A simulation method for electro-thermal coupling effect of substrate integrated waveguide filter

Abstract

The application provides a simulation method for the electro-thermal coupling effect of a substrate integrated waveguide filter, and the method comprises the following steps: S1: extracting the equivalent thermal conductivity of a via array of the substrate integrated waveguide filter; S2: establishing a thermal conduction network of the substrate integrated waveguide filter; S3: obtaining the electromagnetic field distribution of each resonant cavity of the substrate integrated waveguide filter according to the resonant cavity mode theory; S4: based on the electromagnetic field distribution, analytically modeling the electromagnetic loss of the substrate integrated waveguide filter; S5: taking the electromagnetic loss as a heat source, simulating the temperature field distribution of the substrate integrated waveguide filter; and S6: if the deviation of the temperature field calculation results of two times in succession is greater than a set threshold, then the temperature field is updated and optimized, and steps S3 to S5 are repeated until the simulation result converges. The application greatly shortens the time of electro-thermal simulation, and is suitable for the rapid evaluation of the electro-thermal coupling effect of the substrate integrated waveguide filter.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of microwave integrated circuits, in particular to a simulation method for the electro-thermal coupling effect of a substrate integrated waveguide filter. BACKGROUND

[0002] Substrate integrated waveguides have been widely used in the design of microwave filters due to their high integration, high quality factor, low crosstalk and other characteristics. However, as substrate integrated waveguide filters develop towards higher frequencies, higher powers and more compact sizes, their electromagnetic losses also increase, leading to an increase in circuit operating temperature and further causing stress deformation, which brings a series of reliability problems. Therefore, it is necessary to accurately simulate and analyze the electro-thermal coupling effect.

[0003] In order to achieve the above purpose, the electro-thermal coupling simulation of the substrate integrated waveguide filter is usually based on commercial software. There are many commercial software on the market that can realize multi-physical field simulation, such as ANSYS and COMSOL Multiphysics. These software provide complex physical field coupling interface and rich material database, which can perform detailed simulation analysis on the electro-thermal coupling effect of the substrate integrated waveguide filter. However, their main disadvantage is that they require a large amount of computing resources and a long analysis time, especially when performing high-frequency or large-scale simulation. Therefore, it is urgent to propose a new fast simulation method to improve the efficiency of the electro-thermal coupling simulation of the substrate integrated waveguide filter. SUMMARY

[0004] In view of the defects in the prior art, the purpose of the present application is to provide a simulation method for the electro-thermal coupling effect of a substrate integrated waveguide filter, which can efficiently and accurately calculate the temperature distribution of the substrate integrated waveguide filter.

[0005] To solve the above problems, the technical scheme of the present application is as follows:

[0006] A simulation method for the electro-thermal coupling effect of a substrate integrated waveguide filter, characterized in that the method comprises the following steps:

[0007] S1: Extracting the equivalent thermal conductivity of the via array of the substrate integrated waveguide filter;

[0008] S2: Establishing a thermal conduction network of the substrate integrated waveguide filter;

[0009] S3: Obtaining the electromagnetic field distribution of each resonant cavity of the substrate integrated waveguide filter according to the resonant cavity mode theory;

[0010] S4: Analytically modeling the electromagnetic loss of the substrate integrated waveguide filter based on the electromagnetic field distribution;

[0011] S5: Simulate the temperature field distribution of a substrate-integrated waveguide filter by using electromagnetic loss as a heat source;

[0012] S6: If the deviation between the two temperature field calculation results is greater than the set threshold, then optimize and update the temperature field, and repeat steps S3 to S5 until the simulation results converge.

