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⛄ 内容介绍
将直接三维有限差分时域（FDTD）方法应用于各种微带结构的全波分析。 该方法被证明是对复杂的微带电路元件和微带天线进行建模的有效工具。 从时域结果计算出线馈矩形贴片天线的输入阻抗以及低通滤波器和支线耦合器的频率相关散射参数。 制作了这些电路，并将对它们进行的测量与 FDTD 结果进行了比较，结果表明它们非常吻合。
⛄ 部分代码
%% Microstrip low-pass filter analysis using 3D FDTD code with UPML
%% absorbing borders (ABC)
%
% Here we use FDTD 3D with UPML to calculate scattering coefficients S_{11}
% and S_{21} for planar microstrip low-pass filter following by the original
% paper by D. Sheen, S. Ali, M. Abouzahra, J. Kong "Application of the
% Three-Dimensional Finite-Difference Time-Domain Method to the Analysis of
% planar Microstrip Circuits", IEEE Trans. on Microwave Theory and Techniques
% (http://dx.doi.org/10.1109/22.55775).
% Also |S_{21}| dependence can be taken from the paper "Computational
% electromagnetic method for interconnects and small structures" by C.
% Balanis, A. Policarpou and S. Georgakopoulos
% (http://dx.doi.org/10.1006/spmi.2000.0865)
function FDTD_3D_Lowpass
close all; clear; clc;
%% Physical constants
epsilon0 = 8.85418782e-12; mu0 = 1.25663706e-6;
c = 1.0/sqrt(mu0*epsilon0);
%% Gaussian half-width
t_half = 15.0e-12;
%% Microstrip transmission lines parameters
lineW = 2.413e-3;
lineH = 1.0e-3;
lineEr = 2.2;
Z0 = 49.2526;
%% End time
t_end = 1.5e-9;
%% Total mesh dimensions and grid cells sizes (without PML)
nx = 80; ny = 100; nz = 16;
dx = 0.4064e-3; dy = 0.4233e-3; dz = 0.2650e-3;
%% Number of PML layers
PML = 5;
%% Matrix of material's constants
number_of_materials = 4;
% For material of number x = 1,2,3... :
% Material(x,1) - relative permittivity, Material(x,2) - relative permeability,
% Material(x,3) - specific conductivity
% Vacuum
Material(1,1) = 1.0; Material(1,2) = 1.0; Material(1,3) = 0.0;
% Metal (Copper)
Material(2,1) = 1.0; Material(2,2) = 1.0; Material(2,3) = 5.88e+7;
% Substrate material (RT/Duroid 5880)
Material(3,1) = lineEr; Material(3,2) = 1.0; Material(3,3) = 0.0;
% Matched load material is calculated from transmission line parameters
Material(4,1) = 1.0; Material(4,2) = 1.0; Material(4,3) = lineH/(Z0*lineW*dy);
% Add PML layers
nx = nx + 2*PML; ny = ny + 2*PML; nz = nz + 2*PML;
% Calculate dt
dt = (1.0/c/sqrt( 1.0/(dx^2) + 1.0/(dy^2) + 1.0/(dz^2)))*0.9999;
number_of_iterations = ceil(t_end/dt);
%% 3D array for geometry
Index = ones(nx, ny, nz);
%% Define of low-pass filter geometry
% Ground plane
Index((1+PML):(nx-PML), (1+PML):(ny-PML), PML+1) = 2;
% Rectangular patch (one cell thickness)
Index((nx/2-25):(nx/2+25), (ny/2-3):(ny/2+3), PML+5) = 2;
% Transmission line from port 1 to rectangular patch
Index((nx/2-10):(nx/2-5), (PML+1):ny/2, PML+5) = 2;
% Transmission line from rectangular patch to port 2
Index((nx/2+5):(nx/2+10), ny/2:(ny-PML), PML+5) = 2;
% Dielectric substrate between ground plane and filter patch
