1. No category

IEEE TRANSACTIONS
94
Exact
Design
ON
MICROWAVE
of TEM
Using
Summary—Modem
of
synthesis
TEM
network
techniques,
mode
theory
Quarter~Wave
procedures,
are reviewed
microwave
networks
based
for application
using
on Ozaki-
in the design
parallel
coupled
bars
and/or
series and shunt stubs. The circuit equivalences
and identities obtained are theoretically
valid over the entire frequency spectrum and lead to several physical configurations
having identical
response functions. These equivalent circuits often allow simplification of the physicrd circuitry and realization of both broad and narrow bandwidths.
The problem of practical circuit configurations
is
dkcussed from the viewpoint
of bandwidth
and circuit element
values. Neglecting multiple responses, TEM low-pass, high-pass and
band-pass Butterworth
filters are shown to offer steeper bandedge
characteristics
than those of corresponding
lumped element filters.
The use of complementary
filters to match a source and load over a
wide frequency range is outlined and TEM realizations
of these
complements are obtained. A simple procedure for obtaining element
values of Butterworth
complements
is described. An analysis of
parallel coupled filters is made and a simplified equivalent circuit is
obtained. An exact synthesis procedure for parallel coupled bar
filters and their equivalent forms is given. Construction
details and
experimental
results are describe d for two filters which use series
networks
to be treated
networks.
figurations
ment”
T
HE
PURPOSE
account
thesis
TEM
In
past,
have
been
The
depth
afforded
the
designed
by
of
relative
network
of
in
the
existing
This
duces
scope
wherever
parameter
methods.
element
has
not
networks.
modern
circuits
Kuroda’s
and
been
com-
network
and
of
this
paper
and
may
be
reviewed
is presented
transformation
as follows:
that
allows
Section
distributed
II
introTEM
IV
nature
May
17, 1963; revised
manuscript
received
September
t The Bendix
Corporation,
Research Laboratories
Division,
Southfield, Mich.
1 H. Ozaki and J. Ishii, “Synthesis of a class of strip-line filters, ”
IRE TRANS. ON CIRCUIT THEORY, vol. CT-5, pp. 104–109; June,
.Tune. 1958.
z ‘L. Weinberg,
“Network
Analysis
and Synthesis, ” McGraw-Hill
Book Co., Inc., New York, N. Y:; 1962
3 K Reference
Data
for Radio
Engineers,
” published
by International
Telephone
and Telegraph
Corp., 4th ed., pp. 187–1 88; 1957.
4 E. A. Guillemin,
“Synthesis
of Passive Networks,
” John Wiley
and Sons, Inc., New York, N. Y.; 1957.
6 N.
Balabanian,
“Network
Synthesis,
” Prentice-Hall,
Inc.,
Englewood
Cliffs, N.J.; 1958.
this
in
of the
con-
ability
ideal
filter
with
a
LLFPB
it
is one
real
two
of
a
was
ideal
bars.
established
Foster
and
for
by
Brune
in
1930.8
associated
passive
for
realizable
Cauer
realizability
finite,
that,
and
with
stubs,
THEORY
functions
linear,
be
open
coupled
conditions
energy
filter
lines
eliminating
theory
work
exact
series
for
showed
to
for
parallel
and
a function
driving
sufficient
function—where
that
a positive
real
of
point
it
be
a
(p.r.)
that
for
real
values
s = u +ju
a positive
positive
given
bars,
with
stated
is necessary
function
methods
FILTER
the
Brune
variable
variable
the
(lumped,
real”
for
network
of the
network,
“positive
Construc-
are
coupled
impedances
a consideration
ob-
intro-
obtained.
having
with
From
are
are
are
NETWORK
with
of driving-point
filters
a procedure
culminated
circuits
results
lines
networks
modern
impedance,
coupled
equivalent
the
and
for
complex
sections
Parallel
stubs.
mid-1920’s
has
re-
not
preceding
parallel
MODERN
basis
is
is
the
microwave
uniform
outlines
II.
2)
and
series
II
Appendix
1)
circuits
of the
realizations
stubs
bilateral)
lumped
The
problems.
experimental
shorted
was
element
between
an analysis
theory
I demonstrates
transformers
and
the
analyzed
and
of
the
and
to approximate
network
containing
The
response.
characteristics.
condition
network
Appendix
in
microwave
the
physical
circuit
a comparison
of
are
and
shunt
impedance
equivalences.
filter
V applies
synthesis
concludes
of
bandwidths
distributed
interesting
details
filters
and
of equivalent
circuit
Complementary
duced
ele-
element
is given.
filters
tion
the
of network
characteristic
tained.
con-
“unit
section
same
practical
to
Under
type
Section
The
the
contains
petitive
a
* Received
18, 1963.
of
and
of each
The
a discussion
a number
realize
is related
element
image
theories
can
be found
in literaof modern
network
synthesis
are
describes
Achievement
sidered.
various
circuit
are discussed.
that
Section
tabulated,
and
to lumped
for
scaling.
III
structures
values
and
example
frequency
Section
analogous
circuits
as a necessary
identities
a design
A discussion
publications.2,4,6
paper
the
on microwave
networks
theory
merits
syn-
microwave
of lumped
distributed
a uni-
distributed
theory
use of image
to
parameter
network
ture.1,3
The
details
of
is placed
TEM
present
network
design
network
most
by
modern
extended
beyond
than
of understanding
pletely
the
the
Emphasis
rather
is to
modern
derived
is introduced
line
paper
exact
to
networks.
aspects
possible.
an
applicable
mode
circuit
of this
of
in a manner
Equivalent
are
to some
1. INTRODUCTION
Networks
IEEE
element
with
January
TECHNIQUES
Lines*
MEMBER,
stubs.
fied
AND
Microwave
R. J. WENZEL~,
Ishiil
THEORY
real
real
of
the
complex
frequency
and
part
for
values
of
s having
a
part.
GO. Brune, “Synthesis
of a finite 2-terminal
driving
point impedance is a prescribed function
1930.
J. Math. and Phys., vol. 10, pp. 191-236;
network
whose
of frequency, ”
1964
Wenzel:
From
these
if a given
The
two
extension
niques
to
in
pedances
1948.
lumped
network
In
of j
the
fll
where
Richards
showed
of lumped
resistors
lines
can
and
be
transformation
the
are
the
equal
complex
by
o
length.
U3zIL,
transmis-
networks
using
0
a
2f
f/2f
.0
f=
OMAPS
INTO
f.
is the
real
constant
wavelength
As can
be seen,
microwave
.S is periodic
network
mapping
will
properties
illustrated
frequency
and f is the
of
in Fig.
real
in 2fo and
repeat
functions
1 for
at
the
case
which
frequency
the
at
this
of
the
f=
1 is a
variable.
response
.
=l
J
IO MAPS INTO
: = m
J
Fig. I—Mapping
properties
of a function
S/j = tan 7rf/2f0.
of a
interval.
of the variable
The
variable
of a high
4f
0
;=O
lo
~= —MAPSINTO;
2
quarter
3f
.
BAND-PASS, REAL FREQUENCY PLANE FI1.TER
variable
(1)
where
s
.
J
HIGH-PASS S-PLANE FILTER
composed
frequency
1
constant
line
length
as lumped
95
im-
expressed
phase
1 is
Lines
“’m-”’
tech-
the
networks
Iossless
treated
to
line
microwave
Quarfer-Wave
determine
networks,
and
Using
be realized.
(3 is the
line
that
to
Networks
demonstrated
of transmission
tan
TEM
synthesis
was
microwave
transmission
of
is possible
of s can
networks
of sections
as functions
along
of
it
function
microwave
Richards7
sion
conditions,
impedance
Design
S are
pass
S-plane
becomes
appar-
filter.
The
ent
synthesis
from
using
lumped
verted
elements
problem
considered
for
in Fig.
Jones
and
for
the
to
be used
required
in
response
the
S plane
structure
is to
as functions
shown
matrix
1.8 The
to a microwave
remaining
tures
procedure
Fig.
application
is then
plane.
the
microwave
TEM
in
and
in the~
express
of S. One
is synthesized
mode
modern
con-
The
struc-
structure
network
(a)
only
to be
design
is
2.
Bolljahn’
have
configuration
derived
of Fig.
the
impedance
where
2(a).
0 = electrical
The
by
where
even
Ozaki
and
additional
cot 8
z,,
=
z?,,
=
24,
=
–j(zo,
+
zoo)
—
2
212
=
221
=
234
=
Z43
=
– j(zo.
–
cOt e
zoo) —
2
csc
z,, = 23, = z,, ==2,2 = –j(zoe – zoo)—
e
many
hold
unequal
calculating
strip
and
Ishii.l
degree
odd
By
using
strips
follows
the
same
strips.
For
simplicity,
for
the
unequal
analysis
d
S=jtan
—=jtan
csc
=
Z41
=
223
=
.Z32
=
–
j(zoe
+
Zoo)
an
of
unequal
used
for
will
in
width
equal
be car-
Substituting
Ogives
20, + zoo
e
211 =
222
=
233
=
244
=
2s
T
20.
Zlz
=
Z!ll
=
Z34
=
Z43
–
zoo
=
2.s
? P. 1. Richards,
“Resistor
transmission-line
circuits, ” PROC. I RE,
36, pp. 2 17–220; February,
1948.
8 The basic svnthesis
mocedure
used follows
that of Ozaki and
Ishii.1
g E. M. T. Jones and J. T. Bolljahn,
“Coupled-strip-transmission
line filters and directional
couplers, ” IRE
TRANS. ON MICROWAVE
THEORY AND TECHNIQUES, vol. 19, pp. 75-81; April,
1956.
vol.
widths,
2f o
2
Z14
for
given
as required
derivation
case.
