Analysis of Cylindrical Waveguide Structures with Noncircular Cross - PowerPoint PPT Presentation

Analysis of Cylindrical Waveguide Structures with Noncircular Cross Sections Marcos. V. T. Heckler and Achim Dreher Folie 1 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Outline Possible applications of the present formulation Theory Application 1: dielectric elliptical waveguide Application 2: stripline with elliptical top ground Application 3: conformal microstrip antennas Conclusions Folie 2 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Introduction Possible applications Dielectric waveguides with quasi- Transmission lines with quasi- circular cross sections circular cross sections Folie 3 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Wave equation within every layer (normalized to k 0 ) � � � � � � � 2 � � � � � � � � � � � � k z 2 � � , , 0 � � � k � � � � � � 2 � � k � � � � � � k k where 2 2 � r r z k k k � k � or E H z z k k Modal expansions � � � � � � �� � � � � � � � � � � � jk z E z A J k B Y k e ( , , ) ( , ) z � � � � z k k e i k i i k i i k � �� i � � �� � � � � � � � � � � � � � � � jk z H z C J k D Y k e ( , , ) ( , ) z z k � � k � � h i k i i k i i k � �� i Folie 4 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Inner layer Outer layer �� � � � � �� � � � � � � � � � � jk z � � � � � � � � jk z E z A J k e E z A H k e ( 2 ) z ( , , ) ( , ) ( , , ) ( , ) z � � � � z e i z M e i 0 i i M i i M 0 0 i � �� � �� i �� � � � � �� � � � � � � � � � � � jk z � � � � � � � jk z H z C J k e H z C H k e ( 2 ) z z ( , , ) ( , ) ( , , ) ( , ) z � � h i � � z M h i 0 i i M i i M 0 0 � �� i i � �� The other field components may be obtained by � E � � � � � � � � � � � � k � � z r � � � � � H � � � � � � � � � k � E � � � � � � � r z z k 0 j 2 � � � � � � � E � � � � � � � � � � � k H � � � z r z � � 0 � � � H � � � � � � � � � � k � � � � � � � r z 0 Folie 5 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Discretization E z , H � and H � are sampled on the e -lines H z , E � and E � are sampled on the h -lines Each line system has N lines, which governs the truncation of the infinite expansions: i � � � � final � � � � � � � � jk z E ( , , z ) A J k ( , ) e z z � � e i 0 i i 0 0 i i initial with i initial = -( N- 1)/2 and i final = ( N -1) /2 Folie 6 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Matching of the fields in the k -th non-circular interface � � � � � � � � � � � � � � E t E E , , sin , cos � � k k k � � � � � � � � � � � � � � H t H H , , sin , cos � � k k k Using the short notation A C � � � � � � � � E k H k � L A L B � L C L D � � � � z k k k z k k k B D � � � � k k � � T E � � � e � � e � � e E E E where � � ( , ) ( , ) ( , ) z z z i z N 1 k k k k � � A T � A A A � � k k k k i i i inicial final Folie 7 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Arranging the expressions for E t , E z and for H t and H z , results in A A � � � � k k � � � � H � � B E � � B L C L D � � � � A B C D Q Q Q Q 0 0 � � � � z k t � k k � k k k k k � k � � � � k � � � H � � C E � C � G A G B G C G D A B L L � � � � � � � � 0 0 t k z k k k k k k k � � k � � k D D � � � � k k M M E H k k Folie 8 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Adopting the notation A � � k � � E H � � � � B � � t z E � H � k � � k � � k � C E H k k k � C � � � � � z t k k k � � D � � k we can write E � M ! H � M ! k C k C k E k k H k The elimination of the coefficients results in F F � k +1 � k k 1 � k +1 � k H E � � � � k � � k � � M ! M � k 1 1 1 � � � � � k � 1 H E H E � � k k � � � � � k -1 k k ~ ~ � E E k k Y � k -2 k Folie 9 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Considering the boundary condition at the interfaces � � H � H � J E E � � � k k k k k an equivalent circuit can be obtained H � H � H � H � H � H � � 1 � 1 k � 1 k k k k � 1 k E J J J E Y k Y k +1 � k- k k- k k � 1 E 1 1 1 k Folie 10 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

Extension of the DMM to 2D Cylindrical Problems with Quasi-Circular Cross Sections Relation between current and electric field in the interfaces J L E L E L E � � � k k k � k � k k k k k � k � , 1 1 , , 1 1 J � L ! E where L L � � � 0 0 0 J E � � � � 11 12 � � L L L 1 1 � � � � � 0 0 � � 21 22 23 � � � � � � L L L � � � 0 0 � � � � J � J E � E L � 32 33 34 � � k k L L � � � � � 0 0 0 � � 43 44 � � � � � � � � � � � � � � � � � � J E � � � � � � M M L � � � 0 0 0 0 MM Folie 11 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

1st Application – Analysis of Elliptical Dielectric Waveguides Geometry System equation ! E � L 0 where L � Y � Y rod air 2 B � r = 2.37 � 0 For non-trivial solutions rod air � � � L det 0 2 A Folie 12 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

1st Application – Analysis of Elliptical Dielectric Waveguides Modes in elliptical dielectric waveguides 2 A Determinant of the system equation 2 B Normalized propagation constant Modes in ellipical dielectric waveguides with different B / A ratios ( B was kept constant) Folie 13 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

1st Application – Analysis of Elliptical Dielectric Waveguides Numerical results B / A = 0.5 Variation of the propagation constant Variation of the propagation constant with the aspect ratio with the dimension B � B � � � V B rod air Folie 14 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

2nd Application – Stripline with elliptical upper GND Cross-sectional view Electrical Parameters � r = 3.00 � 0 = 10.98 mm � 1 = 11.75 mm t = 0.762 mm 2 A B = 16.23 mm W/t = 5.37 A = 11.94 – 15.28 mm f 0 = 5 GHz 2 B Folie 15 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

2nd Application – Stripline with elliptical upper GND Convergence of the solution with the discretization refinement Folie 16 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt

3rd Application – Conformal Microstrip Antennas Conformal microstrip patch Use of symmetry E-wall H-wall Dokumentname > Heckler_Dreher_COST_2008.ppt

3rd Application – Conformal Microstrip Antennas Discretization scheme Cross section e- lines (sampling of E z ) h- lines (sampling of H z ) Folie 18 > GeMiC 2008 > Marcos Heckler Dokumentname > Heckler_Dreher_COST_2008.ppt