impedance boundary conditions. and The form is as follows; note that these two matrices have one-dimensional line integrals. First, let's... Simplified to:
[0088]
[0089] Consider the one-dimensional line integral in equation (32), according to the formula:
[0090]
[0091] Combining this with equation (23), we can obtain The discretized form; similarly, the following is derived. The discretized form of .
[0092] Step F. Solve the generalized eigenvalue equation.
[0093] Based on step E above, the discretized generalized eigenvalue equation is finally obtained, and the matrix form of each element is as shown in equation (11). Through finite element integration, the global matrix equation of the entire solution domain is finally obtained. There are currently various methods and techniques for solving this matrix equation. Since this method is a well-known technique in this field, different methods can be found, so it will not be described in detail here. Once the generalized eigenvalue equation is solved, the eigenvalues and eigenvectors can be obtained, and then other parameters of the electromagnetic field can be obtained for waveguide port cross-section transmission matching design simulation.
[0094] In summary, this invention starts with the vector wave equation and addresses the waveguide transmission problem by appropriately constructing a hybrid basis function form of scalar and vector basis functions, followed by functional discretization and various boundary conditions. Ultimately, this invention effectively handles waveguide port matching techniques for complex geometries and materials (anisotropic, non-uniform, etc.), especially for lossy physics problems. It can quickly and accurately solve the generalized eigenvalue equations, address waveguide transmission issues, and obtain better S-parameters, facilitating the analysis of corresponding electromagnetic parameters and effectively improving simulation design. It solves the problem that existing port technologies cannot achieve optimized simulation of complex electromagnetic waveguide models and cannot meet the needs of designers. Attached Figure Description
[0095] Figure 1 This is a flowchart of the present invention.
[0096] Figure 2 This is a schematic diagram of the universal electromagnetic waveguide port of the present invention.
[0097] Figure 3 This is a schematic diagram regarding t in this invention, using t0 as an example.
[0098] Figure 4 This is a schematic diagram of a dielectric block waveguide for an embodiment.
[0099] Figure 5 A comparison chart of attenuation constants for the number of port modes calculated for the example dielectric block waveguide. Detailed Implementation
[0100] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.
[0101] Reference Figure 1 A two-dimensional finite element method for calculating the electromagnetic numerical problem of discontinuous port transmission includes the following steps:
[0102] Step A: Based on the physical structure of the target electronic device, and in combination with the working environment and boundary conditions, perform simulation modeling on it.
[0103] For a detailed explanation of the fundamental principles of waveguide physics, please refer to [reference needed]. Figure 2 First, the electric field vector waveguide equation derived from the time-harmonic Maxwell equations in a dielectric waveguide structure is given, as shown below:
[0104]
[0105] Where E is the electric field intensity vector, ε r μ is the relative permittivity of the medium. rLet S be the relative permeability of the medium, k0 be the wave number in vacuum, D be the electric displacement vector, and B be the magnetic flux density. Assuming S is the waveguide cross-section, and often a two-dimensional region, the closed boundary of this cross-section is defined as... In practical waveguide physics problems, there are mainly three types of boundaries:
[0106]
[0107] Where H is the magnetic field strength vector. It is the ideal conductor boundary condition (PEC). It is an ideal magnetic wall boundary condition (PMC). It is the boundary condition for applying the impedance boundary. It is the surface impedance of the conductor material, where μ = μ0μ r Let be the permeability, where μ0 is the permeability in vacuum. f is the operating frequency, and σ is the conductivity of the conductor material. It can be seen that the surface impedance Z of the conductor material... s It is closely related to the operating frequency; once the frequency changes, Z s And so it changes.
[0108] Step B: Use tetrahedral elements to divide the three-dimensional solution domain. Surface discretization and volume discretization must be compatible.
[0109] Using tetrahedral elements to partition the 3D solution domain is a well-known process, therefore this step will not be described in detail here. It is important to note that surface discretization and volume discretization must be compatible.