Index((1+PML):(nx-PML), (1+PML):(ny-PML), (PML+2):(PML+4)) = 3;
% Matched load before port 1
Index((nx/2-10):(nx/2-5), PML+1, (PML+2):(PML+4)) = 4;
% Matched load after port 2
Index((nx/2+5):(nx/2+10), ny-PML, (PML+2):(PML+4)) = 4;
%% 3D FDTD physical (fields) and additional arrays are defined as 'single'
%% to increase performance
Ex = zeros(nx, ny+1, nz+1, 'single');
Gx = zeros(nx, ny+1, nz+1, 'single');
Fx = zeros(nx, ny+1, nz+1, 'single');
Ey = zeros(nx+1, ny, nz+1, 'single');
Gy = zeros(nx+1, ny, nz+1, 'single');
Fy = zeros(nx+1, ny, nz+1, 'single');
Ez = zeros(nx+1, ny+1, nz, 'single');
Gz = zeros(nx+1, ny+1, nz, 'single');
Fz = zeros(nx+1, ny+1, nz, 'single');
Hx = zeros(nx+1, ny, nz, 'single');
Bx = zeros(nx+1, ny, nz, 'single');
Hy = zeros(nx, ny+1, nz, 'single');
By = zeros(nx, ny+1, nz, 'single');
Hz = zeros(nx, ny, nz+1, 'single');
Bz = zeros(nx, ny, nz+1, 'single');
%% FDTD PML coefficients arrays. Here they are already filled with values
%% corresponding to free space
m = 4; ka_max = 1.0; R_err = 1.0e-16;
eta = sqrt(mu0/epsilon0*Material(1,1)/Material(1,2));
k_Ex_c = ones(nx, ny, nz, 'single')*2.0*epsilon0;
k_Ex_d = ones(nx, ny, nz, 'single')*(-2.0*epsilon0);
k_Ey_a = ones(nx+1, ny, nz, 'single');
k_Ey_b = ones(nx+1, ny, nz, 'single')/(2.0*epsilon0);
k_Gz_a = ones(nx+1, ny, nz, 'single');
k_Gz_b = ones(nx+1, ny, nz, 'single');
k_Hy_a = ones(nx, ny, nz, 'single');
k_Hy_b = ones(nx, ny, nz, 'single')/(2.0*epsilon0);
k_Hx_c = ones(nx+1, ny, nz, 'single')*2.0*epsilon0/mu0;
k_Hx_d = ones(nx+1, ny, nz, 'single')*(-2.0*epsilon0/mu0);
k_Bz_a = ones(nx, ny, nz, 'single');
k_Bz_b = ones(nx, ny, nz, 'single')*dt;
k_Gx_a = ones(nx, ny+1, nz, 'single');
k_Gx_b = ones(nx, ny+1, nz, 'single');
k_Ey_c = ones(nx, ny, nz, 'single')*2.0*epsilon0;
k_Ey_d = ones(nx, ny, nz, 'single')*(-2.0*epsilon0);
k_Ez_a = ones(nx, ny+1, nz, 'single');
k_Ez_b = ones(nx, ny+1, nz, 'single')/(2.0*epsilon0);
k_Bx_a = ones(nx, ny, nz, 'single');
k_Bx_b = ones(nx, ny, nz, 'single')*dt;
k_Hy_c = ones(nx, ny+1, nz, 'single')*2.0*epsilon0/mu0;
k_Hy_d = ones(nx, ny+1, nz, 'single')*(-2.0*epsilon0/mu0);
k_Hz_a = ones(nx, ny, nz, 'single');
k_Hz_b = ones(nx, ny, nz, 'single')/(2.0*epsilon0);
k_Ex_a = ones(nx, ny, nz+1, 'single');
k_Ex_b = ones(nx, ny, nz+1, 'single')/(2.0*epsilon0);
k_Gy_a = ones(nx, ny, nz+1, 'single');
k_Gy_b = ones(nx, ny, nz+1, 'single');
k_Ez_c = ones(nx, ny, nz, 'single')*2.0*epsilon0;
k_Ez_d = ones(nx, ny, nz, 'single')*(-2.0*epsilon0);
k_Hx_a = ones(nx, ny, nz, 'single');
k_Hx_b = ones(nx, ny, nz, 'single')/(2.0*epsilon0);
k_By_a = ones(nx, ny, nz, 'single');
k_By_b = ones(nx, ny, nz, 'single')*dt;
k_Hz_c = ones(nx, ny, nz+1, 'single')*2.0*epsilon0/mu0;
k_Hz_d = ones(nx, ny, nz+1, 'single')*(-2.0*epsilon0/mu0);
%% General FDTD coefficients
I = 1:number_of_materials;
K_a(I) = (2.0*epsilon0*Material(I,1) - Material(I,3)*dt)./...
