1. Electromagnetic Modeling of Large and Non-uniform Planar Array Structures using Scale Changing Technique (SCT) Presented by: Aamir RASHID Thesis Adviser: M. HervéAubert 1

2. Presentation Plan Motivation Scale Changing Technique (SCT) Modeling of a Planar Reflector cell Characterization of Electromagnetic Coupling in non-uniform array Scattering from 2-D Planar Arrays Conclusions and Perspectives 2

3. Presentation Plan Motivation Scale Changing Technique (SCT) Modeling of a Planar Reflector cell Characterization of Electromagnetic Coupling in non-uniform array Scattering from 2-D Planar Arrays Conclusions and Perspectives 3

5. Long Execution Times

6. Immense Memory Requirements

7. Numerical conditioning problems

8. Generally used solution

9. Simulations under Infinite array assumption using Floquet Modes

10. Problems with the infinite array approach

11. not applicable to non-uniform arrays4

12. Presentation Plan Motivation Scale Changing Technique (SCT) Modeling of a Planar Reflector cell Characterization of Electromagnetic Coupling in non-uniform array Scattering from 2-D Planar Arrays Conclusions and Perspectives 5

13. Scale Changing Technique W1 W3 W0 Reflectarray Phase-shifter cell RF-MEMS switch W2 W4 Slot loaded by RF-MEMS switches Region below the switch 100 10-1 10-2 10-3 10-4 10-5 Surface of Wi / Surface of W0 RF-MEMS controlled Reflectarray – An example of a multiscale structure 10-6 Simulation of such structures with conventional techniques a Nightmare!! 6

14. Scale Changing Technique Decomposition of the planar surface ------- by introducing artificial boundary conditions Definition of tangential electromagnetic fields on orthonormal modal basis Determination of appropriate number of modes Computation of Scale-changing Networks (SCN) Cascade of all SCNs 7

15. Scale Changing Technique W3 W2 W4 Partitioning of the Discontinuity Plane W1 [Ref] E.Perret et al, “Scale-changing technique for the electromagnetic modeling of MEMS-controlled planar phase-shifters”, IEEE Transactions on Microwave Theory and Techniques, Vol.54, No.9, September 2006, Pages: 3594-3601. 8

17. Perfect Electric

18. Perfect Magnetic

19. A combination of above two

20. Periodic (Floquet)

21. Tangential EM fields expressed on orthonormal modal basis

22. Lower order modes define Large scale field variations

23. Higher order modes define Localized and fine scale variations

24. EM coupling between parent domain and daughter subdomains is largely due to lower order modes ---- active modes

25. Appropriate number of active and passive modes need to be determined --- convergence study

26. EM coupling between a domain at scale ‘s’ and its subdomains at scale ‘s-1’ is represented by a Scale Changing Network9

27. Scale Changing Technique Scale Changing Network (SCN) A multiport network where the ports represent active modes Models EM coupling between successive scales Computation of all SCNs mutually independent --- PARALLEL PROCESSING 10

28. Scale Changing Technique Cascade of all Scale Changing Networks 11

29. Scale Changing Technique Structures already simulated using SCT MEMS controlled Reflector cell Active Patch Antenna [Ref] E.Perret et al, “Scale-changing technique for the computation of the input impedance of active patch antennas”, IEEE Antennas and Wireless Propagation Letters, Vol.4, 2005. Multi-frequency Selective surfaces Prefractal Multi-band scatterer [Ref] D.Voyer et al, “Scale-changing technique for the electromagnetic modeling of planar self-similar structures”, IEEE Transactions on Antennas and Propagation, Vol.54, No.10, October 2006. 12

30. Presentation Plan Motivation Scale Changing Technique (SCT) Modeling of a Planar Reflector cell Characterization of Electromagnetic Coupling in non-uniform array Scattering from 2-D Planar Arrays Conclusions and Perspectives 13

