Electromagnetic field simulation boundary absorption method and system based on triangular prism grid-FDTD

Abstract

The invention discloses an electromagnetic field simulation boundary absorption method and system based on triangular prism grid-FDTD. The method comprises the following steps: constructing an initial triangular prism grid FDTD; virtual points are added in the triangular prism grid FDTD, and MurABC is constructed; and completing boundary absorption in the electromagnetic field simulation process by using MurABC. In the triangular grid, boundary absorption is carried out by intelligently designing the grid structure and the algorithm, the reflection effect can be effectively reduced, and the simulation accuracy is improved. In this way, unnecessary boundary interference can be avoided, and the whole simulation is closer to the actual situation.

Description

Technical Field

[0001] The invention relates to the research field of boundary absorption conditions, and in particular to an electromagnetic field simulation boundary absorption method and system based on triangular prism grid-FDTD. Background Art

[0002] The Yee finite-difference time-domain (FDTD) method has been widely used in the field of computational electromagnetics due to its easy implementation, low computational requirements, and ability to maintain zero divergence, especially when using regular hexahedral grids. However, in three-dimensional electromagnetic simulations, the orthogonal Cartesian grids commonly used by the Yee-FDTD method for objects of arbitrary shape will lead to a decrease in accuracy, especially the step-wise approximation when dealing with curved interfaces. This approximation method will introduce large errors when simulating smooth or complex geometries. To overcome this problem, researchers have proposed a variety of improved methods, such as sub-grid technology, non-uniform grids, conformal grids, face-centered cubic grids, etc. However, most of these methods still rely on traditional Yee grids, limiting the exploration of other grid types.

[0003] In a two-dimensional environment, regular hexagonal meshes have been proven to be an effective alternative to traditional rectangular meshes, and related studies have shown that triangular meshes are superior to traditional Yee meshes in terms of convergence and accuracy. Despite the remarkable results achieved in two dimensions, the FDTD method based on hexagonal lattices still has limitations when dealing with three-dimensional electromagnetic scattering problems. In order to better simulate complex geometric shapes, researchers have proposed three-dimensional meshes including tetrahedral and hexagonal prismatic meshes. However, the existing literature on the boundary conditions of triangular prismatic meshes is relatively limited. Summary of the invention

[0004] In order to solve the above technical problems, the present invention proposes an electromagnetic field simulation boundary absorption method based on triangular prism grid-FDTD, the steps comprising:

[0005] Construct the initial triangular prism grid FDTD;

[0006] Add virtual points in the triangular prism grid FDTD to construct MurABC;

[0007] The MurABC is used to complete the boundary absorption in the electromagnetic field simulation process.

[0008] Preferably, the constructed triangular prism grid FDTD includes:

[0009]

[0010] Among them, H1, H2, H3, H z1 , H z2All represent magnetic field strength; E1, E2, E3, Ez all represent electric field strength; m, n, p represent spatial coordinates; q represents the qth time step; ε represents dielectric constant; μ represents magnetic permeability, Δl represents spatial length; Δt represents the change in time; A R A represents the area of ​​the side rectangle in the triangular prism grid; T represents the area of ​​the small triangle in the triangular grid; Δz represents the spatial step in the z direction; A H Represents the area of ​​a hexagon in a triangular prism grid.

[0011] Preferably, the MurABC constructed comprises:

[0012] First level:

[0013]

[0014] Second order:

[0015]

[0016] Among them, S′ ec Relative Courant number representing the electric field Ez;

[0017] S′ ec The expression is as follows:

[0018]

[0019] S c The expression is as follows:

[0020]

[0021] Among them, μ r Represents relative magnetic permeability; ∈ r Represents the relative dielectric constant; S c represents wave impedance; c represents phase velocity;

[0022]

[0023] S′ hc The expression is as follows:

[0024]

[0025] Among them, S′ hc Relative Courant number expressing the magnetic field in Hz.

[0026] The present invention also provides an electromagnetic field simulation boundary absorption system based on triangular prism grid-FDTD, the system is used to implement the above method, including: a construction module, an addition module and an absorption module;

[0027] The construction module is used to construct an initial triangular prism grid FDTD;

[0028] The adding module is used to add virtual points in the triangular prism grid FDTD to construct MurABC;

[0029] The absorption module is used to complete the boundary absorption in the electromagnetic field simulation process by using the MurABC.

