Mode Matching Analysis of the Coplanar Microstripline on a Layered Dielectric Substrate

Abstract

In this project, mode matching was used to calculate the propagation constant and the characteristic impedance of a coplanar coupled microstrip line. The Striplines are considered to be perfect electric conductors of negligible thickness, and are separated from the ground plane by three layers of dielectric material. The layer with the higher dielectric constant is sandwiched between two layers of lower dielectric constants, such that the field is confined to this middle layer, which is called the 'conducting layer' of the microstrip line. By confining the field to this layer, losses at the metal conductor are minimized.

Full text

MICROWAVE
LABORATORY
REPORT
NO.
89-P-1
MODE
MATCHING
ANALYSIS
OF
THE
COPLANAR
MICROSTRIP
LINE
ONA
LAYERED
DIELECTRIC
SUBSTRATE
TECHNICAL
REPORT
AFRODITI
VENNIE
FILIPPAS
AND
TATSUO
ITOH
APRIL
1989
ARMY
RESEARCH
OFFICE
T
CONTRACT
DAAL03-88-K-0005
C
u
I'jN
1
1989
THE
UNIVERSITY
OF
TEXAS
DEPARTMENT
OF ELECTRICAL
ENGINEERING
AUSTIN, TEXAS
78712
&~
p~k'i~.los
ad
"

UNCLA.SS
I
i'ILD
FOlAS))
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RLI)J
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TITLE
(include
Security
Classification)
Mode
matching
analysis
of a
coplanar
microstrip
line
on a
layered dielectric
a
ii
a
4-
v-=
4- a
12
PERSONAL
AUTHOR(S)
Afroditi
Vennie Filivpas
and
Tatsuo
Itoh
13a.
TYPE
OF
REPORT
t
13b.
TIME
COVERED
14.
DATE
OF
REPORT
(Year,
Month,
Oay),15
PAGE
COUNT
technical
report
FROM
-
TO
-,I_
April
1989
8
-16
SUPPLEMENTARY
NOTATION
The
view, opinions
and/or findings
contained
in
this
report
are
those
of
lhe
authqr($)
and
sh
uld
not
be
constugd
as an
qfficial
Dqpartment
of
the
Army position,
7
COSATI
CODES
18.
,IBJECT
TERMS
(Continue
on
reverse
if
necessary
and
identify
by
block
number)
FIELD
GROUP
SUB-GROUP
coupled
microstrip
line,
coplanar,
mode
matching,
modes,
layered
dielectric, propagation
constant,
characteristic
impedance,
conducting
laver
9-;
BSTP.ACT
(Continue
on
reverse
if
necessary
and
identify
by
block number)
In
this
project,
mode
matching
was
used
to
calculate
the
propagation
constan
and
the
characteristic
impedance
of
a
coplanar
coupled
microstrip
line.
The
Striplines
are
considered
to
be
perfect electric conductors
of
negligi-
ble
thickness,
and are
separated
from
the
ground
plane
by
three
layers
of
dielectric
material.
The
layer with
the
higher
dielectric
constant
is
sandwiched
between
two
layers
of
lower
dielectric
constants,
such
that
the
field
is
confined
to
this
middle
layer,
which
is
called
the
iconducting
layer
of
the
microstrip
line. By
confining
the
field
to
thi
layer,
losses
at
th
metal
conductor
are
minimized.
I,/ /
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OF
ABSTRACT
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iod
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NAME
OF
RESPONSIBLE
INDIVIDUAL
22b
TELEPHONE
(Include Area
Code)
22c
OFFICE
SYMBOL
Tatsuo
Itoh
-
(512)
471-107
2 1
DO
FORM
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MAR
83
APR
edition
may
be
used
until
exhausted
SECURITY
CLASSIFICATION
OF
THIS
PAGE
All
other
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are
obsolete
I
ASS
I]
FD

MICROWAVE
LABORATORY
REPORT
NO.
89-P-1
MODE
MATCHING
ANALYSIS
OF
THE
COPLANAR
MICROSTRIP
LINE
ONA
LAYERED
DIELECTRIC
SUBSTRATE
TECHNICAL
REPORT
AFRODITI
VENNIE
FILIPPAS
AND
TATSUO
ITOH
Accession
--
or
-
APRIL
1989
NTIs
GRAi-
DTIC
TAB
Unannounced
D
Justificatio
By
ARMY
RESEARCH
OFFICE
Distribut
io/
CONTRACT
DAAL03-88-K0005
Availab'lity
Codes
Avail
and/or
i
Dist
Speoa2
THE
UNIVERSITY
OF
TEXAS
DEPARTMENT
OF ELECTRICAL
ENGINEERING
AUSTIN,
TEXAS
78712

Table
of
Contents
Introduction
1
Planar
Transmission
Media
1
Chapter
1.
Mode
Matching
3
Chapter
2.
Analysis
of
the
coupled
microstrip
line
5
2.1.
Parallel-plate
waveguides
5
2.2.
Three-layer
parallel-plate
waveguide
8
2.3.
Four-layer
parallel-plate
waveguide
17
Chapter
3.
Coupled
microstrip
lines
22
3.1.
Propagation
constant
24
3.2.
Characteristic
Impedance
47
Chapter
4.
Numerical
Results
54
4.1.
Convergence
criteria
54
4.2.
Program
verification
57
4.3.
Results-Design
charts
59
Chapter
5.
Conclusions
72
Appendix
A.
Notations
74
Appendix
B.
Field
equation
derivation
75
References
82
A a l i lI

Introduction.
Planar
Transmission
Media
Planar
transmission
media
at
millimeter
wavelengths
provide
a
reasonably
good
performance
and
lend
themselves
to
mass
production
techniques.
There
are
three
categories
under
which
such
lines
may
be
placed;
planar
and
quasi-planar
(i.e.
microstrip
line
and
its
variations,
and
finline), dielectric
guides
(dielectric
slab,
image 'line,
insular
guide,
inverted
image line,
etc.),
and
H-guides
(groove guide, trough
guide, etc.)
[14].
At
millimeter
wavelengths,
the
most
commonly
used
planar
transmi-ssion
lines
are
microstrip
and
microstrip-like
(inverted
and
suspended)
lines.
These lines
are
suitable
for
the
design
of
low-cost,
mass-producible millimeter
wave
integrated
circuits.
For
microwave
integrated
circuits
up
to
110
GHz, the
main
contenders
are
microstrip,
suspended
microstrip,
fin
line
and
image
line.
Some
examples of
these
lines
are
presented
in
figure
1.
Microstrip
and
microstrip-like
transmission
lines
consist
mainly of
a
thin
strip
conductor
on
a
homogeneous
or
inhomogeneous
dielectric substrate
that
is
backed
by
a
ground
plane
of
infinite conductivity.
Many
numerical
and
analytical
techniques
exist
which
are
used
to
analyze
the
behavior
of
these
media
[15].
One
of
the
simplest
of
these methods
is
the
quasi-
static approach
[6].
This
approach, however,
has
a
limited
range
of
validity,
as
the
nature
of
the
mode
of
propagation
in
this
case
is
assumed
to
be
pure
TEM,
and
the
transmission
line
characteristics
are
calculated
from
the
electrostatic
mutual-
and
self
capacitances
and
inductances of
the
structure.
The
quasi-static
analysis
is
therefore
adequate
for
designing
circuits
only
when
the
strip
mmmmm
m m 1

2
width
and
the
substrate
thickness
are
very
small
compared
to
the
wavelength
in
the
dielectric
material.
The
full-wave
approach,
on
the
other
hand,
is
complete
and
rigorous.
It
takes
into account
the
hybrid
nature
of
the
mode
of
propagation,
and
the
transmission
line
characteristics
are
calculated
by
determining
the
propagation
constant
of
the
device.
The
aforementioned hybrid
modes
are
a
superposition
of
TMY
and
TEY
fields
that
may,
in
turn,
be
expressed
in
terms
of
two
scalar
functions,
Xy
and
iq,
respectively.
There
are
several
methods
available
for
calculating
the
propagation constant,
including
the
integral
equation
methods,
finite
difference
methods,
spectral
domain
methods
[2,8,18],
and
mode
matching
[7,10,11,191.
Strip Co"utnwr
ADi~ootho
Sabsaxt
Oromd
Plow
-CCM
*;
(b) -
(C)
Figure
1. Some
examples
of
planar
transmission
lines:
(a)microstrip
line,
(b)inverted
microstrip
line,
(c),
(d)coplanar
waveguide,
and
(e)coupled
microstrip
line.

Chapter
1.
Mode
Matching.
Mode
matching
is
one
of
the
most
frequently
used
techniques for
formulating
boundary
value
problems.
Its
primary
advantage
is
that
it
does
not generate
spurious
solutions.
It
does,
however,
have
a
relative
convergence
problem,
so
the
accuracy
of
the
results
should
be
verified
carefully.
It
is
not
a
very
efficient
method,
and
not
suitable
for
CAD
packages,
but
it
does
afford
an
exact
solution
within
a
logical margin
of
error.
Mode
matching
is
used
when
the
structure
in
question
can
be
identified
as
the
junction
of
two
or
more
regions,
each
belonging
to
a
separable
coordinate
system. For
a
planar
structure,
rectangular
cartesian
coordinates,
which
are
a
separable
system,
are
used
to
describe
the
structure.
One
other
consideration
in
mode
matching
is
that
in
each
region,
there
must
exist
a
set
of
well-defined
solutions
of
Maxwell's
equations
which
satisfy
all
the
boundary
conditions
of
the
structure,
except
the
continuity
conditions
at the
junctions
between
the
regions.
Thus,
the
separation
of
the
structure
into
regions
must
be
done
in
a
well-defined
and
judicious
manner
such
that
a
simple
and
well-
converging solution
may
be
obtained.
The
steps
followed
in
the
mode-matching procedure
are
simple
and
straightforward.
The
first
step
is
to
define
a
certain
set
of
normal
basis
functions
for
each
region
of
the
device,
and
to
expand
the
unknown
fields
in
these
regions
with
respect
to
these
normal
functions.
For
the
sake
of
simplicity,
these
basis
functions
will
be
called
"modes",
although
they
do
not
satisfy
the
source-free
wave
equation
with
all
the
boundary
conditions.
They
do,
however, satisfy
the
wave
equation
in
their
respective
regions,
and
they
will
be
subject
to the
boundary
conditions
of
those
regions.
The
functional
forms
of
these
modes
are
already
known,
3

4
so
the
electric
fields
are
now
actually
defined
by
the
weight
of
each
mode.
In
this
manner,
the
original problem
reduces
to
that
of
determining
the
set
of
modal
coefficients
associated
with
the
field
expansions
in
the
various
regions. This
procedure
leads to
an
infinite
set
of linear
simultaneous
equations
for
the
unknown
modal
coefficients.
To
obtain
the
exact solution
to the
problem
at
hand,
one
must solve this
infinite
set
of
equations,
a
generally
impossible
task. Approximation techniques,
such
as
truncation
or
iteration,
must
therefore
be
applied,
and
herein
lies
the
difficulty
of
mode
matching.
In
a
straightforward
analysis,
the
number
of
modes
retained
will
determine
the accuracy
of
the
solution.
More
modes
would
logically
seem
to
provide
a
more
accurate solution.
However,
computation
time
increases
as
the
square
of
the
number
of
modes
retained.
This
is
one
important
factor
which
necessarily
limits
the
number
of
modes
which
can
be
retained
in
each
region.
Another
source
of
numerical
difficulties
arises
from
the
fact
that
planar
structures
such
as
the
microstrip
have
geometrical
discontinuities
in
the
form
of
sharp edges.
In
this
case,
the
fields
must
be
subjected
to
one
more
physical
condition,
known
as
the
edge
condition,
which
states
that
the
power
of
the
electric
and
the
magnetic
fields
at
the
edges
must
be
finite.
This
extra condition
is
needed
so
ihat
a
unique
solution
to
Maxwell's
equations
may
be
obtained.
In
mode
matching,
this
condition
translates
into
a
relationship
between
the
number
of modes
retained
in
each
region
and
the
actual
physical
dimensions
of
the
regions.
This
relationship
will
become
more
clear
as
the
analysis
of
the
particular
device
under
consideration,
the
coplanar microstrip
line,
progresses.
The
mode
matching
technique
may
be
extended
to
include
cases
of
continuous
spectra.
This
work,
however,
will
not
take
such
a
case
into consideration.