(2.0*epsilon0*Material(I,1) + Material(I,3)*dt);
K_b(I) = 2.0*dt./(2.0*epsilon0*Material(I,1) + Material(I,3)*dt);
K_c(I) = Material(I,2);
Ka = single(K_a(Index)); Kb = single(K_b(Index)); Kc = single(K_c(Index));
%% PML coefficients along x-axis
sigma_max = -(m + 1.0)*log(R_err)/(2.0*eta*PML*dx);
for I=0:(PML-1)
sigma_x = sigma_max*((PML - I)/PML)^m;
ka_x = 1.0 + (ka_max - 1.0)*((PML - I)/PML)^m;
k_Ey_a(I+1,:,:) = (2.0*epsilon0*ka_x - sigma_x*dt)/...
(2.0*epsilon0*ka_x + sigma_x*dt);
k_Ey_a(nx-I,:,:) = k_Ey_a(I+1,:,:);
k_Ey_b(I+1,:,:) = 1.0/(2.0*epsilon0*ka_x + sigma_x*dt);
k_Ey_b(nx-I,:,:) = k_Ey_b(I+1,:,:);
k_Gz_a(I+1,:,:) = (2.0*epsilon0*ka_x - sigma_x*dt)/...
(2.0*epsilon0*ka_x + sigma_x*dt);
k_Gz_a(nx-I,:,:) = k_Gz_a(I+1,:,:);
k_Gz_b(I+1,:,:) = 2.0*epsilon0/(2.0*epsilon0*ka_x + sigma_x*dt);
k_Gz_b(nx-I,:,:) = k_Gz_b(I+1,:,:);
k_Hx_c(I+1,:,:) = (2.0*epsilon0*ka_x + sigma_x*dt)/mu0;
k_Hx_c(nx-I,:,:) = k_Hx_c(I+1,:,:);
k_Hx_d(I+1,:,:) = -(2.0*epsilon0*ka_x - sigma_x*dt)/mu0;
k_Hx_d(nx-I,:,:) = k_Hx_d(I+1,:,:);
sigma_x = sigma_max*((PML - I - 0.5)/PML)^m;
ka_x = 1.0 + (ka_max - 1.0)*((PML - I - 0.5)/PML)^m;
k_Ex_c(I+1,:,:) = 2.0*epsilon0*ka_x + sigma_x*dt;
k_Ex_c(nx-I-1,:,:) = k_Ex_c(I+1,:,:);
k_Ex_d(I+1,:,:) = -(2.0*epsilon0*ka_x - sigma_x*dt);
k_Ex_d(nx-I-1,:,:) = k_Ex_d(I+1,:,:);
k_Hy_a(I+1,:,:) = (2.0*epsilon0*ka_x - sigma_x*dt)/...
(2.0*epsilon0*ka_x + sigma_x*dt);
k_Hy_a(nx-I-1,:,:) = k_Hy_a(I+1,:,:);
k_Hy_b(I+1,:,:) = 1.0/(2.0*epsilon0*ka_x + sigma_x*dt);
k_Hy_b(nx-I-1,:,:) = k_Hy_b(I+1,:,:);
k_Bz_a(I+1,:,:) = (2.0*epsilon0*ka_x - sigma_x*dt)/...
