finite element method for waveguide
TRANSCRIPT
byAnuj
14/MAP/0012 M.Sc. (Applied Physics) Under the guidance of
Dr. Manmohan Singh Shishodia Gautam Buddha University, Greater Noida (U.P.)
Finite Element Method for Analyzing TE Modes of Rectangular Hollow Waveguide
Outlines Review of Different Approaches Based On Finite Element
Method Introduction To Waveguide EM Field Configuration Within The Waveguide
FEM Formulation Homogeneous Hollow waveguide Example MATLAB Code To Calculate Propagation Constant Result and comparison b/w FEM and Analytical Results Overall Summary Future Plan References
REVIEW OF DIFFERENT
APPROACHES
BASED ON
FINITE ELEMENT METHOD
Review of Different Approaches Based On Finite Element Method
FINITE ELEMENT METHOD: finite element method is a numerical technique to solve the ordinary/partial differential equation.1.Weighted Residual Method Boundary value problem
i. Galerkin’s
ii. Least Square
iii. Collocation
‘types of element’
R
jR
j njdguLd )1(1,0)(Re
0Re2 d
Ai
b
a i nidxxx )1(1,0)(Re
SDgLu ,_
Re uL
0.0 0.2 0.4 0.6 0.8 1.0
-0.008
-0.004
0.000
0.004
0.008
0.012
(yT
/ WL
2 )
x/L
Galerkin Least Square Collocation
ERROR PLOT FOR DIFFERENT APPROACHES
0.0 0.2 0.4 0.6 0.8 1.0-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05 Solutions obtained from different methods
(yT/
WL
2 )
x/L
Exact, Collocation Galerkin, Least Square
Review of Different Approaches Based On Finite Element Method
0)(2
2
xwdx
ydT
032
2
Wdx
ydT2
0 Lxfor
02
2
Wdx
ydT LxLfor 2
* Hence the plot shown that the eroor is least in the Galerkin’s approach.
Introduction To Waveguide
A Hollow metallic tube of uniform cross section for transmitting electromagnetic waves
by successive reflections from the inner walls of the tube is called waveguide.
It may be used to carry energy over longer distances to carry transmitter power to an
antenna or microwave signals from an antenna to a receiver.
Waveguides are made from copper, aluminum or brass. These metals are extruded into
long rectangular or circular pipes.
The electric and magnetic fields associated with the signal bounce off the inside walls
back and forth as it progresses down the waveguide.
*Fig. Three-dimensional view of the electric field for the TE₁₀– mode in a rectangular waveguide
*http://www.radartutorial.eu/03.linetheory/Waveguides.en.html
EM Field Configuration Within The Waveguide
In order to determine the EM field configuration within the waveguide, Maxwell’s equations should be solved subject to appropriate boundary conditions at the walls of the guide. Such solutions give rise to a number of field configurations. Each configuration is known
as a mode.
022
2
2
2
zzz Ek
yE
xE022 EkE
ck 22
)()(),( yYxXyxEz
tBEE
,0 t
Ec
BB
2
10
022
2
2
2
2
zEkcyx
xkBxkAxX xx cossin
022
2
2
2
2
XYk
cdyYdX
dxXdY
bnaxmEEz /cos/cos0
2222/
bn
amck
mnbn
amc
22
Scalar Wave Equation
Where
Maxwell’s Equations
*David J. Griffiths, “Introduction to Electrodynamics” Pearson Education, Inc., ISBN-978-81-203-1601-0 (1999),.
Possible Types of modesTransverse Electromagnetic (TEM): Here both electric and magnetic fields are directed components. (i.e.) E z = 0 and Hz = 0
Transverse Electric (TE) wave: Here only the electric field is purely transverse to the direction of propagation and
the magnetic field is not purely transverse. (i.e.) E z = 0, Hz ≠ 0.
Transverse Magnetic (TM) wave: Here only magnetic field is transverse to the direction of propagation and the electric
field is not purely transverse. (i.e.) E z ≠ 0, Hz = 0.
Dimensions of the waveguide which determines the operating frequency range
RECTANGULAR WAVEGUIDE
EM Field Configuration Within The Waveguide
mnbnamc 22 //
Where is the cutoff frequency
The size of the waveguide determines its operating frequency range.
The frequency of operation is determined by the dimension ‘a’ and b.
This dimension is usually made equal to one – half the wavelength at the
lowest frequency of operation, this frequency is known as the waveguide
cutoff frequency.
At the cutoff frequency and below, the waveguide will not transmit energy. At
frequencies above the cutoff frequency, the waveguide will propagate energy. Angle of incidence(A) Angle of reflection (B)
(A = B)
(a)At high frequency
(b) At medium frequency
( c ) At low frequency
(d) At cutoff frequency
EM Field Configuration Within The Waveguide
022 k
dskyx
Fs
2222
21)(
,220
2zr kkk
00
2
k
dSkyx
F eee
N
e A
e
e
2222
21
2
22 ,
e
yx
3
33 ,
e
yx
11
1
, yxe
yxP ,
x
y
Element e
1
2 3
cybxae
FEM Formulation The scalar wave equation for a homogeneous isotropic medium is chosen.
