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Discontinuities in reciprocal and nonreciprocal inhomogeneous waveguides McRitchie, William Kenneth 1975

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DISCONTINUITIES IN RECIPROCAL AND NONRECIPROCAL INHOMOGENEOUS WAVEGUIDES by W. KENNETH McRITCHIE B.Sc, Queen's University, Kingston, 1969 M.Sc, Queen's University, Kingston, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA May, 1975 In presenting th i s thes is in par t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t i on of this thes is fo r f i nanc ia l gain sha l l not be allowed without my written permiss ion. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date J ) l ] ^ i ! X 6 ABSTRACT Four types of waveguide d i s c o n t i n u i t y are i n v e s t i g a t e d : (1) the i n t e r f a c e between r e c i p r o c a l homogeneous and r e c i p r o c a l inhomo-geneous waveguides; (2) the i n t e r f a c e between r e c i p r o c a l homogeneous and nonreciprocal i n -homogeneous waveguides; (3) a t h i n metal diaphragm i n a r e c i p r o c a l inhomogeneous waveguide; (4) a t h i n metal diaphragm i n a nonreciprocal inhomogeneous waveguide. Mode matching i s used to obtain t h e o r e t i c a l s o l u t i o n s f o r d i s c o n t i n u i t i e s of types (l)-(3); experimental r e s u l t s are obtained f o r type (3) and type (4) d i s c o n t i n u i t i e s . D e t a i l e d studies are made of the two types of i n t e r f a c e d i s -c o n t i n u i t y . Both E-plane and H-plane d i e l e c t r i c loading are i n v e s t i g a t e d f o r the r e c i p r o c a l case while the configuration used f o r the nonreciprocal structure i s that of the twin-slab f e r r i t e loaded waveguide. Based on these analyses, the i n t e r f a c e and the two waveguides, homogeneous and inhomogeneous, are described by r e l a t i v e l y simple equivalent transmission c i r c u i t s . In these c i r c u i t s , unique normalized equivalent transmission l i n e c h a r a c t e r i s t i c admittances are defined f o r the inhomogeneous wave-guide. These admittances are shown to be not g e n e r a l l y p r o p o r t i o n a l to the wave admittances, even when such can be defined. In the nonreciprocal case, the c h a r a c t e r i s t i c admittances are nearly the same i n the two d i r e c -tions of propagation although the phase c o e f f i c i e n t s can be very d i f f e r e n t . The two types of diaphragm d i s c o n t i n u i t y are i n v e s t i g a t e d ex-perimentally at a frequency of 8.5 GHz. The experimental procedure i s the same f o r both types and requires the measurement of r e f l e c t i o n and trans-mission c o e f f i c i e n t s of unmatched sections of the inhomogeneous waveguide. T h e o r e t i c a l r e s u l t s are obtained for the r e c i p r o c a l case which i n d i c a t e that the error i n the measurements i s l e s s than ±6%. i i CONTENTS FIGURES . . . . . . . . . . . . . . . . . . . . . Iv TABLES . . . v i SYMBOLS v i i ACKNOWLEDGEMENT . x i i 1. INTRODUCTION 1.1 Background 1 1.2 Objectives . . . . . . . . . 5 2. RECIPROCAL INHOMOGENEOUS INTERFACE 2.1 Introduction. . . . . . . . . . . . . . . . . 7 2.2 E-plane dielectric loading (u - UQO • •• • • 12 2.3 E-plane dielectric loading (p ^ p g ) . 17 2.4 H-plane dielectric loading . . 19 3. NONRECIPROCAL INHOMOGENEOUS INTERFACE 3.1 Introduction 29 3.2 Mode matching 30 3.3 Normalized reflection and transmission coefficients. . . . 35 3.4 Equivalent c i r c u i t representation. -.. . . . . . . . . 38 3.5 Numerical results and discussion . . . 40 4. EXPERIMENTAL INVESTIGATION 4.1 Introduction. . 54 4.2 Method of measurement. . . . . . . . . . . . . . 54 4.3 Derivation of the diaphragm effect from the measurements . . 55 4.4 Measurements. . . . . . . . . . . . . . . . . 61 4.5 Results and Discussion . . . . . . . . . . . . 74 5. CONCLUSIONS . . . 85 Appendix A. REFLECTION AND TRANSMISSION COEFFICIENTS . FOR A RECIPROCAL INTERFACE . . . . . . . . . 87 Appendix B. INTERFACE BETWEEN EMPTY AND H-PLANE DIELECTRIC SLAB LOADED WAVEGUIDES. . . . .89 Appendix C. TE* (n odd) MODES IN THE TWIN-SLAB FERRITE LOADED WAVEGUIDE . . . . . . . . 96 REFERENCES H i 100 FIGURES 1. Nonreciprocal f e r r i t e phase shifter configurations . . . . . . . . 2 2. Cutaway diagram of a four-bit f e r r i t e phase shifter 4 3. Inhomogeneously f i l l e d waveguides 8 4. Junction of two reciprocal waveguides, A and B 9 5. Equivalent lumped-element circuits used to represent the junction between empty and inhomogeneous waveguides 11 6. Junction of empty and E-plane loaded waveguides 14 7. Equivalent circuit parameters for E-plane loading; a/XQ = 0.691 15 8. . Normalized characteristic admittance for E-plane center loading; e r * P r = 4.0, a/XQ =0.691 ...... 16 9. X and B for E-plane center loading; a/Xg = 0.691 18 10. Variation of Y with dielectric thickness; e r = 4.0, u r = 1.0, a/XQ = 0.691, b/X0 = 0.3 . . . . 20 11. Effect on the solution for B of varying the ratios of TE:TM and LSE:LSM modes separately and together; b/XQ = 0.3, 2t/b = 0.7, e r = 8.0, u r = 0.75, total no. of modes = 28 . . . . . 23 12. Convergence of the solution for B with various ratios of TE:TM (=LSE:LSM) modes; b/X_ = 0.3, 2t/b =0.7, e = 8.0, V = 0.75 . V T . . . . . . . 24 r 13. Normalized characteristic admittance for H-plane loading; E • y = 6.0, b/Xn = 0.3 26 r r ' 0 14. X and B for H-plane loading; e *u =6.0, b/X Q = 0.3 27 15. X and B for H-plane loading; y r = 1.0, b/XQ =0.3 . . . . . . . 28 16. Twin-slab fe r r i t e phase shifter model . . . . . . . 31 17. The modes excited by a single mode incident from either A or B . . . . . . . . .32 18. Identification of the reflection and transmission coefficients . 36 19. Equivalent transmission line representation of the junction . . 39 20(a). Normalized phase coefficients and equivalent characteristic admittances of waveguide B versus 2t/a; 2d/a = 0.04, £^=1 . . 42 Iv 20(b). Normalized phase coefficients and equivalent characteristic admittances of waveguide B versus 2t/a; 2d/a = 0.04, £^=16 . • . 43 21. Reflection and transmission coefficient parameters versus 2t/a; 2d/a = 0.04, £ d = 1 45 22(a). Interface 1 equivalent c i r c u i t parameters versus 2t/a; 2d/a = 0.04, e d = 1 46 22(b). Interface 2 equivalent c i r c u i t parameters versus 2t/a; 2d/a = 0.04, e d = 1 47 23(a). Interface 1 equivalent c i r c u i t parameters versus 2t/a; 2d/a = 0.04, e d = 16 . . . . . . . . . . . . . . . . 48 23(b). Interface 2 equivalent c i r c u i t parameters versus 2t/a; 2d/a = 0.04, e d - 16 49 24. Normalized phase coefficients and equivalent characteristic admittances of waveguide B versus 2d/a; 2t/a =0.24, £ ^ = 1 6 . .51 25(a). Interface 1 equivalent ci r c u i t parameters versus 2d/a; 2t/a = 0.24, e d = 16 . . . . . . . . . . . . . '. . . . . . . .52 25(b). Interface 2 equivalent ci r c u i t parameters versus 2d/a; 2t/a = 0.24, e d = 16 53 26. Block diagrams of experimental systems 56 27. Junctions of homogeneous and inhomogeneous waveguides identifying the semi-infinite interface parameters . . . . . . 57 28. Equivalent circuit of a thin metal diaphragm in a nonreciprocal waveguide 62 29. Inhomogeneous loading configurations for the four experimental cases . . . . . . . . . . . . . . 63 30. Waveguide and polystyrene tapers relative to the inhomogeneous section for cases (b) and (c) . . . 66 31. Arrangement of the magnetizing wire . . 69 32. Differential phase s h i f t , »^ and total interface effect, K, versus magnetizing current; lengths of f e r r i t e , & = 1.0 i n and 1.25 in . . . . . . . . . . . 70 33. Diaphragm configuration and transverse position .. . . . . . 75 34. Equivalent circuit parameters of a metal diaphragm in ceramic loaded waveguide . . . . . . . . . . . . . . . . . . . 79 35. Equivalent c i r c u i t parameters of a metal diaphragm in fe r r i t e loaded waveguide; magnetizing current = 15 A. . . . .83 36. Junction of empty and H-plane loaded waveguides 90 v TABLES 1 . Example of mode amplitude decay for H-plane loading 2 2 2 . Assumed characteristics of the f e r r i t e 4 1 3 . Characteristics of the f e r r i t e 6 4 4 . Sources of experimental error 7 2 5 . Effect of errors in interface measurements 7 3 6 . Interface measurements, ceramic ,(Al2C>3) loading 7 6 7 . Semi-infinite interface parameters 7 6 8. Interface equivalent c i r c u i t parameters 7 7 9 . Typical diaphragm wave-transmission matrix, case (a) 7 7 1 0 . Interface measurements, magnetized ferrite loading . . . . . . . . 8 0 1 1 . Semi-infinite interface parameters . t 8 0 1 2 . Interface equivalent c i r c u i t parameters 8 1 1 3 . Typical diaphragm wave-transmission matrix, case (c) . . . . . . 8 2 v i SYMBOLS a = broad dimension of both waveguides A and B = complex amplitude coefficient of the T E ^ Q mode in waveguide A A = reciprocal homogeneous waveguide Aj , A3 = interface wave-transmission matrices A2 = wave-transmission matrix of a length of inhomogeneous wave-guide A^ = diaphragm wave transmission matrix A ,B ,C ,D = amplitude constants for the n t n mode in waveguide B n n n' n r 0 b = narrow dimension of both waveguides A and B bj = complex amplitude coefficient of the L S E _ . Q mode (E-plane loading) or the L S E ^ . . or L S M ^ mode (H-plane loading) in waveguide B , b± = complex amplitude coefficient of the T E TQ mode in waveguide B B = reciprocal or nonreciprocal inhomogeneous waveguide = normalized shunt susceptance in the two-element equivalent circuit of a reciprocal interface = normalized shunt susceptance in the equivalent c i r c u i t of a diaphragm in reciprocal waveguide B~ = normalized shunt susceptances i n the equivalent c i r c u i t of a diaphragm in nonreciprocal waveguide B 2 normalized shunt susceptance in the three-element equivalent circuit of a reciprocal interface B^, B* = normalized shunt susceptances in the equivalent c i r c u i t of a nonreciprocal interface with a wave incident from wave-guide A and waveguide B, respectively c 8 3 distance between the side wall of waveguide B and the i n -homogeneous loading material d = half the spacing between the two f e r r i t e (or dielectric) slabs v i i = distance between the side wall of the waveguide and the i n -side edge of the diaphragm d . = width of one of the metal strips used to form the diaphragm min d = c max = transverse electric f i e l d of the TE^Q mode in waveguide A e„. = transverse electric f i e l d of the L S E . _ mode (E-plane loading) B j or the L S E ^ j o r I S M ^ mode (H-plane loading) i n waveguide B e£. = transverse electric f i e l d of the T E * N mode in waveguide B B j J u total transve respectively E E = rse electric fields in waveguides A and B, A, B n t h empty region of waveguide B h = transverse propagation coefficient of the m n mode in the m = transverse magnetic f i e l d of the TE^Q mode in waveguide A hgj = transverse magnetic f i e l d of the LSE..Q mode (E-plane loading) or the i'SEj.j or ^ sM^ mode (H-plane loading) i n waveguide B = transverse magnetic f i e l d of the TE* mode in waveguide B B j J u H A, H = total transverse magnetic fields i n waveguides A and B, A B respectively H c = coercive force of fe r r i t e material V = 2 i r / X0 K = di f f e r e n t i a l phase shift of an unmatched section of non-reciprocal waveguide due to interface effects I - length of the inhomogeneous section 1 = transverse propagation coefficient of the m*"*1 mode i n wave-in guide B,in the dielectric region for the reciprocal case and in the fe r r i t e region for the nonreciprocal case M, N = numbers of modes used to approximate the total transverse fields at a discontinuity M r = remanent magnetization of the f e r r i t e M g = saturation magnetization of the f e r r i t e v i i i = transverse propagation coefficient of the m t n mode in wave-guide B in the region between the two f e r r i t e (or dielectric) slabs T2_* r2 ~ s e m i ~ ^ n : f l n l t e interface reflection coefficients with a unit amplitude wave incident on the interface R = reflection coefficient of an unmatched section of inhomo-geneous waveguide R^  = semi-infinite interface reflection coefficient R^  = normalized semi-infinite interface reflection coefficient t = half the thickness of the inhomogeneous loading material t^, = semi-infinite interface transmission coefficients T = transmission coefficient of an unmatched section of inhomo-geneous waveguide T* = transmission coefficients of an unmatched section of non-reciprocal waveguide = semi-infinite interface transmission coefficients = normalized semi-infinite interface transmission