Ayan Kumar Bandyopadhyay

Transcription

1 Charaterization of radiating apertures using Multiple Multipole Method And Modeling and Optimization of a Spiral Antenna for Ground Penetrating Radar Appliations Ayan Kumar Bandyopadhyay FET-IESK, Otto-von-Guerike-University, Chair of Mirowave Engineering and Communiation ayan@iesk.et.uni-magdeburg.de

2 Part I Outline 1. Introdution 2. Multiple Multipole Method to haraterize radiating apertures 3. Problem statement 4. Solution approahes: SVD-Angle approah Rank Revealing QR approah 5. Simulation results 6. Conlusions

3 Motivation Advantages of Multiple Multipole Tehnique: Highly flexible. Does not require time onsuming onvergene tests, provides possibility for residual error estimation. The radiation onditions are automatially satisfied (no approximation like infinite flange, absorbing boundaries) Aurate estimation of the bak-sattered field and near field. Cruial point Using appropriate number of multipoles at appropriate positions.

4 1. Introdution The multiple multipole method is a semi-analyti method for eletromagneti field omputation. It is a frequeny domain tehnique. V Aperture (S a ) Known E, H The spae outside the radiating objet is soure free. Metalli Surfae (S ) The field on the aperture is expressed in terms of aperture waveguide modes. Our aim is to ompute the Y matrix desription for the aperture.

5 2. MM Method to Charaterize Radiating Apertures TE (TM) multipoles => osillating magneti (eletri) surfae harge distributions residing on the surfae of an infinitesimal sphere emitting eletromagneti field as spherial wave. Multipole loations Metalli Surfae (S ) Aperture (S a ) The eletromagneti fields radiated by the multipoles loated within the volume V are used to represent the field outside V Sum on multipole loations Sum on multipole orders Unknowns elementary multipole field External field E (ext) (r) = Σ k Σ l Q kl E l (mult) (r-r k ) H (ext) (r) = Σ k Σ l Q kl H l (mult) (r-r k ) multipole loations

6 2. MM Method to Charaterize Radiating Apertures (Contd.) The unknown oeffiients (Q kl ) should be hosen so that the total field math the boundary onditions on S and S a Multipole loations Mathing Points Enforing tangential E-field on (S +S a ) E (r m ) = Σ k Σ l Q kl E l (mult) (r m -r k ) Enforing tangential H-field on (S a ) H (r m ) = Σ k Σ l Q kl H l (mult) (r m -r k ) The field on some mathing points r m (on the surfae of the volume) are enfored in order to find out the expansion oeffiients Q kl Matries ontaining the multipole field omponents at the mathing points Vetor ontaining E and H fields at mathing points v = [C] Q i = [C] Q Vetor ontaining the unknowns

7 2. MM Method to Charaterize Radiating Apertures (Contd.) Modal expansion of the aperture field on the field mathing points (r m ) gives E (r m ) = Σ n V n e n (r m ) H (r m ) = Σ n I n h n (r m ) Matrix form v = [D] V i = [D] I Modal voltages and urrents Matries ontaining field omponents of aperture waveguide modes Equating these fields to the multipole fields [C] Q = [D] V ~ ~ [C] Q = [D] I Q = ([C] H [C] ) 1 [C] H [D] V Inversion [C] must be well onditioned

8 3. Problem Statement We need a dense distribution of multipoles, at the same time having a well onditioned C matrix. The redundant multipoles should be removed. [ C ] = M m n 3 n mn 12 1 n Field omponent m of all multipoles at a partiular point Field omponent of a partiular multipole on all mathing points n = total number of multipoles m = 3 x number of mathing points m>n => Over-determined system

9 3. Problem Statement (ontd.) Effet of redundant multipoles: Mathing points Multipole loations Antenna surfae The field produed by distant multipoles at the adjaent mathing points are almost idential. linearly dependent or nearly dependent olumns leading to ill onditioned or rank defiit C-matrix.

10 4. Solution approahes SVD-Angle approah: a. Estimation of the linear dependeny among the olumns of C matrix using the Singular Value Deomposition (SVD). C = U S V H Unitary (m x m) Diagonal (m x n) Unitary (n x n) Zero or very small singular values depit rank defiieny or ill onditioning i.e. Redundant multipoles. Number of Singular values whih are less than a threshold t svd is equal to the number of redundant olumns (multipoles) (with respet to the orresponding threshold t svd ).

11 4. Solution approahes (ontd.) b. Identifiation of the redundant multipoles: The angle matrix A is alulated as, A ij = os 1 C * j C i C C i j C i = i-th olumn of C matrix i,j = 1,2... n The elements of matrix A provide a measure of the linear dependeny among the olumns of C, hene the redundany among the multipoles. A ij is upper triangular and the diagonal elements are zero. The redundant multipoles are identified aording to the values of A.

12 4. Solution approahes (ontd.). Removal of the olumns A12 A13... A A23... A A = A 1n 2n (n-1)n Repeat the whole proedure till the number of deleted olumns equals the number of linearly dependent olumns deteted previously by SVD. Find out Minimum A 1min A 2min.. A (n-2)min A (n-1) Find out Minimum Note the orresponding olumn numbers and remove the olumn from C whih has smaller normamong them.

13 4. Solution approahes (ontd.) Rank Revealing QR fatorization approah: The matrix C an be deomposed in a rank revealing QR fatorization as: Permutation matrix CP = QR Matrix with orthogonal olumns R11 R = Q 0 R Upper triangular matrix Well onditioned triangular submatrix Triangular matrix having very small norm The threshold t rrqr is used as the upper bound of the ondition number of R 11. From the permutation matrix P, we an get the olumns treated as linearly dependent during the deomposition.

14 5. Simulation results Metalli surfae Y aperture Shemati diagram of the simulated struture X Z Illumination with the fundamental mode only EM symmetry (x-axis eletri wall, y axis magneti wall) The boundary ondition of the problem (eletri field amplitude) Color bar represents the eletri field strength

15 5. Simulation results (ontd.) Aperture mathing points = 400, Mathing points on metalli surfae = 1205, Multipole loations = 199, Total number of multipoles = 1194 The C matrix is severely ill-onditioned Completely wrong field alulation Calulated E-field amplitude without multipole redution Average normalized error at the aperture ~ 0.99 (out of 1.0)!!

16 5. Simulation results (ontd.) SVD-Angle approah: Error in the eletri field alulation for different thresholds (t svd ) The LAPACK routines have been used for the SVD Calulated E-field amplitude after multipole redution with threshold t svd = 10-3 No of deleted multipoles 53

17 5. Simulation results (ontd.) RRQR approah: Error in the eletri field alulation for different thresholds (t rrqr ) The ZGEQPX routines have been used for the RRQR deomposition Calulated E-field amplitude after multipole redution with threshold t rrqr = 10-6 No of deleted multipoles 51

18 5. Simulation results (ontd.) Comparison of the radiation patterns for the elliptial aperture before and after multipole redution (at 4.5 GHz)

19 6. Conlusions The multiple multipole method has been applied to haraterize radiating apertures. Two methods, one involving Singular value deomposition and the other one involving the rank revealing QR fatorization have been applied to identify the redundant multipoles in Multiple Multipole Method. Applying these methods to haraterize a simple radiating aperture, a substantial improvement has been ahieved for the eletromagneti field alulation. These methods are general and an be used for automati multipole setting for any struture.