Transcript Power dividers and couplers part 1x
ECE 5317-6351 Microwave Engineering
Fall 2011
Prof. David R. Jackson Dept. of ECE
Notes 19
Power Dividers and Couplers Part 1 1
Power Dividers and Directional Couplers Three-port networks
P
1 Divider
P
3
P
2
P
1
P
1
P
1
P
2
P
3 Coupler (Combiner)
P
2
P
3 General 3-port:
S
11
S
21
S
31
S
12
S
22
S
32
S
13
S
23
S
33 2
Power Dividers and Directional Couplers (cont.) For all ports matched, reciprocal, and lossless: 0
S
12
S
13
S
0 12
S
23
S
0 13
S
23 (There are three distinct values.) Not physically possible Lossless [
S
] is unitary
S
12 2
S
13 2 1
S
12 2
S
23 2 1
S
13 2
S
23 2 1
S
13
S
12
S
12
S
23 0
S
23 0
S
13 0 These cannot all be satisfied.
(If only one is nonzero, we cannot satisfy all three.) At least 2 of
S
13 ,
S
12 ,
S
23 must be zero.
(If only one is zero (or none is zero), we cannot satisfy all three.) 3
Power Dividers and Directional Couplers (cont.) Now consider a 3-port network that is non-reciprocal , all ports matched, and lossless: 0
S
21
S
0 12
S
13
S
23 “Circulator”
S
31
S
32 0 These equations will be satisfied if: (There are six distinct values.) Lossless
S
21 2
S
31 2 1
S
12 2
S
32 2 1
S
13 2
S
23 2 1
S
31
S
21
S
12
S
32
S
23
S
13 0 0 0 1 2
S
12
S
21
S
23
S
32
S
31
S
13 0 1 or Note that
S ij
S ji
.
S
21
S
12
S
32
S
23
S
13
S
31 0 1 4
1 Circulators 0 1 0 0 0 1 1 0 0 Note: We have assumed here that the phases of all the
S
parameters are zero.
Clockwise (LH) circulator
S
21 2
S
32 1 3
S
13 2 0 0 1 1 0 0 0 1 0 Circulators can be made using biased ferrite materials.
S
12 1 2
S
23
S
31 3 Counter clockwise (RH) circulator 5
Power Dividers T-Junction: lossless divider
Y in
1
Z
02
Z
01
Y i n
1 1
Z
02 1
Z
03
Y in
3
Z
03 To mat ch
Y i n
1 Note, however,
Y in
3 1
Z
01
Z
01
Z
02 | |
Z
03 1
Z
01 1
Z
02
Z
02
Z
02 03
Z
03
Z
03 03 1
Z
02
Z
02 2
Z
03 03 1
Z
03
Z
02
Z
02 2
Z
03 Th us ,
Y in
3 1
Z
0 3
Y in
2 1
Z
0 2 If we match at port 1, we cannot match at the other ports!
6
Power Dividers (cont.) Assuming port 1 matched:
Z
01
Z
02 03
Z
03
P in
1 1 2
V
1 2
Z
0 1
P out
2 1 2
V
1 2
Z
02
P ou t
3 1 2
V
1 2
Z
0 3
Z
01
Z
02
P in
1
Z
01
Z
03
P i n
1 1
Z
01
V
1
Z
02
Z
03
Z
0 3
P i n
1
K P in
1
Z
0 2
Z
02
Z
0 3
P i n
1
K
P i n
1
Z
02 2
Z
03 3 We can design the splitter to control the powers into the two output lines.
