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1/20220 IEEE Antennas and Propagation Magazine,Vol. 56, No. 2, April 201

Testing Ourselves

Levent Sevgi

Dou University

Electronics and Communications Eng. Dept.

Zeamet Sokak, No 21, Acibadem Kadiky

Istanbul - Turkey

E-mail: [email protected], [email protected]

http://www3.dogus.edu.tr/lsevgi

We have introduced many computer codes and virtualtools for electromagnetic modeling and simulation inthe Magazine for nearly a decade. Im very glad to see that

those tools are used in several universities and institutions,

even in national research centers, from USA to Japan, Europe

to Australia and Africa. Most of those tools can be downloaded

from http://modsim.dogus.edu.tr (and may also be requestedfrom the authors), and be used in teaching/training in virtual

undergraduate labs as well in graduate-level research. Now,

were happy to announce they can also be downloaded from my

new Web site (http://leventsevgi.net). The list of the tutorials

we have introduced since February 2007 may also be found

there.

We have received several requests and questions on some

of our Finite-Difference Time-Domain (FDTD)-based virtual

tools from our readers, who have experiencedMATLAB-based

coding/compiling/version problems. I have assigned Miss

Gizem Toroglu, the youngest research and teaching assistantin our department at Dou University, to re-shape (as well as

redevelop) a collection of MATLAB-based core FDTD codes

in two dimensions (2D), without the need for any toolbox

and/or special command/macro, and to present them in her

interdepartmental seminar this semester. I liked the way she

tailored and presented these codes. Although there were a

number of FDTD codes and packages, I therefore decided to

share them with our readers, through the tutorial we prepared

for this purpose in this issue (by the way, the tutorial on novel

RCS measurement approaches by B. Fisher is on the way).

Those codes mentioned in this issues tutorial are already there,

under EM Virtual Tools, at leventsevgi.net. I hope the readers

will enjoy having them and nd them useful.

We have already discussed Statistical Decision Making

[1] and Biostatistics with hypothetical tests on cell-phon

users using statistical decision making [2]. What abou

Strategic Decision Making? The study of strategic decisio

making is called Game Theory. We, engineers, have mostlbeen familiar with game theory after a wonderful movie, A

Beautiful Mind, a 2001 American biographical drama lm

based on the life of John Nash, a Nobel Laureate in Econom

ics. John Nash, who introduced the Nash equilibrium concept

was played by Russell Crowe. Im glad to announce that I hav

nally convinced Prof. Benan Zeki Orbay, former Dean of th

Faculty of Economics and Administrative Sciences, and curren

Chair of the Department of Economics and Finance at Dou

University, to prepare a tutorial on game theory. She is goin

to give a presentation entitled Game Theory and Engineerin

Applications in one of our inter-departmental seminars i

April 2014. Hopefully, well extend it to an interesting tutoria

References

1. L. Sevgi, Hypothesis Testing and Decision Making: Con

stant-False-Alarm Rate, IEEE Antennas and Propagatio

Magazine, 51, 3, June 2009, pp. 218-224.

2. L. Sevgi, Biostatistics and Epidemiology: Hypothetica

Tests on Cell Phone Users, IEEE Antennas and Propagatio

Magazine, 52, 1, February 2010, pp. 267-273.

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2/20IEEE Antennas and Propagation Magazine,Vol. 56, No. 2, April 2014 22

Finite-Difference Time-Domain (FDTD)

MATLABCodes for First- and Second-Order

EM Differential Equations

Gizem Torolu, Levent Sevgi

Electronics and Communications Engineering Department

Dou University

Zeamet Sokak 21, Acbadem Kadky, 34722 Istanbul Turkey

E-mail: [email protected]

Abstract

A set of two-dimensional (2D) electromagnetic (EM)MATLABcodes, using both first-order coupled differential (Maxwel

equations and second-order decoupled (wave) equations, are developed for both transverse-magnetic (TM) an

transverse-electric (TE) polarizations. Second-order MUR type absorbing boundary conditions are used to simulate

free space. Metamaterial (MTM) modeling is also included. Performance tests in terms of computational times, memor

requirements, and accuracies were done for simple EM scenarios with magnetic field, current, and voltage comparisons

The codes may be used for teaching and research purposes.

