Lecture 12 - web page for staff
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Transcript 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
P2aP1
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
unitary that leads to
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