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微波電路講義7-1
Chapter 7 Power dividers and directional couplers
7.1 Basic properties of dividers and couplers
three-port network (T-junction), four-port network
(directional coupler), directivity measurement
7.2 The T-junction power divider
lossless divider, lossy divider
7.3 The Wilkinson power divider
even-odd mode analysis, unequal power division divider,
n-way Wilkinson divider
7.5 The quadrature 90 hybrid
branch-line coupler
7.6 Coupled line directional couplers
even- and odd-mode Zo, single-section and multisection
coupled line couplers
微波電路講義7-2
7.7 The Lange coupler
7.8 The 180 hybrid
rat-race hybrid, tapered coupled line hybrid
7.9 Other couplers
reflectometer
微波電路講義7-3
7.1 Basic properties of dividers and couplers
• N-port network
N-port
networkjkVj
iij
NNNN
N
N
k
V
VS
SSS
SSS
SSSS
SSSS
,0
21
333231
2232221
1131211
1
2
3
4
Discussion
1. matched ports
2. reciprocal network symmetric property
3. lossless network unitary property
0iiS
jiij SS
2 *
1 1
1 , 0 N N
ij ki kj
i k
S j S S i j
N
微波電路講義7-4
• three-port network (T-junction)
divider
or coupler
12
3
P1
P2= a2P1
P3 = b2P1
combiner
or coupler
1
2
3
P1 =a2P2 + b2P3
P2
P3
Discussion
1. Three-port network cannot be lossless, reciprocal and matched
at all ports.
2. A lossless and matched three-port network is nonreciprocal
circulator
1 2
3
010
001
100
微波電路講義7-5
1 2
3
001
100
010
3. A matched and reciprocal three-port network is lossy resistive
divider
4. A matched and lossy three-port network can have ∞ isolation at
two output ports (S23=S32=0) Wilkinson power divider
微波電路講義7-6
(derivation of 1)
For a matched, reciprocal three-port network2 2
*12 13
12 13 13 23 122 2 *
12 23 12 23 12 13 13 23
*2 213 23 12 23 13
13 23
S S 10 S 0 1
0 S S 1 , S 0 if 0, 0 lossy
0 S 0 1 0S S 1
losslessS S S S
S S S S S
S S S S
(derivation of 2)
For a matched, lossless, nonreciprocal three-port network2 2
*21 31
12 13 31 322 2 21 31 23 13 12 32*
21 23 12 32 12 13
21 31 32 12 13 23*2 231 32 21 23
13 23
S S 10 S 0if S =1 S =0,S =0,S =1,S =0,S =1
0 S S 1 , S 0if S =0 S =1,S =0,S =1,S =0,S =1
0 S 0S S 1
losslessS S S
S S S
S S S
0 0 1
1 0 0
0 1 0
0 1 0
0 0 1
1 0 0
1 2
3
1 2
3
微波電路講義7-7
• four-port network (directional coupler)
1 2
3
input port P1 P2 through port
P3 coupled port4
isolated port P4
1 2
34
1 2
3 4
131
3
313 3 1
4 1 4 41
141
4
20 3
1
coupling ( ) 10log 20log (>0 )
Sdirectivity ( ) 10log 10log 20log (>0 )
S
isolation ( ) 10log 20log (>0 )
voltage coupling factor 10 1
direc
C(dB)/
PC dB S dB
P
P P PD dB dB
P P P
PI dB C D S dB
P
VC
V
( ) / 20 3
4
tivity 10 1D dB VD
V
3 4
4 3
a measure of the coupler's ability to isolate forward and backward/port 1 port 3
i/p coupled directivity : waves (or the coupled and isolated ports),/port 2 port 4
the ability to differentiate tw
P P
P P
o input signals from opposite directions
微波電路講義7-8
Discussion
1. Matched, reciprocal and lossless four-port network symmetrical
90 directional coupler or antisymmetrical 180 directional
coupler
2. C=3dB 90 hybrid (quadrature hybrid, symmetrical coupler),
180 hybrid (magic-T hybrid, rate-race hybrid)
00
00
00
00
0j0
00j
j00
0j0
ab
ab
ba
ba
ab
ab
ba
ba
0110
1001
1001
0110
2
1
01j0
100j
j001
0j10
2
1
微波電路講義7-9
(derivation of 1)
For a matched, reciprocal and lossless four-port network
*24
* * *12 13 14 13 23 14 24 (1)
