of 13
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Directional Coupler
4-portNetwork
1 2
3 4
A directional coupler is a 4-port network exhibiting:
• All ports matched on the reference load (i.e. S11=S22=S33=S44=0)
• Two pair of ports uncoupled (i.e. the corresponding Si,j parameters are zero).Typically the uncoupled ports are (1,3) and (2,4) or (1,4) and (2,3)
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Characteristic Parameters
It is here assumed that the uncoupled ports are (1-4) and (2-3). The S
parameters the ideal coupler must exhibit are:
S11=S22=S33=S44=0, S14=S41=S23=S32=0.
In this case the ports (1-2), (1-3), (3-4), (2-4) are coupled.
Let define C (Coupling) as:
C=|S13|2, CdB=-20 log(|S13|)
If the network is assumed lossless and reciprocal, the unitary conditionof the S matrix determines the following relationships:
2 2 2 2 2
11 12 13 14 12 12
2 2 2 2 2 2 2 2
21 22 23 24 12 24 24 132 2 2 2 2
31 32 33 34 34 34 12
1 1 1
1 1
1 1 1
S S S S S C S C
S S S S S S S S C
S S S S C S S S C
Then, exciting the network at port 1 (2), the output power is divided between ports
3 and 2 (4 and 1) according the factors C and 1-C. No power comes out of port 4(3)
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Further implications of lossless
condition 13 3412 24* *
12 24 13 34 12 24 13 34
12 24 13 34
0 j j
S S S S S S e S S e
Assigning 12=34=0 e 13=±/2, will result 24=±/2, then
S12=S34, S13=S24 The network is symmetric
It can be demonstrated that it is sufficient to impose the matching
condition to the four ports of a lossless and reciprocal network to get a
directional coupler
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Parameters of a real directional coupler In a real coupler the matching at the ports is not zeros at all the frequencies. It
is then specified the minimum Return Loss in the operating bandwidth.
The coupling parameter C, it is in general referred to the port with the lowest
coupling.It must be then considered that to the uncoupled port actually arrive a not zero
power. To characterize this unwanted effect the isolation I is introduced:
I=Power to the coupled port/Power to the uncoupled port
For the coupler considered in the previous slides (assuming the port with the
lowest coupling is the 3):
2 213 14 I S S
Note that I for the ideal coupler is infinite
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Use of the directional coupler (1)
Measure of the reflection coefficient
CVi Vr
LLine, Zc
13 24,i L L r L L
r L L L
i L L
V V S j C V V V S j C V
V j C V V
V V j C V
121, 1 1C S C
V+
V-
LV
LV
LV V
1 2
3 4
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Use of the directional coupler (2)
Power divider (C=3 dB)
C=3dB
Pin 1 2
3 4
Pout= Pin /2
Pout= Pin /2
Ports closed on the
reference load
2 1 12 1 2 1 3 1 13 1 3 1
1 1 1 1,
2 2 2 2V V S V P P V V S V P P
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Use of the directional coupler (3)
Power combiner (C=3 dB)
1 2
3 4
Pin/2
Pin/2
Pin
3 2
1 2 12 3 13
2 2
2 3 2
1
2 2
1 1 1
2 2 2 2 2 2 2
in in
in
V V
V V S V S j
V V P PV P
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C=3dB
Vout1
1 2
3 4
V A
VB
Use of the directional coupler (4)
Sum and difference of voltages (C=3 dB)
1 1 12 13 1 ,2
out A B A BV V V S V S V V
12=13= 24=0
34=
0°
0° 0°
°Vout2
2 4 24 34
1
2out A B A B
V V V S V S V V
Ports closed on the
reference load
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Use of the directional coupler (5)
Balanced Amplifier
A
A
0°
90°
C=3 dB
90°
0°
Vin
Vout
Vin/2
jVin/2
2
AV in
2
A jV in
1a 2a
3a 4b
2b
3b
C=3 dB
Gain:
2 3
2 34
2 , 2
2 2
b in b in
b b out out b in
in
V A V V j A V V V P
V V j j A V AP
Reflection:
in
in
1 2 2 3
3 1 3 2
1
1
, 2 , 2 , 2 ,
12 , 2 2
2
0
a in a in a in in a in
a in in a a a in in in in
ain
a
V V V A V V A V V j A V
V j A V V jV V A V A V
V V
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Coupled TEM lines as directional coupler
0 , ,= Zc p c d Z Z
Zcp, Zcd
L
1 2
3 4
We have previously seen that, for having all the ports matched, the following
condition must be satisfied:
This condition also implies that port 4 is uncoupled :
1 4
2 3
0
0
S S
S S
14 1 2 3 41
0
4
S S S S S
The maximum value |S13|2 defines the coupling: C=(|S13|
2)maxIt is obtained for L=/2 :
22 2
2
13 1 2 3 4 1 2max max max
1 1
4 2
cp cd
cp cd
Z Z S S S S S S S
Z Z
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Variation of frequency
Matching and Isolation are frequency independent and equal to zero and
infinity respectively (ideal lossless TEM line).
