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ENE 428
MicrowaveEngineering
Lecture 12 Power Dividers andDirectional Couplers
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Power dividers and directional couplers
Passive components that are used for power division orcombining.
The coupler may be a three-port or a four-port component
Three-port networks take the form of T-junctions
Four-port networks take the form of directional couplers
and hybrids.
Hybrid junctions have equal power division and either 90or a 180 phase shift between the outport ports.
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Types of power dividers and directionalcouplersT-junction power divider Resistive divider
Wilkinson power divider
Bethe Hole Coupler
Quadrature (90) hybrid and magic-T (180) hybrid
Coupled line directional coupler
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Basic properties of dividers and couplers
The simplest type is a T-junction or a three-port networkwith two inputs and one output.
The scattering matrix of an arbitrary three-port network
has nine independent elements
11 12 13
21 22 23
31 32 33
S S S
S S S S
S S S
Divider
or
couplerP1
P2aP1
P3(1-a)P1
Divider
or
coupler
P1=P2+P3P2
P3
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The scattering parameters losslessproperty
The unitary matrix:
This can be written in summation form as
where ij= 1 ifi= jand ij= 0 ifijthusifi= j,
while ifij,
1
tS S
-*
1, ,
N
ki kj ijk
S S for all i j*
1
1,N
ki kik
S S*
1
0.N
ki kj
k
S S*
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It is impossible to construct a three-portlossless reciprocal network. (1)
If all ports are matched, then Sii= 0, and if the network isreciprocal the scattering matrix reduces to
If the network is lossless, the scattering matrix must be
unitary that leads to
12 13
12 23
13 23
0
0 .
0
S S
S S S
S S
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It is impossible to construct a three-portlossless reciprocal network. (2)
Two of the three parameters (S12, S13, S23) must be zeros
but this will be inconsistent with one of eq. (1a-c), implying
that a three-port network cannot be lossless, reciprocal, and
matched at all ports.
2 212 13
2 2
12 23
2 2
13 23
13 23
23 12
12 13
1, (1 )
1, (1 )
1, (1 )
0, (1 )
0, (1 )
0. (1 )
S S a
S S b
S S c
S S d
S S e
S S f
*
*
*
+
+
+
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Any matched lossless three-port networkmust be nonreciprocal. (1) The [S] matrix of a matched three-port network has thefollowing form:
If the network is lossless, [S] must be unitary, which
implies the following:
12 13
21 23
31 32
0
0 .0
S S
S S SS S
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Any matched lossless three-port networkmust be nonreciprocal. (3) This results show that SijSji forij, therefore the device
must be nonreciprocal.
These S matrices represent two possible types of
circulators, forward and backward.
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A lossless and reciprocal three-port network
can be physically realized if only two of its
ports are matched. (1) If ports 1 and 2 are matched ports, then
To be lossless, the following unitary conditions must be
satisfied:
12 13
12 23
13 23 33
0
0 .
S S
S S S
S S S
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A lossless and reciprocal three-port network
can be physically realized if only two of its
ports are matched. (2)
From (3a-b), , so (3d) shows that S13 = S23= 0.
Then |S12|=|S33|=1.
2 2
12 13
2 2
12 23
2 2 2
13 23 33
13 23
12 13 23 33
23 12 33 13
1, (3 )
1, (3 )
1, (3 )
0, (3 )
0, (3 )
0. (3 )
S S a
S S b
S S S c
S S d
S S S S e
S S S S f
*
* *
* *
+
+
+ +
+
+
13 23S S
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A lossless and reciprocal three-port network
can be physically realized if only two of its
ports are matched. (3) The scattering matrix and signal flow graph are shown
below.
If a three-port network is lossy, it can be reciprocal and
matched at all ports.
21
3
S21=e
jq
S12=ejq
S33=ejf
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Four-port networks (Directional Couplers)
Power supplied to port 1 is coupled to port 3 (the coupled
port), while the remainder of the input power is delivered to
port 2 (the through port)
In an ideal directional coupler, no power is delivered to
port 4 (the isolated port).
