lumped components a thesis

FDTD MODELING OF RF AND MICROWAVE

CIRCUITS WITH ACTIVE AND

LUMPED COMPONENTS

by

BHARATHA YAJAMAN BE

A THESIS

IN

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

ELECTRICAL ENGINEERING

Approved

Chairp^r^noftfeômmittee

Accepted

Dean of the Graduate School

August 2004

ACKNOWLEDGEMENTS

I would like to thank everybody who has helped me during my graduate school

years and while I was working on my thesis

I would like to express my sincere gratitude to Dr Mohammad Saed my thesis

advisor for his support and guidance during my graduate studies research work and

thesis preparation This thesis would not have been possible without his continuous help

I would also like to express my sincere appreciation to Dr Jon G Bredeson

for serving on my thesis committee I would like to thank Sharath and Vijay for helping

me out in programming on quite a few occasions

Most of all I would like to thank my parents who have always stood by me and

made it possible for me to pmsue graduate studies My brother has also been a constant

source of motivation

I would like to thank my good friends Anu Dipen Nikhil Sharath Rohini Hari

Kanth Kiran and Swapnil who have constantly supported and encouraged me during my

Masters

To all of you I thank you

11

TABLE OF CONTENTS

ACKNOWLEDGEMENTS u

ABSTRACT vi

LIST OF FIGURES vii

CHAPTER

L INTRODUCTION 1

11 Background 1

12 Scope of the Thesis 3

13 Document Organization 4

IL FDTD ALGORITHM 5

21 Introduction 5

22 Maxwells Equation and hiitial Value Problem 5

23 FDTD Formulation 10

24 Numerical Dispersion 20

25 Numerical Stability 21

26 Frequency Domain Analysis 21

27 Source 22

28 Implementation of Basic FDTD Algorithm 23

m BOUNDARY CONDITIONS 24

31 Inttoduction 24

32 Perfect Electric Conductor 25

33 Perfectly Magnetic Conductor 26

34 Perfectly Matched Layer 28

35 Perfectly Matched Layer Formulation 28

111

IV MODELING LUMPED COMPONENTS 33

41 Inttoduction 33

42 Linear Lumped Components 33

43 Incorporating 2 Port Networks with S-Parameters 37

V FDTD SIMULATION PROGRAM 39

51 Introduction 39

52 Building a Three Dimensional (3D) model 39

53 FDTD Simulation Engine - An Overview 40

54 Software flow 42

55 Important Functions in Simulation 43

VL SIMULATION RESULTS 44

61 Inttoduction 44

62 Microstrip Line 44

621 Modeling of the materials 45

622 Source 45

623 Botmdary Condition 46

624 Results 46

63 Line Fed Rectangular Microstrip Anterma 49

631 Modeling of Materials 51

632 Somce 51

633 Botmdary Conditions 51

634 Results 52

64 Microstrip Low-pass Filter 55

641 Somce 56

642 Boimdary Conditions 56

643 Results 57

65 Branch Line Coupler 59

651 Somce 60

IV


653 Results 61

66 Series RLC Circuit 62

661 Somce 63

662 Boundary Conditions 63

663 Results 63

67 Parallel RL Circuit 65

671 Source 66


673 Results 66

VIL CONCLUSION 68

71 Thesis Highlights 68

72 Futtire Work 69

REFERENCES 70

APPENDIX 72

FORMULATION OF PERFECTLY MATCHED LAYER 73

ABSTRACT

The objective of this thesis is to develop a systematic framework for modeling

electromagnetic wave propagation m RF and Microwave circuits with active and lumped

components using S-parameters using the Finite-Difference Time-Domain (FDTD)

approach which was originally proposed by K S Yee in 1966 The mediod

approximates the differentiation operators of the Maxwell equations with finite

difference operators m time and space This mediod is suitable for finding the

approximate electric and magnetic field in a complex three dimensional stmcture in the

time domain The computer program developed in this thesis can be used to simulate

various microwave circuits This thesis provides the theoretical basis for the method

the details of the implementation on a computer and fmally the software itself

VI

LIST OF FIGURES

21 Discretization of the model into cubes and the position of field components on the grid 11

22 Renaming the indexes of E and H field components

cortesponding to Cube(ijk) 16

23 Basic flow for implementation of Yee FDTD scheme 23

31 FDTD Lattice terminated by PML slabs 25

32 PEC on top surface of Cube (i j k) 26

33 PMC Boundary Conditions 27

51 Major software block of FDTD simulation program 41

52 Flow of FDTD Program 42


62 Distribution of Ez(xyz) at 200 time steps 47


64 Distribution of Ez (xyz) at 1000 time steps 48

65 Amplitude of the input pulse 48

66 Amplitude of the Fourier ttansform of the input pulse 48

67 Line-fed rectangular microstrip antenna 50

68 Distiibution of E2(xyz) at 200 time steps 52


610 DisttibutionofEz(xyz) at 600 time steps 53

611 Return loss ofthe rectangular antenna 54

VI1

612 Amplitudes of the Fourier ttansforms of the input

pulse and the reflected waveform 55

613 Low-pass Filter Detail 55

614 Retum loss of Low-pass Filter 57

615 Insertion loss of Low-pass Filter 58

616 Branch Line Coupler Detail 59

617 Scattering Parameters of Branch Lme Coupler 61

618 Series RLC Circuit Detail 62

619 Amplitude ofthe input and transmitted pulse 64

620 Amplitude of the Fourier transform of the input

reflected and ttansmitted pulse 64

621 Parallel LC Circuit Detail 65

622 Amplitude ofthe input and ttansmitted pulse 67

623 Amplitude of the Fourier ttansform of the input reflected and ttansmitted pulse 67

Vlll

CHAPTER I

INTRODUCTION

11 Background

In 1966 Kane Yee (1966) presented what we now refer to as the Finite-Difference

Time-Domain (FDTD) method for modelmg electtomagnetic phenomenon The Yees

algorithm as it is usually called in the literatme is well known for its robustness and

versatility The method approximates the differentiation operators of the Maxwell

equations with finite-difference operators in time and space This method is suitable for

finding the approximate electric and magnetic fields in a complex three-dimensional

stmcture in the time domain Many researchers have contributed immensely to extend the

method to many areas of science and engineering (Taflove 1995 1998) Initially the

FDTD method is used for simulating electtomagnetic waves scattering from object and

radar cross section measurement (Taflove 1975 Kunz and Lee 1978 Umashankar and

Taflove 1982) In recent years there is a proliferation of focus in using the method to

simulate microwave circuits and printed circuit board (Zhang and Mei 1988 Sheen et al

1990 Sui et al 1992 Toland and Houshmand 1993 Piket-May et al 1994 Ciampolini et

al 1996 Kuo et al 1997 Emili et al 2000) hiitially Zhang and Mei (1988) and Sheen et

al (1990) concenttated on modeling planar circuits using FDTD method Later Sui et al

(1992) and Toland and Houshmand (1993) inttoduced the lumped-element FDTD

approach This enabled the incorporation of ideal lumped elements such as resistors

capacitors inductors and PN junctions Piket-May et al (1994) refined the method

fiirther witii the inttoduction of Eber-Molls ttansistor model and die resistive voltage

source All these components comcide with an electiic field component in the model

Ciampolini et al (1996) introduced a more realistic PN junction and ttansistor model

taking into account junction capacitance Furthermore adaptive time-stepping is used to

prevent non-convergence of the nonlinear solution particularly when there is rapid

change in the voltage across the PN junction In 1997 Kuo et al presented a paper

detailing a FDTD model with metal-semiconductor field effect ttansistor (MESFET)

Recentiy Emili et al (2000) inttoduced an extension ofthe lumped-element FDTD which

enabled a PN junction resistor inductor and capacitor to be combined into a single

component Parallel to the development in including lumped circuit elements into FDTD

formulation another significant advance occmed with the development of frequency

dependent FDTD method (Luebbers and Hunsberger 1992) This development allowed

modeling of system with linearly dispersive dielectric media Further developments by

Gandhi et al (1993) and Sullivan (1996) enabled the inclusion of general dispersive

media where in addition to being dispersive the dielectric can also behave nonlinearly

under intense field excitation A detailed survey in this area can also be found in Chapter

9 ofthe book by Taflove (1995)

FDTD method is performed by updating the Electric field (E) and Magnetic field

(H) equations throughout the computational domain in terms of the past fields The

update equations are used in a leap-frog scheme to incrementally march Electric field (E)

and Magnetic field (H) forward in time This thesis uses the formulation initially

proposed by Yee (1966) to find the approximate electric and magnetic fields in a complex

three-dimensional stiiicture in the time domain to model planar circuits and active and

lumped components

12 Scope ofthe Thesis

The object ofthe thesis is to come up with a systematic framework for modeling

electtomagnetic propagation in RF and Microwave circuits with active and lumped

components using S-parameters The FDTD simulation engine is written in C and a

graphing utility (gnuplot) is used to view the electric and magnetic fields The software

mns on Windowsreg platform The simulation engine is developed in a systematic manner

so that it can be extended in fiiture to incorporate more features A convenient and

flexible notation for describing complex three-dimensional models is also inttoduced

This notation describes the model cube-by-cube using lines of compact syntax With this

notation many different types of problems can be rapidly constmcted without having to

modify the declaration in the FDTD simulation engine The description of the model is

stored in a file the FDTD simulation engine simply reads the model file sets up

necessary memory requirements in the computer and the simulation is ready to proceed

Since it is not possible to embody all the progresses and advancements of FDTD

into the thesis the model considered is limited to the foUowings

bull Active and lumped components include resistors capacitors inductors diodes

and ttansistors

bull All conductors are assumed to be perfect electric conductor

bull Dielectric material is isottopic non-dispersive and non-magnetic (|i =io)

However the permittivity 8 of the dielectric is allowed to vary as a function of

location time

13 Document Organization

This chapter presented a background for the Finite Difference Time Domain

(FDTD) method The following is a brief account ofthe rest ofthe contents ofthe thesis