* * *12 23 24 14 13 24 23
* * *13 23 34 12 23 14 34
* * *14 24 34 14 12 34 23
0 row 1 , 2: 0...(1)
0 row 3, 4 : 0...(2)
0 row 1 , 3: 0...(3)
0 row 2, 4 : 0...(4)
SS S S S S S S
S S S S S S S
S S S S S S S
S S S S S S S
*13
12 34
(2)2 2*
14 13 24
(3) (4)2 2
23 12 34
2 2
12 13 12 3412 132 2
1312 2412 24 13 24
14 23 2 213 34 12 3413 34
2 224 34
24 34
( ) 0
( ) 0
10 0
10 0case 1: 0
0 0 1
0 01
S
S S
jchoose
S S S
S S S
S S S SS S
S eS SS S S SS S
S S S SS S
S SS S
a
b
24
2 2
* * *
12 13 24 34
+ =1
row 2 ,3 0 0
(a) 90 directional coupler with (b) 180 directional coupler wit2
0 0 0 0
0 0 0 0,
0 0 0 0
0 0 0 0
j
j j
o o
S e
S S S S e e
j j
j j
j j
j j
b
a b
a b a b
a b a b
b a b a
b a b a
h 0,
0 0
0 0
0 0
0 0
a b
a b
b a
b a
微波電路講義7-10
*13 24 13 24 23 14
*12 34 12 34 23 14
14 23
14
23
23
14
(1) : ( ) 0case 2 : if
(3) : ( ) 0
( ) 0 directional coupler as case 1
0 0 0
0 0 0( ) 0 , two decoupled two-port
0 0 0
0 0 0
chooseS S S S j j S S
S S S S S S
a S S
S
Sb
S
S
b b
a a
a b
networks
1 2
3
input port P1, 1 P2 through port, a2
P3 coupled port, b2
4
isolated port P4, 0
*24
* * *12 13 14 13 23 14 24 (1)
* * *12 23 24 14 13 24 23
* * *13 23 34 12 23 14 34
* * *14 24 34 14 12 34 23
0 row 1 , 2: 0...(1)
0 row 3, 4 : 0...(2)
0 row 1 , 3: 0...(3)
0 row 2, 4 : 0...(4)
SS S S S S S S
S S S S S S S
S S S S S S S
S S S S S S S
*13
12 34
(2)2 2*
14 13 24
(3) (4)2 2
23 12 34
( ) 0
( ) 0
S
S S
S S S
S S S
微波電路講義7-11
3. directivity measurement
If C=20dB, D=35dB, RL=30dB (require <D=35dB!)
1 2
3 4
13 (C, -20dB)4 (C/D, -55dB)
124 (C, -50dB)3 (C/D, -85dB)
CV1+
C/DV1+
V4-
V-4maxV-
4min
2
4max 1
2
4min 1
2 33 1
4max
4max
4min
1For
with the use of a sliding load
( )
( )
,1
1
1
2
1
j
ΓD
e
CP P C Γ
D
CP P C Γ
D
P DP C P M
P Γ D
Γ DPm
P Γ D
mD M
m
33 1 4 1
4
,V C
V CV D V VV D
2
4 1 1 3 1( 1 ),C
V C V C C V V VD
-20dB > -55dB (C/D) -85dB -50dB (ΓC)
-30dB
4
1
: 3
int :
another solution different circle of V
Ccenter po V D directivity
D
1/D
微波電路講義7-12
1 2
3 4
11 12 13 14 11 12 13 141 1 1 1
12 22 23 24 12 22 23 242 2 2 2 2
13 23 33 34 13 23 33 343 3 3
14 24 34 44 14 24 34 444 4 4
0
0
S S S S S S S Sb a b a
S S S S S S S Sb a b b
S S S S S S S Sb a b
S S S S S S S Sb a b
1 11 1 12 2 2
2 12 1 22 2 2
3 13 1 23 2 2
4 14 1 24 2 2
122 1
22 2
3 1213 23 2
1 22 2
4 1214 24 2
1 22 2
1
1
1
b S a S b
b S a S b
b S a S b
b S a S b
Sb a
S
b SS S
a S
b SS S
a S
2
31
31
41
0.077 69 0.686 0.8 0.684 89 0.114 24
0.686 0.8 0.092 54 0.111 19 0.686 92Given
0.684 89 0.111 19 0.106 72 0.680 0.1
0.114 24 0.686 92 0.680 0.1 0.114 64
0.684, 3.3 , 6,
o o o o
o o o o
o o o o
o o o o
SS C dB D
S
30
202
3 4
1 1
4 4
1 1
15.56
30 ( 15.56 ), 10
mean max0.6840 0.1288
=5.3089, =1.30040.1288 0.0990
max min
26.0022, 20log6=15.56dB
1
j
dB
RL dB D dB e
b b
a aM m
b b
a a
mD M
m
0 100 200 3000
0.1
0.2
0.3
0.4
0.5
0.6
| b 3 / a
1 |
| b 4 / a
1 |
微波電路講義7-13
7.2 The T-junction power divider
• lossless divider
Zo
Z2
Z3
Yin
jB
2 3
1 1 1 0 "not practical"
A lossless divider has mismatched ports.
in
o
Y jB BZ Z Z
1
2
3
Γ2
Γ3Γ1
微波電路講義7-14
Discussion
1. Ex. 7.1 Zo=50, P2:P3=1:2, calculate 2 and 3.
1 2 3
2 3
2 3
2 3
322 3
2 3
1 2 1 1 1: : 1: : : :
3 3
33 150 , 75
2
// 50
50 //15050 // 750.666, 0.333
50 // 75 50 //150
o
o o
in
P P PZ Z Z
Z Z Z Z
Z Z Z
ZZΓ Γ
Z Z
2. It’s a lossless and mismatched three-port divider, and not good
in isolation.
