1. A semi-analytic analysis method of dense metal via holes is applied to a plurality of layers of parallel plates, wherein the plurality of layers of parallel plates are provided with a plurality of radial ports, the plurality of radial ports comprise a plurality of coaxial ports and a plurality of metal through holes, and the plurality of metal through holes between the plurality of layers of parallel plates form a metal via hole array, and is characterized by comprising the following steps:

s1, establishing a local coordinate system by taking the center of any radial port as an origin, obtaining a cylindrical wave function of an electric field perpendicular to the space where the parallel plates are located based on cylindrical waves under the local coordinate system, obtaining an expansion coefficient of an emergent wave and an expansion coefficient of an incident wave based on the cylindrical wave function of the electric field, and respectively converting the expansion coefficient of the emergent wave and the expansion coefficient of the incident wave into an expansion coefficient matrix of the emergent wave and an expansion coefficient matrix of the incident wave;

s2, according to the cylindrical wave addition theorem, obtaining a characterization function of the amplitude of the electromagnetic wave emitted from any radial port of the parallel plates and incident on the other radial port, wherein the characterization function comprises a cylindrical wave addition theorem coefficient;

s3, judging whether a reflected wave from the boundary exists according to the boundary condition of the parallel plates, and solving the cylindrical wave addition theorem coefficient by using a boundary integration method when the reflected wave from the boundary exists;

s4, correlating the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave through the cylindrical wave addition theorem coefficient to obtain a first correlation function relation;

s5, determining a reflection matrix according to the boundary conditions of the metal through holes, obtaining a total reflection matrix after all the metal through holes are combined according to the reflection matrix, and associating the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave with the total reflection matrix according to the reflection matrix relational expression of all the metal through holes so as to obtain a second correlation function relationship;

s6, performing matrix operation on the first incidence function relation and the second incidence function relation to obtain a second radial scattering matrix;

s7, transforming the second radial scattering matrix into an impedance matrix of the parallel plates based on the transformation relation between the second radial scattering matrix and the impedance matrix.

2. The semi-analytic analysis method for dense metal vias of claim 1, wherein step S1 specifically comprises:

s101, establishing a local coordinate system by taking the center of the ith radial port as an origin, obtaining an expression of a cylindrical wave function of an electric field perpendicular to the space where the parallel plates are located based on cylindrical waves under the local coordinate system as follows,

coefficient of expansion of wave, J_mAndrespectively expressed as an mth order Bessel function and a second class of Hankel functions, rhoⁱ、z represents a coordinate value of a point in a local coordinate system centered on the ith radial port, k_nExpressed in terms of wave numbers, and,j₁indicates a virtualUnit, m and n each represent the number of electromagnetic field patterns,andthe sum of the infinite terms expressed being respectively the integer M_iAnd integer N truncation, M ∈ [ -M_i,M_i]，n∈[0,N]H represents the thickness of the medium between the parallel plates;

expansion coefficient of incident waveAnd the expansion coefficient of the emergent waveConversion to number of lines of (N +1) × (2M)_i+1) expansion coefficient matrix a of the incident wave⁽ⁱ⁾And a matrix b of expansion coefficients of the outgoing waves⁽ⁱ⁾And are specifically represented as, respectively,

3. the semi-analytic analysis method for dense metal vias of claim 2, wherein step S2 specifically comprises:

s201, according to the cylindrical wave addition theorem, representing the amplitude of the electromagnetic wave which is emitted from the jth radial port on the parallel plate and enters the ith radial port through the conversion of a Hankel function and a Bessel function, wherein the representation expression is as follows,

in the formula 4, the first and second groups of the compound,representing the coefficients of the cylindrical wave addition theorem,is (2M)_i+1)×(2M_j+1) of the matrix, and,andexpressed as the coordinates of the ith and jth radial ports, respectively, in the local coordinate system.