[0030] Preferably, the constructed triangular prism grid FDTD includes:

[0031]

[0032]

[0033] Among them, H1, H2, H3, H z1 , H z2 All represent magnetic field strength; E1, E2, E3, Ez all represent electric field strength; m, n, p represent spatial coordinates; q represents the qth time step; ε represents dielectric constant; μ represents magnetic permeability, Δl represents spatial length; Δt represents the change in time; A R A represents the area of ​​the side rectangle in the triangular prism grid; T represents the area of ​​the small triangle in the triangular grid; Δz represents the spatial step in the z direction; A H Represents the area of ​​a hexagon in a triangular prism grid.

[0034] Preferably, the MurABC constructed comprises:

[0035] First level:

[0036]

[0037] Second order:

[0038]

[0039] Among them, S′ ec Relative Courant number representing the electric field Ez;

[0040] S′ ec The expression is as follows:

[0041]

[0042] S c The expression is as follows:

[0043]

[0044] Among them, μ r Represents relative magnetic permeability; ∈r Represents the relative dielectric constant; S c represents wave impedance; c represents phase velocity;

[0045]

[0046] S′ hc The expression is as follows:

[0047]

[0048] Among them, S′ hc Relative Courant number expressing the magnetic field in Hz.

[0049] Compared with the prior art, the present invention has the following beneficial effects:

[0050] The present invention can effectively reduce the reflection effect and improve the accuracy of simulation by intelligently designing the grid structure and algorithm for boundary absorption in the triangular grid, thus avoiding unnecessary boundary interference and making the whole simulation closer to the actual situation. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the technical solution of the present invention, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0052] Figure 1 Schematic diagram of a two-dimensional TE mode and TM mode according to an embodiment of the present invention; wherein (a) represents the TE mode; (b) represents the TM mode;

[0053] Figure 2 A schematic diagram of a triangular prism grid according to an embodiment of the present invention;

[0054] Figure 3 Schematic diagram of the absorption boundary and "virtual point" of the present invention; wherein (a) represents the absorption boundary; (b) represents the added "virtual point";

[0055] Figure 4 It is a schematic diagram of the free space result of the present invention;

[0056] Figure 5 It is a schematic diagram of the medium results of the present invention. DETAILED DESCRIPTION

[0057] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0058] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0059] Before describing, the technical concept of the present invention is first introduced:

[0060] The purpose of absorbing boundary technology is to allow the boundaries encountered in simulation or simulation (such as the outer edge of the calculation area) to "absorb" the signal or fluctuation instead of reflecting it back into the calculation area.

[0061] For example, when simulating an electromagnetic field, if the process of electromagnetic wave propagation is simulated, if the boundary of the grid is not specially processed, the electromagnetic wave may be reflected back when it reaches the boundary, interfering with the calculation results. By setting an absorbing boundary, the electromagnetic wave can be made to "disappear" at the boundary and will not be reflected back, making the simulation more realistic.

[0062] Embodiment 1

[0063] This embodiment provides an electromagnetic field simulation boundary absorption method based on triangular prism grid-FDTD, the steps comprising:

[0064] S1. Construct the initial triangular prism grid FDTD.

[0065] This embodiment constructs a triangular prism grid FDTD without loss of generality, and the expressions of Maxwell's equations in linear, lossless, isotropic media include:

[0066]

[0067] Among them, E and H represent the electric field and magnetic field respectively; the parameters of the medium are the dielectric constant (∈) and the magnetic permeability (μ); c represents the phase velocity; l represents the circumference; S represents the area of ​​the integrated region; ∈ represents the dielectric constant; and t represents time.

[0068] Therefore, in this simple medium, the phase velocity c and the wave impedance S c It can be expressed as:

[0069]

[0070] Among them, Δt represents the time change; Δl represents Figure 1Length of space shown.