微波電路講義7-15
• resistive (lossy) divider
Zo
Zo
ZoZin
1
2
3
R
R
R+
V1
_
+
V2
_
+
V3
_
matched ports ( ) //( ) 3
oo o o
ZR Z R Z R Z R
Discussion
1.
1 1
2 3 1
2
2 23 ,0 1 1 22 3 3
11 0 1 3
23 11 1 0
,4 2
3
oo
in o
o
o
oo
ZR Z
Z Z V V V
R ZS
ZV V V V V
ZZ
+
V_
微波電路講義7-16
2.
3. It’s a lossy and matched three-port divider, and not good in
isolation.
212 2 2
1 2 12 3
22 21 1
1 1, @1
22 21 1
2 1, @2,3
( )1 1 1 12, 2 2 2 8 4
2( )
( )1 1 13= =2 2 6 3
3
2 1( )
( )1 1 13 2= 2 2 24 12
3
lossy divider
inin
o o o o
inloss R
o o
inloss R
o o
VPV V V
P P PZ Z Z Z
V VPV V V
PZR Z
V VPV V V
PZR Z
微波電路講義7-17
7.3 The Wilkinson power divider
• basic concept
Zo 1
2Zo
Zo
Zo
2
3
4/,Z2 o
4/,Z2 o
Input port 1 is matched→ Port 2 and port 3 have equal voltage and power.
input port 2 or 3 → Port 2 and port 3 have perfect isolation.
=> a lossy, matched and good isolation (equal phase) three-port
divider
4/,Z2 o
01
= 0 02
0 0
j j
S j
j
微波電路講義7-18
• even-odd mode analysis
A linear, symmetric network
2Zo
R/2
Zo
Zo
4/,Z2 o
4/,Z2 o
R/22Zo
+ |
4V 2V 2V
2V -2V
even- odd-
mode excitation
even mode
O.C.
odd mode
S.C.
+
≡
微波電路講義7-19
2Zo Zo4/,Z2 o
+ |
even-mode
2V21
0 /422 33
2 3
1
1
2 2
4 4 2 22
1
ports 2, 3 matched 0, 0
,symmetry of ports 2 and 3
2 2 12
12 2
(derivation of ) ( )
(1 )
e e
e e
e
j z j z
e
j j j je
e
S S
V V V V
ΓΓ V jV j V
Γ
V V z V e V e
V V e V e V e V e jV V
V V V V
b b
2 21
1 42 2(1 ) 2
1 2 2 2 21
2 2
j V j V j V j V
e
in oZ Z
微波電路講義7-20
Zo4/,Z2 o
+ |
odd-mode 2V21
22 1
22 33
2 3 1
2
2 2 2
2
311 1 1 3 3 3
31
1 112 21 31
2 20, 0
2 ports 2 and 3 matched 02
, , 0
2
22 , 0
0
2 1
2 2L
o o
o o
o o o
e
e o
o
eee o e o
oo
S
RZ R Z S S
V V V V V
V VV V V V
V V
V VV j VV V V j V V V V
V VV
V V j VS j S S
V V V
22 3
13
3 332 23
2 2 0, 0
00
2LS
S
V VS S
V V V
R/2o
in oZ Z
微波電路講義7-21
Discussion
1. A 3dB Wilkinson power divider has equal amplitude and phase
outputs at port 2 and port 3.
2. Ex. 7.2 A 3dB Wilkinson power divider has Zo=70.7 and R=100.
frequency response (p.332, Fig. 7.12)
3. 3dB Wilkinson power combiner2 3
1
2 2
3 3
1( )
0 0 21
0 0 02
0 0 0
j a ab j j
b j a
b j a
2
3
a
a
2 2 2 2 2
2 3 1 2 3 2 2 2 3 2 3
2 2
2 3 1 2 3 2 3
2 2
2 3 2 3 2 3 3 22 3 2 31
2
23 2 3 3 2
2 32 3
1 1 1 1 1if ( ) ( 2 ) ( )
2 2 2 2 2
1 1 1if , ( )
2 2 4
1 1 1( )
2 2 2 2 2 2 22 2/ ,
1 1 =22 2 2 2 4
a a P a a a a a a P P
a a P a a a a
a a a a a a a aa a a a Pa
i p even odda a a a a a a
a a
2 2 2
2 3 2 3
1( )
2a a P P
1b
微波電路講義7-22
(Proof of )
2
3
a
a 1b
2 2 2
1 2 3 2 3 2 3
1 1( )
4 2P a a a a P P
2 2 2* * * *
1 2 3 2 3 2 3 2 3 2 3 2 3
2 2 2 2 2 2 2 2*
2 3 2 3 2 3 2 3 2 3 2 3
2 2 2* * * *
2 3 2 3 2 3 2 3 2 3 2 3
2 2 *
2 3 2 3
2*
2 3 2
1 1 1
4 4 4
1 1 1 1 1 1Re[ ]
4 2 4 4 2 2
2Re[ ] 0
2Re[ ]
P a a a a a a a a a a a a
a a a a a a a a a a P P
a a a a a a a a a a a a
a a a a
a a a
2 2 2*
3 2 3 2 3
1 1Re[ ]
2 4a a a a a
微波電路講義7-23
