CRL711
Quadrature Coupler and
Rat-Race Coupler
1
Quadrature (90o) Coupler
“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.”
Taken from “Microwaves 101”
http://www.microwaves101.com/encyclopedia/Quadrature_couplers.cfm
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.
2
Quadrature Coupler (cont.)
2
1
90o Coupler
4
3
0 j 1 0


−1  j 0 0 1 
[S ] =
2 1 0 0 j 


0
1
0
j


 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.
3
Quadrature Coupler (cont.)
The quadrature coupler is usually used as a splitter:
2
1
90o Coupler
4
+90o out of phase
φ2- φ3 = 90o
-90o out of phase
φ2- φ3 = -90o
3
Note: A matched load is usually placed on port 4.
The signal from port 1 splits evenly between ports 2 and 3, with a 90o
phase difference.
S21 = jS31
Can be used to produce right-handed circular polarization.
The signal from port 4 splits evenly between ports 2 and 3, with a -90o
phase difference.
S24 = − jS34
Can be used to produce left-handed circular polarization.
4
Quadrature Coupler (cont.)
Branch-Line Coupler
A microstrip realization of a quadrature hybrid (branch-line coupler) is shown here.
Z0
1
λg
Z0
4
Z0
4
2
Z0
Z0
λg 4
Z0
2
Z0
2
3
Z0
Circuit of the branch-line hybrid coupler in
normalized form
5
Analysis
Note that superimposing the two sets of excitations produces
the original excitation
Analysis
V + Z0
Port 1 Excitation
“even” analysis
V+
jY0
e
1
V
Z0
V1e Z0
Z0
Z0
2
ls = λg 8
V+
e
2
V
2
λg 4
V2e
Z0
jY0
Z0
Z0
Y0 ≡ 1/ Z 0
λg 4
Z0
Z0
V4e
2
V3e
Z0
Input admittance of open-circuited stub:
Ys = jY0 tan ( β s ls )
V3e = V2 e
V4 = V
e
e
1
= jY0 tan (π / 4 )
= jY0
7
V + Z0
Port 1 Excitation
“odd” problem
V+
V1o
Z0
Z0
Z0
2
- jY0
V2o
Z0
V2o
2
λg 4
Z0
- jY0
Y0 ≡ 1/ Z 0
Z0
l s = λg 8
−V +
V1o Z0
λg 4
Z0
V4o
Z0
2
V3o = −V2o
V4o = −V1o
V3o
Z0
Input admittance of short-circuited stub:
Ys = − jY0 cot ( β s ls )
= − jY0 cot (π / 4 )
= − jY0
8
Z0
Z0
Consider the general case:
.
Z0
2
λg 4
 + for even 
Ys = ± jY0 

for
odd


In general:
1
[ ABCD ]Y = 
Y
0
1 
Shunt load on line

 0
[ ABCD ]λ = 
j 2
4
 Z
 0
jZ 0 
2


0 


cos ( β  )

 ABCD line  = 
 j / Z line sin ( β  )
0 )
(
Quarter-wave line
jZ 0line sin ( β  ) 


D = cos ( β  ) 
Here : Z 0line = Z 0 / 2
β = π / 2
 [ ABCD ] = [ ABCD ]Y [ ABCD ] λ [ ABCD ]Y
4
9
Hence we have:

0

 1 0 
[ ABCD ] = 
 j 2
1
Y


 Z
 0
jZ 0 
2   1 0

 Y 1 
0 

 jZ 0Y

 1 0  2
=

Y 1  j 2
 Z
 0
jZ 0 
2


0 

jZ 0Y


2

=
 jZ 0Y 2 j 2
 2 + Z
0

jZ 0 
2 

jZ 0Y 
2 
j  + for even 
Y = ± jY0 = ±
Z 0  - for odd 
10
Continuing with the algebra, we have:

 j 
 jZ 0  ± 
 Z0 
1 
[ ABCD ] = 
2
2


j
j2
 jZ ±
+
 0  Z 0  Z 0
j (± j)

1 
=
 1  j2
2 − j   +
  Z0  Z0

 1
1 
=
 1 
2  j 
  Z 0 

jZ 0



 j 
jZ 0  ± 
 Z 0  


j ( ± j )


jZ 0
jZ 0 

 1

11
Hence we have:
 1
1 
[ ABCD ]0e =   1 
2 j 
  Z 0 
jZ 0 

 1

Convert this to S parameters (use Table 4.2 in Pozar):

 0
[ S ]0e = 
 1 − j
 2
1 − j 
2 

0 

Note:
We are describing a two-port
device here, in the even and
odd mode cases.
This is a 2×2 matrix, not a 4×4 matrix.
12
Adding even and odd mode cases together:
V1+ = V + + V +
Z0
Z0
V1
2
Z0
V2
V1+
λg
V1-
V4
-
Z0
V1−
S11 = +
V1 a =a =a =0
2
3
4
V2 -