Chapter
2.
Analysis
of
the Coupled
Microstrip
Line
2.1.
Parallel-plate
Waveguides
19
.. . . .. .
z
Figure
2.
Parallel-plate
waveguide.
The
medium
between
the
plates
is
described
by
the
general
dielectric
constant
C(x,y).
A
parallel-plate
waveguide
(figure
2)
is
a
rectangular
structure,
so
it
can
easily
be
described
using
rectangular
cartesian
coordinates.
Without
loss
of
generality,
it can
be
assumed that
the metal
plates of
the
waveguide
lie
parallel
to
the
plane
of two of
the
axes,
and
that
the
direction
of
propagation
lies
on
one
of
these
axes.
The
medium between
the
plates
may
be
homogeneous
or
inhomogeneous.
In
the
case
of
a
homogeneous
medium,
finding
the
field
distribution
is
elementary
and
the
procedure
may
be
found
in
any
textbook.
In
the
latter
case
of
an
inhomogeneous
medium,
the case
where the
medium
is
layered
in
only
one
direction,
parallel
to
the
plane
of
the
metal
plates
of
the
waveguide,
will
be
investigated.
Assuming
that
the
plates
lie
5

6
parallel
',r
the
x-z
plane,
and
that
the
direction
of
propagation
is
along
Lhe
z
axis,
TMY
and
TEY
solutions
to
Maxwell's equations
may
be
constructed.
Using
untilde'd
variables
to
denote
TMY
quantities,
and
tilde'd
variables
to
denote
TEY
quantities,
the
TMY
field
components
may
be
written
as
(see
Appendix
B)
[13]:
2
Ex_
f(x,y)
e
jkz
H,
=j
kzf(xy)
ejk
x
j
co
e(y)
ax
Dy
E,
k
k+T
f(x,y)
H
e
H=0
(2.1.1)
"
j
CO
FE(y)
Ez=-
kz
a
f(x,y)
e-jkz
Hz=
a
f(x,y)
e-jk7z
oe(y)
ay
ax
where
f(x,y)
is
the
TMY
scalar
potential,
e(y)
is
the
y-dependent
permittivity,
k2
=
(o
2g
OF,(y)
is
the
wavenumber,
and
kz
is
the
propagation
constant
in
the
z
direction
[4,7,19].
For
the
TEY
field
components,
the
corresponding
equations
are:
=
-
j
kzf(XY)
eik?
H=
1
f(x,y)
-jk?
Ey=
1
k2
+
f(x,y)
ejkz
(2.1.2)
_____)
kz
f(x,y)
jkj
He
x
y te
TE
sc(Aar
o
pyona
Here,
f(x,y)
is
the
TEY
scalar potential.

7
In
a
parallel
plate
waveguide
with
the
x
dimension
of
the
waveguide
very
large,
or
for
high
frequencies,
the
potential
functions
for
the
TMY
and
the
TEY
fields
may
be
assumed
to
be
a
function
of
y
only.
So,
the
TMY
potential
may
be
written
as:
f(x,y)
=
W(y)
(2.1.3)
and the
TEY
potential
function
may
be
written
as:
f(x,y)
=
V(y)
(2.1.4)
Substituting
(2.1.3)
into
equations
(2.1.1)
for
the
TMY
fields
will
yield:
Ex
=O
H
j
k
z,(y)
e-jkj
E
-1
(k2+
d
x(y)
ei
j k 7
z
Hy
=0
(2.1.5)
Yj
(0
E(y)
2
Ez=
kz
dqf(y)
e-Jkz
Hz =
C
(y)
dy
Similarly,
substituting
(2.1.4)
into
equations
(2.1.2)
for
the
TEY
field
components
will
yield:
E j=-J
k(Y)
e-Jkg
x=0
Ey
1
(k2
+ 2)(y)
e
(2.1.6)
JW4g0 dy]
Ez
=
0H
= -
k
d
W(y)
e-jkz
0
dy

2.2.
Three-Layer
Parallel-Plate
Waveguide:
pec
plate
h2
pec
plate
Figure
2.2.1.
Three-layer
parallel
plate
waveguide.
At
this
point,
it
would
be
advisable
to
separate
the
structure
into
regions.
For
a
three-layer
parallel-plate
waveguide
(a
parallel
plate waveguide
with
three
layers
of
dielectric),
the
structure
can
be
separated into
three
regions,
each
region
characterized
by
its
own
dielectric.
So,
for
the
parallel-plate
waveguide
in
fig(2.2.1),
region
1
will
be
that
region
where O
2
!.y
.
hj,
h,1
being
the
point
where
the
dielectric constant
of
the
medium
changes from
F-
1
to
F-
2.Region
2
will
be
that region
where
hj
1
!-,.y
! h2,
h2
being
the
point
of transition
from
=E
2
to
E=E
3.
Similarly, region
3
will
be
that
region
for which
hV
.h,
h
being the
height
where
the
top
metal
plate
of
the
waveguide
is
situated.
Each
region
will
be
characterized
by
its
own
set
of
TMY
and
TEY
potential
functions,
f(y)
being
the
TMY
potential,
and
t(y)
the
TEY
potential
of region
i,
i=1,2,3.
Both
the
TMY
and
TEY
scalar
potentials
are
solutions
of
the
scalar Helmholtz equation:
8

9
2
d(y) + 2
X(y)
= 0
2
dy
(2.2.1)
a
general
and
complete solution
to
the
above
equation
being:
x(Y)
=
a I
cos
(ICy)
+
X2
sinf
(icy)
(2.2.2)
where
a,
and
a2 are
constants
which
depend
on
the
boundary
and
continuity
conditions
of
the
structure.
The
function
sinf(icy)
is
defined
as
sinf(Cy)=sin(icy)/x
and
its
usefulness
will
be
presented
later
on
in
this
analysis.
The
electric
and
magnetic
fields
must conform
to
certain
boundary
and
continuity conditions.
These
conditions
will
serve
to
specify the unknown
variables
C1,
oX
2,
and
iK
in
equation
(2.2.2).
The
general
solution
to
the
scalar
wave
equation
shown
above
is
modified
such
that these
continuity
conditions
can
be
implemented.
Substituting
X(y)
into
the
TMY
equations,
and
applying
the
conditions
of
zero
tangential electric field
on
the
pec
plates will
yield
the
general form
for
the
potential
for
the
TMY
modes
of
a
three
layer
parallel-plate
waveguide:
acos
( ky1
y)
;0.5
_y5
h1
(y)=
b
cos[
ky2h2-y)]
;h
1:
y
5
h2
(2.2.3)
+
c
sinf[
ky
2(h2
-y)]
d
cos[
ky
3
(h-y)]
;h
2
-<
y:5
h

10
Correspondingly,
substituting
the
appropriately
modified
forms
for
the
corresponding
form
for
the
TEY
scalar
potential
into
the
TEY
field
equations
will
yield:
asinf
(kyly)
0.
<
h,
b
cos[
ky2(h
2
-y)]
h1 -
y
< h2
(2.2.4)
+
c
sinf[
ky2(hE-y)]
cos[
ky
3
(h-y)]
; h2
5
y
_ h
In
general,
the
coefficients
of
the
TMY
and
TEY
scalar
potentials,
as
well
as
their corresponding eigenvalues,
will
not
be
identically
equal.
The
continuity
conditions
between
the
different
regions state that
the
tangential fields
at
a
dielectric
discontinuity
must
be
continuous. Applying
these
conditions,
and
solving
the
resulting
equations,
will
yield
the
eigenvalue
equations
for
the
TMY
and
TEY
y-directed eigenvalues
as
well
as
the
corresponding
expansion
coefficients
of
the
scalar
potentials
in
equations
(2.2.3)
and
(2.2.4).
The
system
is
underdetermined,
so
the four
expansion
coefficients
in
each
case will
be
found within
a
multiplicative
constant.
Following
the
procedure
described
above
will
yield
the
TMY
eigenvalue
equation
for
the
three-layer
parallel-plate
waveguide:

k'tan(k 1
h
I) +
kY
tan[ky
2(h2
-h
I))+
-kY3tanh~ky
3
(h-h2j)
e
le 2e (
2
.2
.5
)
kL-tan(kh1)
-!tan[k
2(h2
-h
1
)]
kI-
a~k
3-
and
the
coefficients
are:
1.
ky
3
real
cos[ky
3
(h-h
)] k3ianry(2.2.6.
a)
C -
C3
ky
3
sin[ky
3
(h-h
2)]
ky
3
real
-2
ky
3
tan[ky
3
(h-h
2
)]
ky
3
imaginary
F-3
(2.2.6.b)
b
=
cos[ky
3
(h-h
2)]
ky
3
real
1
.
ky
3
imaginary
(2.2.6.c)
a=b
cos
[k
y
2
(h
2
-h
1)]
+ c
sinf[k
y
2
(h
2
-h
1)]
One
other
relationship
of
great
importance
in
this
analysis
is the
dispersion
equation.
This
equation
links
the
eigenvalues
k.~,
ky,
and
kz,
with
the
wavenumber
k
in
a
certain
medium.
The
dispersion
equation
states
that:
2
2
2
2
2
kx
+
ky+
kz=k
=~
(o
1(y)
(2.2.7)
for
the
TMY
case,
and
correspondingly:

12
-2
-2 2
2
2
kx
+
ky
+ kz
=k
=CO
go0
(y)
(2.2.8)
for the
TEY
case.
For
the
particular
case
of
the
three-layer
parallel-
plate waveguide,
it
has
been
assumed
that
kxi=0,
i
=
1,
2,
and
3,
so
the
dispersion
relations
for
the
TMY
and
the
TEY
case
correspondingly
become:
k
2 2 2
2
yi
+kz
i =
o0
[/0(2.2.9)
and
-2
2 2
2
ky
+
kz
=
ki
=
o)
g
t0e
i
(2.2.10)
kz
being,
of
course,
the
propagation
constant
of
the
waveguide,
and
k
the
wavenumber.
The
z
axis
(and
therefore,
the
direction
of
the
propagation constant)
are
parallel
to
the
planes of
the
discontinuities,
and
so
continuity
forces
kz
to
be
the
same in
every
region.
This
gives
rise
to
a
very
useful
relationship
between
the
eigenvalues
in
the
y
direction
in
each
region.
This
relationship
is
derived
by
subtracting
the
dispersion
relation
defined
in
the one
region
from
the
corresponding
equation
in the
other
region.
Thus, the
dispersion relationships corresponding
to
two
neighboring regions
of
the
waveguide
reduce
to:
2
2
2
ky
i -
kyj
=o
gO~e
0-
P)
(2.2.11
)
and:
-2 -2 2
kyi
+
kyj
o
go0(e
-
E)
(2.2.12)

13
so
that
the
y
eigenvalues
in
each
region
of
the
waveguide
are
not
independent
variables,
but
are
linked
through
the
dispersion
relation.
The
TEY
eigenvalue
equation
as
well
as
the
coefficients
for
the
TEY
scalar
potential
may
be
derived
in
a
manner
similar
to
that
used
in
the
derivation
of
the
TMY
potential.
The
TEY
eigenvalue
equation
is
thus found
to
be:
ta~yh)
tan
[k
y2(h
27h
1)]
tan[ky3(h-h2)]
kyl_
ky2
_ky3
(..3
tan(kylh
1),, ktak2h-h)
tan[ky3(h-h2)]
kyl
ky3
and
the
coefficients
of
the
TEY
scalar
potential
in
the
waveguide
are:
1.
ky3
real
1.
ky
3
imaginary
cos[ky3
(h-h
2)]
(2.2.14.
a)
-
cos[ky
3
(h-h
2)]
ky
3
real
C- 1.
ky3
imaginary
(2.2.14.b)
sinf[ky
3
(h-h
2)]
ky3real
sinflky3(h-h2)]
-
ky3
imaginary
cos[ky
3
(h-h
2
)]
(2.2.14.c)

