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An investigation of nonreciprocal periodic structures, transverse discontinuities in nonreciprocal waveguides,… Enegren, Terry A. 1979

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AN INVESTIGATION OF NONRECIPROCAL PERIODIC STRUCTURES, TRANSVERSE DISCONTINUITIES IN NONRECIPROCAL WAVEGUIDES, AND AN INHOMOGENEOUS AND MAGNETIZED FERRITE LOADED RIDGED WAVEGUIDE by TERRY A. ENEGREN B.A.Sc.j. University of British Columbia 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE. DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of El e c t r i c a l Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1979 © Terry A. Enegren, 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f L\~ [ e<zA \Co\ ^ * A C ^ V N v\( The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date O r t 2£ J 1^ ABSTRACT T h i s t h e s i s i s m a i n l y concerned w i t h t h e e f f e c t s o f p e r i o d i c l o a d i n g o f n o n r e c i p r o c a l waveguides by r e g u l a r placement o f d i s c o n t i n u i t i e and w i t h the a s s o c i a t e d problem o f the e v a l u a t i o n of a t r a n s v e r s e d i s c o n t i n u i t y i n a n o n r e c i p r o c a l f e r r i t e l o a d e d waveguide. The mode-matching t e c h n i q u e i s used t o a n a l y s e a t r a n s v e r s e d i s c o n t i n u i t y i n a n o n r e c i p r o c a l waveguide. The d i f f i c u l t o r t h o g o n a l i t y r e l a t i o n s are circumvented u s i n g a G a l e r k i n approach. The elements o f an e x a c t t h r e e - e l e m e n t e q u i v a l e n t c i r c u i t f o r an i n f i n i t e l y t h i n m e t a l l i c diaphragm are e v a l u a t e d . Each element has two v a l u e s , one f o r each d i r e c t i o n of p r o p a g a t i o n . The n u m e r i c a l r e s u l t s show the same t r e n d s as those o b t a i n e d e x p e r i m e n t a l l y i n a s i m i l a r c o n f i g u r a t i o n . The p r o p e r t i e s o f a n o n r e c i p r o c a l , f e r r i t e l o a d e d , r e c t a n g u l a r waveguide, which i s p e r i o d i c a l l y l o a d e d by t h i n m e t a l l i c " i n d u c t i v e " diaphragms, a r e i n v e s t i g a t e d e x p e r i m e n t a l l y . The p r o p a g a t i o n c o n s t a n t s o f the s t r u c t u r e s are measured and a r e compared w i t h p r e d i c t i o n s b a s e d on measured v a l u e s o f the s c a t t e r i n g parameters o f a s i n g l e diaphragm i n the n o n r e c i p r o c a l waveguide. The agreement between t h e o r y and experiment i s g e n e r a l l y good e x c e p t f o r the s m a l l e r s p a c i n g s between the diaphragms. T h i s d i s c r e p a n c y i s a t t r i b u t e d t o the e f f e c t s o f h i g h e r mode i n t e r a c t i o n . An i n v e s t i g a t i o n was a l s o made of a magneti z e d f e r r i t e l o a d e d r i d g e d waveguide. A t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n was f i r s t made of the p r e l i m i n a r y problem o f a r e c i p r o c a l , inhomogeneous r i d g e d waveguide and g u i d e d by t h e s e r e s u l t s an approximate, t h e o r e t i c a l t e c h n i q u e was f o r m u l a t e d to a n a l y s e the n o n r e c i p r o c a l f e r r i t e l o a d e d r i d g e d waveguide problem. I t was found t h a t i n t r o d u c t i o n o f the r i d g e had d e t r i m e n t a l " e f f e c t s on the d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s t i c s e x c e p t f o r a few s p e c i a l c a s e s . i i CONTENTS FIGURES . . . v • TABLES 4 a o « « 4 « « e o o o « e « e o » « a -\X SYMBOLS x i ACKNOWLEDGEMENT . . . .. . . . . . . . . .' . . . • 1. INTRODUCTION 1.1 Background . . . . . . . . . . 2 1.2 O b j e c t i v e s . . . . . . . . . . . . . . . . . 9 2. TRANSVERSE DISCONTINUITIES IN NONRECIPROCAL WAVEGUIDES 2.1 I n t r o d u c t i o n . . . . . . . . . . . 11 2.2 T h e o r e t i c a l c o n s i d e r a t i o n s . . . . . . . . . . v . 12 2.3 N u m e r i c a l r e s u l t s . . . . . . . . . . 19 2.4 F o u r i e r s e r i e s as b a s i s f u n c t i o n s i n a p e r t u r e . . . . . . 27 2.5 E x p e r i m e n t a l i n v e s t i g a t i o n . . . . . . . . . . . . 33 2.6 A p p l i c a t i o n to p e r i o d i c l o a d i n g . . . . . . . . . . 37 2.7 H i g h e r mode i n t e r a c t i o n 42 2.8 Summary 48 3.0 AN INVESTIGATION OF NONRECIPROCAL PERIODIC STRUCTURES 3.1 I n t r o d u c t i o n . . . . . . . . 49 3.2 Wave m a t r i x a n a l y s i s o f n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s based on s i n g l e mode i n t e r a c t i o n . . . . . . . . . . 49 3.3 Experiment 57 3.4 Summary . . . . . . . . . . . . . 80 4.0 AN INVESTIGATION OF INHOMOGENEOUS AND MAGNETIZED i -FERRITE LOADED RIDGED WAVEGUIDE 4.1 I n t r o d u c t i o n . . . . . . . . . "84 4.2 Complete n u m e r i c a l s o l u t i o n . . . . . ' .- 86 4.3 Approximate method . . . . . . . . . . . . . . 1 94 4.4 N u m e r i c a l r e s u l t s o f v a r i o u s methods . . . . . . . . 96 4.5 F e r r i t e l o a d e d r i d g e d waveguide . . . . . . . . . . , 104 i i i 4.6 Summary i i 2 5. CONCLUSIONS . . . . . . . . . 114 Appendix 1 < , . . . . iyj REFERENCES . . 119 iv FIGURES 1. Practical toroid designs for nonreciprocal ferrite phase shifter . . . . . . . . 4 2. Cutaway diagram of four-bit phase shifter . ... 6 3. Theoretical twin-slab model for fer r i t e loaded waveguide . . . . 7 4. Spaulding's waveguide geometry 7 5. Theoretical configuration for "inductive" diaphragm in nonreciprocal waveguide . . . . . . . 13 6. General equivalent c i r c u i t for discontinuity in nonreciprocal waveguide-- . . .- . . ; i /• . . . . . > . . 21 7. Amplitude coefficients of modes in waveguide I against N freq = 9.0GHz.; a = 1.143cm.; t d = .046cm.; t f = .255cm. w = t,.; e, = 16.0 r a o fer r i t e magnetized as shown in Fig.5 |Y I < IY I o fer r i t e magnetized in opposite sense |y+| > |y I x fer r i t e unmagnetized = |y | . . . 22 8. Convergence of normalized susceptances freq. = 9.0GHz.; a = 1.143cm.; t d = .046cm.; t = .255cm. w = t^; e d = 16.0 8(a) B + . . . . . 23 8(b) B~ 24 8(c) B . 25 9. Equivalent circuit parameters of the diaphragm as a function of the relative permittivity of the central dielectric freq. = 9.0GHz.; a = 1.143cm.; t d = .046cm.; w = t^ = .255cm. . . 28 10. Equivalent circuit parameters of diaphragm as a function of aperture width freq. = 9.0GHz.; a = 1.143cm.; t d = .046cm.; t = .255cm. 10(a) e d = 16.0 29 10(b) e d = 1.0 . 30 11. Frequency variation of equivalent circuit parameters of diaphragm a = 1.143cm.; t d = .046cm.; w = t f = ,255cm. 11(a) e d = 16.0 31 11(b) e d » 1.0 . . 32 v 12. Experimental configuration section of f e r r i t e loaded waveguide with m e t a l l i c s t r i p . . . "35 13. Experimental r e s u l t s f o r the equivalent c i r c u i t parameters . . . 3g of the diaphragm 14. Periodic loading of t h e o r e t i c a l twin-slab configuration of f e r r i t e by "inductive" diaphragms . . . 38 15. V a r i a t i o n of the propagation constants with the spacing d for the f i r s t "pass region" of the pe r i o d i c structure, (single mode int e r a c t i o n ) f r e q . = 9.0GHz.; a = 1.143cm.; t = .046cm.; w = t =..255cm. + 3 3 f e r r i t e magnetized 3 f e r r i t e unmagnetized 15(a) - 16.0 . . . . . . . 39 15(b) = 1.0 . . . . . . . . . . . . . . . 40 16. V a r i a t i o n of the d i f f e r e n t i a l phase s h i f t per unit l e n g t h , . A 3 , with the inverse of the spacing 1/d 4^ 17. General 2N port nonreciprocal junction representing diaphragm . . 44 18. V a r i a t i o n of propagation constants with spacing d, comparing the analysis based on s i n g l e mode i n t e r a c t i o n with that i n c l u d i n g two higher order modes freq = 9.0GHz.; a = 1.143cm.; t d = .046cm.; t f = .255cm.; w = .352cm.; e d = 1.0 sin g l e mode i n t e r a c t i o n higher mode i n t e r a c t i o n included . . .. . . . 45 19. V a r i a t i o n of A3 with 1/d, comparing the analysis based on s i n g l e mode and that i n c l u d i n g higher mode i n t e r a c t i o n s i n g l e mode i n t e r a c t i o n - - - - - - higher mode i n t e r a c t i o n included 47 20. Experimental configuration . . . 50 21. Nonreciprocal two-port junction representation for unit c e l l of p eriodic structure with spacing d 52 22. Nonreciprocal two-port junction representation f o r loading obstacle . . . . . . . 52 23. Equivalent representation for unit c e l l 56 v i 24.. The.two parts of the measurement section . . . . . . . . . 59 25. The lower part of the measurement se c t i o n showing p e r i o d i c loading of the f e r r i t e - l o a d e d waveguide . . . . . . . . . 60 26. Cross-section of the measurement section i l l u s t r a t i n g s l o t and probe 62 27. Experimental arrangement f o r phase and amplitude measurements . • 63 28. Scale reference for phase and amplitude measurements . . . . . 54 29. Amplitude and phase measurements f o r f e r r i t e loaded section without p e r i o d i c loading. The points "+" and "-" correspond '•• to magnetized f e r r i t e and "o" to unmagnetized f e r r i t e . . . . . 67 30. Amplitude and phase measurements for f e r r i t e loaded section with periodic loading; spacing of the diaphragms i s .22cm. The points "+", "-" and "©" correspond to the two d i r e c t i o n s of magnetized and unmagnetized f e r r i t e , r e s p e c t i v e l y . . . . . 31. Comparison of phase change across f e r r i t e - l o a d e d section of between no loading and p e r i o d i c loading; spacing of diaphragms i s .22cm. . . . . . . . . . . . . . 32. Experimental configuration f o r measuring s c a t t e r i n g parameters of diaphragm i n f e r r i t e loaded waveguide 34(a) freq. * 9.0GHz. 34(b) f r e q . = 9.3GHz. 34(c) f r e q . = 9.6GHz. 34(d) f r e q . = 9.9GHz. 36. Ridged waveguide loaded by twin-slab arrangement of magnetized f e r r i t e 37. Inhomogeneous ridged waveguide 38. D i e l e c t r i c - s l a b loaded ridged waveguide 68 70 72 33. Transmission c o e f f i c i e n t s for "in d u c t i v e " diaphragm. For phase r e s u l t s the points "+", "-"and "o" correspond to the two d i r e c t i o n s of magnetized and unmagnetized f e r r i t e , r e s p e c t i v e l y . In.the magnitude pl o t r e s u l t s are given f o r magnetized f e r r i t e and were found to be independent of the d i r e c t i o n of propagation. ^ 34. Measurement of the pe r i o d i c structures with the predicted values indicated by the s o l i d curves, 76 77 78 79 35. D i f f e r e n t i a l phase s h i f t per unit length as a function of the inverse of the spacing of the p e r i o d i c structures . . . . . 81 85 85 87 v i i •39. Propagation constant vs. t/b for d i f f e r e n t values of e^ £ 2 = u 1 = M 2 = 1.0; freq. = 9.0GHz.; d = ,159cm.; a = 1.143cm. b = ,508cm.; 4 modes were used i n the ridge . . . . . . . 97 40. Propagation constant vs. t/b for d i f f e r e n t values of d/a e 1 = 12.0; E 2 = u = u 2 = 1.0; a = 1.143cm.; b = ,508cm.; fr e q . = 9.0GHz. 4 modes were used i n the ridge . . . . . . . . . . . . gg 41. Comparison of approximate method and f i r s t order s o l u t i o n freq. = 9.0GHz.; t d = 0.0; t f = d = ,265cm.; a = 1.143cm.; b = . 508cm. . 42. Comparison of approximate method and f i r s t order s o l u t i o n freq. = 9.0GHz.; t , = O.C d a = 1.143cm.; b = .508cm. 0 0; t f = ,159cm.; d = ,424cm.; 100 101 43. Comparison of theory with experiment f o r d i e l e c t r i c - s l a b loaded ridged waveguide £ ] L = 7.0; t = ,129cm.; d = .220cm.; a = .287cm.; t = .287cm. b = ,408cm. x, o experimental points measured from two d i f f e r e n t lengths of d i e l e c t r i c - s l a b , loaded ridged waveguide approximate method - - - f i r s t order s o l u t i o n 44. Comparison of theory with experiment f o r d i e l e c t r i c - s l a b loaded ridged waveguide e 1 = 4.71; t = .096cm.; d = .220cm.; a = .715cm.; t = ,490cm. b = .810cm. x, o experimental points approximate method - - - f i r s t order s o l u t i o n . 47. Propagation constant vs. t/b f o r magnetized f e r r i t e loaded ridged waveguide of Fig.36 freq. = 9.0GHz.; e f = 12.0; u = .96; K = -.3 47(a) ( i ) E d = 16.0; t f l = .053cm.; t ^ = d = .265cm, a = 1.143cm. b = .508cm. j 102 103 45. Experimental configuration f o r measuring the propagation constant of a wave propagating i n a d i e l e c t r i c loaded ridged waveguide . . 105 46. Comparison of theory with experimental r e s u l t s f o r configuration of Fig.45 o experiment —: approximate method - - - f i r s t order s o l u t i o n . . . . . . . 106 v i i i ( i i ) e d = 1.0; t d = .053cm.; = d = .265cm.: a = 1.143cm; b. = .508cm. ( i i i ) £_, = 1.0; t J = .053cm; d d t f = d = .159cm.; a = 1.143cm.; b = ,508cm. 107 47(b) D i f f e r e n t i a l phase s h i f t per unit length vs. t/b The frequency, the properties of the f e r r i t e and the parameters f o r cases ( i ) , ( i i ) , ( i i i ) are those l i s t e d i n Fig.47(a) '(iv) e, = 16.0; t , = .053cm. d a t- = d = .159cm. ; a =. 1.143cm.; b = .508cm. 108 48. D i f f e r e n t i a l phase per unit length vs. d for d i f f e r e n t values of t/b freq . = 9.0GHz.; e = 12.0, u= .96. t d = 0.0; t f = d = .265cm.; a = 1.143cm.; b = .508cm. 48(a) K = -.3 . . . . . . . . . . . . . . . . 109 48(b) K = -.6 . . . . . . . . ' . . H O 49. Magnetized f e r r i t e loaded grooved or crossed waveguide . . . . I l l ix TABLES 1. Convergence o f n o r m a l i z e d s u s c e p t a n c e s - example of F i g . 8 „ . .. . . . . . . . . 26 2. Comparison o f b a s i s f u n c t i o n s . . . . . . . 34 3. C h a r a c t e r i s t i c s o f f e r r i t e . . . . .• . . . . . . . . 6 1 4. E x p e r i m e n t a l c o n f i g u r a t i o n o f f e r r i t e l o a d e d waveguide - measured v a l u e s f o r the p r o p a g a t i o n c o n s t a n t s . . . . . 