Keywords: Maxwell equations; finite-difference time-domain; FDTD; wave equation; absorbing boundary conditions;

MUR conditions; transverse electric; TE; transverse magnetic; TM; metamaterials; MTM; MATLAB

1. Introduction

The Finite-Difference Time-Domain (FDTD) method is oneof the most powerful numerical approaches widely used insolving a broad range of electromagnetic (EM) problems since

its rst introduction [1] (a quick Internet search will list tens of

thousands of FDTD studies). A few of the many useful books

written on the FDTD are [2-8]. Information related to the FDTD

may also be found in Wikipedia [9]. The books on the parallel

FDTD [10] and FDTD-based metamaterial (MTM) modeling

[11] are also worth mentioning. We have also presented manyuseful tutorials, and have shared our codes and virtual tools for

a long time [12-19]. Table 1 lists these free FDTD-based virtual

tools, with short explanations. These and many more can be

found in the IEEE Press/John Wiley book recently published

within the Press series on EM Wave Theory [20].

The MATLAB-based codes and virtual tools in [12] use

the one-dimensional FDTD for the plane-wave propagation

modeling and simulation through inhomogeneous media, and

in [13] for voltage/current wave transmission and reection

along a transmission line (TL) under different termination and

impedance-mismatch conditions. The TDRMeter virtual tool

in [13] can be used for the visualization of both transmission

reections and fault identication.

A general-purpose two-dimensional FDTD virtual too

MGL-2D [14], and its modied version, MTM-FDTD [15

can be used in the modeling and simulation of EM waves i

two dimensions. A variety of electromagnetic problems, from

indoor/outdoor radiowave urban/rural propagation to electro

magnetic compatibility (EMC), from resonators to closed/ope

periodic structures, linear and planar arrays of radiators can b

simulated easily withMGL-2D. The beauty ofMGL-2Dcomefrom its visualization power, as well as its easy-to-use desig

steps. Similarly, MTM-FDTDmay be used for the visualiza

tion of EM waves interacting with different metamaterials

Snapshots during these interactions may be taken. Scenario

with normal and oblique incidences, demonstrating focusin

beams in planar metamaterials and the existence of a negativ

refractive angle, respectively, may be observed in the tim

domain. In addition, video clips of wave- metamaterial inter

actions may easily be recorded.

TheMATLAB-based virtual tool WedgeFDTDwas deve

oped to investigate EM scattering on the canonical non-pene

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trable wedge problem with the FDTD method [16]. Diffracted

elds may easily be extracted and compared with the results

of high-frequency asymptotic (HFA) models. Some interesting

applications of the two-dimensional FDTD method were also

discussed in one of our tutorials [17]. There, FDTD-based path

planning and segmentation were modeled and implemented.

Finally, full-wave, 3D-FDTD EM virtual tools have been

prepared and reviewed in tutorials [18] and [19] for realistic

problem modeling and simulations. In [18], MSTRIP was

introduced for the investigation of a variety of microstrip cir-

cuits.MSTRIPis a 3D-FDTD EM simulator that uses the pow-

erful perfectly matched layer terminations (PML) [21]. The

user needs only to render the microstrip circuit via a computer

mouse on a rectangular grid, and to specify basic dimensions

and supply operational requirements, such as the frequency

band and simulation length. The rest is handled byMSTRIP. It

is easy-to-use, strengthened with visualization and video-clip

capabilities, and can handle very complex single- and double-

layer microstrip structures. Time-domain visualization is pos-

sible during the simulations and video clips may be recorded.