coefficients X = normalized series reactance in the two-element equivalent c i r c u i t of a reciprocal interface —aZ = e for a reciprocal section = e a ^ for a nonreciprocal section X* = normalized series reactances in the equivalent c i r c u i t of a diaphragm in nonreciprocal waveguide X^, Xg = normalized series reactances i n the three-element equivalent c i r c u i t of a reciprocal interface X * , X* = normalized series reactances in the equivalent c i r c u i t of a nonreciprocal interface with a wave incident from waveguide A and waveguide B, respectively Y = normalized equivalent transmission lin e characteristic admittance in a reciprocal inhomogeneous waveguide ix y* = normalized equivalent transmission line characteristic ad-mittances in a nonreciprocal waveguide Y. = input admittance in r a = attenuation coefficient in waveguide B for reciprocal loading a- = attenuation coefficients in waveguide B for nonreciprocal loading a = ( a + + cT)/2 3 = phase coefficient in waveguide B for reciprocal loading 3 * = phase coefficients in waveguide B for nonreciprocal loading § = (e + + fT)/2 3 = phase coefficient in waveguide A «B.. Y = the gyromagnetic ratio (= 1.76 x 10^ oe~^- sec--'-) TA' ^B = P r o P a S a t i ° n coefficients of the dominant modes in waveguides A and B, respectively for reciprocal loading t i l = propagation coefficient of the i mode in waveguide B for nonreciprocal loading e = permittivity of the loading material e , = relative permittivity of the di e l e c t r i c material between the d two fer r i t e slabs e£ = relative permittivity of the f e r r i t e CQ = permittivity of free space e r = relative permittivity of the loading material n = cf>2 - 3& for reciprocal loading = $2 - §Jt for nonreciprocal loading 9 = phase angle of the transmission coefficients of a reciprocal interface (semi-infinite case) = phase angle of the transmission coefficients of a diaphragm in reciprocal waveguide x 6* = phase angles of the transmission coefficients of a diaphragm in nonreciprocal waveguide 0^ , 82 = phase angles of the transmission coefficients of a nonrecip-rocal interface (semi-infinite case) K = off-diagonal element of the relative permeability dyadic of the f e r r i t e X Q = free space wavelength u = permeability of the loading d i e l e c t r i c , reciprocal case = diagonal element of the relative permeability dyadic of the fer r i t e u e = effective relative permeability of the fe r r i t e U Q = permeability of free space u r = relative permeability of the loading material II , 11^  = elect r i c and magnetic Hertzian potentials p = magnitude of the normalized reflection coefficients T = magnitude of the normalized transmission coefficients <J> = phase angle of the reflection coefficients of a diaphragm i n reciprocal or nonreciprocal waveguide <(>^, fy^ = phase angles of the reflection coefficients of a reciprocal or nonreciprocal interface (semi-infinite case) t|) = di f f e r e n t i a l phase shift per unit length of nonreciprocal waveguide iji e, TJI^ = elec t r i c and magnetic scalar potentials th if»^  = scalar potential for the i mode in waveguide A = scalar potential for the j*** 1 mode in waveguide B ia = angular frequency x i . ACKNOWLEDGEMENT It has been a valuable experience to work with Dr. M. Kharadly. His interest and enthusiasm were unflagging throughout the course of this project. I am grateful for the financial support afforded by two W.C. Sumner Memorial Fellowships for the years 1971-1973 and an H.R. Mac-Millan Family Fellowship for the year 1973-1974. The project was sup-ported by the National Research Council of Canada under grant A-3344. I wish to thank Mr. C. Chubb and especially Mr. J. Stuber for their excellent work and advice in making the various waveguide compo-nents required for the experimental measurements. t" Finally, I wish to thank Miss S. Lund for doing an excellent j.ob of typing, the manuscript. xii Chapter 1 INTRODUCTION 1.1 Background Electronic beam steering of antenna arrays is one of the more recent major advances in, antenna systems, mainly because of the' develop-ment of variable phase shifters. The nonreciprocal f e r r i t e type of phase shifter has proven to be practical and i s now widely used i n these systems. The usefulness of this type of device was realized theoretically in the early 1950's, with much of the work done to analyse a waveguide containing (1-3) transversely-magnetized f e r r i t e slabs . The twin-slab configuration shown in Fig. 1(a) was found to have significantly different propagation coefficients, depending on the sense of the magnetization. To produce the required transverse magnetization, external magnets were originally used - a somewhat impractical proposition. It was not u n t i l late in the 1950's that the idea of a f e r r i t e toroid with a magnetizing wire passing (4) through the centre was proposed . This actually marked the beginning of the development of present-day nonreciprocal f e r r i t e phase shifters, as v i r t u a l l y a l l such devices since then have employed a toroid configur-(5-9) ation . The most commonly used types are shown diagrammatically in Figs. 1(b) - 1(d). The toroid design had the further advantage of considerably re-duced power requirements for switching the sense of the magnetization. This was achieved through the proper design of the f e r r i t e material so that i t s remanent magnetization could be 75-80% of i t s saturation value. Thus, i t was only necessary to apply a current pulse to the magnetizing wire and the device operated with the remanent magnetization - hence, the name latching f e r r i t e phase shifter. With the proper choice of the length « 2 Fig. 1 . Nonreciprocal f e r r i t e phase shifter configurations. (a) theoretical model; (b)-(d) practical toroid designs. 3 of the ferrite-loaded section, the device could be made to have the desired value of di f f e r e n t i a l phase shift (the d i f f e r e n t i a l phase s h i f t per unit length constitutes the figure of merit of the nonreciprocal type of phase shifter). A typical phase shifter design consisted of a series of d i f -ferent lengths of f e r r i t e toroids, each with i t s own magnetizing wire (see Fig. 2) and providing d i f f e r e n t i a l phase shifts of, for example, 180°, 90°, 45° and 22.5°. This allowed the phase difference between the feeds of each element of an array to be varied d i g i t a l l y , in steps of 22.5°, from 0° to 360°. The large size of the array, which could be of the order of 10,000 elements or more, permitted this d i g i t a l operation to scan the beam over a continuous range. A large number of phase shifters i s used in a single system, since one is required for each element of the array. Thus, any reduction in the size, weight or switching power of the phase shifter would con-stitute a significant improvement in the design of the system. In an (8 attempt to achieve such an improvement a recent development by Spaulding ' involved loading the f e r r i t e phase shifter periodically with thin metallic diaphragms. His experimental investigation showed that such loading could provide significant enhancement of the d i f f e r e n t i a l phase shift over a wide bandwidth. Although the waveguide configuration he used, Fig. 1(d), was quite unconventional, the results of the investigation were note-worthy. A subsequent theoretical analysis of nonreciprocal transmission lines periodically loaded with diaphragms showed similar trends . This indicated that Spaulding's results were not peculiar to the type of waveguide he used but were typical of what could be expected by loading any form of nonreciprocal waveguide. DIFFERENTIAL PHASE SHIFT: Fig. 2. Cutaway diagram of a four-bit f e r r i t e phase shifter. 5 1.2 Objectives The application of periodic loading to simple waveguide structures appeared to be an attractive proposition and provided the incentive to start the work in this thesis. To implement the theory in reference (10), i t was necessary to determine the equivalent c i r c u i t of a single metallic diaphragm in a nonreciprocal waveguide. In attempting to achieve this objective, however, there were some fundamental problems which had to be solved in order to answer certain questions. These questions are: (1) What is the equivalent characteristic admittance of an i n -homogeneous nonreciprocal waveguide? This must be determined before the theory can be applied. (2) In fact, how does one define an equivalent characteristic admittance even in a reciprocal inhomogeneous waveguide where a wave ad-mittance cannot be defined? (There seemed to be l i t t l e unanimity amongst workers on the subject.) (3) Can the effect of an interface between reciprocal homogeneous waveguide and nonreciprocal inhomogeneous waveguide be evaluated? This is also necessary in order to measure the effect of a diaphragm. Because of the lack of fundamental work on discontinuities in inhomogeneous waveguides, the problem i s tackled in three stages: (1) The reciprocal inhomogeneous waveguide interface i s analysed in an attempt to define the equivalent characteristic admittance of an i n -homogeneous waveguide. This work i s presented i n Chapter 2. (2) The properties of a nonreciprocal inhomogeneous waveguide interface are determined and appropriate characteristic admittances for the two directions of propagation are defined. This work i s presented i n Chapter 3. 6 (3) With the insight provided by the work in Chapter 3, the pro-perties of a thin diaphragm in a nonreciprocal inhomogeneous waveguide are determined. This i s done experimentally since the nonreciprocal waveguide configurations used in practice make an analytical solution very d i f f i c u l t , i f not impossible. To assess the accuracy of the results, at least indirectly, a basis for comparison is needed. This is provided by comparing the experimental results for a metal diaphragm in reciprocal inhomogeneous waveguide with numerically obtained results for the same configuration. The experimental work is presented in Chapter 4. Chapter 2 RECIPROCAL INHOMOGENEOUS INTERFACE 2.1 Introduction As discussed in Section 1.2, i t is necessary to consider this problem and determine a unique method of defining the equivalent charac-t e r i s t i c admittance of an inhomogeneous waveguide before tackling the non reciprocal case. The problem of the junction between homogeneous and i n -homogeneous waveguides is important in i t s own right and i s encountered in many practical devices. The most thoroughly examined inhomogeneity has been that of E-plane dielectric-slab loading, Fig. 3(a). Such a junction has been described in terms of a lumped-element c i r c u i t joining two equivalent transmission lines of different characteristic admittances when only the dominant mode propagates in each waveguide^"'"'. The parameters of this c i r c u i t depend on the value of the equivalent trans-mission-line characteristic admittance for the dominant mode in the i n -homogeneously f i l l e d waveguide relative to that of the empty waveguide. The definition of this quantity, the characteristic admittance, has i t -self been subject to various interpretations^^ . To date, neither the interface problem nor the definition of characteristic admittance has been resolved in a manner that i s both physically meaningful and sufficiently general. The situation under consideration is depicted in Fig. 4, where waveguides A and B are reciprocal and are assumed to have characteristic admittances 1 and Y, respectively. The wave matrix analysis of this junction i s straightforward^*^ . Let the wave amplitudes, c and d, be normalized so that they represent average power flows. We may then F i g . 3 . Inhomogeneously f i l l e d waveguides. (a) E-plane d i e l e c t r i c loading; (b) H-plane d i e l e c t r i c loading. 9 F i g . 