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Extended spatial domain and hybrid time-domain analyses for microwave circuits

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U
n iv e r s it y o f
C a l if o r n ia
Los Angeles
EX TEN D ED SPATIAL DOM AIN A N D
H YBRID TIM E-DOM AIN ANALYSES
FOR MICROWAVE CIRCUITS
A dissertation subm itted in partial satisfaction
of the requirem ents for the degree
Doctor of Philosophy in Electrical Engineering
by
D o n g so o Koh
1997
t
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© Copyright by
Dongsoo Koh
1997
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The dissertation of Dongsoo Koh is approved.
Tony F.-C . Chan
Behzad Razavi
Yahya R ahm at-Sam ii
Tatsuo Itoh, C om m ittee Chair
University of California, Los Angeles
1997
u
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To m y family and friends
iii
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T able
of
Contents
1
I n t r o d u c t io n ..................................................................................................
1
2
E x te n d e d sp a tia l d o m a in a n a l y s i s ...................................................
6
2.1
Spatial Green’s functions on the conductor-backed dielectric slab .
2.2
The m ixed-potential integral equation (M PIE) and the m ethod of
moments (MoM) fo rm u la tio n s..........................................................
3
7
21
2.3
Incorporation of lum ped resistors into the spatial domain analysis
36
2.4
Analysis of W ilkinson power divider and c o m b in e r......................
40
H y b rid tim e -d o m a in a n a l y s i s .............................................................
49
3.1
Finite difference tim e-dom ain m e t h o d ...............................................
50
3.2
Hybrid analysis using the FDTD and F E T D ......................................
72
3.2.1
FETD f o r m u la tio n .......................................................................
74
3.2.2
Hybridizing the FDTD analysis and the FETD analysis . .
79
3.2.3
Applications of the hybrid a n a l y s i s .........................................
83
4
C o n c lu s io n .....................................................................................................
97
A
E v a lu a tio n o f e le m e n ta l
m a trix [Ke\ for th e stiffn ess m a tr ix . . 100
B
E v a lu a tio n o f e le m e n ta l
m a trix [F/] for th e load v e c t o r ............. 109
R e f e r e n c e s .....................................................................................................................118
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L is t
of
F ig u r e s
2.1
Conductor-backed dielectric slab..............................................................
2.2
The am plitude of normalized Green’s functions, G
and G q. . . .
20
2.3
Geom etry for the triangular-patch-pair basis function.......................
22
2.4
Normalized area coordinates for a local coordinate system ..............
28
2.5
Geometric param eters associated with the line integral along A jS
7
for calculating Ai and B{............................................................................
32
2.6
Sample points for seven quadrature rule................................................
33
2.7
Voltage gap source applied onto the sth patch....................................
35
2.8
New type of triangular patch pair for the lumped resistor connection. 36
2.9
Example of resistor connection.................................................................
2.10 Schematic diagram for a microstrip power combiner / divider.
. .
36
38
2.11 Surface current on the T-junction with and w ithout resistor. . . .
39
2.12 The analyzed m icrostrip Wilkinson power divider stru ctu re.
41
2.13 New type of triangular patch pair for the matched excitation.
...
. . 42
2.14 M atched excitation and load for the m icrostrip...................................
43
2.15 M agnitude of I x along the microstrip....................................................
43
2.16 Scattering param eters of the Wilkinson power divider......................
46
2.17 Current distribution on the Wilkinson power divider (P o rt 1 exci­
ta tio n ).............................................................................................................
47
2.18 C urrent distribution on the Wilkinson power divider (P o rt 2 exci­
ta tio n ).............................................................................................................
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48
!
3.1
The FDTD analysis.....................................................................................
51
3.2
The FDTD unit cell.....................................................................................
53
3.3
Bondwire interconnect
structure (3-D view)...............................
56
3.4
Bondwire interconnect
structure (top and side view)...............
56
3.5
Frequency responses of a bondwire interconnect structure...............
57
3.6
Bondwire interconnect
58
3.7
M atching stub design for a bondwire interconnect structure.
3.8
R eturn losses of bondwire interconnects with different bondwire
structure with matching stubs.............
...
lengths.............................................................................................................
3.9
59
60
R eturn losses of bondwire interconnects with different bondwire
heights.............................................................................................................
61
3.10 Microstrip-to-waveguide transition structure (3-D view)...................
62
3.11 Microstrip-to-waveguide transition structure (Side, top, and back
view)................................................................................................................
63
3.12 FDTD domain decom position...................................................................
65
3.13 Interface between the region I and the region II in the FD TD
domain decomposition m ethod .................................................................
66
3.14 R eturn loss of the microstrip-to-waveguide transition structure
with 8 mil neck..............................................................................................
68
3.15 R eturn loss of the microstrip-to-waveguide transition structure
with 9 mil neck..............................................................................................
69
3.16 The housing structure for the microstrip circuitry (3-D view).
. .
70
3.17 The housing structure for the microstrip circuitry (Top view).
. .
70
3.18 The spectrum of the housing stru ctu re...................................................
71
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3.19 The prism elem ent.......................................................................
75
3.20 The FD TD and FET D interface region.................................
80
3.21 Com m unication between FD TD and FETD field...............
81
3.22 The regular brick elem ent.........................................................
82
3.23 The via hole grounded m icrostrip...........................................
83
3.24 FET D meshes for the via holes...............................................
85
3.25 FD TD staircasing model for the 0.6 mm diam eter via hole
86
3.26 |S 2 i| of the via hole grounded m icrostrip..............................
87
3.27 IS2 1 I of the via hole grounded m icrostrip..............................
88
3.28 The FD TD modeling for the two via holes..............................
90
3.29 |52i| of the grounded m icrostrip w ith two via holes..............
91
3.30 IS2 1 I of the via hole grounded m icrostrip.................................
92
3.31 The waveguide with an iris of finite thickness.........................
94
3.32 Normalized inductive iris susceptance vs. iris w idth.............
95
3.33 ISnl of the waveguide w ith an inductive iris...........................
B .l
Boundaries of the FETD dom ain...............................................
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L is t
of
T ables
2.1 Surface wave poles and residues (freq. = 30 GHz)...............................
21
2.2 Ai and B{ for complex image term s (freq. = 30GHz)..........................
21
2.3 Param eters for three point quadrature m ethod.....................................
30
2.4 Param eters for seven point quadrature rule...........................................
34
viii
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A cknow ledgm ents
The author would like to express his sincere gratitude to his advisor Professor
Tatsuo Itoh for his guidance, encouragement, and patience during this research.
A debt of gratitude is owed to Professors Yahya Rahmat-Samii, Behzad Razavi,
and Tony F.-C. Chan for serving in the committee.
The author would like to th an k Professor Ruey-Beei Wu at N ational Taiwan
University, who visited UCLA in 1994-95, for his helpful discussions. T he author
is very grateful to Dr. Hong-bae Lee for his fruitful discussions during the course
of this research. The author would like to thank Dr. Jon Gulick a t Hughes
A ircraft Company, for his valuable advice. In addition, the author would like
to express his appreciation to all the fellow students and visiting scholars in
Professor Ito h ’s group, for their assistance and helpful discussions.
The author also thanks TRW for partially funding this research.
A sincere appreciation is given by author to his parents for their constant
support and encouragement.
T he author also would like to extend a special
gratitude to Ms. Esther C. Roe and Mr. Kimin Cha for their sincere help when
he started his graduate study in America.
Finally, the author would like to thank his friends, Mr. George Kondylis and
Ms. A nastasia Karaglani for their friendship during his graduate study in UCLA.
‘
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V
it a
1967
Born, Seoul, Korea
1985-1989
B.S., Electronic Engineering
Sogang University, Seoul, Korea
1989-1991
M.S., Electronic Engineering
Sogang University, Seoul, Korea
1991-1992
Army, Korea
1993-1993
Member of technical staff
Korea Telecom, Seoul, Korea
1993-present
Graduate Student Researcher
Electrical Engineering Departm ent
University of California, Los Angeles
1995-present
Teaching Assistant
Electrical Engineering Departm ent
University of California, Los Angeles
P
u b l ic a t io n s
Dongsoo Koh, Ruey-Beei Wu, and T atsuo Itoh, “Spatial domain analysis of the
full-wave effect of a lumped resistor in m icrostrip power combiners and dividers,”
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Asia-Pacific Microwave Conference Proceedings, pp. 52-55. Taejon, Korea, Oct.
1995.
Dongsoo Koh, Ruey-Beei Wu, and Tatsuo Itoh, “A hybrid spatial domain analysis
of th e W ilkinson power divider,” Proceedings o f the 26th European Microwave
Conference, pp. 760-762, Prague, Czech Republic, Sept. 1996.
Dongsoo Koh, Hong-bae Lee, Bijan Houshmand, and Tatsuo Itoh, “A hybrid
analysis using FDTD and FETD for locally arbitrarily shaped structures,” Pro­
ceedings o f the 13th annual review o f progress in applied computational electro­
magnetics, pp. 119-124, Monterey, CA, Mar. 1997
Dongsoo Koh, Hong-bae Lee, and Tatsuo Itoh, “A hybrid full-wave analysis of
via hole grounds using finite difference and finite elem ent time domain m ethods,”
IEE E M T T -S International Microwave Sym posium , pp. 89-92, Denver, CO, June
1997.
Juno Kim , Dongsoo Koh, and Tatsuo Itoh, “A novel broadband flip-chip inter­
connection,” accepted to be published in the 6th Topical Meeting on Electrical
Performance o f Electronic Packaging, San Jose, CA, Oct. 1997.
Hong-bae Lee, Dongsoo Koh, Tatsuo Itoh, Frank J. Villegas, and H. A. Hung
“O ptim um shape design of matching stubs based on full-wave analysis,” accepted
to be published in Asia-Pacific Microwave Conference Proceedings, Hong Kong,
China, Dec. 1997.
xi
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i
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Dongsoo Koh, Hong-bae Lee, and Tatsuo Itoh, “A hybrid full-wave analysis of
via hole grounds using finite difference and finite element time domain m ethods,”
accepted to be published in IEEE Trans. Microwave Theory and Tech., Dec.
1997.
xii
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fI
A bstra ct
of the
D is s e r t a t io n
EXTENDED SPATIAL DOM AIN AND
H YBRID TIME-DOMAIN ANALYSES
FOR MICROWAVE CIRCUITS
by
D on gsoo K oh
Doctor of Philosophy in Electrical Engineering
University of California, Los Angeles, 1997
Professor Tatsuo Itoh, Chair
The development of integration technology for microwave circuits is based on
accurate and efficient full-wave analysis and design tools. A full-wave spatial
dom ain analysis is developed to analyze arbitrarily shaped planar microwave cir­
cuits. By introducing a lumped element incorporation scheme, the spatial domain
analysis is extended to analyze the microwave circuits having lumped elements
such as Wilkinson power dividers. In addition, the proposed matched load and
excitation scheme replaces the conventional voltage gap source excitation m ethod
in the spatial dom ain analysis. Three-dimensional general structures can be ana­
lyzed more efficiently using the finite difference time-domain method. Bondwire
interconnect and microstrip-to-waveguide transition structures are studied using
the finite difference time-domain method. Finally, a hybrid time-domain analysis
using the finite difference and finite element time-domain methods is proposed to
analyze three-dimensional locally detailed and curved structures. The validity of
this hybrid m ethod is demonstrated using waveguide iris problems and via hole
grounded microstrips.
xiii
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CH APTER 1
Introduction
The development of integration technology for microwave circuits such as mi­
crowave integrated circuits (MICs) and monolithic microwave integrated circuits
(MMICs) has brought the same advantages as those of the low frequency inte­
grated circuits, such as improvement of system reliability, small volume, light
weight, and mass productivity.
This integration technology is based on pre­
dictable and precise analysis and design tools, because optim ization techniques
such as trim m ing and tuning cannot be employed after fabrication [1, 2]. Since
complex integrated circuits cannot avoid surface wave excitations, radiations, and
couplings a t high frequency operations, full-wave analysis and design tools are
necessary.
Full-wave analyses solve the wave equations considering tim e varying elec­
trom agnetic fields with given boundary conditions [3]. A num ber of different
m ethods are available for solving the wave equations, and they can be classified
in many ways, for example, frequency domain or time domain analyses, or inte­
gral equation based or differential equation based formulations.
The finite difference method (FDM) solves the differential forms of the wave
equations directly [4] and the finite element method (FEM) solves the field solu­
tions using the variational forms of the wave equations [5]. The integral equations
can be derived from the application of the vector Green’s theorem on the wave
equations and the m ethod of moments (MoM) has been widely employed to solve
1
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the integral equations in electromagnetic problems [6]. The full-wave analysis
m ethods mentioned above are based on the frequency dom ain, from which the
steady-state responses can be obtained. W ith the rapid development of fast and
large memory digital com puters, th e direct solution m ethods of the differentialform wave equations in the time domain have been introduced. Time-domain
analyses give the transient response information as well as the wide-band fre­
quency response inform ation which can be obtained using the Fourier transform
of the tim e dom ain responses [7]. The finite difference tim e-dom ain (FDTD)
technique is one of the m ost extensively used m ethods due to its numerical effi­
ciency and sim plicity in the form ulation and im plem entation [8, 9].
The integral equation m ethod can be one of the m ost general and rigorous
electrom agnetic analysis m ethods [10]. Usually, electric or magnetic current dis­
tributions on the microwave circuits are unknown param eters in the integral equa­
tions and the m ethod of moments (MoM) technique is used to solve the integral
equations. T he kernel of the integral equation, Green’s function, can be modified
to simlify the integral equation by taking account of the boundary conditions.
Most of the microwave circuits in the MIC / MMIC are modeled using the planar
layered structures. The Green’s functions on the planar layered structures are
derived in closed-forms in the spectral domain [11]. The spatial Green’s functions
can be obtained by taking inverse Hankel transformations of the spectral domain
G reen’s functions, which become th e well-known Sommerfeld integrals [12, 13].
The spatial dom ain analysis needs to calculate the singular integrals when the
field points correspond to the source points, but this m ethod can be applied to
more arbitrarily shaped structures than the spectral dom ain m ethod [14]. In
addition, the num ber of unknown param eters are generally smaller than those of
FDM and FEM , because the unknown parameters are defined only on the surface
of the microwave circuits, while those of FDM and FEM should be defined in the
2
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entire com putational volume including the microwave circuits. Therefore, the
spatial dom ain analysis can be one of most efficient full-wave analysis m ethods
for arb itrarily shaped planar structures.
An extended spatial domain m ethod is proposed to analyze the arbitrarily
shaped planar microwave circuits having lumped elements. T his m ethod makes
it possible to incorporate the lum ped element effects into the spatial dom ain
analysis. The current trend in microwave circuits is making use of lum ped ele­
m ents [1]. W ith the development of photolithography and thin film techniques,
the size of the lumped elements has been reduced to be used in the microwave
frequency band. The spatial domain analysis in this dissertation is based on the
m ixed-potential integral equation (M PIE) and employs triangular patch pair ba­
sis functions which are suitable for arb itrarily shaped structures [15]. This spatial
dom ain analysis is extended to include lum ped element effects by introducing new
triangular patch pair basis functions. In addition, the proposed m atched load and
excitation scheme replaces the conventional voltage gap source excitation m ethod
in the spatial domain m ethod. The new method is applied to analyze the W ilkin­
son power combiner and divider.
The interconnect structures of different substrates like bondwires and the
transition structures between two different guided-wave structures are neces­
sary for the integration of microwave circuits [16].