The coupling varies with the frequency according to the following expression:
maxmax2
max
,
1 (1 ) cot
cp cd
cp cd
Z Z C C C
C Z Z
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f/f0
C / C m a x
Cmax=0.01-0.1
B0.5 dB For Cmax
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Practical Restrictions
They concern mainly the maximum value of Cmax. In fact, increasing Cmax, thelines become closer and closer until the practical implementation is no more
possible with sufficient accuracy. Typically the maximum value of Cmax must be
lower than 0.1 (CdB=10)
Example: Design a stripline coupler with C=0.1 with Z0=50 using thefollowing figures reporting the values of
as a function of S and W (lines separation and width). Frequency: 1 GHz
max 0,
cp cd
cp cd
cp cd
Z Z C Z Z Z
Z Z
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
S (mm)
Cmax
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
Re(Eqn(Cmax))
Output Equations
W=9.5
W=12
W=11.18
S=0.195
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
S (mm)
Z0
45
50
55
60
65
Re(Eqn(Z0))Output Equations W=9.5
10
10.5
11
11.5
12
W=11.18
S=0.195
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Solution:
We draw the line C=0.1 on the first graph and the line Z0=50 on the second graph. A point on each of these lines has to be then found, which must characterized by
the same pair of values (S, W) .
10 mm11.18 mm 11.18 mm
0.19565 mmr =1
Note: If a coupler dimensioned for the requested C but with a different valueof Z0 is available, impedance transforming networks can be used in place of
redesigning a new coupler.
0 2
75 mm
L Lc
L
C=0.1
Z’0
Z0
Z0
Z0
Z0
Z’0
Z’0
Z’0
Z’0
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Couplers with quasi-TEM Lines
If quasi-TEM coupled lines are considered (Microstrip), the phase velocity of theeven and odd modes are not exactly coincident. Strictly speaking, that would
not allow to apply the model here assumed for the characterization of the
directional coupler.
In the practice, until the difference between the two velocities is not too large,
the same phase velocity can be assumed for both modes (equal to the averageof the actual values), assuming the lines as TEM. There are however some
differences comparing the performances with the ideal TEM coupler (a perfect
matching is no more possible)
800 850 900 950 1000 1050 1100 1150 1200
Frequency (MHz)
Microstrip Coupler
-40
-35
-30
-25
-20
-15
-10
-5
0
1000.3 MHz-35.83 dB
999.18 MHz
-26.38 dB
1000.1 MHz-10.03 dB
S21
S31
S41
S11
1.575 mm
2 mm 2 mm
0.4037 mm r =2.33
0
, ,
0, ,
0.1 0.4037 mm
74.12
1.97, 1.71
1.84,2
55.24 mm
cp cd
eff p eff d
eff medio eff medio
m S
Z Z Z
L L
c
L
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Coupler with lumped couplingsTo realize couplers with a large coupling (C
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Branch-line coupler with C=0.5 (3 dB)
The couplers with C=3 dB are identified with the word Hybrid.To realize an hybrid of branch-line type the characteristic impedances of the
lines must be:
0 0 00.5
1 0.5 35.35 , 50 500.5c c
Z Z Z Z Z
Practical restrictions
One can easily verify that, with C tending to 0: Z’cZ0 and Z’’c∞. In the
practice, even with con C=0.1 (10 dB) the corresponding value of Z’’c is
very difficult to realize (Z’’c =3Z0). Usually, C must be between 3 and 6 dB.
Frequency dependence
For this device, both matching and isolation vary with frequency (the
nominal value is obtained at the frequency where the length of the lines is
0/4). Also the coupling depends on f (the max is again at f 0).The bandwidth for a given value of maximum coupling increases with Cmax;
Usually, the frequency variation of matching and isolation is more
pronounced than that of the coupling.