The
are
strip
as that
the
width
strips.
as methods
impedances,
procedure
equal
width
as well
is obtained
The
cases.
equal
mode
of freedom
practical
out
for
widths,
width
ried
line,
impedance.
relations
for
of the
impedance,
mode
above
matrix
length
mode
ZOO= odd
=
(c)
Fig. 2—Parallel coupled stripline four-port. (a) Four-port parameters.
(b) Even mode field configuration.
(c) Odd mode field configura-
20. = even
z,,
(b)
-
20. – zoo
21$ =
231
=
224
=
242
=
Z14
Z41
=
Z23
=
Z32
=
–
s’
2s
20. +
=
——
~1
2s
zoo
—
~1
– S2.
(2)
IEEE TRANSACTIONS
96
To
determine
two-strip
necessary
to
example,
open,
the
impedance
configuration
(1)
The
port
the
lows
that
port
terminal
input
port
conditions
are
(2)
and
it
1,=
properties
For
and
its
output
O and
These
u.e.’s
(3)
is
fol-
TECHNIQUES
are
as lengths
load
or
readily
apparent
of line
between
January
the
Now
consider
S-plane
the
cascade
inductorlo
[Fig.
for
VI and
V4 in
terms
of
11 and
of
zIl–
z14–
11+
—
,,
–—
11, gives
I,
Substituting
(2)
+
’44-
Z22
)
’22
)
=
and
arranging
them
in
(L
<1–s’2
The
corresponding
matrix
–
S2
s
– S2
Comparing
[1V4
lent,
r
2’0.’00
220,’0.<1
S2L + ‘O
s
2’0.’0.<1
s
(’O.+
’O.)s
proceeding
further,
circuit
defined
by
it is necessary
element,
ABCD
the
r
1
/l–s’z
matrix
of the
transmission
line
characteristic
has
the
pedance
1)
network
the
point
as without
the
magnitude
phase
of
the
elements
instead
of between
1)
the
input
and
2)
the
magnitude
the
of the
the
phase
of
the
as that
6 = r~/
series
of
unit
(2~0)
3(b)
and
related
elements
when
of
has
A
inserted
a termination
the
of a
equivalence
been
by
+
.
~9)
’00)
v’L(’o
+ L)
(lo)
+&(’o+L)
im-
list
of
remains
exactly
the
case
procedure
and
all
of
obtained
transfer
impedance
The
remains
impedance
may
be
before
the
and
lumped
its
impedances
the
networks
where
shown
ZO., ZOO, Z,
in
Fig.
and
L are
possible
S-plane
unequal
by
can
equivalent
setting
synthesis
circuit
circuits
strip
in Table
lence
of the
II
(p.
for
may
is given
other
port
be developed.
and
in
Table
equal
width
ZO,U = ZOeb and
Zoo”=
ZOob.
to
98).
be used
As a sample
in Fig.
is based
known
as
equivalence
4 will
Kuroda’s
of the
proof,
A
their
I for
The
procedure
networks
applied
configurations
widths.
of equivalent
circuits
II Kuroda
showed
the
listed
be
circuits
the
be demonstrated
the
case
is
upon
a
idencircuits
equivaby
network
termination,
are changed,
coefficient
coefficient
series
tities.
of
established
(10).
similar
conditions
u.e.’s,
reflection
– ‘0.)2
.
l+;
Thus
reflection
L +
ZO +
—
– Zo –
2’0.
remains
unchanged,
3)
equiva-
r,),
Zozo,
properties
network
transfer
be
ZOOgives
(6)
transfer
are inserted
L =
to
“
impedance
of the
(b)
z,. =zo+L+dL(Zo+
changed.
If unit
ZC, and
‘OO =
1
and
the
two
element”
unchanged,
3)
for
equivalent
driving
the
2(’08
Solving
ZO.
The
same
2)
interesting
a lumped
shows
= ‘~~z~oO
as
length
ZO. A
Z.
to introduce
‘{unit
is the same
electrical
(a)
zo.q
element
impedance
following
between
unit
of
1
Is
11
4
the
matrix
(5)
1
4’0,’00
2s(’0,+200)
additional
(5)
if
(’0.
1
W(zoe – ‘0.)2+
–s2
with
1
(8)
“
–s:
(zoe+zo.)s-
— (Z04+ZOX
this
(7)
“
is
‘0/1
‘0<1
1
The
is
‘0)s
VI
(u.e.),
an
Zo
7
= [
[’]
form
+
matrix
‘0
gives
Before
and
matrix
S2L;1
impedance
(4)
1,.
)(
’22
(
one
driving
element
ABCD
The
Zo
—
)(
’22
z4,
1
its
1
‘21’42
V4=
and
a unit
3(a)].
ZIZZZ4
‘1:’21
(
VI=
of
a network
[1’[:;:1[1
‘3)“’]=4A[;Z
‘:”]
[1
—
solving
thinks
between
network
1
and,
if one
inserted
source.
port.
O. It
AND
THEORY
the
conditions,
is the
V,=
the
WAVE
is only
is grounded,
(4)
then
MICRO
when
as a twoport,
pertinent
that
is the
parameters
is used
apply
assume
ON
is changed.
10 An .$plane
inductor
L represents
the characteristic
impedance
of a transmission
line and has units of ohms. ?
“Derivation
Methods
of Distributed
Constant
11 K. Kuroda,
Filters
from
Lumped
Constant
Filters, ” test for lectures
at Joint
Meeting
of Konsoi
Branch
of Institute
of Elec. Commun.,
of Elec.,
and of Illumin.
Engrs. of Japan, p. 32; October,
1952. (In Japanese. )
Wenzel:
Design
of
TEM
Nefworks
Using
Quarter-Wave
Lines
(a)
(b)
Fig. 3—(a)
inductor.
Cascade connection
of a unit element
and an S-plane
(b) Equivalence
of stripline
and ,S-plane networks.
TABLE
I
EQUIVALENT CIRCUITS FOR UNEQUAL STRIP WIDTHS
~~hf
Element
Equiva~ent Circuit
Circuit
Values
—
+Z
z
Z=oeoo
0
2
~
iFj=
‘-l
—
=ZO+L+~~
z
oe
‘m’
Zo+
L+{-
Z=
00
1++++
z.~
oe
1
z
m
Cz
00
=L-{L
(L-;)
LC>l
00
—
I
(e) ~
2
Y
a+Ya
f
00
oe
.Y
++
2
1
0
oe
Yoe’
b
Y=++$
oe
a+
1
1“32E’2
z
2
2
L
00
.Zt+
2
0
00
=+zoob
Zoe
=+++
1
1
+J0
‘2
b
a+za
oe
+ Yoo’
::Y
1
b
+ zoo
.Z++
2
2-0
2
(d
z
2
oeatza
00
1
.—
2
1 ~-]
L
C+-Z
n
z
a + zoo’
oe
1
.—
2
●L
-i
c
n
n<O
(i)
~
2
~
1
Z.E!E2
Y
a
a+Y
oe
00
L
2
z
0.
.—
1
c1
Yoeb
=
Yoe’
r.
‘+Z
00
++*
2
3
b=
00
Zoe’
+ zoo”
++
3
Yoo”
2
++
3
+
.=
1
z
00
a +
oe
++
1
a+za
2
1
.—
-+++-
2
.
.=
+++
2
3
IEEE
98
TRANSACTIONS
TABLE
ON
MICROWAVE
II
2)12
Perform
KURODA’S IDENTITIES
3)
Original
Cmcuit
I
Equivalent
the
synthesis
niques
to yield
Insert
a series
Jan
by
modern
a lumped
uary
network
tech-
network.
of unit
elements
of characteristic
n
C,rcuit
impedance
its
3EI= EEl=z
l+ZOC
Z.
load
ance
(of
between
z
4)
l++-
is to be preserved.
If the
coefficient,
impedance
front
of the
Use
Kuroda’s
of unit
nated,
input
quantity
can
also
and
imped-
of interest
elements
in Table
From
the
lumped
be
of charinserted
in
a configuration
elements
a cascade
that
of circuits
can
similar
to
1,
L, C and
each
to obtain
and
into
those
13xam$le-It
1++
Z.
network
the
unit
identities
elements
ZOO for
if
network,
be separated
.5)
lumped
ZO),
acteristic
ml-i-:
the
impedance
is a reflection
Zoln (n-l)
m
AND TECHNIQUES
THEORY
20 values
determine
Z08 and
section.
is desired
Butterworth
to design
filter
having
a resistively
a reflection
termicoefficient
0
characteristic
of
values
network,
are
shown
before
the
writing
ABCD
the
matrix.
For
the
network
on
the
left
load
of
source
that
of Fig.
the
of Fig.
and
Fig.
Ii ‘~1
‘4(4)sz~~+J
7
[1
the
circuit
known,
from
@
on the
right
of Fig.
1
(Z2
(12)
41–s’;
The
two
networks
‘:+1
are equivalent
(a) Zl=Zz+L,
n is defined
“
if
(c)~=
(b) C+~=~,
a parameter
L)S
+
1
by
n = 1 +ZIC,
ZIC.
(13)
then
(14)
n
All
techniques
presented.
1)
The
Choose
provides
necessary
procedure
the
the
function
desired
n
for
design
to be followed
have
now
of S to be synthesized
response.
been
is:
that
Adding
load,
the
be added
Z.=
of this
in
element
tables,2,18
of interest
can
source.
the
the
(Table
gives
phase
three
the
unit
after
network
differs
of
the
is
both
1, two
network
II)
circuit
circuit
from
reflection
ZO in
in
the
tables’’-’o
or
element
values
Once
elements
by
have
the
TEM
Table
as shown
5(c).
to the
in Fig.
5(c).
I shows
in Fig.
are obtained
Fig.
of
applied
of realizable
circuits
section
now
is shown
be realized
dimensions
available
are
a cascade
equivalent
each
L and
1! Lumped
‘~1~
.