These structures are not
planar structures any more. The integral equation formulations for the threedimensional structures become very complicated because all the nine elements
of the dyadic Green’s functions need to be calculated. Finding G reen’s func­
tions is not an easy task as well because the Green’s functions cannot generally
be expressed in closed forms [17]. T h e finite difference tim e-dom ain (FD TD )
m ethod and the finite element m ethod (FEM) can be effectively applied to the
three-dim ensional structures [7, 18]. B oth methods are based on the differen-
3
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tial forms of wave equations and do not require Green’s functions. The finite
difference tim e-dom ain (FDTD) m ethod using Super absorbing M ur’s 1st order
absorbing boundary condition (ABC) is developed in this dissertation [19]. In
addition to time dom ain responses, wide-band frequency responses can be ob­
tained using the tim e dom ain d a ta through the Fourier transform in the finite
difference tim e-dom ain (FDTD) analysis [8]. A domain decom position m ethod
for the finite difference tim e-dom ain (FDTD) method is also developed to save
memory in case the com putational dom ain does not have a regular box shape.
T he finite difference tim e-dom ain (FDTD) method has a num ber of advan­
tages in analyzing three-dim ensional microwave structures [20]. M aking use of
the uniform mesh in the finite difference time-domain (FDTD) algorithm does not
require any special mesh generation scheme and storage for the mesh. However
the finite difference time-domain (FD TD ) analysis has difficulty in dealing with
curved structures and needs very fine mesh in the entire com putational dom ain
for locally detailed structures. Several methods have been developed to over­
come these difficulties, including th e FDTD algorithm in curvilinear coordinates
[21]-[23], the discrete surface integral (DSI) method [24], the locally conformed
FD TD algorithm [25], a hybrid F V T D /F D T D algorithm [26] and so on. Recently,
a hybrid m ethod was developed to model the locally curved structures by incor­
porating the finite element m ethod (FEM ) into the finite difference tim e-dom ain
(FD TD ) m ethod [27]. This m ethod utilizes the advantage of the m ore flexible
FEM while retaining all the advantages of the FDTD method. However, the
FD TD and FEM mesh m atching m ethod in the interface region introduces diffi­
culty in the mesh generation of the FEM region. This dissertation proposes a new
finite difference time-domain (FD TD ) and finite element tim e-dom ain (FETD )
hybrid m ethod by introducing an interpolation scheme for com m unicating the
FDTD field and the FETD field. In this method, one can avoid th e effort of
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fitting the FET D mesh to the FDTD cells in the interface. T his hybrid m ethod
is applied to the via hole grounded microstrips and the waveguide w ith an iris of
finite thickness.
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C H A PT E R 2
Extended spatial dom ain analysis
The extended spatial dom ain analysis is proposed to analyze the arbitrarily
shaped planar microwave circuits having lum ped elements.
This m ethod in­
troduces an incorporation scheme of the lum ped element circuit equations into
the sp atial dom ain analysis formulated by the m ixed-potential integral equation
(M PIE). The m ixed-potential integral equation (M PIE) is obtained by apply­
ing th e boundary conditions on the electric field integral representations based
on vector and scalar potentials [14, 28]. T he m ixed-potential integral equation
(M PIE) has an advantage over the standard electric field integral equation (EFIE)
in th a t the M PIE Green’s function has weak singularities due to the introduction
of th e surface charge distribution which is related to the surface current by a
two-dimensional continuity equation [29].
T he approach of the extended spatial dom ain analysis for microstrip mi­
crowave circuits can be divided into four steps:
1. o b tain the spatial Green’s functions on the conductor-backed dielectric slab,
2. derive the mixed-potential integral equation (M PIE) of the current distri­
bution on the conductor by enforcing the boundary condition such th a t the
to ta l tangential electric field is zero on the conductor surface,
3. express the current with suitable basis functions and apply the m ethod of
m om ents (MoM) to obtain a m atrix equation of the expansion coefficients,
6
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»
4. incorporate the lumped element effect into the spatial domain analysis by
m odifying the system of equations based on circuit characteristics.
2.1
S p a tia l G reen ’s fu n ction s on th e con d uctor-b ack ed di­
ele c tr ic slab
• (x,y,z)
conductor
Figure 2.1: Conductor-backed dielectric slab.
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Spatial Green’s functions on the conductor-backed dielectric slab as shown in
Fig. 2.1 can be obtained by taking inverse Hankel transform ations of the spec­
tral dom ain G reen’s functions, which become the well-known Sommerfeld inte­
grals [12, 13]. It is time-consuming to find the spatial G reen’s functions on the
conductor-backed dielectric slab by directly evaluating the associated Sommer­
feld integrals. Recently, an efficient method based on the Sommerfeld identity
and the P rony’s m ethod has been proposed to obtain the closed-form Green’s
functions [30]. In this dissertation, these closed-form G reen’s functions are used
to improve the efficiency of the numerical com putation.
Instead of using the
P rony’s m ethod, more systematic m atrix pencil m ethod [31, 32] is employed to
get asym ptotic exponential functions of the spectral G reen’s functions.
The G reen’s function of an electric dipole on the conductor-backed dielectric
slab can be derived using the electric Hertzian potential fl [12, 13]. Using the
electric H ertzian potential fl and the Lorentz gauge, the magnetic and electric
field intensities can be expressed as
H = juje0€rV x II,
( 2 . 1)
E = (V V • + k2) ri,
( 2 .2 )
and the electric H ertzian potential 11 becomes the solution of the inhomogeneous
vector Helmholtz equation,
(V 2 + fc2) n = - ( j W 0er ) - ‘ / .
(2.3)
In free space, th e electric Hertzian potential n has the sam e direction as the
excitation current density J . The electric Hertzian potential II in the inhomoge-
8
'
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neous planar layered structure has only the vertical component for th e vertical
current excitation. However, the horizontal current excites the H ertzian potential
having the vertical component as well as the horizontal component. Since planar
microstrip structures are the structures of interest, this dissertation focuses on
the horizontal current excitation case.
For the x directed current,
J = xld x '6 (x )6 (y )6 (z — z1),
(2.4)
the Hertzian potential will have x and z components [12]:
II = Uxx + n Zz.
(2.5)
By exploiting the homogeniety of the planar structure along the x an d y direc­
tions, two-dimensional Fourier transform defined as
OO
n=
OO
J J
n
e x p [ - j( k xx + kyy)]dxdy,
— OO — OO
can be applied to the vector Helmholtz equation (2.3) and the result is
where
!
i
9
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(2.6)
ri
k zi = sjk f - k l - fcjj,
Im { k zi) < 0 ,
i = 0,1,
(2.9)
and
k f = u 2ii0e0eri,
I 0 = {jujtQer l ) lIdx'.
( 2 . 10 )
The Fourier-transformed Helmholtz equations (2.7) and (2.8) have the general
solution which satisfies the radiation condition, expressed as
The unknown coefficients in (2.11) and (2.12) can be found by applying the
boundary conditions of electromagnetic fields along the interface between air
and the dielectric substrate as well as the ground plane of the substrate. The
calculated results are shown as
e-jkzo\z-z'\ _
noz — I q
e-jk zo(z+z')
kx (K - 1)(1 - e~^4k:lh)e~:’kz0<‘z+z^
10
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(2.13)
nl2 =
/0
kx (K - 1)(1 - e- l2k--lh)e -jk'-oz' [e>*slS + e- jk'-l(:+2h)]
j K (kzl + kzQ)(k:l + K k z0)(l + f if ^ e - - ' 2*=i/l)(l - jf c jf f ilje-J2* ^ * ):
(2.16)
where
AT=^.
^Or
(2.17)
Now the spatial vector H ertzian potential can be obtained using an inverse
Fourier transform ,
OO OO
H = ^
/
j ^ exp \j(h xx + kyy)\dkxdky,
(2.18)
— OO — OO
which can be transform ed into an inverse Hankel transform using the following
procedure. By the substitutions,
kx = —kpcos£
(2.19)
ky = —kpsinti
and
x = p COS(f)
( 2 . 20 )
y = p sincf)
(2.18) becomes
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
n =
/ / S e - ^ coa^ k pdkpdi
4^2
—7T 0
i 00- r -i *■
=
—
/ ft
27r 7
0
=
—
L
[ e ~ JpkpCOS^ ~ ^ d f
27r J
kpdkp
—7T
1 °° ~
2tt J 11
o
1 00 — ^ fi H ^ \ k pp)kpdkp.
( 2 . 21 )
W ith the relationship between the vector potential .4 and the vector Hertzian
potential ft,
A = juje0erHo n ,
( 2 . 22 )
the x component of the vector potential due to the x-directed electric dipole can
be obtained using (2.13) and (2.21) as
A xx =
ou
Afo f Id x '
Air
j 2 k so
J
, - j k zo \ z - z ' \ _
( k - i - k - n \ , e - j 2 k zlh
\ k z i + k zp j ___________ ^ - j k . 0 ( z + z ' )
l i f k : l - k :0 \ p - j 2 k z l h
1 ^ \ k zi+kz0J e
H g \ k pP)kpdkp.
(2.23)
Therefore, the spatial Green’s function, Gx£ due to the rr-directed electric dipole
of u n it strength becomes [30]
12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
00
G“ =
/
1
jn2 kTz0
(2-24)
— OO
where
f k ; l - k :p \
R te
e ~ j 2 k z lh
\ ------------•
=
1
+ ( f cf f e)
(2.25)
e-m " k
T he scalar potential of a point charge of the x-directed electric dipole can be
derived using [33]
1 dGq _ ju f d G f
j u dx'
k2 [ d x
d G z* '
+ - od zf - •
(2.26)
'
(2 2 7 >
The Fourier transform of (2.26) gives
d < = 7 e { G~f - tk M
2 2 °'
Gxx and G z£ above the substrate can be calculated from (2.13) and (2.14) using
(2.22), and the substitution of Gx£ and G zx into (2.27) gives [30]
G = ——^— \ \ e - jk^ z- z^ +
e0 j 2k zo *-<•
J
4- R qe~jh^ z+z'A ,
J
where
R = _________________ 2A;0(1 - K )( 1_- _e~**klth)_________________
’ (fei +*«o)(fc,i +K k, o)(l +
-
!tT R to e~J2’“'h'>
13
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.28)
By applying the sim ilar procedure as (2.21) to (2.28), the spatial scalar Green’s
function can be obtained as
i
^
l
H
G" = 4 J j2
00
+
(Rte +
Ha’\ kpp)kpdkp.
— OO
(2.30)
Finding the spatial Green’s functions, (2.24) and (2.30) by direct integrations
can be tim e-consum ing because of the oscillatory behavior of the integrands.
However, the spatial Green’s functions can be efficiently calculated by making
use of the Sommerfeld identity [11, 12],
,-jk o r
1_
J2
<JU
/
d h j £ H $ \ k pp )e -* -
(2.31)
where
k2 -
(2.32)
and
r = yj p1 + z2,
p = yjx2 + y 2.
(2.33)
T he Sommerfeld identity has the physical meaning th at the spherical wave can
be the infinite sum m ation of the cylindrical waves propagating in the p direction
m ultiplied by plane waves propagating in the z direction, over kp [11].
The procedure for obtaining the spatial Green’s functions is divided into three
14
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
steps [30]. First, quasi-dynamic image terms can be extracted by assuming zero
frequency condition such as
k0 =
ss 0.
(2.34)
W ith the approximation (2.34), R te and R q in (2.25) and (2.29) become
R teo = - e ~ ]2k-lh « _ e-J2h-oh,
(2.35)
_ P (1 - „ - * * ..* ) __ P( 1 - e-W"*")
q0
1 - P e-i2k--'h ~ 1 - P e~i2kz°h ’
^
'
where
P = f 0r— l l r = 1— ! l .
^Or + e l r
1 +
(2 .3 7 )
Cr
W ith i?riEo and R q0, the quasi-dynamic image term s can be extracted from Gxx
and G q using the Sommerfeld identity, which gives
= GS + ^
1
G " = G ,° + 4 ^
f f f (k ,p ) i k „
/
—oo
°°
/
(2.38)
1
] 2 k f i {RTE + R q ~ R t e 0 " «*> )e ' 2‘ ,0<' +‘') f lo2\ k f p ) d k „
(2.39)
where
S I^
J’
15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2 40)
, - j k 0r0
G,}o =
47ren
e—jkorn '
e ~jkorg
+ P — z - + ' £ P ’,~ '( P 2 - 1)
n=L
r0
(2.41)
and
r'Q
= \jfp- + (z + z' + 2h)2
r'o
=
rn
= \Jfp- -F (z + z' 4- 2nh)2.
\JfP
+ (z
+ z')2
(2.42)
These quasi-dynamic image terms dom inate in the near-field region.
Next, the surface wave terms can be extracted by perform ing the integration
using the residue theorem since the surface wave poles exist along the real axis
of the complex kp plane for lossless spectral Green’s functions [34]. Gxx and G q
can be rewritten as
j
G f = C% + G % „ + ^
~
F
A(k ,)e -ik^ » ^ H S - \ k eP)kedkp,
(2.43)
—oo
G q = Gqo + Gq,sw +
1
°° 1
/ — — Fq(kp) e 'jk^ ^ ’)HSl)(kpp)kpdkp,
47T€o —
Joo jAkzO
(2.44)
using the surface wave terms,
= £ ( - 2 T j) • £
p(TE)
Gq,sw — .
(2-45)
R esqHQ \kppp)kpp,
(—2ttj)
p(TE,TXf)
16
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2-46)
where R es a and Resq are expressed as
e - j k z0( z + z ' )
Res a =
jZ k zQ
lim {kp - kpJRTE,
(2.47)
lim (kp —kpp)(RTE + Rq),
(2.48)
up“__L
+kpp
ft>
""pp kp—
e - j k zQ( z + z f )'
Resq =
j 2 k z0
fCn“tKq
which can be analytically calculated using the L’H opital’s rule. FA and Fq of the
rem aining integrands of G“ and G q are
FA(kp) = R t e — R teq — 5 3
p(TE)
kp
^pp
^pp
R e s Aj 2 k z0ejk^ z+z' \ (2.49)
Fq(kp) =
R t e + Rq — R teo — R qo —
53
p{T E ,T M)
kp
kpp
kp -t- kpp
R e s qj 2 k zOejk:o{z+z').
(2.50)
The surface wave terms affect the far-field along the substrate.
Lastly, the remaining integrals of Gxj£ and Gq can be calculated by asym ptotic
expansions of FA and Fq using exponentials. In other words, if F a and Fq in (2.49)
and (2.50) can be expanded as
N
F * iK ) = ' £ a i e - b‘k- \
(2.51)
i=l
N'
F,{k„) =
(2.52)
i= l
the Sommerfeld identity can be applied to the remaining integrals, which gives
17
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G XX _ (~\XX . /~1XX
A
~
Gq
AO +
I /~1XX
l* A * w +
^-r A , c i i
GqO + GqSW-+- GqlCi,
(2.53)
(2.54)
where
/-in _
N
p - j k o r;
i= l
'«
V'
n = yJfP + iz + z' — jb{)2,
1 * t e~ikari
Gq,a = j47re0
— t-Y,=,l ai IT—
’
r,
ri = 'Jp2
+ (z + ~ ~ M ) 2v
(2.55)
(2-56)
The complex amplitudes, a, and a', as well as the complex distances, r t and r[
can be obtained using numerical m ethods such as Prony’s m ethod [35] and the
m atrix pencil method [31, 32]. By introducing a parametric equation for kzQ as
k zo — Atq - j t +
Q < t< T 0
(2.57)
F .i and Fq in (2.49) and (2.50) can be rew ritten as
N
(2.58)
i= l
Fq(kp) = ' £ A l e *
:=1
(2.59)
with
:
Ai = aie_SiTo/(l+jTo),
(2.60)
B = 6y i+ £ o )
To
(2.61)
18
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Now, Ai and B{ can be obtained from Prony’s method or the m atrix pencil
i
m ethod by uniform samplings of F a and Fq a t t = n A t. The truncation point,
To is set to be larger than
in order to avoid the surface wave effect. The
calculated complex image term s are related to leaky waves and dom inate in the
interm ediate fields.