4. Junction of two r e c i p r o c a l waveguides, A and B. 10 write: r x - p e r 2 » P e ,-J02 tj - t 2 - re (l) 01 + 02 ~ 2 9 _ + TT where r^ and ^  are the complex reflection coefficients of the dominant mode in waveguides A and B, respectively, and t^ and t2 are the corres-ponding normalized complex transmission coefficients. In the above r e l a -tions, there are only three independent (measurable) parameters which must be determined to describe the junction completely. For example, these may be p, 0^  and 6. Thus an equivalent c i r c u i t for the junction can have only three independent parameters. In this respect, there have been two alternatives: (i) Y i s considered an unknown quantity and the effect of the junction i s represented by a two-element c i r c u i t , as in Fig. 5(a); and ( i i ) Y i s assigned a value, Y', according to some criterion and the effect of the junction i s represented by a three-element c i r c u i t , as in Fig. 5(b). The criterion by which Y has been assumed i s essenti-a l l y that i t should be proportional to the wave admittance in the inhomo-geneous section (in an effort to give i t some physical significance). This w i l l be shown to be approximately true in some cases where a wave ad-mittance can be defined, such as the E-plane loaded waveguide of Fig.. 3(a). However, in other cases such as the H-plane loaded waveguide shown i n Fig. 3(b), or where both the permittivity and permeability vary over the cross-section of the waveguide, i t i s not possible to define a unique wave 11 F i g . 5. Equivalent lumped-element circuits used to represent the junc-t i o n between empty and inhomogeneous waveguides, (a) two-element representation; (b) three-element representation. 12 admittance. Either of these representations is mathematically correct since the reflection and normalized transmission coefficients, from which the ci r c u i t elements are derived, are the same in both cases. How-ever, while the f i r s t representation yields an equivalent c i r c u i t that i s physically meaningful in a l l cases, the second does not because i t is based on an a r b i t r a r i l y assumed value for Y. In this chapter a study is made of the properties of the junction between homogeneous and inhomogeneous waveguides with emphasis on the f o l -lowing points: (a) The two-element and three-element equivalent circuits for the interface are compared. It is shown that while the former yields results which can be interpreted physically, the use of the latter cannot be j u s t i f i e d . In addition, by using the two-element equivalent c i r c u i t , a general and meaningful definition of the normalized characteristic ad-mittance of inhomogeneous waveguides is provided. (b) The junction between empty and H-plane dielectric-loaded waveguide i s analysed for the f i r s t time. (c) Computed values are presented for the normalized junction parameters and characteristic admittance to i l l u s t r a t e the properties of inhomogeneous waveguides with various types of di e l e c t r i c loading. 2.2 E-plane dielectric loading (u=yQ) E-plane loading i s considered f i r s t since this type of loading has already been discussed by previous authors, thus providing a basis for comparisons. In addition, a wave admittance can be defined i n this case. C o l l i n u s e d the Rayleigh-Ritz method to solve the interface problem and gave a c i r c u i t representation which i s essentially the same as that shown in Fig. 5(a). The numerical example he used was for a relatively low value of permittivity of the loading d i e l e c t r i c . From his results i t appeared that the normalized characteristic admittance, Y , was equal to the normalized wave admittance (which, in turn, i s equal to the ratio of the propagation coefficients, y^fy.). Chang.^12^ used B A mode matching to solve the same problem. Assuming that Y was in fact equal to the normalized wave admittance, he gave the three-element c i r -cuit representation shown in Fig. 5(b). In this work, the amplitudes of the reflected and transmitted waves in waveguides A and B are determined by means of the mode matching technique described in refs.(17)and(18). Values of the equivalent c i r c u i t parameters are then obtained from expressions relating them to the ref-lection and transmission coefficients (Appendix A). These parameters are plotted in Figs. 7 and 8 for the two types of E-plane loading shown in r Fig. 6. In Fig. 8, Y is plotted as a function of y^/y. and is very nearly, B A but not exactly, equal to the normalized wave admittance. It is also noted that the deviation of the value of Y from y ly i s somewhat larger B A for side loading, Fig. 6(b), than for centre loading, Fig. 6(a). The corresponding two-element parameters, X and B, are plotted as a function of 2t/a in Fig. 7. Note that both X and B are inductive. This i s con-sistent with what one should expect i n this case, since the only evan-escent modes excited are TE modes for which the energy stored i s predom-inantly magnetic. With the three-element equivalent c i r c u i t representa-tion of the interface, where Y' i s assumed equal to y j y h * different results are obtained. The values of these equivalent c i r c u i t parameters, B£ and X.j, are also shown in Fig. 7 for centre and side loading. Note that X^ and B2 are inductive while X^ i s capacitive. Examination of the results i n Fig. 7 shows that X tends to zero when 2t/a approaches 0 or 1 for both centre and side loading, as i t should 14 15 16 F i g . 8. Normalized c h a r a c t e r i s t i c admittance f o r E-plane center loading; e *y = 4.0, a/X- = 0.691. r r • • 0 (i ) e r = 2.0, y r - 2.0 ( i i ) e r = 2.5, u r = 1.6 ( i i i ) e r = 3.2, u r =1.25 (iv) e r = 4.0, u r =1.0 (v) e r = 5.0, vr = 0.8 (vi) e = 6.0, y = 0.66667 r r ( v i i ) e = 8.0, u = 0.5 r r ( v i i i ) side loading, er = 4.0, y r = 1.0 (ix) reference: Y = Y T , / Y A 17 when waveguide B is empty or completely f i l l e d with d i e l e c t r i c . The be-haviour of X^ and X^ is not so predictable, however. With centre loading, Fig. 7(a), X^ and X^ approach zero when 2t/a = 1 but their excessively large values when waveguide B is nearly f u l l can be misleading: i t appears that the junction effect is quite large here, but in fact i s small and the 'effective series reactance', X^ + X^, i s also very small. The behaviour of X^ and X^ i s even more disconcerting for the side loading case, Fig. 7(b). As 2t/a approaches unity their magnitudes do not con-verge to zero but increase rapidly to large values . While i t i s not unreasonable to assume that the values of X^, B^ and at 2t/a = 1 should be approached differently for these two types of E-plane loading (since the f i e l d distributions are different in each case), one would r expect that, at that point, the values should be the same in the two cases; i.e., X^ = ^2 = ^3 = 0. From the above observations, i t appears that an analysis which assumes a value for Y, or indeed for any of the equivalent ci r c u i t parameters, can lead to a physically unjustifiable behaviour of the junction parameters. 2.3 E-plane dielectric loading (u / u 0) In many applications, the permeability, y, of the loading material may be different from y D; e.g. for f e r r i t e s . In such situations a unique wave admittance is not definable. The computed results shown in Figs. 8 and 9 are for centre loading and are given for values of y r be-tween 0.5 and 2. Except in the case of loading with a di e l e c t r i c whose y = y Q, Y i s markedly different from ^ / Y ^ Fig. 8. In a l l cases, Y * The results for X.., B and X plotted in Fig. 7 agree with those given (12) -i by Chang . 19 v a r i e s r a p i d l y with 2t/a f o r t h i n d i e l e c t r i c s and soon reaches a plateau as the d i e l e c t r i c thickness becomes la r g e , F i g . 10. With side loading on the other hand, Y changes very slowly when 2t/a i s small and v a r i e s almost l i n e a r l y when 2t/a i s large. The j u n c t i o n parameters, X and B, are again inductive and are p l o t t e d i n F i g . 9. The maximum value of X i s approximately p r o p o r t i o n a l to the r a t i o of p e r m i t t i v i t y to permeabil-i t y , e/y, of the loading d i e l e c t r i c . The maximum value of B appears to be a more complicated function of e and y, but f o r both X and B, the peak values occur at smaller d i e l e c t r i c thicknesses as e/y i s increased. The e f f e c t of the j u n c t i o n may be estimated by comparing Y, as determined by using the two-element c i r c u i t , with the input admittance at port A ( F i g . 5( a ) ) : v _ 1 " r l Y + jB  i n ~ 1 + r1 (1 - XB) + jXY <2> The maximum dif f e r e n c e between Y. and Y occurs where X and Y are l a r g e s t . i n " For example, with = 8 and y^ = 0.5, these occur at 2t/a = 0.2 and the diffe r e n c e between Y and Y. i s 1.6%. With smaller r a t i o s of e/y the m j u n c t i o n e f f e c t i s smaller: the d i f f e r e n c e i s l e s s than 0.5% with e r = 6, y r = 2/3, and l e s s than 0.06% with = 4, y r = 1. For impedance matching purposes, then, the lumped-element c i r c u i t parameters of the i n t e r f a c e can often be neglected and the j u n c t i o n treated as one between two transmission l i n e s with normalized c h a r a c t e r i s t i c admittances 1 and Y. Thus i t i s important to be able to determine a value f o r Y that i s phys-i c a l l y meaningful. 2.4 H-plane d i e l e c t r i c loading The j u n c t i o n between empty and H-plane d i e l e c t r i c - l o a d e d wave-guides has not been analysed before. This case i s somewhat more 20 3.0 r Y 2t/a or 2t/b Fig. 10. Variation of Y with dielectric thickness; e = 4.0, u = 1.0, a/\Q = 0.691, b / A Q = 0.3. (i) E-plane center loading ( i i ) E-plane side loading ( i i i ) H-plane center loading or side loading 21 complicated than the E-plane case, although the procedure for mode matching is similar. The d i f f i c u l t y here i s in deciding which modes must be used to satisfy the boundary conditions at the interface. The analysis is given in Appendix B from which i t i s seen that the modes excited are: TE.rt + TE, + TM, in waveguide A 10 lm lm & 0 . , m = z ,H,t>,.... L S E l m + L S M l m - l l n w a v e S u i d e B The amplitudes of the TE and LSE modes were found to decay much more ra-pidly with increasing m than the TM or LSM modes (Table 1 i s a typical example). Based on this observation, the ratios of TE to TM and LSE to LSM modes were chosen to be about 1:2. The total number of modes in each waveguide was varied and the convergence found to be adequate with 9 each of the TE and LSE modes and 19 each of the TM and LSM modes. The exact ratios of TE to TM and LSE to LSM modes did not seem to be c r i t i -cal but after the results had been obtained, a systematic study was made to determine the importance of these ratios and their effect on the con-vergence of the results. Figure 11 ill u s t r a t e s the effect on B (in the interface equivalent circuit) of varying the two ratios independently and together. The effect on Y and X i s similar though not as pronounced. This indicates that the ratios should be the same, assuming that the correct answer i s somewhere near 0.12260. Convergence was then checked using several different ratios from 1:6 to 1:1. The calculated values of B are plotted in Fig. 12 as a function of the total number of modes in either waveguide. For a l l ratios, the value of B converges to 0.12272, in this case, as the total number of modes is increased. The most rapid convergence occurs with a ratio of 1:3 or less. However, with a ratio of 1:2, approximately that used for a l l the interface results presented Table 1. Example of mode amplitude decay for H-plane loading. e r = 8; y r = 0.75; 2t/b = 0.105; b/XQ = 0.3 WAVEGUIDE A WAVEGUIDE B m TE, lm TM lm+2 L S Elm+2 LSM lm+1 0 -.42187 -.06642 6.44151 8.71578 -.01110 -.00703 112.90114 -.32990 2 -.00456 .00591 .53971 1.38273 .00002 .00024 9.60478 43.67913 4 .00026 -.00006 -.03760 .01110 .00002 .00007 .31850 .09105 6 .00008 -.00005 -.08434 -.19051 .00001 -.00001 .38655 -.15653 8 -.00000 -.00002 -.04184 -.11722 .00000 -.00001 -.00038 -.06325 10 -.00002 -.00000 -.00245 -.01924 -.00000 -.00001 -.02038 .00110 12 -.00001 .00000 .01415 .03123 .00000 -.00000 -.03948 .01310 14 .01238 .03340 -.03141 .02156 16 .00341 .01236 .00486 -.00043 18 -.00365 -.00723 .00705 -.00213 20 -.00523 -.01352 .01890 -.00948 22 -.00261 -.00800 -.00156 -.00008 24 .00084 .00104 -.00212 .00061 26 .00251 .00621 -.00936 .00409 28 .00187 .00530 -.00063 .00081 30 .00010 .00088 .00090 -.00024 32 -.00123 -.00288 .00309 -.00120 34 -.00131 -.00352 .00308 -.00164 36 -.00042 -.00143 -.00047 .00011 38 .00055 .00119 -.00096 .00034 40 .00089 .00231 -.00309 .00143 . 42 • .00050 .00147 .00022 -.00002 23 0.123Q 0.1226 0.1222 0.1218 LSE/LSM = 9/19 .