1956

lRE TRANSACTIONS

ON MICROWAVE

THEORY

AND

TECHNIQUES

Coupled43trip-Transmission-Line and Directional

Filters

Couplers
J. T. BOLLJAHN~

SummaryTkis paper describes the theory of operation of coupled strip line tilt ers and dhectional couplers, and presents information from which these components may be easily designed. Low-pass, band-pass, all-pass, and all-stop filter characteristics are obtained from these coupled lines either by placing open or short circuits at two of the four available terminal pairs, or by interconnecting two of the terminal pairs. Directional couplers having perfect dk-ectivity and constant input impedance at all frequencies, and for all degrees of coupliig, are obtained by placing equal resistive loads at each of the four terminal pairs.
INTRODUCTION HE exists T gators coupled coaxial been to ELECTROMAGNETIC between used construct open-wire lines.15 This to parallel advantage filters and. COUPLING transmission by a number lines of couplers and can that has

~-
Fig. 1.NTotation used in deriving the impedance of coupled transmission lines. matrix

investifrom coupled where

directional lines, coupling

transmission natural

be This

used transpaper

equally mission derives filters

well line

in

the

construction and from directional which

of shielded couplers. these directional

strip

Z.e = characteristic with Likewise, on the two the equal

impedance currents in the

of

one

wire direction. prc~duce

to

grcmnd

filters

same ~s/2

formulas and

coupled-strip-line couplers may

coupled-strip-line

be designed.
COUPLED AND TRANSMISSION-LINE ADMITTANCE IMPEDANCE MATRICES

%3(2)

= %3(Z) =

jzoez-s
sin

. ki
odd or

(2)

The Iines venient strip matrix coupled two the

general way which to

physical here lines now describe

configuration is illustrated the is by behavior means of

of of

coupled these the the

s+rin ----tcoupled

The anced

current voltages

generators on the

& and lines

i,

produce form
COS

bal-

to be considered transmission will

in Fig.

1. A conimpedance shielded the

of the

k(l kl

Z)

ZJa2(Z)

~b2(z)

jiZ.,,iZ

be derived. lines shown supported are

Consider in Fig.

sin

(3)

transmission lines planes

1, where

and
COS

transmission two ground

are and

midway by end. even voltages

between of five curwhere Zoo = characteristic with equal current 2, 3, and of the the then the impedance currents and of one 7Ja4(z) = %4(Z) =
jzooi4 ~

driven

a set The

kz
)

(4)

constant-current rent mode generators on the

generators ;1/2 structure form energize

at each the

four

sm

kl

or

unbalanced on the two

producing

conductors

of the

wire

to

ground

in opposite voltage at related

directions. each of the ter-

v.,(z)

= n,(z)

Cos

= j.zo.il

k(z z)

>
sin

(1)

kl

The minals

total (1,

4) are easily evenand current

to the

currents Asa

~ Stanford Research Institute Menlo Parkj Calif. I .\. Alford, Coupled networks in radio-frequency circuits, PROC. IRE, vol. 29, pp. 55-70; February, 1941. J. J. Karakash and D. E. Mode, .1 coupled coaxial transmission-line band-pass filter, PROC. IRE, vol 38, pp. 4852; January, 1950. Analysis of transmission line directional s W. L. Firestone, couplers, PROC. IRE, vol. 4?, pp. 1.529-1538; October, 1954. 4 B. M. OIiver, Directional electromagnetic couplers, FROC. IRE, vol. 42, pp. 1686-1692; November, 1954. s R. C. Knechtli, Furtheranalysis of transmission-line directional couplers, PROC. IRE, vol. 43, pp. 867869; JuIv, 1955.

and sume

voltages that

odd-mode in each are

excitation. case is into

positive terminal

terminal;

currents

Il=il+;z
12=ili2 13=i3i4 14 = i?, + i4.

(5)

IRE TRANSACTIONS

ON

MICROWAVE

THEORY

Afdi)

TECHNIQUES

April

FILTER

-u
8c, ; 1LOW PASS z= II [-(~0< L.12

[(zoe

+ 200)2

cosh(a

Cos 6 2 d-.

(Zoe - Zoo)zlk

+ IE)

OR c2
2ZOC ,700 + (Zee z Z*,

Cos !3
+zOO)cosd]x

- zoo)

Cos 1 = Cos 2-Z 6 #

~., z0. -1 .&+l ..

200

1---8---1

+.!J.LLAZ..