Chapter 2 provides some fimdamental concepts of FDTD and inttoduces the Yees FDTD

formulation and its implementation Chapter 3 inttoduces the boimdary conditions and

formulation of a Perfectly Matched Layer Chapter 4 discusses the method to incorporate

lumped elements and devices into FDTD framework Chapter 5 discusses the simulation

and the results to illustrate the effectiveness of the program Finally Chapter 6

summarizes the work and provides a brief conclusion ofthe thesis

CHAPTER II

FDTD ALGORITHM

21 Inttoduction

This chapter inttoduces the procedure for applying finite-difference time domain

(FDTD) method to time-domain Maxwell equations Specifically the Yees FDTD

formulation will be illusttated hi this chapter the Yee FDTD scheme will be restiicted to

a model with linear non-dispersive and non-magnetic dielectric Various considerations

such as numerical dispersion stability ofthe model will be discussed

22 Maxwells Equation and Initial Value Problem

Consider the general Maxwells equations in time domain including magnetic

curtent density and magnetic charge density

VxE = -M-mdashB (221a) dt

VxH = J + mdashD (221b) dt

V3 = p^ (221c)

V5 = pbdquo (22 Id)

M and pbdquo are equivalent somces since no magnetic monopole has been discovered to

the best of die autiiors knowledge The other parameters for (221a)-(221d) are listed

as follows

E - Electiic field intensity

H - Magnetic field intensity

D - Electric flux density

B - Magnetic flux density

p^ - Electric charge density

J - Electric curtent density

Usually we will just call the 1 electiic field and die ~H magnetic field In

(221a)-(221d) if the media is linear we could inttoduce additional constitutive

relations

D = Efl (222a)

^ = ^fH (222b)

where indicates convolution operation in time [3] We would usually call poundf the

permittivity and ii the permeability of the media both parameters are ftinction of

frequency co For a dispersionless medium gf (co) = e and f (co) = i (222a)-(222b)

reduce to the form

3 = eE (223a)

B = HH (223b)

From the Maxwells equations continuity relations for electric and magnetic current

density can be derived For instance from (221a)

V(VxE) = -VM-V[mdashB] = 0 dt )

=gtmdash(v5)=-vi7 =mdashp dt^ dt

^ bull ^ = - ^ P trade (224a)

Note that the vector identity VV x = 0 is used Similarly from (22 lb)

V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)

Notice that in (221a)-(221d) only the cml equations (221a) for pound and (221b) for H

determine how the fields will change over time This provides a clue that perhaps only

these two equations are all that is needed to formulate a FDTD scheme To obtain a

unique solution for the E and H fields of a system described by (221a)-(221d)

additional information in the form of initial conditions and boundary conditions (if the

domain is bounded) is required First assuming an unbounded domain then (221a)-

(22Id) together with the initial conditions for all the field components and sources

iJMp^p^) constitute the initial value problem (IVP) for Maxwells equations

For simplicity suppose the model is a three-dimensional (3D) model with linear

isottopic non-dispersive dielectiics and there is no magnetic charge and magnetic

curtent (M = 0p^ =0) Also assume that initially all field components and sources are

0 Thus the following equations describe the Initial Value Problem for the model

VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0

for all (jcyzt)eR^ and tgt0 (225e)

From (224b)

0 _^

p^(xyzt) = -jVJ(xyzT)dT + C^ where Ci is a constant Using pXxyzt)^^^ =0

and J(xyzt) = 0 we see that Ci =0 Thus =0

p(xyzt) = -^VJ(xyzT)dT

From (225a)

V(Vxpound) = -V dt J dt^ V

Using the initial condition (225e) V5 r=0

= fNH = 0 this implies that (=0

VB = VB = 0 oO (=0

Therefore the divergence equation for ~B is implicit in (225a) hi other words (225d)

can be obtained from (225a) provided initial conditions (225e) apply Now consider

(225b)

(vx )= VJ + V (d-^ mdashD

dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2

Using the initial conditions (225e) VD

Hence from (226)

= euroVE 1=0 t=0

= 0 this implies that C2 =0

VD = -jVJdT = p^

Again this shows that divergence equation for D is implicit in (225b) provided initial

conditions (225e) applies From this argument we see that the divergence equations

(225c) and (225d) are redundant thus die PP of (225a)-(225e) can be reduced to

mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0

In general this is also tme when the dielecttic is dispersive and nonlinear as long

as die initial conditions for magnetic and electiic flux densities fulfill

VZ) = V5 =0 =0

= 0 This can also be extended to the case when magnetic cmrent

density M and magnetic charge density p^ are present

23 FDTD Formulation

Equations (227a)-(227c) form tfie basis of Yees FDTD scheme [10] There are

a number of finite-difference schemes for Maxwells equations but die Yee scheme

persists as it is very robust and versatile Allowing for conduction electiic curtent

density J = crE (227a)-(227b) can be written as

dE 1 mdash CT-mdash = _ V x E dt s e

= V xE dt jLi

Under Cartesian coordinate system these can be further expanded as

dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt

1 (dH dH^ ^ aE dy dx

(23If) y

Let us introduce the notation

^(ijk) = EîAxjAykAznAt)) and so on for Ey Ez and Hx Hy H components In

Yees scheme the model is first divided into many small cubes The edges of each cube

will form the three-dimensional space grid The Yees scheme can be generalized to

variable cube size and non-orthogonal grid The position ofthe E and field components

in the space grid is shown in Figure 21

(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i

Figure 21 Discretization of die model into cubes and the position of field components on the grid

11

The cube in Figme 21 is called the Yee Cell From Figure 21 it is observed that

each E field component is smrounded by four H field components similarly each H field

component is surtounded by fom E field components (If die field components on

adjacent cubes are taken into account) For example the component H 2 2

IS

surtounded hy E ^ E E and pound z(iJM-) z(ij-HM-) y(ij+-k)

The inspiration for

choosing this artangement stems from the cml equations (227a)-(227b) As an example

converting (227a) into integral form after using Stokes Theorem

T T ^ d rr-JEdl = -mdashj^Bds dt

This equation states that a changing magnetic flux will generate a circular electric

field surtotmding the flux tube Similarly the integral form of (227b) also states that a

changing electric flux and an electric cmrent will generate a circular magnetic field

surtounding the flux tube Using the center difference operator to replace the time and

( 1 1 ^ space derivatives at time-step n and space lattice point zAx(y + mdash)Agt(A + mdash)Az on

(231a)

_1_ At

f n+mdash

H 2 - H 2 1 0 rbullbull 1 0 bull 1 2 ) 2 2)

1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az

Repeating this procedure for (33Id) at time step laquo + - and space lattice point

^ 1 (i + -)AxjAykAz

At

(

xi^îk]^ ii+iytj eAy H

V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-

2 +-_+- V i+-jkmdash

I 2 2j 1 2 2J

- - o pound ^ f x(i+-^Jk)

H-1

Substituting E^^^j^-^ with average time step laquo and n+I

Af

^ pn+l _ pn

x i+mdashJk x i+-Jk I 2- J I 2^ j

sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13

After some algebraic manipulation we obtained

^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)

By the same token the update field equations for the other field components can be

derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash

E -E E -Er jMn 4+^7tj zf+iy+ j h-j-k+^

Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J

lt J n^mdash n^mdash

A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)

Examination of (233a)-(233f) shows that all the field components fall on the

locations accounted for by the space grid of Figure 21 Equations (233a)-(233f) are

explicit in natme thus computer implementation does not require solving for determinant

or inverse of a large matrix To facilitate the implementation in digital computer the

indexes of the field components are renamed as shown in Figure 22 so that all the

indexes become integers This allows the value of each field component to be stored in a

three-dimensional artay in the software with the artay indexes cortespond to the spatial

indexes of Figme 22 In the figme additional field components are drawn to improve the

clarity ofthe convention

15

Cubed jk)

t i ( i H i ki

-yiH-lpc)

Figure 22 Renaming the indexes of E and H field components cortesponding to Cube(ijk)

Using the new spatial indexes for field components as in Figure 22 (233a)-

(233f) become

1 n + -

1 nmdash

HxLjk) - ^ x ( A ) -At

Ay Az (234a)

+ - 1 mdash At fj 2 mdash TT 2 _ _

( f _ P P mdash P ^x(iJk-H) ^x(ijk) ^bull( + jk) ^zUjk)

Az Ax

(234b)

ffzijk) - ^z(ijk)

n-- l^t P P P _ P ^y(jk + l) ^y(iJk) ^x(jk ^xUjk)

Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s

n + mdash i + I - - n + I N

^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az

1 1 A IHmdash nHmdash

Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)

Equations (234a)-(234f) form the basis of computer implementation of Yees

FDTD scheme for FVP of Maxwells equations Since (234a)-(234f) compute the new

field components from the field components at previous time-steps these equations are

frequently called update equations Notice that in the equations the temporal location of

the E and H field components differs by half time-step (mdashAr) In a typical simulation

flow one would determine the new H field components at n + - from the previous field

components using (234a)-(234c) Then the new E field components at laquo + l will be

calculated using (234d)-(234f) The process is tiien repeated as many times as required

until the last time-step is reached Because of diis die scheme is sometimes called

leapfrog scheme To show tiiat die divergence equations (225c) and (225d) are implicit

in (234a)-(234f) considermdash(^X)-^(M)) Assuming ^ E and a are

homogeneous and using (234d) diis can be written as

17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2

zjk) ^^zij-k) y(ijk) ^yijk-)

Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A

zi-jk) ^zi-j-k) yi-jk] ^y(-ljk-

Ay Az Ax

Similar terms can be vmtten for Ex and Ey components

(235a)

Ay (pn+l _ pn+i ^ _ ^yiJk) ^y(i]-k))-

_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A

Hj mdash H_i I H mdash H (ij-Xk) ~-x(-IA-l) _ --|-IO -(-l-)

AT AX Av

(235b)

18

^^zijk)-^z(ijk-)r 2s_ aAt

+ -

1 + V 2s

A^

2s

^zijk)-Eiij^k-i))mdash

( 1

1 + aAt

2s V

2s

1 Pf T- mdashPI 2 PI 2 PI 1 bullyijk) ^y(i-ljk) ^x(ijk)-^x(ij-lk)

Ax Ay V

_1_ Az

1 + aAt 2s

)

1 At ( 1 1 laquo+mdash n^mdash n^mdash nmdash

^yuik) - -^ j ( - i j t - l ) ^xiJk)-^x(ij-lk-)

Ax Ay Az

(235c)