[0110] Step C: Combining finite element scalar basis functions and vector basis functions, derive the eigenvalue equations for waveguide structures under general boundaries;
[0111] To solve equation (1), in the three directions x, y, and z of the spatial rectangular coordinate system, it is assumed that the geometry of the waveguide cross-section remains constant along the longitudinal z-axis (e.g., Figure 2 The electric and magnetic fields are divided into transverse and longitudinal fields, as detailed below:
[0112]
[0113] Where the subscripts t and z represent the transverse and longitudinal fields of the field, respectively, γ is the propagation constant, and z represents the z-direction vector; E, E t H and H t It is a vector, E z With H z It is a scalar. Further definition:
[0114]
[0115] in Substituting equations (3) and (4) into the first equation of equation (1), we can decompose it into a transverse field and a longitudinal field, as follows:
[0116] Lateral field:
[0117] Longitudinal field:
[0118] Since the waveguide cross-section is often a two-dimensional region, we can discretize the two-dimensional cross-sectional region into multiple small triangular surface units through step B. Based on the forms in (5) and (6), the vector electric field E is discretized using the vector stacked basis function N. t The scalar electric field E is discretized using the scalar basis function L. z .
[0119] To clearly describe the implementation process, a simple first-order basis function is used below. Note that this method is also applicable to second-order and higher-order basis functions. Consider a triangular surface element u, whose transverse electric field can be expressed as:
[0120]
[0121] The longitudinal electric field is represented as:
[0122]
[0123] in It is a transverse vector electric field The expansion coefficients, expressed as vectors express; It is a longitudinal scalar electric field The expansion coefficients, expressed as vectors Substituting equations (7) and (8) into equations (5) and (6), we obtain the following matrix equation:
[0124]
[0125]
[0126] definition Equations (9) and (10) can then be transformed into the following system of generalized eigenvalue equations:
[0127]
[0128] Where the matrix and All are related to the vector basis function N, matrix It is simultaneously dependent on both vector basis functions N and scalar basis functions L, and the matrix It is related to the scalar basis function L.
[0129] Since the finite element method is based on mathematical functional variation, it is a numerical algorithm that effectively solves partial differential equations by constructing a complete basis function space. Therefore, different functional variation expressions are used for different physical problems. For example, for the eigenmodes of dielectric-filled waveguides, it is necessary to establish longitudinal electric and magnetic fields simultaneously, and also combine transverse electric and magnetic fields, as well as vector potentials, scalar potentials, and other different functional variation expressions. In most existing literature, waveguide problems with general boundary conditions (ideal electric walls and ideal magnetic walls) are mainly considered. This is because the tangential electric field is 0 on an ideal electric wall, and the tangential magnetic field is 0 on an ideal magnetic wall, and the resulting functional expression is well-known and easy to solve. In order to make a comparative analysis with the present invention, the element matrix expression of the triangular surface element in formula (11) under general boundary conditions is given below:
[0130]
[0131]
[0132]
[0133]
[0134] Where the subscripts i∈{0,1,2} and j∈{0,1,2} represent the three nodes on the triangular facet element, and i and j can be the same or different. Based on the node and edge information of each facet element, each facet element is integrated into an element matrix using the well-known finite element method to obtain the matrix form of the entire computational domain. As can be seen from the element matrix expression above, the final obtained matrix of the entire computational domain is... and Both are sparse symmetric matrices. By combining the boundary conditions and solving the generalized eigenvalue equation (11), the eigenvalues and eigenvectors can be obtained.
[0135] Step D: Consider the lossy boundary conditions—impedance boundary conditions—and derive the generalized eigenvalue equations for the waveguide structure.
[0136] In practice, various loss scenarios often occur, including not only dielectric losses but also conductor losses. To further improve the computational efficiency of finite element waveguide eigenvalue analysis for lossy problems, this invention, in this step, details the key techniques for handling impedance boundary conditions in the two-dimensional port problem with losses. Through theoretical derivation, this boundary condition is added to the vector wave equation, thereby obtaining the specific form of the generalized eigenvalue equation.
