CHAPTER
7
Directional Couplers
Stephen Jon Blank and Charles Buntschuh
I. Definitions and Basic Properties
A
directional coupler as treated here is a passive, reciprocal four-port
coupler in which power incident on one port, the input, is split between
two other ports, the coupled and through-ports, and little or no power
emerges from the fourth, isolated, port. With the ports numbered as in
Figure 1, with the scattering matrix [1, 2] configured as
S --
Sll
821
831
841
821
822
832
842
and with P1 as the input power and
the coupling is defined by
831
832
333
343
841
842
843 ,
844
(1)
Pi as the power out of the i th port,
P3
C=
- 1 0 1 O g p 1 = -201oglS311
I=
-101og~
(dB),
(2)
(dB),
(3)
the isolation by
Handbook of Microwave Technology, Volume I
P.
= -201oglS411
199
Copyright © 1995 by Academic Press, Inc.
All rights of reproduction in any form reserved.
200
Blank and Buntschuh
Pl Input
P2 Through
-
Pl Input
-"
P4 Isolated
P2 Through
"
P3 Coupled
X
"
P4 Isolated
P3 Coupled
Figure I. Two common symbols for directional couplers and power flow conventions. Reprinted with
permission from David M. Pozar, Microwave Engineering, © 1990 Addison-Wesley Publishing.
and the direct transmission by
P2
T = - 10log P1
20 log IS21l (dB).
(4)
The directivity is the power out the isolated port relative to the coupled
power and is defined by
P4
D = I - C = - 10 log P3
15411
20 log
iS311
(riB).
(5)
In an ideal directional coupler, no power is delivered to port 4 and
D = I = oo.
This port numbering is commonly used for waveguide couplers. Other
numbering systems are also used. Frequently, on coupled-line couplers,
the coupled, isolated, and through-arms are numbered 2, 3, and 4, respectively.
By proper choice of the phase references, we have S21--543 ~--a,
5 3 1 - - ~ e jO, and $42 = / 3 e j6, where a and /3 are real and 0 and ~b are
phase constants. An ideal lossless directional coupler is perfectly matched
and has infinite directivity: Sii = 0, i = 1, 2, 3, 4, $41 = $32 = 0, and a 2 +
/3 2 = 1. Two common choices for 0 and ~b are (1) symmetric coupler,
0 = ~b = rr/2, and (2) antisymmetric coupler, 0 = 0, ~b = 7r.
The scattering matrix of an ideal, symmetric 3-dB coupler, called a
quadrature hybrid, is, at its center frequency,
0
S=
-j
- 1
0
1
-j
0
0
- 1
-~
-1
0
0
-j
0
-1
-j
(6)
0
whereas that of an ideal antisymmetric 3-dB coupler, called a magic-T or
201
7. Directional Couplers
f
//
/
I
I
,-"/(~i
p/ll
I
J
Figure 2. Waveguide magic T. Adapted with permission from Peter A, Rizzi, Microwave Engineering:
Passive Circuits, © 1987 Prentice-Hall, Inc.
rat-race hybrid, is
1
s =
0
-j
_j
0
-j
0
-j
0
o
o
j
-j
0
j
-j
•
(7)
0
A waveguide magic T is a classic example of an antisymmetric 3-dB
directional coupler. From Figure 2, it is seen to be a combination of Eand H-plane Ts. If a TE10-mode wave is incident at port 1, there are waves
of equal magnitude and waves phase coupled to ports 2 and 3, and no
power is coupled to port 4; incident at port 4, the waves are coupled to
ports 2 and 3 with equal magnitude but of opposite phase, and no power is
coupled to port-1. Magic Ts are available with an isolation of 30 dB or
greater and a coupling balance of 0.1 dB or less over the waveguide
bandwidth. In this chapter we divide directional couplers into three
categories: (1) waveguide aperture, (2) coupled-line, and (3) branch-line
couplers.
2. Waveguide Aperture Couplers
This class of couplers depends on the electromagnetic properties of one or
more apertures cut into the common wall between two waveguides.
202
Blank and Buntschuh
Among this class are the Bethe-hole [3, 4], the multihole [5-7], the Riblet
short-slot [8, 9], the Schwinger reversed-phase [10], and the Moreno crossguide [10] couplers.
Bethe-Hole Coupler
A single small hole in the common broadwall between two rectangular
waveguides, a Bethe-hole, can provide directional coupling [3]. The two
guides of a Bethe-hole may be either parallel or skewed (Figure 3).
For the parallel-guide TE10-mode case, with a circular hole of radius
r 0 and an offset s of the hole from the guide sidewall, the scattering
amplitudes IS311 and 1S411 are given by
IS311 = F( f )r 3
(8a)
IS411 - - n ( f ) r 3,
(8b)
where F ( f ) and B(f), the forward and backward wave intrinsic amplitudes, are functions of frequency, but are assumed to be independent of
the hole radius, and are given by
F(f)
=
2 10120
ab
t
- ~ sin2 a
3Z20 sin2 a + f12a2 cos2
__a
(9a)
B(f)
=
27rfZ10 [ 2E 0
rrs
ab
- ~ sin2 a
+
4~o [
3Z2o ~sin2
7rs
a
"17.2
f12a2
COS2
'?'/"S
a
)
(9b)
®
®
®
Figure 3. Bethe-hole directional coupler. (a) Parallel guides. (b) Skewed guides. Reprinted with
permission from David M. Pozar, Microwave Engineering, © 1990 Addison-Wesley Publishing.
7. Directional Couplers
20~
where
/x0
,c-2a,
/
c
-
~
Zlo
=
EO
,
and
,8=
c
1-
Solving Equation (9a) for s, for F(f) = 0, yields the hole offset for
backward coupling to port 4 and isolation to port 3:
a
s
=
sin - 1
--
(10)
"Jr
_1
f
2
It should be noted that real values of s can be obtained only for a
restricted range of frequencies, f/fc = 1 to V~-, as shown in Figure 4.
°5i . . . . . . . . . . . . . . . . . . . . . . . . .i .. .. .......i.............i........................i.............. .. .. .. .. ..!............i... . . . . . . . .