* TE/TM = 9/19 * 3/25 5/23 7/21 9/19 11/17 13/15 15/13 17/1 Ratio of TE/TM or LSE/LSM Modes Fig. 11. Effect on the solution for B of varying the ratios of TE:TM and LSErLSM modes separately and together; b/XQ = 0.3, 2t/b = 0.7, e r = 8.0, v r = 0.75, total no. of modes = 28. Total No. of Modes i n Each Waveguide Fig. 12. Convergence of the solution for B with various ratios of TE:TM (= LSE:LSM) modes; b/\n = 0.3, 2t/b = 0.7, er = 8.0, y r = 0.75. (i) ratio = 1:6 ( i i i ) ratio = 1:2 (i i ) ratio =1:3 (iv) ratio =1:1 25 in this section, the value of B is within 0.1% of i t s "true" value with a total of 27 or more modes in each waveguide. The general behaviour of the characteristic admittance as a function of the dielectric thickness i s shown in Fig. 10 where i t i s compared with the Y variations for the two types of E-plane loading con-sidered. Because of the nature of the modes excited i n H-plane loading, the next lowest order mode i s the LSE^ o r LSM^ mode. Thus, i t i s pos-sible to use dielectrics with larger values of permittivity than can be used for E-plane loading and s t i l l have only one propagating mode. With E-plane loading, e r " u r must be less than 4.712 but with H-plane loading, e *u can be as big as 11.635. Two variations i n e and y are con-r r & r r sidered here: (i) with the product e • U r equal to 6.0, y r i s varied from 0.5 to 2.0, and ( i i ) with y^ = 1.0, i s varied from 2.56 to 9.0. For the f i r s t variation, Y is plotted as a function of Y /y in Fig. 13. B A The deviation of Y from y ly with y = y n is greater for the H-plane case B A u than for the E-plane case (compare Figs. 8 and 13). The corresponding junction parameters, X and B, are plotted for both variations in Figs. 14 and 15. This time both elements are capacitive, which can be ex-pected since the evanescent modes excited are primarily TM modes, for which the energy stored i s predominantly e l e c t r i c . As shown in Fig. 14, with e *y held constant, B increases rapidly with the ratio e /y while r r r r X remains v i r t u a l l y unchanged. With y = VQ, as £ r i s increased both X and B increase rapidly, although there i s l i t t l e variation i n the maxi-mum value of B when i s greater than 6, as indicated in Fig. 15. 26 Y 1 2 3 3.39 4 Fig. 13. Normalized characteristic admittance for H-plane loading; £ r - u r = 6.0, b/XQ = 0.3. (i) e r = 3.0, y r - 2.0 (iv) e r =8.0, y r = 0.75 ( i i ) e r = 4.0, u r = 1.5 (v) e r =12.0, y r = 0.5 ( i i i ) e r = 6.0, y r = 1.0 (vi) reference: Y = Y B / Y A 29 Chapter 3 NONRECIPROCAL INHOMOGENEOUS INTERFACE 3.1 Introduction The analysis i n this chapter i s concerned with the properties of a nonreciprocal inhomogeneous waveguide interface, and i t follows along the lines of that in Chapter 2. In nonreciprocal waveguides, how-ever, the f i e l d distributions and the phase coefficients can be very d i f -(2) ferent in the two directions of propagation . Accordingly, one would expect the interface effect and the normalized characteristic admittance to be also different in these two directions. In some cases i t i s im-portant to evaluate only the combined effect of these two factors. In others, however, i t i s necessary to isolate the two phenomena, e.g., i n order to carry out the analysis and design of the device , or to mea-sure a discontinuity within a nonreciprocal section of waveguide. The scattering parameters of a section of twin-slab ferrite-loaded waveguide between two reciprocal homogeneous waveguides have been recently deter-(19) mined . The evaluation in ref. (19) did not require the explici t an-alysis of the nonreciprocal interface, and there has been as yet no at-tempt to analyse such an interface f u l l y . As in the reciprocal case, the nonreciprocal interface problem is solved numerically by means of the mode matching technique. The ex-ample chosen here i s that of the twin-slab f e r r i t e loaded waveguide, (2 19 20) often used as a model for nonreciprocal f e r r i t e phase shifters ' ' . Using the solutions obtained for the reflection and transmission coef-ficients due to the nonreciprocal interface, i t i s shown that i t is pos-sible to derive simple equivalent transmission circuits which account for the effect of the interface and the equivalent characteristic 30 admittances of the nonreciprocal inhomogeneous waveguide, separately. 3.2 Mode matching The configuration analysed i s that shown i n Fig. 16. The slabs are symmetrically placed about the centre of the waveguide and magnetized transverse to the direction of propagation. This limits the modes excited at the junction between the homogeneous and inhomogeneous waveguides to TE^Q modes, where n i s odd. The width of the waveguides, a, i s such that only the TE^Q mode may propagate in either waveguide A or waveguide B. For c l a r i t y , the modes in the nonreciprocal waveguide, B, are referred to as TE^Q and TE~Q modes for propagation in the positive and negative direc-tions of z, respectively. Only counterclockwise magnetization need be considered, as in Fig. 16(a), since reversal of the magnetization simply has the effect of interchanging junctions 1 and 2. It is necessary, how-ever, to analyse both junctions because different sets of modes are ex-cited at each. In fact, there are four separate situations to consider: the dominant mode from either waveguide A or waveguide B may be incident on either junction 1 or junction 2, as shown in Fig. 17. The mode matching procedure in this case i s basically the same as for reciprocal j unctions ^  The transverse ele c t r i c and magnetic fields are made continuous over the cross-section of the waveguide at the interface and orthogonality relations for each set of modes are used to separate the continuity conditions on the fields into N equations involving the N unknown amplitude constants of the modes. The solution of these equations yields accurate values of the amplitudes of the dominant modes. The orthogonality relations and their implementation, however, are d i f -(21V ferent in the present case; they are, i n fact, biorthogonality relations , The following simplified form of these relations i s made applicable by 3X Fig. 16. Twin-slab f e r r i t e phase shifter model. (a) cross-section, identifying the loading parameters; (b) longitudinal section, identifying interfaces 1 and 2. B B TE 10 Jn0 TE nO •TE 10 TE nO -TE hO (a) (c) TE hO B J10 TE nO TE 10 TE nO • TE nO (b) (d) Fig. 17. The modes excited by a single mode incident from either A or B. (a) interface 1, wave incident from A; (b) interface 1, wave incident from B; (c) interface 2, wave incident from A; (d) interface 2, wave incident from B. t o 3 3 (22) the restriction to transverse magnetization : £ [h± x e.. + x e±] • a^ds = 0 , Y ± 4 ( 3 ) The integration i s over the cross-sectional area, S, of the waveguide, {e^,h^} i s the transverse component of a waveguide mode with propagation coefficient Y- a n d {e.,h.} i s the transverse component of a waveguide i 3 3 ' mode with propagation coefficient y j • T ne procedure used to separate the continuity conditions on the electric and magnetic fields into two sets of equations i s similar to that which has been successfully employed (23) to analyse an H-plane bifurcation in a plasma-filled waveguide . Modal equations need be developed for only two of the four situations shown i n Fig. 17: either (a) and (b) or (c) and (d). The other two can be treated similarly. Considering case (a), the total transverse fields are approximated by the following par t i a l sums: Waveguide A _ N _ A A l i Ai _ _ N _ H A = h A l " a i h A i (4) (5) • Waveguide B _ N E = Z b+ e j=l - N ^x. H = £ b+.hl ^ j=l 3 B j The continuity conditions on the transverse fields at the interface may be written either as: { [n~ x E A + H A x i - ] • a zds = / [h" « V + W ^ or a s , •i X \ + A X % ] • ¥ z d s -i f h t j x h + «B X < J ] • V S ( 7 ) 34 Using eqn. (3), most terms on the right side of eqns. (6) and (7) may be eliminated since: / [h„ x ej\ + h^. x e~ ] • a ds = 0 , for a l l j and n s Bn Bj Bj Bn z ' J Substituting eqns. (4) and (5) into (6) and (7) now yields the following two sets of eqns: _ N _ _ N _ _ _ t [ \ j X ( 6A1 + ^ 1 * ^ + ( h A l - J ^ A i * X e B ? ' a z d s - 0 (8) ' • • - + • • • _ _ • _ + • _ • N • _ _ _+ -s [ h B j X e A l + h A l X e B j ] , a z d s + .1*1 iHj X e A i " h A i X e B j ] , a Z d s  b+ = _ _ _ i 2 / h+. x e n. • a ds ,Ql* s B J Bj Z (9) The set of eqns. (8) consists of N equations in the N unknowns, i = 1 to N. Once determined, the coefficients a. are then substituted into the set of eqns. (9) to determine the values of frt. In case (b), the total fields are again approximated by partial sums: Waveguide A _ N _ E = E a. e.. A i = 1 i Ai N _ (10) H = - I a., h . A i = 1 i A i Waveguide B *B ~ *Bi + ^fi 7 B 3 N (11) 35 The modal equations are: N (12) •4\H s [ hB~j X e A i ' h A i X e B J ] , a z d 8 = 2 6 j , l i hIlX e B l * a z d s N _ '; _ _ + b + = i E = 1 a l ^ [hB*j X 6 A i " h A i X e B j 3 ' a z d s j _+ Z ' ' (13) 2 A h B j X eBj * a z d s where 6. is the kronecker delta, j ji-l t i s desirable that the expressions for ( e ^ , h^} and (e^", hgT} should have the same form i n order to avoid d i f f i c u l t y when + deriving the equivalent transmission line networks from a^ and b^. Suitable expressions are given in Appendix C, along with the transcen-+ dental equations for the TEnQ modes. 3.3 Normalized reflection and transmission coefficients Once solutions have been obtained for the amplitude c o e f f i -cients, a^ and b^, one can proceed to derive relations for the normal-ized reflection and transmission coefficients for the interface. Refer-ring to Fig. 18 and using conservation of energy, we can write for case 17(a): - (1 ~ Kl2) I A = l ^ l 2 4 (14) where I. = / e., x h* • a ds , A s A l A l z ' *B = s ^ 1 X ( H J l ) * ' V s and a^ and b^ are determined from eqns. (8) and (9). Rearranging, eqn. (14) has the form, 1 - | R l|2 = Y +|T 2|2 = |f 2|2 (15) where R^  = a^ i s the reflection coefficient i n waveguide A Fig. 18. Identification of the reflection and transmission coefficients, (a) interface 1; (b) interface 2. 37 and T- = b1"(IT,/l.) is the normalized transmission coefficient i n c 1 ii A waveguide B. Similarly, for case 17(b): where I" = / x ( h ^ ) * - ^ d s and a^ and b+ are determined from eqns. (12) and (13). Rearranging, eqn. (16) has the form, 1 - £ |R2|2 - i . (17a) 1 - |R2|2 - |T X| 2 (17b) where iL, = b*(Ig/I~)^ i s the normalized reflection coefficient in waveguide B and T.. = a..(IA/.I~) i s the normalized transmission coefficient in wave-guide A. In the above eqns., Y + and Y~ are the equivalent characteristic admittances of waveguide B normalized to that of waveguide A for propaga-tion in the positive and negative directions of z, respectively. The normalized scattering matrix for the interface i s unitary. Thus, the following relations, as implied in Fig. 18, must be satisfied: |R | = |R2| = p (18) | T J = |T2| = x and in addition, 1 - P 2 = T 2 (19) *1 + *2 ~ 91 _ 92 = ± i r The numerical results are consistent with this condition provided a large number of modes (40 or more) are used for the mode matching. However, ac-curate results can be obtained with fewer modes, i f the magnitudes and phase angles of the normalized reflection and transmission coefficients 38 are averaged. Referring to Fig. 18, the numerical results also show that f 3 = f 1 , T 4 = f 2 , R 3 = R 2 and R 4 = R^. 3.4 Equivalent ci r c u i t representation It i s convenient to express the behaviour of the interface i n terms of an equivalent transmission network, as shown in Fig. 19. With this representation, the nonreciprocal nature of the junction i s accounted for by allowing the reactive elements to have different values corres-. ponding to propagation in the positive or negative direction of z, and incidence from waveguide A or waveguide B. Hence, these parameters are identified by the superscripts + or - , and the subscripts A or B. The dashed boxes indicate that the series elements, Xg, appear at this a l -ternative position for some values of f e r r i t e loading parameters. This situation i s signified by the overscript, ~. Waveguide B i s represented by a nonreciprocal transmission line with normalized characteristic ad-mittances, Y*. The input impedance at port A i s thus given by: *» A - ^ T T B I - — i ? , ( 2 0 ) and at port B, the input admittance (or impedance) i s given by: 1 _ ,~.„± 1 Y i n Zf~ B 1 + j Xg (21) „+ 1 + /Yi/Y* p e ^ 2 (22) where Z- = — 7X7 in Y±_ tfFf= pe39z The transmission coefficients, for modes incident from waveguide A, are: T e " 2 = ' Y+ + i Bt- (23) J 6 2 _ (1 - p e 3 * ! ) ^ Y+ + j B£-and T e J 9 1 ~ (1 - p e 3 * * ) ^ (24) ana xe Y~ + j B~ F i g . 19. Equivalent transmission line representation of the junction, (a) interface 1; (b) interface 2. 