4. unequal power division (Wilkinson power divider)
Zo
1 R
Zo
Zo5
2
3
2 ,4
oZ
3 ,4
oZ
Zo4
Zo
Zin2
Zin3
R2
R3
3
2
232
2
2
22
3
2
32
2
3
32
(3)
(2)
//match 1port (1)
inin
inin
inino
ZKZV V
Z
VK
Z
VK
P
P
ZZZ
22
2 3 2
2
2 3 2 3 4 5
22
2 2 2 3 3 3 3
1(1),(3) (1 ) ,
(2) , let , ,
1(1 ) ,
in o in o
o oo o o o
o in o o in o
KZ K Z Z Z
K
Z ZR K R R KZ R Z K Z Z
K K
KZ Z R K K Z Z Z R Z
K
微波電路講義7-24
R
Zo5
Zo4I2 VaZo2
Zo3
Zo
Zo
Zo
+ |
2V
V1
I1’=0 I1 “GND”
IR
1 11 2
3 2 2
2 2 2 211 2
3 3 3
2
2 3
, ,
0 ( )
1, 1 for a 3dB Wilkinson divider
ao
o o
a o o o oR
o o o
o oo
o
VV VI I Z
jZ jZ I
V jZ I Z I Z ZVI I jI
R jZ R jZ R Z
Z Z KR Z K
Z K
1 24.11 2 2
1 21 2 2
2
cos sin 0
sincos 0
tableo o o
o o o
jZ jZ V jZ IV V V
j j jI VI I I
Z Z Z
,oZ 1V 2V
1I 2I
微波電路講義7-25
5. N-way Wilkinson power divider (not in a planar shape)
Zo4/,ZN o
R Zo
NZo
R=Zo
Zo
微波電路講義7-26
7.5 The quadrature 90 hybrid
• branch-line hybrid
Port 2 and port 3 have equal amplitude and 90 phase difference.
4/,2/Zo
4/,Zo 4/,Zo
4/,2/Zo
1 2
34
0 1 0
0 0 11=
1 0 02
0 1 0
j
jS
j
j
• even-odd mode analysis
1 2
34
1 b2
b3b4
b1
1 1,
2 2
1 1,
2 2
微波電路講義7-27
1 11 2 21
3 31 4 41
1 1 1 1,
2 2 2 2
1 1 1 1,
2 2 2 2
e o e o
e o e o
b S b T T S
b T T S b S
1,
4 2
1 2
,18
,18
O.C. O.C.
even-mode
4 3
1
2
1
2e
1 2
S.C. S.C.
odd-mode
1
2oT
1
2
1
2o
4 3
1
2
1
2o
1
2oT
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
b b b b
b b b b
b b b b
b b b b
1
2
1
2e
1
2eT
1
2eT
微波電路講義7-28
4/,Zo
2
table 4.1
cos sin 0
sin cos 0
1 0
1
open-circuit stub8
tan4
short-circuit stub tan8 4
even-mode: o8
o o
o o
oin o in o
in o o in o
jZ jZA B
jY jYC D
A B
C D Y
ZZ j jZ Y jY
Z jZ jZ Y jY
1 0pen-circuit stub
1
1 0odd-mode: short-circuit stub
18
o
o
jY
jY
Y
2
11 212 2
2,o o o o
o o o o o o
AZ B CZ DZ ZS T S
AZ B CZ DZ AZ B CZ DZ
微波電路講義7-29
1 2 3 4
01 0 1 0 112
1 1 122 0
01 0 1 0 112
1 1 122 0
1 10 0 (1 ), (1 )
2 2
10, , , 0
2 2
e o e o
jA B j
C D j j je j
jA B j
C D j j jo j
, ,T j T j
jb b b b
Discussion
1. Unequal power division branch-line coupler uses Za, Zb /4 lines
(prob. 7.18).
2. Ex. 7.5, frequency response (p.346, Fig. 7.25), BW: 10% ~ 20%
3. Multisection branch-line couplers can increase the operation BW.
Zb
Zb
ZaZa
微波電路講義7-30
4. Smith chart consideration
1 2
open short
8/,oZ
4/,2/Zo 1 2
8/,oZ
4/,2/Zo
jyj
zj
ZZ o ,
1, jyjzjZZ o ,,
G
L y=1+j
2
)1( jy
2
)1( jy
jy 1 G
L y=1+j
2
)1( jy
2
)1( jy
jy 1
even-mode odd-mode
微波電路講義7-31
7.6 Coupled line directional couplers
• coupled line theory
W S WC12
C11 C22
C11 C22
2C12
C11 C22
2C12
even-mode excitation
+V +V
H wall
odd-mode excitation
+V -V
11 22
11 12
1
2
1
e
oe
e e
o
oo
o o
C C C
Zv C
C C C
Zv C
E wall
微波電路講義7-32
Discussion
1. In general vevo (∵effe>effo), only for TEM mode ve = vo = v.
2. Zoe, Zoo (W/b, S/b) (p.349, Fig. 7.29 and p.350, Fig. 7.30) are given
for coupled striplines and microstrip lines.