The derived closed-form Green’s functions Gx£ and Gq in (2.53) and (2.54)
were tested using a conductor-backed dielectric slab with the high dielectric con­
stant (er = 12.6) and the thick substrate (h = 1mm) [30]. Fig. 2.2 shows the
calculation results of the closed-form vector and scalar Green’s functions. The
closed-form G reen’s functions include the full-wave effects such as surface waves
and leaky waves. Table 2.1 shows the surface wave poles and the corresponding
residues for the 30 G H z excitation. The vector Green’s function, G1* has one
T E mode surface wave, while the scalar Green’s function, Gq has one T E mode
and one TM m ode surface waves. T he complex image terms take into account
of the leaky wave effect. For the 30 G H z excitation, the leaky wave effect was
considered using two complex image term s for both the vector and scalar G reen’s
functions. Table 2.2 shows the .Aj’s and B,-’s for this case, calculated by the ma­
trix pencil m ethod.
19
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a) Freq. = 10 GHz
(b) Freq. = 10 GHz
o
3
a
<
*
x
x
to
O
o>
o
— Closed-fqrjn
0 Q Numerical integration
•2
0
(c) Freq. = 30 GHz
-1
0
1
(d) Freq. = 30 GHz
o
3
£
—
a
«
o(A
a
<D
a
<
<
«
x
x
«
O’
(0
<2
O)
o
(5
o>
o
•2
0
(e) Freq. = 50 GHz
■2
0
(f) Freq. = 50 GHz
1
o
|3
<a
«
x
x
CO
g
O)
o
0
■2
0
■2
Iog(k0*rho)
!og(kO*rho)
Figure 2.2: The am plitude of normalized Green’s functions, G xj£ and G q.
20
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
r[
i
Green’s function
(30 GHz)
surface wave
mode
pole (kpp)
residue
TE
1.666 k0
+ 0 .3 6 9 + j0.531 x 10~14
TE
1.666 k0
-0.236 - j0.676 x 10“ 14
TM
2.641 k0
-0.351 + j0.436 x 10~14
Gq
Table 2.1: Surface wave poles and residues (freq. = 30 GHz).
G reen’s function
(30 GHz)
G f
Ai
Bi
—1.9922 + j'1.7212
-0.1416 -jO .1 1 4 4
-1.2946 - j ‘0.4864
+0.3669 - j'0.7691
-3.6786 - jO.0567
-0.0086 + j'0.0052
-0.6099 - 70.6894
+0.0180 - j0.0736
Gq
Table 2.2: .4, and Bi for complex image term s (freq. = 30 GHz).
2.2
T he m ixed -p oten tial integral eq u ation (M P IE ) and
th e m eth o d o f m om ents (M oM ) form ulations
Using the mixed potential formulation, E = —ju)A — V V, the mixed-potential
integral equation (MPIE) can be derived by applying the boundary condition
such th a t the tangential electric field on the m icrostrip conductor surface is zero,
where A represents the magnetic vector potential and V represents the electric
scalar potential. The mixed-potential integral equation (M PIE) is expressed as
21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
h x E excitation = n x ju/G a * / -
(G q * V • J )
(2.62)
where C?a and G , are the vector and scalar Green’s functions, respectively, of the
conductor-backed dielectric slab and * represents the convolution integral [36].
W hen applying the m ethod of moments (MoM), the triangular-patch-pair
basis function
r in T+
2.4+
fn ( f) =
2A ^ Pn
0
(2.63)
r in T~
otherwise
is chosen, since it is well-suited for modeling arbitrary conductor shapes [15]. The
basis function (2.63) is defined on the geometry shown in Fig. 2.3.
■-A4
Figure 2.3: Geom etry for the triangular-patch-pair basis function.
22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The unknown current distribution on the conductor surface can be expanded by
(2.64)
n=l
where ln is the length of the common edge in the n th triangular patch pair, In is
the total current flowing across the common edge, and N is th e to tal number of
triangular patch pair criss-crossing the microstrip structures. T his introduction
of the division of
f n [f)
by
ln
gives the unknown coefficient
In
w ith the unit in
Ampere. This new scheme makes it easy to incorporate circuit equations of the
lum ped elements into the spatial domain analysis.
After taking the inner product of (2.62) w ith the same modified current basis
functions as in (2.64), we arrive a t a system of equations,
N
5 3 Z mnln = Kn^ms,
™, = 1, 2, . . . , N ,
(2.65)
n=l
where
Z-mn =
* fn if)) ~
V re?, * V • f n ( f) ) ) ) , (2.66)
and Vm5ms represents the voltage gap source applied onto the s th patch.
The evaluation of the m atrix elements, Zmn’s requires various num erical tech­
niques including singular integrals when the field points are close to th e source
points. Using the basis function, (2.63), the first reaction integral term can be
w ritten as
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(/»(»=), G W » ( d )
=U+4'*■fe 4 B*WU*"+£ 4 B^.)**)*
+2A^It-P"'' (2I+4G
-4(r1r"*^£fa'+2A^4 * '
(2.67)
In order to calculate the second reaction integral term , the surface vector calculus
identity can be first applied [15], which gives
(L(f), v (a, .
V • /> - ) ) ) = -(G , » V •
V . /1(f)).
(2.68)
Next, using
r in T+
V • / m(f) = <
(2.69)
r in T z
0
:
otherwise
(2.68) can be expressed as
=-44 (44
-44
*
+4 4 ( 4 4 G"(rT”)<is' _ 4 4 G’(r1f-)ds') d s (2.70)
24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
When calculating (2.67) and (2.70), the l / r 0 singularities within the G reen’s
functions need to be extracted and calculated separately.
In order to decide
whether or not to extract the singularity, the unitless distance param eter crmn
defined as
m, n € 1, 2 , . . . , N ,
(2.71)
can be used [37], where f£, and f£ represent the global position vectors from
origin to the centroids of the m th and nth patches having areas .4m and .4n,
respectively. Now, by defining the threshold a th and using the unit step function
U(crth — crmn), the 1/ro singularities can be extracted from G reen’s functions,
which gives the reduced Green’s functions :
= r.40 +
G A , s w + G A ,c i,
ho U{aa1
where
^mn)
(2.72)
and
r,
where
=
Gq -
1 U (a th Omn)
47Te0
r0
Tq0 = Gq0 -
1
47Te0
^mn)
r0
(2.73)
25
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The unit step function U (ath — crmn) is nonzero only when amn is sm aller than
(Tth- Using the reduced Green’s functions, (2.72) and (2.73), (2.67) becomes
( fm ( r ) iG A * fn (r))
-i 4 *•(*44 ™ ds'+w4
+Ui 4* -($44
+k 4
+i44 «•fe 4
+ 4
4 *' G44
+£ 4
*
*
*’
(2.74)
and (2.70) can be rewritten as
( / m W , V ( G ,* V - / ; ( r ) ) )
--4 4 (44
+4 4 ^ UJ* ^ ds')ds
+4 4 (44
+4 414^44=^)*
+44 ( 4 4 +4 4 i4£44m
n)JJ'
-ds' ds
"-4-4 J/t’- (\ A4~ h/ n r,(f]f„)*' + i=/
A - hT ~
1
47T£o
a m n)
U ( a th
A
r0
(2.75)
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Since the integral terms having the reduced G reen’s functions are locally smooth,
the double integrals of (2.74) can be reduced to the single int.pgra.ls as shown in
( fm { r ) ,G A * f n (r))
L-mln - c -f
Po^mk
P
o^m k
f
-*f /"
U (& th
r - f
16tt.4+.4+ *Jt1m
+ m Jt +
t*0
( z c +1 —
j
■kf ^ A^i ( ri* i f „ ) ^ d si' +
f
^
^
j ,
pPo^mk
k m k
[f tfr f[
• f ^ A ( f % \ f n)fa d s ' +
+ 4 ^ p" Jk*irt
'r")p’'d s + \16vA+A~
6 i A ± A.k; k Pmp" Itk, ^ m k -c +
(m
Im k —
c—
f
-JJ+ A . ' b
f=r f s c —i ~ \
j„ i
,
^ ( r . M /V * +
U {(T th
°m n)
p - d■s /a ,s
&mn)
---- ^---- p' dsds
Po^mk
p k m k
[ f_p„
r— j fnf U ( ( T t h
Jt
16ttA -.4+ .k Pm<'T+
Po^mk
fL
fL . ^r.4(rTlr
A i r ^ l ^ „)K
P ^ dM
s' +
+ T ^ r p! I <r
j4-4n
k f a •■7r„
K 7'T167T/1-.4lb7T.4m.4n 7r~
,k Pm'
r" *in
& m n ) -*+. , / ,
f t t * d.
jr„ds'ds,
r0
(2.76)
where fjjf represent the global position vectors pointing the centroids of T * . In
the sam e manner, (2.75) can be approximated as
( / l ( 0 , v ( c , . v - /> -} ))
=
- j1A+
t t Jt+
I r q, ( d
lm
■ nkk
f
r . ) d s ' - ■■
u
47re0.4 + .4 +
»-> / - c +1 -« \ jj i .
lI imn k
f
f
[
Jt+ Jt+
f
^ k r<{rMds+t ^ x k k
^■mk
f
'At k
\
, I
^mk
f
r’(r" K)d*+ toSTt* k
/ -xz— | — \ i /
— ^f r,(rm
Irjd, -
Im ^ n
f
f
k
f
r0
tmnn) ) , , / /,
&< m
UU\{G&t thh
—
- - ^ d s ’d s
S -------- d s d s
U {(7th
^mn)
^
& \& th
, /,
dS*
^m n)
, / »
dsds.
(2.77)
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t
!
W hen evaluating the nonsingular integrals in (2.76) and (2.77), it is convenient
to em ploy the local coordinate system [15, 38, 39].
A 2
c
A i
§= 1 —
A n
A 3
A 2
11= ——
An
„
5
=
A3
A n
2
Figure 2.4: Normalized area coordinates for a local coordinate system.
Fig. 2.4 shows the local coordinate system. Using the normalized area coordi­
nates,
the source position vector r ' can be rew ritten as
r' = £fi + r}f2 + <T3,
(2.79)
and th e surface integral over the nth triangular patch can be transform ed as
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[
JT n
F(r')ds' = 2A n [
+ -qf2 + (1 - £ - T])f3]d^drj.
f
JO J 0
(2.80)
Therefore the nonsingular integrals in (2.76) and (2.77) can be calculated as
j L P A(fa^\fa) fa d s ' = 2.4* [(f\ - r 3)/?r 4- (r2 - f 3)/„r + (r3 - r ^ ) /r]
***■n
f
** * fi
^ A( fa ^ \rn) fa d s ' = -2 .4 * [(r4 - f 3) / ?r + (r2 - f 3)/„ r + (r3 ~
Kr]
(2.81)
(2.82)
and
L
^*n
r ,(r S = |r - )ds' = 2 4 j / r
(2.83)
where
/ ' f a tr(f*J?)dt;dr,
V =
JO Jo
V = /
L
(2.84)
(2.85)
Jo Jo
/ r = f l f lr> r ( f a l \r')d£dT).
(2.86)
Jo Jo
In th e evaluation of (2.84), (2.85), and (2.86), a three point quadrature method
is applied using the sample points (fa + ri)/2 , i = 1,2,3. Hence, (2.84), (2.85),
and (2.86) can be calculated as
V = E f r ( e . ’/) » ri
1=1
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.87)
s a m p le p o in t
(^,^7)
( ^ + n )/2
( -3 ’ 6 '
'■
(f£ + r 2) / 2
(I 2)
' 6! 3'
(f£ + T3) / 2
'‘6 ’ 6 '
Wi
I
6
1
6
1
6
Table 2.3: Param eters for three point quadrature method.
( 2 . 88 )
i= l
h = E m ,r iW ,.
(2 .8 9 )
t= l
Table 2.3 shows the sample points and corresponding (£, t j ) coordinates with
weighting coefficients Wvs.
The evaluation of the singular integrals in (2.76) can be performed by dividing
the integrands into the removable singularity term s and the l / r 0 singularity terms
as [40]
U{<Jth ^mn) -4-j / j
------------------ pZds as
ro
ds
= U (ath - crmn) f (? - f £ ) [ f ~ - r - ds + (P - f ^ ) [
ds'
JT+
Jt + r - r'\
Ji
It + \r —r'|_
= & (&th
^mn) ^ 1
^"tiI ' S 2 “I" S 3 + ( r n j • r mi)iS4
{fnl 4“ ^m l) ' ^oj •
(2.90)
Similarly,
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f
U
-4-
f
U { a th
Pm' k
°~mn)
n
, ,
p*
- a m n ) [5 , - f„4 • S 2 + S 3 + (f„ 4 • r m i ) S 4 - (f„ 4 + r m |) ■S s]
= -U io th
(2.91)
f
k
Xf
P m k
— °m n ) - + j i j
^
----- r 0---- «•**
U {& th
@Tnn) [S i
^*nl " *?2 "h ^ 3 “h (^nl
^'m4)‘^4
( ^n l d* ^m4) ’
(2.92)
f
x— f
& i& th
k ^ ' k -
O’mn) — j t j
-----S-----^
U ( ( 7 th
^mn)
^3 “h (j~Ti4 ‘ ^*m4) *^4
^*n4 ‘ *^2
(^n 4 ~F Cm-1)
**5gJ .
(2.93)
Since the integrands of singular integral term s in (2.77) have only l / r 0 singular­
ities, the singular integrals can be calculated as
[ [
Jt ± Jt£
ds'ds = S 4.
(2.94)
tq
S i through S 5 are defined as
N
Si = Y l Wi(Psi ■Ai)
:=1
(2.95)
yv
s3
S 2 = ^ 2 w iA i
i=i
N
= Y ^ wi\T'si\2Bi
i=L
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.96)
(2.97)
source
p oint
h
A :S
Figure 2.5: G eom etric param eters associated with the line integral along Aj S for
calculating A., an d Bi.
32
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AT
S4 =
(2.98)
Y l w i& i
1=1
N
B5 — ^ ' WiBit s{,
(2.99)
1=1
where
1 3
( 2 . 100 )
1
3
-
//+ 4- P ^ '
( 2 . 101 )
and the geometric param eters associated with Ai and Bi are defined in Fig. 2.5.
In calculating Si through 5s, a seven point quadrature rule can be applied [41].
Fig. 2.6 and Table 2.4 show the sample points and the weighting coefficients.
c3
c2
cl
Figure 2.6: Sample points for seven quadrature rule.
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sample point
position
Wi
1
rc
270
1200
2
rc + '/if+ l(rc3 ~ r c)
3
tc +
4
rc + ^ ( r c2 - r e)
5
rc - & = ± (r* ~ r c)
6
rc
7
rc
^ ^ ( r cl
^
^
- r e)
1(rcl - r e)
155—\/l5
1200
l5 5 + \/ l5
1200
1(rc2 ~ r c)
Table 2.4: Param eters for seven point quadrature rule.
The RHS of (2.65). VmSms can be obtained from the moment m ethod proce­
dure using the voltage gap source applied onto the sth patch as shown in Fig.
2.7. When the voltage gap source, V* induces the electric field intensity, E l, the
moment m ethod procedure gives
—
[
2^4+ Jt +
p i - E'ds -
f p i ■E'ds
2A ~ Jt ~
( 2 . 102 )
Since the induced electric field exists only in the gap, the surface integral on the
patch reduces to the surface integral only on the gap. Hence, (2.102) becomes
34
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- o
1
2
I
Vs
A
A
A
Figure 2.7: Voltage gap source applied onto the sth patch.
A / 2 Zm
T.
,
A / 2 Zm
= ~7T
f
A /2t,
A / 2 Zm
f h + ^Td s + T T r
2Am o o
| h +im v; ^ f y r l r n V ,
A+ 2
.4- 2
= K.