4. The semi-analytic analysis method for dense metal vias of claim 3, wherein step S3 specifically comprises:

s301, judging whether a reflected wave from the boundary exists according to the boundary condition of the parallel plates, executing S302 when judging that the reflected wave from the boundary does not exist, and executing S303 when judging that the reflected wave from the boundary exists;

s302, only considering the direct incident wave from the jth radial port to the ith radial port, analyzing the cylindrical wave addition theorem coefficient according to the conversion of the Hankel function and the Bessel function to obtain the following formula,

in equation 5, δ_ijRepresenting the Crohn's function, p_jiIndicating the distance from the jth radial port to the ith radial port;

and S303, solving the cylindrical wave addition theorem coefficient by using a boundary integration method.

5. The semi-analytic analysis method for dense metal vias of claim 4, wherein step S4 specifically comprises:

s401, obtaining a first radial scattering matrix according to the cylindrical wave addition theorem coefficient, obtaining the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave through the first radial scattering matrix according to the cylindrical wave addition theorem,

in the formula 6, the first and second groups,representing a first radial scattering matrix;

s402, assuming that the parallel plates are provided with Np radial ports, wherein the parallel plates comprise P coaxial ports and Q metal through holes, combining the first radial scattering matrixes to form a total first radial scattering matrix of the Np radial ports;

s403, considering the transmission relation of the cylindrical harmonics between the coaxial ports and the metal through holes, converting the total first radial scattering matrix into a radial scattering submatrix,

in the formula 7, S_ssRepresenting a first radial scattering matrix, S, corresponding from coaxial port to coaxial port_svRepresenting a first radial scattering matrix, S, corresponding from the coaxial port to the metal via_vsRepresenting a first radial scattering matrix, S, corresponding from the metal via to the coaxial port_vvRepresenting a first radial scattering matrix corresponding from metal via to metal via;

s404, obtaining the first correlation function relationship as follows according to the radial scattering submatrix and a formula 6,

in the formula 8, a_sAnd b_sA matrix of expansion coefficients of the incident wave and of the exit wave, respectively, of the coaxial port, a_vAnd b_vAnd respectively representing the expansion coefficient matrix of the incident wave and the expansion coefficient matrix of the emergent wave of the metal through hole.

6. The semi-analytic analysis method for dense metal vias of claim 4, wherein step S5 specifically comprises:

s501, determining elements in the reflection matrix as follows according to the boundary conditions of the metal through holes,

in the formula 9, T⁽ⁱ⁾Representing a reflection matrix, T⁽ⁱ⁾(m, n) denotes an element in the reflection matrix, rⁱRepresents the radius of the ith radial port;

s502, assuming that Q metal through holes are arranged on the parallel plates, the total reflection matrix T formed by combining the Q metal through holes passes through the reflection matrix T⁽ⁱ⁾As indicated by the general representation of the,

T＝diag{T⁽ⁱ⁾}，i∈[1,Q]equation 10

S503, based on the microwave network theory, the reflection matrix T⁽ⁱ⁾The expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave are related to obtain a reflection matrix relational expression of the metal through holes,

b⁽ⁱ⁾＝T⁽ⁱ⁾a⁽ⁱ⁾equation 11

S504, substituting the total reflection matrix T into the formula 11 to obtain a second correlation function relationship,

a_v＝T^-1b_vequation 12.

7. The method for semi-resolved analysis of dense metal vias of claim 1 or 6, wherein the port radius of the coaxial port is electrically small.

Background

Metal vias, which are used to connect signals of different layers on multilayer PCBs and packages, are discontinuous structures that are common in high-speed systems and cause very serious problems with signal integrity, power integrity and electromagnetic interference at high frequencies. For example: crosstalk and ringing between signals, power distribution network noise, signal distortion, eye pattern degradation, edge radiation, etc. Therefore, accurate and efficient modeling of metal vias plays a critical role in designing high speed PCB and packaging systems.

The wave guide structure formed by the dense metal via hole array, such as a substrate integrated waveguide or a metal electromagnetic band gap structure, can be regarded as a parallel plate embedded with the metal via hole array, has the advantages of low processing cost and easy integration, keeps the advantages of low radiation, low loss and the like of the traditional metal waveguide, and can realize good interconnection on a high-speed digital circuit. Millimeter wave frequency band resources are used in fifth generation mobile communication (5G), and the millimeter wave antenna usually adopts a metal through hole array feed technology in consideration of loss and electromagnetic radiation brought by microstrip feed in the millimeter wave frequency band.