Svimming up (235a)-(235c) the following equation is obtained

pn+l P^ F+l C^+l 17+ 17+ ^xijk)-^x(i-ijk) ^ ^y(iJk)-^y(iHk) ^ ^z(Uk)-^zijk-)

Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )

p _ p P _ p P p ^x(ijk) ^x(i-jk) ^y(jk) ^y(ij-lk) _^^zjk) ^z(ijk-)

V Ax Ay Az

(236)

From (236) it is observed that if initiallypounddeg( ) = pounddeg(t) = - (y) = 0 for all j k

then

zpn+l i rlaquo+l i r + 77 P _ P^^ ^x(iJk)-^xi-Jk) ^ ^yijk)~^y(iJ-k) ^ ^zijk) ^ziJk-) _

Ax Ay Az = 0 (237)

for all time-steps n It is immediately recognized that (237) is actually the finite-

difference equivalent of the divergence equation (225c) in free space A similar

procedme can also be applied to the H field components obtaining

19

1+i bdquo+i _ i _l 1 PI 2 PI 2 PI2 --^^ ^ xiMik)-t^xijk) ^^(iy-

Ax Ay Az

(+iylt) ^(-yt) ^ ^^(i+it)--^^(-yA) ^ ^zijM)-Ezujk) _ Q

Finally it should be pointed out that die Yees FDTD scheme is second order

accurate due to the center-difference operator employed for approximatmg the

differential operators This means the tiiincation ertor for die center difference scheme is

proportional to (AX^ Ay^ A Z ^ A^ )

24 Numerical Dispersion

The numerical algorithms for Maxwells cml equations as defined by (234a)-

(234f) causes dispersion ofthe simulated wave modes in the computational domain For

instance in vacuum the phase velocity of the numerical wave modes in the FDTD grid

can differ from vacuum speed of light In fact the phase velocity of the numerical wave

modes is a fimction of wavelength the direction of propagation and the size ofthe cubes

This numerical dispersion can lead to nonphysical results such as broadening and ringing

of single-pulse waveforms imprecise canceling of multiple scattered wave and pseudo-

refraction A detailed analysis of this numerical dispersion is presented in Chapter 5 of

Taflove (1995) [1] and dius will not be discussed here It is shown in [1 2] that to limit

the amount of numerical dispersion the edges of each cube must be at least ten times

smaller than the shortest wavelengtii expected to propagate in the computational domain

Of course the numerical dispersion will be suppressed even further if smaller cube size is

maintained However using a cube size that is too small will increase the number of cubes

20

needed to fill die computational domain and hence increase computational demand of die

model The mle-of-thumb of Ax Ay Az lt -^ where A is die wavelengtii cortesponding

to the expected highest significant harmonics in the model is adequate for most purposes

25 Numerical Stability

The numerical algorithm for Maxwells cml equations as defined by (234a)-

(234f) requires that the time increment At have a specific bound relative to die spatial

discretization Ax Ay and Az For an isotropic FDTD analysis based on centtal

differencing to remain stable the following mathematical statement must be tme

At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)

Equation (241) is known as Comant-Freidrichs-Lewy (CFL) Stability Criterion

In an unstable model the computed result for E and H field components will increase

without limit as the simulation progresses

26 Frequency Domain Analysis

The FDTD method is conducted in the time domain In order to obtain frequency

domain generalities such as scattering parameters reflection coefficients and input

impedance Fourier Transform is used to post process the results Oppenheim and Schafer

[12] Zhang and Mei [15] as well as Sheen Ali Abouzahra and Kong [4] present die

details on how such an analysis is performed using Fast Fourier Transform The Fourier

ttansform of an electric field E(t) at frequency fi is calculated using

21

deg (251)

27 Somce

To simulate a voltage source excitation it is necessary to impose an electric field

in a rectangular region Somce can either be sinusoidal or a Gaussian pulse A Gaussian

pulse is desirable as the excitation because its frequency spectrum is also Gaussian and

will provide frequency-domain information from dc to the desired cut off frequency by

adjusting the width of the pulse There are two types of somces the hard somce and the

soft source If a source is assigned a value to E it is referred to as hard source If a value

is added to E at a certain point it is referted to as soft source With a hard somce a

propagating pulse will see that value and be reflected because a hard value of E looks

like a metal wall to the FDTD but with a soft source a propagating pulse will just pass

through

i-gt

28 hnplementation of Basic FDTD Algorithm

Figme 23 illusttates the basic flow of implementing Yees FDTD scheme on a

computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL

bull Compute E fields components at interior

bull Compute E fields components at boundary

r

Compute new H field component values

C 1

Max time-steps (n) reached

^ Yes

n = n+l

i k

^ No

y

Post process Results

Figure 23 Basic flow for implementation of Yee FDTD scheme

23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction

A basic consideration with implementation of FDTD approach on computer is the

amount of data generated Clearly no computer can store an unlimited amount of data

therefore the domain must be limited in size The computational domain must be large

enough to enclose the stmcture of interest and a suitable absorbing boundary condition

on the outer perimeter ofthe domain must be used to simulate its extension to infinity If

the computational domain is extended sufficiently far beyond all somces and scatterers

all waves will be outgoing at the boimdary A suitably formulated boundary condition

will absorb majority ofthe incident wave energy allowing only a small amount to reflect

back into the computational domain This type of boundary condition is known as the

Absorbing Boundary Condition (ABC) Realization of an ABC to terminate the outer

boundary of the space lattice in an absorbing material medium is analogous to the

physical tteatment ofthe walls of an anechoic chamber

Ideally the absorbing medium is a only a few cells thick does not reflect all

impinging waves over their fiill frequency specttami highly absorbing and effective in

near field of a source or a scatterer

24

PML Comer Region

Figure 31 FDTD Lattice terminated by PML slabs

The theories and formulations of some highly effective ABC can be found in

Taflove (1995) and Kunz and Luebbers (1993) Here only the formulation concerning the

ABC used in this thesis is discussed The FDTD algorithm in this thesis uses the Uniaxial

Perfectly Matched Layer ABC terminated by a perfect electric conductor

32 Perfect Electric Conductor

The term PEC is an acronym for perfect electiic conductor and is used to model a

perfectly conductive metal surface The boundary conditions at a perfect electric

conductor require the tangential electric field components to be zero at the boundary

A perfect electiic conductor is modeled by simply setting the tangential electric

field components equal to zero at every time step where die perfect electric conductor is

25

located For example if there is a PEC on one ofthe surface of Cube (ijk) in Figure 32

the following E field components will be zero at all time-steps

^xdjMx) = iy+u+i) = KiijMi) = K(MjMi) =0 (321)

In addition to being used to tertninate the PML layers PEC type conditions can be

assigned to the surface of Yee Cells that are inside the mesh The use of PEC surfaces

inside the mesh allows us to model perfectiy conductive metal surface The use of such

conditions is used to model conductive surfaces and ground planes

PEC surface

Cube(ijk)

Figure 32 PEC on top surface of Cube (i j k)

33 Perfectly Magnetic Conductor

The term PMC is an acronym for perfect magnetic conductor The boundary

conditions at a perfect magnetic conductor require the tangential magnetic field

components to be zero at the boimdary stated mathematically

nxH = 0 where n is a surface normal unit vector

26

It should be clear by examining die figure 22 that there are no tangential

magnetic fields specified on die surface of a Yee Cell To handle this sittiation Li

Tassoudji Shin and Kong recommend use ofthe image method As we shall see using

the image method implies a certain restiiction Shen and Kong [11] provide a clear

inttoduction to the image method Figure 34 part (a) illustrates the interface between

media and a perfect magnetic conductor Part (b) is an illusttation of tiie image problem

copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy

(a) PMC Material Shaded

(b) Image Problem

Figure 33 PMC Boundary Conditions

Assigning image field values that are opposite the values of the cortesponding

fields in the media ensure zero tangential magnetic fields on the cell boundary

PMC and PEC surfaces are often used in the modeling certain problems that

posses symmetry to reduce computational time

27

34 Perfectly Matched Layer

A new fervor has been created in this area by JP Berengers inttoduction of a

highly effective absorbing-material ABC designated as the Perfectiy Matched Layer or

PML [8] The itmovation of Berengers PML is that plane waves of arbitrary incidence

polarization and frequency are matched at the boundary and subsequently absorbed in

the PML layer

The PML formulation introduced by Berenger (split-field PML) is a hypothetical

medium based on a mathematical model In the split-field PML if such a medium were to

exist it must be an anisotropic medium An anisotropic medium was first discussed by

Sacks et al [14] For a single interface the anisottopic medium is uniaxial and is

composed of both electric and magnetic permittivity tensors

35 Perfectly Matched Layer Formulation

The idea of Berengers PML [8] is if a wave is propagating in medium A and it

impinges medium B the amount of reflection is dictated by die inttinsic impedance of die

two media

r = mdashmdashmdash for normal incidence (3-51)

The inttinsic impedance of a medium is determined by tiie dielectric constant e and

permeability i

28

If the value of p changes with e so that ri remains a constant F would be equal to

zero and there would be no reflection but the pulse will continue to propagate in the new

medium So the new medium must be lossy so that the pulse will die out before it hits the

boundary This can be accomplished by making e and i complex Sacks et al [14] shows

that there are two conditions to form a PML

1 The impedance going from the background medium to the PML must be a

constant

^ = 1 (353) = 1 ^Fx

2 In the direction perpendicular to the boundary (the x direction for instance) die

relative dielectiic constant and relative permeability must be inverse of those in

the other directions

(354) poundFX

MFX

Spy

1

^^Fy

And each of die complex quantities will be of die form

^Fm = ^ l F m + 7 ^Hm

(3 i gt) form=xoryorz o--i

s and u are fictitious values to implement the PML and have nothing to do with the

real values of s and n which specify die medium

Following selection of parameters satisfies Equation (354) [7]

29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)

Substittiting Equations (356a)-(356b) in (355) the value in Equation (353) becomes

fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1

This fulfills the above mentioned requirements

The development of the PML is done for two dimensional version and then it is

developed for three dimensions as it closely follows the two dimensional version The

arrival for three dimensional PML equations from two dimensional PML equations is

derived in the appendix A This section only goes as far as presenting the general form of

three dimensional PML equations

curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)

