178
J.B.
~a i io
Acebal,
P.M.
C espo
Bo ill,
E.
de
los
Reyes
Da 6
E.
Ma
s
g au
~6mez.
E.T.S.
Ingenie os
de
Telecomunicaci6n-
Ba celona
-SPAIN
COHN'S
FILTERS
WITH
PARALLEL-COUPLED
TRANSMISSION-LINES
AND
NO
DISCONTINUITIES.
ABSTRACT.- A new
me hod
o
designing
Cohn's
pa allel-coupled
ansmission-line
il e s
is
in oduced.
This
me hod
is
pa icula ly
well
sui ed
o
ob ~in
designs
wi hou
discon inui ies
in
he
junc-
ion
o
he
di e en
ansmission
lines
cons i u ing
he
il e .
I.
INTRODUCTION.-The
main
p oblem
acing
he
designe
o
mic o
__
wa e
il e s
wi h
ansmission
lines,
a e
he
junc ion
discon inui_
ies
be ween
he
di e en ;s
lines.
The
conside a ioas
o
hei
e ec s
in
The
design
esul s
e y
oublesome,
o
which eason,
di e en
me hods,
inco po a ing
co ec ions
o
he
desings
made
wi hou
such
conside a ions,ha e
been
p oposed(1).
Ano he
way
o
o e come
he
p oblem,
is
o
use
in e digi al
s uc u es,
e en
hough
hey
do·no
ad oid
he
discon inui ies
wi h
he
50
0.
inpu -ou pu
lines
and
a e
inadecua e
o
s ip-line
design.
In
his
pape
we
.
employ
Cohn's
s uc u es(2)
wi h
open
ended
couped-lines
/Fi .1-a/,
which
o e s
a
e y
high
numbe
o
deg ees
o
eedom,
and
a
desing
me hod
yelding
con igu a ions
whi ou
in e -lines
discon inui ies,
is
p oposed.
II.
OBTENTION
OF·
EQUIVALENT
REALIZATIONS.-
When
Richa d's
ans o ma ion
(3)
s=
h
j~
1
S
being
he
p opaga ion
cons an
o
he
T.E.M.
mode,
is
aplied
o
he
Cohn's
s uc u e
/Fig.1-a/
he
s-plane
p o o ype
shown
in
Fi~u e
1-b
is
ob ained.
Figu e
2-a
shows
an
in e digi al
s uc u e
which
is
he
dual
o
he
Cohn's
il e .
I
is
easily
shown
ha
he
no malized
s a ic
capaci ance
ma ices
o
he
co esponding
sec ions
sa is y
-
he
ela ionship:
i -i
-1
-
[c i]
=[cp]
1=
1,
...
,m+1
wi h
m
he
o de
o
he
il e .
The
no malized
s a ic
capaci ance
ma ix
CT
o
he
dual
in e digi al
s uc u e
is
gi en
by
C1
D:U
1
CD12
m+.l
CD12
cm+1
U22
179
'
In
hi
s
manne ,
by
in e ing
he
no malized
s a ic
capaci ance
ma ices
o
each
sec ion
o
de
eohn's
s uc u e,
and
compounding
he
0
L ained
ma ices
in
he
way
shown
by
he
p eceied
i
ng
exp ession
[eT/
E],
~~e
ob ain
he
no malizad
s a ic
capaci ance
ma ices
o
an
e
qui alen
in e digi al
s uc u e
in
hei
selec i e
p ope ies.
As
i
is
known
(4),
equi alen
in e digi al
ealiza ions
can
be
de ·i ed
om
hi
s
one
using
he
ans o ma ion
[e~/£1
= l
nl[
eT/EJ[n
l
/1/
·:
·1he e
n
is
a
diagonal
ma ix.
In
o de
o
he
e minal
impedances
o
e
main
unchanged
a e
he
ans o ma ion,
we
mus
ha e
n = n 2=1
1
m+
I ,
as
usual,
we
wo k
wi h
syme ical
s uc u e
s ,
he
equali y
.
m+3'
n1=
nm+J-i
1=2,
••.
N=
.~
mus
hold.
The e o e,
we
ha e
N-1
~eg ees
o
eedom.
In e ing
he
sequence
o
ope a ions
had
led
o
~T/~we
ob ain
om
~~~~an
implemen a ion
equi alen
o
he
o iginal
Cohn's
il e
Along
his
p ocess
new
deg~ees
o
eedom
a e
in oduced
;
--
nam.
ely,
he
pa i ioning
·
o
each
o
he
elemen s
on
he
main
diagonal
o
h
e
ma ix,
.
(c~/E].,I
we
exp ess
1
j:his
pa i ioning
as
cii
=
ai
eii
+ (
l-
a)
cii
he
syme y
o
he
implemen a ion
is
kep
i
1
,ai
=
am+J-i
i=2,
•.•
,N
/2/
( e
11
and
em+ 2,m+2
a e
no
pa i ioned).
When m
is
odd,
condi ion
/2/
is
supplemen ed
by
he
ac
ha
aN=
0.5
,
since
he
wo
cen al
coupled-lines
mus
be
equal.
I
is
easily
sho~n
ha
he
numbe
o
deg ees
o
eedom
now
in oduced
is
M=(m/2]
,
esul ing
hem
a
o al
numbe
,
MT=
N-l+M=m,
o
deg ees
o
eedom
in
he
ob en ion
o
an
equi alen
ealiza ion
o
he
ini ial
design.
