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A multi-mode microwave cavity resonator Fall, Stewart Temple 1963

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A MULTI-MODE MICROWAVE CAVITY RESONATOR by STEWART TEMPLE FALL B.A.Sc, U n i v e r s i t y of B r i t i s h . Columbia, 1955 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department of E l e c t r i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the standards r e q u i r e d from candidates f o r the degree of Master of A p p l i e d Science Members of the Department of E l e c t r i c a l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia APRIL 1963 v-In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall'make i t freely available for reference and study. I further agree that per-mission for extensive copying of this thesis for scholarly • purposes may be granted by the.Head of my Department or by his representatives. It i s understood that copying, or publi-cation of this thesis for f i n a n c i a l gain shall not be allowed without my written permission. Department of E l e c t r i c a l E n g i n e e r i n g The University of B r i t i s h Columbia,. Vancouver 8, Canada. Date A p r i l 29, 1963  ABSTRACT Most multi—mode microwave c a v i t y f i l t e r s e x h i b i t i n s e r t i o n -l o s s c h a r a c t e r i s t i c s which deviate widely from the t h e o r e t i c a l responses on the low frequency side of the passband. This paper d e s c r i b e s an X-barid three-mode f i l t e r which i n h e r e n t l y overcomes the primary cause of the response discrepancy. The design i n c l u d e s a method of i d e n t i f i c a t i o n of the f i e l d c o n f i g u r a t i o n s w i t h i n the c a v i t y and a method of r e l a t i n g the co u p l i n g produced between two degenerate modes of resonance by a probe to the r e l a t i v e frequency s h i f t caused by the probe. A simple method of measuring i n s e r t i o n l o s s i s shown, and both t h e o r e t i c a l and measured i n s e r t i o n l o s s c h a r a c t e r i s t i c s are recorded f o r three baridwidths, 7.4 mc/s, 10.0 mc/s, and 13,6 mc/sj a l l of which show no extreme discrepancy between the t h e o r e t i c a l and measured responses. i ACKNOWLEDGEMENT The author i s indebted to both the Northern E l e c t r i c Company of Canada Ltd* f o r the a s s i s t a n c e r e c e i v e d through a post-graduate f e l l o w s h i p granted i n 1955* and the N a t i o n a l Research C o u n c i l of Canada f o r sponsoring the r e s e a r c h p r o j e c t under Grant Number BT-68. Acknowledgement i s ve r y g r a t e f u l l y given to both Dr. F. Noakesy grantee of the p r o j e c t and Dr. B.A. Auld, under whose guidance t h i s work was accomplished* The author a l s o expresses s i n c e r e a p p r e c i a t i o n to Dr* A»D„ Moore f o r h i s in v a l u a b l e help and guidance i n the completion of the t h e s i s . v i TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS i v ACKNOWLEDGEMENT . v i 1. INTRODUCTION 1 1.1 H i s t o r y of Multi-mode F i l t e r s ....... 1 1.2 The Problem and Proposed Method of S o l u t i o n . 2 2. THEORY OF MULTI-MODE MICROWAVE FILTERS 6 2.1 B a s i c P r i n c i p l e s 6 2.2 P e r t u r b a t i o n Theory ................... 7 2.21 Inter-mode Coupling .•••••••»•<>... 7 2.22 Mode I d e n t i f i c a t i o n 13 3. THEORETICAL DESIGN 15 3.1 Normalized Prototype Design . . e . . . . . . 15 3.2 P r e s c r i p t i o n of I n s e r t i o n - L o s s C h a r a c t e r i s t i c s 18 3.3 Determination of Coupling C o e f f i c i e n t s 23 3.31 Input and Output Coupling ............. 23 ^..^ 3.32 I n t e r n a l Coupling ............. 24 3.4 Tuning the C a v i t y 34 3.5 T h e o r e t i c a l I n s e r t i o n - L o s s C h a r a c t e r i s t i c ... 35 4. PRACTICAL DESIGN 38 4.1 P r e l i m i n a r y C o n s i d e r a t i o n s 38 4.2 C o n s t r u c t i o n of the F i l t e r .................. 40 4.3 Measurement and I d e n t i f i c a t i o n of the Modes 43 of Resonance ..*«•••....... i i Page 4.4 Measurement of E x t e r n a l Coupling and I r i s S i z es . 49 4.5 Measurement of I n t e r n a l Coupling and Probe Depths .. ... 54 4.6 Line—up Procedure ............................... 56 5. EXPERIMENTAL RESPONSE CHARACTERISTICS 58 5.1 The Measuring Technique 58 5.2 Experimental R e s u l t s and C o r r e l a t i o n with Theory 60 5.21 Resu l t s f o r the Butterworth Design (13.6 Mc/s Bandwidth) ....... 60 5.22 R e s u l t s f o r the 10.0 Mc/s Bandwidth Design 62 5.23 Resu l t s f o r the 7.5 Mc/s Bandwidth Design .......................... 64 6. CONCLUSIONS 69 7. BIBLIOGRAPHY 71 i i i LIST OF ILLUSTRATIONS Page Fi g u r e 1 Proposed F i l t e r Shape and Mode D i s t r i b u t i o n . 3 2 E q u i v a l e n t C i r c u i t of n C a v i t i e s Coupled i n Cascade ......<.• .. J 3 E q u i v a l e n t C i r c u i t of a Three-Mode Cavity-F i l t e r 16 4 Low-Pass Prototype E q u i v a l e n t F i l t e r C i r c u i t 21 5 E q u i v a l e n t C i r c u i t of Input or Output S e c t i o n Loaded on One Side Only ..•••••••»«••*•...... 23 6 Symmetrical C a v i t y Supporting Two Degenerate Resonances 25 7 Another P a i r of Degenerate Modes i n a Symmetrical C a v i t y , ...*•....*.. 25 8 E q u i v a l e n t C i r c u i t of the C a v i t y of F i g . 6 wi t h E x t e r n a l C i r c u i t r y ..................... 26 9 E q u i v a l e n t C i r c u i t of the C a v i t y of F i g . 7 wi t h E x t e r n a l C i r c u i t r y .....*•»»«.»... 27 10 Schematic I l l u s t r a t i o n of Coupling Loop P o s i t i o n s f o r Measurement of WQ-J- and w Q J J ••• 33 11 Non—contacting Plunger and I t s E q u i v a l e n t C i r c u i t : 39 12 Phonograph of C a v i t y J u n c t i o n w i t h Plunger Arms Attached to Four Faces «•...••••... 41 13 Test Set-up f o r Measurement of Resonant Frequencies of C a v i t y ....................... 43 i v Figure Page 14 Photograph of Test Set-up f o r Measurement of Resonances and I d e n t i f i c a t i o n of Modes . 45 15 Graph of Resonances Occuring i n Gross-Armed C a v l t y .»«»««». »»«oo..9«o-«»»»^«..»...»»•*«. 47 16 The Four Bas i c Modes of Resonance i n the Cross—Armed C a v i t y .............»..«.»«.... 48 17 Graph of Resonant Frequencies of Modes I and II as Functions of Coupling Probe Depth ... 53 18 Graph of (^Q) As a F u n c t i o n of Coupling Probe Depth ... 55 19 Photograph of Completed Cross-Armed C a v i t y . 57 20 Experimental Test Set-up f o r Measurement of F i l t e r I n s e r t i o n - L o s s C h a r a c t e r i s t i c s .. 58 21 Experimental and T h e o r e t i c a l Butterworth Design Responses .......*......••.. 63 22 Experimental and T h e o r e t i c a l Responses f o r the 10 mc/s Bandwidth Design .............. 65 23 Experimental and T h e o r e t i c a l Responses f o r the 7.4 mc/s Bandwidth Design ............. 67 v 1. INTRODUCTION 1.1 H i s t o r y of Multi-Mode F i l t e r s The constant expansion of communications and r a d i o d e t e c t i o n has brought about a need to i n v e s t i g a t e the a p p l i c a t i o n s of fre q u e n c i e s higher than those a l r e a d y i n use. As fr e q u e n c i e s progress i n t o the microwave r e g i o n , so techniques begin to a l t e r from those used w i t h f r e q u e n c i e s below the microwave r e g i o n . Transmission l i n e s become c o a x i a l l i n e s and waveguides, and lumped element f i l t e r s t r u c t u r e s become resonant c a v i t i e s , and, i n g e n e r a l , the l a t t e r are f a r more complex than t h e i r r e l a t i v e l y simple low—frequency c o u n t e r p a r t s . This complexity of microwave elements can be used to advantage i n the design of c e r t a i n types of resonant c a v i t i e s . The simple resonant c a v i t y w i l l resonate i n an i n f i n i t e number of modes and fr e q u e n c i e s but w i l l e x h i b i t the c h a r a c t e r i s t i c s of some lumped-element resonant c i r c u i t near each resonance. U s u a l l y only the lowest order mode i s considered i n the design of a r e s o n a t o r . C a v i t i e s can al s o be coupled i n cascade to give numerous i n s e r t i o n / l o s s c h a r a c t e r i s t i c s t h a t may be s p e c i f i e d by the d e s i g n e r . Many symmetrical c a v i t y s t r u c t u r e s such as the cube, sphere, or c y l i n d e r have two or three modes of resonance t h a t occur at the same frequency; t h a t i s , degenerate modes of resonance. These degenerate modes have been u t i l i z e d i n the design of m u l t i -mode c a v i t i e s i n which the modes are a c t u a l l y coupled together i n a cascade f a s h i o n , by such p e r t u r b a t i o n s as loops and probes, to make a s i n g l e p h y s i c a l c a v i t y the e l e c t r i c a l e q u i v a l e n t of many 2 single-mode resonators i n cascade. However, these s t r u c t u r e s e x h i b i t a poor i n s e r t i o n - l o s s c h a r a c t e r i s t i c on the low-frequency side of the pass-band. This has been suspected to be due to some extraneous c o u p l i n g across the f i l t e r which i s caused by excessive coupling-probe volumes. I t i s the purpose of t h i s paper to study a three—mode c a v i t y resonator in, which t h i s problem i s avoided by choosing such a s t r u c t u r e t h a t , i n the r e g i o n of each co u p l i n g probe, no dominant mode f i e l d s of the t h i r d or uncoupled mode of resonance w i l l e x i s t . This s t r u c t u r e may be d e s c r i b e d as the j u n c t i o n of three c y l i n d e r s whose axes l i e along the axes of the three—dimensional r e c t a n g u l a r co-ordinate system, and with c l o s e d arms of lengths determined by experiment. 1 2 I t has been shown by Curry and L i n t h a t as many as f i v e degenerate modes of resonance w i t h i n a c a v i t y can be coupled i n a chain to give the same i n s e r t i o n - l o s s c h a r a c t e r i s t i c as a f i v e -c a v i t y ladder-type f i l t e r s t r u c t u r e . The advantages of such a c a v i t y i n the saving of freight and space over conventional chains of coupled c a v i t i e s could be g r e a t . However, the r e s u l t i n g experimental c h a r a c t e r i s t i c s of the three—mode f i l t e r s of Curry and L i n have a c o n s i d e r a b l e d i s c r e p a n c y from the t h e o r e t i c a l i n the a t t e n u a t i o n band on the low-frequency side of resonance. Both concluded that the volumes of the i n t e r -mode c o u p l i n g p e r t u r b a t i o n s exceeded the l i m i t s of the small p e r t u r b a t i o n theory, thus s e t t i n g up some e x t r a c o u p l i n g . 1.2 The Problem and Proposed Method of S o l u t i o n I t i s t h i s d iscrepancy between the t h e o r e t i c a l and e x p e r i -mental c h a r a c t e r i s t i c s encountered i n multi—mode c a v i t y f i l t e r s t h a t has l e d to the present study,, I t can be shown t h a t excessive probe volumes cause a d i r e c t c o u p l i n g between the input and out-put modes of the f i l t e r . I f the f i e l d s of the output mode do not e x i s t at a l l i n the r e g i o n of the probe c o u p l i n g the input and centre modes together, then no such d i r e c t through-coupling can e x i s t * The f a v i t y shape shown i n Fig« 1 f u l f i l l s t h i s r e q u i r e -ment i f a mode of resonance can be found t h a t has f i e l d s i n only f o u r of the s i x arms. Then three such modes can be seen to be degenerate and the r e s u l t i s t h a t no more than two sets of f i e l d s w i l l e x i s t i n any one arm. Coupling Probe #1 Output Coupling Probe #2 Input F i g . 