# Analysis, modelling, and design of microwave planar ferrite devices and antenna circuits using spectral domain method of moments

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Fawzy Elshafiey A Dissertation Presented in Partial Fulfillm ent of the Requirements for the Degree Doctor of Philosophy ARIZONA STATE UNIVERSITY December 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9710349 UMI Microform 9710349 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ANALYSIS, MODELLING, AND DESIGN OF MICROWAVE PLANAR FERRITE DEVICES AND ANTENNA CIRCUITS USING SPECTRAL DOMAIN METHOD OF MOMENTS by Tarief M. Fawzy Elshafiey has been approved November 1996 APPROVED: . Chairperson 5 ■ - 8- Lo-fi* Supervisory Committee :pa:trnO$t Chairperson Dean, G raduate College Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT T he objective of this research is to develop the theoretical and numerical models, based on the method of moments, for the analysis of various ferrite microwave devices and antenna circuits. The transmission m atrix for a normally biased ferrite layer is derived in a closed form. The Green’s function is formulated using the transmission m atrix. A 2-D model is implemented to analyze the edge-guided mode microstrip isolator. The resistive film term ination over one edge of the microstrip, to absorb the backward wave, is considered in this work. The closed form transmission m atrix for any arbitrarily biased ferrite slab is also derived to formulate a very general Green’s function for multi-layer a rb itra rily magnetized ferrite structures. Two numerical models are implemented: phase shifter model and a m agnetostatic surface wave model. Good agreement with the previously published results is achieved. A novel approach to analyze scattering from arbitrarily shaped patches on arbitrarily biased ferrite substrates is presented. In th at approach, the excitation vectors for various single and multi-layer antenna structures are derived in a closed form using the transm ission matrix. A 3-D model is implemented to study the Radar Cross Section (RCS) for several ferrite antennas. A novel cross patch antenna is proposed which results in significant RCS reduction. 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. roT* k ^ok ^ a (Over all Endued with knowledge is One, the All-Knowing) /d L lz ll ASal^. 0A. *111 ^f»V) U i t ) (Those truly fear Allah, Among His Servants who have knowledge) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To th e spirit of m y father and To m y sons, M ahm oud, Abdelrahm an and Ahmed V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to express my deep gratitude to my advisor, Dr. James T. Aberle for his friendship, encouragement, guidance and advice throughout the duration of this research. When you work with someone like Dr. Aberle you feel that you are working with a friend. A special thanks to Dr. Samir M. El-Ghazaly for his sincerity and his brother hood. I do not like to say much about his encouragement and solving many personal problems th at I faced, but I ask the All-mighty ALLAH to bless him and his family. I am also grateful to the members of my committee: Dr. R. Renaut, Dr. R. Grondin, and Dr. B. Welfert who unfortunately was on leave when this work was completed and defended. Deep appreciation is due to my friends and colleagues at the Telecommunications Research Center, in particular, Dr. David Kokotoff for his time discussing MoM and ferrites and Chris Bishop for providing his mesh generator that I used to generate the Cross-Patch mesh, and for his help in revising the manuscript of this work. I can not forget two of my sincere friends, Dr. Osman Ibrahiem and Dr. Khalid Shehata of the Postgraduate Navy School at Monterey, CA. Their constant support and advice especially after I faced many difficulties gave me much strength and confidence. I would like also to take the opportunity to thank many Brothers for their com pany and encouragement, Hussein Mahmoud, Bahader Yildirim, Samir Hammadi, and Sohel Imtiaz. Their friendship and humor has made this process bearable. I wish to express my sincere appreciation to the Egyptian Government who gave me this opportunity and partially supported this program. My appreciation also extends to Gen. Dr. Essam Ibrahiem, one of the very active members of the adm in istration of this program. His understanding and consideration for the difficulties th at I faced gave me the chance to complete this work. vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I would especially like to acknowledge the supplication of my mother and contri bution of my wife. Their enduring love, patience and understanding has extended fax above what could be expected from any person. vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page LIST OF T A B L E S ......................................................................................................... xii LIST OF F I G U R E S ......................................................................................................... xiii CHAPTER 1 IN T R O D U C T IO N ......................................................................................... 1.1 M otiv atio n ............................................................................................. 1 1.2 Previous Work ................................................................................... 4 1.3 M e th o d s ................................................................................................ 11 1.4 2 1 1.3.1 Introduction ..................................................................... 11 1.3.2 M ethod of M o m e n ts ........................................................ 14 Research O b je c tiv e s ......................................................................... 15 GREEN’S FUNCTION FORMULATION FOR A NORMALLY BI ASED FE R R IT E S L A B ............................................................................... 18 2.1 In tro d u c tio n ......................................................................................... 18 2.2 Transmission M atrix F o rm u la tio n .................................................. 19 2.3 Discussion on the Difficulty of Deriving Normally Biased Trans 2.4 mission M a t r i x .................................................................................. 25 Green’s Function F o rm u latio n ......................................................... 26 viii Reproduced with permission of the copyright owner. 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CHAPTER 2.5 3 Page 2.4.1 Green’s Function for a Single Ferrite Substrate 2.4.2 Green’s Function for a Ferrite-Dielectric Substrate . 28 2.4.3 Green’s Function for a General Multi-layer Structure 30 2.4.4 Circuit Model Interpretation of the Green’s Function 26 for a Multi-layer S tru c tu re .............................................. 30 Results and Conclusion...................................................................... 31 2-D FULL-WAVE ANALYSIS OF AN EDGE-GUIDED MODE ISO LATOR ............................................................................................................ 39 3.1 In tro d u c tio n .......................................................................................... 39 3.2 Full Wave F o rm u latio n ....................................................................... 40 3.2.1 Green’s Function F o rm u latio n ........................................ 41 3.2.2 Basis Functions and Resistive Region Tr eat ment . . . 41 3.2.3 Numerical Considerations in the Evaluation of the Spectral Domain In teg ratio n .......................................... 44 Complex Root S earching................................................. 46 Numerical Results and C onclusion.................................................. 46 3.2.4 3.3 4 ... ANALYSIS OF PHASE SHIFTERS AND TRANSDUCERS USING A GENERAL GREEN’S FUNCTION ................................................... 65 4.1 In tro d u c tio n ......................................................................................... 65 4.2 Transmission Matrix for an Arbitrarily-Biased Ferrite Slab . . 65 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH A PTER 4.3 Green’s Function F o rm u la tio n ......................................................... 81 4.4 Varying the Magnetization A n g l e .................................................. 82 4.5 Planar Phase S h ifters......................................................................... 83 4.6 4.7 5 Page 4.5.1 Introduction ...................................................................... 83 4.5.2 Slot Line Phase S h if te r s .................................................. 84 4.5.3 Microstrip Phase S h i f t e r s ............................................... 88 Magnetic Surface Wave T ransducers............................................... 93 4.6.1 Introduction 93 4.6.2 Full-Wave Analysis of the Microstrip Phase Shifters . ...................................................................... Results and D iscussion...................................................................... 93 94 3-D ANALYSIS OF RADAR CROSS SECTION OF A FERRITE PATCH A N T E N N A ........................................................................................114 5.1 In tro d u ctio n ............................................................................................ 114 5.2 Plane Wave Propagation in a Ferrite Medium: an Introduction to the Excitation Vector F o rm u la tio n ..............................................116 5.3 5.4 T h e o ry ......................................................................................................122 5.3.1 Full Wave F orm ulation.........................................................122 5.3.2 Excitation V ecto r.................................................................. 127 5.3.3 Green’s Function F o rm u latio n ........................................... 132 Impedance Matrix In te rp o latio n ........................................................ 134 x Reproduced with permission of the copyright owner. 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CHAPTER 5.5 6 Page Results and C o n clu sio n s............................................................... 137 C O N CLU SIO N ...................................................................................................166 R E F E R E N C E S ............................................................................................................. 168 APPENDIX A THE TRANSMISSION MATRIX OF A NORMALLY BIASED FER RITE S L A B ................................................................................................... 174 B THE TRANSMISSION MATRIX OF A DIELECTRIC SLAB . . . . 185 C THE SEM I-INFINITE SPACE GREEN’S FU N C T IO N ........................... 187 D RESISTIVE MATRIX ELEMENTS IN SPATIAL D O M A IN E GREEN’S FUNCTION FOR A SLOT LINE WITH A FER RITE 190 S U B S T R A T E ................................................................................................ 192 E .l G reen’s Function for a Single Slot Line Ferrite Substrate . . . 193 E.2 G reen’s Function for a Multilayer Slot Line Ferrite Substrate . xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 LIST OF TABLES Table 3.1 Page Percentage of the impedance m atrix f il lin g .............................................. xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 LIST OF FIGURES Figure Page 1.1 Symbolic representation of phase shifter, isolator and circulator with ideal properties................................................................................... 2 1.2 3-D edge-guided isolator with transverse slot l o a d i n g ........................... 5 1.3 3-D edge-guided isolator with resistive film loading.................................. 6 1.4 3-D edge-guided isolator with shorted load................................................. 7 1.5 Microstrip patch antenna on a normally biased ferrite substrate. . . . 9 1.6 Current basis functions for (a) Longitudinal current (b) Transverse c u rre n t................................................................................................. 14 2.1 Geometry of single layer isolator structure.................................... 26 2.2 Geometry of double layer isolator structure................................... 32 2.3 Geometry of drop-in element isolator structure............................ 32 2.4 Circuit representation for a single ferrite layer structure............ 33 2.5 Comparison of the computed Green’s function versus Pozar’s(Imag(Gxx)) {d = 7.62 x 10~4m, e, = 12.0, 4irMs = 2100.0G, H dc = 700.00e, A H = O.OOe, R 3 = 0.00, W = 1.016 x 10~2m, / = 3.6GHz, K x = (110.0, —10.0)).......................................................................... 33 2.6 Comparison of the computed Green’s function versus Pozar’s(Real(Gxx)). The param eters are the same as in Fig. 2.5.................................. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.7 Page Comparison of the computed Green’s function versus Pozar’s (Imag(Gxy)). The param eters axe the same as in Fig. 2.5...................................... 2.8 Comparison of the computed Green’s function versus Pozax’s (Real(Gxy)). The param eters axe the same as in Fig. 2.5...................................... 2.9 34 35 Comparison of the computed Green’s function versus Aberle’s(Imag(Gxx)). (dd = 0.1 x 10-2 m, e<f = 30.0, dj = 0.762 x 10-3 m, e/ = 12.0, K x = 110.0ra d /m , f = 2.0G H z)......................................................... 35 2.10 Comparison of the computed Green’s function versus Aberle’s(Imag(Gyy)). The param eters axe the same as in Fig. 2.9...................................... 36 2.11 Comparison of the computed Green’s function versus Aberle’s (Imag(Gxy)). The param eters axe the same as in Fig. 2.9...................................... 36 2.12 Comparison of the computed Green’s function versus Aberle’s (Imag(Gxx)). (dd = 0.1 x 10-2m , td = 9.8, df = 0.762 x 10-3m, e/ = 12.0, da = 0.1 x 10_2m, ea = 1.0, K x = 110rad/m , f = 2.0G H z ) ......... 37 2.13 Compaxison of the computed Green’s function versus Aberle’s (Imag(Gyy)). The param eters axe the same as in Fig. 2.12.................................... 37 2.14 Compaxison of the computed Green’s function versus Aberle’s (Imag(Gxy)). The param eters are the same as in Fig. 2.12.................................... 38 3.1 Edge-guided isolator with resistive film loading.......................................... 39 3.2 Geometry of single layer structure................................................................. 40 3.3 Geometry of double layer structure............................................................... 40 3.4 Geometry of drop-in element structure......................................................... 41 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 3.5 Current basis functions for (a) Longitudinal current (b) Transverse c u r r e n t............................................................................................................... 43 3.6 Integration contour in the complex plane of either kx or ky..................... 46 3.7 Symmetric current distribution over dielectric m icrostrip........................ 50 3.8 Asymmetric current over ferrite microstrip in forward and backward directions for R 3 = 0.0ft.................................................................................. 50 3.9 Asymmetric current over ferrite microstrip in forward and backward directions for R a = 100.0ft.............................................................................. 51 3.10 The phase constants of forward and backward waves (d = 7.62 x 10-4m, t f = 12.0, 4v M . = 1750.0G, Hdc = 800.00e, A H = 80.00e, R , = 100.0ft. W = 1.016 x l0 -2m )............................................................... 51 3.11 Computed isolation and insertion loss [d = 7.62 x 10-4m, e/ = 12.0, 4 ttM s = 1750.0G, Hdc = 800.00e, A H = 80.0Oe, R 3 = 100.0ft, W = 1.016 x 10-2m )........................................................................................ 52 3.12 Comparison of the insertion loss for three isolator structures (47tA/s = 1750.0G, Hdc = 800.OOe, A H = 80.0Oe, R 3 = 100.0ft, W = 1.016 x 10~2m. For th e single-layer: dj = 7.62 x 10-4m , ej = 12.0. For the double-layer: dd = 2.62 x 10_4m, ed = 3.0. For the triple-layer: dd = 4.0 x 10_4m, ed = 8.9 da = 5.0 x 10-3m, ea = 1.0).......................... 52 3.13 Comparison of the isolation for three isolator structures. The same param eters are as in Fig. 3.12........................................................................ 53 3.14 The effect of the dielectric thickness on the insertion loss (dj = 7.62 x 10-4m, e/ = 12.0, 4;rM. = 2100.0G, Hdc = 700.0Oe, A H = 80.0Oe, R s = 100.0ft, W = 1.016 x 10"2m, td = 30.0)............................................ xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 Figure Page 3.15 The effect of the dielectric thickness on the isolation (<// = 7.62 x 10-4m, e/ = 12.0, 471 M s = 2100.0G, Hdc = 700.0Oe, A H = 80.00e, R, = lOO.Ofi, W = 1.016 x 10-2m, ed = 30.0)....................................... 54 3.16 The effect of the dielectric constant of the dielectric layer on the isolation and the insertion loss, (dj = 7.62 x 10~4m, e/ = 12.0, At M s = 2100.0G, H dc = 700.00e, A H = 80.0Oe, R s = lOO.Ofi, W = 1.016 x 10_2m , Dd = 0.381 x 10-3m, Freq.= 6.0 GHz)............ 55 3.17 The effect of the external DC bias on the isolation, (d = 7.62 x 10_4m, £f = 12.0, AirM3 = 2100.0G, A H = 80.0Oe, R s = 100.0D, W = 1.016 x 10-2m )........................................................................................... 56 3.18 The effect of the external DC bias on the insertion loss, (d = 7.62 x 10_4m, ef = 12.0, 4ttM s = 2100.0G, A H = 80.0Oe, R s = 100.0D, W = 1.016 x 10-2m ).................................................................................. 57 3.19 The frequency behavior of /xe/ / , (47tM„ = 2100.0G, Hdc = 800.OGe). . 58 3.20 The effect of the film resistance on the insertion loss and the isolation, (d = 7.62 x 10-4m, ef = 12.0, ArM s = 1750.0G, Hdc = 800.00e, A H = 80.Ge, Freq.= 5.0 GHz, W = 1.016 x10-2 m )............................... 59 3.21 The effect of the resistive film width on the insertion loss, (d = 7.62 x 10"4m, ef = 12.0, 4ttM s = 2000.0G, Hdc = 700.0Ge, A H = 80.0Ge, R , = 100.0n, W = 1.016 x 10-2m ).............................................................. 60 3.22 The effect of the resistive film width on the isolation, (d = 7.62 x 10-4m, ef = 12.0, 4ttM s = 2000.0G, Hdc = 700.0Ge, A H = 80.0Ge, R a = lOO.OD, W = 1.016 x 10_2m ).............................................................. 61 3.23 3-D edge-guided isolator with resistive film loading.................................. 62 xvi Reproduced with permission of the copyright owner. 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Figure Page 3.24 Compaxison between the numerical and experimental insertion loss (d = 7.62 x 10-4m, ef = 12.0, 4;rM , = 2100.0G, Hdc = 700.0Oe, A H = 80.00e, R s = 100.0ft, W = 1.016 x 10~4m ).................................. 63 3.25 Comparison between the numerical and experimental isolation (d = 7.62 x 10-4m , £/ = 12.0, 47rMs = 2100.0G, H dc = 700.00e, A H = 80.0Oe, R s = lOO.Ofi, W = 1.016 x 10_4m )............................................... 64 4.1 Geometry of single layer structure............................................................... 66 4.2 Magnetization angle (a) an approximate picture (b) a real picture. . . 83 4.3 A hysteresis curve for a ferrite sample........................................................ 85 4.4 Cross-section of basic slot line single-layer ferrite planar phase shifter. 85 4.5 n = 0 Chebychev basis function of the electric field in the slot in transverse directions......................................................................................... 4.6 86 n = 1 Chebychev basis function of the electric field in the slot in longitudinal directions..................................................................................... 87 4.7 Cross-section of basic microstrip single-layer ferrite planar phase shifter. 89 4.8 Cross-section of the phase shifter using oppositely-magnetized ferrite layers................................................................................................................... 89 The odd mode of the dual structure........................................................... 90 4.10 The odd mode representation of the dual structure................................ 90 4.9 4.11 n=0 Chebychev basis function of the electric current on the strip in longitudinal directions..................................................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 Figure Page 4.12 n = l Chebychev basis function of the electric current on the strip in transverse directions........................................................................................ 92 4.13 Geometry of MSSW transducers with microstrip embedded between dielectric ferrite structure............................................................................... 93 4.14 Geometry of MSSW transducers in multilayer practical structure. . .93 4.15 Geometry of MSSW transducers in two-layerstructure............. 94 4.16 Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Imag(Gxx)). (d = 7.62 x 10-4m , ej = 12.0, 4irMs = 2100.0G, Hdc = 700.0Oe, A H = 0.Oe, R , = Oft, W = 1.016 x 10-2m , / = 3.6GHz, K x = (110.0, -1 0 .0 ))................................. 98 4.17 Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gxx)). The param eters are the same as in Fig. 4 .1 6 ............................................................................................................ 98 4.18 Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Imag(Gyy)). The param eters are the same as in Fig. 4 . 1 6 ..................................................................................................... 99 4.19 Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gyy)). The param eters are the same as in Fig. 4 .1 6 ............................................................................................................ 99 4.20 Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Imag(Gxy)). The param eters are the same as in Fig. 4 . 1 6 ........................................................................................................ 100 xviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 4.21 Compaxison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gxy)). The parameters are the same as in Fig. 4 .1 6 ............................................................................................................... 100 4.22 Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Imag(Gyx)). The param eters are the same as in Fig. 4 . 1 6 ........................................................................................................ 101 4.23 Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gyx)). The param eters are the same as in Fig. 4 .1 6 ............................................................................................................... 101 4.24 Comparison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Imag(Gxx)) (d = 7.62 x 10-4m, t j = 12.0, iirM s = 2100.0G, Hdc = 700.00e, A tf = 0.Oe, R 3 = 0Q, W = 1.016 x 10-2m, / = 3.6GHz, K x = (110.0, -1 0 .0 )).................................... 102 4.25 Comparison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Real(Gxx)). The parameters are the same as in Fig.4 .2 4 ..................................................................................................... 102 4.26 Comparison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Imag(Gyy)). The parameters are the same as in Fig.4 .2 4 ..................................................................................................... 103 4.27 Comparison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Real(Gyy)). The parameters are the same as in Fig.4 .2 4 .....................................................................................................103 4.28 Comparison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Imag(Gxy)). The parameters are the same as in Fig.4 .2 4 ..................................................................................................... 104 xix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 4.29 Comparison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Real(Gxy)). The param eters are the same as in Fig. 4 .2 4 ..................................................................................................... 104 4.30 Compaxison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Imag(Gyx)). The param eters are the same as in Fig. 4 .2 4 ..................................................................................................... 105 4.31 Comparison of the computed Green’s function versus Elsharawy’s for transversely biased slab (Real(Gyx)). The param eters are the same as in Fig. 4 .2 4 ..................................................................................................... 105 4.32 Comparison of the calculated differential phase shift versus theoretical and experim ental results for a microstrip single layer phase shifter {df = 0.635 x 10-3m, e, = 12.9, 4ttM s = 2300.0G, = 150.0Oe, S = 0.45 x 10~3m, a = 1.27 x 10“2m, / = 1.52 x 10"2, 9 = 90°, (f>= 90°). 106 4.33 Comparison of the calculated differential phase shift versus theoretical and experim ental results for a microstrip single layer phase shifter {df = 0.635 x 10-3m , e, = 12.9, 4jtM, = 2300.0G, Hdc = 150.0Oe, 5 = 0.45 x 10-3m, a = 1.27 x 10~2m, I = 1.52 x 10-2 , 9 = 90°, <p = 80°). 107 4.34 Comparison of the normalized propagation constants for dual strip phase shifters (d f = 1.0 x 10-3m, t j — 17.5, 47rM , = 1500.0G, Hdc = 0Oe, S = 1.0 x 10 3m , a = 1.0 x 10 2m )........................................................ 107 4.35 Comparison of the calculated differential phase shift for dual strip phase shifters (d f = 1.0 x 10-3m, ef = 17.5, 4 itM 3 = 1500.0G, Hdc = 0Oe, S = 1.0 x 10~3m, a = 1.0 x 10_2m, 9 = 90°, <f>= 80°)........................108 xx Reproduced with permission of the copyright owner. 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Figure Page 4.36 Compaxison of the calculated symmetric propagation constant for the two layer transducer (dd = 1.27 x 10_3m , d f = 2.03 x 10-3m, e/ = 17.5, t d = 10.2, 4irM3 = 2267.0G, Hdc = 144.0Oe, AH = 300.00e, S = 0.3 x 10-3m ).................................................................................................108 4.37 Comparison of the calculated asymmetric propagation constant for the two layer transducer (dd = 1.27 x 10-3m, df = 2.03 x 10-3m, t f = 17.5, ed = 10.2, 4ttM s = 2267.0G, Hdc = 144.0Oe, A H = 300.00e, S = 0.3 x 10-3m )................................................................................................ 