7
Power Dividers (cont.) For each port we have:
Z
01
S
11
V
1
V
1
a
2 0
Z
02
Z
02
Z
03
Z
03
Z
01
Z
01 zero if port 1 is matched
V
1 + +
V
2 -
Z
02
Z
03
S
22
V
2
V
2
a
1 0
Z
01
Z
01
Z
03
Z
03
Z
02
Z
02
S
33
V
3
V
3
a
1
a
2 0
Z
01
Z
01
Z
02
Z
02
Z
03
Z
03 8
Power Dividers (cont.) Also, we have
Z
02
S
21
V
2
Z
02
V
1
Z
01
a
2 0
Z
01 +
V
1 +
V
2 -
V
1
V
1 1
S
21
S
11
S
11 ;
V
2
V
2
V
1
Z
01
Z
02
S
12 Similarly,
S
31
S
13
S
11
Z
01
Z
0 3 and
S
32
S
23
V
2 /
V
1
V V
1 / 1 1
S
11
S
22
Z
02
Z
0 3
Z
03 9
Power Dividers (cont.) If port 1 is matched:
Z
01
Z
02 03
Z
03
S
11 0 ;
S
22
Z
01
Z
01
Z
03
Z
03
Z
02
Z
02 ;
S
33
Z
01
Z
01
Z
02
Z
02
Z
03
Z
03
S
21
S
12
S
13
S
31
S
32
S
23
S
11
S
11
S
22
Z
01
Z
02
Z
01
Z
03
Z
01
Z
02
Z
01
Z
03
Z
02
Z
03 1
S
22
Z
02
Z
03
Z
03
Z
02
Z
02
Z
03
Z
02
Z
03 0
S
21
S
31
S
21
S
22
S
32
S
31
S
32
S
33 The output ports are not isolated.
10
Powers: Power Dividers (cont.)
P
2
P
1
S
2 1 2
Z
02
Z
03
Z
03
P
3
P
1
S
31 2
Z
0 2
Z
02
Z
03 Hence
P
3
P
2
Z
02
Z
03 Check :
P
1
P
2
P
3
Z
02
Z
03
Z
0 3
P
1
Z
0 2
Z
02
Z
03
P
1
P
1 11
Power Dividers (cont.) Summary
Z
02
Z
01
Z
02 03
Z
03 1
Z
01 2
Z
03 3
S
11 0 ;
S
22
Z
01
Z
01
Z
03
Z
03
Z
02
Z
02 ;
S
33
Z
01
Z
01
Z
02
Z
02
Z
03
Z
03
P
3
P
2
Z
02
Z
03
S
21
S
12
S
13
S
31
S
32
S
23
Z
02
Z
03
Z
03
Z
02
Z
02
Z
03
S
22
Z
02
Z
03 The input port is matched, but not the output ports.
The output ports are not isolated.
12
Power Dividers (cont.)
S
11 0
S
22
S
33 1 2
S
21
S
12
S
31
S
13 1 2 1 2
S
32
S
23 1 2 Example: Microstrip T-junction power divider
Z
02
Z
01
Z
01
Z
02
Z
01
Z
03 50 100
Z
03 13
Resistive Power Divider
Z in
1
Z
0 3 4
Z
0 3 4
Z
0 3
Z
0 3 2
Z
0 3
Z
0 Same for
Z in
1 and
Z in
2 All ports are matched.
S
11
S
22
S
33 0
Z
0
Z
0
Z in
1
V
1
Z
0 3 port1
Z
0 3
V Z
0 3
V
3 port3 port2
Z
0 14
Resistive Power Divider (cont.)
S
21
V
2
Z
0
V
1
Z
0
a
2 0
V
1
V
1 1
S
11
V
1
V
2
V
2
V
1
V
1
Z
3 0 2 3
Z
0 2 3
Z
0 2 3 3 1 2
V
1
Z
0
Z
0
Z
0 3
Z
0
Z in
1
V
1 port1
Z
0 3
Z
0 3
V Z
0 3
V
3 port3 port2 By reciprocity and symmetry
Z
0
Z
0
S
21 1 2
S
12
S
3 1
S
13
S
32
S
23 15
Resistive Power Divider (cont.)
P
2 Hence we have
P
1
Z in
1
Z
0 3
Z
0 3
V
2 0 1 1 1 0 1 1 1 0
Z
0
V
1
Z
0 3
V
3 port1 port3
P
1
P in
1 2
V
1 2
Z
0 1 2
a
1 2
P
2
P
3 1 2
b
2 2 1 2 21 2 21 2
P
1 1 2 2
P in
4 All ports are matched, but 1/2
P in
output ports are not isolated.
is dissipated by resistors, and the
Z
0 port2
P
3
Z
0 16
Even-Odd Mode Analysis (This is needed for analyzing the Wilkenson.) Example: We want to solve for
V
.