1. Proposed F i l t e r Shape and Mode D i s t r i b u t i o n , A mode of resonance i s d e f i n e d as a p a r t i c u l a r e l e c t r i c and magnetic f i e l d d i s t r i b u t i o n t h a t w i l l occur w i t h i n the c a v i t y at a p a r t i c u l a r n a t u r a l frequency of resonance. Thus, r e f e r r i n g to F i g , 1, one of the modes, or c o n f i g u r a t i o n s of resonance t h a t 4 coul d occur i s one i n which the f i r s t order f i e l d s are contained i n f our arms of the s t r u c t u r e l y i n g i n one plane. There are three such modes that are i d e n t i c a l because of the s t r u c t u r e ' s symmetry. These three modes of resonance are s a i d to be degenerate* R e f e r r i n g s t i l l to F i g . 1, l e t the mode of resonance w i t h f i e l d s i n the fou r arms l y i n g i n the XZ-plane be c a l l e d mode 1, the mode having f i e l d s i n the ZY-plane arms mode 2, and the mode having f i e l d s i n the XY—plane arms mode 3» Then, i t i s apparent t h a t i n any arm of the c a v i t y the f i e l d s due to only two of the three p o s s i b l e degenerate modes are present. Therefore, i f mode 1 i s coupled to mode 2 by means of a p e r t u r b a t i o n i n arm 2 and i f mode 2 i s coupled to to mode 3 by means of a p e r t u r b a t i o n i n arm 5, the net r e s u l t i s a chain of three degenerate modes coupled i n cascade. Furthermore, there can be no d i r e c t c o u p l i n g from mode 1 to mode 3 due to e i t h e r p e r t u r b a t i o n , because i n n e i t h e r arm c o n t a i n i n g the p e r t u r b a t i o n s do the f i e l d s of modes 1 and 3 e x i s t t o g e t h e r . Mode 1 can be e x c i t e d i n arm 1 by an i r i s from a re c t a n g u l a r waveguide propagating the TE-^ Q mode with the E f i e l d p e r p e n d i c u l a r to the XY-plane, and mode 3 can.be coupled out through an i r i s to a waveguide o r i e n t e d to propagate the TE^ Q mode with i t s E f i e l d p e r p e n d i c u l a r to the XZ—plane. In F i g . 1, the resonant frequencies of the modes w i l l be determined b a s i c a l l y by the arm le n g t h s , once the c y l i n d e r diameters have been s e t . ¥ith the c i r c u l a r waveguide T E ^ mode i n the arms, there are four d i f f e r e n t sets of f i e l d d i s t r i b u t i o n s that w i l l resonate i n the "cross-armed" c a v i t y . Of course there are an i n f i n i t e number wit h other d i s t r i b u t i o n s i n the arms* but only the lowest-order modes are c onsidered i n t h i s work* These f o u r sets are shown i n F i g * 16 with the dominant higher-order mode shown i n the t h i r d p a i r of arms* The non-propagating TEQ^^ -^QM A N <^ ^ 2 M m o ^ e s a r e a l s o present but t h e i r a t t e n u a t i o n i s so much g r e a t e r than that of the dominant modes th a t they are of l i t t l e importance. The dominant mode th a t e x i s t s i n the t h i r d p a i r of arms w i l l not propagate f a r down these arms provided that the diameter of the c i r c u l a r waveguide i s chosen so t h a t the TE-^ mode i s above c u t o f f and a l l others are below c u t o f f . Thus a mode of resonance w i l l e f f e c t i v e l y e x i s t i n only the fo u r arms of the c a v i t y whose axes l i e i n one p l a n e . The HQ mode was chosen f o r use i n the study because the dominant mode produced i n the t h i r d p a i r of arms has the hi g h e s t c u t o f f frequency of a l l the modes produced i n these arms by any of the d i f f e r e n t sets of resonances* Those modes of resonance w i t h f i e l d d i s t r i b u t i o n s higher i n order than T E ^ i n the arms were considered to be outside the scop of t h i s work. However, where such modes c o — i n c i d e with T E ^ - typ resonances* the c a v i t y could p o s s i b l y be made to represent more than three cascaded f i l t e r s . F i g . 15 shows where these occur, f o r example, where resonance 5 meets resonances 1, 2, 3 or 4. 2. THEORY OF MULTI-MODE MICROWAVE FILTERS 2.1 B a s i c P r i n c i p l e s The s o l u t i o n to Maxwell's Equations shows that f o r c e r t a i n geometries a hollow c a v i t y can support simultaneously two or more f i e l d d i s t r i b u t i o n s , at a s i n g l e frequency, without any i n t e r a c t i o n . These f i e l d d i s t r i b u t i o n s are s a i d to be orthogonal and degenerate i n that they a l l represent simultaneous s o l u t i o n s to the equations. In such a c a v i t y there are an i n f i n i t e number of such sets of degenerate resonances j but only the lowest order w i l l be considered here. Such shapes as the sphere, c y l i n d e r , and cube can support many degenerate resonances"*"' ^ ' ^ . The " c r o s s -armed" c a v i t y s t r u c t u r e under present study i s able to support sets of three degenerate modes of resonance. In order to form a f i l t e r chain with these three modes, the modes must be coupled i n cascade i n a l o g i c a l manner so that there are no inherent i n t e r a c t i o n s between them, other than those caused by the c o u p l i n g s t r u c t u r e . Coupling from one mode to the next along the chain may be achieved by i n s e r t i n g a probe or loop i n t o the f i e l d s at a p o i n t where f i e l d s of both modes can be pre s e n t . I f the f i e l d s of three o r more modes e x i s t i n the c a v i t y , the c o u p l i n g p e r t u r b a t i o n must be i n s e r t e d at a' p o i n t where the f i e l d s of only the two modes to be coupled together e x i s t , or where the f i e l d s of a l l other modes are orthogonal to these two, and where, f o r s m a l l p e r t u r b a t i o n s , t h i s c o n d i t i o n a p p l i e s over the volume of the probe. In the cases of such s t r u c t u r e s as the sphere, cube, or c y l i n d e r , i f the c o u p l i n g p e r t u r b a t i o n i s l a r g e , i t w i l l a c t u a l l y perturb the f i e l d s of the 7 t h i r d mode, with the r e s u l t t h a t undesired d i r e c t c o u p l i n g w i l l e x i s t between the input and output modes* A ch a i n of c a v i t i e s coupled i n cascade can be represented by an e q u i v a l e n t lumped—element s t r u c t u r e f o r a range of f r e q u e n c i e s near the pass-bandy The e q u i v a l e n t c i r c u i t of the c h a i n can be represented as shown i n F i g , 2 where inter-mode couplings are s c h e m a t i c a l l y r e p l a c e d by mutual inductances. I t w i l l be shown l a t e r t h a t a system of tuned c a v i t i e s can be e a s i l y d e r i v e d from the corresponding e q u i v a l e n t lumped-element s t r u c t u r e s . Input n 3 Output F i g . 2. E q u i v a l e n t C i r c u i t of n C a v i t i e s Coupled i n Cascade, Because of the f a c t t h a t small probes are to be used i n c o u p l i n g otherwise orthogonal modes^ i t w i l l be necessary to i n v e s t i g a t e the e f f e c t s which such p e r t u r b a t i o n s have upon the f i e l d s . 2.2 P e r t u r b a t i o n Theory 2.21 Inter-Mode Coupling Maxwell's equations* s t a t e d i n d i f f e r e n t i a l form ares c u r l E = -d i v B = 0 St c u r l H = J + fct (2.1) d i v D = f where E i s the e l e c t r i c f i e l d i n t e n s i t y (volts/metre) H i s the magnetic f i e l d i n t e n s i t y (amperes/metre) B i s the magnetic i n d u c t i o n (weber/metre ) D i s the e l e c t r i c displacement (coulombs/metre ) ^ i s the charge d e n s i t y (coulombs/m ) and D = e e,-. E r 0 B =.|i r | i Q H — 1 P where i s the p e r m i t t i v i t y of f r e e space = 8.85 x 10 f/m e^ i s the r e l a t i v e p e r m i t t i v i t y — T U.Q i s the p e r m e a b i l i t y of f r e e space = 4% x 10 h/m \i i s the r e l a t i v e p e r m e a b i l i t y Let E be a s o l e n o i d a l E f i e l d , one of the p o s s i b l e £1 s o l u t i o n s of Maxwell's equations i n a hollow charge-free c a v i t y , and l e t H be the corresponding s o l e n o i d a l H f i e l d . I t w i l l be assumed that E and H are as d e f i n e d by S l a t e r , such that a a k E = c u r l H cL cl £L k H = c u r l E £1 cl 3/ (2.2) r 2 and t h a t they are normalized i n such a way t h a t /E dv = 1 and r 2 7 v a IE dv = 1, where V i s the volume of the c a v i t y . Under the assumption that V i s bounded by the surface S + S 1 (with outer normal "n) where S i s a m e t a l l i c boundary and S' i s an ©pen' c i r c u i t 9 boundary, S l a t e r has shown that and f^l l f a d v + k a / E - E a d v = *0 tt( / J - E a d v ~ / (* X ?)'V a J " k a / (n- x E).H ada (2.3) f%2 / H.H ad y + k a / H ' H a d v = k a ( / J ' E a d v - / ( " X H>-Va " e 0 It / ^  x B > ' H a d a S (2.4) Under the c o n d i t i o n of zero c o n d u c t i v i t y w i t h i n the volume, zero t a n g e n t i a l E f i e l d on the m e t a l l i c boundary S, and zero t a n g e n t i a l H f i e l d on the open .1 boundary S 1 , the r i g h t hand sides of these two equations are zero. Then i f , i n f r e e o s c i l l a t i o n , the mode s o l u t i o n s E and H have time v a r i a t i o n of the form e , where <0 i s the mode 7 a 2 2 frequency, equations (2.3) and (2.4) show that k a = (o^ e^ |O,Q. To consider the a f f e c t of a p e r t u r b a t i o n of the c a v i t y boundary on the f i e l d s , f i r s t consider a c a v i t y e n t i r e l y enclosed by a p e r f e c t conductor, and l e t two degenerate modes be e x c i t e d . L e t t i n g the orthogonal modes be designated by s u b s c r i p t s a and 2 b, the f i e l d i n the c a v i t y i s given by ,s H = A H a + BHfe (2.5) 10 E = C E + DE, (2.6) a b where A = / H.Ha<lv, B = / H.H^dv C = / E.E adv, D = / E*E^dv Now, i f the c a v i t y i s perturbed by pushing a small p o r t i o n of the boundary surface i n t o the c a v i t y , w i t h i n the small volume formed by t h i s p e r t u r b a t i o n the f i n a l E and H f i e l d s are zero. W i t h i n the small volume J i s zero and over the m e t a l l i c boundaries of the p e r t u r b a t i o n , t a n g e n t i a l E i s zero. But there e x i s t s a surface c u r r e n t equal to "n x H over the surface of the p e r t u r b -a t i o n * Thus* f o r a time v a r i a t i o n of the form e*'40^, equation (2.4) becomes, f o r the f i e l d s of mode a (»f - » 2 ) A = - / (n x H).E da (.2.7) and f o r the f i e l d s of mode b - a>2) B = - b / (n x H).E bda (2.8) I f the volume v of the p e r t u r b a t i o n i s s m a l l , the perturbed f i e l d i s v e ry n e a r l y i d e n t i c a l to the o r i g i n a l f i e l d , except at the p e r t u r b a t i o n , and the value of H from the equation (2.5) can be s u b s t i t u t e d . 11 Then the i n t e g r a l on the r i g h t - h a n d side of equation (2.7) becomes Cn" x H).E ada = A f (n x H j . E ^ d a + B f(t x H ^ . t ^ a (2 .9) It 4, 4t But S i m i l a r l y * (n x H ).E da a a n.