As microwave and high-speed systems are developing towards integration, large scale, high speed and broadband, the functions of circuits are more and more, the indexes are higher and higher, the smaller the size is, the more and more complex the analysis and design, and the shorter the design cycle is, the more and more, so the electromagnetic characteristics of the systems need to be accurately and rapidly analyzed and designed. Therefore, modeling dense arrays of vias on parallel plates facilitates analysis and design of high speed and microwave circuits.

In the prior art, full-wave simulation software is mostly adopted to analyze or an analytic method based on a multiple scattering method is used to model a dense through hole array on a parallel plate, but the modeling method is found to be long in simulation time because the whole calculation area needs to be divided into grids by using the full-wave simulation software, and the analytic method based on the multiple scattering method cannot calculate the reflection of the parallel plate boundary, is equivalent to an infinitely large parallel plate and is not accurate enough at higher frequency.

Disclosure of Invention

The application provides a semi-analytic analysis method of dense metal via holes, which is used for solving the technical problems that the modeling time of the existing modeling method is long and the modeling on higher frequency is not accurate enough.

In view of the above, the present application provides a semi-analytic analysis method for dense metal vias, which is applied to multiple layers of parallel plates, where each of the multiple layers of parallel plates is provided with a plurality of radial ports, each of the plurality of radial ports includes a plurality of coaxial ports and a plurality of metal vias, and the plurality of metal vias between the multiple layers of parallel plates form a metal via array, and the method includes the following steps:

s1, establishing a local coordinate system by taking the center of any radial port as an origin, obtaining a cylindrical wave function of an electric field perpendicular to the space where the parallel plates are located based on cylindrical waves under the local coordinate system, obtaining an expansion coefficient of an emergent wave and an expansion coefficient of an incident wave based on the cylindrical wave function of the electric field, and respectively converting the expansion coefficient of the emergent wave and the expansion coefficient of the incident wave into an expansion coefficient matrix of the emergent wave and an expansion coefficient matrix of the incident wave;

s2, according to the cylindrical wave addition theorem, obtaining a characterization function of the amplitude of the electromagnetic wave emitted from any radial port of the parallel plates and incident on the other radial port, wherein the characterization function comprises a cylindrical wave addition theorem coefficient;

s3, judging whether a reflected wave from the boundary exists according to the boundary condition of the parallel plates, and solving the cylindrical wave addition theorem coefficient by using a boundary integration method when the reflected wave from the boundary exists;

s4, correlating the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave through the cylindrical wave addition theorem coefficient to obtain a first correlation function relation;

s5, determining a reflection matrix according to the boundary conditions of the metal through holes, obtaining a total reflection matrix after all the metal through holes are combined according to the reflection matrix, and associating the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave with the total reflection matrix according to the reflection matrix relational expression of all the metal through holes so as to obtain a second correlation function relationship;

s6, performing matrix operation on the first incidence function relation and the second incidence function relation to obtain a second radial scattering matrix;

s7, transforming the second radial scattering matrix into an impedance matrix of the parallel plates based on the transformation relation between the second radial scattering matrix and the impedance matrix.

According to the technical scheme, the invention has the following advantages:

the invention solves the theorem coefficient of the cylindrical wave addition by utilizing the boundary integration method, segments are divided on a one-dimensional boundary, the grid division of the whole area is not needed, the simulation time is greatly shortened, the modeling speed is improved, and meanwhile, the reflected wave from the boundary of the parallel plates can be calculated by considering the reflected wave from the boundary according to the boundary condition of the parallel plates, so that the modeling on higher frequency can be more accurate, and the practicability is improved.

Drawings

Fig. 1 is a flowchart of a semi-analytic analysis method for dense metal vias according to an embodiment of the present disclosure;

FIG. 2 is a schematic top view of a parallel plate according to an embodiment of the present disclosure;

FIG. 3 is another schematic top view of a parallel plate according to an embodiment of the present disclosure;

fig. 4 is a schematic diagram of an equivalent microwave network of a parallel plate with metal vias provided in an embodiment of the present application.

Detailed Description

In order to make the technical solutions of the present application better understood, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.

