Notes 21 ECE 5317 - 6351 Microwave Engineering Fall 2019 Prof. David R. Jackson Dept. of ECE Quadrature Coupler and Rat-Race Coupler 1 Adapted from notes by Prof. Jeffery T. Williams
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Notes 21
ECE 5317-6351 Microwave Engineering
Fall 2019Prof. David R. Jackson
Dept. of ECE
Quadrature Coupler and Rat-Race Coupler
1
Adapted from notes by Prof. Jeffery T. Williams
Quadrature (90o) Coupler
2
“A quadrature coupler is one in which the input is split into two signals (usually with a goal of equal magnitudes) that are 90 degrees apart in phase. Types of quadrature couplers include branchline couplers (also known as quadrature hybrid* couplers), Lange couplers and overlay couplers.”
http://www.microwaves101.com/encyclopedia/Quadrature_couplers.cfm
Taken from “Microwaves 101”
This coupler is very useful for obtaining circular polarization: There is a 90o phase difference between ports 2 and 3.
Note:The term “hybrid” denotes
the fact that there is an equal (3 dB) power split to the
output ports.
3
90o Coupler
1 2
4 3
The quadrature hybrid is a lossless 4-port (the S matrix is unitary ).
All four ports are matched.
The device is reciprocal (the S matrix is symmetric.)
Port 4 is isolated from port 1, and ports 2 and 3 are isolated from each other.
Quadrature Coupler (cont.)
[ ]
0 1 00 0 11
1 0 020 1 0
jj
Sj
j
− =
4
The signal from port 1 splits evenly between ports 2 and 3, with a 90o
phase difference.
The signal from port 4 splits evenly between ports 2 and 3, with a -90o
phase difference.
The quadrature coupler is usually used as a splitter:
90o Coupler
1 2
4 3
+90o out of phaseφ2- φ3 = 90o
-90o out of phaseφ2- φ3 = -90o
21 31S jS=
24 34S jS= −
Can be used to produce right-handed circular polarization.
Can be used to produce left-handed circular polarization.
Quadrature Coupler (cont.)
Note: A matched load is usually placed on port 4.
Branch-Line Coupler
5
A microstrip realization of a quadrature hybrid (branch-line coupler) is shown here.
Notes: We only need to study what happens when we excite port 1, since the structure is physically symmetric.
We use even/odd mode analysis (exciting ports 1 and 4) to figure out what happens when we excite port 1.
Quadrature Coupler (cont.)
1 2
34
An analysis of the branch-line coupler is given in the Appendix.
Summary
[ ]
0 1 00 0 11
1 0 020 1 0
jj
Sj
j
− =
6
Quadrature Coupler (cont.)
The input power to port 1 divides evenly between ports 2 and 3, with
ports 2 and 3 being 90o out of phase.
1 2
34
Note: A matched load is usually placed on port 4.
7
Quadrature Coupler (cont.)A coupled-line coupler is one that uses coupled lines (microstrip, stripline)
with no direct connection between all of the ports.
Please see the Pozar book for more details.
/ 4gλ
Port 1 Port 2
Port 3 Port 4
This coupler has a 90o phase difference between the output ports (ports 2 and 3), and can be used to obtain an equal (-3 dB) power split or another split ratio.
8
Quadrature Coupler (cont.)
One feed port produces RHCP, the other feed port produced LHCP.
Circularly-polarized microstrip antennas can be fed with a 90o coupler.
Note: This is a better way (higher bandwidth) to get CP than with a simple 90o delay line.
Rat-Race Ring Coupler (180o Coupler)
9
http://www.microwaves101.com/encyclopedia/ratrace_couplers.cfm
Taken from “Microwaves 101”
“Applications of rat-race couplers are numerous, and include mixers and phase shifters. The rat-race gets its name from its circular shape, shown below.”
Photograph of a microstrip ring couplerCourtesy of M. D. Abouzahra, MIT
Lincoln Laboratory
10
180o Coupler
1 2
4 3
[ ]
0 1 1 01 0 0 11 0 0 120 1 1 0
jS
−− = −
The rat race is a lossless 4-port (the S matrix is unitary). All four ports are matched. The device is reciprocal (the S matrix is symmetric). Port 4 is isolated from port 1, and ports 2 and 3 are isolated from each other.
Rat-Race Coupler (cont.)
11
The signal from the “sum port” Σ (port 1) splits evenly between ports 2 and 3, in phase. This could be used as a power splitter (alternative to Wilkinson).
The rat race can be used as a splitter:
180o Coupler
1 2
4 3
In phase
Σ
∆ 180o out of phase
21 31S S=
24 34S S= −
Note: A matched load is usually placed on port 4.
Rat-Race Coupler (cont.)
The signal from the “difference port” ∆ (port 4) splits evenly between ports 1 and 2, 180o out of phase. This could be used as a balun.