75 5 . Inhomogeneous r i d g e d waveguide - comparison o f the t h r e e methods f o r d e t e r m i n i n g the p r o p a g a t i o n c o n s t a n t s . . . . . . . . . . . . , 99 x SYMBOLS 2a " = b r o a d dimension o f a l l waveguides a,b,a',b' = a m p l i t u d e s o f r e f l e c t e d and t r a n s m i t t e d f i e l d s i n 'waveguides I and I I r e s p e c t i v e l y , r e l a t e d by wave m a t r i x f o r t h e d i s c o n t i n u i t y a,b,c = v e c t o r s whose elements a r e t h e a m p l i t u d e s o f the r e f l e c t e d , a p e r t u r e and t r a n s m i t t e d f i e l d s , r e s p e c t i v e l y , f o r the diaphragm problem a,a_ = c o e f f i c i e n t s o f the H z e i g e n f u n c t i o n e x p a n s i o n i n the o m r i d g e s e c t i o n o f t h e inhomogeneous r i d g e d waveguide a , b = c o e f f i c i e n t s o f the modal e x p a n s i o n s i n waveguides I and n n I I , r e s p e c t i v e l y , f o r the diaphragm p r o b l e m A = wave m a t r i x f o r u n i t c e l l b a s e d on s i n g l e mode i n t e r a c t i o n 2b = am p l i t u d e o f f o r w a r d p r o p a g a t i n g mode i n t r a n s m i s s i o n l i n e model of m a g n e t i z e d f e r r i t e l o a d e d waveguide A.. = elements o f t h e m a t r i x A A,B^,B,C = amplitude c o n s t a n t s o f mode e x i s t i n g i n f e r r i t e l o a d e d r e c t a n g u l a r waveguide A',B',C' = am p l i t u d e c o n s t a n t s o f mode e x i s t i n g i n f e r r i t e l o a d e d r e c t a n g u l a r waveguide o f b r o a d dimension 2w A,B,C,D = amplitude c o n s t a n t s o f f i e l d s e x i s t i n g i n r i d g e s e c t i o n of f e r r i t e l o a d e d r i d g e d waveguide A^ = wave m a t r i x f o r diaphragm i n c l u d i n g two h i g h e r o r d e r modes = wave m a t r i x f o r u n i t c e l l i n c l u d i n g two h i g h e r o r d e r modes = h e i g h t o f a l l waveguides b = c o e f f i c i e n t s o f E„ e i g e n f u n c t i o n e x p a n s i o n i n the r i d g e m s e c t i o n o f the inhomogeneous r i d g e d waveguide b, c = v e c t o r s a s s o c i a t e d w i t h t h e m a t r i x A = a m p l i t u d e o f backward p r o p a g a t i n g mode i n t r a n s m i s s i o n l i n e model o f f e r r i t e l o a d e d waveguide = shunt s u s c e p t a n c e o f e q u i v a l e n t c i r c u i t r e p r e s e n t i n g diaphragm i n r e c i p r o c a l waveguide x i + B = sh u n t s u s c e p t a n c e s o f e q u i v a l e n t c i r c u i t f o r diaphragm i n n o n r e c i p r o c a l waveguide n m m c o e f f i c i e n t s o f H Z e i g e n f u n c t i o n e x p a n s i o n i n t h e o u t e r r e g i o n o f the inhomogeneuos r i d g e d waveguide c , d = c o e f f i c i e n t s o f the e l e c t r i c and m a g n e t i c f i e l d e x p a n s i o n s , r e s p e c t i v e l y , i n the a p e r t u r e o f the d i s c o n t i n u i t y p l a n e o f t h e diaphragm = s p a c i n g o f p e r i o d i c s t r u c t u r e s d = c o e f f i c i e n t s o f E z e i g e n f u n c t i o n e x p a n s i o n i n the o u t e r r e g i o n o f the inhomogeneous r i d g e d waveguide D = d i a g o n a l m a t r i x whose elements a r e the e i g e n v a l u e s o f the m a t r i x A E = wave t r a n s m i s s i o n m a t r i x f o r 2N t r a n s m i s s i o n l i n e s = e l e c t r i c f i e l d i n t r a n s m i s s i o n l i n e model o f m a g n e t i z e d f e r r i t e l o a d e d waveguide E ,E ,E = components of e l e c t r i c f i e l d x y z F = m a t r i x whose e i g e n v a l u e s y i e l d the p r o p a g a t i o n c o n s t a n t s o f the waves p r o p a g a t i n g i n the inhomogeneous r i d g e d waveguide 2 2 h = Y + e2v2 I,U = u n i t m a t r i c e s H ^ J H ^ J H ^ = components o f ma g n e t i c f i e l d k Q = wave number o f p l a n e wave p r o p a g a t i n g i n f r e e space L = wave o p e r a t o r f o r f e r r i t e l o a d e d r e c t a n g u l a r waveguide M = number o f modes used i n a p e r t u r e f o r diaphragm p r o b l e m = number o f modes u s e d i n r i d g e s e c t i o n o f inhomogeneous r i d g e d waveguide N number of modes used i n f e r r i t e l o a d e d waveguide f o r diaphragm p r o b l e m = number of modes used i n o u t e r r e g i o n of inhomogeneous r i d g e d waveguide p = t r a n v e r s e wave number o f mode i n f e r r i t e l o a d e d waveguide, i n f e r r i t e l o a d e d r e g i o n x i i t r a n s v e r s e wave numbers o f mode e x i s t i n g i n unmagnetized f e r r i t e l o a d e d r e c t a n g u l a r waveguide o f w i d t h 2w fundemental m a t r i x and i t s i n v e r s e f o r the m a t r i x A power i n i n c i d e n t mode to diaphragm power i n r e f l e c t e d mode from diaphragm power i n t r a n s m i t t e d mode from diaphragm elements o f the m a t r i x P t r a n s v e r s e wave number o f mode i n f e r r i t e l o a d e d waveguide, i n a i r - f i l l e d r e g i o n complex r e f l e c t i o n c o e f f i c i e n t o f mode r e f l e c t e d from diaphragm r e f l e c t i o n c o e f f i c i e n t m a t r i x f o r waveguide a, a = I , I I ot elements of the m a t r i x R ' : ' h e i g h t of r i d g e f o r r i d g e d waveguide one h a l f the t h i c k n e s s o f the d i e l e c t r i c s e p a r a t i n g the f e r r i t e s l a b s d i s t a n c e from m a g n e t i c w a l l t o f e r r i t e - a i r i n t e r f a c e f o r f e r r i t e l o a d e d waveguide complex t r a n s m i s s i o n c o e f f i c i e n t f o r mode t r a n s m i t t e d from diaphragm t r a n s m i s s i o n c o e f f i c i e n t m a t r i x f o r waveguide a, a = I , I I ct elements o f t h e m a t r i x T v e c t o r a s s o c i a t e d w i t h the m a t r i x F w i d t h o f a p e r t u r e f o r diaphragm p r o b l e m s e r i e s r e a c t a n c e s f o r e q u i v a l e n t c i r c u i t f o r diaphragm i n n o n r e c i p r o c a l waveguide, f o r the two d i r e c t i o n s o f p r o p a g a t i o n c h a r a c t e r i s t i c a d m i t t a n c e s f o r n o n r e c i p r o c a l t r a n s m i s s i o n l i n e m o d e l l i n g f e r r i t e l o a d e d waveguide c h a r a c t e r i s t i c a d m i t t a n c e o f f r e e s p a c e , = /e /u o o x i i i Zg (=U ) = B l o c h impedances o f n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e f o r the two d i r e c t i o n s o f p r o p a g a t i o n + • + Z = 1/ Y~ c c Z = 1/ Y o o 3 = p r o p a g a t i o n c o n s t a n t f o r wave i n f e r r i t e l o a d e d r i d g e d waveguide + 3 = p r o p a g a t i o n c o n s t a n t s o f B l o c h waves f o r two d i r e c t i o n s o f p r o p a g a t i o n A3 = B~ - 3 + + 3 = p r o p a g a t i o n c o n s t a n t s o f wave p r o p a g a t i n g on u n l o a d e d ° t r a n s m i s s i o n l i n e A3 = 3 + - 3~ o o o 6 = ft+ - f f A = m a t r i x r e s u l t i n g from matching t a n g e n t i a l f i e l d components at x=t^ and x=t^ f o r f e r r i t e l o a d e d r e c t a n g u l a r waveguide Y = p r o p a g a t i o n c o n s t a n t o f wave p r o p a g a t i n g i n inhomogeneous r i d g e d waveguide + Y = p r o p a g a t i o n c o n s t a n t s o f B l o c h waves f o r two d i r e c t i o n s o f p r o p a g a t i o n - t h Yjq = . p r o p a g a t i o n c o n s t a n t s of N modes i n f e r r i t e l o a d e d r e c t a n g u l a r waveguide a = J K / y u e e^,' ^2 ~ r e l a t i v e p e r m i t t i v i t y o f r i d g e s e c t i o n . and o u t e r r e g i o n s , r e s p e c t i v e l y , o f inhomogeneous r i d g e d waveguide = r e l a t i v e p e r m i t t i v i t y o f d i e l e c t r i c s e p a r a t i n g the f e r r i t e s l a b s E ^ = r e l a t i v e p e r m i t t i v i t y o f f e r r i t e E Q = p e r m i t t i v i t y o f f r e e space ..+ 0 = phase a n g l e s o f the t r a n s m i s s i o n c o e f f i c i e n t s f o r the diaphragm f o r the two d i r e c t i o n s o f p r o p a g a t i o n ^ = t r a n s v e r s e wave number o f m term o f t h e e i g e n f u n c t i o n e x p a n s i o n i n the r i d g e s e c t i o n o f the inhomogeneous r i d g e d waveguide x i v o f f - d i a g o n a l component o f the r e l a t i v e p e r m e a b i l i t y t e n s o r o f the f e r r i t e m a t r i x r e s u l t i n g from m a t c h i n g t a n g e n t i a l f i e l d components a t x=t, and x=t_ f o r , f e r r i t e l o a d e d r i d g e d waveguide d f r e l a t i v e p e r m e a b i l i t y o f the r i d g e s e c t i o n and o u t e r r e g i o n s , r e s p e c t i v e l y , o f the inhomogeneous r i d g e d waveguide r e l a t i v e p e r m e a b i l i t y o f the f e r r i t e e f f e c t i v e r e l a t i v e p e r m e a b i l i t y o f t h e f e r r i t e p e r m e a b i l i t y o f f r e e space magnitude o f complex r e f l e c t i o n c o e f f i c i e n t f o r diaphragm t r a n s v e r s e wave number o f n 1" 1 1 term of t h e e i g e n f u n c t i o n e x p a n s i o n i n the o u t e r r e g i o n o f the f e r r i t e l o a d e d and inhomogeneous r i d g e d waveguide t r a n s v e r s e wave number o f mode i n f e r r i t e l o a d e d waveguide, i n d i e l e c t r i c r e g i o n s e p a r a t i n g f e r r i t e s l a b s Y 2 + e 2 V . 2 magnitude o f t r a n s m i s s i o n c o e f f i c i e n t f o r diaphragm T phase a n g l e o f r e f l e c t i o n c o e f f i c i e n t o f diaphragm m a t r i c e s d e f i n e d and used i n the c a l c u l a t i o n o f the m a t r i x F m a t r i c e s t h a t form system o f l i n e a r e q u a t i o n s f o r diaphragm problem a n g u l a r f r e q u e n c y + + e~ - <r X V ACKNOWLEDGEMENT T would like to express my sincere thanks to Dr. M.M.Z. Kharadly for his guidance and assistance throughout the course of this work. I am also grateful for the financial support provided by the H.R. MacMillan Family Fellowship for the year 1976-1977 and by the National Sciences and Engineering Research Council of Canada Scholarship for the years 1977-1979. The project was supported by the National Sciences and Engineering Research Council of Canada under grant A-3344. Finally, I would l i k e to thank Mr. J. Stuber for his excellent workmanship in constructing.the various microwave components and to Mrs. Shih-Ying Hoy for typing the thesis. xvi 1 I! INTRODUCTION Chapter 1 INTRODUCTION 1.1 Background A w a v e g u i d i n g s t r u c t u r e i s s a i d t o be p e r i o d i c a l l y l o a d e d i f t h e r e i s a p e r i o d i c change i n one o r more parameters a l o n g the d i r e c t i o n o f p r o p a g a t i o n . Such changes can c o n s i s t o f p e r i o d i c v a r i a t i o n s i n t h e m a t e r i a l p r o p e r t i e s o f the medium ( p e r m i t t i v i t y , p e r m e a b i l i t y ) , i t s p h y s i c a l d i m e n s i o n s , o r can c o n s i s t o f a r e g u l a r placement o f d i s c o n t i n u i t i e s . The wave t h a t p r o p a g a t e s i n a p e r i o d i c a l l y l o a d e d s t r u c t u r e i s c a l l e d a B i o c h wave and, depending on the n a t u r e o f the l o a d i n g , can have a p r o p a g a t i o n c o n s t a n t t h a t i s g r e a t e r o r l e s s than t h a t f o r the unloaded c a s e . The s t r u c t u r e i s s a i d t o be r e c i p r o c a l i f the p r o p a g a t i o n c o n s t a n t o f the wave i s independent o f the d i r e c t i o n o f p r o p a g a t i o n . R e c i p r o c a l p e r i o d i c s t r u c t u r e s have been the s u b j e c t o f e x t e n s i v e 1 i n v e s t i g a t i o n because they have a v a r i e t y o f i m p o r t a n t p r a c t i c a l a p p l i c a t i o n s . R e c e n t l y , a comprehensive r e v i e w o f the s u b j e c t has been made by E l a c h i [ 1 ] . V a r i o u s methods e x i s t f o r the a n a l y s i s o f these s t r u c t u r e s depending on the n a t u r e o f the l o a d i n g . I f the p e r i o d i c n a t u r e o f the medium can be e a s i l y i n c o r p o r a t e d i n t o the wave e q u a t i o n , then s o l u t i o n s i n v o l v i n g H i l l f u n c t i o n s a r e a p p r o p r i a t e . I n c a s e s such as c o r r u g a t e d s u r f a c e s o r tape h e l i c e s , a n a l y s i s by e x p a n s i o n o f the f i e l d s i n t o F l o q u e t harmonics i s more c o n v e n i e n t , however. The c o n v e n t i o n a l t r a n s m i s s i o n l i n e approach i s a p p r o -p r i a t e when the p e r i o d i c l o a d i n g i s a c c o m p l i s h e d through r e g u l a r placement o f o b s t a c l e s . T h i s r e q u i r e s knowledge o f t h e s c a t t e r i n g parameters o f the o b s t a c l e and the p r o p a g a t i o n c h a r a c t e r i s t i c s o f the unl o a d e d l i n e . 3 N o n r e c i p r o c a l waveguides have the p r o p e r t y t h a t t h e i r p r o p a g a t i o n c h a r a c t e r i s t i c s a r e dependent on the d i r e c t i o n o f p r o p a g a t i o n . T h i s p r o p e r t y has been e x p l o i t e d i n many a p p l i c a t i o n s , o f which the n o n r e c i p r o c a l f e r r i t e s h i f t e r i s an example o f c u r r e n t i n t e r e s t . I n t h i s p a r t i c u l a r c a s e , i t i s d e s i r a b l e t o enhance the n o n r e c i p r o c a l e f f e c t and i t has been shown t h a t t h i s can be e a s i l y a c h i e v e d through p e r i o d i c l o a d i n g o f t h e n o n r e c i p r o c a l medium [ 2 ] , [ 3 ] . S i n c e the d e s i r e t o improve the d i f f e r e n t i a l phase s h i f t c h a r a c t e r -i s t i c s o f the n o n r e c i p r o c a l phase s h i f t e r has p r o v i d e d the main m o t i v a t i o n f o r the i n v e s t i g a t i o n o f n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s , a b r i e f summary of t h a t s u b j e c t w i l l be g i v e n . N o n r e c i p r o c a l f e r r i t e phase s h i f t e r s a r e used t o e n a b l e e l e c t r o n i c s h a p i n g and s t e e r i n g o f antenna phased a r r a y s . Antenna a r r a y s , when c o n t r o l l e d i n t h i s f a s h i o n , can s c a n a t a r a t e t h a t i s much f a s t e r t h a n c o u l d be a c h i e v e d through m e c h a n i c a l means. A t y p i c a l a r r a y c o n s i s t s o f a l a r g e number o f i n d i v i d u a l r a d i a t i n g elements. The c h a r a c t e r i s t i c s o f t h e o v e r a l l beam i s deter m i n e d from the phase o f t h e microwave f e e d to each element. Phase c o n t r o l i s a c c o m p l i s h e d through a phase s h i f t e r w i t h each element o f t h e a r r a y h a v i n g i t s own phase s h i f t e r and a s s o c i a t e d d r i v e r c i r c u i t . Hence due t o t h e l a r g e number o f t h e s e d e v i c e s used, g r e a t s a v i n g s can be a c h i e v e d by r e d u c i n g t h e p h y s i c a l s i z e (weight) o f the phase s h i f t e r and the c o s t o f c o n s t r u c t i o n and o p e r a t i o n o f the phase s h i f t e r and a s s o c i a t e d d r i v e r c i r c u i t . Examples o f c o n v e n t i o n a l f e r r i t e phase s h i f t e r g e o m e t r i e s a r e shown i n F i g . 1. The f e r r i t e r o d i s magnetized t r a n s v e r s e l y by p a s s i n g a c u r r e n t through the m a g n e t i z a t i o n w i r e . Changing the d i r e c t i o n o f m a g n e t i z a t i o n g i v e s r i s e t o d i f f e r e n t i a l phase s h i f t . The use o f ma g n e t i c m a t e r i a l w i t h F i g . l P r a c t i c a l t o r o i d d e s i g n s f o r n o n r e c i p r o c a l f e r r i t e phase s h i f t e r s 5 a near square h y s t e r i s i s loop allows operation at remanant magnetization. Only a pulse of current i s needed to change the d i r e c t i o n of magnetization. The o v e r a l l phase change (neglecting i n t e r f a c e e f f e c t s ) i s given by A3£, where A3 i s the d i f f e r e n c e i n the propagation constants for the forward and backward propagating waves and Jl i s the length of the f e r r i t e . Hence, i t i s clear that to maintain a given phase s h i f t , the length (and hence s i z e , weight) can be reduced by i n c r e a s i n g A g . In p r a c t i c a l configurations, the phase s h i f t e r consists of d i s c r e t e lengths of f e r r i t e , each giving a s p e c i f i e d phase s h i f t , as shown i n F i g . 2. Each length i s i n d i v i d u a l l y c o n t r o l l e d . An a l t e r n a t i v e method that has been proposed [4] i s to use a s i n g l e length of f e r r i t e and change the amount of phase s h i f t by varying the strength of magnetization of the f e r r i t e . T h e o r e t i c a l analysis of the p r a c t i c a l configurations for f e r r i t e . loaded waveguides would be very d i f f i c u l t . Instead, the twin slab model of F i g . 3 i s generally used and has been studied by many workers [5], [6], [7], {8], [9]. An important r e s u l t of t h e i r i n v e s t i g a t i o n s was showing that the d i f f e r e n t i a l phase s h i f t per unit length can be s i g n i f i c a n t l y increased by increasing the p e r m i t t i v i t y of the c e n t r a l d i e l e c t r i c . The use of p e r i o d i c loading for s p e c i f i c a p p l i c a t i o n to nonreciprocal f e r r i t e phase s h i f t e r s was f i r s t considered by Spaulding [2], [3]. Using a rather unconventional waveguide design, F i g . 4, he produced experimental r e s u l t s showing that the d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s t i c s of h i s device can be s i g n i f i c a n t l y improved through periodic loading of the non-r e c i p r o c a l section by inductive or c a p a c i t i v e diaphragms. Also through the use of both types of diaphragms he achieved a f l a t response over a s u b s t a n t i a l frequency i n t e r v a l . Fig.2 Cutaway diaphragm o f f o u r b i t phase s h i f t e r H, H 6 ' 'O d i e l e c t r i c — * ^ f e r r i t e F i g . 