40 Equations (20) - (24) are used to calculate values for the equivalent c i r c u i t parameters from p, x, <f>^, fy^* 8^ and Q^- As compared with the reciprocal case where three c i r c u i t elements are derived from three i n -dependent measurable quantities, here five c i r c u i t elements are derived from four independent quantities. Consequently, the c i r c u i t elements are not independent in this case although they are unique. The reason for the extra circuit element i s the fact that two reciprocal equivalent circuits, each valid in only one direction of propagation, are used to represent a nonreciprocal interface. 3.5 Numerical results and discussion In order to obtain r e a l i s t i c results, the values chosen for the f e r r i t e saturation magnetization and remanence ratio are typical of those of ferrites used at X-band in nonreciprocal phase shifters. The elements of the permeability tensor were derived from these values, based (24) on empirical formulas recently published by Green and Sandy . Table 2 l i s t s the pertinent data. The twin-slab parameters, d, c and t, are identified in Fig. 16 and the loading d i e l e c t r i c i s assumed to have a relative permittivity, E^, of either 1 or 16. The normalized waveguide width, a/A^, is 0.65 for a l l the results. The number of modes used for the mode matching solutions i s 25 i n a l l cases and the accuracy was checked by comparing the results of one case with the corresponding re-sults obtained with 40 modes. This check indicated that the values c a l -culated for the interface parameters are accurate to at least four dec-imal places. Properties of the nonreciprocal waveguide, B, are shown in Fig. 20 where the normalized phase coefficients, 6l7S. and B~/8A, the norma-ls A a A lized equivalent characteristic admittance, Y"1", and the percentage Table 2. Assumed characteristics of the f e r r i t e Saturation magnetization, 4TTM (Gauss) 2000 s Remanence ratio, M /M 0.8 ' r s Y4TTMS/O) at 8.5 GHz 0.66 Permeability tensor: diagonal element, y 0.96 off-diagonal element, K -0.528 Effective relative permeability, y £ 0.67 Relative permittivity, ef 12.0 3 * Phase Coefficients or Characteristic Admittances J-> to . CO *-n Percentage Difference Between Y and Y 44 difference between Y"*" and Y are plotted as a function of 2t/a. It i s seen that although the normalized phase coefficients can be quite d i f -ferent, the difference between Y + and Y~ i s very small indeed, contrary to i n i t i a l (intuitive) expectations. With this result in mind (Y"*" = Y~), one might wonder whether these admittances could be determined from the simpler case of twin-slab dielectric loading. The relative admittance, Y, i n Fig. 20 was obtained for a loading di e l e c t r i c with relative per-mittivity, e^, and relative permeability, u . The two curves, Y and Y + (or Y~), are similar but for most values of the thickness, t, there i s a significant difference between the two. The properties of the interface between waveguides A and B are shown in Figs. 21 - 23. The magnitudes and phase angles of the norma-lized reflection and transmission coefficients, as defined i n Fig. 18, are plotted in Fig. 21 for £^=1 (the curves for £^=16 are similar). The elements of the equivalent ci r c u i t of the interface, determined from these parameters, are plotted in Figs. 22 and 23 for the cases of e ^ = l and 16, respectively. It i s worth noting that there are two possible solutions for each of X* and B* as obtained from eqns. (21) and (22). However, only one solution is shown here. It may be instructive to discuss the behaviour of the inter-face in terms of stored energy. An inspection of eqns. 20 - 22 shows that the reactive part of the interface impedance is positive when <j>^' (or i s between 0 and TT radians, for incidence from waveguide A (or waveguide B). Referring to Fig. 21, the interface impedance from wave-guide A has a positive reactance i f 2t/a i s less than 0.36; i.e., the stored energy is magnetic for small f e r r i t e thicknesses, changing to electric for larger thicknesses. The junction behaviour is somewhat different when the wave i s incident from waveguide B; the change from Fig. 21. Reflection and transmission coefficient parameters versus 2t/a; 2d/a = 0.04, e, = 1. 8* I I I Shunt Susceptance (B) ' • • • . . « • C* *- lo O to ON co • o . - 0 1 Series Reactance (X) 6*7 50 magnetic to electric stored energy occurs at 2t/a = 0.295. This be-haviour i s of course reflected in the values of the elements of the i n -terface equivalent c i r c u i t . The case of twin-slab d i e l e c t r i c loading again provides an interesting comparison. The equivalent c i r c u i t elements, X and B, ob-tained for this case are also shown in Figs. 22 and 23. It can be seen that the stored energy i s magnetic for a l l values of the die l e c t r i c thickness, t, which i s consistent with the results of Chapter 2. Thus, although the nonpropagating modes excited with either d i e l e c t r i c or fer-r i t e loading are a l l TE, the effect of the interface can be markedly d i f -ferent i n the two cases. For the sake of completeness, the dependence of the various parameters on the slab separation, d, was also investigated. A typical case, where 2t/a = 0.24 and e, = 16, i s shown in Figs. 24 and 25. The d difference between Y + and Y i s again found to be very small and is v i r -tually unaffected by changes in d, as shown in Fig. 24. The corresponding reactive elements of the equivalent c i r c u i t are plotted in Fig. 25. Phase C o e f f i c i e n t s or C h a r a c t e r i s t i c Admittance *1 OQ t o O tt a ( X o S H H - 3 r t tt r t i - 1 tt H -3 N O fD fD P . CA >d o 3* M l tt CO fD tt < O CD o OQ fD C M i K - u M ) t o &. H -n> O H - M w (D 3 0 r t fD CO M CO tt c 3 CO O . r o n> a. ,a c tt H ' <! tt t o M r t fD 3 tt r t II O 3* O tt • H t o tt -P» O <• I t (D rn H H« CO II r t H " M O O N o oo I O O o o I U l U l Percentage Difference Between Y and Y~ T9 I.— l l I Shunt Susceptance (B) t o " l s > . O M LO •C Shunt Susceptance (B) i i i i eg Chapter 4 EXPERIMENTAL INVESTIGATION 4.1 Introduction The main objective of this thesis has been stated earlier as being the determination of the effect of a metal diaphragm in a nonre-ciprocal inhomogeneous waveguide. It may be possible to do this analy-t i c a l l y for the theoretical twin-slab f e r r i t e model but i t was thought unlikely for the more complex f e r r i t e toroid configurations used i n prac-tice. The modes in such structures are complicated hybrid modes, for which i t would be very d i f f i c u l t to derive analytical expressions. Hence, i t was decided that the effect of the diaphragm would have to be measured experimentally. Since a theoretical analysis did not appear feasible, the accuracy of the experimental results was d i f f i c u l t to as-sess. This was accomplished by an indirect method: the effect of a metal diaphragm in a reciprocal inhomogeneous waveguide was also deter-mined experimentally, but the configuration was chosen so that theoreti-cal results could be obtained for comparison. The accuracy of the re-sults for the reciprocal case was then used as an indication of the ac-curacy of the results for the nonreciprocal case. 4.2 Method of measurement Before the effect of a diaphragm i n inhomogeneous waveguide could be measured, the problem of either measuring or matching the junc-tion between empty waveguide and the inhomogeneous waveguide was encoun-tered. Impedance matching was considered f i r s t , but this did not seem feasible for the nonreciprocal case. Thus, the effect of the inhomogen-eous waveguide interface was measured so that the diaphragm measurements could be made on unmatched sections of the loaded waveguide. It was 55 then possible to eliminate the effects of the interfaces from the mea-surements as shown in section 4.3. The quantities measured were complex reflection and transmission coefficients. The reflection coefficient, R, was measured by means of a calibrated attenuator and a slotted line, Fig. 26(a). A n u l l bridge, Fig. 26(b), was used to measure the transmission coefficient, T. To determine the interface parameters, R and T were measured for two d i f -ferent lengths of the inhomogeneous section. Measurements of R and T of a section containing the metal diaphragm then yielded the information required to determine the equivalent c i r c u i t of the diaphragm as shown in section 4.3.2. In the nonreciprocal case, R and T were measured for both senses of the fe r r i t e magnetization. 4.3 Derivation of the diaphragm effect from the measurements 4.3.1 Interface wave-transmission matrices For the reciprocal case, the wave-transmission matrices of the interfaces are: A l = re je 1 - p e ^ . . .(25a) A3 = xe je i - p e 3 * 1 pe 3** eJ26 . . .(25b) where pe , pe * and xe are the semi-infinite interface reflection and transmission coefficients shown in Fig. 27(a). Thus, p, T , C|>I, $2 and 6 are to be determined from measurements of R and T (only three of these parameters are independent). In addition, the wave-transmission matrix of a length, £,of the inhomogeneous waveguide i s : A2 = (a + j e H -(a + J3H .(26) 56 matched termination test section slotted line calibrated attenuator isolator 10 dq couple attenuator directional detector sweep generator VSWR meter (digital frequency imeter feedback VSWR meter attenuator probe tuner test section probe tuner 10 dB coupler isolator 10 dB coupler attenuator directional detector sweep generator r idetector [10 dB coupler calibrated phase shifter calibrated attenuator d i g i t a l frequency meter feedback (a) (b) Fig. 26. Block diagrams of experimental systems, (a) reflection coef-ficient measurements; (b) transmission coefficient measurements. 57 xe J02 j*2 re j 9 l ^ p e j $2 xe J 8 l ^ p e j < ! > 1 Jl (b) Fig* 27. Junctions of homogeneous and inhomogeneous waveguides identi-fying the semi-infinite interface parameters, (a) waveguide B is reciprocal; (b) waveguide B i s nonreciprocal. 58 where (a + j3) i s the complex propagation c o e f f i c i e n t i n the inhomogeneous waveguide. The o v e r a l l wave-transmission matrix of the se c t i o n i s simply A 2 * Ag, so R and T may be expressed as follows: R = p e j * l [1 - e j 2 ( * 2 - B & ) e'2al] - P e 3 * 1 [1 X 2 e j 2 T 1 ] 1 - p 2 X 2 e 3 2 * • (27) T = T 2 eJ(26-BA) &-al x _ p 2 eJ2(<t»2-B£) e-2a^ _ -(1-p 2) X e j(<h+n) 1 - P 2 X 2 e j 2 r i (28) where n = <t>2 - 3^  and X = e . Equations (27) and (28) may be separated into r e a l and imaginary parts, so that they represent four equations i n the four unknowns, p, <j>i, X and n . Linear i t e r a t i o n was used to solve for these unknowns f o r each length of inhomogeneous waveguide. This method proved to be r e l i a b l e when a l l simultaneous s o l u t i o n methods f a i l e d : small experimental errors seemed to preclude convergence. Re-su l t s f o r two lengths, then, were expected to y i e l d the same values f o r p, <f>i and a, while the two values f o r n, <f>2 ~ B&i a n ^ <t>2 ~ 3^2> deter-mined <j>2 and p. Thus, the wave matrices f o r the i n t e r f a c e s and f o r the length of inhomogeneous waveguide were determined f o r the r e c i p r o c a l case. From the r e s u l t s of Chapter 3, the wave transmission matrices f o r the two nonreciprocal i n t e r f a c e s are: A l = xe J 6 2 pe - p e 3 * 2 j<h J ( 6 i + e2). •(29a) A3 = xe j e 2 1 -pe 3** p e j * 2 e i ( 9 i + e 2 ) ...(29b) where p e 3 ^ 1 , pe^2, xe^1 and re^2 are the s e m i - i n f i n i t e i n t e r f a c e 59 reflection and transmission coefficients, Fig. 27(b). For the opposite sense of magnetization, Qi and 62 are interchanged. The wave-transmis-sion matrix of a length, I, of the nonreciprocal waveguide i s : A2 * > + + J3 +) -(<*" + jB") • .(30) where ( a + + j3 +) and (a~ + j3~) are the complex propagation coefficients in the forward and reverse directions, respectively. Here, reversing the sense of the magnetization has the effect of interchanging ( a + + j g + ) and ( a - + j3~). The overall wave-transmission matrix of•the section is • A 2 • A^, and i s dependent on the sense of the magnetization. The reflection coefficient, however, is not. The coefficients, R, T + and T~, may be expressed as follows: = nJ*l [ 1 - P J < 2 * 2 - ( 3 + + 3 " )A) Q ~ ( a + + aT)I R = peJ ] 1 - p 2 e J ( 2 < ( . 2 - (3 + + 0 - ) J l ) e - ( a + + cTHj = pe 3+l [1 - X 2 e J 2 n ] 1 _ P 2 X 2 eJ2n • (31) T 2 e J ( 2 9 2 - 3 +£) -ct+* 1 ._ p 2 e J ( 2 < r 2 - (3 + + 3"W) e ~ ( a + + cTH = - ( 1 - p 2 ) X e ^ ^ ^ ) . e J ( x - 7*/2). -e-«4/2 1 - p 2X 2 e j 2 n (32). T~ = x^e 2 j ( 2 6 i - p-A) e-a-A 1 - p 2 e J ( 2 4 > 2 " (3 + + 3~)I) -(a+ + cT)A -a - p 2) X e j ( ^ + ^ . e " 3 ^ - y ^ z ) . e ^ / z 1 - p 2X 2 e j 2 n . • .