[0013] Preferably, in the step of extracting the equivalent thermal conductivity of the via array of the substrate integrated waveguide filter, a via array with radius r, period p, and thickness t of the metal conductor layer inside the via arranged along the x-direction on a dielectric substrate with thickness h can be equivalently represented as a uniform medium with width w and anisotropic thermal conductivity, wherein the thermal resistance of one period in the x-direction is:

[0014]

[0015] Among them κ metal and κ sub Thermal conductivity of metal and dielectric, respectively. The width of the outer edge of the through hole, Let be the width of the air-filled portion within the through-hole; therefore, the equivalent thermal conductivity in the x-direction is .

[0016] Preferably, in the step of establishing the thermal conductivity network of the substrate integrated waveguide filter, the substrate integrated waveguide filter is first structurally partitioned using a hexahedral mesh, and then the thermal conductivity network is established based on the finite volume method. For node (i,j,k), the thermal conductivity between it and its neighboring nodes satisfies the following equation:

[0017] (G x- +G x+ +G y- +G y+ +G z- +G z+ )T i,j,k -G x- T i-1,j,k -G x+ T i+1,j,k -G y- T i,j-1,k -G y+ T i,j+1,k -G z- T i,j,k-1 -G z+ T i,j,k+1 =q i,j,k

[0018] Where T i,j,k and q i,j,k G represents the temperature and heat source of node (i,j,k), respectively. x± Gy± and G z± are the thermal conductances between the node (i,j,k) and its neighbors, which are given by G x± = κ x± (y - +y + )(z - +z + ) / (4x ± ), G y± = κ y± (z - +z + )(x - +x + ) / (4y ± ) and G z± = κ z± (x - +x + )(y - +y + ) / (4z ± ), where κ x± , κ y± and κ z± are the thermal conductivities along the x, y, z coordinate axes in the positive and negative directions, and x±, y±, z± are the grid sizes in the corresponding directions.

[0019] Preferably, in the step of obtaining the electromagnetic field distribution of each resonant cavity of the substrate integrated waveguide filter according to the resonant cavity mode theory of the substrate integrated waveguide, the expression of the electromagnetic field component in the i-th resonant cavity is:

[0020]

[0021] where a and d are the equivalent length and width of the resonant cavity, k m0n = [(mπ / a) 2 +(nπ / b) 2 ] 1 / 2 is the resonant wave number, m and n are the mode numbers in the x and z directions, η = (μ / ε) 1 / 2 is the intrinsic impedance, μ and ε are the magnetic permeability and dielectric constant of the filling medium, and E i is the electric field amplitude of the resonant cavity.

[0022] Preferably, in the step of analytically modeling the electromagnetic loss of the substrate integrated waveguide filter based on the electromagnetic field distribution, the expression of the conductor loss of the substrate integrated waveguide filter is: where T is the temperature, is the surface resistance, σ0 is the metal conductivity at the ambient temperature T amb , and β is the temperature coefficient of resistivity.

[0023] Preferably, in the step of analytically modeling electromagnetic loss of the substrate integrated waveguide filter based on electromagnetic field distribution, the expression of the via array loss of the substrate integrated waveguide filter is: Where H t is the tangential component of the magnetic field passing through the vertical plane where the via array is located, Q via is the quality factor of the SIW resonant cavity only considering the via array loss, Q plane is the quality factor after replacing the via array with a vertical plane.

[0024] Preferably, in the step of analytically modeling electromagnetic loss of the substrate integrated waveguide filter based on electromagnetic field distribution, the expression of the dielectric loss of the substrate integrated waveguide filter is: Where tan delta is the dielectric loss tangent, epsilon (T) = epsilon 0 epsilon r0 [1 + gamma (T - T amb )] is the temperature-dependent dielectric constant of the dielectric, where epsilon 0 is the vacuum dielectric constant, epsilon r0 is the relative dielectric constant of the dielectric substrate at ambient temperature T amb , and gamma is the temperature coefficient of the dielectric constant.