0.4
0.35
0.3
0.2
0.15 .....
0.1
...........................................................................................................................................................................................................................
0.05 - ....
0 i
1
..........................................................................................................................................................................................................................
i
J
t
J
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Normalized Frequency - f/fc
Figure 4. Hole offset s / a vs normalized frequency f / f c for ideally directive Bethe-hole couplers.
204
Blank and Buntschuh
The offset s determines the isolation to port 3. The hole radius r 0
determines the coupling to port 4 and can be found for a given frequency,
offset, and coupling from
1/3
10-(c/2o)
t- 0
--
(11)
IB(f)l
For example, Figure 5 shows r 0 as a function of C for values from 10 to
30 dB for WR-90 waveguide, with a = 2.286 cm, b - 1.016 cm, f =
8.75 GHz, and s = 0.909 cm. The coupling and directivity as a function of
frequency of a Bethe-hole coupler in the same guide with r 0 = 0.431 cm
are shown in Figure 6.
It is possible to find values for s to give forward coupling to port 3 by
solving Equation (9b) for I B ( f ) l - 0. Values of s/a as a function of the
0.65 . . . . . . . . . . . . .
o.e L
i
.
.
0.55
o
.
.
.
....................
0 . 5 ~. . . . . . . . . . .
i
.................
0.45
;
0.4-
.
...........
.
.
.
.
.
.
.
.
............
.
.
.
.
0.35
0.3 ~ . . . . . . . . . .
0.25 L
,
10
.....................
r
15
-
20
Coupling
................. j
25
30
- dB
s - 0 . 9 0 9 , f - 8 . 7 5 GHz
Figure 5. Hole radius vs coupling for Bethe-hole couplers with backward wave coupling.
7. Directional Couplers
205
•
-5
-10
15
,',',
"10
I
20
•~
25
~
3o
"OI~t~t,.-35
~- ....................... " . . . . . . . . .. . . . . . . .....
.c::
0o
-40
. . . . . . . . . . . . . . . . . . . . . .
-45
-50
.....
-55
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-60
L
7
................. i. . . . . . . . . . . . . . . . . . . .
8
:
i............................. , ................................ i
9
10
11
Frequency - GHz
8-0.909
cm, ro,,0.431 cm
Figure 6. Coupling and directivity vs frequency for a single-hole Bethe-hole coupler.
normalized frequency for forward coupling are shown in Figure 4. This
case is generally of less practical interest as it requires coupling holes
relatively close to the guide sidewall.
The hole may be on the centerline of the skewed Bethe-hole coupler,
s = a/2,
and the angle 0 may be adjusted for isolation at port 3. The
skewed geometry, however, is often a fabrication and packaging disadvantage [3].
Multihole
Waveguide
Couplers
The narrow-band directivity performance of the single-hole coupler is
evident from Figure 6. High directivity over a much greater bandwidth can
be obtained using an array of coupling holes offset s from the centerline
and spaced one-quarter wavelength apart at the center frequency f0
Blank and Buntschuh
206
®
Figure 7. Multihole broadwall waveguide coupler.
(Figure 7). With this arrangement, the forward-coupled-wave contributions from each hole are added in phase at port 3, whereas the backward
wave contributions cancel at port 4. The cancellations occur essentially
pairwise from the holes, so it shall be assumed that the total number of
holes, N + 1, is even and that the distribution of hole radii is symmetric,
i.e., r, = rN_ n. The coupling and directivity responses are given by
N
C(f) = -20log F(f)
Ern3
(12)
n=0
N
D ( f ) = - 2 0 l o g B( f ) E r3ne-jn~d~f) - C( f ),
(13)
n=0
where F ( f ) and B ( f ) are given by Equation (9), G is the hole diameters,
and d ( f ) is the ratio of guide wavelengths at f0 and f:
2 _f2
d(f) =
~f
f z _ f2 •
(14)
The design problem is to find the values of rn which achieve a specified
minimum directivity, Dm, over a specified bandwidth (f~, f2). It is usual to
synthesize the directivity response with either a binomial or a Chebychev
design.
207
7. Directional Couplers
Binomial Design
For a specified coupling, minimum directivity, and bandwidth, N is
obtained from
Om
N = - 20 log Icos 011 '
(15)
where
01
= ~d(fl)
= ~-
(fa -fg
fz_f2
7T
0 2 - - 77" - - 01 "-- - ~ - - d ( f 2 ) .
Alternatively, given N, Equation (15) can be used to find fl and f2. The
values of the hole radii rn, n = O, 1 . . . ( N - 1)/2, are given by
1/3
lO-C/2°Cn
rn "--
N
N
,
(16)
IF(fo)lEfn
0
where
Cn
N~
(N-
n)!n!
For example, with C = 20 dB, D m = 40 dB, f0 = 8.75 GHz, f l =
8.24 GHz, and f2 = 9.3 GHz in a WR-90 waveguide with s = a / 4 , we
obtain N = 3, r 0 = 3.16 mm, and r 1 4.55 mm. Inserting these binomial
values for r n into Equations (12) and (13), the coupling, C b ( f ) , and
directivity, D b ( f ) , responses are obtained, as shown in Figure 8. The
directivity, D b ( f ) , deviates from an ideal, maximally flat binomial response due to the effects of the F ( f ) and B ( f ) terms. Examination of the
D b ( f ) response shows that the lower frequency for D m = 40 dB is 8.0
GHz, not 8.24 G H z as calculated for an ideal binomial response. Furthermore, if D m is reduced to 38.5 dB, a bandwidth from f l = 7.5 G H z to
f2 = 9.35 GHz is obtained. The corresponding ideal binomial response,
with F ( f ) , B ( f ) = 1, is shown in Figure 9.
=
Blank and Buntschuh
208
-lUJ"___ ................................................... i ............................................................................................................................
\
•,=,
-2o
~
/
Coupling
__--~
._>
- 3 0 - ........................
"¢....
0
Directivity
.c:
o
L)
Binomial
-40
~ ....................