[
[ h~~ds
m oo
(2.103)
W hen the voltage gap source is excited on the sth patch, (2.103) can be concisely
expressed as VmSms in the system of equation (2.65).
35
i
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.3
In co rp o ra tio n o f lum ped resisto rs into th e sp a tia l do­
m ain an a lysis
To include the lum ped resistor effect into the spatial dom ain full-wave analysis, a
new type of triangular patch pair has been proposed as shown in Fig. 2.8. Since
3
3’
2
2
’
Figure 2.8: New type of triangular patch pair for th e lum ped resistor connection.
Resistor
Figure 2.9: Example of resistor connection.
current flowing out of the edge of one patch m ust flow into the edge of the other
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t
patch, it is reasonable to assum e a new type of triangular patch pair in which the
common edges are connected through the resistor. Fig. 2.9 shows an example of
connecting the new type of triangular patch pair in order to take account of the
resistor effect. The new patch pair become the N + 1th patch pair when N patch
pairs criss-cross the m icrostrip conductor portion.
It is well known from circuit theory th at the induced voltage in the simple
lum ped resistor is related to th e new patch current I^+i by the following equation,
Vn +I = Zrlr ~ (R + j u L ) I tf+ 1-
(2.104)
This circuit equation (2.104) can be directly incorporated into the system of
equations denoted by (2.65). As a result, the new system of equations, including
the current distribution on the conductor and the lumped device current, becomes
jV
+1
5 3 Z'mnIn = VmSms,
m = 1 ,2 ,-----N + 1,
(2.105)
n=l
where the m atrix element Z'mn is the same as th a t defined in (2.66) except th a t
the self element of the new patch now becomes
Zn+i,n+i
— Z n + i , n + i 4- Z r
—
%n + i ,n + i + R + j u L
(2.106)
because of the lumped resistor connected in series.
The resistor effect was studied using a simplified T-junction m icrostrip power
combiner / divider configuration as shown in Fig. 2.10. In order to investigate the
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C )
(d )
Figure 2.10: Schematic diagram for a microstrip power combiner / divider.
(a) In-phase excitation without resistor.
(b) In-phase excitation with resistor.
(c) Out-of-phase excitation without resistor.
(d) Out-of-phase excitation with resistor.
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
(a)
4
2
(wiu)A
0
-2
-4
0
5
10
15
x(mm)
(c)
4
1. 1
i.f
2
t/l
t J
(uuiu)A
.........//I
. * /\\
? „
E 0
............. . l i t !
...................................
0
5
10
. . . »
>
f. \
1,1
x(mm)
-4
\
/
1
f . t
t,t
-2
15
t I r i
..........................
0
5
10
x(m m )
Figure 2.11: Surface current on the T-junction with and w ithout resistor.
(a) In-phase excitation w ithout resistor.
(b) In-phase excitation with resistor.
(c) Out-of-phase excitation w ithout resistor.
(d) Out-of-phase excitation with resistor (R = 10 f2, L = 0.1 nH).
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
15
resistor effect, both in-phase and out-of-phase excitations at the two sym m etrical
arm s were sim ulated. In the case of in-phase excitation, the voltage difference in
the resistor is zero because of the symm etry, and the resistor does not affect the
field distribution. In contrast, the out-of-phase excitation would induce a voltage
difference in the resistor. The resistor effect, thus, would be significant on the
T-junction.
T he T-junction was designed on a substrate with er = 2.33 and thickness
of 20 mil. The voltage gap sources of 10 G H z and Vs —
1
V were applied as
excitation sources as shown in Fig. 2.10. Fig. 2.11 depicts the surface current
on the T-junction. In the case of in-phase excitation, a magnetic wall exists a t
the y = 0 plane due to the even sym m etry. The resistor (R = 10 fi, L = 0.1 nH),
even though present, is open-circuited on the PM C, and consequently, there is no
difference between Fig. 2.11 (a) and (b). In the case of out-of-phase excitation,
the y = 0 plane is equivalent to a perfect electric wall.
If the resistor is in
plase, a large current would flow through the resistor. As a result, the current
distributions shown in Fig. 2.11 (c) and (d) behave differently. Fig. 2.11 (d)
clearly shows th a t current flows through the resistor.
2.4
A n alysis o f W ilk in son p ow er divider and com b in er
The proposed extended spatial dom ain analysis was applied to analyze an actual
W ilkinson power divider. Power combiners and dividers are essential com ponents
in microwave circuit integrations. Among various implementations, the threeport T-junction configuration is the m ost popular. Because all three ports of the
lossless reciprocal junction cannot be m atched at the same time, im plem enting a
lossy sim ultaneous im pedance-m atching junction with a lumped resistor becam e
a well-known technique in planar power combiner and divider designs [42].
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A m icrostrip Wilkinson power divider is modeled as shown in Fig.
V ss2
2.12.
Z s2
w2
Vss i •( \ y \
Z r = R r+ jco L r
V ssI " ( f \ j
0-wHi'
V ss3
Z s3
Figure 2.12: The analyzed m icrostrip Wilkinson power divider structure.
In order to consider the m atched excitation and load, another new type of the
triangular patch pair is introduced in Fig. 2.13, in addition to the triangular
patch pair for the resistor (Fig. 2.8). T he load can be regarded as a special case
of Fig. 2.13 when the voltage source is shorted. The matched excitation and load
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3
_ Vsi
A/
Zi
2
Figure 2.13: New type of triangular patch pair for the matched excitation.
scheme was tested using the microstrip line modeled in Fig. 2.14. Fig. 2.15 shows
the calculation results. The m icrostrip line is designed to have the characteristic
impedance, Z q = oOQ, (w = 1.464 m m ) on the 20 mil thickness substrate having
the relative dielectric constant, er = 2.33. Both an open-ended and a matchedloaded (Rl = 50 Q) microstrip were simulated by applying the 10 G H z sinusoidal
wave of 1 volt amplitude with the source impedance, R s = 50 fi. It verifies the
new matched excitation and load scheme th a t the matched-loaded m icrostrip has
about 1.15 standing wave ratio (SWR) while the open-ended m icrostrip has SW R
close to infinity. The small variation in the current magnitude on the matchedloaded m icrostrip may be due to the imperfect 50
characteristic impedance
of the microstrip. The moment method results were also compared with the
circuit theory results. This comparison shows th a t the new matched excitation
and load scheme can simulate the exact am ount of voltage excitation which is
difficult with the voltage gap source excitation. The small deviations between
the moment m ethod results and the circuit theory results may be caused by the
42
t
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w
'K\}
Vs
R1
Figure 2.14: Matched excitation and load for the microstrip.
(w = 1.464 m m , I = 14.640 m m )
o p en -en d ed
<
Zl = 5 0 (ohm)
E
X
—
Method of Moments
Circuit Theory
x (mm)
Figure 2.15: M agnitude of Ix along the microstrip.
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
negligence of the fringing effect in the circuit theory.
W ith the new m atched excitation and load scheme as well as the lum ped
resistor connection scheme, the application of the method of moments to the
W ilkinson power divider as shown in Fig. 2.12 gives the system of equations,
W +5
£
Z'mJn =
m = 1 , 2 , . . . , N + 5,
K n,
(2.107)
n=L
where N is the number of patch pairs on the conductor part and the additional
five term s come from the four matched excitation patch pairs and the one lum ped
resistor patch pair as shown in Fig. 2.12. Z ' ^ s include the contribution from
the patch current, i.e., Zmn’s expressed as (2.66), and the induced voltage term s
due to the current flowing through the resistor elements, if any. Let the m atched
excitation patch pairs be the s i', s i ” , s2, and s3th patches, and the lum ped
resistor patch pair be the r th patch in the system . The induced voltage is
Vi — Z i l i ,
i = s i', s i ” , s2, s3, and r,
(2.108)
based on the circuit theory. Ztcan be directly incorporated into the corresponding
diagonal term Za due to the equivalence between the physical interpretation of
the impedance m atrix [Z] and Zt, and , consequently,
Z'ii = Z ii + Z i.
(2.109)
The excitation voltage, Vssi’s are positioned on the RHS of (2.107).
T he sim ulation procedure of the Wilkinson power divider of Fig. 2.12 is sim ilar
to the actual measurement procedure [43]. In order to find the S u and S 2 1 , Port
44
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1 is excited while ports 2 and 3 are term inated in m atched loads. Similarly, 5 12,
S 22, and S 32 are characterized by excitation a t port
a t ports
1
2
and matched term inations
and 3. In order to investigate the effect of the lumped resistor on the
Wilkinson power divider, the same structure w ithout resistor was also simulated.
According to theory, the resistor contributes a simultaneous match of all three
ports and provides a strong isolation between ports 2 and 3.
The actual circuit was designed on a substrate of er = 2.33 and thickness t
= 20 mil in X-band. A tiny 100 Q chip resistor,
20
mil by 20 m i l , was used
as the lum ped resitor element. The calculated S-parameters of both cases were
compared w ith measured results in Figs. 2.16 (a) and (b). The simulated and the
experim ental results agree well in both cases, and one can see the improvement of
S 22 and S 32 (isolation) in the power divider w ith resistor over the power divider
without resistor. Fig. 2.17 and Fig. 2.18 show the current distribution on the
Wilkinson power divider when the circuit is excited from port 1 and port 2,
respectively. The resistor effect can be clearly observed in case port 2 is excited.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a) w ithout resisto r
I
S
3
0---O
7
9
8
10
2
T
,
m easu i
11
12
13
freq u en cy (G H z)
(b) with resisto r
- ■ a - - - w - - - 0 ------ 0------ a------
IS12I
IS22I
_ ^)-"0
^
7
-
IS32I
0 -""9 ”
9
8
10
11
12
13
frequ ency (G H z)
Figure 2.16: Scattering parameters of th e Wilkinson power divider,
(wl = vv2 = 1.44 mm, w3 = 0.9 m m , a = 4.896 mm, b = 0.36 m m ,
Z s\> = Z s\" = 100 £7,
^ 2
= %sz ~ 50 Q, Rr — 100 Q, L r = 0.1 n H )
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
6
4
2
"E
jr . 0
>.
-2
-4
-6
~8 0
5
10
15
x (mm)
Figure 2.17: C urrent distribution on the Wilkinson power divider (P ort
tion).
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
excita­
8
6
4
2
eT
E. 0
>*
-2
-4
-6
~8 0
5
10
15
x (m m )
Figure 2.18: Current distribution on the W ilkinson power divider (P ort 2 excita­
tion).
48
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CH APTER 3
Hybrid time-domain analysis
The transient and wide-band responses of the three-dim ensional microwave struc­
tures can be effectively explored using the finite difference tim e-dom ain (FDTD)
m ethod [20]. This dissertation develops the FDTD m ethod in order to analyze
and design bondwire structures and microstrip-to-waveguide transition tructures
by employing Super absorbing M ur’s
1
st order absorbing boundary condition
(ABC) [19]. Especially, for effective analyses of microstrip-to-waveguide transi­
tion structures, a dom ain decomposition m ethod for the FD TD m ethod is de­
veloped, which reduces the memory requirements and the com putational time.
In addition, a hybrid m ethod using the finite difference tim e-dom ain (FDTD)
m ethod and the finite element time-domain (FETD ) m ethod is proposed for
locally detailed and curved structures and applied to via-hole structures and
waveguides with an iris of finite thickness. In the hybrid tim e-dom ain analysis,
the FE T D m ethod is applied to the locally detailed and curved structures only
and the FD T D m ethod is applied to the remaining regular structures. Therefore,
this hybrid m ethod can be an optim um time-domain analysis for locally detailed
and curved structures by taking advantage of both the F E T D flexibility and the
FDTD efficiency.
49
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3.1
F in ite difference tim e-d om ain m eth od
T he formulation of the finite difference time-domain (FDTD) m ethod starts from
Maxwell’s two curl equations in a source free, isotropic, and lossless medium [8 , 9],
dE
1_
f t- = 7 V x H
dH
.
.
(3*1}
.
-
-ST
a t = —n V x
(3-2
E-
After discretizing (3.1) and (3.2) using the central finite differencing both in tim e
and space, electric and magnetic fields can be alternatively calculated based on
the leapfrog scheme [44].
Fig. 3.1 shows the FDTD analysis. The electric field updating equations can
be obtained from the finite difference approximations of the Maxwell’s equation
based on A m pere’s law, (3.1):
E 2 + '( i, j,k )
=
E%(i,j,k)
+ C„ [H«+l/2(i,j + l,k) - H r l/2(i,3,k)]
-
Cz i [H^*'l2( i , j , k + l ) - H ; * ' l 2( i , j , k ) } ,
(3.3)
£%*'(>,},k) =
+ Czi [ H r ' l2( i , } , k + 1) -
-
*)]
+
(3-4)
+ C* [flJ+'/2(>+ lj\k )~ H”+'l2(i, j, A)]
-
C ^ J £ +1* ( i J + l , * ) - i £ +I/ 2 (i,.;,* )] ,
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.5)
FDTD
A n a ly sis
F in ite d ifferen ce ap p roxim ation s
o f M a x w e ll’s equations
D iscretization o f the com p u tation al
d om ain w ith th e F D T D unit c e ll
T im e-step d ecisio n
(stab ility con d ition)
- Program m ing In itialization o f all field s (t= 0 )
G au ssian p u lse ex cita tio n
U p d ate H fie ld (H n+I/2 )
U pdate E field ( E n+I )
A p p ly boundary condition^
n
n-J-1
P ost p ro cessin g (F F T ,...)
for freq u en cy resp o n ses
Figure 3.1: The FDTD analysis.
51
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!
where
r
Cxi ~
At
r
Cyi ~
At
At
C zi ~
(3 ' 6)
and the superscript n represents the tim e-step. Also, finite difference approxim a­
tions of the Maxwell’s equation based on Faraday’s law give the m agnetic field
u p d atin g equations:
H ^* '/2( i , j , k )
m *l'2{i,j,k)
-
D , { E ' ; a , j + i , k ) ~ E';(t.] . k)\
+
A [ £ J ( i , J , * + l ) - £ ” ( i,J ,* ) ] ,
=
k)
-
k + l) - £ ? ( i,;,f c ) ]
+
Ar K (> ' + l , i , * ) - £ ? ( ; , j.fc)],
=
H?-‘'2(ij,k)
-
D,[EZ(i + l , j , k ) - E 2 ( i , j , k ) }
+
Dy[E^{i,j + l , k ) - £ % ( i , j , k ) ] ,
(3.7)
(3.8)
(3.9)
where
Next, the structure modeling is obtained using box-shaped uniform meshes
discretized by the FDTD unit cells. Fig. 3.2 shows the FDTD unit cell where the
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
electric field elements and the magnetic field elements are defined in the unit cell
(i,j, k). The upper limit of the unit cell size is decided by the guided wavelength,
Hx(i,j,k)
node (ij,k ) /
^
l,j
Hz(iJ,k)
Ex(i,j,k)
Hy(i,j,k)
Figure 3.2: The FD TD unit cell.
Ag of the m axim um frequency wave. Generally, Ax, Ay, and A z are chosen
to be smaller th an Aff/10. Also, the decision of the tim e-step is im portant for
the stable FD TD analysis. The maximum time-step is lim ited by the Courant
stability condition [45]:
1 ( 1
1
1 \ -1/2
A t < ------ ( ——j "b — — 2 "b T ~2 )
>
Umax \ A x
Ay
Az J
(3.11)
where vmax is the maximum velocity of wave in the com putational medium.
The FD TD analysis is usually performed by exciting a Gaussian pulse at one
port, expressed as
53
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I
g (t)
=
exp
[-(i -
lo f / T 2] .