14
b
cos[ky
2(h2
-hl)]
+
c
sinf[ky
2(h2
-h
1)]
a--cos(
kyl
hi
I
(2.2.14.d)
Working
in
the
reverse order
now,
the
eigenvalue
equation,
in
conjuction
with
the
modified
dispersion relation,
will
yield
the
eigenvalues
in
the
y-direction
of
the
three-layer
parallel-plate
waveguide.
The
dispersion
relation
will
then
yield
kz,
the
propagation constant
of
the
waveguide
in
the
z
direction:
/2 2
kz=
OV oE i -
ky
i
(2.2.15)
for the
TMY
case,
and
(2
-2
kz=
o
ei -
kyi
(2.2.16)
for
the
TEY
case.
In
addition,
the
ky
i and
ky
i
eigenvalues
serve
to
specify,
within
a
multiplicative
constant,
the
expansion
coefficients
of
the
corresponding
scalar
wave
equations.
These
scalar
wave
equations,
substituted into
the
corresponding
TMY
or
TEY
field equations
will
yield
the
TMY
or
TEY
electric
and
magnetic
field components.
Since
the
coefficients
are
found
only
within
a
multiplicative constant,
only
the
distribution
of
the
E
and
the
H
fields
in
the
waveguide
can
be
found.
When
the
TMY
and
TEY
scalar
potentials
were
defined,
a
new
functional
form
was
introduced.
This form
was
the
function
sinf(icx),
which
was
subsequently
defined
as:
sinflcx)
-
sinicx)
K(2.2.17)

15
From
the
definition
of
the
"sinf"
function
as
stated
above, it
is
evident that this function
has
the
following
properties:
d
{
sinf(
x)}=cos(cx)
dy
(2.2.18)
and
lim
ki
~{sinf(Kx)}=
x
ky
---
40
(2.2.19)
The
"sinf"
function
is
preferred
over
the
simpler
"sin"
because
it
offers
two very
important
advantages.
One
advantage
is
that
sinf(Kx)
offers
the
correct
solution
to
the
scalar
Helmholtz
equation
for
vanishing
K.
Using
sin,
the
general
solution
to
the
scalar
wave
equation
is:
X(Y)
=
C
I
cos(Ky)
+ a2
sin(icy)
(2.2.20)
As
K
approaches
0,
X(y)
becomes:
X(Y)
=
(x 1
(2.2.21)
Although
this
does
provide
a
solution,
it
is
not
a
general
solution
to
Laplace's equation.
With
the
"sinf"
representation,
however,
the
solution
to
Laplace's
equation becomes:
X(Y)
=
aI
cos(Cy)
+ a2
sinf(cy)
(2.2.22)
which,
for
vanishing
Kc
yields:
X(Y)
=
Ctl
+
ct
2
y
(2.2.23)

16
This
latter
form
does
provide
a
general
solution
to
Laplace's
equation.
The
second
very
important
advantage
of
the
"sinf"
function
over
the
"sin" is
found
in
the
nature
of
the
eigenvalues
ic.
For
a
purely
lossless
dielectric
medium,
Kc
will
be
either
real
or
imaginary. For
KC
imaginary,
the
"sin"
function
would also
be
imaginary,
which
fact
might
give
rise
to
a
complex
analysis.
However,
the
"sinf" function remains
real
irrespective
of
whether
'K
is
real
or
imaginary.
This
provides
a
distinct
improvement
over
the
"sin".
For
programming purposes, "sinf"
is
defined
as:
.x)2 (x) 2
sinf(icx)=x
*6.
120.
(2.2.4)
sin(
cx)
IKXI>
0.1
KX
Here, the
asymptotic
expression
for
"sinf"
is
used
for
the
case
II:xI<0.1.

2.3.
Four-layer
parallel-plate
waveguide:
h
pec
plate
i/f /f/ li /If li -i -fe toof/
.e
*. %% 11 . % % % % .
.S
S .. S .
ilet./ to li0. i-d-- i i--
e#ooteift
hI Oll1f/ lI Ill o,2:Illi10Al lif/1.1 f Ie
'S S'
lool eel too Allot'
SS
.S S'
' '
'S S.S'
' '
-------
------ -i /----
------
-------
-- i-- -
parallel-plate davdgude may beiwritten as:
adjl coIdly 0/ii/i. 5 y 5/ h,/l/ddd
b
~.
co~y(r) h,
*. :5 h
W..) +. ..
.ifk2h
) (2..1
d cos~ky3(3.....h2....Y.!5..
+__e
________________
fzo~y(-)
35Y:
1e7pat

18
The
eigenvalue
equation
is
found
using
the
same
procedure
that
was
described
for
the
three-layer
parallel-plate
waveguide.
The
TMY
eigenvalue
equation
is
therefore:
kY
tan(k
yh1) + ky
2
tan[ky
2(h2-h1)]
+
y-3
tan[ky
3(h3_h
2)] +
k_._
tan[ky
4
(h-h
3)]=
F3
E4
kY
tan(kylh1)
62
tan[ky2(h2_h1)]
k y3
tan[ky
3(h3-h2)] +
61
ky2
E3
.
tan(k
yh1)
-__2_
tan[ky
2
(h2.h
1)]
ky4
tan[ky
4
(h-b)]
+
Ek -C4
ky-_
tan(kylhl)
---
tan[ky
3(h3-h2)] .
-4
tan[ky
4
(h-h
3
)]
+
E1
k
y3
F-4
ky--2
tan[ky2(h2-h
I) --
tan[ky3(h3-h2)]
ky4
tan[kyn(h-h3)]
F2
ky3
C4
(2.3.2)
The
coefficients
which
serve to
define
the
TMY
scalar
potential
of
the
waveguide
are,
within
a
multiplicative
constant:
1.
;ky
4
real
f1.
cosI~ky
4
(h-h
3
I
k
C)iainr
(2.3.3.
a)
_C3ky
smtfky
4
(h-h
3]
;ky
4
real
e
E2
63
ky
4
tank
4
(h-h]
k4
imaginary
C2
(2.3.3.b)
d
k
cos[ky
4
(h-h3)]
;
k
y
4
real
1.
ky
4
imaginary (2.3.3.c)

19
c
=
d
cos[ky3
(h3-h
2
)]
+
e
sinffky
3
(h3-h
2
)]
(2.3.3.d)
b=
-d
1-
ky
3
sin[ky
3(h3-h2
)]
+
e
-
cos[ky
3(h3-h2)]
63
-3
(2.3.3.e)
b
cos[ky
2(h2
-h
1)]
+
c
sinf[k
2(h2
-h
1)]
cos(kylh
)
(2.3.3.
f)
The
TEY
scalar
potential
may
be
written
as:
a
cos(kylY)
;
0:<
y
<5h
1
b
cos[ky
2(h2
-y)]
;
hj!<y:5h2
+
c
sinf[ky
2(h2
-y)]
W(y)
=
(2.3.4)
dcos[ky
3(h3
-y)]
+
e
sinf[ky
3(h3
-y)]
f
cos[ky
4
(h-y)]
; h3!!
y
<
h
Using this
expression
for
the
potential,
the
eigenvalue
equation
is
found
to
be:

20
*y ky4
tann
~
3(
yh
2)])
tan[k(hh
]
+ ytnk2hh) +
iYI
ky3(2.3.5)
________)
tan[ky
4
(h-h
3)]
tan(kyl
Ih)
ki
2tan~kk(h2-h1)]
tan[k,
2(h2
-h
IA~
ank
3
hr
)
tan[ky
4
(h-h)]
iy2
k
y4
The
coefficients
of
the
scalar potential
are,
within
a
multiplicative
constant:
1.
;y
4
real
1.
, ky
4
imaginary
cos[ky
4
(h-h)]
(2.3.6.
a)
-
cos~k
y4
(h-h
3]
;ky
4
real
1 .;ky4imaginary(236b
-
Sinf1iky
4
(h-h3)
ky4
real
d=tan[ky
4
(h-h3)]
; k 4 imaginary(236c
b=d COs~iyk
3(h3
-h2)]
+
e
sinqik3('h
3
-h
)I
(2.3.6.d)

21
c
=
-d
ky
3
sin[ky
3(h3
-h)]
+
e
cos[ky3(h3-h2
)]
(2.3.6.e)
b
cos[ky
2(h2
-hl)]
+
c
sinf[ky
2(h2
-hl)]
a=
cos(kylhl)
(2.3.6.f)
Thus,
so
far,
the
TMY
and
TEY
potential
functions
for
the
three-
and
four-layer
parallel-plate
waveguides
have
been
derived.
Their
connection
with
the
coupled
microstrip
line,
which
is
the
main
subject
of
this
paper,
will
become
apparent
shortly.

Chapter
3.
Coupled
Microstrip
Lines
Coupled
microstrip
lines
are used
in
a
number
of
circuit
applications,
principally
as
directional
couplers,
filters,
and
delay
lines.
Mode
matching
will
be
used
to
calculate
the
propagation
constant
of
such
a
line.
A
pair
of
microstrip-like
transmission
lines
as
shown
in
fig.
(3.1)
are
known
to
have
the
property
of
a
broadband
directional
coupler
when
placed
in
parallel proximity
to
each
other.
As
a
result
of
this
proximity,
a
fraction
of
the
power
present
on
the main
line is
coupled
to the
secondary
line.
The
power
coupled
is
a
function of
the
physical
dimensions
of
the
structure
and
the
direction
of
propagation
of
the
primary
power.
|V4
Fiue.1.
Thopldmcrsrpie
In
geeral
thecoupe
le
strctues:hw
nfg
induces
gthe
3.Tcouplwented
twcotrsisin
lne..h
properties
of
the
coupled
structures
may
be
described
in
terms
of
a
suitable
linear
combination
of
these
even and
odd
modes.
For
22

23
geometrically
symmetric
structures,
the
geometrical
plane
of
symmetry
may
also
be
thought
of
as
an
electrical
plane
of
symmetry.
The even mode
has equal
amplitude
and
equal phase
correspondence
with
respect
to
this
plane
of
symmetry,
in
which
case
this plane
takes
the form
of
an
open
circuit.
In
the
even
mode
case,
therefore,
the
plane
of
symmetry takes the form
of
a
perfect magnetic
wall (pmc).
The
odd
mode,
on
the
other
hand,
has
an
equal
amplitude
180
0
-out-of-phase correspondence
with
respect
to
the
plane
of
symmetry,
so
this
plane
acts
like
a
short
circuit
and
is
simulated
by
a
perfect electric
conductor
(pec)
wall.
Thus, the
directional
coupler
can
be
treated
as
a
two-port
network,
and
the
total
response
can be
obtained
by
superimposing
the
responses
calculated
for
the
even
and
odd
mode
excitations.
This
reduces
the
problem
to
about
half
its
original
size, which
is
a
great advantage
of
symmetric
structures.

