Decoupling FeedingNetwork for

Antenna ArraysStudent: Eli RivkinSupervisor: Prof. Reuven ShavitDepartment of Electrical and Computer Engineering

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Table of Contents BGU

Motivation – mutual coupling and its effect on antenna arrays

Eigenmode theoryDecoupling conceptHardware implementation of the

decoupling networkConclusions

Introduction

One of the major problems in antenna arrays is mutual coupling among the elements.

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So What’s the Problem?

Single antenna:

Antenna + friend:

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More Specifically - What’s the Problem with the Mutual

Coupling Again?• Difficulties in designing a predefined radiation pattern.• Gain reduction, especially at scanning.• Power mismatch => losses, reflections.• Matching is possible only for one excitation, not

always.

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Main GoalTo design a feeding network connected to the antenna array so that its input ports are always matched, independently of the mutual coupling.

0inS

Decoupling &Matching

Network(DMN)

Antenna Array

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Geometry

x

z

d

d

d

d

d

d(4,4)

(4,3)

(4,2)

(3,4)

(3,3)

(3,2)

(2,4)

(2,3)

(2,2)

(1,4)

(1,3)

(1,2)

(3,1)

(2,1)

(1,1)

(4,1)

PEC

y

dipole2

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V1,1

I1,1

V1,2

I1,2

.

.

.

h

z

y

PEC

x

(1,1)

(1,2)

Mutual Coupling1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

.................................

...

M M

M M

M M M MM M

V Z I Z I Z I

V Z I Z I Z I

V Z I Z I Z I

1 11 2

,1 ,2 , 1 , , 1 ,... ...a i i i Mi i i i i i i i i i M

i i i i i i

V I II I IZ Z Z Z Z Z Z

I I I I I I

Input impedance at each port )active impedance(:

depends on the excitation!

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1I Z V Y V

11 1

1

M

M MM

Y Y

Y

Y Y

- admittance matrix

1 11 1 12 2 1

1 1 2 2

...

.................................

...

M M

M M M MM M

I Y V Y V Y V

I Y V Y V Y V

11 1

1

M

M MM

Z Z

Z

Z Z

- impedance matrix

V Z I

each element’s current depends on the voltages of all the others!

V1

I1

V2

I2

.

.

.

Radiation Pattern

( , )F

4 4

( 1)sin cos ( 1)cos

1 1

( , ) jkd m nmn

m n

AF I e

E-plane: 90

H-plane: 90

AF (array factor)AF (array factor)

4 4

( 1)sin cos ( 1)cos

1 1

jkd m nmn

m n

I e

EF (element factor)EF (element factor)

cos cos sin sin sin2 2

sin

x

z

d

d

d

d

d

d

PEC

y

r

(x,y,z)

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Examples

1. Difference pattern )Bayliss(2. Sum pattern )Taylor(

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zoom:

Eigenmode Theory[ ]b S aH H

i rP a a P b b

[ ] [ ] [ ]H H H H Hrad i rP P P a a b b a U S S a a H a

[H] is Hermitian )[H]H=[H]( => it can be diagonalized by a unitary matrix:

[ ][ ][ ]HH Q Q

where:

[ ][ ] [ ]HQ Q U )unitary( 1 2{ }

H

Mdiag Q H Q )diagonal(

The columns of [Q] are the eigenmodes of the antenna array. They are orthonormal vectors.0 1i - eigenefficiencies

(eigenvalues)

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[H] - radiation matrix

Antenna Array

[S]

b

a

Eigenmode Theory )cont’d(

If [Q] diagonalizes [H] )as shown previously( then it also diagonalizes [S] via:

*[ ] [ ][ ]HS Q Q

where:

1 2{ }T

Mdiag Q S Q )diagonal(

i- modal reflection coefficients )complex, (- modal reflection coefficients )complex, (0 1i

21i i => energy conservation

[ ][ ][ ]HH Q Q

[ ] [ ] [ ]HH U S S - radiation matrix

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Decoupling Concept

bs

as Antenna Array

[S]