I rn-Xi- - bull 1 Io(iJk + -) = ir(ifk + -) + curl_h

laquo+-- 1 laquo mdash 1 D HiJk + -) = gi3(i)gj3(i)D^ ilJk + -)

gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30

The one dimensional gparameters are defined as

gm2(m) = -mdash where m = iik 1 -I- fml(m)

1-fml(m) gm3(m)= ) where m = ijk

1 + fml(m)

where

fil(i)=^

In calculatingand g parameters the conductivity is calculated using polynomial

or geometric scaling relation Using polynomial scaling

a(x) = ^ xY

7 -

And optimal value of cr is calculated using the relation

m + 1 ôpt =

1507ryjsÂx

For polynomial scaling the optimal value ofm is typically between 3 and 4

Similarly

DrHi + fk) = g3(j)gk3(k)DrHi + jk)

+ gk2(k)gj2(j) ^curl_h + gil(k)r^(i + ^jk)

31

1 1 1

laquo + - 1 laquo--- 1 Dy HhJ + -k) = gi3(i)gk3(k)D^ HiJ + -^k)

gi2(i)gk2(k) curl_h + gjl(j)Il(ij + k)

2 J

HT ii + j k) = fj3(j)fk3(k)H (f + 1 j k)

+ fj2(J)fk2(k)

curl _e + filii)Jbdquo (i + - j k)

1 1 H(ij+-k)^fi3(i)fk3(k)H(ij+-k)

fi2(i)fk2(k) 1 n+-

curl _e + fjl(j)Jffy Hij + -k)

Hf (U j k + ^) = fi3(i)fj3(j)H (i j A + i )

+ fi2(i)fj2(j)

curl _e + fkl(k)Jbdquo^ ^ (i Jk + -)

32

CHAPTER IV

MODELING LUMPED COMPONENTS

41 Introduction

Chapter 2 and Chapter 3 provided the algorithms of the FDTD method for

modeling a three dimensional distributed passive problems hi this chapter die inclusion

of active and passive component models such as resistor capacitor inductor and bipolar

junction transistor (BJT) will be presented The inclusion of resistor capacitor and

inductor was first proposed by [13] The formulation for resistor capacitor and inductor

follows an altemative approach using S-parameters proposed by [16]

42 Linear Lumped Components

In FDTD method a lumped component is assumed to coincide with an E field

component Linear lumped components encompass elements such as resistor capacitor

and inductor Associated with each lumped element is the I-V relation A certain potential

difference imposed across the lumped element will result in a certain amount of curtent

flowing through the lumped element with negligible time delay Since the lumped

element coincide with an E field component the curtent along the element and the rate of

change of the E field across the element determine die values of the H field components

surtounding the E field Thus to include the lumped element into die FDTD model only

the cortesponding E field component needs to be modified to reflect the presence of an

electiic current density due to die lumped element The basis of formulating the update

33

equation for the E field component with the presence of lumped element is Maxwells

curl equation (227b) The following procedures are essentially based on Piket-May et al

(1994) [13] Here it is assumed the lumped element to coincide with E field component

Ezijk) generalization to other E field components can be easily carried out In all cases

the voltage V across the lumped element at time-step n is defined by

F =-pound Az (421)

Resistor

Assuming a z-directed resistor in a dielectric with permittivity e Let the

resistance be R From V = IR the V-I relationship for the resistor at time-step n + mdash can

be written as

i+- _ Az bdquo+] p (A^f) bullz(yt) - 2jj- = (M) (bully)

The cortesponding cmrent density is given by

1 laquo+-

T^2 _hAlplusmn (423) ^ bull ^ bull bullgt A x A y

From Maxwells equation we have

V x = J + mdashpound) dt

Reartanging the equation (424) will lead to an update equation for E^^jj^^ which

coincides with a lumped resistor R

34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy

Similar update equations can be derived for E for a lumped resistor along x and y

direction

Capacitor

1-V relationship for an ideal capacitor is given by

dV I = C-

dt

Let the capacitor with capacitance C be z oriented the I-V relationship at time step n + -

can be written as

+- CAz bdquobdquo+ 2 - z(iyi) A^

Pz(ijk) EzUJk)) (426)


n+- 2 =

2

AxAy (427)

Using the similar procedure we used above die update equation for die associated E field

is given by

^ A ^

Z7+ _ p A ^zUJk) - ^zUjk) ^

1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor

I-V relationship for an ideal inductor is given by

I = jv(T)dT

Let the inductor with inductance L be z oriented the I-V relationship at time step laquo + -

can be written as

n+- AzAt ^ ^zojk)=mdashr-ZEujk) (429)

The cortesponding curtent density is given by

1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy

Using the similar procedme we used above the update equation for the associated E field

is given by

EZ-Kraquo -^vxî -^t^ (laquo-gto)

36

43 hicorporating 2 Port Networks with S-Parameters

Simple networks may have an equivalent lumped circuit element model and they

can be included in the FDTD modeling using the algorithms described in the previous

section In this section a general algorithm proposed by [16] is presented

The electric field time-stepping equation is modified to allow the addition of the

two-port network From Maxwell equation we have

V x = J bdquo + | ^ Z ) (431)

dt

where Jnet is an impressed curtent density through which the network will be

incorporated Let the network be oriented in z-direction so the curtent density is given by

J bdquo = - ^ (432)

AxAy

At port 1 ofthe network

Iea VbdquoeaYuit)^VbdquoenY2(t) bull^ net ~ AxAy AxAy

^ j _ÊbdquobdquoY(t)Êbdquo^2ynit) (4 3 3) ^bull^^ AxAy

where indicates convolution

To incorporate the network characteristics in die time-domain the S-parameters is

converted to Y-parameters and then an IFFT or hiverse Discrete Fourier Transfonn is

performed to ttanslate die Y-parameters in the frequency domain into time domain

The time-marching equation for the z-component ofthe electtic field at Port i (i =

1 2) is given by

37

At 1

EL =E+mdashVXH 2 ^zPorti -zPorti

AtAz

1+-2

z(y)

^ oAxAy Ziî(opoundbdquo(-+Ei^2(opound =0 1=0

zPort2(n-l)

(434)

Similar update equations can be derived for electric field E along x and y

direction

38

CHAPTER V

FDTD SIMULATION PROGRAM

51 Inttoduction

In this chapter the FDTD simulation program will be described in detail The

flow the dynamics and important algorithms of die FDTD simulation engine will be

shown Simulation examples to verify the accuracy of the program are presented in

Chapter 6

52 Building a Three Dimensional (3D) model

A three-dimensional model consists of many characteristics and elements To

describe to the FDTD simulation program the natme of the model obviously a computer

file with certain syntax is required Syntax is developed in this thesis and the file

describing the model This file defining the model is a text file consisting of lines of text

which describe the properties of a single cell or a collection of cells or for all cells in the

3D model

Syntax

Identifier XI Yl Zl X2 yen2 Z2 Material Property

XI = Start coordinate along x-axis

Yl = Start coordinate along y-axis

Zl = Start coordinate along z-axis

X2 - End coordinate along x-axis

39

Y2 - End coordinate along y-axis

Z2 = End coordinate along z-axis

Material Property = Characteristic value of die identifier for die specified range of cells

Identifier = Characteristic of the cell or collection of cells 1 represents Dielectiic

material 2 represents MetalPEC 3 represents PMC 4 represents a Lumped element

Example

2 0 0 0 60 120 3 22

100060120000

4303953041500

The first line declares a range delimited by (0 0 0) and (60 120 3) with

dielectric constant Zr - 22 The second line specifies a range delimited by (0 0 0) and

(60120 0) is a perfect conducting plane (specifying a ground plane) The third line

specifies it is a lumped element port 1 specified by co-ordinates (30 39 5) and port 2

specified by co-ordinates (30 39 5)

53 FDTD Simulation Engine - An Overview

The basic simulation engine is shown in Figure 51 In this chapter only an

overview of the program will be given The actual somce code for the complete software

contains more dian 3000 lines of C code FDTD is coded in C language widi ftittire

upgrade and expansion in mind

The output of the software contains the following files the electiic and magnetic

components and voltages at specified points The output files are dat files (data files)

40

Thus visualizing software is required to view the output file Gnuplot [17] is employed in

visualizing the output file This software can be integrated with the curtent FDTD

software but the C language integration works only on LINUX platform This integration

is not provided in my thesis as it will predominantiy be used on Windows platform

Input File(s)

FDTD Simulation Engine jo bull Read and parse input file(s) bull^^ bull Setup all data-types and

variables bull Preprocessing and

determining computational coefficients

bull Enforce Boundary conditions bull Perform E and H updates

Output Files

Figure 51 Major software block of FDTD simulation program

41

54 Software flow

High level flow of FDTD module is shown in figure 52

Standard FDTD

Routine

Start

hiitialize FDTD cells

Parse the input file(s)

Initialize the Field obiects

Update Boundary Conditions

Update E field Components

Update H field Components

End of Iteration

Yes

Stop

Figure 52 Flow of FDTD Program

No

42

There are more steps in the initialization portion as compared to the basic flow of

Figure 52 as the FDTD simulation engine needs to accommodate different types of

elements with various update functions

55 Important Functions in Simulation

A number of functions are declared for the FDTD Simulation Engine Among the

more important fiinctions are the electric field update function (fncUpdateE) magnetic

field update function (fncUpdateH) and Absorbmg Boundary Condition (fncPML) The

electric field update function contains the variables to store all the E field components in

the model [Equations (233a)-(233f)] Similarly the magnetic field function contains the

variables to store all H field components in the model [Equations (233a)-(233f)] The

Absorbing Boundary Condition function contains some variables and artays which store

the values calculated to implement a PML [Equations (3524a)-(2524c) (3525a)-

(2525c)] The other functions are the Discrete Fourier Transform function (fncDFT)

which is used to obtain the frequency domain quantities of the electric field components

S parameter function (fncsparam) which is used to find Y parameters from an S-

parameter file to model lumped elements Complex number arithmetic operations

functions (f nccomplex) is used to perform all die complex number arithmetic operations

as S-parameter values are complex numbers

43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction

In this chapter several simulation examples using the FDTD software developed

will be shown to verify the accuracy and effectiveness ofthe program Simulation results

of a microstrip line line fed rectangular patch antenna low-pass filter a branch line

coupler series RLC circuit and parallel LC modeled using the FDTD software will be

presented

62 Microstrip Line

The model of the Microstrip line used for this numerical simulation is shown in

Figure 61

L2= 128

246 mm

Figure 61 Microstiip Line

In order to cortectly model the diickness of the substtate Az is chosen so that

tiiree nodes exactly match die diickness hi order to cortectly model the dimensions ofthe