~osh(a

+ j~)=

H
0+1 F--,0 Loc z -1 do
2

Cos ti

2- BANO PASS

CCS d 1

-Cosd=

1:
06

z~

-1

=+1 -7:

F--8--I

cosh 3- BANO PASS

(a + .j,B)

Fig. CASE 7 ,2

2(a)

N
zoo /:0= -1 ,? .0

=+1

Cos e

~:ca
4- BAND PASS z= 11

y-~

I---l
-O
[(z.. -z..)
sm z=12

- (Zo= +2..)
+ ZOO) zoo z IL

msDl

B (Z ., z

I-----el
~r, =
z . . +3 . .
2

3=9

6-

ALL I------O*

PASS z ., -.0 z =

- tanz

. ,,
7ALL PASS

=/c-n

:0s $

..

tan2 d

Fig.

2(b)

/
1956 Jones and Bolljahn: Transmission Line Filters and Directional Couplers

CASE

. ,,I

cl,

,0,

l--t)-,

a=~

8-ALL

STOP

y-+---z
z=
11 +
OC

zoo
zoo

,=

ta.nf+

cosh

z 0.

9-ALL

sToP

1---81
cosh a =

z
_.S_._. z 0.

+ ,Z
-

00

zoo

IO-ALL

STOP
Fig. 2(c) .Image parameters for coupled transmission line filters.

When in terms

these of

equations the terminal il i!

are

solved

for it 12)

the

mode that

currents

where The stituting at each

O is the

electrical matrix

length may

of the also

coupled be derived

wires. by sub-

currents, = = +(11 *(II ;(14 +(14 +

is found

admittance a double-T end for the

configuration double-~ used the

of voltage configuration

generatom of constant

12) + 13) (6) having this all the infinite ineach at

current analysis matrix

generators show are that

above. elements

The of

results the

of

this

i3 = i4 = Since ternal terminal that constant-current impedance voltage terminal. VI are is the Thus = (v.,
(~b, (~b, (Va,

admittance

13).

generators used sum of in

cot Yll =
Y22 = Y33 = F44 = ./woo + y..) -y

derivation, mode voltages

cOt

Y12 = Yn =
+
+ + +

1734

Y43

j(Yoo

Y..)

v.,
Vb, Vb, v.,

+
+ + +

v.,
fib, Vb, v.,

+
+ + +

v.,)
?%l) Vb,) b,)

I.=a
l,=. lz=t 1.=2. (7) Y14 = YH = Y23 = Y13 = Y31 = ~24 = Y42 =

~2 =
~, V4 = =

j(yooy..) ~
j(Yoo + l.,)

Csc

Y32 =

Csc y

e
o (9)

When cients elements operation,

(I)(4) of of

and the

(6)

are substituted 14 are the easily matrix. matrix

in (7), identified

the

coeffias the this

11, 12, 13, and

impedance

Performing elements are cot o

COUPLED

TRANSMISSION-LINE

FIILTERS

Zll

= 222 =

There of coupled

are

ten strip

filters lines Fig

that by pairs,

can placing or by

be obtained open (c), or on

from short ends this

a pair circuits of the and postheir

233

244

j(z..

zoo)

--j-

Cot 0
212 = 221 = 234 = 243 = j(zo. 2..) 2

on various lines the previous coupled

terminal page,

connecting sections

together.

2(a), show
In

(b), single
the

and

page ten with

of the
diagrams output

213

231

224

z42

j(zoe Z.o)

Csc

sible
image coupled

transmission-line

fillters
schematic input small and open seen

together

~
Csc 8

parameters. line are filters,

the by

terminal The into image each of

Zl~

= 241 = Z23 = Z3z =

j(zo.

+ zoo) y

(8)

pairs

designated or

circles. looking

impedance,

admittance,

78
these pair. are nated terminal Open-circuited shown with with while the no pairs is

IRE TRANSACTIONS
also shown pairs in near of the the

ON
the

MICROWAVE
terminal lines by Here while For

THEORY
applying [V] [Z] and and

AND
the [~] [Y] in

TECHNIQUES
appropriate are four-element boundary column square that

April
conditions. matrices, matrices. terminal are boundthat are

terminal connection

coupled

filter pairs

schematic are desig-

are sixteen-element 7 if it is assumed boundary the other

diagram, Reference I.ow-pass, coupled an infinite ples infinite ples of of r,

short-circuited standard 2 shows and

terminal that

example, 4 are

filter

grounding

symbol. it is possible filters low-pass centered about that the is just Fig. the same, half from to obtain a pair of