.......................................7L
Chebychev
-50
-
......
- 6 0 ...........................................................................................................................................................................................................................
7
8
9
10
Frequency - GHz
WR-90
Guide, f 0 - 8 . 7 5 GHz
Figure 8. Performance of four-hole binomial and Chebychev couplers.
Chebychev Design
A
Chebychev
n = 0, 1 . . .
response
(N -
1)/2,
can be
from
N = cosh- 1
obtained
by c a l c u l a t i n g
N
and
[ 10o.o 1
c o s h - 1( s e c
rn,
(17)
01)
and
k
rn ~
w h e r e 01 "-
2",r
( Jo
-~
"rrd(f l)/2,
COS[(N
-- 2 n ) [ ~ ] T N ( S e c
02 --- "IT" -- 01,
10-(C+D,,)/20
k=
[F(fo) I
01 COS 0 ) dO
,
(18)
7. D i r e c t i o n a l
209
Couplers
'
i
-10
2O
Binomial
m
"
........ . . . .
....
"O
I
->'
._>
-30
-
Chebychev
a
L,.
-40
- 50
-60
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
7
~
........ t................................................i
J
8
9
10
11
Frequency - GHz
WR-90
F i g u r e 9.
Guide,
f0-8.75
GHz
Ideal d i r e c t i v i t y vs f r e q u e n c y
f o r ideal f o u r - h o l e
b i n o m i a l and C h e b y c h e v
couplers.
and
TN = c o s ( N cos- 1( sec 01 COS O) )
= N t h order Chebychev polynomial of argument sec 0 a cos 0.
An ideal Chebychev response gives the optimum compromise between
directivity and bandwidth. That is, for a specified directivity, it gives the
maximum bandwidth; alternatively, for a specified bandwidth, it gives the
maximum directivity. However, due to the F ( f ) , B ( f ) terms, the actual
response is only approximately Chebychev. For example, with C =
20 dB, O m -- 40 dB, and f0 = 8.75 GHz in a WR-90 guide with s = a / 4
and N = 3, values of r 0 = 3.25 m m and r I = 4.51 m m are obtained from
Equation 18. For an ideal Chebychev response, bandwidth values of
f l = 8.0 G H z and f2 = 9.6 GHz are obtained.
Inserting these Chebychev values for rn into Equations (12) and (13),
the coupling C t ( f ) and directivity D t ( f ) responses, including the F ( f )
210
Blank and Buntschuh
and B(f) terms, are obtained as shown in Figure 8. The calculated
directivity response, D t ( f ) , provides a somewhat greater bandwidth than
the corresponding Chebychev response shown in Figure 9. For this example the lower frequency of a 40-dB bandwidth, calculated from Dt(f), is
7.5 GHz, versus 8.0 GHz with an ideal Chebychev response. The coupling
response of the binomial and Chebychev designs are nearly identical and
are close to the ideal responses shown.
Optimum Design
The Bethe theory, upon which the above analysis and design techniques are based, is only approximate in that it assumes that the forward
and backward wave responses, F(f) and B(f), are independent of hole
size. It also assumes an infinitely thin common wall. Cohn [4] extended this
theory to include the important effects of finite hole size and finite wall
thickness. Levy [5-7] further refined and modified the theory to a point at
which there is now available a rigorously accurate method of analysis.
Levy's theory deals with the practically significant case of a double row of
coupling apertures offset from the centerline and accounts for the effects
of mutual coupling between the apertures in such a configuration.
These highly sophisticated analysis methods notwithstanding, there
still remains the question of optimizing the design of a multihole coupler,
i.e., of finding the design that optimizes the performance of a coupler
according to some defined criteria. This question has been approached via
numerical search methods. In this approach, the design variables, which
are the hole radii, are represented by a vector p = (r 0, r l , . . . , rn). The
desired directivity response is specified by the function Odes(f) , and the
actual directivity response by D(f, p). The response D ( f , p) can be made
to account rigorously for the very complicated effects of mutual coupling,
intrinsic coupling, hole size, wall thickness, finite conductivity, and dielectric loss.
An error function, e(p), is defined as the normed difference between
the actual and the desired responses, i.e.,
e( p) - Il D ( f
(11 II) represents
, p)
-
Odes(f) 11.
The norm symbol
some numerical measure of a varying
function. A typical choice of norm is the maximum norm, giving
e(p) =
max ] D ( f , p ) - D0es(f)].
(20)
fl<f<f2
An optimum design, p*, is achieved when e(p*) is less than some
specified value. The mathematics literature [11] contains many ingenious
methods that can be used to search in the space of vectors p to find the
21 I
7. Directional Couplers
®
Figure 10. Riblet short-slot coupler. Reproduced with permission from David M. Pozar, Microwave
Engineering, © 1990 Addison-Wesley Publishing.
optimum, p*. The mathematical statement of the problem is
min e ( p ) ~ p*.
p
This approach has been applied to the optimum design of multihole
couplers and other microwave circuits and components and has achieved
significant improvements to designs based on closed form, analytic methods [11, 12].
Riblet Short-Slot Coupler
The Riblet short-slot coupler [8] (Figure 10) consists of two waveguides
with a common sidewall. Continuous coupling takes place in the region in
which part of the common sidewall has been removed. A comprehensive
theory for such continuous coupling has been developed by Miller [9]. For
the case of the configuration in Figure 10, both the even TEl0 and the odd
TE20 modes are excited and can be utilized to cause cancellation at the
isolated port and addition at the coupled port. The overall width of
the interaction region is made less than 2a to prevent propagation of the
undesired TE30 mode. The Riblet short-slot coupler is commonly designed
to provide 3-dB coupling.
Schwinger Reversed-Phase Coupler
The Schwinger reversed-phase coupler [10], as depicted in Figure 11,
consists of two thin slots spaced A/4 apart at the center frequency. This
212
Blank and Buntschuh
®
$
...- T -d
®
®
$
Figure II. Schwinger reversed-phase coupler. Reproduced with permission from David M. Pozar,
Microwave Engineering, © 1990 Addison-Wesley Publishing.
results in coupling in the backward wave direction and isolation in the
forward direction. Directivity is practically independent of frequency,
whereas coupling is very frequency sensitive, the opposite of the multihole
coupler discussed under multihole waveguide couplers.