44

ttansmission line Ax and Ay are chosen such that an integral number of nodes will

exactly fit the ship

The space steps used are Ax = 0389 mm Ay = 04mm and Az = 0265mm The

dimensions ofthe metallic strip are thus 6 Ax x 3 00 Ay The reference plane for the port 1

is lOAy from the edge ofthe FDTD wall

The total mesh dimensions are 60 x 320 x 25 in x y and z directions respectively

and a 10 cell PML is used thus the dimensions ofthe PML are 20 x 20 x 10 in x y and z

directions respectively

The time step used is At = 0441 picoseconds A Gaussian pulse is used for the

source Its half-width is T = 15 picoseconds and time delay to is set to 3T so that the pulse

will start at approximately zero

621 Modeling ofthe materials

In this case we are dealing with only three materials free space the dielectric

material ofthe substtate and the metal The circuit shown in Figme 61 is constmcted on

Dmoid substtate with Sr = 22 and no appreciable loss term The metal is specified by

ensuring the E fields within diose points cortesponding to die metal remain as zero

622 Somce

The source is initialized by a uniform E field between die smface and the ground

plane at a point Vi shown in Figure 61

45

A hard source cannot be used as the main information is the reflection coming

back the opposite way A hard source would look like a metal barrier which will make

the source to decay almost to zero so a soft source is used since we want the propagating

pulse to pass through

pulse = exp(-l X (pow((T -tjl spread20)))

dz[i][J][k] = dz[i][j][k] +pulse

623 Boundary Condition

We are using PML for our absorbmg boundary condition A 10 cell PML is used

in the simulation thus the dimensions of the PML are 20 x 20 x 10 in x y and z


624 Results

The simulation is performed for 6000 time steps The spatial distiibution of E (x

y t) just beneath the microstiip at 200 400 and 1000 time steps is shown in Figure 62-

64 The amplitude of the input pulse and the amplitude of the Fourier ttansform of the

input pulse are shown in the Figme 65 Figme 66

46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez

Figure 62 Distribution of Ez(xyz) at 200 time steps

Ez 400 Stepsdat

300 350

Figure 63 Distiibution of Ez(xyz) at 400 time steps

47

Ez

Ez 1000 Stepsdat

250 300

350

Figure 64 Distiibution of Ez (xyz) at 1000 time steps

00005

00004

1000 2000 3000 4000

Time Steps

5000 6000

Figure 65 Amplittide of die input pulse

48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20

Figure 66 Amplitude ofthe Fourier ttansform ofthe input pulse

63 Line Fed Rectangular Microstrip Antenna

The model of the line fed rectangular patch antenna used for this numerical

simulation is showm in Figme 67 [4] The operating resonance approximately

cortesponds to the frequency where Li = 1245nim = XI2

49

Li = 1245 mm ylt gt

0794 mm

246 mm

Figme 67 Line-fed rectangular microstrip antenna

In order to cortectly model the thickness of the substrate Az is chosen so tiiat

three nodes exactly match the thickness In order to cortectly model the dimensions ofthe

antenna Ax and Ay are chosen such that an integral number of nodes will exactly fit the

rectangular patch


rectangular patch is thus 32Ax ^ 40Ay the length ofthe microstrip from the source plane

is 50Ay and the reference plane for port 1 is chosen 1 OAy from the edge of the FDTD

wall The line width of microstrip line is modeled as 6Ax A 10 cell PML is used and the

total mesh dimensions are 60 x 120 x 25 in x y and z directions respectively

The time step used is At = 0441 picoseconds The Gaussian half-width is T = 15

picoseconds and time delay to is set to 3T so diat the Gaussian will start at approximately

zero

50

631 Modeling of Materials

In this case we are dealing with only three materials free space the dielectiic

material ofthe substtate and the metal The circuit shown in Figure 67 is consttoicted on

Duroid substrate with s = 22 and no appreciable loss term The metal is specified by

ensuring the tangential E fields within those points cortesponding to the metal remam as

zero

632 Source

The source is defined by a uniform Ez field between the anterma surface and the

ground plane at a point Vi shown in Figure 67

A hard source carmot be used as the as the main information is the reflection

coming back the opposite way A hard source would look like a metal barrier so soft

source is used since we want the propagating pulse to pass through

pulse = exp(-l X (pow((T -1^) I spread20)))

dz[i][j][k] = dz[i][j][k] +pulse

633 Boundary Conditions

We are using PML for om absorbing boundary condition A 10 cell PML is used

in the simulation thus the dimensions of die PML are 20x20x 10 in x y and z directions

respectively

51

634 Resuhs

The simulation is performed for 4000 time steps The spatial distribution of Ez (x

y t) just beneath the microstrip at 200 400 and 600 time steps is shown in Figure 69 -

611 which show good agreement with [4]

Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100

Figure 69 Distribution of Ez (xyz) at 400 time steps

Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53

The scattering coefficient |Sii| and the amplitude ofthe Fourier ttansforms ofthe

input pulse and the reflected wave form shown in Figme 611 612 show good

agreement with [4]

8 10 12 14 Frequency (GHz)

16 18

Figure 611 Retum loss ofthe rectangular antenna

54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)

Figure 612 Amplitudes ofthe Fourier transforms ofthe input pulse and the reflected

waveform

64 Microstrip Low-pass Filter

The acttial dimensions ofthe low pass filter to be simulated are shown by Figure-

613 [4]

2032 mm

254 mm

2413 mm

Figme 613 Low-pass filter detail

55

The space steps used are Ax = 04064 mm Ay = 04233nmi and Az = 0265mm

The dimensions of the long rectangular patch are thus 50Ax x 6Ay The reference plane

for port 1 is lOAy from the edge ofthe FDTD wall


and a 10 cell PML is used


picoseconds and time delay to is set to 3T so that the Gaussian will start at approximately

zero

641 Source

The source is initialized by a uniform Ez field between the surface and the ground

plane at a point Vi shown in Figure 613 Soft source is used since we want the

propagating pulse to pass through


We are using PML for our absorbing boundary condition A 10 cell PML is used

in the simulation tiius the dimensions of the PML are 20 x 20 x 10 in x y and z


56

643 Resuhs

The simulation is performed for 6000 time steps The scattering coefficient

results |Sii| (Retum Loss) and IS12I (Insertion Loss) shown in Figure 614-615 show

good agreement with [4]

8 10 12

Frequency (GHz)

14 16

Figure 614 Retum loss of Low-pass filter

18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20

Figure 615 Insertion loss of Low-pass filter

58

65 Branch Line Coupler

The acttial dimensions of the branch line coupler to be simulated are shown by

Figure 616 [4]

975 mm

mm

396 mm 2413 mm

Figure 616 Branch Line Coupler Detail

To model this circuit Ax Ay and Az are chosen to match the dimensions of the

circuit as effectively as possible In order to correctly model the thickness of the

substtate Az is chosen so that three nodes exactly match the thickness The space steps

Ax and Ay are chosen to match the center-to-center distance of 975mm exactly Again

small ertors in the other x and y dimensions occm

The space steps used are Ax = 0406 mm Ay = 0406 mm and Az = 0265nim and

the total mesh dimensions are 60 x 140 x 25 in x y and z directions respectively and a

10 cell PML is used dius the dimensions of the PML are 20 x 20 x 10 in x y and z

directions respectively The center-to-center distances are 24Ax x 24Ay The distance

59

from the somce to the edge of the coupler is 50Ay and the reference planes are chosen

lOAy from the edge ofthe coupler The strip width is modeled as 6 Ay and die wide ship

ofthe coupler is modeled as lOAy



zero

651 Source


plane at a point VI shown in Figme 616 Soft source is used since we want the




in the simulation thus die dimensions of the PML are 20 x 20 x 10 in x y and z


60

653 Results


results shown in Figure 617 show good agreement with [4]

4 5 6 Frequency (GHz)

Figure 617 Scattering Parameters of Branch Line Coupler

61

66 Series RLC Circuit

The layout ofthe series RLC circuit to be simulated are shown by Figure 618

Lumped Element

Ground plane

Figure 618 Series RLC Circuit Detail

In order to cortectly model the thickness of the substtate Az is chosen so that


transmission line Ax and Ay are chosen such that an integral number of nodes will

exactly fit the strip


dimensions of the metallic ship are thus 6Ax x 50Ay and the lumped element is 1 cell

thick The reference plane for die port 1 is lOAy from the edge ofthe FDTD wall


and a 10 cell PML is used thus the dimensions of die PML are 20 x 20 x 10 in x y and

z directions respectively


somce Its half-width is T = 15 picoseconds and time delay to is set to 3T so diat die pulse

will start at approximately zero The cutoff frequency is chosen as 7GHz

62

661 Source


plane Soft somce is used since we want the propagating pulse to pass through





663 Results

The simulation is performed for 1500 time steps The amplitude of the input

pulse ttansmitted pulse and the amplitude ofthe Fourier ttansform ofthe input pulse and

transmitted pulse are shown in the Figure 619 Figure 620

The results do not satisfy the theory and large variations occur between the

simulations and the theory The simulation results show that the series RLC circuit is

behaving like an open circuit

63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps

Figure 619 Amplitude ofthe input pulse and ttansmitted pulse

0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull

DFT_Vincdat DFT_Vrefdat

DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100

Figure 620 Fourier amplittide of die input pulse reflected and ttansmitted pulse

64

67 Parallel LC Circuit

The layout ofthe parallel LC circuit to be simulated are shown by Figme 621

Lumped Element

Ground plane

Figure 621 Parallel LC Circuit Detail


three nodes exacdy match the thickness In order to cortectly model the dimensions ofthe