(2.0*epsilon0*ka_x + sigma_x*dt);
k_Bz_a(nx-I-1,:,:) = k_Bz_a(I+1,:,:);
k_Bz_b(I+1,:,:) = 2.0*epsilon0*dt/(2.0*epsilon0*ka_x + sigma_x*dt);
k_Bz_b(nx-I-1,:,:) = k_Bz_b(I+1,:,:);
end
%% PML coefficients along y-axis
sigma_max = -(m + 1.0)*log(R_err)/(2.0*eta*PML*dy);
for J=0:(PML-1)
sigma_y = sigma_max*((PML - J)/PML)^m;
ka_y = 1.0 + (ka_max - 1.0)*((PML - J)/PML)^m;
k_Gx_a(:,J+1,:) = (2.0*epsilon0*ka_y - sigma_y*dt)/...
(2.0*epsilon0*ka_y + sigma_y*dt);
k_Gx_a(:,ny-J,:) = k_Gx_a(:,J+1,:);
k_Gx_b(:,J+1,:) = 2.0*epsilon0/(2.0*epsilon0*ka_y + sigma_y*dt);
k_Gx_b(:,ny-J,:) = k_Gx_b(:,J+1,:);
k_Ez_a(:,J+1,:) = (2.0*epsilon0*ka_y - sigma_y*dt)/...
(2.0*epsilon0*ka_y + sigma_y*dt);
k_Ez_a(:,ny-J,:) = k_Ez_a(:,J+1,:);
k_Ez_b(:,J+1,:) = 1.0/(2.0*epsilon0*ka_y + sigma_y*dt);
k_Ez_b(:,ny-J,:) = k_Ez_b(:,J+1,:);
k_Hy_c(:,J+1,:) = (2.0*epsilon0*ka_y + sigma_y*dt)/mu0;
k_Hy_c(:,ny-J,:) = k_Hy_c(:,J+1,:);
k_Hy_d(:,J+1,:) = -(2.0*epsilon0*ka_y - sigma_y*dt)/mu0;
k_Hy_d(:,ny-J,:) = k_Hy_d(:,J+1,:);
sigma_y = sigma_max*((PML - J - 0.5)/PML)^m;
ka_y = 1.0 + (ka_max - 1.0)*((PML - J - 0.5)/PML)^m;
k_Ey_c(:,J+1,:) = 2.0*epsilon0*ka_y+sigma_y*dt;
k_Ey_c(:,ny-J-1,:) = k_Ey_c(:,J+1,:);
k_Ey_d(:,J+1,:) = -(2.0*epsilon0*ka_y-sigma_y*dt);
k_Ey_d(:,ny-J-1,:) = k_Ey_d(:,J+1,:);
k_Bx_a(:,J+1,:) = (2.0*epsilon0*ka_y-sigma_y*dt)/...
(2.0*epsilon0*ka_y+sigma_y*dt);
k_Bx_a(:,ny-J-1,:) = k_Bx_a(:,J+1,:);
k_Bx_b(:,J+1,:) = 2.0*epsilon0*dt/(2.0*epsilon0*ka_y+sigma_y*dt);
k_Bx_b(:,ny-J-1,:) = k_Bx_b(:,J+1,:);
k_Hz_a(:,J+1,:) = (2.0*epsilon0*ka_y-sigma_y*dt)/...
(2.0*epsilon0*ka_y+sigma_y*dt);
k_Hz_a(:,ny-J-1,:) = k_Hz_a(:,J+1,:);
k_Hz_b(:,J+1,:) = 1.0/(2.0*epsilon0*ka_y+sigma_y*dt);
k_Hz_b(:,ny-J-1,:) = k_Hz_b(:,J+1,:);
end
%% PML coefficients along z-axis
sigma_max = -(m + 1.0)*log(R_err)/(2.0*eta*PML*dz);
for K=0:(PML-1)
sigma_z = sigma_max*((PML - K)/PML)^m;
ka_z = 1.0 + (ka_max - 1.0)*((PML-K)/PML)^m;
k_Ex_a(:,:,K+1) = (2.0*epsilon0*ka_z - sigma_z*dt)/...
(2.0*epsilon0*ka_z + sigma_z*dt);
k_Ex_a(:,:,nz-K) = k_Ex_a(:,:,K+1);
k_Ex_b(:,:,K+1) = 1.0/(2.0*epsilon0*ka_z + sigma_z*dt);
k_Ex_b(:,:,nz-K) = k_Ex_b(:,:,K+1);
k_Gy_a(:,:,K+1) = (2.0*epsilon0*ka_z - sigma_z*dt)/...