Moreno Crossed-Guide Coupler
The Moreno crossed-guide coupler [10] consists of two waveguides at right
angles, with coupling provided by two apertures in the common broad wall
of the guides (Figure 12). The two apertures are on opposite sides of the
waveguide centerline, placed so that they are at diagonally opposite
corners of a square of side l, with l = A/4 at the design center frequency.
This results in coupling to the port to the left of the input port and
isolation of the port to the right. Both the coupling and the directivity are
®
Figure 12. Moreno cross-guide coupler. Reproduced with permission from David M. Pozar, Microwave
Engineering, © 1990 Addison-Wesley Publishing.
213
7. Directional Couplers
frequency dependent. The apertures are usually crossed slots in order to
provide tight coupling.
3. Coupled-Line Couplers
Two parallel transmission lines in close proximity, sharing a common
ground plane, have directional coupling properties. The simplest, basic
coupler consists of two identical straight lines of common length, uniform
cross-section, and homogeneous dielectric, with each of the four ports
terminating in a resistance, Z 0, as illustrated in Figure 13. (Note the
change in port numbering from that used for waveguides, above.) Two
orthogonal TEM modes may propagate: the even mode, excited by equal
voltage drives on the two lines, E A = E B , and the odd mode, excited by
opposite voltage drives, E A = - E B. The coupling properties, with drive at
a single port, are obtained by superposition of the even and odd modes.
The coupler is completely described by three parameters:
ga ]'
--- even-mode characteristic impedance
Z°e "- -~A ,EA = E B
VA ]
= odd-mode characteristic impedance
Zo° = -~A ,EA=--EB
l
0 = 2 r r f ~ = electrical length of coupler,
/]p
Figure 13. Coupled-line coupler.
i/14
Blankand Buntschuh
where f is the frequency, Up = c / ~ r is the wave phase velocity, and e r is
the medium relative dielectric constant. The impedance level of the
coupler is
(21)
Z k = ~/ZoeZoo.
When Z k = Z o the coupler is perfectly matched and has infinite directivity. The scattering matrix of the ideal matched, lossless coupler is, using
the port numbering of Figure 13,
0
S ~-~
821
0
841
821
0
841
0
0
841
0
821
841
0
821
0
(22)
where
821
=
j k sin
0
¢1 -- k 2 cos 0 + j sin 0
V/1 -
(23)
= direct wave voltage,
(24)
k 2
-
841 -
= coupled wave voltage
v/1 - k 2 cos 0 + j sin 0
and
k -
Z0e - Z0o
= voltage coupling coefficient at midband,
0 = ~-/2.
Zoe + Zoo
(25)
Note that 8 2 1 / 8 4 1 ~-- j k sin 0/V/1 - k 2 is pure imaginary, meaning that the
direct and coupled waves are in phase quadrature at all frequencies and
couplings.
Figure 14 shows the coupling response in decibels for several values of
k, normalized to the midband coupling and to a unit center frequency.
This pattern repeats, ad infinitum, as frequency increases.
The bandwidth over which the coupling remains within a specified
tolerance can be increased by cascading two or more couplers end to end
to form a multisection coupler. For example, Figure 15 shows the coupling
responses and phase differences for symmetrical and asymmetrical threesection 10-dB couplers, specified to have 0.5-dB coupling ripple. Note that
symmetrical couplers, in which the coupling decreases symmetrically from
the center to the ends, have (N + 1)/2 ripples and retain the 90 ° quadrature property of the single section couplers, whereas asymmetrical cou-
7. Directional Couplers
215
L
. . . . .
-1
-!
,
i
. . . . . . . .
!. . . . . .
... ....
-2
30 dB Mid-band Coupling
-3
"o
t-
.
-4
.JO
E
.
.
.
.
.
.
.
,
.
,
-5
O
i.,4tO
.
i
....................
i...........
i ................
-6
!
-7
>
CI
-8
,"
-9
o -10
-11
:
i
-12
-,3 f
-14
.... i
-15
0
0.2
0.4
. . . . . . .
i
..... :
.........
;. . . . . . . . . .
i
i
i
,
0.6
0.8
1
1.2
:
........
i. . . .
i
1.4
1.6
1.8
Normalized Frequency
Figure 14. Coupling responses of idealized coupled-line couplers.
180
-9
Coupling
-10
- 135
-11
90
-12
-45
o~
c-
-13
0
o
-14
rn
I
hase Difference
o
~
8
*._
-45
a(D
u)
..E:
--90
-15
-16
-
\
-17
0
0.2
0.4
0.6
0.8
1
1.2
I
1.4
I
1.6
I
1.8
a_
-135
-180
2
Normalized Frequency
Figure 15. Responses of symmetrical and asymmetrical three-section, 0.2-dB-ripple, coupled-line
couplers.
216
Blank and Buntschuh
piers, in which the coupling decreases from one end to the other, have N
ripples and are not quadrature couplers, but have a greater equal-ripple
bandwidth.
Continuously tapered couplers have a high-pass coupling response;
i.e., the coupling is theoretically fiat to infinite frequency, and the lowfrequency cutoff is determined by the total coupler length. The following
references provide prescriptions for coupling variation along the length for
various multisection and tapered couplers, which may be applied to any of
the coupler types described in this section: stepped asymmetrical [13],
stepped asymmetrical [14-16], tapered asymmetrical [17], and tapered
symmetrical [18-20].
If the dielectric medium is not homogeneous, as in microstrip and
many other multilayer planar structures, the propagation is no longer pure
TEM, with the consequence that the line is dispersive, and the pattern of
Figure 14 does not repeat with increasing frequency. Nevertheless, in most
practical cases, at low enough frequencies, the deviation from TEM is
slightmcalled quasi-TEMmand dispersion is ignored. The even and odd
modes are still orthogonal, i.e., not coupled, but they will have different
phase velocities and effective dielectric constants, with the result that
when superposed they do not cancel to provide a perfect match and
directivity. Also, because of this, multisection and tapered non-TEM
couplers generally do not perform very well over large bandwidths, and
each proposed case must be studied individually.
uI
_
.__
~
f
-.