(E x H )da a a S* d i v ( E x H )dv a a,' H . c u r l E dv - / E . c u r l H dv a a / a a V H a " E a > d v < 2' 1 0) Ci x H b ) . E a d a = k a / H ^ d v - k b / E a . E b d v (2.11) S u b s t i t u t i n g equations (2.10) and (2..11) i n t o equation (2 .9) and the r e s u l t i n t o equation (2.7), co ( l + m ) - c o ' ci + B co p — co co, q a^ a b H = 0 (2.12) where m = / ( H 2 - E 2 ) d v 12 and p = / H a.H bdv q = / E a . E b d v S i m i l a r l y , from equation (2.8). + B 2 2 a> - aC ( l + n) = 0 (2.13) where n = / ( H 2 - E 2 ) d v For degenerate modes of equal amplitude, A = B and w =•*>>. = <ono Then, i f the p e r t u r b a t i o n possesses such symmetries t h a t m = n, equations (2*12.) and (2.13) y i e l d 2 2 u> = «0Q(1 + m + p - q) = » 0 ( 1 + m) 1 + E r a . 1 + m or a> «>0(l + m) 2 i + JB=a 2(1 + m) (2.14) (2.15) i f m, p , q << 1, Comparing t h i s r e s u l t w i t h that of conventional coupled c i r c u i t s tuned to the same resonant frequency^ the two modes have a c o e f f i c i e n t of co u p l i n g given by 1 + m (2.16) T h e r e f o r e , i t has been shown th a t a c o u p l i n g c o - e f f i c i e n t between two modes i n a c a v i t y can be c a l c u l a t e d , but the c a l c u l a t i o n s are a l l based on the assumption t h a t the p e r t u r b a t i o n be s m a l l . 2.22 Mode I d e n t i f i c a t i o n The f i e l d d i s t r i b u t i o n of each mode of resonance i n a c a v i t y of complex shape i s ve r y d i f f i c u l t to c a l c u l a t e . However, i t can be measured and p l o t t e d by i n t r o d u c i n g small m e t a l l i c or d i e l e c t r i c beads i n t o the c a v i t y and no t i n g the r e s u l t a n t s h i f t i n the resonant frequency. For a s i n g l e mode i n a c a v i t y , the terms p and q i n equation (2.12) are equal to zero, so that 2 2 co - co a ~ = m o r a which can be w r i t t e n , f o r Aco = co — co <<co ; EL £t (2.17) | f i . . = f{*l - E 2 ) d v (2.18) For a d i e l e c t r i c bead, the e f f e c t on the H f i e l d s i s zero, i t can be shown t h a t , f o r e » l , the frequency s h i f t becomes E 2 dv (2.19) Et For a m e t a l l i c sphere, equation (2.18) a p p l i e s , except at poi n t s where the E f i e l d s v a n i s h , where the frequency s h i f t becomes 14 (2.20) Therefore, the p o i n t s of maximum E f i e l d are those where e i t h e r a d i e l e c t r i c or a m e t a l l i c bead gives maximum r e d u c t i o n of resonant frequency. P o i n t s of minimum E f i e l d and maximum H f i e l d are those where a m e t a l l i c bead gives maximum inc r e a s e i n resonant frequency. 4 5 6 For a m e t a l l i c sphere the frequency s h i f t i s given by ' *• & = - ^ 0 H 2 " 2 e 0 E 2 > ( 2 ' 2 1 ) where V i s the bandwidth between the half-power p o i n t s . Note that the frequency s h i f t i s zero at p o i n t s where the — — w H and E f i e l d s are r e l a t e d by 15 3, THEORETICAL DESIGN The preceding pages have d e a l t p r i m a r i l y with the general concepts of the problems i n v o l v e d i n the design of multi-mode c a v i t y resonantors with the emphasis on the "cross-armed" c a v i t y s t r u c t u r e i n p a r t i c u l a r . Once the p o s s i b l e f i e l d d i s t r i b u t i o n had been s t u d i e d , a "cross-armed" s t r u c t u r e was c o n s t r u c t e d and the a c t u a l f i e l d d i s t r i b u t i o n s and modes of resonance were i n v e s t i g a t e d * The r e s u l t s of t h i s i n v e s t i g a t i o n are contained i n a l a t e r s e c t i o n where i t i s shown t h a t the theory p e r t a i n i n g to t h i s c a v i t y shape i s i n agreement w i t h experiment. The set of modes chosen f o r the f i l t e r was of the HQ type as shown i n F i g . 16(a) because t h i s mode type sets up f i e l d s i n the "empty" arms which are of a higher order propagation mode than those set up by the three other types. Thus the i n t e r a c t i o n between modes due to the disturbance of t h i s higher order mode by the c o u p l i n g probes would be ax a minimum wit h the HQ type of resonance mode. The c i r c u i t problem, that i s * the i d e n t i f i c a t i o n of a low-pass prototype s t r u c t u r e with the e q u i v a l e n t f i l t e r c i r c u i t , and the p r e s c r i p t i o n of i n s e r t i o n - l o s s c h a r a c t e r i s t i c s , i s b a s i c a l l y 3 that o r i g i n a l l y used by Fano and Lawson * 3.1 Normalized Prototype Design Because the c a v i t y w i l l be operated between matched l i n e s , i t i s found convenient to normalize the elements of the e q u i v a l e n t c i r c u i t with r e s p e c t to the waveguide impedance of the dominant mode of propagation; t h a t i s , the f i l t e r i s to be designed to operate between one-ohm ter m i n a t i o n s and the f i n a l design adjusted to operate between the a c t u a l waveguide impedances. I t i s a l s o 16 found advantageous to normalize the d e s i g n frequency-wise. Then, the band—pass network can be designed as a simple low-pass f i l t e r w ith a c u t o f f frequency of one r a d i a n per second and then t r a n s -formed i n t o a band-pass f i l t e r by the r e l a t i o n x = 0 co 0 0 CD (3.1) where a>^ i s the angular frequency of the pass-band centre and ¥ i s the bandwidth. The e q u i v a l e n t c i r c u i t of the three-mode f i l t e r i s shown i n F i g . 3« The elements of the c i r c u i t are normalized and i t i s assumed that X and X are both much l e s s than one ohm. • • s r Input I r i s Coupling Probe #1 L,C "0 Coupling Probe #2 Output I r i s F i g . 3. E q u i v a l e n t C i r c u i t of a Three-Mode C a v i t y F i l t e r * The inter-mode c o u p l i n g c o — e f f i c i e n t s are r e l a t e d to the co u p l i n g reactances by the equations. X 2 = W 0 M 1 2 = W 0 L k 1 2 X 3 = <o 0M 2 3 = <o QLk 2 3 (3.2) (3,3) 17 I t may be noted at t h i s p o i n t that the eq u i v a l e n t c i r c u i t a c c u r a t e l y represents the microwave c a v i t y s t r u c t u r e over a narrow band of freq u e n c i e s centred at the resonant frequency of the f i l t e r . In t h i s band, the c o u p l i n g s t r u c t u r e s can be represented by constant lumped elements and the modes e x h i b i t c h a r a c t e r i s t i c s s i m i l a r to those Of e q u i v a l e n t L-C tuned c i r c u i t s . To i d e n t i f y the band-pass c i r c u i t with that of the p r o t o -type low-pass c i r c u i t , f i r s t the input impedance i s considered. R e f e r r i n g to P i g . 3, the input impedance i s given bys Z. xn . WLx X l ¥ L x X 2  X 2 ¥L x X 2 x:x z 1 3 Y 2 Y 2 A 1 A 3 2 2 X 2 X 4 (3.4) where x = ^p- I — - — ¥ I <OQ to P i g . 4 i s the low-pass prototype s t r u c t u r e r e p r e s e n t i n g frequency n o r m a l i z a t i o n of the the three-stage band-pass f i l t e r . The input impedance of t h i s network ±s, a f t e r transforming back to the band-pass frequency v a r i a b l e , 18 Z. i n j x L 9 + j x C 3 + | (3.5) Comparison of equations (3.4) and (3.5) y i e l d s the f o l l o w i n g i d e n t i f i c a t i o n s . 2 2 v 2 _ L n r 2 = L 2 C 1 T 2 _ L 2 ¥ 2 3 = L 2 C 3 X 2 C ^4 RC^ = JJQ"~ where R = 1 ohm (3.6) Now the s p e c i f i c a t i o n of an i n s e r t i o n - l o s s f u n c t i o n f o r the low-pass prototype w i l l u n i q u e l y set the values of C^, L^, C^ and hence the parameters of the e q u i v a l e n t c i r c u i t of the c a v i t y . 3.2 P r e s c r i p t i o n of I n s e r t i o n - L o s s C h a r a c t e r i s t i c s I n s e r t i o n l o s s i s d e f i n e d as the d e c i b e l measure of the power d e l i v e r e d to the l o a d , when the network i s i n s e r t e d between the source and l o a d , r e l a t i v e to the power d e l i v e r e d to the lo a d without the network. I t i s given by the e x p r e s s i o n : L = 10 l o g 1 0 - y * - (3.7) L where PQ i s the power i n t o the l o a d without the network and P-^ i s the power i n t o the loa d w i t h the network between the source and the l o a d . The r a t i o p— can be expressed i n terms of ABCD parameters L of the network as p p ^ = j | A + B + C + D | 2 (3.8) L i n which the assumption i s made th a t the source and loa d impedances are i d e n t i c a l and equal to u n i t y . For a l o s s l e s s n e t -work* A and D are p u r e l y r e a l and B and C are p u r e l y imaginary, so t h a t °- = \ [(A + D ) 2 - (B + C ) 2 ] But, since AD - BC = 1, the above expre s s i o n can he r e w r i t t e n as £ = 1 + \ [(A - D ) 2 - (B - C ) 2 ] (3.9) In the case of a symmetrical network^ A = D, so that ^ = 1 - \ (B - C ) 2 (3.10) The r e s u l t s of the i n s e r t i o n - l o s s theory of f i l t e r d e s ign show that maximum stop—band a t t e n u a t i o n f o r a given pass-band t o l e r a n c e can be 20 obtained by p r e s c r i b i n g the i n s e r t i o n - l o s s f u n c t i o n i n terms of the Chebyshev polynomials. Thus L •= 10 l o g 10 1 + h2 T 2 (x) (3.11) where T n ( x ) i s the Chebyshev polynomial of order n, and h i s a constant that determines the magnitude of the r i p p l e or pass-band t o l e r a n c e . The e q u i v a l e n t low-pass f i l t e r i n the present study i s of the t h i r d order, so L = 10 l o g 10 1 4 h2 T 2 ( •3 where T 3 ( x ) = 4xJ> - 3x (3.12) (3.13) P i g . 4 shows t h i s low-pass e q u i v a l e n t prototype network having parameters A = D = 1 - x 2 L 2 C 1 B = j x L 0 C = 2 .xC, 3 1 3 2 (3.14) S u b s t i t u t i n g these values i n t o equation'(3.10) y i e l d s L = 10 l o g 1 0 - j l t | x 3 L 2 C 2 - x ( 2 C 1 - L 2 ) (3.15) 21 C. = C- f o r a symmetrical 1 J f i l t e r . - F i g. 4. Low-Pass Prototype E q u i v a l e n t F i l t e r C i r c u i t . From equations (3.12), (3*13) and (3.15), L 2 C ^ •=. 8h and 2G 1 - L 2 = 6h (3.16) Thus, s p e c i f i c a t i o n of a pass—band t o l e r a n c e w i l l u n i q u e l y determine the lumped-element values and L 2 f o r the optimum or Chebyshev response. Another d e s i r a b l e response shape i s the Butterworth, or m a x i m a l l y - f l a t , response. This i s a c t u a l l y a s p e c i a l case of Chebyshev response, i n which the pass-band t o l e r a n c e becomes zero and the f u n c t i o n becomes L = 10 l o g (1 + x b ) (3.17) Then, from equations (3.15) and (3*17)» 2 C 1 " L 2 = 0 a n d L 2 C 1 = 2 (3.18) so t h a t , C^ = 1 f a r a d and L 2 = 2 h e n r i e s (3.19) 22 In p r a c t i c e some d i s s i p a t i o n i s always present, causing the a c t u a l c h a r a c t e r i s t i c s of the f i l t e r to d i f f e r somewhat from the t h e o r e t i c a l r e s u l t . Assuming that the l o s s e s are u n i f o r m l y d i s t r i b u t e d between the inductancesaand capacitances, the i n c i d e n t a l d i s s i p a t i o n can be accounted f o r by adding a small constant r e a l f a c t o r to the frequency v a r i a b l e p = jco so t h a t p = & + j«>§ where 07 = ^ . i - ( 3 . 2 0 ) *u I f , i n equation ( 3 . 1 5 ) , the transformed frequency v a r i a b l e i s taken to be ( £ + j x ) , a general e x p r e s s i o n f o r the i n s e r t i o n l o s s can be found, i n c l u d i n g the e f f e c t s of i n c i d e n t a l d i s s i p a t i o n : L = 1 0 l o g ^ [ 2 1 ^ (£ 2 - x 2 ) + L 2 C 2 £ ( S 2 - 3x 2) +£(L 2+ 2 C 1 ) + 2 J 2 4 1 ^ 0 ^ + L 2 C 2 ( 3 S 2 - x 2 ) + ( L 2 +• 2 ^ ) ^ / (3.21) + x 2 This suggests a f u r t h e r method of c a l c u l a t i n g the value of the l o s s f a c t o r • At the centre of the pass—band, where x = 0 , the i n s e r t i o n l o s s becomes L = 1 0 l o g i [ L 2 C 2 S 3 + 2 L 2 C 1 S 2 + ( L 2 + 2 0 ^ )g +2] 2 (3.22) and by equating t h i s to the experimental value of L at the centre of the pass-band, the value of S can be determined. 