Decoupling Network

[SD]

b

a

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11 1

1

M

M MM

S S

S

S S

,21

,21

0T

D

D

D

SS

S

,21 ,21[ ]T

S D DS S S S

from the theory: 1

2

0 0

0

0

0 0

T

M

Q S Q

SS

,21[ ] [ ]DS Q

SS

1

2

0 0

0

0

0 0 M

Decoupling Concept - Conclusions

• It is reciprocal and lossless.• Its input and output ports are matched and

decoupled.• Power is transferred according to the matrix of

eigenmodes [Q].• Each input port excites a different eigenmode =>

every excitation is a superposition of the orthonormal eigenmodes.

• All the input ports are independent of each other, so now it is possible to match each port individually.

0

T

D

QS

Q

• The decoupling network is described by: ]( Q -[matrix of eigenmodes)

this network matches the system regardless of the excitation!

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Numerical Results BGU

z

x1234

5678

9101112

13141516difference pattern)Bayliss(

sum pattern)Bayliss(

Hardware Implementationof the DMN

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Objective: implementation of with passive microwave elements. 32 32

0

0

T

D x

QS

Q

Special case: array of 2 antennas with , so that: .11 22s s 11 12

2 212 11

x

s sS

s s

In this case, will diagonalize [S] and decouple the 2-element array. 1 1

1 12

jQ

0 0 1 1

0 0 1 1

1 1 0 02

1 1 0 0

D

jS

Directional Coupler )Magic-T Hybrid(

Our case: symmetric rearrangement of the elements in [S] leads to:

11 128 8 8 8

16 1612 118 8 8 8

x xsym x

x x

S SS

S S

According to the special case, our [Q] can be written in block matrix notation:

(1) 8 8 8 8

16 168 8 8 82x x

xx x

U UjQ

U U

8 Magic-T Hybrids

8 8 8 816 16

8 8 8 8(1)

32 328 8 8 8

16 168 8 8 8

0

20

x xx

x xD x

x xx

x x

U U

U UjS

U U

U U

#1

#2

#3

#4

Hardware Implementationof the DMN )cont’d(

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So what does “symmetric rearrangement of the elements in [S]” mean?z

x1234

5678

9101112

13141516

symmetryplane

symmetric rearrangement

z

x1234

5678

9101112

13141516

original arrangement

pairs: 1 & 9, 2 & 10 …

Each of these pairs should be connected by a coupler.

After this step the system matrix will be:

12 11(1) (1) (1) 8 8 8 8

16 1616 16 16 16 16 1612 118 8 8 8

0

0

Tx x

S sym xx x xx x

S SS Q S Q

S S

no coupling between the 2 groups!

Hardware Implementationof the DMN )cont’d(

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How will it look like so far?

(1)

32 32D xS

#9

#1#1

#2

#3

#4

#1'

#9'

#10

#2#2'

#10'

#16

#8#8'

#16'

#1

#2

#3

#4

#1

#2

#3

#4

.

.

.

11 128 8 8 8

16 1612 118 8 8 8

x xsym x

x x

S SS

S S

12 11(1) 8 8 8 8

16 1612 118 8 8 8

0

0x x

S xx x

S SS

S S

Hardware Implementationof the DMN )cont’d(

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z

x1234

5678

9101112

13141516

z

x1265

3487

11121615

9101413

symmetryplane

What’s next?

• Symmetric rearrangement of [SS)1(] )division of each group of

8 into 2 symmetric groups of 4 elements(.• Connection of 8 more Magic-T Hybrids )4 for each group(.• After this there will be no coupling between the 4 groups of 4.

Same procedure as before:

Hardware Implementationof the DMN )cont’d(

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How will it look like so far?

1 4 44 4

8 824 4 4 4(2)

16 163 4 44 4

8 844 4 4 4

00

0

00

0

xx

x

x xS x

xx

x

x x

S

SS

S

S

#1

#2

#3

#4

#1''

#5''

#4''

#8''

#1’

#5’

#4’

#8’

#1

#2

#3

#4

#12''

#16''

#12’

#16’

#1

#2

#3

#4

.

