1 power dividers and directional couplers divider or coupler divider or coupler power division power...
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1
Power Dividers and Directional CouplersDivider or
coupler
Divider orcoupler
Power division Power combining
ports. allat matched and ,reciprocal lossless, becannot port threeA
zero. bemust )S,S,(Sparameters three theof least twoAt
0 , 0 , 0
1S , 1S , 1S
matrixIdentity S ,
0
0
0
231312
13*1212
*2323
*13
2
23
2
13
2
23
2
12
2
13
2
12
*
2313
2312
1312
SSSSSS
SSS
S
SS
SS
SS
S T
2
coupler ldirectionaa in results which
0
0S , 0S
0
0S , 0S Lossless
0
0
0
0
2314
2
34
2
1223
23*3412
*1434
*1423
*12
2
24
2
13*14
23*2413
*1424
*1423
*13
342414
342313
242312
141312
SS
SSS
SSSSSS
SSS
SSSSSS
SSS
SSS
SSS
SSS
S
Four-Port Network (Directional Couplers)Assume all ports are matched
3
Coupler ricalAntisymmet (The
00
00
00
00
Coupler) lsymmetrica (The
00
00
00
00
1
20S
, e , S S
,
1S , 1S
1S , 1S
22
34*2413
*12
24j
133412
34122413
2
34
2
24
2
34
2
13
2
24
2
12
2
13
2
12
S
j
j
j
j
S
nSSS
eSS
SSSS
SS
SS
j
4
InputThrough ordirect
CoupledIsolated
Directional Coupler
1
4 3
2
dB CDI
dB log20P
P10log IIsolation
dB S
log20P
P10log DyDirectivit
dB log20P
P10log CCoupling
144
1
144
3
3
1
S
5
0110
1001
1001
0110
2
1
4.port at fed when3 and 2 ports
between differnce phase 180a has race-rator hybrid T magic The
010
100
001
010
2
1
1.port at fed when3 and 2 ports
betweenshift phase90a has hybrid quadrature The .2
1
dB. 3 isfactor coupling the wherecoupler,
al Directionof cases special are couplers Hybrid
o
0
S
j
j
j
j
S
6
Directional Couplers
1 2
34
b
f
f
i
P
P
P
P
log10D
:is Dy directivit The
log10C
:is C coupling The
Pi
PbPf
Pt
7
00
00
00
00
0
0 , 0
0
0
00
00
0
0Scoupler idealFor
3424
3413
2412
1312
4433
*4424
*3313
*
443424
343313
2412
1312
2211
2314
SS
SS
SS
SS
S
SS
SSSS
USS
SSS
SSS
SS
SS
S
SS
S
t
8
,
1
1
0 , 0
213112
2
24
2
12
2
13
2
12
3412
2413
34241312
34132412
*3424
*1312
*3413
*2412
jCSCS
SS
SS
SS
SS
SSSS
SSSS
SSSSSSSS
00
00
00
00
12
12
21
21
CjC
CjC
jCC
jCC
S
9
Directional Couplers
Two-hole Waveguide Couplers
Port 1
Port 3Port 4
Port 2a1 0
b2
b4
b3
Kf a1-L
Kb a1-2L
Kf a1-L
Kf a1 0
a1 -L
L
and reverse aperture coupling coeffiecients
Kf and Kr are the forward
Pi
PbPf
Pt
10
array. with theassociatedy directivit a plus aperture
single theofy directivit theof sum theisy directivit The
seclog20log20
coslog20
1
2log20D
:is Dy directivit The
2log-20C
:is C coupling The
2
LK
K
LK
K
eK
K
K
r
f
r
f
Ljr
f
f
11
Multi-element CouplersTo achieve good directivity over a band of frequencies, couplers With many apertures may be used.