If
4,
only
can
ZOOfor
responding
network
before
transformed
the
above
ZO. and
the
the
to obtain
The
Consulting
for
quantity
impedance
identities
capacitors
elements.
and,
the
elements
response
5 (a)
normalized
by consulting
Since
unit
after
one
The
Kuroda’s
shunt
(11)
and
The
coefficient.
4,
“~[:s
5 (a).
characteristic
5(b).
of
three.
obtained
coefficient,
elements
identity.
order
in Fig.
a reflection
=+III==E
Fig. 4—Kuroda
for this
from
ZO, and
can
method
been computed
5(d).
the
cor-
ZC~ are
be
found
given
by
and
tabu-
lated for Butterworth
and Chebyshev
filter characteristics
?,’i The
tabulated
element values must be chosen for the proper characteristic.
For example,
element
values for a Butterworth
transfer
impedance
which assumes an ideal current
source are not the same as those for a
network
having a Butterworth
transmission
coefficient which assumes
a matched
source. In general, steps 1) and 2 ) can be omitted
by consulting the many excellent
tables available,
‘3 L. Weinberg,
“Network
design
by use of modern
synthesis
techniques
and tables,”
PYOC. Natl. Electronics
Con .f... vol. 12. -“
DD.
704-817 ; 1956.
‘4 S. B. Cohn,
“Shielded
coupled-strip
transmission
line, ” IRE
TRANS ON MICROWAVE THEORY AND TECHNIQUES, vol. MTT-3,
pp.
29-38;
October,
1955.
‘5 S. B.
Cohn,
“Parallel-coupled
transmission-line
resonator
filters, ” IRE TRANS. ON MICROWAVE THEORY AND TECHNIQUES, vol.
MTT-6.
DU. 223–231: Am-ii. 1958.
“ W. J. Getsinger,
“C&pled
rectangular
bars between
parallel
plates, ” IRE TRANS. ON MICROWAVE THEORY AND TECHNIQUES, pp.
65–72; January,
1962.
17 W. , J Getsinger,
VA coupled
strip-line
configuration
using
minted-clrcmt
construction
that allows verv close counlimz. ” I RE
TRANS. ON MICROWAVE
THEORY AND TECH~IQUES,
vol~ 9, ~p. 535–
544; November,
1961.
‘8 S. B. Cohn,
“Characteristic
impedances
of broadside-coupled
strip transmission
lines, ” IRE TRANS ON MICROWAVE TEEORY AND
TECHNIQUES, vol. 8, pp. 633–637;
November,
1960.
N S$ B. Cohn ‘> Thickness
correction
for capacitive
obstacles
and
7,1
strip conductors,
” IRE TRANS. ON MICROWAVE THEORY AND TECHNIQUES, vol. 8, pp. 638-644;
Novermber,
1960.
20 ~. D. Horgan,
“Coupled
strip transmission
lines with rectangular inner conductors,
” IRE
TRANS. ON MICROWAVE THEORY AND
TECHNIQUES, vol. 5, pp. 92-99; April,
1957.
Wenzel:
1964
Design
of TEM
Networks
Using
Quarter-Wave
multiplied
by
99
Lines
a constant
in
the
above
normalization
procedure.
The
j
(a)
2
—
oZo.
,
2.=
of a renormalization
rf/2j0
is to change
response
of the
between
O and
bandwidth
v .. .
0-
network
filters.
As
an
the
of
real
variable
the
band-edge
frequency
interval
of this
100
per
which
type
are
the
S
to
mapping
(c)
1). This
about
the
in
new
points
are
loj
s
s’
i
loj
.— —_
.
5—(a)
Third-order
Butterworth
filter.
(b) Butterworth
filter
with unit elements added. (c) Equivalent
circuit
after application
of Kuroda’s
identities.
(d) Microwave
network
that realizes a
third-order
Butterworth
response.
“’”~
renormalized
and
responses
of the
bandwidth
of
by
teristic
t13456789
101112
!,,,,,
a change
10S.
of
a response
is made
The
pertinent
into
f
=
— 1
maps
into
j
= 0,938~0,
maps
into
j
= fo.
m
O,
original
the
this
properties
are shown
in Fig.
filter.
of
it
TEM
the
S’
the
is (seen that
the
is controlled
and
change
component
plane
with
variable
is to
the
the
filter
S-plane
renormalization
of
in
6, together
Thus
distributed
impedance
TEM
that
has
maps
a renormalization
of
and
S’=
mapping
in the ~ plane
effect
f~
lo’j
.i
The
point
filter
has a bandwidth
0
s
s’
—.—.————.
(d)
filter
variable
S-plane
now
s
s’
-— .—
j
line
to a band-pass
2f0. Assume
the
distributed
a normalized
transforms
(see Fig.
cent
in TEM
consider
which
is periodic
from
factors
example,
filter
in the f plane
o
in
S-plane
location
fo. Renormalizations
controlling
high-pass
(b)
Fig.
in the
the
I
\
N .. .
effect
tan
the
the
charac-
ele]ments
in
the
structure.
(a)
II 1. EQUIVALENT
Id’
18
When
‘,
I
~
:
/
1
:
,1
‘,
been
+
2 f.
,0
3,.
46.
I
obtained,
Y-plane
j-plane
high-pass
filter.
filter.
take
ment
Ozaki
and
To
make
Ishii
in the
above
a load
impedance
frequency
and
fo
lengths
obvious
tan
which
used
that
frequency
the
by
designed
as
with
A.
impedance
Frequency
pacitances,
multiplying
all
achieved
by
TEM
change
variable
of
anaIogous
range
cannot
an
structure.
S-plane
can often
do
not
limitations
arrangement.
only
by
in
For
di-
on ele-
However,
greatly
j udi-
simplify
the
constant
For
line
l/C
of the
a function
of line
[see
20.
width,
the
example,
by
By
be
available
an
the
S-plane
of open
cir-
7(a)].
with
the characteristic
consulting
21‘Zz it
ca-
can
consider
of
a Iengt’h
Fig.
is associated
line
circuit
10b: 1
realization
is represented
transmimion
of
circuits,
circuit
of element
graphs
becomes
apparent
of 20 as
that
the
21 (~The Microwave
multiIt
S is
limited.
TEM
inductors,
rarlge
lurnpedl
ranges
microwave
and
the
With
value
more
which
capacitors
in mind
realized.
In
impedance
lumped
by
a constant.
variable
a
be
element
is much
The
S-plane
to keep
can
frequency
cuited
S-plane
a change
is accomplished
by
presented
practical
have
network
device.
with
easily.
capacitor
associated
normalization
to
~
ZO and
real
can be accomplished
the
the
Values
that
S-
networks
The
the
important
normalized
as that
7rf/2f0.
circuit
working
it is very
ZO.
manner
of
Element
When
band-edge
values
by
is tantamount
in
a
of S-plane
same
therefore,
a frequency
the
of
inductance
values
elements.
S plane,
elements,
and
in the
is Z = jL
in
level
normalization
lumped
inductor
plying
as possible,
are
values
impedance
capacitance
Frequency
line
networks
the
of the
values
of constructing
techniques
account
methods
element
stripwidths.
of 1 Q and a normalized
resistance
all
change
unequal
to 20 is accomplished
be accomplished
in the
of
as simple
most
level
the
normalized
with
case
of co= 1.
network
dividing
the
computations
example,
Changing
plane
for
filter
problem
STRICTURES
by the
pertinent
use of the techniques
construction
FILTER
designed
the
design
into
values
cious
and
the
The
rectly
6—(a) Renormalized
(b) Corresponding
has been
section
remains.
(b)
Fig.
a filter
preceding
TEM
wave, Inc., Eh-ookline,
is
not
zz R. H. T. Bates,
Engineers
Handbook,
Mass.; 1962.
“The characteristic
” Horizon
impedance
Home- Ivfkroof the shielded
IRE TRANS. ON MICROWAVE THEORY AND TECHNIQUES,
1956.
vol. 4, pp. 28–33; January,
slab
line,”
100
IEEE TRANSACTIONS
FREf2u
ENcy
PLANE
MICROWAVE
ON
THEORY
AND
TABLE
s - PLANE
,_mc
,.
z
,.
III
EQUIVALENT CIRCUITS FOR ELEMENTS WITH SERIES AND SHUNT
OPEN AND SHORT CIRCUITED LINES
G-J
.1
January
TECHNIQUES
TEM
Equivalent
Circuit
S-Plane
Circuit
Sc
“=HEE22=+
(a)
Z3
n
z
(c)
(b)
Fig.
7—(a)
Stripline
realization
of .S-plane capacitor.
network.
(c) Stripline
realization.
(b)
z
1
z
S-plane
.
3
z
‘1
for
ZO <10
the
lines
they
become
range
of Zo, and
30:1.
As might
tion
on the
circuit
may
with
more
ceeding
with
the
Assume
that
1, that
realizes
in
cuits
is desired
this
to
useful
sections
physical
forms
a TEM
shown
as found
network.
by
for
in
line
Fig.
a
a series
unit
the
re-
the
can be
of
two
as shown
in
1/C
for
dielectric
unit
is the
of the
such
large
can
Application
the
shunt
importance
to
inner
series
stub
of
the
if one
considers
by
an S-plane
same
first
as a shunt
of
the
inner
conductor,
structure
is filled
will
are
series
equivalent
followed
of Kuroda’s
For
band,
coaxial
fit
useful
f.
for
the
range
the
realization
inductor
identity
capacitor
a
odd
series
the
can
be
of this
L =2.
Then
to
line
with
of this
to have
it
realized
shunt
no
as
that
level
Z2 = 3 and
of 50 Q gives,
C, Zg = 150 Q and
are
stop
values
within
the
Z, = 75 Q.
realizable
problem.
coupled
impedances
a broad
Assume
follows
an impedance
magnitude
present
parallel
type
of L or C is required.
Normalizing
mode
line
realization,
the
even
and
are
the
in realiz-
Zo,
= Zo+
L+
~L(Zo+
L)
=
5.45
Con-20.
zoo =
of
= 0.55.