3 and

connected, 12= 11. In are or

the all

conditions filters the

to Fig. band-pass,

V3 = V1 and ary conditions

all-stop The bands filters

V = O or open-circuited, Zr,

I = O at terminals
respectively. admittance, are Y calculated by the

transmission number while It the number

lines. of pass band-pass bands

filter about odd

1 has multian multi-

short-circuited the the matrix image image transfer elements Z or of

Second,
YI,

impedance, constant the admittance

or image (CZ+j@) matrix

and from

2, 3, and

4 have bandwidth while that shows

of pass

centered to note

resulting

four-element

impedance following

of 7r/2. the three

is interesting band-pass

filters filter of

is

the of the that

relations

bandwidth band-pass there ances image filters the equal pedance metrical to are of

of the filters. close these

low-pass Inspection

2 also the the the

Zrl =

relationships filters, 8 are 2 and In each =21, and image

among particular, Zr~ of equal

image product

impedof the

zllz12~ 1/2 .Z112 , 222 )

21, =

2222122 112 2222 zll~ ) ( (12) 212

impedances 1, 4, and of the

unsymmetrical of the dual 10 are image 7 is equal symalso imto Yrl = or

(z, ,2,2) /2

to ZO,ZO.. Furthermore, 6, and square 9 and of filter the

product

impedances the

filters of the

3, 5 and

ZO.ZOO. Finally, symmetrical

all-pass

Zo.zoo.
lWter ever, infinite and be it 8 does does image 10 are made not into elements circuit shunt series not have interesting at by filters much practical utility; of possessing Filters but they howan 9 can rethe of in have property all

=(y22-y2n ( 1:2)12
Y1l~

Y(ll1722)1/2
cosh (a + j~) = Y12 .

(13)

attenuation very useful

frequencies.

themselves, by adding For of the

band-pass to The their of filter arms arm (v.. is

suitable example, inductances the

PHYSICAL

LAYOUT

OF

COUPLED-STRIP-LINE

FILTERS

active equivalent of each ance

terminals.

In the

most basic the

applications filter desired the using way strip and

it is necessary shown basic in from filter performance these line output strips and

to cascade Fig. the 2 in filter.

several order Fig. might to .3 be that at

9 is a symmetrical is Y.e cot O, while

r composed suscept-

sections

inductances. of the of the

susceptance

achieve illustrates cascaded where

sections It is

techniques. of (i.e., output basic

seen occur

the

input

sections

l..) ~

cot o .

opposite and ever, at the ~ections The is either pedance to connect the

ends the

of the input (i.e.,

as in filters may of a single 4, 7, and

1, 2, 3, 5, 6, Howoccur two section 10) only

8) any when same may image of

number end,

of sections as in filters

be cascaded.

The circuit

addition of filter Each cot

of series type of the 0, while of

capacitances band-pass

converts filter. The has T network capacitance

filter

9 into

a well-known itances. of Zoo of ance

equivalent of capaca reactance has a react-

be cascaded. impedance or isolated filters the to lower strip. strips in filter an the of the than coupled the strip it line filters im-

10 is a symmetrical series the capacitances shunt

higher

characteristic

Therefore, having order For to

is necessary widths the when misthe

different reduce the

(Zo. zoo) cot

than match image

coupled at

strips, of the

2 In this
filter The Fig. filters case, series inductances a band-pass design in from the or the [v] or [1] = [Y][v] (11) = following admittance four network must be added to this in the the

loss

terminals.

example,

impedance impedance wider impedance impedance made lines is

is less than strip, the strips strip, width filters way and

characterstrip the charcon3. At the all-

istic is image

of an isolated than the of the filter The for

connecting and, than the in of when the the

to convert 2 are

it into

filter. equations manner. matrix equations presented First, of

made

coupled

image-parameter obtained impedance

is greater proper the the be

acteristic line necting

of an isolated

connecting Fig.

four-element

narrower. is illustrated of Fig. 9 and

is obtained

[2][1]

(lo)

the stop filters

bottom filters by

3 is shown 10 series may

in which to series

converted

band-pass inductors,

adding

capacitors

respectively.

1956

Jones

and

BoHjahn:

Transmission

Line

Filters

and

Directional

Couplers

.
. * -L J

!-l

. 5

BAND PASS
2 4-SECTIONS

BAND

PASS

4 SECTIONS

BAND

PASS -v+r+~qiz .

2-SECTIONS

*
(b] ODD EXCITATION

ALL 5

PASS

4SECTIONS

~
-1 J. .-L .

:s

+==Jqiz

. = (c)

EVEN

EXCITATION with even and odd excitation.

ALL

PASS

Fig.