The S parameters are automatically calculated, and may be

displayed online.

In [19], a three-dimensional FDTD-based RCS prediction

virtual analysis tool (MGL-RCS) was introduced. It can be used

to design any kind of a PEC target using basic blocks, such as

a rectangular prism, cone, cylinder, sphere, etc. A collection of

pre-designed surface and air targets stored in 3DSformat les,

are also supplied. Time-domain near scattered elds can be

simulated around the object under investigation, and transients

can be recorded as video clips. Far elds are then extrapolated,

and RCS as a function of frequency and RCS as a function of

angle plots can be produced (FORTRANsource codes of this

package may also be found in [4]).

2. The Two-Dimensional FDTD Models

The FDTD method [1] discretizes Maxwell equations b

replacing derivatives with their nite-difference approxima

tions, directly in the time domain. It is simple, easy to code

but has the open-form (iterative) solution. It is therefore con

ditionally stable: one needs to satisfy a stability condition. Th

FDTD volume is nite, and therefore may model only close

regions. Free-space simulation is an important task in FDTD

and various effective boundary terminations have been developed for the last two decades (see [22] for the second-orde

MUR-type terminations used here). Broadband (pulse) excita

tion is possible in the FDTD, but inherits the numerical-disper

sion problem. Finally, only near elds can be simulated aroun

the object under investigation; far elds can be extrapolate

using the Equivalence Principle (e.g., the Stratton-Chu equa

tions) [4].

2.1 First-Order Coupled Equations

The assumption of a continuous translational symmetralongzlets us reduce the three-dimensional problem into tw

dimensions on the xy plane. Maxwell equations in such a

environment are characterized with three parameters (the per

mittivity, ,permeability, ,and conductivity, ):

H

Et

=

, (1)

E

H Et

= +

. (2)

Table 1. Free FDTD-based EM Virtual Tools presented in the IEEE Antennas and PropagationMagazine.

Virtual Tool Explanation

1DFDTDAMATLAB-based 1D FDTD simulation of plane wave propagation in time domain through single, double

or three-layer media. EM parameters are supplied by the user [12].

TDRMeterA virtual time-domain reectometer virtual tool. It is used to locate and identify faults in all types of

metallic paired cable. Fourier and Laplace analyzes are also possible [13].

MGL2DA general purpose 2D FDTD package for both TE and TM type problems. Any 2D scenario may be created

by the user by just using the mouse [14].

MTM-FDTDModied version ofMGL-2Dto simulate cylindrical wave propagation through MeTaMaterials (MTM)

[15].

WedgeFDTD*A 2DMATLAB-based simulator for the modeling of EM diffraction from a semi-innite non-penetrable

wedge using high frequency asymptotics and FDTD [16] (*published in ACES).

MSTRIP

A 3D FDTD-based EM simulator for the broadband investigation of microstrip circuits. The user only needs

to picture the microstrip circuit via computer mouse on a rectangular grid, to specify basic dimensions and

operational needs such as the frequency band, simulation length [18].

MGL-RCS

A 3D FDTD-based EM simulator for RCS prediction. The user only needs to locate a 3D image le of

the target in 3DS graphics format, specify dimensions and supply other user parameters. The simulator

predicts RCS vs. angle and/or RCS vs. frequency [19].

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These reduce to two sets of scalar equations (i.e., TMz and

TEz) in two dimensions under the assumption 0z , and

can be given as [23]

SET #1: TMz( 0zH )

x zH E

t y

=

, (3a)

y z

H E

t x

=

, (3b)

y xz

z

H HEE

t x y

=

, (3c)

SET #2: TEz( 0zE )

x z xE H

Et y

=

, (4a)

y z

y

E HE

t x

=

, (4b)

y xz

E EH

t x y

=

. (4c)

As observed, knowing the zE ( zH ) component is enough to

derive all the other eld components for the TMz( TEz) prob-

lem. The discretized FDTD iteration equations then reduce to

SET #1: TMz

( 0z

H )