Table
l+;
of a unit
(see Table
(Table
element
value
and
For
with
inductors.
circuits
a filter
Z1 = 1 and
C=
configuration
8.
a large
Impedances
Z1 is
within
This
in Fig.
is self-explanatory.
be seen
be the
short
impedance
of
element.
shown
case of an inductor
coax
stubs
or small
stubs
and
single
outer
coaxial
the
cir-
by using
The
the
inner
10. Series
capacitors
of the
element
to
that
a
impedance
to L for
The
length
series
struction
The
equal
open
planes.
respect
a capacitor.
element
shunt
or
ground
with
impedance
equivalent
are constructed
element
numerically
S-plane
structure
flat
structure,
made
“WE
-
7(b).
the circuit
element
and
stubs
coaxial
unit
of
series
series
between
coaxial
and
with
The
double
structure
‘==’=
m
Table
as a combination
a number
elements
lines.
Za of
K-L+
unit
lists
E121E=zl
1
.
to
and
that
consulting
However,
network
and
obtain
network
circuit,
a TEM
III
either
this
is
line
7(c).
circuited
III
T
pro-
z
Elements
S-plane
capacitors
Table
ing
it
the
realized
Fig.
it
TEM
.a
response
Before
problem,
different
the
response
same
i
+1-
Fortunately,
the
3
Z
a given
restriction,
configuration.
of some
possible,
is no simple
shunt
with
impossible.
realization
of
restric-
a specified
to realize
physical
TEM
of
There
a severe
this
z
order
responses,
Equivalent
alization
of
with
if not
behavior
where
similar
places
Because
realizable
on the
‘=z’2ElEc”k
ZIEc=k
2
m
ZO >300
for
the
be obtained
of a filter
one
the
obtain,
realize
can
it is possible
than
investigate
or
that
and
Thus,
this
to be difficult
cases,
wide
C, is only
be expected,
responses
appear
B.
consequently
realization
in many
very
narrow.
configuration.
practical
to
become
extremely
“= ‘3EIE2=Z2
2
Normalizing
11) shows
followed
+;
UZO+L)
I).
by
a
Attempts
to
at
50 Q gives
realizing
Zo. = 272 Q and
impedances
of
ZOO= 27.5
this
order
Q.
in
1964
Wenzel:
~:w.m
parallel
coupled
pling.
8—Stripline
lines
very
easier
limited
it
having
coupled
This
to
of TEM
achieve
element
is a direct
close
may
impedance
values
in
example,
I(c).
This
Inspection
of
values
of Elements
Classification
from
Bandwidth
prove
filters.
On the
other
(Table
III)
tions
tion.
For
of the
a filter,
the
case
limited
of
range
very
important
form
applicable
considera-
of all
synthesis
of
will
have
the
filters
centered
aboutfo
have
and
the
(K>
O) by substituting
tion.
This
will
have
TEM
at f 0/2.
of
2.,
realize
the
Sin
the
band-edge
respect
A.
to
malized
(19)
%bandwidth
2(fo –
then
the
The
ideal
does
changes
the
elements
narrow
wide
the
bandwidths
.
(20)
)
decreases
filter
bandwidth
For
the
tion
requires
case
and
(a practical
‘3 The
“bandwidth
(K>
1), small
large
L’s
(K<
1).
range
and
S plane,
its
identical
Butterwort’h
re-
to its
filter,
for
response
u in its
T EM
with
realiza-
7rw/2c00
plane
or
response
filter
provides
low-pass
1:
Filters
Butterworth
a better
high-pass
plane
of
whose
a
nor-
frequency
approximation
filter
(characteristic
Butterworth
filter
whose
O <OJ <WO.
Low-pass
filter
Butterworth
characteristic
with
band-edge
expression
that
cutoff
at
approximates
is
(21)
The
for
ideal
<uO/2.
(frequency
characteristic
being
L and
This
L’s
variable)
‘Z “
new
that
small
of TEM
line
lines,
for
and
>
for
ZW <150
Q and
realizaclose
to 20
1
1
C’s are
configurations.
ZOO be reasonably
range
rang-e,
therefore
7r&l 2“
l+@2n”
rw/2u0
provides
(22)
tan~
()
establishing
practical
of 1 in the
for
C’s are required
a criterion
a value
frequency
tanz<w
20Jo
S
C to the
in this
it
K <1
variable
means
and
large
coupled
and
the
has
Further,
1+
give
properties
2..
for
elements
C’ = C/K.
of parallel
that
S’
and
considerations
the
be de-
1
T
variable
original
bandwidth
These
can
\tl’=
l–3tan–1A7
bandwidth
whereas
the
is not
variable
a frequency
becomes
(
L’ =L/K
required,
capabilities
III
a Butterworth
Low-Pass
increases.
Substituting
equa-
be diffi-
AND LUMPED
in
frequency
is co for
ideal
O <u
the
will
FILTERS2~
S-plane
1 +
=2
K >1,
values.
the
f.)
j!l
For
I (f j,
I and
plane
have
frequency
is tan
the
than
~.
An
S-plane
Proof
fractional
without
element
section
OF TEM
frequency
and
w = COo/2. The
The
equivalent
manner.
not
High-Pass
to
K.
tan-l
is
stop-band
bandwidth
is synthesized
real
to the
Theorem:
funcfo/2
Then
X
the
Thus
and
tion.
cent
S/K,
filter
the
does
variable
=
in Tables
response.
variable
f,
C’s
COMPARISON
a filter
in
example,
changes
fc from
sponse
S-plane
cutoff
S’=
desired
small
L.
that
L <1
lband
Table
this
shown
in a similar
When
the nor-
to
cutoff
~.f e
—
=
2f ~
tan
for
stop
the
in
that
for
the
of
L is small.
stub-line
a wide
element
value
narrow,
in achieving
ELEMENT
are
per
in ?fo. For
for
f,,
ZOO show
only
its shunt
ZOOshow
sections
and
realizing
hand,
of the
and
IV.
realization
100
S is changed
S’ for
the
their
general,
in
Table
about
the
when
in
it is
that
line
In
be periodic
variable
changes
Filters
bandwidthszt
will
to
physical
consideration
filter.
basis
cutoff
will
bandwidth,
bandwidth
S-plane
corresponding
a band-edge
malized
in
the
on a normalized
1 and
the
case
termined
because
values,
choose
the
2..
realizable
be realized
bandwidth.
for
the
filters,
impedance
designer
a given
the
S/j=
the
for
point
at
TEM
of practical
starting
synthesized
is a major
microwave
that
The
is
bandwidth
can
by
for
sections.
element
centerec[
to 20 only
use
necessary
of these
controlled
difficulty
for
cult
designing
close
of the
Considera-
is
is practically
encountering
information
strip-line
equations
for
tions
In
the
element
obtainable.
the
which
a good
well
the
capabilities
has a stop-band
are
section
101
I provide
bandwidth
section
of
this
Lines
consider
bandwidth
For
C.
Table
the
For
these
cou-
a narrow
consequence
Quarter-Wave
to evaluate
configurations
extremely
is desired
parallel
construct.
range
to impractical
Using
relations
equivalents.
strips
if
the
to
lead
small
Conversely,
stop-band,
of TEM Networks
+--J--J+p
Fig.
requiring
Design
Thus
the
variable
proximation
variable
to
u.
In
tan
the
the
ideal
characteristic
frequency
range
a. better
than
wJ2!
ap-
does
the
<OJ <CO,
the
ZOO> 20 Q). The
term
“bandwidth”
will be taken
as synonymous
with
about $,” and can be either a stop band or a IXMS band.
O <w< @ofor the
21This comparison is made only over the interval
case of low-pass and high-pass
filters and over the range O <co< 2a0 for
band-pa:s
filters. Thus, the repetititve
nature of the distributed
filter is
not considered
in the comparison.
ON MICROWAVE
IEEE TRANSACTIONS
102
ideal
characteristic
tan
7ru/2wO>U
has
and
a value
of
zero.
range,
this
In
THEORY
For
a bandwidth
fractional
1+
bandwidth
1+
6P”
[see
T&l\
—
2wl
1/
j tan
the
with
the
approximation
variable
Proof
variable
tan
7ru/2oJ0
the
characteristic
than
The
band-edge
proof
cutoff
for
the
case of a high-pass
at w = uO/2 follows
The
worth
K
)
1
= /t/;E.,=
~
1+
,.
tan
~~;
1
for
K
.
1
[
in an analogous
gives
low-pass
lumped
normalized
of
order
n
the
has
Butter-
element
charac-
transmission
or
’4’---)
I t];EM =
I t(jfl)
]’ =
unnormalized
low-pass
filter
of
u.
bandwidth
then
tan
characteristic
jfl2
101
1
t—=
~c
low-pass
to band-pass
Q2??”
1+
—
()WC
The
response
filter
is symmetric
the
TEM
be made
transformation4
of
and
the
TEM
about
the
identical
is
@c
[1
1+
The
1 – ~ tan-l
T
Filters
filter
has the
(
filter
teristic
An
2
K
Solving
Band-Pass
M =
(.00
12
t
does
manner.
B.
()
provides
to
W.
II:
u = UO,
(2o) ], then
2CO0
again,
=
(23)
tan~
()
once
of U. about
1
<
Zn
Therefore,
January
TECHNIQUES
thus
1
a better
AND
2W0
realization
element
provided
that
(26)
~
2W0
w =Wo. The
lumped
.
h
of
the
S-plane
band-edges
of both
band-pass
filters
can
~2=uo+2
2
Wc
——
2
@l=wo
where
tif —wI = u, the
U1W2‘WL2
filter
the
ment
bandwidth,
geometric
WIQ
center
of
the
lumped
ele-
pass-band,
WQ= upper
WI= lower
band-edge
band-edge
Defining
frequency,
stricted
frequency.
range
the
O <u
teristic
Then
B = w,/cJO,
to
WC2
= W02 — —
= (.01.2.
