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MICROWAVE
LABORATORY
REPORT
NO.
89P1
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
DAAL0388K0005
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))
,'} '('' .'(0
RLI)J
WCTION
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Park,
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277092211
11
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
SUBGROUP
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,/ /
20
DISTRIBUTION/AVAILABILITY
OF
ABSTRACT
21.
ABSTRACT
SECURITY
CLASSIFICATION
OIUNCLASSIFIED/UNLIMITED
0
SAME
AS
RPT
0
DTIC
USERS
Urnc
lass
if
iod
22a
NAME
OF
RESPONSIBLE
INDIVIDUAL
22b
TELEPHONE
(Include Area
Code)
22c
OFFICE
SYMBOL
Tatsuo
Itoh

(512)
471107
2 1
DO
FORM
1473,84
MAR
83
APR
edition
may
be
used
until
exhausted
SECURITY
CLASSIFICATION
OF
THIS
PAGE
All
other
editions
are
obsolete
I
ASS
I]
FD
MICROWAVE
LABORATORY
REPORT
NO.
89P1
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
DAAL0388K0005
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.
Parallelplate
waveguides
5
2.2.
Threelayer
parallelplate
waveguide
8
2.3.
Fourlayer
parallelplate
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.
ResultsDesign
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
quasiplanar
(i.e.
microstrip
line
and
its
variations,
and
finline), dielectric
guides
(dielectric
slab,
image 'line,
insular
guide,
inverted
image line,
etc.),
and
Hguides
(groove guide, trough
guide, etc.)
[14].
At
millimeter
wavelengths,
the
most
commonly
used
planar
transmission
lines
are
microstrip
and
microstriplike
(inverted
and
suspended)
lines.
These lines
are
suitable
for
the
design
of
lowcost,
massproducible 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
microstriplike
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
quasistatic
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
fullwave
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
welldefined
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
welldefined
and
judicious
manner
such
that
a
simple
and
well
converging solution
may
be
obtained.
The
steps
followed
in
the
modematching 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
sourcefree
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.
Parallelplate
Waveguides
19
.. . . .. .
z
Figure
2.
Parallelplate
waveguide.
The
medium
between
the
plates
is
described
by
the
general
dielectric
constant
C(x,y).
A
parallelplate
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
xz
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)
ejkz
Hz=
a
f(x,y)
ejk7z
oe(y)
ay
ax
where
f(x,y)
is
the
TMY
scalar
potential,
e(y)
is
the
ydependent
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)
ejkj
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)
eJkz
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)
eJkg
x=0
Ey
1
(k2
+ 2)(y)
e
(2.1.6)
JW4g0 dy]
Ez
=
0H
= 
k
d
W(y)
ejkz
0
dy
2.2.
ThreeLayer
ParallelPlate
Waveguide:
pec
plate
h2
pec
plate
Figure
2.2.1.
Threelayer
parallel
plate
waveguide.
At
this
point,
it
would
be
advisable
to
separate
the
structure
into
regions.
For
a
threelayer
parallelplate
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
parallelplate
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
parallelplate
waveguide:
acos
( ky1
y)
;0.5
_y5
h1
(y)=
b
cos[
ky2h2y)]
;h
1:
y
5
h2
(2.2.3)
+
c
sinf[
ky
2(h2
y)]
d
cos[
ky
3
(hy)]
;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(hEy)]
cos[
ky
3
(hy)]
; 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
ydirected 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
threelayer
parallelplate
waveguide:
k'tan(k 1
h
I) +
kY
tan[ky
2(h2
h
I))+
kY3tanh~ky
3
(hh2j)
e
le 2e (
2
.2
.5
)
kLtan(kh1)
!tan[k
2(h2
h
1
)]
kI
a~k
3
and
the
coefficients
are:
1.
ky
3
real
cos[ky
3
(hh
)] k3ianry(2.2.6.
a)
C 
C3
ky
3
sin[ky
3
(hh
2)]
ky
3
real
2
ky
3
tan[ky
3
(hh
2
)]
ky
3
imaginary
F3
(2.2.6.b)
b
=
cos[ky
3
(hh
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
threelayer
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(hh2)]
kyl_
ky2
_ky3
(..3
tan(kylh
1),, ktak2hh)
tan[ky3(hh2)]
kyl
ky3
and
the
coefficients
of
the
TEY
scalar
potential
in
the
waveguide
are:
1.
ky3
real
1.
ky
3
imaginary
cos[ky3
(hh
2)]
(2.2.14.
a)

cos[ky
3
(hh
2)]
ky
3
real
C 1.
ky3
imaginary
(2.2.14.b)
sinf[ky
3
(hh
2)]
ky3real
sinflky3(hh2)]

ky3
imaginary
cos[ky
3
(hh
2
)]
(2.2.14.c)
14
b
cos[ky
2(h2
hl)]
+
c
sinf[ky
2(h2
h
1)]
acos(
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
ydirection
of
the
threelayer
parallelplate
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.
Fourlayer
parallelplate
waveguide:
h
pec
plate
i/f /f/ li /If li i fe toof/
.e
*. %% 11 . % % % % .
.S
S .. S .
ilet./ to li0. id 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 
parallelplate 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
threelayer
parallelplate
waveguide.
The
TMY
eigenvalue
equation
is
therefore:
kY
tan(k
yh1) + ky
2
tan[ky
2(h2h1)]
+
y3
tan[ky
3(h3_h
2)] +
k_._
tan[ky
4
(hh
3)]=
F3
E4
kY
tan(kylh1)
62
tan[ky2(h2_h1)]
k y3
tan[ky
3(h3h2)] +
61
ky2
E3
.
tan(k
yh1)
__2_
tan[ky
2
(h2.h
1)]
ky4
tan[ky
4
(hb)]
+
Ek C4
ky_
tan(kylhl)

tan[ky
3(h3h2)] .
4
tan[ky
4
(hh
3
)]
+
E1
k
y3
F4
ky2
tan[ky2(h2h
I) 
tan[ky3(h3h2)]
ky4
tan[kyn(hh3)]
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
(hh
3
I
k
C)iainr
(2.3.3.
a)
_C3ky
smtfky
4
(hh
3]
;ky
4
real
e
E2
63
ky
4
tank
4
(hh]
k4
imaginary
C2
(2.3.3.b)
d
k
cos[ky
4
(hh3)]
;
k
y
4
real
1.
ky
4
imaginary (2.3.3.c)
19
c
=
d
cos[ky3
(h3h
2
)]
+
e
sinffky
3
(h3h
2
)]
(2.3.3.d)
b=
d
1
ky
3
sin[ky
3(h3h2
)]
+
e

cos[ky
3(h3h2)]
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
(hy)]
; 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
(hh
3)]
tan(kyl
Ih)
ki
2tan~kk(h2h1)]
tan[k,
2(h2
h
IA~
ank
3
hr
)
tan[ky
4
(hh)]
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
(hh)]
(2.3.6.
a)

cos~k
y4
(hh
3]
;ky
4
real
1 .;ky4imaginary(236b

Sinf1iky
4
(hh3)
ky4
real
d=tan[ky
4
(hh3)]
; 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(h3h2
)]
(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
fourlayer
parallelplate
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
microstriplike
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
outofphase 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
twoport
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
xy
and
xz
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
leftmost
boundary
becomes
either
a
pmc
or
a
pec
wall,
according
to
whether
the
search
is
conducted
for
the
oddmode
or
evenmode
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