S41-Direct coupling
.
-10
-20
-30
-40
0
2
4
6
8
10
12
14
16
Frequency - GHz
Figure 16. Wideband responses of a 10-dB microstrip coupler on 0.020-in. alumina, centered at
I GHz.
7. Directional Couplers
217
T h e s c a t t e r i n g p a r a m e t e r s of the i n h o m o g e n e o u s
dielectric c o u p l e r
are
J( e2 -- 1)sin 0 e
all
-
-
-
j(Z2o - 1)sin 0 o
-
m
4Z e COS 0 e q- 2 j ( z e2 + 1)sin 0 e
4z o cos 0 o + 2 j ( z o2 + 1)sin 0 o
(26)
j ( Z e2 -- 1)sin
S21 --
0e
j ( Z o2
+
4 z e COS 0 e + 2 j ( Z e2 + 1)sin
--
1)sin 0 o
4 z o cos 0 o + 2 j ( z o2 + 1)sin 0 o
0e
(27)
2Z e
531 =
2z o
m
4 Ze
COS 0 e "{-
2 j ( Z e2 -F 1 )sin 0 e
4Zo
COS 0 o +
2 j ( z o2 + 1)sin 0 o
(28)
2Z e
541
2z o
4 z e c o s 0 e -t- 2 j ( z e2 + 1 )sin 0 e + 4Zo cos 0 o + 2 j ( Z o2 + 1)sin 0 o '
(29)
w h e r e z e -- Zoe/Zo,
z o - " Z o o / Z 0 are n o r m a l i z e d i m p e d a n c e s a n d w h e r e
0 e - - ~ol/vpe a n d 0 o = a~I/vpo are the electrical l e n g t h s of e a c h m o d e . W e
still have Z k = 1/ZoeZoo a n d k = (Z0e -- Z o o ) / ( Z o e + Zoo) , b u t the coupler is not m a t c h e d for any Z k , a n d the m a x i m u m coupling, which occurs
C12
II
C0
j ¢0
Figure 17. Coupled-line interelectrode capacitances
A
o)
I-w.-I. I-w-
#/i/ill/i/////ill/ill//,
b I t'w-I'~'ffw'l ±
h)
.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.•.:.
b)
j/ / I / l l I / l + i i l l t l l l l l l
~wobw~
j)
h
L, ;;. ;"-. :," . ," ;. ;.-. ;.;. ;J
•ii}iiii}].ii ii./.iiii.iiiiii].iiiiiii.ii}ii../ii.i.]]iiiiiii
2ill'il)lTIJliiJl21il'll'l
c)
Y/ill//i/ill/ill//////,
~llllJ
vllllA
i/ill///ill/i////ill///
k)
ZiiiT--;;iiii
i
iT:
"llllllllllllllllillii 4
d)
Z/i/I//Ill///////Ill/h
m
m)
I/i/i/i/ill////////////
........................../......./...
.........................
......................~.......~..........
..............~ .........~.....
,'///////I/////i///////i
////I////ll//ll///////l
@ @
......-,......
;.], ;'], ;..,.......-,...
"////////I/i/I/i///i///,
"I////II//I///////////I,
"/I////////H/I////////,
f)
o)
II/////I/I///IIII//////
I////I/I//////////////;
I/////I///I///////////I
Y//I////I/////I///////,
...........-....................
t///IllI//I/llllll/lll/.
Figure 18. Common coupled-line coupler cross-sections. (A) Homogeneous dielectrics of the relative
dielectric constant Er. (a) Edge-coupled stripline, t -0 [11], and coupled rectangular bars, t > 0, [21].
(b) Offset-coupled striplines [22]. (c) Broadside-coupled stripline [23]. (d) Vertical broadside-coupled
stripline [24]. (e) Coupled round rods [25]. (f) Slot-coupled stripline [26]. (g) Reentrant coupled lines
[27]. (B)Inhomogeneous dielectrics. (h) Microstrip coupled lines [11]. (i) Interdigitated (Lange) coupler
[28]. (j)Microstrip with dielectric overlay [29]. (k)Coplanar waveguide coupler [30]. (I)Coupled slotlines
[30]. (m) Microstrip reentrant coupler [31]. (n) Broadside-coupled suspended substrate lines [32].
(o) Double-registered edge-coupled suspended substrate lines [33]. (p) Edge-coupled suspended
substrate coupler [34].
7. Directional Couplers
219
w/b vs coupling for Zk = 48, 50, 52 ohms,
Er = 1.0, 2.2
a
1.6
Zk = 48
:
14
..........................
•
1.2
.0
!...................
!.......................................
........................................................................................
....................................
IE r = 1 . 0
~5--0
s2
....................
..........................
!
......... .........
i
...................................... !..................................
1
=
i ...................................... i
....................................................
08
!
............................ ...................................
Z ...... i. . . . . ! ......
k=48
............................. ~
~
~
....................;
52.i. - - n
Er= 2.2
0.6
......................................................................................
~ ....................................
i ....................................
i ..............................!...........
~.................................................
0.4
5
10
15
20
25
30
50
40
Coupling - dB
b
s/b vs coupling for Zk = 48, 50, 52 ohms,
1.0, 2.2
Er =
10 I
-_
'
1
,.,
0.1
"~
52
!
!
!
15
20
25
.
!
0.01
0.001
0.0001
5
10
30
40
50
Coupling - dB
Figure 19. Edge-coupled stripline parameters for Z k - - 4 8 , 50, and 52 ~, and for e r - 1 . 0 ,
(a) w / b vs coupling. (b)
s/b
vs coupling.
and 2.2.