4the
filter
range
< 2u0,
fractional
O <B
allows
to be written
<1,
the
and
bandwidth
X =u/coO,
lumped
element
rein
the
charac-
as
1
(27)
\tI;=
1
l_x2_2
2“
[
4
1+
BX
1
The
terworth
about
TEM
filter
line
realization
of a high-pass
can be used as a band-pass
w = wO. For
the
normalized
S-plane
filter
S-plane
But-
symmetric
and
the
TEM
1+
1
tan
~:
2W0 1
as
1
I
filter,
It(’tanw=, ‘i
characteristic
high-pass
J’I;EM=
1+
‘n
Cot:
1
cot+
[
“ ‘2’)
A comparison
the
characteristic
for
fractional
(28)
2??
of the
slopes
bandwidths
two
filters
is made
at the low
in the
by evacuating
band-edge
range
O <B
frequency
<1.
Wenzel:
1964
The
slope
X=1
of
–B/2
the
TEM
is given
Des;gn
filter
of TEM Networks
response
at
band-edge
Using
an
exact
synthesis
identities
by
lem.
~l~lhM
dX
Quarter-Wave
=
Application
mz
(29)
()
X=1–B 12
2sin7r
basic
gives
each
out
The
slope
of the
lumped
element
filter
is
former
and
process
dx
leads
and
—2“(’”-3-1[1+31-:)3
[1+(1-3”T ‘
the
nals
(30)
The
to
circuit
A = 1 –B/2,
the
ratio
of slopes
is given
by
well
SlopeTEM
7r(l
.-
Slope~
— A)[l
+
–
A +
slope
ratio
apparent
at
is plotted
that
the
band-edge
slope
lumped
as the
same
limiting
TEM
filter
element
of the
(31)
One
1] sin 7rA
slope
filter
than
filter,
for
narrow
the
the
a large
number
The
above
cascade
of
transformer
equivalent
of
10 gives
termi-
dual
discussion
applies
10(a)
is
anal-
equivalent
following
procedure
of Fig.
circuit
a procedure
the
The
synthesis
the
to
turns
formed
and
ratio
n;
and
Fig.
equally
its
dual
Fig.
The
stepped
impedance
such
of
C.
For
transformation
that
these
lines
has only
n =1.
one
the
can
This
networks
with
IV+2
the transformer
capacitor
the
a manner
transformation
10(f),
elements,
the
networks,
in
preceding
the N unit
the
allows
in
the
series
case
be per-
or
the
form
of
shunt
ele-
ment,
On
band-
the
2N+
of sections.
of the
circuit,
of freedom:
realization
lumped
wide
degrees
result
equivalent
practical
have
(B~O).
than
the
of symmetric
other
hand,
1 parameters
polynomial
ture.
14
‘LOpE
general
of the
to the input
its
11 (b).
circuit
apparent
with
becoming
Both
moderate
it is
corresponding
factor
slope
for
graph,
characteristic
bandwidths
cutoff
especially
for
TEM
the
is increased.
has a sharper
and
9. From
greater
element
bandwidth
filter,
widths
slope
is always
of the
greater
in Fig.
with
Fig.
Fig.
the
transtrans-
same
a single
Application
in
in
both
is that
The
along
to an exact
to
the
the
the
n(a).
A2’]2
.
– 4A2n–l[A2
one
of
10(e).
by
is
10(f).
11 (a).
shown
pertinent
has
can be transformed
network
the
sides
of Fig.
section
transforming
Continuation
replaced
in Fig.
fi ~ter
between
element
circuit
be
capacitor
in Fig.
ogous
unit
10(b).
transformer
by
right
prob-
in Fig.
end
The
network
circuit.
can
dual
shown
last
to the
as shown
to
The
original
transformers
I X= I-B/z
Defining
the
as the
synthesis
terminals
left
10(d).
the
10(c).
accessible
c)f Kuroda’s
the
is shown
to
Fig.
from
Fig.
form
identity
of
component
former,
into
to be used
this
to
Application
insight
identity
circuit
brought
1–+
some
of
the
103
procedure.
provide
The
Lines
as Fig.
In
fact,
the
but
for
circuit
has the
10(f)
many
of Fig.
same
and
is thus
practical
10(a)
order
contains
characteristic
a redunc[ant
filter
struc-
structures,
the
N
TEM
unit
S,OPEL
/
11
element
impedance
of the
provide
10
of
values
enough
the
can be set equal
system
and
degrees
desired
to the
the
N+
of freedom
network
characteristic
1 capacitors
to allow
Also,
function.
still
realization
the
parallel-
8
. .
4
coupled
valid
6
/
.=2
/
type
/
k
/
2<
1- .
-
~
-
—
over
attempt
4-
—
.—
02
03
04
05
06
by
L-C
frequency
an exact
of network
08
and
B
10
exact
and lumped
for two- and
for
(f)
using
synthesis
V.
Equivalence
common
type.
ladder
equivalent,
spectrum.
synthesis
an L-C
Thus,
procedure
ladder
any
for
this
prototype
will
of both
filters
is shown
circuit.
in
Cohn15
of
methods
filter
of the
and
line
Fig.
obtained
by
use
preceding
of
sult
Parallel
Coupled
configuration
10(a)
along
a
design
image
sections
An
Line
used
with
its
can
many
equivalent
procedure
parameter
Filters
in
for
theory.
be used
that
example
important
of the
the
have
the
circuits
of
procedure
an
this
1)
From
lel
points
above
of Fig.
L-C
is not
ladder,
method
is
the
10(a)
same
S-pli~ne
giverl
as
proto-
in
Appen-
the
viewpoint
bar
Fig.
11 (a)],
and
quired
The
elements.
been
should
be emphasized
as a re-
discussion:
coupled
this
to obtain
but
I.
Two
AppLIcATIOIis
of Stub
TEM
is possible,
networks
dix
type
S plane,
entire
to obtain
An
07
9—Ratio
of low band-edge
slopes for the TEM
elements
band-pass
filters vs fractional
bandwidth
four-section
Butterworth
network.
A
the
no
b
01
0
A.
has
fail.
—-
0
Fig.
filter
are
shown
C’s,
of exactly
filters
[of
the
only
L’s,
Furthermore,
to
require
ideal
synthesizing
the
type
S-plane
in
elements
transformers
and
all filters
of this
only
L
one
paralFig.
or
type
one
10(a)
reunit
have
C and
IEEE TRANSACTIONS
104
ON
MICROWAVE
THEORY
AND
TECHNIQUES
January
1
I
0
I
5F--II-I5-I-I
I
=
=
(a)
(a)
3mEI:mmEm
.,=,
(b)
+__L_
z.OcN+l
(b)
Fig.
1 l—(a)
Stripline
circuit
over
3! C’N+,
(c)
~.,
,
the
width
“1<
circuit
and equivalent
circuit.
without
parallel
coupled bars.
entire
frequency
limitation
limitations
in
that
arise
An
bility
The
parallel
line
with
uniform
impedance
10(f),
realize
(e)
B.
the
In
many
Fig.
10—(a)
Stripline
circuit
and equivalent
circuit.
(b) Kuroda
identity.
(c) Application
of Kuroda
identity
to end section
of
filter.
(d) Network
obtained
by bringing
the transformer
to
accessible
terminals.
(e) Equivalent
circuit
after
r-z cycles of
transformation.
(f) Equivalent
circuit
with single transformer.
situations,
the
input
impedance
Z1. Then,
if another
entire
can
always
nate
forms
be
realized
which
do
theoretically
not
require
in
the
parallel
alter-
coupled
By
carrying
out
in Appendix
of element
microwave
shown
in
the
I for
values
a large
procedure
number
can be obtained
engineer
Figs.
synthesis
to
10 and
design
of cases,
which
filters
11 with
outlined
known
tableszs
enable
of
the
the
type
responses
n The
many
tables
now
available
do
not
contain
the
eletnent
values obtained
by the above synthesis
procedure.
This is a consequence
of the fact that these networks
do not have an S-plane,
L-C ladder equivalent.
The insertion
loss function
of these networks
is of a form different
than those with L-C equivalents
and thus leads
to different
element
values.
See Appendix
I (41) and (42) and the
following
discussion.
the
match
range.
that
2,2(S),
and
input
that
admittances
that
are
that
network
in series
is
such
impedance
situation
such
will
Assume
be provided
dual
band
source
coupling
of frequency
replacing
has
has
The
the
respectively,
is
over
is the
the
addi-
Y1 +
Y. is con-
add
to
said
where
network,
give
a
to be com-
terminated
a Butterworth
a driving
point
a complementary
in a Iossless
network
L’s
a thorough
min,4 p. 476.
ladder
realizes
be realized
1 Q resistor.
% For
band.
can
a lossless
a specified
driving
over-all
The
A Iossless
always
can
a given
practical
26
1 Q resistor
pedance
the
Fig.
values.
Z, is added
in parallel
or
the
load
a perfect
independent
Theorem:
that
of the
Impedances
which
for
more
to design
the
impedance
frequency
plementary.
be
over
outside
of admittances
stant.
Fig.
or
element,
element
that
is constant,
and
constant
bars.
2)
tion
a way
matched
21+2,
stubs;
However,
resistances
in such
properly
resistive
series
it is desirable
two
be
that
one
filter,
Filters
to couple
of frequencies
re-
realiza-
bar
open
of realizable
Complementary
physical
series
may
The
physical
values.
of
response.
is no band-
to
coupled
configuration
standpoint
due
problem
with
same
one
network
(f)
filter
the
bandwidth,
the
Equivalent
procedure.
element
of
the
be
required
appreciation
stepped
from
the
is important.