109 4.38 Comparison of the calculated attenuation constant for the three layer transducer (d\d = 2.5 x 10-4m, d2d = 2.5 x 10-4m, df = 0.5 x 10-4m, t f = 15.0, eld = 9.8, eu = 10.0, 4ttM s = 1780G, Hic = 600.0Oe, A H = 45.0Oe, S = 0.5 x 10-4m ).....................................................109 4.39 Comparison of the calculated phase constant for the three layer trans ducer ( d u = 2.5 x 10-4m, d2d = 2.5 x 10-4m, df = 0.5 x 10-4m, t f = 15.0, eu = 9.8, eld = 10.0, 4?rM s = 1780G, Hdc = 600.0Oe, A H = 45.0Oe, S = 0.5 x 10~4m ).....................................................110 4.40 The effect of number of basis functions on the attenuation constant 4.41 The effect of number of basis functions on the phase constant . . . . . Ill Ill 4.42 Comparison of the insertion loss along a MSSW transducer (dd = 1.27 x 10_3m, d f = 2.03 x 10_3m, ef = 17.5, ed = 10.2, 4irMa = 2267.0G, H dc = 144.0Oe, A H = 490.00e, 5 = 0.3 x 10"3m, I = 12.7 x 1 0 '3m )..................................................................................................... 112 xxi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 4.43 The effect of the magnetization angles on the insertion loss along a MSSW transducer (dd = 1.27 x 10-3 m , d f = 2.03 x 10~3m , cj = 17.5, t d = 10.2, 4ttM s = 2267.0G, Hdc = lU.OOe, A H = 44O.O0e, S = 0.3 x 10_3m , I = 12.7 x 10~3m )........................................................................112 4.44 The effect of the 3-dB line width on the insertion loss along a MSSW transducer (dd = 1.27 x 10-3m, d f = 2.03 x 10-3m, e/ = 17.5, ed = 10.2, 4:irMs = 2267.0G, Hdc = 144.O0e, S = 0.3 x 10~3m, I = 12.7 x 1 0 '3m )....................................................................................................................113 5.1 Incident wave on a normally biased single ferrite layer............................... 119 5.2 Geometry of single ferrite layer with a general incident wave.................... 122 5.3 Microstrip patch antenna on a normally biased ferrite substrate 5.4 RWG basis function and corresponding edge connectivity....................... 126 5.5 Geometry of a patch antenna with a biased ferrite as a cover layer . . 141 5.6 Fields and currents of a patch antenna with a biased ferrite as a cover . . . 123 l a y e r ......................................................................................................................141 5.7 Compaxison of RCS for a microstrip patch antenna (Ms = H0 = 0, eT = 13.0, d = 1.3 x 10-3m, L = W = 1.3 x 10"2m, 0{ = 30°, <?,• = 45°) 142 5.8 Comparison of RCS for a microstrip patch antenna (M3 = Ho = 0, = 13.0, d = 1.3 x 10-3m, L = W = 1.3 x 10-2m, 0t- = 30°, fa = 45°) 142 5.9 Comparison of RCS for a microstrip patch antenna (Ms = H0 = 0, er = 4.0, d = 3.0 x 10~4m, L = 1.25 x 10"2m, W = 2.5 x 10_2m, Qi = 45°, fa = 0 ° ) ...............................................................................................143 XXI1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 5.10 Comparison of RCS for a microstrip patch antenna (Ma = H q = 0, er = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10-2m, W = 0.4 x 10~2m, 0{ = 60°, fa = 45°) ............................................................................................143 5.11 Comparison of RCS for a microstrip patch antenna (Ms = Ho = 0, tr = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10_2m, W = 0.4 x 10~2m, 0i = 60°, fa = 45°) ............................................................................................144 5.12 Comparison of RCS for a microstrip patch antenna, the bias field is in the y-direction, (47rMj = 1780.0G, Ho = 360.00e, tr = 12.8, d = 6.0 x 10-4m, L = 0.55 x 10-2m, W = 0.4 x 10~2m, 0,- = 60°, <f>i = 4 5 ° )...............................................................................................................144 5.13 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47rMs = 1780.0G, Ho = 360.00e, ^ = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10_2m, W = 0.4 x 10_2m, 6; = 60°, 4>i = 4 5 ° )...............................................................................................................145 5.14 Comparison of RCS for a microstrip patch antenna, the bias field is in th e x-direction, (47rM a = 1780.0G, H0 = 300.00e, = 40.0G, Crd = 2.2, eTf = 13.0 dd = 1.3 x 10-3m, d j = 0.5 x 10-3m L — 4.0 x 10-2m, W = 3.0 x 10-2m, 9{ = 30°, fa = 4 5 ° ) .................................. 145 5.15 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47rM s = 1780.0G, H 0 = 300.OGe, AH = 40.0G, Crd = 2.2, trf = 13.0 dd = 1.3 x 10_3m, dj = 0.5 x 10~3m L — 3.0 x 10“2m, W = 4.0 x 10"2m,0{ = 30°, fa = 4 5 ° ) .................................. 146 xxiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 5.16 Comparison of RCS for a microstrip patch antenna, the bias field is in th e y-direction, (4t M 3 = 1780.0G, Ho = 360.OOe, er = d = 6.0 x 10-4m, L = 0.55 x 10~2m, W 12.8, = 0.4 x 10"2m, x 10-2m, x 10-2m, x 10_2m, x 10~2m, <t>i = 4 5 ° ) ...............................................................................................................146 5.17 Comparison of RCS for a microstrip patch antenna, the bias field is in the y-direction, (47rMa = 1780.0G, Ho = 360.OOe, tr = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10-2m, W = 0.4 4>i = 4 5 ° ) ...............................................................................................................147 5.18 Comparison of RCS for a microstrip patch antenna, the bias field is in the y-direction, (4irMs = 1780.0G, H0 = 36O.O0e, er = 12.8, d = 6.0 x 10-4m, L = 0.55 x 10_2m, W = 0.4 fa = 4 5 ° ) ...............................................................................................................147 5.19 Comparison of RCS for a microstrip patch antenna, the bias field is in the r-direction, (4ttM s = 1780.0G, Ho = 360.OOe, tr = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10-2m, W — 0.4 (f>i = 4 5 ° ) ...............................................................................................................148 5.20 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47tM, = 1780.0G, H0 = 360.00e, er = d = 6.0 x 10-4m, L = 0.55 x 10-2m, W 12.8, = 0.4 <f>i = 4 5 ° ) ...............................................................................................................148 5.21 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (AirMa = 1780.0G, Ho = 360.00e, er = 12.8, d — 6.0 x 10_4m, L = 0.55 x 10_2m, W = 0.4 x 10~2m, 0,- = 60°, <f>i = 4 5 ° ) ...............................................................................................................149 xxiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 5.22 Comparison of RCS for a microstrip patch antenna, the bias field is in th e x-direction, (A tM s = 1780.0G, H q = 300.OOe, A H = 40.0G, trd = 2.2, trf = 13.0 dd = 1.3 x 10_3m, df = 0.5 x 10-3m L = 4.0 x 10~2m, W = 3.0 x 10"2m, 0,- = 30°, fa = 4 5 ° ............................. 150 5.23 Comparison of RCS for a microstrip patch antenna, the bias field is in th e x-direction, (47rM s = 1780.0G, H0 = 300.OOe, A H = 40.00, eTd = 2.2, Crf = 13.0 dd = 1.3 x 10-3m, dj = 0.5 x 10_3m L = 4.0 x 10-2m, W = 3.0 x 10-2m, d{ = 30°, ^, = 4 5 ° ............................. 151 5.24 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4ttM 3 = 1780.0G, H0 = 300.OOe, A H = 40.00, erd — 2.2, erj = 13.0 dd = 1.3 x 10-3m, dj = 0.5 x 10-3m L = 4.0 x 10~2m, W = 3.0 x 10_2m, 9{ = 30°, & = 4 5 ° ............................. 152 5.25 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47tM 3 = 1780.0G, H q = 300.OOe, A H = 40.00, erd = 2.2, 6rf = 13.0 dd = 1.3 x 10-3m, df = 0.5 x 10-3m L = 3.0 x 10~2m, W = 4.0 x 10_2m, 9{ = 30°, fa = 4 5 ° ............................153 5.26 Comparison of RCS for a microstrip in the x-direction, (47tM s = 1780.0G, patch antenna, H q the bias fieldis = 300.OOe, A H = 40.00, trd = 2.2, Crf = 13.0 dd = 1.3 x 10~3m, dj = 0.5 x 10~3m L = 3.0 x 10-2m, W = 4.0 x 10-2m, 0,- = 30°, <f>i = 4 5 ° ............................154 5.27 Comparison of RCS for a microstrip in the x-direction, (47rMa = 1780.OG, patch antenna, H q the bias = 300.OOe, A H = 40.0G, erd = 2.2, trf = 13.0 dd = 1.3 x 10-3m, dj = 0.5 x 10-3m L = 3.0 x 10-2m, W = 4.0 x 10-2m, 0,- = 30°, fa = 4 5 ° ............................155 xxv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fieldis Figure Page 5.28 Comparison of RCS for a microstrip patch, antenna, the bias field is in the x-direction, (4xM s = 1780.0G, H q = 300.00e, A H = 40.0G, 6rd = 2.2, er/ = 13.0 dj = 1.3 x 10~3 m, dj = 0.5 x 3.0 x 10- 2m, W = 4.0 x 10- 2m, 1 0 ~3m L = = 30°, fa = 4 5 ° ........................ 156 0,- 5.29 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4irM3 = 1780.0(7, H q = 300.00e, A H = 40.0(7, eTd = 2.2, trf = 13.0 dd = 1.3 x 10_3 m, df = 0.5 x 10~3m L = 3.0 x 10“ 2m, W = 4.0 x 10“ 2 m, 0t- = 30°, fa = 4 5 ° ........................ 157 5.30 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47rMs = 1780.0G, H0 = 300.00e, A H = 40.0(7, trd = 2.2, tr/ = 13.0 dd = 1.3 x 10- 3 m, df = 0.5 x 3.0 x 10- 2m, W = 4.0 x 10_ 2 m, 0,- 1 0 ~3m L = = 30°, ^, = 4 5 ° ........................ 158 5.31 Geometry and dimensions of a cross patch compared to the full patch antenna...................................................................................................................158 5.32 Comparison of RCS for a microstrip patch antenna (M s = H eT = 12.8, d = 6.0 x 10_4 m, L = 0.55 x 10- 2 m, W = 0.4 x q = 0, 1 0 - 2 m, 0,- = 60°, fa = 4 5 ° ............................................................................................... 159 5.33 Comparison of RCS for a microstrip patch antenna (M3 = Ho = 0, tr = 12.8, d = 6.0 x 10- 4 m, L = 0.55 x 10- 2 m, W = 0.4 x 10- 2m, 0{ = 60°, fa = 4 5 ° ............................................................................................... 159 5.34 Comparison of RCS for a microstrip patch antenna (M„ = H0 = 0, tr = 12.8, d = 6.0 x 1 0 _4 m, L = 0.55 x 10_ 2m, W = 0.4 x 10- 2m, 0,- = 60°, fa = 4 5 ° ............................................................................................... 160 xxvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 5.35 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4irMs = 1780.0G, Ho = 360.00e, tr = 12.8, d = 6.0 x 10- 4 m , L = 0.55 x 10_2m, W = 0.4 x 10- 2m , 0,- = 60°, 4>i = 4 5 ° ...............................................................................................................160 5.36 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4xiVfa = 1780.0G, H q = 360.00e, tr = 12.8, d = 6.0 x 1 0 _4 m , L = 0.55 x 10_2m, W = 0.4 x 10_ 2m , 0,- = 60°, <f>i = 4 5 ° ............................................................................................................... 161 5.37 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4irMa = 1780.0G, H q = 360.OOe, tr = 12.8, d = 6.0 x 1 0 _4 m , L = 0.55 x 10“ 2m, W = 0.4 x 10- 2m, 0,- = 60°, (f>i = 4 5 ° ............................................................................................................... 162 5.38 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47rMa = 1780.00, H q = 360.OOe, tr = 12.8, d = 6.0 x 10- 4 m , L = 0.55 x 1 0 ~2m, W = 0.4 x 10- 2m , 0; = 60°, <f>i = 4 5 ° ............................................................................................................... 163 5.39 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, ( 4 ^ ^ , = 1780.0G, Ho = 360.OOe, tT = 12.8, d = 6.0 x 1 0 - 4m , L = 0.55 x 1 0 ~2m, W = 0.4 x 10- 2m , 0,- = 60°, <f>i = 4 5 ° ............................................................................................................... 164 5.40 Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4 ^ ^ ^ = 1780.OG, H0 = 360.00e, tr = 12.8, d = 6.0 x 10_4m , L = 0.55 x 10"2m, W = 0.4 x 10~2m , 0; = 60°, (pi = 4 5 ° ............................................................................................................... 165 A .l Geometry of single layer structure................................................................... 175 xxvii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page C .l Geometry of single layer isolator structure................................................... 188 D .l Current basis functions for (a) Longitudinal current (b) Transverse c u rre n t.................................................................................................................. 191 E .l Geometry of single layer slot line structure.................................................. 193 E.2 Geometry of single layer slot line structure.................................................. 195 xxviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION 1.1 M otivation T he objective of this research is the development of numerical models suitable for the analysis of a variety of microwave ferrite devices such as isolators, phase shifters, M agnetostatic Surface-Wave (MSSW) transducers and circulators. The main reason behind this work is the extensive progress in the epitaxial growth of ferrite substrates. M icrostrip is one of the most widely used transmission lines in the design of microwave integrated circuits (MIC’s). Although microstrip circuits have relatively high line losses and low power capability, they tire easy to fabricate, have small footprint and weight, exhibit large bandwidth, facilitate realization of passive circuits, and allow for good integration of chips, ferrites, and lumped elements. Ferrite materials are widely used in conjunction with planar structures for many microwave applications. Their high resistivity enables an electromagnetic wave to penetrate the m aterial so that the m agnetic field component of the wave can interact with the magnetic moment of th e ferrite. The interaction is distinguished by the remarkable behavior of the microwave permeability of ferrite. The permeability shows a clear resonance at a frequency which is simply related to the strength of the applied magnetic field within the ferrite. Another useful property of the microwave perm eability of a magnetized ferrite is th a t it causes cross coupling of linearly, circularly or elliptically polarized waves which would remain uncoupled in an isotropic dielectric medium. Because of this coupling, the propagation of electromagnetic waves through a ferrite medium can be nonreciprocal. Many microwave devices have been implemented based on the nonreciprocal phenomena. Among the most im portant of these devices are phase shifters, isolators, MSSW transducers and circulators. These devices have found an im portant place in microwave receivers, transm itters and duplexers. The symbolic representations of a phase shifter, an isolator and a circulator are shown in Fig. 1.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 (c) fl» (■) -A A A A Isolator Gyrator Circulator Fig. 1.1: Symbolic representation of phase shifter, isolator and circulator with ideal prop erties. In these devices, the microwave signal level is kept low such th at nonlinear effects in the ferrite axe negligible. Thus, a linear (small signal) analysis can be used to sim ulate most ferrite devices. One of the most useful microwave ferrite devices is the isolator, which is a twoport device having unidirectional transmission characteristics. The scattering m atrix for an ideal isolator has the form [1 ] o o 0 1 1 ( 1.1) 1 [s indicating that both ports are matched, but transmission occurs only in the direction from port 1 to port 2. Since th e S m atrix is not unitary, the isolator m ust be lossy. And, of course, [S] is not symmetric, since an isolator is a nonreciprocal component. A common application uses an isolator between a high-power source and a load to prevent possible reflections from damaging the source. An isolator can be used in place of a matching or tuning network, but any power reflected from the load will be absorbed by the isolator as opposed to being reflected back to the load, as the case when a matching network is used. Although there are several types of ferrite isolators, we concentrate here on the field displacement isolator. Field displacement isolators are simple and compact microwave ferrite devices with good electrical performance. The principle of operation of these devices is based on the field displacement effect; i.e, the microwave field configurations of the forward and backward propagating waves are different from each other. If an absorbing resistive film is placed at the regions of low electric field of forward propagating waves and high electric field of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. backward waves, then different attenuation of these two waves occurs, and an isolator is realized. T he circulator is another im portant ferrite device. It is a three-port device that can be lossless and matched at all ports. The scattering m atrix of such an ideal device can be w ritten as [1]. [s]. 0 0 1 1 0 0 0 1 0 The S-m atrix shows th at power flow can occur from ports 1, ( 1.2 ) 1 to 2, 2 to 3, and 3 to but not in the opposite direction. Thus, complete transmission between adjacent arms takes place in one sense of circulation only. A circulator with any number of arms can be built. M agnetostatic surface waves (MSSW) axe potentially im portant for carrying out signal processing directly at microwave frequencies because of their low propagation loss, ease of excitation, and possibility for electrically variable delay. Operations of MSSW delay lines have been dem onstrated from 1 to 15 GHz [2]. MSSW transducers have potential applications in several signal processing devices, such as delay lines, filters, resonators, and oscillators. The magnetostatic waves are efficiently excited by simple microstrip transducers. Ferrite materials have also found wide applications in a class of microwave devices called phase shifters. These are two-port devices that perm it the passage of guided waves with very little attenuation but with a variable phase delay controlled by changing the bias field of the ferrite. There are many types of ferrite phase shifters, both reciprocal (same phase shift in either direction) and nonreciprocal, and they find use in a variety of laboratory test equipment. The most significant use is in phased array antennas where the antenna beam can be steered in space by electronically controlled phase shifters. This work comprises investigations of the edge - guided Wave microwave devices including isolators and MSSW transducers and phase shifters, and in addition, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 radax cross section (RCS) of ferrite patch antennas. The spectral domain m ethod of moments (MoM) is used in our work. 1.2 Previous Work Over th e past twenty years, more than one hundred papers have been published on the theory and application of edge-guided wave (EGW) devices. During this period, EGW multi-octave isolators [3, 4, 5, 6 ], circulators [7, 8 ] and phase shifters [9, 10] were constructed and used in operational systems. All of the previous work reported in this area has comprised either experimental implementations or highly simplified analyses. To the best of our knowledge, the work presented here represents the first full-wave analysis for the EGW devices. In this section, we summarize m ajor aspects of previous work in this area. Kane [11], in his experimental work on edge-guided isolators, introduced a trans verse slot discontinuity located at one edge of the upper conductor as shown in Fig. 1.2 to attenuate the backward wave. Hines [4] presented an approximate analysis and a physical description of wave propagation in a wide microstrip line printed on a normally magnetized ferrite sub strate. He divided the microstrip line into three zones and then solved Maxwell’s equations for the propagation constant under some approximations. Hines concluded th at th e dominant mode resembles the TEM mode, except that there is a strong transverse field displacement causing wave energy to be concentrated along one edge of the line. He also reported experimental work to show the performance of a lossy term inated isolator. In his experiment, Hines constructed a device consisting of a normally magnetized ferrite slab over a ground plane and under a wide microstrip conductor as shown in Fig. 1.3. Waves incident from the port 1 are guided by the straight edge of the microstrip and propagate with relatively low attenuation. Waves incident from port 2 are guided toward the other edge of the conductor where they suffer high attenuation in the resistive film. Hence, nonreciprocal behavior is ob- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S22 S12 S21 Ho S li Fig. 1.2: 3-D edge-guided isolator with transverse slot loading Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 tained. e,^i S21 S12 Ha n X/4 S ll Fig. 1.3: 3-D edge-guided isolator with resistive film loading. Hines increased the bandwidth and improved the performance of the device by adding capacitance to the low loss edge to compensate for the inductive susceptance of the fringing fields. Similar devices have been reported which operate in other frequency bands [5, 6, 12], Cortucci [13] introduced magnetic losses into Hines’ model and accounted for the finite curvature of the guiding edges. He performed an experim ent to check the accuracy of his theoretical results and found th a t the overall behavior of the experimental results are in accordance with the theoretical results. However, he found that the theoretical value of the spacing between adjacent resonances of the EG wave structure is greater than the experimental value by a factor of 1 .6 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. De Santis [14], using an equivalent model, evaluated the fringing field effects in edge-guided waves propagating along ferrite microstrip devices. He used the fringingfield param eter to evaluate the ratio between the reactive power stored in the fringing fields and th e RF power in the ferrite under the strip conductor in a disc resonator. Araki [15] investigated the transmission characteristics of a ferrite substrate stripline in comparison with an ordinary dielectric substrate stripline. His studies included reflection problems of ferrite striplines with one edge shorted to the ground (short-end) and wide striplines with a transverse slot (open-end) using the eigenmode expansion m ethod. He confirmed his analytical results by constructing an EG mode isolator with short-end as shown in Fig. 1.4. He showed experimental results for the device for which the insertion loss is about 2.5 dB, isolation is about 35 dB, and the usable frequency bandwidth is about 0.6 GHz. However, the characteristics of this isolator are not as good as another type of EG mode isolator previously described by Araki [16]. A Shorted to the gound plane TOP VIEW Conductor Ferrite Slab Ho Ground Plane Fig. 1.4: 3-D edge-guided isolator with shorted load. In a very good review, De Santis [17] covered the basic concepts of EGW, includ ing the partially magnetized state of the ferrite substrate and higher order modes of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 propagation th at m ay exist. The transverse field displacement effect (TFDE) modes as well as dispersion diagram of the directional waves in various geometries have been widely reported [4], [18] - [20]. In 1976, Courtois et al [21] introduced the first very high frequency (VHF) edge-mode isolator. They used a high applied dc field to construct a wide bandwidth isolator in the VHF range. An isolator with an operating range of 225-400 MHz is described. Lin [22] suggested a new method to enhance the performance of field-displacement isolators by putting th e ferrite slab of m oderate thickness along the axis of the rectangular waveguide. She verified her theoretical analysis by experimental results, indicating that the bandwidth may be increased to 20 percent. Dydyk [23, 24], in a two-part series, examined the propagation characteristics of the edge-guide isolator. Shively [25], investigated the effects of resistive materials on microstrip antennas as well as other microstrip structures. We shall use resistive film loading of one of the microstrip conductor edges to simulate the performance of the EG mode isolator. Recently, researchers have begun to use the spectral domain approach for analysis of a variety of ferrite geometries with dc magnetic bias. Yang [26] studied the mi crostrip open-end discontinuity on a nonreciprocal ferrite substrate using a full-wave MoM analysis. In his work, an exact Green’s function and a careful numerical inte gration scheme axe used to determine the properties of an open-end for three biasing directions: longitudinal bias, transverse bias and normal bias. Pozar [27] described the radiation and scattering characteristics of microstrip antennas on normally bi ased ferrite substrates using spectral domain method of moments. He concluded that the extra degree of freedom offered by the biased ferrite can be used to obtain a number of novel characteristics, including switchable and tunable circularly polarized radiation from a microstrip antenna having a single feed point, dynamic wide-angle impedance matching for phased arrays of microstrip antennas, and a switchable radar cross section (RCS) reduction technique for microstrip antennas. Fig. 1.5 shows the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 geometry considered by Pozar. d Fig. 1.5: Microstrip patch antenna on a normally biased ferrite substrate. Alexopoulos et al [28] also studied microstrip patches on ferrite substrates. They compared the RCS for the case of an arbitrary biased ferrite to the case of an unbiased ferrite. They found that the resonant frequencies of a patch vary significantly with the change of the bias field except for those resonant modes with a dominant magneticfield component in the direction of the bias field. They also found that the magnetic loss affects RCS at resonance significantly. Their analysis is based on a full-wave integral equation formulation in conjunction with the method of moments. In spite of a reasonable effort and many publications about the scattering from printed antennas on a biased ferrite substrates, an efficient and versatile formulation of the RCS is not available yet. W ith the exception of Yang [29], multi-layer structures are not yet discussed. However, neither Yang nor anyone else show the excitation vector expression or a detailed derivation of the RCS formulation. It is very im portant for anyone working in the area of ferrites to deeply under stand the physics of magnetostatic surface and volume wave mode in ferrite m aterial. There are two types of modes th at may exist in a ferrite sample. The first type is the pure electromagnetic mode or the dynamic mode. This mode does not interact appreciably with the electron spin in the ferrite material. It acts as if the ferrite ma terial is isotropic with an effective permeability. The second type is the extraordinary mode, also called the magnetostatic wave (MSW) mode. Magnetostatic modes are Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 10 a class of wave phenomena that appear in ferrimagnetic m aterials and are situated between the electromagnetic waves at the long wavelength end of the spectrum and the spin waves at the short wavelength end. In contrast to norm al electromagnetic waves, this decrease in the wavelength does not require a proportional increase in the frequency of the wave, but can be obtained by a change in the value of the DC field th at magnetizes the ferrite. Auld [30] demonstrated that in m agnetostatic waves, the electric field components become less and less im portant for determ ining the propa gation, and thus, Amperes