3. Zoe > Zoo, W/b Zoe Zoo , S/b Zoe Zoo and
Zoe, Zoo Zo.
4. Ex.7.6 derive Zoe, Zoo of coupled striplines (p.347, Fig.7.26(b))
b S
W
11 122 2
11 11 12
2 2
2 2
4,
( ) 2 ( ) 2
, 2
1
4
1 1
2 [2 /( ) 1/ ]
e o
oe o
re
oo o
ro
W W b W WC C
b S b S b S S
C C C C C
b SZ Z
vC bW
Z ZvC W b b S S
微波電路講義7-33
• coupled line coupler
1 2
3 4
S
W
20 1 0
21 0 0
20 0 1
20 1 0
matched port
,2
1 1design equations: , ,
1 1
o oe oo
oe oo
oe oo
oe o oo o
j C C
j C C
C j C
C j C
Z Z Z
Z ZC
Z Z
S
C CZ Z Z Z S W
C C
微波電路講義7-34
Discussion
1. Design procedure: given C and Zo, calculate Zoe and Zoo, then
use Fig. 7.29 or 7.30 to find W/b and S/b for stripline or
microstrip.
2. even-odd mode analysis
1 2
3 4
| +
2V
+ +
V3, I3 Zoe, Zoo I4, V4
+ I1 I2 +
V1 V2
even odd
V -V
V V
1 3 2 4 1 3 2 4
1 3 2 4 1 3 2 4
even mode: , , ,
odd mode: , , ,
e e e e e e e e
o o o o o o o o
I I I I V V V V
I I I I V V V V
微波電路講義7-35
1 1 1 2 2 2 3 1 1 4 2 2
1 1 1 2 2 2 3 1 1 4 2 2
,,, , ,
, 1 1, ,
,
1 1 1
1 1 1
, , ,
, , ,
tan, ,
tan
e o
e o e o e o e o
e o e o e o e o
e oo oe oe o e o e oin
in oe o e o e o
oe o o in o in o
e o
in oe o
I I I I I I I I I I I I
V V V V V I V V V V V V
Z jZ Z VZ Z V V I
Z jZ Z Z Z Z
V V VZ Z
I I I
2
2
,,
1
3 1 1
32
2( )...(1)
2
( ) ...(2)
tan...(3)
2 ( ) tan
( ) tan
2 ( ) tan
tan
1 tan
e o
in in o
e o
in in o
e o
o o in in oe oo
o oe oe o
o oe oo
e o oe oo
o oe oo
oe oo
oe oo
Z Z Z
Z Z Z
Z if Z Z Z Z Z
Z jZV V
Z j Z Z
j Z ZV V V V
Z j Z Z
Z Z jCC V V
Z Z C j
...(4)
微波電路講義7-36
1 1 1
1 1 1
2
2
derivation of (1) :
( ) ( ) ( 2 ) 2 2
2 2
2( )
e o
in in
e oe o
in o in oin e o
e o
in o in o
e o o e e o o e
in in o in in o o in in o in in o
e o e o
in in o in in o
e o
in in oo e o
in in
Z ZV V
Z Z Z ZV V VZ
V VI I I
Z Z Z Z
Z Z Z Z Z Z Z Z Z Z Z Z Z
Z Z Z Z Z Z
Z Z ZZ
Z Z
2
2
tan tanderivation of (2) :
tan tan
tan 1/ tan 1/
tan 1/ tan 1/
ta
oe oo o
o
e o o oe o ooin in oe oo
oe o oo o
let Z Z Zoe oo oe oe oe oo oo oo
oe oo
oe oe oo oe oo oe oo oo
oo oe
oe oo
Z
Z jZ Z jZZ Z Z Z
Z jZ Z jZ
Z Z jZ Z Z Z jZ ZZ Z
Z j Z Z Z Z j Z Z Z
Z j ZZ Z
2
n tan: i/p match
tan tan
oe oo
oe oo o in o
oe oo oo oe
Z j ZZ Z Z Z Z
Z j Z Z j Z
微波電路講義7-37
1
2
2 2
3 1 1
tan
tanderivation of (3):
tan
tan
tan 1/ tan
2 tan tan 1/ 2 tan tan
( ) ta
o oeoee
e in oe o
eo oein o
oe o
oe o
oe o oe oe o oe
oe o oe o oe o oe oo
e o oe oo
Z jZZ
Z Z jZV V V
Z jZZ ZZ Z
Z jZ
Z Z jZ Z Z jZV V
Z Z jZ jZ Z Z jZ jZ
j Z ZV V V V
3
2 2
2 2
n
2 ( ) tan
tantan
derivation of (4):2 2
tan tan
( ) ( ) 2 21 1 ( )
o oe oo
oe oo
oe oo oe oo
o ooe oo
oe oo oe oo
oe oo oe oo oe oooe oo o
oe oo oe oo oe oo oe oo