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Branch Line: a summary
11 33 22 44 13 24
2 2
14 23
2 2
12 34
0, 0
1
S S S S S S
S S C
S S C
Design equations:
S parameters obtained:
14 23 12 34, 1S S C S S j C
For C=0.5 (3dB), it has (assuming Z0=1/Y0=50):
Conditions to be imposed:
0 0
1,
11c c
C Y Y Y Y
C C
0 0 00.5
1 0.5 35.35 , 50 500.5c c Z Z Z Z Z
=0180°
-90°cY
cY
cY cY
11 22
33 44
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800 850 900 950 1000 1050 1100 1150 1200
Frequency (MHz)
Branch-line C=3dB
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
DB(|S(1,1)|)
Schematic 1DB(|S(2,1)|)Schematic 1
DB(|S(3,1)|)
Schematic 1DB(|S(4,1)|)Schematic 1
Adattamento
Isolamento
Accoppiamento
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Rat-race coupler
cY cY
1 3
2 4
cY
cY
0 4
0 4 0 4
03 4
There is only one symmetry axis (vertical)
It is anyhow possible to still use the eigenvalues method for finding the
dimensioning equations
P1P1
P2P2
P3P3
P4P4
cY
cY
cY
c
Y
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Conditions to be imposed:
11 33 22 44 23 14
2 2
13 24
2 2
12 34
0, 0
1
S S S S S S
S S C
S S C
Design equations:
0 01 ,
c cY Y C Y Y C
S parameters obtained:
13 24 12 34, , 1S j C S j C S S j C
11
22 44
33
=0
cY cY
cY
cY
-90°
-90°
11
22 44
33
=0
cY cY
cY
cY
-90°
90°
For C=0.5 (3dB), we obtain (assuming Z0=1/Y0=50):
0 0 0 070.707 , 70.707
1 0.5 0.5
c c
Z Z Z Z Z Z
C C
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Parameters dependence on f
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
Frequency (GHz)
Rat-race C=3dB
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
DB(|S(1,2)|)RatRace
DB(|S(1,4)|)RatRace
DB(|S(1,3)|)RatRace
DB(|S(1,1)|)RatRace
Accoppiamento
Adattamento
Isolamento
• The bandwidth is larger than that of the branch-line
• With the decreasing of the coupling the bandwidth increases
• La practical feasibility limits the coupling between about 3 and 8 dB
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3-port lossless networks A 3-port reciprocal lossless network cannot be matched at
the 3 ports (i.e. S11
=S22
=S33
=0 not possible).
In fact, imposing the unitary of S:
2 2
12 13
2 2 2 2 2
12 23 12 13 23
2 2
13 23
1
1 0.5
1
S S
S S S S S
S S
*
13 23
*
12 23 12 23 23
*
12 13
0
0 =0 or 0 or 0
0
S S
S S S S S
S S
These conditions are
incompatible so it is
impossible to haveS11=S22=S33=0
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Is it still possible to realize a power splitter
with a 3-port?
Possible solution: a 3 dB hybrid with the uncoupled
port closed on a matched load
C=3dB
Pin
Z0
2
4 3 Pout= Pin /2
Pout= Pin /2
Circuito a 3 porte
Portadisaccoppiata
The 3-port network is lossy
due to the presence of Z0.This resistor however does
not dissipate power
because the port 4 is
uncoupled. Pin is then split
between ports 2 and 3without losses
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A 3-port divider: the Wilkinson network
c Z
1
2
3
RW
0 4
0 4
c Z
0 Z
0 Z
0 Z
Due to the presence of RW the 3-
port network is lossy, so the
condition S11=S22=S33=0 can be
imposed
2-port network obtained by closing port 1 with Z0:
2 3
0 4 0 4
c Z
0 Z
c Z
RW
S22=S33 and S23 can be computed thorugh the eigenvalues of this network
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Zp
0 4
02 Z
c Z Zd
0 4
2
W R
c Z
2
3
04
04
c Z
02 Z
c Z
02 Z
2
W R
2
W R
2
3
04
04
c Z c Z
02 Z
c Z c Z
02 Z
2
W R
2
W R
Even Network
2 2 2
0
2 2
0 0
2,
2 2
c c p p
c
Z Z Z Z
Z Z Z
0
0
2,
2 2
W W d p
W
R R Z Z
R Z
22
0 0
0 0, 0
2
2 , 2
p d
p d
c W
S
Z Z R Z
Also S23=0:
Per Z0=50 Zc=70.7, RW=100
Odd Network
23 02
p d S
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Frequency response
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
Frequency (GHz)
Divisore di Wilkinson
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
DB(|S(1,1)|)Ideale
DB(|S(2,1)|)Ideale
DB(|S(3,1)|)
Ideale
Adattamento
Trasmissione
• The bandwidth for RL=20 dB is about 40% of f 0• Transmission (|S21|=|S31|) is independent on frequency
• Dissipation in RW is zero provided that the load at ports 2 and 3 is the
same
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Microstrip implementation
RW
The very small size of Rw (pseudo
lumped component) must be
accounted for. The two output linesmust be enough close to allow the
connection of Rw.
On the other hand is not advisable to have
the output lines too close each other
because an unwanted coupling may arise.
So diverging lines are often used.