10(a);
(d)
of
There
synthesis
will
35r-FL_”kr’Nn’
alizability
range.
the
(b)
and
for
C’s
C=
discussion
of
l/L,
and
ZI(S)
impedance
ZC(S)
can
21(S)
a
im-
impedance
ladder
Z,(S)
in
transfer
with
terminated
be obtained
C’s
and
in a
by
L’s,
L = I/C.
of complementary
filters, see Guille-
1964
Wenzel:
As
of
an example,
Fig.
12(b),
12 (c).
The
Proof:
if ZI(S)
then
For
the
is dissipated
is realized
Z,(S)
network
Design
network
in the
are
of Fig.
1 Q resistor,
I 1,12
Re
TEM Nefworks
in the
is realized
definitions
of
ladder
as shown
shown
12(a),
Quarter-Wave
in
Fig.
12(a).
input
power
=
105
“f:)-+ii~tv’
so that
[z,(@)]
Lines
form
in Fig.
the
Using
I V~l’.
z,2(s).
V2
—
z,
(s]
.
11
VI
—
xl
Thus,
(a)
Re [Z,(j.o)]
=
I Z&@)
1’ =
~ +1W2W
,,_”----”+’”’r____
c1
(32)
L..,
---yr-
C2
for
a Butterworth
response.27
complementary
filters,
Re
From
it follows
[ZC(@)]
=
the
definition
of
1 –
(h)
I
z12(jO)
.c<o~~-----+~
(33)
Re [Zl(@)]
—
1 –
?]
0
that
,
—
1’
1
=1–
1 +
W2”
(c)
1
.
1
1+
Fig.
(34)
‘m
12 —(a) Network
parameters.
network.
(c) Complementary
tuting
u’=
1/0
Butterworth
ladder
ladder network.
—
() CIJ
Subst
(b)
.
gives
1
Re
[Zc(jk)]
(35)
=
1 +
Thus,
the
and
the
and
C of Z(S)
network
change
of
Z.(S)
in variable
into
(o’)z~
has
results
a C and
“
the
same
geometry
in converting
every
L
(a)
L where
c=+-
L=;.
,,4
The
been
prototype
elements
tabulated
of the
above
totypes
can
By
using
possible
theorem,
the
to
the
practical
microwave
required
for
not
mittance
lumped
worth
the
as the
complements.
The
the
TEM
prototypes
line
Section
II.
served,
unit
realization
Since
the
elements
the
parallel
III
series
is
connec-
is in most
complementary
-i
Butter-
13(a).
follows
that
for
of the
admittance
be
is
inserted
obtaining
example
to
in
be
pre-
before
the
(d)
basis
filters
are connected,
their common
input
is purely
resistive
(R) and provides a matched
load to a generator
with
internal
resistance
equal
to R.
28 p. R. &ffe
‘f Computer
prepared
tables enable design of Ultra” E~ectronic
Design,
‘1?
(c)
admittance
‘ZTNote that this filter is desigued on a transfer impedance
and assumes a current source. When the two complementary
flat networks,
1
of ad-
a third-order
procedure
only
it
filters.
connection
the
input
can
Section
impedances
in Fig.
VI,
z
pro-
microwave
for
are shown
admittance
network
in
realization,
as desirable
response
For
described
complementary
element
have
as a result
inspection.
complementary
For
filters
28 and,
complementary
by
methods
design
Butterworth
references’,
be obtained
tion
cases
of
in several
“.
vol. 8, pp. 48-51;
August31,
1960.
Fig.
(,)
13—(a)
Admittance
complements.
(b) Realization
of YI, (c)
Realization
of Y.. (d) Realization
of Y1 without
parallel
coupled
lines. (e) Realization
of Y, with a reduced
number
of parallel
coupled
bars.
IEEE TRANSACTIONS
106
load.
The
line
Y.
can
be
general,
can
equivalent
obtained
final
along
There
are
two
in Figs.
tions
with
using
but
TECHNIQUES
with
the
major
13(b)
shorted
filter
of
narrow
stop
bands
cussed.
These
in
(a)
illustrated
transformerless
equivalent
realization,
is shown
drawbacks
to the
and
(c):
1) the
ends
are
difficult
Fig.
13(b)
due
can
to the
difficulties
TEM
parallel
to
provide
only
Lmw......TT...\\\.\:\\\\\~.\\\\\~
and
0250
Tic’’’’’’’’’’’’’’’’’””
relatively
previously
be reduced
(b)
sec-
construct
restriction
can
circuits
coupled
if the
;0
‘~~
~\
\mo,No
dis-
Table
in a different
III.
Thus,
as shown
circuit
in (d) 29 and
in (e) .30 By
using
reduce
by
the
circuits
13(b)
identities
can
RASS
SIB
in Table
parallel
in
““’-X
be realized
of (c) can be realized
identities
with
using
of Fig.
that
the
filters
alternate
C.
the
form
elements
to
of
filters
discussed
of open
coupled
bars
paper
tails
by
of parallel
To
details
of
ments,
one
of this,
a band
in
Fig.
responding
that
used
but
the
rather
from
the
low-pass
S-plane
responds
to
29 The
13(d)]
reader
or
to
filters
employing
the
element
de-
by
these
other
a band
for
The
band-pass
values
reflection
3 and
the
fact
reciprocals
that
the
response.
of
A
of
1, 2,
band
S-plane
the
networks
those
of
100
stop
the
network
these
pass
filter.
in Fig.
about
The
13(b)
The
f.
and
(d)
pass microwave
filters on the insertion
MICROWAVE THEORY AND TECHNIQUES,
is seen
to
filter
corto
was
10SS basis, ” IRE TRANS. ON
vol. 8, pp. 580–593; Novem-
one
on either
details
VSWR
of
with
the
theoretical
small.
db and
The
shown
side
five-section,
II 1,
cascade
and
of
a
another
were
required.
cir-
of the
in the
than
16(a).
0, l-db
in
the
The
ripple
50 Q
15(a)
and
element
along
the
filter
in
degree
con-
to which
response
these
bands
for
response
of
multiple
by
Con-
The
purpose
15(a),
pass
brass
two
Fig.
the
improved
prototype
The
using
photograph
Fig.
center
2 Gc.
conductor.
primary
alter
be further
rather
A
planes
was
14(c).
to establish
from
loss
Fig.
shown
The
was
by
center
in
is
15(b).
ground
a 50 Q round
frequency
shown
would
Insertion
in Fig.
after
equivalent
between
with
values.
filter
effects
S-plane
the
stub
obtained
filter
is evident
conductors
and
Table
transformations
was
the
Fig.
could
as
inductor
center
are
and
As
(See
reflection
before
realizable
spacing
design
stubs,
Gc.
element
value.
Consulting
constructed
inductor
junction
series
].
25$2
this
series
50 L! gives the final
ZO =
shunt
structing
be
No
The
in
added
network.
stub.
was
is shown
each
re-
dual
a specified
was
a shunt
a 0.250-inch
struction
a
reciprocal
provide
stub,
14(b)
conductor.
a
to
the
to
[Fig.
having
This
low-pass
capacitor
cuit
cent.
provide
series
and
with
element
of
Normalization
is not
values.
1 values
is
value
was
part
capacitor
network
by
602.)
filter
series
normalized
element
element
a unit
Iossless
cor-
per
the
coefficient,
the
char-
distribution
p.
ele-
filter
are
coefficient
a bandwidth
1, 2, 1 element
usual
a high-pass
~~uivalence
the
element
shunt
a
shunt
a reciprocal
dual
Since
re-
of
each
with
a
Guillemin,4
litera-
been
replacing
the
construction
have
transformation
quires
parallel
in the
no
high-pass
discussed by Matthaei
and Schiffman
at the 1963 PTGMTT
National
Symposium
in Santa Monica,
Calif. Their paper, to be published
in
the IEEE
TRANSACTIONS under the title “Exact
Design of Band-Stop
Microwave
Filters, ” provides
information
relative
to constructing
these networks.
JOThis
network
can be constructed
entirely
without
parallel
coupled bars by using series stubs. See Section V-C.
31 G. L, Matthaei
and B. M. Schiff man, “Exact
Design of BandStop Microwave
Filters, ” presented
at 1963 PTGMTT
National
Symposium,
Santa Monical
Calif., May 20-22.
32 G, L.
Matthaei,
~~De51gn
of wide-band
(and narrow-band)
band.
ber, 1960.
f)
all
con-
Construction
stubs
jo,
in
is referred
series
about
prototype
14(a).
of order
Note
arises
pass
tested.
stop-band
are available
to a Butterworth
acteristic
the
knowledge,
two
and
be described.
S-plane
shown
filters
employing
view
will
fo,
,,
14—(a)
S-plane band-pass
prototype.
(b) Prototype
after addition of two unit elements
and renormalization
to Z,= 50 Q. (c)
Construction
details of three-section
Butterworth
filter.
network
obtained
Schiffman.31
author’s
filters
In
The
coupled
the
built
[Fig.
], the
of
was
of
stubs
13(b)
and
the
ported.
about
details
Matthaei
ture.82
been
theory
shunt
[Fig.
most
have
with
construction
figurations
E .
(c)
employing
agreement
For
!MPEOAN.
BLOCK
configurations
cases,
COAXIAL
SHOR11t4G
Fig.
Excellent
f
1.!+,.
an
Results
number
ROD
T
TEFLON
FILLED
form.
Experimental
A
/~,~
7
as shown
II 1, it is possible
coupled
PM.,,
(A LUM!...)
—50
are realized
to
January
‘m’
of
will,
transformers
a technique
a TEM
AND
a TEM
realization
These
THEORY
13(c).
shown
2)
Y.,
The
A
manner,
transformers.
II.
with
13(b).
a similar
by
for
Fig.
in
ideal
along
in Fig.
be eliminated
Appendix
circuit
circuit
is shown
involve
often
in
in
final
realization
ON MICROWAVE
was
using
at
effects
6
were
less than
silver
0.4
plated
aluminum.
the
stop-band
values
Chebyshev
are
filter
those
reflection
for
coeffi-
is
a
1964
Wenzef:
of TEM Networks
Design
Using
Quarter-Wave
L“onv”-1
t
LESS THAN
107
Lines
~LE$$
TMAN
(0,4 DB _
LESS THAN
FREQUENCY.