3 T h e o r e t i c a l t w i n - s l a b model f o r n o n r e c i p r o c a l f e r r i t e phase s h i f t e r f e r r i t e Fig.4 S p a u l d i n g ' s waveguide geometry ' * 8 The q u e s t i o n a r i s e s o f how dependent a r e S p a u l d i n g ' s r e s u l t s on h i s p a r t i c u l a r waveguide geometry. Can o t h e r g e o m e t r i e s show s i m i l a r r e s u l t s ? A l s o under what c i r c u m s t a n c e s w i l l the d i f f e r e n t i a l phase s h i f t be i n c r e a s e d , as n o t a l l l o a d i n g schemes r e s u l t i n an improvement. To answer t h e s e ques-t i o n s , w hich a r e o f g r e a t importance f o r d e s i g n p u r p o s e s , a n a l y t i c a l t o o l s a r e needed t o h a n d l e the problem o f p e r i o d i c l o a d i n g o f a g e n e r a l n o n r e c i p r o c a l waveguide. D i r e c t s o l u t i o n o f the wave e q u a t i o n o r methods based on f i e l d e x p a n s i o n s a r e g e n e r a l l y d i f f i c u l t to a p p l y due to c o m p l i c a t e d l o a d i n g c o n f i g u r a t i o n s a n d - c o u p l i n g o f the wave e q u a t i o n i n a n i s o t r o p i c media. But f o r p e r i o d i c l o a d i n g by d i s c r e t e o b s t a c l e s , an approach by K h a r a d l y [ 1 0 ] , which w i l l be c a l l e d the t r a n s m i s s i o n l i n e a pproach, can be a p p l i e d w i t h comparative ease. In t h i s approach the n o n r e c i p r o c a l medium i s r e p r e s e n t e d by a t r a n s m i s s i o n l i n e h a v i n g d i f f e r e n t p r o p a g a t i o n c o n s t a n t s f o r the two d i r e c t i o n s o f p r o p a g a t i o n . The d i s c o n t i n u i t y i s r e p r e s e n t e d by a n o n r e c i p -r o c a l two p o r t j u n c t i o n which may then be used to d e r i v e an e q u i v a l e n t c i r c u i t whose elements have two s e t s o f v a l u e s , one f o r each d i r e c t i o n o f p r o p a g a t i o n . Through u s i n g h e u r i s t i c e s t i m a t e s o f the s c a t t e r i n g parameters o f the d i s c o n t i n u i t y , K h a r a d l y was a b l e to a c c o u n t f o r t h e i m p o r t a n t a s p e c t s o f S p a u l d i n g ' s e x p e r i m e n t a l r e s u l t s . A p r o p e r a p p l i c a t i o n o f t h i s method, however, r e q u i r e s an a c c u r a t e d e t e r m i n a t i o n of the s c a t t e r i n g p a r a m e t e r s o f the d i s c o n t i n u i t y i n the n o n r e c i p r o c a l waveguide. The a p p l i c a t i o n o f t r a n s m i s s i o n l i n e t h e o r y to n o n r e c i p r o c a l p e r i o d i c l o a d i n g i n i t i a t e d , i n p a r t , i n t e r e s t i n the p r e l i m i n a r y p r o b l e m o f the i n t e r -f a c e between a r e c i p r o c a l homogeneous waveguide and a magnetized f e r r i t e -l o a d inhomogeneous waveguide. 9 T h i s problem was f i r s t i n v e s t i g a t e d t h e o r e t i c a l l y by Bernues and B o l l e [11] who s o l v e d an i n t e g r a l f o r m u l a t i o n o f the p roblem u s i n g a R i t z G a l e r k i n t e c h n i q u e . I n t h e i r work they r e p l a c e d the h i g h e r o r d e r modes t h a t e x i s t i n the magnetized f e r r i t e l o a d e d waveguide by those e x i s t i n g i n the " E q u i v a l e n t d i e l e c t r i c g u i d e " t h a t i s i n a waveguide where the f e r r i t e has been r e p l a c e d by a d i e l e c t r i c o f t h e same p e r m i t t i v i t y . T h i s was done i n o r d e r to a v o i d h a v i n g to f i n d the complex p r o p a g a t i o n c o n s t a n t s o f the modes e x i s t i n g i n the magnetized f e r r i t e l o a d e d waveguide. They d e r i v e d the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s f o r the i n t e r f a c e and f o r a s e c t i o n o f n o n r e c i p r o c a l waveguide. M c R i t c h i e and K h a r a d l y [12] used the mode-matching t e c h n i q u e to s o l v e t h i s problem u s i n g the p r o p e r s e t o f modes t h a t e x i s t e d i n t h e n o n r e c i p r o c a l f e r r i t e l o a d e d waveguide a n d d e r i v e d the e q u i v a l e n t c i r c u i t f o r the i n t e r f a c e . The i n t e r f a c e problem was a l s o i n v e s t i g a t e d e x p e r i m e n t a l l y as p a r t o f M c R i t c h i e ' s Ph.D. d e s e r t a t i o n work. An i m p o r t a n t r e s u l t o f t h e i r e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i g a t i o n s was showing t h a t the c h a r a c t e r -i s t i c a d m i t t a n c e s o f the n o n r e c i p r o c a l t r a n s m i s s i o n l i n e r e p r e s e n t i n g the magnetized f e r r i t e l o a d e d waveguide a r e almost e x a c t l y t h e same f o r the two d i r e c t i o n s o f p r o p a g a t i o n . 1.2 O b j e c t i v e s I n t h i s t h e s i s , work i s u n d e r t a k e n to a p p l y , t h e o r e t i c a l l y and e x p e r i m e n t a l l y , the t r a n s m i s s i o n l i n e approach to the a n a l y s i s o f n o n r e c i p -r o c a l p e r i o d i c s t r u c t u r e s . A l s o d i e l e c t r i c and f e r r i t e l o a d e d r i d g e d wave-gui d e s a r e i n v e s t i g a t e d . The t h e s i s i s summarized as f o l l o w s : (a) In Chapter 2 a t h e o r e t i c a l a n a l y s i s of the s c a t t e r i n g p r o b l e m o f an i n f i n i t e l y t h i n m e t a l l i c diaphragm i n a m a g n e t i z e d f e r r i t e l o a d e d waveguide i s undertaken u s i n g a mode matching t e c h n i q u e . E x p e r i -m e n t a l measurements a r e made on a s i m i l a r , b u t not i d e n t i c a l , c o n f i g u r a t i o n t o a l l o w comparison o f the t r e n d s o f the r e s u l t s . The t h e o r e t i c a l r e s u l t s a r e then a p p l i e d t o a p a r t i c u l a r type o f n o n r e c i p r o c a l s t r u c t u r e c o n s i s t i n g o f the t w i n s l a b c o n f i g u r a t i o n o f f e r r i t e p e r i o d i c a l l y l o a d e d by " i n d u c t i v e " diaphragms. A l s o an attempt i s made to acco u n t f o r t h e e f f e c t s o f h i g h e r mode i n t e r a c t i o n . I n Chapter 3 the p r o p e r t i e s o f n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s c o n s t r u c t e d by r e g u l a r placement o f " i n d u c t i v e " diaphragms a r e measured e x p e r i m e n t a l l y and a r e compared w i t h p r e d i c t i o n s based on the measurements o f one diaphragm. A l s o d i r e c t measurements o f the f i e l d s w i t h i n n o n r e c i p r o c a l waveguides a r e g i v e n . I n Chapter 4 a t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n o f an inhomogeneous r i d g e d waveguide i s un d e r t a k e n . The t h e o r e t i c a l a n a l y s i s i s extended t o a magneti z e d f e r r i t e l o a d e d waveguide through an approximate method. 11 C h apter 2 TRANSVERSE DISCONTINUITIES IN NONRECIPROCAL WAVEGUIDES 2.1 I n t r o d u c t i o n In t h i s c h a p t e r the s c a t t e r i n g p r oblem o f a t h i n m e t a l l i c diaphragm i n a n o n r e c i p r o c a l waveguide i s i n v e s t i g a t e d t h e o r e t i c a l l y and the e q u i v a l e n t c i r c u i t i s e v a l u a t e d . The n u m e r i c a l r e s u l t s show the same t r e n d s as t h o se o b t a i n e d e x p e r i m e n t a l l y i n a s i m i l a r , but n o t i d e n t i c a l , c o n f i g u r a t i o n . The r e s u l t s f o r one diaphragm a r e then used to a n a l y s e p e r i o d i c l o a d i n g o f th e n o n r e c i p r o c a l waveguide by an a r r a y o f diaphragms. The a n a l y s i s o f the non-r e c i p r o c a l p e r i o d i c - s t r u c t u r e s i s i n i t i a l l y based on s i n g l e mode i n t e r a c t i o n between the diaphragms. An e x t e n s i o n i s t h e n made to i n c l u d e the e f f e c t s o f h i g h e r mode i n t e r a c t i o n . P o w e r f u l n u m e r i c a l methods e x i s t f o r a n a l y z i n g t r a n s v e r s e d i s c o n t i -n u i t i e s i n r e c i p r o c a l homogeneous [13] and inhomogeneous [14] waveguides. These t e c h n i q u e s i n v o l v e expanding the r e f l e c t e d and t r a n s m i t t e d f i e l d s i n t o p a r t i a l sums o f waveguide modes and then s a t i s f y i n g the boundary c o n d i t i o n s a t the d i s c o n t i n u i t y . ' The modal c o e f f i c i e n t s a r e found by u t i l i z i n g the o r t h o g o n a l i t y r e l a t i o n s f o r the waveguide modes. So f a r , i t has been d i f f i c u l t t o e x t e n d t h i s approach to d i s c o n t i -n u i t i e s i n n o n r e c i p r o c a l f e r r i t e l o a d e d waveguides because the o r t h o g o n a l i t y r e l a t i o n s f o r the n o n r e c i p r o c a l waveguide modes, a l t h o u g h they e x i s t , a r e c o m p l i c a t e d and cannot be r e a d i l y a p p l i e d . I n t h i s work a G a l e r k i n approach [15] i s t a ken t o c i r c u m v e n t t h i s problem and a l l o w the f i e l d m a t c h i n g t e c h n i q u e to p r o c e e d i n s i m i l a r manner to the r e c i p r o c a l c a s e . 2.2 T h e o r e t i c a l C o n s i d e r a t i o n s 2.2.1 F e r r i t e Loaded Waveguide The c o n f i g u r a t i o n i n v e s t i g a t e d i s shown i n F i g . 5. The t w i n - s l a b arrangement o f the f e r r i t e r e p r e s e n t s a s i m p l i f i e d model f o r the t o r o i d geometry o f c e r t a i n n o n r e c i p r o c a l f e r r i t e phase s h i f t e r s . The f e r r i t e s l a b s a r e m agnetized t r a n s v e r s e l y . O nly the dominant modes p r o p a g a t e . The f i e l d components have no y v a r i a t i o n and can be d e r i v e d from the s i n g l e component o f the e l e c t r i c f i e l d , E^. The p l a n e x=0 behaves as a m agnetic w a l l and the f i e l d components a r e e x p r e s s e d a s : + Y Z E = 4>~(x) e Y y -1 + G dx • K d$ (x) 1 y 2 H - - ^ z jwu r d$ (x) _ Y <  1 dx J + (1) (2) (3) The s u p e r s c r i p t s "+", "-" denote p r o p a g a t i o n i n p o s i t i v e and n e g a t i v e z d i r e c t i o n s , r e s p e c t i v e l y . The p e r m e a b i l i t y t e n s o r c o r r e s p o n d i n g to magnet! z a t i o n i n the n e g a t i v e y d i r e c t i o n i s g i v e n by: P 0 also, u = (u 2 - K 2 ) / u 0 1 0 0 y = p e r m i t t i v i t y o f f e r r i t e = p e r m i t t i v i t y o f c e n t r a l d i e l e c t r i c $(x) i s a s o l u t i o n o f the wave e q u a t i o n L$ = 0, where inductive diaphragm y inductive diaphragm Z=0 ferrite dielectric; £ Fig.5 T h e o r e t i c a l c o n f i g u r a t i o n f o r " i n d u c t i v e " diaphragm i n - n o n r e c i p r o c a l waveguide 14 L .= d x 2 d ° d 2 + y + I dx' 0 1 t < t d ^ — 5 - + Y 2 + y e £ k 2 t , < x < t _ d x 2 e f 0 d - - f t,. < x < a r — — T a k i n g advantage o f the symmetry about x = 0, s o l u t i o n s s a t i s f y i n g the boundary c o n d i t i o n s a t x = 0 and x = a a r e g i v e n by A. cos Ix 0 < x < t , — — a $(x)= ' cos px + B s i n px t d _< x _< t ^ C s i n q(a-x) t f 1 x — a where & 2 ,2 Y + e,k P = d' o Y 2 + u e r k 2 e f o Y 2 + k 2 Imposing the c o n d i t i o n o f c o n t i n u i t y o f the t a n g e n t i a l f i e l d components a t x = t d and x = t ^ y i e l d s an e i g e n v a l u e r e l a t i o n f o r the p r o p a g a t i o n c o n s t a n t s . The a m p l i t u d e c o n s t a n t s A, B and C can then be d e t e r m i n e d . D e t a i l s o f t h i s d e r i v a t i o n can be found i n Appendix 1. F o r the purpose o f t h i s i n v e s t i g a t i o n , the dimensions o f the waveguide have been chosen to c o r r e s p o n d to X-band v a l u e s . T y p i c a l v a l u e s a r e chosen f o r the p e r m i t t i v i t y and p e r m e a b i l i t y o f t h e f e r r i t e ; E^ = 12.0, p. = 0.96. The v a l u e o f K i s s e t e q u a l t o ±.3. T h i s g i v e s a d i f f e r e n t i a l change i n the p r o p a g a t i o n c o n s t a n t s comparable to e x p e r i m e n t a l l y measured v a l u e s . 15 2.2.2 F i e l d M a t c h i n g a t D i s c o n t i n u i t y C o n s i d e r a wave i n c i d e n t from waveguide I . The r e f l e c t e d f i e l d i n waveguide I and the t r a n s m i t t e d f i e l d i n waveguide I I a r e approximated by the f o l l o w i n g p a r t i a l sums o f n o n r e c i p r o c a l waveguide modes. Waveguide I E y H x Waveguide I I E + + YjZ N _ Y n z = e + T a <J> e (6) 1 « n n v ' i • i • d<i> Y-,2 = _ J ± _ J ( + $ + _ IE _ J _ ) e T l j w y e * v Y l 1 y dx ' N d<J» y z ^ , n ' n n J y d x / ' v / n=l N , yz I b * ! e n (8) y i n n n=l + + . N , , d$ y z H = -ZL_ I b $+ _ -j < -JL ) e n ( 9 ) x jwy i*, n n n y dx J e n=l The f i e l d s i n the a p e r t u r e a r e expanded i n a p a r t i a l sum o f a complete s e t o f f u n c t i o n s t h a t s a t i s f y the boundary c o n d i t i o n s a t x = 0 and x = a. The b a s i s f u n c t i o n s chosen a r e the waveguide modes e x i s t i n g i n an i d e n t i c a l l y l o a d e d waveguide e x c e p t w i t h w i d t h w i n s t e a d o f a and t h e f e r r i t e unmagnetized (K = 0). The b a s i s f u n c t i o n s a r e w r i t t e n as A' cos V 'x 0 < x < t , — — d cos p'x + B' s i n p'x t ^ <^  x _< t ^ 0-0) C' s i n q'(w-x) t ^ <^  x <_ w These e q u a t i o n s a r e s o l v e d i n the same manner as done e a r l i e r t o o b t a i n the amp l i t u d e c o n s t a n t s A', B', C' and the t r a n s v e r s e wave numbers V, p', q' f o r the r e q u i r e d number o f modes. Thus the e x p a n s i o n i n the a p e r t u r e i s w r i t t e n as; M E = y c Y y u . m m m=l M H = J d ¥ x •* m m m=l (11) (12) The f i e l d m atching p r o c e d u r e r e q u i r e s t h a t the e l e c t r i c f i e l d i n the gap be expanded i n terms o f the waveguide modes. Thus N M f I c * 1 u . m m m=l $ , + y a $ = 1 « n n n=l N 7. b * + = u n n M I I c Y  1 u . m m m=l n= l 0 < x < w w < x < a 0 < x < w w < x < a (13) (14) I n the r e c i p r o c a l c a s e , o r t h o g o n a l i t y o f the modes i s imposed t o f i n d t h e c o e f f i c i e n t s a and b e x p l i c i t l y . I n the case o f magnetized n n f e r r i t e , the o r t h o g o n a l i t y r e l a t i o n s a r e [ 1 6 ] : dO Y $ d$ Y $ a,. , m , ' m m . , , , n n n . , , _ n t I * (o — j — + ) + $ (a — - j — + ) ] dx = 0 ./ n v dx u mN dx u ' -a e e (15) where a = JK/UU The p e r m i t t i v i t y , p e r m e a b i l i t y and K a r e c o n s i d e r e d as f u n c t i o n s o f x. These orthogonality relations cannot be easily utilized in the conventional mode-matching procedure except with perhaps very awkward manipula-tions of the modal equations. Instead a Galerkin approach is considered. The set of equations (13) and (14) are multiplied by <K and respectively and integrated from x=0 to x=a. This yields: Ya - ftc = -f (16) and Tt fi'c (17) where a = N b = N c = The elements of the matrices ti, T, ti1 and the vector f are given by: in a -A -/ $. $ dx o l n i=l,,..,N n=l,... ,N lm w —* f <S>. V dx o I m i=l,.,.,N m=l,...,M (18) in / $ dx o i n i=l,. ... ,N n=l,. . . ,N i ft:!, lm fw .+* ... j / $. Y dx o l m i=l,. , .,N. m=l,...,M -a .+* .+ . / $. 0, dx o l 1 i=l , . . . ,N The asterisk * denotes complex conjugate. The usual procedure [13], [14] i s followed for matching the magnetic fields at the aperture to obtain: r a - T'b = - f (19) 18 where mt /.w ,,, , - $n . K .• d$n N , m - " / * . ( - Y J "* T~ ) d x 1=1 .. N n=l N i n o' * i x 'n p J p p dx e e + d$ + r'. = / W H ' . ( - Y + — - j — - r n - ) d x i = l , . . . , N n = l , . . . , N / m o i n p pp dx ' ' ' e e . <I> d$ f = / V.(-Y, 3 j ) dx i = l ... N l o i '1 p J p p dx x X , . . . , I N e e E q u a t i o n s ( 1 6 ) , (17) and (19) form a system o f l i n e a r e q u a t i o n s from which'the modal c o e f f i c i e n t s can be de t e r m i n e d . The r e f l e c t i o n c o e f f i c i e n t , R, and the t r a n s m i s s i o n c o e f f i c i e n t , T, a r e o b t a i n e d from the modal c o e f f i c i e n t s a^ and b ^ r e s p e c t i v e l y and a r e n o r m a l i z e d w i t h r e s p e c t t o power. where , - , j£ X - . , X ( 2 2 ) i I \ = power i n i n c i d e n t mode P r = power i n r e f l e c t e d mode = power i n t r a n s m i t t e d mode t <j)+ = a r g ( a 1 ) 6 + = a r g ( b x ) The case o f a mode i n c i d e n t from waveguide I I i s e q u i v a l e n t t o r e v e r s i n g the d i r e c t i o n o f m a g n e t i z a t i o n . 