This document analyzes the resonant frequencies of different modes in a cylindrical waveguide of varying lengths. It calculates the resonant frequencies of transverse magnetic (TM) and transverse electric (TE) modes for different values of m, n, and l. It plots the resonant frequencies versus cylinder length for different modes and overlays the operating frequencies of a magnetron. The conclusion is that several modes will be available in the 2-3 GHz range to use in the cylindrical waveguide, but neglecting the dielectric material will make tuning the cavity difficult.

1. 1
solution to cylindrical waveguide
By Kurt Zeller 4/17/15
clc
clear
clear all
% close all
c = 299792000; %m/s
% Permittivity and Permeability of air
mu = 1.00000037;
ep = 1.00058986;
% taken from http://www.diva-portal.org/smash/get/diva2:354559/FULLTEXT01.pdf
% skip to page ~15
% m n l
% For transverse Magnetic
TMmn = [2.405 3.832 5.136 5.520]; %01 11 21 02
% For transverse electric
% m= 0 1 2
TEmn = [1.84 3.05 3.832 4.201]; % 11 21 02 31
% more availble at: http://faculty.uml.edu/cbaird/all_homework_solutions/
%Jackson_8_4_Homework_Solution.pdf
a = 4.25*.0254/2; % inch diamter pipe to radius in meters
d = linspace(.1,.3048,1000); % lengths of 10 cm to 30.48 cm (12in )
l = [1 2 3 4 5]; % mode in the length direction
% Calculate resonant frequency for TM (m,n,l)
for k = 1:length(l)
for i = 1:length(d)
ftm1(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TMmn(1)/a)^2 + (l(k)*pi/d(i))^2); %01L
ftm2(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TMmn(2)/a)^2 + (l(k)*pi/d(i))^2); %11L
ftm3(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TMmn(3)/a)^2 + (l(k)*pi/d(i))^2); %21L
ftm4(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TMmn(4)/a)^2 + (l(k)*pi/d(i))^2); %02L
end
end
% Calculate resonant frequency for TE (m,n,l)
for k = 1:length(l)
for i = 1:length(d)
fte1(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TEmn(1)/a)^2 + (l(k)*pi/d(i))^2); % 11L
fte2(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TEmn(2)/a)^2 + (l(k)*pi/d(i))^2);%21L
fte3(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TEmn(3)/a)^2 + (l(k)*pi/d(i))^2); %01L
fte4(i,k) = c/(2*pi*sqrt(mu*ep))*sqrt((TEmn(4)/a)^2 + (l(k)*pi/d(i))^2); %31L
end
end
d = 100*d/2.54; %convert length to inches

2. 2
%convert Hz to Ghz
ftm1 = ftm1/(10^9);
ftm2 = ftm2/(10^9);
ftm3 = ftm3/(10^9);
ftm4 = ftm4/(10^9);
color = 'brgmc';
%Upper and Lower for LG Mag
BU = 2.47;
BL = 2.45;
%
figure(1)
for k=1:length(l)
plot(d,ftm1(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)
grid on
title('TM 01L')
xlabel('Cylinder Length (in)')
ylabel('Resonant Frequency (GHz)')
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','eas
figure(2)
for k=1:length(l)
plot(d,ftm2(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)
title('TM 11L')
grid on
xlabel('Cylinder Length (in)')
ylabel('Resonant Frequency (GHz)')
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','eastout
figure(3)
for k=1:length(l)
plot(d,ftm3(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)
title('TM 21L')
grid on
xlabel('Length (in)')
ylabel('resonant freq (GHz)')
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','east
figure(4)

3. 3
for k=1:length(l)
plot(d,ftm4(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)
title('TM 02L')
grid on
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','eastout
hold off
%%TE plots
%convert Hz to Ghz
fte1 = fte1/(10^9);
fte2 = fte2/(10^9);
fte3 = fte3/(10^9);
fte4 = fte4/(10^9);
color = 'brgmkc';
figure(5)
for k=1:length(l)
plot(d,fte1(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)
grid off
title('TE 11L')
xlabel('Cylinder Length (in)')
ylabel('Resonant Frequency (GHz)')
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','eastou
figure(6)
for k=1:length(l)
plot(d,fte2(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)
title('TE 21L')
grid on
xlabel('Length (in)')
ylabel('Resonant freq (GHz)')
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','eastout
figure(7)
for k=1:length(l)
plot(d,fte3(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)

4. 4
title('TE 01L')
xlabel('Length (in)')
ylabel('Resonant freq (GHz)')
grid on
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','eastou
figure(8)
for k=1:length(l)
plot(d,fte4(:,k),color(k))
hold on
end
plot(d, BU,'k','linewidth',2)
plot(d, BL,'k','linewidth',2)
title('TE 31L')
xlabel('Length (in)')
ylabel('Resonant freq (GHz)')
grid on
legend('L = 1','L = 2','L = 3','L = 4','L = 5','Magnetron Freq','location','eastout
hold off
% Conclusion:
% This analyis indicates that several modes will be available for TM and TE
% using a frequency range of 2~3 GHz. The effect of the dielectric was
% neglected because its affect on the resonant frequency is unknown.
% This fact will make tuning the cavity much more difficult but
% will yield experimental results that may contribute to an increased
% understanding of EM waves within dielectric materials.

5. 5

6. 6

7. 7

8. 8

9. 9
Published with MATLAB® R2013a