3L!L-SL
0,234
5,,
GC
(a)
*GC
(a)
(b)
Fig.
17—(a) Insertion
loss and VSWR
band-stop
filter with series stubs.
cient
characteristic.
per cent
band
frequencycent,
of
one,
the
stop
O.1-db
K,
bandwidth
III-C.
band
designed
to have
is
where
K
100
required
per
is greater
in accordance
The
a 30
band-edge
bandwidth”
a constant
in Section
30 per cent
was
Chebyshev
filter.
fo. For a normalized
the
!2 by
reduces
discussion
filter
about
Q =1,
Dividing
than
The
stop
of five-section
(b) Band-stop
with
constant
the
for
a
is
mO.85j0
K=tan
4,16.
—=
2ja
(b)
Fig.
15—(a)
Application
Insertion
loss and VSWR
of third-order
Butterworth
filter, (b) three-section
Butterworth
filter.
to
of Kuroda’s
ZO = 50 Q gives
identities
the
final
and
renormalization
equivalent
circuit
of
Fig.
double
coaxial
16(b).
The
filter
structure.
1.ld7
1,975
The
1.147
response
17(a)
‘m’
23., (Z
,8,, Q
and
Little
e.,n
,$,,$2
IN,,,
VSWR
are
of the
a photograph
1J.
of
series
shunt
was
the
a
shown
filter
filter
encountered
series
stubs.
stubs
were
found
stubs,
especially
TEM
~
—/~”
with
In
fact,
to
when
VI.
//
,,.,10.,
ASE
sCREWED TOGETHER
difficulty
filters
COAXIAL
. . .. \.
in Fig.
16(c).
is shown
in
Fig.
is ~shown
in
Fig.
in
constructing
junction
be
less
low
effects
significant
impedance
the
for
the
than
values
for
were
\
,“..,.
AREAS ,,
TEFLON
PREsENT
S,,,”,,
networks
(c)
16—(a)
5-section
0, l-db Chebyshev
filter prototype.
(b) Filter
circuit
after application
of Kuroda’s
Identities
and renormalization to 50 Q. (c) Construction
of stop-band
filter
with
series
shorted
stubs. The impedance
of the unit element
is realized
by
using the correct
dl/dn ratio and that of the inductors by using the
correct inner coaxial impedance.
CONCLUSIONS
designed
by
techniques
have
been
shown
advantages
over
those
based
1)
The
over
Fig.
and
using
details
required.
(b)
I
constructed
17(b).
(a)
8.90
was
Construction
2)
3)
equivalent
the
circuits
entire
Different
physical
can often
tainable
are
with
network
exhibit
the
on approximate
used
frequency
responses
Limitations
modern
to
theory
following
methods:
are theoretically
valid
range.
structures
that
realize
identical
be obtained.
readily
a given
found
physical
on the
response
ob-
configuration.
w A Chebyschev
filter characteristics
as a 3-db point whose bcations depend on the number of elements.
For Q =1, the characteristic
differs from the ideal by a value equal to the specified
ripple
and
thus serves as a convenient
bandwidth
reference point.
IEEE TRANSACTIONS
108
4) The
networks,
number
5) It
is
possible
tainable
in
in
Types
minimum
The
impedance
factor
filter
in
the
values
has
response
ob-
configurations.
been
physical
frequency
to
or
be better
element
filters.
band-pass
filter
in
the
high-pass
the
of the
of
filters
was
plane,
filter.
In
microwave
used
Complementary
in
microwave
providing
a wide
an
frequency
the
elements
obtained.
formers
in
was
shown
than
the
this
to
cor-
comparison,
filter
were
parallel
simple
Following
shown
will
for
with
not
basic
network
plane
The
line
enable
filters
these
circuit
elements
configurations
the
be
PN+I
– S’p’
be
of this
= Q2(1 +
type
Q2)N
(36)
Pjv+, (w)
inductors
and
ideal
trans-
and
shunt
used
to
scribed.
of
III
is
microwave
the
coupled
basic
not
restricted
basic
following
example
procedure
applicable
the
and
components
is not
with
the
discussion
values
A
convenient
=1.
The
first
tain
a suitable
Letting
step
choice
in
form
Q = tan,
the
can
be
in Section
can
is to
set
synthesis
demonstrates
Z1 = Zz =
. . . ZN
is to
ob-
in
sin2 e
(37)
P~+l(tan2
=
0)
is a polynomial
ferent
Now,
a
Q~+,(cos20)
of
coefficients
letting
degree
than
x = cos 8, (37)
N+
those
1 having
dif-
of PN+I.
becomes
X2—1
Iflz=
Q~+,(z2)
(.x2 –– 1) –
=
GN+l(x2)
Sand
where
a neces-
QN+I(X2) = GN+I(X’) +
(1 – X2)
or
described
structures
other
1
1,12=
de-
1 –
com-
and
de-
an
exact
syn-
to
(38)
F,v+,(x’)
where
The
next
step
proximation
coupled
bars
with
open
coupled
bars
with
shorted
3)
stepped
ends,
4)
stepped
lished
ends,
sections
with
one
series
quarter-wave
sections
with
one
shunt
stub,
spaced
series
open
defines
the
This
reader
synthesis
that
the
is,
is referred
GN+1(X2)
~, _ ~ “
procedure
is, a criterion
manner
in
which
to approximate
in
general,
to
is the
must
the
the
literature
function
desired
a difficult
z,A,s for
ap-
be estabI t 12
problem
detailed
discussions.
A
quarter-wave
the
is to be chosen
characteristic.
and
in
problem;
that
F~+l(x2)
quarter-wave
with
N
as-
O gives
I
parallel
line
the
I t[2.
for
FAT+,(X’) =
parallel
stub,
V-A,
arbitrarily
procedure
tanz 0(1 + tanz O)N
ItI’=
Q~+l
are
analyzed
1)
uniform
be
W.
methods.
2)
shorted
1 in
where
elements.
many
N +
physical
theory
to
transformers
use of these
paper
a different
network
APPENDIX
open
elements
of degree
impedance
obtain
transformers
bar
allows
the
Couplers,
by
is a polynomial
element
synthesizing
this
ideal
involving
Application
parallel
for
in
capacitors,
networks
for
Section
of unit
In accordance
are
be designed
required
it
realization
= number
unit
procedure
practical
to
N
be
procedure
discussed
however,
5)
a network
1–X2
The
The
6)
to
load
synthesis
can
element;
thesis
to
to
in obtaining
filters
elements.
mon
shown
source
capacitors
exact
unit
signed
of
eliminating
series
sary
in
for
I’lf+,(-s’)
manner.
The
4) and
18(a).
function
–s’(1
signed.
con-
complements
to be useful
coupled
that
band.
Techniques
are
are
match
Butterworth
configurations.
tables
filters
exact
of
networks
inductors
for
Fig.
2),
prototype
It)’=
as a
sidered.
circuit
in
and
basic
lumped
filter,
band-edge
element
responses
realizations
of corresponding
frequency
at
lumped
multiple
easily
is shown
transmission
The
as improve-
Butterworth
that
S-plane
cutoff
responding
given,
equivalent
duals.
where
to allow
as well
of TEM
low-pass
than
An
steeper
over
5) are
Equiva-
demonstrated
circuitry
response
high-pass
useful
3) and
is”
realizable
have
.lanuary
TECHNIQUES
respective
considered
realize
reflection
response.
TEM
in
that
impedance,
1),
their
AND
response.
S-plane
have
of
elements
real
shown
a
networks
input
be a limiting
several
simplification
The
construct
desired
range
to
network
ment
employ
THEORY
are
or phase
small
shown
lent
to
a
coefficient
The
general,
MICROWAVE
of elements.
exactly
been
in
ON
common
sponse
approximation
which
is obtained
is the
by setting
maximally
all
but
the
flat
(N+
reI)st
stubs,
6)
uniform
line
shorted
stubs.
with
quarter-wave
spaced
shunt
34 This
network.
can
be
obtained
by
multiplying
the A B CD matrices
for the
7964
Wenzel:
Design
of 7EM
Networks
2a2mim>Nm1
Udng
Quarter-Wave
The
synthesis
which
of the
theorem,7
(a)
to
mlmz
‘~i
Zi.
(b)
unit
terion
of
(x’)
FN~
equal
to each
elements.3s
Z’i.(S)
Under
coefficient
common
depending
is desired.
This
procealure,
theorem
a unit
the
is used
states
element
with
upon
Richard’s
of
that
if
impedance
remaining
impedance
beirw
for 1) parallel coupled bars with shorted
Line with quarter-wave
short circuited
stubs. (b)
for third-order
Butterworth
filter.
ends, 2)
Prototype
is different
I .S‘- — O, then
can be removed
(1)
109
configurations
—nltiz
18—(a) Prototype
Fig.
procedure
possible
basically
remove
Lines
to
zero.
Applying
this
cri-
these
denominator
Z’i.(S)
gives
of
the
the
can
one
even
respectively,
of
parts
nlnz
S2,.(S)
–
Zin(! ) ‘
both
degree
parts
the
the
lower
of
than
is given
by
numerator
factor
The
and
by
mlm2
nlnz.
A
prod-
denominator,
and
one
the
product
stub
network
by
applying
=
4S2(4S2 + 12) — (S$ + 7S)2 Isql = O,
]S=l
Richard’s
and
(S’ – 1) and
Zi.(S).
numerator
is given
be obtained
m1m2 –
Zi.(S)
contain
of Zi. (S)
odd
–
conditions,
of Z’,.(S)
is thus
uct
.S’2,.(1)
= Zi*(l)
theorem
to
(40).
where
G
is
the
(N+
I)st
coefficient
of
the
A unit
polynomial
The
Giv+,(x2).