( ) ( ) ( ) ( )1, , , , 1n n n nx x z zt

H i j H i j E i j E i jy

=

,

(5a)

( ) ( ) ( ) ( )1, , , 1,n n n ny y z zt

H i j H i j E i j E i jx

= +

,

(5b)

( ) ( )12

, ,2

n nz z

tE i j E i j

t

+ = +

( ) ( ), 1,2

2

n ny yH i j H i jt

t x

+

+

( ) ( ), , 122

n nx xH i j H i jt

t y

+ ,

(5c)

SET #2: TEz( 0zE )

( ) ( )12

, ,2

n nx x

tE i j E i j

t

=+

( )

( ) ( )2

, , 12

n nz z

tH i j H i j

t y

+

,

(6a)

( ) ( )12

, ,

2

n nz z

tE i j E i j

t

=+

( )

( ) ( )2

, 1,2

n nz z

tH i j H i j

t x

+ +

(6b)

( ) ( )1, ,n nz zH i j H i j=

( ) ( )

0

, 1,n n

y yE i j E i jt

x

+

( ) ( )

0

, , 1n n

x xE i j E i jt

y

.

(6c)

2.2 Second-Order Decoupled Equations

Two of the three eld components in Equations (3) and (4

can be eliminated, and a second-order differential (wave

equation with a single eld component can be obtained. Fo

example, the following wave equation for the TMzproblem

can be directly obtained from Equation (3c) using Equa

tions (3a) and (3b):

2 2 2

2 2 2

10zE

tx y t

+ =

. (7)

This equation, dened for

0; 0 , 0max maxt x X y Y , (8)

together with the boundary conditions

( ) ( )10, , ,zE y t g y t= for 0, 0 maxx y Y= , (9a)

( ) ( )2,0, ,zE x t g x t= for 0, 0 maxy x X= , (9c)

( ) ( )3, , ,z maxE X y t g y t= for , 0max maxx X y Y= (9b)

( ) ( )4, , ,z maxE x Y t g x t= for , 0max maxy Y x X= ,

(9d)

and, the initial conditions

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( ) ( )1, ,0 ,zE x y f x y= , (10a)

( )

( )2, , 0

,zE x y

f x yt

=

, (10b)

are enough to solve for zE and the other eld components.

Equation (7) can therefore also be used in the FDTD modeling

and simulations. The FDTD discretized form of Equation (7) is

( ) ( )

( ) ( )1 14 1

, , ,n n nz z zp q t

E i j E i j E i jg g

+ =

( ) ( )2

1, 1,n n

z z

pE i j E i j

g + + +

( ) ( )2

, 1 , 1n nz zq

E i j E i jg

+ + +

(11)

where

2v t

p

x

,

(12a)

2v t

qy

2

2g v t+ ,

(12b)

2

2t v t +

1

v

= . (12c)

Note that the dispersion and stability conditions, as well as the

source injection in time, are handled just like the rst-order

coupled FDTD equations. On the other hand, the values at the

rst two time instants of zE (i.e., ( )0 ,zE i j and ( )

1 ,zE i j ) must

be supplied for the spatial source injection.

2.3 Basic Features of the FDTD Equations

The observations listed below are important for thenumerical implementation of the rst-order coupled (FOC)

FDTD model:

There are three eld components ( xH , yH , zE for

TMzand xE , yE , and zH for TEz) in each cell,

and they are distinguished by the ( ),i j label for the

rst-order coupled FDTD model.

The discretization steps are ,x y , and t , and the

physical quantities are calculated from x i x= ,y j y= , and t n t= .

Since the FDTD equations are iterative (i.e., open-

form solutions), they are conditionally stable. The

Courant stability condition, which states that the

time step cannot be arbitrarily specied once the

spatial discretization is done, must be satised.