[0025] Preferably, in the step of simulating the temperature field distribution of the substrate integrated waveguide filter by taking the electromagnetic loss as a heat source, the matrix equation to be solved in the thermal simulation is GT = Q, where G is the thermal conductivity matrix composed of the thermal conductivities between any two nodes in the thermal conduction network, Q is the heat source vector whose specific value is determined by the electromagnetic loss, and T is the temperature vector to be solved.

[0026] Preferably, in the step of optimizing and updating the temperature field if the deviation between the results of the previous and subsequent temperature field calculations is greater than the set threshold, repeating steps S3 to S5 until the simulation results converge, the Aitken dynamic relaxation method is used to optimize and update the temperature field, thereby improving the iteration efficiency of the electro-thermal coupling simulation.

[0027] Preferably, in the step of optimizing and updating the temperature field if the deviation between the results of the previous and subsequent temperature field calculations is greater than the set threshold, repeating steps S3 to S5 until the simulation results converge, T k-1 and T k are the temperature fields obtained in the (k-1)th and kth iterations, respectively, and if AT = ||T k -T k-1 ||2 is less than the convergence error delta, the iteration is terminated and the temperature field T k is output; otherwise, the temperature field is updated as the input for the next iteration by the formula , where lambda k is the relaxation factor.

[0028] Compared with the prior art, the present application has the following beneficial effects:

[0029] 1、The application fully considers the electro-thermal coupling effect of the substrate integrated waveguide filter under the material temperature variation characteristics, and ensures high accuracy of the electro-thermal analysis result;

[0030] 2、The application reduces the complexity of the original model by extracting equivalent thermal parameters of the via array, which is beneficial to subsequent efficient simulation;

[0031] 3、The application introduces the Aitken dynamic relaxation method for iteration acceleration in the electro-thermal coupling analysis, greatly shortens the time of electro-thermal simulation, and is suitable for rapid evaluation of the electro-thermal coupling effect of the substrate integrated waveguide filter. BRIEF DESCRIPTION OF DRAWINGS

[0032] Other features, objects and advantages of the application will become more apparent from the following detailed description of non-limiting embodiments, made with reference to the accompanying drawings:

[0033] Figure 1 The flow chart of the simulation method for the electro-thermal coupling effect of the substrate integrated waveguide filter according to the application;

[0034] Figure 2 The schematic diagram of equivalent thermal parameter extraction for a via array;

[0035] Figure 3 The schematic diagram of grid partitioning of the substrate integrated waveguide filter based on the finite difference method;

[0036] Figure 4 The schematic diagram of the structure and geometric parameters of the substrate integrated waveguide filter;

[0037] Figure 5 The S parameter of the substrate integrated waveguide filter calculated based on the normalized coupling matrix;

[0038] Figure 6 The temperature rise profile of the substrate integrated waveguide filter calculated by the method;

[0039] Figure 7 The temperature rise profile of the substrate integrated waveguide filter calculated by the multi-physics simulation software. DETAILED DESCRIPTION

[0040] The application will be described in detail below with reference to specific embodiments. The following embodiments will help those skilled in the art to further understand the application, but do not limit the application in any form. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the application. These all belong to the protection scope of the application.

[0041] Specifically, the application provides a simulation method for the electro-thermal coupling effect of a substrate integrated waveguide filter, as shown inFigure 1 As shown, the method comprises the following steps:

[0042] S1: Extracting the equivalent thermal conductivity of the via array of the substrate integrated waveguide filter;

[0043] Specifically, for a via array with a radius of r, a period of p, a thickness of t of the metal conductor layer in the via, and a thickness of h of the dielectric substrate arranged along the x direction, it can be equivalent to a uniform medium with a width of w and an anisotropic thermal conductivity, as shown. Figure 2 Where the thermal resistance of one period length in the x direction is:

[0044]

[0045] Where κ metal and κ sub are the thermal conductivities of the metal and the dielectric, respectively, the width of the outer edge of the via, the width of the air-filled part in the via.