4.Strip

line directional

coupler

2-SECTIONS

COUPLED-STRIP-LINE When the four lines terminal illustrated

DIRECTIONAL pairs of the

COUPLERS coupled strip

r-i

1-8 ALL sTop

transmission

in

Fig. the

1 are input. energy if

terminated behaves impedance and for all is coupled is fed emerges emerges

in and

the

proper

constant coupler directivity o In of

resistance, with at this all constant coupler Hence,

device

4-SECTIONS

as a directional infinite of degrees backward into one the the

frequencies,

coupling. instead terminal adjacent

forward. pair, the pair

a signal signal

coupled and no

ALL

STOP

2-SECTIONS

from from The

terminal opposite of this from will the best

signal

diagonally rigorously its behavior gives coupler is shown Z, arguments,

terlminal directional the

pair. coupler here picture from that of can matrix. certain such the a dibe impedance

characteristics

determined However,
10 ALL STOP 2SECTIONS

be derived physical

symmetry presentation rectional

because

it is believed

J t

operation. schematic at at the each diagram top of Fig. terminal. of The the directional resistance is ex-

mMl -@_.lH~9, CONVERTEO EANO PASS TO ,., CONVERTED TO BAND PASS

A coupler

simplified

4 with

terminations

coupler

Fig.

3.Physical

layout

of coupled

strip

line

filters.

6 Oliver and Knechtl~, 10C. cit., have also recently transmission-line directional couplers and shown designed to have infinite directivity and constant for all degrees of coupling.

analyzed coupled that they can be input impedance

80
cited bottom in two tion, with a voltage 2, the states

IRE TRAAfSACTiONS

ON

MICROWAVE
l,

THEORY
= J, e

AND
J.. v~ino

TECHNIQUES

April

2 IT in series same

with

terminal coupler In the in odd

1. At

the

of Fig. different out-of-phase 1 and voltages of of in is seen strips the with that

directional are the with applied even these it may

is shown excitawith =;
J

of excitation.

[~<~ 00

voltages 2, while

series

d-l

terminals in-phase the

excitation terminals. be with seen

applies Through that the 2 V from of

in series superposition directional with even the

principle

2COS+SWE+?21 18)
V30=0
V4. V40

behavior applied its It these the even behavior

coupler 1 can voltage odd odd

voltage

(19)

series

terminal and in the characteristic

be obtained excitations.

V4 = .

impedance excitation

of one

to ground

is Zoo, while in the coupler The

characteristic excitation a perfect

impedance is Zo,. match

of a strip for the

to ground directional

0+ maximum
is a from

s[>2coupled
quarter

E!] 20)
voltage,
wavelength

In order

to have
that the the

at all frequencies
Z;n be it equal is seen to

it is necessary
Z.. that Applying the input

V2,

occurs (;.e.,

when

the

long

0 = 90).

(18) as

a maximum

coupling

coefficient

principal

impedance can be written

of

the as

directional

coupler

terminated

in

Z.

may

be defined

z Oe
1 v,

z 00 =k=-

. 1

(21)

V
(14)

With
written

this
as

substitution,

the

voltage

at terminal

2 can

be

where (ZO + jZOO tan 0) V,=v (15) jk sin 0 ~1kcos9+jsin0 at terminal Vlll V4= (16) Eqs. When that (15) and (16) are substituted in (14), it is found 01iver.4 that for Zlfi = Z. when (22) and (23) to _
~lkcos O+~sin O

(22)

Zlo = zoo Zoo + jZo tan e

while and

the

voltage

4 becomes P
(23)

are his

the Fig.

same 6 or

as those to (22) will

obtained above operate while

by shows over for

Reference small

values

of k the band even

coupler 3 db bandwidths

20 = (2.20.)
This Zao)lfz, width characteristic is always strip lines wider weaker the at of the less line than than impedance than of the the the same

1/2. of width. coupler strips. the change the coupler, Therefore, is required Although in width

(17) (Zoe the to for is

a 3 to stronger

1 frequency couplings

between wider

points,

of operation

are obtained.
COUPLED TRANSMISSION LINE IMPEDANCES Zoe AND

characteristic

impedance

of a single be slightly couplings negligible. Under appearing voltages the length either voltage

ZOO
in-

connecting

to the coupled

The lated in the Cohns thick In operated down of the strips,

characteristic thin by for and at high rigorously form paper strips applications rf is minimized Approximate conductors

impedances coupled Cohn7 strip who The

Zoe lines presents reader values

and have the is also

ZOO of been information referred

finitesimally

calcuto

15 db,

of a nomogram. approximate strips printed potentials, by using have where

condition terminal

that 1 of Vs.= analysis

Z.= the of

(ZoeZo~)l/, coupler is

the

voltage

of Zo. and sheets.