Let
V S e
V S o
1 2
V S
using even/odd mode analysis
V S
2 Obviously,
V
V S
2
V
2 1
V S
3
V S e
2 POS 2
V e
2 “even” problem Plane of symmetry
V S e
V S o
2 POS 2 V 0 2
V
V e
V o
“odd” problem
V S o
17
V S
2
V S e
2 Even-Odd Mode Analysis (cont.) “Even” problem POS POS 2
I
0 2 2
V e
2
V S e V S
2 4
V e
4 Open circuit (OC) plane of symmetry 2
V S
2 4
V e
V e
V S
2 4 2 4
V S
3 18
V S o
Even-Odd Mode Analysis (cont.) “Odd” problem POS POS 2
V o
2 2
V S o V S
2 2 4
V o
0 4 2
V S
2 short circuit (SC) plane of symmetry 2
V S
2 4
V o
V o
0 19
V S
2 Even-Odd Mode Analysis (cont.) 2 2
V S
V
2 2
V e
2 2
V S
2
V S
2 2
V o
2 2 “even” problem By superposition:
V
V e
V o
V S
3
V
V S
3 0 “odd” problem
V S
2 20
Wilkenson Power Divider Equal-split (3 dB) power divider
Z
0 1 g / 4 2
Z
0 2 g / 4 2
Z
0 3 2
Z
0
Z
0
Z
0 All ports matched (
S
11 =
S
22 =
S
33 = 0 ) Output ports are isolated (
S
23 =
S
32 = 0 ) Note: No power is lost in going from port 1 to ports 2 and 3.
S
21 2
S
31 2 1 2 2 0 1 1 0 1 0 1 0 0 Obviously not unitary 21
Z
01 Wilkenson Power Divider (cont.) Example: Microstrip Wilkenson power divider
Z
02
Z
0
T
Z
0
T
R
Z
03 22
Wilkenson Power Divider (cont.) • Even and odd analysis is required to analyze structure when port 2 is excited.
To determine
S
22 ,
S
32 • Only even analysis is needed to analyze structure when port 1 is excited.
To determine
S
11 ,
S
21 The other components can be found by using symmetry and reciprocity.
23
Wilkenson Power Divider (cont.) Top view
Z
0
V
1
V
1
g
/ 4 2
Z
0
g
2
Z
/ 4 0 A microstrip realization is shown.
Z
0 2
Z
0
Z
0
V
2
V
2
V V
3 3 Plane of symmetry Split structure along plane of symmetry (POS) Even Odd voltage even about POS voltage odd about POS place OC along POS place SC along POS 24
Wilkenson Power Divider (cont.)
Z
0
V V
1 1
g
/ 4
g
2
Z
0 / 4 2
Z
0
Z
0 2
Z
0
Z
0
V V
2 2 Plane of symmetry
V V
3 3 How do you split a transmission line? (This is needed for the even case.) top view
I
/ 2
Z
0 POS Voltage is the same for each half of line (
V
) Current is halved for each half of line (
I
/2 )
I
/ 2 (magnetic wall)
Z
0 microstrip line
Z
0
h
I V
2 2
Z
0 For each half 25
Wilkenson Power Divider (cont.) “Even” Problem 2
Z
0
V
1
e
g
/ 4 2
Z
0
V
2
e
Z
0 OC
Z
0 2
Z
0
V
1
e
OC 2
Z
0
Z
0
g
/ 4
V
3
e Z
0 Note :
V
3
e
V
2
e V
V
e
Ports 2 and 3 are excited in phase.
V
V
e
Note: The 2
Z
0 resistor has been split into two
Z
0 resistors in series.
26
Wilkenson Power Divider (cont.) “Odd” problem 2
Z
0 2
Z
0
V
1
o
g
/ 4 2
Z
0
V
2
o
Z
0
Z
0
V
V
o
Note: The 2
Z
0 resistor has been split into two resistors in series.