12
The signal from the sum port Σ (port 1) is the sum of the input signals 1 and 2.
The rat race can be used as a combiner:
12 13S S=
42 43S S= −
180o Coupler
1 2
4 3
Σ
∆
Signal 1 (V1)
Signal 2 (V2)
( )1 2 12V V S+
( )1 2 42V V S−
Rat-Race Coupler (cont.)
The signal from the difference port ∆ (port 4) is the difference of the input signals 1 and 2.
[ ]
0 1 1 01 0 0 11 0 0 120 1 1 0
jS
−− = −
13
A microstrip realization is shown here.
Rat-Race Coupler (cont.)
2
1
3
4
Σ
∆
An analysis of the rat-race coupler is given in the Appendix.
[ ]
0 1 1 01 0 0 11 0 0 120 1 1 0
jS
−− = −
Magic T
14
A waveguide realization of a 180o coupler is shown here, called a “Magic T.”
“Magic T”
Note the logo!
IEEE Microwave Theory and Techniques Society
Note: Irises are usually used to obtain matching at the ports.
Monopulse Radar
15
Rat-Race couplers are often used in monpulse radar.
A B C DΣ = + + +
( ) ( )( ) ( )
AZ
EL
B C A D
C D B A
∆ = + − +
∆ = + − +
Four antennas
https://www.microwaves101.com/encyclopedias/monopulse-comparator-networks
Σ
∆
Rat-Race:
Input from AInput from B
Input from C Input from D
A B C DΣ = + + +
( )AZ B C A D∆ = + − +
( )EL C D B A∆ = + − +
Σ∆
Σ
∆
Σ
Σ∆∆
Monopulse Radar (cont.)
16
The difference signals are used to determine the azimuth and elevation of the target.
https://www.microwaves101.com/encyclopedias/monopulse-comparator-networks
( )( )( )
/2 /2 /2
/2
1
2 sin / 2
j
j j j
j
A B e
e e e
e j
φ
φ φ φ
φ φ
− ∆
− ∆ + ∆ − ∆
− ∆
− = −
= −
= ∆sinL D θ∆ =
D = separation
0 sink Dφ θ∆ =
A
B
The difference between the two antenna signals maps into the phase difference ∆φ, which maps into the angle θ.
Port 1 Excitation“even” analysis
3 2
4 1
e e
e e
V VV V
=
=
17
( )( )
0
0
0
tan
tan / 4s s sY jY l
jYjY
β
π
=
=
=
0 01/Y Z≡
Appendix A
OC OCOCOC
OC
Here we analyze the quadrature coupler.
Input admittance of open-circuited stub:
Port 1 Excitation“odd” problem
3 2
4 1
o o
o o
V VV V
= −
= −
18
Appendix A (cont.)
( )( )
0
0
0
cot
cot / 4s s sY jY l
jYjY
β
π
= −
= −
= −
0 01/Y Z≡
SC SCSCSC
SC
Input admittance of short-circuited stub:
Consider the general case:
0
+ - sY jY
= ±for evenfor odd
[ ] [ ]0
4
0
021 0
1 2 0Y
jZ
ABCD ABCDY j
Z
λ
= =
[ ] [ ] [ ] [ ]4
Y YABCD ABCD ABCD ABCDλ⇒ =
19
Quarter-wave lineShunt load on line
.
( ) ( )
( ) ( ) ( )
0
0
cos sin
/ sin cos
line
line
line
jZ
ABCDj Z D
β β
β β
= =
In general:
0 0 / 2/ 2
lineZ Zβ π
==
Here :
Y Y
2-port
Appendix A (cont.)
[ ]0
0
0 0
0
0 0
20 0
0
021 0 1 0
1 12 0
2 21 01 2 0
2 22
2 2
jZ
ABCDY Yj
Z
jZ Y jZ
Y jZ
jZ Y jZ
jZ Y j jZ YZ
=
=
=
+
20
Hence we have:
00
+ -
jY jYZ
±
= ± =
for evenfor odd
Appendix A (cont.)
[ ]
( )
( )
0 00
2
0 00 0 0
0
0 0
0
0
12 2
11 2
2
11
1 12
jjZ jZZ
ABCDj j jjZ jZ
Z Z Z
j j jZ
jj j jZ Z
jZ
jZ
± = ± + ±
± = − + ±
=
21
Continuing with the algebra, we have:
Appendix A (cont.)
[ ]0
0
0
11
1 12e
jZABCD
jZ
=
[ ]0
102
1 02
e
j
Sj
−
= −
22
Hence we have:
Convert this to S parameters (use Table 4.2 in Pozar):
Note: We are describing a two-port device here, in the even and
odd mode cases.
Appendix A (cont.)
This is a 2×2 matrix, not a 4×4 matrix.