[0137] Since the magnetic field vector H exists in the expression of the impedance boundary (the third equation in formula (2)), but there is no magnetic field vector in the generalized eigenvalue equation set (11), it is necessary to eliminate the magnetic field vector. Considering that the magnetic field vector can be derived from the electric field vector, that is:
[0138]
[0139] Substituting formulas (3) and (4) into the above equation, we have
[0140]
[0141] make Then H = zH z +H t Where z represents the longitudinal vector along the z-axis, E z It is the longitudinal electric field, E t It is a transverse electric field, H z It is a longitudinal magnetic field, H t It is a transverse magnetic field. Substituting equation (3) into the expression for the impedance boundary, we get:
[0142]
[0143] Substituting equation (17) into equation (18), the impedance boundary condition can be further derived as follows:
[0144]
[0145] Where η0 is the wave impedance in vacuum.
[0146] Because of the introduction of impedance boundary conditions, unlike the previous PEC and PMC boundaries, the impedance boundary conditions contribute to the surface integral calculation, thus affecting the matrix in step C. and It will change; through derivation, we have:
[0147]
[0148] in and It is an intermediate variable in the derivation, specifically in the form of:
[0149]
[0150] Similarly, the generalized eigenvalue equations (11) can be obtained. Once the transmission coefficient γ is solved, other electromagnetic parameters can be obtained.
[0151] Step E: Using hybrid basis functions, the generalized eigenvalue equations of the discrete waveguide structure are derived.
[0152] This step is the core step of the generalized eigenvalue equation and is also the innovation of this invention. For different physical models, the final mathematical model needs to be discretized to facilitate programming implementation. In fact, the selection of appropriate basis functions often plays a crucial role in the convergence and stability of the entire numerical solution. From the specific form of the electromagnetic field, i.e., equation (3), it can be seen that the transverse field is a vector and the longitudinal field is a scalar. Considering that the use of vector basis functions can effectively avoid spurious solutions in the problem of discontinuity of complex structures, this invention uses vector stacked basis functions for the vector field and general scalar basis functions for the longitudinal field. The following will use this hybrid basis function to perform the generalized eigenvalue equation (11) of the discrete waveguide structure.
[0153] To illustrate the structure of basis functions more clearly, Figure 3 The specific form of the two-dimensional vector stacked basis function is given. In the following step, combined with the two-dimensional mixed basis function, the discretization form of each matrix in (11) will be derived respectively.
[0154] For clarity, this step only considers simple first-order vector basis functions; however, it also applies to second-order and higher-order basis functions. Note: In this invention, the nodes and edges of the triangular surface element are numbered starting from 0. First, the three edges l on the triangular surface element are defined. i The local index numbers are shown in the table below:
[0155] Table 1 Local numbering of edges on face units
[0156] 0 1 2 1 0 2 2 0 1
[0157] Define the three nodal scalar basis functions of the triangular face element as L i (i = 0, 1, 2), define the vector basis functions of the three edges as follows:
[0158]
[0159] For the gradient term in the above equation The present invention adopts the following form, which avoids the tedious calculation of traditional scalar basis function differentiation.
[0160]
[0161] Where t i The magnitude of is the length of the i-th edge, and the vector direction is perpendicular to the i-th edge and points towards the interior of the face triangle, specifically as follows: Figure 3 As shown. S is the area of the triangular facet unit.
[0162] First, derive equation (12). In the matrix According to the formula:
[0163]
[0164] but:
[0165]
[0166] Similarly, for In the matrix Then we have:
[0167]
[0168] Combined with the following known formula (27):
[0169]
[0170] Equation (26) can be solved, and then equation (13) can be obtained.
[0171] The following derivation of equation (14) is based on equation (23). In the matrix
[0172]
[0173] According to the formula:
[0174]
[0175] Then equation (28) becomes:
[0176]
[0177] The following derivation of equation (15) is based on equation (23). Matrix, where Combining formula (27), we get:
[0178]
[0179] The discrete forms of all general boundary condition generalized eigenvalue equations have now been discussed.
[0180] Next, consider step D, where the matrix in equation (21) of the eigengeneralized equation system includes impedance boundary conditions. and The form is as follows; note that these two matrices have one-dimensional line integrals. First, let's... Simplified to:
[0181]
[0182] Consider the one-dimensional line integral in equation (32), according to the formula:
[0183]
[0184] Combining this with equation (23), we can obtain The discretized form of [the expression] can be derived similarly; The discretized form of .