Z
0 Ports 2 and 3 are excited 180 o out of phase.
V
1
o
2
Z
0
Z
0
V
g
/ 4
V
3
o Z
0
V
o
Note :
V
1
o
0,
V
3
o
V
2
o
27
Wilkenson Power Divider (cont.) Even Problem Port 2 excitation
V
1
e
g
/ 4
Z e in
2 ,
S e
22
V
2
e
2
Z
0 OC
Z
0
V
V
e
Port 2
e Z in
2
e S
22 2
Z
0 2 2
Z
0
e Z in
2
Z in e
2
Z
0
Z
0
Z
0 0 2
Z
0 Also, by symmetry,
e S
33 0 28
Wilkenson Power Divider (cont.) Odd Problem Port 2 excitation 2
Z
0
o Z in
2 ,
o S
22
V
2
o
g
/ 4 2
Z
0
V
1
o V
1
o
short 0
Z
0
Z
0
V
V
o
Port 2
Z in o
2
o S
22
Z
0
Z Z in o
2
Z in o
2
Z
0
Z
0 0 0 Also, by symmetry,
o S
33 0 29
S
22
V
2
V
2 Wilkenson Power Divider (cont.) We add the results from the even and odd cases together: 0
S
22
V V
e
V
o
V
V
e
V
o
2
V
1 2
e S
22
o S
22 2 1
S
33 0 (by symmetry) 0
S
32
V
3
V
2
a
1
a
0
S
32
V
e V
V
o
V
V
e
2
V
V
o
1 2
e S
22
o S
22 2 1
S
2 3 0 (by reciprocity) Note: Since all ports have the same
Z
0 , we ignore the normalizing factor
Z
0 in the
S
parameter definition. In summary,
S
22
S S
33 32 0 0
S
23 0 0 30
Wilkenson Power Divider (cont.)
Z
0 Port 1 excitation Port 1
Z
0
V
1
V
1
g
/ 4
g
2
Z
/ 4 0 2
Z
0
Z
0 2
Z
0 When port 1 is excited, the response, by symmetry, is even. (Hence, the total fields are the same as the even fields.)
V
2
V
3 Even Problem
Z
0
Z
0
V V
1 1
g
/ 4 2
Z
0
g
/ 4 2
Z
0 2
Z
0
Z
0 O.C. symmetry plane 2
Z
0
V V
1 1
g
/ 4 2
Z
0 OC
Z
0 31
Wilkenson Power Divider (cont.) Even Problem Port 1 excitation
Z e in
1 2
Z
0 2 2
Z
0
Z
0
S e
11
e Z in
1
Z e i n
1 2
Z
0 2
Z
0 0
S
11
V
1
V
1
a
2
a
3 0
V
1
V
1
e e a
2 0
e S
11 0 2
Z
0 2
V
1
V
1
e e
g
/ 4 2
Z
0 1
Z e in
1 ,
e S
11 OC
Z
0 Hence
S
11 0 32
Wilkenson Power Divider (cont.) Even Problem
V
1
V
1 Port 1 excitation
S
21
V
2
V
1
a
2
g
/ 4 2
Z
0
z
0 2
Z
0 OC
Z
0
V
1
V
1
V
2
V
2 1
V
2
S
11
V
1
S
21
V
2
V
1
V
2
V
1
j
1 1 2
j
2 2
S
21 2
j
S
12 (reciprocal) Along
g
/4 wave transformer:
V e
0 1
e
j
2
z
V
2
V
1
z
distance from port 2
V
V
g
V
0 / 4 1
V
0
j
1
Z
0
Z
0 2
Z
0 2
Z
0 1 1 2 2
V
2 33
Wilkenson Power Divider (cont.) For the other components: By symmetry:
S
31
S
21
j
2 By reciprocity:
S
13
S
31
j
2 34
Wilkenson Power Divider (cont.) 2 0 1 1 0 1 0 1 0 0
Z
0
Z
0
g
/ 4 2
Z
0
g
2
Z
0 / 4 2
Z
0
Z
0
S
11
S
22
S
33 0
S
32
S
23 0 All three ports are matched, and the output ports are isolated.
35
Wilkenson Power Divider (cont.)
Z
0 2 0 1 1 0 1 0 1 0 0
Z
0
g
/ 4 2
Z
0
g
2
Z
0 / 4 2
Z
0
Z
0
S
21
S
31
j
2
S
12
S
13
j
2 When a wave is incident from port 1, half of the total incident power gets transmitted to each output port (no loss of power).
When a wave is incident from port 2 or port 3, half of the power gets transmitted to port 1 and half gets absorbed by the resistor, but nothing gets through to the other output port. 36
Wilkenson Power Divider (cont.) Figure 7.15 of Pozar Photograph of a four-way corporate power divider network using three microstrip Wilkinson power dividers. Note the isolation chip resistors.
Courtesy of M.D. Abouzahra, MIT Lincoln Laboratory.
37