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 !!!