23 3.3 Determination of Coupling C o — e f f i c i e n t s 3.31 Input and Output Coupling The input and output c o u p l i n g s t r u c t u r e s are to be i r i s e s i n the c a v i t y w a l l * one at each end of a co-line-ar p a i r of arms. The s i z e s of the i r i s e s determine the amount of coup l i n g between the e x t e r n a l c i r c u i t r y and the c a v i t y . Because of the inherent mathematical complexity of the "cross-armed" c a v i t y , d i r e c t e v a l u a t i o n of e x t e r n a l c o u p l i n g co-e f f i c i e n t s i s not p o s s i b l e . Thus the i r i s s i z e s are determined by r e l a t i n g the amount of c o u p l i n g r e q u i r e d to the e x t e r n a l Q's of the input and output f i l t e r s e c t i o n s . Assume f i r s t t h a t the measurement of Q g i n v o l v e s only the input e x t e r n a l c i r c u i t r y , the i r i s and the f i r s t f i l t e r section? t h a t i s * the f i r s t mode of resonance i s o p e n - c i r c u i t e d a t i t s output s i d e , and thus i s loaded on one side only as shown s c h e m a t i c a l l y i n F i g . 5. - A / W Y -1. i r i s F i g . 5. E q u i v a l e n t C i r c u i t of Input or Output S e c t i o n Loaded on One Side Only, 24 The e x t e r n a l impedance, 1 + jX * i s coupled i n t o the c a v i t y c i r c u i t s S 1 ¥L/C 1 'coupled •"" 1 + j X — 1 + jX x - s. (3.23) Since X s « 1, Z c o u p l e d « WL/ C l (3,24) A l s o , by d e f i n i t i o n * coupled co 0L C 1  e e = ¥ L 7 c ^ = W 0 I T ( 3 ' 2 6 ) Thus, since G-^  i s f i x e d by the s p e c i f i c a t i o n of the response shape and <DQ i s f i x e d b a s i c a l l y by the p h y s i c a l s i z e of the c a v i t y . Q G i s a f u n c t i o n of pass—band width* I f a c e r t a i n bandwidth i s s p e c i f i e d ^ Q G i s u n i q u e l y determined f o r the g i v e n response c h a r a c t e r i s t i c s * Then the procedure f o r determining the i r i s s i z e i s by measurement of Q E f o r v a r i o u s i r i s s i z e s u n t i l the d e s i r e d Q i s reached, e 3.32 I n t e r n a l Coupling The i n t e r n a l c o u p l i n g c o - e f f i c i e n t s f and k 2 3 are equal f o r a symmetrical f i l t e r * These couplings are to be e f f e c t e d by the i n s e r t i o n of screws or probes i n t o the C a v i t y w a l l * one i n an arm c o n t a i n i n g the f i e l d s of the input and centre modes, and the other i n an arm c o n t a i n i n g the f i e l d s of the centre and output modes. The i n t e r n a l c o u p l i n g c o - e f f i c i e n t s are expected to r e q u i r e probe depths t h a t exceed the l i m i t s of small p e r t u r b a t i o n s . The f o l l o w -i n g method of c a l i b r a t i n g probe depth to c o u p l i n g c o - e f f i c i e n t i s unhampered by these l i m i t s * F i r s t c o n s i d e r a c y l i n d r i c a l c a v i t y that w i l l support two degenerate modes of resonance as shown i n F i g . 6. The "cross-armed" c a v i t y w i l l a l s o support two such orthogonal degenerate modes so that t h i s d e r i v a t i o n w i l l apply e q u a l l y w e l l to the "cross-armed" c a v i t y . f* Mode 1 Mode 2 F i g . 6. Symmetrical C a v i t y Supporting Two Degenerate Resonances. This c a v i t y i s coupled^ by means of i d e n t i c a l loops l o c a t e d d i a m e t r i c a l l y opposite each other* to c o a x i a l input and output l i n e s . The planes of the loops are p a r a l l e l . An a d j u s t a b l e screw i s p l a c e d i n the c a v i t y w a l l at 45° to the diameter between the cou p l i n g l o o p s . There are an i n f i n i t e number of degenerate mode p a i r s of t h i s type i n a c y l i n d r i c a l c a v i t y , but i n the "cross-armed" c a v i t y only one other p a i r of t h i s type e x i s t s as shown i n F i g . 7. 26 Since* i n F i g . 6, both loops couple to one and the same mode, the i n t e r n a l c o u p l i n g between mode 1 and mode 2 i s e f f e c t e d by the screw, which a l s o tunes both modes by equal amounts. Because modes 1 and 2 have the same unloaded Q and the same resonant frequency* f ^ , and because the input and output couplings are i d e n t i c a l , the e q u i v a l e n t c i r c u i t of the c a v i t y w i t h e x t e r n a l networks becomes t h a t shown i n F i g . 8. •juuim-m L", C, r Mode 2 4 ' R 0 F i g . 8. E q u i v a l e n t C i r c u i t of the C a v i t y of F i g . 6 with E x t e r n a l C i r c u i t r y . For the modes shown i n F i g . 7, the loops couple e q u a l l y to both and the screw tunes each by a d i f f e r e n t amount, but does not cause any c o u p l i n g between the modes* Here mode I and mode I I w i l l not n e c e s s a r i l y have the same resonant frequency; however, i t w i l l be assumed t h a t the Q's are equal* With these p o i n t s i n mind, the equ i v a l e n t c i r c u i t f o r t h i s c a v i t y * i n c l u d i n g the e x t e r n a l networks, becomes t h a t shown i n F i g , 9. 27 F i g . 9. E q u i v a l e n t C i r c u i t of the C a v i t y of F i g . 7 with E x t e r n a l C i r c u i t Because the c i r c u i t r e p r e s e n t a t i o n s of the two cases are e q u i v a l e n t , they w i l l y i e l d i d e n t i c a l t r a n s f e r f u n c t i o n s . By c a l c u l a t i n g and comparing the t r a n s f e r f u n c t i o n s f o r the two cases, the tu n i n g i n t r o d u c e d by the screw i n F i g . 7 can be r e l a t e d to the c o u p l i n g i n t r o d u c e d by the screw i n F i g . 6. To c a l c u l a t e the t r a n s f e r f u n c t i o n f o r the network on F i g . 8, i t w i l l f i r s t be assumed t h a t the Q's are not a f f e c t e d by the presence of the screw. The self-impedance of meshes 2 and 4 i s : '22 = Z 44 r + J w L + j*C = r + jfi>0L CO 0 CO, 1 . /CO Q 3 U 0 (3.27) 28 Since the unloaded Q's are l a r g e , and the bandwidth of i n t e r e s t i s r e l a t i v e l y s m a l l , a l l but f i r s t - o r d e r terms i n the expansion of ~ <o^ ) ' u s i n S  a - wo c a n ^ e n e g l e c t e d , Thus Z 22 = Z 4 4 = » 0 L + i ^ ) <3-28' W r i t i n g the mesh equations f o r the c i r c u i t of F i g . 8 , and s o l v i n g f o r 1^  y i e l d s ^ 1 ^ ) 2 « 0 L ( | + 3 2 ^ ) z 3 = - / - ; : -TT - m < 3 \ 2 9 > U 0 + j<^) (R Q+ j o ) i ) ( « , 0 L ) 2 ( | + 32 | ^ + 2(<oM)2 « 0 L ^ + j2 fe) + (cpm) 2(R 0 + j j j / Now introduce the f o l l o w i n g approximations: 1« Neglect the v a r i a t i o n of the c o u p l i n g reactances w i t h frequency*; so t h a t «OM««>Q M and « m » <0Q m. 2* Assume t h a t R Q + J « ^ » R Q . T h i s w i l l be s u f f i c i e n t l y accurate i f the loops are so l i g h t l y coupled to the c a v i t y t h a t the frequency of the coupled mode i s n e g l i g i b l y a f f e c t e d . Let K 2 = gy k 2 = ^ (3.30) 29 Then, a f t e r d i v i d i n g numerator and denominator of expression (3.29) by j j « 0 L ) 2 » Q i ] , I 3 be, i c o m e s But h = Vf-2($ + 3"2x) R, '0 + j 2 x ) 2 + 2K 2 (I + J 2 x ) + k 2 ^ (3.31) R 0 <* 0L 0* (co0Mr/R0 2 W 0 L K 2 K = TT-^ = Q K (• e R coupled (3.32) where Q g i s the e x t e r n a l Q of mode 1 f o r one l o o p . Therefore, I 3 R 0 = V Q = -+ j2x) Qe(i + j 2 x ) 2 + 2 ( i + j2x) + k 2Qc (3.33) Turning to the network of P i g . 9, the self-impedance of mesh 2 i s (3.34) where <0j = » - 40^ ^ and a s i m i l a r r e s u l t a p p l i e s f o r mesh 4. I t i s found convenient to r e f e r a l l f r e q u e n c i e s to f 0 I + f 0 I I f 0 = (3.35) 30 Thus, 2 2 (3.36) The self-impedance of meshes 2 and 4 t h e r e f o r e bee ome fa . M o i - " o n (3.37) and Z44 * * 0 L (g + 3 " % (3.38) The approximations that - — » 1 and » 1 w i l l only c o n t r i b u t e • « 2 » — ~'0 "0 a small e r r o r to the value of Q since the d i f f e r e n c e between co and <0QJJ i s only a f r a c t i o n of one per cent Now l e t 01 6* co "o A, A I I = = x «oi - % • o i l - % (0 0 (3.39) where » u i s the resonant frequency of modes I and I I when the screw 31 i s f l u s h w i t h the c a v i t y w a l l . W r i t i n g the mesh equations f o r t h i s case, and s o l v i n g f o r I-j y i e l d s : \ +l+ j [ 2 x - (Aj - A n ) ] } , \ (3.40) '(RQ + j«X) j(R 0 + jcoX)(co0L)2[(i + j 2 x ) 2 4 (Aj - A J J ) 2 ] +. 4(»1)2 « 0 L ( j j + j2x) -y Now introduce the same approximations as b e f o r e , 1, « ^Q^-2. co X « R 0 and d i v i d e numerator and denominator by («0QL)2 C*Q^» l e t t i n g 1/2 rh? (3.41) so that Io = -2 V 1 ^ 2 ( ^ + 3 ' 2 x ) /R \ Ro;<r§r ( s + j 2 x ) 2 + 4k2{h + d2x) v. 0 2? R. (3.42) 32 But. B 0 « 0 L X 2 . y x 3 2 ^ = / „ a ^ 2 / D = R , = 9 e * ( 3 ' 4 3 > '(T- (co0?h.)^/R0 coupled where Q g i s the e x t e r n a l Q of one mode f o r one loop. Therefore^ V r ( | + j2x) 3 0 0 *e ,1 x2 ,1 v * e , x2 f - + j 2 x ) ^ + 2 ( | + j2x) + ^ - ( A j - A Z I ) Z (3.44) Comparison of t h i s equation w i t h equation (3.33) shows t h a t % = 2Q e (3.45) k = (Aj - A J J ) (3.46) k / 0 1 ; W Q I 1 (3.47) 0 Thus the c o u p l i n g c o - e f f i c i e n t can be c a l i b r a t e d a g a i n s t c o u p l i n g probe depth merely by measurement of &QJ and ">QJJ at each screw depth* This procedure w i l l y i e l d a graph of k versus probe depth. E x p r e s s i o n (3.45) i s f u r t h e r e x p l a i n e d by the f a c t t h a t Q g v a r i e s i n v e r s e l y w i t h the square of the magnetic f i e l d i n t e n s i t y f o r a small loop and i s t h e r e f o r e twice as great f o r the l a t t e r case as f o r the f i r s t since the f i e l d i n t e n s i t i e s f o r the two cases are i n the r a t i o of /~2~ at the screw. 