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