law simplifies in the magnetostatic lim it to V x H = 0. In specimens w ith finite dimensions, there are also surface dipoles, so th a t the propaga tion can be quite different from th at in the infinite medium. Walker [31] has devel oped a general theory of magnetostatic modes in ellipsoidal samples. Experimental verification of this theory was first made by W hite and Solt [32] with absorption experiments. Auld showed th at these modes can exist under certain conditions in specimens of any size, and th at they gradually merge with the electromagnetic waves. Damon and Eshbach [33, 34] extended Walker’s theory to slabs and studied the mag netostatic modes in ferrite materials. Their studies established the existence of three types of m agnetostatic waves (MSW): magnetostatic surface waves (MSSW’s), mag netostatic forward volume waves (MSFVW’s) and m agnetostatic backward volume waves (MSBVW ’s). The orientation of the internal bias field relative to the ferrite structure and to the propagation direction determine which particular wave type can exist. M SFVW ’s propagate perpendicularly to the applied m agnetic field which is itself normal to the ferrite substrate. MSBVW’s propagate perpendicularly to the applied magnetic field which is in the longitudinal direction of the substrate plane. MSSW propagates in the same direction as the applied magnetic field. Both are in the longitudinal direction of the substrate plane. In our analysis for a normally biased ferrite structure, we deal prim arily with m agnetostatic forward volume waves. However, the surface wave mode may also be excited depending on the operating frequency range as described by Pozar [27]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 The last subject of this review concerns units. In the Gaussian system of units, the magnetization is measured in Gauss (lG au ss = 10“ 4 W e6er/m 2), and the field strength is measured in Oersteds (4tt x 10-3 O ersfed = 1A / m ) . Thus, fio = 1G/Oe in Gaussian units, implying th a t B and H have the same numerical values in a nonmagnetic m aterial. Saturation magnetization is usually expressed as AirM3 Gauss. In Gaussian system of units, the lamor frequency can be expressed as fo = u m/2ir = Ho^ H o/ I k = (2.8MHz/Oersted) (H0Oersted), and f m = wm/2 tt = fi0j M s/2ir = (2.8M H z (Oersted), (AirMsGauss). On the other hand, In the MKS systems of units, the permeability for free space fig is equal to 4?r x 10~7F / m and the dielectric constant for free space e0 = ( 3^ ) x lO~9H / m . It follows that ^ = = is equal to 3 x 108 m /s , the velocity of light in free space, and ^/no^o — 377fl is th e impedance of free space. The conversion factors from Gaussian to MKS units are given by [35], B {G ) x 10“ 4 = 1.3 1.3.1 B ( W b / m 2) 4trM{G) x 79.5 = M (A/m ) H(Oe) x 79.5 = H(A/m) Methods Introduction The most widely used method for analyzing microstrip and printed line structures is the spectral domain approach. This method is essentially a Fourier-transformed version of the spatial domain integral equation method. The method was first intro duced by Yamashita and M ittra [36] for com putation of the characteristic impedance and the phase velocity of a microstrip line based on a quasi-TEM approximation. As the operating frequency is increased, dispersion characteristics of the microstrip become im portant for an accurate design. This requirement has led to the full wave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 analysis of microstrip lines, represented by the work of Denlinger [37], who solved the integral equations using a Fourier transform technique. His technique is strongly dependent on the assumed current distribution. To avoid this difficulty, another method was introduced by Itoh and M ittra [38], now commonly called the spectral domain approach (SDA). In SDA, Galerkin’s method is used to form ulate a homo geneous system of equations to determine the propagation constant and the current distribution from which the characteristic impedance is derived. In this m ethod the Fourier transform is taken along the direction parallel to the substrate and perpendic ularly to the strip. The use of the Fourier transform domain analysis and Galerkin’s m ethod leads to several im portant features in SDA [39]: • Easy formulation in the form of a pair of algebraic equations • Variational nature in determ ination of the propagation constant • Ability to identify the physical nature of the modes corresponding to each solution The SDA method is computationally efficient due to significant analytical pre processing. This method has the following limitations: • SDA assumes infinitesimal thickness for the strip conductor • It is difficult to treat a strip having finite conductivity • The substrate is assumed to be infinite in the transverse direction In spite of these limitations, SDA is a very popular and widely used numeri cal technique. However, the formulation of a Green’s function represents a major complexity in th at approach. Solving for a Green’s function as a boundary condi tion problem is quite difficult for magnetic substrates and becomes more difficult for multi-layer structures. In addition, Green’s functions determ ined using such a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 procedure axe useful only for the specific structures for which they are derived. A more flexible approach is to find the transmission m atrix of the medium and then use this m atrix to find the Green’s function. The transmission m atrix of a medium does not depend on the geometry of the total structure. Thus, it is possible to form a “software” library of transmission matrices for several m aterial layers and to use this library for th e analysis of different multi-layer structures. In this work, we follow the approach used by El-Sharawy [40] to derive the transmission m atrix for a lon gitudinally biased ferrite slab. W ith this approach, the transmission m atrix for any material, ferrite or dielectric, can be derived in closed form. Thus, a considerable improvement in the flexibility of the numerical technique is achieved. Dyadic Green’s functions axe concise representations of vector-input, vectoroutput systems. For our analysis, they axe the relationships between vector elec tromagnetic fields and vector current sources. For instance, Gxy is the electric field in the x direction due to a current source in y direction. In a m atrix representation of the dyadic G reen’s function, Gxy is just one of four elements in a 2 x 2 matrix. The tangential current vector is related to the vector electric or magnetic field by the dyadic Green’s function. To expand th e electric current densities in the strip, piecewise linear basis func tions axe used to model both longitudinal and transverse currents as shown in Fig. 1.6. To apply these basis functions in the spectral domain, the Fourier transform is found analytically. Once the Green’s function is obtained and the basis functions are se lected, Galerkin’s technique is used to fill the impedance m atrix. The complex prop agation constant is the variable that forces the determinant of the impedance matrix to zero. The isolation and the insertion loss axe calculated using the attenuation constant. The current basis function coefficients correspond to the eigenvector of the impedance m atrix for a given propagation constant. There axe many numerical considerations which are investigated in detail including integration limits, integra tion intervals, num ber of integration points, and the integration path modification Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 to avoid the m agnetostatic wave poles. ft) Fig. 1.6: Current basis functions for (a) Longitudinal current (b) Transverse current 1.3.2 Method of Moments The Method of Moments (MoM) is a powerful tool for solving linear integral-differential equations, such as the determ inistic problem £ ( /) = « (1.3) where L is a linear operator, / is the unknown function, and g is the function resulted from the application of L on / . In the moment method, the unknown function / (in our analysis, the current distribution function) is replaced by an approximate function f a which is a linear combination of a series of known functions (basis), the coefficients of which are to be determined in the process / “ = £<■ „/„ (i-4) n Here, the moment m ethod uses a finite number of terms (functions) to represent an unknown function. Thus th e original problem becomes 6 + J 2 anL(fn) = g Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1.5) 15 where 6 is the error (residual) due to the introduction of / “. Another series of functions, called testing or weighting functions, wm, in the range of L, is used to take the inner product with Equation 1.5. The moment methods lets the inner product < wm,6 > be zero. Comparing equation 1.5 with the original problem Equation 1.3, it is noted th at by letting inner product < wm,S > be zero, MoM approximates the exact solution / with / “ in a sense that equates the projections of L ( f ) and L ( f a) on where C{wm) is the space spanned by the wm. Since the error 6 is orthogonal to the projections (inner product is zero), it is of second order, and the moment method solution minimizes the error 6 [41]. If the dimensions of the linearly independent basis functions is equal to or greater than the dimensions of the solution space, the moment method is exact. The moment method is not only used in the spectral domain, but also widely used in th e spatial domain, where a wider variety of basis functions can be used [42]. 1.4 Research Objectives This research attem pts to address some of the current issues involving numerical modeling and design optimization of microwave ferrite devices and antennas. The impetus behind the work is to develop theoretical and numerical techniques which allow rigorous analysis of various microwave ferrite devices with different m agnetiza tion directions. Software tools such as Maple V are extensively utilized. Maple V is an interactive computer algebra system which is useful in evaluating complex alge braic expressions such as the derivation of the transmission m atrix for normally and arbitrarily biased ferrite slab. In addition, a variety of data visualization programs such as GNUPLOT and GLE are utilized during our work. This is the first effort to present a full-wave analysis of edge-mode microstrip isolator. This also is the first attem pt to study the effect of the magnetization angle on the performance of microwave ferrite devices, phase shifters and m agnetostatic surface wave transducers. The effect of the magnetization angle cannot be stud Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 ied without the formulation of the general Green’s function that has two degree of freedom (6 and <f>). The generalization of exsisting methods is the main contribution of this work. This work provides a cleax and versatile formulation for the scattering from a biased ferrite antenna. This formulation includes general plane wave incident angles, general biasing directions, and multi-layer ferrite and dielectric structures. The Green’s function is formulated using the closed form transmission m atrix. The excitation vectors axe also derived in closed form. The expression of the excitation vectors is general for any incident angle, any number of layers, and any magnetization angle. This study of ferrite microwave devices and antennas addresses various topics. In chapter 2, the Green’s function for a normally biased ferrite slab is evaluated. A validation of the Green’s function is also presented by comparison with dielectric Green’s function w ith a Green’s function for a single ferrite slab. Chapter 3 presents the full-wave analysis of the edge-guided microstrip isolator using the spectral domain approach, the moment method, and microwave circuit theory. In this analysis, we present the figures of merit for three isolator structures, discuss the effect of adding a dielectric layer underneath the ferrite layer, and determine the optimum value and location of the resistive layer that is added to absorb the backward wave. We compare our results with published experimental results. Chapter 4 presents the derivation of the general Green’s function for arbitrarily biased ferrite structures. Two special cases (the transversely biased Green’s function and the normally biased Green’s function) dem onstrate the validity of the Green’s function. We also present a versatile model using the general Green’s function to analyze various microwave ferrite devices. As an example, we present the results for differential phase shifters and magnetostatic surface wave transducers. A compaxison with the available results in the literature shows the efficiency and the versatility of the proposed model. In chapter 5, we study th e scattering from arbitrarily biased ferrite patch antennas. We present a unique and versatile approach for evaluating the excitation vectors using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 th e generalized transmission m atrix th at we derive in chapter 4. Our results agree well w ith many previously published results. In chapter 5, we also propose a novel cross-patch antenna and describe its advantages over the rectangular patch antenna. Finally, Chapter 6 summarizes the dissertation and gives recommendations for future work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER 2 GREEN’S FUNCTION FORMULATION FOR A NORMALLY BIASED FER RITE SLAB 2.1 Introduction Many people use ferrite materials extensively in microwave devices such as phase shifters, isolators, circulators, and antennas. The most popular technique for analyz ing planar structures is the spectral domain approach, which can be adopted for a wide variety of geometries. However, the spectral Green’s function for a biased ferrite substrate does not have a simple analytic form. Solving for a Green’s function as a boundary condition problem is quite difficult for magnetic substrates and becomes more difficult for multilayer structures. In addition, Green’s functions developed us ing conventional methods axe useful only for the particular configuration for which they were derived. A more flexible approach is to find the transmission m atrix of the medium and then use this m atrix to obtain the Green’s function numerically. The transmission matrices of different layers are combined to form the Green’s function for a multilayered structure. The transmission m atrix approach used in our paper was developed by El-Sharawy [43]. This approach yields the Green’s function for general multi-layered structures in a more efficient and versatile way than other approaches. For instance, Krowne’s method [44], [45] is based on eigenvector matrices th at have to be constructed to find the transmission m atrix. As a result, in contrast to our work, closed form transmission matrices are not generally available. The approach of Morgan et al [46] and Berman [47], like Krowne’s, requires numerically evaluated matrices and is lim ited to the treatm ent of propagation through a single anisotropic layer bounded either on both sides by isotropic media or an isotropic medium on one side and a ground plane (PEC) on the other. The work of Tsalamengas [48] is based on solving a boundary value problem. The eigenvectors are determined numerically, then the Green’s function is formulated through a lengthy procedure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Thus, Tsalamengas’ approach is much less versatile than our approach. While the Green’s function for a multilayer structure is derived in closed-form by Hsia and Alexopoulos [49], their analysis is lim ited to one particular structure. In contrast, our approach can be used for any multi-layer structure. Lee and Harackiewicz [50] treated the case of in-plane biased ferrite substrate using the transmission m atrix approach by El-Sharawy [43]. In [50], the authors refer to the work of Pozar [51] and Yang, et al. [52]. In these papers, the Green’s function is valid only for a single grounded ferrite layer. Also in [50], the capability to handle multilayer structures is not dem onstrated. It should also be noted th at the in-plane biased case is much easier to treat than the normally biased ferrite, as may be readily deduced by examining the permeabil ity tensor for each case. To the best of our knowledge, the closed form transmission m atrix for a normally biased ferrite layer has not been presented in the literature. The derivation of this m atrix would probably be intractable without the use of a com puter algebra system. While the scope of this chapter is limited to the normally biased case, this case has numerous applications for antennas and microwave devices. We hope th at this work will represent a useful reference for those attem pting to de velop efficient numerical models of general multilayer microwave and antenna ferrite circuits. 2.2 Transmission Matrix Formulation The transmission m atrix T for a m aterial layer is a 4 x 4 matrix w ritten as [43] _ __ E2 - f . . . ____ ' =E Ei t ^ . —™ =T z Ei =T =J . Y T . =E =T = r - j where T , Z , Y , T are 2 x 2 submatrices of T, Ei, E? are the tangential electric field at the boundaries of the layer, J \ and J i are the tangential surface currents Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 defined by J n = z x H n, where H n is the tangential magnetic field at the n th surface of the layer, and denotes the spatial Fourier transform defined as E ( K X, K y) = f I E ( x , y)e~iKzXe~iKyy dx dy J —COJ —oo (2.2) The transmission m atrix for a ferrite layer is derived in Appendix A. Results for a dielectric slab can be found easily by taking the limit of the ferrite formulation when th e magnetization is set to zero and can be found in Appendix B [40]. The elements of the the transmission m atrix for a ferrite layer are given by T\i — ------ r [(n 2 + n i n 2 2(n2 —n i) = (—ni —n xn 2 + T12 = 1 1 - n x) cosh(Rxd) + + n 2) cosh(i? 2^)] f f 2 = —— ------ -[(n 2 + n in 2 + i[n2 — ni) 1 + nx) cosh(Rxc/) + (—ni — riin2 — 1 —n 2) cosh(/? 2d)] Tii = ZT = -3[(Tii + l ) ( D s - D e ) s i n h ( R i d ) + (« 2 7 i4 = + l)(Ds — D 7) sinh(R2^)] Z 12 = -^-[(ni + 1 )(Z?5 + Dq) sinh(Rxd) + (n2 + 1)(Z?8 + D7) sinh(i? 2 ^)] T2i = T2i = —— r[(—n 2 + n xn 2 + Z{n2 —ni) 1 —n x) cosh(Rxd) + (n x —n xn 2 — 1 + n 2) cosh(i? 2d)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 ?22 ff2= = tt, — r [ ( ~ n 2 + n in 2 - 1 + ni) cosh (Rid) + Z{ri2 — ni) (nx —riiTi 2 + 1 — n 2) cosh(i22<0] T 23 = Z 21 = + ! ) ( A - A>) sinh(/?1rf) + (—n2 + l)(Ds —Dt) sinh(i?2^)] T24 = Z 22 = ^ [(- n i + 1 )(^ 5 + A>) sinh(Rid) + (—n 2 + 1)(A* + D j) sinh(i?2</)] fzx = 2{ri2 - n i ) [(n2 ~ 1){Dl ~ ° 2) sinh^ 1<f) + = (ni - 1)(Z)3 —D 4) sinh.(R2d)] fz2 = ? i 2 = 2 ('n 2 _ n ) [(n 2 + ! ) ( A ~ D )sm h(Rid) 2 + («i + 1)(Z)3 — Z?4)sinh(i?2^)] ^33 = f 1Jl = ^[(Dl - D 2 )(D5 - D 6)cosh(Rl d) + (D4 —Dz)(D8 —D7) cosh.(R2d)] T34 = f ^2 = -J [ ( D i - D 2 )(D5 + D6)cosh(R1d) + (D 4 - D 3 )(D 8 + D 7) cosh(i?2cO] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 = 2 K - m ) [(n2 ~ 1)(£>1 + ° 2) siah(<R ^ ?41 = + (—n i + 1)(£>4 + ^ 3 ) sinh(i? 2<i)] ?42 = ^22 = 2 { n 2 ■— M l) ^ n 2 — (nj + 1)(^4 + ^ 3 ) sinh(i?2^)] £3 = Tj1 = ^ - [ ( D l + D2)(D5 - D 6)cosh(R1d) + (D4 + D 3){D s - Dr) cosh (R2d)} T4A = f ^ = ^[(Dl + D 2)(D5 + D6)cosh(Rl d) + (D4 + D3)(D8 + Dr) cosh(i? 2 ^)] where Dl = n = 5 ‘•"2 we K+KA r fi2(- i — 2 * if2 + " 2- f ) uj2e2R 2R i ( n 2 — n \ ) ^ Z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 u>2e2R.2Ri(n2 —ni) u 262R 2Ri(n2 — ni) -------------D2 u 2e2R 2 Ri{n 2 —n x) 9i + 9 i + R 2 9 2 2k2er (fi2 91 _ 92 - K-K+(n + 2 (fi + k ) k 2) k + 1) I<1(H + K — 1) 2{n + k ) k ±2 = K 2 + K 2 — u)2hqc([j, =F k ) Ax = u 2fio en (K l + K \ - u 2n 0e - ----- — ) fi K+ = K. K s+ jK t = K x - jK, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 and 2k$er([l2 — K2) — K - K + (fi + K + 1) ^ = 2(fi -f- /c) 2k$6r([i2 - _ h 1 ~ K2) k K + 2kler(fi2 - I) k k) - K ^K +(fi - K2) K K) K \ K t ( n — k — 1) ( f i + 4(/z — K)(fi + k ) where 2(n - + - K - K +(fi + 2 {fi + 2kQ6r(fi2 — K2) — K - K + ( f l — k + 1) 2 (h - k K + I) ) ] + 1) and fi are the elements of the permeability tensor and will be defined later. Transmission matrices have the following useful properties f(o) = 7 T { a + b) = T{a)T{b) f {-d) = f (2.3) \d) where / is the identity m atrix. The first property means th a t for zero thickness, the field and current components axe equal satisfying the boundary conditions. VVe can think of the second property as dividing a single ferrite slab into two sub-layers with the same ferrite param eters but with different thickness. We can prove easily th at the transmission m atrix for the slab is the multiplication of the transmission matrices of the sub-layers. The third property simply means th at when we reverse the reference axis from one side of the slab to the other side weget the the transmission m atrix. inverse of In general, the three properties are useful forchecking the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 computer routines written to com pute the transmission m atrix. We have shown that the transmission m atrix defined, by equations (2.3) satisfies these identities. 2.3 Discussion on the Difficulty of Deriving Normally Biased Transmission Matrix To understand the reason of the difficulty of deriving the transmission m atrix for a normally biased ferrite slab compared to the other transmission matrices of in-plane biased ferrite slab, either in th e x or y direction, we can refer to the permeability tensor for each case. For a ferrite magnetized in the 2 -direction, the permeability tensor is given by f* = fl - jk jk ft 0 0 0 0 1 (2.4) For a ferrite magnetized in the y-direction, the permeability tensor is given by fl 0 0 1 0 JK - jk 0 (2.5) fl For a ferrite magnetized in the x-direction, the permeability tensor is given by '1 0 fi = O ' 0 fi 0 JK - jk ( 2 .6 ) fl We can notice th at the top left 2 x 2 submatrix in each tensor is responsible for defining the relation between the field components in x and y directions. Because of the zeroes in the submatrix, th e field components in the x and y are not coupled. Therfore, the wave equation in the case of in-plane biasing is a second order differ ential equation, while the wave equation in the case of normal bias is a fourth order differential equation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 2.4 Green’s Function Formulation Using the transmission m atrix, Green’s functions can be form ulated in the spectral domain for single and multi-layer structures. The Green’s function relates the tan gential electric field on one surface to the surface currents on the same or another surface. This relation has the form, E 3(kx, k y) = G (kx,ky)Js(kx, k y) (2.7) where GXX Gly G = 2.4.1 Gyx ( 2 .8 ) Gyy Green’s Function for a Single Ferrite Substrate E J Air Ground Planes 1 I E J Fig. 2.1: Geometry of single layer isolator structure. The Green’s function is formulated at the plane of the source, which is the ferrite-air interface of Fig. 2.1. The electric surface current at this plane, J s, is split into two -= + equivalent currents J — — and J as follows 13 = zx(T-T) = T +7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 2.9) where (-) denotes the tangential fields just below the plane of source (on the ferrite side of the interface) and (+ ) denotes the tangential fields just above the plane of source (on the air side of the interface). From the transmission m atrix equation, we can write the following equation for the upper air region, -----1- 1 E _ T = Ta " -=ru ’ E J . ' =E =T ' * -=ru " E T a z na -=u =T =J J Ya f ° . ( 2 . 10 ) In this equation, E and J represent the field and the current at a distance dj_ from the air-ferrite interface. We force E = 0 by placing a ground plane at distance dj from the source plane. This results in the following equation E = =T=J~1_ + Z af a J = gT (2.11) where the superscript (—1), here and throughout this work, means the m atrix inver sion. Gais a semispace Green’s function, which is calculated by taking the limit of the dielectric Green’s function when the distance ddgoes to infinity and the dielectric constant goes to unity, see Appendix C. Ga = lim Gd dd—oo Ed-1 (2.12) and Gd is formed using the dielectric transmission m atrix derived by El-Sharawy [40]. For the lower region, we have Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 .T ’ =E =T ' — -| E E -=i = TJ r ! f -=i . J . . J . L *1 T , . • r infs' II — _ E . i (2.13) -=i where E and J represent the field and current at the other side of the ferrite slab. We force E = 0 by placing a ground plane at distance dj from the source plane. This results in the following equation _ _ _____ =t =j ~1__ E = Z jT f J = (2.14) G SJ where G j is th e ferrite region Green’s function, which can be calculated using the derived ferrite transmission m atrix. From equations (2.9), (2.11) and (2.14) we obtain J, = where E s, E and E T +J = (Go. +GS W (2.15) are equal due to the continuity condition of the electric field in the plane of source. The total Green’s function, which is the sum of the upper and lower semispace components, is given by Es = = 2.4.2 =J =T~l = -1 (Tf Z f + G a )-V , =MS_ G J, (2.16) Green’s Function for a Ferrite-Dielectric Substrate In this section, we present the Green’s function for two additional structures. The first structure is similar to that depicted in Fig. 2.1 except for the addition of another Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 dielectric layer under the ferrite layer as shown in Fig. 2.2. The only difference in the derivation of the Green’s function of this structure from the previous one is the transm ission m atrix of the lower region. We modify the transmission m atrix of the lower region to include the dielectric region as follows = (2.17) [?A [h The Green’s function is found to be, = where Z = -1 =J=T~l G= (T Z + Ga T 1 (2.18) =j and T are the elements of [Tnew\. The second structure, called a “drop-in element,” is a structure compatible with Monolithic Microwave Integrated Circuits (MMIC). In this structure, a dielectric substrate with relative perm ittivity of 9.8 is used. To form an EG isolator, we place a piece of ferrite with a resistive thin film on top of the dielectric strip as shown in Fig. 2.3. In this structure, the source is at the ferrite-dielectric interface. We divide th e structure into two regions. The upper region includes the ferrite and air layers. The lower region is the dielectric region. We can consider the total adm ittance Green’s function as the sum of two parallel admittances, one for the upper region and another for the lower region. The two semispace Green’s functions are formulated using the transmission m atrix as previously described. The transmission m atrix for the upper region is given by [ ? J = (?/][?•] (2.19) where the air transmission m atrix is the dielectric transmission m atrix with er= l. The final form of the Green’s function can be found as = = J= T G = (f,Z „ ~1 =J=T ~l + f dZd )-■ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.20) 30 2.4.3 Green’s Function for a General Multi-layer Structure For a general planar structure with any combination of ferrite and dielectric layers and any arbitrary location of the source, the Green’s function can be easily formu lated following the procedures described in the preceding sections. 