In the prior art, full-wave simulation software is mostly adopted to analyze or an analytic method based on a multiple scattering method is used to model a dense through hole array on a parallel plate, but the modeling method is found to be long in simulation time because the whole calculation area needs to be divided into grids by using the full-wave simulation software, and the analytic method based on the multiple scattering method cannot calculate the reflection of the parallel plate boundary, is equivalent to an infinitely large parallel plate and is not accurate enough at higher frequency.

For easy understanding, please refer to fig. 1, the semi-analytic analysis method for dense metal vias provided by the present invention is applied to multilayer parallel plates, each of the multilayer parallel plates is provided with a plurality of radial ports, each of the plurality of radial ports includes a plurality of coaxial ports and a plurality of metal vias, and the metal vias between the multilayer parallel plates form a metal via array, including the following steps:

s1, establishing a local coordinate system by taking the center of any radial port as an origin, obtaining a cylindrical wave function of an electric field perpendicular to the space where the parallel plates are located based on cylindrical waves under the local coordinate system, obtaining an expansion coefficient of emergent waves and an expansion coefficient of incident waves based on the cylindrical wave function of the electric field, and respectively converting the expansion coefficient of the emergent waves and the expansion coefficient of the incident waves into an expansion coefficient matrix of the emergent waves and an expansion coefficient matrix of the incident waves;

s2, according to the cylindrical wave addition theorem, obtaining a characterization function of the amplitude of the electromagnetic wave emitted from any radial port on the parallel plate and incident to the other radial port, wherein the characterization function comprises a cylindrical wave addition theorem coefficient;

s3, judging whether a reflected wave from the boundary exists according to the boundary condition of the parallel plates, and solving a cylindrical wave addition theorem coefficient by using a boundary integration method when the reflected wave from the boundary exists;

s4, correlating the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave through a cylindrical wave addition theorem coefficient to obtain a first correlation function relation;

s5, determining a reflection matrix according to boundary conditions of the metal through holes, obtaining a total reflection matrix after all the metal through holes are combined according to the reflection matrix, and associating the expansion coefficient matrix of outgoing waves and the expansion coefficient matrix of incoming waves with the total reflection matrix according to the reflection matrix relation of all the metal through holes so as to obtain a second incidence function relation;

s6, performing matrix operation on the first incidence function relation and the second incidence function relation to obtain a second radial scattering matrix;

and S7, transforming the second radial scattering matrix into an impedance matrix of the parallel plates based on the transformation relation of the second radial scattering matrix and the impedance matrix.

It should be noted that, using a full-wave simulation tool requires to divide the entire region into grids, and therefore, the simulation time is long, while the present embodiment solves the theorem coefficient of cylindrical wave addition by using the boundary integration method, divides the segments on the one-dimensional boundary, and does not need to divide the entire region into grids, thereby greatly shortening the simulation time and increasing the modeling speed, and meanwhile, the reflected wave from the boundary of the parallel plates can be calculated by considering the reflected wave from the boundary according to the boundary condition of the parallel plates, thereby enabling the modeling on higher frequencies to be more accurate and improving the practicability.

The following is a detailed description of a specific embodiment of a semi-analytic analysis method for dense metal vias provided by the present invention.

Referring to fig. 2, fig. 2 is a schematic top view of a parallel plate having coaxial ports and a metal via array, where a plurality of radial ports are formed in each of a plurality of layers of parallel plates, the plurality of radial ports include a plurality of coaxial ports and a plurality of metal vias, and the metal vias form the metal via array, and a fence formed by the metal via array between the layers of parallel plates has an arbitrary shape, a dielectric thickness between the parallel plates is h, and a positive division of a relative permittivity and a loss angle is e, respectively_rAnd tan δ.

