notes 20 - power dividers and couplers part 2

7/29/2019 Notes 20 - Power Dividers and Couplers Part 2

1/37

Notes 20

ECE 5317-6351

Microwave Engineering

Fall 2011

Prof. David R. Jackson

Dept. of ECE

Power Dividers and Couplers

Part 2

1


2/37

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 S

S S S S

S S S

14 23 0S S

The device is called a directional coupler if:

If lossless [S] is unitary.

2

There are 6 degrees of freedom

(independentS

parameters).


3/37


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

3

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).


4/37


Directional coupler symbol:

4

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


5/37


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 34S S

ugate)

(Dot column 2 dotted with its conjugate, subtract from column 3 dotted with its conjugate,

and use the with previous result above)

5

12 13

12 24

13 34

24 34

0 0

0 0

0 0

0 0

S S

S SSS S

S S

From the unitary property of the matrix, we have the following results:


6/37

12 34

13 24,j j

S SS e S e

2 2 2 2

12 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.)

6

The first equation is really just a choice of reference planes on ports 2 and 3,

which makes these parameters real.

(usually, n = 0)


7/37

( / 2)

0 0

0 0

0 00 0

j

jS

jj

0;

0 0

0 0 -

0 0

0 - 0

S

Two possible choices:

1) Symmetrical coupler

2) Anti-symmetrical coupler


7

Example: 90

o

quadrature hybrid coupler

Example: 180o rat-race hybrid coupler


8/37

-

111

1

-10log -20logdBP

RL SP

1

3 31

110log 20log -20log

dB PC

P S

Assume a signal into port 1

Return loss

Coupling8


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 SDP S S

1

4 41

110log 20logdB

PI

P S

Directivity

Ideally + dB

Isolation

Ideally + 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:

311

; 3 dB2

dBS C


9/37

90o (Quadrature) Hybrid Coupler

9

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 ofquadrature 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.


10/37

10

90o hybrid

1 2

4 3

The quadrature hybrid is a lossless 4-port (the Smatrix is unitary ).

All four ports are matched.

The device is reciprocal (the Smatrix 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


11/37

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


Can be used to produce right-handed circular polarization.

Can be used to produce left-handed circular polarization.


12/37

0 1 0

0 0 1-1

1 0 02

0 1 0

j

jS

j

j

0Z

4

g

1 2

34

4g

02Z

02Z

plane of symmetry

(POS)

0Z

0Z

0Z

0Z0Z

(Branch-line coupler)

12

A microstrip realization is shown here.


Note: We only need to study what

happens when we excite port 1, sincethe structure is symmetric.

We use even/odd mode analysis

(exciting ports 1 and 4) to figure out

what happens when we excite port 1.


13/37

Even Analysis

3 2

4 1

e e

e e

V V

V V

0Z

0Z 0Z

1

eV2

eV

3

eV4

eV

V

V

0 2Z

0 2Z

8s gl

4g

OC OC

OCOC

OC

0Z

0Z0Z

13


0

0

0

tan

tan / 4

s s sY jY l

jY

jY

0 01 /Y Z

0Z

0

Z0Z 2V

4g

1

eV

2

eV

0jY 0jY


14/37

Odd Analysis

3 2

4 1

-

-

o o

o o

V V

V V

0Z

0

Z0Z 2V

4g

1

oV

2

oV

0- jY0- jY

14


0Z

0Z 0Z

1

oV2

oV

3

oV4

oV

V

V

0 2Z

0 2Z

8s gl

4g

SC SC

SCSC

OC

0Z

0Z0Z

0

0

0

cot

cot / 4

s s sY jY l

jY

jY

0 01 /Y Z


15/37

Consider thegeneral case:

0

+

-Y jY

foreven

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

15


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:


16/37

0

0

0 0

0

0 0

2

0 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

16


Hence we have

0

+

-

jY

Z

for even

for odd


17/37

0 0

0

2

0 0

0 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 j

Z Z

jZ

jZ

17


Continuing with the algebra, we have


18/37

0

0

0

11

112

e

jZ

ABCDj

Z

0

1-

0 2

1-0

2

e

j

Sj

18


Hence we have

Convert this to Sparameters (use Table 4.2 in Pozar):


19/37

2 3 4

-

111

1 0a a a

VS

V

11 0S

- - - - - -

1 1 1 1 1 111

11 11

1

2 2

10 02

e o e o e o

e o

V V V V V V S

V V V V V

S S

0Z

0Z 0Z

0 2Z

0 2Z

4

g

4g

1V 2V

4V

3V

-

3V

-

2V

-

1V

1V

-

4V

0Z

0Z0Z

19


Hence

11 22 33 440S S S S By symmetry:


20/37

- - - -

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:

20


2 3 4

-

221

1 0a a a

VS

V

0Z

0Z 0Z

0 2Z

0 2Z

4

g

4g

1V

2V

4V

3V

-

3V

-

2V

-

1V

1V

-

4V

0Z

0Z

0Z


21/37

2 3 4

-

331

1 0a a a

VS

V

31 13 24 42

-1

2S S S S By symmetry and reciprocity:

21


- - - - - -

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

4

g

4g

1V 2V

4V3V

-

3V

-

2V

-

1V

1V

-

4V

0Z

0Z0Z

d b d l ( )


22/37

2 3 4

-

441

1 0a a a

VS

V

41 14 23 32 0S S S S By symmetry and reciprocity:

22


0Z

0Z 0Z

0 2Z

0 2Z

4

g

4g

1V

2V

4V

3V

-

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

d b d l ( )


23/37

Summary

0 1 0

0 0 1-1

1 0 02

0 1 0

j

jS

j

j

23


0Z

4

g

1 2

34

4g

0 2Z

02Z0Z

0Z

0Z

0Z

0Z

The input power to port 1divides evenly between ports 2

and 3, with ports 2 and 3 being

90o out of phase.

b l ( b )


24/37

180o Hybrid Coupler (Rat-Race Ring Hybrid)

24

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 Pozar

Photograph of amicrostrip ring hybrid.Courtesy of M. D. Abouzahra, MIT

Lincoln Laboratory

b d ( )


25/37

Rat-Race Ring Hybrid (cont.)

25

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 Smatrix is unitary).

All four ports are matched.

The device is reciprocal (the Smatrix is symmetric).

Port 4 is isolated from port 1 and ports 2 and 3 are isolated from each other.

i b id ( )


26/37

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


21 31S S

24 34S S

R R Ri H b id ( )


27/37

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)


12 13S S

42 43S S

1 2 12V V S

1 2 42V V S

R R Ri H b id ( )


28/37

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

4g

3 / 4g

0Z

0Z

0Z

28

A microstrip realization is shown here.


180o H b id C l (M i T)


29/37

0 1 1 0

1 0 0 -1-

1 0 0 12

0 -1 1 0

jS

180o Hybrid Coupler (Magic T)

29

A waveguide realization of a 180ohybrid 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.

R t R Ri H b id ( t )


30/37

30


2

0Z

02Z

/ 4g

1

3

4

4g

4g

3 / 4g

Plane of

symmmetry

0Z

0Z

0Z

1 2

3 4

4

g 3

4

g

02Z0Z

2V

1V

V

4V

3V

0Z

/ 4g

/ 4g

02Z

02Z

0Z 0Z02Z

Layout Schematic



31/37

Even Analysis

1 2

3 4

8

g 3

8

g

02Z0Z

2

eV

1

eV

V

4

eV

3

eV

V

0Z

OC

OC

OC

OC

02Z

02Z

0

2

jY0

2

jY

0Z 0Z02Z

31


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



32/37

1 2

3 4

8

g

3

8

g

02Z

0Z

-

2

oV

-

1

oV

V

-0

4V-3

oV

-V

SC

SC

SC

SC

0Z

0Z 0Z

32


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



33/37

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 Z

ABCD jY jY j

Z

j Z

jYj

Z

j Z

jZ

Proceeding as for the 90o coupler, we have:

33




34/37

0

0

0

1 2

21

e

j Z

ABCDj

Z

0

1 1-

1 12e

jS

Converting from the ABCD matrix to the Smatrix, we have

34


Table 4.2 in Pozar

Rat Race Ring H brid (cont )


35/37

21 12 34 43

-

2

( )

jS S S S

symmetry and reciprocity

2 3 4

-

111

1 0a a a

VS

V

2 3 4

-

221

1 0a a a

VS

V

35


For the Sparameters coming from port 1 excitation, we then have:

- -

1 111 11 11

1

2 2

1 -

2 2 2

0

e oe oV V

S S SV

j j

- -

2 221 21 21

1

2 2

1 - -

2 2 2

-

2

e oe oV VS S S

V

j j

j

11 330

( )

S S

symmetry

Rat Race Ring Hybrid (cont )


36/37

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

-

331

1 0a a a

VS

V

Similarly, exciting port 2, and using symmetry and reciprocity, we have the

following results:

36


- -

3 331 11 11

1-

2 21 -

-2 2 2

-

2

e oe oV VS S S

Vj j

j

Rat Race Ring Hybrid (cont )


37/37

0 1 1 0

1 0 0 -1-1 0 0 12

0 -1 1 0

jS

2

0Z

0

2Z

/ 4g

1

3

4

4g

4g

3 / 4g

0Z

0Z

0Z

37

Summary


notes 20 - power dividers and couplers part 2

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