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SÜLEMAN DEMİREL UNIVERSITY

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Mühendislik fakültesi

ELEKTRONİK VE HABERLEŞME

MÜHENDİSLİĞİ

Biological Effect of Electromagnetic Waves

COURSE OFFERED

Dr. Selçuk Comlekçi

Electromagnetic Analysis using FDTD Method

Submitted by

MSc. Student

Mohammed Mahdi AboAjamm

Student No. 1330145006

Computational Electromagnetics

Maxwell’s equations can be given in differential or integral form.

Maxwell’s equations can be given in time domain or frequency domain.

Co

Commercial software packages

The Finite-Difference Time-Domain Method:

Finite-difference time-domain (FDTD) is a numerical analysis

technique used for modeling computational electrodynamics

(finding approximate solutions to the associated system of

differential equations). Since it is a time-domain method, FDTD

solutions can cover a wide frequency range with a single

simulation run, and treat nonlinear material properties in a

natural way.

The FDTD method belongs in the general class of grid-based

differential numerical modeling methods (finite difference

methods). The time-dependent Maxwell's equations (in partial

differential form) are discretized using central-difference

approximations to the space and time partial derivatives. The

resulting finite-difference equations are solved in either software

or hardware in a leapfrog manner: the electric field vector

components in a volume of space are solved at a given instant in

time; then the magnetic field vector components in the same

spatial volume are solved at the next instant in time; and the

process is repeated over and over again until the desired

transient or steady-state electromagnetic field behavior is fully

evolved.

When Maxwell's differential equations are examined, it can be

seen that the change in the E-field in time (the time derivative)

is dependent on the change in the H-field across space (the curl).

This results in the basic FDTD time-stepping relation that, at

any point in space, the updated value of the E-field in time is

dependent on the stored value of the E-field and the numerical

curl of the local distribution of the H-field in space.

The H-field is time-stepped in a similar manner. At any point in

space, the updated value of the H-field in time is dependent on

the stored value of the H-field and the numerical curl of the

local distribution of the E-field in space. Iterating the E-field

and H-field updates results in a marching-in-time process

wherein sampled-data analogs of the continuous electromagnetic

waves under consideration propagate in a numerical grid stored

in the computer memory.

This description holds true for 1-D, 2-D, and 3-D FDTD

techniques. When multiple dimensions are considered,

calculating the numerical curl can become complicated. Kane

Yee's seminal 1966 paper proposed spatially staggering the

vector components of the E-field and H-field about rectangular

unit cells of a Cartesian computational grid so that each E-field

vector component is located midway between a pair of H-field

vector components, and conversely. This scheme, now known

as a Yee lattice, has proven to be very robust, and remains at the

core of many current FDTD software constructs.

To implement an FDTD solution of Maxwell's equations, a

computational domain must first be established. The

computational domain is simply the physical region over which

the simulation will be performed. The E and H fields are

determined at every point in space within that computational

domain. The material of each cell within the computational

domain must be specified. Typically, the material is either free-

space (air), metal, or dielectric. Any material can be used as

long as the permeability, permittivity, and conductivity are

specified.

Maxwell’s Equations

The basic set of equations describing the electromagnetic world:

FDTD Overview – Cells

A three-dimensional problem space is composed of cells.

FDTD Overview- The Yee Cell

The FDTD (Finite Difference Time Domain) algorithm was first

established by Yee as a three dimensional solution of Maxwell's

curl equations.

Three scalar equations can be obtained from one vector curl

equation.

Leap-frog Algorithm.

SAR Analysis Using FDTD

Recent progress in computer technology enables us to use

FDTD method to numerically calculate the EM interactions of a

inhomogeneous, realistic human head and mobile phone models.

In FDTD, spatial and time derivatives in Maxwell's equations

are replaced with their central difference approximations in

specially organized unit cells. In the unit cell, made up by xy z ,

six components of electromagnetic fields are arranged to

minimize the computational storage needs. The entire

computation domain is obtained by stacking these unit cells into

a larger rectangular volume. FDTD is very easy to implement

and to trace the wave phenomen a and can handle complex

geometries with inhomogeneous materials, either conductors or

lossy dielectrics. FDTD has been applied to a large amount of

EM problems, including planar microstrip analysis, scattering

and inverse scattering problems, antenna simulation together

with near-to-far field transformations, etc. Its comparison with

other powerful time-domain techniques has also been presented.

One-dimensional FDTD with MATLAB Example

%Scott Hudson, WSU Tri-Cities %1D electromagnetic finite-difference time-domain (FDTD) program. %Assumes Ey and Hz field components propagating in the x direction. %Fields, permittivity, permeability, and conductivity %are functions of x. Try changing the value of "profile". %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

close all; clear all;

L = 4; %domain length in meters N = 200; %# spatial samples in domain Niter = 400; %# of iterations to perform fs = 300e6; %source frequency in Hz ds = L/N; %spatial step in meters dt = ds/300e6; %"magic time step" eps0 = 8.854e-12; %permittivity of free space mu0 = pi*4e-7; %permeability of free space x = linspace(0,L,N); %x coordinate of spatial samples

%scale factors for E and H ae = ones(N,1)*dt/(ds*eps0); am = ones(N,1)*dt/(ds*mu0); as = ones(N,1); epsr = ones(N,1); mur= ones(N,1); sigma = zeros(N,1);

for i=1:N epsr(i) = 1; mur(i) = 1; w1 = 0.5; w2 = 1.5; if (abs(x(i)-L/2)<0.5) epsr(i)=4; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

ae = ae./epsr; am = am./mur; ae = ae./(1+dt*(sigma./epsr)/(2*eps0)); as = (1-dt*(sigma./epsr)/(2*eps0))./(1+dt*(sigma./epsr)/(2*eps0));

%plot the permittivity, permeability, and conductivity profiles figure(1) subplot(3,1,1); plot(x,epsr); grid on; axis([3*ds L min(epsr)*0.9 max(epsr)*1.1]); title('relative permittivity'); subplot(3,1,2); plot(x,mur); grid on; axis([3*ds L min(mur)*0.9 max(mur)*1.1]); title('relative permeabiliity');

subplot(3,1,3); plot(x,sigma); grid on; axis([3*ds L min(sigma)*0.9-0.001 max(sigma)*1.1+0.001]); title('conductivity');

%initialize fields to zero Hz = zeros(N,1); Ey = zeros(N,1); figure(2); set(gcf,'doublebuffer','on'); %set double buffering on for smoother

graphics plot(Ey); grid on;

for iter=1:Niter

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%% %The next 10 or so lines of code are where we actually integrate

Maxwell's %equations. All the rest of the program is basically bookkeeping

and plotting. %"smooth turn on" sinusoidal source Ey(3) = Ey(3)+2*(1-exp(-((iter-1)/50)^2))*sin(2*pi*fs*dt*iter); Hz(1) = Hz(2); %absorbing boundary conditions for left-propagating

waves for i=2:N-1 %update H field Hz(i) = Hz(i)-am(i)*(Ey(i+1)-Ey(i)); end Ey(N) = Ey(N-1); %absorbing boundary conditions for right

propagating waves for i=2:N-1 %update E field Ey(i) = as(i)*Ey(i)-ae(i)*(Hz(i)-Hz(i-1)); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%% figure(2) hold off plot(x,Ey,'b'); axis([3*ds L -2 2]); grid on; title('E (blue) and 377*H (red)'); hold on plot(x,377*Hz,'r'); xlabel('x (m)'); pause(0); iter end