In step S1, the method specifically includes:

s101, establishing a local coordinate system by taking the center of the ith radial port as an origin, obtaining an expression of a cylindrical wave function of an electric field vertical to the space where the parallel plates are located based on cylindrical waves under the local coordinate system,

in the formula 1, the first and second groups of the compound,which is indicative of the electric field,andexpressed as the expansion coefficient of the incident wave and the expansion coefficient of the emergent wave, J, respectively_mAndrespectively expressed as an mth order Bessel function and a second class of Hankel functions, rhoⁱ、z represents a coordinate value of a point in a local coordinate system centered on the ith radial port, k_nExpressed in terms of wave numbers, and,in which w represents the angular frequency, ε₀、μ₀、ε_rAre all constants, j₁Representing imaginary units, m and n each representing a number of electromagnetic field modes,andthe sum of the infinite terms expressed being respectively the integer M_iAnd integer N truncation, M ∈ [ -M_i,M_i]，n∈[0,N]H represents the thickness of the medium between the parallel plates;

it should be noted that, in the following description,andthe sum of the represented infinite terms is an integer, i.e. it can be truncated by an integer, so that more complex electromagnetic field patterns can be calculated to improve at higher frequenciesModeling accuracy at rate.

The outgoing wave represents the outgoing cylindrical harmonic, and the incoming wave represents the incoming cylindrical harmonic.

Expansion coefficient of incident waveAnd the expansion coefficient of the emergent waveConversion to number of lines of (N +1) × (2M)_i+1) expansion coefficient matrix a of the incident wave⁽ⁱ⁾And a matrix b of expansion coefficients of the outgoing waves⁽ⁱ⁾And are specifically represented as, respectively,

step S2 specifically includes:

s201, according to the cylindrical wave addition theorem, representing the amplitude of the electromagnetic wave which is emitted from the jth radial port on the parallel plate and enters the ith radial port through the conversion of a Hankel function and a Bessel function, wherein the representation expression is as follows,

in the formula 4, the first and second groups of the compound,representing the coefficients of the cylindrical wave addition theorem, m' representing the number of electromagnetic field modes,is (2M)_i+1)×(2M_j+1) of the matrix, and,andexpressed as the coordinates of the ith and jth radial ports, respectively, in the local coordinate system.

Step S3 specifically includes:

s301, judging whether a reflected wave from the boundary exists according to the boundary condition of the parallel plates, executing S302 when judging that the reflected wave from the boundary does not exist, and executing S303 when judging that the reflected wave from the boundary exists;

s302, only considering the direct incident wave from the jth radial port to the ith radial port, analyzing the cylindrical wave addition theorem coefficient according to the conversion of the Hankel function and the Bessel function to obtain the following formula,

in equation 5, δ_ijThe function of the kronecker is expressed,representing the Hankel function, p_jiIndicating the distance from the jth radial port to the ith radial port;

it should be noted that when it is determined that there is no reflected wave from the boundary, it is determined that the parallel plate is an infinite parallel plate, that is, a parallel plate at the boundary of an ideal matching layer (PML), and there is no reflected wave from the boundary, so that only a direct incident wave from the jth radial port to the ith radial port needs to be considered.

And S303, solving the cylindrical wave addition theorem coefficient by using a boundary integration method.

It should be noted that, in practical applications, it is common to determine that there is a reflected wave from a boundary, for example, if there is a finite parallel plate with an ideal electrical conductor (PEC) boundary or an ideal magnetic conductor (PMC) boundary, the reflected wave from the edge of the parallel plate needs to be calculated, and a boundary integration method is used to solve the cylindrical wave addition theorem coefficient, and the specific solving process is as follows:

(1) for parallel plates of the PEC boundary, the cylindrical wave additive theorem coefficients can be written as:

in the formula 5.1, the first step,representing the coefficients of the cylindrical wave addition theorem,representing a direct incident wave, which is equal to that in equation 5 Which represents the reflected wave from the boundary and,a matrix of the method of moments is represented,a cylindrical harmonic representing a segment of the cylindrical harmonic emerging from the jth radial port incident on the PEC boundary,indicating that the equivalent current on the PEC boundary segment is converted to a cylindrical harmonic that is transmitted to the ith radial port.

