Notes 21

ECE 5317-6351 Microwave Engineering

Fall 2019Prof. David R. Jackson

Dept. of ECE

Quadrature Coupler and Rat-Race Coupler

1

Adapted from notes by Prof. Jeffery T. Williams

Quadrature (90o) Coupler

2

“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 of quadrature 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 coupler is very useful for obtaining circular polarization: There is a 90o phase difference between ports 2 and 3.

Note:The term “hybrid” denotes

the fact that there is an equal (3 dB) power split to the

output ports.

3

90o Coupler

1 2

4 3

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

All four ports are matched.

The device is reciprocal (the S matrix is symmetric.)

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

Quadrature Coupler (cont.)

[ ]

0 1 00 0 11

1 0 020 1 0

jj

Sj

j

− =

4

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 coupler is usually used as a splitter:

90o Coupler

1 2

4 3

+90o out of phaseφ2- φ3 = 90o

-90o out of phaseφ2- φ3 = -90o

21 31S jS=

24 34S jS= −

Can be used to produce right-handed circular polarization.

Can be used to produce left-handed circular polarization.

Quadrature Coupler (cont.)

Note: A matched load is usually placed on port 4.

Branch-Line Coupler

5

A microstrip realization of a quadrature hybrid (branch-line coupler) is shown here.

Notes: We only need to study what happens when we excite port 1, since the structure is physically symmetric.

We use even/odd mode analysis (exciting ports 1 and 4) to figure out what happens when we excite port 1.

Quadrature Coupler (cont.)

1 2

34

An analysis of the branch-line coupler is given in the Appendix.

Summary

[ ]

0 1 00 0 11

1 0 020 1 0

jj

Sj

j

− =

6

Quadrature Coupler (cont.)

The input power to port 1 divides evenly between ports 2 and 3, with

ports 2 and 3 being 90o out of phase.

1 2

34

Note: A matched load is usually placed on port 4.

7

Quadrature Coupler (cont.)A coupled-line coupler is one that uses coupled lines (microstrip, stripline)

with no direct connection between all of the ports.

Please see the Pozar book for more details.

/ 4gλ

Port 1 Port 2

Port 3 Port 4

This coupler has a 90o phase difference between the output ports (ports 2 and 3), and can be used to obtain an equal (-3 dB) power split or another split ratio.

8

Quadrature Coupler (cont.)

One feed port produces RHCP, the other feed port produced LHCP.

Circularly-polarized microstrip antennas can be fed with a 90o coupler.

Note: This is a better way (higher bandwidth) to get CP than with a simple 90o delay line.

Rat-Race Ring Coupler (180o Coupler)

9

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.”

Photograph of a microstrip ring couplerCourtesy of M. D. Abouzahra, MIT

Lincoln Laboratory

10

180o Coupler

1 2

4 3

[ ]

0 1 1 01 0 0 11 0 0 120 1 1 0

jS

−− = −

The rat race is a lossless 4-port (the S matrix is unitary). All four ports are matched. The device is reciprocal (the S matrix is symmetric). Port 4 is isolated from port 1, and ports 2 and 3 are isolated from each other.

Rat-Race Coupler (cont.)

11

The signal from the “sum port” Σ (port 1) splits evenly between ports 2 and 3, in phase. This could be used as a power splitter (alternative to Wilkinson).

The rat race can be used as a splitter:

180o Coupler

1 2

4 3

In phase

Σ

∆ 180o out of phase

21 31S S=

24 34S S= −

Note: A matched load is usually placed on port 4.

Rat-Race Coupler (cont.)

The signal from the “difference port” ∆ (port 4) splits evenly between ports 1 and 2, 180o out of phase. This could be used as a balun.

12

The signal from the sum port Σ (port 1) is the sum of the input signals 1 and 2.

The rat race can be used as a combiner:

12 13S S=

42 43S S= −

180o Coupler

1 2

4 3

Σ

∆

Signal 1 (V1)

Signal 2 (V2)

( )1 2 12V V S+

( )1 2 42V V S−

Rat-Race Coupler (cont.)

The signal from the difference port ∆ (port 4) is the difference of the input signals 1 and 2.

[ ]

0 1 1 01 0 0 11 0 0 120 1 1 0

jS

−− = −

13

A microstrip realization is shown here.

Rat-Race Coupler (cont.)