The space steps used are Ax = 0386 mm Ay = 04nim and Az = 0265mm The

dimensions of the metallic strip are thus 6Ax x 50Ay and the lumped element is 1 cell

thick The reference plane for the port 1 is lOAy from the edge ofthe FDTD wall




65

The time step used is At = 0441 picoseconds A Gaussian pulse is used for die



671 Source


plane Soft source is used since we want the propagating pulse to pass through





673 Results



transmitted pulse are shown in the Figure 622 Figme 623


simulations and the dieory The simulation results show that the parallel LC circuit is

behaving like a short circuit

66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps

Figure622 Amplitude ofthe input pulse and transmitted pulse

004

0035

003

0025

002

0015

001

0005

DFr_Vincdat DFr_Vrefdat

DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100

Figure623 Fourier amplittide ofthe input pulse reflected pulse and ttansmitted pulse

67

CHAPTER VII

CONCLUSION

This chapter brings this thesis to a close by drawmg from all odier chapters m

this thesis to summarize the nuggets of knowledge that we revealed A few of die

highlights from this diesis and suggestions for fiiture research are discussed

71 Thesis Highlights

A systematic framework for modeling electtomagnetic propagation in RF and

Microwave circuits with active and lumped components using Finite-difference Time-

domain (FDTD) approach has been developed in this thesis Chapter 2 provided the

algorithms to approximate a three-dimensional model and all the update equations

needed to find the E and H fields in the model as a function of time-step Chapter 3

provided the algorithms to implement a three-dimensional PML absorbing boundary

condition Chapter 4 provided algorithms to include the presence of lumped

components using S-parameters in the diree-dimensional FDTD model Chapter 5

introduced a convenient notation to describe a complex three-dimensional model cube-

by-cube This notation is very flexible and compact with ample capacity to include

new feattires in the fiiture In Chapter 5 die algoridims presented in Chapter 2

Chapter 3 and Chapter 4 are systematically converted mto a computer program The

architecttire of die software is also illustrated clearly in Chapter 5 Aldiough the

algorithm can be implemented in any programming language it was coded using C

68

programming in this thesis Verification ofthe effectiveness ofthe FDTD framework is

performed in Chapter 6 where many simulations were carried out

72 Futtire Work

The following are some remarks for futme improvement

bull The present work uses Cartesian co-ordinate system so solving problems in other

co-ordinates system can be implemented

bull Include radiation characteristics for antermas

bull Include formulation of dispersive and magnetic materials

bull More verifications for active circuits

bull Improve the speed ofthe simulation and optimizing the code further

bull Provide a GUI

Finally it should be cautioned that as with all computer simulations the results are

only as good as the models This means diat if the models are not accurate or do not

reflect the actual physics of the components then the result will not match well with

measurements

69

REFERENCES

[I] Allen Taflove Advances in computational electrodynamics The Finite-Difference Time-Domain method 1995

[2] Allen Taflove Computational electtodynamics The Finite-Difference Time-Domain method 1998

[3] Constantine A Balanis Advanced Engineering Electtomagnetics Wiley Text Books New York 1989

[4] David M Sheen and Sami L Ali Application ofthe Three-Dimensional Finite-Difference Time-Domain Method to the Analysis of Planar Microstiip Circuits IEEE Trans Microwave Theory Tech vol38 No7 July 1900

[5] Dermis M Sullivan A simplified PML for use with the FDTD mediod IEEE Microwave and Guided Wave Letters Vol 6 Feb 1996 pp 97-98

[6] Dennis M Sullivan An unsplit step 3-D PML for use widi the FDTD method IEEE Microwave and Guided Wave Letters Vol 7 July 1997 pp 184-186

[7] Dermis M Sullivan Electromagnetic Simulation using the FDTD Method IEEE Press

[8] J P Berenger A perfectly matched layer for the absorption of electtomagnetic bullwaes^ Journal of Computational Physics vol 114 no 2 pp 185-200 1994

[9] KS Kunz and RJ Luebbers The Finite Difference Time Domain Method for Electromagnetics Boca Raton FL CRC Press 1993

[10] Kane S Yee Numerical Solution of hiitial Boundary Value Problems Involving Maxwells Equations in Isottopic Media IEEE Transactions on Antennas and Propagation vol 14 no 3 May 1966

[II] L Shen and J Kong Applied Electtomagnetism PWS Publishers Boston Massachusetts 1983

[12] Oppenheim A V Schafer R W Discrete-time signal processing Prentice Had 1989

[13] Piket-May M J Taflove A Baron J (1994) FD-TD modeling of digital signal propagation in 3-D circuits witii passive and active loads IEEE Trans Microwave Theory and Techniques Vol 42(No 8) 1514-1523

70

[14] Sacks Z S D M Kingsland R Lee and J-F Lee A perfectiy matched anisottopic absorber for use as an absorbing boundary condition IEEE Transactions on Antennas and Propagation Vol 43 No 12 1460-1463 December 1995

[15] X Zhang and K K Mei Time-Domain Finite Difference Approach to the Calculation of the Frequency-Dependent Characteristics of Microstrip Discontinuities IEEE Transactions on Microwave Theory and Technique vol36 no 12 December 1988 pp 1775-1787

[16] Xiaoning Ye James L Drewnaik Incorporating Two-Port Networks with S-Parameters into FDTD IEEE Microwave and Wireless Components Vol 11 No2 Febmary2001

[17] httpwvv^wgnuplotinfo Gnuplot

71

APPENDIX

72

This appendix contains die derivation of three dimensional PML equations we

used The development ofthe PML is done for two dimensional version and dien it is

developed for three dimensions as it closely follows the two dimensional version

We begin by using Maxwells equation

^ = V x i (Ala) dt

D(CD) = Sosl(co)E(o)) (A lb)

dH 1 ^ ^ = Vxpound (Alc)

dt ju

where D is the electric flux density We will now normalize the above equations using

E^ Ê (A2a)

D= -^D (A2b)

Using (A2a)-(A2b) in (Ala-Alc) will lead to

^ = mdash L - V x (A3a) dt is^pi

D(CO) = SI(CO)F(CO) (A-3b)

dH I -VxE (A-3c) dt V ôMo

In three-dimensional case we will be dealing with six different fields E^ Ey pound- H Hy

H hi two-dimensional simulation diere are two groups of vectors (1) the ttansverse

73

magnetic (TM) mode which is composed ofE^ Hx Hy (2) ttansverse electiic (TE) mode

which is composed of Ex Ey_ Hx

We will work here with TM mode and equation (A3) is reduced to

dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)

We now have eliminated s and n from the spatial derivatives for the normalized

units Instead of putting them back to implement the PML we can add fictitious dielectric

constants and permeabilitiesf^^ JUFX and i^^[1315]

(dH dH joD^Sp^(x)sl^(y) = Co

D(CO) = SI((O)I(O))

H(co)MFxix)-MFx(y) = -^c

Hy(co)Mry(x)MFyiy) = ô

dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74

We will begin by implementing PML only in x direction Therefore we will retain

only the x dependent values of sp and p^ in equations (A5a) - (A5d)

joD^sp^(x) = c^ dH dH^

V dy dx

Hx((o)pp(x) = -c^ dE^

dt

dE ^yi(0)MFy(x) = Cbdquo^

Using the values of equation (355)

Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)

The penneability of Hx in equation (A6b) is the inverse of H in equation (A6c)

which satisfies the second die requirement of PML mentioned in equation (354)

Equations (A6a) - (A6c) have to be put in FDTD fortnulation Moving equation

(A6a) to the time domain along with the spatial derivatives we get

75

1 nmdash

D (ij) = gi3(i)D^ Hij)

f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)

where the parameters gi2 and gi3 are given by

1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)

Almost identical treatinent of equation (A6c) gives

Hy=fi3(i + ^)H(i + j)

+ fi2(i + Âtcbdquo E ~Hi + lj)-E^ Hij)

Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)

Equation (A6b) needs a different tteattnent so it is rewritten as

jaH^ = -cbdquo dE^â^(x) 1 dE^

_ dy jo)s^ jo) dy

The spatial derivative will be written as

1 1 n + - n+-

dE^ Ê^ îj + l)-E^ HiJ)^ curl_e dy Ay Ay

Implementing this into FDTD formulation gives

Hf(ij+^)-H(ij+^)

A = -c curl e a^ (x) J^ curl e

Ay n=o Ay

1 1 HT (i J + -) = H (i j + -) + cÂt mdash - mdash +

curl_e cÂt a[j(x) n^ 1

^y ^y poundo IHXHIJ + T)

Equation (A6b) is implemented as the following series of equations

curl_e = E^ ~Hhj)-Ez HUj + l) (A 11a)

1 1 -- 1

Jfix (i^J + ^) = hx iiJ + 2^ + ^^-^ (A lib)

Hf a j^b=H a y -F i )+ - ^+ fHiDO (u j+) (A lie)

77

with filii) = oii)At

2s


or geometric scaling relation Using polynomial scalmg

aix) = 11 ^-

And optimal value of cris calculated using the relation

w-i-1 ôpt =

150nJsAx


So far we have shown the implementation in the x direction Obviously it must

also be done in the y direction Therefore we must also go back and add y dependent

terms from equations (A5) we set aside So instead of equations (A6) we have

JO) 1-F-o-z)(^)Yi ôiy)

JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo

dE HAo)) = -cbdquo-^

dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)

Using the same procedme as before equation (A7) will be replaced by

78

1 nmdash

A (^7) = gi^(i)gj3(i)DrHiJ)

+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax

In y direction Hy will need an implementation which is similar to the one used for

Hv in the x direction

1 n+-curl_e = E^ Ai + lj)-E^ Hij) (A 13a)

1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)

1 curl pound bullmdash 1

+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)

Finally the Hx in the x direction becomes

curl e = E^ ~HiJ)-E Hij + )

n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)

^^ l^curl e ^^r-si^^- bull U + i2(7 + - ) ^ = - + MO (27 + - )

We will now implement the PML for three directions the only difference being

we will be dealing with three directions instead of two [14]

So equation (A 12a) will now become

r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)

= Jmdashcurl_h is an integration when it goes to the time domain and equation (A 15) JO)

now becomes

JO) Y

V Jogtpoundo ) Jo)So J D=Cr bdquo h+^Î

Dz ô J

curl h mdash

Hy(i^fk^]^-H](i-fk^^-^

-H(ij^k^^^H(ij-k^--^^

(A 16)

Il(ifk^) = r ih j ^ + ) + curl _ h (A 17)