33 Measurement of «0QJ and i s r e a d i l y accomplished by i n s e r t i n g the loops so t h a t they couple to e i t h e r of modes I or I I as shown i n F i g * 10 . Mode II —vwv*—1|— Fig*, 10* Schematic I l l u s t r a t i o n of Coupling-Loop P o s i t i o n s f o r Measurement of <K>QJ and <^Q^» I f the loop p e n e t r a t i o n i s s u f f i c i e n t l y small t h a t < o £ « R Q > the frequency p u l l i n g i s n e g l i g i b l e and WQ-J- and « Q T T can be measured d i r e c t l y from the peaks of the t r a n s m i s s i o n through the c a v i t y . A l s o note H?ha+> the loops are not i n the same plane as the screw* This process a l s o gives the r e l a t i v e detuning by the screw of modes 1 and 2 » a,* _ to 0 u A I + A I I to 0 (3.48) where »1>Q i s the resonant frequency q.f modes I and 2 w i t h the probe i n s e r t e d i n t o the c a v i t y * 34 3.4 Tuning the C a v i t y The f r e q u e n c i e s of the three modes of resonance w i l l change with i n s e r t i o n of the c o u p l i n g probes and i r i s e s due to t h e i r phy-s i c a l presence i n a d d i t i o n to any frequency s h i f t due to the coup-l i n g to adjacent c i r c u i t s . T h e r e f o r e , a method of r e - t u n i n g the modes must be p r o v i d e d . This i s done by means of screws or tuning probes i n s e r t e d a x i a l l y i n t o the end w a l l s of the two arms th a t c o n t a i n n e i t h e r i r i s e s nor co u p l i n g probes. The resonant f r e q u e n c i e s of both modes i n the arm c o n t a i n i n g a tuning probe w i l l s h i f t with a change i n depth of the probe, and i f both probe depths are changed the frequency of the centre mode of resonance w i l l be changed by an amount equal to the a l g e b r a i c sum of the frequency s h i f t s experienced by the other two modes. Thus! Y±l = F. + fi12 t 1 2 (3.49) V = F 0 + ^23*23 ( 3 ' 5 0 ) V = F c + S l 2 t 1 2 + '23*23 ( 3 - 5 l ) where F^ = frequency of input mode before t u n i n g , F ^ = frequency of input mode a f t e r t u n i n g , FQ = frequency of output mode before t u n i n g , FQ"^ frequency of output mode a f t e r t u n i n g , F C = frequency of centre mode before t u n i n g , ^ c^= frequency of centre mode a f t e r t u n i n g , S^2= depth change of probe i n t o c a v i t y , a f f e c t i n g the i n p u t and centre modes, S22= depth change of probe i n t o c a v i t y , a f f e c t i n g the centre and output modes, 35 *12 = ^ u n i n S r a t e (frequency s h i f t per u n i t i n c rease i n probe depth) of the probe between the input and centre modes, *23 = ^ > 1 i* Li n& r a t e (frequency s h i f t per u n i t i n c rease i n probe depth) of the probe between the centre and output modes. Then i f the fr e q u e n c i e s of the three modes are to be e q u a l i z e d at a frequency F: F = F i + ^2*12 = F 0 + S23 i23 = F c + ^12 +12 t ^23*23 or F - F ^2 " \ ^ < 3 - » > F. - F and g = ~~~~7 - (3.53) 23 t 2 3 3.5 T h e o r e t i c a l I n s e r t i o n Loss C h a r a c t e r i s t i c s The general t h e o r e t i c a l i n s e r t i o n — l o s s c h a r a c t e r i s t i c i s given by equation (3.21), The l o s s f a c t o r , S 4 may be found by equation (3*20) but since Q u was measured before the t u n i n g probes were added, t h i s i s l i k e l y to y i e l d a value f o r S lower than the a c t u a l . Thus, the value of £ was found from equation (3.22) a f t e r an experimental i n s e r t i o n - l o s s c h a r a c t e r i s t i c had been measured. Since Q g and k are the f a c t o r s t h a t are independently v a r i a b l e i n t h i s f i l t e r , i t i s found more convenient to r e w r i t e the general i n s e r t i o n - l o s s f u n c t i o n i n terms of these f a c t o r s r a t h e r than i n terms of L 2 and C^» For the low-pass prototype design to be transformed to the band-pass design, the equations (3.6) must be a p p l i e d , and from F i g * 5 f o r X « 1, s <o0L *1 For a symmetrical f i l t e r i n the c o n f i g u r a t i o n of F i g . 3, 36 X X k12 = k23 = k = ^ L = ^ L (3.55) Equations (3*6) become, f o r the symmetrical f i l t e r , X l 2 = C~f = X 4 2 (3.56) and v 2 _ L 2 ¥ 2 _ Y 2 2 - C 1 L 2 - A 3 \ S u b s t i t u t i o n of equations (3.56) i n t o equations (3.54) and (3.55) y i e l d s 2 ~" 2 *e 0 and ( 3 i 5 7 ) Q V C l = 5T-x 0 Nov by s u b s t i t u t i n g equations (3.57) i n t o equations (3.21), and l e t t i n g the frequency v a r i a b l e be <0 y = xV = I * P-17 0 | » 0 (0 y (3.58) and. the l o s s f a c t o r be ^ = 8W, (3.59) the general i n s e r t i o n - l o s s f u n c t i o n can be w r i t t e n i n terms of Q and k as t 37 L = 10 l o g j u ( » 0 k ) ' (<o0k) co0 (<^2 - 3y 2) (• 0k)' 2 + <^k + 22e k> + 2 ] + y J 0 *e J O * 2 " y2) + A + 2B.k)T (<o0k) » 0 0 *e J (3.60) P i g . 18, to be exp l a i n e d l a t e r , p rovides the value of k f ^ f o r each c o u p l i n g probe depth used* By s e t t i n g y = 0 i n the above equation and equating L to the minimum l o s s i n the pass-band, the l o s s f a c t o r <f can be found and t h e r e f o r e the value of 8 can be obtained from equation (3.59). I f A « ^ « Q , s u b s t i t u t i o n of to = o>0 + A». i n t o equation (3.58) y i e l d s y a s 2 A » = 4% (Af) (3.61) 38 4* PRACTICAL DESIGN 4,1 P r e l i m i n a r y C o n s i d e r a t i o n s The f i r s t t h i n g to be considered i s a general p l a n of a t t a c k on the problem* This must i n c l u d e a v e r s a t i l e c o n s t r u c t i o n adaptable t o f i n a l - t e s t s as w e l l as the p r e l i m i n a r y t e s t s . Since the dimensions of t h i s c a v i t y cannot be c a l c u l a t e d on a t h e o r e t i c a l b a s i s , p r e l i m i n a r y t e s t s must be made to determine the p o s s i b l e types of modes of resonance and the arm lengths r e q u i r e d f o r any given frequency. This r e q u i r e s that the c a v i t y f i r s t be made w i t h arms c o n t a i n i n g s h o r t - c i r c u i t s t h at are a d j u s t a b l e over a r e l a t i v e l y wide range of d i s t a n c e s from the c a v i t y c e n t e r . Once an arm l e n g t h has been decided upon, the a d j u s t a b l e s h o r t - c i r c u i t s can be r e p l a c e d by f i x e d — l e n g t h arms f i t t e d w i t h narrow-range tuning probes f o r f i n a l alignment. Before a c t u a l c o n s t r u c t i o n can be s t a r t e d , both technique and dimensions must be decided upon* The most c r i t i c a l dimension at the outset i s the i n s i d e diiameter of the c y l i n d r i c a l waveguide arms. This must be as small as p o s s i b l e i n order that the a t t e n u a t i o n of any second and higher order modes of propagation be as great as p o s s i b l e , y e t i t must be l a r g e enough to support the c y l i n d r i c a l T E ^ mode of propagation* Another c o n s i d e r a t i o n i n the choice of diameter of the c y l i n d r i c a l guide i s due> to the use of the a d j u s t a b l e s h o r t - c i r c u i t s * In order t h a t l o s s e s be kept to a minimum, these must be non—contacting type plungers as shown i n F i g . 1 1 ( a ) . 39 ab be A / B L C D \ \ \ \ A \ \ \ \ \ \ \ \ / \ \ \ \ \ \ \ N \ \ \ \ \ \ \ \ \ \ \ NA. Z i n ab Lbc cd i n R V T \ \ \ \ \ \ \ T T T T T T T T T T T T (a) ab be (b) P i g , H i Non-Contacting Plunger and I t s E q u i v a l e n t C i r c u i t * Assuming that a l l the l o s s i n the contact l e n g t h CD. occurs at C, the e q u i v a l e n t c i r c u i t of the plunger i s t h a t shown i n F i g . 11(b); i n terms of the c h a r a c t e r i s t i c impedances of the plunger s e c t i o n s * the impedance Z^ n becomes i n ab  5bc R (4.1) The c h a r a c t e r i s t i c impedance of a c o a x i a l l i n e i s given by! '0 k L o g g (a/b) (4.2) where a = the i n s i d e diameter of the outer conductor and b = the diameter of the i n n e r conductor. Therefore, equation (4.1) becomes ' i n " W e ( d / d , c ) 2 0 (4.3) 40 showing t h a t Z. O i f d, ^ d , . 1X1 C DC £tD The TE^^ f i e l d s i n the c a v i t y induce f i e l d s i n the c o a x i a l s e c t i o n s of the plunger t h a t can be considered as the T E 2 Q mode i n c o a x i a l l i n e . Study of the e q u i v a l e n t c i r c u i t of the plunger gap f o r t h i s mode shows t h a t a resonance may occur w i t h i n the gap. Thus the dimension dj d and d ^ c shown i n F i g * 11 must be chosen such that Z. i s much smaller than R i n order to have low l o s s and a l s o ' i n c such that the frequency of the T E 2 Q resonance i n the gap l i e s out-sid e the o p e r a t i n g frequency range. To meet these requirements, as w e l l as those p r e v i o u s l y mentioned f o r propagation of the TE-^ c i r c u l a r waveguide mode and a t t e n u a t i o n of any higher order modes, the i n s i d e diameter of the c y l i n d r i c a l waveguide was chosen to be 0.950 i n c h * w i t h d . = 0.930 in c h and d, = 0.750 i n c h . The ab be c o n t a c t i n g s e c t i o n of the plunger was made to be a snug s l i d i n g f i t . Each s e c t i o n l e n g t h was designed to be a quarter of a guide wave-l e n g t h f o r the TE-^Q c o a x i a l mode, n e g l e c t i n g the f o r e s h o r t e n i n g e f f e c t of the d i s c o n t i n u i t y c a p a c i t a n c e s * 4.2 C o n s t r u c t i o n of the F i l t e r The j u n c t i o n of the c y l i n d r i c a l arms i s obvious-Ly a compli-cated shape, but was quite r e a d i l y made by f i r s t c a r e f u l l y shaping a cube of brass* and then b o r i n g through each face so that the axes of the three bored holes meet at the same p o i n t at the centre of the cube* To each face of the cube was b o l t e d a c y l i n d r i c a l arm, four of which were f i t t e d w i t h the s h o r t - c i r c u i t i n g a d j u s t a b l e plungers and the remaining two l e f t open* The assembled c a v i t y i s shown i n F i g * 12. Two of the plunger arms were d r i l l e d c l o s e to the cube* and p e r p e n d i c u l a r to t h e i r axes to allow the i n s e r t i o n of 41 c o u p l i n g loops i n t o the c a v i t y . F i g . 12. Photograph of C a v i t y J u n c t i o n w i t h Plunger Arms Attached to Four Faces. With two of the arms l e f t open only one of the three degenerate modes of each type of resonance can e x i s t because, with c o u p l i n g to the other two modes, they can propagate out the open arms; as a r e s u l t , the Q f o r those modes w i l l be so low that they w i l l not a f f e c t any measurements made on the e x i s t i n g one. Thus with the above arms i n place on the cube, a s i n g l e degeneracy can be s t u d i e d without p o s s i b l e i n t e r f e r e n c e and i n t e r a c t i o n due to the other two degeneracies. Before the f i n a l c o u p l i n g probe arms, tuning probe arms, 42 and input and o u t p u t - i r i s arms were designed, and f i t t e d to the cube faces f o r f i n a l t e s t i n g of the f i l t e r $ t h i s c a v i t y was t e s t e d f o r resonant f r e q u e n c i e s and f i e l d d i s t r i b u t i o n s , as w i l l be d e s c r i b e d l a t e r * The two arms c o n t a i n i n g the input and output i r i s e s had quarter-wave s e c t i o n s of waveguide d i r e c t l y s o l d e r e d to the e x t e r n a l w a l l s of each i r i s to minimize any e f f e c t s due to the waveguide flange connections. The f i r s t i r i s e s used were holes d r i l l e d through the end w a l l s of c l o s e d c y l i n d r i c a l arms^ but i t was found that holes l a r g e r i n diameter than the narrow dimension of the e x t e r n a l r e c t a n g u l a r waveguide were r e q u i r e d to give the d e s i r e d value of e x t e r n a l Q» To enlarge the i r i s e s the holes were m i l l e d to a re c t a n g u l a r shape wi t h l / 8 i n c h r a d i u s corners u n t i l they were larg e enough. Each i r i s arm was al s o b u i l t with a l / l O i n c h c y l i n d r i c a l l i p t h a t f i t t e d i n t o a c y l i n d r i c a l s l o t i n the cube face i n order t h a t the arms could' be r o t a t e d f o r alignment* Each of the two arms c o n t a i n i n g the c o u p l i n g probes was made wit h a f i x e d s h o r t - c i r c u i t at the endj and w i t h a screw prober f i t t e d w i t h a l o c k i n g nut, such that the a x i s of the probe was pe r p e n d i c u l a r to the axis of the arm clos e to the cl o s e d end of the arm. The mounting holes were d r i l l e d through the flange so t h a t , w i t h the arm attached to the op r r e c t face of the cube, the probe a x i s was at 45 degrees to the e l e c t r i c f i e l d s of each of the two degenerate modes of resonance w i t h i n t h a t arm of the c a v i t y . E x t r a holes were a l s o d r i l l e d i n the f l a n g e so t h a t the probe could a l s o be set e i t h e r p a r a l l e l t o or p e r p e n d i c u l a r to the e l e c t r i c f i e l d of one of the modes f o r purposes of measurement of the c o u p l i n g c o - e f f i c i e n t s as d e s c r i b e d i n s e c t i o n 3.3(b). The remaining p a i r of arms "were b u i l t w i t h screw probes i n s e r t e d a x i a l l y through the end w a l l s f o r purposes of f i n a l t u n i n g * Each of the s i x arms was fastened by s i x s c r e w s through f l a n g e to a cube f a c e . The arms c o n t a i n i n g the i r i s e s were on opposite f a c e s ^ w i t h the r e c t a n g u l a r ports r o t a t e d 90° r e l a t i v e t one another* Each coupling-probe arm was b o l t e d to a cube face* opposite to t h a t h o l d i n g a tuning—probe arnu The connection of each arm to the cube face was made as t i g h t as p o s s i b l e by us i n g a 0.010 i n c h r a i s e d l i p on the f l a n g e . Placement of the screws was also such than no screw holes would meet w i t h i n the cube. 4.3 Measurement and I d e n t i f i c a t i o n of the Modes of Resonance The c a v i t y w i t h four plunger arms and two open arms was set up i n the way shown i n P i g . 