III.-
DESLGN
METHOD.-
The
p oposed
design
me hod
is
based
in
he
p ocess
o
ob en ion
o
he
equi alen
ealiza ions
xe.pose?,
and
he
use
o
he
deg ees
o
eedoms
ha
ha e
appea ed
in
he
design
o
a
con igu a ion
wi hou
disc9n inui ies
•
. • ' '
Once
one
pa i ion
o
(CT/£J
has
been
chosen,
each
pai
o
coupled
lines
ha
se
up
he
i s
hal
o
he
il e ,
is
indepen_
den
o
he
emaining
wi h
he
excep ion
o
he
cen al
one
when
m
is
e en.
This
is
so
because,
when
we
numbe
hose
ansmission
lines
om
he
closes
o
he
end
o
he
u hes ,
he
no malized
s a ic
capaci ance
ma ix
~i/£~ o
each
o
hem
is
a ec ed
by
an
elemen
..
J
· .
.:
180
ni
o
[nJ
no
appea ing
p e iusly.
Thus
we
can
ake
o
each
ma ix
he
alue
o
ni
ha
elimi_
na es
he
discon inui y
in
he
jun ion
o
h
cu en
pai
wi h
he
p e ious
one.
I
m
is
e en,
o
adjus
he
pai
o
cen al
coupled
lines
we
mus
use
aN
in
an
i e a i e
p ocess.
IV.-
PRACTICAL REALIZATION. - A
band-pass
il e
has
been
des
i
g_
1
ned
wi h
he
ollowing
·
speci ica ions:
·-cen al
equency
0 =
3.
9 GHz.
- ela i e
bandwi h
Bw 5 %
- il e
o de
m=3
- e minal
impedances
z0=
50
0,
The
il e
is
cons uc ed
in
s ip-line
e< uology
wi h
1/16"
1
2200
Rexoli e
subs a
(E =
2.56).
F om
he
Bu e wo h
(3)
ap
o
xima ion
a
p o o ype
in
he
s-plane
/Fig.2-b/
o
he
in e digi al
s uc u e
is
ob eined,
leading
o
he
ollowing
no maliced
s a ic
capaci ance
ma ix
o
he
i s
hal
o
he
s uc u e
whe e
. 1
, •
no
1
TI~
=7£
l1
/2
I'
0
o<a<
1
1 0
2 6 . 4 4 7 7
3a
+ 2 6 • 4 4 7 7 3 (
1-a
) 1 j.
1
25.47734
ep esen s
he
pa i ion
ha
has
o
be
done
on
c22
and
cons i u s
he
emaining
deg ee
o
eedom
in
he
design.
Taking
a=
0.4825
and
using
he
ans o ma ion
111 we
ob ain
[
11
L•1.osso26s
_1
-8.0850265
CP~
=
-0.850265
n2
O.o8So265
whe e
n;
1
has
o
be
chosen
such
ha
he
1 1id h
-1
]
n2-1
n2
o
he
line
o
he
pai
coupled
conec ed
o
he
500.
e minal
line
has
he
same
alue
ha
his
one,
which
o
he
p oposed
ma e ial
is
w/b
=
0.7366771
wi h
b
he
sepa a ion
be ween
s ip
line
planes.
Using
he
exp esion
o
he
dimensions
o
a
pai
o
coupled
lines
in
s ip
line
con i
gu a ion
as
a
unc ion
o
he
s a ic
capaci ance
ma ix(S),
n;
1=
2.646267
is
ob ain
by
means
o
succesi e
ap oxima ions.
-1
In
a
simila
way,
n3 =
4.461639
is
ob ained
o
he
second
-
line.
Table
I
summa izes
he
dimensions
o
he
lines
o ming
he
-
il e .
Figu e
3
shows
he
esponse
o
he
cons uc ed
il e .
181
REFERENCES
(1)
J.Malhe be,A.
S eyn
"
'l'he
compensa ion
o
s ep
discon inui ies
in
T.E.M.-Mode
ans
mision
lines"
M.T.T/16,pp.889-885,No embe
1978.
( 2 )
S.
B.
COHN
"Pa alled-coupled
ansmission-line- esona o
il e s"
M.T.T.-6
pp.223
.-
231,
Ap il
1958
(3)
R.J.
Wenzel
"Exac
design
o
T.E.M.
Mic owa w
Ne wo ks
using
qua e -wa e
lines"
M.T.T.
12
pp.94-111,
Janua y
1964
(4)
R.J.
Wenzel
"Theo e ical
and
p ac ical
applica ions
o
capaci ancema ix
ans
o ma ions
o
T.E.M.
ne wo k
design"
M.T.T.-14
pp.635-647,Decembe
1966
(5)
S.B.
Cohn
"Shielded
coupled-
-
s ip
ansmission
line"
M.T.T.-3
pp.29-38,
Oc obe
1955
AUTHOR'S
ADDRESS.-
E.T.S.I.
de
Telecomunicacion.
Ap do.
30.002.
Ba celona-
SPAIN
Lines
1
and
4
Lines
2
and
3
_W
A1
W.B4
~-
. i
I-
.
-WA_2
____
W_B_3
---.-0-.
~
:·- •
-b-=
-b--
I
0.
74
-b--
-b-
WB1
WA4
0.19
WB2
_wA3
0.48
--b-
=-b-
-b-
--b-
/
s/b
0.12
s/b
10.66
1
b=
g ound
planes
spacing
T A B L E I
l
182
a.)
[c!/el
[c} ]
[c'!E}·~
~~1
J
c'l
1~
VzL
c!ll
.
c!
Yza
c:
IJ

18}
en
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