power dividers and couplers part 2
DESCRIPTION
power dividerTRANSCRIPT
1
Notes 20
ECE 5317-6351 Microwave Engineering
Fall 2011
Prof. David R. JacksonDept. of ECE
Power Dividers and Couplers Part 2
2
Directional Couplers
Four-Port Networks
Consider a reciprocal 4-port network with matched ports:
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
14 23 0S S
The device is called a “directional coupler” if:
If lossless [S] is unitary.
There are 6 degrees of freedom (independent S parameters).
3
Directional Couplers
Directional coupler:
12 13
12 24
13 34
24 34
0 0
0 0
0 0
0 0
S S
S SS
S S
S S
There are now 4 degrees of freedom.
All ports are matched.
No power flows to port 4 from in the input port 1 (i.e., port 4 is “isolated”).
Ports 2 and 3 are isolated (power incident on port 2 does not get to port 3 and vice versa).
4
Directional Couplers
Directional coupler symbol:
All ports are matched.
No power flows to port 4 from in the input port 1 (i.e., port 4 is “isolated”).
Ports 2 and 3 are isolated (power incident on port 2 does not get to port 3 and vice versa).
1 2
34
Input Through
Isolated Coupled
5
Directional Couplers
2 2
12 13
* *12 13 24 34
13 24
1
0
S S
S S S S
S S
(Dot column 1 with its conjugate)
(Dot column 2 with the conjugate of column 3)
(Dot column 1 dotted with its conjugate, subtract from column 2 dotted with its conj
12 34 S S
ugate)
(Dot column 2 dotted with its conjugate, subtract from column 3 dotted with its conjugate,
and use the with previous result above)
12 13
12 24
13 34
24 34
0 0
0 0
0 0
0 0
S S
S SS
S S
S S
From the unitary property of the matrix, we have the following results:
6
12 34
13 24,j j
S S
S e S e
2 2 2 212 13
* *12 13 24 34
1 1
0 0
1 0
2
j j
jj
S S
S S S S e e
e e
n
Choose
Directional Couplers (cont.)
The first equation is really just a choice of reference planes on ports 2 and 3, which makes these parameters real.
(usually, n = 0)
7
( / 2)
0 0
0 0
0 0
0 0
j
jS
j
j
0;
0 0
0 0 -
0 0
0 - 0
S
Two possible choices:
1) Symmetrical coupler
2) Anti-symmetrical coupler
Directional Couplers (cont.)
Example: 90o quadrature hybrid coupler
Example: 180o rat-race hybrid coupler
8
-1
111
-10log -20logdB PRL S
P
1
3 31
110log 20log -20logdB P
CP S
Assume a signal into port 1
Return loss
Coupling
Directional Couplers (cont.)
Input Through
Isolated Coupled
1 2
34
1P
3P
2P
4P
1P
Ideally + dB
Assume ports 2-4 are matched.
3 31
4 41 41
10log 20log 20logdB P SD
P S S
1
4 41
110log 20logdB P
IP S
Directivity
Ideally + dB
IsolationIdeally + dB
A “hybrid” coupler is one for which the coupling is 3 dB (equal power split at the output ports 2 and 3):
dB dB dBI D C Note :
31
1; 3 dB
2dBS C
9
90o (Quadrature) Hybrid 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.”
http://www.microwaves101.com/encyclopedia/Quadrature_couplers.cfm
Taken from “Microwaves 101”
This is very useful for obtaining circular polarization: There is a 90o phase difference between ports 2 and 3.
10
90o hybrid
1 2
4 3
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.
Quadrature Hybrid Coupler (cont.)
0 1 0
0 0 1-1
1 0 02
0 1 0
j
jS
j
j
11
The signal from port 1 splits evenly between ports 2 and 3, with a 90o phase difference.
The signal from port 4 splits evenly between ports 2 and 3, with a -90o phase difference.
The quadrature hybrid is usually used as a splitter:
90o hybrid
1 2
4 3
+90o out of phase -90o out of phase
21 31S jS
24 34S jS
Quadrature Hybrid Coupler (cont.)
Can be used to produce right-handed circular polarization.
Can be used to produce left-handed circular polarization.
12
0 1 0
0 0 1-1
1 0 02
0 1 0
j
jS
j
j
0Z
4g
1 2
34
4g
0 2Z
0 2Z
plane of symmetry (POS)
0Z
0Z
0Z
0Z0Z
(Branch-line coupler)
A microstrip realization is shown here.
Quadrature Hybrid Coupler (cont.)
Note: We only need to study what happens when we excite port 1, since the structure is symmetric. We use even/odd mode analysis (exciting ports 1 and 4) to figure out what happens when we excite port 1.
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Even Analysis
3 2
4 1
e e
e e
V V
V V
0Z
0Z 0Z
1eV 2
eV
3eV4
eV
V
V
0 2Z
0 2Z
8s gl
4g
OC OCOCOC
OC
0Z
0Z0Z
Quadrature Hybrid Coupler (cont.)