(33) where X = e , n = d>2 - g£, 2a = a + a , 2B = 3 + + 8~, x = 6 2 - 0 y = 3 + - 3~ and z v= a + - a~. If a mean value of the transmission co-efficients is defined such that T~ , then T = -g - P2) . x . a J ( * i + n) ^  Equations (31) and (34) have the same form as eqns. (27) and (28) so the procedure for determining the wave-transmission matrices of the interfaces i s the same for the nonreciprocal case as i t is for the re-ciprocal case. The terms x, y and z are readily determined since £ i e j ( 2 * - y * ) e - z * _ < ( 3 5 ) 4.3.2 Diaphragm wave matrices and equivalent circuits Once the interfaces had been measured, a thin metal diaphragm was placed at the center of an unmatched section of the inhomogeneous waveguide. The wave matrix then measured was A: A = A x • A 2 • A d • A 2 • A 3 .'. .(36) where A^ and A 3 are the interface wave matrices, defined i n eqns. (25) and (29) for the reciprocal and nonreciprocal cases, respectively, A 2 is the matrix of half the length of transmission line between the i n -terfaces and A^ is the wave-transmission matrix of the diaphragm. Since A^, A 3 and A 2 are known from the interface measurements and are a l l easily inverted, i t is now a simple matter to determine A^. For a thin metal diaphragm of the type considered here, the matrix A^ has the forms: 61 A d = 1 e x £ e x 1 e x £ e' x -j9 j(<r-6) -g_ e x 1 e x je -£ e ' x j (<j>-e+) j(<J>-e+) 1 e x , reciprocal case . . .(37a) , nonreciprocal case . . .(37b) In the reciprocal case, the diaphragm may be represented by a single shunt element equivalent c i r c u i t , although there appear to be two independent measurable quantities, p and 6. In the nonreciprocal case, there are apparently three independent measurable quantitiesi p, 0 + and 6~. It was decided that in this case the diaphragm should be represented by a two-element equivalent c i r c u i t as shown in Fig. 28, with each element, X and B, assuming different values for a different sense of magnetization (or direction of propagation). These four quan-t i t i e s , X~ and B^, are not independent although they are unique. 4.4 Measurements 4.4.1 Preliminary work Measurements were made to determine the effect of a metal d i -aphragm in three different configurations of inhomogeneously loaded waveguide. Cases (a) and (b), Figs. 29(a) and (b), were reciprocal, with symmetrical E-plane ceramic loading. These cases served as a means of evaluating the accuracy of the measurements since theoretical results could be obtained for these cases. The third configuration was the non-reciprocal case, (c), shown in Fig. 29(c), and involved magnetized fer-r i t e loading. The properties of the f e r r i t e are l i s t e d i n Table 3. A fourth case, shown in Fig. 29(d), was also reciprocal, with * dense Alumina, A1 20 3 62 J X + Y(=Y-) Fig. 28. Equivalent ci r c u i t of a thin metal diaphragm in a nonreciprocal waveguide. 63 F i g . 29. Inhomogeneous loading configuration for the four experimental cases, (a) and (b) ceramic loading, e r = 9.37; (c) f e r r i t e loading; (d) polystyrene loading, e r = 2.53. Table 3. Characteristics of the f e r r i t e . Saturation magnetization, 4TTM (Gauss) s Remanence ratio, M /M r s Dielectric loss tangent Relative permittivity, Coercive force, H £ (Oersteds) Type of fer r i t e Material composition Supplier 1400 0.35 <0.002 12.0 5.0 microwave spinel Nickel, Chromium, Iron Marconi, England 65 symmetrical E-plane polystyrene loading. This case was investigated earlier to provide confidence in the theoretical technique used to de-(25) termine the effect of a diaphragm in inhomogeneous waveguide . The measurement procedure was somewhat different but is worth noting: the inhomogeneous section was matched to the empty waveguide and the effect of the diaphragm measured directly. This procedure s t i l l required the determination of the effect of the homogeneous - inhomogeneous waveguide interface in order to design the quarter-wave transformers. However, the method could be applied to nonreciprocal cases, although the design of the quarter-wave transformers would be more complicated, and i t might be preferable in some instances to the "unmatched technique" generally used in this type of work. The inhomogeneous waveguides in cases (a)-(c), Fig. 29, are a l l smaller than standard X-band size. Two waveguide tapers were made to join each size of waveguide to the standard 0.9" x 0.4" waveguide at both ends of the inhomogeneous section. The tapers were made by elec-troforming copper onto polished stainless-steel forms, which gave the inner surfaces of the tapers a very smooth f i n i s h . The total length of each taper was approximately 5 in . In cases (b) and (c) the smallest waveguide cross-section, i f a i r - f i l l e d , would have been beyond cut-off at the measurement frequency of 8.5 GHz, so this region was f i l l e d with polystyrene. The polystyrene loading was then tapered to a point in the center of the waveguide cross-section so as to match the polystyrene-f i l l e d , small size waveguide to the empty, standard size waveguide as shown in Fig. 30. Reflections from the polystyrene and waveguide tapers of cases (b) and (c) were measured and found to have a maximum value of about 0.07 over the frequency range from 8.25 - 9.5 GHz. The reflections polystyrene tapers waveguide taper reference plane polystyrene spacer ferrite or ceramic Fig. 30. Waveguide and polystyrene tapers relative to the inhomogeneous - section for cases (b) and (c). ON ON 67 from the waveguide tapers used for case (a) were less than 0.02. The effect on the results of these reflections could not be predicted, but i t was found by considering the accuracy of the results for cases (a) and (b), section 4.5.1, that this was not serious. Several precautions were taken to ensure that the results would be repeatable. Aluminum spacers were made to locate the ceramic or fer-r i t e in the center of the waveguide to ±0.001 cm. The polystyrene i n -serts for cases (b) and (c) were glued in place at the side walls of the waveguide so that they would exert pressure on the ceramic or f e r r i t e and eliminate the possibility of a i r gaps at the interfaces. The waveguide sections which contained the inhomogeneous loading material were machined from brass blocks so that the tops could be removed. The dimensions for case (c) were such that pressure was exerted on the f e r r i t e so there would be no gap between the fe r r i t e and the broad walls of the waveguide. 4.4.2 Preliminary measurements on nonreciprocal waveguide sections In order to magnetize the f e r r i t e , wires were inserted into the waveguide through holes in the side walls, through slots in the poly-styrene and through the center of the f e r r i t e toroid. The wires and the slots i n the polystyrene were found to have a very small effect on the measured quantities: less than 1° and 0.3 dB. From the l i s t of properties of the f e r r i t e i n Table 3, i t i s seen that the remanent magnetization was only a third of the saturation magnetization. By plotting curves of di f f e r e n t i a l phase shift versus magnetizing current for various lengths of f e r r i t e , i t was apparent that a constant magnetizing current would have to be used for the measurements in order to ensure that the f e r r i t e loaded waveguide was substantially nonreciprocal. A current of 15 amps was thus used, which meant that 68 the ends of the fer r i t e toroid, where the wire passed across to the wave-guide wall, could be magnetized to a greater extent than the rest of the fe r r i t e (normally, this would not be a problem since the device i s used as a latching device). I n i t i a l l y , the arrangement of the magnetizing wires was as shown in Fig. 31(a): the wire passed through one side of the waveguide, through the f e r r i t e and back out the same side, causing a non-symmetric increase in the magnetization at the ends of the f e r r i t e . It was thought that this nonsymmetry might excite a mode that i s possible with only one ferrite slab positioned off-center i n the waveguide. Thus, the magnetizing current was s p l i t so that half came from each side of the waveguide, Fig. 31(b). However, the results for the two arrangements were vir t u a l l y the same. From eqn. (35), the measured phase difference between T"*" and T~ of an unmatched section containing magnetized f e r r i t e i s : /T + - /J2 = 2(6 2 - 9i) + (3 - - 3 +H + K\. . .. .(38) where K' is the added di f f e r e n t i a l phase shift due to the increased mag-netization at the ends of the f e r r i t e , (3~ - 3+) is the true differen-t i a l phase shift per unit length and 2(8 2 - 9j) i s the theoretical effect of the interfaces. Equation (38) may be rewritten in the form: /T+ - JTT_ " + K • • • (38a> where iji i s the d i f f e r e n t i a l phase shift per unit length and K i s the total interface effect. These two terms were determined from measure-ments on two lengths of f e r r i t e , 1.0 in and 1.25 i n , and are plotted i n Fig. 32 versus magnetizing current. As the current i s increased, the regions at the ends of the f e r r i t e approach saturation so that the d i f -ference in magnetization between the ends of the toroid and the centre areas of increased Fig. 31. Arrangement of the magnetizing wire. (a) i n i t i a l arrangement; (b) f i n a l symmetric arrangement. 71 becomes smaller. This i s reflected in the behaviour of K in Fig. 32. Although K would apparently be smaller with a higher magnetizing current, a current of 15 amps was chosen to avoid overheating the f e r r i t e . Various sources of experimental error are l i s t e d in Table 4, along with indications of their effects on the accuracy of the results where possible. The maximum combined effect of these errors was estimated to be ±2° and ±1 dB in the measured values of reflection and transmission coefficients. 4.4.3 Interface measurements In order to determine the properties of the homogeneous-inhomo-geneous waveguide interfaces and the propagation coefficients in the i n -homogeneous section, R and T were measured for two different lengths of the section. For each length, values were then calculated for p , <|>i, n •and X as described in section 4.3.1. The effect on these parameters of small errors in the measured values of R and T was investigated. This was done by choosing a hypothetical case with typical values of p , <j>i, n and X and calculating the corresponding values of R and T. The magnitudes and phases of R and T were changed about their correct values and the ef-fect on "the derived values of p , <J>i, n and X was observed. Typical re-sults are presented in Table 5 where i t i s seen that small errors in R and T can cause large errors in p , <j>i, n and X. Small errors in R and T were, however, inevitable. Thus, the measured angles were changed sl i g h t l y (but within the estimated experimental error of 2°) i n order to obtain approximately the same values of p, <|> i and a (= -In X) for the two speci-Z men lengths used. Typical changes required were 1° - 1.5°. 72 Table 4. Sources of experimental e r r o r . Source of erro r Maximum e r r o r E f f e c t on r e s u l t s Centering of f e r r i t e +0.001 cm Po s i t i o n i n g of f e r r i t e along the waveguide ±0.003 cm ±0.4° Magnetization of f e r r i t e ±1°, ±0.1 dB Temperature variance of f e r r i t e ±0.5°, ±0.02 dB Po s i t i o n i n g of metal diaphragm ±0.005 cm Frequency d r i f t ±20 KHz O s c i l l a t o r amplitude d r i f t ±0.03 dB Reading accuracy ±0.05°, ±0.03 dB Mismatch errors i n bridge ±1°, ±0.1 dB Ref l e c t i o n s from tapers (magnitude) 0.02-0.07 Table 5. Effect of errors in interface measurements. Errors in measured quantities Derived parameters |R| /R (deg.) |T| H (deg.) $i(rad.) -n(rad ) x P no error 3.002 5.703 0.900 0.301 .005 3.008 5.713 0.902 0.309 .010 3.015 5.724 0.904 0.318 .015 3.023 5.736 0.907 0.327 1 3.052 5.761 0.902 0.325 2 3.096 5.814 0.905 0.350 3 3.135 5.862 0.908 0.377 .005 3.025 5.730 0.906 0.311 .010 3.049 5.759 0.911 0.322 .015 3.074 5.789 0.917 0.335 1 3.034 5.761 0.902 0.325 2 3.061 5.814 0.905 0.350 3 3.083 5.862 0.908 0.377 .005 1 .005 1 3.107 5.853 0.914 0.376 -.005 -1 -.005 -1 2.865 5.528 0.889 0.247 74 4.4.4 Diaphragm measurements The diaphragm was made from two strips of 0.003 in. thick brass f o i l and held in place by pieces of polyfoam (e r = 10.3). Each metal strip was bent in the shape of a bracket, [, so that the f l a t ends would make proper contact with the broad walls of the waveguide and would at the same time f i t over the polyfoam pieces for positioning. The dia-phragm was placed half way between the two interfaces and i n a l l cases, (a) -(c), measurements were made for several transverse positions, d, of the diaphragm in this plane, Fig. 33. 