1 2
34
Input Through
CoupledIsolated
1 2
34
Input Through
CoupledIsolated
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Basic properties of directional couplers aredescribed by four-port networks.(1)
12 13 14
12 23 24
13 23 34
14 24 34
0
0.
0
0
S S S
S S SS
S S S
S S S
The [ S ] matrix of a reciprocal four-port network matched
at all ports has the above form.
If the network is lossless, there will be 10 equations result
from the unitary condition.
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Conditions needed for a lossless reciprocalfour-port network (2) Then the self-products of the rows of the unitary [S] matrix
yield the following equations:
which imply that |S13|=|S24|and that |S12|=|S24|.
2 2
12 13
2 2
12 24
2 2
13 34
2 2
24 34
1, (6 )
1, (6 )
1, (6 )
1, (6 )
S S a
S S b
S S c
S S d
+
+
+
+
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Symmetrical and Antisymmetrical coupler (2) If 2is ignored, we yield
1. The symmetrical coupler: q= f= /2.
2. The antisymmetrical coupler: q= 0, f= .
0 0
0 0.0 0
0 0
j
jS j
j
a
a
a
a
0 0
0 0.
0 0
0 0
S
a
a
a
a
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Symmetrical and Antisymmetrical coupler (3) The two couplers differ only in the choice of the reference
planes. The amplitudes aand are not independent, eq
(6a) requires that
a2+ 2=1.
Another way for eq. (4) and (5) to be satisfied is if|S13|=|S24| and |S12|=|S34|.
If phase references are chosen such that S13=S24=aand
S12=S34=j, two possible solutions are given. FirstS14=S23=0, same as above.
The other solution is fora= =0, which implies
S12=S13=S24=S34=0, the case of two decoupled two-port
network.
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Power supplied to port 1 is coupled to port 3 (the coupledport) with the coupling factor
The remainder of the input power is delivered to port 2 (the
through port) with the coefficient
In an ideal coupler, no power is delivered to port 4 (the
isolated port).
Hybrid couplers have the coupling factor of 3 dB ora= = The quadrature hybrid coupler has a 90 phase shift
between ports 2 and 3 (q= f= /2) when fed at port 1.
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Directional couplers characterization (1)2 2
13 .S
2 2 2
12 1 .S -a
1/ 2.
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Coupling = C= = -20log dB,
Directivity = D = = 20log dB,
Isolation = I= = -20log|S14
| dB.
The coupling factor indicates the fraction of the input
power coupled to the output port.
The directivity is a measure of the couplers ability toisolate forward and backward waves, as is the isolation.
These quantities can be related as
I= D + CdB. 22
Directional couplers characterization(2)1
310log
P
P
3
4
10logP
P
1
4
10logP
P
14S
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The ideal coupler would have infinite directivity andisolation (S14 = 0).
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Ideal coupler
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Lossless divider (1)
A lumped susceptance, B, accounts for the storedenergy resulted from fringing fields and higher ordermodes associated with the discontinuity at the junction.
In order for the divider to be matched to the input lineimpedance Z0, and assume a TL to be lossless, we willhave
1 2 0
1 1 1
.inY Z Z Z +
jB
Z1
Z2
Z0
+V0-
Yin
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Lossless divider (2)
The output line impedances Z1 and Z2 can then beselected to provide various power division ratios.
In order for the divider to be matched to the input line
impedance Z0, and assume a TL to be lossless, we willhave
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Ex1 A lossless T-junction power divider has a
source impedance of 50 . Find the output
characteristic impedances so that the input poweris divided in a 3:1 ratio. Compute the reflection
coefficients seen looking into the output ports.
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Resistive divider
A lossy three-port divider can be made to matched at allports, although the two output ports may not be isolated.
Z0/3
Z0
Z0
+
V
-
Zin
Z
Z0+
V2-
+V3-
Z0/3
Z0/3+
V1-
P1
Port 1
Port 2
P2
P3
Port 3
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