31. Infinite Reflectarray Modeling Planar Reflector Cell under infinite array conditions y Floquet modal expansion to simulate infinite array conditions kinc Plane-wave Excitation (Oblique Incidence) θ φ b2 b1 b0 Nine geometric configurations simulated varying dimensions b1 and a2 a2 z a1 Objective: To find the phase of reflection coefficients for TE00 and TM00 modes a0 Periodic BC Magnetic BC Electric BC a0=15mm b0=15mm Dielectric thickness = 4mm Dielectric=Air a1=12mm b2=1mm b1 and a2variable 14

32. Infinite Reflectarray Modeling Application of Scale Changing Technique (SCT) Partitioning of planar geometry at three scale-levels Cascade of SCNs 15

33. Infinite Reflectarray Modeling Convergence Study (Normal Incidence) Convergence curves between modes in domains D1(2) and D1(3) (discontinuity domain) Convergence curves between active modes in domains D1(2) and D1(1) (slot domain) Modes for convergence 2500 [D1(3)] 600 [D1(2)] 80 [D1(1)] 16

34. Infinite Reflectarray Modeling Numerical Results (Normal Incidence) Phase results for the reflection coefficient of TM00 mode Convergence study performed at the centre frequency of 12.1GHz for each case ____ SCT x xx HFSS 17

35. Infinite Reflectarray Modeling Numerical Results (Oblique Incidence) at 12.1 GHz (φinc = 0⁰) Phase S11 (Degrees) SCT solid lines HFSS broken lines Variation in the phase results of the TM00 mode with the change in the incidence angle (θ) 18 Phase S22(Degrees) Variation in the phase results of the TE00mode with the change in the incidence angle (θ)

36. Presentation Plan Motivation Scale Changing Technique (SCT) Modeling of a Planar Reflector cell Characterization of Electromagnetic Coupling in non-uniform array Scattering from 2-D Planar Arrays Conclusions and Perspectives 19

38. Artificial boundary conditions are introduced around each unit-cell domain

39. A SCN can then be used to combine two unit-cell domains into a single domain modeling the mutual interactions between them

40. Appropriate number of modes need to be determined in each domain

41. An iterative cascade of these SCNs is used to model the entire array20

42. Electromagnetic Coupling Characterization Bifurcation Scale Changing Network Multiport Representation of Bifurcation SCN (Characterized by an impedance or admittance matrix ) Equivalent Electric Network (for only TEM excitation in each domain) 21

43. Electromagnetic Coupling Characterization Scattering from two planar half-wave dipoles 22

44. Electromagnetic Coupling Characterization Scattering from two planar half-wave dipoles RCS ratio in the absence of mutual coupling For separation D >> λ RCS Ratio λ 23

45. Electromagnetic Coupling Characterization Electromagnetic coupling modeled by Bifurcation SCN IE3D SCT |Er| in the absence of mutual coupling (IE3D SCT) [Ref] A.Rashid, H.Aubert , “Modeling of Electromagnetic Coupling in Finite Arrays Using Scale-changing Technique”, Progress In Electromagnetics Research Symposium (PIERS), 5-8 July 2010, Cambridge, USA. 24

46. Modeling Linear Arrays Modeling of a non-uniform linear (1-D) array of metallic stripes A non-uniform array of lossless metallic strips A unit-cell of the array Dimensions: a=10 mm b=9mm x=2mm Freq=5 GHz Partitioning process of an eight cell array 25

47. Modeling Linear Arrays Modeling of a non-uniform linear (1-D) array of metallic stripes Modeling of entire array by an iterative cascade of Bifurcation SCNs 26

48. Modeling Linear Arrays Modeling of a non-uniform linear (1-D) array of metallic stripes 27

49. Modeling Linear Arrays Modeling of a non-uniform linear (1-D) array of metallic stripes Execution Time (Normalized) Iteration = Number of Bifurcation SCNs used Array size (no of unit-cells) = 2^(Iteration) 1. [Ref] A.Rashid, H.Aubert, H.Legay “ Modélisation Electromagnétique d’un Réseau Fini et Non-Uniforme par la Technique par Changements d’Echelle”, JNM 2009, Grenoble 28