dA
Bb BfF0
B0A
A
Let the aperture coupling in the forward direction be Cn
And the reverse coupling in the reverse direction be Dn
N
nn
Ndj FAeF0
: is guideupper wave
thein waveforward total theaperture Nth theof position At the
Input
Isolated
Through
Coupled
n=0 n=1
B1A F1 A
12
ndjN
nn
ndjN
nn
ndjN
nn
eBC
eB
F
BD
A
F
eBAB
2
0N
0nn
2
0
N
0nn
2
0
log20
F
log20log20
F -20loglog2C :arey directivit and coupling The
: is guideupper wave
thein wavebackward total theaperturefirst theof plane At the
13ts.coefficien binomial is constant, is K ,
choosingby achieved be can sticcharacteri passbandflat maximally A
er. transformvequarter wa section-N fromt coefficien reflection
for the expression theas same (factor Array :
log20log20
dB log20log20log20log20
aperture. allfor same theare that tscoefficien coupling backward
and forward for theconstant are , , B,F
aperture. nth theof radius thebeing r withguide, theof edge theto
relative position identical withholes round are apertures that theAssume
3
2
0
3
0
3
2
0
3
3n
3n
n
Nn
Nnn
ndjN
nn
N
nnf
bndj
N
nnb
bfnbnf
CKCr
erS
rKC
SKCerKCD
KKrKrK
14
In order to obtain an equal-ripple characteristics in the pass bandThe array factor F is made proportional to Chebyshev polynomial.
/2. whereband theofcenter thein
C coupling desired thegivesit that such K Choose passband. the
of edgeslower andupper at the value theis sec and
)cos(sec)2cos(2S Choose
evenN for 2/)(
oddN for 2/)1(
)2cos(2
then,..., Choose
m
0
0
23
110
d
TKnNr
NM
NM
dnNrerS
rrrr
mN
M
nn
M
nn
jnn
NN
Chebyshev Response
15
)(bandwidth determined be can then, DGiven
dB )(seclog20
.1)cos(sec
whenoccursfactor array by the dcontribute
, passband thein Dof valueminimum the, small very is
, cos(sec)(sec since
,1)cos(sec occur when will Dof aluesmallest v The
dB )cos(sec
)(seclog20
K
K-20loglog20K20log-D
m min
min
min
b
f0b
mN
mN
b
f
mNmN
mN
mN
mN
TD
T
K
K
TT
T
T
TSC
16
Branch-Line Coupler
Z02
Z02
Z01 Z01
ll
Z0
Z0 Z0
Z0
Port 4 Port 3
Port 2Port 1Even-mode pair
Odd-mode pair
a1/2
a1/2 -a1/2
a1/2
b4
b1
b2
b3
DCBAT
DCBA
DCBA
abTT
ab
TTa
ba
b
oeoe
oeoe
2 ,
)(2
, )(2
)(2
, )(2
14
13
12
11
17
Open circuits Short circuits
Even-mode excitation Odd-mode excitation
/4 , 11
mode evenfor is 1
01
0
0 1
01
02
012
022
01
01
0102
01
0201
01
02
Z
Z
ZZZj
ZjZ
Z
Z
j
Z
jZj
Z
jDC
BA
18
4)port power to (no 0b
3) to1port fromshift phase180- power, (half 2
1b
2) to1port fromshift phase90- power, (half 2
b
matched is 1port 0b
mode oddfor )1(2
1T , 0
modeeven for )1(2
1T , 0Then
2/ ZImpedance thechoose weIf
4
o3
o2
1
oo
ee
0102
j
j
j
Z
19
Parallel-Coupled Lines Directional Coupler
Input
Coupled Port 3
Direct Port 2
Isolated Port 4
Port 1
L
Input
Coupled Port 3
Direct Port 2
Isolated Port 4
Port 1
L
H wall
a1/2
a1/2
Even Mode excitation