24
3.1.
Propagation
Constant
The
structure
shown
in
figure
(3.1)
is
symmetric
with
respect
to
a
plane
drawn
perpendicular
to the
x-y
and
x-z
planes
and
placed
halfway
between
the
two metal
strips.
Thus,
the
structure
under
consideration
is
modified
to
that
shown
in
figure
(3.1.1).
h-
top
plate
pe¢
h
or
-**"**3"
piC
zx
Wl
W2
Figure
3.1.1.
Symmetry
as
applied
to
the
coupled
microstrip
line
As
was
mentioned
earlier,
the
left-most
boundary
becomes
either
a
pmc
or
a
pec
wall,
according
to
whether
the
search
is
conducted
for
the
odd-mode
or
even-mode
propagation

25
constant.
There
is no
perpendicular
boundary
to
the
right
of
the
microstrip,
so
propagation
in
the
positive
x
direction
for
large
x
must
take
the
form
exp(-jkxx),
such
that
there
is
an
exponential
decay
for
x->co.
This
serves
to
put
a
lower
bound
on
the
search
for
kz,
since
this
exponential
decay
is
observed
only
for
the
case
where
kx
is
imaginary.
The
dispersion relation
states
that:
2
2 2 2
2
kx+ky+kz=k
-)
g~0
OF-.1
so,
for
kx:
2 2
2
2
kX-"
= .) [to -
ky-
kz
(3.1.2)
The
condition for
imaginary
kx
yields:
2
2
2
(0
g.oe
-
ky
_
kz <
0
(3.1.3)
or:
/2 2
k
-(3.1.4)
An
upper
bound
may
also
be
placed
on
kz,
since
it
may
never
be
larger
than
the
unbounded
propagation
constant
in
the
medium
with the
highest
relative
dielectric constant.
Thus:
kz
<
o5
/
tIOinax(e'
(3.1.5)
The
structure
must
further
be
separated into regions
for
modal
expansion.
Two
more
bounding
planes
may
be
drawn
parallel
to
the
plane of
symmetry,
at each
end
of
the
metal
strip.
Also,
since
the
structure
must
be
bounded,
a
top
plate
must
be
placed
above the
structure
at
such
a
distance
that
it
does
not
interfere
with
the
computation
of
the
propagation
constant
of
the
actual
open
structure
(this
top
plate
was
also
included
in
figure

26
3.1.1).
The two
lateral
bounding planes
may
be
thought
of
as
perfect
magnetic
conductor
walls, since
the
tangential electric
and
magnetic
fields
must
be
continuous
across
them. The
original
structure
has
thus
been
separated
into four
distinct
regions,
shown
in
figure
(3.1.2).
h
top
plate
~~onV
pec
h13
or
--
Region
1
Region 4
piMC
h2
wi
w2
Figure
3.1.2.
The
four
distinct
regions
of
the
coupled
microstrip
line
As
was
stated
earlier,
a
set
of
normal
modes
must
be
defined
for
each
region,
and
the
unknown
fields
in
each
must
be
expanded
with
respect
to
these
modes.
Each
region
may
be
thought
of
as
a
multilayered
parallel-plate
waveguide,
consisting
of
two
parallel
metal
sheets placed
at
a
certain
distance
from,
and
parallel
to,
each
other,
with
a
layered
dielectric
medium between
them.
Thus,
regions
1
and
4
are
four-layered
parallel-plate

27
waveguides, region
2
is
a
three-layered
parallel-plate
waveguide,
and
region
3
is
a
one-layered
parallel-plate
waveguide.
This
latter
case
has
been
studied extensively
in
the
literature,
and
has
not
been
presented
here.
The
normal
modes
for
the three-
and
four-layered
parallel-plate
waveguides
have been
defined
in
chapters
2.2
and
2.3,
respectively.
It
now
remains
to
establish
the
equations
which
define
the
electric field components
with
respect
to
these
modes.
Let
it
be
noted
that
the
following
modal
expansions
and
field
representations
are
not
unique,
but
just
constitute
one
possible
solution.
Using
the
expressions
derived
for
the
TMY
and
TEY
modes
of
the three-
and
four-layer
parallel-plate
waveguides,
the
corresponding expressions
for
the
electric
and
magnetic
field
expansions
may
be
derived.
As
shown
in
Appendix
B,
the
electric
and
magnetic
fields
in
each
region
due
to the
TMY
mode
potentials
are
given
by:
2
(fi(x,y)e'k
(i(x,y)eJkJ
E
xi
= -.. H xi= - ,
Yi
ax
ay
aZ
i + f ~~~xHi
a
(f
i(x,y)ek?
')
(Ji(x,y)e-k
)
Ezi
= ^
Hzi
=-
Yi
ay
az
ax
and
the
electric
and
magnetic
fields
due
to
the
TMY
fields
are:

28
-k2-
-jkl,
alfi(x,y)e
-JJ
~
-
i~(f
1
(x,y)e)
az
z
ax
Dy
2
E=Oi -
+
k
2
(i(x,y)ek)
(3.1.7)
D
(f
i(x,y)eijk)
14
a
(fi(x,y)e&ik)
ax
z
ay
az
A A
where
yi
=
jocl
and
Z=
jcoj~i[l3],
and
subscript
i=1,2,3,or
4,
correspondingly,
for
each
region.

29
The
electric
and
magnetic
potentials
in
each
region
are:
Region
1:
flm(xY)
=
AlmNflm(Y)if(klxmx)
even
sinf
eve(3.1.8.a)
flm(~y)--
inf
-
odd
f
1mkxY)
A=
4i
m(Y)
cos
(k
1xmx)
even
Re2ion
2:
f
2
m(X,Y)
=
V2
r(Y){
A2
mCOS[kx
2
m(X-W
1)1
+
B
2
msinf[k
x2
m(x-w )]}
(3.1.8.b)
J2
m(X,Y)
=V2m(Y)
I X2
mCOS[kx
2
m(X'W
01
+ B2
msinf[kx
2
m(x-wl)]}
Region
3:
f
3
m(X,Y)
=
V3m(Y)
{ A3mcos[k, 3m(X
-
W
1)]
+
B
3
mSiflflkx
3
m(X-W
1A
1
(3.1.8.c)
f3m(X,y) =
V3(y)
{
A3
mCOS[kx
3
m(X
-
W
1)]
+ B3
msinfkx
3
m(XW
1)]
egion
4:
f4m(X,Y)
=
"V4m(Y)
exp[-jk
4
xm(X-W
2)]
(3.1.8.d)
f4
m(X,Y)
=
V4m(Y)
exp[-jk
4
xm(x-w
2)]
where
Wi(y)
and
ji(y)
are
the
TMY
and
TEY
potential
functions
for
the
corresponding
cases of
the
one-,
three-,
and
four-layer
parallel-plate
waveguides.
For
a
suppressed z-dependence,
the
equations
for
the
electric
and
magnetic
field components
yield,
for
each
region:

30
REGION
1
Electric
Field
Components:
1
1
d~4im(y)
(kxm
sin
El.=-
YIkxlx l
jix
1(y)
m=1
dy
1I O
M,
~sinf
jkzyX
1'im(Y)
Cs(klxmx)
Alm
m=i
Ely= (k.29ka2
(+klxmINlm(y)
Cosf(klx) Alm(319a
j O) 1
(y)
m
,sn
M,
El---4
1
1\Mo)
s-
-kxx
l
I
'VIM(Y)Ik
x)
sin
(1
X
i

31
Magnetic
Field
Components:
Hix~jkz
Y
Vm(y)
sjn(klxmx)
Aim
+
1
dl4imY
1
cos(km)
X
E~
dy
1x
sin
JWtJ
m=l
x
(3.1
.9.b)
1
M
2
-2
sinf
-
Hy=
X
(k
+klxn)NVim(y)
(kixn)
Alm
j(09
0
m=1
zCos
Hlz=
wiy
Csn(klxmx)
Alm
k
ld'~fim(Y)
sinf
-lmx
l
wg~oh1I
dy
Cos

32
REGION
2
Electric
Field
Components:
*E
2.M2
dW4
2
m(Y)
IAm~mi[k,~-
)
( Y
)
d
y
{ A k x
n S
f
[
( x m X )
+
B2
m..cos[(k2xm,(X-W
i)])+
M2..
jk,.Y
NV2m(Y)
{A
2
mCOS~k
2
xm(XW)I
M=1
+ 2
nsifllk
2
xm,(X-W
0)
E~
jE(y)
m=
(k
+kxn)V4
2
m(Y)
( A2
mCOS(k
(3.(1X1i))a
+
B
2
mSifl1(k
2
xm(xw
1)J)
(..Oa
-k
z
M2
dNI
2
m(Y)
f
A m
01k
x(
-)
E2
,,,
z
-
dy
~AmO[kx(~)
(t)6-
2
(y)
m=1
+
B
2
mnsjff[(k
2
xm(XWi
01-
M2
X:
Nf2m(Y)
I A242
,nsi[(k
2
xm(X-W
1)
M=1
+
B
2mnCOS[(k2xm(X-W
0)

33
Magnetic
Field
Components:
H2x=jkz
Y,
NVim(y)
(
A2
nc0s[k
2
xm(X-WI)]
m=1
+
B
2
nif[k
2
m(X-W0)
1+
1 2dV4 2
m,(y)
Ecp
0
=
dy
I
A2mk2xmcos[k
2xm(X-W
i)
jC0M1+
ii
2
mSiflf[k
2
xm(X-W)
01
(3.
1.
1O.b)
H
=
Y,
(kz+k
2
xffNf
2
m(Y)
{
A2
mcO5[k
2
xm(X-W
1)
JW0
t
m=1
+
Bi
2
msiff
2
xm(X-W1)I
M2
H2.
I' 2
m(Y)
I
-A
242
xm5sfl[k
2
xm(X-W1)]
M=1
+
B2mCOS[k2xm(x-wl))
kzM
2
dIJm)
dVi~m1
d
-
A2niK2xmcOS~k2xm(XW
01]
+ 92
mSinflfk
2
xm(X-W1)I
The
equations
for
region
3
are
identical
to
those
for
region
2
with
subscript
"2"
substituted
by
subscript
"3"1.
REGION
4
Electric
field
components:
E~x1 M4
dWV~m(y)
jk
4
xmeXP[-jk
4
xm(X-W
2)]
Am
M4m
jkz
W44m(y)
exp[-jk4xm(X-w
2)] A4m
m=l

34
E~1
(k
+k4ip)W4
m(y)
exp[-jk
4
xm(X-W
2
Yl
A4.
i(J)F-4Y) M=1(3. 1.
11.
a)
E~z=
z
114dIJ 4(Y)
exp[-jk
4
xm,(X-w
2
)]
A4
m,
+
=-()e-
4(Y)
rn-
dy
x
V4m(y)
jk
4
xre-XP[(-jk
4
xrn(X-W
2)]A4
m=4m
Magnetic
field
components:
Hx=
jkz
Y,
W4m(y')
exptk
4
xm(x-w
2
lA
4
m-
M=1
E__
d4m,(Y)
jk
4
xrrgXP[-jk
4
xm(X-W
2
)JA4n
jcPot
rn=1
dy
H 1 M4
2
-2
H4y
E
(kz+k
4
xri)14
4
m(Y)exp[-jk
4
xm(X-W
2
)]A4n
H~
1
xr4rn(Y)
jk
4
xmeXPll-jk
4
xm(X-W,)]A
4m -
M=1
kz-
dMy)
exp[-jk
4
xm(X-W
2
)]A
4m
In
the
above
equations,
the sums
have
been
truncated
at
M,
modes
for
the
fields
in
region
1, M2
modes for
those
in
region
2,
etc.
For
the
solution
of
this
system
of
equations
to
be

35
unique,
these
equations
must
form
a
square matrix,
which
means
that
:
2
M2
+2M
3
=M
I+M4
(3.1.12)
The
relationship
in
eq.
(3.1.12)
ensures that
the
system of
equations
will
have
a
unique
solution.
It
does
not,
however,
ensure convergence
of
the
system. This
is
because
one
very
important
geometrical factor
has
not
been
considered
yet.
All
planar
structures
have
discontinuities
in
the
form
of
sharp
edges.
It
was
noted
earlier
in
this
chapter
that
these
discontinuities
must
conform
to
the
edge
condition,
that
is,
the
power
of
the
electric
and
magnetic
fields
at
these points
must
be
finite.
This
condition
was
found
to
be
adequately
met
when
the
number
of modes
retained
in
each
region
were
such
that:
M1 M4
_
h
M3 M3
h-h
3
(3.1.13)
and
M1 M4 h
M3 M3
h-h
3
(3.1.14)
Thus,
the
edge
condition
will
yield
a
relationship
between
the
number of
modes which
should
be
retained
in
each
region,
and
their
relative
dimensions.
Matching
the
tangential
fields
at
the
interface
x=w1
and
0_yh3
yields:

36
E
Y(X=W
1)
=
E
2
y(X=W
1)
,2
2
Cos(k
)
jcOe-
2)I
(zklmWm
sinf
1
imw)Am
(3.1.15.
a)
1
M2
2
2
I(kz+klxnmjJlm(y)A
2m
E
12
x=w
1) =
E
2(x=w
1) -
kNf
m
dkJ
I
my)o
(kixm
)Alm
m=1
(Yml
y
sif m
X
~
(3.1.15.)
-
1
M2
2
)
A
-
x
I
kztkxnV
2m
W)
2
my)m
(j
CO-2(y
m 1 d