2.4.4 Circuit Model Interpretation of the Green’s Function for a Multi-layer Struc ture Consider the structure shown in Fig. 2.1. In th at structure, the surface current J3 at th e plane of source can be split into two currents, J + and J - , which implies th at the two layers, ferrite in the lower region and air in the upper region, are in parallel. Thus, we can use the circuit model shown in Fig. 2.4 as an analogy to the structure. From the definition, the Green’s function of a microstrip structure has the units of an impedance. The total impedance of the analogous circuit is given by the following equation, Z to ta l = { -y 6 f + ( 2 .2 1 ) 6a where Z a is given in equation 2.11 and Z j is given in equation 2.14 for the air and ferrite layer, respectively. Notice the sim ilarity between equation 2.21 and equa tion 2.16. For th e multilayer structure shown in Fig. 2.2, we can simply apply the same procedure. By investigating equation 2.18, we conclude th at we added two adm ittance term s for the upper and lower regions and then inverted to get the total impedance of the structure, and hence the Green’s function. Generally speaking, the adm ittance of th e region above the plane of the source is added to the adm ittance of the region underneath the plane of the source, and the total summation is inverted to get an expression for the total impedance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 2.5 Results and Conclusion We compared the Green’s function of the structure shown in Fig. 2.1 derived us ing the transm ission m atrix approach with the Green’s function derived by Pozar [27] using th e boundary condition method. Excellent agreement is achieved as is clear from Figures 2.5-2.8. No Green’s function is available in the literature for the dielectric-ferrite structure shown in Fig. 2.2. However, we replaced the ferrite with dielectric layer to form a multi-layer dielectric structure and compared its Green’s function with th a t formulated by Aberle [53]. Excellent agreement is clear from Figures 2.9-2.11. We did a similar comparison with Aberle’s results for the drop-in element structure shown in Fig. 2.3. Again, excellent agreement is clear from Figures 2.12-2.14. In comparing our results with Pozar, we demonstrate the validity of the ferrite transmission m atrix. In comparing our results with Aberle, we dem onstrate our ability to obtain Green’s functions for multi-layer structures. The case of normally magnetized ferrite slab is considered to be significantly more difficult than the cases of transversely or longitudinally magnetized ferrite slab. One can reach this conclusion simply by comparing the permeability tensors for each case [54]. As a result of this complication, no closed form for the transmission m atrix of a normally magnetized ferrite medium has been available to date. The Green’s function is th e core of the spectral domain MoM and is used extensively to analyze planar microwave structures as well as antenna problems. Normally biased ferrite medium is widely used in many microwave devices such as isolators, circulators, and phase shifters, and more recently in antenna applications [27], [28]. Our technique is based on the exact derivation of the transmission m atrix, and no approximation has been made in the Green’s function formulation. In addition, the Green’s function is formulated in a way to increase the numerical efficiency in the MoM applications. Our next step is to generalize the derivation of the normally biased ferrite slab transmission m atrix to the arbitrary biased ferrite slab. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t 1 E J Fig. 2.2: Geometry of double layer isolator structure. Ground Planes E 1J 1 Fig. 2.3: Geometry of drop-in element isolator structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 J, TTTT 7777" Fig. 2.4: Circuit representation, for a single ferrite layer structure. — Thi* w o rk ' — Polar 3060 B 2550 O CQ c 2040 S ao 6 1530 1020 -280 -140 0 140 280 Ky/k0 Fig. 2.5: Comparison of the computed Green’s function versus Pozar’s (Imag(Gxx)) (d = 7.62 x 10-4 m, ef = 12.0, 4k M s = 2100.0G, Hdc = 700.00e, A H = 0.0Oe, R 3 = 0.00, W = 1.016 x 10-2m, / = 3.6GHz, K x = (110.0, -10.0)). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 400 — This work Pozar 350 (§ 2 5 0 200 150 -280 -140 0 Ky/kO 140 280 Fig. 2.6: Comparison of the computed Green’s function versus Pozar’s (Real(Gxx)). The parameters are the same as in Fig. 2.5. — This work — Pozar 2480 -1240 -2480 -280 -140 0 Ky/kO 140 280 Fig. 2.7: Comparison of the computed Green’s function versus Pozar’s (Imag(Gxy)). The parameters are the same as in Fig. 2.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 240 120 -120 -240 -280 -140 0 Ky/kO 140 280 Fig. 2.8: Comparison of the computed Green’s function versus Pozar’s (Real(Gxy)). The parameters are the same as in Fig. 2.5. This work Aberle o.o -4.5 0 S' as c '5b a c -9.0 -13.5 -18.0 -480 -240 0 240 480 Ky/kO Fig. 2.9: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxx)). (.dd = 0.1 x 10"2 m, ed = 30.0, df = 0.762 x 10~3 m, ef = 12.0, K x = HO.Orad/m. / = 2.0 GHz). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 18000 Thii work. Aberle / 14400 & 10800 S ' 7200 3600 -480 -240 0 Ky/kO 240 480 Fig. 2.10: Comparison of the computed Green’s function versus AberLe’s (Imag(Gyy)). The parameters are the same as in Fig. 2.9. Thw work Aberle I S3 c ’& S3 s I—I -40 -80 -480 -240 0 Ky/kO 240 480 Fig. 2.11: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxy)). The parameters are the same as in Fig. 2.9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 This work Aberle -480 -240 0 240 480 Ky/kO Fig. 2.12: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxx)). (dd = 0.1 x 10_2m, ed = 9.8, dj = 0.762 x 10~3m, e/ = 12.0, da = 0.1 x 10~2m, €a = 1.0, K x = 110r a d / m , f = 2.0G\ff2). 12000 This work — Aberle 9600 <3 7200 « 4800 2400 0 -480 -240 0 Ky/kO 240 480 Fig. 2.13: Comparison of the computed Green’s function versus Aberle’s (Imag(Gyy)). The parameters are the same as in Fig. 2.12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 This work Aberle Fig. 2.14: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxy)). The parameters are the same as in Fig. 2.12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 CHAPTER 3 2-D FULL-WAVE ANALYSIS OF AN EDGE-GUIDED MODE ISOLATOR 3.1 Introduction The isolator is one of the most widely used magnetic devices. The principle of operation of these devices is based on the field displacement effect; i.e, the microwave field configurations of the forward and backward propagating waves are different. If an absorbing resistive film is placed at one edge of the conductor, then different attenuations of these two waves occur and an isolator is realized. Fig. 3.1 shows the geometry of an isolator with a resistive thin film. Experimental work on this type of isolator has been widely reported in the literature [4] - [11]. Approximate theoretical analyses have also been reported [3] - [6]. However, no full-wave analysis of this structure has been reported to date. C on d u ctor Strip R esistiv e F ilm F errite S u h strate d G round P lan e Fig. 3.1: Edge-guided isolator with resistive film loading. Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 40 3.2 Full Wave Formulation We investigated three different isolator structures. The first structure comprises a single normally magnetized ferrite substrate as shown in Fig. 3.2. In the second structure, we added another dielectric layer underneath the ferrite layer as shown in Fig. 3.3. The th ird structure, called “drop-in elem ent”, is an isolator structure com patible with Monolithic Microwave Integrated Circuits (MMIC). In this structure, a dielectric substrate w ith relative perm ittivity equal to 9.8 is used. To form an EG isolator, we place a piece of ferrite with a resistive thin film on top of the dielectric as shown in Fig. 3.4. « « E I d4 Ground Plane dt Fig. 3.2: Geometry of single layer structure. mimmimmmmmmmmm r t B J E J f - Fertile Fig. 3.3: Geometry of double layer structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 E J Ground Plane E J Fig. 3.4: Geometry of drop-in element structure. 3.2.1 Green’s Function Formulation As outlined in Chapter 2, the Green’s function for the three isolator structures are formulated using the transmission m atrix approach. The details of the derivation and the Green’s function expressions are mentioned in the previous chapter and will not be repeated here. 3.2.2 Basis Functions and Resistive Region Treatm ent One very im portant step in any MoM numerical solution is the selection of basis functions. In general, one chooses the set of basis functions that has the ability to accurately represent the unknown physical quantity while minimizing the com puta tional effort required to employ it. Theoretically, there are many possible basis sets. However, only a lim ited number are used in practice. These sets may be divided into two general classes, the entire domain basis functions and the subdomain basis functions. Using th e subdomain basis functions to represent the current distributions in solving the problem on hand is necessary, in order to properly treat the resistive region. Due to the edge effect, we cannot use piecewise constant (pulse) basis functions in the y direction. Instead, we can combine piecewise linear basis functions in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Table 3.1: Percentage of the impedance matrix filling nx 3 5 7 9 ny 1 3 5 7 n = (nx + ny) 4 8 12 16 Pulse-Tri angle 75 43.75 30.5 23.4 Triangle-Triangle 93.75 73.43 54.8 43.35 x direction with a half triangle at the edges. An alternative is to use piecewise linear basis functions in both x and y directions. Differentiating between the two alternatives, we find th at the number of non-zero elements in the impedance m atrix which utilizes triangles in x and pulses in y to be ( N x N —( N —2)2). For the case of triangles in x and y, the number of elements is ( N x N — ( N — 4)2) — 1. The following table gives us a feeling for the difference between the two choices in terms of the percentage of the impedance m atrix we have to fill. Although th e filling percentage of the impedance m atrix using pulse-triangle com bination is less than the filling percentage of triangle-triangle combination, the nu merical effort required to evaluate the latter is less than the numerical effort required to evaluate th e former (the Fourier transform of pulse-triangle contains sine while the the Fourier transform of triangle-triangle contains sine2). We selected piece-wise linear basis functions to represent the current on the microstrip as shown in Fig. 3.5. The current in the y-direction (transverse) is zero at the edges of the conductor and the current in th e x-direction (longitudinal) is maximum at the edges to satisfy the edge conditions. Five basis functions in the y-direction and seven basis functions in x-direction are found to be sufficient for convergence of the solution. In m atrix notation, the system m atrix can be w ritten as ' Er' _ / J y . " l Z xx Zyx Zxy Zyy . + Rxx 0 0 Ryy \ ‘ Jr ' J J ,. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 Resistive Layer R* 2L (») Conductor 2L (b) Fig. 3.5: Current basis functions for (a) Longitudinal current (b) Transverse current We separate the resistive m atrix from the impedance matrix, and integrate the resistive m atrix in the spatial domain in closed form. We also exploit the block Toeplitz symm etry of the impedance m atrix. This means that only the first two rows and the first two columns of each subm atrix must be calculated. The remaining term s of each subm atrix can be filled by using the terms of the first two rows and first two columns, thus reducing the time needed to calculate the impedance m atrix. The elements of the resistive m atrix can be derived in a closed form in the spatial domain. As an example, r2d 2d / 1 . J 1J/ = JR Jo 3 Ryy( 1,1) = R, Ryy( 1,2) = Rs #yy(l,3) = 0.0 d J x - J 2d l = R sJo 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 R a j “ j 2 . J 2dl = R ^ R *,( l ,l ) = Ra / Ji ■J xdl = *■3 to R * * ( 1,2) = Rs f 2d d j J\ • J2dl = fl>6 to R * x ( 1,3) = 0.0 R x z i 2,2) = R, o 2d It r2d f t . = 10s Ryy{ 2,2) - r -t where 2d is the width of each basis elements. The entire resistive m atrix for the case of five basis functions in the x-direction and three basis functions in the ydirection is given in Appendix D. A similar m atrix can simply be derived for different combinations of basis functions in the x and y directions and for different resistive region widths. 3.2.3 Numerical Considerations in the Evaluation of the Spectral Domain Integra tion The spectral domain integrals required in the evaluation of the impedance m atrix elements are evaluated numerically using sixteen-point Gaussian quadrature. The numerical integration must be carried out to a sufficiently large value of (3 for the integral to converge. As the subdomain size becomes smaller, this upper lim it must increase due to the spectral properties of the Fourier transform of the subdomain basis functions. The numerical integration must be performed with enough intervals to adequately model any rapid changes of the integrand. As the two subdomain basis functions in the integrand become physically farther apart, the integrand becomes more oscillatory requiring more intervals to assure accurate results. The integrand Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 has poles corresponding to the propagation constants of magnetostatic modes. A pole extraction m ethod was used [56], [57], [58]- [60] in the past, which in addition to the numerical integration, requires the calculation of residues and the application of the Cauchy principle at the singularity points. For a multilayer or anisotropic structure, the location of the singularities usually introduces more complexity, if not more difficulty, into the problems. A nice way to avoid this singularity problem is to deform the integration path in a flexible m anner to account for a different range of m agnetostatic modes. The deformed path should be applied to both propagation directions, kx and ky. Fig. 3.6 shows the deformed integration path. Although the deformed integration path can be arbitrarily selected, care must be taken in choosing the proper p ath for numerical integration. The integrand increases exponentially as the distance from the real axis increases; therefore, if the contour is too far from the real axis, numerical problems can occur. Also, if the path is too close to the real axis (where th e singularities are), the integrand is not a smooth function [61]. T he following numerical expressions are used in the course of our simulation, • Maximum integration limit • Number of integration points N { = 1 + [C2A/3{S max\ where C i ,2 — numerical constants determined experimentally lmin = S mcLX = minimum edge length in the discretization maximum dimension of the structure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 A/3,- = w idth of the if ft. region Im (kx or Icy) ■\ Real(kxorky) \ \ / / Fig. 3.6: Integration contour in the complex plane of either kx or k y. 3.2.4 Complex Root Searching The propagation constant is a complex value th at makes the determ inant of the impedance m atrix equal to zero. There is no input or excitation to the problem, i.e. the governing equation is homogeneous. The solution is found when the determ inant of the coefficient m atrix is zero or less than a given small value. This condition can be achieved by an iterative root searching process. If there is no loss in the ferrite layer or the strip, a simple Newton’s method or interval-halving should be sufficient; however, a complex root searching algorithm should be employed to compute the phase velocity and the attenuation for the device. It is found th at M uller’s threepoint method is an efficient root searching algorithm. It can be used for complex roots, and it converges relatively quickly. Usually a convergence can be reached within seven iterations. 3.3 Numerical Results and Conclusion We compared our Green’s function for the single layer structure shown in Fig. 3.2 with the Green’s function derived by Pozar [27] using the boundary condition method. An excellent agreement between these two methods was achieved. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Since the Green’s function of a multi-layer structure which includes a normally bi ased ferrite substrate is not available in the literature, we compared the limiting case of the ferrite with th e Green’s function derived by Aberle for multi-layer dielectric structures [53]. Again excellent agreement was achieved. We constructed a 2-D MoM code for simulating an EG mode isolator with resistive loading as shown in Fig. 3.1. First, we examined the limiting case of ferrite with zero ferrite parameters, which is essentially the dielectric case, and compared our results to the widely published results for dielectric microstrip. The computed current distribution in the longitudinal direction over the conductor is shown in Fig. 3.7. As expected, the current is symmetric. In addition, very good agreement with the dielectric case is obtained for the propagation constant. For the ferrite case, Fig. 3.8 shows the asymmetric longitudinal current distribu tion over the conductor for no surface resistance, and Fig. 3.9 shows the longitudinal current distribution for a surface resistance equal to 100 Q over half of the strip. The phase constants for forward and backward waves are shown in Fig. 3.10. The computed insertion loss and isolation axe given in Fig. 3.11 A preliminary analysis of the three isolator structures indicates th at the best electrical performance is given by the double-layer structure shown in Fig. 3.3. While the performance of the triple-layer structure shown in Fig. 3.4 is not as good as the other two structures, its advantage is that it can be compatible with MMIC. Fig. 3.12 compares the insertion loss of the three isolator structures and Fig. 3.13 compares the isolation of the three structures. For a single layer ferrite isolator, the field ellipticities at the upper and the lower boundaries with air counteract each other. If the air at one of these boundaries is replaced by a dielectric layer, as in Figures 3.3 and 3.4, one of the counteract ing ellipticities is replaced by a co-acting ellipticity which leads to increase in the nonreciprocity and the isolation as well [40]-- [63]. The effect of the dielectric thickness of the ferrite-dielectric structure shown in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Fig. 3.3 is studied, Figures 3.14 and 3.15 give the results. We note th at decreasing the dielectric thickness will increase the isolation and decrease the insertion loss which results in an improvement in the isolator performance. It is worthwhile to note th a t the isolation and the insertion loss do not seem to depend significantly on the dielectric constant of the dielectric layer as shown in Fig. 3.16. T h e normally biased ferrite structure excites the magnetostatic forward volume wave which has the frequency range / & < / < / # where H 0, £irMs, and 7 [33, 34], where h = 7 H0 fa = 7 y/H oiffo+ teM .) axe the applied DC magnetic field, the magnetization of the ferrite, and the gyro-magnetic ratio, respectively. It is shown from Figures 3.17 and 3.18 th at the peaks move forward when we increased the external DC bias, H a, following the limits of the volume wave. The isolation begins after the cut-off lim it of the volume wave, f a - The frequency range for negative fj,ef / is u < 7 H0(H0 + At M 3) < (H 0 + 4irM3) as shown in Fig. 3.19. The upper limit of the isolation is when Heff = 0, which is the second peak in Fig. 3.11 and it occures at frequency 47rMs). 7 (H0 -j- That peaks limit the operating range (the band width) of an isolator has been pointed out and experimentally demonstrated in [1 1 ]. T he optim um resistance of the film is determined from Fig. 3.20. We note th at both th e isolation and the insertion loss are equal when the resistance is zero, which is physically true. When the resistance is around twenty times the characteristic im pedance of the line, we find that the isolation returns to its value when th e resis tance is zero. A possible explanation is th at the high resistance acts like an open circuit and the backward wave will pass through the resistance free region. T he inser tion loss is not a strong function of the resistance since the forward wave propagates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 m ainly in the resistance free region. T he optim um width of the resistive film may be deduced from Figures 3.21 and 3.22. Increasing the width of the resistive film increases both insertion loss and isolation. When we covere the entire conductor with a resistive film the isolation and insertion loss become equal and the nonreciprocity vanishes. Finally, we compared our results for the insertion loss and the isolation for the 2-D structure shown in Fig. 3.1 with the experimental results published in [11] for the structure shown in Fig. 3.23. Fair agreement is clear from Figures 3.24 and 3.25. The difference between the numerical and the experimental structures is the difference between Fig. 3.1 and Fig. 3.23. In Fig. 3.1 we assumed m atched ports while in Fig. 3.23 matched ports are assumed only at the center frequency. Thus, the best agreement with the experimental results is in the middle of the frequency range. In addition, in our analysis, the external bias field is assumed to be exactly perpendicular to the substrate which is not necessarily true in the practical case. This problem suggests it is essential to have a flexible tool to be able to handle ar bitrary biased ferrite slab, the topic of the next chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 1.0 D ttkctricC j 0.96 uh 0.92 0.9 0.88 g 0.86 Number of the Basis Function on the Conductor Fig. 3.7: Symmetric current distribution over dielectric microstrip. 1.0 — Forward Wava, RasO.O Ohm* Backward Wave* RasO.O Ohm* .5 0.8 0.6 O 0.4 0.0 Number of the Basis Function on the Conductor Fig. 3.8: Asymmetric current over ferrite microstrip in forward and backward directions for R s = 0.0ft. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 1.0 Forward Wat*, Ra*100 Oho* —— Backward Watt, Ras 100 Ohna •S 0.8 0.6 O 0.4 0.0 Number of the Basis Function on the Conductor Fig. 3.9: Asymmetric current over ferrite microstrip in forward and backward directions for R s = 100.Oft. 14.00 Forward Wave ----Backward Wave ----12.00 10.00 o * o O I 8.00 6.00 4.00 2.00 0.00 2J0Q 3.00 4.00 5.00 6.00 7.00 Frequency GHz 9.00 11.00 Fig. 3.10: The phase constants of forward and backward waves (d = 7.62 x 10-4 m, €/ = 12.0, 4t M s = 1750.0G, H dc = 800.00e, A H = 80.0Oe, R s = lOO.Ofi, W = 1.016 x 10_2m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.00 Insertion Loss Isolation -5.00 -15.00 a■a a© So - 20.00 -25.00 -35.00 -40.00 too 3.00 4.00 6.00 7.00 Frequency GHz 9.00 11.00 Fig. 3.11: Computed isolation and insertion loss (d = 7.62 x 10-4 m, e/ = 12.0, 4 irM3 = 1750.0G, H dc = 800.00e, A H = 80.00e, R , = 100.0ft, W = 1.016 x 10"2m). 0.00 •5.00 a - 10.00 J o ■g 8 5 -15.00 - 20.00 -25.00 200 3.00 4.00 6.00 7.00 Frequency GHz 9.00 11.00 Fig. 3.12: Comparison of the insertion loss for three isolator structures (4ttM3 = 1750.0G, Hdc — 8OO.O0e, A H = 8O.O0e, R3 = 100.0ft, W = 1.016 x 10-2 m. For the single-layer: dj = 7.62 x 10-4 m, c/ = 12.0. For the double-layer: dd = 2.62 x 10-4 m, ed = 3.0. For the triple-layer: dd = 4.0 x 10_4m, ed = 8.9 da = 5.0 x 10~3m, ea = 1.0). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 0.00 Single-layer ----- - 10.00 - 20.00 Double-layer ----Triple-layer----- w -30.00 om •50.00 •60.00 100 3.00 6.00 4.00 7.00 Frequency GHz 11.00 9.00 Fig. 3.13: Comparison of the isolation for three isolator structures. The same parameters are as in Fig. 3.12. i 1 r Dd=0.381d-3 m Dd=0.635d-3m Dd=1.143d-3 m oj-2.4 5.4 7.2 9.0 10.8 Frequency GHz Fig. 3.14: The effect of the dielectric thickness on the insertion loss (dj = 7.62 x 10-4 m, ef = 12.0, 4wMa = 2100.0G, Hdc = 700.00e, A H = 80.0Oe, R s = lOO.Ofi, W = 1.016 x 10-2 m, ed = 30.0). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 Dd=0 J8 1 d -3 m Dd=0.635d-3 m Dd=1.143d-3 m 5.4 7.2 Frequency GHz Fig. 3.15: The effect of the dielectric thickness on the isolation (df = 7.62 x 10-4 m, ej = 12.0, 471-Jlf, = 2100.0G, H dc = 700.0Oe, A H = 80.00e, R s = 100.0(2, W = 1.016 x 10~2m, ed = 30.0). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 0.0 i i----------------1 i---------------- 1----------------1 i---------------- 1— Insertion LossIsolation -4.2 PQ T3 - -8.4 12.6 -16.8 e' I _____ !_____ i_____ I_____ I_____ I_____ I_____ !_____ I_____ L 9.6 19.2 28.8 38.4 48.0 Dielectric Constant Fig. 3.16: The effect of the dielectric constant of the dielectric layer on the isolation and the insertion loss, (df = 7.62 x 10~4m, ej = 12.0, 4 t M s = 2100.0G, H dc = 700.00e, A H = 8O.O0e, R a = lOO.Ofi, W = 1.016 x 10"2m, D d = 0.381 x 10~3m, Freq.= 6.0 GHz). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 Ho=700 Oe ----- Ho=800 Oe Ho=900 Oe -19 -a S-38 -76 Ha 3.6 5.4 7.2 9.0 10.8 Frequency GHz Fig. 3.17: The effect of the external DC bias on the isolation, (d = 7.62 x 10_4m, ej = 12.0, 4ttM 3 = 2100.0G, A H = 80.00e, R s = 100.0D, W = 1.016 x 10“ 2m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 o.o Ho=700 Oe Ho=800 Oe Ho=900 Oe -3.6 n CO § -7.2 C o "S -10.8 0) 09 ai i— -14.4 -18.0 3.6 5.4 7.2 9.0 10.8 Frequency GHz Fig. 3.18: The effect of the external DC bias on the insertion loss, (d = 7.62 x 10-4 m, ef = 12.0, 4ttM 3 = 2100.0G, A H = 80.0Oe, R a = 100.0ft, W = 1.016 x 10"2m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 8 ^eff 4 0 -4 8 2 3 4 5 6 7 8 9 10 11 Frequency GHz Fig. 3.19: The frequency behavior of Me//> (4 tMs = 2100.0G, Htc = 800.00e). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 Insertion Loss Isolation 0.0 -5.4 -16.2 - 21.6 0 40 80 120 160 200 Ohms Fig. 3.20: The effect of the film resistance on the insertion loss and the isolation, (d = 7.62 x 10-4m, e/ = 12.0, 4ttM3 = 1750.0G, Hdc = 800.0Ge, AH = 80.Oe, Freq.= 5.0 GHz, W = 1.016 x 10-2m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 25 Rs width =0.25 w Rs width =0.50 w Rs width =0.75 w Rs width=1.00 w 0 2-25 _ J _________ I_________ I_________ 1_________ !_________ 1_________ 1_________ i_________ !_ 3.6 5.4 7.2 9.0 10.8 Frequency GHz Fig. 3.21: The effect of the resistive film width on the insertion loss, (d = 7.62 x 10-4 m, €/ = 12.0, 4x114, = 2000.0G, Hdc = 700.00e, A H = 80.00e, R s = 100.0Q, W = 1.016 x 10-2m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 — Rs width =0.25 w~ Rs width =0.50 w Rs width =0.75 w Rs width=1.00 w -21 C -42 -84 3.6 5.4 7.2 9.0 10.8 Frequency GHz Fig. 3.22: The effect of the resistive film width on the isolation, (d = 7.62 x 10-4m, cj — 12.0, 4xMa = 2000.0G, Hdc = 700.00e, AE = 80.00e, Ra = 100.0ft, W = 1.016 x 10-2 m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.23: 3-D edge-guided isolator with resistive film loading. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 Numerical Experimental" -14 -28 2-42 -56 3.6 7.2 Frequency GHz 5.4 9.0 10.8 Fig. 3.24: Comparison between the numerical and experimental insertion loss (d = 7.62 x 10-4 m, ef = 12.0, 4ttM5 = 2100.0G, H dc = 700.00e, A H = 80.00e, R 3 = lOO.Ofi, W = 1.016 x 10"4m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 0.00 -5.00 -10.00 -15.00 § § -20.00 I -25.00 a -30.00 -35.00 -40.00 -45.00 240 3.00 4.00 540 6.00 7.00 Frequency GHz 9.00 U.00 Fig. 3.25: Comparison, between the numerical and experimental isolation (d = 7.62 x 10-4 m, e/ = 12.0, At M s = 2100.0G, Hdc = 700.0Oe, A H = 80.0Oe. R s = lOO.Ofi, W = 1.016 x 10~4m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 CH APTER 4 ANALYSIS OF PHASE SHIFTERS AND TRANSDUCERS USING A GENERAL GREEN’S FUNCTION 4.1 Introduction A rigorous analysis of multilayer planar ferrite structures can best be carried out using the spectral domain method of moments. To formulate the integral equation, the Green’s function for the structure is needed. Over the past few years, several techniques have been employed to formulate the Green’s function for a single ferrite slab with a given bias direction [43], [27], [64] and [65]. In addition, Green’s func tions for multilayer structures comprising both ferrite and dielectric layers have been investigated [66], [67], [68], [69] [70] and [71]. In this paper, we present a Green’s function which overcomes the lim itations of previously presented Green’s functions and allows us to treat structures which include any number of dielectric and ferrite layers (possibly with different bias directions). Previous spectral domain analyses for anisotropic materials have assumed that the bias field is along one of the principal axes of the ferrite slab. However, in many practical applications, the bias axis may not be perfectly aligned along the principle axis due to misalignment of the bias magnet. This deviation in the bias angle may alter th e electromagnetic characteristics of the structure. Using the derived Green’s function, we can study the effect of bias angle deviation on circuit performance. 4.2 Transmission M atrix for an Arbitrarily-Biased Ferrite Slab In this section, the transmission m atrix is formulated in the spectral domain for a m agnetic substrate of thickness d as shown in Fig. 4.1. The transmission m atrix T is a 4 x 4 m atrix written as [40] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where T e , Z t , Y t , T are 2 x 2 submatrices of T, j denotes the spatial Fourier transform defined as E ( kx, k y) = r f°° E ( x , y ) e - jk**e-jkyydkxdky (4.2) J — OO «/—OO E i , E 2 axe the tangential electric fields at the boundaries of the layers and J 1 and J 2 axe the tangential surface currents defined by «/,- = z x H n where H n is the tangential magnetic field at th e n th surface of the layer. z A z= 0 h e 2 z ' ' y d Fig. 4.1: Geometry of single layer structure. To find the transmission m atrix for a ferrite slab, we start from Maxwell’s equa tions —j u p H = V x E V - ( [ J - ■H ) = 0 (4.3) jueE = V x. H eV • E = 0 where ^ is the permeability tensor of the ferrite. For an arbitrary magnetized ferrite slab, the permeability tensor is given by [72] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Mil M= M21 _ M31 Ml2 Ml3 M22 M23 M32 M33 where Mu = M+ (Mo —y)smOcas <f> Ml2 = (Mo M13 = Mo —M — - — sin2 0 cos (p —jKsmd sin <p /X21 = Mo —M • 2 — - — sin 0 sin 2 <?i —jk cos 0 M22 = M+ (Mo —y)sm 2 9sm2<p M23 = Mo —M — „— sin20 sin <p + j k sin 6 cos <p /131 = Mo —M — - — sin20 cos <p + jk sin 9 sin <p /132 = Mo —M — - — sin 20 sin <f>— jk sin 0 cos <p M33 = ^^sin 2gsin 2 ^ -(- j/c cos 0 Mo ~ (Mo ~ M)sin2^ / 1 . u o w7n \ M= ( l + -T2----- 7 ) U/o - w ‘ K= -2 w0 - U20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.4) 68 7 Um = U= U)0 + —, U0 = ~fH0 (jJq is the precession frequency, Ho is a z-directed impressed DC magnetic field, the gyromagnetic ratio, T = 7 is is the relaxation tim e, and AH is the 3-dB line width. From Maxwell’s equations, we can write Vx(/i -V x E) — k%erE = 0 (4 .5 ) where k0 = Uy/eo. M anipulation of (4.5) yields three scalar equations { j K y d” ( J^y d f*5 q z 2 ^ ^ E y ~Qz d2 {J.gKyKx 4- fl4 Q^ 2 d* d ~ ) Ey 4 - d d ~ (- f i 7 K 2 4- fi&KyKx - jii 4 K y — + jfisK x ^ j ) E z= 0 d ( - j f i SK x- ^ - fl9 K XKy - d ^m i K x ~dz + d2 ~ d2 d ~ - j ^ K y - ^ ) Ex + d +J/i3KxdZ ~ Ey + d d (,(i7 K xK y - nsl<l + JUiK y-^ - j/jl2 K x — ) E z= 0 r\ O ^m s K x d z + H K x K y ~ ~ Ex + — f*6 K 2 + JUlKy— + flZK x Ky) Ey + ( - f i 4 K y K x + fi 5 I<l + yLXK j2 - jjL2 K xKy - kleT) E z= 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 Eliminating E z from the three equations and rearranging we obtain d 2E x dEx ~ d 2E v dEv ~ w i - q^ T + W2~&T + Wz x +W4~ d z ^ + W5~ d f + We y= d 2 Ex dEx W7~q~T + Wa~ dT + ~ 109 d2E v dEv +Wl01 h 2’ + Wll~ d T + Wl2 Wi = fi5Cl + K 2 (fi 2 fi4 - MiMs) w 2 = jK y in e P i - M7 M2 ~ M4M3 + MsMi) + , v ^4’6^ 0 ^4'7^ E y~ 0 j K xK l { - n e f i 2 + M7Hs - MsM4 + HsHz) - jCiKy{fi6 + Ms) wz = K l K : ( f j 9fi5 ~ PaVe) + K xK 2(fi8fi3 - ( j , 9fi4 - [ i 9fi 2 - H7iie) (M9Mi — M7 M3 ) ~ C i K 2 (fi9 + mi) + C \ K xK y{y. 4 + M2 ) — P2 C 1 K w4 = - K xK y(fi4 (i2 - msMi) - M4 C 1 w5 = ~ j K l K y ( f l 7 f l s + M5 M3 ~ M8M4 — M6M2) 2 —j K x K 2 { —f i 7f i 2 — M4M3 + M sM i + M6M1) + J f r C i K y + j f i e C i K x w6 = - K 2 K l ( / i s f i z + M7 Me - M9M2 - M9M4) - K x K y ( f i 9fx 1 - y. 7 li z ) —K ^ K y ( —fisfi 6 + M9 M5 ) + fi9 C \ K XKy IV7 = M 2C 1 + K XK y { f l 2 fi 4 - M1M5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. + C\ 70 ws = ]K xK * ( - i i 7fi2 + MsMi + MiMe - M 3 M4 ) + jK lK y { n 7fi5 - H 2 M6 ~ M s M 4 + M sM s) = —W6 Wg ww = - K l ( n 2 H4 ^11 = - j K l K y(flIfle - fi7f*2 - f*4ll3 + HaHl) - - M1 M5 ) - P\C\ j K x i - f * 8M4 - M2 M6 + M7M5 + M5M3 ) + 3CiKx {fiz + fl7) W12 = ~ K y K x ( —/J.7fl3 + M9Ml) —K x K y(—fJ.gfl4— flgfi2 + M8M3 + M7M6) + -^r(M8M6 —^ 9 /^5 ) + ClKl(fig + ^ 5 ) —C \ K xK y(n4 + fl2) + fl\CiKy — C\ Cl = UJ2Cgtr Ml = (M22M33 — M 2 3 M 3 2 ) / A / 1 M2 = M3 = (M12M23 — M l 3 M 2 2 ) / A / l M4 = ( ~ M21M33 + M 2 3 M 3 l ) / A / l Ms = (M 11 M33 — M i 3 M 3 i ) / A m M6 = ( — M llM 23 + M i 3 M 2 i ) / A m M7 = (M21M32 — M 2 2 M 3 l ) / A M ( —M12M33 + M l 3 M 3 2 ) / A / Z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -jfisC iK 71 A fl8 = (—/*11^32 + Ml2/*3l)/A^ P9 = (l*11P22 — /*12/*2l)/AM f l = f l l l f l 2 2 [ i 3 3 ~ f & l l 1 * 2 3 ^ 3 2 ~ f c l l t U f t S S + f * 2 l f i l 3 f 1 3 2 + ^ 3 1 ^ 1 2 ^ 2 3 ~ H 3 1 ^ 1 3 ^ 2 2 Rewrite equations (4.6) and (4.7) in the following form w i D 2 E x + \V2 D E x + w 3 Ex + wliD 2 E y + W 5 D E y + w 6Ey = 0 (4.8) W7 D 2 E x + w gDEx + wqEx + w\qD 2 Ey + w \ \ D E y + w\ 2 E y = 0 (4-9) or (w \D 2 + W2 D + 11)3 ) E x + (w\D 2 -f-11)5 D + wo)Ey = ( 1V7 D 2 + w$D + iv^)Ex + (w\qD2 + W\\D + w\ 2 )Ey 0 (4.10) — 0 (4.11) where D 2 = Jp- and D = J j. Equating the operational determ inant of the coupled equations (4.11) and (4.11) to zero, we obtain the following fourth order equation C 01D 4 + C02 D 3 + C03 D 2 + C04 D + Cos = 0 where C 01 = wiW\q —UJ7UJ4 C02 Wi Wu + W2 Wiq — WjWs — w8 w 4 = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.12) 72 C q3 = W1W12 + W 2W n — W 7W 6 — W sW s — w s w 4 C04 = W Co5 = W 3 W i 2 — W qW q 2 W 12 + W3Wn — W 8W6 — W 9W 5 By investigating equation (4.12), we found th at solving for the four roots analytically is quite difficult due to the fact th at this equation contains the full coefficients. At this step and knowing the five coefficients, we find the four roots numerically. As a check, the four roots have to satisfy the following relations R iR 2R3R 4 = R \R 2R3 + R1R1R4 + R \R 3Rj4 -f- R 2R 3R 4 = R i R 2 + R \R 3 + R1R4 + R 2R z + R 2R4 + R3R4 = i?i + R\ + R\ + R\ = C04 Cqi Cp3 Cqi Cq2 The solution of the coupled partial differential equations (4.6) and (4.7) is given by Ex = Crie x p (i? i 2 ) + C2 exp(R 2 z) + C3 exp(R 3 z) + C ^ e x p ^ z ) (4.13) Ey = Di exp(i?iz) + D2 exp(iZ2-j) + D3 exp(R 3 z ) + D4 exp(R 4 z) (4.14) At this point we appear to have eight arbitrary constants in the solutions (4.13) and (4.14). But it follows from [73] that the general solution of a system of two second order equations involves only four arbitrary constants. That is also clear from the order of the operational determ inant, which is four. There must be some relation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 between the eight constants. We can discover it by substituting the solutions in (4.13) and (4.14) into either of the original equations in (4.6) and (4.7). On substituting in the first equation, we get Dx = N XCX X?2 = N 2 C2 D3 = N3C3 D4 — N4C4 where N _ jy _ 2 _ WXR% + W2 R 2 + W3 W4 RI + W3 R 2 + w6 w xR l + w2R 3 + w3 W4 RI + W5 R 3 + w6 3 _ 4 wxR\ + w2R x + w3 w 4R x + w5R x + w6 wxR\ + W2 R 4 + w3 W4R I + W5R4 + we Again, the solution in (4.13) and (4.14) can be written as Ex = Cx exp(Rxz) + C2 exp(R 2 z) + C3 exp(i?3z) + C4 exp(R^z) (4.15) E y = N XCXexp{Rxz) + N 2 C 2 exp(R 2 z) + N 3 C3 exp(R 3 z) + N 4 C4 exp(R 4 z) (4.16) From Maxwell’s equations, we can write the relationship between the components of the magnetic field and the electric field in matrix form as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 Hx Hy Hz uj2eAh h u h\2 h 21 h.22 h 23 ^31 ^32 ^33 (4.17) u e ( K yE x — K x E y) where ^11 — f^33-HxK y + UJ2t(fl33lil2 — ^ 13 ^ 3 2 ) hi 2 — —fl3 3 K^ + U2e(fl23fl22 ~ ^23/^32) /ll3 — —K x (lliz Kx + fl 23Ky) + u 2e(fli3fj,22 — ^ 23/^32) ^21 — (^33-Hy — W2e([lufl33 ~ ^31^13) ^22 — P33KXK y + U)2e(ll33fi21 ~ H23f*3l) h 23 — —K y(fli 3 K x + H23Ky) ~ W2e(/il3/*21 —/*ll/*23) ^31 — —Ky(fi32Ky + fi3i K x ) + u)2e(finfi32 —^ 31^ 12) ^32 — E x (fi 32K y + fi 3l K x ) — U)2e(fJ,2X^32 ~ (*31^22) ^33 — ^i(/^12-^y + /^ll^Cr) + ^y(/^21^Cr + /i22-^y) + w2^(^21^12 ~/^ll/^22) Ah — K x Ky(fl33fli2 — ^13^32 + /*2l/*33 ~ ^3X^23) +^y(/*33/*22 ~ ^23^32) + K 2(llfl33 — fi3Xf*X3) +o;2e(^u (//23M32 ~ ^ 33/^22) + ^ 12(^ 21/^33 — V3 1 H2 3 ) + ^ 13(^ 22/^31 —H21 H3 2 )) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Appling (4.17) to the solutions (4.15) and (4.16) and puting z — 0, we can evaluate --- 1 1 the constants C \, C 2 , C 3 and C4 in m atrix form as follows 1 1 ' \ c x] 1 1 Nx n 2 iV3 N 4 c2 Dx d 2 Dz D 4 C3 . D s De d 7 Ds . lc<. ' Eyi HX1 . (4.18) The inverse of equation (4.18) can be written as E Xl c 2 E y i II o ' f— < 1 1 C X 1 c 3 . C 4 (4.19) K . . K . where r Mxx Mx 2 A/ 14 ‘ M 2l M 22 M 23 A/24 M 31 M 32 A/33 A/34 M 42 A/43 M 1 A m x 3 a x --- 1 £ . M and D\ = <jJ€{jh.nRi + j h i z R i N i + h i3 ( K y — N i K x)) D2 — ide(jhuR2 + D3 = u e ( ]h u R 3 + J^i27?3^3 + h 1 3 ( K y — N 3 K X)) D4 = w e(j^ n i ?4 + jh w R i N ^ + hi3(Ky — N^ K x)) D5 = w€.(]h2iR\ + ]h-22 R \N \ + h,23(Ky —N \ K X)) Dq = a;e(j/i2ii?2 + ^ ^ 22 ^ 2 ^ 2 + ^23(7vy — iV2/ f x)) + hi3(Ky — N 2 K 2;)) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 D? = u e { j h 2\ R z + ]h 2i R z N z + h,2z ( K y — N z K x )) Ds = ue(]fi2iR4 + jh22R-AN^ + h.2a{Ky — N ^ x)) and — N2^D zD s — D4D7) + N z^DqD^ — 7 ) 2 75s ) + -/V4 ( 7 ) 2 D7 — Mu M 12 = 7)3( 75$ — 7)8) + 7) 2 ( 7 )8 —D 7) + 7) 4(757 — 75$) •A7i 3 = ^ 2 ( 7 ) 7 — Ds) + Nz{Ds — Ds) + N ^ D s — D7) Mi 4 —( ^ 2 ( 7)3 —7) 4 ) + ^ 3 ( 7)4 —7) 2 ) + -^4 (7)2 — 7) 3 )) = M 21 M 22 D qD^) = - ( ^ ( 7 ) 3 7 ) 8 - 7 ) 47 ) 7 ) + iV3( 7 ) 5 7 ) 4 - 7 ) 17)8) + ^ 4 ( 7 ) 47 ) 7 - 7 )5 7 ) 3 )) = —(7)i(7 ) 8 —7 ) 7 ) + 7)3(755 —7)8) -F 7) 4(757 — 7) 5 )) M 23 = —( N i ( D 7 —7)8) + TV3 (7)s —7 ) 5 ) + N^(Ds — D 7)) M 24. = Ni(Ds — 7 )4 ) + ^ 3 ( 7)4 —7)i) + ^ 4 ( 7)4 —7) 3 ) M 31 = TVX( 7 ) 2 7)8 — D 4De) + ^ { D s D ^ — DiD s) + N ^ (D i Ds — D5D2) M 32 = D i ( D g —75$) + 7 ) 2 (7 5 5 —7)8) + 7 )4 ( 7 5 $ — 7 ) 5 ) -AT33 = Vl(7)$ ■M34 = 7)8) + TV2 (7)8 —7 ) 5 ) + ^ 4 ( 7 ) 5 —75$) —(Vx(7)2 —7) 4 ) + TV2 (7)4 —7)i) + iV4(Di — 7) 2 )) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 M 44 = —(Ni(D2D7 — D3Dq) + N2{D$D3 — D \D t) -+• N^^DiDs — D 5 D 2 )) M 42 = Di(Dq — D 7 ) + ^ M 43 = N i ( D 7 — D&) + N 2 {Ds — D 7 ) + Ns(Ds — D 5 ) M 44 = —{N \( D 3 — D 2 ) + - ^ (D i — D3) + Nz(D 2 — Di)) Ajvr = N \ ( D 2(Ds — D 7 ) + D3(Ds — Ds) + D 4 (D 7 — 7^6)) 2 (^ 7 —D 5 ) -f- D3{Ds — D&) +iV2(Di(£>7 — 7}g) + D z( D s — D$) + D 4(D$ — £?7)) +jV3(£ M £>8 — -^6) + D 3{Dg — Ds) + D 4(Dq — £?s)) +7V4(£)i(£)8 — £^7) + £>2(£>r — £^5) + D 3(D$ — Ds)) Following the same procedure for z = —d, we get an expression relating the field components at the second surface of the slab to the arbitrary constants th a t appear in the solution. ' ' Eyi HX2 .H y i . rcii c2 = M (-d) c3 .CA. (4.20) where exp (—Rid) Ni e x p ( - R i d ) M {-d) = D\ exp( Rid) . £ )5 exp(—Rid) exp(—R 2d) exp (—R 3d) exp(R4d) N 2e x p ( - R 2d) N 3 exp(—R 3d) N 4exp(—R 4d) D2 exp(—R 2d) D3 e x p ( - R 3d) D4e x p ( - R 4d) £>6 e x p ( - R 2d) D 7e x p ( - R 3d) D4 exp(—R 4d) Convert the m agnetic fields to the electric currents at both surfaces of the slab using the following relations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 ’ E Xl ' E X1 E yi H Xl . K Eyi = [5] . 4 . (4.21) Jx l . E X2 Ey 2 H X2 .Hy> Eyi = [5] . . 4 4 (4.22) . where i ■■ o [5] = o 0 0 1 0 0 0 ' l 0 0 0 0 1 1 0 (4.23) Using equations (4.19), (4.20), (4.21) and (4.22) we can write the final expression th at relate the electric field and current components at both sides of the ferrite slab is given by E: E 3/1 I E *2 : i I X2 I 3/2 = [5 ]- 1 [M (-d)][M (0)]- 1 [5] J L 4 (4.24) j From equation (4.24) we conclude the transmission m atrix for arbitrary biased ferrite slab f ( d ) = [ 5 ] - 1[M (-d)][M (0)]" 1 [5] where the elements of the transmission m atrix are given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.25) 79 Tn = T ^ = M u e x p ( -R id ) + M 2i exp( - R 2d) + A/31 exp(—R3d) + iV/41 exp(—iLjc?) /\2 = f f 2 = Afi2 exp( — + A/22 exp(—R 2d) + A/32 exp(—R3d) + A/42 exp(—/?4 (f) r 13 = Z ^ = —M u exp (—Rid) —A/ 24 exp(—/?2</) — A/34 exp(—i?3 </) —A/44 exp(—/?4 cZ) /\4 = = A/13 exp(—/?!</) + A/ 23 exp(—R 2 d) + A/33 exp(—/?3 rf) + A/43 exp(—i?4 </) /2 1 = T2l = N i M u exp {—Rid) + JV2A/21 exp(—R 2d) + NzMzi exp(—R 3d) + A 4 M 41 exp(—R Ad) T 22 = T22 = iViAfi2 exp(—i?ic/) + A 2A/22 exp(—/?2</) + A/3 A/ 32 exp(—i?3 c/) + N 4 M 42 exp(—R 4 d) T23 = Z 21 = —N i M u exp(—Rid) — N 2M 24 exp(—R 2 d) — N 3 M 34 exp(—R 3 d) — N 4 M 44 exp(—R 4 d) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 f 24 = Z22 = N i M i 3 e x p ( - R i d ) — N 2 M 23 exp(—R 2 d) — N 3 A/33 exp( - R 3 d) —N 4 M 43 e x p ( - R 4 d) T31 = = —D 5 M u exp {—Rid) —DqM2i exp (—R 2 d) — D 7 M 31 exp(—R 3 d) — D 8 M 4i exp(—R 4 d) Z32 = ?X2 = —D 5 M l 2 exp(—R i d ) —D 6 M 2 2 e x p ( - R 2 d) - D 7 M 3 2 exp(—R 3 d) — DSM 42 exp(—R 4 d) T33 = T/x = D 5 M 14 exp(—/?xcf) + D 6 M 24 exp(—R 2 d) + D 7 M 3 4 exp(—R 3 d) + D 8 M 4 4 exp(—R 4 d) Tm = f (2 — - D 5 M 13 e x p ( - R i d ) - D 6 M 23 e x p ( - R 2 d) — D rM^e xjpi—Rsd) — D 8 M 43 exp(—R 4 d) T41 = F2T = A M u exp(—i?x<f) + D 2 M 2i e x p ( ~ R 2 d) + D 3 M 3i exp(—R 3 d) + D 4 M 41 exp(—R 4 d) r 42 = = D \ M x2 e x p ( - R i d ) + D 2 M 22 exp(—R 2d) + D 3 M 32 exp(—R 3 d) + D 4 M 42 exp(—i?4t/) T43 = f/x = —D 5Mx4exp(-i?x<f) — D 6M24ex p (-/? 2 ^ )— Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 6 xp(—R^d) —D 8 M 44 exp(— T 44 = T22 — D\M \z exp (~ R \ d) + Di M 23 exp (— + D 3 M 33 exp(—i ?3 J) + D 4 M 4 3 exp(—iltd) Transmission matrices have the following properties f(o) = 7 T ( a + b) = f ( a ) f ( 6) (4.26) f (-d) = f (d) where I is the identity m atrix. These properties make transmission m atrices ex tremely convenient for deriving Green’s functions for multilayered geometries. 4.3 Green’s Function Formulation Using the transmission m atrix, Green’s functions can be formulated in the spectral domain for single and multi-layer structures. The Green’s function relates the tan gential electric field on one surface to the surface currents on the same or another surface. This relation has the form, E 3( k x , k y ) = G (kx , k y ) J s(kr ,ky) (4.27) where G = Gx Gyx Gyy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.2S) 82 Details of the Green’s function formulation is given in chapter 2. The Green’s func tion for the grounded ferrite slab as shown in Fig. 2.1 is given by = m s G = T =J = t-i = (Tf Zf = -1 +Ga T 1 (4.29) =J where Z j and T f are the elements of the derived ferrite transmission m atrix and Ga is a semispace Green’s function, which is calculated by taking the lim it of the dielectric Green’s function when the distance dj, goes to infinity and the dielectric constant goes to unity. The Green’s function for the multilayer structure shown in Fig. 2.3 is given by = =J =T~l - r - 1 G = (T UZU + T i Z i )-' (4.30) Following the procedures outlined in [65], the Green’s function for any general multilayer structure can be easily constructed. 4.4 Varying the Magnetization Angle Most researchers assume that the field bias is along one of the cartesian principle axes of the ferrite sample. These bias directions yield permeability tensors th at contain four zero elements out of nine, which simplifies the analysis. However, in practical devices, the DC magnet used to bias the ferrite slab may not be exactly aligned along the required axes. This situation is depicted in Fig. 4.2. This variation in the magnetization angle can change the electrical characteristics of the ferrite device. In addition to treating unintentional deviation of the bias angle from the principle axis of the ferrite sample, our approach allows us to discover interesting behaviors th at may occur when th e ferrite sample is magnetized along oblique angles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 1 Ho (a) (b) Fig. 4.2: Magnetization angle (a) an approximate picture (b) a real picture. 4.5 Planar Phase Shifters 4.5.1 Introduction Ferrimagnetic materials have found wide application in a class of microwave com ponents called phase shifters. Phase shifters are two-port devices that perm it the passage of a guided wave with very little attenuation but with a variable phase delay controlled by the external bias field of the ferrite. Phase shifters generally operate in two states, denoted collectively by M , and M = ± M r, where M r is the rem nant m agnetization. The difference between the phase shifts which occur in these two states is called the differential phase shift. To reduce power requirements associated w ith m aintaining an external magnetic field, ferrite devices are commonly operated in a rem nant state. When the ferrite is initially demagnetized and the bias field is off, both M and H 0 are zero. An external magnetic field in the desired direction is applied long enough to magnetize the ferrite to near-saturation. When this external field is removed, the ferrite magnetization drops to a remnant value, and the ferrite remains magnetized in that direction until an external field is reapplied. Fig. 4.3 shows a typical hysteresis curve. This figure illustrates the variation in m agnetiza tion, Af, with bias field, H0. The two most important figures of m erit used to describe ferrite phase shifters Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 axe related to the total differential phase shift, A <£, and the bandwidth, A /. The first figure of m erit, F , can be defined in terms of the total differential shift per unit length by [74] F = A <j>(deg/cm). (4.31) The bandw idth of a phase shifter, A /, is defined to be the range of operating fre quencies over which the insertion loss is less than some specified value (typically 3 dB). T he bandwidth, when expressed as a percentage, is found from A/ and the center frequency, f a, using B W = ^ r - x 100 (4.32) Jo 4.5.2 A. Slot Line Phase Shifters Introduction The direction of magnetization of the ferrite substrate for a phase shifter is given by 9 = 90° and <f>= 90°. The case analyzed here is shown in Fig. 4.4. This structure features a single-layer transversely magnetized ferrite slot line. B. Full-Wave Analysis of the Slot Line Phase Shifter The propagation constant of an infinitely long slot line structure is found using a full-wave spectral domain analysis based on Galerkin’s method of moments [75] [74]. This m ethod uses the fact th at the tangential electric field on the conducting plane and the surface current in the slot are zero. Following the approach of [40], the unknown electric field components in the slot, E x(y) and E y(y), are expanded in the slot using entire domain basis functions as Ny Fy(y) = (4.33) 71=0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 Ms Mr Ho -Mr -Ms Fig. 4.3: A hysteresis curve for a ferrite sample. a W , ♦ ■ . s\ ss-’\sss Ferrite «.% df PF.C Fig. 4.4: Cross-section of basic slot line single-layer ferrite planar phase shifter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 M y) = £ ‘W - n=0 The expansion functions fyn and f m are given by /„ = (- (4.34) /= . = ( - 1 ) “£ « § ? ) y ^ l - ( ^ ) 2 (4.35) where W is the width of th e slot and the functions, Tn and Un, are Chebychev polynomials of the first and second kinds, respectively. We select n to be odd or even according to the physical behavior of the electric fields in the slot.Thus, only the even values of n are used for f yn and only the odd valuesare used for f m . Ey(y) o -W/2 W/2 Fig. 4.5: n = 0 Chebychev basis function of the electric field in the slot in transverse directions. Applying Galerkin’s approach yields an adm ittance m atrix of the form Y= J- x x x xy Y *yx Y x yy with elements given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.36) 87 Ex(y) 0 -W/2 W/2 Fig. 4.6: n = 1 Chebychev basis function of the electric field in the slot in longitudinal directions. Y ™= 1. £ ^ k ti) G „ ( - l 3 , k ^ ) F ^ ( k yi) (4.37) {=—oo,even The indices (p,q) refer to the subm atrix of Y , and the indices (l,m) refer to the testing and expansion functions, respectively [74]. The functions Fpi(ky) are the Fourier transforms of the functions in (4.34) and (4.35), and they are available in closed form as Fyn = 3n^ J n ( ^ Y ~ ) (4.38) - ^ ( » (4.39, + For enclosed structures like the one on hand, ky takes the discrete values ky{ = Y, where i is an even integer [43] and a is the spacing between the conducting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 sidewalls. The propagation constant /3 is the value of kx th at makes the determ inant of th e adm ittance m atrix zero. Reversing the magnetic bias, we get the propagation constant in th e reverse direction. The difference between the propagation constants of the forwaxd and reverse waves is used to find the nonreciprocal phase shift per unit length, given by A <£ = £ / - 0r (4.40) We can control the total differential phase shift by adjusting the length of the phase shifter. 4.5.3 A. Microstrip Phase Shifters Introduction Microstrip phase shifters, shown in Fig. 4.7 for a single ferrite layer structure and in Fig. 4.8 for a double ferrite layer structure, axe compatible with coaxial connectors. Dual ferrite oppositely-magnetized layers have been found to increase nonreciprocity [74]. Thus, the nonreciprocal phase shift of the structure shown in Fig. 4.8 is ex pected to be higher than that of the structure shown in Fig. 4.7. In practical cases, a thin dielectric layer can be inserted between the two ferrite layers shown in Fig. 4.8 to prevent m agnetic leakage from one ferrite layer to the other. In addition, the di electric layer param eters give us some control over the bandwidth and nonreciprocity of the structure. The two strips of Fig. 4.8 can be excited independently. Thus, two mode distri butions axe possible, the even and the odd modes. However, in this paper, we shall limit our analysis to the odd mode shown in Fig. 4.9 and compare our results to ones available in the literature. Due to sym m etry of the odd mode of th e structure shown in Fig. 4.9 with respect to the x-y plane, we can simplify the analysis by placing a perfect electric conductor (PEC) between the ferrite layers [74]. The equivalent structure shown in Fig. 4.10 is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 PEC Fig. 4.7: Cross-section of basic microstrip single-layer ferrite planar phase shifter. PEC Fig. 4.8: Cross-section of the phase shifter using oppositely-magnetized ferrite layers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Fig. 4.9: The odd mode of the dual structure. then easier to analyze. v.., ; >jci M M i PEC Fig. 4.10: The odd mode representation of the dual structure. B. Full Wave Analysis The analytical procedure used for the microstrip is very similar to the procedure applied in the previous subsection for the slot line. For the microstrip, the unknown is the electric current on the strip instead of the electric field in the slot. In addi- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 tion, an impedance m atrix formulation is used for the microstrip in contrast to the adm ittance m atrix formulation for the slot line. The electric current on the strip in both directions is expanded using Chebyshev polynomials as Nx Jx{y) = ' ^ c n hxn (4.41) n=0 Ny Jy{y) ~ dnhyn n=0 where the expansion functions hm and hyn are defined as hrn = ( - i r n t f ) / J 1 - ( ^ ) 2 (4.42) hm = ( - 1 ) " ( / „ ( f ) y i - ( | ) ’ (4.43) where S is the w idth of the strip. Based on the physical behavior of the electric currents on the strip, we use only even values of n with hxn and only odd values of n with hyn. Galerkin’s approach yields an impedance m atrix of the form Z= ZXX Zxy Zyx Zyy (4.44) with elements given by (4.45) i=—oo,odd The solution for th e propagation constant is the value of (3 that forces the determ inant of the impedance m atrix to zero. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 o -W/2 W/2 Fig. 4.11: n=0 Chebychev basis function of the electric current on the strip in longitudinal directions. Jy(y) o -W/2 W/2 Fig. 4.12: n = l Chebychev basis function of the electric current on the strip in transverse directions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 4.6 4.6.1 Magnetic Surface Wave Transducers Introduction M agnetostatic surface wave (MSSW) transducers are another class of microwave ferrite devices. They are widely used in delay lines. MSSWs exist when we magnetize the ferrite slab in the plane of the slab, and the direction of wave propagation is along th e magnetization direction [76]. We investigate this case by setting 6 = 90° and <f> = 180°. In our analysis, Galerkin’s technique is used to find the complex propagation constant of an infinitely long microstrip transducer in various multilayer structures shown in Figures 4.13, 4.14 and 4.15. The results are verified where possible by comparison to previously published ones. The effect of the m agnetization angle on the predicted results is also investigated. Fig. 4.13: Geometry of MSSW transducers with microstrip embedded between dielectric ferrite structure. D2d or m sm m G JB« 3 fariu- Did Fig. 4.14: Geometry of MSSW transducers in multilayer practical structure. 