[0046] Therefore, the equivalent thermal conductivity in the x direction is The equivalent thermal conductivities in the y and z directions can be obtained by a similar method.

[0047] S2: Establishing the thermal conduction network of the substrate integrated waveguide filter;

[0048] Specifically, first, the substrate integrated waveguide filter is structurally divided by using a hexahedral mesh, and then a thermal conduction network is established based on the finite volume method, as shown. Figure 3 For node (i,j,k), the thermal conduction between it and the adjacent nodes satisfies the following equation:

[0049]

[0050] Where T i,j,k and q i,j,k are the temperature and heat source of node (i,j,k), respectively. G x± , G y± , and G z± are the thermal conductivities between node (i,j,k) and the adjacent nodes, and their expressions are G x± = κ x± (y - +y + )(z - +z + ) / (4x ± ), G y± = κ y± (z - +z + )(x - +x +) / (4y ± ) and G z± = κ z± (x - +x + )(y - +y + ) / (4z ± ), where κ x± , κ y± and κ z± are the thermal conductivities along the x, y, z coordinate axes in the positive and negative directions, and x±, y± and z± are the grid sizes in the corresponding directions.

[0051] In addition, for the nodes on the outer surface of the substrate integrated waveguide filter, since the outer surface is in a convective heat transfer condition, the heat conduction equation is modified as:

[0052]

[0053] where h conv is the convective heat transfer coefficient, and Δs = (x - +x + )(y-+y + ) / 4 is the grid area surrounding the node.

[0054] S3: According to the resonant cavity mode theory of the substrate integrated waveguide, the electromagnetic field distribution of each resonant cavity of the substrate integrated waveguide filter is obtained;

[0055] Specifically, the expression of the electromagnetic field component in the i-th resonant cavity of the substrate integrated waveguide is:

[0056]

[0057] where a and d are the equivalent length and width of the resonant cavity, respectively, k m0n = [(mπ / a) 2 +(nπ / b) 2 ] 1 / 2 is the resonant wave number, m and n are the mode numbers in the x and z directions, η = (μ / ε) 1 / 2 is the intrinsic impedance, μ and ε are the magnetic permeability and dielectric constant of the filling medium, respectively, and E i is the amplitude of the resonant cavity electric field. Therefore, the energy stored in the resonant cavity can be represented as:

[0058]

[0059] According to the coupled resonant circuit model of the filter, the energy stored in the i-th (i = 1, 2, 3, …, N, N is the total number of resonant cavities) resonant cavity is:

[0060]

[0061] where P in is the input power, ω i is the resonant angular frequency of the ith resonant cavity, FBW is the relative bandwidth of the filter, and A is the coupling resonator model matrix of the filter:

[0062] A = -jR + ΩW + M (7)

[0063] where M is the normalized coupling matrix of the filter, R and W are matrices related to the filter parameters, and Ω is the frequency of the low-pass prototype filter

[0064]

[0065] W = diag [0, 1, 1,..., 1, 0] (9)

[0066]

[0067] where Q i is the quality factor of the ith resonant cavity, and ω0 is the center frequency of the filter.

[0068] By combining (5) and (6), the amplitude of the electric field of the ith resonant cavity is:

[0069]

[0070] S4: Based on the electromagnetic field distribution, the electromagnetic loss of the substrate integrated waveguide filter is analytically modeled.

[0071] Specifically, the conductor loss expression of the substrate integrated waveguide filter is:

[0072]

[0073] where T represents the temperature, is the surface resistance, σ0 is the metal conductivity at the ambient temperature T amb , and β is the temperature coefficient of resistivity.