[0071] E Z and H Z The update equations in three dimensions are similar to those in two dimensions for TM and TE modes (two modes such as Figure 1 The update equation under (shown in Figure 2) is very similar, with the addition of index p to represent the spatial step length Δz in the z direction. The expression is as follows:

[0072]

[0073] Among them, m, n, p represent spatial coordinates; A T Indicates Figure 1 The area of ​​the small triangle in the triangular prism grid shown; A H Indicates Figure 1 The area of ​​the entire hexagon in the triangular grid shown in the figure; μ represents magnetic permeability; E1, E2, E3, and Ez all represent electric field strength; H1, H2, H3, and H z1 , H z2 Both represent the magnetic field strength.

[0074] like Figure 2 As shown, the tangent plane parallel to the z-axis consists of a rectangular grid, resulting in an update equation similar to the traditional Yee grid. After applying Maxwell's integral equations to H1(m,n,p), the four surrounding electric field expressions are as follows:

[0075]

[0076] Among them, the area of ​​the side rectangle of the triangular prism grid is A R A R =ΔlΔz.

[0077] Expanding the above equations, we get the update equations for H1(m,n,p):

[0078]

[0079] Apply Maxwell's integral equations to H2, H3, E1, E2, E3 at the point (m,n,p) and process the same steps to obtain the updated equations for the other electromagnetic fields:

[0080]

[0081]

[0082] S2. Add virtual points in the triangular prism grid FDTD to construct MurABC.

[0083] like Figure 3As shown, the ABC (boundary absorption condition) of the Yee grid is derived from the advection equation. The advection equation of the electric field EZ is:

[0084]

[0085] in, represents the electric field component E Z Directional derivative along the direction of propagation, perpendicular to the boundary.

[0086] Considering the electric field component E Z The present embodiment deals with the different behaviors of E1, E2, and E3. Specifically, compared with the traditional Yee grid, the triangular prism grid leads to unequal numbers of discrete points between adjacent layers. Compared with the first layer, the second layer has one more EZ field point value; to solve this problem, the present embodiment selects the two field values ​​in the second layer that are closest to the electric field discrete points in the first layer. The average of these two points represents the value at the midpoint of the line connecting them. Finally, the advection equation is applied to the midpoint value in the second layer and the value closest to the midpoint of the first layer, for example, the electric field E at the point (1,1,p) Z The expression is as follows:

[0087]

[0088] Therefore, the electric field component E Z The update equation for (1,1,p) is given by:

[0089]

[0090] Among them, μ r Represents relative magnetic permeability; ∈ r Represents the relative dielectric constant; S c Indicates wave impedance.

[0091] Following a similar approach, the goal of this example is to apply ABC to E1, E2, and E3. However, directly copying these steps leads to divergence. Further analysis shows that in the E1, E2, and E3 mode, the electric field in the second layer requires four time steps to propagate to the first layer, while the electric field observed in the traditional Yee grid algorithm mode requires two time steps. This difference can be effectively resolved by introducing "virtual points" outside the grid; when considering magnetic fields instead of electric fields, H Z1 (1,1,p) propagates to H in two time steps Z2 (0,0,p), and then updated by E1(1,1,p). Considering the similarity of the electric field and magnetic field in form and parameters, the advection equation is applied to the two magnetic field discrete points. Through similar simplification steps, H Z2 The update equation for (0,0,p) is:

[0092]

[0093] Next, solve ABC along the z-direction. Consider the tangent plane parallel to the z-axis, such as Figure 2 As shown, it consists of a rectangular grid. Therefore, the update equation of ABC along the z direction is similar to the update equation of the traditional Yee grid. For the sake of simplicity, this embodiment proposes the update equations of the electric field E1, E2, and E3 at the point (m, n, 1):

[0094]

[0095] After establishing the first-order ABC update equation, the second-order update equation is derived based on it. As with the traditional Yee grid, the second-order ABC can be obtained by using two advection operators:

[0096]

[0097] Considering the similarity of processing between second-order and first-order ABC, applying the same method, we obtain:

[0098]

[0099] Among them, S′ ec Represents the relative Courant number of the electric field Ez;.

[0100] S′ ec The expression is as follows:

[0101]

[0102] Among them, S′ hc Relative Courant number expressing the magnetic field in Hz.