[0185] Step F: Solve the generalized eigenvalue equation.
[0186] Based on the above steps, the discretized generalized eigenvalue equations are finally obtained, with the matrix form of each element as shown in equation (11). Through finite element integration, the global matrix equations of the entire solution domain are finally obtained. There are various methods and techniques for solving these matrix equations. Since this method is a well-known technique in this field, different methods can be found, so they will not be described in detail here. Once the generalized eigenvalue equations are solved, the eigenvalues and eigenvectors can be obtained, and other parameters of the electromagnetic field can be obtained for waveguide port cross-section transmission matching design simulation.
[0187] In this embodiment: with Figure 4 The propagation characteristics of the two-dimensional port of the dielectric block waveguide (unit: mm) are analyzed using this model. The model employs copper as a finite conductor with loss. Using this loss boundary condition, the attenuation constants for the first 10 modes are obtained using HFSS software and the EMLAB algorithm of this invention. Figure 5 As shown, Figure 5 A comparison of attenuation constants (in mm) calculated for the number of port modes in a dielectric block waveguide. Figure 5 It can be seen that, in the first 10 modes calculated using the present invention, the attenuation constant at the two-dimensional port is consistent with the calculation results of commercial HFSS software, and the maximum relative error is 0.18%, which demonstrates the reliability of the results.
[0188] As can be seen from the above embodiments, this invention starts from the vector wave equation and addresses the waveguide transmission problem. It constructs a hybrid basis function form of scalar and vector basis functions, then applies functional discretization and various boundary conditions. Ultimately, this invention can effectively handle waveguide port matching techniques with complex geometries and materials (anisotropic, non-uniform, etc.), especially for lossy physical problems. It can quickly and accurately solve the generalized eigenvalue equations, handle waveguide transmission problems, obtain better S-parameters, facilitate the analysis of corresponding electromagnetic parameters, and effectively improve simulation design. This solves the problem that existing port technologies cannot achieve optimized simulation of complex electromagnetic waveguide models and cannot meet the needs of designers.
Claims
1. A finite element electromagnetic numerical method for transmission matching of port cross-sections in waveguide design, characterized in that, Includes the following steps: Step A: Based on the physical structure of the target electronic device, and in conjunction with the working environment and boundary conditions, perform simulation modeling; derive the electric field vector waveguide equation for the time-harmonic Maxwell equations in the dielectric waveguide structure; Step B: Use tetrahedral elements to divide the three-dimensional solution domain. Surface discretization and volume discretization must be compatible. Step C: Combining finite element scalar basis functions and vector basis functions, derive the eigenvalue equations for waveguide structures under general boundaries; Assuming the waveguide cross-section geometry is along the longitudinal direction With the axis remaining constant, the electric and magnetic fields are divided into transverse and longitudinal fields, as detailed below: （1）； in It is the electric field intensity vector. It is the magnetic field strength vector, subscript The subscript indicates the transverse field, including the x and y axes; Represents the longitudinal field of the field. It is the propagation constant. express Direction vector, , , and It is a vector. and It is a scalar; further definition: （2）； in Based on the time-harmonic Maxwell equations, and combined with the universal boundary conditions for ideal electric boundary (PEC) and ideal magnetic boundary (PMC), vector basis functions are used. Discretizing the field equations with the nodal scalar basis functions L, we derive the generalized eigenvalue equations: （3）； Where the matrix and Both with vector basis functions Related to, matrix , With vector basis functions Scalar basis functions Simultaneously related, matrix With scalar basis functions related; In formula (3), the element matrix expression for the triangular face element is: （4）； （5）； （6）； （7）； Subscript , This represents the three nodes or edges on a triangular face element, and and They can be the same or different; based on the node and edge information of each surface element, each surface element is integrated into an element matrix using the well-known finite element method to obtain the matrix form of the entire computational domain. As can be seen from the unit matrix expression above, the final matrix obtained is the matrix of the entire computational domain. , and All are sparse symmetric matrices; by combining