Although the same notations, ( ),nxE i j and

( ),nzH i j , are used, their locations are different inthe classical Yee cell [1] (see Figure 1), and there is

a half-time-step difference between the E and H

eld computation times. That is, the magnetic-eld

components are calculated at time steps 2t t= ,

3 2t , 5 2t , ..., but the electric elds are calcu-

lated at time steps , 2 ,3 ,...t t t t = .

Figure 1. The Yee cells for the (a) TMzand (b) TEzprob

lems.

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Only neighboring magnetic-eld values and

( )1 ,nzE i j are required to update ( ),nzE i j . Simi-

larly, neighboring electric-eld values and

( )1 ,nxH i j are required to update ( ),nxH i j .

Both magnetic- and electric-eld components in any

cell may be moved to the origin by just cell

averaging. This is accomplished via

( ) ( ) ( ), 0.5 , 1,

x x xH i j H i j H i j= + +

for mag-

netic elds, but four electric-eld components are

required for this purpose:

( ) ( ) ( ), 0.25 , 1,z z zE i j E i j E i j= + + ( ) ( ), 1 1, 1z zE i j E i j+ + + + + .

Any object may be modeled by giving , , and

. Two of these, and , appear in the electric-

eld components, and the third, , appears in the

magnetic-eld components.

Three different and values may be assigned for

three electric-eld components, so that different

objects may be located within the Yee cell. Simi-

larly, different values may be given for H-eld

componentsfor the same purpose.

The important aspects of the second-order decoupled

(SOD) FDTD model are as follows:

There is only one eld component, and its location

may be anywhere in the unit cell.

The models and discrete equations are identical for

the TMzand TEzproblems.

The past two values are needed in every cell.

FDTD iterations yield only zE ( TMz) or zH

( TEz). One therefore needs to write down another

discrete (Maxwell) equation for the other two com-

ponents, i.e., xH , yH ( TMz) or xE , yE ( TEz).

2.4 Absorbing Boundary Conditions

To make it simple in this tutorial, the second-order MUR

terminations [22] are used. Table 2 lists equations that must

be satised along the boundaries (see Figure 2). The discrete

iteration equations will then be

At 0x= ( xN N= )

( ) ( )1 11, 2,n nz zE j E j+ =

( ) ( )1 12, 1,n nz zc t x

E j E jc t x

+ + + +

( ) ( )2

2, 1,n n

z z

xE j E j

c t x

+ + +

( )

( ) ( ) ( ) ( )

2

2 2, 1 2 2, 2, 12

n n nz z z

c t x

E j E j E jy c t x

+ + + +

( )

( )( ) ( ) ( )

2

21, 1 2 1, 1, 1

2

n n nz z z

c t xE j E j E j

y c t x

+ + + +

(13)

At maxx X= ( xN N= )

( ) ( )1 1, 1,n nz zE N j E N j+ =

( ) ( )1 11, ,n nz zc t x

E N j E N j

c t x

+ + + +

( ) ( )2

1, ,n nz zx

E N j E N jc t x

+ + +

( )

( )( )

2

21, 1

2

nz

c t xE N j

y c t x

+ + +

( ) ( )2 1, 1, 1n nz zE N j E N j +

( )

( )( ) ( ) (

2

2, 1 2 , , 1

2

n n nz z z

c t xE N j E N j E N j

y c t x

+ + + +

(14)

Table 2. Differential equations for the second-order MUR

terminations.