13* This t e s t set-up i s al s o shown i n the photograph of F i g . 14 i n which the caps on the open arms of the c a v i t y were f o r supporting small beads used f o r measuring the f i e l d d i s t r i b u t i o n s w i t h i n the c a v i t y . H.V, K l y s t r o n Power Supply P i l . t. Beam D*C* Block C.R.O. •X-13 Detector K l y s t r o n Pad Wavemeter Transducer F i g . 13* Test Set-Up f o r Measurement of Resonant Frequencies of C a v i t y * 44 Sawtooth modulation was a p p l i e d to the k l y s t r o n to sweep i t s output across a range of f r e q u e n c i e s * This same sawtooth was a l s o a p p l i e d to the h o r i z o n t a l p l a t e s of the o s c i l l o s c o p e so t h a t s y n c h r o n i z a t i o n problems were e l i m i n a t e d * The k l y s t r o n was i s o l a t e d by at l e a s t s i x d e c i b e l s of l o s s i n the pad to prevent frequency p u l l i n g of the k l y s t r o n by the e x t e r n a l c i r c u i t r y , A c o a x i a l l i n e was coupled i n t o the c a v i t y by a loop through a hole i n the w a l l of one of the plunger arms. Power was coupled out of the c a v i t y by another loop to a diode d e t e c t o r and then to the v e r t i c a l p l a t e s of the o s c i l l o s c o p e * In t h i s way, any t r a n s m i s s i o n through the c a v i t y showed on the o s c i l l o s c o p e face as a hump i n the t r a c e * The k l y s t r o n was then tuned f o r maximum power at each resonance and the wavemeter p i p centred on the peak df the t r a c e to measure the frequency of each resonance. Before the frequency was measured, the loopswere withdrawn to a p o i n t beyond which:-:no s i g n i f i c a n t change i n resonant frequency o c c u r r e d . By t h i s method, f r e q u e n c i e s of a l l resonances o c c u r r i n g w i t h i n the X—band 8.2kmc/s to 12.4 kmc/s, were measured f o r a s e r i e s of plunger s e t t i n g s chosen so t h a t the plunger f a c e s were a l l e q u i d i s t a n t from the cube c e n t r e . I t was noted a l s o whether or not any f i e l d s e x i s t e d i n the open arms by d e t e c t i n g any p u l l i n g of the resonant frequency when a conductor or d i e l e c t r i c was i n s e r t e d i n t o the open arms. The r e s u l t of t h i s t e s t showed t h a t the f i e l d s i n the open arms were of v e r y small amplitude and i t was found l a t e r that those t h a t d i d e x i s t were of the TE.^ type, and t h e i r presence was a t t r i b u t e d to small mechanical asymmetry of the plunger f a c e s and j u n c t i o n . Photograph of Test Set-Up f o r Measurement of Resonances and I d e n t i f i c a t i o n of Modes. 46 The r e s u l t s of the resonant—frequency measurement are shown i n F i g . 15 where resonant frequency was p l o t t e d as a graph a g a i n s t the d i s t a n c e from the cube centre to the plunger f a c e s . The curves 1^ 2^ 3 and 4 were i d e n t i f i e d w i t h the modes of F i g . 16 by a study of the e f f e c t s of small conducting and d i e l e c t r i c p e r t u r b a t i o n s i n t r o d u c e d i n t o the f i e l d s at and near the j u n c t i o n . From the the o r y of p e r t u r b a t i o n s (Chapter 2 ) . the frequency s h i f t due to the presence of a small m e t a l l i c sphere of r a d i u s r was found to be and the frequency s h i f t f o r a small d i e l e e t r i c sphere, since the magnetic f i e l d s are u n a f f e c t e d , i s Ato 0 3 0 Thus a d i e l e c t r i c bead w i l l tend to reduce the resonant frequency of the c a v i t y * except of course at p o i n t s where the e l e c t r i c f i e l d s v a n i s h * A m e t a l l i c sphere w i l l ^ on the other hand, i n c r e a s e the frequency where the magnetic f i e l d s are at a maximum and decrease i t where the e l e c t r i c f i e l d s are at a maximum; i t w i l l have no e f f e c t where ^ — ^ 6 6 ^0 47 1.4 1.6 1.8 2.0 2.2 2.4 2.6 . 2.8 Distance from c a v i t y centre to plunger faces ( i n . ) P i g . 15. Graph of Resonances Occurring i n Cross-Armed C a v i t y . 48 (c) mode with T E 2 1 mode i n (d) E 1 mode v i t h T E 2 1 mode i n t h i r d p a i r of arms. t h i r d . p a i r of arms. F i g * 16. The Four B a s i c Modes of Resonance i n the "Cross-Armed" C a v i t y . With these p o i n t s i n mind, metal and d i e l e c t r i c beads were i n t r o d u c e d i n t o the c a v i t y by lowering them on a very t h i n n y lon thread* which i t s e l f had n e g l i g i b l e e f f e c t * i n t o the open arms. The e f f e c t on the resonant frequency produced by each bead at d i f f e r e n t p o s i t i o n s i n the j u n c t i o n was noted and thus the higher order modes i n the open arms were e a s i l y i d e n t i f i e d . Checks on the p o l a r i z a t i o n of the T E ^ modes i n the plunger arms were made by i n s e r t i n g a - s t r a i g h t wire through the d i a m e t r i c a l holes i n an arm. A f u r t h e r check on the i d e n t i f i c a t i o n of the EQ mode of resonance was found when t h i s mode no longer resonated above 9 5 0 0 me/s* This d i sappearance was due to the f a c t t h a t the T M m mode of propagation i n the open arms i s above c u t o f f beyond t h i s frequency* and thus can propagate down the arms with the r e s u l t t h a t the Q of the EQ mode of resonance drops so low t h a t the resonance i s no longer d e t e c t a b l e above 9 5 0 0 mc/s* In the higher—frequency p o r t i o n of the X-band, the four modes of resonance repeated themselves* There were also a few other resonances noted t h a t were not s t u d i e d c l o s e l y since they were above the frequency range of i n t e r e s t but one of these was suspected to c o n t a i n TMQ-^ f i e l d s i n the plunger arms. 4 . 4 Measurement of E x t e r n a l Coupling and I r i s S i z e s As d e s c r i b e d i n s e c t i o n 3 . 3 ( a ) * the i r i s s i z e i s determined by the e x t e r n a l Q r e q u i r e d to meet the s p e c i f i e d i n s e r t i o n - l o s s c h a r a c t e r i s t i c s * At t h i s p o i n t * the Centre frequency of the f i n a l c a v i t y was Chosen to be 8 4 0 0 me/s u s i n g the HQ type of resonance. This was a r r i v e d at by c o n s i d e r a t i o n of two s p e c i f i c a t i o n s , one 'that the centre frequency be at l e a s t 100 me/s separated from the 50 next mode of resonance, and the other t h a t the lengths of the arms be great enough to ensure t h a t any probes i n s e r t e d i n the arms f o r c o u p l i n g or tuning do not i n t e r f e r e w i t h the f r i n g i n g f i e l d s near the j u n c t i o n . A l s o * the HQ type of resonance was chosen because i t has higher order f i e l d s i n the open arms than do the other three types, and thus the f i e l d s t rengths at the c o u p l i n g probes w i l l be l e a s t f o r t h i s mode* 8 By u s i n g the p h a s e - s h i f t method to measure the unloaded and loaded Q's* the e x t e r n a l Q was found from the equation *e *L *u To measure Q by the p h a s e — s h i f t method, the p o s i t i o n , of the minimum i n the standing—wave p a t t e r n must be p l o t t e d as a f u n c t i o n of frequency near resonance* For overcoupled systems, the s h i f t i n the standing—wave minimum r e l a t i v e to the minimum p o s i t i o n at A , resonance, corresponding to a frequency s h i f t of j—- = fQ/2Q^, i s e g i v e n by ^ "= f tan - 1 (-S 0) (4.7) g where i s the s h i f t i n the s tanding—wa,ve minimum p o s i t i o n , ta n (*"*^ o^  ^ S a s e c o n < ^ quadrant angle, and SQ i s the SWR at resonance* From the p l o t of standing-wave minimum p o s i t i o n as a f u n c t i o n of frequency, and A^ c a l c u l a t e d from equation (4.7), the value of the frequency s h i f t £^ can be found, and then = fQ'/2^« S i m i l a r l y the unloaded Q can be measured by 51 a p p l i c a t i o n of the equation u 1 , —1 . 0 - I , 0 \ g ( S 0 - 2) where A i s the s h i f t i n standing-wave minimum p o s i t i o n to give a frequency s h i f t of £ u = fQ/2Q u* I f A ^ i s too small to give reasonable accuracy i n determining the ; value of Qu* the f o l l o w i n g e x p r e s s i o n may be used f o r overcoupled systemss G u = QL I1 + so> < 4- 9> To c a r r y out the measurements^ the arms c o n t a i n i n g the i r i s e s were f a s t e n e d to the cube faces that o r i g i n a l l y h e l d the open arms* For each i r i s s i z e , both and Q u were measured and the i r i s e s were enlarged u n t i l the d e s i r e d Q g was reached. A f t e r each i r i s enlargement* a check on the centre f r e q u e n c i e s of the three resonances was made, and the plungers adjusted to e q u a l i z e the three by a p p l i c a t i o n of equations (3.52) and (3.53). The end r e s u l t was t h a t the resonant frequency of the f i l t e r centred on about 8310 mc/s. and the i r i s arms were a c t u a l l y a l i t t l e s h o r t e r than the other f o u r . The o r i g i n a l o b j e c t i v e i n the design of the f i l t e r was f o r a Chebyshev response w i t h a passband t o l e r a n c e of h = l/2 and a bandwidth of 10 mc/s* Equations (3.16) de f i n e and u n i q u e l y f o r these c o n d i t i o n s ; that i s , L 2 C l 2 = 4 a n d 2 C 1 ~ L 2 = 3 (4.10) from which C-^  = 2 farads and 1^ = 1 henry. Equations (3.26) d e f i n e Q g f o r the s p e c i f i e d resonant frequency and bandwidth; t h a t i s . f o r a resonant frequency o f 8310 mc/s, Q e = T~ °! = 1660 (4.11) The c i r c u i t elements f o r a Butterworth response are g i v e n by equation (3.19). For a bandwidth of 10 mc/s» Q e =830 (4.12) A c t u a l l y * c a l i b r a t i o n of i r i s s i z e versus Q g was c a r r i e d to the p o i n t where Q g was 580. Thus, f o r the Chebyshev case, co C ¥ = -°p^ = 28*6 mc/s (4.13) • V-e which r e q u i r e d v e r y l a r g e probe depths t h a t were suspected to have i n v a l i d a t e d some of the assumptions p r e v i o u s l y made. The experimental response f o r t h i s case was found to be of l i t t l e v alue so the Chebyshev case was abandoned at t h i s p o i n t . Three cases were considered f o r the present study, one being the Butterworth and the other two both being of narrower bandwidth than the Butterworth* Q remained constant and k was e reduced by withdrawing the c o u p l i n g probes from the Butterworth p o s i t i o n s to reduce the bandwidth, while the tuning probes were adjust e d to e q u a l i z e the unloaded resonant f r e q u e n c i e s of the three modes. The probe depth f o r the Butterworth case was 0.154 i n c h ; i t was 0^140 i n c h and 0.128 i n c h f o r the other two. 53 8316 8298 0 0.05 0.10 . 