2 3 4
111
1 0a a a
VSV
−
+= = =
=
11 0S =
( )
- - - - - -1 1 1 1 1 1
11
11 11
12 2
1 0 02
e o e o e o
e o
V V V V V VSV V V V V
S S
+ + + + +
+ += = = + +
= + = +
23
Hence
⇒
11 22 33 44 0S S S S= = = =By symmetry:
1V V V+ + += +
Appendix A (cont.)Adding even and odd mode cases together:
( )2 2 2 221 21 21
12 2
1 1 12 2 2
2
e o e oe oV V V VS S S
V V Vj j
j
− − − −
+ + +
+ += = = +
+ − − − = +
−=
21 12 43 34 2jS S S S −
= = = =By symmetry and reciprocity:
24
2 3 4
221
1 0a a a
VSV
−
+= = =
= ⇒
Appendix A (cont.)
1V V V+ + += +
2 3 4
331
1 0a a a
VSV
−
+= = =
=
31 13 24 4212
S S S S −= = = =By symmetry and reciprocity:
25
( )3 3 3 3 2 231 21 21
12 2 2
1 1 12 2 2
12
e o e o e oe oV V V V V VS S S
V V V Vj j
− − − − − −
+ + + +
+ + −= = = = −
+ − − − = − −
=
⇒
Appendix A (cont.)
1V V V+ + += +
2 3 4
441
1 0a a a
VSV
−
+= = =
=
41 14 23 32 0S S S S= = = =By symmetry and reciprocity:
26
( )4 4 4 4 1 141 11 11
1 02 2 2
e o e o e oe oV V V V V VS S S
V V V V
− − − − − −
+ + + +
+ + −= = = = − =
+⇒
Appendix A (cont.)
1V V V+ + += +
27
Layout Schematic
2
1
3
4
Plane of symmmetry
1 2
3 4
Here we analyze the Rat-Race Ring coupler.
Appendix B
Port 1 Excitation“even” problem
28
( )
( ) ( )1 0 1 1
0
0
tan
/ 2 tan / 4
/ 2
s s s sY jY l
j Y
jY
β
π
=
=
=
0 0
0 0
1/
/ 2s
Y Z
Y Y
=
=
( )
( ) ( )2 0 2 2
0
0
tan
/ 2 tan 3 / 4
/ 2
s s s sY jY l
j Y
jY
β
π
=
=
= −
1 2
3 4
Appendix B (cont.)
#1 #2
29
( )
( ) ( )1 0 1
0
0
cot
/ 2 cot / 4
/ 2
s s s sY jY l
j Y
jY
β
π
= −
= −
= −
( )
( ) ( )1 0 2
0
0
cot
/ 2 cot 3 / 4
/ 2
s s s sY jY l
j Y
jY
β
π
= −
= −
=0 0
0 0
1/
/ 2s
Y Z
Y Y
=
=
1 2
3 4
Appendix B (cont.)Port 1 Excitation“odd” problem
#1 #2
[ ]0
0 00
0
0
0
0
0
0
1 0 1 00 211 10
2 22
1 0 1 211 0
2 2
1 2
2 1
e
j ZABCD jY jYj
Z
j ZjY j
Z
j Z
jZ
= ±
± = ±
± =
Proceeding as for the 90o coupler, we have:
30
Appendix B (cont.)
[ ]0
0
0
1 2
2 1e
j ZABCD
jZ
± =
[ ]0
1 11 12
ejS± −
=
Converting from the ABCD matrix to the S matrix, we have:
31
Use Table 4.2 in Pozar
Note: We are describing a two-port device here, in the even and
odd mode cases.
Appendix B (cont.)
This is a 2×2 matrix, not a 4×4 matrix.
21 12 34 43 2( )
jS S S S −= = = =
symmetry and reciprocity
2 3 4
111
1 0a a a
VSV
−
+= = =
=
2 3 4
221
1 0a a a
VSV
−
+= = =
=
32
For the S parameters coming from port 1 excitation, we then have:
( )1 111 11 11
12 2
12 2 20
e oe oV VS S S
Vj j
− −
+
+= = +
− = +
=
( )2 221 21 21
12 2
12 2 2
2
e oe oV VS S S
Vj j
j
− −
+
+= = +
− − = +
−=
11 33 0( )S S= =symmetry
Appendix B (cont.)
31 13 -2
( )
jS S= =
symmetry
22 44
23 32 14 41
24 42
00
2
S SS S S S
jS S
= == = = =
= =
2 3 4
331
1 0a a a
VSV
−
+= = =
=
Similarly, exciting port 2, and using symmetry and reciprocity, we have the following results (derivation omitted):
33
( )3 331 11 11
12 2
12 2 2
2
e oe oV VS S S
Vj j
j
− −
+
+= = −
− = −
−=
Appendix B (cont.)
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