(3.12)
A Gaussian pulse (3.12) has a sm ooth transition in the time domain, which can
reduce the numerical errors due to the finite difference approximations in the
FD T D analysis. In addition, the spectrum of a Gaussian pulse has a Gaussian
pulse form, which gives a salient advantages in studying frequency responses of
microwave circuits. According to th e maximum frequency of interest, / max, T is
determ ined as
(3.13)
Also, a m odulated Gaussian pulse can be used as an excitation, which results in
shifting the center frequency from D C to / c:
9m(t) = g{t) ■sin(u>ct).
(3.14)
The field quantities are updated by the magnetic field updating equations
((3.7) - (3.10)) and the electric field updating equations ((3.3) - (3.6)) alter­
natively, and repeated until the responses at the observation points have zero
steady states. For the efficient analysis, the coefficients, (3.6) and (3.10) can be
calculated before the field updating routine is started. The coefficients, (3.6) can
be precalculated including the dielectric material information (relative dielectric
constants) an d stored in memory to reduce numerical operations.
There are two boundary conditions to apply in the field updating routine.
F irst, the tangential electric fields on the conductor should be set to zero. Sec-
54
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I
I
ondly, absorbing boundary condition (ABC) should be implemented for sim ulat­
ing the infinite real space with the finite com putational domain. This dissertation
employs Super absorbing M ur’s 1 st order absorbing boundary condition (ABC),
in which the error correction procedure is added to the M ur’s 1st order absorbing
boundary condition (ABC) [19, 46].
Due to the point-wise formulation, the finite difference tim e-domain (FD TD )
m ethod can be easily applied to general three-dimensional structures. The bondwire structure which interconnects the microwave circuits on different substrates
as shown in Fig. 3.3 and Fig. 3.4 can be studied using the FDTD m ethod ef­
ficiently. The different substrate height, finite substrates, and vertical portions
of the bondwire structure make it difficult to analyze the bondwire interconnect
structure using integral equation based analyses such as spectral domain and
spatial domain analyses, which require Green’s function calculations.
The bondwire structure connecting two 50 fi microstrip lines on two separate
substrates with er = 2.33 was designed for the S-band operation. Fig. 3.5 shows
b oth measured and simulated return losses of the bondwire interconnect struc­
ture. The FD T D m ethod gives a good prediction of the bondwire characteristics
which resembles a series inductor.
The FD TD m ethod was employed to design impedance m atching stubs for the
bondwire structure connecting two 50 Q microstrip lines on two separate Alum ina
substrates (er = 9.6). The bondwire structure is for Q-band operation with the
center frequency, f c = 44 G Hz. Fig. 3.7 (a) shows the frequency responses of
the bondwire structure without m atching stubs. In the required frequency band
from 40 to 45 GHz, the calculated return loss is about 12 dB. Based on the re­
tu rn loss data, impedance matching stubs can be designed as shown in Fig. 3.6
using Smith ch art or commercial softwares such as LIBRA. Fig. 3.7 (b) shows
th e improved return loss of the bondwire structure with the m atching stubs.
55
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Figure 3.3: Bondwire interconnect structure (3-D view).
S id e V ie w
«■
h b -----
erl
T o p V ie w
Figure 3.4: Bondwire interconnect structure (top and side view).
56
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S21
S11
-10
-1 5
-2 5
-3 0
-3 5
FDTD calculation
.—
-4 0
m easurem ent
-4 5
frequency (GHz)
Figure 3.5: Frequency responses of a bondwire interconnect structure.
(erl = er 2 = 2.33, h i = h2 = 92 mil, hb = 40 mil,
lb = 120 mil, lm = 80 mil, lg = 40 mil,
w l = w2 = 92 mil, vvb = 80 mil)
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.6: Bondwire interconnect structure with matching stubs.
58
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(a)
frequency (GHz)
(b)
0
•10
•20
■30
-40
•50
-60
30
35
40
45
50
55
60
frequency (GHz)
Figure 3.7: Matching stub design for a bondwire interconnect structure,
(a) W ithout m atching stubs, (b) W ith matching stubs.
(er i = er 2 = 9.6, h i = h2 = 5 mil, hb = 2 mil,
lb = 8 mil, lm = 6 mil, lg = 3 mil,
w l = w2 = 5 mil, wb = 3 mil,
w pl = wp2 = 14 mil, lpl = lp2 = 5 mil)
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The sensitivity analyses for param eter changes in the bondwire structure were
Bondwire Interconnection
IS11I: lb: = 8 mil
- - IS11I :l b = 1 2 mil
-10
-1 5
-20
-2 5
freq(GHz)
Figure 3.8: R eturn losses of bondwire interconnects with different bondwire
lengths.
also perform ed using the FDTD method. Fig. 3.8 shows the return losses of
bondwire structures w ith two different bondwire lengths, lb's. As lb increases,
the retu rn losses become worse due to the inductance increase. The larger dif­
ferences of the return losses in higher frequencies support this observation. The
bondwire height effects were studied in Fig. 3.9. Changing of the bondwire height
will m ostly affect the capacitance between the bondwire and the ground plane.
60
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As the bondwire height increases, the characteristic im pedance of the bondwire
will be increased due to th e capacitance decrease. Because the capacitance is
Bondwire Interconnection
IS111: hb = 2 mil
—
IS111: hb = 3 mil
- -
IS11i: hb = 4 mil
-10
-1 5
-20
-2 5
Figure 3.9: R eturn losses of bondwire interconnects w ith different bondwire
heights.
reversely propotional to th e distance between two conductors, larger difference
in the return loss is observed between 2 mil and 3 mil high bondwire structures
th an between 3 mil and 4 mil high bondwire structures.
A nother structure of interest is the microstrip-to-waveguide transition struc­
ture as depicted in Fig. 3.10 and Fig. 3.11. Due to the complex three dimen-
61
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microstrip
dielectric
waveguide
Figure 3.10: Microstrip-to-waveguide transition stru ctu re (3-D view)
62
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12
Figure 3.11: Microstrip-to-waveguide transition structure (Side, top, and back
view).
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sional geometry, it is quite challenging to find Green’s functions for this struc­
ture. Therefore, the differential equation based FD TD m ethod is more suitable
for analyzing this structure th a n the integral equation based analyses. Since the
microstrip section as well as the waveguide section are shielded, the com puta­
tional domain of the microstrip-to-waveguide transition structure can be divided
into two regions by sharing an aperture as shown in Fig. 3.12. If the electromag­
netic fields in region I and region II are calculated separately, b u t communicate
each other through the ap erture fields, the memory size and th e com putational
time can be reduced dram atically compared with the regular F D T D m ethod of
which com putational domain should be the dotted rectangular box.
In other
words, the memory saving as much as the portion excluding the dom ain I and II
in the dotted rectangular box can be obtained since the regular FD TD m ethod
requires box-shaped com putational domain due to the convenience of the array
allocations.
The domain decomposition algorithm can be explained using Fig. 3.13. Fig.
3.13 shows the interface between region I and region II, which corresponds to
the aperture in Fig. 3.12. T he electric field and the m agnetic field components
in region I and region II are assigned to the different arrays, b u t the interface
electric field components are shared by correspoding two arrays for region I and
region II. For example, the sam e value is assigned for E y ( i , j , k ) for region I
and for E™ (i', j ' , h') for region II. When updating the electric field in region I,
E y ( i , j , k — 1 ), E zl ( i , j — l , k ) , and E zl ( i , j , k ) can be updated by A m pere’s law
using the magnetic field of region I only. Also, the electric field in region II,
E +
1
), E ^ i i ^ j ' — 1 , k' +
1
), and E [ ! ( i' , j ', k' + 1 ) can be updated us­
ing the magnetic field of region II only. However, the interface electric field,
E y ( i , j , k ) { E y { i ' , j \ k')) cannot be updated using the magnetic field in region I
(II) only. In order to update E y ( i , j , k ) (E™(i1, j ' , k’) ) , H xl ! { i ' , j ' , k ' 4- 1 ) should
64
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Figure 3.12: FDTD domain decom position
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
n
X
Figure 3.13: Interface between the region I and the region II in the FDTD domain
decomposition method.
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
be employed with H * ( i,j ,k ) . Once E y ( i , j, k ) and E y ! {i',j', k’) are updated cor­
rectly, the magnetic fields in region I and region II can be updated by Faraday’s
law. This m ethod can be extended to n domains an d used for the parallel com­
putation algorithm .
The dom ain decomposition method for FD TD was employed to study the
microstrip-to-waveguide transition structure. Fig. 3.14 shows the return loss of
the microstrip-to-waveguide transition structure w ith
8
mil neck. The design
center frequency is 44 GHz. The calculated results show good agreement with
the HFSS (High Frequency Structure Simulator) results. The center frequency
of the HFSS sim ulation with the patch length, pi = 44.5 mil falls into the middle
of two FD TD results with the patch lengths, pi = 44 mil and pi = 45 mil. The
return loss of the microstrip-to-waveguide transition stru ctu re with the 9 mil neck
length was also depicted in Fig. 3.15. The FDTD results show good agreement
with the HFSS result. As the patch length (pi) an d the neck length (fl) were
increased, the center frequencies were shifted toward high frequencies.
The FDTD m ethod can also be employed to stu d y the resonance characteris­
tics. Using the Discrete Fourier Transform (DFT) of the tim e response [47], the
wide band frequency spectrum can be obtained. Fig. 3.16 and Fig. 3.17 show the
housing structure for the microstrip circuitry. The housing structure is connected
to the waveguides through the microstrip-to-waveguide transition structure. In
order to find the resonance behaviors of the housing structure, both the source
and the observation points were positioned inside th e housing structure. Fig.
3.18 shows the spectrum of the housing structure.
67
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Microstrip-to-Waveguide Transition
-10
-1 5
3 -20
co - 2 5
-3 0
-3 5
FDTD (pi = 44 mil)
-4 0
- - FDTD (pi = 45 mil)
--
HFSS (pi =44.5 mil)
-4 5
freq(GHz)
Figure 3.14: R eturn loss of the microstrip-to-waveguide transition structure with
8
mil neck.
(er=5.9, dt=7.4m il, hlh=94.4m il, h2h=47.2mil, hlw = 47m il,
a=224m il, b=112mil, cw=11.3mil, fl= 8 mil,
pl=44(45)mil, pw=47mil, pi=0mil)
68
v
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Microstrip-to-Waveguide Transition
-10
-15
£"-20
*o
£0-25
-30
-35
FDTD (pi =44 mil)
-40
- - FDTD (pi = 45 mil)
- -
HFSS (pi = 44.5 mil)
-45
freq(GHz)
Figure 3.15: R eturn loss of the microstrip-to-waveguide transition structure w ith
9 mil neck.
(er=5.9, dt= 7.4m il, hlh=94.4m il, h2h=47.2m il, hlw = 47m il,
a=224m il, b=112mil, cw=11.3mil, fl=9m il,
pl=44(45)mil, pw=47mil, pi=0m il)
69
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1
housing
waveguides
Figure 3.16: The housing structure for the microstrip circuitry (3-D view).
( u n i t : in)
0.03
0.045
0.090
0.150
0.195
0.245
0.280
0.045
0.240
0.580
0 145
0.790
0.885
0.825
0.985
0.205
Figure 3.17: The housing structure for th e microstrip circuitry (Top view).
70
I
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Spectrum of housing structure
5
4
3
2
1
0
U
50
60
70
80
110
90
100
frequency (GHz)
120
130
140
Figure 3.18: The spectrum of the housing structure.
71
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150
I
3.2
H ybrid an alysis using th e F D T D and F E T D
The finite difference tim e-domain (FDTD) m ethod has a number of advantages
in analyzing three-dimensional microwave structures due to its simplicity and
numerical efficiency [8 , 9]. The conventional FDTD algorithm makes use of the
uniform meshes and does not require any special mesh generation scheme and
storage for the mesh. However, the use of box-shaped Cartesian coordinate uni­
form meshes in the conventional FDTD algorithm causes difficulties while dealing
w ith curved structures and locally detailed structures. Typically, curved struc­
tures have been analyzed using the staircasing approximations [48], which requires
finer meshes and dram atic increases in memory size as well as longer com puta­
tional tim e caused by using the smaller tim e-step size to satisfy the Courant sta­
bility condition [45]. The same problems can be encountered in employing veryfine meshes in the entire com putational dom ain for the locally detailed struc­
tures.
Several m ethods have been developed to overcome these difficulties in the
FDTD m ethod. The nonorthogonal FD TD algorithm has been introduced to
solve uniform, uncurved, but oblique structures [2 1 ], and improved later to ana­
lyze three dimensional structures including curved shapes [22, 23]. Using covari­
an t and contravariant components of E and H fields to obtain a finite difference
approxim ation of the integral forms of Maxwell’s equations gives an im portant
advantage in th a t the nonorthogonal FD TD algorithm can have the same form
as the conventional FDTD algorithm. However, this method still requires longer
com putational time and larger memory size. The DSI (discrete surface integral)
m ethod has been also proposed to analyze general structures [24]. This m ethod
can be regarded as a generalization of the regular FDTD m ethod since it reduces
to the conventional FDTD m ethod when structured orthogonal hexahedral grids
72
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are used. However, this m ethod needs large memory size due to the require­
m ent for the dual grid as well as the prim ary grid. A locally conformed FD TD
algorithm has been studied for analyzing locally arbitrarily shaped structures ef­
ficiently [25]. This m ethod performs th e conventional FDTD leapfrog scheme and
then corrects the field in order to take into account the m etal structures which do
not conform to the FD TD mesh, by using the integral form of Maxwell’s equa­
tions. Different geometries require different correction procedures, which can be
a disadvantage of this m ethod.
T he finite element time-domain (FETD ) m ethod has also been developed to
improve flexibility in modeling structures by retaining the advantage of tim e do­
main analysis [49]-[51]. Although this m ethod can have no geom etric lim itations
in m odeling structures, it can be less efficient than the FD TD m ethod because
it requires the system of equations to be solved for each tim e-step. Recently,
a hybrid m ethod incorporating the F E T D m ethod into the FD T D m ethod was
developed and applied to the electrom agnetic scattering problem of two dim en­
sional circular shaped dielectric cylinders [27]. By conforming only the circular
structures using FETD while applying FD T D elsewhere, a trade-off between the
FETD flexibility and the FDTD numerical efficiency can be obtained. However,
the F D T D and FETD mesh m atching m ethod in the interface region can give
difficulty in the mesh generation in the FETD region.
This dissertation proposes a new FD T D and FETD hybrid m ethod by intro­
ducing an interpolation scheme for com m unicating between the FD T D field and
the FE T D field in the interface region. In this method, the effort of fitting the
FETD mesh to the FDTD cells in the interface can be avoided. The hybrid anal­
ysis proposed here employs the stan d ard FD TD method w ith Super absorbing
M ur’s 1st order ABC (absorbing boundary condition) [19] and the FE T D m ethod
using the 2nd order vector prism element [52, 53].
73
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This hybrid m ethod is applied to analyze single and multiple via hole grounds
in m icrostrip as well as the waveguide with an iris of finite thickness. The via
hole grounded microstrip stru ctu re is a three-dimensional problem having both
cylindrical and rectangular boundaries, as described in detail later. Applying
the FETD to the part of the FD T D region including via hole grounds and the
FDTD elsewhere preserves the advantages of both FDTD and FE T D . In addition,
changing the structure inside the FETD region does not require any change in
the FDTD variables, which can be a great advantage in design sensitivity analysis
because different structures can be analyzed by changing the FET D meshes only.
In the same manner, when analyzing the waveguide with an iris of finite thick­
ness, FETD is applied to the iris region while FDTD is applied to th e remaining
waveguide sections. This application shows th a t the hybrid analysis can handle
locally detailed structures efficiently.
3 .2 .1
F E T D fo rm u lation
S tarting from the source-free Maxwell’s two curl equations in a linear isotropic
region, the vector wave equation can be obtained as
&2E
V x V x l + Me— — — 0 .