0.15 0.20 Coupling Probe Depth ( i n . ) P i g . 17* Graph of Resonant Frequencies of Modes I and JI as Functions of Coupling Probe Depth. 54 4.5 Measurement of I n t e r n a l Coupling and Probe Depths Two arms were const r u c t e d whose lengths were equal to those o found f o r the plunger arms a f t e r the i r i s e s had been cut to s i z e , and a l l tliree resonant f r e q u e n c i e s e q u a l i z e d . These arms were provided w i t h c o u p l i n g probes and mounting holes were arranged i n the fl a n g e s so that the probe axes could be set at 6 , 45°, 90° to the e l e c t r i c f i e l d s w i t h i n the arms. The tuning-probe arms were fa s t e n e d to the cube fa c e s opposite to the coupling-probe arms. The t u n i n g probes were adjusted f o r e q u a l i z a t i o n of the 0 three resonant f r e q u e n c i e s . Then, as d e s c r i b e d i n s e c t i o n : 3 . 3 ( b ) , the c o u p l i n g loops were i n s e r t e d as shown i n P i g . 12, and the fr e q u e n c i e s <OQJ and ®QJJ of equation (3.47) were measured f o r v a r i o u s coupling-probe depths. P i g . 17 shows a p l o t of the resonant f r e q u e n c i e s of modes I and I I of F i g . 7 as f u n c t i o n s of probe i n s e r t i o n * The d i f f e r e n c e between these two curves gives the value of kfQ f o r any gi v e n probe depth by r e f e r e n c e s to equation (3*47)* and the detuning of the modes i s given by equation (3.48)* A graph of k f ^ as a f u n c t i o n of probe depth i s shown i n F i g * 18. The i n t e r n a l c o u p l i n g c o - e f f i c i e n t i s given by k = ^ - « I T 1#i (4.14) which can be evaluated i n terms of probe depth by use of equation (3.47); thus* f o r the Butterworth c o n d i t i o n , k f o = ^ 0 = 1 = = - = 1 0 a m c / s ( 4 * 1 5 ) / L2 C l 55 56 Then the probe depth r e q u i r e d to give t h i s value of - £QJJ i s tha t r e q u i r e d to give the d e s i r e d f i l t e r c h a r a c t e r i s t i c s , and t h i s i s obtained from P i g . 18, which i s a p l o t of k f Q as a f u n c t i o n of probe depth* P i g . 19 i s a photograph of the f i n a l c a v i t y s t r u c t u r e showing the f i x e d - l e n g t h arms and the tuni n g and co u p l i n g screws. 4.6 Line-up Procedure The resonant f r e q u e n c i e s of the three modes i n the f i n a l c a v i t y were e q u a l i z e d as before by use of equations (3.52) and (3.53). The centre frequency of each of the input and output modes of resonance was found by p l o t t i n g the VSWR across resonance, the minimum VSWR o c c u r r i n g at the centre frequency. The frequency of the centre mode was measured, as were the f r e q u e n c i e s of the resonances d e s c r i b e d i n s e c t i o n 4.12, by use of an o s c i l l o s c o p e and by r e t r a c t i n g the c o u p l i n g loops u n t i l the wavemeter p i p on the t r a c e was no longer a f f e c t e d by loop p e n e t r a t i o n . The three f r e q u e n c i e s were measured and e q u a l i z e d with the c o u p l i n g probes set f l u s h w i t h the i n s i d e w a l l of the c a v i t y . With c o r r e c t probe i n s e r t i o n , the three f r e q u e n c i e s s h i f t e d by an amount s p e c i f i e d by equations (3.39) and (3.48): Then t h i s s h i f t was compensated f o r by re-adjustment of the tuni n g probes. An attempt to a l i g n the i r i s e s e l e c t r i c a l l y f o r a minimum of s t r a i g h t - t h r o u g h t r a n s m i s s i o n was made but t h i s method was found u n s a t i s f a c t o r y and they were f i n a l l y a l i g n e d m e c h a n i c a l l y . F i g . 19. Photograph of Completed "Cross-Armed" Cavity. 58 5. EXPERIMENTAL RESPONSE CHARACTERISTICS 5.1 The Measuring Technique The t e s t set-up used f o r measuring the i n s e r t i o n - l o s s c h a r a c t e r i s t i c s of the c a v i t y i s t h a t shown i n P i g . 20. This i s a bridge network which gives a n u l l type of measurement. Vave-meter C.R.O. A, Square-law Detector + P r e c i s i o n Attenuators , 20 db 3 db K l y s t r o n Phase (1000 cps D i r e c t i o n a l : R N I o h i i t e r sq. wave Couplers modulation) Twist • 10 db S e c t i o n . D i r e c t i o n a l Coupler P i g . 20. Experimental Test Set-Up f o r Measurement of F i l t e r I n s e r t i o n - L o s s C h a r a c t e r i s t i c s . In the b r i d g e , some microwave power i s taken from the main l i n e by the 3 db d i r e c t i o n a l coupler to the side arm; a f t e r undergoing l o s s i n the attenuator A-^ , t h i s * power i s recombined wi t h the power i n the main l i n e through the 10 db d i r e c t i o n a l c o u p l e r . The r e s u l t a n t power i s measured by the d e t e c t o r and . meter. I f the c o n t r i b u t i o n to f i e l d i n t e n s i t y due to input, from the side arm i s equal to and 180° out of phase with the c o n t r i b u t i o n from the main arm at the d e t e c t o r , then a n u l l i s seen on the meter. Under these c o n d i t i o n s , the l o s s e s measured by the two a t t e n u a t o r s w i l l give the l o s s through the f i l t e r by the f o l l o w i n g c a l c u l a t i o n s ? 59 The e l e c t r i c f i e l d i n t e n s i t y * at the det e c t o r i s g i v e n by E = E R + (5.1) where E^ i s the c o n t r i b u t i o n to the i n t e n s i t y of the e l e c t r i c xi f i e l d at the d e t e c t o r from the main arm, E^ i s the c o n t r i b u t i o n to the i n t e n s i t y * o f the e l e c t r i c f i e l d at the d e t e c t o r from the side arm, 0 i s the phase angle between E^ and E^. Then N = J | E H | 2 + K l 2 + 2 I E H I I E J C O S * (5.2) Thus, f o r a n u l l at, the d e t e c t o r E H = E L , 0 = 180° (5.3) Then, when A-^ , A 2 and 0 are adjusted f o r a n u l l , the f i l t e r i n s e r t i o n l o s s i s giv e n by L = A L - A 2 + 1 0 - . C (5.4) where A^ i s the l o s s i n A-^  i n Fig u r e 21 A 2 i s the l o s s i n A 2 i n Fig u r e 21 C i s the c a l i b r a t i o n c o r r e c t i o n to account f o r i n c i d e n t a l l o s s e s . C was measured by r e p l a c i n g the f i l t e r by a piece of waveguide and 60 a t w i s t s e c t i o n , so t h a t L « 0; then C; = - A 2 + 10 (5.5) The measurements of the f i l t e r i n s e r t i o n l o s s were made f o r the three coupling—probe depths mentioned above. I t may be noted at t h i s p o i n t that A 2 should be kept to a minimum as f a r as p o s s i b l e . For low f i l t e r i n s e r t i o n l o s s e s t h i s value i s approximately 10 db«, but as L i n c r e a s e s A 2 should be decreased to keep the b r i d g e s e n s i t i v i t y as high as p o s s i b l e . The method used to measure the f i l t e r i n s e r t i o n l o s s at a gi v e n frequency was to a l t e r n a t e l y a d just A-^  and the phase s h i f t e r f o r a n u l l at the d e t e c t o r . Then a f t e r n o t i n g the readings of A-^  and A 2, the l o s s through the f i l t e r was c a l c u l a t e d from equation (5.4 ,) • The i n s e r t i o n — l o s s c h a r a c t e r i s t i c of the f i l t e r was p l o t t e d a f t e r a s e r i e s of measurements were made at s e v e r a l f r e q u e n c i e s w i t h i n the band of i n t e r e s t . 5.2 Experimental Results and C o r r e l a t i o n with Theory 5.21 R e s u l t s f o r the Butterworth Design (13.6 Mc/s Bandwidth) The experimental i n s e r t i o n - l o s s c h a r a c t e r i s t i c f o r the Butterworth design i s shown i n F i g , 21. Sev e r a l response runs were made but the general response was the same i n each. The experimental mid-band frequency was found to be 8309 mc/s. The o r i g i n a l design o b j e c t i v e was about 8400 mc/s, but i n t r o d u c t i o n of the input and output i r i s e s lowered the resonant frequencies of the input and output modes, and to b r i n g the centre mode down to equal the outer two i n v o l v e d lowering the outer two by a d j u s t i n g the p l u n g e r s . This lowered the centre mode frequency by an amount 61 equal to the sum of the frequency changes experienced by the outer modes as shown mathematically by equations (3.52) and (3.53). T h e o r e t i c a l l y , there should be no t r a n s m i s s i o n through the f i l t e r when the c o u p l i n g probes are set f l u s h w i t h the i n s i d e w a l l s of the c a v i t y . Experimentally> the i s o l a t i o n was found to be gr e a t e r than 25 db i n the pass-band* The major f a c t o r i n determining t h i s value was accuracy of alignment of the input and output waveguides at 90° to one another* Alignment was attempted e l e c t r i c a l l y but t h i s was found to be u n s a t i s f a c t o r y because i t appeared to be f r e q u e n c y — s e n s i t i v e * so mechanical alignment was f i n a l l y c a r r i e d out. The l o s s i s great enough* however* to support the theory t h a t three degenerate modes were a c t u a l l y e x c i t e d w i t h i n the c a v i t y * I n s e r t i o n — l o s s f u n c t i o n s were measured u s i n g each i r i s i n t u r n as the input* and the r e s u l t s were i d e n t i c a l w i t h i n the experimental e r r o r . For the Butterworth case, the bandwidth has been taken to be the frequency d i f f e r e n c e between the half-power p o i n t s of the f i l t e r ; the experimental bandwidth was 15*4 mc/s, compared to the t h e o r e t i c a l value of 13.6 mc/s. This discrepancy i s suspected to be due i n part to spurious s t r a i g h t - t h r o u g h c o u p l i n g d e s c r i b e d above* The l o s s at pass—band centre was 2.7 db. In t h i s case k f ^ = 10.1 mc/s; f o r zero frequency d e v i a t i o n (y = 0)* equation (3.60) y i e l d s L = 2.7 db = 10 Log j ^2.82xl0" 6<f 3 + 5*07xl0""4<r 2 + 4*50x10"* V + 2 J (5.6) from which < = 13.9 (5.7) Now, s u b s t i t u t i o n of the valu e s of kfQ*' QEt<f» and (OQ i n t o equation (3.60) y i e l d s L = 10 L o g j l . 9 2 x l O ~ 2 y 6 + 1 . 0 8 x l 0 ~ 8 y 4 + 6,60x10" 5y 2 + 1.86^ (5.8) and from equation ( 3 . 6 l ) , y = 4n;(Af), so t h a t L = 10 Log|jT.5xlO~ 6(Af ) 6 + 2 * 6 8 x l 0 ~ 4 ( A f ) 4 + 1.04xlO~ 2(Af) + 1.86J (5.9) f o r the t h e o r e t i c a l Butterworth response. F i g . 21 shows i n s e r t i o n l o s s p l o t t e d against d e v i a t i o n from the centre frequency, with the t h e o r e t i c a l c h a r a c t e r i s t i c superimposed on the experimental. The reason f o r the asymmetry of the experimental curve i n t h i s case was not uniquely determined but i s suspected to be due to some small amount of s t r a i g h t -through c o u p l i n g , caused mainly by large probe volumes, between the hjgher—order f i e l d s i n the "empty" arms of each degeneracy. I t may a l s o be due to the f a c t that some l i m i t s have been exceeded by the amount of i n t e r n a l c o u p l i n g r e q u i r e d . Otherwise the experimental r e s u l t s agree w e l l with the t h e o r e t i c a l . 