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Sergei P . Skobelev JSC “Radiophyzika” , Moscow, Russia 2007

A Hybrid Projection Method for Analysis of Waveguide Array Antennas with Protruding Dielectric Elements. Sergei P . Skobelev JSC “Radiophyzika” , Moscow, Russia 2007. Contents. Introduction : a review of the methods and results available

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Sergei P . Skobelev JSC “Radiophyzika” , Moscow, Russia 2007

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  1. A Hybrid Projection Method for Analysis of Waveguide Array Antennas with Protruding Dielectric Elements SergeiP. Skobelev JSC“Radiophyzika”, Moscow, Russia 2007

  2. Contents • Introduction: a review ofthe methodsand results available • 2-D problems: algorithms for the cases of Е- and Н-polarization, validation, results • 3-D problem for an array of circular waveguides with cylindrical-conical rods:algorithm, validation, results • Conclusions

  3. Phased-Array Antennas with Dielectric Elements (Examples) A 9-element array(Tanaka, et al, 1968) A laboratory breadboard, 1955(O. G. Vendik, Yu. V. Egorov, 2000) An array-antenna module(A. A. Tolkachev,et al., 1996) S300-PMU

  4. Phased-Array Antennas with Dielectric Elements (Examples) APAA with DBF for Communication System (“APEX” - Russia, ETRI - Republic of Korea, 2003)

  5. Methods of Numerical Analysis • Mode-Matching Method • - Lewis, Hessel, Knittel, 1972 • Vinichenko, et al., 1972-1975 • Krekhtunov and Tyulin, 1983 • Incomplete Galerkin’s Method • Il’inskiy and Kosich, 1974 • Il’inskiy and Trubnikov, 1980;Trubnikov, 1988 • Davidovitz, 2001 • Electric/Magnetic Wall Cavity Method • Kovalenko, 1974 • Method of Surface Integral Equations • Method of Auxiliary Sources • Skobelev and Mukhamedov, 1991-1993; Skobelev, 2003 • Method of Integral Equations for Polarization Currents • Skobelev and Nikitin, 1998;Skobelev, 2004-2005 • Method of Finite Elements • Herd, D’Angelo; 1998, McGrath, 2000 • Commercial Codes LikeHFSS and Microwave Studio

  6. d y z x d y 0 x 0 x Mode-Matching Method Lewis, Hessel, Knittel, 1972 – 2D-caseof longitudinally uniform protrusions Vinichenko, et al., 1972 (2D), 1975 d y z x Krekhtunov, Tyulin, 1983 d y Main difficulty: calculation of theeigenmode parameters for theprotrusion area 0 x 0 x

  7. Incomplete Galerkin's Method Il'inskiy, Kosich, 1974 – formulation only Il’inskiy, Trubnikov, 1980 - uniform only Trubnikov, 1988– small conical Unit cell z 0 x Result: System of ordinary differential equationsfor apq(z)and bpq(z). Similar approach for 2D-case:M. Davidovitz, "An approach to analysis of waveguide arrayswith shaped dielectric insets and protrusions", IEEE Trans. on Microwave Theory and Tech., vol. 40, no. 2, 2001.

  8. Electric/Magnetic Wall Cavity Method Kovalenko, 1974 – no numerical realization Unit cell z 0 x Result: System of linear algebraic equations for apqland bpql.

  9. z J1leiu J1l J1le-iu J0l 0 x x e-iu eiu 1 z Method ofSurface Integral Equations Difficult to realize because of the necessity of account for all the equivalent surface current components. There are no results. 0 x Method of Auxiliary Sources Skobelev and Mukhamedov, 1991-1993 – 2D-case for both TE and TM modes Integral Eqs for Polarization Currents z Especially simple and promising for the case of cylindrical protrusions. 0 x Skobelev and Nikitin, 1998;Skobelev, 2004-2005

  10. Finite-Element Method Herd, and D’Angelo; 1998 Commercial Codes LikeHFSS and Microwave Studio Powerful, however there are some problems

  11. Conclusions from the Overview • Shortcomings or restrictions of the existing methods. • No results for long elements necessary for shaping the flat-topped element pattern. • No results for elements with sharp ends for providing good array matching to free space.

  12. z t1 b h e t0 x 0 ea a Е-polarization. Statement of the Problem Array Geometry Excitation of waveguides: in the ТЕ10modes with uniform amplitude and linear phase distributions;u=kbsinq

  13. z b h e ea x a 0 Method of Solution Representation of the Fields Central Waveguide: Above the Protrusions: Region Containing Protrusions :

  14. z b h e ea x a 0 Method of Solution Matching ofEyatz=0, -b/2<x<b/2: Matching ofHxatz=0, -a/2<x<a/2:

  15. z b h e ea x a 0 Method of Solution Region Containing Protrusions: Projection onexp(-ipx)/b and substitutionresult in ODE:

  16. Method of Solution System of ODE: Finite-Element Method:

  17. Method of Solution Matrix Elements:

  18. M + P + 0 0 Energy Balance: Element Pattern: Method of Solution System of Linear Algebraic Equations:

  19. z b h e x 0 ea a Н-polarization. Statement of the Problem Array Geometry Excitation of waveguides: in the ТЕMmodes with uniform amplitude and linear phase distributions;u=kbsinq

  20. z b h e ea x a 0 Н-polarization. Method of Solution Representation of the Fields The central waveguide: Above the protrusions:

  21. z b h e ea x a 0 Н-polarization. Method of Solution The region with protrusions 0z h: Matching of Hyatz=0, -a/2< x< a/2: Matching ofHyatz=h, -b/2  x  b/2:

  22. Н-polarization. Method of Solution Field in the region with protrusions, 0z h: Projection onexp(-ipx)/b: After integration by parts over x:

  23. Н-polarization. Method of Solution Finite element method. Projection on fm(z): Integration by part overzand account for give:

  24. Н-polarization. Method of Solution Substitution of gives:

  25. Н-polarization. Method of Solution Boundary condition forExatz=0, -b/2 x  b/2: on the flange Boundary condition for Exatz=h, -b/2 x  b/2:

  26. Energy balance: Array element pattern: Н-polarization. Method of Solution System of linear algebraic equations: Order of the system: M 0 0

  27. Fragment of the Output Data File: sinq |R1| |F|PRPT 1-PR-PT 0.00000 0.27974 0.96008 0.078255 0.921745 0.000000 0.01000 0.27915 0.96022 0.077926 0.922074 0.000000 0.02000 0.27851 0.96034 0.077567 0.922433 0.000000 0.03000 0.27781 0.96042 0.077180 0.922820 0.000000 Implementation of the Algorithm (Е-polarization) A computer С++ code Input Data File: 15 MB – number of Floquet modes kept 0.6 b – element spacing in wavelengths 10 MA – number of waveguide modes kept 0.4 a – waveguide width in wavelengths 2.0000 epsi_a – permittivity of the guide filling 20 MH – number of nodes over protrusion 0.6 h – protrusion height in wavelengths 0.4 t0 – protrusion width at the base 0.000001 t1 – protrusion width at the top 2.0000 epsi – protrusion permittivity

  28. z b h e ea x a 0 Validation of the Code (Е-polarization) Convergence overN: a = 0.4l, b = 0.6l e = ea= 2.0, h = 0.6l, t0 = 0.4, th = 0.2l P = 31, M = 20 Element pattern Reflection coefficient

  29. z b h e ea x a 0 Validation of the Code (Е-polarization) Convergence overP: a = 0.4l, b = 0.6l e = ea= 2.0, h = 0.6l, t0 = 0.4, th = 0.2l N = 30 Element pattern Reflection coefficient

  30. z b h e ea x a 0 Validation of the Code (Е-polarization) Convergence overN: a = 0.3l, b = 0.6l e = ea= 6.0, h = 0.6l, t0 = 0.4, th = 0.2l P = 31, M = 15 Element pattern Reflection coefficient

  31. z b h e ea x a 0 Validation of the Code (Е-polarization) Convergence overP: a = 0.3l, b = 0.6l e = ea= 6.0, h = 0.6l, t0 = 0.4, th = 0.2l N = 30 Element pattern Reflection coefficient

  32. z t’ e 0 x Validation of the Code (E-polarization) Comparison of the Results: N. Amitay, V. Galindo, C. P. Wu, Theory and Analysis of Phased Array Antennas, NY: Wiley, 1972, Fig. 6.12. |R1| a = b = 0.5714l e = 3.0625 Phase, degrees

  33. z b h e ea x a 0 Validation of the Code (E-polarization) Comparison of the Results: S. P. Skobelev,et al, “Analysis of a waveguide array with protruding dielectric elements by the method of integral equation for electric field in the protrusion”, Electromagnetic Waves and Electronic Systems, vol. 10 no. 3, pp. 31-35, 2005. a = 0.4l, b = 0.6l e = ea= 2.0, h = 0.5l, t0 = th = 0.3l

  34. Fragment of the Output Data File: sinq |R0| |F|PRPT 1-PR-PT 0.00000 0.27974 0.96008 0.078255 0.921745 0.000000 0.01000 0.27915 0.96022 0.077926 0.922074 0.000000 0.02000 0.27851 0.96034 0.077567 0.922433 0.000000 0.03000 0.27781 0.96042 0.077180 0.922820 0.000000 Implementation of the Algorithm (H-polarization) A computer С++ code Input Data File: 15 MB – number of Floquet modes kept 0.6 b – element spacing in wavelengths 10 MA – number of waveguide modes kept 0.4 a – waveguide width in wavelengths 2.0000 epsi_a – permittivity of the guide filling 20 MH – number of nodes over protrusion 0.6 h – protrusion height in wavelengths 0.4 t0 – protrusion width at the base 0.000001 t1 – protrusion width at the top 2.0000 epsi – protrusion permittivity