(3.15)
Applying th e weak form form ulation, or the Galerkin’s procedure to (3.15) gives
r
r
f
r
J (V x E) • (V x W ) d v + J » e W • - ^ - d v = J W • (V x E x n)ds,
*
(3.16)
where W is the weighting function defined using the 2nd order vector prism ele­
m ent [52, 53].
74
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T he prism element is composed of 30 edge elements as shown in Fig. 3.19.
25
16
28
30
26
14
29
27
12
10
.
22
24
20
a
23
Figure 3.19: The prism element.
W ith the shape function, the electric field inside the prism element can be inter­
polated as
3
E(r, t)
=
3
[.
Y l i Y l L iT(-bj 2 + c3£ ) f y ^ ( . t ) i +6{l-i)
i=i t = i
^
U
^3 ^Lj--(
biZ + Ci£} fyi£te(t)i+3+6(i—i)]
i=i
^
75
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a
3
1=4
t= l
— l)fyl£ye(t)i+6(l-l)
+
3
fyl^ye{i)i+3+6(l—l) ] ? / i
"b Xrf
(3.1 ( )
x= l
where
£te(t)k
= E ( r , t ) - t k,
k = 1 , 2 ,...,18,
£ye(t)k
= E ( r , t ) - y k,
k = 19,20, ...,30,
(3.18)
and
fy 1
=
L y l ( 2 L y l — 1)
fy 2
=
^ L yi L y 2
fy 3
=
L y 2{2Ly2 — 1)
fy4
—
Ly 1
/t/5
=
Ly2 .
(3.19)
In (3.18), ik and yk are the unit vectors of the kth edge in the prism element
(Fig. 3.19). Therefore £te{t)k and £ye(t)k represent the state variables of the fcth
edge in FETD , which can be also interpreted as electric fields. In (3.19), Li s are
the barycentric coordinates of a triangle, and Ly, ’s are the first order Lagrange
interpolation polynomials between upper and lower triangular faces, which are
expressed as [52, 53]
a.i +
U
=
+ CiX
'
76
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( 3
' 2 0 )
(Li
ZjXfc
bi
Xj
Zfz'Xj
Xfc
(3.21)
Ci = zk - Zj,
and
Lyi —
V u-y
2
Ly2 =
ly
y - vi
2L '
(3.22)
In (3.20), A represents the area of the triangular face, which can be calculated
as
- 4
1
1
1
Zi
22
23
Xi
X2 ^3
(3.23)
The variables, yu and yi in (3.22) are the y coordinates of the upper triangle
and the lower triangle in the prism element and 2 ly corresponds to the difference
between yu and yiIn the finite differencing of (3.16) in time, the unconditionally stable backward
difference is used [51], which gives:
AT'
£ ( [ J f ] [ £ T = K l l B 1]’ + 2 K
e=l
)
[
- K H B '] " - 2) ,
where
77
/
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(3.24)
Kfj
=
| ( V x Wf) • (V x W f ) d v +
Ff
=
J w ? ■(V x
F‘
=
w S . * i -* !* > ’
W f ■W ' d v
x h)d s
(3.25)
N e is the num ber of the total edge elements and [Ee\n represents the state variable
vector a t the tim e-step n. In (3.25), W f ’s represent shape functions for the 2nd
order vector prism element, expressed as
h
Li — (bjZ H~ Cji'jfyi
^ i + 6 (/—1)
li
—L)
"h Ci£)fyl
for i = 1 ,2 ,3 ,
Wi+6 (i-i)
I,^ii+3+6(Z—l)
=
and
/ = 1,2,3,
and
1 = 4,5.
Li(2Li — 1 ) f yi
^LiLjfyl
for* = 1,2, 3,
(3.26)
The evaluation of the elemental m atrix [.K e] is shown in Appendix A.
78
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3 .2 .2
H y b rid izin g th e F D T D an alysis and th e F E T D an alysis
The RHS of (3.24), the load vector of the FETD requires the knowledge of the
electric field values for two previous time-steps, as well as the boundary values at
the present time-step. Appendix B shows the evaluation of the elemental m atrix
[Ff] for the load vector. The boundary values calculated from FDTD become
the Dirichlet boundary conditions on the FETD boundary, and are used to solve
the inner field of the FETD region. The FETD region is chosen to be a brick
replacing the part of the FDTD region and includes the locally arbitrarily shaped
structures. This choice of FETD region gives a great advantage when different
arbitrarily shaped structures need to be analyzed because only the F E T D mesh
change will be required without affecting the FDTD variables.
Fig. 3.20 shows the FDTD and FETD interface region in the two dim en­
sional view.
One cell size of FDTD region is to be overlapped in the FE T D
region. In the FD TD time-marching procedure, E y n{ i , j , k —1 ), E z n(i,j, k), and
E z n( i , j — 1 , k) are updated, but the FDTD boundary value, E y n(i:j , k ) can not
be updated because
k + 1 ) does not exist. The updated interface
values are employed in the FETD through the surface integrals of (3.25) to cal­
culate the inner field in the FET D domain. The calculated inner field is used to
update the FD TD boundary value, E y n( i , j , k ) . Once E y n( i , j , k ) is updated, the
H x n+^(i, j, k) can be calculated from H field updating procedure of FD TD . This
completes one time-marching procedure of the hybrid method.
This hybrid m ethod makes use of an interpolation scheme in com m unicating
between the FETD field and the FDTD field at the interface. Fig. 3.21 explains
this communication method. After the FETD calculation, the FDTD interface
field is interpolated using the FETD prism element shape function (3.17). O n the
other hand, the FETD interface field can be interpolated using the regular brick
79
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FDTD
r
i
FETD
*
Hx+L(ij-l,k)
Ez (ij-l,k )
x
G-
Ez (ij,k)
n + l/2
Hx (ij,k)
Figure 3.20: The FDTD and FET D interface
80
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A . FETD —
FDTD
FDTD field is interpolated
FDTD cell
using the FETD prism element
shape function.
FETD mesh
B. FDTD
FETD
FETD field is interpolated
FDTD cell
using the brick element shape
function.
FETD mesh
Figure 3.21: Comm unication between FDTD and FETD field.
81
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elem ent shape function with the FDTD interface field. Fig. 3.22 shows the brick
N z2
N y3
N y4
N z4
N x3
N xl
N zl
Nx2
N yl
4
N x4
N y2
8
N z3
y
Figure 3.22: The regular brick element.
elem ent [18]. By substituting the FD TD interface field to the edge elements of
the regular brick element, the electric field inside the cell can be interpolated as
E(r, i) = i £ N itE ii + y ' £
1=1
+ z£ N
i= l
t= l
where N ^ ’s are defined as
/Ve
■
/vii
(yu - y)(zu - z)
l e le
Lylz
82
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'
,
(3.27)
an d
can be derived from (3.28) using the cyclical relationships of x, y,
z. T he Z®, Z®, and Z® correspond to the A x, Ay, and A z of the FDTD cell. Now,
the F E T D interface field can be easily calculated using (3.18). This scheme is
m ore general th an the FD TD and FE T D mesh matching m ethod [27] because
the m esh m atching effort is avoided.
3 .2 .3
A p p lic a tio n s o f th e hybrid an alysis
FETDregion
m r ; .r
Wm.
h
0
Figure 3.23: The via hole grounded microstrip.
83
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w
This hybrid m ethod was applied to characterize the cylindrical via hole grounds
in m icrostrip. The via hole region is replaced by the FETD region as shown in
Fig. 3.23. Since the microstrip and the ground plane coincide with the top and
the bo tto m boundaries of the FETD region, the Dirichlet boundary conditions
are applied to the top and the bottom as well as the via hole cylinder wall. This
reduces the m atrix size in the FETD analysis and increases the com putational
efficiency.
T he param eters of the first analyzed via hole grounded microstrip structure
are as follows: The via hole diam eter is 0.6 mm. The microstrip width is 2.3
mm.
T he substrate thickness is 0.794 mm.
Lastly, the substrate has a low
dielectric constant (er = 2.32). Fig. 3.24 (a) shows the cross-sectional view of
the F E T D mesh for the 0.6 mm diam eter via hole, which was employed in the
hybrid m ethod. For the good quality triangular meshes, the Delaunay tessellation
algorithm was used. The same structure was also simulated using the FD TD
staircasing approximations. Fig. 3.25 depicts the cross section of the FDTD
staircasing model of the via hole. In order to get the resolution in Fig. 3.25, the
2.3 m m wide microstrip was divided into 40 cells. In the hybrid method, only
6
cells were used for the microstrip in the FDTD region and 4 x 3 x 4 FDTD cells
were replaced by the FETD region among the to tal 60 x 20 x 100 FDTD cells.
Fig. 3.26 compares the |Soil’s of this via hole grounded microstrip calculated by
this hybrid method, the mode m atching m ethod [54], and the FDTD staircasing
approxim ations. A very good agreement was observed between the hybrid m ethod
d a ta and the mode matching data. In the next step, the via hole grounds with
0.4mm diam eter as shown in Fig. 3.24 (b) were analyzed and the result is shown
in Fig. 3.27. The hybrid m ethod gives a good prediction of |S 2 i| for various via
hole diam eters.
Practically, the ground effect of a large diam eter via hole can be obtained
84
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0.5
0.5
-0.5
-0.5
(a)
(b)
0.5
0.5
-0.5
-0.5
0.5
-0.5
0.5
-0.5
0.5
-0.5
0.5
-0.5
(c)
(d)
Figure 3.24: FETD meshes for the via holes.
((a) 2r = 0.6 mm, (b) 2r = 0.4 mm, (c) 2r = 0.3 mm, (d) 2r = 0.3 mm)
85
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- i..
0.0575m m
-y
Figure 3.25: FDTD staircasing model for the 0.6 mm diam eter via hole.
86
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Or
-10
CM
-1 5
-20
this method(FDTD + FETD)
o
—
M ode Matching
FDTD staircasing
-2 5
freq(GHz)
Figure 3.26: IS 2 1 I of the via hole grounded microstrip.
(er = 2.32, w = 2.3 mm, h = 0.794 m m , 2r = 0.6 mm)
87
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Or
x
-5
x
-10
m
13
CM
CO
-15
this method(2r = 0.6mm)
-20
—
this method(2r = 0.4mm)
o
Mode Matching(2r = 0.6mm)
x
Mode Matching(2r = 0.4mm)
-2 5
8
10
freq(GHz)
12
14
16
18
Figure 3.27: |5 2 i| of the via hole grounded m icrostrip.
(er = 2.32, w = 2.3 mm, h = 0.794 mm)
88
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
using multiple small diam eter via holes. To begin with, the two via hole problem
was chosen as shown in Fig. 3.24 (c). Both via holes have the same diam eter ( 2 r
= 0.3 mm). For reference, the two via hole grounds with the square via holes (a =
b = 0.3 mm) were also analyzed using the FDTD method only. Fig. 3.28 shows
the square via holes in the FD TD cells. The l u l l ’s of these via hole grounds
are shown in Fig. 3.29. The result of the via hole grounded m icrostrip with
two circular via holes is very close to th at of the via hole grounded m icrostrip
with one circular via hole having larger diameter size ( 2 r =
0 .6
m m ). The via
hole grounded m icrostrip w ith two square via holes shows lower IS2 1 I than one
with circular ones (Fig. 3.29). This result is shown to be reasonable because the
effective via hole area of the square via holes is larger than the circular via holes.
In the next step, three via holes were analyzed for the grounded m icrostrip. Fig.
3.24 (d) shows the cross-sectional view of the three via hole grounds. IS^il of the
three via hole grounds is a t least 3 dB less than IS2 1 I of the two via hole grounds
over a wide frequency band (Fig. 3.30).
At this moment, it is worthwhile to mention th at only the FE T D meshes
were changed for analyzing four different via hole grounds w ithout affecting the
FDTD variables. In Fig. 3.24, one can notice th at the outer boundaries of all
the FETD meshes are fixed and correspond to 4 x 4 FDTD cells. For staircasing
approximations, the cell size and the A t size need to be changed for different
structures. Therefore, the hybrid m ethod is more suitable for investigating the
design sensitivity of locally arbitrarily shaped structures. For exam ple, one can
analyze circuit performance according to the variations of the design param eters,
such as structure geometries.
This hybrid m ethod makes use of the QMR (quasi minimal residue) iterative
method [55] to solve the system of equations in FETD for each tim e-step. Solving
the system of equations for every tim e-step can be inefficient. However, the FET D
89
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0.1mm
0 .1mm
■
t
r=- - b -
I
-=>1
Figure 3.28: The FD T D modeling for the two via holes.
(er = 2.32, w = 2.3 m m , h = 0.794 mm, a = b = 0.3 mm)
90
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-1 0
-15
-20
—
this method(two Was, 2r = 0.3mm)
—
this method(one w'a, 2r = 0.6mm)
—- FDTD(two was, a=b=0.3mm)
-25
freq(GHz)
Figure 3.29: IS2 1 I of the grounded microstrip with two via holes.
(er = 2.32, w = 2.3 mm, h = 0.794 mm)
91
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-10
-1 5
-20
—
two via holes
three via holes
-2 5
freq(GHz)
Figure 3.30: |f>2 i| of the via hole grounded microstrip.
(er = 2.32, w = 2.3 mm, h = 0.794 mm)
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
region in this hybrid m ethod takes only small part of the entire dom ain and the
overall com putational efficiency is not affected. For example, the num ber of edge
elements for the 0.6 mm diam eter via hole ground is 448. The hybrid m ethod
requires 23 Mbytes and takes
8 .1
seconds for one time-step running on a Sun
SPARC station 20. In the mean time, the FDTD staircasing approxim ations
(Fig. 3.25) using total 140 x 40 x 180 cells require 54 Mbytes and takes 7.2
seconds for one tim e-step running in the same machine. Since the A t size of the
FD TD staircasing approxim ations is chosen to be 2.5 times less than the A t size
of the hybrid m ethod (A t = 0.25 x 10- 1 2 Sec.), the com putational tim e of the
FD TD staircasing is a t least two times longer than th at of the hybrid m ethod.
The hybrid m ethod was also applied to the waveguide with an iris as shown in
Fig. 3.31. The iris region is analyzed by FETD so that the iris thickness can be
considered regardless of the FD T D Az size, while the remaining waveguide regions
are characterized by FDTD. For simulation, the standard WR90 waveguide (0.9
inches x 0.4 inches) was chosen. The FETD volume corresponds to 14 x 32 x 4
com putational domain of FD TD .
Fig. 3.32 depicts the normalized inductive iris susceptance versus the iris
w idth when the iris thickness is zero. The normalized iris susceptance can be
calculated from S u using
Y
CTo ~
2S n e ^ L
~ l +
Sn e> W L '
( 3
’2 9 )
where L is the distance from the iris to the reference plane for S u [56]. T he
calculated results by the hybrid m ethod show very good agreement with the
M arcuvitz’s curve [57], the m ethod of moments solutions [58], and the FD TD
data.
W hen the iris has a finite thickness, the flexible FETD m ethod can be effec­
tively employed, while the FD TD Az size is not affected. Fig. 3.33 shows the
93
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| S n | of the waveguide w ith an iris of different thickness. The waveguide of an
iris of 20 mil thickness was analyzed while the Az size of the FDTD domains is
28 mil.
Figure 3.31: T h e waveguide with an iris of finite thickness.
94
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Marcuvitz curve
Method of Moments
FDTD
FDTD+FETD
0.8
aoc
SQ.
©
a
3 0.6
«
(O
0.4
0.2
0.05
0.1
0.15
0.2
Iris Width (inches)
0.25
Figure 3.32: Normalized inductive iris susceptance vs. iris width.