Zo, for be

VI=

V. The
from line of at
by a

on dielectric these components the round been possibility

vz~ = VI, and 0, and end characteristic by be an

VA may impedance

be determined Z~~ terminated

must

straightforward

a transmission

of breakinstead ZOO for Honey. S and by

conductors for Z.. derived

impedance the of length by voltages from

(Z~~Z~~)lt2 and
a similar

formulas

V. Likewise line terminated show that

VZO= VM and analysis characteristic The results

V30 = of impedof this a

round

V4. may transmission ance Z..

determined

6 and (Z#o~)l2.

analysis

7 S. B. Cohn, Shielded MTT-3, pp. 2938; October, S R. C. Honey, Stanford tions.

coupled-strip line, TRANS. IRE, vol. 1955. Research Institute, private communica

1956 He finds
that

Maxweli

and

Leon:

Absolute

Measurement

of Receiver
where Coo = one half

Noise

Figures

at

UHF

the

capacity (p@/cm),

between

the

two

center

(24)

conductors Co. = one half the of

capacity the two

between center

the

parallel

comand

bination

conductors

(25)

ground Czz = coefficient center

(~~f/cm), self capacity (ppf/cm) of either of thr,

conductors, of induction (ppf/cm) (a negative measured when between cluantity). as the the capacity strip lhalf the the center conductors CZZ is easily strip the and

where diameter, dielectric ~ For

s = center constant

to center plane of

spacing

of the surrounding lines possible

wires, the that to

d = wire wires. not

b = ground
configurations to analysis ZO. from Thus it

C~3 = coefficient two The between

spacing, coupled

and

e,= relative are

medium of

capacity either

amenable 2., the and

is always

determine capacity of

ground

other one of

measurements

of the

static

is grounded, the two capacity center

while

capacity the and

Cz~ -r Czs is just combination

strips.

between conductors

parallel grouncl.

lood; zoo = 3C00

loo& (26) = 3 [c,? C,3]

33(038)-7850 Research and D.4

sponsored Center, 36-039 and

Ioo{z z., = 3coe

Ioovc
(27)

DA by

-sc-64t525

sponsc)red

= 3 [C22

C23]

Engineering

!Laboratories.

Measurement
E.

of Receiver
MAXWELL~ AND B. J.

Noise
LEON?

Figures

at U13F*

Srmrmary-Absolute measurements of noise-figures in the UHF range are described, using hot and cold thermal sources as standards. It was found that the noise temperature of the T-5 6 watt fluorescent tube is 16.1 kO.6 db above k TAv. Noise diodes were found to be in error at these frequencies by approximately 1 db. lNTRODUCTION HE fined T Noise power X output noise of receiver power the from from source power of noise techand be noise as figure of a receiver is conventionally de-

determined.1 calibrated usually liable

The noise

second generator,

technique, is simpler precision to

which apply

uses and

a is

capable standard which are

of greater of noise the

provided Two and types and the at

that

a re-

is available. source

of noise selfdiode. tempower in

generators calibrating, The k TAv principle thermal to perature,

are primary thermal is simply is capable

standards, a resistor of delivering Although

hence noise some noise simple

source T, which

some

external

device. restricted output. diode. a shot It The

it is obviously low noise

to laboratory diode noise

use and source that is the

F=
Receiver It is measured by of the generator, known that the gain available by with but

has relatively a temperature diode current square means

limited contains value is

can noise

be Shownz component

whose

either receiver or

comparing with an a signal external signal The

noise

produced signal power nique requires

a calibrated iz =
in and the frequency interval 2eIAv Av. e is the electronic charge

source generator of the

intensity. (noise

is straightforward

difficult

to do precisely receiver

I the dc diode R the

current. combination At the

If the

diode

is shunted generator high may enough, not

by with

a so

bandwidth

resistance available that the

is a noise frequencies electrons

2eIRAv.
time of

* The research in this document was supported jointly by the .Army, Navy, and Ak Force under contract with the M.I.T. This paper was presented at the URSI-IRE symposium held in Washington, D. C.; May 2-5; 1956. t Massachusetts Institute of Technology, Cambridge, Mass.

be neg

~ G. E. Valley McGraw-Hill Book a Ibid., P. 701.

and H. Wallman, Vacuum Tube Co., New York, p. 695; 1948.

Amplifiers,