example 3.1.3
ZIP with MATLAB scripts and note:
example 3.1.3 notes:
Rectangular waveguide:
Find the lower TE mode cut-off frequencies for a rectangular waveguide of section a=1.07cm b=0.43cm filled with Teflon er=2.08 real conductor tand=.0004.
Directly applying the cut-off frequency formula from the above table:
pozar_03_example_01_3.m
clear all;close all;clc
f0=15e9; % choose operating frequency [Hz]
a=1.07e-2;b=0.43e-2; % choose waveguide cross-section dimensions [m]
c=10*a; % choose waveguide length [m]
er=2.08; % Teflon % choose filling material
dx=min(a,b)/50;dy=dx;dz=dx;; % set space resolutions
c0=299792458;
M_range=[0:1:5];N_range=[0:1:5]; % modes
for k=M_range
for s=N_range
fc(k+1,s+1)=c0/(2*pi*er^.5)*((M_range(k+1)*pi/a)^2+(N_range(s+1)*pi/b)^2).^.5;
end
end
fc % cut-offs
fMN=zeros(max(M_range)+1,max(N_range)+1,3);
fMN(:,:,3)=fc;
fMN(:,:,2)=meshgrid(M_range,N_range);
fMN(:,:,1)=meshgrid(M_range,N_range)';
L=[0 0 0];
for k=1:1:(M_range(end)+1)
for s=1:1:(N_range(end)+1)
L=[L;fMN(k,s,1) fMN(k,s,2) fMN(k,s,3)];
end
end
L=uint64(L)
TEM doesn't propagate inside rectangular waveguides, removing all null lines off L
k_all_nulls=[];
for k=1:1:size(L,1)
if L(k,1)==0 && L(k,2)==0 &&L(k,3)==0
k_all_nulls=[k_all_nulls k]
end
end
L(k_all_nulls,:)=[]
% sorting cut-offs in frequency ascending order
fc=L(:,3);
[n,v]=sort(fc);
L=L(v,:)
fc_sorted=L(:,3);
% modes that get through, for a given input frequency f0
% f0=f_cutoff doesn't get through, it has to be f0>f_cutoff
% safety_factor=1
% safety_factor=1 means no safety band,
% safety_factor=1.5 means at least half carrier above cutoff
k=1;
while L(k,3)<=f0 && k<size(L,1)
k=k+1;
end
k
L_through=L([k:end],:)
fc_through=fc_sorted([k:end],:)
0 2.417077773416413 4.834155546832826
0.971349011746783 2.604953716549509 4.930778716754183
1.942698023493566 3.101022504470010 5.209907433099017
L =
0 0 0
0 0 0
0 1 24170777734
0 2 48341555468
1 0 9713490117
1 1 26049537165
1 2 49307787168
2 0 19426980235
2 1 31010225045
2 2 52099074331
L =
0 1 24170777734
0 2 48341555468
1 0 9713490117
1 1 26049537165
1 2 49307787168
2 0 19426980235
2 1 31010225045
2 2 52099074331
L =
8×3 uint64 matrix
1 0 9713490117
2 0 19426980235
0 1 24170777734
1 1 26049537165
2 1 31010225045
0 2 48341555468
1 2 49307787168
2 2 52099074331
k = 2
L_through =
2 0 19426980235
0 1 24170777734
1 1 26049537165
2 1 31010225045
0 2 48341555468
1 2 49307787168
2 2 52099074331
fc_through =
9713490117
19426980235
24170777734
26049537165
31010225045
48341555468
49307787168
52099074331
[POZAR] takes the previous TE TM fields simulations inside ideal infinite rectangular waveguides from
[RAMO] chapter 8 named 'Waveguides with cylindrical conducting boundaries'. Despite the name of the chapter before any cylindrical waveguide result the complete analytical study for rectangular waveguides precedes cylindrical.
Some Standard waveguides shorter table [POZAR]
As for the parallel plates table, the following script helps develop values of rectangular waveguide parameters table in page 117.
And these are the field simulations inside ideal rectangular waveguides available in early Microwave Journal Handbooks
As clearly explained as it is in [SDK] the analytical solutions are one way to attempt simulating how E H fields are going to behave on, through, around, matter structures, but the usefulness of the results, as well as the difficulty to obtain meaningful results, both limit the analytical approach to certain well known basic structures.
