### Lecture 12 - web page for staff

```ENE 428
Microwave
Engineering
Lecture 12 Power Dividers and
Directional Couplers
1
Power dividers and directional couplers
 Passive components that are used for power division or
combining.
 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 90
or a 180 phase shift between the outport ports.
2
Types of power dividers and directional
couplers
 T-junction power divider
 Resistive divider
 Wilkinson power divider
 Bethe Hole Coupler
 Quadrature (90) hybrid and magic-T (180) hybrid
 Coupled line directional coupler
3
Basic properties of dividers and couplers
P1
Divider
or
coupler
P2aP1
P1=P2+P3
P3(1-a)P1
Divider
or
coupler
P2
P3
 The simplest type is a T-junction or a three-port network
with two inputs and one output.
 The scattering matrix of an arbitrary three-port network
has nine independent elements
 S11
 S    S21
 S31
S12
S 22
S32
S13 
S 23 
S33 
4
The scattering parameters’ lossless
property
 The unitary matrix:
S 

 
t -1
 S 
 This can be written in summation form as
N

S
S
 ki kj   ij ,
k 1
for all i, j
where ij = 1 if i = j and ij = 0 if i  j thus
if i = j,
N

 S ki S ki  1,
k 1
while if i  j ,
N

 S ki S kj  0.
k 1
5
It is impossible to construct a three-port
lossless reciprocal network. (1)
 If all ports are matched, then Sii = 0, and if the network is
reciprocal the scattering matrix reduces to
0
 S    S12
 S13
S12
0
S 23
S13 
S 23  .
0 
 If the network is lossless, the scattering matrix must be
6
It is impossible to construct a three-port
lossless reciprocal network. (2)
S12  S13  1,
(1a)
S12  S 23  1,
(1b)
S13  S 23  1,
(1c)
S13 S23  0,
(1d )

S23
S12  0,
(1e)
S12 S13  0.
(1 f )
2
2
2
2
2
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.
7
Any matched lossless three-port network
must be nonreciprocal. (1)
 The [S] matrix of a matched three-port network has the
following form:
0
 S    S21
 S31
S12
0
S32
S13 
S 23  .
0 
 If the network is lossless, [S] must be unitary, which
implies the following:
8
Any matched lossless three-port network
must be nonreciprocal. (2)
S12  S13  1,
(2a )
S 21  S 23  1,
(2b)
S31  S32  1,
(2c)

S31
S32  0,
(2d )

S 21
S 23  0,
(2e)
S12 S13  0.
(2 f )
2
2
2
2
2
2
 Either of these followings can satisfy above equations,
or
S12  S23  S31  0,
S21  S32  S13  1,
(3a )
S21  S32  S13  0,
S12  S23  S31  1.
(3b)
9
Any matched lossless three-port network
must be nonreciprocal. (3)
 This results show that Sij  Sji for i  j, therefore the device
must be nonreciprocal.
 These S matrices represent two possible types of
circulators, forward and backward.
10
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
0
 S    S12
 S13
S12
0
S 23
S13 
S 23  .
S33 
 To be lossless, the following unitary conditions must be
satisfied:
11
A lossless and reciprocal three-port network
can be physically realized if only two of its
ports are matched. (2)
S12  S13  1,
(3a)
S12  S23  1,
(3b)
2
2
2
2
S13  S23  S33  1,
(3c)
S13 S23  0,
(3d )
2
2
2

S12 S13  S23
S33  0,
(3e)


S23
S12  S33
S13  0.
(3 f )
 From (3a-b), S13  S23 , so (3d) shows that S13 = S23 = 0.
Then |S12|=|S33|=1.
12
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.
S21=ejq
1
jq
2
S12=e
S33=ejf
3
 If a three-port network is lossy, it can be reciprocal and
matched at all ports.
13
Four-port networks (Directional Couplers)
Input
1
2
Isolated
4
3
Input
1
2
Isolated
4
3
Through
Coupled
Through
Coupled
 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).
14
Basic properties of directional couplers are
described by four-port networks.(1)
0
S
12

S

  S
13

 S14
S12
S13
0
S 23
S 23
0
S 24
S34
S14 
S 24 
.
S34 

0 
 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.
15
Conditions needed for a lossless reciprocal
four-port network (1)
 The multiplication of row 1 and row 2, and the
multiplication of row 4 and row 3 can be arranged so that
S14 ( S13 - S 24 )  0.