19 2.2.3 Equivalent circuit After determining the reflection and transmission coefficients the diaphragm is then modelled by the three-element equivalent c i r c u i t [10] , shown in Fig. 6. The characteristic admittance of the nonreciprocal trans-mission and the ci r c u i t elements have two sets of values one for each direction of propagation or magnetization. It has been shown theoretically [12] and verified experimentally in this work that Y* i s almost exactly equal to Y . Hence for the purposes of this investigation the characteristic admittances of the nonreciprocal transmission line w i l l be assumed to be equal. The cir c u i t parameters are normalized with respect to the characteristic admittances, + - '• which is equivalent to setting Y^ = 1 = Y^. The positive superscript on the ci r c u i t elements corresponds to propagation in the positive z direction with the ferrite slabs magnetized as shown in Fig. 5. The magnitude of the propagation coefficient of the incident mode in waveguide I in this case i s less than that of the dominant reflected mode. The negative superscript corresponds to the case where the direction of magnetization is reversed or the wave i s incident from waveguide II. No superscript indicates the unmagnetized case. 2.3 Numerical Results In choosing the number of aperture and waveguide modes used, both the edge condition and the convergence of the solution must be considered. The ratio (M/N) is taken to be equal to the ratio of aperture width to waveguide width (w/a). In the reciprocal case this method of choosing (M/N) was found to yield the most reliable result for both homogeneous [13] and inhomogeneous [14] waveguides. The edge condition places a constraint on the higher order modal decay rate. Since the higher order modes for the cases of magnetized and unmagnetized f e r r i t e more c l o s e l y resemble each o t h e r as the mode number i n c r e a s e s i t appears r e a s o n a b l e to assume t h a t the b e s t c h o i c e o f the r a t i o (M/N) i s the same f o r the magnetized c a s e . A l o g - l o g p l o t o f the modal am p l i t u d e c o e f f i c i e n t s i n waveguide I a g a i n s t mode number, F i g . 7, show t h a t the decay r a t e s a r e the same f o r both the c a s e s o f magnetized and unmagnetized f e r r i t e . A more r i g o r o u s check on the c o r r e c t c h o i c e o f r a t i o (M/N), however, would r e q u i r e a t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n o f the same c o n f i g -u r a t i o n , as was done i n th e r e c i p r o c a l inhomogeneous c a s e [ 1 4 ] . I n F i g u r e s 8a-c a r e p l o t s o f t h e n o r m a l i z e d shunt s u s c e p t a n c e s a g a i n s t the number o f waveguide modes used. The number o f a p e r t u r e modes i s g i v e n by the i n t e g e r p a r t o f the q u a n t i t y N(w/a). I n t h i s c a s e w/a i s s l i g h t l y g r e a t e r than 2/9. The most r a p i d convergence i s seen to be when N i s a m u l t i p l e o f 9 as t h e s e v a l u e s y i e l d a l m o s t e x a c t i n t e g e r r e s u l t s from the q u a n t i t y N(w/a). T h i s o p t i m a l c h o i c e o f N i s the same as t h a t r e p o r t e d f o r the r e c i p r o c a l homogeneous case [ 1 3 ] . The r a t e o f convergence appears t o be the same r e g a r d l e s s o f whether the f e r r i t e i s m a g n e t i z e d o r unmagnetized. The r e s u l t s show t h a t f o r a l l c a s e s <J)+ and p a r e i n d e p e n d e n t of the d i r e c t i o n o f m a g n e t i z a t i o n and the u n i t a r y c o n d i t i o n s f o r the s c a t t e r i n g m a t r i x o f t h e diaphragm a r e s a t i s f i e d . p 2 +. T 2 = 1 2<j,+ - 0+ - e" = 180° (23) where 0 i s phase o f the t r a n s m i s s i o n c o e f f i c i e n t when the d i r e c t i o n o f m a g n e t i z a t i o n i s r e v e r s e d . T h e o r e t i c a l r e s u l t s a r e g i v e n f o r s e v e r a l s i t u a t i o n s . F i g . 9 shows the dependence o f the c i r c u i t parameters on the p e r m i t t i v i t y o f the 21 Fig.6 G e n e r a l e q u i v a l e n t c i r c u i t f o r d i s c o n t i n u i t y i n n o n r e c i p r o c a l waveguide 22 1.0-1 5 10 20 40 Mode number Fig.7. Amplitude c o e f f i c i e n t s of modes i n waveguide I against N fre q . = 9.0GHz.; a = 1.143cm.; t , = .046cm.; t f = .255cm. w « t £ ; e J = 16.0 d * t a • f e r r i t e magnetized as shown i n F i g . | y + l < l y I o f e r r i t e magnetized i n opposite sense | y + | > | y I x f e r r i t e unmagnetized | y + l = | y I 2 3 24 -.620 <D o C O QL u V) -610 I - 6 0 0 a E i _ o -.590 -.580 Fig.8c TABLE 1 Convergence of normalized susceptances - example of Fig.8 Number of modes used B + 3= 5.02979 (rad./cm.) B 0= 5.56583 (rad./cm.) B~ 0= 6.00337 (rad./cm.) 9 -.77072 -.58019 -.50410 18 -.76942 -.57966 -.50102 27 -.76634 -.57837 -.50028 36 -.76593 -.57790 -.50106 45 -.76554 -.57773 -.50119 c e n t r a l d i e l e c t r i c , e, with w = t ^ . As the p e r m i t t i v i t y increases the values of the c i r c u i t parameters decrease. This i s a t t r i b u t e d to greater concentra-t i o n of the f i e l d s i n the c e n t r a l part of the waveguide. In Figures lOa-b the c i r c u i t parameters are p l o t t e d as a function of the aperture width w f o r e = 1.0 and e = 16.0. The values of the c i r c u i t parameters decay very quickly from the f e r r i t e boundary. The e f f e c t i v e area of the diaphragm l i e s i n a region close to the f e r r i t e . The frequency dependence of the c i r c u i t parameters over a frequency range of 8.3 to 10.0 GHz. i s shown i n Figures l l a - b with w=tf. In a l l cases the dominant parameters for the nonreciprocal d i s -+ continuity are the shunt susceptances B and these behave i n a manner s i m i l a r to the unmagnetized case. The s e r i e s elements are r e l a t i v e l y small. The + - + -r e s u l t s seem to i n d i c a t e that X^ = -X^ and = X^ to w i t h i n l e s s than 1%. 2.4 Fourier Series as Basis Functions i n Aperture In the preceeding analysis the basis functions chosen to represent the f i e l d s i n the aperture are those e x i s t i n g i n the equivalent loaded wave-guide of width d and the f e r r i t e unmagnetized. In theory any complete set of functions s a t i s f y i n g the boundary conditions can be used. I t i s merely a question of convergence. In t h i s section,.the aperture f i e l d s are expanded i n a Fourier s e r i e s . This choice of basis functions reduces the amount of c a l c u l a t i o n s involved i n the mode-matching procedure. The f i e l d s i n the aperture are written as: M „ r mirx E = > c cos -—— y m d "M 1 (24) „ . r , mirx H = > d cos —7— x L. m d m=l 28 -Ah L _ — „ J L _ _ _ J L 4.0 8.0 12.0 16.0 • £ Fig.9 Equivalent c i r c u i t parameters of the diaphragm as a function of the r e l a t i v e p e r m i t t i v i t y of the c e n t r a l d i e l e c t r i c f r e q . = 9.0GHz.; a = 1.143cm.; t d = .046cm.; w = t f = .255cm. 29 w=tf . 3 5 w (cm.) "• Fig.10 E q u i v a l e n t c i r c u i t parameters of diaphragm as a f u n c t i o n of aperture w i d t h f r e q . = 9.0GHz ; a = 1.143cm ; t = .046cm ; t = .255cm (a) e = 16.0 o c u a (D .20 *D <L> ^ .10 o o •"X,, ~X2 o a » C L CL» O I/) 3^ *o <i> N *6 E o 8.5 9.5 Frequency (GHz.) 9.0 Frequency (GHz.) 10.0 .11 Frequency v a r i a t i o n of equivalent c i r c u i t parameters of diaphragm a'= 1.143cm.: t ^ = .046cm.; w-::t = .255cm. (a) e d = 16.0 <L> Frequency (GHz.) F i g . l i b e - 1.0 33 The same procedure is used for matching the fields at the dis-continuity. Numerical results for a particular case are given in Table 2. This example was chosen as the Fourier series was required to represent the fields in three distinct regions; diel e c t r i c , ferrite and a i r - f i l l e d ; while at the same time maintaining reasonably high values for the c i r c u i t elements. Convergence is seen to be slightly faster using the basis functions derived from the inhomogeneous waveguide. However, the difference in the results when 28 or 35 modes are used is small. 2.5 Experimental Investigation An experimental investigation has been carried out on the ferrite that toroid configuration of Fig. 12, This configuration is not identical with that investigated theoretically and in this context, the experimental results only serve to i l l u s t r a t e trends and give orders of magnitude of the various parameters. The width of the metallic strip is one half of that of the dielectric / spacers. Increasing the width adds l i t t l e to the total effect. The measure-ment procedure involves matching the interfaces between the ferrite-loaded and empty waveguides to allow direct measurement of reflection and transmission coefficients of the metallic strip. The details of the experimental investiga-tion are given in the next chapter. The experimental results for the ci r c u i t parameters are shown in Fig. 13 over a frequency range of 8.3 to 10.0 GHz. The experimental points show the same trends as the theoretical results with the dominant parameters being the shunt susceptances. The measured values for X* , X_ are too small to plot individually and l i e within the solid dot. TABLE 2 Comparison of basis functions Number of modes 13 26 39 Basis functions I F I F I F -.00911 -.00968 -.00995 -.00995 -.00975 -.00975 <-• X2 -.05851 -.05839 -.06059 -.06061 -.05978 -.05979 B* -.42433 -.42162 -.42844 -.42866 -.42740 -.42749 B -.34830 -.34534 -.34850 -.34870 -.34848 -.34856 X l .05842 .05790 .06038 .06039 .05957 .05958 X2 -.00963 -.00904 -.00996 -.00996 -.00972 -.00972 B~ -.30822 -.30522 -.30795 -.30813 -.30844 -.30851 e = 1.0 ; e f = 12.0 : = .96 ; K - -.3 t^ = .046 cm. ; t f = .255 cm. ; d = .888 cm. ; w = .352;cm. w/a = 4/13 : frequency = 9.0 GHz. F = Fourier series as basis functions I = basis functions derived from inhomogeneous loaded waveguide dielectric s p a c e r s , £^=1.5 metallic strips 493 cm. hole for magnetization wire .279 cm. 1143 cm. Fig.12 E x p e r i m e n t a l c o n f i g u r a t i o n S e c t i o n o f f e r r i t e - l o a d e d waveguide w i t h m e t a l l i c s t r i p -1.5 U c •BL-1.0 o •3 N E O z: -.5 o c a 7 •-Xx, X, Frequency C S E O \ \ V B . B B" " 8.5 9.5 Frequency (GHz.) Fig.13 E x p e r i m e n t a l r e s u l t s f o r e q u i v a l e n t c i r c u i t parameters o f diaphragm 37 2.6 Application to Periodic Loading Consider periodic loading of the twin-slab configuration for ferrite by regular placement of thin "inductive" diaphragms, as shown in Fig. 14. The eigenvalue equation for the nonreciprocal periodic structure (which i s derived in the next chapter), based on single mode interaction, i s cosh ( yd + Jf. ) = 7 cos ( > <25> where 6 = fi+ - i f o+ = e~ _ g"d o Q~ = e" - g~d o d = spacing of periodic structure 6*, 8 q are the propagation constants of the waves propagating i n the ferrite loaded waveguide 0 +, 0 are the phase angles of the transmission coefficients for the diaphragm The two examples analysed in the previous sections are considered. A plot of the propagation constants against the spacing d i s shown i n Figs. 15a-b. For the magnetized case, at smaller spacings just before cutoff is .. reached, there i s , in effect, unidirectional propagation. This i s a non-physical result and indicates that the analysis based on single mode inter-action i s breaking down due to higher mode interaction between the diaphragms. A plot of the differential phase sh i f t aganist 1/d i s shown in Fig. 16. There i s a significant increase in the differential phase shift d i e l e c t r i c f e r r i t e / m e t a l l i c d i a p h r a g m s Fig.14 P e r i o d i c l o a d i n g o f t h e o r e t i c a l t w i n - s l a b c o n f i g u r a t i o n o f f e r r i t e by " i n d u c t i v e " diaphragms 39 i I l Fig.15 Variation of the propagation constants with the spacing d for the f i r s t "pass region" of the periodic structure.(single mode interaction) freq. = 9.0GHz.; a = 1.143cm.; t d = ,046cm.; w = t f = .255cm. 3 , 3 ' f e r r i t e magnetized 3 f e r r i t e unmagnetized ' 15(a) £ j = 16.0 d Propagation constant (rad/cm.) o O F i g . 1 6 V a r i a t i o n o f the d i f f e r e n t i a l phase s h i f t p e r u n i t l e n g t h , A B , w i t h the i n v e r s e o f the s p a c i n g 1/d 42 per u n i t l e n g t h and i t i n c r e a s e s l i n e a r l y u n t i l c u t o f f i s re a c h e d . I t i s exp e c t e d , however, t h a t as the s p a c i n g becomes s m a l l , h i g h e r mode i n t e r a c t i o n w i l l a l s o a f f e c t the d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s t i c s . An attempt to e v a l u a t e the e f f e c t s o f h i g h e r mode i n t e r a c t i o n i s p r e s e n t e d i n the nex t s e c t i o n . 2.7 H i g h e r Mode I n t e r a c t i o n . When the s p a c i n g o f the diaphragms becomes c l o s e , the ev a n e s c e n t modes e x c i t e d a t one diaphragm w i l l n ot have decayed to n e g l i g i b l e v a l u e s when they have r e a c h e d the next diaphragm. U s i n g a t r a n s m i s s i o n l i n e t h e o r y t h a t c o n s i d e r s the dominant mode as the o n l y mode t h a t i s a b l e t o p r o p a g a t e the e f f e c t s o f t h e d i s c o n t i n u i t y i s no l o n g e r v a l i d . I t i s now n e c e s s a r y t o i n t r o d u c e a s e p a r a t e e q u i v a l e n t t r a n s m i s s i o n l i n e f o r each i n t e r a c t i n g mode The t r a n s m i s s i o n l i n e s a r e c o u p l e d by the network r e p r e s e n t i n g the d i s c o n t i n u i t y . The approach p r e s e n t e d h e r e i s an e x t e n s i o n o f t h a t g i v e n i n r e f e r e n c e [16] f o r the r e c i p r o c a l c a s e . The d i s c o n t i n u i t y i s r e p r e s e n t e d by the 2N p o r t j u n c t i o n shown i n F i g . 17, where N i s the number o f i n t e r a c t i n g modes (dominant and evanes-c e n t ) . The r e f l e c t i o n c o e f f i c i e n t m a t r i c e s R*, R"^ and the t r a n s m i s s i o n c o e f f i c i e n t m a t r i c e s T^", T** a r e d e f i n e d as f o l l o w s R?. = r e f l e c t i o n c o e f f i c i e n t f o r 1^ mode w i t h j 1 " * 1 mode i n c i d e n t from waveguide a. T?. = t r a n s m i s s i o n c o e f f i c i e n t f o r i * " * 1 mode w i t h i * " * 1 mode i n c i d e n t from waveguide a . i> j = 1, N a = I , I I 43 The wave m a t r i x f o r the d i s c o n t i n u i t y , r e l a t e s the v e c t o r s a and b to a' and t>', where the elements o f t h e s e v e c t o r s a r e the mode amp l i t u d e s a t the i n p u t and ou t p u t t e r m i n a l s . o f the j u n c t i o n , as shown i n F i g . 17. The wave m a t r i x A d i s g i v e n by A d = i n v ( T ) R 1 i n v ( T 1 ) - i n v ( T 1 ) R 1 1 T 1 1 - R 1 i n v ( T 1 ) R 1 1 where i n v ( ) denotes i n v e r s e . Hence the wave m a t r i x f o r the u n i t c e l l i s g i v e n by A T = E A d E (26) where E = -y^d/2 <d/2 Y 2d/2 • V / 2 e y- and y ^ r e t n e p r o p a g a t i o n c o n s t a n t s f o r t h e n mode p r o p a g a t i n g i n the •n 'n forwa r d and backward d i r e c t i o n s r e s p e c t i v e l y . Fig.17 G e n e r a l 2N p o r t j u n c t i o n r e p r e s e n t i n g diaphragm 45 The e i g e n v a l u e r e l a t i o n f o r the p r o p a g a t i o n c o n s t a n t , y, o f a p a r t i c u l a r B i o c h wave i s where U i s the u n i t m a t r i x There a r e 2N s o l u t i o n s t o t h i s e q u a t i o n c o r r e s p o n d i n g t o N f o r w a r d and N backward p r o p a g a t i n g B i o c h waves. m a t r i c e s were o b t a i n e d by making a few s i m p l e adjustments i n the e x i s t i n g computer program d e v e l o p e d f o r the mode-matching p r o c e d u r e . The r e l a t i v e amount o f e x c i t a t i o n o f the i n d i v i d u a l e v a n e s c e n t modes was found t o depend on the w i d t h of the a p e r t u r e . The r e s u l t s f o r a p a r t i c u l a r case a r e shown i n F i g . 