  35. z b h e ea x a 0 Н-polarization. Validation of the Code Convergence overN: a = 0.3l, b = 0.6l e = ea= 2.0, h = 0.6l, t0 = 0.4, th = 0.2l P = 31, M = 15 Element pattern Reflection coefficient

  36. z b h e ea x a 0 Н-polarization. Validation of the Code Convergence overP: a = 0.3l, b = 0.6l e = ea= 2.0, h = 0.6l, t0 = 0.4, th = 0.2l N = 30 Element pattern Reflection coefficient

  37. z b h e ea x a 0 Н-polarization. Validation of the Code Convergence overN: a = 0.3l, b = 0.6l e = ea= 6.0, h = 0.6l, t0 = 0.4, th = 0.2l P = 31, M = 15 Element pattern Reflection coefficient

  38. z b h e ea x a 0 Н-polarization. Validation of the Code Convergence overP: a = 0.3l, b = 0.6l e = ea= 6.0, h = 0.6l, t0 = 0.4, th = 0.2l N = 30 Element pattern Reflection coefficient

  39. z b t’ e 0 x a Н-polarization. Validation of the Code Comparison of the results: N. Amitay, V. Galindo, C. P. Wu, Theory and Analysis of Phased Array Antennas, NY: Wiley, 1972, Fig. 6.15. |R0| b = 0.5714l, a = 0.85b e = 3.0625 Phase, degrees

  40. z b h e ea x a 0 Н-polarization. Validation of the Code Comparison of the results: Lewis R. L., Hessel A., Knittel G. H., "Performance of a protruding-dielectric waveguide element in a phased array," IEEE Trans. Antennas Propag., vol. AP-20, Nov. 1972, pp. 712-722. h=0.5l, N=15 h=l, N=30 b = 0.535l, a = 0.432b e = ea= 2.56, t0 = th = a P = 21, M=10

  41. z b h e ea x a 0 z b h e ea x a 0 z b h e ea x a 0 Effect of the Protrusion Shape on the ArrayPerformance (Е-polarization) • a = 0.4l • b = 0.6l • = ea= 2.0 h = 0.5l, • t0 = 0.4l F th = 0.4l *** th = 0.2lO O O R th = 0 D D D

  42. z h b e ea x a 0 a ha a0 Shaping of the Flat-Topped Element Pattern.Е-polarization >>>> F a = 0.78l, b = 0.8l, h = 2.1l, t0= 0.21l,th= 0, e= ea= 2.56 P = 15, M = 12, N = 50 ---- a = 0.78l, b = 0.8l, a0= 0.618l, ha = 0.25l R2 R1 Ideal element pattern:-----

  43. z h b e ea x a 0 Shaping of the Flat-Topped Element Pattern.Е-polarization >>>> F a = 0.7l, b = 0.91l, h = 3.1l, t0= 0.2l,th= 0, e= ea= 2.0 P = 21, M = 19, N = 60 ---- R a = 0.89l, b = 0.91l Ideal element pattern:-----

  44. Shaping of the Flat-Topped Element Pattern.Е-polarization z >>>> h b e ea x a 0 F a = 1.05l, b = 1.1l, h = 4.5l, t0= 0.35l,th= 0, e= ea= 1.5 P = 21, M = 19, N = 90 ---- R a = 1.05l, b = 1.1l Ideal element pattern:-----

  45. z h b e ea x a 0 ha at ht a0 Shaping of the Flat-Topped Element Pattern.H-polarization >>>> F a = 0.7l, b = 0.8l, h = 2.3l, t0=0.42l, th= 0.2l, e= ea= 2.0 ---- R a = 0.78l, b = 0.8l, ha = 0.3l, a0= 0.4l, at= 0.568l, ht= 0.22l Ideal element pattern:-----

  46. z h b e ea x a 0 ha at ht a0 Shaping of the Flat-Topped Element Pattern.H-polarization >>>> F a = 0.68l, b = 0.7l, h = 2.5l, t0= 0.55l,th= 0.2l, e= ea= 2.0 ---- R a = 0.68l, b = 0.7l, ha = 0.4l, a0= 0.4l, at= 0.525l, ht= 0.23l Ideal element pattern:-----

  47. z t1 b he q t0 h0 e 0 x  a h t at ht i ai e–2iU e–iU 1 eiU Account for the Stepped Transitions

  48. Shaping of the Flat-Topped Element Pattern.E-polarization b = 0.91l, h = 2.2l,e= 2 b = 1.1l, h = 4.9l,e= 1.5

  49. Shaping of the Flat-Topped Element Pattern.H-polarization b = 0.8l, h = 2.3l,e= 2 b = 0.96l, h = 3.6l,e= 2. Computation time:<49 sec for 101 points at 1.53 GHz

  50. y dx dy eiV  1 0 x e–iV 1 e–iU eiU 2r1 z he 2r0 ee h0 x 0 e 2a h et 2at ht ei 2ai 1 e–2iU e–iU eiU Three-Dimensional Case. Statement of the Problem Array Geometry and Excitation Excitation: quasi-periodic, in the ТЕ11modes of RH/LH circular polarization

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