20
Z0
Z0
Z0
Z01
3
2
4Z0e Z0oI1
I3 I4
I2
oein
e
ooin
o
oein
eine
ooin
oino
o
oooin
e
eeein
oe
oe
ZZ
VI
ZZ
VI
ZZ
ZVV
ZZ
ZVV
jZZ
jZZZZ
jZZ
jZZZZ
II
VV
I
V
11
11
00
000
00
000
11
11
1
1in
,
,
tan
tan ,
tan
tan
Z
V
21
1
1 ,
1
1
1 , Then 2For
sincos1
1
0 , tan1
tan
C e Defin, tan)(2
tan)(
then, ZZLet
2
)()(
00
223
2
2
222
2244423
00
00
000
003
20
oin
ein
C
CZZ
C
CZZ
CjV
VC
V
V
jC
CVVVV
VVVVVjC
jCVV
ZZ
ZZ
ZZjZ
ZZjVV
ZZZZZ
ZZZ
ZZZZZZZ
oooe
oe
oeoe
oe
oe
oe
oe
oinoooe
ooin
ein
ooin
eino
ein
oin
in
22
Hybrid Junctions Magic T
0110
1001
1001
0110
00
00
00
00
2424
2412
2412
1212
S
SS
SS
SS
SS
S
23
Microstrip Hybrid Ring
1
2 3
4
The hybrid ring (rat-race)
a
a
Ports 1 and 3 are uncoupled, ports 2 and 4 are uncoupled
Z1
Zc
Va+
Va+
or -Va+
Vb+
or -Vb+
Vb+
Even Odd
1/2 1/2
Even Odd
1/2 -1/2
24
jB2jB1
1
Y1 YcYc
Va+ Vb
+
Equivalent circuit forone half of Hybrid ring
b
a
scsc
scsc
b
a
scsc
scsc
b
a
occ
ococ
b
a
ococ
ococ
V
V
SS
SS
V
V
V
V
SS
SS
V
V
V
V
SS
SS
V
V
V
V
SS
SS
V
V
,
:excitation oddFor
,
:excitationeven For
2221
1211
3
4
2221
1211
2
1
22021
1211
3
4
2221
1211
2
1
25 S ,S ,
)(2
1S , )(
2
1
)(2
1)(
2
1S , ) (
2
1 )(
2
1
)(2
1S , )(
2
1
)(2
1)(
2
1S , ) (
2
1 )(
2
1
2
1
2
1
0
2V,2V ionsuperposit Use
213422331144
121242111141
222232oe121231
222222121221
121212oe111111
2
1
22222121
12121111
3
4
2
1
22222121
12121111
2
1
43
21
SSSS
SSSSS
TTSSSSS
SSSSS
TTSSSSS
V
V
SSSS
SSSS
V
V
V
V
SSSS
SSSS
V
V
VV
VV
scocscoc
oescocscoc
scocscoc
oescocscoc
scocscoc
scocscoc
scocscoc
scocscoc
ba
26
0S , 2
S , 2
S
matched. is 1port i.e.
2
T , 2
, 2
T , 2
12
21 ,
12
21
313121
11
ooee
jj
oS
jjjj
j
j
DC
BA
j
j
DC
BA
oe
3
1/2
O.C
O.C
2 1
Te 2
2
2
3
1/2
S.C
S.C
2 1To
2
2
2
Even Mode Odd Mode
Port 2
Port 1 Port 1
Port 2
27
0101
1010
0101
1010
2
jS
28
Power Dividers
Z1
Z3
Z2
P3
P2
P1
A lossless three port juntion
.
1Y
Y
matching impedancefor
)1( , 2
1
2
1
3
2
321
1312
2
13
2
121
YYY
PPPP
VYVYP
29
resistor.
in thecurrent no is thereimpedance, loadcorrect in d terminateare ports
output When theresistor. by theby absorbed isport output thefrom
power reflected the3,port and 2port between resistor connect weIf
3.port into couplenot
does 2port at power reflected the, 0 S have todesired isIt
ports.output ebetween th islolation no 0, S
. ports outputs
matched havenot lldivider wipower losselss of typeThis
23
23
R ZL2
ZL3
Z2
Z3
Zc
30
,Y ,Y
: 1port At
)(
Y 1port at load matchedFor
K
: require weratiopower obtain the order toIn .Then
and Zload matched in the d terminateare 3 and 2port when
Rin current zeromaintain and
23
3in,32
2
2in,2
23
22
322
23
2
23
322
2
in
232
3
2
3
2
2
22
32
3L2
22
3
Z
Z
Z
Z
Z
ZZZZZK
YZ
Z
Z
ZY
Y
ZZKZ
V
Z
V
VV
Z
PKP
LL
cL
cLL
in
c
LLLL
L
31
.impedances
load matchedin their teminatedare 3 and 2port as