[0103] S′ hc The expression is as follows:

[0104]

[0105] S3. Use MurABC to complete boundary absorption in the electromagnetic field simulation process.

[0106] Embodiment 2

[0107] Two numerical experiments were conducted in this embodiment to verify the accuracy and efficiency of the proposed method. By comparing with the results of the traditional FDTD algorithm, the numerical results prove the effectiveness of MurABC based on triangular prism grid finite difference time domain. The FDTD ABC based on triangular prism grid is simple and easy to implement, which is conducive to the wide application of triangular prism grid FDTD method.

[0108] Evaluate electromagnetic wave propagation in free space and in media using absorbing boundaries. Figure 4 The value of the electric field at an observation point in free space as a function of time. The analytical solution represents the field before the reflected wave reaches the observation point. Figure 5 In the presence of a medium, the electric field value at the observation point changes. The dimensions of the simulation area are as follows:

[0109] E z (t)=cos(ω(t-t0))exp(-((t-t0) / τ) 2 ),

[0110] t0=1.4984×10 -15 s,

[0111] τ=4.9947×10 -16 s,

[0112] ω=1π×10 -15 , where ω represents the angular frequency; t0 represents the center time or delay time of the signal; τ represents the pulse width parameter (half-width parameter).

[0113] The spatial step size was set to 5 nm and the time step size was set to 4.1279 × 10 -18 s.

[0114] Figure 4 It is shown that the second-order ABC of the present invention exhibits better absorption compared with the first-order ABC.

[0115] In this example, free space with medium is considered to verify the accuracy of the proposed method. The geometry of the simulation model and the medium size are as follows: Figure 5 As shown, the relative dielectric constant of the medium ∈ r =4.1, add power at the center of 80nm above the dielectric layer, the power expression is:

[0116] E z (t)=cos(ω(t-t0))exp(-((t-t0) / τ) 2 ),

[0117] Where, t0 = 1.4984 × 10 -15 s; τ = 4.9947 × 10 -16 s;ω=1π×10 15 .

[0118] The value of the electric field was recorded at 80 nm below the dielectric. The absorption effect of the first-order and second-order absorption boundaries was compared, and the results showed that the second-order has a better absorption effect than the first-order.

[0119] Embodiment 3

[0120] This embodiment also provides an electromagnetic field simulation boundary absorption system based on triangular prism grid-FDTD, including: a construction module, an adding module and an absorption module; the construction module is used to construct an initial triangular prism grid FDTD; the adding module is used to add virtual points in the triangular prism grid FDTD to construct MurABC; the absorption module is used to use MurABC to complete the boundary absorption in the electromagnetic field simulation process.

[0121] The following will describe in detail how the present invention solves technical problems in real life in conjunction with this embodiment.

[0122] First, an initial triangular prism grid FDTD is constructed.

[0123] This embodiment constructs a triangular prism grid FDTD without loss of generality, and the expressions of Maxwell's equations in linear, lossless, isotropic media include:

[0124]

[0125] Among them, E and H represent the electric field and magnetic field respectively; the parameters of the medium are the dielectric constant (∈) and the magnetic permeability (μ); c represents the phase velocity; l represents the circumference; s represents the area of ​​the integrated region; ∈ represents the dielectric constant; and t represents time.

[0126] Therefore, in this simple medium, the phase velocity c and the wave impedance S c It can be expressed as:

[0127]

[0128] Among them, Δt represents the time change; Δl represents Figure 1 Length of space shown.

[0129] E Z and H Z The update equations in three dimensions are similar to those in two dimensions for TM and TE modes (two modes such as Figure 1 The update equation under (shown in Figure 2) is very similar, with the addition of index p to represent the spatial step length Δz in the z direction. The expression is as follows:

[0130]

[0131] Among them, m, n, p represent spatial coordinates; A T Indicates Figure 1 The area of ​​the small triangle in the triangular prism grid shown; A H Indicates Figure 1 The area of ​​the entire hexagon in the triangular grid shown in the figure; μ represents magnetic permeability; E1, E2, E3, and Ez all represent electric field strength; H1, H2, H3, and H z1 , Hz2 Both represent the magnetic field strength.