Z j Z Z
Z Zj
Z Z Z Z jCC V V V
Z ZZ Zj j
Z Z Z Z
Z Z Z Z Z ZZ Z ZC
Z Z Z Z Z Z Z Z
V
32
tan
1 tan
jCV
C j
微波電路講義7-38
, ,,1 2
, ,,1 2
,2
2
2 2 22
4 2 2
cos sin
sin cos
...(5)2 cos ( )sin
2 1
2 cos ( )sin 1 cos sin
0
e o e ooe o
e o e ooe o
oe o
o oe oo
oe o
o oe oo
e e
jZV V
jYI I
ZV V
Z j Z Z
Z CV V V V V
Z j Z Z C j
V V V
3. frequency response at port 2 and port 3
0 2
1
1-C2
C2
|S21|2
|S31|2
微波電路講義7-39
, , , ,, ,1 2 2 1
, , , ,, ,1 2 2 1
,,1
derivation of (5)
cos sin cos sin
sin cos sin cos
tan(3)
2 ( ) tan
e o e o e o e ooe o oe o
e o e o e o e ooe o oe o
o oe o oe o
o oe oo
jZ jZV V V V
jY jYI I I I
Z jZ ZV V V
Z j Z Z
,
1
1
2 2
cos sin
2 cos ( )sin
tan,
tan
tan
tan ( tan ) ( tan )
tan
tan
2 ( ) ta
oe o
o oe oo
o oee e
oeine
o oe oin
oe oe
e o oeo oe o oe o oe oinoe o
oe o
oe o
oe o oe o
jZ
Z j Z Z
V Z jZI Z Z
Z Z Z jZ
V V Z jZI V
Z jZZ Z Z Z jZ Z Z jZZ Z
Z jZ
Z jZV
Z Z j Z Z
, ,,1
, , ,,2 1 1
,
1/ 1 / tan cos / sin
n 1/ 2 ( ) tan 2 cos ( )sin
cos / sin
2 cos ( )sin
cos sin
( cos sin )cos
2 c
oe oo oe oo oe
oe o oe oo o oe oo
oo e oe oe o
o oe oo
e o e o e ooe o
o oe o
o
Z j Z Z j Z ZV V
Z Z j Z Z Z j Z Z
j Z ZI V
Z j Z Z
V V jZ I
Z jZV
Z
, , ,
2 2, ,
sin (cos / sin )
os ( )sin 2 cos ( )sin
cos sin cos sin cos sin
2 cos ( )sin
2 cos ( )sin
oe o oo e oe o
oe oo o oe oo
o oe o oe o oe oo
o oe oo
o
o oe oo
jZ j Z ZV
j Z Z Z j Z Z
Z jZ jZ Z ZV
Z j Z Z
ZV
Z j Z Z
微波電路講義7-40
4. selection of line length
coupler quadrature difference phase 90 have V and V , 2
0 0
0 ,
1 )1(,1
1 1
2
42
o
32
44
31
2
23
21
2
2
2
2
11
VV
VPCPCV
VPCPCjV
VV
ll
5. In general vevo eo, coupled line coupler is suited for a
weakly coupled coupler.
6. C=3dB Zoe=120.7, Zoo=20.7 S becomes not practical
and coupled line theory is not applicable Lange coupler
微波電路講義7-41
20/ 2010 0.1 55.28 , 45.23
from p.349, Fig. 7.29, / 0.809, / 0.306 0.098
frequency response (p.356, Fig. 7.34)
oe ooC Z Z
W b S b S cm
7. Ex.7.7 design a 20dB coupled stripline coupler, b=0.32cm,
r=2.2, Zo=50, f=3GHz.
• multisection coupled line coupler
12
34
C1 C3C2 CN
微波電路講義7-42
2 2( 1)31 2
1
1 2 1
2( 1) 2 2( 2) ( 1)31 2
1
1 2
multisection sin sin .... sin
if , ,....
sin [ (1+ ) ( + ) .... ]
2sin [ cos( 1) cos( 3)
j j j j j N
N
N N
j j N j j N j N
M
jN
VjC e jC e e jC e e
V
C C C C
Vj e C e C e e C e
V
j e C N C N
3 3
1 1
1 1.... ],
2 2
( ), ( ) ,2
M
n n n
NC M
V VC C S W
V V
3
21
2
2
21
1, :
tan tansingle section sin
1 tan1 tan
1 1
cos sin1 cos sin
j
j
C N odd
V jC jCjC e
V jC j
V Ce
V jC j
微波電路講義7-43
Discussion
1. Ex. 7.8 design a 20dB coupler with binomial response, N=3,
Zo=50, fo=3GHz.