As shown in FIG. 3, assume that the boundary Γ of the parallel plates is divided into N_eSegment by segment, and then, by the boundary integral equation, N can be calculated_e×N_eMatrix of the moment methodIn order to realize the purpose,

in the formula 5.2, the first step,γ denotes a constant, γ 1.781072418, s and t each denote an arbitrary segment on the boundary Γ, s, t 1,2, …, N_eVector ofAndpointing to the midpoints, w, of segments t and s, respectively_sDenotes the length, p, of the segment s_tsRepresents the distance from segment t to segment s;

in equation 5.1Is N_e×(2M_j+1) a matrix representing the cylindrical harmonic of the segment incident on the PEC boundary from the cylindrical harmonic emerging from the jth radial port, whose elements are,

in the formula 5.3, the first step,vectorThe position of the jth radial port is indicated,representing vectorsSum vectorThe included angle between them;

in equation 5.1Is N_e×(2M_j+1) matrix, representing the conversion of the equivalent current on the PEC boundary segment into a cylindrical harmonic that is transmitted to the ith radial port, with the elements,

in the formula 5.4, the first step,vectorIndicating the location of the ith radial port;

(2) consistent with the above solution process for the cylindrical wave addition theorem coefficients of the parallel plates of the PEC boundary, for the parallel plates of the PMC boundary, the cylindrical wave addition theorem coefficients can be written as:

in the formula 5.5, the first step,representing a direct incident wave, which is equal to that in equation 5 Is shown toA reflected wave from the boundary is reflected from the boundary,indicating that the equivalent magnetic flow on the PMC boundary segment is converted to a cylindrical harmonic that is emitted to the ith radial port,representing a moment method matrix;

wherein, the matrix of the moment methodThe element (b) is a compound of (b),

in the formula 5.6, the first step, represents a unit vector;

the element in (A) is as follows,

in the formula 5.7, the first step,a unit vector representing a vertical direction;

step S4 specifically includes:

s401, obtaining a first radial scattering matrix according to the cylindrical wave addition theorem coefficient,

in the formula 6.1, the first step,a first radial scattering matrix is represented which,and the coefficient matrix of the cylindrical wave addition theorem between the ith radial port and the jth radial port under the N-order higher-order mode is shown, wherein N is 2.

It can be understood that the cylindrical wave addition theorem coefficientsCombining cylindrical wave addition theorem coefficients of all radial ports to obtain a first radial scattering matrix in a formula 6.1, wherein the cylindrical wave addition theorem coefficients correspond to the ith and jth radial ports;

according to the cylindrical wave addition theorem, the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave are obtained by correlating the first radial scattering matrix,

in the formula 6, the first and second groups,representing a first radial scattering matrix;

s402, assuming that Np radial ports are arranged on the parallel plates, wherein the Np radial ports comprise P coaxial ports and Q metal through holes, combining the first radial scattering matrixes to form a total first radial scattering matrix of the Np radial ports;

wherein the total first radial scattering matrix is defined as S_PPThen, the result is obtained,

s403, considering the transmission relationship between the cylindrical harmonics and the coaxial ports and the metal vias, converting the total first radial scattering matrix into a radial scattering submatrix,

in the formula 7, S_ssRepresenting a first radial scattering matrix, S, corresponding from coaxial port to coaxial port_svRepresenting a first radial scattering matrix, S, corresponding from the coaxial port to the metal via_vsRepresenting a first radial scattering matrix, S, corresponding from the metal via to the coaxial port_vvRepresenting a corresponding first radial scattering matrix from metal via to metal via.

It is understood that the transmission relationship between the coaxial port and the metal via for the cylinder harmonic can include four cases, specifically, transmission from the coaxial port to the metal via, transmission from the metal via to the coaxial port, transmission from the coaxial port to the coaxial port, and transmission from the metal via to the metal via, and therefore, in order to distinguish different transmission relationships, the total first radial scattering matrix is converted into a radial scattering sub-matrix as shown in formula 7.

S404, obtaining a first correlation function relationship according to the radial scattering submatrix and the formula 6,

in the formula 8, a_sAnd b_sA matrix of expansion coefficients representing the incident wave and the emergent wave of the coaxial port, respectively_vAnd b_vAnd respectively representing the expansion coefficient matrix of the incident wave and the expansion coefficient matrix of the emergent wave of the metal through hole.