4.5 Results and discussion 4.5.1 Reciprocal inhomogeneous loading The results of the interface measurements for cases (a) and (b) , Fig. 29, are shown in Tables 6 and 7. The theoretical values for the interface parameters and propagation coefficients are l i s t e d for comparison in Table 7. The values of p, <j> \ and 8 are in good agreement with the theoretical values, but those of $2 and 6 are not. The rela-tively large error in the last two parameters i s due to the fact that $2 i s small and is calculated from the small difference between two large values of n; the transmission angle, 6, i s derived from $2 and <f>i by eqn. (1). This error affects the values of the equivalent c i r c u i t parameters of the interface significantly as shown in Table 8, although the error in the characteristic admittance i s less than 5%. The wave-transmission matrix of the diaphragm i s given by eqn. (37a). Experimentally determined values of the elements of this matrix are presented in Table 9 for a typical situation; i.e., for the inhomo-geneous loading of case (a), a diaphragm width, d ^ n = 0.25 cm and posi-tion, d = 0.78 cm, where d . and d are defined in Fig. 33. Because of 75 ceramic ( A l 2 0 3 ) or max F i g . 33. Diaphragm configuration and transverse p o s i t i o n . Table 6. Interface measurements, ceramic ( a i 2 o 3 ) loading. Case (a) Case (b) I = 2.0" I = 2.15" a = 2.0" I = 2.15" ^(rad.) 3.030 3.032 3.161 3.129 -n(rad.) 22.572 24.245 22.434 24.098 X(=e ) 0.977 0.960 0.981 0.964 P 0.583 0.583 0.423 0.424 Table 7. Semi-infinite interface parameters. Case (a) Case (b) Expt. Theor. Expt. Theor. P 0.583 0.591 0.424 0.426 •1 (rad.) 3.031 3.0216 3.145 3.1005 (rad.) -0.310 0.2015 -0.306 0.0724 6 (rad.) -0.2103 0.0408 -0.1513 0.0157 t3 (rad./in) 11.124 11.223 11.064 11.106 a (nep./in) 0.019 — 0.011 — 77 Table 8. Interface equivalent c i r c u i t parameters. Case (a) Case (b) Expt. Theor. Expt. Theor. Y 3.672 3.848 2.412 2.478 B 0.629 -0.329 0.370 -0.069 X 0.097 0.034 0.061 0.006 Table 9. Typical diaphragm wave-transmission matrix, case (a): d . =0.25 cm, d = 0.78 cm. min ' . . Matrix element Theoretical form Experimental values A d l l 1 -je ± e T 1.0269/-10.4180 Adl2 _ £ e J ( < M » T 0.1627/-80.9450 Ad21 £ e j ( * - e ) T 0.1627/99.055° Ad22 l e j 9 T 0.9983/10.876° 78 measurement e r r o r s , the various elements of the matrix are i n c o n s i s t e n t ; e.g., 0 i s not the same i n and ^22' 4" ~ ^ i s not ±90° i n A^^ 2 a n c * Ad21' a n d S ° ° n " t* i e v a r-*- o u s diaphragm p o s i t i o n s , the magnitude, p/x, and the two values of 6 showed the most consistent behaviour. Thus, the values of p/x and 9 were taken as being the most correct and were used to determine p , x , <J> and 6 from the u n i t a r y conditions. The values of these parameters were then used to c a l c u l a t e the values of the equiv-alent c i r c u i t elements of the diaphragm. The r e s u l t s f o r cases (a) and (b) are p l o t t e d i n F i g . 34 along with the t h e o r e t i c a l curves. The agree-ment of B with the t h e o r e t i c a l values i s w i t h i n ±6%, and most values are within ±3%. Considering the o v e r a l l accuracy obtained, i t appears that the errors i n (f>2 and 0 (of the i n t e r f a c e measurements) have l i t t l e e f f e c t on the diaphragm measurements. 4.5.2 Nonreciprocal inhomogeneous loading The r e s u l t s of the i n t e r f a c e measurements f o r case ( c ) , F i g . 29, are presented i n Tables 10 and 11 and the equivalent c i r c u i t parameters are l i s t e d i n Table 12. Values of the elements of the diaphragm wave-transmission matrix, eqn. (37b), are l i s t e d i n Table 13 f o r a t y p i c a l s i t u a t i o n : the width of the diaphragm, d ^ = 0.25 cm and the diaphragm p o s i t i o n , d = 0.547 cm. As i n the r e c i p r o c a l cases, the magnitude, p/x, and the angles, 0, were used to derive the values of p, x , <J>, 0 + and 0~ from the unitary conditions. The equivalent c i r c u i t parameters, B* and X*-, are p l o t t e d i n F i g . 35 as functions of d. Comment on the values of X* and X~ I t i s seen from F i g . 35 that r e l a t i v e to B +, X + i s of the order of 15-20% and r e l a t i v e to B~, X~ i s l e s s than 9%. Thus, there i s some 79 F i g . 34. Equivalent ci r c u i t parameters of a metal diaphragm in ceramic loaded waveguide. theory; ooo experiment. Table 10. Interface measurements, magnetized fe r r i t e loading. A » 1.0" I = 1.25" 4>l (rad.) 3.087 3.089 -n (rad.) 10.393 12.976 X(=e- f iV 0.903 0.874 P 0.321 0.359 T+/T" (dB/deg.) .20/74.9° .22/92.9° Table 11. Semi-infinite interface parameters. p 0.340 • l (rad.) 3.088 $2 (rad.) -0.062 6i (rad.) -0.0706 02 (rad.) -0.046 e + (rad./in) 9.70 3" (rad./in) 10.96 « + (nep./in) 0.093 0J~ (nep./in) 0.117 Table 12. Interface equivalent c i r c u i t parameters. Y+ 1.942 Y~ 1.938 *t 0.064 B I 0.111 »t 0.070 B i 0.102 X +  XA 0.037 X I 0.050 0.026 0.058 82 Table 13. Typical diaphragm wave-transmission matrix, case (c): d . = 0.25 cm, d = 0.547 cm. min Matrix element Theoretical form Experimental values A dl l I e 1.0270/-16.144' T A d l 2 _ £ e J ( < } )~ e + ) 0.2033/-95.272c » „ J(<r-6 +) 0.2033/84.728'  Ad21 £• e A d 2 2 1 e J 8 1.0168/9.688' 83 0.65 d(cm) F i g . 35. Equivalent c i r c u i t parameters of a metal diaphragm in f e r r i t e loaded waveguide; magnetizing current = 15 A. question as to the s i g n i f i c a n c e of X (or X7") , although the un i t a r y conditions f o r the diaphragm wave matrix cannot be s a t i s f i e d by B"*" and B~ alone. This was investigated by r e c a l c u l a t i n g the diaphragm equiv-alent c i r c u i t elements f o r the r e c i p r o c a l cases using the nonreciprocal c i r c u i t representation. I d e a l l y , one would expect X + and X~ to be zero and B + and B~ to have the same values. The c a l c u l a t i o n s showed that the values of B + and B - were n e g l i g i b l y d i f f e r e n t from those of B and that the values of X* were of the order of 3-7% of B. This i n d i c a t e s that the values of X+ and X - determined f o r the nonreciprocal case may be greatly i n e r r o r . However, they are small compared with the values of B + and B and may be ignored f o r some purposes. 85 Chapter 5 CONCLUSIONS The main objective of this work was to determine an equivalent circuit representation for a thin metallic discontinuity in a nonrecip-rocal inhomogeneous waveguide. The configuration chosen for the non-reciprocal structure i s typical of those found in latching nonreciprocal fe r r i t e phase shifters. Because of the complexity of this configur-ation, a theoretical analysis did not seem possible. The effect of a metal diaphragm was determined experimentally but before the diaphragm could be represented by an equivalent c i r c u i t i t was necessary to be able to derive values for the characteristic admittances of the nonreciprocal waveguide. The experimental technique involved the measurement of re-flection and transmission coefficients of unmatched sections of the non-reciprocal waveguide, which introduced a further problem: i t was neces-sary to determine the effect of a nonreciprocal inhomogeneous waveguide interface. Before considering the diaphragm discontinuity, the related dis-continuities of reciprocal and nonreciprocal inhomogeneous waveguide i n -terfaces were analysed theoretically. In both cases, the numerical tech-nique mode matching was used. A two-element equivalent c i r c u i t represen-tation of a reciprocal interface led to a useful definition of the norm-alized equivalent characteristic admittance of an inhomogeneous waveguide. This was an important step towards the definition of the characteristic admittances in a nonreciprocal waveguide. Computed values of the norma-lized equivalent c i r c u i t elements and characteristic admittances were obtained for both Emplane and H-plane di e l e c t r i c loading over a wide range of parameters. 86 A similar representation was used for a nonreciprocal inhomo-geneous waveguide interface. In this case, the equivalent c i r c u i t elements and the characteristic admittance of the nonreciprocal section assumed different values depending on the direction of propagation. The nonreciprocal structure was a twin-slab f e r r i t e loaded waveguide which i s a theoretical model for f e r r i t e toroid configurations. Con-trary to expectation, the characteristic admittances for the two direc-tions of propagation were fcund to be nearly the same over a wide range of parameters, although the difference between the two phase coefficients could be quite large. The analysis of these two interface problems provided the un-derstanding required to determine the effect of a diaphragm i n a non-reciprocal waveguide from experimental measurements. The diaphragm was represented by a two-element equivalent c i r c u i t , each element having different values for the two directions of propagation. Measurements made at 8.5 GHz showed that the series elements are an order of magnitude smaller than the shunt elements. The accuracy of these results could not be checked by comparison with theoretical results so an indirect method was used. The effect of a diaphragm in a reciprocal inhomogeneous waveguide was measured, with the inhomogeneous loading configuration chosen so that theoretical re-sults could be obtained. The accuracy of the results for the reciprocal case then provided an indication of the accuracy of the results for the nonreciprocal case. The maximum error in the measured quantitites i s estimated to be ±2° and ±1 dB, while the error in the derived values of the shunt elements i s estimated to be less than ±6%. The percentage er-ror i n the values of the series elements may be much larger but for many purposes these elements are negligible. 87 Appendix A REFLECTION AND TRANSMISSION COEFFICIENTS FOR A RECIPROCAL INTERFACE Mode matching y i e l d s values f or the normalized c o e f f i c i e n t s of the r e f l e c t e d and transmitted waves (dominant modes) at the i n t e r f a c e . The r e f l e c t i o n c o e f f i c i e n t i s obtained d i r e c t l y from these r e s u l t s ; i . e . , r l ^ a l * ^ e t r a n s m - ' - s s i o n c o e f f i c i e n t i s obtained by equating the power flowing i n the p o s i t i v e z - d i r e c t i o n i n the two waveguides, A and B: r 2 K 2 1 * 1 ( 1 - y •> i hi X HA1 • VS = 1 ^ 1 i %1 X HB1 ' 5 z d s i . e . , b l l 2 / s Hi r * x h B l ' ' a ds z I 5A1 _ * x h M • a ds z 2 r. U , g A 11—-, d U S _ 1 _ = B 1 B l z A | ,2 r — * — 0 b, 2 J x h T 1 1 • a ds ... t * = ( - ^ ) -SL-Bl 5 1 5_ . . . ( 3 9 ) * a l / e A 1 x h * . a d s s A l A l z where s i s the waveguide c r o s s - s e c t i o n a l area. In the above expressions, e ^ and h ^ represent the dominant mode i n A while e ^ and h ^ represent the dominant mode i n B. The equivalent c i r -c u i t parameters, F i g s . 5(a) and 5(b), may then be defined i n terms of r-j/a^ and t 2 > Two-element c i r c u i t z < „ i - / / / = + i n 1 - r 1 / a 1 Y + jB . . . ( 4 0 ) (1 - r./a.) /Y t „ o ± 2 Y + jB 88 Three-element c i r c u i t 1 + jX Y' Z 4„ = jX- + in J " l Y1 (1 - X 3B 2) + j B 2 (1 - r ^ ) /Y7" t2 = Y' (1 - X 3B 2) + j B 2 . .(41) 89 Appendix B INTERFACE BETWEEN EMPTY AND H-PLANE DIELECTRIC SLAB LOADED WAVEGUIDES The steps outlined here for the mode-matching solution follow the general procedure in refs. 17 and 18. Considering Fig. 36, wave-guides A and B are matched so that the only reflections are those caused by the interface and the dimensions are such that only one mode may pro-pagate in either A or B. If the TE^Q mode i s incident on the junction from waveguide A, the total transverse fields at z = 0 may be approximated by sums of mode f i e l d s : Waveguide A M E A = ( a i + r i ) i M + ^  a.e A. . . .(42) H A= (a, - r i ) h A 1 - ^  a.h A i • • -(43) Waveguide B N E_ = I b.e . . . (44) j=l 2 B J N H = E b.h . . . (45) j=l J B j In the above equations, e ^ and h ^ are the transverse ele c t r i c and mag-netic f i e l d components of the i-th mode excited in waveguide A and e^j and h„, are corresponding terms for waveguide B. Using the orthogonality Bj relations for modes in each waveguide and satisfying the boundary condi-tions at the interface we have: N b M / e x h.. • I ds r l / e_ x h.. • a ds + I -± E 8. J * 3 J-r- • / e_ x h,, • I ds — 5 Bn Al z . . . a r . „ / 5.. x h A. • a ds s Bn A i z a^ j=l 1 i=2 s Ax A i z b + — T e x hL * a ds = ^ e. x h A 1 • a ds . . . (46) a^ s Bn Bn z s Bn Al z 36. Junction of empty and H-plane loaded waveguides. 