18. I n t h i s example the w i d t h o f the a p e r t u r e was s l i g h t l y g r e a t e r t h a n the w i d t h o f the f e r r i t e . T h i s produced a s i g n i f i c a n t , a p p r o x i m a t e l y e q u a l , c o n t r i b u t i o n from the f i r s t two e v a n e s c e n t modes w h i l e k e e p i n g the c o n t r i b u t i o n from the t h i r d and h i g h e r o r d e r modes r e l a t i v e l y s m a l l e r . E v e n t u a l l y , as the s p a c i n g d e c r e a s e s , t h e s e modes would have t o be taken i n t o a c c o u n t . However, as the f i r s t s t e p i n a c c o u n t i n g f o r the e f f e c t s o f h i g h e r mode i n t e r a c t i o n , and to g i v e an i n d i c a t i o n o f the t r e n d s , the f i r s t two e v a n e s c e n t modes a r e s u f f i c i e n t . h i g h e r than those p r e d i c t e d u s i n g s i n g l e mode i n t e r a c t i o n and t h a t the s p a c i n g o f the p e r i o d i c s t r u c t u r e can be made s m a l l e r b e f o r e c u t o f f o c c u r s . A comparison of the d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s i t c s i s shown i n F i g . 19. G e n e r a l l y , t h e d i f f e r e n t i a l phase s h i f t p e r u n i t l e n g t h i s r e d u c e d when the e f f e c t s o f h i g h e r mode i n t e r a c t i o n a r e i n c l u d e d . (27) The elements o f the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t The r e s u l t s show t h a t the p r o p a g a t i o n c o n s t a n t s a r e g e n e r a l l y 46 Fig.18 Variation of propagation constants with spacing d, comparing the analysis based on single mode interaction with that including two higher order modes freq. = 9.0GHz.: a » 1.143cm.; 1: ~ .046cm.; t = ,255cm. : w « .352cm. 5 Ed = 1 ' 0 " • ' . . , ,. single mode interaction ' . i - — — h i p t i e r mode interaction included 47 5.0 10.0 1/d (cm.-1) Fig.19 V a r i a t i o n o f Ag w i t h 1/d, comparing the a n a l y s i s based on ' - s i n g l e mode and t h a t i n c l u d i n g h i g h e r mode i n t e r a c t i o n y • s i n g l e mode i n t e r a c t i o n m. <_ <•» m, «. am h i g h e r mode i n t e r a c t i o n i n c l u d e d 2.8 Summary In summary, the f i e l d matching techniques developed f o r transverse d i s c o n t i n u i t i e s i n r e c i p r o c a l waveguides can be extended to nonreciprocal f e r r i t e loaded waveguides where the d i f f i c u l t orthogonality r e l a t i o n s can be circumvented by a Galerkin approach. The choice of the number of aperture modes used was determined from the r a t i o of aperture width to waveguide width. I t was not, of course, p o s s i b l e to v e r i f y t h i s choice experimentally as the experimental and t h e o r e t i c a l configurations were not i d e n t i c a l . However, i t i s s i g n i f i c a n t that t h i s method of choosing the number of aperture modes gives the most rapid convergence of the s o l u t i o n as i n the r e c i p r o c a l case. The s i g n i f i c a n t elements of the equivalent c i r c u i t for the d i a -+ phragm i n the nonreciprocal waveguide, the shunt susceptances, B , behave i n a manner s i m i l a r to the r e c i p r o c a l case. The s e r i e s elements are r e l a t i v e l y small and are needed to s a t i s f y the unitary conditions of the s c a t t e r i n g o matrix for the diaphragm. The analysis of the t h e o r e t i c a l nonreciprocal p e r i o d i c structures show that the d i f f e r e n t i a l phase s h i f t per unit length, AB, i s s i g n i f i c a n t l y increased. The analysis based on s i n g l e mode i n t e r a c t i o n shows that AB increases l i n e a r i l y as the inverse of the spacing. When the e f f e c t s of higher mode i n t e r a c t i o n were included i t was found that the growth of AB with 1/d was, for t h i s p a r t i c u l a r example, notably slower. S t i l l , the increase i n AB for d = .660 cm. was approximately 300% over that for no p e r i o d i c loading. 49 C h a p t e r 3 AN INVESTIGATION OF NONRECIPROCAL PERIODIC STRUCTURES 3.1 I n t r o d u c t i o n I n t h i s c h a p t e r the p r o p e r t i e s o f n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s a r e i n v e s t i g a t e d e x p e r i m e n t a l l y . The n o n r e c i p r o c a l medium c o n s i s t s o f a r e c t a n g u l a r waveguide s y m m e t r i c a l l y l o a d e d by a t r a n s v e r s e l y magnetized f e r r i t e t p r o i d as shown i n F i g . 20. Only the dominant modes p r o p a g a t e . P e r i o d i c l o a d -i n g i s a c c o m p l i s h e d through r e g u l a r placement o f t h i n m e t a l l i c " i n d u c t i v e " diaphragms. A s i m p l e e i g e n v a l u e e q u a t i o n , based on s i n g l e mode i n t e r a c t i o n , i s d e r i v e d f o r the n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e r e q u i r i n g t h e r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s o f the diaphragm i n the magnetized f e r r i t e l o a d e d s e c t i o n o f waveguide. An e x p e r i m e n t a l method i s d e v e l o p e d which p e r m i t s d i r e c t measurement of the f i e l d s i n the n o n r e c i p r o c a l waveguide and easy d e t e r m i n a t i o n o f t h e s c a t t e r i n g parameters o f the diaphragm. These r e s u l t s a r e then a p p l i e d to the n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s . 3.2 Wave M a t r i x A n a l y s i s o f N o n r e c i p r o c a l P e r i o d i c S t r u c t u r e Based on S i n g l e Mode I n t e r a c t i o n 3.2.1 P r o p a g a t i o n C o n s t a n t s The a n a l y s i s p r e s e n t e d h e r e i s s i m i l a r t o t h a t i n r e f e r e n c e [10], However, u s i n g the u n i t a r y c o n d i t i o n s f o r the s c a t t e r i n g m a t r i x o f the l o a d -i n g o b s t a c l e , a r e l a t i v e l y s i m p l e e i g e n v a l u e e q u a t i o n can be d e r i v e d . The b a s i c u n i t c e l l o f a n o n r e c i p r o c a l t r a n s m i s s i o n l i n e p e r i o d i c a l l y l o a d e d w i t h d i s c r e t e l o a d i n g o b s t a c l e s , may be r e p r e s e n t e d as shown i n F i g . 21. The l o a d i n g o b s t a c l e may be r e p r e s e n t e d by a two p o r t j u n c t i o n whose r e f l e c t i o n d i e l e c t r i c s p a c e r s , ' £ = 1 . 5 C r o s s - s e c t i o n o f e x p e r i m e n t a l c o n f i g u r a t i o n 51 and transmission coefficients are defined in Fig. 22 and i t s wave amplitude transmission matrix is thus given by 1 -j' — e J T P e j ( * + - e+) _ P ej(<j) - e ) T i e J ( X (28) where p 2 + x 2 = 1 <j>+ + <j»~ - e+ - e~ = ± 180' (29) The wave amplitude transmission matrix for the nonreciprocal section of + - « transmission line with propagation constants g Q and B q i s given by J - iV " 2 (30) The positive and negative superscripts denote propagation i n forward and backward directions respectively. Thus the overall matrix for the unit c e l l is A = I e -j(e+ - /) x £. e 2 ^ + ^ - 2 ™ x ) i „ j(e - i> ) r) (31) where / = 3+d * - B D d (32) nonreciprocal t ransmiss ion l ine 52 0 0 loading obstacle F i g . 2 1 N o n r e c i p r o c a l two-port -junction r e p r e s e n t a t i o n f o r u n i t c e l l o f p e r i o d i c s t r u c t u r e w i t h s p a c i n g d te re Fig.22 N o n r e c i p r o c a l two-port j u n c t i o n r e p r e s e n t a t i o n f o r l o a d i n g o b s t a c l e 53 The s o l u t i o n for Bioch waves requires det (A - e Y d I); = 0 (33) where I i s the unit matrix and y i s the propagation c o e f f i c i e n t for the forward propagating Block wave. Condition (33) y i e l d s the following eigen-value equation cosh (yd + j — ) = - cos ( - — — - ) (34) where ~+ „+ ,+ Q = 9 - i> u~ = e~ - i|T 6 = U+ - $T (35) + -For propagating waves, y = j g . The two solutions 3 and 3 f o r forward and backward propagating Bioch waves are given by D+ f 6 . - 1 , 1 ( f t + + »") w . , 3 = t - 2 + cos ( — cos ) d e - • r 6 . - 1 / 1 (^+ + n") . , . 3 = I T + c o s ( — c o s o ) j ' d (36) 2 T 2 Thus the differ e n c e i n the propagation constants, i s given by A3 = 3~ - 3 + = 6/d (37) Substituting from (32) and (35) in t o (37) y i e l d s A3 = A3 Q + ( 6 + - 9~)/d where A3Q i s the di f f e r e n c e i n the propagation constants of the unloaded l i n e A3„ = 3~ - 3* o o o 54 - + + -F o r 3 Q > 3 Q the q u a n t i t y [0 - 0 ] i s p o s i t i v e f o r " i n d u c t i v e " l o a d i n g and hence the d i f f e r e n c e i n the p r o p a g a t i o n c o n s t a n t s i s i n c r e a s e d . 3.2.2 Bioch-wave Impedance The Bloch-wave impedance i s d e f i n e d as the impedance n e c e s s a r y to t e r m i n a t e a p e r i o d i c s t r u c t u r e such t h a t t h e r e i s no r e f l e c t e d Bloch-wave. The Bloch-wave impedance i s not unique b u t depends on the c h o i c e o f t e r m i n a l p l a n e s . R e f e r r i n g to F i g . 22, l e t the u n i t c e l l be matched by c o n n e c t i n g an + + impedance Z to t e r m i n a l s o-o . "L i s t h e n e q u a l t o t h e i n p u t impedance s e e n a t t e r m i n a l s i - i . S i m i l a r i l y the i n p u t impedance a t t e r m i n a l s o-o y i e l d s Z B when the u n i t c e l l i s t e r m i n a t e d by Z a t t e r m i n a l s i - i . The Bloch-wave impedances a r e n o r m a l i z e d t o the c h a r a c t e r i s i t c impedances o f the unloaded ± + -n o n r e c i p r o c a l t r a n s m i s s i o n l i n e , Z . I t w i l l be assumed t h a t Z = Z = 1 . c c c The wave m a t r i x f o r the u n i t c e l l , A, can be decomposed as f o l l o w s A = PDP -1 (38) The m a t r i x D i s a d i a g o n a l m a t r i x whose elements a r e the e i g e n v a l u e s o f A. ,S+d 0 : 3~d 0 (39) where 3 and 3 a r e the p r o p a g a t i o n c o e f f i c i e n t s f o r the f o r w a r d and backward p r o p a g a t i n g Block-waves. The columns o f the fundamental m a t r i x P a r e the e i g e n v e c t o r s o f A. The m a t r i x P can be w r i t t e n as U~ - 1 2(U + i f ) U + + l U + - l U + 1 (40) 55 where A l l ~ A 2 2 A — — + s + 12 2 A l l A 2 2 . A — 2 § + A 1 2 U = A l l A 2 2 _ A 2 1 " 2 ~ § ' A A H - A 2 2 , , < 4 1> A 2 1 2 + § a n d . / ( A + A ) Z § = / 2 + . < A 1 1 A 2 2 -where A.. are the elements o f t h e m a t r i x A. The u n i t c e l l i s e q u i v a l e n t t o t h e network shown i n F i g . 23. The n o n r e c i p r o c a l s e c t i o n o f t r a n s m i s s i o n l i n e o f l e n g t h d s u p p o r t s the B l o c h -waves. The m a t r i c e s P and P ^ can be i n t e r p r e t e d as d e s c r i b i n g t h e i n t e r f a c e e f f e c t s between t h i s s e c t i o n and the unloaded n o n r e c i p r o c a l t r a n s m i s s i o n l i n e . L e t the matching impedance be con n e c t e d t o the t e r m i n a l s o-o. The i n p u t a impedance a t the t e r m i n a l s i - i can be found by c o n n e c t i n g t h e s e t e r m i n a l s t o the unloaded l i n e . An i n c i d e n t wave e x c i t e s a for w a r d p r o p a g a t i n g B l o c h wave. S i n c e a r e f l e c t e d B l o c h wave i s n o t prod u c e d , the r e f l e c t i o n c o e f f i c i e n t , R, a t the t e r m i n a l s i - i can c a l c u l a t e d from the wave m a t r i x P. R . . l a . («) 11 U + 1 Hence the Bloch-wave impedance, , i s g i v e n by B 2 B - f r i - D + <"> F o l l o w i n g a s i m i l a r p r o c e d u r e f o r Z„ y i e l d s B 56 nonreciprocal t ransmiss ion line o o P P F i g . 2 3 E q u i v a l e n t r e p r e s e n t a t i o n f o r u n i t c e l l 57 R e f e r r i n g to the wave matrix f o r the l o a d i n g o b s t a c l e (28), i f = - ^ 2 1 ' a s ^ S t^ i e c a s e ^ o r a n " i n d u c t i v e " diaphragm i n a n o n r e c i p r o c a l f e r r i t e loaded waveguide, then the Bioch impedances f o r the two d i r e c t i o n s of propagation are equal; = Z . This r e s u l t has p r a c t i c a l importance f o r the magnetized f e r r i t e case as the matching scheme designed to match the n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e to the a i r - f i l l e d waveguide w i l l be v a l i d f o r both d i r e c t i o n s of magnetization. 3.3 . Experiment The main o b j e c t i v e s of the experimental i n v e s t i g a t i o n are ( i ) to measure the s c a t t e r i n g parameters of a t h i n i n d u c t i v e diaphragm i n a n o n r e c i p r o c a l waveguide and ( i i ) to measure the propagation constants of a n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e . Measurement of the diaphragm i n a n o n r e c i p r o c a l s e c t i o n of wave-guide r e q u i r e s accounting f o r the e f f e c t s of the i n t e r f a c e s between the r e c i p r o c a l and n o n r e c i p r o c a l waveguides. One approach i s to measure the p r o p e r t i e s of the i n t e r f a c e and then account f o r them i n the diaphragm measurements. The a p p l i c a t i o n of t h i s approach has r e s u l t e d i n c o n s i d e r a b l e e r r o r i n the s c a t t e r i n g parameters f o r the diaphragm. In t h i s work another approach i s taken where the i n t e r f a c e s are matched thus a l l o w i n g the s c a t t e r -i n g parameters of the diaphragm to be d i r e c t l y obtained through measurement of the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s , as discussed i n s e c t i o n 3.3.3. This i s made experimentally f e a s i b l e through the use of a s p e c i a l l y - c o n s t r u c t e d waveguide s e c t i o n that permitted d i r e c t sampling of the e l e c t r i c f i e l d along the n o n r e c i p r o c a l waveguide. This arrangement a l s o allowed easy and accurate determination of the propagation c h a r a c t e r i s t i c s of the n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s . 58 3.3.1 The Measurement Section A photograph of the measurement se c t i o n i s shown i n F i g . 24. The c e n t r a l part accommodates the f e r r i t e toroids and the l i n e a r tapers provide a gradual t r a n s i t i o n to standard X-band waveguide. The maximum VSWR produced by the tapers at the frequencies at which measurements were taken was 0.2 db. The f e r r i t e t o r o i d was centered by d i e l e c t r i c (expanded polystyrene) spacers; = 1.5. The transverse symmetry ensured that only the dominant modes prop-agated. The length of the f e r r i t e rod was 4.57 cm. The f e r r i t e was magnetized to s aturation by passing 15 amp., through a magnetization wire. To eliminate errors due to the magnetization wire, the f e r r i t e was magnetized e x t e r n a l l y and then placed i n t o the block without the wire. The type of f e r r i t e used i s TT1-390 and i t s properties are given i n Table 3. In the frequency range of i n t e r e s t , the losses due to the f e r r i t e were found to be small and have been neglected i n the i n v e s t i g a t i o n . ' P e r i o d i c loading of t h i s c o n f i guration was accomplished by weaving aluminum f o i l through and f i l l i n g t h i n s l o t s cut at regular i n t e r v a l s i n the d i e l e c t r i c spacers, as shown i n the photograph of F i g . 25. The thickness of the s l o t was 0.15 mm and i t s width was one h a l f of that of the d i e l e c t r i c spacers. The f o i l made good contact with the top and bottom of the waveguide. Rather crude d i e l e c t r i c matching sections were used to reduce the r e f l e c t i o n s between the f e r r i t e - l o a d e d and a i r - f i l l e d waveguides. In the upper part of the measurement se c t i o n a s l o t was cut to allow i n s e r t i o n of a c a p a c i t i v e probe to sample the e l e c t r i c f i e l d at 0.30 mm above the top face of the f e r r i t e . A cross s e c t i o n of the measurement section i l l u s t r a t i n g the s l o t and the probe i s shown i n F i g . 26. I t was F i g . 2 4 The two p a r t s o f t h e measurement s e c t i o n On •O F i g . 2 5 The lower p a r t o f the measurement s e c t i o n showing p e r i o d i c l o a d i n g o f the f e r r i t e - l o a d e d waveguide cr-O / TABLE 3 _^  ^ Characteristics of Type Ma t e r i a l Composition R e l a t i v e P e r m i t t i v i t y , e f(9.4 GHz) D i e l e c t r i c Loss Tangent (9.4 GHz) Saturation Magnetization (4nM ) f e r r i t e '. TT 1-390 Magnesium - Manganese 12.7 ±5% <.00025 2150 ±5% gauss @ 23°C 62 Fig.27 E x p e r i m e n t a l arrangement f o r phase and a m p l i t u d e measurements. ON dielectric matching sect ions —H <r- £38 cm ' I l I ! ! I L _ -1.0 0.0 1.0 2.0 3.0 4 0 5.0., 6.0 . Probe displacement (cm.) S c a l e - r e f e r e n c e f o r phase and a m p l i t u d e measurements. ascertained that the probe had l i t t l e e f f e c t on the f i e l d s and that r a d i a t i o n from the s l o t was n e g l i g i b l e . The probe was secured i n a brass carriage which allows a t r a v e l of 7.5 cm. The microwave s i g n a l from the probe was fed i n t o a phase-locked receiver where the magnitude and phase of the s i g n a l are measured. 