4.6.2 Full-Wave Analysis of the M icrostrip Phase Shifters The MSSW tranducers can be analyzed by following an approach similar to the one outlined in section 4.5.3. We note th at the expression for the impedance m atrix is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.15: Geometry of MSSW transducers in two-layer structure. no longer a discrete sum m ation b ut rather an infinite integral given by 4.7 Results and Discussion To validate our approach, we obtained the Green’s function of the single layer ferrite structure shown in Fig. 2.1 for two special cases - normally magnetized and transversely magnetized (6 (6 = 0°) = 90°, <j> = 90°). In the normally magnetized case, we compared our Green’s function to the one derived by Pozar using the boundary condition method [27]. Excellent agreement is achieved as is clear from Figures 4.164.23. We notice that in our results there axe some noise for laxge values of K y. In our expressions for the transmission m atrix we have an exponential terms th at may cause this noise for large arguments. Using a normalization method, this noise is simply avoided. In the transversely magnetized case, we compared our Green’s function with the Green’s function derived by Elsharawy [40]. Again, excellent agreement is achieved as shown in Figures 4.24-4.31. To verify the present theory for the single layer microstrip phase shifter shown in Fig. 4.7, we compared our predicted differential phase shift A <f>with the numerical result of Elsharawy [40] and the experimental result of Riches et. al. [77]. Excellent agreement with [40] and fair agreement with [77] is shown in Fig. 4.32. The discrep ancy between the numerical and experimental results is attributed to the fact that the m agnet’s fringing field is probably not strong enough to magnetize the entire Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 length of the ferrite substrate in the experimental case. We also examined the effect of magnetization angle deviation by changing the magnetization angle from <f>= 90° to <f>= 80°. The result is shown in Fig. 4.33. While this change in magnetization angle decreased slightly the discrepancy between our theoretical result and the ex perim ent, it does not seem th at the magnetization angle deviation entirely explains the discrepancy between theory and experiment in this case. Further validation for our theory is obtained by comparing our results for the forward and backward propagation constants and differential phase shift of the struc ture shown in Fig. 4.8 with those of Koza [74]. Excellent agreement is achieved as shown in Fig. 4.34 for the normalized propagation constants and in Fig. 4.35 for the differential phase shift. In our analysis of the MSSW transducers in Figures 4.13, 4.14 and 4.15, we used two different kinds of basis functions, piecewise linear subdomain basis functions and entire domain basis functions (Chebyshev polynomials). Both basis sets are conver gent with a reasonable number of basis functions. In this dissertation, the presented results are obtained using the entire domain basis set. The complex propagation constant is computed and compared to the results given in [40] for both symmetric and asymmetric cases. In the symmetric case, we represent the electric current dis tribution by even symmetric basis functions in the longitudinal direction and odd symm etric basis functions in the transverse direction. For MSSW transducers, this assum ption is considered to be questionable. A better assumption is to use both even and odd basis functions to represent the current in both directions. This is referred to as the asymmetric case. Fig. 4.36 indicates good agreement with Elsharawy’s symm etric case, and Fig. 4.37 shows fair agreement with his asymmetric case. We also analyzed the three layer structure shown in Fig. 4.14. This structure is considered to be a practical one, in which a layer of ferrite is deposited on the top of an alum ina dielectric with dielectric constant 9.8. It has the advantage of compatibil ity with MMIC circuits. Comparison of the attenuation constant computed using our Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 approach with the result of [40] is given in Fig. 4.38 for both symmetric and asym m etric current distributions. Comparison of the phase constants is given by Fig. 4.39. MSSW can exist over the range of frequencies given by + u>0u>m < uj < u Q+ ~um. The first peak in Fig. 4.38 corresponds to the second limit of MSSW frequency range, 4.172 GHz. For the symmetric case in Fig. 4.38, there is only one attenuation m ax im um at 4.18 GHz. However, for the asymmetric case, there are two maxim a, the first one at 4.18 GHz and the second one at 4.609 GHz, which is due to the coupling to th e MSSW. We varied the number of the basis functions from 5 to 13. Figures 4.40 and 4.41 show th at the results are almost the same after 9 basis functions are used. The structure shown in Fig. 4.13 is used to model the experiment carried out by Elsharawy [40]. The comparison of the symmetric and asymmetric theoretical cases with th e experimental result is shown in Fig. 4.42. A peak of S 21 = 23 dB occurs in the measured curve at 4.34 GHz while a peak of about 20.3 dB at 4.11 GHz occurs in Elsharawy’s theoretical result. The peak in our curve is about 21.4 dB at 4.35 GHz. The bandwidth of the device can be defined as the frequency range over which the insertion loss is within 3-dB of its peak value. The bandwidth of Elsharawy’s theoretical result is about 0.94 GHz, while the bandwidth of our theoretical result is about 0.70 GHz. The experimental result exhibits a bandwidth of 0.76 GHz. Elsharawy attributed the discrepancy between the measured and calculated results to the lack of accurate knowledge of magnetization magnitude and angle and to the line width at the surface. Using our Green’s function, we studied the effect of the m agnetization angle, and the results are shown in Figures 4.43 and 4.44 for several different values of 0 and <£, respectively. Varying 9 has very slight effect on the results, while varying <f> has a clear effect on the results and improvement in the agreement with the experiment is observed. We studied another factor th a t may also has an impact on the results, the 3-dB line width. Wheu we decreased the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 3-dB line width, a better agreement with the experim ental results is achieved at A H = 420.0Oe as shown in Fig. 4.45. The measured 3-dB line width for the ferrite m aterial under the experiment was 490.OOe ± 2% a t 9.0G H z. Since the 3-dB line width depends on the operating frequency, using A H = 420.0Oe for the comparison in the frequency range of 2.5 : 5.0G H z is acceptable. We noticed also th at increasing the material loss decreases th e attenuation which agrees with what Elsharawy stated in his dissertation. This work Pozar 3000 J 2400 aB* a a 1800 s 1200 600 -320 -160 0 Ky/k0 160 320 Fig. 4.16: Comparison of the computed Green’s function versus Pozar’s for the normally bi ased slab (Imag(Gxx)). (d = 7.62x 10-4m, £/ = 12.0,4 k M s = 2100.0G, Hdc = 700.0Ge, A H = O.Oe, R 3 = Ofi, W = 1.016 x 10~2m, / = 3.6GHz, K x = (110.0, -10.0)). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 350 This work Pozar 300 250 150 100 -320 -160 160 320 Ky/kO Fig. 4.17: Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gxx)). The parameters are the same as in Fig. 4.16 12000 — This work •— Pozar . 9600 & a 7200 4800 2400 -320 -160 0 Ky/kO 160 320 Fig. 4.18: Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Imag(Gyy)). The parameters are the same as in Fig. 4.16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 os m 15 d ' i2° -180 -240 -320 -160 0 Ky/kO 160 320 Fig. 4.19: Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gyy)). The parameters are the same as in Fig. 4.16 This work Pozar 3200 ^ 1600 CJ b 2 0 S d CO S |- < -1600 -3200 -320 -160 0 Ky/kO 160 320 Fig. 4.20: Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Imag(Gxy)). The parameters are the same as in Fig. 4.16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 This work — Pozar 320 160 CQ £ -160 -320 -320 -160 0 Ky/kO 160 320 Fig. 4.21: Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gxy)). The parameters are the same as in Fig. 4.16 — This work Pozar 3200 * 1600 I O c o -1600 -3200 -320 -160 0 Ky/kO 160 320 Fig. 4.22: Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Imag(Gyx)). The parameters are the same as in Fig. 4.16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 This work Pozar 320 160 I -160 -320 -320 -160 0 Ky/kO 160 320 Fig. 4.23: Comparison of the computed Green’s function versus Pozar’s for the normally biased slab (Real(Gyx)). The parameters are the same as in Fig. 4.16 This work Elsharawy 3000 2400 'So 1800 1200 600 -320 -160 0 Ky/kO 160 320 Fig. 4.24: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Imag(Gxx)) (d = 7.62 x 10-4 m, e/ = 12.0, 4 irMs = 2100.0G’, H dc = 700.00e, A H = O.Oe, R s = 0Q, W = 1.016 x 10~2m, / = 3.6 G H z . K s = (110.0,-10.0)). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 This work Elsharawy' 288 180 144 -320 -160 0 Ky/kO 160 320 Fig. 4.25: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Real(Gxx)). The parameters are the same as in Fig. 4.24 This work Elsharawy 12000 9000 3000 -320 -160 0 Ky/kO 160 320 Fig. 4.26: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Imag(Gyy)). The parameters are the same as in Fig. 4.24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 -32 « -64 -96 -128 -320 -160 0 Ky/kO 160 320 Fig. 4.27: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Real(Gyy)). The parameters are the same as in Fig. 4.24 3200 1600 I IC *5 C3 S 0 -1600 -3200 -320 -160 0 Ky/kO 160 320 Fig. 4.28: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Imag(Gxy)). The parameters are the same as in Fig. 4.24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 This work Elsharawy 252 126 I CQ £ -126 -252 -320 -160 0 Ky/kO 160 320 Fig. 4.29: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Real(Gxy)). The parameters are the same as in Fig. 4.24 This work Elsharawy . 3200 1600 I £ e •N bo es g -1600 -3200 -320 -160 0 Ky/kO 160 320 Fig. 4.30: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Imag(Gyx)). The parameters are the same as in Fig. 4.24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 260 130 -130 -260 -320 -160 0 Ky/kO 160 320 Fig. 4.31: Comparison of the computed Green’s function versus Elsharawy’s for trans versely biased slab (Real(Gyx)). The parameters are the same as in Fig. 4/24 This Work-90-90 Riches-Experimeot. ELaharawy-Theo ry 7 8 9 10 Frequency GHz 11 12 Fig. 4.32: Comparison of the calculated differential phase shift versus theoretical and ex perimental results for a microstrip single layer phase shifter (df = 0.635 x 10-3 m. ff = 12.9, 4x M a = 2300.0G, Hdc — 150.0Oe, S = 0.45 x 10_3m, a = 1.27 x 10-2 m. / = 1.52 x 10"2, 9 = 90°, <f>= 90°). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 Thi# Woric-90-80 Richei-Experiment.. E I*harawy-Theo ry cu 20 7 8 9 10 11 12 Frequency GHz Fig. 4.33: Comparison of the calculated differential phase shift versus theoretical and ex perimental results for a microstrip single layer phase shifter (df = 0.635 x 10-3m, €/ = 12.9, 4irMs = 2300.0G, Hdc — 150.00e, 5 = 0.45 x 10-3m, a = 1.27 x 10-2m. I = 1.52 x 10"2, 9 = 90°, <t>= 80°). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 i * i This Wort-Forward Roza-Forward This Wort-Backward Koza-Backward -2 7.2 9.6 12.0 14.4 16.8 Frequency GHz Fig. 4.34: Comparison of the normalized propagation constants for dual strip phase shifters (df = 1.0 x 10-3 m, e/ = 17.5, 47rMs = 1500.0G, Hdc = 0Oe, S = 1.0 x 10"3m, a = 1.0 x 10~2m). 40 — This Wort — Roza 35 O 30 -S-25 20 15 7.2 9.6 12.0 14.4 16.8 Frequency GHz Fig. 4.35: Comparison of the calculated differential phase shift for dual strip phase shifters (df = 1.0 x 10-3m, ef = 17.5, 4 xM s = 1500.0G, H dc = 0Oe, 5 = 1.0 x 10-3m. a = 1.0 x 10-2 m, 0 = 90°, <p = 80°). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 • This Work-Real Elsharawy*ReaI - This Wbrk-Imag. EUharawy-lmag. 6.4 S . 3.2 es <2 - o.o bo |- 3 .2 -6.4 r 2.4 3.2 4.0 4.8 5.6 Frequency GHz Fig. 4.36: Comparison of the calculated symmetric propagation constant for the two layer transducer (dd = 1.27 x 10-3m, df = 2.03 x 10-3m, e/ = 17.5, q = 10.2, 4ttM s = 2267.0G, Hdc = 144.0Oe, A H = 300.00e, S = 0.3 x 10"3m). This Work-Real Elsharawy*Real — This Wdrk-Imag. EUharawy-lmag. 6.4 $5_ 3.2 73 ,- c 0.0 a -3.2 r -6.4 2.4 3.2 4.0 Frequency GHz 4.8 5.6 Fig. 4.37: Comparison of the calculated asymmetric propagation constant for the two layer transducer (dd = 1.27 x 1 0 '3m, df = 2.03 x 10_3m, ef = 17.5, e<* = 10.2. 4-iV/s = 2267.0G, Hic = 144.0Oe, A H = 300.00e, S = 0.3 x 10-3m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 60 -i 1----- 1----- 1----- s----- r “T — 4.0 4.25 This Work-Symm. Elshajawy-Symm. This Work-Asym. Elsharawy-Asym. 4.5 Frequency GHz Fig. 4.38: Comparison of the calculated attenuation constant for the three layer transducer (i n = 2.5 x 10-4m, d.2d = 2.5 x 10-4m, df = 0.5 x 10-4 m, ej = 15.0, eu = 9.8, exd = 10.0, 4trMs = 1780G, Hdc = 600.00e, A H = 45.0Oe, S = 0.5 x 10~4m). 8.0 — — — - 6.4 3 This Work-Symm. Elsharawy-Symm.. This Work*Asym. EIsharawy-Asym. 3.2 1.6 0.0 3.5 3.75 4.0 4.25 4.5 4.75 5.0 Frequency GHz Fig. 4.39: Comparison of the calculated phase constant for the three layer transducer (d u = 2.5 x 10_4m, did = 2.5 x 10_4m, df = 0.5 x 10-4 m, e/ = 15.0, eu = 9.8. eld = 10.0, 4ttM s = 1780G, Hdc = 600.0Ge, AH = 45.0Oe, S = 0.5 x 10_4m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 45 u 3.5 3.75 4.0 4.25 4.5 4.75 5.0 Frequency GHz Fig. 4.40: The effect of number of basis functions on the attenuation constant 8 This Work-BF=5 This Work-BF=9 This Work-BF=13 6 /-— V 4 2 0 3.5 3.75 4.0 4.25 4.5 4.75 5.0 Frequency GHz Fig. 4.41: The effect of number of basis functions on the phase constant Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 I Thu Work-Symm. Elsharawy-Symm This Work*Asym. Elsharawy-Asym. EUharawy-Exp, 3.5 4.0 Frequency GHz Fig. 4.42: Comparison of the insertion loss along a MSSW transducer (dd = 1.27 x 10~3m. df = 2.03 x 10“ 3m, ef = 17.5, ed = 10.2, A vM . = 2267.0G, Hdc = 144.0Oe, A H = 490.0Oe, S = 0.3 x 10"3m, I = 12.7 x 10"3m). 30 1------ rThis Work*fe90f £=00 This Work*$=95, £=00 This Work-JblOO, £=00 Elshar&wy-Exp. 24 ffl 18 CM — !2 2.5 3.0 3.5 4.0 4.5 5.0 Frequency GHz Fig. 4.43: The effect of the magnetization angle, 9, on the insertion loss along a MSSW transducer (dd = 1.27 x 10-3 m, df = 2.03 x 10- 3 m, e/ = 17.5, ed = 10.2, 4 ~ M S = 2267.0G, H dc = 144.0Oe, A H = 440.00e, 5 = 0.3 x 10~3m, / = 12.7 x 10- 3 ,m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 30 24 -i r This Work-#=90, *=00 This Work-#=90, *s05 This Work fa DO, *slO Elsharawy-Exp. -I PQ 1 8 -a — 12 2.5 3.0 3.5 4.0 4.5 5.0 Frequency GHz Fig. 4.44: The effect of the magnetization angle, o, on the insertion loss along a MSSW transducer (dd = 1.27 x 10-3 m, df = 2.03 x 10-3 m, ej = 17.5, e<* = 10.2, 4 ttM 3 = 2267.0G, Hdc = 144.0Oe, A H = 44O.O0e, S = 0.3 x 10"3m, I = 12.7 x 10“ 3m). 30 This Work*AH=490 Oe This Work-&H=450 Oe This \Vork-AH=420 Oe — 24 — Elsharawy-Exp. !2 2.5 3.0 3.5 4.0 4.5 5.0 Frequency GHz Fig. 4.45: The effect of the 3-dB line width on the insertion loss along a MSSW transducer (idd = 1.27 x 10~3m, df = 2.03 x 10~3m, ef = 17.5, ed = 10.2, 4;rA/a = 2267.0G. H dc = 144.0Oe, S = 0.3 x 10_3m, I = 12.7 x 10_3m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 CHAPTER 5 3-D ANALYSIS OF RADAR CROSS SECTION OF A FERRITE PATCH ANTENNA 5.1 Introduction The basic configuration of a microstrip antenna consists of a sandwich of two parallel conducting layers separated by a single thin dielectric substrate. The lower conductor functions as a ground plane, and the upper conductor acts as a radiating element which may be a patch of regular shape, a dipole, or a monolithically printed array of patches or dipoles and the associated feed network. The patch conductor typically has some regular shape, for example, rectangular, circular or elliptical. The feed is often a coaxial probe or a microstrip transmission line. Microstrip antennas exhibit all the advantages of the microstrip devices: a) they are light weight, have small size, and exhibit low profile planar configurations which can be made conformal; b) they are inexpensive to build and ideally suited for large scale production by printed circuit techniques; c) they are compatible with modular designs (solid state devices such as oscillators, amplifiers, phase shifters, etc., can be added directly to the antenna substrate board); d) their feed lines and matching networks can be fabricated simultaneously with the antenna structure so th at discontinuities due to connectors can be minimized. All these advantages compensate, at least in part, some of the disadvatages: a) simple microstrip antennas have narrow bandwidths; b) their gain is low; c) they have a small power handling capability; d) their dielectric losses reduce th e radiation efficiency; e) unwanted surface waves may cause spurious radiation at th e edges of the microstrip patch [78]. Recently, there has been great interest in incorporating ferrites in patch antenna designs to incorporate phase shifting, impedance matching, frequency tuning, and nonreciprocal effects in the operation of antennas. It has been reported th at by including ferrite materials, the main beam from an antenna array can be scanned. Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 114 and th a t the radiation frequency of a microstrip antenna can be tuned by varying the dc bias. Ferrite patch antennas, similar to the dielectric antennas, consist of metalic patches deposited on ferrite substrates. The substrates are electrically grounded by placing them on a m etal surface. There are m any possibilities and combinations for using ferrite materials in printed antenna systems. Bias fields can be applied in different directions; ferrite mate rials can be used as single substrates, in multilayer substrate configurations with dielectrics, or as cover layers for printed antennas; ferrites can also be used inhomogeneously with dielectric materials [27]. In spite of a reasonable effort and a large number of publications concerning the scattering from the printed antennas on a biased ferrite substrates, an efficient and versatile formulation of the RCS is not available yet. In addition, the multilayer structure is not discussed yet except by Yang [29]. However, neither Yang nor any one else has presented the excitation vector expression or detailed derivation. This work presents a clear and versatile formulation of such problems. The gen eralization is th e m ain contribution of this chapter. The Green’s function is formu lated using the closed form transmission matrix. The excitation vectors are given in a closed form too. T he expression of the excitation vectors is general for any incident angle, any num ber of layers, and any magnetization angle. The I E E E dictionary of electrical and electronics terms defines R C S as a mea sure of reflective strength of a target defined as 4tt times the ratio of the power per unit solid angle scattered in a specified direction to the incident power per solid an gle. More precisely, it is the limit of that ratio as the distance from the scatterer to the point where the scattered power is measured approaches infinity [79]: \E s c a t\2 a = lim 47rr2T -r—oo \Etnc 2 (5.1) where E scat is the scattered electric field and E 'nc is the field incident to the target. Three cases are distinguished: monostatic or backscatter, forward scattering, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 bistatic scattering. The units for radar cross section are square meters. T he crosssectional area does not relate directly to the physical size of a target. Due to the large dynamic range of RCS, a logarithmic power scale is most often used with the reference value of aref = lm 2. crdBsm = crd B m 2 = 10logl0^ ~ = 10 lo g x f i f e f (5.2) 1 Two unit notations axe used. The dB sm notation is very common w ithin the aca demic, government, and industrial organizations. The d B m 2 notation is less used, although it is sometimes seen in radar system design literature. 5.2 Plane Wave Propagation in a Ferrite Medium: an Introduction to the Excitation Vector Formulation The first step in our analysis is to derive the propagation constant of a wave propa gating at an arbitrary angle ( 9 \ <f>1) to the direction of the dc magnetization in ferrite. W ithout loss of generality, we can consider the incident wave to be traveling in the 2 -direction in an arbitrary magnetized ferrite medium. The full perm eability tensor of an arbitrarily magnetized ferrite medium is given by [72] ft = ftn ft\ 2 ft\3 ft2 l ft22 ft2Z . ftzi ft32 ft33 . where [*n = f t 12 = (ti3 = ft + ( f t o — f t ) s i n 29 c o s 2 <£ s i n 2 0 s in 2 < £ + jk cos 9 (to — ft — -— sin20 cos 6 —j k sin 9 sin 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 uo — U 9 f i 2i = — -— sin H22 = y + {yo Uo o 0 sin 2 6 — jk cos 0 —y ) sin 20 sin V —u y.22 = — ~— sin 2 0 sin 6 + JK sin 0 cos 6 y.31 = u0 — a — -— sin 20 cos 6 + JK sin » sin 6 yz 2 = — -z— sin 20 sin 6 ~ JK sin 0 cos 6 y& = yo —y y o - {yo - y ) sin 20 U0 U}m (* = (! + ^Ll]2 ----—r.j*7 ) K = um UJUJm w2 —W2 = 747riV/s, w= w0 + ^, w0 = 7 Ho wo is the precession frequency, #0 is a z-directed impressed dc magnetic field, 7 is the gyromagnetic ratio, T = — ^27? is the relaxation time, and AH is the 3-dB line -fAH width. The four Maxwell’s equations are given by —j u y -H ju e E = = V x E = V x H V • ( y -H) _ eV • E = 0 (5.4) = 0 If the wave is propagating in the z-direction with a propagation constant j3 and its angular frequency, w, and if the conditions for a plane wave ^ = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 are 117 substituted into equation (5.4), that equation becomes ueEx (5.5) = u;eEx (5.6) = Ez (5.7) j(3Ey = —jujB x (5.S) -jfiE x = -ju jB y (5.9) 0 = -j0 B z (5.10) /3Hy = fiHy 0 Equation (5.4) becomes It is interesting to note that, because it is a plane wave, the longitudinal component of the m agnetic flux density, fiBz, is zero, but there still may be a finite longitudinal (normal) component of the magnetic field intensity, H z. After substituting equation (5.3) into Maxwell’s equations and after some ma nipulation we find Bi P P2 ^ = T ± \ / w2e 2 V -4 r - 0 (5-ID p (5. 12) where = M33 Q = ( 5 . 13) ^33 Ml — ~M22M33 + M23M32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5-14) 118 = /^3 = ^21/^33 —^23^31 (5.16) = (5-17) P 4 ~ 1 1 ^3 3 + (5-15) (J -2 f i H 13/^31 ^ 1 2 ^ 3 3 — ^1 3 /^ 3 2 As a validation, we set the parameters in equation 5.11 to the case of wave normally incident on a norm ally biased ferrite slab, 0 = 0°, <f>= 0°. We obtain the following expression for th e propagation constant for this case, P± = u y / e ( p ± K (5.18) ) The resulted propagation constant given by equation 5.18 is exactly the result reached by Fuller for this case [80]. Ground plane Fig. 5.1: Incident wave on a normally biased single ferrite layer. For a single grounded ferrite substrate which is normally biased, the field com ponents outside and inside the ferrite region as seen in Fig. 5.1 can be written as e]k°yv' ejk°' cos s‘ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The field components inside the ferrite slab consist of right and left-handed circularly polarized partial waves propagating along positive and negative directions, and are expressed as ^ = [ A + eJ^ +xut e-,/3+yv' eJ/3+' cos e ‘ + B + e -jP + x u ' e ~]0+yv ‘e - j 0 + - c o s 8‘ j (a* - ja#) + [A_eJ/3-xut ej(3~yvCejf3~~ cos e‘ + 5 _ e~j/3~zu‘e t ~ji3- - cos e‘] { a e + ja t) (a t+ ja g ) + Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 } _ ^ _ e j P - x u c e J 0 - y v c e J p - z c o s S c _|_ e - j P - x u c g - J 0 - y v c g - j 0 - z cos5 ' j n~ (a t-ja g ) E[ [A+eJ0+xv,t e]l3+yvt e~3t3+zcos6t + B +e~jt3+xu' e~JP+yv‘eJ*3+zcoset]. = (5.19) (—ag + ja,j,) + ^ _ e J 0 - x u ‘ £ j 0 - y v c e ~ ] 0 - z c o s 9 C _|_ g _ e - j 0 - x u ‘ e ~ ] 0 - y v ' & j 0 - z c o s S ' j (a g + j a t ) 1TX = — |r a r x E[ V _ L _ (5.20) e ] 0 + x u ‘ e j/3+ y v ‘ e ~ ] 0 + : c o s 8 ‘ _|_ Q ^ e - ] 0 + x u ( e ~ ] 0 + y v ‘ £ j 0 + z c o s 6 ‘ -l T] + (-it-jag) + + _ ^ _ [ A _ e 3 0 -x v ‘ e J 0 -y v l e -tf-z c o s O * g ^ e - ] 0 - x u ‘ e - j ( 3 - y v c e j ( 3 - z c o s e ‘ -l v~ ( - a t + jag) The generalized Snell’s law, which comes from matching the phases of E and H at the air-ferrite interface, leads to the following expression for the transmission angle . Sm k 0 sin 6 l P± Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5'21) 121 The total field in each region is the sum of the incident and reflected components. First, we apply the continuity of the electric and magnetic fields at the interface, z = d. Then, we solve for the vector reflection coefficients, Rg and R ^ . This procedure results in a tedious and complex expression for single normally biased ferrite layer. The procedure for an arbitrarily biased ferrite layer are even more complex, and for an multi-layer arbitrarily biased imulti-layer structure, they may be impossible to derive. 5.3 Theory 5.3.1 Full Wave Formulation First we consider the single ferrite layer given in Fig. 5.2 with a general incident wave with either soft or hard polarization. m e ' j ’ 1 1 E 2 J 2 — 2 Fig. 5.2: Geometry of single ferrite layer with a general incident wave. The transmission m atrix T for a ferrite layer is a 4 x 4 matrix written as [43] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 - II'Fh II J l = E = T = T . = - = h - ^2 - =J where T , Z , Y , T ' -i =E =T f z = T =J Y T ' e . h 2 (5.22) . = axe 2 x 2 submatrices of T , E l , E 2 axe the tangential electric field at the boundaries of the layer, J x and J 2 are the tangential surface currents defined by J n = z x H n, where H n is the tangential magnetic field at the n th surface of the layer, and denotes the spatial Fourier transform defined as E {K x, K y) = f 0 E { x ,y ) e - jK*xe - jKyy dx dy J —OOJ —CO (5.23) Ferrite Substrate d \ G round Plane Fig. 5.3: Microstrip patch antenna on a normally biased ferrite substrate The structure, shown in Fig. 5.3, is illuminated by an incident wave, E l. In the absence of the conducting patches, E T is reflected into the air region by the ground plane and the ferrite substrate. The total field E° in this case is the summ ation of these two components. W ith the presence of the patch, a surface current is induced on the patch conductor which causes a scattered field, E s. The total field in the air region is the vector sum of the incident, the reflected, and the scattered fields, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 El = E ' + E r + E3 Hl = W + H' + H3 (5.24) (5.25) Rewriting equations (5.24) and (5.25), we get E l = E° + E 3 = H° + H 3 (5.26) (5.27) The integral equation is a statem ent of the boundary conditions on the patch that the total tangential field on the surface of the conducting patch must be zero, that is a~ x E 3 = —dz x (E °) (5.28) Equation (5.28) represents an integral equation since E 3 is given by = J G ' J . ds (5.29) where J s is the surface current density on the patch and G is the Green’s function derived in our previous works [65] and [81]. The method of moments is essentially a technique for transforming integral equa tions into m atrix algebric equations th at can be solved numerically using a computer. Applying the Galerkin’s approach to equation (5.29), using the basis functions as weighting functions leads to a system of linear algebric equations which can be solved for the unknown coefficients of the basis functions. In a m atrix form we can write [Zij][Ij] = N Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.30) 124 where [Z\ is known as the impedance m atrix with elements ^ Z{j — ~ ro o 2 J ro o — x J k y ) ’ G" [ k x j z|z ) • J — k y ' j d k x dky (o.31) and [e,-] is the excitation vector defined by < = I E ° J ’J s (5-32) where u is either z or y and E° is the tangential total electric field at the air-ferrite interface with the absence of the conducting patch. The patch currents in the integral equations (5.28) and (5.29) are expanded in term s of a set of knownfunctions. Analysis of arbitrarily shaped patches requires the use of subdomain basis functions. Entire domain basis functions are difficult to implement on a complex structure and may take many modes to describe fine electric current details accurately. A convenient choice is a linearly-varying basis function with triangular support. This set of basis functions ensure current continuity from triangle to triangle, and it has been introduced by Rao, Wilton and Giisson (RWG) to evaluate the electromagnetic scattering by surfaces of an arbitrary shape using the EFIE formulation [82]. Assuming th a t a suitable triangulation has been defined to approximate the mi crostrip antenna, we note th at each basis function is associated with an interior (nonboundaxy) edge and vanishes everywhere except in the two triangles attached to th at edge. Thus, within the triangle X) shown in Fig. 5.4, the surface current can be approximated as 3 ___ J *{x >y ) = J L h Jj ( x ’ y ) ( 5 -3 3 ) i= i where Ij is a coefficient to be determined via MoM and Jj is the piece of the basis function on the triangle J2 associated with j th edge. The Jj are given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 J j (r) = ± 2 A~ ( r ~ ri) where A ^ is the area of triangle (5-34) lj is the magnitude of the j th edge vector given by (j ^'mo<f(j-(-l,3)+l T 'm o d .