.

.

(2)

32 32D xS

#9''

#13''

#9’

#13’

#1

#2

#3

#4

.

.

.

(1)

32 32D xS

#9

#1#1

#2

#3

#4

#1'

#9'

#16

#8#8'

#16'

#1

#2

#3

#4

.

.

.

#12

#4#1

#2

#3

#4

#4'

#12'

#13

#5#5'

#13'

#1

#2

#3

#4.

.

.

12 11(1) 8 8 8 8

16 1612 118 8 8 8

0

0x x

S xx x

S SS

S S

Hardware Implementationof the DMN )cont’d(

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So what do we have so far?• 4 independent sub-arrays of 4 elements each.• Each sub-array has known [S] and [Z] matrices, calculated

in MATLAB.• No more symmetry planes have left, so it’s impossible to

use the same method again.

A different method will be used to decouple each of the sub-arrays.The method is based on diagonalizing the imaginary and the real partsseparately.

[S]...

[SA]...

in inZ or S Z or S

columns of [A] are orthonormal realvectors, then besides ,also: .

0

0

T

A

AS

A

T

inS A S A

T

inZ A Z A

Theorem: if , and the

Hardware Implementationof the DMN )cont’d(

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[S1]

-jx1

-jx2

-jx3

-jx4

[SA]

0

0

T

A

AS

A

[SB]

0

0

T

B

BS

B

1 1 1Z R j X

1

21 1 1 1

3

4

0 0 0

0 0 0'

0 0 0

0 0 0

T T T

A

x

xZ A Z A A R A j A R A j

x

x

1 1 1' ' 'Z R j X

The columns of [A] are the eigenvectors of [X1]. Since they are real andorthonormal, the theorem can be used:

1 '''Z

1 1 1''T T

A AZ A R A j j A R A

1 1'T

AZ A R A j

1 1''T

Z A R A

The columns of [B] are the eigenvectors of [A]T[R1][A]. Since they are real and orthonormal, the theorem can be used again:

1

21 1

3

4

0 0 0

0 0 0'''

0 0 0

0 0 0

T T

B

r

rZ B A R A B

r

r

B

the input impedance matrix is diagonal -decoupling accomplished!

Hardware Implementationof the DMN )cont’d(

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Last thing left to do is to implement [SA] and [SB].

4 4 4 48 8

4 4 4 4

0

0

T

x xA x

x x

AS

A

Using Givens Rotations, [A] can be expressed as:where each one of [Ai] is a matrix which represents a directional coupler )an arbitrary one, not Magic-T as before(.

1 2 6...A A A A

[SA] and [SB] are implemented by 6 cascaded couplers each.

Decoupling a 4-element sub-array requires 12 couplers.

The 4 sub-arrays which were left after the first method require 48 couplers.

ConclusionsBGU

The suggested decoupling network achieves the goal- the system will be always matched.

It can be implemented both in software and hardware.

Software implementation requires connecting the antenna array to a computer which does all the matrix calculations )after translating the signal to baseband and sampling(.

Hardware implementation requires 64 directional couplers )16 for the first steps with the first method, 48 for the last step with the second method(. All the parameters of the couplers were calculated in MATLAB.

Using only the second method, which is a general one )not depending on symmetries(, would require 320 couplers.

ReferencesBGU

[1] Volmer, C., Weber, J., Stephan, R., Blau, K. and Hein, M.A., "An Eigen-Analysis of Compact Antenna Arrays and Its Application to Port Decoupling", IEEE Transactions on Antennas and Propagation, Vol. 56, No. 2, February 2008.

[2] Geren, W.P., Curry, C.R. and Andersen, J., "A Practical Technique for Designing Multiport Coupling Networks", IEEE Transactions on Microwave and Techniques, Vol. 44, No. 3, March 1996.