Z0
4
λg 4
V4
Z0
S11 = 0
2
V3-
V3
Z0
V1- e + V1-o V1- e + V1-o
1  V1- e V1-o 
S11 =
=
=  + + +
+
+
+
V +V
2V
2V
V 
=
Hence
Z0
1 e
S11 + S11o ) = 0 + 0
(
2
By symmetry:
S11 = S22 = S33 = S44 = 0
13
Z0
+
1
+
V = V +V
2
Z0
V2
V1+
+
V4
-
Z0
V2 −
S21 = +
V1 a =a =a =0
3
4
V2 -
λg
V1-
2
Z0
V1

Z0
4
V4
Z0
λg 4
Z0
2
V3V3
Z0
V2 − e + V2 − o V2 − e + V2 − o 1 e
o
S21 = +
=
=
S
+
S
(
21
21 )
V +V +
2V +
2
1   −1 − j   1 − j  
= 
+

2  2   2  
−j
=
2
−j
By symmetry and reciprocity: S21 = S12 = S43 = S34 =
2
14
Z0
Z0
V1
2
Z0
V2
V1+
V1+ = V + + V +
V1V4 -
Z0
V3−
S31 = +
V1 a =a =a =0
2
3
4

V2 -
λg
Z0
4
V4
Z0
λg 4
Z0
2
V3V3
Z0
V3− e + V3− o V3− e + V3− o V2 − e − V2 − o 1 e
o
S31 = +
=
=
= ( S21 − S21
)
+
+
+
V +V
2V
2V
2
1  −1 − j   1 − j  
= 
−
2  2   2  
−1
=
2
By symmetry and reciprocity:
−1
S31 = S13 = S24 = S42 =
2
15
Z0
V1+ = V + + V +
Z0
V1
2
Z0
V2
V1+
V1-
V4 -
Z0
V2 -
λg
Z0
4
V4
Z0
λg 4
Z0
2
V3-
V3
Z0
V4 −
V4 − e + V4 − o V4 − e + V4 − o V1− e − V1− o
1 e
o
S41 = +
S41 =
=
=
=
S
−
S

(
11
11 ) = 0
+
+
+
+
V +V
2V
2V
2
V1 a =a =a =0
2
3
4
By symmetry and reciprocity:
S41 = S14 = S23 = S32 = 0
16
Quadrature Coupler (cont.)
Summary
Z0
1
λg
Z0
4
Z0
Z0
4
0 j 1 0


−1  j 0 0 1 
[S ] =
2 1 0 0 j 


j
0
1
0


2
Z0
λg 4
Z0
Z0
2
2
3
Z0
The input power to port 1 divides
evenly between ports 2 and 3, with
ports 2 and 3 being 90o out of phase.
Note: A matched load is usually placed on port 4.
17
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.
Port 1
Port 3
Port 2
λg / 4
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.
18
Quadrature Coupler (cont.)
Circularly-polarized microstrip antennas can be fed with a 90o coupler.
One feed port produces RHCP, the other feed port produced LHCP.
Note: This is a better way (higher bandwidth) to get CP than with a simple 90o delay line.
19
Rat-Race Ring Coupler (180o Coupler)
“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.”
Taken from “Microwaves 101”
http://www.microwaves101.com/encyclopedia/ratrace_couplers.cfm
Photograph of a
microstrip ring coupler
Courtesy of M. D. Abouzahra, MIT
Lincoln Laboratory
20
Rat-Race Coupler (cont.)
2
1
180o Coupler
4
0 1

− j 1 0
[S ] =
2 1 0

0 −1
3
1 0
0 −1

0 1

1 0
 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.
21
Rat-Race Coupler (cont.)
The rat race can be used as a splitter:
Σ
2
1
In phase
180o Coupler
Δ
4
180o out of phase
3
Note: A matched load is usually placed on port 4.
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).
S21 = S31
 The signal from the “difference port” Δ (port 4) splits evenly between ports 1
and 2, 180o out of phase. This could be used as a balun.
S24 = − S34
22
Rat-Race Coupler (cont.)
The rat race can be used as a combiner:
(V1 + V2 ) S12
Σ
180o Coupler
Δ
(V1 − V2 ) S42
2
1
Signal 1 (V1)
4
3
Signal 2 (V2)
The signal from the sum port Σ (port 1) is the sum of the input
signals 1 and 2.
S12 = S13
 The signal from the difference port Δ (port 4) is the difference of
the input signals 1 and 2.
S42 = − S43
23
Rat-Race Coupler (cont.)
A microstrip realization is shown here.
Σ
0 1