31 2 1 2
1
1 2 1
220/ 20
1 2 2 12
2 2
1 3
2
1 3
1 1 12sin [ cos 2 ] 2 (sin 3 sin ) sin
2 2 2
sin 3 ( )sin
0, 10 0, ( ) C -2C 10 0.12
0.0125 1 1,
0.125 1 1
50.63 ,
oe o oo o
oe oe
VC C C C C
V
C C C
dC d CC C C
d d
C C C CZ Z Z Z
C C C
Z Z Z
1 3
2 2
49.38
56.69 , 44.1
frequency response (p.359, Fig. 7.37)
oo oo
oe oo
Z
Z Z
微波電路講義7-44
7.7 The Lange coupler (interdigitated coupler)
1
2
4
3
14
23
1 2
3 4
/4
Unfolded Lange coupler
微波電路講義7-45
1 2
3 4
1 2
3 4
Ze4, Zo4
Ze4, Zo4
Cex
Cm CmCm
Cin Cin
Cex Cex
Cm
Cex
four-wire coupled line model
approximate two-wire coupled line model
nearest
neighbor
coupling
～Cex
even + + + +
odd + - + -
( in series with )ex in m ex
ex min ex
ex m
C C C C
C CC C
C C
4
4 3 2
e ex in
o ex in m
C C C
C C C C
2
e ex
o ex m
C C
C C C
微波電路講義7-46
4 4
4 4
4 4
4 44 4
4 4
4-line model : , 6
2-line model : , 2
(3 ) (3 ), ...(1)
1, ...(2)
3 3
,
e ex in o ex in m
e ex o ex m
e e o o o ee o
e o e o
oo oe oo oeo e oe o oo
oo oe oe oo
e oo e o
e o
C C C C C C C
C C C C C
C C C C C CC C
C C C C
Z Z Z ZZ Z Z Z Z
vC Z Z Z Z
Z ZZ Z Z C
Z Z
2
.7.27
2
4 3 9 8
2 (1 ) /(1 ),
4 3 9 8
2 (1 ) /(1 )
oe oprob
oo o
C CZ Z
C C CW S
C CZ Z
C C C
Discussion
1. Lange coupler is suitable for wideband 3dB 90 hybrid, and MMIC
design uses air bridges instead of bond wires.
微波電路講義7-47
4
2 2 2
4
4
derivation of (1) : 4-line : 2
2-line : , 22
2 2 (3 )22 2
2
ex min ex
ex m
C CC C
C Cex m
e ex in ex
ex m
o ee ex o ex m m
o ee
e o e e e o e o e e e oe e e
o e e o e o e oe
o ex i
C CC C C C
C C
C CC C C C C C
C CC
C C C C C C C C C C C CC C C
C C C C C C C CC
C C C
2 2 2
4
4
(3 ) 3 3 3 (3 )6 3( )
1derivation of (2) :
1/ 1/ 1/1 1 1 1
(3 ) 3 1/ 3/ 1/ 3
e e o e e o o e o o en m o e
e o e o e o
o
e o e o o e oo oee oe
e e oe e e o e o e o e oo oe
e o
C C C C C C C C C C CC C C
C C C C C C
ZvC
C C vC C vC vC Z ZZ Z
C C CvC vC C C vC C vC vC vC Z Zv
C C
4
4
1/ 1/ 1/1 1 1 1
(3 ) 3 1/ 3/ 1/ 3
e o e o o e oo oeo oo
o o eo o o e e o o e o oe oo
e o
C C vC C vC vC Z ZZ Z
C C CvC vC C C vC C vC vC vC Z Zv
C C
微波電路講義7-48
7.8 The 180 hybrid
• rat-race (ring) hybrid
4/3
oZ2
oZ
4/
1
()2
3
4
()
0110
1001
1001
0110
2
j
(1) input port 1
port 1 match 2 2
3 1 port 4 "GND"
4 4 2
or isolated port
(2) input ports 2 and 3
port 1: port, port 4 : port
o o oZ Z Z
4/
4/
微波電路講義7-49
Discussion
1. even-odd mode analysis
1/2 1/2
1/2 -1/2
2,4
1 2
2,8
2,
8
3
1/2
eTe
1 2
2,8
2,
8
3
1/2
oTo
O.C.
O.C.
S.C.
S.C.
1
2
3
4
1 1
2 2
1 1
2 2
1 1
2 2
1 1
2 2
e o
e o
e o
e o
b ,
b T T
b ,
b T T
1
2
3 4
2,4
微波電路講義7-50
4/,Zo
stub /83circuit -short mode-odd 1
01
stub /8circuit -short mode-odd 1
01
stub /83circuit -open mode-even 1
01
stub /8circuit -open mode-even 1
01
0
0
o
o
o
o
o
o
jY
jY
jY
jY
DC
BA
jY
jZ
DC
BA
Y
微波電路講義7-51
1 2 3 4
1 0 1 00 21 2
1 10 2 12 22
1 0 1 00 21 2
1 10 2 12 22
,2 2 2 2
0, , ,2 2
e o e o
jA B j
j jjC D je
jA B j
j jjC D jo
j j j j, ,T T
j jb b b b
0
2. input port 2 ports 1, 4 :180 phase difference, port 3 isolated port
input port 4 ports 2, 3 :180 phase difference, port 1 isolated port
微波電路講義7-52
3. Ex. 7.9 3dB rat-race hybrid Zo=70.7, BW 20~30% (p.367, Fig.7.46)
4.
