minimum complexity decoupling networks for arbitrary coupled loads

Minimum Complexity Decoupling Networks for ArbitraryCoupled Loads

Ding Nie, Bertrand Hochwald and Erik Stauffer

University of Notre DameBroadcom Cooperation

[email protected]@[email protected]

July 8, 2014

Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 1 / 13

Overview

1 Introduction to Decoupling Networks

2 Systematic Design of Decoupling Networks

3 Minimum Complexity Decoupling Networks

4 Summary


Coupled Loads

RF mutual coupling

Coupled antennas in MIMOcommunications

RFIC coupled microstrip lines ⋯

Coupling is undesirable

Introduces power reflection

Mixes useful signals with unwanted signals

Solution: decoupling network

Achieves perfect impedance matching at the design frequency

Maximizes the power efficiency of the RF system


Introduction to Decoupling Networks

Two-port matching network is used to match single source to a singleload

Decoupling network is used to match uncoupled sources to coupledloads

Transforms the coupled impedance of the loads into the uncoupledcharacteristic impedance of the sources

2N-port matching network

N coupled loads

.

.

.

.

.

.

.

.

.

0Z

0Z

Two-port matching network

LZ 0Z0Z

0Z

0Z

.

.

.


Complexity of Decoupling Networks

The decoupling networks arecomplicated in general

But the realization is not unique. . .

. . .1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N

Our Contribution

Systematic and unified decouplingnetwork design for arbitrary coupledloads

Decoupling network design method withminimum complexity

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N


Examples of Minimum Complexity Decoupling Networks

Two and Three Dipoles (at 2.4 GHz)[0.17 + 0.48j 0.36− 0.36j0.36− 0.36j 0.17 + 0.48j

] 0.18 + 0.45j 0.42− 0.34j −0.10− 0.10j0.42− 0.34j 0.16 + 0.60j 0.16− 0.26j−0.10− 0.10j 0.16− 0.26j 0.37 + 0.35j

10λ

2λ

5λ

2λ

10λ

(a) (b)

1.39 nH

2.74 pF

2.15 pF

1.09 nH

3.81 pF

1.61 nH

10.03 nH

1

2

3

4

11c14c 44c

12c

13c

33c

23c 22c

25c

55c

36c

66c

1

2

3

4

5

6

c11 32.06 pFc12 9.20 pFc13 2.24 nHc14 29.66 pFc22 3.46 pFc23 4.26 pFc25 3.96 pFc33 3.13 pFc36 5.41 pFc44 0.29 nHc55 2.12 nHc66 2.14 nH


Properties of Decoupling Networks

Definition of Decoupling Network

A decoupling network for Ndissipative reciprocal loads withS-matrix SL is a lossless, reciprocal,2N-port network S that satisfiesSLM = 0, where

SLM = S11 + S12SL(I − S22SL)−1S21

0Z

2N-port matching network

0

Z

.

.

.

LSLMS

N-port loads

1a

1b

2a

2b

.

.

.

Output ports N

+1~2N

Input ports 1~N

11 12

21 22

S SS S

0

Non-uniqueness of Decoupling Networks

Set of decoupling networks for SL

S := {S ∈ C2N×2N : S22 = SHL ,S

HS = I ,ST = S}S has N2 degrees of freedom

All S-matrix in S has the same performance, but different realizationcomplexity


Network Synthesis with Generalized Π-Network

Generalized Π-Network

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N

11c

22c

33c

44c

12c

13c

24c

34c

14c

23c

1

2

3

4

(b)

22c21

12c

(a)

(c)

11c

Π-network:Y =[c11 + c12 −c12

−c12 c12 + c22

]Generalized 2N-port Π-network:Y =∑2N

i=1 c1i −c12 · · · −c1(2N)

−c12∑2N

i=1 c2i · · · −c2(2N)...

.... . .

...

−c1(2N) −c2(2N) · · ·∑2N

i=1 ci(2N)


Systematic Decoupling Networks Design Steps

Systematic Decoupling Networks Design Steps

1 Find a S-matrix S that belongs to the set of decoupling networks S,such that the number of impedance is minimized

2 Compute the admittance matrix of the decoupling network usingCayley transform

Y =1

Z0(I − S)(I + S)−1

3 Realize Y using generalized Π-network

The minimum number of impedances needed to realize a decouplingnetwork is N2 + N


Minimum Complexity Decoupling Networks Design

Minimum Complexity Decoupling Networks Design

For arbitrary coupled loads, we obtain the following decoupling networkstructure with N2 + N impedances, which is the minimum numberachievable.

Y? =

× × × · · · × × × 0 0 · · · 0 0× × × · · · × × 0 × × · · · × ?× × × · · · × × 0 0 × · · · × ×...

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.× × × · · · × × 0 0 0 · · · × ×× × × · · · × × 0 0 0 · · · 0 ×× 0 0 · · · 0 0 × 0 0 · · · 0 00 × 0 · · · 0 0 0 × 0 · · · 0 00 × × · · · 0 0 0 0 × · · · 0 0

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.. . .

.

.

.

.

.

.0 × × · · · × 0 0 0 0 · · · × 00 ? × · · · × × 0 0 0 · · · 0 ×

. . .

. . .

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N(a) (b)

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N


Minimum Complexity Decoupling Networks for SymmetricLoads

Symmetric loads are SL that has the form

SL =

µL + ξL ξL · · · ξLξL µL + ξL · · · ξL...

.... . .

...ξL ξL · · · µL + ξL

Apply the systematic design method, we get

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N

. . .

2

3

1N −

N

1N +

2N +

2 1N −

2N

3N +

13c

4c2c

1c

Only 4N impedances are needed


Summary

Systematic design of decoupling networks for arbitrary coupled loads

Decoupling networks realization using N2 + N components, theminimum possible

Examples of two-, three-antennas and symmetric loads


References

J. B. Anderson and H. H. Rasmussen, “Decoupling and descatteringnetworks for antennas,” IEEE Transactions on Antennas andPropagation, vol. 24, no. 6, pp. 841-846, Nov. 1976.

J. C. Coetzee and Y. Yu, “Design of decoupling networks for circulantsymmetric antenna arrays,” IEEE Antennas and Wireless PropagationLetters, vol. 8, pp. 291-294, 2009.

D. M. Pozar, Microwave Engineering 4th ed., John Wiley & Sons,2011.

D. Nie, B. Hochwald and E. Stauffer, “Systematic design of large-scalemultiport decoupling networks,” IEEE Transaction on Circuits andSystems I: Regular Papers, vol. 61, no. 7, pp. 2172-2181, July 2014.


minimum complexity decoupling networks for arbitrary coupled loads

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