220
Blank and Buntschuh
w/h vs coupling for s/h = .0375, .0725, .135
Zo = 50 ohms, E r = 2.2
0.8
'-
0.6
0.4
.0375
i
0.2
0
5
10
15
20
25
30
35
40
25
30
35
40
Coupling - dB
Wo/h vs c o u p l i n g for s/h = .0375, . 0 7 2 5 , . 135
Z o = 50 o h m s , E r = 2.2
1.8
-
1.6
1.4
1.2
¢-
1
0.8
0.6
-
0.4
s/
.0375- ~ ~ ~
0
0
5
10
15
20
Coupling -dB
Figure 20. Offset-coupled stripline parameters for s / h = 0.0375, 0.0725, and O. 135, for Z 0 = 50 /),,
and for Er = 2.2. (a)
w/h vs
coupling. (b)
Wo/h vs
coupling.
7. Directional Couplers
221
w/h vs coupling for Z k = 48, 50, 52 ohms,
E r -- 10.0; Eel f for 5 0 - o h m s
7.5
1.2
Eeff - e ~
_
0.8
6.5
0.6
6
0.4
5.5
0.2
0
5
10
15
20
25
I
I
I
I
30
35
40
45
5
50
Coupling - d B
b
~ vs coupling for z k = 48, 50, 52 ohms,
f'r =" I 0 . 0
m
"ID
I
~, 10
............ I
.~
•
::
1
0.01
0.1
1
10
s/h
Figure 21. Microstrip coupled line parameters for Z k - 48, 50, and 52 ~ and for ~r - I0. (a) w/h and
Eeff for 50 II vs coupling (b) s/fl vs coupling.
222
Blank and B u n t s c h u h
w/h vs coupling for Z k = 48, 50, 52 ohms,
Er ,,= 10.0; E~f for 50 ohms
0.2
6.5
0.15
:
..........~ z~ = 48-~
~50---0.1
5.5
........
0.05
_
0
0
1
I
I
I
I
I
I
2
3
4
5
6
7
I
8
4.5
9
10
Coupling - d B
s/bvscoupling for 7_k = 48, 50, 52 ohms,
b
Er= 10.0
~::::..::::::i:::::::::::::::i:-:
.......................... 4 ............................................i
=
I
!!!i!!i! !!! :!i!:!!!iii.!i!: :!:!i !!i!i ! !!!!!!!!! ! !i!! !i !i! ii:::i!!:!i!!!i!!!.::::i:i!:ii!!:::~,::::::,,::,:,i : :!!
.........................~-.........................................
i..........................................
i ............................... 4...........................................i ............................... i ..........
; ..............
....:.................................
i .......................................
~
O.01
-52 .................~......................................
~........................................
4.............................................
i............................................
i ....................................
i • ..........
................................. ~ ..............
~ _ : : : : : - ~
.............==================================
........................................
~.............................................
~ ......................................
i ..............................................
~.............................................
~...................................
0,001
0
1
2
3
l
~
4
5
L
6
7
8
9
10
Coupling - dB
Figure 22.
Interdigitated coupled line parameters for Z k = 48, 50, and 52 D, and for ~r -
and ~eff for 50 D, vs coupling. (b) s / h vs coupling.
10. (a)
w/h
7. Directional Couplers
223
when ( 0 e -t-" 00)/2 = rr/2, is slightly less than k, due to the reflection and
directivity losses. Figure 16 shows the computed coupling response of a
typical 10-dB coupler on alumina, including the effects of unequal mode
velocities and dispersion.
The design problem is to determine the physical dimensions of a
desired coupler type for a given Z~ and k (or Z0e and Zoo) and dielectric
constant e r. This is a formidable mathematical problem, even for the
simplest geometries. Typically, for TEM and quasi-TEM formulations,
the capacitances per unit length between the two lines and from each to
the ground, (Figure 17) are calculated from electrostatics, and the even-
Z12
Q
f
---
~/4
®
Z12
Z1
;./4----
- - - M4----Z2
Z1
.
Figure 23. Planarbranch-line coupler topology.
Figure 24. Waveguide branch-guide coupler.
®
Blank and Buntschuh
224
and odd-mode impedances determined from the relations
1
Z0e =
1
vpCo
Z0o --
vp(Co + 2C12 )
.
(30)
There are numerous computer programs commercially available which
compute the impedances and effective dielectric constants for virtually
any arbitrary cross-sectional geometry. Also, the popular microwave
C A D / C A E packages have routines for both synthesis (dimensions from
impedances) and analysis (impedances from dimensions) for stripline and
microstrip lines. Coupled-line geometries which have been treated in the
literature and for which some design data or formulas have been published are shown in Figure 18. Space precludes reproducing design data
for all of them. Data for the four most common and important configurations are given in Figures 19 through 22, for specific values of the
dielectric constant. All curves are for the zero-thickness approximation.
Those for edge-coupled stripline and the microstrip lines are plotted for
0
!
!
i
~
0
i
-10
20
- 9 0 ~,
3O
- 40
L .........................
i..............................................
i.......................................................
i
180
......................................................................................................................
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Normalized Frequency
Figure 2.5. Coupler performance for a two-branch, 50-,0,, 3-dB branch-line hybrid.
7. DirectionalCouplers
225
= 48, 50, and 52 f~, to provide a rough indication of the sensitivity of
50-~ couplers to variations in the parameters. The data for offset-coupled
50-f~ stripline couplers are given for three useful ratios of spacer thickness
to ground plane spacing.
Zk
4. Branch-Line, Branch-Guide, and Rat-Race Couplers
Branch-line and branch-guide couplers are formed by coupling two main
transmission lines together with two or more quarter-wavelength-long
transmission lines, spaced by quarter-wavelengths along the coupled lines.
For branch-line couplers, i.e., coax and planar configurations, the coupling
lines are shunt connected, as illustrated for a three-branch coupler in
Figure 23. The branches of a waveguide branch-guide coupler are normally series connected to the main line with E-plane Is, as sketched in
Figure 24, also for a three-branch coupler.