37
H
2
x~w
1)
=
H~z(x=w
1)
~
k
dW
1
,1(y)
Sind
Z
I
dy
COS
(klxmw)Alinf
X
V2
m(Y)B32m
kz
Nfmy;2

38
Matching
tangential
electric
and
magnetic
fields
at
x=w2
for
O:5y:h3
yields:
E2
$x=w
2) = E4
$X=W
2) 4--
jOF
=
(kz+k
2
xm)XV
2
m(Y)
[A
2
mCOSlk
2.m(
W2
7W
01
+B 2
nmsiff[k
2
xm(W
2
-W
I)
(3.1.16.a)
1
4
2
2
E(kz+k
4
xm)~f
4
m(y)A
4m
E~z(x.=w
2
)-=.E-
4
x=w
2)
4
-
kM
2
kz
dWmy)fA
2
mCO5[k
2
xm(WZ-Wi)1
coE
2
(y)M
1
j
dy
M,+B
2
m
inf[k
2
xm(W-W1)
I-
(3.1.16.b)
W
M2m(Y)
f1 -A2
mk
2
xffnSif[k
2
xm(W
zW
1A]
m=1
+B2mCOS[i2xm(W-1W
)I
k 4dxV4M(v')
A
M4-
EZ-
I
''
4m
+
2.
V4m(y)jk4xmA4m
(o) -4
(y)
m =
dy
j=
H2
IX=W
2)
=
H4
X=W
2) *
j(40=
(k
z+k2
xO)
\2m(Y)
I
A
2
rnCOS[k
2xm(
W2
W
01J
JCOJL~mI(3.1.16.c)
+B2m-,Siffk2xm(WT-WIl
1 M4
2
-2
i()LO=
(kz+k
4
xft11
4
my)A
4m

39
Ha(x=w
2) =
H4
(x=w)
4
M2
Y
XV2m(Y)
{-A
2mk2
xmSink
2
xm(W
2
"W
1)]
m=1
+B 2
mCOS[k
2
xm(W
2-W) I-
(3.1.16.d)
M1 d2
m(Y)
{2rnCOS[2xm(WTW1)]
kzm1
dy
+B 2
mSinf[k
2
xm(W
2
Wl)]
I
=
M4
+ zM4
-
W
W
4
m(Y)jk
4
xnA4m
Z
dI4m(Y)Aft
m=l
(o0t0
go dy
The
equations for
x=wi
and
x=w2
with
h3_<y_<h
are
identical
to
the
corresponding equations
for
0_y<_h3
with
the
subscript
"3"
substituted for subscript
"2".
As
stated
earlier,
the
orthogonality
of
the
modal
expansion
functions
x
im(y)
and
'jtim(y)
take the
form:
f
I
Virn(Y)Vin(Y)dy
for
man
over
one
region
Ei(y)
(3.1.17)
and:
lover
one
region
imf
(or
i
n(y)dy
(3.1.18)
So,
orthogonality
can be
utilized
here
to
reduce
the
complexity of
the
continuity
equations.
By
multiplying
each
equation
by
the
appropriate
orthogonal function,
and
integrating
over
the
appropriate
region
(from
0
to
h3),
the
equations
for
0_<yh3
may
be
rewritten
as:

40
M'2
2
e2
cos
I
(kz+k
lxr
m
llnflf
(k
lxmw
i)Ailm
Ml2
2
e2
(3.1.18.a)
(k= ln)2n~
MI
1z
I
h2
cos
k
x
w Al
Om=l
1msn
M
'
h2
1
osy (3
1 1
8 .b)
M=l
k
M2
Ih2
h2
-xY
I3;nA2m-14nnB2fl
M,
2
-
h2
sinf
-
2
-2
h2
-
=(kz+k2xn)I
4
nnA2n
I
el{k1x
Cs(k
ixmwi)Aim
II3mn
(kixmwlvinr
3i.8d
(O)XOmi
Cos
e2
kz
2 -
I2nnB
2m
-XIl4mnA
2m
O)01O

41
(2
2
e2
nOl2.(2WO
+B2
nSinffk
2
xn(W
2
-WlO]
I
(3.1.18.e)
M42
2 e2
Y,
(kz+k
4
xrp)IimnA
4m
k
M2
h2
-y
1
3mn
IA
2
mCOSlk
2
xm(W
2
Wl)]+B2mSiffk2xm(WZ-WI)]
I
-
0
m~l
h2 (3118f
J4nnfAlk~{
n~~x(W
1
)]+B
2
nCOS[k
2
xn(WZ-W
1)]
3.
1
kM
h2
y
lh2
I
ImnA4m
+ ,
2mnJk4xfl4ml
2 -2
h2-
(k~~n
~nA2
Xcosrk
2
xn(W
2
-W )]+B
2
nsinf[k
2
xn(W
2-W '
-(..1Ig
M4
2
-
h2
-
I (kz+k4
xOnI
2
nrnA
4m
M=l
e2
12nn
{
A2
nk
2
xrrSiflk
2
xf(W2W)]+B
2
nCoslk
2
xn(W
2
7W
I)]
I
k
z
I
e2
-
1
14m.{
A2
iCOS[k
2
xm(
WZ-W
)]+B
2
n-sinlf[k2xm(W-W
I)I
=
(3.
1.18
8)
('41
0
m=l
M
4
e
2 + -
k
z
4
i
2
r
I
1
Imnik4xmA
4m
-
:
I3mnm
m=l
W9
where:

42
h3h
I
n~o-,1
dW
1
m(Y)VnYd
1o
d~(YV
m~yd
-W
d
Y
Vin(Y)d
y
=
8Y
h3
I~y
d
my)h
I
W
mY
I3mn--=f
Fi-i(y)Wd
in(Y)dy
=
LNIe(y)
dy
Wn(
Y)d
ei
h3
1 -
d
i
(Y__
_ _ _
_
5mn~
--
IVimY)V~(Y
~ d
0
4(I4m(Y)dy
mn
0
POD)
d
iYJ
(y)
d
h3
1
dvlm(y)
I
3 d44
m(Y)
le-
In~(Y)d
f
XV
jin(Y)d
y
6m
0f
e(y)
d
y
e(y)
dy
hi
mY)~in(Y~dY
Vi(=
y
JmYNi(~
1
O
mn
8
(y)
dyey) d
I2mnfo
Wim(Y)Vin(Y)dy
=
f
Nf4m(Y)Vin(Y)d
y
h3
1
d~fim(y)
-hd~~mY
Jo~f
8,(y)
d
y
Vifn(Y)YJ
0( 4(~

43
where
i=2,3,
for
regions
2
and
3,
correspondingly.
Thus,
an
identical
set
of
equations
as
those
derived
for
h<_y<_h3
may
be
derived
for
the
case
h3
<y5h.
For
this case,
it
is
sufficient
to
substitute
i=3
instead
of
2
in
the
superscript
in
the
orthogonalization integrals,
and
substitute
3
for
2
in
the
subscript
of
the
remaining functions.
By
developing
the
above
equations
further,
the
final
form
of
the set
of
eigenvalue equations
is
derived
for
0<y_<h3:
MM
h2
e2
1
k,
h2
COS
2 2
'3pilmp
icos
lnlnsinf(
kIxmw
I)-(k
z
+k
T
e2
Jsinf(klxm
w
l)
Alm+
CO
m=1
{K zz+K2x12pp}
h2
( 1
")COSl.
.(kz+klxrn)
sinf
.WW
m
1ixmW)
2 ... .
lxmW)k
2
xnFOtI
2
xn(W2-W
)}
Alrn
rn=1
kim)sin
2
-2
COS
2
' 1
-11
I2m n A4
mO0
(3.1.1
9.
a)
2,2 2
e2
lx
in
k
xm
I)-
( z+k
1~
Ix)coS
kx
cos[k
2
xn(W
2
7WI)I
Alim,
{M(kzkxm)
Sih2 1 (km-
2
(3..1}
a
1 fi I2
COsinfkx
2 2
"weh2
]-
m=m M)l (kz+k2xn)
NI4 M
e2
2
Se2
-kmOsin
()
2.os...b)
3m
n1co
(k l xmw l)
zkln
2 -2 h2f S
kxW-l)]A1j
m=l P(k
kz+k2xpp
M, 2 2.2 .h
(kz+k4xm)
e2
1
.,m_
3119b
e'
I 1i1il m l l
mill
(3119b

44
M122
M,
(kz+kxm)
e2
COS
-e2
lmnsfS(k
l
xmwl)Alm
r+
m=l
(kz+k2xr)
14pn
2rap
e2~
sinfljl
2
xn(W2_
iV')]A
4
m=0
I (kzk~xpIapp(3.1.
19.c)
M,
2
(ke+kil,)
h2
sinfi
x
,"-
2m
2
212mnos
(KixmWl)/Irn-
m==
(kz+k2xr)
kM4
2=
23pe2
h2snf[k
2
x(W
2-
w
)]-h2nnkx(
1
\A4m
1
(kz+k4x os
2
x(
)
+JmSinf[f2xn(W2Wl)]4=
0
10
2mn
2
2
m=l
(kz+k
2
xn)
(3.1.19.d)
As
before,
substituting
the
corresponding
variables
for
region
3
will
yield
the
corresponding equations
for
h3
_y<h.
The
above
set of
equations
may
be
expressed
as a
homogeneous
matrix
equation
[A]x=0,
where
[A]
is
the
matrix
of
the
coefficients
of
Aimo,
Aim,
A4m,
and
A4m
x
is
the
vector
containing
Aim,
Aim,
A4m,
andA
4
m.
This
homogeneous equation
has
a
nontrivial
solution
only
for
the
case
det(A
)=O.
The
coefficients
of
the
matrix
[A]
have
only
one
unknown
variable,
kz.
so
solving
the
equation
det(A
)=O
yields
the
solution
for
the
MI 2zIjI

45
propagation constant.
By
substituting
this
value of
k,
back
into
the
set
of
equations
[A]x=O,
an
underdetermined
system
is
derived
for
determining
the
coefficients
Ai,
Aim,
Alm,andA4m
Of
the
electric
and
magnetic
fields of
the
structure.
After solving
for
the
coefficients
of
the
fields
in
regions
1
and
4,
the
remaining
electric
and
magnetic
field
coefficients
are
determined
by:
MI
2
2
ei
AimXY
(
k
-k,
p)_I
Ipm
Cos
(k,
xpw
)A
I
2 2
ei
sinf
l
P=I
(k
-kjyrp)
12mm
(3.1.20.a)
Mel
M1
3p
f-
sin
(k p A, i
Bim
pp~)
Cos
k
~
k
MI
2e i2
ei
hi
04~Op
~
O
os~
1=
hi2S..
I
2
-2 hii h
A i
nEX
(k
-klyp)
I2
pmsinf
(k
jx~
)Alp
P
2_j2
h
i
COS
pI(k
kiyn
114mm
(3.1
.20.c)

46
k
M1 1
hi
B
k
Z p
(k
IP
os
1
k.
w1)A1i-
hi
sinf
tO
--
4mm
k
MI
2
2
COS
2
hi
ei
1
sinf
ei
2 2
A[
ta)
p
l
1=1I211(k
-kiyl)
MI
hi
(I
iCos~
MI
hi
k
1
sinS(i.
1)
,1
X'
2pnl
_
)lp~
n
\lp
P=I
XP)(3.1.20.d)
The
coefficients
determined
through
the
matrix
equation
are
determined
within
a
multiplicative
constant,
so
in
fact
the
electric
and
magnetic field
distribution
in
each
region,
if
not
their
true amplitude,
may
be
derived.