[0132] like Figure 2 As shown, the tangent plane parallel to the z-axis consists of a rectangular grid, resulting in an update equation similar to the traditional Yee grid. After applying Maxwell's integral equations to H1(m,n,p), the four surrounding electric field expressions are as follows:

[0133]

[0134] Among them, the area of ​​the side rectangle of the triangular prism grid is A R A R =ΔlΔz.

[0135] Expanding the above equations, we get the update equations for H1(m,n,p):

[0136]

[0137] Apply Maxwell's integral equations to H2, H3, E1, E2, E3 at the point (m,n,p) and process the same steps to obtain the updated equations for the other electromagnetic fields:

[0138]

[0139] Then add a module to add virtual points in the triangular prism grid FDTD to construct MurABC.

[0140] like Figure 3 As shown, the ABC (boundary absorption condition) of the Yee grid is derived from the advection equation. The advection equation of the electric field EZ is:

[0141]

[0142] in, represents the electric field component E Z Directional derivative along the direction of propagation, perpendicular to the boundary.

[0143] Considering the electric field component E Z The present embodiment deals with the different behaviors of E1, E2, and E3. Specifically, compared with the traditional Yee grid, the triangular prism grid leads to unequal numbers of discrete points between adjacent layers. Compared with the first layer, the second layer has one more EZ field point value; to solve this problem, the present embodiment selects the two field values ​​in the second layer that are closest to the electric field discrete points in the first layer. The average of these two points represents the value at the midpoint of the line connecting them. Finally, the advection equation is applied to the midpoint value in the second layer and the value closest to the midpoint of the first layer, for example, the electric field E at the point (1,1,p) Z The expression is as follows:

[0144]

[0145] Therefore, the electric field component E Z The update equation for (1,1,p) is given by:

[0146]

[0147] Among them, μ r Represents relative magnetic permeability; ∈ r Represents the relative dielectric constant; S c Indicates wave impedance.

[0148] Following a similar approach, the goal of this example is to apply ABC to E1, E2, and E3. However, directly copying these steps leads to divergence. Further analysis shows that in the E1, E2, and E3 mode, the electric field in the second layer requires four time steps to propagate to the first layer, while the electric field observed in the traditional Yee grid algorithm mode requires two time steps. This difference can be effectively resolved by introducing "virtual points" outside the grid; when considering magnetic fields instead of electric fields, H Z1 (1,1,p) propagates to H in two time steps Z2 (0,0,p), and then updated by E1(1,1,p). Considering the similarity of the electric field and magnetic field in form and parameters, the advection equation is applied to the two magnetic field discrete points. Through similar simplification steps, H Z2 The update equation for (0,0,p) is:

[0149]

[0150] Next, solve ABC along the z-direction. Consider the tangent plane parallel to the z-axis, such as Figure 2 As shown, it consists of a rectangular grid. Therefore, the update equation of ABC along the z direction is similar to the update equation of the traditional Yee grid. For the sake of simplicity, this embodiment proposes the update equations of the electric field E1, E2, and E3 at the point (m, n, 1):

[0151]

[0152] After establishing the first-order ABC update equation, the second-order update equation is derived based on it. As with the traditional Yee grid, the second-order ABC can be obtained by using two advection operators:

[0153]

[0154] Considering the similarity of processing between second-order and first-order ABC, applying the same method, we obtain:

[0155]

[0156] Among them, S′ ec Represents the relative Courant number of the electric field Ez;.

[0157] S′ ec The expression is as follows:

[0158]

[0159] Among them, S′ hc Relative Courant number expressing the magnetic field in Hz.

[0160] S′ hc The expression is as follows:

[0161]

[0162] Finally, the absorption module uses MurABC to complete the boundary absorption during the electromagnetic field simulation.

[0163] Two numerical experiments were conducted in this embodiment to verify the accuracy and efficiency of the proposed method. By comparing with the results of the traditional FDTD algorithm, the numerical results prove the effectiveness of MurABC based on triangular prism grid finite difference time domain. FDTDABC based on triangular prism grid is simple and easy to implement, which is conducive to the wide application of triangular prism grid FDTD method.