0
0
0
tan
tan / 4
s s sY jY l
jY
jY
0 01 /Y Z
0Z 0Z0Z 2V
4g
1eV 2
eV
0jY 0jY
14
Odd Analysis
3 2
4 1
-
-
o o
o o
V V
V V
0Z 0Z0Z 2V
4g
1oV 2
oV
0- jY0- jY
Quadrature Hybrid Coupler (cont.)
0Z
0Z 0Z
1oV 2
oV
3oV
4oV
V
V
0 2Z
0 2Z
8s gl
4g
SC SCSCSC
OC
0Z
0Z0Z
0
0
0
cot
cot / 4
s s sY jY l
jY
jY
0 01 /Y Z
15
Consider the general case:
0
+
- Y jY
for even
for odd
0
4
0
021 0
1 20
Y
jZ
ABCD ABCDY j
Z
Y
0Z 0Z0 2Z
4g Y
2-port
4
Y YABCD ABCD ABCD ABCD
Quadrature Hybrid Coupler (cont.)
Quarter-wave lineShunt load on line
.
0
0
cos sin
/ sin cos
line
line
line
jZ
ABCD
j Z D
In general:
0 0 / 2
/ 2
lineZ Z
Here :
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0
0
0 0
0
0 0
20 0
0
021 0 1 0
1 120
2 21 0
1 20
2 2
2
2 2
jZ
ABCDY Yj
Z
jZ Y jZ
Y j
Z
jZ Y jZ
jZ Y j jZ Y
Z
Quadrature Hybrid Coupler (cont.)Hence we have
0
+
-
jY
Z
for even
for odd
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0 00
2
0 00 0 0
0
0 0
0
0
1
2 2
11 2
2
11
112
jjZ jZ
ZABCD
j j jjZ jZ
Z Z Z
j j jZ
jj j jZ Z
jZ
jZ
Quadrature Hybrid Coupler (cont.)
Continuing with the algebra, we have
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0
0
0
11
112
e
jZ
ABCDjZ
0
1-0
21-
02
e
j
Sj
Quadrature Hybrid Coupler (cont.)
Hence we have
Convert this to S parameters (use Table 4.2 in Pozar):
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2 3 4
-1
111 0a a a
VS
V
11 0S
- - - - - -1 1 1 1 1 1
11
11 11
1
2 2
10 0
2
e o e o e o
e o
V V V V V VS
V V V V V
S S
0Z
0Z 0Z
0 2Z
0 2Z
4g
4g
1V 2V
4V3V
-3V
-2V
-1V
1V
-4V
0Z
0Z0Z
Quadrature Hybrid Coupler (cont.)
Hence
11 22 33 44 0S S S S By symmetry:
20
- - - -
2 2 2 221 21 21
1
2 2
1 -1- 1-
2 2 2
-
2
e o e oe oV V V V
S S SV V V
j j
j
21 12 43 34
-
2
jS S S S By symmetry and reciprocity:
Quadrature Hybrid Coupler (cont.)
2 3 4
-2
211 0a a a
VS
V
0Z
0Z 0Z
0 2Z
0 2Z
4g
4g
1V 2V
4V3V
-3V
-2V
-1V
1V
-4V
0Z
0Z0Z
21
2 3 4
-3
311 0a a a
VS
V
31 13 24 42
-1
2S S S S By symmetry and reciprocity:
Quadrature Hybrid Coupler (cont.)
- - - - - -
3 3 3 3 2 231 21 21
1-
2 2 2
1 -1- 1--
2 2 2
-1
2
e o e o e oe oV V V V V V
S S SV V V V
j j
0Z
0Z 0Z
0 2Z
0 2Z
4g
4g
1V 2V
4V3V
-3V
-2V
-1V
1V
-4V
0Z
0Z0Z
22
2 3 4
-4
411 0a a a
VS
V
41 14 23 32 0S S S S By symmetry and reciprocity:
Quadrature Hybrid Coupler (cont.)0Z
0Z 0Z
0 2Z
0 2Z
4g
4g
1V 2V
4V3V
-3V
-2V
-1V
1V
-4V
0Z
0Z0Z
- - - - - -
4 4 4 4 1 141 11 11
1- 0
2 2 2
e o e o e oe oV V V V V V
S S SV V V V
23
Summary
0 1 0
0 0 1-1
1 0 02
0 1 0
j
jS
j
j
Quadrature Hybrid Coupler (cont.)
0Z
4g
1 2
34
4g
0 2Z
0 2Z0Z
0Z
0Z
0Z0Z
The input power to port 1 divides evenly between ports 2 and 3, with ports 2 and 3 being 90o out of phase.
24
180o Hybrid Coupler (Rat-Race Ring Hybrid)
http://www.microwaves101.com/encyclopedia/ratrace_couplers.cfm
Taken from “Microwaves 101”
“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.”
Figure 7.42 of PozarPhotograph of a
microstrip ring hybrid.Courtesy of M. D. Abouzahra, MIT
Lincoln Laboratory
25
Rat-Race Ring Hybrid (cont.)