50. Presentation Plan Motivation Scale Changing Technique (SCT) Modeling of a Planar Reflector cell Characterization of Electromagnetic Coupling in non-uniform array Scattering from 2-D Planar Arrays Conclusions and Perspectives 29

51. Modeling 2-D Planar Arrays Bifurcation Scale Changing Network in 2-D Partitioning of 2-D planar array Phase results for the reflection coefficient of TM00 mode Cascade of SCNs 30

52. Modeling 2-D Planar Arrays Formulation of the Scattering Problem Surface Equation for electromagnetic scattering from a planar surface When an equivalent current is assumed at the whole discontinuity plane In spectral domain Using Galerkin’s method 31

53. Modeling 2-D Planar Arrays Normal Plane-wave Incidence Uniform Array Simulation Results (Normal Plane-wave Incidence) HFSS SCT 8x8 uniform patch array 32

54. Modeling 2-D Planar Arrays Normal Plane-wave Incidence Non-uniform Array Simulation Results (Normal Plane-wave Incidence) IE3D SCT HFSS 8x8 non-uniform patch-slot array 33

55. Modeling 2-D Planar Arrays Normal Plane-wave Incidence Uniform Array Simulation Results (Normal Plane-wave Incidence) 16x16 uniform patch array 34

56. Modeling 2-D Planar Arrays Normal Plane-wave Incidence Non-uniform Array Simulation Results (Normal Plane-wave Incidence) 16x16 non-uniform patch array 35

57. Modeling 2-D Planar Arrays Horn Antenna Excitation Uniform Array Simulation Results (Horn antenna Excitation) Horn antenna has been modeled analytically by a radiating aperture in an infinite ground plane 36

58. Modeling 2-D Planar Arrays Horn Antenna Excitation Uniform Array Simulation Results (Horn antenna Excitation) 8x8 uniform patch array SCT FEKO SCT FEKO H-Plane E-Plane 37

59. Modeling 2-D Planar Arrays Horn Antenna Excitation Non-uniform Array Simulation Results (Horn antenna excitation) SCT FEKO 8x8 non-uniform patch-slot array H-Plane 38

60. Modeling 2-D Planar Arrays Execution Times to compute surface impedance of the array 8x8 uniform array 8x8 non-uniform array 39

61. Presentation Plan Motivation Scale Changing Technique (SCT) Modeling of a Planar Reflector cell Characterization of Electromagnetic Coupling in non-uniform array Scattering from 2-D Planar Arrays Conclusions and Perspectives 40

63. SCT models mutual coupling effects between the elements of non-uniform arrays

64. The unique formulation avoids the direct computation of large structures with high aspect ratios – thus prevents numerical and convergence errors

65. Inherent modular nature of SCT allows the parallel execution of SCNs allowing execution times to increase linearly with the exponential increase in array size

66. In case of a modification in geometry at a given scale only two SCNs need to be recalculated ---- an essential feature for a good PARAMETRIC and OPTIMIZATION TOOL 41

68. A general implementation framework for grid-computing

69. Usage in hybrid with other techniques and methods for the simulation of multi-layered structures42

70. Thank You All 43

71. Publications A. Rashid et al, "Modeling of Infinite Passive Planar Structures using Scale-Changing Technique"IEEE-APS July 5-11, 2008, SanDiego, USA. A. Rashid et al, "Modélisation Electromagnétique d’un Réseau Fini et Non-Uniforme par la Technique par Changements d’Echelle"JNM 2009, Grenoble, France. A. Rashid et al, "Modeling of finite and non-uniform patch arrays using scale-changing technnique "IEEE-APS June 1-5, 2009, Charleston, USA. A. Rashid et al, "Scale-Changing Technique for the numerical modeling of large finite non-uniform array structures " PIERS, 2009, Moscow, Russia. E.B.Tchikaya et al, "Multi-scale Approach for the Electromagnetic Modeling of Metallic FSS Grids of Finite Thickness with Non-uniform Cells "APMC, 2009, Singapore. F. Khalil et al, "Application of scale changing technique-grid computing to the electromagnetic simulation of reflectarrays "IEEE-APS June 1-5, 2009, Charleston, USA. 44