[0164] Evaluate electromagnetic wave propagation in free space and in media using absorbing boundaries. Figure 4 The value of the electric field at an observation point in free space as a function of time. The analytical solution represents the field before the reflected wave reaches the observation point. Figure 5 In the presence of a medium, the electric field value at the observation point changes. The dimensions of the simulation area are as follows:

[0165] E z (t)=cos(ω(t-t0))exp(-((t-t0) / τ) 2 ),

[0166] t0=1.4984×10 -15 s,

[0167] τ=4.9947×10 -16 s,

[0168] ω=1π×10 15 ,

[0169] Wherein, ω represents the angular frequency; t0 represents the center time or delay time of the signal; and τ represents the pulse width parameter (half-width parameter).

[0170] The spatial step size was set to 5 nm and the time step size was set to 4.1279 × 10 -18 s.

[0171] Figure 4 It is shown that the second-order ABC of the present invention exhibits better absorption compared with the first-order ABC.

[0172] The embodiments described above are only descriptions of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A method for electromagnetic field simulation boundary absorption based on triangular prism grid-FDTD, characterized in that the steps include: Construct the initial triangular prism grid FDTD; Add virtual points in the triangular prism grid FDTD to construct MurABC; The MurABC is used to complete the boundary absorption in the electromagnetic field simulation process.

2. The electromagnetic field simulation boundary absorption method based on triangular prism grid-FDTD according to claim 1 is characterized in that: The constructed triangular prism grid FDTD includes: Among them, H1, H2, H3, H z1 , H z2 All represent magnetic field strength; E1, E2, E3, Ez all represent electric field strength; m, n, p represent spatial coordinates; q represents the qth time step; ε represents dielectric constant; μ represents magnetic permeability, Δl represents spatial length; Δt represents the change in time; A R A represents the area of ​​the side rectangle in the triangular prism grid; T represents the area of ​​the small triangle in the triangular grid; Δz represents the spatial step in the z direction; A H Represents the area of ​​a hexagon in a triangular prism grid.

3. The electromagnetic field simulation boundary absorption method based on triangular prism grid-FDTD according to claim 2 is characterized in that: The MurABC constructed includes: First level: Second order: Among them, S′ ec Relative Courant number representing the electric field Ez; S′ ec The expression is as follows: S c The expression is as follows: Among them, μ r Represents relative magnetic permeability; ∈ r Represents the relative dielectric constant; S c represents wave impedance; c represents phase velocity; S′ hc The expression is as follows: Among them, S′ hc Relative Courant number expressing the magnetic field in Hz.

4. An electromagnetic field simulation boundary absorption system based on triangular prism grid-FDTD, the system is used to implement the method described in any one of claims 1 to 3, characterized in that: include: Building modules, adding modules, and absorbing modules; The construction module is used to construct an initial triangular prism grid FDTD; The adding module is used to add virtual points in the triangular prism grid FDTD to construct MurABC; The absorption module is used to complete the boundary absorption in the electromagnetic field simulation process by using the MurABC.

5. The electromagnetic field simulation boundary absorption system based on triangular prism grid-FDTD according to claim 4 is characterized in that: The constructed triangular prism grid FDTD includes: Among them, H1, H2, H3, H z1 , H z2 All represent magnetic field strength; E1, E2, E3, Ez all represent electric field strength; m, n, p represent spatial coordinates; q represents the qth time step; ε represents dielectric constant; μ represents magnetic permeability, Δl represents spatial length; Δt represents the change in time; A R A represents the area of ​​the side rectangle in the triangular prism grid; T represents the area of ​​the small triangle in the triangular grid; Δz represents the spatial step in the z direction; A H Represents the area of ​​a hexagon in a triangular prism grid.

6. The electromagnetic field simulation boundary absorption system based on triangular prism grid-FDTD according to claim 5 is characterized in that: The MurABC constructed includes: First level: Second order: Among them, S′ ec Relative Courant number representing the electric field Ez; S′ ec The expression is as follows: S c The expression is as follows: Among them, μ r Represents relative magnetic permeability; ∈ r Represents the relative dielectric constant; S c represents wave impedance; c represents phase velocity; S′ hc The expression is as follows: Among them, S′ hc Relative Courant number expressing the magnetic field in Hz.

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