(Waveguide dimensions : a = 0.9” , b = 0.4” , Freq. = 9.375 GHz)
95
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0.3
FDTD (t=Omil)
- - FDTD + FETD (t=20mil)
- - FDTD (t=28mil)
-10
-1 5
6.5
7.5
8.5
freq(GHz)
9.5
10.5
Figure 3.33: |5 u | of the waveguide with an inductive iris.
(Waveguide dimensions : a = 0.9” , b = 0.4” , w /a = 0.25)
96
i
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CHAPTER 4
Conclusion
The im portance of full-wave analysis for microwave circuits has been increasing as
microwave circuits employ integration technology such as microwave integrated
circuits (MICs) and monolithic microwave integrated circuits (MMICs), which
requires predictable and precise analysis and design tools. T he full-wave analyses
solve the wave equations considering time varying electric and magnetic fields
with given boundary conditions, and include all th e full-wave effects such as ra­
diations, couplings, and surface waves.
The full-wave analyses can be divided into two categories: integral equation
based formulations and differential equation based formulations. The integral
equation based full-wave analyses are formulated using G reen’s functions and
their unknown variables are defined only on the microwave structures themselves.
Since G reen’s functions for free space or layered structures are well-known, the
integral equation m ethods can be efficiently used for microwave circuits designed
in free space or layered structures. However, for general structures having com­
plex boundary conditions, finding Green’s functions is not an easy task and the
integral equation formulations become very complicated. On the other hand,
the differential equation based formulations do not require G reen’s functions and
can deal with general structures relatively easily due to their point-wise formu­
lations. However, the entire computational dom ain needs to be discretized by
meshes defining unknown field values, which require large memory in general.
97
i
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Both integral equation and differential equation based full-wave analyses were
studied in this dissertation. Among various integral equation analyses, the spa­
tial dom ain analysis was chosen to analyze arbitrarily shaped m icrostrip circuits.
Two differential equation based full-wave analyses, finite difference tim e dom ain
(FDTD) and finite element tim e domain (FETD) methods were developed for
three dimensional general structures. Also, by combining the FDTD and the
FETD methods, a hybrid m ethod was proposed to analyze locally detailed and
curved structures.
The spatial dom ain analysis has an advantage over the spectal domain analy­
sis in th a t it is more flexible in modeling arbitrarily shaped m icrostrip structures.
The triangular patch pair basis functions were used for the spatial domain anal­
ysis.
By introducing new triangular patch pair basis functions, the extended
spatial dom ain analysis was proposed to incorporate lumped elements. In ad­
dition, the matched excitation and load scheme were also introduced, which is
superior to the conventional voltage gap source excitation m ethod. The valid­
ity of the extended spatial dom ain analysis was shown by analyzing m icrostrip
T-junction, open and loaded microstrips, and microstrip W ilkinson power divider.
Three-dimensional microwave circuits such as the bondwire interconnect stru c­
tures and the microstrip-to-waveguide transition structures were studied using
the finite difference tim e-dom ain (FDTD) m ethod employing Super absorbing
M ur’s 1 st order absorbing boundary condition (ABC). The bondwire intercon­
nect structures have finite substrates with different heights, of which G reen’s
functions become complex. In addition, the vertical com ponents of the bond­
wire gives difficulty in the integral equation formulation.
waveguide transition structures have similar problems.
T he m icrostrip-to-
Therefore, differential
equation based finite difference time-domain (FDTD) method is better for these
structures. Besides, wideband responses and transient responses can be obtained
98
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since the FD TD m ethod is based on the tim e dom ain analysis. Especially for
microstrip-to-waveguide transition structures, a dom ain decomposition scheme
was developed, which resulted in saving memory and com putational time. This
domain decomposition algorithm can be employed for th e parallel computation
for FDTD.
The finite difference time-domain (FDTD) m ethod is numerically efficient,
but it has difficulty in modeling arbitrarily shaped structures because of using
box-shaped uniform meshes. In order to overcome these difficulties and model
the locally arbitrarily shaped structures efficiently and accurately, a hybrid fullwave time-domain analysis was introduced by incorporating the finite element
time-domain (FETD ) m ethod into the finite difference tim e-dom ain (FDTD)
method. The finite element time-domain (FETD ) m ethod is the tim e domain
formulation of the finite element m ethod which is based on the differential equa­
tion formulations, and has flexibility in modeling general structures. Therefore,
the application of the FE T D method to the locally arb itrarily shaped structure
and the application of the FDTD method to the rem aining regular structure will
give advantages of both the FETD flexibility and the FD T D efficiency. This hy­
brid time-domain m ethod was verified by analyzing via hole grounded microstrip
structures and the waveguide structures with an iris with finite thickness.
Analyses of complex microwave integrated circuits require full-wave analyses
in order to predict the circuit characteristics precisely, because the full-wave ef­
fects can be included in the analyses. Since full-wave analyses require rigorous
formulations and lengthy com putational time in general, th e proper choice of the
full-wave m ethod depending on the microwave circuit stru ctu re is very' im portant
for efficient and successful analysis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A P PE N D IX A
Evaluation of elem ental matrix
for the
[ K e]
stiffness matrix
The elemental m atrix, [K e] is composed of 3 x 3 small m atrices as shown in (A .l),
and has to tal 30 x 30 m atrix elements.
[K'} =
K n
-f-A i i
A £
A 10
+K
22
K n
+A 21
K f,
12
k
+A 22
K
l3
+ k 7
A l3
+ A 23
+K&
+ A 23
K n
+ K 21
K 12
+ K 22
K n
+ K 2l
+K
Kr>
+I<t
A 13
+ A 23
4 -A 'T o
K n
+ K 21
+ A 22
Kyy
+ K 22
Kj->
K
"F A 2 3
+ k72
+K
KTt
K'fo
K ir
K Ts
K l
K is
K
K
12
13
K J0
&T>
&To
Kjn
A 12
22
A l3
+K
23
A n
A 12
+A 21
+A 22
KTs
K fr
KTs
13
23
K f9
K jo
KTs
k
To
Kn
K 18
K n
K is
A ' 13
+A 23
a 19
A 10
ATig
A 10
A '[2
- t - A >2
K 17
A ' 18
K n
K is
A 13
A ig
A 10
K 19
K io
K n
+ K 21
+A 22
K t
+ kT
2
K n
+ K 21
i <t
A '12
2
+ a >3
K n
+ K 21
K lt
+ K 22
Kn
A ts
K n
K is
k T>
+ K 22
K X3
+ A >3
K \o
A ho
K
K io
K u
+A 24
A '1 5
+A 25
KT
+ K js
+
+
K14
+ K '24
A 'is
+Ao5
A m
+A 24
+ k7
KTt
KTs
K yr
KTs
k
T
KTo
KTs
KTo
K \
+K%
K 16
K26
K
16
+A 26
A ,4
+A 24
k
a
A 16
|
+A 26
K h
. Evaluation of f v(V x WT) ■(V x W ^ d v .
100
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A 15
+A 25
T
+KJ5
Each elements in (A .l) can be calculated as following.
1
19
K is
+A 25
K is
+K26
I<u (t =
1
, 2 , 3, I =
/(V x
1
, 2 , 3) {if =
1
i)) • (V x
^ ( 1 + fe ) ( w
, 2 , 3, V =
1
, 2, 3)
i) )c/i; =
+ 4^) /
+X /
(A.2)
iT 12 (i =
1
, 2, 3, Z = 1, 2, 3) (i' =
1
, 2, 3, Z' = 1, 2, 3)
/ ( V x Wi+eft-!)) • (V x W ii+a+6(lt_l))dv =
+
/
df a J ~d y d y +
/
f y ‘f y i ’d 'y
(A-3)
A" 17 (i =
1
, 2, 3, I =
1
/ ( V X
, 2, 3) (i# =
1
, 2 , 3, /' = 4, 5)
W v + 6 ( i - l ) ) ’ ( V X W V + 6 ( / / _ l ))rZu =
d fy t
dy
(A.4)
• AT18 (z = 1, 2, 3, Z = 1, 2, 3) (z' = 1, 2, 3, Z' = 4, 5)
/(V x
tf i+ e p .! ) ) •
(V x
' + 3 + 6 ( / '- l ) ) « Z u =
- ^ [ ( i + &w)(bjbj> + CjCf ) + (1 + Sij’W j b i '
+ CjCi»)] / - ^ - f y i ' d y
(A.5)
• K n [i = 1, 2, 3, Z =
1
, 2, 3) (*# =
1
, 2, 3, Z' =
1
, 2, 3)
/ ( V x Wi+3 + 6 (/_i)) • (V x l'V'i/+3+6(//_ 1))cZu =
^ - ( i + * * ,)( < * * + w
/
(A-6 )
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
• K w (Z = 1, 2, 3, I = 1, 2, 3) (Z7 = 1, 2, 3, V = 4, 5)
/ ( V X Wi+3+6(/-l)) - (V x Wi
dfyl
■^SjAciCe +bibi>) / ^ f 1jVdy
dy
(A.7)
• K l0 [i =
/ ( V X
1
, 2, 3, Z =
1
, 2, 3) (Z7 =
t'V’i + 3 + 6 ( / _ i ) ) • ( V
, 2 , 3 , Z7 =
X V V t'+ 3 + 6 ( / ' _ l ) ) d u
5-jji) (bibi> + aa>) +
^ •[(1 +
1
(1
+
, 5)
4
=
+ qcj/)]
J -gjj-fyi'dy
(A-8 )
• t f 14 (i = 1, 2, 3, Z = 4, 5) (Z7 = 1, 2, 3, Z7 = 4, 5)
/" (V
X PVrt + 6 ( / - l ) ) ■ ( V
X W V + 6( ,/_ i) ) e fi; =
1 (4 ^ -1 )
•(CjCi< + 6j6,/)
4A
J fyifyi'dy
(A.9)
AT1S (i =
1
, 2 , 3, Z = 4, 5) (Z7 = L 2, 3, Z7 = 4r 5)
/(V
X
^^
Wf-H6 (/—1 )) • (V
W'ri/+3+6(//_1))cZu =
X
\djj’(CjCj>4“
4"
^it'(CiCj' 4~
6 j 6 j ') ] /
fy ify i'd y
(A-10)
#16 (Z = 1, 2, 3, Z = 4, 5) (Z7 = 1, 2, 3, Z7 = 4, 5)
/ ( V x Wi-j-3 -i-6 (z—i)) • (V x Wii+z+e(i>-i))dv =
2^[(1
+ d jj> )(c iC i> + bib{>)
4 - ( l 4-
^j't)(ci'Cj
4-
4- (1 4-
bj
Z)j/ ) 4- (1
) (c,-Cj< 4- &,&■/')
4-
^it')(cjC_,' 4-
bjby)} J fyifyi'dy
( A - 11)
102
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(
2. Evaluation of Jp- f v W t ■W ^dv.
K 2l (i =
1
, 2 , 3, / =
1
, 2 , 3) (*' =
1
, 2 , 3. V =
1
, 2, 3)
/iC
f —
->
if!
"72 J ^»'+6(/-i) • W V+e^'-ijdi; =
S t2
/ie lil?
+ SH ' ) ( bj bj ' + cJ cj ' )
J
fyifyi'dy
(A.12)
•
#22
(i =
1
, 2, 3, Z = 1, 2, 3) (i' = 1, 2, 3, I' = 1 , 2, 3)
fie f f
if!
"72
j W»+6(l-l) •
St2
1^1ft
S ^ l +
—1)cto =
+ c J Ci' ) J f y i f y i ' d y
(A-13)
• K 2Z (i = 1, 2, 3, Z = 1, 2, 3) (i' = 1, 2, 3, Z' =
if!
A , y
1
, 2, 3)
^ i + 3 + 6 ( i - l ) • W V + 3 + 6 (/'-i)d i; =
St2
/ie ZjZj/
[(1
+6jj>)(bibii + aci>) J fyifyi’dy
(A .14)
• K 2A (i = 1, 2, 3, Z = 4, 5) (t' =
fie
f
-
1
, 2. 3, Z' = 4, 5)
->
772 / ^ i + 6 ( i - l ) • W i'+ 6(Z '-l)C fr> =
Ot
*/t;
(A-15)
• AT25 (* =
1
, 2, 3, Z = 4, 5) (*' = 1, 2, 3, I' = 4, 5)
fJLE f
~S? J v
—
—
i+ 6 (/ _ 1 ) ’ ^
+3+6(Z' —1) d v =
/ie A
^ 2
45 (— 1
+ <%' or j'))
J
fyifyi'dy
(A .16)
103
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K 26 (z = 1, 2, 3, / = 4, 5) (i ' = 1, 2, 3, I' = 4. 5)
fie
J ?
f
Jv
^
+ 3 + 6 (, _ l ) ’
delta
^^2
delta =
=
J fyi fyii dy
4
: if z= z'
2
: if z= j ' or z' = j
(A-17)
: if z 7^ z' and j 7 ^ /
1
Evaluation of
3 + 6 ( i '- l ) ^
f fyifyi'dy.
- I = 1. /' = 1
/
~ 4£yi + l)dy — — /j,
(A.18)
- / = 1. /' = 2
J
- I = l, /' =
4 ( 2 L y l L y2
—
L y l L y 2 ) d y — - ^ z ly
(A.19)
3
j LylLy2(4LylLy2 ~ 2Lyl ~ 2Ly2 + 1)^Z/ —~ ~ £
(A.20)
- I = 1, /' = 4
1
/<
2 -^ y l —
^ y l ) d y = ~ly
- / = l, /' = 5
J (2 ^ i^ y 2 —LyiLy2)dy —0
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A .21)
- 1 = 2.1' = 2
/ 1^L2ylL]2dy = ^
-
1 =
2 ,1 '=
3
J
-
1 =
-
I =
2. V
=
4 ( 2 L y i L y2
L y i L y2) d y — ^ l y
J
hy
2 , I' =
I
=
3 .
/ '
=
3
-
I
=
3 .
/ '
=
4
J
=
3 ,
/ '
=
^ liL y2 d y =
/
=
4
.
I'
=
A L y i L y2d y
( “^ L y i L ^
( A
. 2 5 )
—
- / j ,
( A
. 2 6 )
L y\L y2)dy
—
0
( A
. 2 8 )
( A
. 2 9 )
( A
. 3 0 )
5
/
-
. 2 4 )
5
-
/
( A
4
J
-
(A.23)
( 2
^
2
^
2)
^
2/
—
4
J
L yid y
—
1 0 5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
j L
M
ly
(A .31)
3
— / =
5, /' =
5
/
L y2 2 d y =
-ly
(A .32)
Evaluation of / = ^ ~ £ - d y .
- I = 1. /' = 1
J
-
Z=
1. /' =
C1
~ 8 L j,i + 16Z -^)rf7/ =
—
(A .33)
2
J
- i = l. /' =
^2
J2 ( ^ l ~
+ ^ L y iL y 2 )d y =
(A .3 4 )
3
I
~ *~
^ L y i L y 2 + 4 L y l)dy =
—
(A .3 5 )
7y-
(A .36)
- I = 2,1' = 2
J
~ 2 L y iL y 2 + ^
"y
2)
^
=
%ly
- 1 = 2. I' = 3
J
j2^ L y i L y
2
— L y i — 4 L 22 + L y 2 ) d y =
- —
106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A .3 7 )
- / = 3, /' = 3
J
Evaluation of / ^
/ (1 6 iy2 - SL y2 + 1)dy = —
4 2
y
6ly
(A.38)
f yUd y.