3.2.2 F i e l d s i n Nonreciprocal Waveguides A block diagram of the experimental arrangement for measuring the f i e l d s i n the nonreciprocal s e c t i o n i s shown i n F i g . 27. The following two cases were investigated at the frequency of 9.0 GHz. (1) no p e r i o d i c loading with the f e r r i t e rod magnetized and unmagnetized (2) p e r i o d i c loading by "i n d u c t i v e " diaphragms of spacing .22 cm. with the f e r r i t e rod magnetized and unmagnetized The scale reference f or the f i e l d p l o t s was established as shown i n F i g . 28. The zero point f o r the probe displacement i s taken as the point where the probe i s d i r e c t l y over the i n t e r f a c e nearest the generator. Thus, the p o s i t i v e values correspond to displacement of the probe towards the matched termination. The f i e l d plots f o r case 1, no p e r i o d i c loading, are shown i n F i g . 29. For t h i s p a r t i c u l a r case the crude matching sections were not present. The magnitude p l o t for unmagnetized f e r r i t e i s s h i f t e d upward for c l a r i t y . The standing wave patterns f or magnetized f e r r i t e are e s s e n t i a l l y the same for both d i r e c t i o n s of magnetization. Their p e r i o d i c i t y i s s l i g h t l y d i f f e r e n t from that for unmagnetized f e r r i t e . 66 The phase plots show c l e a r l y the three s i t u a t i o n s ; the f e r r i t e rod magnetized i n opposite senses and the unmagnetized f e r r i t e rod. The f i e l d p lots for the p e r i o d i c structure are shown i n F i g . 30. The magnitude pl o t s f o r the two magnetizations are seen to be s l i g h t l y s h i f t e d r e l a t i v e to each other. The p e r i o d i c i t y of the standing wave patterns i s d i f f e r e n t from that for unmagnetized f e r r i t e . For both the r e c i p r o c a l and nonreciprocal p e r i o d i c structures the higher order Floquet harmonics appear to contribute l i t t l e to the f i e l d s t r u c t u r e . The phase pl o t s show considerably less r i p p l e due to the presence of the crude matching sections. The information i n these p l o t s may be inte r p r e t e d by modelling the f e r r i t e - l o a d e d s ection of waveguide by a transmission l i n e equivalent c i r c u i t . The t o t a l f i e l d i s expressed as E = Ae -1* +Z + B e ~ ^ ' Z < 4 5 ) where the f i r s t term i s the forward propagating mode and the second term i s the r e f l e c t e d mode. The constants A and B depend on the properties of the medium and the i n t e r f a c e e f f e c t s . The magnitude of the t o t a l f i e l d i s given by |E| = |A| e J ( f 3 + + e"> Z| (46) and phase by + - z arg(E) = arg(A) - 3 +z + a r g ( l + | e j ( 3 + 3 ) ) (47) In the standing wave pattern the distance between the minima i s + -given by ir/g where 3 = — — ^ ^ • The phase v a r i a t i o n along the se c t i o n av av 2 r + *^* B i (3 *f* B ) z consists of a l i n e a r term 3 z and an o s c i l l a t o r y term a r g ( l + — e J ) The slope of the phase p l o t y i e l d s the propagation constant 3 + of the forward propagating wave. In p r a c t i c e , greater accuracy of the slope measurement i s 67 30* T J • e • e . • • • • • X J 0 _ + _ • • i, + + + • + t • 1 • + i + — - + -30 0.0 2.0 4.0 P r o b e D i s p l a c e m e n t (cm.) a) /,00[ a» CL» ; <D 10 a Q_ 800 • 4 S i * t J t - - " a + + 0 + + + + 0 , — O I 9 • , 1200 + +. 4) & + • • a 9 F i g . 2 9 Amplitude and phase measurements f o r f e r r i t e l o a d e d s e c t i o n w i t h o u t p e r i o d i c l o a d i n g . The p o i n t s "+" and "-" c o r r e s p o n d t o magnetized f e r r i t e and ' V t o unmagnetized f e r r i t e 68 • • • TD • •• CU TD -»—» C cn O .+.* + + + -20 _ + + + + * + -1 - 4 * + ^ U 0.0 2.0 4.0 • + + P r o b e D i s p l a c e m e n t (cm.) 9 — » CD 1— cn CD TD 400 + + + •<- + + CD I/) a C L 800 •Fig.30 A mplitude and phase measurements f o r f e r r i t e l o a d e d s e c t i o n w i t h p e r i o d i c l o a d i n g : s p a c i n g o f diaphragms i s ,22cm. The p o i n t s "+",' "-" and ' V c o r r e s p o n d to the two d i r e c t i o n s of magnetized and unmagnetized f e r r i t e , r e s p e c t i v e l y 69 o b t a i n e d i f the c o e f f i c i e n t B i s s m a l l . F o r a n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e , E - A e " j f 3 + 2 + B e ^ ~ z , c o r r e s p o n d s t o f o r w a r d and r e f l e c t e d B i o c h waves. M e a s u r i n g the s l o p e o f t h e phase change a c r o s s the f e r r i t e - l o a d e d s e c t i o n o f the waveguide p r o v i d e d an easy means o f o b t a i n i n g t h e p r o p a g a t i o n c o n s t a n t o f the f o r w a r d p r o p a g a t i n g wave. The c r u d e d i e l e c t r i c m a t c h i n g s e c t i o n s h e l p e d to i n c r e a s e the a c c u r a c y o f the measurements. As a check, the p e r i o d i c i t i e s o f t h e s t a n d i n g wave p a t t e r n were compared w i t h t h o s e c a l c u l a t e d from t h e p r o p a g a t i o n c o n s t a n t s found u s i n g t h i s method and were found to be i n agreement. To i l l u s t r a t e the e f f e c t s o f p e r i o d i c l o a d i n g c l e a r l y , a c o m p a r i s i o n o f the phase v a r i a t i o n a l o n g the f e r r i t e l o a d e d s e c t i o n o f waveguide f o r no l o a d i n g and f o r p e r i o d i c l o a d i n g i s shown i n F i g . 31. Two i m p o r t a n t p o i n t s a r e noteworthy: (1) t h e phase change o v e r the l e n g t h o f the s e c t i o n f o r b o t h cases o f magnetized and unmagnetized f e r r i t e i s r e d u c e d w i t h p e r i o d i c l o a d i n g , as would be e x p e c t e d from t h e i n d u c t i v e n a t u r e o f t h e l o a d i n g (2) f o r the m a g n e t i z e d f e r r i t e , t h e d i f f e r e n t i a l phase s h i f t i s s u b s t a n t i a l l y i n c r e a s e d w i t h p e r i o d i c l o a d i n g , thus a g r e e i n g w i t h the f i n d i n g s o f S p a u l d i n g [2], [3] and K h a r a d l y [10]. 70 Probe D i s p l a c e m e n t (cm.) 1.0 2.0 3.0 40 Fig.31 Phase change a l o n g f e r r i t e l o a d e d s e c t i o n o f waveguide, .comparing no l o a d i n g w i t h p e r i o d i c l o a d i n g ; s p a c i n g o f diaphragms i s .22cm. 71 3.3.3 Measurement of the Scattering Parameters of a Diaphragm in a Nonreciprocal Waveguide The experimental arrangement for measuring the scattering parameters of a diaphragm is shown in Fig. 32. The interfaces between the f e r r i t e -loaded and empty waveguides were matched using the crude dielectric matching sections as well as additional tuning elements. After matching the inter-faces, the scattering parameters of the diaphragm were obtained from the reflection and transmission coefficients of the ferrite loaded section with the diaphragm in place. The reference points for the transmission measure-ments were obtained with the diaphragm removed. The transmission measurements were done using the null method in a bridge arrangement. The reflection measurements were used to verify that the phase angle of the reflection coefficient was indeed independent of the direction of propagation (or magnetization) as predicted by the mode-matching solutions of the last chapter. The interfaces were matched in the following straightforward manner. With the wave incident as shown in Fig.32, tuning element 2 was adjusted unt i l the standing wave pattern in the ferrite-loaded section of waveguide, as measured by the probe, was minimized. Interface 1 could then be matched by adjusting tuning element 1. In order to gain confidence in the accuracy of this method, the scattering parameters of the diaphragm were f i r s t measured with the fe r r i t e unmagnetized. In this case the equivalent ci r c u i t for the diaphragm is a simple shunt susceptance and can be determined from either the magnitude or the phase of the transmission coefficient. Both quantities were measured and yielded consistent results. The scattering parameters of the diaphragm with the fe r r i t e magnetized were then measured. The transmission coefficients for the diaphragm are shown in Fig.33 over the frequency range of 8.3 to 9.0 GHz. " induct ive" diaphragm F i g . 3 2 E x p e r i m e n t a l c o n f i g u r a t i o n f o r measuring s c a t t e r i n g p a r a m e t e r s o f diaphragm i n f e r r i t e - l o a d e d waveguide. 73 AO cu 20 U) O J C Q . 10 .95 .85 •5- + 8.5 9.0 9.5 10.0 Frequency (GHz.) 0 © F i g . 3 3 T r a n s m i s s i o n c o e f f i c i e n t s f o r i n d u c t i v e diaphragm. F o r phase r e s u l t s t h e p o i n t s "+", "-" and "o" c o r r e s p o n d to the two d i r e c t i o n s o f magnetized and unmagnetized f e r r i t e , r e s p e c t i v e l y . In the magnitude p l o t s r e s u l t s a r e g i v e n f o r magnetized f e r r i t e • and were found to be independent of the d i r e c t i o n o f p r o p a g a t i o n . 74 Smooth c u r v e s a r e f i t t e d to t h e e x p e r i m e n t a l p o i n t s . The e x p e r i m e n t a l p o i n t s , e s p e c i a l l y t h o se f o r the phase measurements, appear t o o s c i l l a t e s l i g h t l y about the c u r v e s . T h i s i s p r o b a b l y due to s m a l l r e f l e c t i o n s from the f e r r i t e -a i r i n t e r f a c e s . The f e r r i t e was magne t i z e d such t h a t the p r o p a g a t i o n c o n s t a n t o f the f o r w a r d - t r a v e l l i n g wave was l e s s than t h a t o f the b a c k w a r d - t r a v e l l i n g wave. The r e s u l t s show t h a t t h e t r a n s m i s s i o n a n g l e f o r the d i s c o n t i n u i t y i s l a r g e r f o r the forward d i r e c t i o n o f p r o p a g a t i o n , as may be e x p e c t e d i n t u i t i v e l y . The c i r c u i t s parameters shown i n F i g . 13 were c a l c u l a t e d from the smooth cur v e s o f F i g . 33. 3.3.4 Measurement on N o n r e c i p r o c a l P e r i o d i c S t r u c t u r e s P e r i o d i c s t r u c t u r e s were c o n s t r u c t e d w i t h s p a c i n g s r a n g i n g from .22 cm t o .51 cm. The measurements were c a r r i e d out a t f o u r f r e q u e n c i e s 9.0 GHz., 9.3 GHz., 9.6 GHz. and 9.9 GHz. w i t h the f e r r i t e m a g n e t i z e d and unmagnetized. The p r o p a g a t i o n c o n s t a n t s o f the f e r r i t e - l o a d e d s e c t i o n w i t h no ^ p e r i o d i c l o a d i n g were measured a t these f r e q u e n c i e s , u s i n g t h e method d e s c r i b e d i n s e c t i o n 3.3.2, and a r e shown i n T a b l e 4. With t h e s e measurements and the s c a t t e r i n g parameters o f the " i n d u c t i v e " diaphragm, found as d e s c r i b e d i n s e c t i o n 3.3.3, the v a r i a t i o n o f the p r o p a g a t i o n c o n s t a n t s w i t h t h e s p a c i n g o f the p e r i o d i c s t r u c t u r e s , was c a l c u l a t e d a t the f o u r f r e q u e n c i e s . These c a l c u l a t i o n s a r e r e p r e s e n t e d by the s o l i d c u r v e s i n F i g u r e s 34 a-d. The measurements on these s t r u c t u r e s a r e r e p r e s e n t e d by the p o i n t s i n t h e s e f i g u r e s . The agreement i s seen t o be b e t t e r a t the h i g h e r f r e q u e n c i e s and f o r the l a r g e r s p a c i n g s . As the f r e q u e n c y and s p a c i n g d e c r e a s e , t h e d i s c r e p a n c y between the e x p e r i m e n t a l and t h e o r e t i c a l v a l u e s i n c r e a s e s , and TABLE 4 E x p e r i m e n t a l c o n f i g u r a t i o n o f f e r r i t e l o a d e d waveguide - measured v a l u e s f o r the p r o p a g a t i o n c o n s t a n t s F r e quency (GHz) (rad/cm) £ o (rad/cm) (rad/cm) 9.0 9.3 9.6 9.9 4.02 4.24 4.45 4.68 4.11 4.41 4.66 4.94 4.71 4.91 5.12 5.40 76 5.0 . 4.0 £ o c 3.0 a U) c 8 c . o a 2 0 a> O Q . e OL 1.0 1 1 0.1 0.2 0.3 Spacing (cm.) 0.4 0.5 Fig.34 ' Measurement of the p e r i o d i c structures with the predicted -• .values indicated by the s o l i d curves. 34(a) f r e q . = 9.0GHz. 77 1 1 1 I I 0.1 0.2 0.3 O A 0.5 S p a c i n g (cm.) Fig.34b f r e q . = 9.3GHz. I 301 a 2.01 o CL O 1.0 1 1 0.1 0.2 0.3 Spacing (cm.) Fig.34c f r e q . = 9.6GHz. 79 80 i t appears that the effects of higher mode interaction between the diaphragms become more important. Higher mode interaction could be accounted for by increasing the dimensions of the wave matrix to include the modes which appreciably interact, as was done in the theoretical analysis of the last chapter. In practice, however, i t would be d i f f i c u l t to determine a l l the matrix elements. Finally, the differential change in the propagation constants i s plotted as a function of the inverse of the spacing between loading elements, Fig. 35. The solid line is theoretically obtained by using the measured scattering parameters of the diaphragm. For the smaller spacings the differential phase shift per unit length i s seen to be less than the predicted values, which are made on the basis of single mode interaction. This i s in agreement with the theoretically predicted trends of section 2.6. It i s clear that the differential change in the propagation constant i s significantly increased using periodic loading. For the smallest spacing used this increase is apprximately 260% over that for no periodic loading. 3.4 Summary An experimental investigation has been made of a certain type of nonreciprocal periodic structure. Periodic loading was accomplished through regular placement of thin "inductive" diaphragms. This type of loading is easy to implement and provides a significant improvement in the differential phase shift characteristics and could be of special interest in the design of nonreciprocal ferrite phase shifters. The main contributions in this work may be summarized as follows: 81 J „ 1 _ _ _L_ J_ 1.0 2.0 3.0 4.0 1 / s p a c i n g ( e r a 1 ) Fig.35 D i f f e r e n t i a l phase s h i f t p e r u n i t l e n g t h as a f u n c t i o n o f the i n v e r s e qf the s p a c i n g o f t h e p e r i o d i c s t r u c t u r e s The wave t r a n s m i s s i o n l i n e approach has been shown t o be a p p l i c a b l e f o r the a n a l y s i s o f n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s where the p e r i o d i c l o a d i n g i s a c c o m p l i s h e d through r e g u l a r placement o f t h i n o b s t a c l e s . The v a l u e s o f t h e p r o p a g a t i o n c o n s t a n t s o b t a i n e d e x p e r i m e n t a l l y agree q u i t e w e l l w i t h t h o s e p r e d i c t e d a n a l y t i c a l l y ( u s i n g measured v a l u e s o f the s c a t t e r i n g parameters) e x c e p t f o r the s m a l l e r s p a c i n g s between the l o a d i n g elements. The d i s c r e p a n c y a t the s s p a c i n g s i s a t t r i b u t e d t o h i g h e r mode i n t e r a c t i o n , between the diaphragms and appears t o be o f t h e same o r d e r f o r b o t h r e c i p r o c a l and n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s . The e f f e c t s o f h i g h e r mode i n t e r a c t i o n o b s e r v e d e x p e r i m e n t a l l y f o l l o w the same t r e n d s as t h o s e p r e d i c t e d t h e o r e t i c a l l y u s i n g the t w i n - s l a b c o n f i g u r a t i o n o f f e r r i t e . A d i r e c t e x p e r i m e n t a l method has been used t o measure t h e magnitude and phase o f the e l e c t r i c f i e l d a l o n g a f e r r i t e -l o a d e d s e c t i o n o f waveguide. T h i s i s a c h i e v e d t h r o u g h the use o f a s p e c i a l l y c o n s t r u c t e d s l o t t e d s e c t i o n . The s l o p e o f the phase v a r i a t i o n thus o b t a i n e d y i e l d s d i r e c t l y t h e p r o p a g a t i o n c o n s t a n t o f the forw a r d p r o p a g a t i n g wave. I t a l s o c l e a r l y i l l u s t r a t e s the e f f e c t o f p e r i o d i c l o a d i n g on the d i f f e r e n t i a l phase s h i f t . The magnitude p l o t s i n d i c a t e o n l y a s l i g h t dependence on the d i r e c t i o n o f m a g n e t i z a t i o n . A l s o , the h i g h e r o r d e r F l o q u e t harmonics appear t o c o n t r i b u t e l i t t l e t o the f i e l d s t r u c t u r e . 