3.2.
Characteristic
Impedance
Theoretically,
three
different
expressions
may
be
used
to
calculate
the
characteristic
impedance
of
any
device.
When
applied
to
the
coupled
microstrip
line,
these
three
expressions
define
the
relationship
between
the
characteristic
impedance
of
the
coupled
microstrip line,
the
time-averaged
power
flow
in
the
strip,
the
complex
voltage
of
the
strip calculated
at
the
center
of
the
strip,
and
the
current of
the
strip.
The
power-voltage
relationship
gives
the
characteristic
impedance
of
the
coupled
microstrip
line
as
a
function
of
the
time-averaged
power
flow
in
the
strip,
Pavg,
and
the
complex
voltage
V
of
the
strip, calculated
at
the
center
of
the
strip.
In the
case
of
the
coupled
microstrip
line,
the
characteristic
impedance
would then
be:
I *
vv
Pavg
(3.2.1)
The
power-current
relationship
gives
the
characteristic
impedance
of
the
coupled
microstrip
line
as
a
function
of
Pavg,
and
the
complex
current
I
on
the
strip,
and
is
given
by:
ZO
avg
II
(3.2.2)
The
voltage-current
relationship
gives
the
characteristic
impedance
of
the
coupled
microstrip
line
as a
function of
the
complex
voltage,
V,
and
the
complex
current,
I,
of
the
strip,
and is
given
by:
47

48
V
°-
(3.2.3)
Although
experiments
have
shown
the
power-current
calculation
to
be
more
valid
in
the case
of
the
microstrip
line,
all
three
calculations
will
be
presented
in
this
work.
The
power,
Pavg
is
calculated
using
the
Poynting vector,
which
is
defined
as:
P=-LRe
{ExH*-azldxdy
S
Using
the
rectangular
compoents
of
the
electric
and
magnetic
fields,
the
Poynting
vector
may
be
written
as:
P=-LRe
t
ExH;-EyHxldxdy
S
Using
the
relationships
in
(3.1.8)
through
(3.1.11)
to
express
the
fields
in
terms
of
the
scalar
potential
functions
in
each
region,
a
Poynting
vector
may
be
calculated
for
each
region
of
the
coupled
microstrip
line.
The
power
in
region
1
is
thus
calculated
as:

49
P,=-2Refod
Yf
dx
(E,,H*.-E,,H*,)
M
1 ~
0
M
1 ixn1
2~
W
(k
4yn)~n 6 1Ow imOkx,)i~n
I0
m=1
n=1
1zM
2-
12
-2
-OWk
klym)I5mnm
(O,wi,klxm,O,klxm,O)Aim
+
gm=1
11
kz
M
2 2 M~le m
11(O,Wi,klxm,O,klxm,O)A
m
+
CO
I
(k
-kiym511m+
M=1
12
M1 M
1
1
11
X2_~nIn
2
)l
(O,Wi,klxm,O,klxn,O)AiMAin
(.0o1
m=1
n=1
-k
1
xn
12
where
the
functions
11-13
are
defined
as:
I3
(l1Iul,a,b,c,d)=jU
sf[a(x-b)]-sinf[c-(x-d)]-dx
and
the
other
parameters
have
been
defined
in
section
3.1.
The
power
in
regions
2
and
3
are
calculated
as:

50
Pj=-LRefdyJ
dx(Ei,,H*
-EiyH*
LX
(k2_
h
i
02
P0m=ln=
1
LE(k
kyn)~imLNiTim+
k
4E
k
2 -2 ei
~~~l
.dym
2
mm'T3mm+
M=
1
m~ln=l
where
i=2
or
3,
for
region
2
or
3,
correspondingly.
The
functions
I:NTI
through
INT4
are
defined
as:
INT~nn=Aim
kl~mA
inI3(Wl,W
2
,kixn,Wl,kixm,wl)+
B
imA
in'
1(w1
,w2,kjxm,w
1
,kixn,w
1)-
imxm
inl2(WliW
2
,kixm,wI
,kixn,Wl)+
Bim~n1
3
(wI
,W2
,kjxm,wjiixn,w0)
1T.'mn=Aim
Ai1l(Wl,w2,kixm,wl,kixn,Wl)+
AimBinl
3
(W
,W2
,
kjxm,w
1'
nl)
BimA
1
nI
3
(wl1,w
2
,
kixn,Wl1'kjxm,wl
1
BimBinI2(w
1,w2,kixm,wj,kixn,wI)

51
ThNTZn=AimAinI
l(wl,W2,kixm,wi,kixn,Wl)+
B
imAin13(W
1,W2,kixnW
I
kixm,wi1)+
AimB*
in 3
(W
1,W2,kixm,w
I
kixn,w
0+
B
imBin12(W
1,W2,kixm,wi,kixn,Wi1)
1'NT4~n=-AimAinkixn13(W
l,w2,kixm,wil,kixn,wl0+
Ajjinl
1
(W
T,W2,kjxm,wj
1kjxn,W
0-
Bjm~kjnkixnI2(W
1,W2,kjxm,w
1,kixn,WI)+
BjmBinI3(W
1
Wkxn
,kixm,w
1)
The
power
in
region
4
is:
P4
=-Reldyldx (ExH4-~-*x
2
JO
w2
2_M2
fm
k4,xm
]
2
1
(k
k
yn)1n.
6
A4
mA
4
n-
(0 k'
M=1
n=1
I.k4xm+k4xnJ
M1
2 2
I{
2
k4
xm}
kz
(
22
y)emm{2
x}4
~
(k
1
ynIAnm+
c
~
_ky
2o
m=
4m
Am~
The total power
in
the
device
is
equal
to
the
sum
of
the
powers
calculated
for
each
regyion
of
the
line.
It
still
remains,
however,
for
the
voltage
across
the
strip
to be
calculated.
In
this
case,
the
assumption
is
made
that
this
voltage
may
be

52
approximated
by
the
voltage
at one
point,
the
center
point,
of
the
strip. Thus,
the
voltage
may
be
defined
as:
where:
E
____
Y
Y,
(k
k2
Yf)XV
2
m(y).
(A
2
rCOS[k
2
xm(
W/2))+B
2
msinf[k
2
xm(
w/2)]
Integrating
from
0
to
h3
yields:
-
V=
LY,
(k
2
-k
ym)
{A
2
mc05s[k
2
xm(w/2)1+13
2
msinf[k
2
xm(w/2)]1
{
Lsff(k
2
ymI
h
I
)
+sinf[k
2
ym
2
(h
2-
h01+
C
1
-cos[k
2
ym
2(h2
hi)]
}+-aco
s[kym
h-2)
e2k2ym2
where
A,
B,
C,
and
D
are
the
expansion
coefficients
defined
in
eqs.
(2.2.6.a-2.2.6.d)
for
the
TMY
potential
for
the
three-layer
parallel-
plate
waveguide.
The
current,
1,
on
the
strip
is
calculated
from
the
transverse
magnetic
field
on
the
strip. For
a
strip
of
zero
thickness,
the
current
I
is
given
by:

53
Ifstri
.H
transversc d
S
W2
+ Wi
M2
I{jikzd-(A 2
m-siff(k
2
xm-w)+2-B
2
mnslff
2(k2
xm-W/2)I+
M=1
1d-[A
2
m-(1
cos(k
2
x-W))-B
2
m-slff(k
2
xW)
}
1{j-k~cos
[k
3
ym(h-h
3
)][A
3
m-siff(k
3xm-w)+
m=
1
2-B
3
m-sinf
2 (k3
xmw/2)+--COSlk
3
ym'(hh
3
)1-
Jco
IA
3
m'(1
-COS(k
3
xmw))-B
3m
*sif(k
3
xm'W)I
The
equations
are
now
all
in
place
to
complete
the
analysis
of
the
three-layered
coupled
microstrip
line
through
the
calculation
of
the
propagation
constant
and
the
characteristic
impedance of
the
line.

54
Chapter
4.
Numerical
Results
The
formulas derived for
the
calculation
of
the
propagation
constant
and
the
characteristic
impedance
of
the
coupled
microstrip
line
were
implemented
in
Fortran
77.
The
Fortran
program
was
run
in
double
precision
on
the
VAX
or
in
single
precision
on
the
CRAY,
with
the
same
results.
Some
characteristic
results
are
presented
in
this
chapter.
4.1.
Convergence
Criteria
The
exact
electric
and
magnetic
field
representation
for
each
region
of
the
waveguide
described
in
this
thesis
was
given
by
an
infinite
sum
of
expansion
functions,
which were
characteristic
for
each
region
of
the
device.
Due
to
subsequent
truncation
of
each
infinite
sum,
the
propagation
constant
which
was
derived
for
this
structure
was
not
exact,
and
the
error
present
in
the
calculation
not
exactly
known.
However,
as
was
presented
in
chapter
2,
care
was
taken
so
that
the
relative
number
of
expansion functions retained for
each
region
was
such
that
the
truncation
of the
infinite
series
would
lead
to
a
convergent
system.
The
accuracy
of
the
results
was
determined
via
convergene
plots,
such
as
those
shown
in
figs.
(4.1.1)
and
(4.1.2).
In
these
plots,
the
normalized
propagation
constant
was
plotted
against
the
number of
modes,
M2,
retained
in
region
2
for
the
calculation
of
that particular
value for
the
propagation
constant.
As
M2
increases,
the
value
of
the
normalized
propagation
constant
converges
to
a
particular
value.
So,
after
setting
a
certain
error
bracket
as
acceptable,
the
minimum
M2
required
to
calculate
the
propagation
constant
within
that
error
bracket
is
found
from
the
convergence plots,
and
the
corresponding
normalized
propagation
constant
is
accepted
as
the
solution.

55
This
test
cannot, however,
be
conducted
for
each
data
point,
as
the
time taken
to
do
this
would
be
prohibitive.
The
procedure followed
is
to
calculate
the
smallest
M2
which
satisfies
the
convergence
conditions
at
the
extreme data
points.
These
points
in
this
particular
case
would
be
smalles
and
largest
structure
size,
and
lowest
and
highest
frequency.
It
is
safe
to
accept
that
this
M2
will
then
satisfy
the
same
convergence
criteria
for
any
other
point
within
the
boundaries
set
by
these extreme
points.
The
error
bracket
for
the
convergence plots
are
0.5%
for
the
propagation constant,
and
1.5%
for
the
characteristic
impedance.
3.8
,
,
3.7
3.6
3.5
3.4
"
3.3
0 2
4
6
8
10
#
modes
Figure
4.1.1.
Convergence
plot of
the
even
mode
propagation constant
versus
the
number
of
modes
retained
in
region
2.
Another
important
numerical
consideration,
which
has
been
touched
only
briefly
so
far,
is
the
height
at
which
the
top
plate
must
be
placed,
such
that
it
does not
perturb
the
solution.
It

56
was
suggested
in
[191
that
placing
the
top
plate
at
a
height
double
the
height
at
which the metal
strips
are
placed,
would
be
enough
to
ensure
an
unperturbed solution.
This
is
the
height
used
for
all
the
calculations.
The same
program
was
run
for
greater
heights,
but
it
was
deduced
that
this
only
added
unnecessarily
to
the
computation
time,
without
producing
more
accurate
results.
3.8
3.7
3.6
3.5
3.4
3.3
-
3.2
0 2 4
6
8
10
#
modes
Figure
4.1.2.
Convergence
plot
of
the
odd
mode
propagation constant
versus
the
number of
modes
retained
in
region
2.

4.2.
Program
Verification
The
accuracy
of
the
programs
was
verified
by
selecting
the
permittivities
of
the
dielectrics
and
the
geometry
of
the
structure
such
that it
would
be
reduced
to
structures
for
which
results
were
available.
Both
the
propagation
constant
calculation
and
the
characteristic
impedance
calculations
for
the
single-layer,
coupled
microstrip
line
were
verified
against
the
results
obtained
by
[2].
The
calculation
of
the
propagation
constant
for
the
single-line
case
was
achieved
by
placing
the
striplines
at
a
sufficiently
large
distance
from
each
other.
The
results
obtained for
this
case
for
the
single-layer
case
were
verified
against
[2],
and
for
the
multiple-layer
case
against
[19
(p.
57)].
The
calculation
of
the
propagation constant
for
the
three-
layer
coupled
microstrip
line
was
verified
with
[19
(p.
57)].
A
comparison
of
the
mode-matching
technique
versus
the
spectral-domain
technique
was
achieved through
the
cooperation of
Mr.Yu-De
Lin, whose
analysis
may
be
found
in
[18].
The two
different
programs
were
run
for
the same
coupled
microstrip
line,
and
the
results
are
presented
in
the
graphs
in
fig.
4.2.1.
The
results
obtained
from
the
mode
matching
method agree
quite
well with
those obtained
through
application
of
the
spectral-
domain
technique.
57

58
3.4
eve
mde-
mode
machng
eve
mod
-spectral domain
5
8
.8mm
2.8.9
2.6 0
10
20
Frequency
(GHz)
400
Z-
mode
matching
II
300
een
mode
200
100
•odd
mode
01
0
10
20
Frequency
(GHz)
Figure
4.2.1.
Propagation constant
and
characteristic
impedance
(power-current
relationship)
verification
for
the
coupled
microstrip
line.

4.3.
Results-Design
Charts
The
program
was run
for several
cases
of
coupled
microstrip lines.
Each
graph
shows
a
family of
curves,
calculated
for
a
specific
microstrip
width
and
dielectric
layering,
but
for
different
strip separations,
or
for
a
variable
dielectric constant
in
the
conducting
layer.
In
each
case,
for
wider
strip
separation,
the
results
converge
to the
case
of
a
single
microstrip
line.
Also,
in
the
case
of
a
variable
dielectric constant
value
in
the
conducting
layer,
the
results
converge
to
the
single-layer
solution
when
the
dielectric
constant
of
the
conducting
layer approaches
the
value
of
the
dielectric
constant
of
the
insulating layers.
Many
curves
were
graphed
versus
a
normalized
frequency,
where the
normalization
took
the
form
27th
3/X0,
where
X0
is
the
free-space
wavelength,
The
characteristic
impedances,
however,which
are
calculated
using
the
power-voltage,
Z0
=V.V*/P,
and
the
power-current,
Z0
=P/I.I*,
expressions,
exhibit
discontinuities
at
regular intervals,
which
seem
to
correspond
approximately
to
the
frequency
where
h3=n)Ld/4,
where
n=1,2,3,etc.,
and
Xd
is
the
free-
space
wavelength
in
a
medium
of relative
dielectric constant
equal
to
the
relative
dielectric
constants
of
the
layered
medium.
This
behavior
is
more
evident
where
the
characteristic
impedance
is
plotted
against
h3
/)Ld.
As
the
relative
dielectric constant
of
the
middle
layer
is
increased
with
respect
to the
outer layers,
this
discontinuity
becomes
more
pronounced,
as
may
be
witnessed
in
figs.
4.3.2(b),
and
(c).
This
behavior
seems to
signify
the
"turning
on"
of
higher-order
modes,
which
introduce
a
numerical
instability
around
that
region
of
frequencies.
In
figs.
4.3.6(b),
and
(c),
the
characteristic
impedances
are
plotted
over
a
wider
range
of
normalized
frequencies,
and
the
almost
"periodic" behavior
of
these
discontinuities
is
more
evident.
The
calculation of
Z0
using
Z0
=V/I
does
not
exhibit
any
discontinuous
behavior.
59

60(
4.3.1.
Coupled
microstrip
line
design
charts
for
EI=E3=2.5,
E2
=3.9,
and
w=3.O3mm.
1.70.5 evenmode0.5w S
s=W
0.5
1.5
1.41
0
-2
3
Normalized Frequency
Figure
4.3.1(a).
kz/ko
versus
21ch
3
/Xo
0.
300
-
even
mode
0.25w
0.5
200
S=
100
odd
mode
0.W
V"
0.25w
0 12 3
Normalized,
Frequency
Figure
4.3.1(b).
Z0
=P/II*
versus
2hfO

61
300
250
-
even
mode
200
-I025
15
-j
0.5w
00.5w
50
4
.25
odd modeJ
0
0
12 3
Normalized
Frequency
Figure
4.3.1(c).
Z0
=VV*/P versus
27ch
3
I10.
200
0.25w
evenmode0.5w
100
odd
mode
~
S=w
0.5w
0.25w
0
p -
0 12 3
Normalized Frequency
Figure
4.3.4.
Z0
=V/I
versus
2nh
3
P/.
0.

62
4.3.2.
Coupled
microstrip
line
design
charts
for
eI=e-
3
=1O,
E2
=119
12,
and
13,
and
w=12.7mm.
a
3.4
E
2-13
N
.2even
mode
12
E2--13
12
3.0
1.61
22.78
2.67m
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Frequency
Figure
4.3.2(a).
kz/ko
versus
21ch
3
AX
0.
200
*
E2--13
13
E2-12
*
E2--12
100-
A
E2--11
0
T
*0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized
Frequcncy
Figure
4.3.2(b).
Z0
=P/III
versus
27Eh
3
1X
0
O.

63
100
evnmd
80
60
0E-1
4013 E-1
20
odmoe0AE-I
01
U.0
0.2
0.4
0.6
0.8
1.0
1.2
Normnalized
Frequency
Figure
4.3.2(c)
ZO=VV*/P
versus
2irh
3
IXO.
100
1
N 0
60
0E-1
40
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Frequency
Figure
4.3.2(d)
Z0
=V/I
versus
27rh
3/k0.

64
4.3.3.
Coupled
microstrip
design
charts
for
El=Fs
3
=11.5,
F,
2
==12.9,
and
w=1.4mm.
3.4
S=0.8
1.6
2.0
3.2
-s=2.0
1.6
0.8
3.03m
...........
2.11
mm
2.8
-1.
0.0
0.2
0.4
0.6
0.8 1.0
1.2
Normalized
frequency
(27cth3~/
%
0
Figtire
4.3.3(a).
kz/k
0
versus.21rh
3
/XO.
250
~
I
a
S=2.omm
I-
*
s=2.Omm
~-
200
*s=1.6mm
N *
s=1.6mm
1;()*
s=-0.8mm
*s=-0.8mm
50I
0
1 1
0.0
0.2
0.4
0.6 0.8
1.0
1.2
Normalized
frequency
(27C
h
3 0
)
Figure
4.3.3(b).
Z0
=PIII*
versus
2irh
3
/?k
0.

65
300
M
S=2.ornm
*
s=2.Omm
0
s=1.6mm
20a
s=1.6mm,
200
*
s=-0.8mm
even
mode
*s=-0.8mm
100L
odd
mode
0
1
L
-I - -I
U.0
0.2
0.4
0.6 0.8
1.0
1.2
Normalized
frequency
(2%
h3~/
X O)
Figure 4.3.3(c).
Z0
=VV*/P
versus
2nth
3
fX
0.
160
-1
0
s=2.Omm
O~140
even
mode
*s=2.0mmn
*s=1
.6mm
N 120s=1.6mm *
S=0.8mm
100
*S=-0.8mm
80
60
0.0
0.2
0.4 0.6
0.8
1.0
1.2
Normalized frequency
(27C
h 3 X 0)
Figure
4.3.3(c). Z0
=VV*/P
versus
2nth
3/X
0.

66
4.3.4.
Coupled
microstrip
design
charts
for
*E
1
=E
3
=3.5,
e2
==4.9,
and
w=3.03mm.
2.0
s=.757mm
Q
0
1.515mm
14
3.03mm
N
1.9
even
mode
4.545mm
s=4"545mm
3.03mm
1.8
1.5
15mm
2S
3.03
7575rnim
~I
3.03mm
1.7
3.5
1
..............
2.11m
od
mode
6.9
1.6
.......
3.5
.....
0.92mm
1.5
1 1 1 1
0.0
0.1
0.2
0.3
0.4
0.5
Normalized
Frequency
(h
3 /
Xd)
Figure 4.3.4(a).
kz/ko
versus
h3/Xd.
300
*
s=4.545mm
*
s--4.545mrn
0
s=3.03mm
,
200
a
s=3.03mm
Q
s=1.515mm
e
s=1.515mm
o
s=0.7575mm
100_
a
00090@="L1I1
odd mode
0 I
I
I I
0.0
0.1
0.2
0.3
0.4
0.5
N,;rmalized
Frequency
(h
3 /
X
d)
Figure
4.3.4(bi.
Zo=P/II*
versus
h3/kd.

67
250-
_______
M