- I = 1, /' = 4
(A.39)
- / = 1, /' = 5
/■ I
1
J ^ ~ ( ^ y 2 — ^ L y i L y 2 ) d y = —-
(A.40)
- 1 = 2.1' = 4
(A.41)
- I = 2.1' = o
J J ~ ( ^ y i ^ 2 — L y 2 ) d y = ——
r^
2
2
(A.42)
f 1
1
J ^~(4-£'2/i£y2 ~ L y i ) d y = - -
(A.43)
- I = 3, /' = 4
J 2ly {AL2y2 ~ Ly2)dy ~ I
107
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.44)
In the evaluation of the above integrals, the following integral formula can be
employed :
J L
(A.45)
/W -O T ^ T I)!'
where Ai and A2 are defined as Ai = 'thLf 21 and A2 = U~ UL.
108
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( A -46 )
I
A PPE N D IX B
Evaluation of elem ental matrix [Fse] for the load
vector
The elemental matrix, [F/] is calculated from the surface integrals. It is also
composed of 3 x 3 small matrices as shown in (B .l), and has total 30 x 30 m atrix
elements.
[Ff]
=
1*3 1
F32
Fn
F32
Fzi
F32
F37
F38
F37
Fl8
Fla
F36
Fza
Fzb
Fza
Fzb
f
Fzh
f
39
F 3 /1
Fn
F32
Fn
F32
Fzi
F32
Fit Fl8
F 37
F38
Fl„
F3&
F}a Fzb Fza Fzb Fi3
F zh
Fig Fi/i
Fn
F32
F 31
Fzi
Fzs
F 37
F32
F 32
39
F 37
F38
Fla Fn Fza Fzb Fza Fzb F3g Fzh Fig F3A
Fu F3i Fzi Fii Fzi Fzj Fzd Fze Fzd Fie
Fzk
Fzi
Fzk
Fzi
F3 k
Fzi
Fzm
Fzn
Fzm
F3„
Fa
Fzj
Fzi
F*j
Fzi
Fzj
Fzd
Fze
Fzd
Fie
Fn Fzi Fzk Fzi Fzk Fzi Fzm Fzn Fzm Fin
Fig.
B .l shows the boundaries of th e FDTD dom ain.
The surface integrals
become different from face to face. This Appendix will show the element calcu­
lations based on the Face I, III, and V.
109
t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
* Evaluation of / Wi - (V x W ,-/ x n)ds.
Face V
Face ID
Face I
Face II
Face VI
Face IV
Figure B .l: Boundaries of the FETD domain.
1. Face I : h = —z
• F n (i = 1 , 2 , 3,1 = 1, 2, 3) (z' = L 2, 3, /' = 1, 2, 3)
/
Wi+6(/—i) • (V x
x {-z))d s =
-C j~- J
Lifyifyi'ds (B.2)
• FZ2 (i = 1, 2, 3 , I = 1, 2, 3) (i1 = 1, 2, 3, f = 1, 2, 3)
^
/■///*
IVj+6(z_i)-(VxJ'F’t-/+3+6(j/_1) x ( —z))ds = —cj ~^2 J Lifyifyi'ds (B.3)
• F 37 (i = 1, 2, 3, I = 1, 2, 3) (*' = 1, 2, 3, /' = 4, 5)
J IFi+6(i_i) • (V x Wi*+6(z'_i) x ( —z))ds = 0
110
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(B.4)
• F38 (i = 1 ,2 ,3 ,1 = 1, 2, 3) (z' = 1, 2, 3, /' = 4, 5)
J ^t+6(i-i)
(V x Wr,;/_(_3 +6(//_l) x ( —z))ds = 0 (B.5)
•
• F3a (z = 1 ,2 ,3 ,1 = 1, 2, 3) (i' = 1, 2, 3, V = 1, 2, 3)
/
• F3b
/
•"*
_»
I'l't f
^ + 3 +6(/_1) • (V x l'Vri/+6(i/_ 1) x (~z))ds = C j ~ J L j f yi f yi>ds (B.6)
(i = 1 ,2 ,3 , I = 1, 2, 3) (i' = 1, 2, 3, V = 1, 2, 3)
-*
-*
I I' f
# I+3+6 ( i - i ) - ( V x ^ +3W _ 1)x ( - i ) ) r f s = a - £ J L jfyifyvd s (B.7)
• F3g (z = 1 ,2 ,3 , I = 1, 2, 3) (i' = 1, 2, 3, I' = 4, 5)
J W'i+3+6(i—l) • (V X l'Vj'+6(r-l) X {—z))ds
•
(B.8)
PPj+3+6(/-l) * (V X HV-f3+6(J'-L) x {—z))ds = 0
(B.9)
F3h(i = 1, 2, 3, / = 1, 2, 3) (ii' = 1, 2. 3, I' = 4. 5)
J
•
= 0
F3i(* = 1, 2, 3, / = 4, 5) (z' = 1, 2, 3, I' = 1, 2, 3)
J ^ i+6(;_ 1)- ( V x ^ w
. l) x ( - i ) ) r f s = ^
j
L t f L i - V U 'f y ^ d s
(B-10)
F3j (i = 1 ,2 , 3,1 = 4, 5) (i ' = 1, 2, 3, I' = 1, 2, 3)
J
W i + 6 (1- 1) • (V X PF,/+3 +6 (Z'-l) x ( —z))ds =
J j f c j L _(2L__ 1)L jif^ d j j ^ ds
(B n )
F3d (z = 1, 2, 3, I = 4, 5) (*' = 1, 2, 3, I' = 4, 5)
J W i+ (Z- ) • (V X PPV+ (Z'- ) x (—z))ds =
6
1
6
1
~ 2 K I Li{2Li ~ 1)(4Li' ~ W y 'fv 1"*8
111
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B -12)
•
F3e(i = 1, 2, 3, 1 = 4, 5) (i' = 1, 2. 3, I' = 4, 5)
J
Wi+ 6 (l-l) ■(V X WV+3+6(/'-l)
—
(-£))(fe =
X
J Li(2Li — 1){bi>Lj> + L i'bj')fyi f yi'ds
(B.13)
• F3k {i = 1, 2, 3,1 = 4, 5) (t# = 1, 2, 3, I' = 1, 2, 3)
J H W « - i ) • (V x
x (~z))ds = ^
J U L .U f y P - ^ d s
(B.14)
• F3l (i = 1 ,2 ,3 , I = 4, 5) (i' = 1, 2, 3, /' = 1. 2, 3)
J ^i+3+6(i-l) • (V X Wi'+3+6(f'-t) X{ —z ) ) d s =
_ 4 l ^ j L .L j L f f J h ' - ds
(B. 15)
• F3m {i = 1 , 2 , 3 , 1 = 4, 5) {%' = 1, 2, 3, V = 4r 5)
J
Wi+3+6(l-l) • (V
X
X
J L iL ji4Le •
(—z ) ) d s =
1) f ylf yPds
(B.16)
F3n(i = 1 ,2 ,3 ,1 = 4, 5) (i! = 1, 2, 3, V = 4, 5)
J Wi+3+6(i-L) • (V
X
WV+3+6(/'-l) X (—z ) ) d s =
—— J LiLj{bi>Lj’ + L i'bj')fyifyi'ds
(B.17)
2. Face III : fi = —y
• F31 (i = 1 ,2 , 3,1 = 1, 2, 3) {i! = 1, 2, 3, V = 1, 2, 3)
-*
-
/ IFl+6({_1)-(Vxl'FI/+6(/-_i)
x(-y))ds =
lit
( 1 +£«') (fyby 4- Cj Cj>)Sn
t
(B-18)
112
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
;
•
• F32 (i = 1, 2, 3, I = 1, 2, 3) (i' = 1, 2, 3, /' = 1, 2. 3)
J Wi+
6 ( /_ i) •
~8
• F 37
(z
(V
X W V + 3 + 6 ( /' - l) X
^+
(—y ) ) ds =
+
(B.19)
= 1, 2, 3, I = 1, 2, 3) (»' = 1, 2, 3, /' = 4, 5)
J ^i+ 6(/-i),( V x ^ i '+6(jf.1) x ( - y ) ) ds =
(B.20)
• F38 (i = 1, 2, 3, * = 1, 2, 3) (z' = L 2, 3, /' = 4, 5)
J Wi+efz-L) • (V x Wi'+3+6(Z'-i) x ( —y))ds =
g ^ [ ( l + M P A + cJct') + (1 + $ii')(bjbj> + CjCj')]6nSi' 4
(B.21)
• F*. (z = 1, 2, 3, / = 1, 2, 3) (*' = L 2. 3, /' = 1, 2, 3)
‘
_ 8 A /” (^
X ^ i'+ 6(Z '-l) x
{-y))ds
=
+ CiCj>)S[iSi>i
(B.22)
• F3b (z = 1, 2, 3, Z = 1, 2, 3) (*' = 1, 2, 3, /' = 1, 2, 3)
J Wz+ + (z-i) • (V x l^i'+ + (Z'-i) x (—y) ) ds =
3
6
3 6
g ^ ' (1 + 5jj')(bibi> + c,Cj')<5a^z'i
(B.23)
• F3g (i = 1, 2, 3, Z = 1, 2, 3) (z' = 1, 2, 3, Z' = 4, 5)
J ^t+3+6(z-i) • (V x Wi'+6(z'-i) x ( —y) ) ds =
k
— Q^Sjii(bibi> + CiCi')SiiSi'4
113
/
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B.24)
• F3h (i = 1, 2, 3, / = 1, 2, 3) (i' = 1, 2, 3, /' = 4, 5)
/ ^+3+6(/-i) • (v x W v+ z+ w -i) x (~ y ))d s =
+ City) + (1 -F Sji')(bibjt + CiCj')]5n di>4
~ g ^ [(! +
(B.25)
• F3i- (i = 1, 2, 3, f = 4, 5) («' = 1, 2. 3, Z' = 1, 2, 3)
/ ^K+6(i-i) • (V x WV+ef*'-!) x (—y))ds = 0
(B.26)
• F3j (i = 1, 2, 3, Z = 4, 5) (i' = 1, 2, 3, Z' = 1, 2, 3)
J ^t+6((-i) • (V x M'V+3+6(i/_ l) x (—y))ds =
0
(B.27)
• Fzd (i = 1 ,2, 3,1 = 4, 5) (i' = 1, 2. 3, V = 4, 5)
/p W
d
• (V x VFi/+6(/<_L) x ( —y))ds = 0
(B.28)
• Fze (i = 1, 2, 3, Z = 4, 5) (i' = 1, 2, 3, Z' = 4, 5)
J
Wi+6(/-i) • (V x $V +3 +6(/'-i) x (—y))ds = 0
(B.29)
• Fzk (i = 1, 2, 3, I = A, 5) (*' = 1, 2, 3, Z' = 1, 2, 3)
J
Wi+3+6(j-i) • (V x WV+efJ'-i) x ( —y))ds = 0
• Fzi (i = 1, 2, 3, I = 4, 5)
/l W
- 1 ,'
(B.30)
(i' = 1, 2, 3, V = 1, 2, 3)
x ^'+ 3 + 6 (^ -i) x (—y))ds = 0
(B.31)
• F3m (i = 1 ,2 ,3 ,1 = 4, 5) (i' = 1, 2, 3, Z' = 4, 5)
/ * W
i , ‘
x ^z'+6(i'-i) x ( —y))ds = 0
114
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B.32)
• F 3„ (z = 1, 2, 3, / = 4, 5) (i' = L 2, 3, V = 4, 5)
J
Wi+3+6(f-L) • (V
X
^ +3+6(/'-l) X ( —y))ds = 0
(B.33)
3. Face V : h = x
• F 31 (i = 1, 2, 3, / = 1, 2, 3) (z' = 1, 2, 3, I' = 1, 2, 3)
t
/
• F32
(
z
J
I~l ~i c
f
W ^ + sji-d •
(V
X W i / + 6 (p _ i )
X i)d s
(z
j
J L ifylf yl'ds
(z
(
z
(B.36)
= 1, 2, 3, I = 1, 2, 3) (*' = 1; 2, 3, /' = 4. 5)
/ ^i+6(z-i) • (V x ^i'+3+6(i'-i) x %)ds = 0
. F3a
(B.35)
= 1, 2, 3, / = 1, 2, 3) (*' = 1, 2, 3, /' = 4, 5)
J ^ i+ 6{i-i) • (V x PFi»+6(//_1) x i)rfs = 0
• F38
(B.34)
= 1, 2, 3, I = 1, 2, 3) (z' = 1, 2, 3, Z' = 1, 2, 3)
• (V x Wi'+3+6{ir-i) x i) d s = - b
• F37
J L ify ify i'd s
=
(B.37)
= 1, 2, 3, I = 1, 2, 3) (*' = 1, 2, 3, /' = 1, 2, 3)
J Wi+3+6(z—i) • (V x PFi/+6(z'-i) x x)d s = bi~^2 J L j f y[fyiids
(B.38)
• F36 (i = 1, 2; 3, / = 1, 2, 3) (z' = 1, 2, 3, /' = 1; 2, 3)
/
• F 3j?
_*
lli r
^Fi+3+6(z-i) • (V x W'i;+3+6^/_l) x x )d s = bi-^£ J Ljfy[fy[>ds (B.39)
(
z
= 1, 2, 3, I = 1, 2, 3) (z' = 1, 2, 3, Z' = 4, 5)
J Wi+ + (Z-l) • (V
3
6
X
WV+6 (Z'-l) x x )d s = 0
115
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B.40)
• F3h (Z = 1, 2, 3, I = 1, 2, 3) (i' = 1, 2, 3, Z' = 4, 5)
J
^ i+ 3 + 6 (t-l)
’ (V
IV? + 3 + 6 (i'-i) x i) d s = 0
X
(B.41)
• F3i (Z = 1, 2, 3, Z = 4, 5) (i' = 1, 2, 3, Z' = 1, 2, 3)
J ^:+ 6(f-i)-(V xP F t'+6(Z'-i) x x ) d s = —
J Li{2Li —l) L i'fyl- ^ - d s
(B.42)
•
F3i (Z= L 2, 3, Z = 4, 5) (Z' = 1, 2. 3, Z' = 1, 2, 3)
J
W i+6 (/_ i) •
(V
X PVV+3+6(Z'-L) X £ ) d s =
J Li(2Li - l)L r f J t e - d a
^
(B.43)
• F3d (i = 1, 2, 3, Z = 4, 5) (Z' = 1, 2, 3, Z' = 4, 5)
J
W i+ ey-D •
X ^V +6(t'-l) X *)<& =
g - J L i( 2 L i - 1)(4L , - l ) f vlf yl'ds
(B.44)
F3e (Z= 1, 2, 3, Z = 4, 5) (Z' = 1, 2, 3, V = 4, 5)
•
J
W i+6(l-l)
AI
' ( V X P F j/+ 3 + 6 ( i '_ i ) X x)d s
~
=
Li'cj')fyifyi'ds
(B.45)
. F3k (i = 1, 2, 3, Z = 4, 5) (*' = 1, 2, 3, V = 1, 2, 3)
I
^ + 3 + 6 ( 1 —!)
• (V
x x )d s =
X
J L iL jL v fy ^ d s
(B.46)
• F3Z (Z = 1, 2, 3, Z = 4, 5) (Z' = 1, 2, 3, Z' = 1, 2, 3)
y ^i+3+6(z-i) • (V x Wi'+3+6(t'-i) x x )d s = - ^
J L iL jL ffy i- ^ -ds
(B.47)
116
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
FZm (i = 1, 2, 3, I = 4, 5) (i' = 1, 2, 3, /' = 4, 5)
J
^ i+ 3 + 6 (/-I) •
(V
X
W i ' +6( l ' - 1) X f ) t / s =
J L iL j(4 L i- ~
1
)fy lfy l'd s
(B.48)
FZn (i = 1, 2, 3, I = 4, 5) (*' = 1, 2, 3, /' = 4, 5)
J
^ i +
3+ 6 ( / - l ) •
J
8 r
A
(V
X
H'V+3+6(r _ 1)
X
x)d s =
L i ' cj ' ) f , j i f , j i ' d s
117
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B.49)
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