2

1

3

4

Σ

∆

An analysis of the rat-race coupler is given in the Appendix.

[ ]

0 1 1 01 0 0 11 0 0 120 1 1 0

jS

−− = −

Magic T

14

A waveguide realization of a 180o coupler is shown here, called a “Magic T.”

“Magic T”

Note the logo!

IEEE Microwave Theory and Techniques Society

Note: Irises are usually used to obtain matching at the ports.

Monopulse Radar

15

Rat-Race couplers are often used in monpulse radar.

A B C DΣ = + + +

( ) ( )( ) ( )

AZ

EL

B C A D

C D B A

∆ = + − +

∆ = + − +

Four antennas

https://www.microwaves101.com/encyclopedias/monopulse-comparator-networks

Σ

∆

Rat-Race:

Input from AInput from B

Input from C Input from D

A B C DΣ = + + +

( )AZ B C A D∆ = + − +

( )EL C D B A∆ = + − +

Σ∆

Σ

∆

Σ

Σ∆∆

Monopulse Radar (cont.)

16

The difference signals are used to determine the azimuth and elevation of the target.

https://www.microwaves101.com/encyclopedias/monopulse-comparator-networks

( )( )( )

/2 /2 /2

/2

1

2 sin / 2

j

j j j

j

A B e

e e e

e j

φ

φ φ φ

φ φ

− ∆

− ∆ + ∆ − ∆

− ∆

− = −

= −

= ∆sinL D θ∆ =

D = separation

0 sink Dφ θ∆ =

A

B

The difference between the two antenna signals maps into the phase difference ∆φ, which maps into the angle θ.

Port 1 Excitation“even” analysis

3 2

4 1

e e

e e

V VV V

=

=

17

( )( )

0

0

0

tan

tan / 4s s sY jY l

jYjY

β

π

=

=

=

0 01/Y Z≡

Appendix A

OC OCOCOC

OC

Here we analyze the quadrature coupler.

Input admittance of open-circuited stub:

Port 1 Excitation“odd” problem

3 2

4 1

o o

o o

V VV V

= −

= −

18

Appendix A (cont.)

( )( )

0

0

0

cot

cot / 4s s sY jY l

jYjY

β

π

= −

= −

= −

0 01/Y Z≡

SC SCSCSC

SC

Input admittance of short-circuited stub:

Consider the general case:

0

+ - sY jY

= ±for evenfor odd

[ ] [ ]0

4

0

021 0

1 2 0Y

jZ

ABCD ABCDY j

Z

λ

= =

[ ] [ ] [ ] [ ]4

Y YABCD ABCD ABCD ABCDλ⇒ =

19

Quarter-wave lineShunt load on line

.

( ) ( )

( ) ( ) ( )

0

0

cos sin

/ sin cos

line

line

line

jZ

ABCDj Z D

β β

β β

= =

In general:

0 0 / 2/ 2

lineZ Zβ π

==

Here :

Y Y

2-port

Appendix A (cont.)

[ ]0

0

0 0

0

0 0

20 0

0

021 0 1 0

1 12 0

2 21 01 2 0

2 22

2 2

jZ

ABCDY Yj

Z

jZ Y jZ

Y jZ

jZ Y jZ

jZ Y j jZ YZ

=

=

=

+

20

Hence we have:

00

+ -

jY jYZ

±

= ± =

for evenfor odd

Appendix A (cont.)

[ ]

( )

( )

0 00

2

0 00 0 0

0

0 0

0

0

12 2

11 2

2

11

1 12

jjZ jZZ

ABCDj j jjZ jZ

Z Z Z

j j jZ

jj j jZ Z

jZ

jZ

± = ± + ±

± = − + ±

=

21

Continuing with the algebra, we have:

Appendix A (cont.)

[ ]0

0

0

11

1 12e

jZABCD

jZ

=

[ ]0

102

1 02

e

j

Sj

−

= −

22

Hence we have:

Convert this to S parameters (use Table 4.2 in Pozar):

Note: We are describing a two-port device here, in the even and

odd mode cases.

Appendix A (cont.)

This is a 2×2 matrix, not a 4×4 matrix.