− j 1 0
[S ] =
2 1 0

0 −1
1 0
0 −1

0 1

1 0
λg
4
Z0
Z0
2Z 0
λg 4
Z0
3λg / 4
λg / 4
Z0
Δ
24
Analysis
2Z 0
Z0
λg
4
Z0
V+
Z0
2Z 0
λg 4
Z0
λg / 4
V2−
λg / 4
V1−
2Z 0
Z0
2Z 0
λg
3λg
4
4
V3−
λg / 4
Z0
2Z 0
V4−
3λg / 4
Z0
Layout
Z0
Schematic
25
Port 1 Excitation
“even” problem
2Z 0
Z0
+
V
V1− e
OC
V
OC
+
2Z 0
Z0
#1
Z0
λg
3λg
8
8
Z0
−e
2
V
#2
jY0
2
−
jY0
2
2Z 0
OC
OC
2Z 0
V3− e
Y0 = 1/ Z 0
Y0 s = Y0 / 2
Ys1 = jY0 s1 tan ( β s ls1 )
(
)
= j Y0 / 2 tan (π / 4 )
V4− e
= jY0 / 2
Ys 2 = jY0 s 2 tan ( β s ls 2 )
(
)
= j Y0 / 2 tan ( 3π / 4 )
= − jY0 / 2
26
Port 1 Excitation
“odd” problem
2Z 0
Z0
V+
λg
8
-o
1
V
SC
SC
V2-o
3λg
Y0 = 1/ Z 0
Y0 s = Y0 / 2
#2
−
jY0
2
jY0
2
8
SC
SC
Ys1 = − jY0 s1 cot ( β s ls )
-0
4
Z0
Z0
#1
Z0
-V +
V3-o
2Z 0
Z0
(
)
= − j Y0 / 2 cot (π / 4 )
V
= − jY0 / 2
Z0
Ys1 = − jY0 s 2 cot ( β s ls )
(
)
= − j Y0 / 2 cot ( 3π / 4 )
= jY0 / 2
27
Proceeding as for the 90o coupler, we have:
 1
[ ABCD ]0e =  ± jY0
 2
0  0

1
1  j
  2 Z 0
j 2Z0   1

jY
0   0
 
2
 1
=  ± jY0

 2
0  ±1

1


1 j
  2 Z 0
j 2Z0 

0 

 ±1

= 2
j Z
0

j 2Z0 


1 

0

1

28
Converting from the ABCD matrix to the S matrix, we have:
 ±1

[ ABCD ]0e =  2
j Z
0

j 2Z0 


1 

Note:
We are describing a two-port
device here, in the even and
odd mode cases.
This is a 2×2 matrix, not a 4×4 matrix.
Use Table 4.2 in Pozar
− j  ±1 1 
[ S ]0e = 

2  1 1
29
For the S parameters coming from port 1 excitation, we then have:
V1− e + V1− o
1 e
o
S11 =
=
S
+
S
(
11
11 )
+
2V
2
1 −j
j 
= 
+

2 2
2
=0
V1−
S11 = +
V1 a =a =a =0
2
3
4
S11 = S 33 = 0
( symmetry )
V2 −
S21 = +
V1 a =a =a =0
2
3
4
S 21 = S12 = S34 = S 43 =
−j
2
( symmetry and reciprocity )
V2 − e + V2 − o
1 e
o
S21 =
=
S
+
S
(
21
21 )
+
2V
2
1 −j −j 
= 
+

2 2
2
−j
=
2
30
−
V3
S31 = +
V1 a =a =a =0
2
S31 = S13 = (symmetry)
3
4
j
2
V3− e + V3− o
1 e
o
S31 =
=
S
−
S
(
11
11 )
+
2V
2
1 −j
j 
= 
−

2 2
2
−j
=
2
Similarly, exciting port 2, and using symmetry and reciprocity, we have
the following results (derivation omitted):
S22 = S44 = 0
S23 = S32 = S14 = S41 = 0
j
S24 = S42 =
2
31
Magic T
A waveguide realization of a 180o coupler is shown here, called a “Magic T.”
“Magic T”
Note: Irises are usually used to obtain matching at the ports.
IEEE Microwave Theory and Techniques Society
0 1

− j 1 0
[S ] =
2 1 0

0 −1
1 0
0 −1

0 1

1 0
Note the logo!
32
Monopulse Radar
Rat-Race couplers are often used in monpulse radar.
Σ = A+ B +C + D
Σ
Δ
Δ AZ = B + C − ( A + D )
Δ EL = C + D − ( B + A )
Σ
Input from B
Δ
Σ
Four antennas
Δ EL = ( C + D ) − ( B + A )
Δ
Δ
Input from C
Σ = A+ B +C + D
Δ AZ = ( B + C ) − ( A + D )
Input from A
Input from D
Σ
Rat-Race:
Σ
λg
4
Z0
Z0
2Z 0
λg 4
Z0
3λg / 4
λg / 4
Z0
Δ
https://www.microwaves101.com/encyclopedias/monopulse-comparator-networks
33
Monopulse Radar (cont.)
A
A − B = 1 − e − jΔφ
D = separation
Δφ = k0 D sin θ
B
ΔL = D sin θ
The difference signals are used to determine the
azimuth and elevation of the target.
= e − jΔφ /2 ( e + jΔφ /2 − e − jΔφ /2 )
= e − jΔφ /2 ( 2 j sin ( Δφ / 2 ) )
The difference between the two
antenna signals maps into the
phase difference Δφ, which maps
into the angle θ.
https://www.microwaves101.com/encyclopedias/monopulse-comparator-networks
34