^ j, (5.35) 3 )+ l r is the position vector to an arbitrary point in the triangle and r j is the position vector to node j . The sign multiplying lj is determined by the connectivity of the mesh. The Fourier transform of the piece of the basis function on triangle J2 along the j th edge is given by 7 j$ ) = j j J ] ( r ) e - * TdA (5.36) where k = xkx + yky (5.37) The Fourier transform of this basis functions has been evaluated in a closed form by M clnturff and Simon [83]. (x4,]T4) <*i Fig. 5.4: RWG basis function and corresponding edge connectivity The impedance m atrix elements in the spectral domain MoM solution of an arbi trary microstrip structure using RWG basis functions th at has x and y-components for each set may be written as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 J Z {j — 4-ti^ j to o J "hJ r t ( ^ -n “f*J yt ( ^xi " I” J yi ( to o — zi ^“*5 xi( — ^ y ) ' G x x (.^ x 7 ^*y) ’ J x j i j ^ x i k y ) (5.38) k y ) ' G x y {,^xi k y ) ’ J y j ^ x t k y ) ) ‘ G y x ( ^ n ^-y) * J x j ( j ky ) G yy ( kx 7 ^ y ^ y ) } ^*277 f c y ) • ) * *JIjj ( ^X 7 dfcr dl\*y Notice the difference between the two im pedance equations, (5.31) and (5.39). Equa tion (5.39) is a special case of (5.31) where we use one set of basis functions having both x and y components, which is the case of RWG basis functions. 5.3.2 A. Excitation Vector Soft Polarized Incident Wave As mentioned above, the total tangential electric field in the absence of the conduct ing patch is E°, which is given in equation 5.26 as the summation of the incident and reflected waves at the air-ferrite interface. Applying the boundary conditions on both sides of the ferrite layer shown in Fig. 5.2, we can write E+ = E[ + E[ (5.39) E* = Ef (5.40) E2 = 0 (5.41) Since there is no current source at surface 1, J = + ./f = 0, hence, ./f = —J* . T he current above surface 1 in terms of the magnetic fields is given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 J+ (5.42) = zxHf = z x ( H i + H[) = z x (— —an x H{ + — an x H[) f]0 TjO where an is th e normal vector of a general incident wave which has the following form On = oxu' -f ayv' + az cos 9l (5.43) ul = sin 9' cos 4>l (5.44) vl = sin 9l sin <f>* (5.45) After m anipulation, equation 5.43 becomes cos 9l J t = -------(E\ - E l) lo (5.46) Rewriting equation 5.22 we get —r ' El + El ^ ( e x- X ) -E =T f =T z ' _ _ 0 =J Y f (5.47) J 2 Solving equation 5.47 for E ' x and E Ti we can write = & i = J = T ~ l cos 9'I + rjoT Z = J = T ~ l cos 9'I — Tj0 T Z Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5 .4 8 ) 12S The total tangential electric field at the ferrite-air interface is the sum of the incident and the reflected electric field components which leads to the following expression for E + 1 2 cos 6 XI =J = T~l (5.49) E\ . cos O'I + t\qT Z In general, we can write equation 5.49 in matrix form as C0 [£ + i] = r ”t> ri4> ri<t> 11 °21 u 12 (5.50) [&l] '-'22 J The general incident soft-polarized plane wave can be expressed as E[ = a^ejkQXU' ejhzu' e3kaZCOsd' (5.51) Substituting equation 5.51 into equation 5.50 and the result into equation 5.32, the excitation vector elements are found to be = ( - C f i sin <?+ Cf 2 COS <l>)e3k°dcos6' j * { - k Qu \ - k 0 v l) (5.52) ey = (—C2i sin <j>+ C%i cos <f>)ej k o d c o s O 1 Jq(—kQu \ —kov') (5.53) < where J* and Jq are th e Fourier transforms of the basis functions components. The same procedures are used to find the excitation vector for the double layer structure shown in Fig. 5.5, a conducting patch on the top of a dielectric grounded plane and a ferrite cover on the top of both. The total tangential electric field in terms of the soft-polaxized incident field is found to be = T -J = T ~ l =E £ += - ( Y , - T f Z, fj) f j i , Ci + u )+ — no Ci - u ) E \ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.54) 129 where subscript f stands for ferrite, the superscript (—1) throughout this work means m atrix inversion, / is the unity matix, and contains the reflection coefficients is given from equation 5.48 as = j cos I+ j]0 T U - Z Zj V t-i COS 0 ‘ 7 — 7/0 =J =T~l {o.zz) T Z =T T and Z are the submatrices of the new transmission m atrix which results from the multiplication of the ferrite times the dielectric transmission matrices. In a m atrix form, equation 5.54 can be w ritten as [£>] = Wl 11 cti C< f> [£'u] 11 (5.56) In final form, the excitation vectors axe given by exq = ( - C f x sin 0 + C ?2 cos Q)ejho{d'i+d' ]cosa'J~*{-k 0 u \ - k Qv l) (5.57) e* = ( - C ^ s m t + C ^ c o s ^ e ^ ^ + ^ ^ ' J Z i - k o u ^ - k o v 1) (5.58) where dd and dj are the thicknesses of the dielectric and the ferrite layers, respectively as seen in Fig 5.6. B. Hard Polarized Incident Wave For the single ferrite layer shown in Fig. 5.2, the difference between the soft and the hard-polarization case is the expression of the incident plane wave which is given by E[ = dg ejk°xu' ejk°xu' ejh°' cos 8' (5.59) Following the same procedures as the soft-polarization case, we can prove the follwing relations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 2/ E+1 c o s O' = E\ =J = T~l (5.60) sfsr + ^ ^ In a m atrix form, equation 5.60 can be writen as . C8 no 11 12 riO riB U21 0 22 (5.61) The excitation vector elements for a hard polarized plane wave incident on a single ferrite layer axe given by ef = (Cf, cos 9 + CI,J sin . / H - i o u '. - t o i '1) (5.62) (5.63) For the double layer structure shown in Fig. 5.5, the final expression is found to be = t —j = r~ 1 = E+ = - ( Y f - Tf Zf e =J =T~l f f ) Tf Zj ( I + T g) + — s - ( / - r () E \ (5.64) COS t / l T]o In a m atrix form, equation 5.64 can be written as [E+] = CO 11 O C21 r'Q °12 (5.65) r^O u 22 The reflection coefficent Fg is found to be =j = t - i f•L.e i :_ d F + r>o T Z =J =T~ 1 TF-nof z cos 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5 .6 6 ) 131 =T =J where T and Z are the submatrices of the new transmission m atrix resulting from the multiplication of the ferrite transmission m atrix by the dielectric transmission m atrix. The final form of the excitation vectors is = {C9 n cos <f>+ CU sin <j>)ejko(dd+df ) cosB' J = ( - k Qu \ - k Qv l) (5.67) eyq = {Ce2l cos d>+ C e22 sin ^ )eJM«*«+<*/)cos6' J $ ( - k 0 u \ - k Qv') (5.68) < 5.3.3 A. Green’s Function Formulation General The detailed derivation of the Green’s functions for a single ferrite layer and for some multi-layer dielectric- normally biased ferrite structures using the transmission m atrix approach are outlined in chapter 2 and for arbitrary biased ferrite structures chapter 4. We will summarize the Green’s functions for the structures analysed in this work. For the single structure shown in Fig. 5.2, the Green’s function is found to be =J=T ~l Es = {Tf Z f - - i _ + G a ) " 1J S = MS_ = G Js (5.69) Notice th at the Green’s function given in equation 5.69 is not limited for certain bias direction. The elements Z j and T j are elements of any transmission m atrix derived for any bias direction. Ga is a semispace Green’s function, which is calculated by taking the limit of the dielectric Green’s function when the thickness goes to infinity and the dielectric constant goes to unity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B. Derivation of the Green’s Function for a Two-Layer Structure Patch Antenna The dielectric layer in Fig. 5.6, the lower region, is represented by the following relation E = Td T Ei Ji =E —T Td Zd = T =J Ya Ei (5.70) Ji f, We set E[ = 0 by placing a ground plane at distance dd from the source plane. This results in the following equation = J = J = T ~1__ T dZ d E (5.71) For the ferrite layer, the cover region, we have (5.72) Applying the boundary conditions at the air-ferrite interface leads to the following relation _+ E = . J h K Ju . - =E ■ r -1 If If — . —T j f El U (5.73) X. f,_ After some manipulations we get = T J = j = -l =E =T = -1 = ( Y f + T f Ga ) ~i \/ T f + Z s Ga )E (5.74) The total current at the source plane, the conducting patch, is the sum of the two current components above and below the surface, J J3 = J and J +J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.75) 133 ' =T =J = -l =E =T = - 1 = J = T~l ( Y f + T f Ga ) ( Tf + Z f Ga )~1 + f dZ d E3 (5.76) From, the definition, the Green’s function relates the current to the electric field, hence the final expression of the Green’s function is obtained in the following form =T ' E, = J = -1 =E =T = - l =J = T~l = ( Y f + T f Ga ) ( Tf + Z f Ga )~ 1 + T dZ d = = \rs_ G J3 -l J3 (5-77) (5.78) Following similar procedures, the Green’s function for any structure is simply obtained. » 5.4 Impedance M atrix Interpolation To improve the computational efficiency of the MoM code, we follow the same tech nique outlined in Aberle’s notes [84], where a cubic spline algorithm is used to inter polate the impedance m atrix at intermediate frequencies. A so-called natural spline is used. The cubic spline algorithm used here differs from conventional cubic spline algorithms because we are interpolating matrices and so do not wish to store them in memory. The MoM code computes the impedance matrix at a set of discrete frequencies given by fm — f st art 4" (m 1) 771 = 1, ..., iVy (o. f9) where A/ = f “° [ ~ (5.80) IS j — 1 and N j is the number of frequency points at which the impedance matrix is calcu lated. We denote the impedance m atrix at frequency f m by Zm The first task in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 the development of the cubic spline interpolation algorithm is to compute the second derivative matrices at each f m. We denote the second derivative m atrix at frequency fm by Z"m. We have Z'[ = 0 (5.81) Nf - 1 Z"m = £ (5.82) n=2 Z N, = (5.83) 0 where T is m atrix whose inverse is an N j — 2 by N j —2 tridiagonal m atrix given by T -i = a = 6 = b a 0 a 0 ••• 0 0 b a • ■■ 0 0 a b ••• 0 0 0 0 0 ••• a b ( A /) 2 (5.84) (5.85) 2(A f f (5.86) and J3n is a m atrix given by -Sn — Zn+x — 2Zn + Zn_! (5.87) To avoid storing more than a few Zn’s in memory at any one time, the impedance matrices are read from disk as needed, and the second derivative matrices are written to disk as computed. The algorithm for computing the B n matrices is based on writing B n as Bn — Qn Q n— 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.8S) 135 where Qn = Zn+l - Zn (5.89) Thus, the algorithm for computing the B n is Qi — Z i — Z\ n = 2,..., Nf —1 for Bn — Zn+1 Zn Qn—1 Qn — B n + Q n-1 end (5.90) The algorithm can be implemented by storing three matrices at a given tim e (Q n- 1 , Zni 3*nd Bn)- Once the Z"m have been obtained the cubic spline interpolated value of the impedance m atrix at frequency / can be obtained as Z ( /) a Z m+ f3Zm+1 + 7Zm + = 8 Zm+l (5.91) where a = ~ i f 1 (S-92) /3 = I-a (5.93) 7 = ^ Q ( a 2 - l) (A / ) 2 (5.94) s = i/j(/32 - l ) ( A I f (.5.95) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 m = f - f st art A/ + 1 (5.96) and [x] is the largest integer less than or equal to x, i.e., fm < f < fm+l 5.5 (5-97) Results and Conclusions To validate the results of our approach, we compared the numerical values of the RCS components, for the dielectric to those in [85], [86], and [87]. The dielectric case is the limiting case of ferrite when we set the ferrite parameters to zero. Figures 5.7, 5.8, 5.9, 5.10 and 5.11 indicate the results are identical. For the in-plane biased cases, x-biased and y-biased, we compared our results of <70# for the two cases against those in [52], and very good agreement is clear from Figures 5.12 and 5.13. To make sure th at our approach is also correct and valid for the multi-layer structure, we compared our results for the case of a ferrite cover on the top of a dielectric patch against those in [29], the only case available in the literature for a multi-later strucure. Figures 5.14 and 5.15 show good agreement. In reference [52], the thickness of the ferrite cover is not mentioned; therefore, we assumed it to be 0.5mm. If we knew the exact thickness, we might obtain a better agreement. Figures 5.16, 5.17 and 5.18 show the other elements of the RCS with and without the bias field for the case given in [52], a patch on a ferrite substrate. In these figures, the bias field as well as the propagation direction is the ^-direction. It is clear that the first resonance peak is the same for the biased and unbiased cases. The invari ant location of the first resonant peak can be explained by investigating Maxwell’s equation for the T M \ q cavity mode. The longitudinal magnetic field component, Hy, and th e transverse electric field component, E z are almost unaffected by the bias field. Therefore, the first patch resonance does not depend on the bias field [52]. This concept of analysis is also true for the fourth resonance, which is considered the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 resonance of T M 20 mode. In general, the resonances of T Mm0 modes are unaffected by the bias field. For the resonance of the other modes, the bias field increases their resonant frequency, but the RCS levels rem ain almost the same. Figures 5.19, 5.20 and 5.21 show the other elements of the RCS with and with out the bias field for th at case given in [52]. In these figures, the bias field is the x-direction while the propagation direction is still in y-direction. W ith the same ar gument as for the previous case, those resonances corresponding to the T M 0n modes axe unaffected by the bias field. Yang et al. also explains this phenomenon [52]. We can notice also th at the RCS level is below —60 d B s m for frequencies less than 8 GHz . Figures 5.22, 5.23 and 5.24 show the elements of the RCS with and without a ferrite cover on the top as shown in Fig. 5.5 for the case given in [29]. The bias field is assumed in the x-direction. Using the cavity model concept, the first resonance is assumed to be for the T M \ 0 mode. W ith the ferrite cover, the first resonance is eliminated. The third resonance has less RCS level compared with no ferrite cover case. Under the assumption th a t the bias field and the propagation direction are parallel, that structure can excite m agnetostatic surface wave in the frequency range when fief / is negative. The frequency range for the negative fie/ / is give by [29] \JUJq{u1q + UJm) < U! < UJq + U!m (5.98) In our case, the frequency range of the m agnetostatic surface wave is in the range of 2.2G H z < f < 5.8GHz. As a m atter of fact, the eliminated first resonance frequencyi occurs in the cut-off region. In the frequency range of the magnetostatic surface wave, the RCS level is decreasing with increasing the frequency [29]. Figures 5.25, 5.26 and Fig. 5.27 show elements of RCS with and without a ferrite cover as seen in Fig. 5.5. The param eters are the same as in the prevous case except the patch dimensions are interchanged, L = 3.0cm and W = 4.0cm. We can look for this case from another view, keep the patch dimensions as it was before and change Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 the bias field direction, from x to y. Using this argum ent, we can better understand the new mode configurations. As in the case of the single biased ferrite patch, when the bias field is along the propagation direction, the first resonance is barely affected by the bias field. The surface wave has a strong effect on the second resonance. Therefore, the second resonance is eliminated by adding the ferrite cover. Using the phenomenon of RCS elimination due to the property of the magneto static surface wave, Yang in [29] suggests to design a switchable antenna. At one bias level, the surface wave is a propagating wave, the antenna is “on” near the res onant frequency. At the other biased level, the m agnetostatic surface wave is highly attenuating, and the antenna is “off”. We studied the previous case, but the structure is inverted, ferrite underneath the patch and dielectric layer as a cover. To define the effect of that reversal, we use the same material param eters as in the previous case. Because of the fact that the resonant frequencies of a patch are mainly determined by the patch’s size and the m aterial underneath (not above) it, the RCS changes. Figures 5.28, 5.29, and 5.30 show plots of the RCS due to different polarizations. As a novel antenna structure, we propose the cross patch shown in Fig. 5.31. Using the linearly varying basis functions enables us to analyze such configurations. The cross-patch dimensions are kept constant for all the results, Lc = Wc = 0.05cm. The comparisons of the RCS of the cross patch for the dielectric case versus the RCS of the full patch are shown for the co-polarized case in Figures 5.32 and 5.33 and for the cross-polarized case in Fig. 5.34. The same comparison is repeated for the ferrite case, the results are shown in Figures 5.35, 5.36 and 5.37. The effect of the bias field is insignificant as seen from Figures 5.38, 5.39 and 5.40. In general, we notice that the first resonance is almost at the same frequency for both patches, the cross and the full. However, the RCS level is reduced. The cross patch excites less number of modes than the case of full patch. That this is true can be seen by the reduction in RCS resonances when the cross patch is used. The fact th at the RCS decreases may Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 be due to the smaller physical area of the cross patch. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 Fertile layer Dielectric layer Df ^ Ground plane Fig. 5.5: Geometry of a patch antenna with a biased ferrite as a cover layer Free space Df Ferrite layer Dd Dielectric layer Ground plane Fig. 5.6: Fields and currents of a patch antenna with a biased ferrite as a cover layer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 -10 This work Abe tie Pozar -20 -30 S3 -40 -60 -70 -80 Frequency (GHz) Fig. 5.7: Comparison of RCS for a microstrip patch antenna (M s = d = 1.3 x 10-3 m, L = W = 1.3 x 10-2m, 0,- = 30°, <pt- = 45°) H q = 0, er = L3.0, -10 Thia work Aberle -20 -30 -60 -70 -80 2 3 4 5 6 7 8 Frequency (GHz) Fig. 5.8: Comparison of RCS for a microstrip patch antenna ( M d = 1.3 x 10“ 3m, L = W = 1.3 x 10“ 2m, = 30°, & = 45°) 3 = H q = 0, er = 13.0, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 -10 -20 « -30 -50 -60 2.9 3.0 3.1 3.2 3.3 Frequency (GHz) Fig. 5.9: Compaxison of RCS for a microstrip patch, antenna (Ms = Hq = 0, eT = 4.0, d = 3.0 x 10- 4 m, L = 1.25 x 10 - 2m, W = 2.5 x 10 - 2 m, 0,- = 45°, <?,- = 0°) -20 ThU work Aberle -30 -50 -70 -80 4 8 12 16 20 Frequency (GHz) Fig. 5.10: Comparison of RCS for a microstrip patch antenna (Ms = Hq — 0 , er — 12.8, d = 6.0 x 10- 4 m, L = 0.55 x 10- 2 m, W = 0.4 x 10- 2 m, 0; = 60°, <?>,• = 45°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 -30 This wort Aberle -40 -50 -70 -80 4 8 12 20 16 Frequency (GHz) Fig. 5.11: Comparison of RCS for a microstrip patch antenna ( M s = H q = 0, er = 1‘2.8, d = 6.0 x 10_ 4 m, L = 0.55 x 10- 2m, W = 0.4 x 10"2 m, 0,- = 60°, 4>{ = 45°) -20 Thi* work Yang et al. -30 -50 CO 05 -60 -70 -80 4 8 12 Frequency (GHz) 16 20 Fig. 5.12: Comparison of RCS for a microstrip patch antenna, the bias field is in the y-direction, (4ttM 3 = 1780.0G, H0 = 360.0Oe, er = 12.8, d = 6.0 x 10- 4 m, L = 0.55 x 10_ 2 m, W = 0.4 x 10_ 2 m, 0,- = 60°, <f>i = 45°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 144 -20 -30 -40 O S -60 -70 -80 Frequency (GHz) Fig. 5.13: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47rM, = 1780.0G, Ho = 360.00e, er = 12.8, d = 6.0 x 10 - 4 m, L = 0.55 x 10“ 2m, W = 0.4 x 10"2 m, 0{ = 60°, & = 45°) -10 -20 -30 CQ -40 -o -60 -70 -80 1 2 3 4 5 Frequency (GHz) Fig. 5.14: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4itM 3 = 1780.0G, H q = 300.00e, AH = 40.0G, er(f = 2.2, er/ = 13.0 d& — 1.3 x 10_ 3 m, dj = 0.5 x 10 -3m L = 4.0 x 10 - 2m, W = 3.0 x 10- 2 m. 0,• = 30°, 4>i = 45°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 -10 Thi* work with a biued ferrite cover Yang with a biued ferrite cover -20 -30 -60 -70 -80 1 2 3 4 5 Frequency (GHz) Fig. 5.15: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (AwMs = 1780.0G, Ho = 300.00e, AH = 40.0G, erd = 2.2, er/ = 13.0 dd = 1.3 x 10- 3 m, df = 0.5 x 10-3m L = 3.0 x 10~2m, W = 4.0 x 10- 2 m, 8 { = 30°. 4>i = 45°) -20 Biased Ferrite Unbiased Ferrite • -30 -40 -50 Ias -60 -70 -80 4 8 12 16 20 Frequency (GHz) Fig. 5.16: Comparison of RCS for a microstrip patch antenna, the bias field is in the y-direction, (4ttMs = 1780.0G, H0 = 360.0Ge, eT = 12.8, d = 6.0 x 10_ 4 m, L = 0.55 x 10- 2m, W = 0.4 x 10~2 m, 0; = 60°, fa = 45°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 -30 Biased Ferrite Unbiased Ferrite * -40 -50 CQ -60 -80 -90 -100 Frequency (GHz) Fig. 5.17: Comparison of RCS for a microstrip patch antenna, the bias field is in the y-direction, (4xM s = 1780.0G, Ho = 360.OOe, er = 12.8, d = 6.0 x 10-4m, L = 0.55 x 10-2m, W = 0.4 x 10-2m, 0; = 60°, <p,- = 45°) -30 Biased Ferrite — Unbiased Ferrite • -40 -60 BJ -70 -80 -90 4 8 12 Frequency (GHz) 16 20 Fig. 5.18: Comparison of RCS for a microstrip patch antenna, the bias field is in the y-direction, (4irMs = 1780.0G, H0 = 36O.O0e, er = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10-2m, W = 0.4 x 10_2m, 0; = 60°, o,- = 45°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 Biaxed Ferrite Unbiased Ferrite os -60 Frequency (GHz) Fig. 5.19: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4 t M s = 1780.0(7, So = 360.0(7e, er = 12.8, d = 6.0 x 10-4m, L = 0.55 x 10_2m, W = 0 A x 10~2m, = 60°, fa = 45°) -30 Biased Ferrite Unbiased Ferrite -40 « -60 4 8 12 Frequency (GHz) Fig. 5.20: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4irM 3 = 1780.0(7, S 0 = 360.00e, er = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10-27n, W = 0.4 x 10~2m, 6 { = 60°, <j>i = 45°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 — Biased Ferrite — Unbiased Ferrite -40 03 QQ -60 Frequency (GHz) Fig. 5.21: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47nV/a = 1780.0(7, Ho = 360.00e, er = 12.8, d = 6.0 x 10-4 m, L = 0.55 x 10-2 m, W = 0.4 x 10_2m, 0,- = 60°, <£,- = 45°) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 -10 with a biased ferrite cover without a ferrite cover -20 -30 -50 -60 -70 -80 1 2 3 4 5 Frequency (GHz) Fig. 5.22: Comparison, of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4ttM 3 = 1780.0G, H 0 = 300.0Oe, A H = 40.0G, erd = 2.2, er/ = 13.0 dj, = 1.3 x lO-3 m, df = 0.5 x 10_3m L = 4.0 x 10_2m, W = 3.0 x 10-2 m, 9{ = 30°. <Pi = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 -10 with a biased ferrite cover without &ferrite cover -20 -30 -60 -70 -80 1 2 3 4 5 Frequency (GHz) Fig. 5.23: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4irM3 = 1780.0G, Ho = 300.00e, AH = 40.0(7, erd = 2.2, er/ = 13.0 dd = 1.3 x 10_377i, df = 0.5 x 10-3m L = 4.0 x 10"2m, W = 3.0 x 10“2m, 9{ = 30°, 4>i = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 -10 vrith a biased ferrite cover without &ferrite cover -20 -30 -60 -70 -80 Frequency (GHz) Fig. 5.24: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4tcM„ = 1780.0G, H q = 300.00e, AH = 40.0G, er(* = 2.2, erj = 13.0 dd = 1.3 x 10-3m, dj = 0.5 x 10-3m L = 4.0 x 10-2m, W = 3.0 x 10-2m, 0,- = 30°. 4>i = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 -10 with a biased ferrite cover without a biased ferrite cover -20 -30 ID 3 & r r ? -5 0 CQ -40 -60 -70 -80 Frequency (GHz) Fig. 5.25: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, ( 4 t M a = 1780.0G, H0 = 300.00e, A H = 40.0G, ertf = 2.2, erf = 13.0 di = 1.3 x 10- 3 m, d f = 0.5 x 10-3 m L = 3.0 x 10-2 m, W = 4.0 x 10_2m, 0,- — 30°, 4>i = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 -10 with a biased ferrite cover without a biased ferrite cover' -20 -30 CQ -40 -60 -70 -80 Frequency (GHz) Fig. 5.26: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4ttMs = 1780.0(7, Ho = 300.0(?e, A H = 40.0(7, er(f = 2.2, er/ = 13.0 di = 1.3 x 10-3m, dj = 0.5 x 10-3m L = 3.0 x 10-2m, 17 = 4.0 x 10-2m. 6 { = 30°, 4>i = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 -10 with a biased ferrite cover without a biased ferrite cover' -20 -30 CQ -40 -60 -70 -80 Frequency (GHz) Fig. 5.27: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4 ttM s = 1780.0C?, H q = 300.00e, A H = 40.0G, erd = 2.2, er/ = 13.0 dd = 1.3 x 10-3 m, dj = 0.5 x 10_3m L = 3.0 x 10~2m, W = 4.0 x 10- 2 m, 0,- = 30°. 4>i = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 -10 -20 -30 -40 ^-50 s? cn O -60 -70 -80 -90 1 2 3 4 5 Frequency (GHz) Fig. 5.28: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (AkM, = 1780.0(2, H0 = 300.00e, AH = 40.0G, erj = 2.2, ery = 13.0 di = 1.3 x 10_3m, dj = 0.5 x 10-3m L = 3.0 x 10-2 m, W = 4.0 x 10“2m, 0{ = 30°, <f>i = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 -10 -20 3 -30 -40 « -60 -70 -80 -90 Frequency (GHz) Fig. 5.29: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4 kM s = 1780.0G, H q = 3OO.O0e, A H = 40.0(7, €rd = 2.2, er/ = 13.0 dd = 1.3 x 10-3 m, d f = 0.5 x 10_3m L = 3.0 x 10-2 m, W = 4.0 x 10_2m, 0,- = 30°, = 45° Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 -10 — Ferrite-Dielectric cover •— Dielectric-Ferrite cover ' -20 -30 CO -40 ■o -60 -70 -80 2 3 4 5 Frequency (GHz) Fig. 5.30: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4wMs = 1780.0G, Ho = 300.00e, A H = 40.0G, erd = 2.2, er/ = 13.0 dd = 1.3 x 10-3m, df = 0.5 x 10_3m L = 3.0 x 10~2m, IF = 4.0 x 10"2m, 6 { = 30°, < P i = 45° Ferrite ___________; :................................................................ / r ^_____ Ground plane Fig. 5.31: Geometry and dimensions of a cross patch compared to the full patch antenna. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 Unbiased Ferrite-Cross patch Unbiased Ferrite-Full patch -30 -40 CQ -50 4 8 12 20 16 Frequency (GHz) Fig. 5.32: Comparison of RCS for a microstrip patch antenna ( M s = H q = 0, er = 12.8. <f = 6.0x 10-4 m, L = 0.55 x 10“ 2m, W = 0.4 x 10~2m, 0{ = 60°, & = 45° ) Unbiased Ferrite-Cross patch Unbiased Ferrite-Full patch -40 co CQ -60 4 8 12 16 20 Frequency (GHz) Fig. 5.33: Comparison of RCS for a microstrip patch antenna (Ms = Hq = 0, er = 12.8, d = 6.0 x 10-4 m, L = 0.55 X 10-2m, W = 0.4 x 10~2m, 9; = 60°, 4 > i = 45° ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 -30 Unbiased Ferrite-Cross patch Unbiased Ferrite-Full patch -40 -50 -70 -80 -90 -100 Frequency (GHz) Fig. 5.34: Comparison of RCS for a microstrip patch antenna (Ms = Ho = 0, er = 1‘2.8, d = 6.0 x 10_4m, L = 0.55 x 10-2m, W = 0.4 x 10-2m, 8 { = 60°, <pi = 45° ) -20 Biued Ferrite-Cross patch Biased Ferrite'FuIl patch -30 -40 -50 CO a s -60 -70 -80 Frequency (GHz) Fig. 5.35: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47rMs = 1780.0G, Hq = 360.00e, er = 12.8, d = 6.0 x 10-4m, L = 0.55 x 10_2m, W = 0.4 x 10_2m, 9{ = 60°, <?>,• = 45° ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -30 — Biased Ferrite-Cross patch — Biased Ferrite-Full patch -40 -50 « -60 -80 -90 -100 4 8 12 Frequency (GHz) 16 20 Fig. 5.36: Comparison of RCS for a microstrip patch antenna, the bias field is in the ar-direction, (4t M s = 1780.0(7, Ho = 360.0d?e, er = 12.8, d = 6.0 x 10-4 m, L = 0.55 x 10~2m, W = 0.4 x 10" 2m, 9{ = 60°, & = 45° ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 -30 Biased Ferrite-Cross patch Biased Ferrite-Full patch -40 -50 CQ •o -60 -70 -80 -90 -100 Frequency (GHz) Fig. 5.37: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4icMs = 1780.0G, H q = 360.00e, er = 1*2.8, d = 6.0 x 10~4m, L = 0.55 x 10-2m, W = 0.4 x 10~2m, 0{ = 60°, <?,- = 45° ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 -30 Biased Ferrite-Cross patch Unbiased Ferrite-Cross patch - -40 -50 cn -60 02 05 -70 -80 -90 Frequency (GHz) Fig. 5.38: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (47rMj = 1780.0G, H q = 360.00e, er = 12.8, d = 6.0 x 10-4 m, L = 0.55 x 10-2m, W = 0.4 x 10"2m, 6 { = 60°, <p{ = 45° ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 -40 Biased Ferrite-Cross patch Unbiased Ferrite-Cross patch. -50 -60 -80 -90 Frequency (GHz) Fig. 5.39: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4irMs = 1780.0G, H q = 360.00e, er = 12.8, d = 6.0 x 10~4m, L = 0.55 x 10-2m, W = 0.4 x 10-2m, 0,- = 60°, <f>i = 45° ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 -30 Biased Ferrite-Cross patch Unbiased Ferrite-Crass patch - -40 -50 -60 OS -70 -80 -90 4 8 12 16 20 Frequency (GHz) Fig. 5.40: Comparison of RCS for a microstrip patch antenna, the bias field is in the x-direction, (4xM3 = 1780.0(7, H q = 360.00e, er = 12.8, d = 6.0 x 10“4m, L = 0.55 x 10_2m, W = 0.4 x 10-2m, 0,- = 60°, fc = 45° ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 CHAPTER 6 CONCLUSION A num ber of issues involving planar structures printed on biased ferrite are inves tigated in this dissertation. First, the transmission m atrix of the normally biased ferrite substrate is derived in a closed form. Using the transmission m atrix, the G reen’s functions for normally biased ferrite structures is formulated. The perfor m ance of three edge-guided microstrip isolators is studied. A full-wave MoM based on Galerkin’s technique is utilized in this analysis. The resistive film termination on one side of the microstrip to absorb the backward wave is considered in this work. The effects of the resistive film, resistance value, and resistance width on the inser tion loss and isolation are studied. The insertion loss does not depend strongly on the resistance since the forward wave propagates mainly in the resistance free region. Second, the transmission m atrix for an arbitrarily biased ferrite substrate is also derived in closed form. The generalized Green’s function for a multi-layer arbitrarily biased ferrite structures is formulated. As an application of the generalized Green’s function, two microwave ferrite devices are studied, a phase shifter biased in the transverse direction and a magnetostatic surface wave transducer, biased in the lon gitudinal direction. Excellent agreement with previously published numerical and experim ental results is achieved. The effect of magnetization angle is also studied. T hird, the RCS of a biased ferrite antenna is evaluated. A 3-D full-wave MoM based on Galerkin’s technique is utilized. The RCS formulation is done in a very efficient and versatile manner. The transmission m atrix of the arbitrarily biased ferrite substrate is used to derive the excitation vectors in a closed form. This novel approach enabled us to analyze a variety of complex antenna geometries. Last, a novel cross-patch antenna is presented. The RCS is computed and com pared to th at of the full-patch antenna. A reduction in both the number of resonant peaks and RCS values is noted. In conclusion, this work makes five major contribu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 tions to the study of microwave ferrite devices 1. Development of a closed form transmission m atrix for a normally biased ferrite substrate 2. Full wave analysis of edge-guided mode microstrip isolator 3. Development of a closed form for arbitrarily biased ferrite substrate 4. RCS formulation for arbitrarily-shaped patches on arbitrarily-biased ferrite sunstrates using the trasmission matrix approach 5. Analysis of a novel cross-patch antenna. The future work includes but is not limited to the following areas, • analysis of more complex microwave devices such as three-dimensional circula tors, isolators, phase shifters, and transducers • analysis of antenna radiation patterns and input impedance. • analysis the plasma materials that has a perm ittivity tensor instead of the permeability tensor with the transmission m atrix technique. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES [1] D. M. Pozar, Microwave Engineering. Addison-Wesley, 1990. [2] K. G. Achintya and C. W. Denis, “Microstrip excitation of magnetostatic surface waves: Theory and experiment,” IEEE Transactions on Microwave Theory and Tech niques, vol. MTT-23, pp. 998-1006, Dec. 1975. [3] Peripheral Mode Isolator Operates From 3.5 to 11 GHz. Microwaves, Apr. 1969. [4] M. E. Hines, “Reciprocal and nonreciprocal modes of propagation in ferrite stripline and microstip devices,” IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 442451, May 1971. [5] B. Chiron and G. Forterre, “Emploi des ondes de surace electromagnetiques pour la realisation de dispositifs gyromagnetiques a tres grande largeur de band,” in Dig., Int. Semin. Microwave Ferrite Devices, 1972. [6 ] M. Blanc, L. Dusson, and J. Guidevoux, “Etude de la fonction isolation a tres large band utilisant des materioux ferrites,” in Dig., Int. Semin. Microwave Ferrite Devices. 1972. [7] P. D. Santis and F. Pucci, “The edge guided wave circulator,” IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 516-519, June 1975. [8 ] P. D. Santis, “Symmetrical four-port edge-guided wave circulators,” IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 10-18, Jan. 1976. [9] M. E. Hines, “Ferrite phase shifters and multiport circulators in microstrip and stripline,” in IEEE G-MTT International Microwave Symposium, pp. 108-109, 1971. [10] M. E. Hines, “A new microstrip isolator and its application to distributed diode am plification,” in IEEE International Microwave Symposium, pp. 304-307, 1970. [11 ] R. C. Kane and T. Wong, “An edge-guided mode microstrip isolator with transverse slot discontinuity,” in IEEE MTT-S Digest, pp. 1007-1010, 1990. [12] M. E. Hines, “Ferrite transmission devices using the edge-guided mode of propaga tion,” in IEEE G-MTT Int. Microwave Symp. Dig., 1972. [13] G. Cortucci and P. De Santis, “Edge-guided waves in lossy ferrite microstrips,” in Proc. European Microwave Conference, 1973. [14] P. De Santis, “Fringing-field effects in edge-guided wave devices,” IEEE Trans. Mi crowave Theory Tech., vol. MTT-24, pp. 409-415, July 1976. [15] K. Araki, T. Koyama, and Y. Naito, “Reflection problem in a ferrite stripline,” IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 491-498, Aug. 1976. [16] K. Araki, T. Koyama, and Y. Naito, “A new type of isolator using the edge-guided mode,” IEEE Trans. Microwave Theory Tech.. vol. MTT-23 (Letter), p. 321. March 1975. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 [17] P. De Santis, “A unified treatment of edge-guided waves,” Tech. Rep., Naval Research Laboratory, 1978. [18] B. Chiron, G. Forterre, and Rannou, “Nouveau dispositifs non reciproques a tres grande largeur de bande utilisant des ondes de surface electromanetiques,” Onde Electrique, vol. 51(9), pp. 816-818, Oct. 1971. [19] P. De Santis and R. Roveda, “Magnetodynamic boundary waves,” in European Mi crowave Conference, pp. C 5/2:l-C 5/2:4,1971. [20] L. Courtois, G. Declercq, and M. Peurichard, “On the nonreciprocal aspect of gyromagnetic surface waves,” in Proc. Seventh Annual Conference on Magnetism and Magnetic Materials, pp. 1541-1545, 1972. [21] L. 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Further reproduction prohibited without permission. 171 [61] H. Yang, A. Nakatani, and J. A. Castaneda, “Efficient evaluation of spectral integrals in the moment method solution of microstrip antennas and circuits,” IEEE Transac tions on Antennas and Propagation, vol. AP-38, pp. 1127-1130, July 1990. [62] G. Bock, “New multilayered slot-line structures with high nonreciprocity,” Electronics Letters, vol. 19, pp. 966-968, Nov. 1983. [63] T. I. M. Geshiro, “Analysis of double-layered finlines containing a magnetized ferrite,” IEEE M TT Symp., vol. MTT-S 87, pp. 743-744, June 1987. [64] N. E. Buris, T. B. Funk, and R. S. Silverstein, “Dipole arrays printed on ferrite substrate,” IEEE Trans. Antennas Propagat., vol. AP-41, pp. 165-175, Feb. 1993. [65] T. F. Elshafiey, J. T. Aberle, and E.-B. A. El-Sharawy, “Green’s function formula tion for multilayer normally biased ferrite structures using the transmission matrix approach,” IEEE AP Symp. Digest, vol. AP-S 96, pp. 330-334, July 1996. [6 6 ] J. A. Kong, Electromagnetic Wave Theory. New York, Wiley, 1990. [67] W. C. Chew, Waves ans fields in inhomegeneous media. New York, Van Nostrand, 1990. [68 ] L. W. Li, J. A. Bennet, and P. L. Dyson, “The coefficient of scattering dyadic green’s function in arbitrary multi-layered media,” Proc. 3rd Asia-Pac Microwav Conf. Tokyo, Japan, vol. APMC’90, pp. -, Sep. 1990. [69] J. K. Lee and J. A. Kong, “Dyadic green’s functions for layered anisotropic medium,” Electromagnetics, vol. 3, pp. 111-130, Apr. 1983. [70] S. M. Ali, T. M. Habashy, and J. A. Kong, “Spetral domain dyadic green’s functions in layered chiral media,” J. Opt. Soc. Am. A, vol. 9, pp. 413-423, March 1992. [71] L. Y. Hsia, H. Y. Yang, and N. G. Alexopoulos, “Basic properties of microstrip circuit elements on nonreciprocal substrate-suberstrate structures,” Journal of Electromag netic waves and applications, vol. 5, pp. 465-476, March 1991. [72] G. Tyras, “The permeability matrix for a ferrite medium magnetized at an arbitrary direction and its eigenvalues,” IRE Transactions on Microwave Theory and Tech niques, vol. MTT-7, pp. 176-177, Jan. 1959. [73] C. H. Edwards and D. E. Penney, Elementary Differential Equations with Applications. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1985. [74] C. J. Koza, Planar structures using oppositely-magnetized ferrite layers for broadband, high nonreciprocity phase shifters. Master’s thesis, Arizona State University, 1991. [75] T. Itoh, “Spectral domain immittance approach for dispersion characteristics of gener alized printed transmission lines,” IEEE Trans. Microwave Theory Tech.. vol. MTT28, pp. 733-736, July 1980. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [76] J. Parekh, C. K.W., and T. H.S., “Propagation characteristics of magnetostatic waves,” Circuits Systems Signal Process, vol. 4, pp. 9-39, Jan. 1985. [77] R. E. Riches, P. Brennan, P. M. Brigginshaw, and S. M. Deeley, “Microstrip ferrite devices using surface field effects for microwave integrated circuits,” IEEE Trans, on Magnetics, vol. MAG-6 , pp. 670-673, Sep. 1970. [78] W. K. Bing, Mutual Coupling Analysis for Conformal Microstrip Antennas. PhD thesis, Ohio State University, 1984. [79] F. K. Eugene, F. S. John, and T. T. Michael, Radar Cross Section. Artech House, 1992. [80] F. Baden, Ferrites at Microwave Frequencies. Peter Peregrinus, 1987. [81] T. F. Elshafiey and J. T. Aberle, “Green’s function for multilayer arbitrary biased anisotropic structures: An application on phase shifters, transducers and magnetiza tion angle effect,” IE E E M T T Symp. Digest, vol. to be submitted to MTT-S 97, pp. -, June 1997. [82] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 409-418, May 1982. [83] K. Mclnturff and P. S. Simon, “The fourier transform of linearly varying functions with polygonal support,” IEEE Trans. Antennas Propagat., vol. AP-39, pp. 14411443, Sep. 1991. [84] J. T. Aberle, “Hyperid spectral/spatial method of moments for solving antenna prob lems,” Apr. 1996. Unpublished Notes. [85] E. H. Newman and D. Forrai, “Scattering from a microstrip patch,” IEEE Trans. Antennas Propagat., vol. AP-35, pp. 245-251, March 1987. [8 6 ] D. M. Pozar and D. H. Y. Yang, “Correction to ’’radiation and scattering charac teristics of microstrip antennas on normally biased ferrite substrates”,” IEEE Trans. Antennas Propagat., vol. AP-42, pp. 122-123, Jan. 1994. [87] J. T. Aberle, “Analysis of arbitrary microstrip planar structures,” Apr. 1995. Unpub lished Notes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A TH E TRANSMISSION MATRIX OF A NORMALLY BIASED FER R ITE SLAB Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 Jl Et z=0 z=-d Fig. A.l: Geometry of single layer structure. In this Appendix, the transmission m atrix is derived in the spectral domain for a magnetic substrate of thickness d as shown in Fig. A.I. The symbolic manipulator of Maple Vis extensively utilized to alleviate the huge algebraic burden and to obtain the solution of a fourth order differential equation. To find the t r ansmission m atrix for a ferrite slab, we start from Maxwell’s equations —jupo ft -H = V x E V • (pQft ■H ) = 0 (A.l) juieE = V x H eV • E = 0 where ft is the permeability tensor of the ferrite. For a ferrite magnetized in the z-direction, the permeability tensor is given by fl f1 = Pa —jk 0 JK ft 0 0 0 1 where K = Wm = tZ)2 o —up 74 7TiV/s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.2) 175 u = Uo = u 0 +. -3 7 Ho wo is the precession frequency, Ho is a z-directed impressed DC magnetic field, the gyromagnetic ratio, T = 7 is is the relaxation time, and AH is the 3-dB line width. To simplify the formulation, the wave equation is expressed in term s of fields th at axe perpendicular to the direction of magnetization. From Maxwell’s equations, we can write ' V x £ ) - k ltTE = 0 Vx(/i (A.3) where ko = u}y/fi0 e0. Manipulation of (A.3) yields three scalar equations r\2 r\ 2 A a ~ k 0 ^ A ) E* + { - K x K y - JK— ) Ey +(-K l<y— + j f i Kx — ) E z= 0 (A.4) Q 2 ^ a2 ^ A A ( - K xK v + j k — ) E x + (I< 1 -n — - k le r & ) E y +{]nKy— + nK x — ) E z= 0 (A.5) d ^ d d (jI<xf i + K yK— ) Ex +(-I<xK— + j K yn — ) Ey + (/i/^ + /x /^ -fc o £ r A) E z= 0 (A. 6 ) Eliminating E z from the three equations and rearranging such th at and are separated, we obtain d 2 Ex . W9 q z 2 + (w 7 w 2 — w3 ws)Ex + (W7 W4 — w3 w 2 )Ey = 0 (A.7) d '2E — w9~q^T + ( ^ 5 ^ 4 - wiw 2 )E y + {w5 w 2 - wiw 6 )Ex = 0 (A.8 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 where u>i = K x K y([i2 — k 2) — jC ik . w2 = CxK xK y - iiK x K y( K 2x + K 2) wz = C\n —K x2{fi2 — k 2 ) wA = fiKKKl + K D - ^ K l + K D + C ^ - K l ) w5 = Cin — K y2(fi2 — k 2) we = fiK y{K l + K 2) — w7 = K xK y(/j.2 — k2) + jClK Wg = Cl = k%erA A = + K 2) + Cx{Ci - I<1) W 7W \ — W 2 W 5 (J2 — k 2 Next, wedefine elliptical polarization components as E+ = Ex + j E y (A.9) E~ Ex —jE y (A .10) = Noting th at Ex = \{ E + + E~) and E y = -^-(E+ —E ~ ). we can rewrite (A.7) and (A .8) as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Decoupling equations (A .11) and (A.12) generates the following fourth order differ ential equations dAE + ^ - d2E + + A d zA ^ - + A ^ - o (A .13) = 0 dz2 (A .14) where , _ 2kltr{n 2 - K2) - K_K +(fl + K + 1) 1 f 2Arger (/i2 - K2) — A ./v + (ft — K + 1) 2 ( / i + /c) = . 2k%£r(fl2- K2) - 2 K - K + (fl + K + 1) 2 ( f i -(- K 2+ K 2_ { p - 2(p - k k ) 2fc026r ( ^ 2 - K2)- K - K + j f l - K + 1) ) 2{y. — k ) K - l){fl + K + I) 4 (fJL — K,)(fl + K) K+ = K x + jK y (A .15) K - = K x — jA 'y (A .16) The solutions of (A .13) and (A. 14) can be w ritten as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 E + = A \s in h (R \z ) + B ico sh(R iz) + A 2 sin h (R 2 Z) + B 2 Cosh(R2 z) (A .17) E (A .18) = A zsinh(R xz) + B zcosh(R \z) + A 4 sin h (R 2 z) + B \cosh(R 2 z) where the constants A’s and B’s are determ ined from the boundary conditions, and R i and i ?2 are the roots of the differential equations given by Ri Ri = ~ = \- 2 W + V% £ (A.19) ~ - h (A.20) a From Maxwell’s equations, we can write the relationship between the elliptical com ponents of the magnetic field and the electric field in m atrix form as ' H+ tr fC=K± ue ' = a T _ k -2 KL 2 ■Z= £ + k+* ' a£+ ' L dj: 8E~ dz J (A.21) where k±2 = Ai K 2 + K 2 - u } 2fi0e { fi^ K ) — uj2 fi0ef i ( K2 + K 2 — u)2floe- --------- Substitution of (A.17) and (A.18) into (A.21) yields after some manipulation H + = D i[A icosh(R iz) + B isin h (R iz)] + D 2 [A2 cosh{R 2 z) + B 2 sin h (R 2 z)] (A.22) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 H = D 2 [AiCosh(Riz) + B \s in h { R \z )] + D 3 [A2 Cosh(R 2 z) + At z B 2 sin h (R 2 Z)] (A.23) = 0,we can evaluate the constants Ax, A 2 , Bx and B 2 as follows Ax = D 5 H + + D 6 H~ (A.24) A2 = DsH + + D 7 H~ (A.25) Bx = B2 = U2 - E + + ----- — E- 77.2 — 711 712 — ^1 ---- — E+ + ---- 1-----1 712 —^1 ^2 — (A.27) where 1 = u e „ ,K + K a T * ' (^ — we „ , K 2 „ d 2 = D3 = , , , K l, + "'f ) K+K- — , V ~ + k ' ) ni ) ^ R 2( - ^ + ( ~ K ± ^ + k+2 )n2) 4 = u e n ,K + K 2(^ . _2 5 uj2e2R2Ri(n2 —rii ) (A -26) /fL + n2_2 ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ISO ________ Ax________ Dr = u 2 e2 R 2 R i(n 2 — ni) -A x Do u 2 e2 R 2 R i{n 2 —n x) D* = T li Di 9i + -^1 = 92 9i + R 2 n2 = 92 9\ = 92 = 2k ltr (n 2 — K2) — K - K + ( f l + k + 1) 2 {fi + k) K+(p + K — 1) 2 (n -f K) Equations (A.25)-(A.27) can be w ritten in m atrix form as follows -Ai a2 = M -1 (0) Bi . b2 . ' E+ ' bt (A.28) Ht where 0 0) = _n2_ Tl2— Tli 0 0 -i f l 2 “ Tlx 1__ „ 712 7 1 2 — TU Ds Ds 0 0 Dr 0 0 Following the same procedure for z = —d, we obtain El b2 Ht .h 2 ' ' . A\ a2 = M (-d ) Bi . b2 (A.29) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LSI where —sinh(R id) —nisin h (R id ) Dicosh(Rxd) D 2 cosh(R\d) M {-d ) = cosh(Rid) cosh(R 2 d) n 2 cosh(R 2 d) ■riicosh(Rid) —D isinh(R id) —D \sin h {R 2 d) —D 2 sin h (R id ) —D zsinh{R 2 d) —sin h (R 2 d) —n 2 sin h (R 2 d) D icosh(R 2 d) Dzcosh{R 2 d) From equations (A.28) and (A.29) we can write ' Et e2 Ht h2 ' E f a2 Ef = M (- d ) = M ( - d ) M ~ l {0) B! Ht . b2_ .H r Et El {d) Ht . nr ' ' =f (A.30) where =± (A.31) We use the following relations between th e elliptical and cartesian field and current components at both sides of the slab ' E f Er = [5] Ht .n r . E X2 - Ejn Jx 2 Jy? . (A.32) ' E f Eyi Er = [5] Jxi Ht .n r . . Jyi . ' (A.33) where [5] = J 1 -J 0 0 0 0 0 1 o 1 0 0 -J J 1 1 (A.34) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 Using equations (A.30), (A.32) and (A.33) we can write EXl Eyi JXx ’K ' Tyi = [S]-1^ (d)[S] J 12 . 4 (A.35) . Jyi . . From th e above equation we can write the transmission m atrix for normally biased ferrite slab =± T(d) = [S]-l T (<f)[S] (A.36) where th e elements of the transmission m atrix are given by T \\ = T-jf = —---------- -[(n2 + nxn2 — 1 —ni)cosh(Rid) + (—ni —/ix/i2 + 1 + ri2 )cosh(Rul)] Z{n2 — ni) 2(n2 T\z [(n2 + 7ixn2 + 1 + ni)cosh(Rid) + (—Tix —n \n 2 — 1 —n 2 )cosh{R 2 d)\ T lx ) —D s)sinh(R id) + (n 2 + 1)(Z?8 — D 7 )sin h (R 2 d)\ = T \4 = Z l2 = ~^ [(tix + l)(f ^5 + D o)sinh(Rid) + (n 2 + 1)(-D8 + D 7 )sin h (R 2 d)] T2i T* = = T 22 = T 23 = 2( ti2 - Tlx) [(—712 + 71x712 + 1 —ni)cosh(R id) + (nx —riin 2 — 1 + n 2 )cosh{R 2 d)\ -1 T 22 = ^- n _ n j [(—:n 2 Z 2l = T 24 — Z 22 — - [ ( - n x + 1 )(A > + 71x 712 - — 1 + ni)cosh(R id) + (rii —n xn 2 + 1 - n 2 )cosh(R 2 d)\ D e)sinh(R id) + ( - n 2 + 1 )(D 8 - D 7 )sin h (R 2d ) ] Tix + l)(-^5 + D 6 )sin h (R ld) + (—n2 + 1)(-D8 + D 7 )sinh(R 2 d)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 ^31 = Yu = 2(n 2 — Tii) ^ ^32 = Yyi = ^ ~ l ^ Dl ~ Di) s in h (R i d) + (n i “ 1)(-°3 - D 4 )sin h (R 2 d)] n ”) [(” 2 —^ 2 )sinh(R id) + (ni -f-1 )(.0 3 —D 4 )sin h (R 2 d)] f 1Jl = ^[ ( D 1 - D 2 ){D 5 - D 6 )cosh(Rl d) + (D 4 - D faa = T34 =f 241 = ^42 = n2 (2 = 5^2 3 )(Ds - D 7 )cosh(R 2 d)} J-[{D t - D 2 ){D 5 + D 6 )cosh(Rl d) + (D 4 - D 3 )(DS + D 7 )cosh(R 2 d)} = 2^ --- n "")^772 _ = 777---------- r [(n 2 Z{n2 — n 4) + D 2 )sinh(R id) + (—ni -f- 1 )(Z?4 + D 3 )sin h (R 2 d)] + D 2 )sinh(R id) — (ni + 1)(-O4 -t- D 3 )sin h (R 2 d)] Taz = T i = y [(£>! + D 2 )(D 5 - De)cosh(Rl d) + (D 4 + D 3 )(D 8 - D 7 )cosh(R 2 d)} T44 = T 22 — + D 2 ){D$ + Ds)cosh(Rid) + (D 4 + D 3 )(D 8 + Dr)cosh(R 2 d)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B THE TRANSMISSION MATRIX OF A DIELECTRIC SLAB Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 The transmission m atrix of a dielectric slab can be w ritten as [40], =E ~ J T — T — cosh(kd) Z 1 0 0 1 (B .l) = yr^-sinh(kd) kkj. kl ~ kd k*ky kxky ky - k\ = Y^-sinh{kd) rCKfl kl ~ kd (B.2) Y Y k x ky kx ky kl — kj (B.3) where, zc = y= (B.4) kd = x /^ o (B.5) k = + fc»2 - k i (B.6) where d, is the dielectric slab thickness, er , is the dielectric constant, rj is the freespace intrinsic im pedance and to, is the free-space wave number. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX C TH E SEMI-INFINITE SPACE GREEN’S FUNCTION Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 187 E J Air Ground Planes Ferrite t I E J Fig. C.l: Geometry of single layer isolator structure. ___ + E . T *ZU = T — -1 a J . ' =E = fa - T = T ' ■ =J ^ ■ E SUL J (C .l) We force E = 0 by placing a ground plane at distance dj, from the source plane, as shown in Fig. C .l. This results in the following equation = T = J ~ l !_ E = ZaTa = GaJ J (C .2 ) In the above equation, Ga has units of impedance. Most of the semi-infnite G reen’s functions has the units of admittance. Therfore, the admitance Green’s function can be w ritten as = J —T ~ l —+ J = T aZ a E (C.3) Using the elements of the dielectric trasmission m atrix, we can write Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1SS = -1 G„ = JYC kkdtanh{kd) kj - kj - k xky - k xky kj-kj |t£—>00 (C.4) After taking th e limit, the adm ittance Green’s function has the following form r a kkd kj~kj - k xky ~ k Xky k j - kj (C.o) where, Yc v (C.6) kd = IjJy/JtQt (C.7) y/kj + kj - kj (C.S) k = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX D RESISTIVE MATRIX ELEMENTS IN SPATIAL DOMAIN Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 I 2 [3 4 5 Conductor (») !y Resistive L ay er Rs (b) Fig. D.l: Current basis functions for (a) Longitudinal current (b) Transverse current We selected the piece-wise linear subdomain basis function as shown in Fig. D .l. The conductor is subdivided into N equal overlapped segments. The elements of the submatrices of th e total resistance m atrix given in equation 3.1. R xx and Ryy, are found to be i f R y y — R$d> 00 3 1 6 0 1 6 1 3 0 0 0 2 3 1 6 0 1 6 1 3 0 0 0 Similar matrices can be easily derived for a different number of basis function and for a different location of the resistance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX E GREEN’S FUNCTION FO R A SLOT LINE W ITH A FERRITE SUBSTRATE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. E.l: Geometry of single layer slot line structure. E .l Green’s Function for a Single Slot Line Ferrite Substrate The Green’s function for a ferrite slot line relates the slot currents to the slot fields as Jx = Gs l E x (E .l) The electric field is continous on surface (1) and (2) Et = E~=EX E+ = E~ = E 2 (E.2) (E.3) (E.4) where E xand E 2 are the tangential electric fields at the boundaries of the layer, J * and J } are the currents due to the electric fields in the upper and lower semispaces (above surface (1) and below surface (2), respectively), these currents are defined as J+ = G 1 E+ = Gl E l (E.5) J <2 = G 2 E 2 = G2 E 2 (E.6) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 (E.T) where G\ and G 2 are semi-infinite space Green’s functions above surface (1) and below surface (2), respectively. The total surface currents at both sides of the slab can be written as J x and J 2 Jx = aB x ( f f + - f f + ) = J + - J f (E.8) J2 = dn x { H t - H t ) = J t ~ J 2 (E.9) (E.10) Note the an is +ve in the z-direction for surface (1) and -ve in the z-direction for surface (2). The surface current at surface (1) is due to the slot electric field while there is no current source at surface (2) hence, J 2 = 0 J2 — —J 2 — —G 2 E 2 Jx = J t —J\ — G \E \ — J\ (E .ll) (E.12) The transmission m atrix relates the electric fields and the electric currents just inside the slab on both sides. r ____ e . J ^ 2 " 2 . ' =E -T f z =T =J Y T ' _ —_ Ex From equations E.12, E.12 and E.13 we can write Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 194 = E = T =E __ =T _____ = T _ E 2 = T E x ~h Z J\ = T E \ + Z G \E \ — Z J \ = J2 = t _ = j ___ = t __ = j (E.14) _____ - J _ - G 2 E 2 = Y E 1 + T J x = Y E i + T G \E \ - T Jx (E.15) S ubstitute by E 2 from equation E.15 to E.15 and after some manipulation we get -T J, = [Gx + {G2Z = E.2 - J _ -E + T )\--l/\ G" 2T =T +Y)]El = Gs l E x (E.16) (E.17) Green’s Function for a Multilayer Slot Line Ferrite Substrate Jl. Fig. E.2: Geometry of single layer slot line structure. Before the second layer is added, we have that the current and the electric field below surface 2 are related by J 2 = G2 E 2 (E.1S) After adding the second layer, the relation between J 2 and E 2 is no longer simple. At surface (3), there is no current source so we can write J 3 = 0 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 (E-19) J 3 = G3E 3 The transmission m atrix of the new added layer can be w ritten as —_ e 2 ' =E — . ^2 . T =T Z = T =J Y f ' _ e3 —+ (E.20) . ^3 . From equation E.19 and E.20 we can find the new relation between J 2 and E 2 as J2 = G2 E 2 _ =r = j_ = e = t_ G 2 = (Y + T G3)(T + Z G 3 ) - 1 (E.21) (E.22) The above expression for G 2 replaces the G 3 in the main equation E.17 and G 3 becomes the new semi-infinite free space Green’s function below the newlayer num ber (3). Noticealso th at the elements of the new layer transmsition m atrix are quite different than the transmission matrix of the m ain ferrite layer, but because of the simplicity we used the same symbols without any extra notation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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