Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19...

29
Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE Quadrature Coupler and Rat-Race Coupler 1

Transcript of Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19...

Page 1: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

Notes 19

ECE 5317-6351 Microwave Engineering

Fall 2017

Prof. David R. Jackson Dept. of ECE

Quadrature Coupler and Rat-Race Coupler

1

Page 2: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

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.

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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 1-1

1 0 020 1 0

j

j

S

j

j

=

Page 4: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

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 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.

Quadrature Coupler (cont.)

Page 5: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

Branch-line coupler

5

A microstrip realization of a branch-line coupler is shown here.

Notes: We only need to study what happens when we excite

port 1, since the structure is symmetric.

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

This analysis is given in the Appendix.

Quadrature Coupler (cont.)

1 2

34

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Summary

[ ]

0 1 00 0 1-1

1 0 020 1 0

j

j

S

j

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

Page 7: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

Rat-Race Ring Coupler (180o Coupler)

7

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 coupler

Courtesy of M. D. Abouzahra, MIT Lincoln Laboratory

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180o Coupler

1 2

4 3

[ ]

0 1 1 01 0 0 -1-1 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.)

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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 Wilkenson).

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.

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.)

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The signal from the sum port Σ (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:

12 13S S=

42 43S S= −

180o Coupler

1 2

4 3

Σ

Signal 1 (V 1)

Signal 2 (V 2)

( )1 2 12V V S+

( )1 2 42V V S−

Rat-Race Coupler (cont.)

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[ ]

0 1 1 01 0 0 -1-1 0 0 120 -1 1 0

jS

=

11

A microstrip realization is shown here.

Page 12: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

[ ]

0 1 1 01 0 0 -1-1 0 0 120 -1 1 0

jS

=

Magic T

12

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.

Page 13: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

Even Analysis

3 2

4 1

e e

e e

V V

V V

=

=

13

( )( )

0

0

0

tan

tan / 4s s s

Y jY l

jY

jY

β

π

=

=

=

0 01 /Y Z=

Appendix Here we analyze the quadrature coupler.

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Odd Analysis

3 2

4 1

-

-

o o

o o

V V

V V

=

=

14

Appendix (cont.)

( )( )

0

0

0

cot

cot / 4s s s

Y jY l

jY

jY

β

π

= −

= −

= −

0 01 /Y Z=

Page 15: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

Consider the general case:

0

+ -

Y jY

=

±for even

for odd

[ ] [ ]0

4

0

021 0

1 2 0Y

jZ

A B C D A B C D

Y j

Z

λ

= =

[ ] [ ] [ ] [ ]4

Y YA B C D A B C D A B C D A B C Dλ⇒ =

15

Quarter-wave line Shunt load on line

.

( ) ( )

( ) ( ) ( )

0

0

cos sin

/ sin cos

lin e

lin e

lin e

jZ

A B C D

j Z D

β β

β β

= =

In general:

0 0 / 2/ 2

lin e

Z Z

β π==

Here :

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[ ]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

A B C D

Y Yj

Z

jZ Y jZ

Y j

Z

jZ Y jZ

jZ Y j jZ Y

Z

=

=

=

+

16

Hence we have:

00

+ -

jY jY

Z

±

= ± =

for evenfor odd

Appendix (cont.)

Page 17: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

[ ]

( )

( )

0 00

2

0 00 0 0

0

0 0

0

0

12 2

11 2

2

11

1 12

jjZ jZ

Z

A B C D

j j jjZ jZ

Z Z Z

j j jZ

jj j j

Z Z

jZ

j

Z

± = ± + ±

± = − + ±

=

17

Continuing with the algebra, we have:

Appendix (cont.)

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[ ]0

0

0

11

1 12e

jZ

A B C D

j

Z

=

[ ]0

1-02

1- 02

e

j

S

j

=

18

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 (cont.)

Page 19: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

2 3 4

-1

111 0a a a

VS

V+

= = =

=

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 VS

V V V V V

S S

+ + + + +

+ += = = + +

= + = +

19

Hence

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

1V V V+ + += +

Appendix (cont.)

Page 20: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

( )- - - -

2 2 2 221 21 21

12 2

1 -1- 1-2 2 2-2

e o e o

e oV V V V

S S S

V V V

j j

j

+ + +

+ += = = +

+ = +

=

21 12 43 34-2j

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

20

2 3 4

-2

211 0a a a

VS

V+

= = =

= ⇒

1V V V+ + += +

Appendix (cont.)

Page 21: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

2 3 4

-3

311 0a a a

VS

V+

= = =

=

31 13 24 42-12

S 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 2-12

e o e o e o

e oV V V V V V

S S S

V V V V

j j

+ + + +

+ + −= = = =

+ =

=

1V V V+ + += +

Appendix (cont.)

Page 22: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

2 3 4

-4

411 0a a a

VS

V+

= = =

=

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

22

( )- - - - - -

4 4 4 4 1 141 11 11

1 - 02 2 2

e o e o e o

e oV V V V V V

S S S

V V V V+ + + +

+ + −= = = = =

+⇒

1V V V+ + += +

Appendix (cont.)

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Layout Schematic

2

1

3

4

Plane of symmmetry

1 2

3 4

Here we analyze the Rat-Race Ring coupler.

Appendix (cont.)

Page 24: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

Even Analysis

24

( )

( ) ( )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 (cont.)

Page 25: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

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Odd Analysis

( )

( ) ( )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 (cont.)

Page 26: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

[ ]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 Z

A B C D jY jYj

Z

j Z

jYj

Z

j Z

j

Z

= ±

± = ±

± =

Proceeding as for the 90o coupler, we have:

26

Appendix (cont.)

Page 27: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

[ ]0

0

0

1 2

2 1e

j Z

A B C D

j

Z

± =

[ ]0

1 1-1 12

e

jS

± =

Converting from the A B C D matrix to the S matrix, we have

27

Table 4.2 in Pozar

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

Appendix (cont.)

Page 28: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

21 12 34 43-2

( )

jS S S S= = = =

symmetry and reciprocity

2 3 4

-1

111 0a a a

VS

V+

= = =

=

2 3 4

-2

211 0a a a

VS

V+

= = =

=

28

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

( )- -

1 111 11 11

12 2

1 -2 2 20

e o

e oV V

S S S

V

j j

+

+= = +

= +

=

( )- -

2 221 21 21

12 2

1 - -2 2 2-2

e o

e oV V

S S S

V

j j

j

+

+= = +

= +

=

11 33 0( )S S= =symmetry

Appendix (cont.)

Page 29: Prof. David R. Jackson Dept. of ECEael.chungbuk.ac.kr/lectures/undergraduate/microwave...Notes 19 ECE 5317-6351 Microwave Engineering Fall 2017 Prof. David R. Jackson Dept. of ECE

31 13 -2

( )

jS S= =

symmetry

22 44

23 32 14 41

24 42

00

2

S S

S S S S

jS S

= == = = =

= =

2 3 4

-3

311 0a a a

VS

V+

= = =

=

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

29

( )- -

3 331 11 11

1 -2 2

1 - -2 2 2-2

e o

e oV V

S S S

V

j j

j

+

+= =

=

=

Appendix (cont.)