(2.0*epsilon0*ka_z + sigma_z*dt);
k_Gy_a(:,:,nz-K) = k_Gy_a(:,:,K+1);
k_Gy_b(:,:,K+1) = 2.0*epsilon0/(2.0*epsilon0*ka_z + sigma_z*dt);
k_Gy_b(:,:,nz-K) = k_Gy_b(:,:,K+1);
k_Hz_c(:,:,K+1) = (2.0*epsilon0*ka_z + sigma_z*dt)/mu0;
k_Hz_c(:,:,nz-K) = k_Hz_c(:,:,K+1);
k_Hz_d(:,:,K+1) = -(2.0*epsilon0*ka_z - sigma_z*dt)/mu0;
k_Hz_d(:,:,nz-K) = k_Hz_d(:,:,K+1);
sigma_z = sigma_max*((PML - K - 0.5)/PML)^m;
ka_z = 1.0 + (ka_max - 1.0)*((PML - K - 0.5)/PML)^m;
k_Ez_c(:,:,K+1) = 2.0*epsilon0*ka_z + sigma_z*dt;
k_Ez_c(:,:,nz-K-1) = k_Ez_c(:,:,K+1);
k_Ez_d(:,:,K+1) = -(2.0*epsilon0*ka_z - sigma_z*dt);
k_Ez_d(:,:,nz-K-1) = k_Ez_d(:,:,K+1);
k_Hx_a(:,:,K+1) = (2.0*epsilon0*ka_z - sigma_z*dt)/...
(2.0*epsilon0*ka_z + sigma_z*dt);
k_Hx_a(:,:,nz-K-1) = k_Hx_a(:,:,K+1);
k_Hx_b(:,:,K+1) = 1.0/(2.0*epsilon0*ka_z + sigma_z*dt);
k_Hx_b(:,:,nz-K-1) = k_Hx_b(:,:,K+1);
k_By_a(:,:,K+1) = (2.0*epsilon0*ka_z - sigma_z*dt)/...
(2.0*epsilon0*ka_z + sigma_z*dt);
k_By_a(:,:,nz-K-1) = k_By_a(:,:,K+1);
k_By_b(:,:,K+1) = 2.0*epsilon0*dt/(2.0*epsilon0*ka_z + sigma_z*dt);
k_By_b(:,:,nz-K-1) = k_By_b(:,:,K+1);
end
%% Main 3D FDTD+UPML routine (operates with 'singles' to increase speed)
hhh = waitbar(0, 'Calculations in progress...');
tic;
for T=0:(number_of_iterations-1)
%% Calculate Fx -> Gx -> Ex
I = 1:nx; J = 2:ny; K = 2:nz;
Fx_r = Fx(I,J,K);
Fx(I,J,K) = Ka(I,J,K).*Fx(I,J,K) + Kb(I,J,K).*...
((Hz(I,J,K) - Hz(I,J-1,K))/dy - (Hy(I,J,K) - Hy(I,J,K-1))/dz);
Gx_r = Gx(I,J,K);
Gx(I,J,K) = k_Gx_a(I,J,K).*Gx(I,J,K) + k_Gx_b(I,J,K).*(Fx(I,J,K) - Fx_r);
Ex(I,J,K) = k_Ex_a(I,J,K).*Ex(I,J,K) + k_Ex_b(I,J,K).*...
(k_Ex_c(I,J,K).*Gx(I,J,K) + k_Ex_d(I,J,K).*Gx_r);
%% Calculate Fy -> Gy -> Ey
I = 2:nx; J = 1:ny; K = 2:nz;
Fy_r = Fy(I,J,K);
⛄ 运行结果
编辑
⛄ 参考文献
[1] Sheen D M , Ali S M , MD Abouzahra, et al. Application of the three-dimensional finite-difference time-domain method to the analysis of planar microstrip circuits[J]. IEEE Transactions on Microwave Theory & Techniques, 1990, MTT-38(7):849-857.
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