2 3 4

111

1 0a a a

VSV

−

+= = =

=

11 0S =

( )

- - - - - -1 1 1 1 1 1

11

11 11

12 2

1 0 02

e o e o e o

e o

V V V V V VSV V V V V

S S

+ + + + +

+ += = = + +

= + = +

23

Hence

⇒

11 22 33 44 0S S S S= = = =By symmetry:

1V V V+ + += +

Appendix A (cont.)Adding even and odd mode cases together:

( )2 2 2 221 21 21

12 2

1 1 12 2 2

2

e o e oe oV V V VS S S

V V Vj j

j

− − − −

+ + +

+ += = = +

+ − − − = +

−=

21 12 43 34 2jS S S S −

= = = =By symmetry and reciprocity:

24

2 3 4

221

1 0a a a

VSV

−

+= = =

= ⇒

Appendix A (cont.)

1V V V+ + += +

2 3 4

331

1 0a a a

VSV

−

+= = =

=

31 13 24 4212

S S S S −= = = =By symmetry and reciprocity:

25

( )3 3 3 3 2 231 21 21

12 2 2

1 1 12 2 2

12

e o e o e oe oV V V V V VS S S

V V V Vj j

− − − − − −

+ + + +

+ + −= = = = −

+ − − − = − −

=

⇒

Appendix A (cont.)

1V V V+ + += +

2 3 4

441

1 0a a a

VSV

−

+= = =

=

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

26

( )4 4 4 4 1 141 11 11

1 02 2 2

e o e o e oe oV V V V V VS S S

V V V V

− − − − − −

+ + + +

+ + −= = = = − =

+⇒

Appendix A (cont.)

1V V V+ + += +

27

Layout Schematic

2

1

3

4

Plane of symmmetry

1 2

3 4

Here we analyze the Rat-Race Ring coupler.

Appendix B

Port 1 Excitation“even” problem

28

( )

( ) ( )1 0 1 1

0

0

tan

/ 2 tan / 4

/ 2

s s s sY jY l

j Y

jY

β

π

=

=

=

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

β

π

=

=

= −

1 2

3 4

Appendix B (cont.)

#1 #2

29

( )

( ) ( )1 0 1

0

0

cot

/ 2 cot / 4

/ 2

s s s sY jY l

j Y

jY

β

π

= −

= −

= −

( )

( ) ( )1 0 2

0

0

cot

/ 2 cot 3 / 4

/ 2

s s s sY jY l

j Y

jY

β

π

= −

= −

=0 0

0 0

1/

/ 2s

Y Z

Y Y

=

=

1 2

3 4

Appendix B (cont.)Port 1 Excitation“odd” problem

#1 #2

[ ]0

0 00

0

0

0

0

0

0

1 0 1 00 211 10

2 22

1 0 1 211 0

2 2

1 2

2 1

e

j ZABCD jY jYj

Z

j ZjY j

Z

j Z

jZ

= ±

± = ±

± =

Proceeding as for the 90o coupler, we have:

30

Appendix B (cont.)

[ ]0

0

0

1 2

2 1e

j ZABCD

jZ

± =

[ ]0

1 11 12

ejS± −

=

Converting from the ABCD matrix to the S matrix, we have:

31

Use Table 4.2 in Pozar

Note: We are describing a two-port device here, in the even and

odd mode cases.

Appendix B (cont.)

This is a 2×2 matrix, not a 4×4 matrix.

21 12 34 43 2( )

jS S S S −= = = =

symmetry and reciprocity

2 3 4

111

1 0a a a

VSV

−

+= = =

=

2 3 4

221

1 0a a a

VSV

−

+= = =

=

32

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

( )1 111 11 11

12 2

12 2 20

e oe oV VS S S

Vj j

− −

+

+= = +

− = +

=

( )2 221 21 21

12 2

12 2 2

2

e oe oV VS S S

Vj j

j

− −

+

+= = +

− − = +

−=

11 33 0( )S S= =symmetry

Appendix B (cont.)

31 13 -2

( )

jS S= =

symmetry

22 44

23 32 14 41

24 42

00

2

S SS S S S

jS S

= == = = =

= =

2 3 4

331

1 0a a a

VSV

−

+= = =

=

Similarly, exciting port 2, and using symmetry and reciprocity, we have the following results (derivation omitted):

33

( )3 331 11 11

12 2

12 2 2

2

e oe oV VS S S

Vj j

j

− −

+

+= = −

− = −

−=

Appendix B (cont.)