2
2
(4)
 The multiplication of row 1 and row 3, and the
multiplication of row 2 and row 4 can be arranged so that
S 23 ( S12 - S34 )  0.
2
2
(5)
 If S14 = S23 = 0, a directional coupler can be obtained.
16
Conditions needed for a lossless reciprocal
four-port network (2)
 Then the self-products of the rows of the unitary [S] matrix
yield the following equations:
S12  S13  1,
(6a)
S12  S24  1,
(6b)
S13  S34  1,
(6c)
S 24  S34  1,
(6d )
2
2
2
2
2
2
2
2
which imply that |S13|=|S24|and that |S12|=|S24|.
17
Symmetrical and Antisymmetrical coupler (1)
 The phase references of three of the four ports are chosen
as S12 = S34 = a, S13 = ejq, and S24 = ejf, where a and 
are real, and q and f are phase constants to be determined.
 The dot products or rows 2 and 3 gives

S12 S13  S24
S34  0
which yields a relation between the remaining phase
constant as
q + f =  2n.
18
Symmetrical and Antisymmetrical coupler (2)
 If 2 is ignored, we yield
1. The symmetrical coupler: q = f = /2.
0
a
 S    j 

0
a
0
0
j
j
0
0
a
0
j  
.

a

0
2. The antisymmetrical coupler: q = 0, f = .
0
a
 S    

0
a

0 0
0 0
- a
0 
-  
.
a 

0 
19
Symmetrical and Antisymmetrical coupler (3)
 The two couplers differ only in the choice of the reference
planes. The amplitudes a and  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=a and
S12=S34=j, two possible solutions are given. First
S14=S23=0, same as above.
 The other solution is for a =  =0, which implies
S12=S13=S24=S34=0, the case of two decoupled two-port
network.
20
Directional coupler’s characterization (1)
 Power supplied to port 1 is coupled to port 3 (the coupled
2
2
port) with the coupling factor S13   .
 The remainder of the input power is2 delivered to port 2 (the
2
2
S

a

1

.
through port) with the coefficient 12
 In an ideal coupler, no power is delivered to port 4 (the
isolated port).
 Hybrid couplers have the coupling factor of 3 dB or a = 
= 1/ 2. The quadrature hybrid coupler has a 90 phase shift
between ports 2 and 3 (q = f = /2) when fed at port 1.
21
Directional coupler’s characterization (2)
P1
= -20log dB,
P3
P
dB,
Directivity = D = 10 log 3 = 20log
P4
S14
P1
Isolation = I = 10 log = -20log|S14| dB.
P4
Coupling = C =
10 log
 The coupling factor indicates the fraction of the input
power coupled to the output port.
 The directivity is a measure of the coupler’s ability to
isolate forward and backward waves, as is the isolation.
These quantities can be related as
I = D + C dB.
22
Ideal coupler
 The ideal coupler would have infinite directivity and
isolation (S14 = 0).
23
The T-junction power divider
 The T-junction power divider can be implemented in any
type of transmission line medium.
24
Lossless divider (1)
+
V0
-
Z0
Z1
jB
Z2
Yin
 A lumped susceptance, B, accounts for the stored
energy resulted from fringing fields and higher order
modes associated with the discontinuity at the junction.
 In order for the divider to be matched to the input line
impedance Z0, and assume a TL to be lossless, we will
have
1
1
1
Yin  
 .
Z1 Z 2 Z 0
25
Lossless divider (2)
 The output line impedances Z1 and Z2 can then be
selected 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 will
have
26
Ex1 A lossless T-junction power divider has a
source impedance of 50 . Find the output
characteristic impedances so that the input power
is divided in a 3:1 ratio. Compute the reflection
coefficients seen looking into the output ports.
27
Resistive divider
 A lossy three-port divider can be made to matched at all
ports, although the two output ports may not be isolated.
Port 2
P1
Z0/3
Z0/3
Port 1
+
V1
-
Z0
Zin
P2
+
VZ
-
+
V2
Z0/3 +
V3
-
Z0
Z0
Port 3
P3
28
The Wilkinson power divider
 The lossless T-junction divider cannot be matched at all
ports and does not have any isolation between output
ports.
 The resistive divider can be matched at all ports but the
isolation is still not achieved.
 The Wilkinson power divider can be matched at all ports
and isolation can be achieved between the output ports.
29
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