1 83. (3) The s c a t t e r i n g parameters o f an " i n d u c t i v e " diaphragm i n a n o n r e c i p r o c a l waveguide were measured a c c u r a t e l y and were used t o d e r i v e a t h r e e - e l e m e n t e q u i v a l e n t c i r c u i t . 84 Chapter 4 AN INVESTIGATION OF INHOMOGENEOUS AND MAGNETIZED FERRITE LOADED RIDGED WAVEGUIDE 4.1 Introduction In searching f o r new configurations to improve the nonreciprocal f e r r i t e phase s h i f t e r , the f e r r i t e loaded ridged waveguide,. modelled i n F i g . 3 6 , : i s considered. The ridged waveguide geometry o f f e r s the advantage of increasing the bandwidth separation between the cutoff frequencies of the fundamental and the next highest order mode. This separation has been shown to be further increased when the ridge i s loaded with a d i e l e c t r i c [17]. This could be of importance for the f e r r i t e phase s h i f t e r as t h i s would permit a higher p e r m i t t i v i t y for the c e n t r a l d i e l e c t r i c , and hence a greater d i f f e r e n t i a l phase s h i f t per unit length, before multimode prop-agation occurs. The analysis of the magnetized f e r r i t e loaded ridged waveguide i s a formidable problem. In the homogeneous case s o l u t i o n s to the wave equation consist of d i s t i n c t TE or TM modes. However, when the ridged waveguide i s loaded by d i e l e c t r i c or a n i s o t r o p i c material the solutions to the wave equation w i l l , i n general, have both TE and TM components. I t w i l l be shown i n t h i s chapter that the propagation constant of the fundamental mode of c e r t a i n configurations of an inhomogeneous loaded ridged waveguide can be determined to acceptable accuracy by assuming that i t s t r i c t l y a TE mode. A n a l y t i c a l work has been done on the inhomogeneous loaded ridged waveguide shown i n Fig.37 to f i n d the c u t o f f frequencies of the predominantly TE modes e x i s t i n g i n the structure [17]. The problem of j u s t determining 9 f ferrite ; e , y, dielectric F i g . 3 6 Ridged waveguide l o a d e d by t w i n - s l a b arrangement o f m a g n e t i z e d f e r r i t e 86 i • the cutoff frequencies is much simpler than finding the propagation constants as there is no coupling between the TE and TM modes. However, unlike the homogeneous case, there is no direct way of obtaining the propagation constant from the cutoff frequency. The author was able to show that dielectric load-ing of the ridge further increased the bandwidth separation between the modes. One approach that has been proposed [18] for determining the propagation constant, y = JB> f° r the fundamental mode of the di e l e c t r i c loaded ridged waveguide of Fig.38 is to use the transverse resonance technique when 8 < k , where k is the free space wave number, and a f i r s t order approx-o o imation when 3 > k Q. For most cases of loading the f i r s t order solution applies. In this chapter a complete numerical solution i s undertaken of the inhomogeneous loaded ridged waveguide shown in Fig.37 using a Ritz Galerkin technique. This technique has been used previously for the solution of the homogeneous ridged [19] and crossed [20] waveguide problems and i n reference [17]. On the basis of this analysis, a simple, what w i l l be denoted as the "approximate", method was formulated to analyse the magnetized fe r r i t e loaded ridged waveguide of Fig.36, and i t s di f f e r e n t i a l phase shift characteristics were explored. The f i r s t order solution of reference [18] is a special case of the approximate method. 4.2 Complete numerical solution Referring to the configuration of Fig.37, the analysis presented here w i l l be to derive the propagation constants of those modes that would be excited by the TE^Q mode of an empty waveguide of width 2a.and height 2b. This form of excitation along with the symmetry of the configuration allows the problem to be reduced to the quadrant; x > 0, y > 0, with the planes x=0 and 87 F i g . 3 8 D i e l e c t r i c - s l a b l o a d e d r i d g e d waveguide 88 y=0 b e h a v i n g as magnetic and e l e c t r i c w a l l s r e s p e c t i v e l y . The f i e l d s i n the quadrant a r e expanded i n terms o f the e i g e n f u n c t i o n s o f t h e r e g i o n s 0<x<d and d<x<a and the t a n g e n t i a l E and H components a r e matched a t x=d. The f i e l d s e x p a n s i o n s a r e w r i t t e n as f o l l o w s Ridge r e g i o n 0<x<d M . ,• „ „ a s i n A x cos miry + a s i n A x Hz= E m m ~r- o o (48) m=l cos A d X cos A d X m m o o K = Z b m c o s X m x s i n E_X. (49) z m=l cos A d fc m M 1 b Ey= M -y M m + J u n z „ m } cos A_x cos mtry 2 m=l t cos A d cos A d i m m (50) c 2 cos A d ° • j u ^ Z a + 1 o o cos A x1 M V Z { Y — ' + J e i Y « x 2 bm } s i n A x s i n nnry y .2 m=l t cos A m d l o r n C Q s x d m m m t m (51) where „2 2 , 7. -2 ( ,2 (52) A = H - ( mir ) T Z q = f r e e space impedance k = f r e e space wave number o -yz The p r o p a g a t i o n term, e , i s s u p p r e s s e d . 89 Outer region d<x<a N c P H = E n n cos p (a-x) cos niry + c p cos p (a-x) z i — ; n . / o o o n=l sin p w b — n sm p w ° (53) N d E n n sin p^(a-x) sin rnry (5 A) •n=l sin p w b n 1 N d 2 E = E { -y nir n P n i JM„Z p c P n } sin p (a-x) cos rnry y i T~ : i / o n n —: n : ,2 n=l b sin p w sin p w - b h n n p n : (55) + JU„Z p e p . , J 2 o o o o sin p (a-x) 2 sin p w ° h ° 1 N c H = 2 { Y BIL °n P n + j e y d p . , . '. y v2 n=l b sin p w 2 ° n 5 > cos p (a-x) sin mrv_ n n sin p w b n (56) where i, 2 2 ^ e2 y2 2 2 2 p n = Y + e2 y2 - ^ HI ) b Y q = free space admittance (57) 90 The boundary c o n d i t i o n s f o r the e l e c t r i c f i e l d a r e imposed a t x=d . ( i ) m a t c h i n g E^ a t x=d y i e l d s M £ b s i n miry i m=l t N £ d s i n nTry n = l b 0 < y < t (58) t < y < b u s i n g the o r t h o g o n a l i t y o f the sine' s e r i e s , the c o e f f i c i e n t s d a r e s o l v e d n f o r : M d = E b 2 S n , m T- mn m=l b (59) where ^mn = ^ S ^ n m T r y s ^ n 1 1 d y 91 r ( i i ) m a t c hing E a t x=d y i e l d s 1 N 2 — Z { -y nn d + i y ^ Z p c } cos niry ,2 , — n 2 o n n — h n=l b b + j P 2 Z o P o C o (60) h 2 0 < y < t t < y < b i o l v i n g f o r the c o e f f i c i e n t s c and c y i e l d s : • o n J c = h 2 Hi t i a £ 2 P 2 b p 2 o 92 c = n M E m=l a mn m M a K t 1Y y £ ^ b o n 1 o , mn m m=l (62) where mn b 2 p. 2 mn n 2 2 Y = { h imr C - nir S } mn — — mn — mn / , .2 t b p/ b u A n 2 (63) 5 = 2 1 y l h sin mrt 'n b P 2 ^2 I 2 n n£ b t C = f cos nnry cos niTy dy mn >Q V " The t a n g e n t i a l magnetic f i e l d , H and Hz__, i s continuous f o r 0<y<t, Equating (51) to (56) and (48) to (53), s o l v i n g f o r the c o e f f i c i e n t s a Q , a m and b and s u b s t i t u t i n g from (59) (61) and (63) y i e l d s the f o l l o w i n g , m 1 { Y M a tan A d + je..Y b A tan A d '} t —- 1 — m m_ J 1 o m m m y ,; (64) £ A m M N N = v a £ nn S ft p cot p w + y a E :.nu S £ p cot p w — . p .. — mn pn n n —=• o , ~ r mn n n n 2 p=l ^ n=l b ^ h2 n=l b P N 2 + 1Y E b E S p cot p w { e_ 2 S + y nn ¥ } J o , p , mn n 2 r- pn — r pn — - p=l r n=l b b h 93 ( i i ) H ; f i r s t o r d e r term a t tan X d ; o o 2 2 u a { h t 1 c o t p w O —7T T 0 I1 b y 2 — p c o . N E 5 P c o t p w s i n niTt } , n n n — : — •* n = l b _ m r -b p N ; * + Z a Z fi p c o t p w s i n nirt . p n pn n n — r — p = l v n=l r b b ( 6 5 ) M N + ' 1Y v E b E T p c o t p w s i n nirt o' , p - pn n rw —T— p = l n = l b _ t H i g h e r o r d e r terms M N N - V d " f r ! ^ nf j. V Cmn p n C O t P n W + % ^ 5 n V p n C O t V m M N + j Y Y E b E Y C p c o t p w J o , P , pn mn n n p = l n = l ( 6 6 ) E q u a t i o n s ( 6 4 ) , (65) and (66) can be r e w r i t t e n as F v = 0 where F i s a square m a t r i x o f o r d e r 2M + 1 and ( 6 7 ) v = b. "Ml 94 S o l v i n g det(F ) = 0 y i e l d s the p r o p a g a t i o n c o n s t a n t y . T h i s e q u a t i o n i s s o l v e d by t h e the d i s c r e t e form of Newton-Raphson's method used i n Appendix 1 . A low o r d e r case i s s o l v e d f i r s t and t h i s v a l u e o f y i s used as t h e i n i t i a l v a l u e f o r the s o l u t i o n o f h i g h e r o r d e r c a s e s . 4.3 Approximate method T h i s method assumes t h a t the fundamental mode o f the inhomogeneous l o a d e d r i d g e d waveguide i s s t r i c t l y TE. T h e . e i g e n f u n c t i o n e x p a n s i o n s i n the r i d g e and o u t e r r e g i o n s a r e d e r i v e d from the H component o f the magnetic f i e l d . Only t h e lowest o r d e r term-of. the e x p a n s i o n ; l s used i n the r i d g e s i n c e i t does not p o s s e s s any y dependence. Any y v a r i a t i o n o f t h e f i e l d s i n the r i d g e a r e a would, f o r the case o f m a g n e t i z e d f e r r i t e , r e s u l t i n c o u p l i n g o f the TE and TM wave e q u a t i o n s . In t h e o u t e r r e g i o n , N terms o f the e i g e n f u n c t i o n e x p a n s i o n a r e used R e f e r i n g t o Fig.36, t h e symmetry o f the c o n f i g u r a t i o n i s u t i l i z e d and the problem i s reduced t o one q u a d r a n t . The f i e l d s a r e expanded as f o l l o w s - J B z A cos £x e - j 3 z E = y L(B cos px + C s i n px ) e •-je? f(D s i n qx + cos qx ) e q 0 < x < t fcd < X < ^ t f < x < d (68) - j B z -1 A (-£) s i n 2.x e jwu {B[ -p_ s i n px + B K C O S px] + C [ 1_ cos px + Bjc s i n px ]} e P W& P -1 '{ D cos qx - q s i n qx } e jwu - j B z 0 < x < t 'd < X < ^ t f < x < d (69) matching f i e l d components a t x = t ^ and x = t ^ y i e l d s 95 A b = c where (70) A= cos &t j -cos pt , d d cos p t •I s i n It - [ - p s i n p t + 3K_ cos p t ] VP, [-p s i n p t + j3tc cos p t f ] y y . - s i n p t , d P s i n pt^. - s i n q t ^ •[cos p t ^ + j3j< s i n p t ^ ] yy„ r y_ e p [cos p t ^ + _6j< s i n p t j ] -cos q t ^ yy e p c 0 cos q t ^ 0 -q s i n q t ^ V a l u e s f o r the c o e f f i c i e n t s A,B,C and D can be o b t a i n e d by s p e c i f y i n g a v a l u e f o r 3. F o l l o w i n g t h e f i e l d m a t c h i n g p r o c e d u r e o f t h e p r e v i o u s s e c t i o n t o match the t a n g e n t i a l e l e c t r i c and magnetic f i e l d components at;; the plane_x=d, y i e l d s t h f o l l o w i n g e i g e n v a l u e e q u a t i o n whose s o l u t i o n y i e l d s the p r o p a g a t i o n c o n s t a n t (3. t { D cos qd - q s i n qd } D s i n qd + cos qd q 2—2 -h t c o t p w b ~ N (71) 2 2 - I 2_ h c o t w ( s i n rnrt ) n = l b n b S e t t i n g N=l y i e l d s the f i r s t o r d e r s o l u t i o n o f r e f e r e n c e [ 1 8 ] . 96 4.4 Numerical r e s u l t s f or the various methods The r e s u l t s of the complete numerical s o l u t i o n are shown i n Fig.39 and Fig.40 f o r several s i t u a t i o n s . The plots show that, i n i t i a l l y , the propagation constants increase gradually as the r a t i o t/b decreases from unity. The increase becomes more rapid when t/b i s small. The complete, approximate and f i r s t order solutions are compared i n Table 5 for a p a r t i c u l a r case. It was found that when the r e l a t i v e permeability was appreciably d i f f e r e n t from unity the f i r s t order s o l u t i o n became l e s s accurate. It i s c l e a r that i n t h i s case that the f i r s t order s o l u t i o n does not produce s a t i s f a c t o r y r e s u l t s while the d i f f e r e n c e i n the approximate and complete methods i s small. For the approximate method, c o n t i n u a l l y employing; more terms of the eigenfunction expansion i n the outer region does not increase the accuracy of the r e s u l t . The cases t/b=.5 and t/b=.75 show that there seems to be an optimum value for N. Comparisons of t h e . f i r s t order s o l u t i o n and the approximate method f o r s t r i c t l y d i e l e c t r i c loading are . shown i n Fig.41 and Fig.42. The f i r s t order s o l u t i o n was proposed as a v i a b l e method for the d i e l e c t r i c loaded ridged waveguide of Fig.38 because i t was f e l t that when the p e r m i t t i v i t y was high the decay of the f i e l d s from the d i e l e c t r i c would make the e f f e c t s of the ridge small. In Fig.43 and Fig.44 the f i r s t order s o l u t i o n and the approximate method are compared with experimental measurements made on two d i e l e c t r i c loaded waveguide configurations. The experimental points are obtained from reference [18 ]. Both methods appear to agree s a t i s f a c t o r i l y with experiment. In t h i s work an experimental study was made on the d i e l e c t r i c loaded ridged waveguide shown i n Fig.45, which i s asymmetrical i n the y d i r e c t i o n . 97 98 >Fig.40 Propagation constant vs. t/b f o r , d i f f e r e n t values of d/a • e ] L = 12.0; e 2 = y 1 = y 2 = 1.0; a = 1.143cm. ; b = .508cm. ; freq. = 9.0GHz. 4 modes were used i n the ridge . TABLE 5 Inhomogeneous ridged waveguide - comparison of the three methods f o r determining the propagation constants — t/b F i r s t order. . 3 (rad./cm.) Approximate method 6 N (rad./cm.) Complete s o l u t i o n 3 M N (rad./cm.) .1 6.16437 5.60372 .. 10 5.60022 20 5.59950 30 5.60022 2 20 5.59950 3 30 5.59925 4 40 .25 5.81359 5.37674 4 5.37123 8 5.36950 20 5.36933 30 5.36960 5 20 5.39929 8 32 5.36925 10 40 .50 5.46686 5.25455 2 5.24696 4 5.24403 20 5.24396 30 5.24469 4 8 5.24417 7 14 5.24403 10 20 5.24400 11 22 .75 5.25570 5.20207 2 5.18539 4 5.18375 20 5.18372 30 5.18415 6 8 5.18415 9 12 1.0 5.11158 5.11158 1 5.11158 2 2 t, = 0 t , = d = .318cm. a = 1.143cm. b = .508cm. d . f e ; L = 4.0 vx = 3.0 e 1 = 1.0 u = 1.0 frequency = 9.0 GHz. 100 101 102 Frequency (GHz.) Fig.43 Comparison of theory with experiment for dielectric-slab loaded ridged waveguide • e 1 = 7.0; t f =.129cm.; d = .220cm.; t = .287cm.; b = .408cm. x, o experimental points measured from two different lengths of dielectric-slab loaded ridged waveguides — — ; — approximate method - - - f i r s t order solution 6.0 £ C o V) c o u c o cn O C L o i _ Q_ 4;o 2.0 8.0 12.0 16.0 Frequency (GHz.) Fig.44 Comparison of t h e o r y w i t h experiment f o r d i e l e c t r i c - s l a b l o a d e d r i d g e d waveguide x,o e x p e r i m e n t a l p o i n t s approximate 'method f i r s t o r d e r s o l u t i o n 104 However, a f t e r u t i l i z i n g the symmetry i n the x d i r e c t i o n , t h i s c onfiguration reduces to that studied previously. The brass bar and the d i e l e c t r i c are positioned by two polyfoam supports, one at each end. The e n t i r e structure i s secured i n a brass block of which only the bottom section i s shown i n the diagram. The propagation constant of the wave propagating i n t h i s structure was measured i n the following manner: a short c i r c u i t was placed at one end of the waveguide and the phase of the r e f l e c t i o n c o e f f i c i e n t was measured as a function of the l o n g i t u d i n a l p o s i t i o n of the diaphragm. This yielded a periodic .variation. The distance between successive minima yielded the propagation constant. Comparison of the t h e o r e t i c a l and experimental r e s u l t s are shown i n Fig.46. Agreement between experiment and the approximate method i s good while the t h e o r e t i c a l r e s u l t s of the f i r s t order s o l u t i o n are c l e a r l y too high. 4.5 F e r r i t e loaded ridged waveguide An i n v e s t i g a t i o n i s made of the f e r r i t e loaded ridged waveguide of Fig.36 using the approximate method described i n section 4.3 . Results are shown i n Figures 47a-b for several s i t u a t i o n s . The propagation constants show a s i m i l a r dependence on the r a t i o t/b regardless of whether the f e r r i t e i s magnetized or unmagnetized. Generally i t was found that the d i f f e r e n t i a l phase s h i f t per unit length decreased as the r a t i o t/b decreased. However, there were instances, as the pl o t s of Figures 48a-b show, where A3 increased . with onset of the ridge. For these cases there was no d i e l e c t r i c gap between the f e r r i t e slabs and the width of the ridge and f e r r i t e was small. The t h e o r e t i c a l r e s u l t s could not be v e r i f i e d experimentally i n t h i s work as the experimental configuration i s somewhat d i f f e r e n t . Recently, however, an i n v e s t i g a t i o n has been c a r r i e d out on a r e l a t e d c o n f i g u r a t i o n , the magnetized f e r r i t e loaded grooved or crossed waveguide shown i n Fig.49. [21] The authors analysed t h i s structure using e s s e n t i a l l y a f i r s t order one of two dielectric plugs; £=1.5, to support structure (other one not shown) moveable diaphragm T ~~704cm, 1.016cm... 4 ,312cm.! dielectric, £=9.37 -1.143 cm. Fig.45 E x p e r i m e n t a l c o n f i g u r a t i o n f o r measuring the p r o p a g a t i o n c o n s t a n t o f a wave p r o p a g a t i n g i n a d i e l e c t r i c l o a d e d r i d g e d waveguide 106 P 40 c B c o o c o '•6 g.2.0 o CL 1 9.0 10.0 Frequency (GHz.) F i g . 