The scalar wave equation has many applications. It can be used to analyse problem such as the propagation of plane waves. It can be used to analyse the TE and TM modes in waveguides/weakly guiding optical
fibers.The FEM solution of the above scalar equation by minimisation of a corresponding functional is given by
The function at a point inside the triangle may be approximated as , the linear terms:-
e
Fig. A typical first order triangular element.
111 cybxae 222 cybxae 333 cybxae
the solution of these equations gives
3
1
3
1
3
1 ieii
ieii
ieii ccbbaa
cba
yxyxyx
e
e
e
111
33
22
11
3
2
1
3
2
1
212113133232
123123
211332
21
e
e
e
xyyxxyyxxyyxxxxxxx
yyyyyy
Acba
111
2
1
33
22
11
yxyxyx
A
FEM Formulation
where
……………………………………………..(1)
23132132321 21,
21,
21 xx
Acyy
Abxyyx
Aa
31113213132 21,
21,
21 xx
Acyy
Abxyyx
Aa
12321321213 21,
21,
21 xx
Acyy
Abxyyx
Aa
Hence, we can write ei
iie u
3
1
321 eeee 321 uuuu 321 bbbb 321 cccc
Using row vectors
dSuukc
cbbF
e
e
N
e At
e
tte
te
tte
te
tte
1 221
eN
e
teee
teee QkPF
1
2
21
FEM Formulation
eN
e
teee
teee QkPF
1
2
21 ccbbAP tt
ee
211121112
12e
eA
Q
Using eqn (1)
Homogeneous Hollow Waveguide Example
The example of a homogeneous hollow waveguide (WR-90) is taken to calculate the value of
propagation constant.[WR-90 waveguide :- frequency range:- 8.2GHz to 12.4GHz]
The dimensions of the waveguide is 2.286cm*1.016cm.
First we will dicretize the domain (waveguide) using “pde-toolbox” , this process is known as
meshing,.Then we create three matrices....
1. For triangle node numbers:- A file element which has three node numbers of each triangle,
with rows arranged to correspond to triangle number in ascending order.
2. Coordinates of nodes:- A file coord which has two columns containing x coordinate in the
first column and y coordinate in the second, with rows arranged to correspond to node
numbers in ascending order.
3. Boundary node numbers:- A file bn with one column containing the boundary node numbers
in ascending order.
function [pe,qe] = triangle1(x,y)
ae=x(2)*y(3)-x(3)*y(2)+x(1)*y(2)-x(2)*y(1)-
x(1)*y(3)+x(3)*y(1);
ae=abs(ae)/2;
b=[y(2)-y(3),y(3)-y(1),y(1)-y(2)];
c=[x(3)-x(2),x(1)-x(3),x(2)-x(1)];
b=b/(2*ae);
c=c/(2*ae);
pe=(b.'*b+c.'*c)*ae;
qe=[2,1,1;1,2,1;1,1,2];
qe=qe*(ae/12);
end
MATLAB Code To Calculate Propagation ConstantFunction
clcclear allformat long g% load the data file element=xlsread('element.xlsx');coord=xlsread('coord.xlsx');bn=xlsread('bn.xlsx');% find the total no. of elements and nodesne=length(element(:,1));nn=length(coord(:,1));% set up pull global matricespg=zeros(nn,nn);qg=zeros(nn,nn);% sum over trianglesfor e=1:ne; % Get the three node no. of triangle no. e node=[element(e,:)]; % Get the coordinates of each node and form row vectors x=[coord(node(1),1),coord(node(2),1),coord(node(3),1)]; y=[coord(node(1),2),coord(node(2),2),coord(node(3),2)]; % Calculate the local matrix for triangle no. e [pe,qe]=triangle1(x,y); % Add each element of the local matrix to the appropriate lement of the % global matrix for k=1:3; for m=1:3; pg(node(k),node(m))=pg(node(k),node(m))+pe(k,m); qg(node(k),node(m))=qg(node(k),node(m))+qe(k,m); end endendksquare=eig(pg,qg)
Script
Result and comparison b/w FEM and Analytical Results
No mode 9.5748 (9.56119)
38.4625 (38.24479)
1.8891(1.8886)
11.4693 (11.4498)
40.3694 (40.1334)
7.5638(7.5545)
17.159 (17.1157)
46.11235 (45.7993)
m
n0
0
1
1
2
2
Overall Summary Learned fundamentals of PDEs useful for scientists and engineers
(e.g., elliptic, parabolic & hyperbolic, scalar wave eqn).
Studied waveguide and learned its different fundamental mode i.e.
TE.
Calculated values of propagation constant for different modes.
Future Plan In future, we will calculate the propagation constant for TM mode.
we will solve the problems on waveguide using ANSYS.
We will move towards optical wave guide and plasmonic waveguide
and study the different properties with the help of ANSYS .
1. Erik G. Thompson, “An Introduction To The Finite Element Method”, John Willey &
Sons, ISBN: 978-81-265-2455-6 (2005)
2. Radhey S. Gupta, “Elements of Numerical Analysis”, Macmillan India Ltd.,ISBN: 446-
521 (2009),.3. David J. Griffiths, “Introduction to Electrodynamics” Pearson Education, Inc., ISBN-
978-81-203-1601-0 (1999),.
References
Thank
You !!!