long asfrequency any at Rin current no is There
1
tan
tanY
tan
tanYfrequency any At
V2
1V
2
1
322
22
33
22
3
2
2,2
33
333in,3
22
222in,2
32
2
2,
2
12
3,
2
1
LL
LL
LL
LL
L
L
inL
L
L
L
inin
ZIK
ZI
ZI
ZI
V
V
YKjYY
jYYY
jYY
jYYY
ZKZ
YKY
32
23
c
22
222
222
2222
22
222
2,2
in,3in,2
YG if uncoupled are 3 and 2Port
Y impedance load ain
1 terminate3, and 2between coupling theanalyze order toIn
0)1( Choose
tan)1(2)1(
)1(
)1(YY
inc
cc
c
inc
incin
in
YKY
YYKjYKY
YKY
YY
YY
YK
33
02
323
3
2
3323
2322
3
2
3
VV
IY
V
V
YY
YY
I
I
-Y23
-Y22+Y23 -Y33+Y23
G
V2V3
I2
I3
Yc
Y3 Y2
sc V2
I3
z3 z2
Equivalent circuit with port2 and 3 excited
Equivalent circuit betweenPort 2&3 with port 3 sc and R removed
34
1
, 1
1 Z, ,
Y /2When
2sin)()2cos1(
2
232
2
2
3232
3223
32
3223
K
RZ
K
RKZ
RZK
RZKZZ
ZZR
Y
YY
YYjY
YYY
LL
ccc
c
c
35
Passive Microwave Devices
AttenuatorsR1
R2
R1
ZcZc
Vg
gc
TH
ccin
c
cin
VZRR
RV
ZRRRZR
ZRR
ZRRRR
21
2
2211
21
121
)2(For
)(
R2
R1R1
sectionT section
36
circuit theofout andin it switch to
ion configuratsection - in the R with parallelin used becan diode PIN
.4.141R and ,58.8R attenuator dB-3For 1
2R ,
1
1R
R n,attenuatioPower
8822
1
:is load the todeliveredpower The
2
21
221
in
2
21
22
2
2
22
21
2
2
cc
cc
c
g
c
g
cc
c
THL
ZK
KZ
K
K
ZZRR
RK
Z
VK
Z
V
ZRR
RZ
Z
VP
37
Phase shifters
OC
Bias Current Input
BiasCircuitGround
Bias Current Input
BiasCircuitGround
Length l2
l1
Incremental-line-type phase shifteroc
input output
/)(/)( 1212 pv
38
BiasCurrentInput
jB jB
l=V+
1
V-1
V+2
V-2
V+3
V-3
V+4
V-4
OC
A phase shifter using switched reactive elements
39
21
2
221
__
__
1
Bj
Bj
Bj
Bj
A
Transmission matrix of a normalized shunt suceptance jB:
Transmission matrix of a section of transmission line ofElectrical length :
j
j
e
eA
0
02
40
4
4
2221
1211
4
4321
3
321
2
21
1
1
V
V
AA
AA
V
VAAA
V
VAA
V
VA
V
V
Relationship among wave amplitudes:
Choose V-4 = 0, then V+
4 = V+1 /A11 Thus T14 = 1/ A111
_2
2_
14 421
jj e
Be
BjT
If tanthen :
0
14)(
14
53.14 1.107, 1BFor
/2 , /4 2,BFor
2)(
1 ,
TeeT jj
41
small VSWRinput thekeep to
order in statesbetween shifts phase small o limited isshifter phase This
mismatch. large 92.224. 0.2 as large as B of A value
B2 B-B and BB when phasein change The
2/tan2/ of Phase
1)1(T ,1When
)1(
21
421
jsuch that stubs two theChoose
01
111
114
122
14
1
12
1
2/
_2
2/
2_
14
12
is
BBT
BB
Bjj
BBjje
Be
BjT
BjB
jj
42
P P’ dd1
A phase shifter using open circuited stubs spaced Apart. P and P’ are switched into the circuit when theDiodes are off and on, respectively.
43
211
2332
01
12
0131014
012
010012
01
201
0102
21
and since , 1
1log10 Coupling
1 and
1 and 1 , 1
1 :or then
modes, odd and even bothfor DA , 0
decoupled is 2port 0 , matched is 1port 0
aPbPZ
aZbaZjb
ZjZTZjZTBA
T
Z
ZZCB
DCBA
bb
e
oe