32
32
3
2
4
3
2
1
aa
0
0
aa
2
j
0
a
a
0
0110
1001
1001
0110
2
j
b
b
b
b
5. Unequal power division rat-race coupler uses Za, Zb lines
aZ
aZ
bZbZ
微波電路講義7-53
• tapered coupled line hybrid
12
3
4
oZ
)( ),( zZzZoooe
o
o
o
kZ
Z
k/Z
oZ
zL 2 L
1/2 -1/2
1/2 1/2
1
2
3
4
1 1
2 2
1 1
2 2
1 1
2 2
1 1
2 2
e o
e o
e o
e o
b T T
b
b T T
b
( )oeZ z
( )ooZ z
tapered
coupled line
uniform
uncoupled line
factor coupling
10 k
even odd
input at port 4
2( ) ( )oe oo oZ z Z z Z
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
b b b b
b b b b
b b b b
b b b b
1
微波電路講義7-54
L,Zo b L,Zo b
1:kEven mode
L,Zo b L,Zo b
k:1Odd mode
k/Zo
okZ
2 2
2 2
0cos sin cos sin
(p.190, Table 4.1)1sin cos sin cos0
1 1cos sin ( )sin cos
1 1 2( )sin cos sin cos
e
e
kA B j j
C D j jek
A B C Dk j k
k k A B C D
j k k TA B C Dk k
4
4
3
3
微波電路講義7-55
2 2
2 2
2
10cos sin cos sin
sin cos sin cos0
1 1cos sin ( )sin cos
1 1 2( )sin cos sin cos
1 1
1
o
o
j
e o
A B j jk
C D j jo k
A B C Dk j k
k k A B C D
j k k TA B C Dk k
k ke ,
k k
2 2 2
2 2
1 14 2 24
2 2
3 34 4 44
2 2
2 2,
1 1 1
10 , ,
1
2, 0
1
1
j j j
e o
j j
j j
k ke ,T e T e
k k
kb S b e e S
k
kb e e S b S
k
a
b
a b
1
2
3
4
1 1
2 2
1 1
2 2
1 1
2 2
1 1
2 2
e o
e o
e o
e o
b T T
b
b T T
b
微波電路講義7-56
12 34 22 44 32 14 42 24
23 32 41 14
2 2
2 2
1 11 3 31
input at port 2 input at port 4 , , ,
reciprocal ,
1 1input at port 1
1 1
1 1 1( ) 0 , ( )
2 2 1
input at
j j
e o
j j
e o e o
S S S S S S S S
S S S S
k ke , e
k k
kb S b e e S
k
a
13 31
2
port 3 input at port 1
0 0
0 0
0 0
0 0
j
S S
S e
b a b a
a b
a b
微波電路講義7-57
• waveguide magic-T
1 , H-plane arm
4 , E-plane arm
32
2 3
4
input at port 1
port 4: 0
port 2 and port 3: equal
amplitude and phase
2 3
4
input at port 4
port 1: 0
port 2 and port 3: 180 phase
difference
E H
微波電路講義7-58
7.9 Other couplers
• reflectometer
3 4
212
1
43
C<<1
Vi Vr
Vi Vr
C C
max min
1
as
1
jθ j
i r
r r
i i,
C CV C e ,V C e
D D
V VD , D V V
D
C/D C/D
load
微波電路講義7-59
Prob. 7.4 Two 90 8.34dB couplers are connected in cascade, find
2 2
3' 4' 1' 3 2 1 1
1 1
2' 1' 4' 2 3 1 1
2 2
1 1
4
8.34 20log 0.383, 1 0.924
( ) ( )
=2j 0.707 90
( ) ( )
=( ) 0.707
0
3 quadrat
o
C dB
V V j V V j V j V j V
V V
V V j V V j V V j j V
V V
V
dB
b b a b a
a b a b a b b a
ab
a b a b a a b b
a b
ure coupler
connection problem in microstrip circuits
1 2
34
1’ 2’
3’4’
2'1 3'1S ,S
a
jb jb
jb
a
a
1
2
3 4
1’ 2’
3’ 4’
微波電路講義7-60
Prob. 7.18 Design an unequal power branch-line coupler
Zb
Zb
ZaZa
baa
baa
O.C.
(even mode)
S.C.
(odd mode),a b
o o
Z Za b
Z Z
2
2
2
2
1 0 0 1 0 /
/ 1 / 0 / 1 / / /
1 0 0 1 0 /
/ 1 / 0 / 1 / / /
( 1/ / ) ( 1/ /
2 / ( 1/ / )
o
e o
A B jb b a jb
C D j a j b j a j b jb a b ae
A B jb b a jb
C D j a j b j a j b jb a b a
j b b b a j b b bΓ ,Γ
b a j b b b a
2
2
1 2 2
2 2
2 3 2 3 2 32 2 2
2
) 1 1, ,
2 / ( 1/ / ) / /
1 1 1input match ( ) 0 1 0...(1)
2
1 1 1/( ) , ( ) , 1
2 (1 1/ ) 2 (1 1/ )
(1) ,1 11
e o
e o
e o e o
a o b o
aT T
b a j b b b a b a jb b a jb
bb a
j ab T T b T T P P b b a
b a b a a
ab Z Z Z Z
a
a a a a
a a a
a a
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