Decoupling Feeding Network for Antenna Arrays Student: Eli Rivkin Supervisor: Prof. Reuven Shavit Department of Electrical and Computer Engineering BGU

Table of Contents BGU  Motivation – mutual coupling and its effect on antenna arrays  Eigenmode theory  Decoupling concept  Hardware implementation of the decoupling network  Conclusions

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

So What’s the Problem? Single antenna: Antenna + friend: BGU

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. BGU

Main Goal To design a feeding network connected to the antenna array so that its input ports are always matched, independently of the mutual coupling. Decoupling & Matching Network (DMN) Antenna Array BGU

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 dipole BGU V 1,1 I 1,1 V 1,2 I 1, h z y PEC x (1,1) (1,2)

Mutual Coupling Input impedance at each port (active impedance): depends on the excitation! BGU - admittance matrix - impedance matrix each element’s current depends on the voltages of all the others! V1V1 I1I1 V2V2 I2I

Radiation Pattern E-plane: H-plane: AF (array factor) EF (element factor) x z d d d d d d PEC y (x,y,z) BGU

Examples 1. Difference pattern (Bayliss)2. Sum pattern (Taylor) BGU zoom:

Eigenmode Theory [H] is Hermitian ([H] H =[H]) => it can be diagonalized by a unitary matrix: where: (unitary) (diagonal) The columns of [Q] are the eigenmodes of the antenna array. They are orthonormal vectors. - eigenefficiencies (eigenvalues) BGU [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: where:(diagonal) - modal reflection coefficients (complex, ) => energy conservation - radiation matrix BGU

Decoupling Concept bsbs asas Antenna Array [S] Decoupling Network [S D ] b a BGU from the theory:

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. The decoupling network is described by: ([Q]- matrix of eigenmodes) this network matches the system regardless of the excitation! BGU

Numerical Results BGU z x difference pattern (Bayliss) sum pattern (Bayliss)

Hardware Implementation of the DMN BGU Objective: implementation of with passive microwave elements. Special case: array of 2 antennas with, so that:. In this case, will diagonalize [S] and decouple the 2-element array. Directional Coupler (Magic-T Hybrid) Our case: symmetric rearrangement of the elements in [S] leads to: According to the special case, our [Q] can be written in block matrix notation: 8 Magic-T Hybrids #1 #2 #3 #4

Hardware Implementation of the DMN (cont’d) BGU So what does “symmetric rearrangement of the elements in [S]” mean? z x symmetry plane symmetric rearrangement z x 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: no coupling between the 2 groups!

Hardware Implementation of the DMN (cont’d) BGU How will it look like so far? #9 #1 #2 #3 #4 #1' #9' #10 #2 #2' #10' #16 #8#8' #16' #1 #2 #3 #4 #1 #2 #3 #

Hardware Implementation of the DMN (cont’d) BGU z x z x symmetry plane What’s next? Symmetric rearrangement of [S S (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 Implementation of the DMN (cont’d) BGU How will it look like so far? #1 #2 #3 #4 #1'' #5'' #4'' #8'' # 1’ #5’ #4’ # 8’ #1 #2 #3 #4 #12'' #16'' #12’ #16’ #1 #2 #3 # #9'' #13'' #9’ #13’ #1 #2 #3 #

Hardware Implementation of the DMN (cont’d) BGU 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 parts separately. [S] [S A ] columns of [A] are orthonormal real vectors, then besides, also:. Theorem: if, and the

Hardware Implementation of the DMN (cont’d) BGU [S 1 ] -jx 1 -jx 2 -jx 3 -jx 4 [S A ][S B ] The columns of [A] are the eigenvectors of [X 1 ]. Since they are real and orthonormal, the theorem can be used: The columns of [B] are the eigenvectors of [A] T [R 1 ][A]. Since they are real and orthonormal, the theorem can be used again: the input impedance matrix is diagonal - decoupling accomplished!

Hardware Implementation of the DMN (cont’d) BGU Last thing left to do is to implement [S A ] and [S B ]. Using Givens Rotations, [A] can be expressed as: where each one of [A i ] is a matrix which represents a directional coupler (an arbitrary one, not Magic-T as before). [S A ] and [S B ] 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.

Conclusions BGU  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.

References BGU [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 [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.