[0074] The via array loss expression of the substrate integrated waveguide filter is:

[0075]

[0076] where H t is the tangential component of the magnetic field passing through the vertical plane where the via array is located, Q via is the quality factor of the SIW resonant cavity only considering the via array loss, and Q plane is the quality factor after replacing the via array with a vertical plane. The ratio of the quality factors satisfies the approximate empirical formula

[0077] The medium loss expression of the substrate integrated waveguide filter is:

[0078]

[0079] Where tanδ is the medium loss tangent, ε(T) = ε0εr(T) is the medium dielectric constant, and T is the ambient temperature. r0 [1 + γ(T - T0)] is the temperature-dependent dielectric constant of the medium, where ε0 is the vacuum dielectric constant, εr(T0) is the relative dielectric constant of the medium substrate at ambient temperature T0, and γ is the temperature coefficient of the dielectric constant. amb r0 amb

[0080] S5: Take the electromagnetic loss as a heat source to simulate the temperature field distribution of the substrate integrated waveguide filter.

[0081] Specifically, the specific matrix equation to be solved by the thermal simulation is:

[0082] GT = Q (15)

[0083] Where G is the heat conduction matrix composed of the heat conduction between any two nodes in the heat conduction network described in step S2; Q is the heat source vector, the specific value of which is determined by the electromagnetic loss described in step S4; and T is the temperature vector to be solved.

[0084] S6: If the deviation between the results of the temperature field calculation before and after is greater than the set threshold, update the temperature field and repeat steps S3 to S5 until the simulation result converges.

[0085] Specifically, since the dielectric constant of the substrate integrated waveguide resonant cavity will change with the increase of temperature, taking the temperature field result of the substrate integrated waveguide filter as input will cause the resonant frequency to shift. The shift amount of the resonant frequency can be solved according to the perturbation method, and the expression is:

[0086]

[0087] Then, update the electromagnetic field distribution according to step S3, update the electromagnetic loss according to step S4, and re-calculate the temperature field according to step S5. Let T k-1 and T k be the temperature fields obtained by the k-1th and kth iterations, respectively. If ΔT = ||T k -T k-1 ||2 is less than the convergence error δ, terminate the iteration and output the temperature field T k ; otherwise, update the temperature field as the input of the next iteration according to the formula , where λ k is the relaxation factor. According to the Aitken dynamic relaxation method, the expression of the relaxation factor is:​​​

[0088]

[0089] According to the simulation method of the electro-thermal coupling effect of the substrate integrated waveguide filter, a specific embodiment is analyzed.

[0090] This embodiment is a four-order substrate integrated waveguide filter based on the PCB process, the center frequency of which is f0=20.5 GHz, the relative bandwidth is FBW=0.034, and the structure and geometric parameters of the filter are shown in the schematic diagram as Figure 4 The material of the metal conductor of the filter is copper, the conductivity of which is: σ0=5.8×10 7 S / m, the resistivity temperature coefficient is: β=0.0039K -1 , the thermal conductivity is: κ metal =400W·m -1 ·K -1 , the thickness is: t=0.018mm; the dielectric substrate is Rogers RT5880, the relative dielectric constant of which is ε r0 =2.2, the dielectric constant temperature coefficient γ=-125ppm / K, the loss tangent is tanδ=0.0009, the thermal conductivity is κ sub =0.2W·m -1 ·K -1 , and the thickness is h=0.508mm. The via radius is r=0.25mm, and the via array period is p=1mm. The normalized coupling matrix of the filter is as follows:

[0091]

[0092] In the temperature field simulation, the upper surface of the filter is placed in the natural convection boundary condition, the convection coefficient of which is h conv_top =10W·m -2 ·K -1 , and the lower surface is placed in the forced convection boundary condition, the convection coefficient of which is h conv_bot =100W·m -2 ·K -1 .

[0093] In addition, the ambient temperature is T amb =300K, and the input power is P in =20W. According to the normalized coupling matrix, the S parameters considering the temperature variation effect and ignoring the temperature variation effect are calculated, respectively, as Figure 5It can be seen that the center frequency of the filter increases by 0.15 GHz after considering the temperature variation effect, because the dielectric constant of the Rogers RT5880 medium decreases with the increase of temperature. In addition, the results of the present method are basically consistent with those of the commercial multi-physical field software when considering the temperature variation effect, which proves the accuracy of the present method.