80

1 n+mdash

A HiJk + ^) = gi3(i)gj3(i)D Hijk + ^)

gi2(i)gj2(i) curl_h + gkl(k)Jl(ijk + -)

(A 18a)


gm2(m) = 1

1 + fml(m) where m = i j k

- 1 - finl(m) gm3(m)^-mdashmdashmdashmdash where m = ijk

1 + fml(m)

Similarly

DT (f -F i j k) = gj3(j)gk3(k)Dr ( + i j k)

-F gk2(k)gj2(j) curl_h + gil(k)Jl(i + -Jk)

(A 18b)

D~HiJ + k) = gi3(i)gk3(k)DrHij + k)

+ ga(i)gk2(k) 1 ^

curl__h + gjliJ)JaiiJ + -k)

(A 18c)

Hi + jk) = fj3(j)fk3(k)H(i + jk)

+ fj2(j)fk2(k) curl_e + fil(i)jljii + -jk)

(A 19a)

81

KhJ + k) = fi3(i)fk3(k)H(ij + -k) bull 2

-F fi2(i)fk2(k) 1

curl _e + fjl(j)Jny i j + -k)

(A 19b)

Hf (i j c +1) = fi3(i)fj3(j)H a j k + i )

fi2(i)fj2(j) ( 1

cur _ e + fkl(k)Ibdquo^ ^ (i j k + -) (A 19c)

82

PERMISSION TO COPY

In presenting this thesis m partial fulfilhnent ofthe requirements for a masters

degree at Texas Tech University or Texas Tech University Healdi Sciences Center I

agree tiiat the Library and my major department shall make it freely available for

research purposes Permission to copy this thesis for scholarly purposes may be

granted by die Dhector of the Library or my major professor It is understood that any

copying or publication of this thesis for fmancial gam shall not be allowed widiout my

further written permission and that any user may be liable for copyright infrmgement

Agree (Pemussion is granted)

Student Sigdature Date

Disagree (Permission is not granted)

Stiident Signature Date

ACKNOWLEDGEMENTS

I would like to thank everybody who has helped me during my graduate school

years and while I was working on my thesis

I would like to express my sincere gratitude to Dr Mohammad Saed my thesis

advisor for his support and guidance during my graduate studies research work and

thesis preparation This thesis would not have been possible without his continuous help

I would also like to express my sincere appreciation to Dr Jon G Bredeson

for serving on my thesis committee I would like to thank Sharath and Vijay for helping

me out in programming on quite a few occasions

Most of all I would like to thank my parents who have always stood by me and

made it possible for me to pmsue graduate studies My brother has also been a constant

source of motivation

I would like to thank my good friends Anu Dipen Nikhil Sharath Rohini Hari

Kanth Kiran and Swapnil who have constantly supported and encouraged me during my

Masters

To all of you I thank you

11

TABLE OF CONTENTS

ACKNOWLEDGEMENTS u

ABSTRACT vi

LIST OF FIGURES vii

CHAPTER

L INTRODUCTION 1

11 Background 1



IL FDTD ALGORITHM 5

21 Introduction 5






27 Source 22



31 Inttoduction 24





111


41 Inttoduction 33




51 Introduction 39



54 Software flow 42



61 Inttoduction 44



622 Source 45


624 Results 46



632 Somce 51


634 Results 52


641 Somce 56


643 Results 57


651 Somce 60

IV


653 Results 61


661 Somce 63


663 Results 63


671 Source 66


673 Results 66

VIL CONCLUSION 68


72 Futtire Work 69

REFERENCES 70

APPENDIX 72


ABSTRACT











VI

LIST OF FIGURES





















VI1















Vlll

CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












TABLE OF CONTENTS

ACKNOWLEDGEMENTS u

ABSTRACT vi

LIST OF FIGURES vii

CHAPTER

L INTRODUCTION 1

11 Background 1



IL FDTD ALGORITHM 5

21 Introduction 5






27 Source 22



31 Inttoduction 24





111


41 Inttoduction 33




51 Introduction 39



54 Software flow 42



61 Inttoduction 44



622 Source 45


624 Results 46



632 Somce 51


634 Results 52


641 Somce 56


643 Results 57


651 Somce 60

IV


653 Results 61


661 Somce 63


663 Results 63


671 Source 66


673 Results 66

VIL CONCLUSION 68


72 Futtire Work 69

REFERENCES 70

APPENDIX 72


ABSTRACT











VI

LIST OF FIGURES





















VI1















Vlll

CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY













41 Inttoduction 33




51 Introduction 39



54 Software flow 42



61 Inttoduction 44



622 Source 45


624 Results 46



632 Somce 51


634 Results 52


641 Somce 56


643 Results 57


651 Somce 60

IV


653 Results 61


661 Somce 63


663 Results 63


671 Source 66


673 Results 66

VIL CONCLUSION 68


72 Futtire Work 69

REFERENCES 70

APPENDIX 72


ABSTRACT











VI

LIST OF FIGURES





















VI1















Vlll

CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY













653 Results 61


661 Somce 63


663 Results 63


671 Source 66


673 Results 66

VIL CONCLUSION 68


72 Futtire Work 69

REFERENCES 70

APPENDIX 72


ABSTRACT











VI

LIST OF FIGURES





















VI1















Vlll

CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












ABSTRACT











VI

LIST OF FIGURES





















VI1















Vlll

CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












LIST OF FIGURES





















VI1















Vlll

CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY


























Vlll

CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












CHAPTER I

INTRODUCTION

11 Background










































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY



































lumped components

















and ttansistors




location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY














location time










CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












CHAPTER II

FDTD ALGORITHM

21 Inttoduction











V3 = p^ (221c)

V5 = pbdquo (22 Id)



as follows









relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY




















relations

D = Efl (222a)

^ = ^fH (222b)




reduce to the form

3 = eE (223a)

B = HH (223b)





^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY














^ bull ^ = - ^ P trade (224a)


V(V X i ) = VJ + v | - ^ 5 1 = 0 dt )

dt^ dt^

V J = - - P (224b)













VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY














VxE = B dt

WxH=J+mdashD dt

VD = p^

VB = 0

E(xyzt)

(225a)

(225b)

(225c)

(225d)

=0 = H(xyzt)

1=0 = J(xyzt)

1=0 = 0 and p^(xyzt)[^^ =0


From (224b)

0 _^




From (225a)




VB = VB = 0 oO (=0



(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY














(225b)


dt J

^ = 0 =gt -VJ = V

( d-mdashD

dt

VD = -jVJdT + C2


Hence from (226)

= euroVE 1=0 t=0


VD = -jVJdT = p^




mdashbull a

VxE = B dt

VxH = J + mdashD dt

(227a)

(227b)

E(xyzt) 1=0

= H(xyzt) 1=0

= J(xyzt) = 0 1=0

Pe(xyZt) = 0 1=0



VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












VZ) = V5 =0 =0



23 FDTD Formulation






= V xE dt jLi


dH^

dt

dH^

dt

dH^

dt

dt

(SE^_^^ dy dz

ifdE^ dE

u dz dx

ifdE^ dE^]

p dy dx

I fdH^ dH^

pound dy dz crE^

(231a)

(231b)

(231c)

(23Id)

10

dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












dt

I fdH^ dH ^ ^ -oE

dz dx (23 le)

dE^

dt


(23If) y







(i4-ijk+l)

^ ^

(ijk-t-l)

(i7Jl-Tgt t (iikiL

X

Cube(ijk)

i

^ H

xîAxA)

t Ûl -^jt l

ne^yi-i bullU-)-^X)

(i+ljk)

(ij+lk+li

M

i^

i l - l k)

Av (i+1 j + l k i


11





IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY
















IS


The inspiration for









(231a)

_1_ At

f n+mdash


1_ Ay E E

V f^^-^îJ V-2jj i] __

Az

(233a)

12

H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












H 2

x lJ+

n

= H ^ A^

poundbull - pound

Ay Az


^ 1 (i + -)AxjAykAz

At

(


V I 2

1

- 2

2 j 1 2 )

sAz H 2

1 n+-


I 2 2j 1 2 2J


H-1


Af

^ pn+l _ pn


sAy H

z + -V I 2

-

2 J L 2 ^ 2 j

sAz

1

2

V

- H 2

1 1 mdashamdash s 2

-n+l r 1 p

xii+^jkj xî+-jk]

13


^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY













^n+l

bullbull]

( o-At^

2s

1 + cjAt

V 2s

At_

2s

pn+

J ( 1

H 2

1 + oAt

2s

-H 2 H 2 (bullv4) (H -] ^(4-^-4] +4^--i)

Ay Az

V

(233b)


derived

1 1 nmdash

= H 2 H 2 =H 2 - mdash


Az Ax

(233c)

1 n+-

H 2

Ê At

2 4] = (44)

y iJ+^M -E E -E

i i ) [^y] x[jk]

Ax Av

(233d)

14

poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












poundbull 1-

cjAt

~2i 1 + crAt

2s

pn+

y[u-k]

A^

2s

1 + aAt

n+- n+- 2 _ W 2

Az 2s

1 1 n + - n+mdash

Ax

(233e)

pn+ 1 -

aAt ~2s

aAt 1-1-

V 2s At_

Ys

71+1

z(4)

1 + aAt

J


A^-i^2] lt V-^ Ax

2s

1 1

H ^ - H 2

Agt

(233f)










15

Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












Cubed jk)

t i ( i H i ki

-yiH-lpc)



(233f) become

1 n + -

1 nmdash


Ay Az (234a)



Az Ax

(234b)

ffzijk) - ^z(ijk)


Ax Ay (234c)

pn+ ^x(lJk)

_oA^^ 2s

1 + crAr 2s J

E+

At_

2s

(

1 + 2s


^jk) ^ - ( V l ) ^ i ( A ) ^ v ( A

Av Ar

(234d)

16

rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












rn+l ^y(uk)

1-oAt ~2s

1 + aAt 71+1

^(bullM)+

A^

2s

1 + aAt 2s

1

n + -

^x^jk)

+ -

- 2 x(-7-l)

Az


Hz(i^jk)-H(Xjk)

Ax

(234e)

71+1

2 ( i J )

1 -aAt

~2s aAt 1 +

V 2s

71+1

z(JA) + bull

A^

2pound

1 + aAt 2s

1 IHmdash

^yljk)