( )2 2 2

2 2

0 10, , 0

0 2z

max

x cE y t

y Y x t c t y

= + =

( )2 2 2

2 2

1, , 0

0 2

maxz max

max

x X cE X y t

y Y x t c t y

= + =

( )2 2 2

2 2

0 1,0, 0

0 2z

max

y cE x t

x X y t c t x

= + =

( )2 2 2

2 2

1, , 0

0 2

maxz max

max

y Y cE x Y t

x X y t c t x

= + =

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At 0y= ( yN N= )

( ) ( )1 1,1 , 2n nz zE i E i+ =

( ) ( )1 1, 2 ,1n nz zc t y

E i E ic t y

+ + + +

( ) ( )2

, 2 ,1n n

z z

yE i E i

c t y

+ + +

( )

( )( ) ( ) ( )

2

2 1, 2 2 , 2 1, 2

2

n n nz z z

c t yE i E i E i

x c t y

+ + + +

( )

( ) ( ) ( ) ( )

2

2 1,1 2 ,1 1,12

n n nz z z

c t y

E i E i E ix c t y

+ + + + (15)

At maxy Y= ( yN N= )

( ) ( )1 1, , 1n nz zE i N E i N+ =

( ) ( )1 1, 1 ,n nz zc t y

E i N E i Nc t y

+ + + +

( ) ( )2

, 1 ,n nz zy

E i N E i Nc t y

+ + +

( )

( )( )

2

2 1, 1

2

nz

c t yE i N

x c t y

+ + +

( ) ( )2 , 1 1, 1n nz zE i N E i N +

( )

( )( ) ( ) ( )

2

21, 2 , 1,

2

n n nz z z

c t yE i N E i N E i N

x c t y

+ + + +

(16)

2.5 Parameter Selection in

FDTD Simulations

FDTD modeling and simulations are usually preferre

because of the ability for handling complex EM environment

and broadband behavior. Running simulations require

parameter optimization. The spatial mesh sizes, x and y

the time step, t , the total simulation period ( maxT n t= ), th

source bandwidth,B, and the pulse duration are characteristi

parameters that should be optimally selected prior to thsimulation [4].

FDTD simulations are generally performed in obtainin

the frequency characteristics of a given EM structure, fo

example, the radiation characteristics or input impedance of a

antenna structure, the RCS behavior of a chosen target, th

transmission and/or reection characteristics of a microstri

network, the propagation characteristics of a waveguide, th

resonance frequencies of a closed enclosure, the shieldin

effectiveness of an aperture, etc. One therefore needs to star

with thefrequency requirements(the minimum/maximum fre

quency of interest, minf / maxf , and the frequency resolutionf ). The time-domain discrete simulation parameters ( x

y , t , maxT n t= , source bandwidth B, etc.) are the

accordingly specied.

Suppose the problem was to nd the frequency characteris

tics of reections from a free-space/dielectric interface, from

dc to 1 GHz with 50 MHz frequency steps. Starting from th

frequency-analysis requirements and sampling criteria, th

parameter-optimization steps can be listed as follows:

Choose the source waveform with a duration that

contains the maximum frequency of interest.

According to the properties of the fast Fourier trans-

form (FFT), the maximum frequency determines the

minimum time step, i.e., ( )1 2FFT maxt f = . This

is the hard limit for the frequency analysis. A 1 GHz

maximum frequency corresponds to a 0.5 ns FFTt .

There are two important points in choosing the

maximum simulation (observation) time. First, the

frequency sensitivity, f , which determines the

observation time should be 1 maxf T = . Second,

the simulation should continue until all the tran-

sients are over. Therefore, maxT is chosen to satisfy

both requirements. Since f was given as 10 MHz,

maxT was determined to be 100 ns. The number of

time steps, n, will then be 200. If all transients decay

after 200 time steps, then this will be enough for the

simulation time. If the structure under investigation

is some kind of resonant structure, which

Figure 2. The boundary cells used in MUR terminations.

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corresponds to ringing effects in the time domain,

then a much longer observation period will be

required.

Two important issues in the time-domain simula-

tions are the Courant stability criteria and numerical

dispersion.

The spatial mesh sizes, x and y , are chosen

according to numerical dispersion requirements.

This is nothing but satisfying the Nyquist sampling

criteria in the spatial domain. The minimum wave-

length, min , must be sampled with at least two

samples, i.e., { }max , 2minx y . In practice, at

least 10min is required for acceptable results.