91 and, r l - - - N b1 -— / e... x h 4 n • a ds - Z — / a_ . x h A 1 • a ds = - / e.._ x h A 1 • a ds a^ s Al Al z a^ s Bj Al z s Al Al z • • .(47) where s i s the waveguide cross-sectional area. Equations (46) and (47) represent a set of N + 1 equations in the N + 1 b l b2 b n r l unknowns, — , — »..., — , — • The amplitude coefficients of the modes a l a l a l a l reflected in waveguide A may be found from: a. N b. / e„. x h,. * a ds — = E -1 Z — , i + 1 . . .(48) a l 1=1 a l / e A. x h A. • a ds s A i Ai z To decide which modes to use to approximate the fields E"A» HA> E_ and H„, consider the integrals, • f e_. x h.. • a ds. We start with B B . ' s Bj Ai z the TE^Q mode, the (only) mode incident on the junction from waveguide A. This mode is of the type TE^Q (n odd) which we denote by the superscript I. The f i r s t step i s to examine the integrals • f I . x h ^ • a ds for a l l the different types of modes that may exist in waveguide B. This reveals that the only modes excited directly by the TE^Q mode are LSM^m modes, m = 1,3,5,...; these are also denoted by the superscript I . Next, we determine which modes are excited i n waveguide A by these LSM modes through an examination of the integrals J" e * x h. . • a ds. These i n -s uj A I Z tegrals are zero for a l l but TE^Q, TE^ m and TM^m modes, m = 2,4,6,.... The TE^ m and TM^ m modes are denoted by the superscripts I I and I I I , res-pectively. Now, for the corresponding modes excited in waveguide B, we find that / e „ . x h**'*** • a ds = 0 for a l l but LSE- modes, m = 2,4, s Bj A i z lm * ' 6,..., denoted by the superscript II. Finally, we find that / ^Bj^ X h . . • a ds = 0 for a l l but TE, and TM, modes, m = 2,4,6 which are Ai z lm lm ' * ' ' ' already excited by the. L S t i ^ modes as shown above. Thus, no other modes are exicted at the interface. These results may be summarized as follows Waveguide A TE l m, m = 0,2,4,... Waveguide B TM l m, m = 2,4,6,... LSEj^, 111=2,4,6,... LSM, , m = 1,3,5,... lm Expressions for the fields in both waveguides are derived from similar equations. For TE or LSE modes: ^ nm nm E = - j tou'V x n h H = k 2 n h + vv • n h where = a z \ ( x > y ) e Y Z f o r T E n o d e s = l y i^(x,y)e Y Z for LSE modes and ^ satisfies the relation V t *h + ( y 2 + k 2 ) \ = 0 For TM or LSM modes: nm nm E = k 2n + VV • n e e H = joi'e' V x n J e where S = a I/J ( x , y ) e _ Y Z for TM modes e z e = l y i}»e(x,y)e Y Z for LSM modes and ip satisfies the relation e In the above relations, T,2 2 , , k = 0 ) e u - e r(y)e 0, Vy)po i n waveguide A in waveguide B The resulting equations for the fields are summarized below. Waveguide A TE modes: nm - I . nir . nu _ e . = -jcou- -— sin — x a J H 0 a a y r I n i r . n i r -h, = Y ~ ~ s i n — x a A n m a a x _ II . irrrr , n iT mir , _ e . = j u y n — [ c o s — x s i n —g y ] a x ' 0 b nir 0 a s i n — x c o s — r - y j a mir r II n i r , . n i r mir , _ h . = Y — " [ s m — x c o s — y ] a A n m a a b x mir r n i r mir , _ + Y T " [ c o s — x s i n —7- y ] a ' n m b a b J -y TM modes: nm - I l l n i r r n i r . n m , _ - Y — I c o s — x s i n —r- y j a ' n m a a b x nm b mir r , mr mir n _ —r- [sin — x cos —r- y] a a b ^ y *^ XXX , LU II hA = J u e o T mir r . nir mir . _ — r [sin — x cos —r- yj a b a b x . nir r nir . mir , _ -loie^ — [cos — x sin ~r yJ a J 0 a a b 7 J y 9 4 Waveguide B LSM modes (m odd) nm — I nir , r . , mr n _ e^ = h [sin h y cos — xl a JJ a m m a x 2 , , . nir , _ -Yn Icos h y sin — xl a . 'Om L m"' a y 0 < y < b/2-t = — A £ [sin £ (b/2-y) cos — x] a a m m m a x - Y^ A [cos £ (b/2-y) sin x] I Um m m a y b/2-t i y 1 b/2 (52) r I . r , . nir , -n„ = jo)e r tY Icos h y sin — xl a T$ J 0'nmL m7 a x = iu)£ Y A [cos £ (b/2-y) sin — xl a nm m m a x 0 < y < b/2-t b/2-t 1 y 1 b/2 h sin h (b/2-t) . cos h (b/2-t) ^ _ m m _ x_ m m £ sin £ t e cos £ t m m r m n^m = hm + O 2 - K 0 2 nm m a U 02 , .nir. 2 . 2 m a r r U LSE modes (m even) nm _ II e B " = -juiUoY^tsin h my cos 21 X] 5 -jwy Y_ A [sin £ (b/2-y) cos ^ x ] a nm m m a x 0 < y < b/2-t b/2-t < y < b/2 r- II nir , r , . nir , — h = h [cos h y sin — x] a B a m m a x 2 , . . nir •, -- Yn_ [sin h y cos — x] a 1 Om -nJ 0 " m 0 < y < b/2-t (53) — A £ [cos £ (b/2-y) sin — x] a a m m m y a x - Y/v? A [sin £ (b/2-y) cos — x] a 'Om ml m J a y b/2-t < y < b/2 95 , s i n h (b/2-t) h cos h (b/2-t) ^ _ 1 in _ m m m y s i n It I cos I t r m m m 2 2 ,mr,.2 ,2 nm m a U . 2 ,mK 2 , 2 m x a' r r 0 96 Appendix C TE± (n odd) MODES IN THE TWIN-SLAB FERRITE LOADED WAVEGUIDE nO For the twin f e r r i t e slab configuration of Fig. 16, and magne-tization in the y-direction, the permeability tensor has the form: y = y 0 y 0 0 1 ±JK 0 +3< 0 y . . .(54) where the upper signs correspond to magnetization in the positive direc-tion of y. The T E * Q modes have only E ^ , H X and H Z f i e l d components: E = ik. e' y B n +YnZ H = x ±JUY t l L , • + K - „ J 'nrBn yBn +Y n z 2 2 6 a)yQ(y - K ) TT = " B n Bn +YnZ z ' 2 2, a)yQ(y - K ) • • .(55) . . .(56) • • -(57) •3*Bn + where tf i ^ = ^ and the upper signs correspond to TE^ n modes. Super-Jn0 scripts, ± , have been omitted from and Y n f° r simplicity. In eqns. (56) and (57) K = 0 and y = 1 in the a i r and diel e c t r i c regions of the waveguide. For odd modes, ^ n has the form: = f A sin h x rBn ( n n 0 < x 1 c B sin SL (x-c) + C sin I (c+t-x) c < x < c+t n n n n D^cos q n(a/2 - x) c+t < x < a/2 (58) and satisfies the wave equation: *» n + (Y* + k 2) hn•- 0 . . .(59) 97 where k = "0 ,2 k0 £ f y e k 2 e K0 e d 0 < x < c c < x < c + t c + t £ x £ a/2 2 2 y - K y The modal expressions may be summarized as follows: Waveguide B hj = ±j — — A sin h x a Bn toy^  n n x 'n S± = A sin h x a Bn n n y 0 £ x < c (60) 'n hr = ±j —-^-[B sin £ (x-c) + C sin £ (c+t-x)]a Tin J a)y Qy e l n n n n x </y [B I cos £ (x-c) - C I cos £ (c+t-x)]a ~ ~ n n n n x u > y0 ye n n e± = [B sin £ (x-c) + C sin £ (c+t-x)]a Bn L n n v ' n n / J y c < x < c+t (61) n hBn ± 3 toy D cos q (a/2 - x)I n T I x e* = D cos q (a/2 - x)a Bn n ^n j c+t < x < a/2 (62) Waveguide A r . T i . .ITT ,_ h. . = 3 s m ( — x)a Ai J u>yn a x , i lT. e. , = s i n ( — x)a Ai a y . .(63) ,1* In the empty waveguide, = s i n ( — x). Thus, the constants A , B , C and D should be chosen so that i s real in a l l three regions of the n Bn waveguide B cross-section, regardless of whether h n, £^ and q^ are real 98 or complex. This is done by normalizing the constants to D^. Satisfying the boundary conditions at x = c and x = (c+t) yields the following: D = 1 n = cos q^d/sin Z^t C = [-W <L sin q d sin Z t + I cos I t cos q d n e ^n TI n n n Hn +1 — Y sin I t cos q d]/[£ sin % t] J y n n ^n n n A = C sin I t/sin h c n n n n The eigenvalue equation i s obtained from these boundary conditions. If the subscript n i s omitted, this has the form: -q y e sin qd cos he S ± n £ £ t + (^ - T 2) cos Zt ± j ^ yO^  + T 2) **™J* , 1 ,,K .2 n 2 n sin he , sin It n + ~ [ ( - Y ) -A ] — J J - - cos qd — — = 0 (64) (65) where T^ = cos he cos qd T 2 = jJ- sin he sin qd and the upper sign corresponds to TE^ Q modes. Equation (59) may be written more e x p l i c i t l y for the three regions in the forms 2 C0 2 i ,2 ,Y = h - k. 2 2 2 £ • = ti + ( e f y e - 1) k £ 2 2 2 q = li + ( e d - 1) k^ ...(66) ...(67) ...(68) With eqns. (66)-(6.8), solutions may be found for h from eqn. (65) . For propagating modes,y i s imaginary and a solution exists for h which i s either pure real or pure imaginary. For nonpropagating modes, however, Y i s not pure imaginary and the solutions for h are complex. Thus, y, I and q are also complex. One simplification arises, however, since the TE+Q and TE~ Q modes form a conjugate p a i r ^ 2 6 ^ ; i . e . , i f y* = +. j 8±, 99 then y~ = a ± - j $ ±. The eigenvalue equation was solved by Newton's method, h = a + jg is a solution, then eqn. (65) may be written as: F(h) -'R(h) + jl(h) = 0 and, f=£&> = R + j I = I_ — j R ' dh a J a 6 J B I f .(69) • (70) Newton's iterative method: "i+l = a i + 6 i + l = h+ .(71) . -(72) Equations (71) and (72) converge provided the starting values, + JBQ, are sufficiently close. It was found that for a l l solutions, g << a and suitable starting values were obtained by setting BQ = 0 and finding CIQ by bisection such that R(CXQ) = 0 . 1 0 0 REFERENCES 1 . Van T r i e r , A.A.Th.M.: "Guided electromagnetic waves i n a n i s o t r o p i c media", Appl. S c i . Res., 1953, 3B, pp. 305-371. 2. Lax, B., and Button, K.J.: "Microwave f e r r i t e s and ferrimagnetics", (McGraw-Hill, 1962). 3 . Gurevich, A.G. (A. Tybulewicz, t r a n s . ) : " F e r r i t e s at microwave f r e -quencies", (Boston Technical Publishers, Inc., 1965). 4. Treuhaft, M.A. and S i l b e r , L.M.: "Use of microwave f e r r i t e toroids to eliminate external magnets and reduce switching power", Proc. IRE, 1958, 46, (8), p. 1538. 5. Siekanowicz, W.W., S c h i l l i n g , W.A., Walsh, T.E., Bardash, I. and Gordon, I.: "Design and performance of a 20-kilowatt l a t c h i n g non-r e c i p r o c a l X-band f e r r i t e phase s h i f t e r " , R.C.A. Review, 1965, 26, (12), pp. 574-586. 6. Frank, J . , Kuck, J.H. and Shipley, C.A.: "Latching f e r r i t e phase s h i f t e r f o r phased arrays", Microwave Journal, 1967, 10, (3), pp. 97-102. 7. Querido, H., Frank, J . and Cheston, T . C : "Wide band phase s h i f t e r s " , I.E.E.E. Trans., 1967, AP-15, (3), p. 300. 8. Spaulding, W.G.: "A p e r i o d i c a l l y loaded, l a t c h i n g , n on-reciprocal f e r -r i t e phase s h i f t e r " , presented at the I.E.E.E. G-MTT i n t e r n a t i o n a l microwave symposium, Da l l a s , U.S.A., 1969. 9. Spaulding, W.G.: "The a p p l i c a t i o n of p e r i o d i c loading to a f e r r i t e phase s h i f t e r design", I.E.E.E. Trans., 1971, MTT-19, (12), pp. 922-928. 1 0 . Kharadly, M.M.Z.: " P e r i o d i c a l l y loaded non-reciprocal transmission l i n e s f o r phase s h i f t e r a p p l i c a t i o n s " , I.E.E.E. Trans., 1974, MTT-22, (6), pp. 635-640. 1 1 . C o l l i n , R.E.: " F i e l d theory of guided waves", (McGraw-Hill, 1960). 12. Chang, C.T.M.: "Equivalent c i r c u i t f o r p a r t i a l l y d i e l e c t r i c - f i l l e d rectangular-waveguide junctions", I.E.E.E. Trans., 1973, MTT-21, (6), pp. 403-411. 1 3 . Waldron, R.A.: " C h a r a c t e r i s t i c impedances of waveguides", Marconi Rev., 1967, 30, (3rd Q t r . ) , pp. 125-136. 14. Chatterjee, S.K., and Chatterjee, R.: " D i e l e c t r i c loaded waveguides -a review of t h e o r e t i c a l s o l u t i o n s , Pt. I l l " , The Radio and E l e c t r o n i c Engineer, 1965, 30, (6), pp. 353-364. 15. C l a r r i c o a t s , P.J.B., and McBride, J.M.W.: "Properties of d i e l e c t r i c -rod junctions i n c i r c u l a r waveguide", Proc. I.E.E., 1964, 111, (1), pp. 43-50. 101 16. Lengyel, B.A.: "A note on reflection and transmission", J. Appl. Phys., 1951, 22, (3), pp. 263-264. 17. Clarricoats, P.J.B., and Slinn, K.R.: "Numerical solution of wave-guide-discontinuity problems", Proc. I.E.E., 1967, 114, (7), pp. 878-886. 18. Wexler, A.: "Solution of waveguide discontinuities by modal analysis", I.E.E.E. Trans., 1967, MTT-15, (9), pp. 508-517. 19. Bernues, F.J. and Bolle, D.M.: "The ferrite-loaded waveguide discon-tinuity problem", I.E.E.E. Trans., 1974, MTT-22, (12), pp. 1187-1193. 20. Ince, W.J., and Stern, E.: "Nonreciprocal remanence phase shifters in rectangular waveguide", I.E.E.E. Trans., 1967, MTT-15, (2), pp. 87-95. 21. Walker, L.R.: "Orthogonality relation for gyrotropic wave guides", J. Appl. Phys., 1957, 28, (3), p. 377. 22. Villeneuve, A.T.: "Orthogonality relationships for waveguides and cavities with inhomogeneous anisotropic media", I.R.E. Trans., 1959, MTT-7, (10), pp. 441-446. 23. Mittra, R. and Lee, S.W.: "Mode matching method for anisotropic guides", Radio Sci., 1967, 2, (8), pp. 937-942. •24. Green, J.J. and Sandy, F.: "Microwave characterization of pa r t i a l l y magnetized fe r r i t e s " , I.E.E.E. Trans., 1974, MTT-22, (6), pp. 641-645. 25. McRitchie, W.K., Kharadly, M.M.Z. and Corr, D.G.: "Field-matching solution of transverse discontinuities i n inhomogeneous waveguides", Electron. Lett., 1973, 9, (13), pp. 291-293. 26. Bresler, A.D., Joshi, G.H. and Marcuvitz, N.: "Orthogonality proper-ties for modes in passive and active uniform wave guides", J. Appl. Phys., 1958, 29, (5), pp. 794-799. ADDITIONAL REFERENCE McRitchie, W.K. and Kharadly, M.M.Z.: "Properties of interface be-tween homogeneous and inhomogeneous waveguides", Proc. I.E.E., 1974, 121, (11), pp. 1367-1374. 

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