4 6 Comparison of t h e o r y w i t h e x p e r i m e n t a l r e s u l t s f o r c o n f i g u r a t o f F i g . 4 6 xon experiment-, approximate method f i r s t o r d e r s o l u t i o n 6.0 •g.40 c o c o o c o o cn g.2.0 e Q L 107 t / b 1.0 Fig.47 Propagation constant vs. t/b for magnetized ferrite loaded ridged waveguide. • " • 47(a) freq. = 9.0GIIz.; e = 12.0; y = .96; K ~ -.3 (i) e d = 16.0; t^ = .053cm.; t f = d =.265cm.; a = 1.143cm.; b = .508cm. ( i i ) = 1.0; t^ = .053cm.; = d = .265cm.; a = 1.143.cm. b = .508cm. 1 ( i i i ) e d = 1.0; t d = .053cm.; t f = d = .265cm.; a = 1.143cm.; b = .508cm. 108 Differential phase sh i f t per unit length vs. t/b The frequency, the properties of the ferrite and the parameters for cases (i) , ( i i ) , ( i i i ) are those l i s t e d in Fig.47(a) (iv) e, = 16.0; t, = .053cm.; t^ = d = .159cm.; d d r a = 1.143cm.; b = .508cm.. 1 0 9 d (cm.) Fig.48 D i f f e r e n t i a l phase s h i f t per unit length vs. d f o r d i f f e r e n t values of t/b freq. = 9.0GHz.; e = 12.0;' u = .96 t , = 0.0;" t = d = .265cm.; a = 1.143cm.; b = .508cm. a r ( 48(a) K ? -.3 110 dielectric ferrite F i g . 4 9 M a g n e t i z e d f e r r i t e - l o a d e d grooved o r c r o s s e d waveguide 112 a p p r o x i m a t i o n and found the t h e o r e t i c a l r e s u l t s t o agree s a t i s f a c t o r i l y w i t h e x p e r i m e n t . They a l s o found t h a t the d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s t i c s improved when the r a t i o t/b was d e c r e a s e d from u n i t y . T h i s c o n f i g u r a t i o n has the advantage o f d e c r e a s i n g t h e RF f i e l d s t r e n g t h i n f e r r i t e l o a d e d r e g i o n 4.6 Summary T h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n s have been made o f p a r t i c u l a r c o n f i g u r a t i o n s o f r e c i p r o c a l , inhomogeneous r i d g e d waveguides. A t h e o r e t i c a l i n v e s t i g a t i o n has been made o f a magnetized f e r r i t e l o a d e d r i d g e d waveguide u s i n g an approximate t e c h n i q u e . The main p o i n t s were: y. (a) A n u m e r i c a l method based on the R i t z - G a l e r k i n t e c h n i q u e i s p r e s e n t e d t o s o l v e ' a p a r t i c u l a r c o n f i g u r a t i o n o f a r e c i p r o c a l , inhomogeneous r i d g e d waveguide. I t i s found t h a t f o r t h e fundamental mode, s a t i s f a c t o r y r e s u l t s can be o b t a i n e d from a s i m p l i f i c a t i o n o f t h i s a pproach, the " a p p r o x i m a t e " method. (b) A magnetized f e r r i t e l o a d e d r i d g e d waveguide i s a n a l y s e d u s i n g the approximate method. I t i s found t h a t the d i f f e r e n t i a l phase s h i f t p e r u n i t l e n g t h , A3, g e n e r a l l y d e c r e a s e s w i t h o n s e t o f the r i d g e e x c e p t f o r a few s p e c i a l c a s e s . T h i s , a l o n g w i t h the f a c t t h a t t h i s c o n f i g u r a t i o n c o n c e n t r a t e s the RF f i e l d s i n the r i d g e and hence i n th e f e r r i t e thus d e c r e a s i n g the amount o f power t h a t can be pr o p a g a t e d i n the waveguide b e f o r e t h e o n s e t o f spin-wave i n s t a b i l i t i e s , may make t h i s c o n f i g u r a t i o n u n a t t r a c t i v e f o r phase s h i f t e r a p p l i c a t i o n s . However, t h e r i d g e d waveguide does have t h e advantage t h a t i t i n c r e a s e s the bandwidth s e p a r a t i o n between the c u t o f f f r e q u e n c i e s o f the fundamental and the ne x t h i g h e r o r d e r 113 mode. A l s o the c o n c e n t r a t i o n o f the f i e l d s i n the r i d g e c o u l d enhance the e f f e c t s o f m e t a l l i c d i s c o n t i n u i t i e s such as i n d u c t i v e o r c a p a c i t i v e diaphragms and when used f o r n o n r e c i p r o c a l p e r i o d i c l o a d i n g , i n l i g h t o f S p a u l d i n g ' s r e s u l t s u s i n g a s i m i l a r waveguide geometry, may p r o v i d e d e s i r a b l e f r e q u e n c y r e s p o n s e c h a r a c t e r i s t i c s . (c) The method proposed i n r e f e r e n c e [18] o f u s i n g t r a n v e r s e r e s o n a n c e when 8 < k and a f i r s t o r d e r s o l u t i o n when 3 > k i s f o r most . o o c a s e s i n a d e q u a t e . The t r a n s v e r s e method r a r e l y a p p l i e s because f o r most c a s e s o f l o a d i n g B > k i n which case the f i e l d s i n t h e o a i r - f i l l e d r e g i o n s a r e evan e s c e n t and t h e f o r m u l a s t h a t a r e used t o e v a l u a t e the e f f e c t s o f the r i d g e [22] no l o n g e r a p p l y . The f i r s t o r d e r s o l u t i o n was found t o be adequate o n l y f o r s p e c i a l c a s e s o f d i e l e c t r i c l o a d i n g . 1 C h a p t e r 5 CONCLUSIONS In t h i s work the d e s i r e to improve the d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s t i c s o f n o n r e c i p r o c a l f e r r i t e phase s h i f t e r s has l e d to , i n v e s t i g a t i o n s i n the f o l l o w i n g a r e a s : (a) A t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n o f p e r i o d i c l o a d i n g o f n o n r e c i p r o c a l f e r r i t e l o a d e d waveguides. (b) An i n v e s t i g a t i o n o f a magnetized f e r r i t e l o a d e d r i d g e d waveguide I t was known a t the o u t s e t o f t h i s work t h a t p e r i o d i c l o a d i n g o f n o n r e c i p r o c a l waveguides c o u l d , i n many i n s t a n c e s , improve i t s d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s t i c s . The major g o a l o f t h i s work was t o q u a n t i f y and t o extend the e x i s t i n g e x p e r i m e n t a l r e s u l t s , which a p p l i e d t o p a r t i c u l a r waveguide geometry, so t h a t p e r i o d i c l o a d i n g o f a g e n e r a l n o n r e c i p r o c a l waveguide, by r e g u l a r placement o f d i s c o n t i n u i t i e s could, be e v a l u a t e d . I t was f e l t t h a t the t r a n s m i s s i o n l i n e method would be the most v i a b l e approach as i t r e q u i r e d o n l y the s c a t t e r i n g parameters o f the d i s c o n t i n u i t y and the p r o p a g a t i o n c o n s t a n t s o f the u n l o a d e d n o n r e c i p r o c a l waveguide, q u a n t i t i e s which can always be measured and i n some i n s t a n c e s , c a l c u l a t e d . In t h i s work a t h e o r e t i c a l a n a l y s i s i s u n d e r t a k e n o f the s c a t t e r i n g problem o f an " i n d u c t i v e " diaphragm i n a t w i n - s l a b f e r r i t e l o a d e d waveguide, the s t a n d a r d model f o r p r a c t i c a l c o n f i g u r a t i o n s o f f e r r i t l o a d e d waveguides. The f i e l d m a t ching t e c h n i q u e t h a t has been s u c e s s f u l l y employed i n the r e c i p r o c a l c o u n t e r p a r t o f t h i s problem was u s e d. F o r the c a s e o f magnetized f e r r i t e , the d i f f i c u l t o r t h o g o n a l i t y c o n d i t i o n s f o r t h e waveguide modes were overcome u s i n g a G a l e r k i n approach. The c onvergence 115 and b e h a v i o r o f the s o l u t i o n s were found t o be s i m i l a r whether t h e f e r r i t e was magnetized o r unmagnetized. The r e s u l t s f o r one diaphragm were a p p l i e d t o p e r i o d i c l o a d i n g o f the t w i n - s l a b c o n f i g u r a t i o n o f f e r r i t e u s i n g the t r a n s m i s s i o n l i n e method. The a n a l y s i s based on s i n g l e mode i n t e r a c t i o n showed a s u b s t a n t i a l i n c r e a s e i n the d i f f e r e n t i a l phase s h i f t per u n i t l e n g t h . When t h e e f f e c t s o f h i g h e r mode i n t e r a c t i o n were i n c l u d e d i t " was i n d i c a t e d t h a t the d i f f e r e n t i a l phase s h i f t p e r u n i t l e n g t h would be reduced b u t s t i l l s i g n i f i c a n t . An e x p e r i m e n t a l i n v e s t i g a t i o n was u n d e r t a k e n o f a p r a c t i c a l c o n f i g u r a t i o n o f f e r r i t e l o a d e d r e c t a n g u l a r waveguide p e r i o d i c a l l y l o a d e d by " i n d u c t i v e " m e t a l l i c diaphragms. The main e x p e r i m e n t a l problems t h a t were e n c o u n t e r e d were: (a) Measurement o f the s c a t t e r i n g parameters o f the diaphragm i n t h e f e r r i t e l o a d e d waveguide (b) Measurement o f the p r o p a g a t i o n c o n s t a n t s o f the n o n r e c i p r o c a l p e r i o d i c s t r u c t u r e s These problems were overcome through c o n s t r u c t i o n o f a s p e c i a l measurement s e c t i o n which p e r m i t t e d s a m p l i n g o f the e l e c t r i c f i e l d a l o n g the n o n r e c i p r o c a l s e c t i o n o f a f e r r i t e l o a d e d waveguide. The s l o p e o f the phase v a r i a t i o n y i e l d e d the p r o p a g a t i o n c o n s t a n t o f the fo r w a r d p r o p a g a t i n g wave. The measurements o f the p e r i o d i c s t r u c t u r e s compared f a v o u r a b l y w i t h the p r e d i c t i o n s made on t h e b a s i s o f the measurement o f the s c a t t e r i n g parameters o f one s t r i p . The d e v i a t i o n s were a t t r i b u t e d t o the e f f e c t s o f h i g h e r mode i n t e r a c t i o n and t h e s e f o l l o w e d t h e same t r e n d s as t h o s e p r e d i c t e d a n a l y t i c a l l y u s i n g the t w i n - s l a b c o n f i g u r a t i o n o f f e r r i t e . L a s t l y , a t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n was made on a r e c i p r o c a l , inhomogeneous r i d g e d waveguide. I t was found t h a t s a t i s f a c t o r y r e s u l t s f o r t h e p r o p a g a t i o n c o n s t a n t o f the fundamental mode c o u l d be 116 O b t a i n e d by assuming t h a t i t ' was s t r i c t l y TE and u s i n g an " a p p r o x i m a t e " method o f . . s o l u t i o n . T h i s method was then extended to a n a l y s e the magnetized f e r r i t e l o a d e d r i d g e d waveguide and i t was found t h a t i n t r o d u c t i o n o f t h e r i d g e had d e t r i m e n t a l e f f e c t s on t h e d i f f e r e n t i a l phase s h i f t c h a r a c t e r i s t i c s except f o r a few s p e c i a l c a s e s . 117 APPENDIX I SOLUTION OF EIGENVALUE PROBLEM FOR WAVEGUIDE MODES Using symmetry, the f i e l d s i n the waveguide region x=0 to x=a are expressed as E = A cos £x e y H = r r ^ i - A(-Jl) s i n £x e ~ Y Z 0 < x < t , — — Q z jwy 0 -yz E ={B, cos px + B s i n px}e y 1 H «= {B, (— s i n px + 3££ cos px) z ja>y0 1 ye r yy e P 3 Y K -» Y Z + B ( — cos px + s i n px)}e y e vy c fcd - X ~ 'f E = D s i n q(a-x)e y H = D(-q) cos q(a-x) £ z J^yo t , < x < a f — — (24) Imposing c o n t i n u i t y of the f i e l d s at x=t d and x=t f y i e l d s A v = 0 (25) where v = A B l B C and the matrix A i s given by cos Ht, -cos pt, -sin pt, 0 d d a ) cos p t f sin p t f - s i n qd " i sin £t d sin p t < 1 + ^  cos p t ^ cos p t d + ^ sin p t d ] 0 0 [ ~ sin p t f + JJJS c o s p t f ] [ - 2 c o s p t f + sin p t f ] q cos qd - ( 2 6 ) A nontrivial solution for v requires the determinant of the matrix A to be zero. This condition yields the propagation constants. det {A(Y)) = 0 (27) For propagating modes y i s pure imaginary and solutions can be easily found using the method of bisection. When the fe r r i t e is magnetized, the higher order modes have a complex y• l n this case, solutions were found using the following iterative scheme (based on New ton-Rap hs on's method).. Y ± + 1 - Y ± - A ( Y l ) [ T . - Y ^ ] [Hy.) - A ( Y . _ 1 ) ] The i n i t i a l value for y was obtained by solving (27) with K=0. In this case, y i s pure real and can be found using bisection. Having determined the propagation' constant, the constants A,B,C are found by solving the problem A v = 0 with set to 1. (28) REFERENCES 119 1. ELACHI, G. - "Waves in Active and Passive Periodic Structures; A Review" Proc. IEEE, vol.64, no.12, Dec.1976. 2. SPAULDING, W.G. - "A Periodically Loaded, Latching, Nonreciprocal Ferrite Phase Shifter", presented at the IEEE. G-MTT International Microwave Symposium, Dallas, U.S.A., 1969. 3. SPAULDING, W.G. - "The Application of Periodic Loading to a Ferrite Phase Shifter Design", IEEE Trans. MTT-19 (1971 Symposium issue) vol.19, pp. 922-928, Dec.1971. 4. DiBARTOLO, J.; INCE, W.J.; TEMME, D.H. -"A Solid State 'Flux Drive' Control Circuit for Latching-Ferrite-Phasor Applications" Microwave Journal, vol.15, no.9, Sept..1972. 5. INCE. W.J.; STERN, E -"Nonreciprocal Remanence Phase Shifters in Rectangular Waveguide" IEEE Trans., 1967, MTT-15, (2), pp. 87-95. 6. SCHLOMANN, E. -"Theoretical Analysis of Twin Slab Phase Shifters in Rectangular Waveguide" IEEE Trans.,. MTT-14, pp.15-23, Jan. 1966. 7. LAX, B; BUTTON, K.J.; ROTH, L.M. -"Ferrite Phase Shifters in Rectangular Waveguide", J. Appl. Phys., vol.25, pp. 1413-1421, Nov. 1954. 8. LAX, B; BUTTON, K.J. -"Microwave Ferrites and Ferrimagnetics", New York: McGraw-Hill, 1962, pp. 323-354. 9. von AULOCK, W.H. -"Ferrite Devices for Microwave Applications" Englewood C l i f f s , N.J.; Prentice-Hall. 10. KHARADLY, M.M.Z. -"Periodically Loaded Nonreciprocal Transmission Lines for Phase Shifter Applications", IEEE Trans., MTT-22, (6), 1974, pp. 635-640. 11. BERNUES, F.J.; BOLLE, D.M. -"The Ferrite Loaded Waveguide Discontinuity Problem", IEEE Trans. MTT-22, (12), 1974, pp. 1187-1193. 12. McRITCHIE, W.K.; KHARADLY, M.M.Z. -"Numerical Solution for Nonreciprocal Inhomogeneous Waveguide Interface", Proc. IEE, 1976, vol. 123, (4), pp.291-297. 13. MASTERMAN, P.H.; CLARRICOATS, P.J.B. -"Computer-Field Matching Solution of Waveguide Transverse Discontinuities", Proc. 1971. 118, (1), pp.51-63. 14. McRITCHIE W.K.; KHARADLY, M.M.Z.; CORR, D.G. -"Field-Matching Solution of Transverse Discontinuities in Inhomogeneous Waveguides", Electronic Lett., 1973, 9, (13), pp.291-293. 15. HARRINGTON, R.E. -"Field Computation, by Moment Methods", MacMillan, 1968 16. COLLIN, R.E. -"Field Theory of Guided Waves", McGraw-Hill, i960. 120 17. MAGERL, G. -"Ridged waveguides with Inhomogeneous Dielectric-Slab Loading", IEEE Trans. MTT-26, (6), 1978, pp. 413-416. 18. YOUNG, C.W. -"TE-Mode Solutions for Dielectric Slab Center Loaded Ridged Waveguide", Naval Reasearch Laboratory, Washington, D.C, NRL Report 8105, April 29, 1977. 19. MONTGOMERY, J.P. -"On the Complete Eigenvalue Solution of Ridged Waveguide", IEEE Trans. MTT-19, (6), 1971, pp. 547-555. 20. LIN, F.C. -"Modal Characteristics of Crossed Rectangular Waveguides", IEEE Trans.-MTT-25, (9), 1977, pp. 756-763. 21. MIZOBUCHI, A.; KUREBAYASHT, H. -"Nonreciprocal Remanence Ferrite Phase Shifters Using the Grooved Waveguide", IEEE Trans. MTT-26, (12), 1978, pp. 1012-1016. 22. MARCUVITZ, N. -"Waveguide Handbook" , MIT Radiation Laboratory Series, McGraw-Hill, New York, 1951 

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