Depending on the problem at hand, as much as

100min to 120min may be required in order to

get rid of numerical-dispersion effects. Since min

was 30 cm, 1x y = = cm may be chosen if

30min is good enough for eliminating numerical-

dispersion effects.

The time step, FDTDt , may be directly chosen

from the Courant stability criteria. Since x y =

and this is equal to 1 cm, t may be chosen to be

( )2x c , where cis the speed of light. This gives24t ps. In general, FDTDt is much less than

FFTt , andtherefore FDTDt is taken into account.

With this time step, the simulation time was

5000n= .

3. Tests and Comparisons

SimpleMATLABcodes were developed for the rst-orde

coupled FDTD (FOC-FDTD) and the second-order decouple

FDTD (SOD-FDTD) models, for both the TE and TM prob

lems. Table 3 lists these codes and their explanations (vis

http://leventsevgi.net for these codes).

Tests with rst-order coupled FDTD and second-orde

decoupled FDTD were done in terms of memory requirementand computational times. Table 4 shows some numerical result

for these comparisons. As observed, the computational time

were of the same order, but the second-order decoupled FDTD

was slightly faster. Note that the classical loop philosophy use

inMATLABcoding drastically slowed down the computation

This means that the use of For/End loops had a signican

impact on the computation time (two loops almost doubled th

computation time of one loop). The rst-order coupled FDTD

had three loops, one inside the other, whereas one loop was use

for the second-order decoupled FDTD. The rst-order couple

FDTD lasted roughly three times longer than the second-orde

decoupled FDTD with the classical coding approach. Thei

computational times were almost the same when the For

End loops were removed. (For example, observe in the tabl

that simulations in a 400 400 FDTD area lasted 8.85 s an

8.56 s with the rst-order coupled FDTD and second-orde

decoupled FDTD models, respectively. On the other hand

these values were 523 s and 202 s, respectively, if the classica

For/End loops were used in the MATLABcodes). Howeve

the memory allocation of the second-order decoupled FDTD

was considerably higher than for the rst-order coupled FDTD

because of the requirements of the two past time values of th

elds at every cell.

Table 3. The first-order coupled FDTD and SOD_FDTD MATLABcodes.

FrstOrder_TM_FDTD_MUR.m2D-FDTDMATLABcodes for TM problem under MUR terminations ( xH , yH , zE )

FrstOrder_TM_FDTD_MUR_INH.m 2D-FDTD MATLABcodes for TM problem under MUR terminations ( xH , yH , zE )

having a rectangular lossy layer

FrstOrder_TM_FDTD_MUR_MTM.m 2D-FDTD MATLABcodes for TM problem under MUR terminations ( xH , yH , zE )

having a rectangular MTM layer

FrstOrder_TE_FDTD_PEC.m2D-FDTDMATLABcodes for TE problem under PEC terminations ( xE , yE , zH )

FrstOrder_TE_FDTD_MUR_MTM.m 2D-FDTD MATLAB codes for TE problem under MUR terminations ( xE , yE , zH )

having a rectangular MTM layer

ScndOrder_TM_FDTD_MUR.m2D Second order FDTDMATLABcodes for TM problem under MUR terminations

( zE )

ScndOrder_TM_FDTD_MUR_INH.m2D Second order FDTDMATLABcodes for TM problem under MUR terminations

( zE ) having a rectangular lossy layer

ScndOrder_TE_FDTD_MUR.m2D Second order FDTDMATLABcodes for TE problem under MUR terminations

( zH )

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3.1 Current Computations

Alternating currents produce surrounding magnetic elds

and can be calculated using Amperes Law. This means cur-

rents can be extracted from known magnetic elds. Assume an

innite thin wire is located in the simulation area (see Figure 3).The current passing through this wire can be calculated using

Amperes Law:

I Hdl Hdl= =

( ) ( )2 2

1 1

1 2, ,x y

x yi x j y

H i y dx H x j dx= =

= +

( ) ( )2 2

1 1

2 1, ,

x y

x yi x j y

H i y dx H x j dx= =

(17)

Short scripts were added to the rst-order coupled and

second-order decoupled FDTD codes for the computation of

a current owing on a thin wire. The sample scenario prepared

for this purpose is pictured in Figure 4. At 300 MHz, a 25 m

25 m simulation area was assumed. A PEC rectangular object

(5 m 2.5 m) was placed at Node (200,225). A hard pulse line

source was injected from Node (350,200). An innitely long,

thin wire was located in the simulation area. The number of

time steps was 500.

Figure 5 shows a current as a function of time comparison

of the rst-order coupled FDTD and second-order decoupledFDTD models. The computational times are also given on the

graph. The wire was enclosed by a rectangle sized four cells

along the xdirection and ve cells along the y direction. The

magnetic elds were calculated along this arbitrary loop. The

yH components were considered for the right and left edges of

the rectangle, whereas the xH components were used for the

top and bottom edges. The sum of the magnetic elds was

multiplied by the cell size. Finally, the owing current was

obtained.

Table 4. The Time and memory requirements for the first-

order coupled FDTD and second-order decoupled FDTD

models ( 500n= , FOC: xH , yH , zE ; SOD: zE ).

Simulation

Area

Time (s) Memory (MB)

FOC SOD FOC SOD

300300 7.19 5.27 2.11 345

400400 8.85 8.56 3.71 920

600600 17.3 13.4 8.3 1384

10001000 47.48 37.48 22.9 3845

Figure 3. An application of Amperes Law.

Figure 4. A sample scenario for current simulations.

3.2 Voltage Computations

Potential differences cause electric elds. This means tha

the voltage induced by EM elds between any terminals ma

also be computed during the FDTD simulations. Faradays Law

may be used for this purpose. The necessary equation for th

voltage computations in the scenario shown in Figure 6 is

( )

2

1

,y

z aj y

V Edl E x j dy=

= =

. (18)

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Here, two parallel PEC plates were inserted horizontally, andthe voltage between them was simulated (see the scenario

in Figure 7a). Two thin parallel plates, of a size of 1 100

segments, were inserted into the simulation area. They were

separated from each other by 50 cells. At a selected node (the

150th cell), the voltage value was computed. A hard pulse

source was then applied at node (250,250). Figure 7b shows

an instant (a screen capture) during the FDTD simulations. The

wave components marked 1, 2, 3, and 4 corresponded to the

cylindrical incident eld, reections from the top plate, the top-

edge diffracted elds, and the bottom-edge diffracted elds,

respectively. All theEelds from the 150th to the 200th cells

Figure 5. The current as a function of time obtained with

both models.

Figure 6. An application of Faradays Law.

along theydirection were added, and the sum was multiplied

by the cell size. A comparison of the voltage as a function o

time is given in Figure 8.

Note that the codes listed in the Appendix were used fo

the scenario given in Figure 7, and produced the results in Fig

ure 8. The codes in Appendices 6.1 and 6.2 generated the sam

Figure 8 using the rst-order coupled FDTD and second-orde

decoupled FDTD models, respectively. The code in Appendix

6.3 could be used after the other two. It loaded the recorde

FDTD data, applied the FFT, and compared the two models in

both the time and frequency domains. (All these codes and th

others may also be downloaded from leventsevgi.net.)

4. Metamaterial (MTM) Modeling

The EM response of a material is determined to a larg

extent by its electrical properties. A material/medium with both

permittivity and permeability greater than zero ( 0> , 0>

is called double positive (DPS). A medium with permittivit

less than zero and permeability greater than zero ( 0 ) is designated as -negative (ENG). Materials with

permittivity and permeability both less than zero ( 0

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