180o hybrid
1 2
4 3
0 1 1 0
1 0 0 -1-
1 0 0 12
0 -1 1 0
jS
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.
26
The signal from the “sum port” (port 1) splits evenly between ports 2 and 3, in phase.
The signal from the “difference port” (port 4) splits evenly between ports 1 and 2, 180o out of phase.
The rat race can be used as a splitter:
180o hybrid
1 2
4 3
In phase
180o out of phase
Rat-Race Ring Hybrid (cont.)
21 31S S
24 34S S
27
The signal from the sum port (port 1) is the sum of the input signals 1 and 2.
The signal from the difference port (port 4) is the difference of the input signals 1 and 2.
The rat race can be used as a combiner:
180o hybrid
1 2
4 3
Signal 1 (V1)
Signal 2 (V2)
Rat-Race Ring Hybrid (cont.)
12 13S S
42 43S S
1 2 12V V S
1 2 42V V S
28
0 1 1 0
1 0 0 -1-
1 0 0 12
0 -1 1 0
jS
2
0Z
02Z
/ 4g
1
3
4
4g
3 / 4g
0Z
0Z
0Z
A microstrip realization is shown here.
Rat-Race Ring Hybrid (cont.)
29
0 1 1 0
1 0 0 -1-
1 0 0 12
0 -1 1 0
jS
180o Hybrid Coupler (Magic T)A waveguide realization of a 180o hybrid coupler is shown here, called a “Magic T.”
“Magic T”
Note the logo!
IEEE Microwave Theory and Techniques Society
Please see p. 361 of Pozar.
30
Rat-Race Ring Hybrid (cont.)
2
0Z
02Z
/ 4g
1
3
4
4g
3 / 4g
Plane of symmmetry
0Z
0Z
0Z
1 2
3 4
4g 3
4g
02Z0Z
2V
1V
V
4V
3V
0Z
/ 4g
/ 4g
02Z02Z
0Z 0Z02Z
Layout Schematic
31
Even Analysis
1 2
3 4
8g 3
8g
02Z0Z
2eV
1eV
V
4eV
3eV
V
0Z
OC
OC
OCOC
02Z
02Z
0
2
jY0
2
jY
0Z 0Z02Z
Rat-Race Ring Hybrid (cont.)
1 0 1 1
0
0
tan
/ 2 tan / 4
/ 2
s s s sY jY l
j Y
jY
0 0
0 0
1/
/ 2s
Y Z
Y Y
2 0 2 2
0
0
tan
/ 2 tan 3 / 4
/ 2
s s s sY jY l
j Y
jY
32
1 2
3 4
8g
3
8g
02Z0Z
-2oV
-1oV
V
-04V-
3oV
-V
SC
SC
SCSC
0Z
0Z 0Z
Rat-Race Ring Hybrid (cont.)
Odd Analysis
0
2
jY 0
2
jY
0Z 0Z02Z
1 0 1
0
0
cot
/ 2 cot / 4
/ 2
s s s sY jY l
j Y
jY
0 01 /Y Z
1 0 2
0
0
cot
/ 2 cot 3 / 4
/ 2
s s s sY jY l
j Y
jY
33
0
0 00
0
0
0
0
0
0
1 0 1 00 2
11 10
2 22
1 0 1 2
11 0
2 2
1 2
21
e
j ZABCD jY jY
jZ
j Z
jYj
Z
j Z
jZ
Proceeding as for the 90o coupler, we have:
Rat-Race Ring Hybrid (cont.)
34
0
0
0
1 2
21
e
j Z
ABCDjZ
0
1 1-
1 12e
jS
Converting from the ABCD matrix to the S matrix, we have
Rat-Race Ring Hybrid (cont.)
Table 4.2 in Pozar
35
21 12 34 43-
2( )
jS S S S
symmetry and reciprocity
2 3 4
-1
111 0a a a
VS
V
2 3 4
-2
211 0a a a
VS
V
Rat-Race Ring Hybrid (cont.)For the S parameters coming from port 1 excitation, we then have:
- -
1 111 11 11
1
2 21 -
2 2 2
0
e oe oV V
S S SVj j
- -
2 221 21 21
1
2 21 - -
2 2 2
-
2
e oe oV V
S S SVj j
j
11 33 0
( )
S S symmetry
36
31 13 -2
( )
jS S
symmetry
22 44
23 32 14 41
24 42
0
0
2
S S
S S S S
jS S
2 3 4
-3
311 0a a a
VS
V
Similarly, exciting port 2, and using symmetry and reciprocity, we have the following results:
Rat-Race Ring Hybrid (cont.)
- -
3 331 11 11
1-
2 2
1 --
2 2 2
-
2
e oe oV V
S S SV
j j
j
37
0 1 1 0
1 0 0 -1-
1 0 0 12
0 -1 1 0
jS
2
0Z
02Z
/ 4g
1
3
4
4g
3 / 4g
0Z
0Z
0Z
Summary
Rat-Race Ring Hybrid (cont.)