The
prototype
is shown
in
There
are
for
Fig.
a third-order
maximally
flat
filter
element
input
of impedance
impedance
Z’i*(S)
SZi.(1)
—
SZi*(S)
= Zi~(l)
18(b).
two
unit
elements;
therefore,
l–——
N=
2. Then
The
common
Y’i.(S)
= l/Z’i.(S)
moved
by
S2 – 1
has
taking
out
=;
can
remaining
be removed,
network
is
– Zi.(S)
s’ + 2s
– Zin(l)
= 4s’ T 14s + 24 ‘
factor
has
a pole
a shunt
at
been
zero
cancelled.
that
can
be
re-
inductor,
12
—
~2_l
,s
L=;;
Since
%2 .
Zi.(l)
of the
2s + S-124
+
14s
+
4s2
1
_—
24+
12s
1–s”
2s +
I t 1’= ——+-
The
remaining
after
1 – S2(1
_
S2)’2
Richard’s
and
of
theorem
impedance
or
impedance
cancellation
is Z“in
the
to Z“in(S)
consists
of
a 142 resistance.
shows
a unit
The
(S)=
common
(S+2)/(4S+2)
fact: or.
that
of
circuit
Applying
the
element
final
4S2.
remaining
impedance
is shown
~
in
Fig.
19(a).
]pl’==1–
The
stant
[j12=____–___
G – s?(l
bandwidth
G;
using
allows
(39)
of the
filter
of the
S =-i
filter
tan
to be solved
is determined
r.,/2f0
for
and
G. The
(39)
– sZ)~
by
setting
the
con-
0.5
I P 12 =
fractional
bandwidth
obtain
the
equivalent
unit
To
elements
of
impedance
pole
(shunt
remaining
This
is
2(fo
J
– f
ance
.
For
input
insures
removed
——
inductor)
and
with
of
three-stub
Yin(S)
impedance
that
the
thus
reduces
Richard’s
example,
using
network
ZO = 1, partialJy
= l/Zi.(S)
Z’i.
unit
the
so that
(.S) satisfies
element
order
with
with
remove
Z’i.
20=
of Zi,l(S)
(l)
1 can
a
the
= 1.
be
in accord-
theorem.
(4o),
fo
The
general
and
it
input
the
network
is sufficient
impedance
details
For
of this
G = 36, the
Zi.(S)
configuration
for
that
of the
synthesis
purposes
corresponds
procedure,
bandwidth
filter
to the
to
is known
obtain
given
see Guillemin,4
is 70 per
cent
.9 + 4s’ + 7s
= s, + ~Sz + 7S + 12 “
the
I p I ‘. For
p. 458.
and
a
—
s
_——
7S+4S2+S3112+7S
+4S’+S8
7a + 4aS + as’
(12 – 7a) + (7 – 4a)S + (4 – a)S2 + S3.
(40)
35 This theorem
is discussed further
by Grayze!,
“.~ synthesis
Procedure for transmission
line net works, ” IRE
1 RANS.
ON CIRCUIT
TEIEORY, Vol. CT-5, pp. 172-181;
September,
1958.
IEEE TRANSACTIONS
110
ON
MICROWAVE
‘a511ED1
THEORY
[see Fig.
the
AND
19(c)].
This
transformer
January
TECHNIQUES
gives
n’ =49
or n = 7’. Removal
of
yields
(a)
I
I
Z“in(s)
=
Ts’
1(5s
+
“
1
+
(b)
Application
,{~jml
of
network
impedance
resistor.
(c)
Fig.
19—(a) One-stub
third-order
Butterworth
filter.
(b) Three-stub
third-order.
Butterworth
filter.
(c) Third-order
Butterworth
filter
synthesized
in the equivalent
form of Fig. 11(b).
“in(s)
=
S3
The
Z’in(l)
The
=1,
unit
Z’i.(S).
element
The
remaining
Fig.
therefore,
L=l/a=
of ZO = 1 can
above
process
impedance.
(12 – 7a) “
1.
now
is now
The
the
shunt
form
of the
final
inductor,
elements
network
must
have
repeated
from
using
network
is
the
shown
in
can be obtained
Zi.(S)
and
a cascade
in Section
V-A.
with
of two
or with
can
inductor
(pole
of
Yin
or
are
have
S-plane,
an
equation
not
for
same
L-C
whereas,
tion
flat
for
the
filters
can
be shown
have
steeper
ladder
that
“
by
lim Z’i*(S)
8-0
= n’
proto-
(41)
the
basic
equa-
(42)
+
Q’)”–I
can
serve
as the
given
vicinity
of jo.
that
Whereas
filter,
“
ladder
their
prototype
(42),
entire
where
shown
possible
to
by
the
the
YC. Adding
20=
20(a).
Application
20(b).
Transforming
of the
can
second
lower
three
1 to
Fig.
of
the
to
half
unit
the
lead
to
a response
range.
Kuroda’s
capacitors
and
transformers
input
can
or
in a section
cases,
gives
as
it is
an
13(a)
exthat
of characteristic
the
circuit
identities
capacitor
such
As
in Fig.
be
output
however,
transformers.
circuit
elements
13(a)
transformer,
~ in
transformers
the
many
ideal
4/3
realize
ideal
be realized
In
response
for
may
series
These
I (g).
eliminate
consider
impedance
side
they
to
have
transposed
in Table
the
only
Thus
networks
II
will
realizations.
and
but
of the
(42).
for
frequency
with
generally
TEM
to
by
whose
pass band
approximation
networks
inductors
those
equal
be used
the
described
prototype
this
over
type
than
In the
by
it cannot
is specified
(42).
nearly
of the
ample,
I
that
an L-C
0’”
of the
becomes
follows
that
T++
be obtained
with
slopes
a characteristic
realizes
noting
filters
realized
terminals
= 7S’ + 28S + 49
=
can
basic
above,
Q’(1
band-edge
are given
(fl>>l),
consolidated
transformer
The
1
(41)
in
of the
that
equivalent.
1
=
characteristics
shunt
Then
ratio
these
networks
discussed
filters
S-plane
L=;.
turns
a “maxibut
of
filter
APPENDIX
I
by
(38).
gives
4S2+S3
7S2+16S+1
the
manner
to
to have
as those
ladder
a maximally
or
16S
using
a Cheb yshev
a similar
said
The
with
is
that
S2+
by
with
characteristic,
1 +
a suitable
The
are
Ip\’=
with
S’+4S+7
Fig.
is
an L-C
Z’in(S)
in
a line
criterion
above
the
as
bars
in
“Chebyshev”
responses
It
7S+
1-Q
equivalents.
coupled
approximation
discussed
flat”
(41)
7S+4S2+S3112+
of
a
unit
For
at zero)
in
is shown
realized
be obtained
a
s’ + 4s’ + 7s
= s, + 4sz + 7S + 12 ‘
shunt
remaining
elements
terminated
for filters
1 +
of the
the
unit
exact
be
parallel
a suitable
filters
19 are
can
1. Elements
Ip/,
removal
two
circuit
by noting
circuit
shows
of
Z = ~
Fig.
19(b)
in Table
mally
be removed
an equivalent
a transformer
as demonstrated
of
Fig.
stubs
type
network
equivalent
of
applying
19(b).
A third
that
a=l;
final
networks
network
The
For
and
characteristic
(4 – ~)sz + (7 – 4a)S +
+
theorem
a cascade
Z = 1/14
The
relations
ss.+~2+7S
of
19(C).
shunt
Then
Richard’s
consists
to the
consolidating
gives
of Fig.
Fig.
right
hand
the
trans-
1964
Wenzel:
Design
of TEM
Networks
Using
Quarter-Wave
Lines
111
+2
‘)’=T3EIED
(a)
Fig.
can
(c)
in another
ing
formers
hand
and
side
The
by
former
with
the
resultant
can
is divided
capacitor
formed
of ideal
transforming
of the
transformer
capacitor
the
20—Elimination
be
to
Kuroda’s
a 2:1
inductor
eliminated,
a series
the
third
turns
unit
gives
if
the
to the
right
Fig.
20(c).
series
combination,
element,
identity,
ratio
2/3
so that
when
yields
a
transtrans-
procedure
circuit
gives
the
OL
in Fig.
Fig.
20(e).
of terminals
inductance
au’
For
is divided
so that
function
circuit
to
back
to
the
21 (b).
Dividing
the
circuit
of
can
of Fig.
ideal
as shown
inductor
the
the
into
required
be realized
unchanged,
transformers
region
the
also
20 is left
in Fig.
of two
Fig.
21 (a).
between
inductor,
21 (c) which
elements
of the
I (g) where
n is chosen
to make
The
realization
is shown
TEM
author
Research
L
his
Thanks
in
back
cascade
and
is illustrated
is shown
right
of
turns
Transformtransformers
so that
can
be realized
form
the
shown
realization
in
Fig.
as
in Table
practical.
21 (d).
ACKNOWLEDGMENT
.“. C=2.
ing
of
The
Fig,
The
The
admittance
or
L
transformers.
transformers.
transformer
into
closest
1/4
the
ideal
transformers.
of inductors
n is introduced
the
to the
without
inductor,
way,
a pair
gives
with
results.
above
ratio
sections
circuit
combination
ratio
The
using
to the
of a shunt
turns
Fig.
(d)
a realization
a parallel
(e)
(c)
2 l-—Realization
obtain
case
if
(b)
be applied
to
(d)
(a)
20(d)
and
A similar
the
result-
procedure
of Bendix
comments.
wishes
to thank
Laboratories
assistance
are also
due
Research
for
in
the
his
M.
C. Horton
many
helpfu
preparation
of
V. R. Schoep
and
W.
Laboratories
for
their
of Bendix
1 suggestions
this
paper.
P. Harokopus
constructive
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