From [SDK] the general field inside an ideal waveguide of dimensions x [0 a] y [0 b] z [0 c] with charge distribution e f(x,y,z) (e=e0*er or epws0*er) and the boundary condition V(0,y,z)=V(x,0,z)=V(x,y,0)= V(a,y,z)=V(x,b,z)=V(x,y,c)=0 ((all waveguide metal properly grounded) has to comply with
Assuming a solution with the following form
the coefficients are
On the above graphs of |E| |H| from [POZAR], the amplitudes of E and H are peak values and qualitative only.
As shown in the table on the right hand side [POZAR] above, while [SDK] shows the potential of the field along the waveguide having shape sin(x)*sin(y)*sin(z) the field equations for TE and TM modes produce basically the same shape sin(x)*sin(y)*exp(1j*beta*z).
If beta is real the wave is in propagation mode. If beta is complex only the wave doesn't propagate, only attenuates.
Worth recalling that all these fields and potentials are phasor notation, the real measureable expressions being ther real parts of the right hand side expressions [POZAR] pg22:
Following, |E| in TE10 TE11 and TE21 modes, assuming f(x,y,z) constant, null or far away on a side of the waveguide, ideal conductor.
TE ignoring f(x,y,z):
f0=2e10 % carrier frequency [Hz]
a=1.07e-2;b=0.43e-2; % waveguide cross-section dimensions [m]
length_waveguide=5*b; % waveguide length [m]
er=2.08; % waveguide fill material: Teflon
dx=min(a,b)/50;dy=dx;dz=dx; % set space resolutions
c0=299792458;
lambda0=c0/f0;lambda=lambda0/er^.5;beta=2*pi/lambda;
dx=a/50;x_range=[0:dx:a];y_range=[0:dx:b];
z_range=[0:dx:length_waveguide];
[X,Y,Z]=meshgrid(x_range,y_range,z_range);
m=1;n=0; % TE10 Electric field
E10x=cos(m*pi*X/a).*sin(n*pi*Y/b).*exp(1j*beta*Z);
E10y= sin(m*pi*X/a).*cos(n*pi*Y/b).*exp(1j*beta*Z);
E10z=zeros(size(E10x));
absE10=((abs(E10x)).^2+(abs(E10y)).^2).^.5;
xslice=x_range(1);yslice=y_range(1);zslice=z_range(50);
colormap('jet');shading interp
daspect([1 1 1]);axis tight;
figure(1);h1=slice(X,Y,Z,absE10,xslice,yslice,zslice);
ax1=gca;
ax1.DataAspectRatio=[1 1 1]; % avoid deformation if moving point of view
h1(1).EdgeColor='none';h1(2).EdgeColor='none';h1(3).EdgeColor='none';
colormap(ax1,'jet');shading interp
daspect(ax1,[1 1 1]);axis tight;
xlabel(ax1,'x:[0 a]');ylabel(ax1,'y:[0 b]');
zlabel(ax1,'z:[0 length waveguide');
title(ax1,'TE10 |E|');
campos(ax1,[0.047 0.083 -.104]);
m=2;n=1; % TE21 Electric field
E21x=cos(m*pi*X/a).*sin(n*pi*Y/b).*exp(1j*beta*Z);
E21y= sin(m*pi*X/a).*cos(n*pi*Y/b).*exp(1j*beta*Z);
E21z=zeros(size(E21x));
absE21=((abs(E21x)).^2+(abs(E21y)).^2).^.5;
xslice=x_range(1);yslice=y_range(1);zslice=z_range(50);
colormap('jet');shading interp
daspect([1 1 1]);axis tight;camlight
figure(3);h3=slice(X,Y,Z,absE21,xslice,yslice,zslice);
ax3=gca;
ax3.DataAspectRatio=[1 1 1]; % avoid deformation if moving point of view
h3(1).EdgeColor='none';
h3(2).EdgeColor='none';
h3(3).EdgeColor='none';
colormap(ax3,'jet');shading interp
daspect(ax3,[1 1 1]);axis tight; %camlight('headlight')
xlabel(ax3,'x:[0 a]');ylabel(ax3,'y:[0 b]');
zlabel(ax3,'z:[0 length waveguide');
title('TE21 |E|');
campos(ax3,[0.047 0.083 -.104]);
Standard rectangular waveguides [RAMO]
Some Standard rectangular waveguides [POZAR]
References:
[SDK]
Numerical Techniques in Electromagnetics with MATLAB, Third Edition, by Matthew Sadiku
[RAMO]
Fields and Waves in Communication Electronics, 3rd edition, by S. Ramo, T. R. Whinnery, and T. van Duzer,John Wiley & Sons, New York, 1994.