O-
~
Direct
~~=1-
iO
-10
~,
"~ ~
~ 1 1 "
$41"'s°lati°n / /
ReturnI ° ~ ~
I
u'J
=o
-2o
-90 a
-30
- 40
L
J
J
0.2
0.4
0.6
........... .........i.................... i.............................i........................~ - 180
0.8
1 1.2 1.4 1.6 1.8
NormalizedFrequency
Figure26. Couplerperformancefor atwo-branch,50-~, 10-dBbranch-linecoupler.
Blank and Buntschuh
226
The coupling value and the matching conditions uniquely determine
the impedances of the branches and interconnects of two-branch couplers.
For three or more branches, the additional degrees of freedom are used to
increase the coupling bandwidth. Branch-line and -guide couplers have a
much narrower band than coupled-line or waveguide couplers of the same
number of sections or holes. Moreover, the impedance ranges involved are
rather large, such that branch-line couplers of more than three sections or
either branch-line or branch-guide couplers looser than 10 dB become
impractical. Consequently, branch-line and -guide couplers find their
widest use in tight-coupling applications, especially as 3-dB hybrids, in
which coupled lines and multihole waveguide couplers become difficult to
fabricate because of the narrow gaps or large holes required.
Branch-Line
Couplers
Figures 25 through 28 show the computed responses for two-branch 3- and
10-dB couplers and three-branch Butterworth and Chebychev 3-dB hy-
0
i
0
$31"~Coupling
jj
$21- Direct
$11- Return loss !
............................................
-10
~i
...............
,
$4~- Isolation
!
¢/)
G)
IZI
-o
-20
-90
..........................................................
~=
<D
o
)$21
i
.
i
- 30 .............................~........
. . . . . . . .
i ...............................................
1
i
i
- 40 ..............i...............J........
..
J
0.2
0.4
0.6
0.8
.
.
.
.
.
. . . . . .
J......................................
J......................................
i..........................
u - 18 0
1.2
1.4
1.6
1.8
Normalized Frequency
Butterworth Response
Figure 27. Coupler performance for a three-branch 50-~,, 3-dB Butterworth branch-line hybrid.
7. Directional Couplers
227
brids. Note that the couplers are bandwidth limited in coupling, match,
and directivity by about the same amount. Although branch-line couplers
are not quadrature at all frequencies, the phase difference between the
coupled and the direct arms is within 2° over the useful bandwidth, which
is close enough for most applications. The two-branch 3-dB hybrids have a
useful bandwidth of about 10%; three-branch Butterworth couplers have
about 30% and Chebychev designs have over 50%, although the coupling
imbalance is as great as 1.4 dB at midband.
Young [35] has given a formulation to design couplers with two to
eight branches, in which the impedances for three or more branches are
determined by an optimization procedure. Levy and Lind [36] have given
tables of immitances for couplers of three to nine branches, derived
by exact synthesis for maximally flat responses and almost exact for
Chebychev, characteristics. The branch and main line impedances are
plotted in Figure 29 as a function of the coupling for two- and three-branch
couplers, from References [35] and [36], and for 50-~ shunt-branch
couplers for impedances up to 200 ~.
/ S21 - Direct
-10
¢/)
20
-90
D
-30 -
_40 L
0.2
J
0.4
0.6
l
0.8
1
1.2
1.4
l .....
1.6
-J - 1 8 0
1.8
Normalized Frequency
Chebychev
Response
Figure 28. Coupler performance for a three-branch, 50-~,, 3-dB Chebychev branch-line hybrid.
228
Blank and B u n t s c h u h
200
/
180
i
Z 1 - 3 Branch
'Butterworth
' Chebychev
160
140
Z 1 - 2 Branch
E
120
tO
I
1/1
®
100
0
Z 2 - 3 Branch
c
"
"lO
,-,
E
80
Chebychev
~
60
nch .....................................................................
Butterworth
40
.................i
20
- 2 Branch ..................................................................
~.........................................
..........................................................................................................................
i
Z12 - 3 Branch
Zl 2
Chebychev
I_
2
4
6
8
10
12
Coupling - dB
Figure 29. Impedances for t w o - and three-branch, 50-D,, branchline couplers vs coupling.
Branch-Guide
Couplers
In a waveguide, the impedance values required for each branch are
determined by the guide heights, h i . It is often convenient to maintain the
main line impedance, Z 0, in the main and coupled lines. This requires
three or more coupling branches. Both coupling and directivity are frequency sensitive. To obtain tight coupling and high directivity over a wide
frequency band, more than three branches may be required, and 10 or
more branch designs are not uncommon. The design problem is to choose
the branch line impedances, as determined by the h i , in order to achieve
the required coupling and directivity over a specified frequency band.
Designs based on binomial and Chebychev distributions are given by
Lormer and Crompton [37], Patterson [38], and Levy [39]. T-junction
effects are accounted for in these references. Improvements in bandwidth
performance can be achieved by using the main and coupled line
impedances as design variables. Young has considered this case in Reference [39], and further improvements in performance are presented in
References [36] and [40].
7. Directional Couplers
229
q)
®
xt
® .....
~,
_',"f
®
\
Z3
Figure 30. Planar rat-race coupler topology. Adapted with permission from Peter A. Rizzi, Microwave
Engineering: Passive Circuits, © 1987 Prentice-Hall, Inc.
0
90
S42
S12= S21
-10
"o
- 2 0 ..........
...... ~ - 9 0
-30 -
--180
~S42- ~$12
-40
L
0.2
i
l
0.4
J
0.6
0.8
1
1.2
1.4
1.6
J - 270
1.8
Normalized Frequency
Figure 31. Coupler performance for a 3-dB, 50-~, conventional rat-race hybrid.
o~
¢b
a
230
Blank and Buntschuh
Rat-Race
Couplers
A rat-race coupler, or hybrid ring, consists of a 1 ~1 wavelength ring-shaped
transmission line with four feed lines entering radially at points spaced as
shown in Figure 30, in which the ports are numbered such that the
scattering matrix of a 3-dB hybrid, at midband, is given by Equation (7).
The feed lines are shunt connected in planar transmission media and
normally series connected in a waveguide, although waveguide and coax
rings are rarely implemented.
