FDTD MODELING OF RF AND MICROWAVE CIRCUITS WITH ACTIVE AND LUMPED COMPONENTS by BHARATHA YAJAMAN, B.E. 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^noftfe^ommittee Accepted Dean of the Graduate School August, 2004
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Transcript
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^noftfe^ommittee
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
652 Botmdary Conditions 60
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
672 Boundary Conditions 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
61 Microstrip Line 44
62 Distribution of Ez(xyz) at 200 time steps 47
63 Distribution of Ez(xyz) at 400 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
69 Distribution of Ez(xyz) at 400 time steps 53
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
652 Botmdary Conditions 60
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
672 Boundary Conditions 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
61 Microstrip Line 44
62 Distribution of Ez(xyz) at 200 time steps 47
63 Distribution of Ez(xyz) at 400 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
69 Distribution of Ez(xyz) at 400 time steps 53
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
652 Botmdary Conditions 60
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
672 Boundary Conditions 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
61 Microstrip Line 44
62 Distribution of Ez(xyz) at 200 time steps 47
63 Distribution of Ez(xyz) at 400 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
69 Distribution of Ez(xyz) at 400 time steps 53
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
652 Botmdary Conditions 60
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
672 Boundary Conditions 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
61 Microstrip Line 44
62 Distribution of Ez(xyz) at 200 time steps 47
63 Distribution of Ez(xyz) at 400 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
69 Distribution of Ez(xyz) at 400 time steps 53
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
652 Botmdary Conditions 60
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
672 Boundary Conditions 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
61 Microstrip Line 44
62 Distribution of Ez(xyz) at 200 time steps 47
63 Distribution of Ez(xyz) at 400 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
69 Distribution of Ez(xyz) at 400 time steps 53
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
61 Microstrip Line 44
62 Distribution of Ez(xyz) at 200 time steps 47
63 Distribution of Ez(xyz) at 400 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
69 Distribution of Ez(xyz) at 400 time steps 53
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
61 Microstrip Line 44
62 Distribution of Ez(xyz) at 200 time steps 47
63 Distribution of Ez(xyz) at 400 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
69 Distribution of Ez(xyz) at 400 time steps 53
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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) = E^iAxjAykAznAt)) 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
x^iAxA)
t ^Ul -^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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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^^-^^iJ 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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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^^ik]^ 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 x^i+-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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
^E 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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
-^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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
^^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
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
1+i bdquo+i _ i _l 1 PI 2 PI 2 PI2 --^^ ^ xiMik)-t^xijk) ^^(iy-
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^ ^E (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 ^oMo
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) = ^o
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 + ^Atcbdquo 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^^a^(x) 1 dE^
_ dy jo)s^ jo) dy
The spatial derivative will be written as
1 1 n + - n+-
dE^ ^E^ ^ij + 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 + -) + c^At mdash - mdash +
curl_e c^At 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
In calculatingand g parameters the conductivity is calculated using polynomial
or geometric scaling relation Using polynomial scalmg
aix) = 11 ^-
And optimal value of cris calculated using the relation
w-i-1 ^opt =
150nJsAx
For polynomial scaling the optimal value ofm is typically between 3 and 4
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 ^oiy)
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^^o
(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^^o 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+^^I
Dz ^o 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)
The one dimensional gparameters are defined as
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
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