the boundary conditions and solving the generalized eigenvalue equation system (3), the eigenvalues and eigenvectors can be obtained; Step D: Consider the lossy boundary conditions—impedance boundary conditions—and derive the generalized eigenvalue equations for the waveguide structure. For impedance boundaries, i.e. It has a magnetic field vector However, the generalized eigenvalue equation system (3) does not contain a magnetic field vector, so the magnetic field vector needs to be eliminated. Considering that the magnetic field vector can be derived from the electric field vector, that is: （8）； in It is angular frequency. It is the permeability, the value of which is derived from the permeability in vacuum. relative permeability of the medium Coming together; Substituting the field decomposition, differential operator, and magnetic field expression into the impedance boundary conditions, we derive: （9）； in It is the wave impedance in a vacuum; Because of the introduction of impedance boundary conditions, unlike the previous PEC and PMC boundaries, the impedance boundary conditions contribute to the surface integral calculation, thus affecting the matrix in step C. and After derivation, the following changes occur: （10）； in and It is an intermediate variable in the derivation, specifically in the form of: （11）； Similarly, the generalized eigenvalue equation system (3) can be obtained, once the propagation constant is solved. This allows for the acquisition of other electromagnetic parameters; Step E: Using hybrid basis functions, the generalized eigenvalue equations of the discrete waveguide structure are derived. Step F: Solve the generalized eigenvalue equation.
2. The finite element electromagnetic numerical method for port cross-section transmission matching of waveguide design as described in claim 1, characterized in that, In step E, the generalized eigenvalue equation of the hybrid basis function discrete waveguide structure is adopted, and the specific process is as follows: For the vector field, vector stacked basis functions are used, and for the longitudinal field, scalar basis functions are used; the generalized eigenvalue equations of the discrete waveguide structure are performed (3). Consider first-order vector basis functions. For vector stacked basis functions, the nodes and edges of the triangular face element are numbered starting from 0, with nodes being 0, 1, and 2 respectively. Define three edges on the triangular face element. The local index numbers are: l0 has nodes 1 and 2, with 1 being the start node and 2 being the end node; l1 has nodes 0 and 2, with 0 being the start node and 2 being the end node; l2 has nodes 0 and 1, with 0 being the start node and 1 being the end node. Define the three nodal scalar basis functions on the triangular face element as follows: Define the vector basis functions for the three edges as follows: （12）； For the gradient term in the above equation It takes the following form: （13）； in The modulus is the first The edge length of the strip, the vector direction is perpendicular to the first... The edges point inwards towards the interior of the triangular face. Let be the area of the triangular face unit; First, derive equation (4). In the matrix According to the formula: （14）； but: （15）； Similarly, for In the matrix Then we have: （16）； Combining with the existing formula (17): （17）； Equation (16) can be solved, and then equation (5) can be obtained; The following derivation of equation (6) is based on equation (13). In the matrix ; （18）； According to the formula: （19）； Then equation (18) becomes: （20）； The following derivation of equation (7) is based on equation (13). Matrix, where Combining with formula (17), we get: （21）； This completes the discrete form of all general boundary condition generalized eigenvalue equations; Next, consider step D, where the matrix in equation (11) of the eigengeneralized equation system includes impedance boundary conditions. and In the form of these two matrices, there is a one-dimensional line integral; firstly, ... Simplified to: （22）； Consider the one-dimensional line integral in equation (22), according to the formula: （23）； Combining this with equation (13), we can obtain The discretized form; similarly, the following is derived. The discretized form of .
3. The finite element electromagnetic numerical method for port cross-section transmission matching of waveguide design as described in claim 1, characterized in that, In step F, the generalized eigenvalue equation is solved, and the specific process is as follows: According to step E, the discretized generalized eigenvalue equation is finally obtained, and the matrix form of each element is as shown in equation (3); through finite element integration, the global matrix equation of the entire solution domain is finally obtained; Once the generalized eigenvalue equations are solved, the eigenvalues and eigenvectors can be obtained, and other parameters of the electromagnetic field can be obtained, enabling the design and simulation of waveguide port cross-section transmission matching.