1 i + -

mdash W 2

1 1 n + - +mdash

t^x^jk)-Hx(Lj-k)

Ax Ay

(234f)













17

-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












-^Kljk)-Kt^jk))=

_a^^ 2pound_

aAt 1 +

V 2s At f

K(iik)-K(ij-k) Ax

2s

1 + aAt

2s

A

2s

n+-mdash M 2 _ pf 2


Ay Az

( 1

1 + aAt

2s

1 n+-

1 A

_1_

Ax

J

1 A


Ay Az Ax


(235a)


_aAty 2s

1 + aAt

2s )

WyiJk)-K[iJ-k)) Ay

2E

1 + crA^

2s

V

2E

I 1 n^mdash fHmdash

^x(ijk) ~^x(jk-) ^lk) ~ ^-U-jk)

Az Ax

__

Ay

A f bdquol

1 + crA

2e

1 n+mdash

I A


AT AX Av

(235b)

18


+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY













+ -

1 + V 2s

A^

2s


( 1

1 + aAt

2s V

2s


Ax Ay V

_1_ Az

1 + aAt 2s

)



Ax Ay Az

(235c)



Ax ^y Az

1 -crA

2s ( TT

aAt 1 +

V 2s )


V Ax Ay Az

(236)


then


Ax Ay Az = 0 (237)




19


Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY













Ax Ay Az




















20









At lt C((AJC) +(Ay)^ + (AZ) ) 2 C = - ^ (241)











21

deg (251)

27 Somce










through

i-gt



computer

8 H

E = H = 0

t - n + 05

t = n + l

Tni ti al 1 zati on

1 r

Set excitation

1

^ iL



r


C 1


^ Yes

n = n+l

i k

^ No

y



23

CHAPTER III

BOUNDARY CONDITIONS

31 Introduction
















24

PML Comer Region












25








PEC surface

Cube(ijk)







26







copy - bull x

Boundary

Hbdquo

copy Ex

^

Hbdquo copy

Ex

V

Hbdquo

copy Ex

-Hbdquo copy

Hbdquo copy

Ex

-H

copy


(b) Image Problem






27






the PML layer









two media



permeability i

28







constant

^ = 1 (353) = 1 ^Fx




(354) poundFX

MFX

Spy

1

^^Fy


^Fm = ^ l F m + 7 ^Hm





29

^Fm =MFn=^ (356a)

Dm Hm

o

CTr (356b)


fin = M Fx

bullFx

1 + -7raquo^0

| l + - ^ I laquo^0

= 1







curl h =

H(i+^fk+^)-H(i-^jk+^)

-H(ij+^k+^)+H(ij-^k+^)



gi2(i)gj2(if (^^^^ _ ^^j^^)^^ ( ^ 1)^

30




1 + fml(m)

where

fil(i)=^



a(x) = ^ xY

7 -


m + 1 ôpt =

1507ryjsÂx


Similarly



31

1 1 1



2 J


+ fj2(J)fk2(k)



fi2(i)fk2(k) 1 n+-



+ fi2(i)fj2(j)


32

CHAPTER IV


41 Introduction


















33






F =-pound Az (421)

Resistor



be written as



1 laquo+-






34

^z(ijk) -

AtAz

2RsAxAy

AtAz 1- 2RsAxAy

F +

At^

s

1 + AtAz V X ^ Z ( 0 (425)

2ipoundAxAy


direction

Capacitor


dV I = C-

dt


can be written as




n+- 2 =

2

AxAy (427)


is given by

^ A ^


1 + CAz

poundAxAy ^

1 n + -

Vxz(^y (428)

35

Inductor


I = jv(T)dT


can be written as



1 laquo + -

1 2 -2 _ ^z(ijk)

AxAy


is given by


36







V x = J bdquo + | ^ Z ) (431)

dt



J bdquo = - ^ (432)

AxAy









1 2) is given by

37

At 1


AtAz

1+-2

z(y)


zPort2(n-l)

(434)


direction

38

CHAPTER V


51 Inttoduction




Chapter 6







3D model

Syntax






39






Example

2 0 0 0 60 120 3 22

100060120000

4303953041500













40





Input File(s)





Output Files


41

54 Software flow


Standard FDTD

Routine

Start







End of Iteration

Yes

Stop


No

42



















43

CHAPTER VI

SIMULATION RESULTS

61 Inttoduction





presented

62 Microstrip Line


Figure 61

L2= 128

246 mm




44

















622 Somce



45











624 Results





46

Ez

Ez 200 Stepsdat

- 200 -- 150 100

250 300

350

60

Ez


Ez 400 Stepsdat

300 350


47

Ez

Ez 1000 Stepsdat

250 300

350


00005

00004

1000 2000 3000 4000

Time Steps

5000 6000


48

u u o

0028

0026 bull D

1 0024 pound

t 0-022 3

pound 002

0018

0016

n m I

rmdash--___ _ -

-

-

-

Outputdat -

~

-

bull

-

bull

8 10 12

Frequency (GHz)

14 16 18 20






49

Li = 1245 mm ylt gt

0794 mm

246 mm





rectangular patch








zero

50






zero

632 Source











respectively

51

634 Resuhs




Ez 200 Stepsdat

Ez

120 100


52

Ez

Ez 400 Stepsdat

120 100


Ez 600 Stepsdat

Ez

120 100

50 - g5 0


53



agreement with [4]


16 18


54

003

o I 0015

pound 001

0005

Inputdat Outputdat

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)


waveform



613 [4]

2032 mm

254 mm

2413 mm


55








zero

641 Source








56

643 Resuhs




8 10 12

Frequency (GHz)

14 16


18

57

T3

fM

-30

-40

8 10 12 14

Frequency (GHz)

16 18 20


58



Figure 616 [4]

975 mm

mm

396 mm 2413 mm











59






zero

651 Source








60

653 Results





61



Lumped Element

Ground plane















62

661 Source







663 Results







63

00005

00004

00003

00002

00001

-00001

-00002

Vtotdat Vtransdat

0 200 400 600 800 1000 1200 1400 1600

Time Steps


0035

003

0025 01 TJ 3

t 002 E lt

^ 0015 3

001

0005 h

bull


DFT Vtransdat

0 10 20 30 40 50 60 70 80 90 100


64



Lumped Element

Ground plane












65




671 Source







673 Results







66

00006

00005

00004

00003

00002

00001

0

-00001

-00002

-00003

-00004

-00005

V

Vtotdat Vtransdat

200 400 600 800 1000 1200 1400 1600

Time Steps


004

0035

003

0025

002

0015

001

0005


DFT Vtransdat

- L 1mdash

0 10 20 30 40 50 60 70 80 90 100


67

CHAPTER VII

CONCLUSION



















68



72 Futtire Work








bull Provide a GUI




measurements

69

REFERENCES














70





71

APPENDIX

72





^ = V x i (Ala) dt



dt ju


E^ Ê (A2a)

D= -^D (A2b)







73




dD^

dt = -Cr

dH dH^

dy dx

D((o) = s^(o))E(co)

dt

dH^

dt

Co =

dt

dt

1

(A4a)

(A4b)

(A4c)

(A4d)





D(CO) = SI((O)I(O))



dy dx

dE^

dt

dE^

dt

(A5a)

(A5b)

(A5c)

(A5d)

74




V dy dx


dt



Jed 1 +

joy

J(o

1 +

o-o(x)

J03poundo )

o-p(^)

D=c dH^JJ[A dy dx

H^(co) = -c^ dE

dy

HAco) = cbdquo dE^

dx

(A6a)

(A6b)

(A6c)





75

1 nmdash


f 1

-f gi2(i)AtCQ

H(i + -J)-H(i--j)

Ax

1 ^

2

1 1 H(ij+-)-H(ij--)

Ax

(A7)


1 gi2(i) =

1 + o-rf(0-A

2poundn

(A8a)

1 -oAi)At

gm) = ^- 1 +

2S

(A8b)


Hy=fi3(i + ^)H(i + j)


Ax

(A9)

where

fi2(i + ) = 1

a(i + -)At 1 +

2S 0 y

(A 10a)

76

a(i + -)At 1 2

fi3(i +-) = ^fo 2 1

a(i + -)At l + -2Sn

(A 10b)



_ dy jo)s^ jo) dy


1 1 n + - n+-



Hf(ij+^)-H(ij+^)


Ay n=o Ay






1 1 -- 1



77


2s



aix) = 11 ^-


w-i-1 ôpt =

150nJsAx






JO)

JO)

1 +

JO)S )

q- f l (^ )

JO)So

1+-J0)pound

D =Cr d_H^_dH^

dy dx ^

1 + jo)poundo


dy

Y

Jo)poundo A J^ô

(A 12a)

(A 12b)

(A12c)


78

1 nmdash


+ gi2(i)gj2(i)Atc

1 H(i + -j)-HUi-^j)

Ax

H(ij+^)-H(ij-^)^

Ax




1 1 _i 1 Cx ( + -^) = ^ 1 (i + -J) + curl_e (A 13b)

Hri + ^j) = fi3(i + ^)H(i + ^j)


+ y 2 ( - F i ) ^ ^ + i i ( 0 ^ ( - F - y )

(A 13c)



n mdash 1

2

79

HT(iJ^)=mj^)H(ij^)





r JO)

a^(x) r 1 +

V J^ô J

O-D(gt) (

(

JO)

(

JO) V JO)So

1 + V Jo)poundo J

Y

A

1 + O-D(^) D=Cr

1 + go(j)

Jo)e

JOgtpoundo J

(

dH dH

D =Cr 1-F

V

1-F JO)Sbdquo

D = Cr curl h + Cr

^ dy dx

a(z)YdH dH^^

jo)poundo A ^ dx ^

o-(z)

(A14)

curl h (A 15)


now becomes

JO) Y


Dz ô J

curl h mdash


-H(ij^k^^^H(ij-k^--^^

(A 16)


80

1 n+mdash



(A 18a)


gm2(m) = 1



1 + fml(m)

Similarly



(A 18b)


+ ga(i)gk2(k) 1 ^


(A 18c)



(A 19a)

81


-F fi2(i)fk2(k) 1


(A 19b)


fi2(i)fj2(j) ( 1


82

PERMISSION TO COPY












lumped components a thesis

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