[0094] Then, the temperature rise profile at z = 0.254 mm is calculated using the present method under the condition of considering the temperature variation effect, as shown in FIG. 6, where the hot spot is located at the center of the second resonant cavity, and the temperature rise is about 130 K. For comparison, the results of the commercial multi-physical field software are shown in FIG. 7. Figure 6 Figure 7 It can be seen that the calculation results of the present method are in good agreement with the software results.

[0095] The maximum temperature rise of each resonant cavity of the filter is shown in Table 1. The results show that the temperature rise of the second resonant cavity is the highest, while the temperature rise of the fourth resonant cavity is the lowest. In addition, the results of the present method are in good agreement with those of the multi-physical field software when considering the temperature variation effect, and the relative error between them is not higher than 1.91%. On the contrary, when ignoring the temperature variation effect, the relative error can be as high as 12.52%. These phenomena show that it is necessary to consider the material temperature variation effect in the process of electro-thermal analysis.

[0096]

[0097] Table 1

[0098] Table 2 summarizes the relevant parameters when using the multi-physical field simulation software and the present method to analyze the electro-thermal coupling of the substrate integrated waveguide filter. The present method calculates the electromagnetic loss by an analytical method, thereby avoiding time-consuming electromagnetic full-wave simulation. In addition, the number of thermal simulation grids of the present method is much smaller than that of the multi-physical field simulation software, because the equivalent thermal parameters of the via array are extracted. Due to these two advantages, the total CPU time of the present method is only 15.8 seconds, while the total CPU time of the multi-physical field simulation software is as high as 884 seconds, from which the efficiency of the present method can be seen.

[0099]

[0100]

[0101] Table 2

[0102] The specific embodiments of the present application are described above. It should be understood that the present application is not limited to the above specific embodiments, and those skilled in the art can make various changes or modifications within the scope of the claims, which does not affect the essential content of the present application. The embodiments of the present application and the features in the embodiments can be arbitrarily combined with each other without conflict.​

Claims

1. A simulation method for electro-thermal coupling effect of a substrate integrated waveguide filter, characterized in that, The method comprises the following steps: S1: extracting the equivalent thermal conductivity of the via array of the substrate integrated waveguide filter; S2: establishing a thermal conduction network of the substrate integrated waveguide filter; S3: obtaining the electromagnetic field distribution of each resonant cavity of the substrate integrated waveguide filter according to the resonant cavity mode theory, and the expression of the electromagnetic field component in the u-th resonant cavity is: where a and b are the equivalent length and width of the resonator, k m0n = [(mπ / a) 2 +(nπ / b) 2 ] 1 / 2 is the resonant wave number, m and n are the mode numbers in x and z directions, η = (μ / ε) 1 / 2 is the intrinsic impedance, μ and ε are the permeability and permittivity of the filling medium, respectively; S4: based on the electromagnetic field distribution, the electromagnetic loss of the substrate integrated waveguide filter is analytically modeled, and the conductor loss expression of the substrate integrated waveguide filter is: Wherein, T is temperature, R is surface resistance, σ0 is the metal conductivity at ambient temperature T amb , and β is the temperature coefficient of resistivity; the dielectric loss expression of the substrate integrated waveguide filter is: Wherein, ω u is the resonance angular frequency of the u-th resonant cavity, tanδ is the dielectric loss tangent, ε(T) = ε0ε r0 [1+γ(T-T amb )] is the temperature-dependent dielectric constant of the medium, wherein ε0 is the vacuum dielectric constant, ε r0 is the relative dielectric constant of the dielectric substrate at ambient temperature T amb , and γ is the temperature coefficient of the dielectric constant; S5: simulating the temperature field distribution of the substrate integrated waveguide filter by taking the electromagnetic loss as a heat source; S6: If the deviation of the temperature field calculation results of the previous and the next two times is greater than the set threshold, the temperature field is updated and optimized, and steps S3 to S5 are repeated until the simulation result converges, and T N-1 and T N are the temperature fields obtained in the N-1th and Nth iterations respectively, if ΔT = ||T N -T N-1 ||2is less than the convergence error δ, the iteration is terminated and the temperature field T N is output; otherwise, the temperature field is updated as the input of the next iteration through the formula , wherein λ N is a relaxation factor.