Figure 31 shows the principal responses of a conventional 50-f~, 3-dB
rat-race coupler in which Z 1 = Z 2 = Z 3 = 70.7 l~. For input into port 1,
the power splits between port 2 and port 3, with port 4 isolated. The
outputs are in phase at midband, and the phase difference deviates by
several degrees at the edges of the useful bandwidth. For input at port 2,
the power splits between port 1 and port 4, with port 3 isolated. At
midband the phase difference between outputs is 180°, again with several
degrees of variation over the useful bandwidth. The isolations, $41 and
90
S21
S31
-10
)S31- ~S21
S22
¢/)
a)
Sll
"o
-20
....
- 3 o -
-90
................... i
.....
•-
.........
i
i
L_.............. i ..................~............................... .................i . . . . . . .
0.2
- 180
i
i
-40
=o
a
0.4
0.6
0.8
1
1.2
i....
1.4
.....
1.6
~ - 270
1.8
Normalized Frequency
Figure 32. Coupler performance for a 3-dB, 50-~, modified rat-race hybrid.
7. Directional Couplers
231
$32 , are not shown in the figure; they are both similar to the port 1 return
lOSS, Sll. The useful bandwidth, over which the isolation and return loss
are better than 20 dB and the coupling imbalance better than 1 dB, is
about 30%.
The coupling, defined here by $21 = $12, may be varied by appropriately altering the ring segment impedances Z 1 and Z 2. As the coupling is
made looser, the bandwidth with regard to return loss, isolation, and
coupling deviation increases slightly, but the directivity bandwidth decreases substantially. For instance, for a conventional rat-race coupler in
which the 3A segment has a uniform impedance, Z 3 = Z 1 [41], a 10-dB
coupler has a bandwidth of about 34% for a 1-dB coupling deviation and
> 20 dB of isolation, whereas the bandwidth for 20 dB of directivity is
only 14%. Agrawal and Mikucki [42] have analyzed a modified design, with
Z 3 = Z 2, in which the coupling and isolation bandwidths are significantly
broadened at the expense of relatively low directivity across the band.
Figure 32 shows the characteristics of such a modified 10-dB rat-race
coupler. Figure 33 provides the impedances for the ring segments vs the
coupling for 50-12 rat-race couplers of both types.
200
=
180
i
160
......................
i
+
!
i
Conventional
. . . . . . . . . .
~ ..............
Z3 = Z1
~
~ ..............................4
I
/
140 ...........................................i............. t
E
¢-
+
0
Ioj
N
120
]
z, ....Z ..................
i
. . . . .
/
. . . . . . . . . . . . . . . .
,'~
...........
N
i.
//
.
.
Modified
.
.
i ..........................
,
100
80
j Z 2 - Conventional
60
ModifiedZ3= Z 2 J
-
~'
i
10
12
4O
2
4
6
8
14
16
Coupling- dB
Figure 33. Impedances for conventional and modified 50-,Q rat-race couplers.
232
Blank and Buntschuh
The broadbanding technique of March [43] replaces the 3A segment
with a 3-dB, h/4-coupled line coupler of impedance Z1, with normally
coupled and direct ports shorted to the ground. This scheme provides a
3-dB hybrid with a 1-dB imbalance bandwidth of almost 3.5:1, 20 dB of
return loss over 40%, and isolation always greater than 24 dB. Although
this coupler has excellent performance, it is rarely used due to its difficult
construction. Ashoka [44] has described a deft technique for extending the
practically achievable ranges of coupling and bandwidth of both branch-line
and rat-race couplers by using impedance and admittance inverters in
place of extremely high- and low-transmission line impedances.
5. Developments
The design of basic microwave directional couplers is a mature field, such
that most work has concentrated on refined mathematical analyses or
reports of new couplers of novel structure or materials, which fall outside
the scope of this chapter. One area, however, does deserve mention. With
the steady growth in monolithic microwave integrated circuit technology,
there has been concomitant interest in achieving the tight coupling and
reduced size of microstrip-type couplers. Tight coupling is being achieved
via multilayer couplers, i.e., couplers as in Figure 18h, but with one strip
on top of the other, separated by another dielectric layer. Prouty and
Schwarz [45], Tran and Nguyen [46], and Tsai and Gupta [47] provide
much useful design information as well as a rich list of references. At the
lower microwave frequencies, substantial size reduction can be achieved
by employing lumped elements to some extent. Vogel [48] is an excellent
starting point for applying this technique.
References
[1] R. E. Collin, Foundations of Microwave Engineering. New York: McGraw-Hill, 1966.
[2] D. M. Pozar, Microwave Engineering. Reading, MA: Addison Wesley, 1990.
[3] H. A. Bethe, Theory of diffraction by small holes, Phys. Rev., Vol. 66, pp. 163-182,
1944.
[4] S. B. Cohn, Microwave coupling by large apertures, Proc. IRE., Vol. 40, pp. 697-699,
1952.
[5] R. Levy, Directional couplers, in Advances in Microwaves (L. Young, ed.), Vol. 1. New
York: Academic Press, 1966.
[6] R. Levy, Analysis and synthesis of waveguide multiaperture directional couplers, IEEE
Trans. Microwave Theory Tech., Vol. MTT-16, pp. 995-1006, Dec. 1968.
[7] R. Levy, Improved single and multiaperture waveguide coupling theory, including
explanation of mutual interaction, IEEE Trans. Microwave Theory Tech., Vol. MTT-28,
pp. 331-338, Apr. 1980.
7. Directional Couplers
233
[8] H. J. Riblet, The short-slot hybrid junction, Proc. IRE., Vol. 40, pp. 180-184, Feb. 1952.
[9] S. E. Miller, Coupled wave theory and waveguide applications, Bell Syst. Tech. J., Vol.
33, pp. 661-719, May 1954.
[10] T. N. Anderson, Directional coupler design nomograms, Microwave J., Vol. 2, pp.
34-38, May 1959.
[11] K. C. Gupta, R. Garg, and R. Chadha, Computer-Aided Design of Microwave Circuits.
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