It can be understood that, considering the distinction between the coaxial port and the metal through hole, it is necessary to distinguish a first correlation function relationship between the coaxial port and the metal through hole, where the first correlation function relationship includes a correlation relationship between an expansion coefficient matrix of an outgoing wave and an expansion coefficient matrix of an incoming wave that are correlated with the coaxial port by the radial scattering submatrix, and a correlation relationship between an expansion coefficient matrix of an outgoing wave and an expansion coefficient matrix of an incoming wave that are correlated with the metal through hole by the radial scattering submatrix.

Step S5 specifically includes:

s501, determining elements in the reflection matrix as follows according to the boundary conditions of the metal through holes,

in the formula 9, T⁽ⁱ⁾Representing a reflection matrix, T⁽ⁱ⁾(m, n) denotes an element in the reflection matrix, rⁱRepresents the radius of the ith radial port;

it should be noted that, since each via hole is a metal through hole, the PEC boundary is used as a reference boundary, and for the parallel plate via hole of the PEC boundary, there are no magnetic field and electric field inside, and the boundary condition of the metal through hole is,

then, the formula 9 can be obtained by substituting the formula 9.1 into the formula 1.

S502, assuming that Q metal through holes are arranged on the parallel plates, the total reflection matrix T formed by combining the Q metal through holes passes through the reflection matrix T⁽ⁱ⁾As indicated by the general representation of the,

T＝diag{T⁽ⁱ⁾}，i∈[1,Q]equation 10

S503, based on the microwave network theory, converting the reflection matrix T⁽ⁱ⁾The relation formula of the reflection matrix of the metal through hole is obtained by correlating the expansion coefficient matrix of the emergent wave and the expansion coefficient matrix of the incident wave,

b⁽ⁱ⁾＝T⁽ⁱ⁾a⁽ⁱ⁾equation 11

It should be noted that, in the equivalent microwave network including metal vias with coaxial ports, as shown in fig. 4, the metal vias are regarded as loads, so as to obtain coupling between the coaxial ports, where, assuming that there are P coaxial ports and Q metal vias, the Q metal vias can be regarded as loads of parallel plates.

S504, substituting the total reflection matrix T into the formula 11 to obtain a second correlation function relationship,

a_v＝T^-1b_vequation 12

Step S6 specifically includes:

s601, carrying out matrix operation through the formula 8 to obtain the following formula,

a_v＝S_vsb_s+S_vvb_vequation 13

S602, substituting the formula 12 into the formula 13 to obtain an expansion coefficient matrix b of the emergent wave of the coaxial port_sAnd a matrix b of expansion coefficients of outgoing waves of the metal via holes_vThe relation of (A) is as follows,

b_v＝(T^-1-S_vv)^-1S_vsb_sequation 14

S603, substituting equation 14 into equation 8, obtains the following equation:

a_s＝[S_ss+S_sv(T^-1-S_vv)^-1S_vs]b_sequation 15

S604, characterizing formula 15 by a second radial scattering matrix based on formula 6, and obtaining the following formula:

in the formula 16, the first and second phases,representing a second radial scattering matrix, obtaining

Step S7 specifically includes:

s701, based on the transformation relation between the second radial scattering matrix and the impedance matrix, the following formula can be obtained through calculation by the second radial scattering matrix:

in the formula 17, the process is as follows,representing the impedance matrix of the parallel plates H₀、J₀、H₁、J₁All are P × P diagonal matrices, specifically:

J₀＝diag{J₀(k₀r_i)h}

in the above formula, k₀Represents the wave number k_nN in (1) is a wave number at 0; ω represents angular frequency and μ represents magnetic permeability.

It should be noted that, for the coaxial port, the port radius of the coaxial port of the present embodiment is electrically small, and therefore, the sum of the infinite terms in equation 1 may be set to 0, that is, only the isotropic parallel plate mode with 0 order (i.e., m and n in equation 1 are both 0) is considered. Thus, the coupling between coaxial ports can be characterized by the impedance matrix of the parallel plates.

After the impedance matrix of the parallel plates is obtained, it can be converted into a generalized scattering parameter matrix to characterize the coupling between the coaxial ports.

The above embodiments are only used to illustrate the technical solutions of the present application, and not to limit the same; although the present application has been described in detail with reference to the foregoing embodiments, it should be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions in the embodiments of the present application.