2. The method for simulation of electro-thermal coupling effects in a substrate integrated waveguide filter according to claim 1, characterized in that, In the step of extracting the equivalent thermal conductivity of the via array of the substrate integrated waveguide filter, for a via array with a radius of r, a period of p, and a metal conductor layer thickness of t in the x direction on a medium substrate with a thickness of h, the via array can be equivalent to a uniform medium with anisotropic thermal conductivity and a width of w, wherein the thermal resistance of one period length in the x direction is: wherein, κ metal and κ sub are the thermal conductivities of the metal and the dielectric, respectively, is the width of the outer edge of the via, is the width of the air-filled portion of the via, and thus, the equivalent thermal conductivity in the x-direction is 3. The method for simulation of electro-thermal coupling effects in a substrate integrated waveguide filter according to claim 1, characterized in that, In the step of establishing the thermal conduction network of the substrate integrated waveguide filter, first, the substrate integrated waveguide filter is structurally divided by using a hexahedral grid, and then a thermal conduction network is established based on the finite volume method, and for node (i, j, k), the thermal conduction between it and the adjacent nodes satisfies the following equation: (G x- +G x+ +G y- +G y+ +G z- +G z+ )T i,j,k -G x- T i-1,j,k -G x+ T i+1,j,k - G y- T i,j-1,k - G y+ T i,j+1,k - G z- T i,j,k-1 - G z+ T i,j,k+1 = q i,j,k where T i,j,k and q i,j,k are the temperature and heat source of node (i,j,k) respectively, G x± , G y± and G z± are the heat conductance between node (i,j,k) and its adjacent nodes, which are expressed as G x± = κ x± (y - +y + )(z - +z + ) / (4x ± ), G y± = κ y± (z - +z + )(x - +x + ) / (4y ± ) and G z± = κ z± (x - +x + )(y - +y + ) / (4z ± ), where κ x± , κ y± and κ z± are the thermal conductivities along the x, y and z coordinate axes in the positive and negative directions, and x ± , y ± and z ± are the grid sizes in the corresponding directions.

4. The method for simulation of electro-thermal coupling effects in a substrate integrated waveguide filter according to claim 1, characterized in that, In the step of analytically modeling electromagnetic loss of the substrate integrated waveguide filter based on the electromagnetic field distribution, the expression of the via array loss of the substrate integrated waveguide filter is: Wherein, H t is the tangential component of the magnetic field through the vertical plane where the via array is located, Q via is the quality factor of the SIW resonant cavity only considering the via array loss, Q plane is the quality factor after replacing the via array with a vertical plane.

5. The method for simulation of electro-thermal coupling effects in a substrate integrated waveguide filter according to claim 1, characterized in that, In the step of simulating the temperature field distribution of the substrate integrated waveguide filter by taking the electromagnetic loss as a heat source, the matrix equation to be solved in the thermal simulation is GT=Q, wherein G is a thermal conduction matrix composed of the thermal conduction between any two nodes in the thermal conduction network; Q is a heat source vector, and the specific value is determined by the electromagnetic loss; and T is a temperature vector to be solved.

6. The method for simulation of electro-thermal coupling effects in a substrate integrated waveguide filter according to claim 1, characterized in that, In the step of updating the temperature field and repeating steps S3 to S5 until the simulation result converges, the temperature field is updated by using the Aitken dynamic relaxation method.

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