s--4.545mm
*
s=-4.545mm
200
*
s=3.03mm
0a
s=3.O3num
150
-even
mode
*
s=1.515mni
100
-s077m
50-
U.0
0.1
0.2
C.3
0.4
0.5
Normalized
Frequency
(h
3 /
Xd)
Figure
4.3.4(c).
Z0
=VV*/P
versus
h3/)Xd.
160
__a
s=4.545mm
140
-*vnmd
s--4.545mm
*s=3.03mm
o
120
's=3.03mm
100
s=1.515mm
80
-0
s=-0.7575mm
60
40
20
0.0
0.1
0.2
0.3
0.4
0.5
Normalized
Frequency
(h3 /
X
d)
Figure
4.3.4(d).
Z0
=V/I
versus
h3/kd.

68
4.3.5. Coupled
microstrip
line
design
charts
for
Cl=C
3
=15s
E2
=4.9,
w=1.4.
1.9
s-.7mm
1.4mm
even mode
21u
s=2.lmn
1.8
.m
1.7.4
1.6
.....
21
m
0.0
0.1
0.2
0.3
0.4
0.5
Normalized
Frequency
(h
3
/
X
d)
Figure
4.3.5(a).
kz/ko
versus
h3
/Xd.
~40000 im,
N
300
-ee
oes07m
200-0
-03m
100
0.0
0.1
0.2
0.3
0.4
0.5
Normalized Frequency
(h3~/
X
d)
Figure
4.3.5(b).
Z0
=P/I"*
versus
h3
/kd.

69
400
0
s=2.lmm.
300
s=1.4mm.
even
mode
£s=.4nm
*
s=0.7mm
200-
0.m
100
U.0
0.1
0.2
0.3
0.4
0.5
Normalized Frequency
(h3
X
~d)
Figure
4.3.5(c).
Zo=VV*/P
versus
h3
/Xd.
300
S-i
I
m
m
even
mod
s=2.mni
N
200
=.m
100
0
U.0
0.1
0.2
0.3
0.4
0.5
Normalized Frequency
(h3
X
~d)
Figure
4.3.5(d).
Z0
=
V/I
versus
h3
/Xd.

70(
4.3.6.
Observation
of
the
behavior
of
the
discontinuities
in
the
characteristic
impedance
calculation
over
a
wide
range
of
frequencies.
3.6
-
3.4
0.4
3.2
-
mm
1.515
3.0
.odd
mode
12.91
..........
...............
0.92mmn
2.8
1
0
12 3
Normalized Frequency
(27c
h
3 0 )
Figure 4.3.6(a).
kz/ko
versus
27rh
3/X
0
o.
400
~300-evnmd
200
0
0
1 2 3
Normalized
Frequency
(27E
h
3
/X 0)
Figure
4.3.6(b).
Z0
=P/Il*
versus
27th
3
IXQ.

71
200
"V"even
mode
°
150
100
d.
o
0
50
odd
mode
0
1, -1 -
0
1
2
3
Normalized
Frequency
(27t
h
3
/
0)
Figure
4.3.6(c).
Zo=VV*/P versus
2nh
3
/Lo.
200
a
00 d
N
Sodd
mode
0
0
1
2
3
Normalized Frequency
(2nth
3
/A
0)
Figure
4.3.6(d).
Zo=V/I
versus
21h
3
/?o.

Chapter
5.
Conclusions
In
this
thesis,
it
was
shown
how
a
quasi-planar
structure
like
the
coupled
microstrip
line
on
a
layered
dielectric
substrate,
could
be
analyzed
using
a
variation
of
the
mode
matching
technique.
Mode
matching
is
a
powerful
tool
for analyzing planar
and
even
quasi-planar
structures.
The
results
obtained
through
mode
matching
have
a
high
degree
of
accuracy,
especially
the
propagation
constant.
This
technique,
however,
is
unsuitable
for
CAD
packages,
as
it
is
very
inefficient.
It
is
also
susceptible
to
numerical errors,
as
the
system
which
is
solved
to
calculate
the
field
coefficients
is
defined
by
a
singular
matrix.
It should
be
noted,
as
a
matter of
fact,
that
this
singularity
condition
has
actually
been
imposed
on
the
matrix,
in
order
to
solve
for
the
propagation constant.
The
behavior
of
the
even
and
odd
mode
propagation
constants
of
the
coupled
microstrip
line
on
a
layered
substrate
does
not
differ
from
that
of
its
single
substrate
counterpart,
except
for
a
slight
increase
of
kz/ko
with
an
increase
in
the
conducting
layer
dielectric constant
with
respect
to
the
insulating
layer dielectric constants,
as
can
be
seen
in
fig.
4.3.2(a).
As
far
as
the
impedance
is
concerned,
it
does
not
seem
to
change noticeably
with
the
change
in
dielectric
constants.
The
change becomes
more
noticeable,
however,
at
higher
frequencies.
The
effect of
the
discontinuity
becomes
much
more
noticeable
for
a
higher
conducting layer
dielectric
constant.
Its
position
does
not
change
much,
however,
as
the
values used for
the
dielectric constants
were
of
the
same
order
of
magnitude.
The
impedance
curves
seem to
be
relatively
flat
between
singularities.
The
discontinuities
themselves
follow
a
72

73
very
consistent
pattern
which
seems
to
be
dictated
by
the
relationship
used
to
calculate
the
impedance
(power-voltage,
or
power-current). Irrespective of
the
relationship
used
to
calculate
Z0,
however,
the
first
singularity
occurs
at
the
same
normalized
frequency,
h3
/Xd=_0.
25.
This
seems
to
be
indicative
of
a
cutoff
frequency,
above
which
higher
order
modes
start
propagating,
and
around
which
the
calculations of
the
characteristic
impedance
become
numerically unstable.
It
is
not practical,
in
any
case,
to
fabricate
microstrip
devices of
the
order
of
magnitude of
Xd/
4,
so
only the
portion
of
the
curve before
the
discontinuity
occurs
is
significant.

Appendix
A.
Notation
A
consistent
notation
scheme
is
used
throughout
the
text
where
subscripts
and
superscripts
are
used
to
denote region,
direction, dielectric,
mode
number,
mode
type,
or
to
differentiate
between two
similar functions.
A
variable
"z" may
take the
form
Zabm
where
"a"
would
be
the
cartesian coordinate
(x,
y,
or
z),
"b"
would
be
a
region
number
(1,
2,
3,
or
4),
and
"m"
would
be the mode
number.
A
function
"f"
may
take
the
form
fbcramn
where
"a"
would
be
a
function
number,
"m",
and
"n"
would
be
mode
numbers,
and
"c"
would
be
the
region
number:
c=2 means
the
inner
product
was
performed
with
respect
to
a
normal
mode
of
region
2,
and
c=3
means
the
inner product
was
performed
with
respect
to
a
normal
mode
of
region
3.
Superscript
"b"
is
used
only
in
the
orthogonalization
integrals:
b=e means the
inner product
was
performed
with
respect
to
a
TM
mode,
and
b=h
means
the
inner
product
was
performed
with
respect
to
a
TE
mode.
The
following
general
notation
was
used
to
build
the
desired variables:
Symbol
Interpretation
__
permittivity
9.
permeability
k
propagation
consta
t
E
electric
field
H
magnetic
field
A
tilde
over
any
of
the above
variables
would denote
TE
quantities,
whereas
no
tilde
would
denote
TM
quantities.
74

Appendix
B.
Derivation
of
Field
Equations
A
wave
is
a
field
that
is
a
function of
both
time
and
space.
Electric
and
magnetic
fields
that
vary
in
time
and space
are
governed
by
physical
laws which
are
expressed
in
four
equations,
known
as
Maxwell's
equations. For
a
wave
travelling
in
a
medium
characterized
by
a
certain
permittivity,
e,
and
a
certain
permeability,
pt,
These
equations
are:
D H
VxE=
-p
V
x
H
=F
D-- + J
VD
p
V.
B=O
(B.1)
The
electromagnetic
field
equations
above
are
expressed
in
terms
of
six
quantities:
E,
the
electric
intensity
(in
volts
per
meter)
H,
the
magnetic
intensity
(in
amperes
per
meter)
D,
the
electric
flux
density
(in
coulombs
per
meter)
B,
the magnetic
flux
density
(in
coulombs
per
square
meter)
J,
the
electric
current
density
(in
amperes
per
square
meter)
p,
the
electric
charge
density
(in
coulombs per
square
meter)
The
boldface script
is
used
to
denote complex
quantities.
The
ultimate
sources
of
an
electromagnetic
field
are
the
current
J,
and
the
charge
p.
75

76
The
continuity
equation,
which
is
based
on
the
principle
of
conservation
of
charge,
is
implicit
in
equations
(B.1),
and
simply
states that:
SV
J=_P
a
t
(B.2)
Maxwell's
equations
are
complemented
by
the
so-
called
constitutive
relationships,
which
incorporate
the
characteristics
of
the
medium
in
which the
field
exists.
These
equations define
the
electric
flux
density,
D,
the
magnetic
flux
density,
B,
and
the
current density,
J,
with
respect
to
the
electric,
E, and
magnetic,
H,
intensities.
D=D(E,H)
B=B(E,H)
J=
J(E,
H)
(B.3)
Maxwell's
equations
along
with
the
constitutive
relationships
serve
to
fully
describe
a
wave
travelling
in
a
known
mne
di
um.
In
a
source
free,
linear
medium,
the
constitutive
relationships
take
the
form:
D=
E
B=p.H
J
=
0
(B.4)
Here,
e
and
.are
constants,
where
,
is
the
capacitivity
or
permittivity,
and .
is
the
permeabillity
of
the
medium.
Using
-0
and
go
to
denote
the
corresponding
variables
in
vacuum,
for
a
perfect
dielectric
(Y=0),
one
has
E=CrC
0,
where
Er
is
the
dielectric
constant,
or
the
relative
capacitivity, of
the
medium,
and
g=pt
0
for
most
linear
matter.

77
The
present
analysis
will
be
concerned
only
with
source free,
linear
problems,
where
the wave
has
a
steady-state
sinusoidal
time
dependence.
In
this
case,
the
complex
field
equations
read:
-VxE=z(cm)H
V
xH=y
(o)3)
E
V.D=O
V.
B
=0
(B.5)
A
A
where
y(w)=jco,
and
z(O)=j3O1.t
0
in
nonmagnetic material.
The
above
representation
of
the
field
equations
gives
rise
to
the
definition
of
the
parameter
k,
the
wavenumber
of
the
medium.
The
wavenumber
is
defined
as:
k
=
-y(ct)
z(cO)
(B.6)
The
physical
meaning
of
the
wavenumber
is
that
1/k
is
the
velocity
of
propagation
of
an
electromagnetic disturbance
in
an
open space
filled
with
perfect
dielectric
material
with
permittivity
e and
permeabillity
0.
Taking
the curl
of
equations
(B.5),
and
using the
above
representation
of
k,
the
complex
wave
equations
become:
VxVxE
-k
2E=0
V
x
V
x
H-
k
H=O
(B.7)
In
these
equations,
it
is
implicit
that:
V.E=O
and
V.H=O
(B.8)

78
so
that
a
simplified
form
for
the
vector
wave
equations
may
be
derived:
V2E
+k 2
E
=
0
(B.9)
and:
V2
H+k
2
H=O
(B.
10)
The
geometry
of
planar structures
allows
us
to
work
with
rectangular
cartesian
coordinates.
In
this
case,
the
rectangular
components
of
E
and
H
satisfy
the
complex
scalar
wave
equation:
2 k2
V
2E i
+k
E=0
i=x,y,z
2
k2
V
Hi+k
2
Hi=O
(B.1
1)
To
construct
a
pliable
solution
to
the
above
equations,
the
field
is
expressed
as
a
magneti-
vector
potential
A
and
an
electric
vector
potential
F,
as
shown
below:
E=-VxF+.IVxVxA
y
E=
VxA+I-VxVxF
z
(B.12)
These
expressions
for
E and
H
give
rise
to
a
very
useful
classification
of
the
solutions
of
the
wave
equation.
In
this
classification,
axial
uniformity
(the
cross-sectional
shapes
of
the
wavcguide
do
not
vary
in
the
direction
of
propagation)
is
assumed.
In
addition,
this
classification
is
for
fields
conforming
to
the
homogeneous vector
Helmholtz
equations
(source
free

79
problem).
Propagation
is
assumed
to
be
in
the
z
direction,
and
the
z
dependence
is
assumed
to
be
of
the form
exp(+j
3z).
The
Helmholtz
equation
is
separable,
so
a
solution
of scalar
Helmholtz
equation
of
the form
f(zg(x,y)
is
sought.
Under this
classification,
the
vector
magnetic
and
electric
potentials,
A
and
F
respectively,
may
be
assumed
to
be
directed
along
one
coordinate
only.
Choosing
A=uixV where
xV
is
a
scalar
wave
potential
(a
solution
of
the
scalar
Helmholtz
equation),
will
yield
an
electromagnetic
field
given
by:
;2
E 1±
Ey
y
Jx
y
bz
2
Ez_
a
W'l
HZ=-
a
ya
x
Dz
Dy
-1
'(B.13)
This
choice
of
a
magnetic
vector
potential
will
yield
Transverse
Electric
to
x
(TEx)
modes. The main
characteristic
of
these
modes
is
that
Ez=O
and
Hz*O.
All
TEx
field
components
may
be
derived
from
the
axial
component
Hz
of
the
magnetic
field.
All
field
components
may
be
derived
from the
axial
component
Hz
of
the
magnetic
field.
Similarly,
choosing
F
=
u
x
xV
will
yield
an
electromagnetic
field
given
by

80
2-
z
(B.14)
This
choice
of
an
electric
vector
potential
will
yield
Transverse
Magnetic
to
x
(TMX)
or
E
modes.
The
main
characteristics
of these
modes
are
that"
_~ =.
'ZO
and
Hz=O
(B.15)
All
TMx
field
components
may
be
derived
from
the
axial
component
Hz
of
the
magnetic
field.
The
third
group
in
this
classification
are
the TEM
modes,
or
the
transverse
electromagnetic
waves.
The
waves
belonging
to
this
classification
have
no
Ez
or
Hz
component.
In
this
case, the
electric field
may
be
found
from
the
gradient
of
a
scalar
function
xV(x,y),
which
is
a
function
of
the
transverse
components
only,
and is
a
solution
of
the
two-dimensional
Laplace
equation.
True
TEM
waves,
however,
will
be
found
to
occur
in
very
few
cases
(i.e.,
free-space,
parallel-plate
waveguide,
etc.).
In
addition, true
TE
and
TM
modes
will
usually
not
be
sufficient
to
satisfy
all
the
boundary
conditions
of
most
structures.
In
such
cases,
however,
linear combinations
of
TE
and
TM
modes
will
provide
a
complete
and
general
solution.
As
might
be
anticipated,
TEx and
TMx
are
not
the
only
existing
TE
and TM
solutions
to
a
specific problem.
By
assuming

81
the
magnetic
vector
potential
to
have
the
form
A=uyv,
the
following equations for
the
fields
wil
be
derived:
.2
Ex=
-bllx
H
--j
y
ax
ay
E
--
+k
H
H=
0
-1
b2
'
z
'1
E
=-
I
y b =f
H
x
(B.16)
This
choice
of
A
will
yield
Transverse Electric
to
y
(TEY)
modes,
whose
main
characteristics
are
that
Ey#O
and
Hy=0.
Similarly, taking
F=uy
xV
will
yield:
2-
Fwx
bH
-1
a2V
Ex=
fx
=(B.18)
This
choice
of
electric
vector
potential
will
yield
Transverse
Magnetic
to
y
(TMY)
modes,
whose
i
n
characteristics
are
that
Ey=O
and
Hy2+.
Similar
cases
may
be
derived
for
TEz and
TMz.
M
neic
to
I
ITY
moes
whs
m-.: chrceitc
are
Ithat

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