5.22 Results f o r the 10 mc/s Bandwidth Design F i g . 18 y i e l d s a value f o r k f Q of 7*8 f o r the 0.140 i n c h probe depth used i n the i p Mc/s design* S o l v i n g f o r 4" as i n the Butterworth case, L = 4.2 db - 10 l o g i^4.62xl0" 6< 348.33xl0" 4<- 2+5.98xl0~ 2 <r + 2^ (5.10) from which <f = 16.6 (5.11) 63 -50 -40 -30 -20 -10 0 10 20 30 40 50 A f (mc/s) « 21» Experimental and T h e o r e t i c a l Butterworth Design Response 64 Therefore, equation (3.60) y i e l d s the t h e o r e t i c a l i n s e r t i o n - l o s s e x p r e s s i o n L = 10 Lo! L 2 . 1 1 x l O " 5 ( A f ) 6 + 1 . 7 4 x l 0 ~ 3 ( A f ) 4 + 5-.45xlO~ 2(Af) + 2.62J (5.12) P i g . .22 shows t h i s p l o t t e d as a graph and gives the t h e o r e t i c a l 3 db bandwidth as 10.0 mc/s. The experimental response curve f o r t h i s case i s a l s o shown i n F i g . 22. The bandwidth between the 3 db p o i n t s i s 12.8 mc/s as compared wi t h the t h e o r e t i c a l value of 10*0 mc/s. For t h i s case, and f o r the 7.4 mc/s bandwidth case, e q u a l i z a t i o n of the resonant f r e q u e n c i e s of the three degeneracies i s suspected to have an a p p r e c i a b l e e f f e c t on the response shape and bandwidth, the e f f e c t becoming more apparent w i t h narrowing bandwidth and e q u a l i z a t i o n becoming more c r i t i c a l . The response i n t h i s case i s symmetrical whereas i n the Butterworth case i t was not. This p o i n t s to the p o s s i b i l i t y t h a t some l i m i t s may have been exceeded by the amount of i n t e r n a l c o u p l i n g used to produce the Butterworth response. The l o s s at pass—band centre was 4*2 db. 5.23 R e s u l t s f o r the 7.4 Mc/s Bandwidth Design The response curve f o r t h i s case i s p l o t t e d i n the graph of F i g * 23. Here the experimental bandwidth i s 10.0 mc/s as compared w i t h the t h e o r e t i c a l 7*4 mc/s* Here again the response shape i s symmetrical f o r the most p a r t . The l o s s at pass-band centre was 4.9 db. F i g * 18 y i e l d s a value f o r k f Q of 6.2 f o r the 0.128 i n c h 65 probe depth used i n t h i s case. With y = 0* [ 2 7.34xl0"V 3+1.32xl0^V 2+8.16xl0~«<' + 5J (5,13) from which <T = 14.9 Therefore*- equation (3.60) y i e l d s , f o r t h i s case, L .= 10 logj^«31xlO~ 5(Af,) 6+5.45xlO"" 3(Af ) 4 + »167(Af) 2+ 3.O9J (5.15) F i g . 23 shows t h i s p l o t t e d g r a p h i c a l l y and give s the 3 db band-width as 7*4 mc/s. The d i s c r e p a n c y between the experimental and t h e o r e t i c a l curves above the pass—band, e s p e c i a l l y f o r the broader-band-width cases, i s due to the f a c t t h a t another mode of resonance, the E-^  mode d e s c r i b e d i n S e c t i o n 4.12, s t a r t s to have the e f f e c t of reducing the i n s e r t i o n l o s s as i t i s approached u n t i l , event-u a l l y , the l o s s a c t u a l l y becomes l e s s as the frequency i s swept i n t o the r e g i o n of the E^ resonances* I t may be noted at t h i s p o i n t t h a t the E^ resonance gives a response s i m i l a r to that of the HQ mode* but study of the E^, H^* and EQ modes of resonance i s outside the scope of t h i s work. The experimental r e s u l t s obtained i n t h i s work v e r i f y the statements p r e v i o u s l y made about the i n s e r t i o n - l o s s c h a r a c t e r i s t i (5.14) 67 -50 -40 -30 -20 -10 0 10 20 30 40 50 Af (mc/s) F i g * 23* Experimental and T h e o r e t i c a l Responses f o r the 7.4 Mc/s Bandwidth .Design. 68 below the pass—band, as compared w i t h those of p r e v i o u s l y designed multi-mode c a v i t y resonators* In other words, the spurious c o u p l i n g between input and output, caused by the excessive Coupling p e r t u r b a t i o n s used i n other degenerate-mode microwave f i l t e r s 1 2 such as those of Curry and L i n , does not occur i n t h i s f i l t e r . The absence of t h i s spurious c o u p l i n g i s e x h i b i t e d by a l l three cases s t u d i e d here, even by the Butterworth case which has a r e l a t i v e l y wide bandwidth, and t h i s absence i s demonstrated by the experimental r e s u l t s which show no s i g n of the poor a t t e n u a t i o n c h a r a c t e r i s t i c s on the low-frequency side of resonance t h a t i s e x h i b i t e d by the f i l t e r s of Curry and L i n w i t h lar g e c o u p l i n g -probe i n s e r t i o n s . The phase s h i f t was not measured p r e c i s e l y , but i s d i r e c t l y r e l a t e d to the r e a d i n g of the phase s h i f t e r used i n the t e s t s e t -up f o r measurement of i n s e r t i o n l o s s , and r e s u l t s i n d i c a t e d t hat the phase c h a r a c t e r i s t i c s of the f i l t e r are r e l a t i v e l y l i n e a r throughout the pass-band* The pass—band i n s e r t i o n l o s s of the cross-armed c a v i t y was found to be g r e a t e r than t h a t of e i t h e r Gurry*s or L i n ' s c y l i n d r i c a l c a v i t y . The i n c r e a s e d l o s s i s suspected to be due mainly to the l o s s at the j u n c t i o n s between the cube faces and the s i x arms. A l s o the i n t e r n a l surface area of the cross-armed c a v i t y i s r e l a t i v e l y greater than t h a t of the s p h e r i c a l and c y l i n d r i c a l c a v i t i e s and t h i s too w i l l tend to i n c r e a s e the r e s i d u a l i n s e r t i o n l o s s . S i l v e r - p l a t i n g of the i n s i d e surface of the c a v i t y may reduce the d i s s i p a t i o n * ; 69 6. CONCLUSIONS Studies of multi-r-mode c a v i t y f i l t e r s have shown how these s t r u c t u r e s can be designed and c o n s t r u c t e d to c l o s e l y r e a l i z e c e r t a i n p r e s c r i b e d i n s e r t i o n - f l o s s c h a r a c t e r i s t i c s . However, as mentioned e a r l i e r i n t h i s t h e s i s , a l l such f i l t e r s have shown a tendency toward a departure from the t h e o r e t i c a l i n s e r t i o n l o s s on the low^frequency side of t h e i r pass—bands and t h i s departure becomes more prominent as the bandwidth i s i n c r e a s e d . The f i l t e r under study i n t h i s work* namely the "cross—armed" c a v i t y , d i d not e x h i b i t t h i s u n d e s i r a b l e behaviour even at r e l a t i v e l y l a r g e bandwidths* The c o n c l u s i o n drawn from t h i s r e s u l t i s that much l e s s d i r e c t c o u p l i n g from input to output can occur i n the "cross-armed" c a v i t y than can occur i n the p r e v i o u s l y s t u d i e d geometries such as the c y l i n d e r or sphere* In the l a t t e r type of c a v i t y s t r u c t u r e the l i m i t s of small p e r t u r b a t i o n s are exceeded f o r l a r g e bandwidths, and thus t h i s " s t r a i g h t - t h r o u g h " c o u p l i n g i s e f f e c t e d ^ but i n the "cross-armed" c a v i t y no f i r s t - o r d e r f i e l d s of the output s e c t i o n e x i s t i n the r e g i o n of any p e r t u r b -a t i o n to the i n p u t s e c t i o n . Thus the l i m i t s of small p e r t u r b -a t i o n s may be exceeded without causing much d i r e c t c o u p l i n g from i n p u t to output w i t h consequent degradation of the i n s e r t i o n — l o s s c h a r a c t e r i s t i c below resonance. •The measuring technique proved to be s u c c e s s f u l and convenient to use. Some d i f f i c u l t y was experienced i n o b t a i n i n g a good n u l l when the f i l t e r l o s s was s m a l l . This was suspected to be due to a small amount, of slope on the square—wave modulation 70 a p p l i e d to the k l y s t r o n causing some spurious f r e q u e n c i e s to be present* An important p r a c t i c a l l i m i t a t i o n i n the use of the " c r o s s -armed" c a v i t y l i e s i n the f a c t t h a t the second mode of resonance l i e s only some 100 mc/s above the one considered i n t h i s study. Thus, i n a p r a c t i c a l a p p l i c a t i o n , such as d i p l e x i n g t r a n s m i t t e r s and r e c e i v e r s on d i f f e r e n t f r e q u e n c i e s , care would have to be taken to ensure t h a t the frequency to be r e j e c t e d does not c o i n c i d e w i t h t h i s second resonance, or any of the others o c c u r r i n g f o r a gi v e n arm l e n g t h i n P i g , 15, Seve r a l problems appeared d u r i n g the present study t h a t were considered outside the scope of the work, but which may y i e l d u s e f u l r e s u l t s i f i n v e s t i g a t e d . One of these i n v o l v e d the use of unequal arm lengths to e f f e c t the i n t e r n a l c o u p l i n g r a t h e r than c o u p l i n g probes. This would i n v o l v e an extensive i n v e s t i g a t i o n i n t o the t h e o r e t i c a l aspects of the problem* F i n a l l y , the w r i t e r b e l i e v e s t h a t , i n s o l v i n g the b a s i c problem of d i r e c t c o u p l i n g from i n p u t to output c h a r a c t e r i z e d by multi—mode f i l t e r s , enough i n f o r m a t i o n has been gathered to f u r t h e r the i n v e s t i g a t i o n i n t o a l e s s complicated s t r u c t u r e t h a t w i l l e x h i b i t the same b a s i c behaviour of the "cross-armed" c a v i t y with improvements i n the areas of c o n s t r u c t i o n and l i n e - u p , 71 BIBLIOGRAPHY 1. Curry, Malcolm, Microwave F i l t e r s , U t i l i z i n g Degenerate Modes i n a S p h e r i c a l C a v i t y . Department of E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of C a l i f o r n i a , B erkeley, 1952. 2. L i n , Wei—Guan, "Microwave F i l t e r s Employing a S i n g l e C a v i t y E x c i t e d i n More than One Mode"* J o u r n a l of A p p l i e d  P h y s i c s . August, 1951, pp* 989-1001. 3. S l a t e r , J*C», "Microwave E l e c t r o n i c s " * Reviews of Modern P h y s i c s , October, 1946, pp* 441-512. 4. H a l l , G*L* and Parsen, P., "Measurement of Resonant C a v i t y C h a r a c t e r i s t i c s " 'y Proceeding of the I n s t i t u t e of  Radio E n g i n e e r s . December* 1953, pp. 1769-1773. 5. Maier, L„C» J r . , and S l a t e r , J . C , " F i e l d S t rength Measurements i n Resonant C a v i t i e s " , J o u r n a l of A p p l i e d  P h y s i c s , January, 1952, pp* 68—77. 6. K i t c h e n , S«W. and Schelberg, A.D*, "Resonant C a v i t y F i e l d Measurements", J o u r n a l of A p p l i e d P h y s i c s , May, 1955, pp* 618-621. 7. Fano, R.M* and Lawson, A*W., "Microwave F i l t e r s Using Quarter-Wave Cou p l i n g s " , Proceedings of the I n s t i t u t e of Radio  E n g i n e e r s . November, 1947, pp* 1318-1323. 8. Montgomery, C a r o l G* ed. Technique of Microwave Measurements, M«I,T. R a d i a t i o n L a b o r a t o r y S e r i e s , New York, Toranto and London, McGraw-Hill, 1953, Volume 11, pp. 336-340. 9. Ragan, George L. ed* Microwave Transmission C i r c u i t s , M.I.T. R a d i a t i o n Laboratory S e r i e s , New York, Toronto and London, McGraw-Hill, 1953* Volume 9. 

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